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7d15e9a355d8547f	ATD 219-242 Page 219 they would have little clue . . . their more or less ambushed keesters One of half a dozen Pynchonian circumlocutions for "wouldn't know [blank] if it bit them in the ass." The Tetractys True Worshippers of the Ineffable Tetractys The Tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row. As a mystical symbol, it was very important to the followers of the secret worship of the Pythagoreans, Kabbalists, and nutbars of other affiliations since. It has all kinds of symbological meaning, including the four elements, the organization of space, the Tarot, etc. Wikipedia entry; In the Pythagorean tetractys — the supreme symbol of universal forces and processes — are set forth the theories of the Greeks concerning color and music. The first three dots represent the threefold White Light, which is the Godhead containing potentially all sound and color. The remaining seven dots are the colors of the spectrum and the notes of the musical scale. The colors and tones are the active creative powers which, emanating from the First Cause, establish the universe. The seven are divided into two groups, one containing three powers and the other four a relationship also shown in the tetractys. The higher group — that of three — becomes the spiritual nature of the created universe; the lower group — that of four — manifests as the irrational sphere, or inferior world. [1] This division (three/four) has to be related to the "trivium" (grammar, rhetoric, logic) and "quadrivium" (arithmetic, geometry, music, astronomy) of the Medieval liberal arts. More effably, if you flip the Tetractys left to right, it gives the positions of the pins in ten-pin bowling. The acronym T.W.I.T is most appropriate: a twit is an ineffectual buffoon. Neville and Nigel are certainly twits. I believe the above really misses the Big Symbol, i.e., Pynchon's linking of T.W.I.T. with the vagina, i.e., the female sex organ. "T.W.I.T." sounds like — no, is — a cross between "clit" and "twat." And, natch, it's headed up by Nookshaft. And, let's face it, that tetractys is surely an inverted beaver, yes? (See "Beavers of the Brain"). Its male counterpart is Candlebrow U., to be encountered down the road apiece (and that ain't no spoiler!). "The Tetractys" is also the name of a poem by the Quaternionist prophet Hamilton. I can't imagine Pynchon didn't find it fairly interesting reading. Read it for yourself here Chunxton Crescent Invented by Pynchon. "Crescent" is a female symbol in many mythologies and cultures, and it reinforces T.W.I.T.'s association with the female sex. But "Chunxton"? Eliphas Levi's Baphomet The crescent is also said "to represent silver (the metal associated with the moon) in alchemy, where, by inference, it can also be used to represent qualities that silver possesses." (Alchemy and Symbols, By M. E. Glidewell, Epsilon.) Additionally, the crescent was an important symbol for Eliphas Levi, occultist, magician, and spiritual antecedent to the Hermetic Order of the Golden Dawn, and, in turn, the T.W.I.T. Chunxton may be derived from "chunk stone" or "chunk(s) town." I'm inclined to favor the first. "Chunk stone" has two main meanings: (1) stone that's quarried in chunks instead of blocks, slabs or crystals; (2) a magical stone that figures in some American Indian stories. Turquoise and amethyst chunk stones are often made into jewelry as-is, or larger chunks of (say) marble can be used as decoration. Here are links to two Indian stories in which people use chunk stones in finding or tracking: first, second. Of course it's also possible that "chunk" is the verb meaning "throw," in which case there ought to be a "glass houses" connection somewhere; I can't find it. Pure speculation here, but our own moon is a giant "chunk" of "stone". And how did that "chunk" get there? Well, this being Thomas Pynchon's universe, sometime early in the solar system's history, this proto-planet called Orpheus comes along and smacks into the Earth so violently that it not only creates the moon, but at the same time expels enough water and gas to make "it possible for life on Earth to evolve as we currently know it." Seems to me like something worthy of Occultist reverence In CoL49, TRP states at least twice that the Pacific Ocean is "the hole left by the moon's tearing-free and the monument to her exile." (The Crying of Lot 49, p.41) "Tyburnia occupies the ground on the north side of Hyde-park and Kensington-gardens, and stretches from Edgware-road on the east to about Inverness-terrace on the west. This is not, strictly speaking, a fashionable quarter; but it is not absolutely unfashionable, and is a very favourite part with those — lawyers, merchants, and others—who have to reside in town the greater part of the year." Charles Dickens (Jr.), Dickens's Dictionary of London, 1879. Sir John Soane (1753 – 1837) was an English architect who specialised in the Neo-Classical style. Wikipedia entry Madame Blavatsky Helena Petrovna Blavatsky (1831-1891), Russian-born founder of the Theosophical Society. Madame Blavatsky claimed that all religions were both true in their inner teachings and false or imperfect in their external conventional manifestations. Wikipedia Theosophical Society Seal Theosophical Society The Theosophical Society was founded in New York City, USA, in 1875 by H.P. Blavatsky, Henry Steel Olcott, William Quan Judge and others. Its initial objective was the investigation, study and explanation of mediumistic phenomena. After a few years Olcott and Blavatsky moved to India and established the International Headquarters at Adyar, Madras (Chennai). There, they also became interested in studying Eastern religions, and these were included in the Society's agenda. Wikipedia entry "Its post-blavatskian fragments" refers to the schism that occured between some of the founding members after the passing of H.P. Blavatsky in 1891. Society for Psychical Research The Society for Psychical Research (SPR) is a non-profit organization which started in the United Kingdom and was later imitated in other countries. Its stated purpose is to understand "events and abilities commonly described as psychic or paranormal by promoting and supporting important research in this area" and to "examine allegedly paranormal phenomena in a scientific and unbiased way."[1] It was founded in 1882 by a group of eminent thinkers including Edmund Gurney, Frederic William Henry Myers, William Fletcher Barrett, Henry Sidgwick, and Edmund Dawson Rogers. The Society's headquarters are in Marloes Road, London. Wikipedia entry Rosy Cross of the Golden Dawn Order of the Golden Dawn The Hermetic Order of the Golden Dawn (or, more commonly, the Golden Dawn) was a magical order of the late 19th and early 20th centuries, practicing a form of theurgy and spiritual development. William Wynn Westcott, also a member of the Theosophical Society, appears to have been the initial driving force behind the establishment of the Golden Dawn. See also the aforementioned schism within the Theosophical Society. Wikipedia entry of whom there seemed an ever-increasing supply Supply of seekers, not of "arrangements." (Well, this contributor read it wrong . . . twice.) century had rushed . . . out the other side An instant of zero, not a whole year, because they aren't yet "out the other side" of 1900. ??? A century is 100 years. The one referred to here lasted from 1800-1899 and, since it's 1900, it has "rushed to its end." Missing the point. The image focuses on the zero. And please, let's not have that sterile argument about when a century begins! Don't know if this is of any significance, but in the Tarot the Fool (or Jester), says Wikipedia, is "often numbered 0." [2] Page 220 not even if that tartan were authentic It's a solecism in England, but is (or was—at least until well up in the 19th century) a prosecutable offense in Scotland, to wear the tartan of a clan one doesn't belong to. At the time of the action, Lew's offense against taste is not to wear tartan (see below in this entry) but to wear a tartan he isn't entitled to wear. The previous statement doesn't quite jibe. In the late 17 cent. it was prosecutable for any Scot (read Highlander) to wear a tartan. Those tartans we see ascribed to clans were creations made to please Queen Victoria. Tartans and the Kilt are from Scottish and Irish Clans; from the oppressed. Thus, the fun in the line comes from the fact that an authentic tartan was false to begin with, but that doesn't keep Nigel from lording the fact that Lew's argyle sox are not up to snuff. Kilts came from an earlier garment which covered more of the body than today's piece, and those in plaid were called Breacan, meaning partially colored or speckled. The plaids also came in trews (trousers), and ruanas (shawls). Many had uniformity in design, but probably because those were the colors available and thus recognized as part of a family, clan or sept. Caen stone A cream-colored limestone for building, found near Caen, France. a primitive wind instrument consisting of several parallel pipes bound together; panpipes. an ancient form of harp, so syrinx and lyre are like flute and harp. A famous Concerto for flute and harp is the work of G. F. Handel, who also composed the Messiah. Ten sideshow acts for one admission. Wikipedia Also, a description of the Tetractys. masses of shadow . . . bright presences We've had suggestions, at least, that shadow is more hospitable than brightness. humans reincarnated as cats, dogs, and mice Do the T.W.I.T. members just take the word of the creatures, or do they have some way to be sure? Nicholas Nookshaft Grand Cohen Nicholas Nookshaft's name reinforces the linking of T.W.I.T. to the female sex organ, "Nooky shaft" being a vulgarism for the vagina. Interestingly, "shaft" is both a rod or pole, or penis, as well as a vertical passageway, thus its connations are bisexual. Anyone familiar with Ceremonial Magick is aware of Aleister Crowley. Crowley was famously bisexual, responsible for one of the most famous Tarot Decks — the "Thoth" deck — and was involved in spycraft for British Intelligence and, it is rumored, was a double agent for the Germans as well. Nicholas Nookshaft is a parody of Crowley. Actually, given the chronology and the alliterative name, this is much more likely a parody of MacGregor Mathers. Mathers was the head of the Golden Dawn from 1896 or so until 1900--Crowley never was. Furthermore, Tarot references in AtD do not follow the names from Crowley's Thoth deck; Crowley renamed certain cards, and those names are not the ones used in AtD (i.e. in the Thoth deck, the "Temperance" card is renamed "Art"). Grand Cohen 'Cohen' is Hebrew for 'priest'. Page 221 Couldn't have been the same world as the one you're in now We can infer that Lew got blown up in one world and shifted to another. A review of the explosion episode, particularly with the annotations to p. 188, will be worthwhile. "Lateral world-sets, other parts of the Creation, lie all around us, each with its crossover points or gates of transfer from one to another, and they can be anywhere, really." Could this be the explanation for some of the most inexplicable scenes from the book thus far: Lew Basnight's mysterious offense, causing him to lose his wife, and his first encounter with the Drave group (around page 39); and Hunter Penhallow's escape from the mysterious creature (around page 154)? Parallel worlds? Yashmeen Halfcourt Her initials YH are the first half of the Tetragrammaton -- YHVH or YHWH in English. seventeenth degree Adept Masonic and other esoteric mystery schools have differing number of degrees. Attaining a degree shows that one has sufficiently mastered the material, undergone the tests and passed through any initiations involved with that degree. The Masonic system has three degrees. These are extended to 32 in the Scottish Rite and a 33rd degree is the ultimate akin to a Distinguished Service award. By comparison, the Golden Dawn has 11 degrees divided in three orders; and the Order of the Temple of the East (Order Templi Orientis, O.T.O) has 12. In TWIT, the 17th appears to be the final degree where one becomes a Master TWIT or a Grand TWIT, I suppose. Why 17 degrees? Other than 17 being prime, there seems to be no symbolic or geometric significance to 17. Since the Crowley-associated systems do not reach 17, whereas the Masonic system does, looking to the Masonic A & A Scottish Rite 17th degree we find it is the "Knight of the East and West" which teaches that loyalty to God is man's primary allegiance, and the temporal governments not founded upon God and His righteousness will inevitably fall. Compare this to the Bogomils later in AtD. On the other hand, T.W.I.T. is centered on Tarot cards, so the relationship between number and any correspondences to the Tarot would be very much to the point. In this case, the Major Arcana assigned to the number 17 is the Star. The Crowley-associated system for Tarot consists of the Thoth Tarot deck, along with Crowley's "explanatory" 'Book of Thoth': The full text can be found at's site on the Thoth "Star" card, albiet with the wrong card illustrated, in this case atu 18, "The Moon". Symbolic and Cultural Meanings of 17: Because 17 has no symbolic significance, it does! In The Illuminatus! Trilogy, the symbol for Discordianism includes a pyramid with 17 steps because 17 has "virtually no interesting geometric, arithmetic, or mystical qualities." The number of special significance to Yellow Pig's Day and Hampshire College Summer Studies in Mathematics. and on and on..... A righteous Jew. Wikipedia "One whose merit surpasses his iniquity." The Talmud says that at least 36 anonymous tzadikim are living among us at all times; they are anonymous, and it is for their sake alone that the world is not destroyed. The common theme between the Masonic 17th degree and Tzaddik seems to be righteousness. Page 222 The Tetractys isn't the only thing round here that's ineffable Schoolyard joke. "F" a euphemism for fuck, so "ineffable" = unfuckable also describes Yashmeen. squadron commander A squadron of hussars would number 100-200 troopers commanded by a major. (The linked page concerns Baden-Powell's regiment—the 13th, not the 18th—in the South African War.) Auberon Halfcourt Auberon means royal or noble bear. Punning, "Au" is the chemical symbol for gold, thus, "Golden Bear", mascotte of UC Berkeley. Eighteenth Hussars Prestigious British cavalry regiment. Stationed in India 1864-76 and 1890-98; Halfcourt's secondment must have taken place at one of these times. Summer capital of the British Raj in India in the Himalayas. Wikipedia. A terminus of the Kalka-Simla railway line (built 1906) aka the "British Jewel of the Orient." Named for the goddess Shyamala Devi, an incarnation of the Hindu Goddess Kali. Smartly taken at silly point A cricketing reference. Silly point is a fielding position very close to the batsman. examples There are dozens of named fielding positions, but those called 'silly' (silly mid on, silly mid off, and silly point) are all close to the batsman, and therefore dangerous - fielders in these positions often wear protective helmets. The (very British) concept of sillyness was much explored by Monty Python's Flying Circus. To know, to dare, to will, to keep silent Mystical formula. examples The four precepts of Western Magick, extensively discussed in the writings of Aleister Crowley. In the States, "detective" doesn't mean— . . . An agent who solves criminal cases. The major "detective" bureaus hired personnel out as bodyguards and muscle. "There is but one 'case' which occupies us" This echoes the famous quote from Wittgenstein's Tractatus Logico-Philosophicus: "The world is all that is the case." (See the full text of the Tractatus here.) This quote also factors in heavily in V. (Specifically, in two places: there's the P's and Q's love song, and also in Captain Weissman's repeating, encoded, hallucinated message over the telegraph in Africa.) The Number 22 I found it interesting that the significance of the number 22 was first brought up on page 222. might be nothing, really. 22 is the number of cards in the Major Arcana of the Tarot deck, the section of the deck that has been removed from the modern playing deck which only has the suits (elements) and the Court cards. The 22 Major Arcana are numbers 0 to 21 and move from The Fool card to the Universe. Purportedly and symbolically, the progression of cards tell a tale of the evolutionary path of the Soul in its course. The 22 cards also, in some systems, map onto the 22 paths that connect the spheres of the Kabalistic Tree of Life (which also is mentioned in this chapter). An understanding of the Tarot cards cannot be achieved with an understanding how they relate to the Tree of Life. They are the relationships between the Sephiroths which in turn at 10 in number, just like the Tetractys and portray the energies that flow from the highest monad of Divinity (Kether) down into the manifested world (Malkuth). Pynchon makes use of both the Tarot and the Kabalah in Against the Day as well as Gravity's Rainbow. See also the novel The Greater Trumps by Charles Williams for a similar intrusion of the characters of the Major Arcana into everyday English life. 22 is two times two, so a quaternion... Page 223 "And the crime... just what would be the nature of that?" Might Lew himself be one of the 22 suspects? Perhaps the ineffable crime is what made people treat him like a pariah earlier in the book. Page 224 "'walking out'" A walking date. the veil of maya In Hinduism, maya is the phenomenal world of separate objects and people, which creates for some the illusion that it is the only reality. In Hindu philosophy, maya is believed to be an illusion, a veiling of the true, unitary Self. Many philosophies or religions seek to "pierce the veil" in order to glimpse the transcendent truth. Arthur Schopenhauer used the term "Veil of Maya" to describe his view of The World as Will and Representation. Wikipedia entry the ancient London landscape . . . known to the Druids Peter Ackroyd's recent London, the Biography devotes many pages to sacred and magical features of the city. "Druid". London's royal barbers since 1875. site And what other barber would you mention in a passage about the Greater Trumps . . . . A sentiment echoed in the first sentence of Pynchon's December 2006 letter written in defense of novelist Ian McEwan: "Given the British genius for coded utterance..." Image of Letter crosswords in newspapers The first crossword to appear in a newspaper was in 1913. Cryptic crosswords in British newspapers certainly match Pynchon's description. See, for example, the Listener crossword. Page 225 Girton College Of Cambridge University, for women, founded 1869. history Next they'll be letting you folks vote. Women over the age of 30 were, subject to certain qualifications, granted the right to vote in the UK by the Representation of the People Act 1918. The Representation of the People (Equal Franchise) Act 1928 granted women the vote on the same basis as men (i.e. from the age 21). "the vast jangling thronged somehow monumental London evening" This kind of eschewing of punctuation might be expected in Joyce but it's not typical of Pynchon and seems to serve no special purpose here. A typo? Purposive or no, that ain't no typo. First, numerous compound adjectives reminiscent of Faulknerian portmanteau words are sprinkled throughout the book. Second, this particular deployment of zero-degree punctuation and massing of modifiers jibes with TRP's obvious delight in tripping us readers up and sending us back into sentences for another looksee. Finally, the musicality of this phrase sounds properly Pynchonlike t'me. Pamela Colman Smith Illustrator of the Rider-Waite-Smith Tarot deck "Wikipedia". Arthur Edward Waite Occultist and co-creator of the Rider-Waite Tarot deck. Wikipedia four stone 56 pounds. Ucken-fay is "pig latin" for 'fucken'. gaver du visage A literal translation of "stuff one's face", though this is not how it is said in French (it would be se gaver or se baffrer). cite A smoking salon (divan) for cigar smokers. Interestingly, a work by Robert Louis Stevenson, from 1903, entitled The Dynamiter begins with a "Prologue of the Cigar Divan". Page 226 Seven Dials bad area in London, see Wikipedia entry The Devil by Colman-Smith Four-wheeled carriage drawn by four horses. Supplanted by the Hansom cab. Renfrew at Cambridge and Werfner at Göttingen Note that each Professor's name is the other's spelled backward. Also notice the theme of dual natures or forces. The two professors are "bound and ... could not separate even if they wanted to." They become rivals within the broader conflict of the 'Great Game' -- the political rivalry over Central Asia being played out by the various European powers, but especially by Great Britain and the Russian Empire. Pynchon toys with the idea that World War I was really just the extension of an academic rivalry. This secret scholastic conspiracy also references the role supposedly played in the US policy establishment by neoconservatives [3] (or "neocons") in the run up to the US invasion of Iraq in 2003. Just as Pynchon's professors held great influence over a number of their students, "[s]ome of whom found employment with the Foreign Services", etc., neoconservative professors such as Leo Strauss [4] had a number of disciplines who came to occupy key positions in government and business (for example, Deputy Secretary of Defense (2001-2005) Paul Wolfowitz [5]). This interpretation is further bolstered by the geographic positioning of the "Bagdad" (sic) railway, and the Ottoman territories as the region "where Renfrew and Werfner have often found their best opportunities to make mischief". Cambridge University is one of the oldest and the best universities in the world. In 2009 it will be celebrating its 800th Anniversary. In its early day, Cambridge was a center of the new learning of the Renaissance and of the theology of the Reformation; in modern times it has excelled in science. It is now a confederation of 31 Colleges (such as King's, Girton, St.John, Trinity and others mentioned in ATD), consists of over 100 departments and faculties, and other institutions. Since 1904, 81 affiliates of Cambridge have won Nobel Prize in every category: 29 in Physics, 22 in Medicine, 19 in Chemistry, 7 in Economics, 2 in Literature and 2 in Peace. Göttingen University, one of the most famous universities in Europe, founded in Göttingen, Germany, in 1737 by King George II of England in his capacity as Elector of Hanover. At the end of the 19th century, it became world famous because of its Departments of Mathematics and Physics and rivaled Cambridge for eminence. The reputation of the university was founded by many eminent professors who are commemorated by statues and plaques all over the campus. It claimed 44 Nobel Laureates. But it suffered from the 1933 Great Purge of the Nazi crackdown on "Jewish Physics" and never recovered its original fame. David Hilbert, one of the greatest mathematicians of the 20th century and a professor at Göttingen, was asked in the 40s about the state of mathematics there now that the Jewish influence had been purged; he replied that there was no mathematics left at Göttingen now that the Jewish influence had been purged. Berlin Conference of 1878 Divided Balkans after Russo-Turkish War. Wikipedia bickering-at-a-distance A play on the idea of "action at a distance" theories in physics, a topic that came under much scrutiny at the time of AtD owing to its pertinence in the theories of electromagnetism and gravitation. See wikipedia for a further discussion and its relevance in quantum mechanics. English, . . . , Japanese—not to mention indigenous—components Not to mention them was exactly the point as the Great Powers sorted out the Ottoman possessions. Page 227 "The Great Game" The Great Game was a term used to describe the rivalry and strategic conflict between the British Empire and the Tsarist Russian Empire for supremacy in Central Asia. The term was later popularized by Rudyard Kipling in his novel, Kim. The classic Great Game period is generally regarded as running from approximately 1813 to the Anglo-Russian Convention of 1907. Wikipedia entry Also the name of Padzhitnoff's airship. I believe the great game stands for Espionage in the Age of Gentlemen, the substance of Pynchon's Under the Rose. mamluk lamps A mosque lamp from the mamluk era. ...the Kabbalist Tree of Life, with the names of the Sephiroth spelled out in Hebrew, which had brought her more than enough of that uniquely snot-nosed British anti-Semitism... Kabbalah is the ancient study of Jewish mysticism, long shrouded in mystery and kept from all but a devout few of the most dedicated Talmudic scholars. The Tree of Life is one of the central symbols of Kabbalah, supposedly a physical representation of the path of enlightenment from the most base knowledge of the physical world (at the bottom), to the highest spiritual planes of understanding (at the top). The Sephiroth are the nodes of the Tree, representing the various "stages" of understanding. Of course, tihs is all a very gross oversimplifcation and hardly does justice to the term itself. The "Quabbalah" or "Cabalah" being studied by Madonna and others in Hollywood is a secularized and co-opted form of the original Kabbalah, which is deeply connected to the Torah and Jewish life. In Medieval Europe, Kabbalist scholars wore amulets and other symbols on their clothing, and were often misunderstood to be magicians or wizards (think Merlin). The common magician's expression "abra cadabra" has Kabbalistic origins. "Eskimoff . . . I say what sort of name is that?" Tiptoeing around the real question, "Is she Jewish?" English Rose The phrase "English Rose" or "Bonnie English Rose" when applied to a woman means her skin is unblemished, her coloring subtle, her temper sweet. Madame Eskimoff, in short, is a beauty in a traditional English style. (Incidentally, an officially unrecognized designation of roses.) Page 228 Oliver Lodge English physicist, inventor and writer (1851-1940) involved in the development of wireless telegraphy and radio. After the death of his son in 1915, Lodge became interested in spiritualism and life after death and wrote several books on the subject. Lodge conducted research on lightning, electricity, electromagnetism and wrote about the aether, themes that are repeated throughout ATD. Wikipedia entry. William Crookes English chemist and physicist (1832-1919) who worked in spectroscopy and whose work pioneered the construction and use of vacuum tubes. Like Oliver Lodge, Crookes was also a spiritualist, which appears to be Pynchon's reason for grouping him with others in this passage, although his experiments in electricity and light also tie in with these themes in ATD. Wikipedia entry. Mrs. Piper Probably Leonora Piper, 1857-1950. Wikipedia entry. Eusapia Palladino (1854-1918) Famous italian spiritualist medium. Wikipedia entry. It's fair to say she was often caught cheating. W.T. Stead William T. Stead (1849-1912), British writer, poet, social crusader, and spiritualist. He went down with the Titanic. Wikipedia entry. Mrs. Burchell The Yorkshire Seeress, investigated by WT Stead. cite Trouble with the time here. Lew's timeline points pretty strongly to autumn 1900. A séance that's "about to" go on Mme. Eskimoff's résumé, however, leads the murder of the Serbian king and queen by three months, and the murder itself occurred in June 1903, which seems to imply March of that year. This seems as good an instance as any to question the insistence of some here to pin down the exact date (and season?). Pynchon doesn't knock it to the wall, doesn't find cause to bother and I think the reason for that is obvious... the ambiguity lends a freer hand with which to paint. So don't fuck with the butterfly on the wheel. Alexander and Draga Obrenovich, the King and Queen of Serbia According to Wikipedia the assassination occured on 11 June 1903, so the seance at which Mrs. Burchell "witnessed" it, should have taken place in March 1903. Parsons-Short Auxetophone pic and info. The Auxetophone appears to have been a sound amplification device, not a recorder. Parsons did not enter the picture till 1903, so the apparatus would not have this name in 1900, but Short demonstrated it as early as 1898. electros of the original wax impressions A thin film of metal was electroplated onto the wax, then peeled off and wrapped around a new cylinder. "Bagdad" railway Page 229 A term used in both engineering and psychology. Psychology: "Characterized by a high degree of emotional responsiveness to the environment." Electricity: "Of or relating to two oscillating circuits having the same resonant frequency." The syntonic comma, a small interval in the frequency ratio of 81:80, is a problem in musical temperament. the Russo-Turkish War The Russo-Turkish War (1877-1878), the latest Russo-Turkish War of many fought between these two contries since 16th century as a result of Russian attempts to find an outlet on the Black Sea and to conquer the Caucasus, dominate the Balkan Peninsula, gain control of the Dardanelles and Bosporus straits, and retain access to world trade routes. The last Russo-Turkish War came as a result of the anti-Ottoman uprising (1875) in Bosnia and Herzegovina and Bulgaria. On Russian instigation, Serbia and Montenegro joined the rebels; after securing Austrian neutrality, Russia openly entered the War in 1877. The War ended in 1878 resulted in the Treaty of San Stefano which so thoroughly revised the map in favor of Russia and her client, Bulgaria, that the European powers called a conference (the Congress of Berlin) to revise its terms by the Treaty of Berlin. kilometric guarantee Money offered by the government to building companies. Apparently, the railroad companies fooled the Ottoman Empire building trails which were much longer than needed. Google books citation Page 230 King's... Girton King's College is one of the most famous and historic colleges at Cambridge, founded in 1441. Girton College, Cambridge, was established in 1869 as the first residential college for women in England. Michaelmas term The fall term, starting early October (1900 here). Wikipedia A between-maid. Edward Oxford attempted to shoot Queen Victoria and her husband, Prince Albert, at the time of her first pregnancy (1840).Wikipedia had the young Queen died then without issue Nookshaft posits two scenarios: (1) The implicit, unmentioned, and not as "interesting" possibility that everything is actual, as it "appears" to be in the "real" world, surrounding Queen Victoria; that she is simply an old, vain regent. (2) "the 'real' Vic is elsewhere," and the current, aged Victoria is a ghostly stand-in. Nookshaft implies that this figure is a proxy or puppet of Ernst-August. If this were "the case," then the question shifts to the following: (a) Is the ruler of the underworld, who holds the "real," eternally young Victoria captive in cahoots with Ernst-August in the "real" world? or: (b) Is the ruler of the underworld, who holds the "real," eternally young Victoria captive NOT in cahoots with Ernst-August, who nevertheless ascends to the throne with real-Vic out of the way, and imposes the stand-in? In which case: What would be the motivation of the underworld-entity third-party? And who, or what, specifically, is it? sixty years ago One event of 1840, the attempt on Victoria's life, is referred to as sixty years ago; another, the issue of the first adhesive stamps, as more than sixty years ago. If it weren't for these nagging problems in Lew's timeline, we could peg the date as 1900. Salic law originated in the Late Roman Empire as Germanic tribes invaded and their law codes were translated into Latin and written down. Salic Law was that of the Franks who settled in present-day northern France and the law code of Charlemagne. Over the course of the Middle Ages it was largely replaced by Roman Law. For examples, see [6]. However, Salic Law continued to be used in a number of European areas to decide matters of noble inheritance. Specifically, Salic Law stated that no female could inherit rulership (above by Owl of Minerva 18:03, 4 April 2007 (PDT)) and, indeed, a royal or noble title could be inherited only through the "male line." When King William IV, ruler of both the United Kingdom and Hanover, died, the Crowns separated. Hanover practiced Salic law, while Britain did not. King William's niece Victoria ascended to the throne of Great Britain and Ireland, but the throne of Hanover went to William's brother Ernest Augustus, Duke of Cumberland. Wikipedia entry Tory despotism Not necessarily-- it describes Ernest himself. "The Duke of Cumberland had a reputation as one of the least pleasant of the sons of George III. Politically an arch-reactionary, he opposed the 1828 Catholic Emancipation Bill proposed by the government of the Prime Minister, the Duke of Wellington." Wikipedia entry It can describe Ernst August and still be an allegory of Thatcher. The description of Ireland fits that of some world-views during her time. All parallels between past and present are worth considering. They don't have to be direct references. The present-day Ernst August - famous for pissing on the Turkish Pavilion at EXPO 2000 - carries on the family tradition. Someone famously cited James Joyce as proof that Catholics shouldn't get university educations. Page 231 Orange Lodges Lodges of the Orange Order, a protestant fraternal organisation based predominantly in Northern Ireland and Scotland. Wikipedia entry. The Orange Order was founded to subvert the United Irishmen of Wolfe Tone by agitating against Protestant and Catholic community. It was hostile to the idea of Irish Home Rule or independence. In the 1880's it developed the Ulster Unionist Party to politically parry Parliamentary attempts at Home Rule for Ireland. "from the first to the twelfth of July, anniversaries of the Boyne and Aughrim." i.e. anniversaries of the Battle of the Boyne and the Battle of Aughrim of the Williamite War in Ireland. This was and still is known as "Marching Season" in Northern Ireland; the time when 'parades' are traditionally a source of fear and violence. Nearly all the parades are organized by the Orange Lodges and hence anti-Catholic. The first adhesive stamp, 1840 "the first adhesive stamps of 1840" This stamp has come to be called the Penny Black. Wikipedia entry Penny Black is also the name of a character (p.18) "immune to Time, [...] neither of them aging" Cf Oscar Wilde's only novel The Picture of Dorian Gray, in which Dorian Gray remains young while his portrait ages. Cf Stray's pregnancy, a "dreamy thing" (page 201). The definition of springtide is springtime. Page 232 Éliphaz Lévi A/K/A Eliphas Levi, nom de plume of Alphonse Louis Constant (1810-1875), French occultist and writer who pioneered a revival of Magick in the 19th Century, and was an influence on A.E. Waite, the Order of the Golden Dawn, and Aleister Crowley. An acquaintance of novelist Edward ("It was a dark and stormy night") Bulwer-Lytton. Wikipedia entry. Punter is being used in the sense of someone who bets, someone who is taking a chance. Or more probably in the common extended sense meaning merely "customer" Greek: things heard. Good information under "A" in the alpha index. number twenty-four Or 25? etext (According to a Greek version, number 4 in the etext above is not included in Iamblichus' list. If my source is correct, Pynchon is right.) (ca. 245 - ca. 325, Greek) was a neoplatonist philosopher who determined the direction taken by later Neoplatonic philosophy, and perhaps western Paganism itself. He is perhaps best known for his compendium on Pythagorean philosophy.Wikipedia Make-up, cosmetics; the application of make-up (especially in heavy or theatrical fashion).[2] Page 233 Inflammation of a mucous membrane; usually restricted to that of the nose, throat, and bronchial tubes, causing increased flow of mucus, and often attended with sneezing, cough, and fever; constituting a common ‘cold’.[3] Collis Brown's Mixture Contained morphine, chloroform, and caramel, among other things. Full ingredients (Previous link not working. For info try here.) Xylene abuse is similar to "glue sniffing"-- xylene is a strong solvent able to cause several damages to health, especially to the brain. wikipedia a thousand pounds a year Over $100,000 today. cite Condy's fluid is pink to purple. Methylated spirits is a kind of denatured alcohol: 95% ethyl alcohol, 5% methyl alcohol. "Pinky" would have a variety of effects, very possibly including blindness. Page 234 Condy's fluid A disinfectant used to treat and prevent Scarlet Fever, among other things. Wikipedia tonight's the night Considering the content here, probable reference to Neil Young's drug-addled album and its title song, "Tonight's The Night" from 1975. Wiki an important market street in the City of London. A street originally for stabling; but in modern times often converted into houses/apartments. Coombs de Bottle "comes the bottle" ? Russian duck Duck is strong, untwilled linen or cotton, lighter and finer than canvas. Russian duck is coarse, heavy and unbleached but softer than English duck. Page 235 sensitive flames Cf GR p.29-32, 715. extractors . . . distillation columns Separatory apparatus. An extractor works on differences in solubility, a distillation column differences in volatility. tremblers and timers A trembler is a kind of motion detector used in both bombs and alarms; one kind has a flexible stem with a heavy contact on the free end so that disturbing the package it contains causes a trigger circuit to close. A timer uses a clocklike mechanism to bring two contacts together. proper solvent procedures Famous 1960s "Anarchist Cookbook" was infamously inaccurate. Amazon w/author's note Page 236 Breathless hush in the close tonight Dr. De Bottle quoting from Henry Newbolt's poem "Vitaï Lampada," which makes school games a metaphor and model for martial bravery. The Gentleman Bomber of Headingly Cf Hornung's 'Gentleman Thief' and cricket player, Raffles. info Reminds me of the Krikkit Robots in Douglas Adams' Life, The Universe, and Everything, where a bomb is put in place of a Cricket Ball at a match between Britain and Australia. Also, acronymically, the GBH=Grievous Bodily Harm, the British term for felonious assault. Here and elswhere the spelling of the cricket ground should be 'Headingley'. The Ashes An international cricket series between England and Australia dating back to 1882. dates A number of references in this chapter relate to this rivalry. For example, on this page the English cricket ball is compared to the Australian "kookaburra". Kookaburra is the brand name of the balls used in Australia, in England it's Duke. The properties of the English ball was one of the keys to England's success in the summer of 2005. Was Pynchon's writing here influenced by the hype in the UK at the time? A poison gas used in World War I.Wikipedia Source of red dye. Wikipedia A helper, assistant. [4] Misspelling of exhilaration. Page 237 beige substance Presumably Cyclomite. Happy Birthday! . . . Gemini Ordinarily you would think this tagged the date as 21 May to 20 June Wikipedia. But other evidence in the text points to deepening autumn. One of two possible explanations: 1. The T.W.I.T. is perhaps using an ascendent or lunar based astrological system rather than the solar-based system commonly used in the West. This resolves the apparent contradiction of a Gemini in autumn since the ascendent travels through all signs every 24 hours and the moon travels through the entire zodiac once a month. For example, Vedic astrology looks primarily to the ascendent, then the moon, and lastly the sun to study respectively the body, the mind and the spirit of the native. Basnight does have a mind that operates on two planes -- hence a moon in Gemini reading. 2. The explosion carried Lew to a place on the other side of the Sun. Deep autumn would then be November 23 to December 21th, our sign of Sagittarius. get the Ashes back . . . next year On page 236 the Ashes (Test Matches, cricket competitions between England and Australia) are "in progress." At some time previous to this conversation Mme. Eskimoff said England will regain the trophy "next year" provided they use the young bowler Bosanquet (next entry). Test Matches took place in (a) December 1901 to March 1902, Australia victorious; (b) May to August 1902, Australia again; (c) December 1903 to March 1904, England bringing back the Ashes and Bosanquet figuring as a key bowler. If Mme. Eskimoff has foreseen aright, "next year" is 1904 and the time of the action is 1903. The conflict in dates is troubling: In a matter of weeks and a few pages, Lew just misses the 1900 Hurricane and gets information that definitely points to 1903. (And he proves to be a Gemini with an autumn birthday!) I don't think there is anything accidental—or negligible—about the discrepancies. Another Ashes reference. Bernard Bosanquet invented the bosie (or googly), as described here, around 1900. A major factor in England's 2005 Ashes success was reverse swing, another type of delivery whose physical dynamics are poorly understood. Check out the "Cricket in Against the Day article by Peter Vernon, which is an in-depth look at, well, cricket in Against the Day. A somewhat derogatory term for a British person, commonly used in Australian English. Also Pommy or Pommie. Hebrew letter Shin Obviously a nod to the Vulcan greeting in Star Trek, with the distinctive hand sign and the phrase, "Live long and prosper." Perhaps also to the Jewish faith of Leonard Nimoy, who played Spock. See The Jewish origin of the Vulcan Salute Pynchon placed one of these in Mason & Dixon, as well: Dixon discovers "The Rabbi of Prague, headquarters of a Kabbalistick Faith, in Correspondence with the Elect Cohens of Paris, whose private Salute they now greet Dixon with, the Fingers spread two and two, and the Thumb held away from them likewise, said to represent the Hebrew letter Shin and to signify, 'Live long and prosper.'(M&D p.485) Might there be a further connection between The Cohen of T.W.I.T., the "Cohens of Paris" and these backwoods Kabbalists? Also, note the hand on the devil tarot card above. Shin "also stands for the word Shaddai, a name for God. Because of this, a kohen (priest) forms the letter Shin with his hands as he recites the Priestly Blessing. In the mid 1960s, actor Leonard Nimoy used a single-handed version of this gesture to create the Vulcan Hand Salute for his character, Mr. Spock, on Star Trek."[7] ...and if we look back to the Devil tarot card we see the shin hand sign and the inverted pentagram. Thus through Eliphas Levi and then Coleman-Smith/Waite a connection is created between shin and the inverted pentagram. And then we can make connections with the Jeshimonians and the TWITsters. This might be why "the cure grows right next to the cause" in Jeshimon. They are under the winged protection of God-the-Destroyer. British term indicating complete ordinariness. Possible Etymology (Dog's Bollocks, British Or German Standard): Wiktionary Page 238 Second Law of Thermodynamics The law of entropy... "The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium." (Rudolf Clausius) [9] There's no such thing as a perfectly efficient engine, i.e., a box that does work by taking in heat from where there is lots of heat (e.g., combustion chamber) and throwing off heat where there is not much (exhaust pipe). Something always gets lost. Similarly, the transfer of money from where there is plenty (bank) to where there isn't much (Europe) is never perfectly efficient. "He began then, bewilderingly, to talk about something called entropy. The word bothered him... But it was too technical for her. She did gather that there were two distinct kinds of this entropy. One having to do with heat engines, the other to do with communication... The two fields were entirely unconnected, except at one point: Maxwell's Demon. As the Demon sat and sorted his molecules into hot and cold, the system was said to lose entropy. But somehow the loss was offset by the information the Demon gained about what molecules were where... Entropy is a figure of speech, then, a metaphor. It connects the world of thermodynamics to the world of information flow." The Crying of Lot 49 (Pages 84 - 85) morsus fundamento Latin: A bite on the ass? The meaning is that he wouldn't know metaphysics if it bit him in the ass. Like "octogenarihexation" ("86"-ing) in Vineland--the vulgar faux fancied up. three-percent consols British "consolidated" bonds, for many years the conservative investment par excellence. wikipedia Page 239 Not mental as in "of the mind" but mental as in "mad". "You're mental, you are" is a common british playground taunt. Colney Hatch London lunatic asylum. Wikipedia Out of the dust . . . beam of morning sunlight I.e., sometimes your horse wins. An encyclical is a letter circulated by the pope or other figure of high authority in a body of believers. A comprehensive Wikipedia article explains and adds a list of papal encyclicals. An encyclical usually takes its first 2 or 3 words as its title (Multi et Unus in this case). Of course, the Vatican would strongly protest that McTaggart, an atheist, should send out an encyclical! Seems to refer to a historical logician joke. explanation Professor McTaggart was, perhaps, the most famous philosopher who argued that Time did not exist as we seem to experience it. W.H. Hardy was a very famous Cambridge mathematician who knew all the famous philosophers in England. John McTaggart Ellis (J. M. E.) McTaggart (1866-1925), British philosopher. He was born in London and educated at Clifton College, Bristol and Trinity College, Cambridge. He lectured Philosophy at Trinity College from 1897 to 1923. His brilliant commentaries and studies on Hegel's dialectic (1896), cosmology (1901) and logic (1910) were preliminaries to his own constructive system-building in Nature of Existence (3 vols., 1921-1927). In his 1908 essay "The Unreality of Time" he argued that our perception of time is an illusion (Cf page 412: dismissing . . . the existence of Time). Godfrey Harold Hardy (1877-1947), English mathematician. He was a lecturer at Cambridge (1906-1919), professor at Oxford (1919-31) and Cambridge (1931-47). Concurrently with Wilhelm Weinberg developed Hardy-Weinberg law (1906) describing genetic distribution and dequilibrium in large populations. He was also known for contributions to complex analysis, Diophantine analysis, Fourier series, distribution of prime numbers, etc. Multi et Unus Many and One. Is the grafitti in Cambridge another cricketing reference? Dukes are the balls used in England (cf. p236). Chucking (or bending the arm when bowling) is an emotive topic in cricket that arises from time to time. It first arose around 1900 [10]. In 2005 it caused administrators to change the rules of the game [11]. "Create More Dukes" has a second meaning, suggested by the odd choice of verb. Duke in Britain refers to the highest rank of nobility, and fittingly there are not many of them. At present only about a dozen people hold the title. Since sometime in the 1870s new dukes have been created (by decree of the monarch) only in the royal family. Most recently at the time of the action, Queen Victoria had promoted a run-of-the-mill marquess to the dukedom of Fife to set the stage for his marriage to one of her granddaughters. If some group of activists thought the nation needed to beef up its peerage, they might adopt the slogan found here as a graffito. Here is a colorful summary of UK dukes today and through history, although it is unsound on coats of arms and such. This site has more names and fewer pictures, listing all the titles (from dukes to lowly barons) created since the year Dot. the Laplacian, a relatively remote mathematicians' pub A little Pynchonian joke? The Laplacian operator is a component of the Schrödinger equation, the basis of quantum mechanics. Quantum mechanics was famously rejected by Albert Einstein (many references on the net but see Stephen Hawking), known for his theories of relativity. Moreover, quantum mechanics deals with the very small and relativity with the very large (this is a simplification of course), so the Laplacian is indeed remote from relativity! No such pub during my stay in Cambridge (1998-2000). Also not today, according to this list. Obviously more than a little joke. Refers to Pierre-Simon, marquis de Laplace (1749-1827)Wikipedia Entry, aka "the French Newton", probably the greatest mathematician and astronomer of his time. Most of the scientific principles derived from his findings are explored in AtD (from lumineferous ether to the existence of black holes). Laplace was also instrumental in the advancement of the science of probabilities. The ever quotable Laplace, much loved by atheists worldwide, famously replied to Napoléon, when he asked why there was no mention of God in his treatise on astronomy: "Sir, there was no need for that hypothesis". Also responsible for what is known as the Laplacian principle: "The weight of evidence for an extraordinary claim must be proportioned to its strangeness." The literal translation of Laplace is "The Place". The connection of the Laplace operator to the Schrödinger equation and quantum mechanics is a bit of a stretch -- the Laplace operator is ubiquitous, appearing in the heat equation, the wave equation and the Navier-Stokes equations. Page 240 Worse than Gordon at Khartoum Refers to Charles George Gordon, British Major-General, whose attempted defense of Khartoum versus Arabi rebels in 1884-85 ended with his beheading. Wikipedia cf. Basil Dearden's 1966 film Khartoum, in which the role of Gordon is played by Charlton Heston. Page 241 "You recognize him?" As, presumably, Webb. How can that be? Webb is dead, there's nothing to suggest he went to England, the costume is not right for him, and—most tellingly—his medium is dynamite, not phosgene. Who might Lew recognize in the photo? The "suspects" are Neville, Nigel, the Grand Cohen, Dr. Coombs De Bottle, Clive Crouchmas and Professor Renfrew. If Prof. Werfner looks much like Prof. Renfrew, he goes on the list too. If the "Gentleman Bomber" could possibly be female, add Yashmeen and Mme. Eskimoff. We haven't met anyone else (except members of the Icosadyad, who don't have faces). Suppose we rule out the ladies and Werfner. Neville or Nigel wouldn't be able to hide their identities with a suit of white flannels. Renfrew is sitting right there when Lew sees the picture, but Lew's reaction (his stomach sinks) does not seem Lew-like if it's Renfrew he has recognized, plus Renfrew himself wants to meet the Bomber. That leaves the Cohen, De Bottle and Crouchmas. Would Lew experience dread on spotting Crouchmas? He doesn't know much about C.C. at this point, so it isn't clear why he would suppress that recognition. Seeing the Cohen might lead to this gastric reaction: Lew might think he's on the fringe of an anarchist group again (and look where it got him the last time). The Cohen stays on the list. Dr. De Bottle not only follows cricket but bets on it; he speaks almost with reverence about phosgene; he knows a nonobvious fact about the bombs; and he dresses like a gentleman. None of these points applies to the Cohen. And recognizing De Bottle would give Lew that sinking feeling because D.B. is purportedly fighting against bombers on behalf of the government. De Bottle goes to the top of the short list. Alternately, there's no clear answer and not enough clues (especially considering the role of time, forces beyond anyone's control, double agents, etc.). This Gentleman Bomber can be any person from Lew's past or a deja vu from the future. The G. Bomber seems to be England's answer to the Kieselguhr Kid, a nebulous personality working against the forces of history. The important thing about this situation is not the Bomber's identity, but the fact that Lew is being thrown into an assignment much like his last one in America (and we know how that ended...) He's obviously not very happy about it, and not inclined to tell anyone what he knows, or might know. For what it's worth, my take was also Webb, especially in the context of all the bilocation business. It isn't "Webb" but evil alter-land "Webb"! (no dig on ya'll Brit folk intended, although the " stay at Cambridge" bit was just wonderful :) --There is a suspect that was left off that list, who occurred to me before any of the others--Lew himself. If we're dealing with bilocation, doubles, and the possibility that 'our' Lew was brought here via the explosion in the creek bed, couldn't the G.B.H. be this world's Lew? Recall that just prior to the explosion, Lew had resolved to choose a side in the Anarchist/plutocrat battle, and had come down on the side of the people. Did this world's Lew make the same choice, somehow ending up in Britain... Also conspicuous is that it is Renfrew showing him the photo with a sly expression, Renfrew who's own double, Werfner, is his very own nemesis? That's how I read it anyway. But, much like the Kieselguhr Kid, we the readers never actually know the identity of this renegade bomb-lobber. A bosie from a beamer More cricket! A bosie is now more commonly known as a googly (cf. p237). A beamer is a full-pitched delivery that reaches the batsman above waist height. Page 242 The northern hemisphere German: uncanny, sinister. 1. From The Secret Teachings of All Ages by Manly P. Hall (1928) 2. The Oxford English Dictionary. 2nd ed. 1989 3. Def.3. The Oxford English Dictionary. 2nd ed. 1989. 4. Def.1. The Oxford English Dictionary. 2nd ed. 1989. Annotation Index Part One: The Light Over the Ranges Part Two: Iceland Spar Part Three: Part Four: Against the Day Part Five: Rue du Départ Personal tools
28342aa42c1b8994	It seems that quantum computing is often taken to mean the quantum circuit method of computation, where a register of qubits is acted on by a circuit of quantum gates and measured at the output (and possibly at some intermediate steps). Quantum annealing at least seems to be an altogether different method to computing with quantum resources1, as it does not involve quantum gates. What different models of quantum computation are there? What makes them different? To clarify, I am not asking what different physical implementations qubits have, I mean the description of different ideas of how to compute outputs from inputs2 using quantum resources. 1. Anything that is inherently non-classical, like entanglement and coherence. 2. A process which transforms the inputs (such as qubits) to outputs (results of the computation). The adiabatic model This model of quantum computation is motivated by ideas in quantum many-body theory, and differs substantially both from the circuit model (in that it is a continuous-time model) and from continuous-time quantum walks (in that it has a time-dependent evolution). Adiabatic computation usually takes the following form. 1. Start with some set of qubits, all in some simple state such as $\lvert + \rangle$. Call the initial global state $\lvert \psi_0 \rangle$. 2. Subject these qubits to an interaction Hamiltonian $H_0$ for which $\lvert \psi_0 \rangle$ is the unique ground state (the state with the lowest energy). For instance, given $\lvert \psi_0 \rangle = \lvert + \rangle^{\otimes n}$, we may choose $H_0 = - \sum_{k} \sigma^{(x)}_k$. 3. Choose a final Hamiltonian $H_1$, which has a unique ground state which encodes the answer to a problem you are interested in. For instance, if you want to solve a constraint satisfaction problem, you could define a Hamiltonian $H_1 = \sum_{c} h_c$, where the sum is taken over the constraints $c$ of the classical problem, and where each $h_c$ is an operator which imposes an energy penalty (a positive energy contribution) to any standard basis state representing a classical assignment which does not satisfy the constraint $c$. 4. Define a time interval $T \geqslant 0$ and a time-varying Hamiltonian $H(t)$ such that $H(0) = H_0$ and $H(T) = H_1$. A common but not necessary choice is to simply take a linear interpolation $H(t) = \tfrac{t}{T} H_1 + (1 - \tfrac{t}{T})H_0$. 5. For times $t = 0$ up to $t = T$, allow the system to evolve under the continuously varying Hamiltonian $H(t)$, and measure the qubits at the output to obtain an outcome $y \in \{0,1\}^n$. The basis of the adiabatic model is the adiabatic theorem, of which there are several versions. The version by Ambainis and Regev [ arXiv:quant-ph/0411152 ] (a more rigorous example) implies that if there is always an "energy gap" of at least $\lambda > 0$ between the ground state of $H(t)$ and its first excited state for all $0 \leqslant t \leqslant T$, and the operator-norms of the first and second derivatives of $H$ are small enough (that is, $H(t)$ does not vary too quickly or abruptly), then you can make the probability of getting the output you want as large as you like just by running the computation slowly enough. Furthermore, you can reduce the probability of error by any constant factor just by slowing down the whole computation by a polynomially-related factor. Despite being very different in presentation from the unitary circuit model, it has been shown that this model is polynomial-time equivalent to the unitary circuit model [ arXiv:quant-ph/0405098 ]. The advantage of the adiabatic algorithm is that it provides a different approach to constructing quantum algorithms which is more amenable to optimisation problems. One disadvantage is that it is not clear how to protect it against noise, or to tell how its performance degrades under imperfect control. Another problem is that, even without any imperfections in the system, determining how slowly to run the algorithm to get a reliable answer is a difficult problem — it depends on the energy gap, and it isn't easy in general to tell what the energy gap is for a static Hamiltonian $H$, let alone a time-varying one $H(t)$. Still, this is a model of both theoretical and practical interest, and has the distinction of being the most different from the unitary circuit model of essentially any that exists. Measurement-based quantum computation (MBQC) This is a way to perform quantum computation, using intermediary measurements as a way of driving the computation rather than just extracting the answers. It is a special case of "quantum circuits with intermediary measurements", and so is no more powerful. However, when it was introduced, it up-ended many people's intuitions of the role of unitary transformations in quantum computation. In this model one has constraints such as the following: 1. One prepares, or is given, a very large entangled state — one which can be described (or prepared) by having some set of qubits all initially prepared in the state $\lvert + \rangle$, and then some sequence of controlled-Z operations $\mathrm{CZ} = \mathrm{diag}(+1,+1,+1,-1)$, performed on pairs of qubits according to the edge-relations of a graph (commonly, a rectangular grid or hexagonal lattice). 2. Perform a sequence of measurements on these qubits — some perhaps in the standard basis, but the majority not in the standard basis, but instead measuring observables such as $M_{\mathrm{XY}}(\theta) = \cos(\theta) X - \sin(\theta) Y$ for various angles $\theta$. Each measurement yields an outcome $+1$ or $-1$ (often labelled '0' or '1' respectively), and the choice of angle is allowed to depend in a simple way on the outcomes of previous measurements (in a way computed by a classical control system). 3. The answer to the computation may be computed from the classical outcomes $\pm 1$ of the measurements. As with the unitary circuit model, there are variations one can consider for this model. However, the core concept is adaptive single-qubit measurements performed on a large entangled state, or a state which has been subjected to a sequence of commuting and possibly entangling operations which are either performed all at once or in stages. This model of computation is usually considered as being useful primarily as a way to simulate unitary circuits. Because it is often seen as a means to simulate a better-liked and simpler model of computation, it is not considered theoretically very interesting anymore to most people. However: • It is important among other things as a motivating concept behind the class IQP, which is one means of demonstrating that a quantum computer is difficult to simulate, and Blind Quantum Computing, which is one way to try to solve problems in secure computation using quantum resources. • There is no reason why measurement-based computations should be essentially limited to simulating unitary quantum circuits: it seems to me (and a handful of other theorists in the minority) that MQBC could provide a way of describing interesting computational primitives. While MBQC is just a special case of circuits with intermediary measurements, and can therefore be simulated by unitary circuits with only polynomial overhead, this is not to say that unitary circuits would necessarily be a very fruitful way of describing anything that one could do in principle in a measurement-based computation (just as there exists imperative and functional programming languages in classical computation which sit a little ill-at-ease with one another). The question remains whether MBQC will suggest any way of thinking about building algorithms which is not as easily presented in terms of unitary circuits — but there can be no question of a computational advantage or disadvantage over unitary circuits, except one of specific resources and suitability for some architecture. • 1 $\begingroup$ MBQC can be seen as the underlying idea behind some error correcting codes, such as the surface code. Mainly in the sense that the surface code corresponds to a 3d lattice of qubits with a particular set of CZs between them that you then measure (with the actual implementation evaluating the cube layer by layer). But perhaps also in the sense that the actual surface code implementation is driven by measuring particular stabilizers. $\endgroup$ – Craig Gidney Sep 17 '18 at 15:51 • 1 $\begingroup$ However, the way in which the measurement outcomes are used differ substantially between QECCs and MBQC. In the idealised case of no or low rate of uncorrelated errors, any QECC is computing the identity transformation at all times, the measurements are periodic in time, and the outcomes are heavily biased towards the +1 outcome. For standard constructions of MBQC protocols, however, the measurements give uniformly random measurement outcomes every time, and those measurements are heavily time-dependent and driving non-trivial evolution. $\endgroup$ – Niel de Beaudrap Sep 17 '18 at 15:57 • 1 $\begingroup$ Is that a qualitative difference or just a quantitative one? The surface code also has those driving operations (e.g. braiding defects and injecting T states), it just separates them by the code distance. If you set the code distance to 1, a much higher proportion of the operations matter when there are no errors. $\endgroup$ – Craig Gidney Sep 17 '18 at 16:07 • 1 $\begingroup$ I would say that the difference occurs at a qualitative level as well, from my experience actually considering the effects of MBQC procedures. Also, it seems to me that in the case of braiding defects and T-state injection that it is not the error correcting code itself, but deformations of them, which are doing the computation. These are certainly relevant things one may do with an error corrected memory, but to say that the code is doing it is about the same level as saying that it is qubits which do quantum computations, as opposed to operations which one performs on those qubits. $\endgroup$ – Niel de Beaudrap Sep 17 '18 at 16:21 The Unitary Circuit Model This is the best well-known model of quantum computation. In this model one has constraints such as the following: 1. a set of qubits initialised to a pure state, which we denote $\lvert 0 \rangle$; 2. a sequence of unitary transformations which one performs on them, which may depend on a classical bit-string $x\in \{0,1\}^n$; 3. one or more measurements in the standard basis performed at the very end of the computation, yielding a classical output string $y \in \{0,1\}^k$. (We do not require $k = n$: for instance, for YES / NO problems, one often takes $k = 1$ no matter the size of $n$.) Minor details may change (for instance, the set of unitaries one may perform; whether one allows preparation in other pure states such as $\lvert 1 \rangle$, $\lvert +\rangle$, $\lvert -\rangle$; whether measurements must be in the standard basis or can also be in some other basis), but these do not make any essential difference. Discrete-time quantum walk A "discrete-time quantum walk" is a quantum variation on a random walk, in which there is a 'walker' (or multiple 'walkers') which takes small steps in a graph (e.g. a chain of nodes, or a rectangular grid). The difference is that where a random walker takes a step in a randomly determined direction, a quantum walker takes a step in a direction determined by a quantum "coin" register, which at each step is "flipped" by a unitary transformation rather than changed by re-sampling a random variable. See [ arXiv:quant-ph/0012090 ] for an early reference. For the sake of simplicity, I will describe a quantum walk on a cycle of size $2^n$; though one must change some of the details to consider quantum walks on more general graphs. In this model of computation, one typically does the following. 1. Prepare a "position" register on $n$ qubits in some state such as $\lvert 00\cdots 0\rangle$, and a "coin" register (with standard basis states which we denote by $\lvert +1 \rangle$ and $\lvert -1 \rangle$) in some initial state which may be a superposition of the two standard basis states. 2. Perform a coherent controlled-unitary transformation, which adds 1 to the value of the position register (modulo $2^n$) if the coin is in the state $\lvert +1 \rangle$, and subtracts 1 to the value of the position register (modulo $2^n$) if the coin is in the state $\lvert -1 \rangle$. 3. Perform a fixed unitary transformation $C$ to the coin register. This plays the role of a "coin flip" to determine the direction of the next step. We then return to step 2. The main difference between this and a random walk is that the different possible "trajectories" of the walker are being performed coherently in superposition, so that they can destructively interfere. This leads to a walker behaviour which is more like ballistic motion than diffusion. Indeed, an early presentation of a model such as this was made by Feynmann, as a way to simulate the Dirac equation. This model also often is described in terms of looking for or locating 'marked' elements in the graph, in which case one performs another step (to compute whether the node the walker is at is marked, and then to measure the outcome of that computation) before returning to Step 2. Other variations of this sort are reasonable. To perform a quantum walk on a more general graph, one must replace the "position" register with one which can express all of the nodes of the graph, and the "coin" register with one which can express the edges incident to a vertex. The "coin operator" then must also be replaced with one which allows the walker to perform an interesting superposition of different trajectories. (What counts as 'interesting' depends on what your motivation is: physicists often consider ways in which changing the coin operator changes the evolution of the probability density, not for computational purposes but as a way of probing at basic physics using quantum walks as a reasonable toy model of particle movement.) A good framework for generalising quantum walks to more general graphs is the Szegedy formulation [ arXiv:quant-ph/0401053 ] of discrete-time quantum walks. This model of computation is strictly speaking a special case of the unitary circuit model, but is motivated with very specific physical intuitions, which has led to some algorithmic insights (see e.g. [ arXiv:1302.3143 ]) for polynomial-time speedups in bounded-error quantum algorithms. This model is also a close relative of the continuous-time quantum walk as a model of computation. • 1 $\begingroup$ if you want to talk about DTQWs in the context of QC you should probably include references to the work of Childs and collaborators (e.g. arXiv:0806.1972. Also, you are describing how DTQWs work, but not really how you can use them to do computation. $\endgroup$ – glS Mar 26 '18 at 17:23 • 2 $\begingroup$ @gIS: indeed, I will add more details at some point: when I first wrote these it was to quickly enumerate some models and remark on them, rather than give comprehensive reviews. But as for how to compute, does the last paragraph not represent an example? $\endgroup$ – Niel de Beaudrap Mar 26 '18 at 20:05 • 1 $\begingroup$ @gIS: Isn't that work by Childs et al. actually about continuous-time quantum walks, anyhow? $\endgroup$ – Niel de Beaudrap Oct 29 '18 at 20:22 Quantum circuits with intermediary measurements This is a slight variation on "unitary circuits", in which one allows measurements in the middle of the algorithm as well as the end, and where one also allows future operations to depend on the outcomes of those measurements. It represents a realistic picture of a quantum processor which interacts with a classical control device, which among other things is the interface between the quantum processor and a human user. Intermediary measurement is practically necessary to perform error correction, and so this is in principle a more realistic picture of quantum computation than the unitary circuit model. but it is not uncommon for theorists of a certain type to strongly prefer measurements to be left until the end (using the principle of deferred measurement to simulate any 'intermediary' measurements). So, this may be a significant distinction to make when talking about quantum algorithms — but this does not lead to a theoretical increase in the computational power of a quantum algorithm. • 2 $\begingroup$ I think this should go with the "unitary circuit model" post, they are both really just variations of the circuit model, and one does not usually really distinguish them as different models $\endgroup$ – glS Mar 26 '18 at 17:20 • 1 $\begingroup$ @gIS: it is not uncommon to do so in the CS theory community. In fact, the bias is very much towards unitary circuits in particular. $\endgroup$ – Niel de Beaudrap Mar 26 '18 at 20:00 Quantum annealing Quantum annealing is a model of quantum computation which, roughly speaking, generalises the adiabatic model of computation. It has attracted popular — and commercial — attention as a result of D-WAVE's work on the subject. Precisely what quantum annealing consists of is not as well-defined as other models of computation, essentially because it is of more interest to quantum technologists than computer scientists. Broadly speaking, we can say that it is usually considered by people with the motivations of engineers, rather than the motivations of mathematicians, so that the subject appears to have many intuitions and rules of thumb but few 'formal' results. In fact, in an answer to my question about quantum annealing, Andrew O goes so far as to say that "quantum annealing can't be defined without considerations of algorithms and hardware". Nevertheless, "quantum annealing" seems is well-defined enough to be described as a way of approaching how to solve problems with quantum technologies with specific techniques — and so despite Andrew O's assessment, I think that it embodies some implicitly defined model of computation. I will attempt to describe that model here. Intuition behind the model Quantum annealing gets its name from a loose analogy to (classical) simulated annealing. They are both presented as means of minimising the energy of a system, expressed in the form of a Hamiltonian: $$ \begin{aligned} H_{\rm{classical}} &= \sum_{i,j} J_{ij} s_i s_j \\ H_{\rm{quantum}} &= A(t) \sum_{i,j} J_{ij} \sigma_i^z \sigma_j^z - B(t) \sum_i \sigma_i^x \end{aligned} $$ With simulated annealing, one essentially performs a random walk on the possible assignments to the 'local' variables $s_i \in \{0,1\}$, but where the probability of actually making a transition depends on • The difference in 'energy' $\Delta E = E_1 - E_0$ between two 'configurations' (the initial and the final global assignment to the variables $\{s_i\}_{i=1}^n$) before and after each step of the walk; • A 'temperature' parameter which governs the probability with which the walk is allowed to perform a step in the random walk which has $\Delta E > 0$. One starts with the system at 'infinite temperature', which is ultimately a fancy way of saying that you allow for all possible transitions, regardless of increases or decreases in energy. You then lower the temperature according to some schedule, so that time goes on, changes in state which increase the energy become less and less likely (though still possible). The limit is zero temperature, in which any transition which decreases energy is allowed, but any transition which increases energy is simply forbidden. For any temperature $T > 0$, there will be a stable distribution (a 'thermal state') of assignments, which is the uniform distribution at 'infinite' temperature, and which is which is more and more weighted on the global minimum energy states as the temperature decreases. If you take long enough to decrease the temperature from infinite to near zero, you should in principle be guaranteed to find a global optimum to the problem of minimising the energy. Thus simulated annealing is an approach to solving optimisation problems. Quantum annealing is motivated by generalising the work by Farhi et al. on adiabatic quantum computation [arXiv:quant-ph/0001106], with the idea of considering what evolution occurs when one does not necessarily evolve the Hamiltonian in the adiabatic regime. Similarly to classical annealing, one starts in a configuration in which "classical assignments" to some problem are in a uniform distribution, though this time in coherent superposition instead of a probability distribution: this is achieved for time $t = 0$, for instance, by setting $$ A(t=0) = 0, \qquad B(t=0) = 1 $$ in which case the uniform superposition $\def\ket#1{\lvert#1\rangle}\ket{\psi_0} \propto \ket{00\cdots00} + \ket{00\cdots01} + \cdots + \ket{11\cdots11}$ is a minimum-energy state of the quantum Hamiltonian. One steers this 'distribution' (i.e. the state of the quantum system) to one which is heavily weighted on a low-energy configuration by slowly evolving the system — by slowly changing the field strengths $A(t)$ and $B(t)$ to some final value $$ A(t_f) = 1, \qquad B(t_f) = 0. $$ Again, if you do this slowly enough, you will succeed with high probability in obtaining such a global minimum. The adiabatic regime describes conditions which are sufficient for this to occur, by virtue of remaining in (a state which is very close to) the ground state of the Hamiltonian at all intermediate times. However, it is considered possible that one can evolve the system faster than this and still achieve a high probability of success. Similarly to adiabatic quantum computing, the way that $A(t)$ and $B(t)$ are defined are often presented as a linear interpolations from $0$ to $1$ (increasing for $A(t)$, and decreasing for $B(t)$). However, also in common with adiabatic computation, $A(t)$ and $B(t)$ don't necessarily have to be linear or even monotonic. For instance, D-Wave has considered the advantages of pausing the annealing schedule and 'backwards anneals'. 'Proper' quantum annealing (so to speak) presupposes that evolution is probably not being done in the adiabatic regime, and allows for the possibility of diabatic transitions, but only asks for a high chance of achieving an optimum — or even more pragmatically still, of achieving a result which would be difficult to find using classical techniques. There are no formal results about how quickly you can change your Hamiltonian to achieve this: the subject appears mostly to consist of experimenting with a heuristic to see what works in practise. The comparison with classical simulated annealing Despite the terminology, it is not immediately clear that there is much which quantum annealing has in common with classical annealing. The main differences between quantum annealing and classical simulated annealing appear to be that: • In quantum annealing, the state is in some sense ideally a pure state, rather than a mixed state (corresponding to the probability distribution in classical annealing); • In quantum annealing, the evolution is driven by an explicit change in the Hamiltonian rather than an external parameter. It is possible that a change in presentation could make the analogy between quantum annealing and classical annealing tighter. For instance, one could incorporate the temperature parameter into the spin Hamiltonian for classical annealing, by writing $$\tilde H_{\rm{classical}} = A(t) \sum_{i,j} J_{ij} s_i s_j - B(t) \sum_{i,j} \textit{const.} $$ where we might choose something like $A(t) = t\big/(t_F - t)$ and $B(t) = t_F - t$ for $t_F > 0$ the length of the anneal schedule. (This is chosen deliberately so that $A(0) = 0$ and $A(t) \to +\infty$ for $t \to t_F$.) Then, just as an annealing algorithm is governed in principle by the Schrödinger equation for all times, we may consider an annealing process which is governed by a diffusion process which is in principle uniform with tim by small changes in configurations, where the probability of executing a randomly selected change of configuration is governed by $$ p(x \to y) = \max\Bigl\{ 1,\; \exp\bigl(-\gamma \Delta E_{x\to y}\bigr) \Bigr\} $$ for some constant $\gamma$, where $E_{x \to y}$ is the energy difference between the initial and final configurations. The stable distribution of this diffusion for the Hamiltonian at $t=0$ is the uniform distribution, and the stable distribution for the Hamiltonian as $t \to t_F$ is any local minimum; and as $t$ increases, the probability with which a transition occurs which increases the energy becomes smaller, until as $t \to t_F$ the probability of any increases in energy vanish (because any of the possible increase is a costly one). There are still disanalogies to quantum annealing in this — for instance, we achieve the strong suppression of increases in energy as $t \to t_F$ essentially by making the potential wells infinitely deep (which is not a very physical thing to do) — but this does illustrate something of a commonality between the two models, with the main distinction being not so much the evolution of the Hamiltonian as it is the difference between diffusion and Schrödinger dynamics. This suggests that there may be a sharper way to compare the two models theoretically: by describing the difference between classical and quantum annealing, as being analogous to the difference between random walks and quantum walks. A common idiom in describing quantum annealing is to speak of 'tunnelling' through energy barriers — this is certainly pertinent to how people consider quantum walks: consider for instance the work by Farhi et al. on continuous-time quantum speed-ups for evaluating NAND circuits, and more directly foundational work by Wong on quantum walks on the line tunnelling through potential barriers. Some work has been done by Chancellor [arXiv:1606.06800] on considering quantum annealing in terms of quantum walks, though it appears that there is room for a more formal and complete account. On a purely operational level, it appears that quantum annealing gives a performance advantage over classical annealing (see for example these slides on the difference in performance between quantum vs. classical annealing, from Troyer's group at ETH, ca. 2014). Quantum annealing as a phenomenon, as opposed to a computational model Because quantum annealing is more studied by technologists, they focus on the concept of realising quantum annealing as an effect rather than defining the model in terms of general principles. (A rough analogy would be studying the unitary circuit model only inasmuch as it represents a means of achieving the 'effects' of eigenvalue estimation or amplitude amplification.) Therefore, whether something counts as "quantum annealing" is described by at least some people as being hardware-dependent, and even input-dependent: for instance, on the layout of the qubits, the noise levels of the machine. It seems that even trying to approach the adiabatic regime will prevent you from achieving quantum annealing, because the idea of what quantum annealing even consists of includes the idea that noise (such as decoherence) will prevent annealing from being realised: as a computational effect, as opposed to a computational model, quantum annealing essentially requires that the annealing schedule is shorter than the decoherence time of the quantum system. Some people occasionally describe noise as being somehow essential to the process of quantum annealing. For instance, Boixo et al. [arXiv:1304.4595] write Unlike adiabatic quantum computing[, quantum annealing] is a positive temperature method involving an open quantum system coupled to a thermal bath. It might perhaps be accurate to describe it as being an inevitable feature of systems in which one will perform annealing (just because noise is inevitable feature of a system in which you will do quantum information processing of any kind): as Andrew O writes "in reality no baths really help quantum annealing". It is possible that a dissipative process can help quantum annealing by helping the system build population on lower-energy states (as suggested by work by Amin et al., [arXiv:cond-mat/0609332]), but this seems essentially to be a classical effect, and would inherently require a quiet low-temperature environment rather than 'the presence of noise'. The bottom line It might be said — in particular by those who study it — that quantum annealing is an effect, rather than a model of computation. A "quantum annealer" would then be best understood as "a machine which realises the effect of quantum annealing", rather than a machine which attempts to embody a model of computation which is known as 'quantum annealing'. However, the same might be said of adiabatic quantum computation, which is — in my opinion correctly — described as a model of computation in its own right. Perhaps it would be fair to describe quantum annealing as an approach to realising a very general heuristic, and that there is an implicit model of computation which could be characterised as the conditions under which we could expect this heuristic to be successful. If we consider quantum annealing this way, it would be a model which includes the adiabatic regime (with zero-noise) as a special case, but it may in principle be more general. Your Answer
b29a0adc68298e6e	Quantum Physics, Faith, and Observation Dr. Stephen Barr, an astrophysicist at the University of Delaware, wrote a profound article titled “Faith and Quantum Theory” that has been on my mind as of late for a few reasons. The first is that as a theist much of my time is spent thinking about how to reconcile what can be seen with what cannot be seen and what to believe about both. I have found Dr. Barr’s piece to be a fantastic justification for the belief in the metaphysical. His argument has some moving parts, but goes fundamentally something like this: the most widely accepted theory on quantum physics is dependent on probabilities, and if probabilities are to have any meaning at all to anyone, they must eventually culminate in a real event. Real events must be realized by a mind, and therefore for the natural world to have any meaning at all a Mind must have existed. This Mind is what we call God. In order to explore Dr. Barr’s complex argument more deeply, I am going to write a little bit about quantum theory, only enough to get our feet wet with the idea. If you are already familiar with the fundamental theories about quantum physics, feel free to skip down to Barr’s solution. Disclaimer: I am not a physicist or a mathematician. Though this is a topic of interest to me, I don’t claim to be an authority on this subject. Read at your own risk…. 🙂 Wave Particle Duality Paradox Though there have been several Wave-Particle experiments, perhaps the most famous is Thomas Young’s experiment with electrons in 1927. It essentially demonstrated that electrons can behave as both waves (like sound waves or waves moving in water) and particles (pieces of matter). Adding further to the mystery of wave-particle duality is that once the electrons were studied more closely with measurement devices, they behaved differently than when they were not observed. The key to this experiment was that the observer of the electron made a difference in the outcome of the experiment. Schrodinger’s Equations Schrodinger developed equations that essentially predicted how likely things are likely to behave like particles vs waves. His equations were remarkably accurate, and the short and sweet of his findings is that the bigger things get, the more likely they are to behave like particles, and the smaller things get, the more likely they are to act like a wave. If Schrodinger’s equations interest you, here is a good video to watch. The important thing to grasp about Schrodinger’s equations is that his equations yield probabilities, not definite realities. If Schrodinger’s equations are accurate, then they have, in Barr’s eyes, a profound philosophical implication on the very nature of Being, and would, therefore, be of ontological interest to us. Barr says, “It starts with the fact that for any physical system, however simple or complex, there is a master equation—called the Schrödinger equation—that describes its behavior. And the crucial point on which everything hinges is that the Schrödinger equation yields only probabilities. (Only in special cases are these exactly 0, or 100 percent.) But this immediately leads to a difficulty: There cannot always remain just probabilities; eventually there must be definite outcomes, for probabilities must be the probabilities of definite outcomes. “ Barr goes on to explain that because physical systems are described in probabilistic terms, there must be a Mind to make any sense of anything at all: As long as only physical structures and mechanisms are involved, however complex, their behavior is described by equations that yield only probabilities—and once a mind is involved that can make a rational judgment of fact, and thus come to knowledge, there is certainty. Therefore, such a mind cannot be just a physical structure or mechanism completely describable by the equations of physics…A probability is a measure of someone’s state of knowledge or lack of it…” Barr goes on to explain that there are differing views of quantum theory that may circumvent the dependence on probability to explain the micro-world (Einstein, for example, believed that there were missing variables that would lead to definite outcomes rather than probabilities). However, if the Copenhagen Interpretation of Quantum Theory is correct, I think Barr is correct in his reasoning that a Mind is required to make relevant and meaningful ontological statements about reality. In conclusion, Barr believes that a Mind is necessary to make sense of the most widely accepted Copenhagen interpretation of quantum theory because Schrodinger’s equations does not yield definite results, only probabilities. For those that have read this far, you may conclude two things: quantum physics can be used as a compelling argument for both free will, as probabilities best explain quantum reality, and for a Mind (God) who observes these realities. Leave a Reply WordPress.com Logo Google photo Twitter picture Facebook photo Connecting to %s
679de1235089a0ab	Dismiss Notice Join Physics Forums Today! I How does the London dispersion force really work? 1. Feb 13, 2017 #1 I wonder about this. The explanation that I keep finding is that "dipoles" occur "randomly" when "electrons move" to different sides of the atom. Yet I find this difficult to reconcile with what I understand about quantum mechanics -- so I must be missing something, on either side or both. In particular, in quantum mechanics electrons are not "moving" like classic particles as seems to be suggested by this "explanation", not unless you subscribe to those theories like Bohm's or similar, but rather are described by wave functions and the Schrodinger equation, and as far as I can tell these do not "randomly" "concentrate" in some fashion. How is this effect explained in a proper quantum-mechanical treatment, in a more mainline view of quantum theory? I've tried searching around but have not found anything satisfying. 2. jcsd 3. Feb 13, 2017 #2 User Avatar Staff: Mentor That explanation is indeed (semi-)classical, although still useful for visualizing the process. The full QM approach would be to simply solve the Schrödinger equation for two atoms at a certain distance from each other, where you would see that the wave function does show a polarization. More simply, you can a hydrogen atoms in its ground state, take a second atom as a perturbation and see how the wave function of the first atom is modified by the presence of the other. 4. Feb 13, 2017 #3 User Avatar Science Advisor Suppose, for simplicity, that dipole can have only two polarizations, namely ##\uparrow## and ##\downarrow##. And suppose that the polarization is random. This means that the wave function is a superposition of two polarizations, i.e. something like Now suppose that you have two dipoles, both of which are random and mutually independent. Then the wave function is $$(\|\uparrow\rangle+\|\downarrow\rangle) (\|\uparrow\rangle+\|\downarrow\rangle)$$ In this case there is no London force. Finally, suppose that you have two dipoles which are random but not independent. Instead of being independent, they are correlated so that they always point to the same direction. In this case the wave function is $$\|\uparrow\rangle \|\uparrow\rangle+\|\downarrow\rangle\|\downarrow\rangle$$ In such a case there is a London force. 5. Feb 14, 2017 #4 User Avatar Science Advisor In QM you can calculate the mean value of the dipole moment, which vanishes for closed shell atoms, and also the variance of the dipole moment, i.e. the expectation value of ##d^2## or more generally the time dependent autocorrelation function ##< d(0)d(t)> ## which do not vanish. From these expressions it is clear that the dipole moment is fluctuating also in QM. Furthermore, in the classical limit, the autocorrelation function will converge against the classical expression. 6. Feb 15, 2017 #5 Thanks, these explanations definitely help a lot. Have something to add? Draft saved Draft deleted
e31a85654fda64da	Jan Scholten 0.5.3 Discussion Phases Division 7 Most clades are divided into 7, for example the Fabidae are divided into 7 orders. Each order is again divided into 7 families. In most cases the 7 groups are given by the Apg classification. Sometimes there are more or fewer groups. In those cases some groups have been split into 2 or 3 groups, for instance the Asparagales are split in Orchidales and the rest of the Asparagales. Sometimes groups, or families, are taken together. The 7 Phases have a similar quality to that of the stages of the Carbon series and Silicon series. But they are a bit more abstract. I’ve named them Phases in order not to get confused with the 18 Stages. The numbers are not running parallel. Division in 8, effectively 7 One can divide a group in a number of subgroups: 2, 3, 4.0 , 7, 10, or 100. In this book the basic division is in 7 groups. The number 7 is connected to the idea of a process, the cycle of life and death. We find the number 7 also strongly in the periodic system of the elements. There we see the 7 series, rows of the periodic system. And we see the 7 columns, especially in the 2nd and 3rd row, the Carbon and Silicon series. The number 7 seems to be given by nature here. Originally the division was in 8, as there are 8 columns in the Carbon and Silicon series. But the last column, Phase 8 or Phase 0, is a kind of denial state, a denial of what exists. It seems to be that this Phase does not exist in the Plant kingdom. It seems to be that only positive expressions are developed in nature. We see a similar thing happening with the series, the rows of the periodic system. There we see only 7 series as series, 8 or 0 does not exist. Divisions in 2, effectively in 1 In the periodic system we also see a division into 2 in the Hydrogen series. We could even say it is only one division when we leave out Phase 0. Then there is no real division, things are just seen as one. This happens frequently in the Apg classification, for instance the order of Acorales has only one family, Acoraceae. It is especially apparent in the introduction clades: Amborellales, Acorales, Arales, Ceratophyllales, Halogarales. Divisions in 18, effectively in 17 The third division that we can see in a periodic system is that in 18 Stages, the 18 columns of the Iron and Silver series. One can leave out Stage 18 or 0 from the 18 stages. 17 Stages are left. These 17 stages are used to differentiate the members of families. Divisions in 32. The division in 32, the one we see in the Gold and Uranium series will not be used for the Plant kingdom. Other divisions. One could argue that divisions in 3, 12 or 24 would give better results. It is my experience that the divisions in 2, 8, 18 work fine. One reason for this is that they are given by nature. We can see that in the Periodic system. The mathematical formula behind the periodic system, the Schrödinger equation, leads automatically to those divisions of 2, 8 and 18 and 32. So the above divisions are more general than just limited to the Mineral kingdom and can easily be used for the Plant and Animal kingdom.
c6970e45f5ddd6ef	Open main menu Wikipedia β Force (in units of 10,000 N) between two nucleons as a function of distance as computed from the Reid potential (1968).[1] The spins of the neutron and proton are aligned, and they are in the S angular momentum state. The attractive (negative) force has a maximum at a distance of about 1 fm with a force of about 25,000 N. Particles much closer than a distance of 0.8 fm experience a large repulsive (positive) force. Particles separated by a distance greater than 1 fm are still attracted (Yukawa potential), but the force falls as an exponential function of distance. Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The potential well is a minimum at a distance of about 0.8 fm. With this potential nucleons can become bound with a negative "binding energy." The nuclear force (or nucleon–nucleon interaction or residual strong force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. Since protons have charge +1 e, they experience an electric force that tends to push them apart, but at short range the attractive nuclear force is strong enough to overcome the electromagnetic force. The nuclear force binds nucleons into atomic nuclei. The nuclear force is powerfully attractive between nucleons at distances of about 1 femtometre (fm, or 1.0 × 10−15 metres), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. At distances less than 0.7 fm, the nuclear force becomes repulsive. This repulsive component is responsible for the physical size of nuclei, since the nucleons can come no closer than the force allows. By comparison, the size of an atom, measured in angstroms (Å, or 1.0 × 10−10 m), is five orders of magnitude larger. The nuclear force is not simple, however, since it depends on the nucleon spins, has a tensor component, and may depend on the relative momentum of the nucleons.[2] The nuclear force is not one of the fundamental forces of nature. The nuclear force plays an essential role in storing energy that is used in nuclear power and nuclear weapons. Work (energy) is required to bring charged protons together against their electric repulsion. This energy is stored when the protons and neutrons are bound together by the nuclear force to form a nucleus. The mass of a nucleus is less than the sum total of the individual masses of the protons and neutrons. The difference in masses is known as the mass defect, which can be expressed as an energy equivalent. Energy is released when a heavy nucleus breaks apart into two or more lighter nuclei. This energy is the electromagnetic potential energy that is released when the nuclear force no longer holds the charged nuclear fragments together.[3][4] A quantitative description of the nuclear force relies on equations that are partly empirical. These equations model the internucleon potential energies, or potentials. (Generally, forces within a system of particles can be more simply modeled by describing the system's potential energy; the negative gradient of a potential is equal to the vector force.) The constants for the equations are phenomenological, that is, determined by fitting the equations to experimental data. The internucleon potentials attempt to describe the properties of nucleon–nucleon interaction. Once determined, any given potential can be used in, e.g., the Schrödinger equation to determine the quantum mechanical properties of the nucleon system. The discovery of the neutron in 1932 revealed that atomic nuclei were made of protons and neutrons, held together by an attractive force. By 1935 the nuclear force was conceived to be transmitted by particles called mesons. This theoretical development included a description of the Yukawa potential, an early example of a nuclear potential. Mesons, predicted by theory, were discovered experimentally in 1947. By the 1970s, the quark model had been developed, by which the mesons and nucleons were viewed as composed of quarks and gluons. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. While the nuclear force is usually associated with nucleons, more generally this force is felt between hadrons, or particles composed of quarks. At small separations between nucleons (less than ~ 0.7 fm between their centers, depending upon spin alignment) the force becomes repulsive, which keeps the nucleons at a certain average separation, even if they are of different types. This repulsion arises from the Pauli exclusion force for identical nucleons (such as two neutrons or two protons). A Pauli exclusion force also occurs between quarks of the same type within nucleons, when the nucleons are different (a proton and a neutron, for example). Field strengthEdit At distances larger than 0.7 fm the force becomes attractive between spin-aligned nucleons, becoming maximal at a center–center distance of about 0.9 fm. Beyond this distance the force drops exponentially, until beyond about 2.0 fm separation, the force is negligible. Nucleons have a radius of about 0.8 fm.[5] At short distances (less than 1.7 fm or so), the attractive nuclear force is stronger than the repulsive Coulomb force between protons; it thus overcomes the repulsion of protons within the nucleus. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. The nuclear force has a spin-dependent component. The force is stronger for particles with their spins aligned than for those with their spins anti-aligned. If two particles are the same, such as two neutrons or two protons, the force is not enough to bind the particles, since the spin vectors of two particles of the same type must point in opposite directions when the particles are near each other and are (save for spin) in the same quantum state. This requirement for fermions stems from the Pauli exclusion principle. For fermion particles of different types, such as a proton and neutron, particles may be close to each other and have aligned spins without violating the Pauli exclusion principle, and the nuclear force may bind them (in this case, into a deuteron), since the nuclear force is much stronger for spin-aligned particles. But if the particles' spins are anti-aligned the nuclear force is too weak to bind them, even if they are of different types. The nuclear force also has a tensor component which depends on the interaction between the nucleon spins and the angular momentum of the nucleons, leading to deformation from a simple spherical shape. Nuclear BindingEdit To disassemble a nucleus into unbound protons and neutrons requires work against the nuclear force. Conversely, energy is released when a nucleus is created from free nucleons or other nuclei: the nuclear binding energy. Because of mass–energy equivalence (i.e. Einstein's famous formula E = mc2), releasing this energy causes the mass of the nucleus to be lower than the total mass of the individual nucleons, leading to the so-called "mass defect".[6] The nuclear force is nearly independent of whether the nucleons are neutrons or protons. This property is called charge independence. The force depends on whether the spins of the nucleons are parallel or antiparallel, as it has a non-central or tensor component. This part of the force does not conserve orbital angular momentum, which under the action of central forces is conserved. The symmetry resulting in the strong force, proposed by Werner Heisenberg, is that protons and neutrons are identical in every respect, other than their charge. This is not completely true, because neutrons are a tiny bit heavier, but it is an approximate symmetry. Protons and neutrons are therefore viewed as the same particle, but with different isospin quantum number[clarification needed]. The strong force is invariant under SU(2) transformations, just as are particles with intrinsic spin. Isospin and intrinsic spin are related under this SU(2) symmetry group. There are only strong attractions when the total isospin is 0, which is confirmed by experiment.[7] Our understanding of the nuclear force is obtained by scattering experiments and the binding energy of light nuclei. A Feynman diagram of a strong protonneutron interaction mediated by a neutral pion. Time proceeds from left to right. The nuclear force occurs by the exchange of virtual light mesons, such as the virtual pions, as well as two types of virtual mesons with spin (vector mesons), the rho mesons and the omega mesons. The vector mesons account for the spin-dependence of the nuclear force in this "virtual meson" picture. The nuclear force is distinct from what historically was known as the weak nuclear force. The weak interaction is one of the four fundamental interactions, and plays a role in such processes as beta decay. The weak force plays no role in the interaction of nucleons, though it is responsible for the decay of neutrons to protons and vice versa. The nuclear force has been at the heart of nuclear physics ever since the field was born in 1932 with the discovery of the neutron by James Chadwick. The traditional goal of nuclear physics is to understand the properties of atomic nuclei in terms of the 'bare' interaction between pairs of nucleons, or nucleon–nucleon forces (NN forces). Within months after the discovery of the neutron, Werner Heisenberg[8][9][10] and Dmitri Ivanenko[11] had proposed proton–neutron models for the nucleus.[12] Heisenberg approached the description of protons and neutrons in the nucleus through quantum mechanics, an approach that was not at all obvious at the time. Heisenberg's theory for protons and neutrons in the nucleus was a "major step toward understanding the nucleus as a quantum mechanical system."[13] Heisenberg introduced the first theory of nuclear exchange forces that bind the nucleons. He considered protons and neutrons to be different quantum states of the same particle, i.e., nucleons distinguished by the value of their nuclear isospin quantum numbers. One of the earliest models for the nucleus was the liquid drop model developed in the 1930s. One property of nuclei is that the average binding energy per nucleon is approximately the same for all stable nuclei, which is similar to a liquid drop. The liquid drop model treated the nucleus as a drop of incompressible nuclear fluid, with nucleons behaving like molecules in a liquid. The model was first proposed by George Gamow and then developed by Niels Bohr, Werner Heisenberg and Carl Friedrich von Weizsäcker. This crude model did not explain all the properties of the nucleus, but it did explain the spherical shape of most nuclei. The model also gave good predictions for the nuclear binding energy of nuclei. In 1934, Hideki Yukawa made the earliest attempt to explain the nature of the nuclear force. According to his theory, massive bosons (mesons) mediate the interaction between two nucleons. Although, in light of quantum chromodynamics (QCD), meson theory is no longer perceived as fundamental, the meson-exchange concept (where hadrons are treated as elementary particles) continues to represent the best working model for a quantitative NN potential. The Yukawa potential (also called a screened Coulomb potential) is a potential of the form where g is a magnitude scaling constant, i.e., the amplitude of potential, is the Yukawa particle mass, r is the radial distance to the particle. The potential is monotone increasing, implying that the force is always attractive. The constants are determined empirically. The Yukawa potential depends only on the distance between particles, r, hence it models a central force. Throughout the 1930s a group at Columbia University led by I. I. Rabi developed magnetic resonance techniques to determine the magnetic moments of nuclei. These measurements led to the discovery in 1939 that the deuteron also possessed an electric quadrupole moment.[14][15] This electrical property of the deuteron had been interfering with the measurements by the Rabi group. The deuteron, composed of a proton and a neutron, is one of the simplest nuclear systems. The discovery meant that the physical shape of the deuteron was not symmetric, which provided valuable insight into the nature of the nuclear force binding nucleons. In particular, the result showed that the nuclear force was not a central force, but had a tensor character.[1] Hans Bethe identified the discovery of the deuteron's quadrupole moment as one of the important events during the formative years of nuclear physics.[14] Historically, the task of describing the nuclear force phenomenologically was formidable. The first semi-empirical quantitative models came in the mid-1950s,[1] such as the Woods–Saxon potential (1954). There was substantial progress in experiment and theory related to the nuclear force in the 1960s and 1970s. One influential model was the Reid potential (1968).[1] In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. The nuclear force as a residual of the strong forceEdit An animation of the interaction. The colored double circles are gluons. Anticolors are shown as per this diagram (larger version). The same diagram as that above with the individual quark constituents shown, to illustrate how the fundamental strong interaction gives rise to the nuclear force. Straight lines are quarks, while multi-colored loops are gluons (the carriers of the fundamental force). Other gluons, which bind together the proton, neutron, and pion "in-flight," are not shown. The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. The strong interaction is the attractive force that binds the elementary particles called quarks together to form the nucleons (protons and neutrons) themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge, but of far greater strength. Quarks, gluons and their dynamics are mostly confined within nucleons, but residual influences extend slightly beyond nucleon boundaries to give rise to the nuclear force. The nuclear forces arising between nucleons are analogous to the forces in chemistry between neutral atoms or molecules called London forces. Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. Similarly, even though nucleons are made of quarks in combinations which cancel most gluon forces (they are "color neutral"), some combinations of quarks and gluons nevertheless leak away from nucleons, in the form of short-range nuclear force fields that extend from one nucleon to another nearby nucleon. These nuclear forces are very weak compared to direct gluon forces ("color forces" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. Nevertheless, they are strong enough to bind neutrons and protons over short distances, and overcome the electrical repulsion between protons in the nucleus. Sometimes, the nuclear force is called the residual strong force, in contrast to the strong interactions which arise from QCD. This phrasing arose during the 1970s when QCD was being established. Before that time, the strong nuclear force referred to the inter-nucleon potential. After the verification of the quark model, strong interaction has come to mean QCD. Nucleon–nucleon potentialsEdit Two-nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. Such systems can be described by attributing a potential (such as the Yukawa potential) to the nucleons and using the potentials in a Schrödinger equation. The form of the potential is derived phenomenologically (by measurement), although for the long-range interaction, meson-exchange theories help to construct the potential. The parameters of the potential are determined by fitting to experimental data such as the deuteron binding energy or NN elastic scattering cross sections (or, equivalently in this context, so-called NN phase shifts). The most widely used NN potentials are the Paris potential, the Argonne AV18 potential ,[16] the CD-Bonn potential and the Nijmegen potentials. A more recent approach is to develop effective field theories for a consistent description of nucleon–nucleon and three-nucleon forces. Quantum hadrodynamics is an effective field theory of the nuclear force, comparable to QCD for color interactions and QED for electromagnetic interactions. Additionally, chiral symmetry breaking can be analyzed in terms of an effective field theory (called chiral perturbation theory) which allows perturbative calculations of the interactions between nucleons with pions as exchange particles. From nucleons to nucleiEdit The ultimate goal of nuclear physics would be to describe all nuclear interactions from the basic interactions between nucleons. This is called the microscopic or ab initio approach of nuclear physics. There are two major obstacles to overcome before this dream can become reality: • Calculations in many-body systems are difficult and require advanced computation techniques. • There is evidence that three-nucleon forces (and possibly higher multi-particle interactions) play a significant role. This means that three-nucleon potentials must be included into the model. This is an active area of research with ongoing advances in computational techniques leading to better first-principles calculations of the nuclear shell structure. Two- and three-nucleon potentials have been implemented for nuclides up to A = 12. Nuclear potentialsEdit A successful way of describing nuclear interactions is to construct one potential for the whole nucleus instead of considering all its nucleon components. This is called the macroscopic approach. For example, scattering of neutrons from nuclei can be described by considering a plane wave in the potential of the nucleus, which comprises a real part and an imaginary part. This model is often called the optical model since it resembles the case of light scattered by an opaque glass sphere. Nuclear potentials can be local or global: local potentials are limited to a narrow energy range and/or a narrow nuclear mass range, while global potentials, which have more parameters and are usually less accurate, are functions of the energy and the nuclear mass and can therefore be used in a wider range of applications. See alsoEdit 1. ^ a b c d Reid, R.V. (1968). "Local phenomenological nucleon–nucleon potentials". Annals of Physics. 50: 411–448. Bibcode:1968AnPhy..50..411R. doi:10.1016/0003-4916(68)90126-7. 2. ^ Kenneth S. Krane (1988). Introductory Nuclear Physics. Wiley & Sons. ISBN 0-471-80553-X. 3. ^ Binding Energy, Mass Defect, Furry Elephant physics educational site, retr 2012 7 1 4. ^ Chapter 4 NUCLEAR PROCESSES, THE STRONG FORCE, M. Ragheb 1/30/2013, University of Illinois 5. ^ Povh, B.; Rith, K.; Scholz, C.; Zetsche, F. (2002). Particles and Nuclei: An Introduction to the Physical Concepts. Berlin: Springer-Verlag. p. 73. ISBN 978-3-540-43823-6. 6. ^ Stern, Dr. Swapnil Nikam (February 11, 2009). "Nuclear Binding Energy". "From Stargazers to Starships". NASA website. Retrieved 2010-12-30. 7. ^ Griffiths, David, Introduction to Elementary Particles 8. ^ Heisenberg, W. (1932). "Über den Bau der Atomkerne. I". Z. Phys. 77: 1–11. Bibcode:1932ZPhy...77....1H. doi:10.1007/BF01342433. 9. ^ Heisenberg, W. (1932). "Über den Bau der Atomkerne. II". Z. Phys. 78 (3–4): 156–164. Bibcode:1932ZPhy...78..156H. doi:10.1007/BF01337585. 10. ^ Heisenberg, W. (1933). "Über den Bau der Atomkerne. III". Z. Phys. 80 (9–10): 587–596. Bibcode:1933ZPhy...80..587H. doi:10.1007/BF01335696. 11. ^ Iwanenko, D.D., The neutron hypothesis, Nature 129 (1932) 798. 12. ^ Miller A. I. Early Quantum Electrodynamics: A Sourcebook, Cambridge University Press, Cambridge, 1995, ISBN 0521568919, pp. 84–88. 13. ^ Brown, L.M.; Rechenberg, H. (1996). The Origin of the Concept of Nuclear Forces. Bristol and Philadelphia: Institute of Physics Publishing. ISBN 0750303735. 14. ^ a b John S. Rigden (1987). Rabi, Scientist and Citizen. New York: Basic Books, Inc. pp. 99–114. ISBN 9780674004351. Retrieved May 9, 2015. 15. ^ Kellogg, J.M.; Rabi, I.I.; Ramsey, N.F.; Zacharias, J.R. (1939). "An electrical quadrupole moment of the deuteron". Physical Review. 55: 318–319. Bibcode:1939PhRv...55..318K. doi:10.1103/physrev.55.318. Retrieved May 9, 2015. 16. ^ Wiringa, R. B.; Stoks, V. G. J.; Schiavilla, R. (1995). "Accurate nucleon–nucleon potential with charge-independence breaking". Physical Review C. 51: 38. arXiv:nucl-th/9408016 . Bibcode:1995PhRvC..51...38W. doi:10.1103/PhysRevC.51.38. • Gerald Edward Brown and A. D. Jackson, The Nucleon–Nucleon Interaction, (1976) North-Holland Publishing, Amsterdam ISBN 0-7204-0335-9 • R. Machleidt and I. Slaus, "The nucleon–nucleon interaction", J. Phys. G 27 (2001) R69 (topical review). • E.A. Nersesov, Fundamentals of atomic and nuclear physics, (1990), Mir Publishers, Moscow, ISBN 5-06-001249-2 • P. Navrátil and W.E. Ormand, "Ab initio shell model with a genuine three-nucleon force for the p-shell nuclei", Phys. Rev. C 68, 034305 (2003). Further readingEdit
5f242df9ccd1d83b	Dismiss Notice Join Physics Forums Today! Stationary states for even parity potential 1. Apr 1, 2010 #1 Hi I know that stationary states in a system with an even potential energy function have to be either even or odd. Why does the ground state have to be even, and not odd???? This is asserted in Griffiths, page 298. 2. jcsd 3. Apr 2, 2010 #2 User Avatar Science Advisor Basically, in order for the state to be odd, it has to pass through zero. In order to do this, it needs to change more quickly, which means higher energy. So a state with fewer nodes (places where the wave funtcion goes to zero) always has lower energy than a state with more nodes. Typically the ground state has no nodes, so it can't be odd. 4. Apr 3, 2010 #3 I suppose it makes sense that stationary states with higher energy have a larger (magnitude of) second spatial derivative on average given the form of the time independent Schrödinger equation. But that would be true in general. Why would the ground state have an even wavefunction given an even potential? 5. Apr 3, 2010 #4 User Avatar Science Advisor As phyzguy said ... the energy of a state is proportional to the "curvature" of the wavefunction (the true mathematical description of curvature is a bit more complicated that just the 2nd derivative, but that simpler version will suffice for this discussion). Thus, states with fewer nodes have lower energies. So you just have to ask youself, what is the smallest number of nodes a wavefunction can have? It's zero, isn't it? So, zero nodes means a wavefunction that doesn't pass through zero, and thus must have even symmetry. 6. Apr 3, 2010 #5 Errr nevermind let me think about this
3dd7e0e4394ed0da	Bohmists & segregation of primitive and contextual observables What is the pilot wave theory? And this construction is actually very unnatural because it picks "X" as a preferred observable in whose basis the wave vector should be (artificially) separated into the probability densities and phases: we will talk about the unequal treatment of different observables throughout this article. And because there are way too many similar classical equations appearing in this rewriting of one-particle quantum mechanics, it is very "easy" for an inexperienced physicist to assign these quantum objects with an incorrect, classical interpretation. EPR problems Einstein never liked the probabilistic interpretation of quantum mechanics but de Broglie's theory was so awkward that Einstein called it an "unnecessary superconstruction". More concretely, it is inconsistent with modern physics in many ways, as we will see. You might think that the experiments that have been made to check relativity simply rule out a fundamentally privileged reference frame. Well, the Bohmists still try to wave their hands and argue that they can avoid the contradictions with the verified consequences of relativity. I wonder whether they actually believe that there always exists a preferred reference frame, at least in principle, because such a belief sounds crazy to me (what is the hypothetical preferred slicing near a black hole, for example?). But can they avoid the flagrant contradictions if you allow their theory to be arbitrarily unnatural? Generically, they certainly can't: a theory of their kind will violate the relationships based on relativity by 100% or so. With a huge amount of fine-tuning, they can surely make some statistical predictions "look" relativistic even though they are fundamentally not. The contradiction between relativity and semi-viable Bohmian models (that violate Bell's inequalities, and they have to in order not to be ruled out by experiments) is a very profound problem of these models. It can't really be fixed. But I want to look at another reason why these models are fundamentally wrong, and it is the following: Life before the segregation of observables What do I mean? Orthodox quantum mechanics is ontologically politically correct with respect to all measurable quantities, known as "observables", such as the position, velocity, momentum, angular momentum, spin, isospin, electric field at point (x,y,z), the gluon field's energy density elsewhere, the amplitude of the third harmonic wave on a string, and so on. All of these observables are represented by Hermitean, linear operators. All of these operators have some eigenstates and eigenvectors. The wave functions can be decomposed into superpositions of eigenvectors associated with different eigenvalues. The squared absolute values of the coefficients give us the probabilities that the corresponding eigenvalue will be measured if the measurement is made right now. I have essentially sketched the postulates of quantum mechanics - the principles that every quantum theory shares, regardless of its detailed dynamical rules (such as the Hamiltonian). You may dislike the probabilistic character of quantum mechanics but I guess that you must agree that the "philosophical democracy" between all observables is pleasing and natural. Segregation begins But things must work differently in Bohmian mechanics. Recall that the wave function always becomes a real wave in this framework. However, one must also supplement it with some additional classical observables - the positions of particles. What about the spin? Does it have some classical value, besides the individual components of the wave function (which may be e.g. a spinor)? The answer is obviously a resounding No. The Bohmists assume that "X(t)" objectively exists. But if the "j_z" existed at the same moment, and had one of the allowed values "+1/2" or "-1/2", it would inevitably mean that "z" is a preferred axis and the rotational symmetry is heavily broken (much like the Lorentz invariance whose breaking by the Bohmian picture was sketched earlier). Moreover, the equation controlling the discrete function of time, "j_z(t)", couldn't be continuous. You could also think that the "real particle" has pre-determined binary polarizations of the spin with respect to any axis in space. Well, that's no good, either. Such a picture could never predict simple facts, for example the fact that if a particle is prepared in the "spin-up" state with respect to one axis, the probability of "spin up" with respect to another axis must be "cos(theta/2)^2" where "theta" is the angle between the two axes. Denying the existence of spin So what do the Bohmists do? They simply deny that the spin exists in the same way as the positions or velocities do. While the wave functions - now real waves - still depend on the spin (because the wave functions must clearly be spinorial for an electron), there is no additional degree of freedom associated with them. That's a tragic tumor, a seed of self-destruction, inserted into the very heart of the pilot wave theory but the Bohmists must obviously try to transform this tumor into a virtue. While Bell would say that the spin and similar observables are "contextual", Shelly Goldstein et al. write: ... referred to as contextuality, but we believe that this terminology, while quite appropriate, somehow fails to convey with sufficient force the rather definitive character of what it entails: Properties which are merely contextual are not properties at all; they do not exist, and their failure to do so is in the strongest sense possible. How nicely radical these people are in writing the very opposite of the actual truth :-) - which says that the ontological reality and the degree of existence of the spin is, in the strongest possible sense, equivalent to the ontological reality of the position or any other observable. And even the previous sentence fails to convey the key point that those who fail to recognize the equal philosophical status of all observables shouldn't have passed their introductory quantum mechanics courses. :-) In order to celebrate the Martin Luther King Jr Day, I will dedicate the rest of the text to a fight against the segregation of observables. :-) So my statement is very modest - that observables can't be segregated into the "real" primitive ones and the "fictitious" contextual ones - a fact that trivially rules out all theories (such as the Bohmian ones) that are forced to do so. There are many arguments to see why I am right: Effective theories automatically remove segregation Imagine a world where not all observables (which are linked to Hermitean operators in quantum mechanics) are "fundamental" at a hypothetically deeper (Bohmian) level. Only a subset of the "primitive" ones can be associated with real "properties" (additional classical degrees of freedom that accompany the wave function). The position and the velocity are "real properties" and so is the orbital angular momentum - which is just the cross product of the position and the momentum. However, the spin can't be a "real property", as the Bohmists (and I) explained: it must be just "contextual", otherwise the Bohmian theory collapses immediately. However, the spin and the orbital angular momentum are just two terms that add up to the total angular momentum. And the separation into the two terms is partly unphysical. For example, it depends on the resolution with which we describe the phenomena, or the characteristic scale of our effective theory, if you wish. What does it mean? If you describe the atoms in a molecule as a chemist, you may want to describe the atoms as "indivisible" objects whose internal angular momentum is simply classified as a discrete spin. In such an approach, the whole angular momentum is "contextual". However, a physicist may know that there are electrons inside the atom. She can study their motion and identify the orbital angular momentum as an important part of the total angular momentum. Suddenly, most of the angular momentum becomes "primitive". One can go even deeper and study shorter distances. The spin of elementary particles in the stringy tower of states can be derived from (and reduced to) internal vibrations and oscillations of a string. If you believe that the positions of all "string bits" along the string are "primitive" properties, i.e. observables equipped with additional classical degrees of freedom, you may argue that the spin is "primitive", too. Of course, you need to consider superspace with spinor-valued coordinates to be able to obtain half-integral spins out of the internal "orbital" motion. Decoherence, and not racial prejudices, decides about the effective segregation in the real world That's my second point and it is a positive one because we are actually going to talk about the emergence of the classical limit in the correct picture. Clearly, some quantities in the real world look more classical than others. But what are the rules of the game that separates them? The Bohmists assume that everything that "smells" like "X" or "P" is classical while other things are not. Usually, position eigenstates produce radiation etc. described by microstates that are almost exactly orthogonal to each other, after a short time, for different choices of the position. That makes the basis of position eigenstates "effectively preferred", much like the basis {dead, alive} of the Schrödinger cat. On the other hand, the state "alive+dead" of the Schrödinger cat doesn't decohere from "alive-dead", even though they are orthogonal to one another, because these two states of the cat don't emit microstates of radiation (imprints in the environment) that would be orthogonal to one another. So these two states of the cat never become good descriptions of "0" and "1" in a classical bit. However, it is not a universal fact that the decohering eigenvectors are eigenvectors of a quantity labeled by the letter "X". They can be eigenvectors of "Omega" or any other letter and the geometric representation of "Omega" can be very different from a position. Quantum computer: a contextual model of the whole world The most extreme example of this fact is a quantum computer. Note that with a big enough quantum computer, you may emulate any quantum system with a satisfactory accuracy, by brute force. That includes quantum systems which have particles with positions "X" and momenta "P". However, the quantum computer is entirely made out of spins or other discrete "qubits"! So according to the Bohmists, there are no additional classical degrees of freedom similar to "X". They only have the wave function as the quantum mechanicians do: the only difference is that they equip it with a wrong, deterministic interpretation. Because any additional "X"-like degrees of freedom are absent, they can never make a "measurement" in their system. If the Bohmian theory is compatible enough with the observations, it must also be compatible enough with the conventional quantum theory because the latter seems to match the experiments. If it is so, it is very likely that the quantum predictions for the quantum computers are valid. One of them says that the events inside a particular quantum computer can be a perfect model of other quantum systems: they are really indistinguishable. There can exist brains and people constructed out of quantum observables in such as quantum computer. However, they will never be able to realize themselves or measure anything. They will stay in the linear superpositions forever. The "measurement problem" is completely unsolved in such a picture. Of course, the correct way to solve the "problem" is to accept the probabilistic interpretation of the wave function and of the quantum mechanical postulates. If you do so, there is really no problem left. In order to simplify their imagination, the Bohmists imagined the existence of additional classical objects - the classical positions. But only in the simplest case of spinless 1-particle quantum mechanics, such a picture could be argued to survive the basic tests. Whatever you add, it makes the framework collapse. And you can make physics of the spins so complicated that virtually all of physics becomes "contextual", i.e. composed out of "non-real properties". In that case, your wave functions will spread indefinitely, which should force you either to accept their probabilistic character, or to try to invent new levels of a physical way to imagine what's happening during the "collapse". I assure you that if you choose the latter approach, you will be failing forever. There can't be any "deeper" unexplained mechanism behind the random outcomes of experiments. They're genuinely random and they must be random, as the free-will theorem guarantees. I was talking about a quantum computer but we know all kinds of dualities where one discrete quantity becomes continuous in a limit, while another quantity becomes discrete. For example, in the BMN limit of the CFT, i.e. the Penrose limit of the AdS space, the number of particle excitations in a chain (the BMN operator) becomes interpreted as a spin of a particle (graviton or a low-lying stringy excitation) in the AdS space. The Bohmists would almost certainly choose the number of particles to be a real, "primitive" quantity, while the spin is just "contextual". That's clearly wrong because these two types of observables can really be the very same thing: the difference is often purely linguistic in character. The physics may be identical, and it is just a matter of terminology whether we call a certain quantity "the spin" or "the number of some particles". I could make things even harder for the Bohmian framework by looking into quantum field theory. What are the real, "primitive" properties in that case? Clearly, they want some quantities that often behave classically in classical limits. Except that there is no simple, universal description what they should be. If you have low-frequency electromagnetic waves, it is the electric and magnetic vectors at each point, "E(x)" and "B(x)", that appear as natural classical quantities in a classical limit. That's why the classical electromagnetic field was the first historically understood limit of QED. On the other hand, typically for higher frequencies when "hf" is pretty high, we describe physics in terms of photons and we would prefer to imagine that its position is the real, "primitive" property that accompanies the wave function. Note that our choices would be very different in these two situations - but the two situations only differ by a boost because a boost is enough to change the frequency of an electromagnetic wave! Absorption and "primitive" information loss We have just explained that there are dualities (equivalences) that relate observables that are "primitive" with those that are "contextual" according to the Bohmian racist picture. ;-) However, dualities are not necessary to see the tension: evolution is good enough. In fact, observables that look "primitive" to the Bohmists can evolve from the "contextual" ones, and vice versa. More generally, the amount or percentage of "primitive" (vs "contextual") information about the physical system is not constant in time. There are all kinds of inconvenient processes that demonstrate this fact. For example, new coordinates "X(t)" of particles must be "invented" whenever new particles are produced at the colliders. Inconveniently enough, a collider can smash two protons or antiprotons and produce 10 of them. It often does so. The Bohmian theories can have no self-consistent rules to "invent" the new classical values of all these new hypothetical classical hidden observables. The new particles could either be born at a random place, violating the local charge conservation, or they could be created at the same point. In both cases, you immediately face contradictions. If you fundamentally violate the local charge conservation, you will be able to amplify this effect and obtain macroscopic violations of the charge. On the other hand, if you create all the new particles at the same point where the wave function happens to have some mysterious property, you will fundamentally violate the verified, microscopic T-reversal symmetry of the laws of particle physics because it is virtually impossible for several particles to meet at the same point which would be needed for them to annihilate (the reverse process: yes, be sure, in the real world, the number of particles can drop, too). All these comments are huge question marks about a general problem that was understood to be a defect of the pilot wave theory from the very beginning: no one could ever produce a sensible explanation what happens with their "classical" wave function and with their classical hidden variable "X(t)" when a particle is absorbed. Although the pilot wave theory was motivated entirely by its goal to "shed light" on the measurement procedure, it has really no sensible, new picture to offer. When a particle is created, one faces the T-reversal image of the same problems. They can't be solved. Reduction of spin measurements to positions: not possible in general There are many other arguments to see that the Bohmist segregated picture is indefensible. We have seen some of them but I want to sketch a similar argument from a different viewpoint. By saying that the spin is not a real property, they also say that all the spin measurements (and the measurements of other contextual observables) must always reduce to the measurement of real, "primitive" properties such as the position and velocity: for example, the Stern-Gerlach devices change the position of a particle in the magnetic field, depending on the projection of its spin, and measure the position. Except that it's not the case in general. In the quantum computer example, everything is made out of spins. Nevertheless, measurements must exist in that world, as long as the quantum computer emulates the environment causing the decoherence, too. Obviously, these measurements of certain functions of the spins (qubits in the quantum computer) can't be reduced to the measurement of any fundamentally "primitive" quantities because there are no fundamentally "primitive" quantities in such a quantum computer! One doesn't have to consider a quantum computer. A simple polarizer is a gadget that can measure the x/y linear polarization of a photon - its form of an internal, discrete, spin-like degree of freedom - without changing the location of the photon (which would be pretty hard to do). You can do similar things with the electron, too. In all these cases, the Bohmian theories have no chance to explain why the photon is sometimes absorbed and sometimes it is not. The Stern-Gerlach device is not the only method to feel the spin! Bohmian mechanics doesn't imply quantization of observables There is one more key problem with the Bohmian interpretation which is a bit different from the problems discussed above but I will mention it, anyway: it doesn't predict the correct rules of quantization (discreteness) of various quantities. For the sake of concreteness, consider the orbital angular momentum of the Hydrogen atom. That's too bad because this is an inherently quantum condition and you may only explain the allowed "monodromies" (behavior under the trip around the string) naturally in terms of "psi". In the Bohmian picture, the crucial region that decides about the single-valuedness of "S" is the "string" (co-dimension two locus) in space where "R=0" (i.e. "psi=0"). To explain what can happen with "S" when you make a circular trip around this string, you really need at least "old quantum mechanics" - Bohr's ad hoc quantization rule for the integer number of waves - or another principle that inserts the "quantization" itself. So even though the Bohmian mechanics stole the Schrödinger equation from quantum mechanics, the superficially innocent step of rewriting it in the polar form was enough to destroy a key consequence of quantum mechanics - the discreteness of many physical observables. More generally, something very singular seems to be happening near the "R=0" strings in the Bohmian model of space. But quantum mechanics correctly says a completely different story: there is nothing "stretched" around the "R=0" i.e. "psi=0" strings at all. They're just the regions where the particle won't be observed but nothing really goes to infinity over there. The very special role of the "R=0" as imagined by the Bohmian picture is just another consequence of its incorrect preference of the position "X" over other quantities. For example, in the momentum "P" space, we would get completely different loci where "psi(p)=0". They wouldn't be special, either. If you take a superposition of two functions "psi(x,y,z)", the location of the string completely changes, too. The Bohmian mechanics thinks that this location is physical, singular, and important. But it is easy to see that in the actual world, as described by proper quantum mechanics, the location is physically uninteresting, smooth, regular, and unimportant. Get real: it is just an ordinary interference minimum. Just because the electron is not located somewhere, doesn't mean that the place is filled with mysterious, ambiguous (undetermined "S"), and singular dragons: in most places of the Universe, the electron is not located, after all. That's why a careful physicist shouldn't use the "R exp(iS)" form of the wave function "psi" in cases when it routinely falls to zero - for example in cases with interference, small quantized quantities, or elsewhere, far away from the classical limit. The separation of the phase from the absolute value is only useful in the WKB, semiclassical approximation. The more likely and important the "psi=0" value becomes (i.e. the more quantum-ish regime we consider), the more defective and misleading the polar form of "psi" becomes. It is therefore very important that we treat all observables with the same philosophical tools, even though the mathematical details (such as the spectrum) clearly depend on the particular operator we study. The segregation of the observables into the "primitive" and "contextual" ones is unphysical, contradicts our freedom to consider effective theories and to construct computer models, disagrees with the existence of dualities (equivalences) between the different descriptions, behaves discontinuously under time evolution, leads to fundamental microscopic violations of the time-reversal symmetry, amplifies the problems of the Bohmian theories to survive the successful tests of relativity, introduces new singular regions of space that require a special (quantum) treatment to restore the quantization rules, and is surely guaranteed to give us no correct predictions that couldn't be extracted from quantum mechanics even if you imagined that all the inconsistencies above would be miraculously cured. While most of us have surely hated the probabilistic nature of quantum mechanics at some point - I surely did 20 years ago - it is an established fact and mature physicists should be able to see it. The attempts to return physics to the 17th century deterministic picture of the Universe are archaic traces of bigotry of some people who will simply never be persuaded by any overwhelming evidence - both of experimental and theoretical character - if the evidence contradicts their predetermined beliefs how the world should work. If someone still believes this Bohmian nonsense, he should at least reduce his arrogant suggestions that he is ahead of the other physicists. In fact, he is still mentally living in the 17th century and is unable to grasp the most important revolution of the 20th century science. He may overwhelm others with his strong belief that all the problems with spins, fields, quarks, renormalization, dualities, and strings can be overcome in the Bohmian framework. And that's the memo. Add to Digg this Add to reddit snail feedback (0) :
98a476c07d12a214	January 31, 2014 People don't really challenge each other to learn, nor do we challenge ourselves enough... for instance, I don't think anyone in my entire life has ever asked me once what my thoughts are on Quantum Mechanics, or anything to do with quantum mechanics such as quantum superposition, the Schrödinger equation, and honestly, thank god... because I know so very little about it, I feel like a complete idiot just trying to think about it... and I tend to think of quantum mechanics in terms of metaphysical relation... such as, energy is neither created or destroyed, to me, means reincarnation. Equal and opposite reaction? I hear that, and I hear, Karma. I need to take a class, badly. January 30, 2014 What is freedom? (A super serious postiness) Freedom is when you knew when the Superbowl was, and spent your whole life getting to the point that you no longer know or care when the Superbowl is... if you just didn't know ever it could be ignorance, this way, it's truly freedom. So when George Bush said they hate us for our freedom, what he's really saying is the terrorists hate us for our ability to both create and forget fundamental problems in our society that we probably never should have invented in the first place, and that my friends is what we call... closure. January 29, 2014 Cats are notorious for getting underfoot, and I think I figured out why. Cats know that we will feel bad for accidentally kicking them so they do it on purpose, they take one for the team, because they know ultimately it will lead to guilt treats, and not just right then, but for weeks to come... this is the mind of a cat... I'm 103% certain of it! I believe my cat Luna has a complex. She doesn't believe she's being loved unless she's being fed.. this may be why she weighs 18 lbs. Sexual Energy is a well-spring, and we are farmers of well-being... when our state of mind is entertained and challenged by life, it draws from the well and raises it up through the filters and passageways of our being to be painted upon the world with love's intention, growing our perceptions and cultivating our state of emotional and mental balance... we should care for this wellspring of creative and potential energies, for we choose to spend it like currency for a show, or save it along the banks of our lives, and in beauty we grow. Whatever teaches you the fundamental lessons in life... of love, and tolerance, acceptance, and letting go... whatever it is, whether it be religion, or atheism, a tv show, a symphony, an ascetic giving away of all unnecessary possessions... whatever it is for you in that moment... embrace it, no matter what anyone says... the world has its critics, but they cannot live you, they cannot breathe you, seek you, or die you... so embrace you, and believe in that, support that... and support that part of everyone you meet, and appreciate that all the days of your life. If people really believed in God, when something happens they don't understand, why would they ask others why God does what he does... why would they not ask him? Do they feel they cannot hear him if he is there? Whatever you seek you shall find, if it's God you seek, you will find God, and though this doesn't prove God exists, it could be your mind defining an experience as "God", but it will still apply the answer to the question you are carrying... like a wound looking for a bandaid... and whatever you label it, a rose by any other name is just as sweet... so worry not what you call it, worry only that you don't open your heart and mind to life, and that life passes you by because you were too concerned with what to call things, to seek them. January 28, 2014 Trying to face the reality of The Holocaust, the specifics of those who died and those who survived, the front end face of Jews in orchestras, to mask the endless death hidden from view... the absolute Hell on Earth, and waiting for death... it is perhaps impossible to fully digest... it is like trying to swallow the sun... it lies somewhere beyond my comprehension the ability for mankind to do this to another... this is why i call the survivors of the Holocaust the strongest people to walk the Earth. I bow my head and pray to the human spirit that overcomes this unimaginable, undigestible suffering... may their stories live on and deepen your compassion and your strength and your humility. I am glad my mom told me about this special being on TV tonight, it was well worth watching, no matter how much you think you know about that time period... listening to these survivors is so powerful. January 27, 2014 I believe one should study all religions so as to get along with others in this time... but follow none of them, and it is my hope that one day religion like training wheels will be put aside and mankind walk side by side without religion. January 24, 2014 People don't really believe in freedom of speech, they believe in speech that they agree with, and that's generally a very narrow corridor of thoughts... Now someone would be offended by my saying that, but it's not offensive... if I changed people to women, then it would be offensive, but women will read it as if it does say women... because women are fucking nuts. January 23, 2014 I apologize to all my friends but you should know by now, that nearly all of the mainstream topics you find value in, I find none in... like politics, and sports, and organized religion, except where I feel like a doctor vaccinating people with Reason against all of your indoctrinated nonsense that I feel is destroying the entire world... so don't be offended, I just think you are part of the problem, but not out of any real fault of yours, you were born into an obligatory observation of these things and for now it works for you... so if I don't respond to these topics, it's not because I don't love you... I just hate on a visceral fundamental level, everything you stand for. Thank you. ~Jonathan It matters not if you always speak the truth, for the only truth others will recognize is the truth that aligns with their own. This makes the truth your own, and you must appreciate it like an invisible butterfly that has landed in your open palm... it is your heartbeat, birth, breath, and death. January 22, 2014 Cultivating compassion is like filling a beaker of water... one doesn't have to choose to love a few, or just your family, some say you can't love everyone, but think of the beaker... when you fill it with water, there isn't parts of the beaker filling lower than others, the level remains the same throughout the entire beaker... in this way, as we cultivate compassion, the way we perceive all things collectively becomes infused with compassion. January 20, 2014 Putting your shoes on a table is not bad luck, it's just mind control to brainwash you away from doing something unhealthy... but ultimately it's about the same as letting the cat walk on the table... it's about germs, and in ancient history, when science was in its infancy, we didn't know about germs... and so we had superstition. It's time to let go of the ignorant conclusions, and realize, we don't live in that world anymore... we understand more, we know more about the universe... let go of fear and silly superstition, and embrace Reason and common sense... and put your mind and what you've learned to good use, rather than indulge magic and ignorance. Doubt is a mark of intelligence, right? Those who do not doubt, don't learn. This is why when asked if he ever doubts himself, Obama's grinning and serious, "No", concerned me so greatly. I don't doubt gravity, I accept it as a fact of life, and so I learn nothing new about gravity. I doubt lots of things in the world, which is why i explore them... an inability to doubt is an inability to learn. The most important lessons to learn in life and the most challenging for me, is to love and to let go. When I make a post on Facebook, I check the post time so I can update my website after, but I've been doing it for so long, that now my brain wonders if the exact time isn't pointing the way to a page or verse in a book, signs along the path, vibrations in the sky.... Our worries do not vanish through figuring something out, they vanish when we cultivate compassion for others. January 19, 2014 To those who do not understand, illusions are magic To those who do not understand, technology is terrifying To those who do not understand, faith is a comfort To those who do not understand, reason is godly Once we understand the process, we are no longer enamored or entrenched in fear by what we witness in the world. This is not to undermine the very real danger in the illusory, technological advancements, religion, or logic... but to point out a knot that exists and a method by which to unravel it. January 17, 2014 America is addicted to utilitarian concerns, and that's okay to an extent... we work on the organization, when we're not comfortable with the reality of endless change... but this leads to neglecting the utilitarian, laundry, folding clothes, cleaning, for there is a void in our spiritual life, and turning to the utilitarian concern can only help for so long... eventually one must make their spiritual life a priority, then utilitarian concern and spiritual practice become harmonized, and there becomes no differentiation between utilitarian and spiritual living, there is simply living. I can open soda bottles on the edge of my PC... it hurts my hand and probably confuses my computer, but it makes me feel really manly, and there's little in my life that does that, so I do it anyway. I'm probably testosterone deficient, I wonder if there's a trend of that in romantic poets? I wonder if anyone has asked that question before? January 14, 2014 It is far more interesting knowing what you believe about what you think, than what you think about what others believe. We have barely scratched the surface of human potential... perhaps we may even transcend being human and become another kind of butterfly... but we toil away in the mud, creating new illusions to chase, because that's where we are... we do not need these attachments... chasing money... but we will someday outgrow even these ideologies... and when we do... and oh how I wish I were around to witness it... a new age will arrive, an age without day or night, but of endless flight, where we experience our own gravity, the fullness and depth of space will be no deeper than us, and not because we limit it, but because we recognize that we are the cosmos, we are the expanding universe, and the interdependent and co-mingled experiential consciousness that we are, become an ocean in the palm of your hand, eyes of moon and star, pores of glimmering lights in the heavens, shoulder horizon, with an atmosphere of hair flowing around us... giving birth to the Cosmos, forever... January 13, 2014 For the young who live within a broken system, to thy master conform or complain. For unless you pursue that which truly inspires, you are nothing but a pawn in the game. I had started accidentally breaking things, not because I didn't worry about possessions but because I had grown so certain of their inevitable demise, and what could have been a valuable lesson in impermanence had become the echo of lonely cries. The lesson is always thus: Balance and Simplify. January 9, 2014 [Posted at backwards 237 o'clock AM] Love for all people, I don't mean conventional love, I don't mean friendship love, I don't mean intimate love, I mean unconditional love... is possible, for it is not about maintaining a list, or trying to be open... it's about changing your view of humanity as a whole... once you uncover the heart, it recognizes the heart in everyone. It is the smaller mind that categorizes and separates and differentiates and judges, it is our greater mind that contains the multitudes, like the bird in the sky you long for... you are not he who chases, you are the sky which embraces. Hey that rhymed, so it's.... probably true. The problem with hitting rock bottom... once you reach it, and pull yourself up, you are simultaneously digging down deeper into truth, which means your rock bottom is also as never-ending as your sky. This makes YOU the expanding Universe. Everything we look at is the world talking to us... when you meet someone who hears a similar message in the world... hold on to them... they should be in your band. But what about those who hear something else? We all hear many things in everything... recognizing that all things are speaking an individualized message to us all, all the time... Perhaps that is nonjudgment. January 7, 2014 I think the one thing I never really got enough of in life was the support of people who care reminding me to stick with it, whatever it is, school, my music, comedy, I met more people who were eager to tear me down than people who were loving and supportive, who wanted to see me succeed, and I needed someone to give me that encouragement. My dad left when I was 3, my mom was overwhelmed with 5 kids, I was the youngest, I left home young. I was on my own too early and didn't get a foundation of confidence. Wherever you go, the Yeti is sure to follow.... for he is the shadow of your subconscious, the externalized uncertainties that you merely get a glimpse of now and then in your life... and it is through not recognizing these things as our own, that we externalize them, and the level of fear associated creates the manifestation experienced by you.... on the end of greater fear we have multi-dimensional beings, demons, ghosts, yetis.... but ultimately they are all the same... they are you as seen through the prism of your own fear and desire. January 6, 2014 Sometimes I look at my cat when he's got that demanding, disconcerted, wtf look on his face, and I point at him and I give him the truth, and I say, "Hey, you ain't got no money, and you ain't got no shoes." People who play records make better lovers, they've cultivated a gentle touch that CD players could never appreciate or understand. In today's disposable culture... a return to art, to attention to the details each moment of our lives, they're valuable in fact they're the most precious thing we possess, simply because they cannot be possessed. It's more about rediscovering what's always been, so we can create what can always be. January 4, 2014 Every time you masturbate, you are technically making porn for aliens. The ego has strong defenses against the embrace of reality, even though that reality is light is truth is liberation... there is an intense fear of such because we have identified with our attachment illusions... Cannabis is like putting the ego defenses in a boxing ring with a more powerful opponent... and each time I partake, my defenses weaken, until it is simply a matter of me, and reality... and only one of us will survive... and I have a sinking feeling.. it's not going to be me. January 3, 2014 The green hit of herb is like oxygen for the soul. Part of the problem with Cannabis is that people abuse it creating the stigma we know of today, which creates the kneejerk reaction against Cannabis by good people. On top of that is the corrupt agenda by the FDA, by those who want to keep people buying their medicines, and their controlling, corporate, system that is becoming obsolete, more and more every day... we are evolving beyond the structure of society that is all around us today... the treatment of animals, gay marriage, abortion... people are waking up to the reality that there are real issues facing humanity, and these cosmetic distraction issues are not worthy of our attention. This is why so many feel voting is pointless. Changing the system from within... where the corruption is so dank and dark and deep that whatever you do as an individual gets filtered by the billionaire corporations calling the shots. So I get it, and I don't believe in shaming people who out of disillusionment stopped voting. Life's too short to fight a current taking us away from everything we believe in, and our nation is being steered by those captains of industry.... we were meant for more than this. Get off the boat and take the scenic route. I don't smoke Cannabis to get high, I smoke it to get balanced. I have found that smoking a very small amount really brings my roller coaster down to a comfortable speed. As a kid, I believe I got negative effects from Cannabis simply by not knowing my limits, but now I'm older, I'm wiser, I've cultivated balance through meditation, exercise, diet, and all areas of my life, and yet the roller coaster continues... Cannabis has thus been very helpful at that point... and I would not mind if I could take Cannabis for that balance without getting high at all... I don't need the high, I need the balance, but right now I can't get one without the other. But then I must wonder what affect on my poetry, my music, the high has... things to ponder... Sometimes I ponder on all of God's creation, and maybe I walk by a flower, and I stop, look and think... "Eh, it's not that great." I immediately sneeze, shrug, giggle look up and softly say, "Just kidding..." and God sheathes his lightning bolt. Where tax dollars support everything I don't believe in... being true to myself becomes a crime. January 2, 2014 I get those cascading deja vu experiences, but I find them easy to pop out of by not putting too much value on my thoughts... emphasis on "too much" ... I think perhaps thoughts are the chickens running around after their heads have been cut off... January 1, 2014 If you ever feel like the universe has set you up to fail, prove it wrong. The universe will likely take that moment to make you pee on yourself just to prove how powerful it is, but you know just laugh it off, wash it off too, and you keep showing the universe what love and beauty is possible.... I think the universe is feeling a loss at all those black holes eating up its things! So you keep loving, and make the universe smile. Perhaps it's our job to teach the universe how to love.
3d90cf31258cc6c5	Bruce R. Johnson Recent rotator from Rice University to the National Science Foundation Distinguished Faculty Fellow, Department of Chemistry Research Activities Wavelet Technology in Quantum Mechanics and Nanophotonics Earlier efforts in the group centered on making orthogonal wavelets a practical means for solving the large-amplitude vibrational Schrödinger equation and Maxwell's equations near plasmonic nanoparticles. The need for large-amplitude descriptions of vibrational motion arise in many problems including specifically Dissociative Resonance Raman Spectroscopy (DRRS). While a number of algorithms exist for solving the needed stationary and time-dependent Schrödinger equations, many either waste effort in spatial regions where not much is happening or else require considerable human coaxing. Similarly, the determination of the near EM fields surrounding general nanostructures and nanoparticles is challenging and in need of accurate and efficient methods that can bridge different scales. Compact support wavelets can be used to approach a wide variety of problems with systematic accuracy, as has been a focus in work of this group and is the basis of an object-oriented software program, MultiWavePack (not updated since I began serving at NSF). Efficiency is a current focus, and is important if wavelets are to provide a rational platform for the development of automated methods. Many of the algorithms we develop for our specific problems, such as overcoming Gibbs-phenomenon interference in satisfying EM boundary conditions, are really of interest in a wide variety of applications. There are many papers and books introducing use of orthogonal wavelets, but it is often difficult for the newcomer to figure out basic tasks such as creating a plot or projecting a known function in a wavelet basis. Over the years of working with REU students at RQI who came in with cursory knowledge of wavelets, we have developed our own introductory package in Mathematica (OW.m) and an accompanying tutorial notebook (TutorialOW.nb) to get people past this stage quickly. The current version of the tutorial (Mathematica 8) was finished owing to an NSF IR/D grant which is gratefully acknowledged. Surface-Enhanced Raman Scattering from Metal Nanoparticles This work aims at calculating molecular near-field response for molecules such as p-mercaptoaniline on silver nanoshells, including vibrational dynamics of multiple Raman-active modes, directionality, field polarization, and polarizability tensors derived from small molecule-metal cluster ab initio calculations. A density matrix formalism is used to include both radiative and non-radiative relaxation processes in spectral simulations of Surface-Enhanced Raman Scattering (SERS), and special attention is paid to the effects of near-resonance in the Raman processes. Activity Links
731416346488bd5d	Time independent schrodinger’s equation What is Schrodinger equation? i = imaginary unit, Ψ = time-dependent wavefunction, h2 is h-bar, V(x) = potential and H^ = Hamiltonian operator. Also Read: Quantum Mechanical Model of Atom. Time-independent Schrödinger equation in compressed form can be expressed as; OR. Time-independent-Schrödinger-nonrelativistic-equation. Why is there an I in the Schrodinger equation? The imaginary constant i appears in the original Schroedinger article (I) for positive values of the energy, which therefore are discarded by Schrödinger, who wants real eigenvalues and requires negative energy. Is the cat alive or dead? What are the applications of Schrodinger equation? Schrödinger’s equation offers a simple way to find the previous Zeeman–Lorentz triplet. This proves once more the broad range of applications of this equation for the correct interpretation of various physical phenomena such as the Zeeman effect. What is de Broglie’s hypothesis? De Broglie’s hypothesis of matter waves postulates that any particle of matter that has linear momentum is also a wave. De Broglie’s concept of the electron matter wave provides a rationale for the quantization of the electron’s angular momentum in Bohr’s model of the hydrogen atom. What is the energy of a free particle? A free particle is not subjected to any forces, its potential energy is constant. Set U(r,t) = 0, since the origin of the potential energy may be chosen arbitrarily. How is information extracted from a wave function? To make predictions about the outcome of all measurements, at any time, one has to “do” something to the wave function to extract information. One has to perform some mathematical operation on it, such a squaring it, multiplying it by a constant, differentiating it, etc. Did Schrodinger own a cat? Erwin Schrödinger doesn’t appear to have personally owned a cat. He did however own a dog. His great-aunt owned several cats. Why is the cat alive and dead? Until the box is opened, an observer doesn’t know whether the cat is alive or dead—because the cat’s fate is intrinsically tied to whether or not the atom has decayed and the cat would, as Schrödinger put it, be “living and dead in equal parts” until it is observed. (More physics: The Physics of Waterslides.) Leave a Reply Equation of vertical line Bernoulli’s equation example
55fcb7551bad22f0	The following comment by Wildcat made me think about whether density functional theory (DFT) can be considered an ab initio method. @Martin-マーチン, this is sort of nitpicking, but DFT (where the last "T" comes from "Theory") can be considered as an ab-initio method since the theory itself is built from the first principles. The problem with the theory is that the exact functional is unknown, and as a result, in practice we do DFA calculations ("A" from "Approximation") with some approximate functional. It it DFA which is not an ab-initio method then, not DFT. :) I always thought that ab initio refers to wave function based methods only. In principle the wave function is not necessary for the basis of DFT, but it was later introduced by Kohn and Sham for practical reasons. The IUPAC goldbook offers a definition of ab initio quantum mechanical methods: ab initio quantum mechanical methods Synonym: non-empirical quantum mechanical methods Methods of quantum mechanical calculations independent of any experiment other than the determination of fundamental constants. The methods are based on the use of the full Schroedinger equation to treat all the electrons of a chemical system. In practice, approximations are necessary to restrict the complexity of the electronic wavefunction and to make its calculation possible. According to this, most density functional approximations (DFA) cannot be termed ab initio since almost all involve some empirical parameters and/or fitting. DFT on the other hand is independent of any of this. What I have my problems with is the second sentence. It states, that treatment of all electrons is necessary. This is technically not the case for DFT, because here only the electron density is treated. All electrons and the wavefunction are implicitly treated. An earlier definition of ab initio can be found in Leland C. Allen and Arnold M. Karo, Rev. Mod. Phys., 1960, 32, 275. By ab initio we imply: First, consideration of all the electrons simultaneously. Second, use of the exact nonrelativistic Hamiltonian (with fixed nuclei), $$\mathcal{H} = -\frac12\sum_i{\nabla_i}^2 - \sum_{i,a}\frac{Z_a}{\mathbf{r}_{ia}} + \sum_{i>j}\frac{1}{\mathbf{r}_{ij}} + \sum_{a,b}\frac{Z_aZ_b}{\mathbf{r}_{ab}}$$ the indices $i$, $j$ and $a$, $b$ refer, respectively, to the electrons and to the nuclei with nuclear charges $Z_a$, and $Z_b$. Third, an effort should have been made to evaluate all integrals rigorously. Thus, calculations are omitted in which the Mulliken integral approximations or electrostatic models have been used exclusively. These approximate schemes are valuable for many purposes, but present experience indicates that they are not sufficiently accurate to give consistent results in ab initio work. This definition obviously does not include DFT, but this is probably due to the fact it was published before the Hohenberg-Kohn theorems. But in general this definition is still largely the same as in the goldbook. Another point which confuses me are titles like: "Potential Energy Surfaces of the Gas-Phase SN2 Reactions $\ce{X- + CH3X ~$=$~ XCH3 + X-}$ $\ce{(X ~$=$~ F, Cl, Br, I)}$: A Comparative Study by Density Functional Theory and ab Initio Methods" Liqun Deng , Vicenc Branchadell , Tom Ziegler, J. Am. Chem. Soc., 1994, 116 (23), 10645–10656. And then again we have titles like: "Ab Initio Density Functional Theory Study of the Structure and Vibrational Spectra of Cyclohexanone and its Isotopomers" F. J. Devlin and P. J. Stephens, J. Phys. Chem. A, 1999, 103 (4), 527–538. Unfortunately Koch and Holthausen, who wrote the probably most concise book on DFT, A Chemist's Guide to Density Functional Theory, never really refer to DFT as ab initio or clearly draw the line. The closest they come is on page 18: In the context of traditional wave function based ab initio quantum chemistry a large variety of computational schemes to deal with the electron correlation problem has been devised during the years. Since we will meet some of these techniques in our forthcoming discussion on the applicability of density functional theory as compared to these conventional techniques, we now briefly mention (but do not explain) the most popular ones. But that does not really answer my question. Throughout the book they use the term only in the form of conventional ab initio theory or in combination of explicitly stating wave function and variations thereof. In my quite extensive research about DFT selection criteria I never came about the term 'ab initio DFT'. So the question remains: Is density functional theory an ab initio method? First note that the acronym DFA I used in my comment originates from Axel D. Becke paper on 50 year anniversary of DFT in chemistry: Let us introduce the acronym DFA at this point for “density-functional approximation.” If you attend DFT meetings, you will know that Mel Levy often needs to remind us that DFT is exact. The failures we report at meetings and in papers are not failures of DFT, but failures of DFAs. Axel D. Becke, J. Chem. Phys., 2014, 140, 18A301. So, there are in fact two questions which must be addressed: "Is DFT ab initio?" and "Is DFA ab initio?" And in both cases the answer depend on the actual way ab initio is defined. • If by ab initio one means a wave function based method that do not make any further approximations than HF and do not use any empirically fitted parameters, then clearly neither DFT nor DFA are ab initio methods since there is no wave function out there. • But if by ab initio one means a method developed "from first principles", i.e. on the basis of a physical theory only without any additional input, then • DFT is ab initio; • DFA might or might not be ab initio (depending on the actual functional used). Note that the usual scientific meaning of ab initio is in fact the second one; it just happened historically that in quantum chemistry the term ab initio was originally attached exclusively to Hartree–Fock based (i.e. wave function based) methods and then stuck with them. But the main point was to distinguish methods that are based solely on theory (termed "ab initio") and those that uses some empirically fitted parameters to simplify the treatment (termed "semi-empirical"). But this distinction was done before DFT even appeared. So, the demarcation line between ab initio and not ab initio was drawn before DFT entered the scene, so that non-wave-function-based methods were not even considered. Consequently, there is no sense to question "Is DFT/DFA ab initio?" with this definition of ab initio historically limited to wave-function-based methods only. Today I think it is better to use the term ab initio in quantum chemistry in its more usual and more general scientific sense rather then continue to give it some special meaning which it happens to have just for historical reasons. And if we stick to the second definition of ab initio then, as I already said, DFT is ab initio since nothing is used to formulate it except for the same physical theory used to formulate HF and post-HF methods (quantum mechanics). DFT is developed from the quantum mechanical description without any additional input: basically, DFT just reformulates the conventional quantum mechanical wave function description of a many-electron system in terms of the electron density. But the situation with DFA is indeed a bit more involved. From the same viewpoint a DFA method with a functional which uses some experimental data in its construction is not ab initio. So, yes, DFA with B3LYP would not qualify as ab initio, since its parameters were fitted to a set of some experimentally measure quantities. However, a DFA method with a functional which does not involve any experimental data (except the values of fundamental constants) can be considered as ab initio method. Say, a DFA using some LDA functional constructed from a homogeneous electron gas model, is ab initio. It is by no means an exact method since it is based on a physically very crude approximation, but so does HF from the family of the wave function based methods. And if the later is considered to be ab initio despite the crudeness of the underlying approximation, why can't the former be also considered ab initio? • $\begingroup$ I read and referred to that paper, too. I love that quote. I agree with you. (I still leave the accepting part for next week, to encourage more people to vote.) Would you say that the IUPAC definition should be updated, i.e. include the electron density explicitly in the last sentence? $\endgroup$ – Martin - マーチン Jul 9 '15 at 12:13 • 1 $\begingroup$ @Martin-マーチン, it depends on how do you interpret this second sentence. In fact, I do not see any problem here with respect to DFT being ab initio method in accordance with this definition. Do we use the Schrödinger equation to treat the electrons? Yes, we do; rather indirectly, but we use it. We don't solve the Schrödinger equation, but we use it a starting point in the development of DFT: the key constituent, electron density, is defined in terms of the solution of the Schrödinger equation. $\endgroup$ – Wildcat Jul 9 '15 at 12:26 • 1 $\begingroup$ @Martin-マーチン, now, for the second part of this sentence that insists on treating "all the electrons", I again see no problem with DFT. We indeed treat all the electrons. Yes, we do so in a tricky way: throughout all the process we use only one-electron density without constructing any many-electron entity that describes our many-electron system as a whole (not like in HF where we construct the many-electron wave function out of one-electron function). But we do so because we already proved that many-electron systems can be treated in such a way: one-electron density is enough. $\endgroup$ – Wildcat Jul 9 '15 at 12:30 • $\begingroup$ Yes, I had no problem with the second one, since HK clarifies that quite nicely. I was referring to the last sentence with limitations, considering that for example B3LYP is not ab initio. $\endgroup$ – Martin - マーチン Jul 9 '15 at 12:37 • $\begingroup$ @Martin-マーチン, got it. So, let us say that DFT is ab initio. Now the situation with DFA is indeed a bit more involved. From the same viewpoint a DFA method with a functional which uses some experimental data in its construction is not ab initio. So, yes, DFA with B3LYP would not qualify as ab initio, since its parameters were fitted to a set of some experimentally measure quantities. $\endgroup$ – Wildcat Jul 9 '15 at 13:00 The convention used by many is that ab initio refers solely to wave-function based methods of various sorts and that first principles refers to either wave-function or DFT methods with little approximation. I can't find a citation at the moment, but I know this convention is fairly widely used in, e.g., J. Phys. Chem. journals. The IUPAC gold book doesn't have "first principles," but Google Scholar gives over 224,000 hits for "first principles DFT". • 4 $\begingroup$ In physics, "from first principles" is a synonym for "ab initio"; just an English equivalent of the Latin phrase. In quantum chemistry we habitually avoid using "ab initio" term with DFT since it still can potentially cause some needless terminological battles due to historical strict meaning of "ab initio" term. $\endgroup$ – Wildcat Jul 9 '15 at 15:41 • 2 $\begingroup$ But you're perfectly right, of course. In QC the convention is to avoid calling DFT methods "ab initio" and exclusively use "from first principles" term for them, while both terms can be used for wf-based methods. $\endgroup$ – Wildcat Jul 9 '15 at 15:59 Your Answer
15e3ff4f470c48a2	NuSol — Numerical solver for the 3D stationary nuclear Schrödinger equation Published: 01-01-2016\| Version 1 \| DOI: 10.17632/83crdhyz4j.1 Timo Graen, Helmut Grubmüller This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2018) Abstract The classification of short hydrogen bonds depends on several factors including the shape and energy spacing between the nuclear eigenstates of the hydrogen. Here, we describe the NuSol program in which three classes of algorithms were implemented to solve the 1D, 2D and 3D time independent nuclear Schrödinger equation. The Schrödinger equation was solved using the finite differences based Numerov’s method which was extended to higher dimensions, the more accurate pseudo-spectral Chebyshev c... Title of program: NuSol Catalogue Id: AEXO_v1_0 Nature of problem Solving the 3D nuclear Schrödinger Equation Versions of this program held in the CPC repository in Mendeley Data AEXO_v1_0; NuSol; 10.1016/j.cpc.2015.08.023
7998fca52c0b8286	UCAS Personal Statement and Examples Guides / UK The Universities and Colleges Admissions Service (UCAS) Personal Statement is the main essay for your application to colleges and universities in Great Britain. UCAS gives a nice explanation here, but in short, this is your chance to stand out against the crowd and show your knowledge and enthusiasm for your chosen area of study. You’ve got 4,000 characters and 47 line limit to show colleges what (ideally) gets you out of bed in the morning. How long is that, really? Use your “word count” tool in Google or Word docs to check as you go along, but 4,000 characters is roughly 500 words or one page. Think they’re the same? Think again. Here are some key differences between the UCAS and the US Personal Statement: 1. When you apply to UK schools, you’re applying to one particular degree program, which you’ll study for all, or almost all, your time at university. Your UCAS personal statement should focus less on cool/fun/quirky aspects of yourself and more on how you’ve prepared for your particular area of study. 2. The UCAS Personal Statement will be read by someone looking for proof that you are academically capable of studying that subject for your entire degree. In some cases, it might be an actual professor reading your essay. 3. You’ll only write one personal statement, which will be sent to all the universities you’re applying to, and it’s unlikely you’ll be sending any additional (supplemental) essays. Your essay needs to explain why you enjoy and are good at this subject, without reference to any particular university or type of university. 4. Any extracurricular activities that are NOT connected to the subject you’re applying for are mostly irrelevant, unless they illustrate relevant points about your study skills or attributes: for example, having a job outside of school shows time-management and people skills, or leading a sports team shows leadership and responsibility. 5. Your personal statement will mostly focus on what you’ve done at high school, in class, and often in preparation for external exams. 80-90% of the content will be academic in nature. This may be obvious, but the first step to a great UCAS Personal Statement is to choose the subject you’re applying for. This choice will be consistent across the (up to) five course choices you have. Often, when students struggle with a UCAS personal statement, it’s because they are trying to make the statement work for a couple of different subjects. With a clear focus on one subject, the essay can do the job it is supposed to do. Keep in mind you’re limited to 47 lines or 4000 characters, so this has to be concise and make efficient use of words. To work out what information to include, my favourite brainstorming activity is the ‘Courtroom Exercise’. Here’s how it works: The Courtroom Exercise Imagine you’re prosecuting a case in court, and the case is that should be admitted to a university to study the subject you’ve chosen. You have to present your case to the judge, in a 47 line or 4,000 character statement. The judge won’t accept platitudes or points made without evidence–she needs to see evidence. What examples will you present in your statement? In a good statement, you’ll make an opening and a closing point. To open your argument, can you sum up in one sentence why you wish to study this subject? Can you remember where your interest in that subject began? Do you have a story to tell that will engage the reader about your interest in that subject? Next, you’ll present a number of pieces of evidence, laying out in detail why you’re a good match for this subject. What activities have you done that prove you can study this subject at university? Most likely, you’ll start with a class you took, a project you worked on, an internship you had, or a relevant extra-curricular activity you enjoyed. For each activity you discuss, structure a paragraph on each using the ABC approach: A: What is the Activity? B: How did it Benefit you as a potential student for this degree course? C: Link the benefit to the skills needed to be successful on this Course. With three or four paragraphs like these, each of about 9 or 10 lines, and you should have the bulk of your statement done. Typically two of these will be about classes you have taken at school, and two about relevant activities outside of school. In the last paragraph, you need to demonstrate wider skills that you have, which you can probably do from your extracurricular activities. How could you demonstrate your time management, your ability to collaborate, or your creativity? Briefly list a few extracurricular activities you’ve taken part in and identify the relevant skills that are transferable to university study. Finally, close your argument in a way that doesn’t repeat what you’ve already shared. Case closed! • What if I’m not sure what I want to study? Should I still apply? There are a number of broader programs available at UK universities (sometimes called Liberal Arts or Flexible Combined Honours). However, you should still showcase two or three academic areas of interest. If you are looking for a broader range of subjects to study and can’t choose one, then the UK might not be the best fit for you. • What if I haven’t done much, academically or via extracurriculars, to demonstrate that I’ll be able to complete the coursework for my degree? Should I still apply? You certainly can, but you will need to be realistic about the strength of your application as a result. The most selective universities will want to see this evidence, but less selective ones will be more willing to account for your potential to grow in addition to what you’ve already achieved. You could also consider applying for a Foundation course or a ‘Year 0’ course, where you have an additional year pre-university to enable you to develop this range of evidence. • If I’m not accepted into a particular major, can I be accepted into a different major? It’s important to understand that we are not talking about a ‘major,’ as what you are accepted into is one entire course of study. Some universities may make you an ‘alternative offer’ for a similar but perhaps less popular course (for example you applied for Business but instead they offer you a place for Business with a Language).At others, you can indicate post-application that you would like to be considered for related courses. However, it’s not going to be possible to switch between two completely unrelated academic areas. • What other information is included in my application? Will they see my extracurricular activities, for example? Is there an Additional Information section where I can include more context on what I’ve done in high school? The application is very brief: the personal statement is where you put all the information. UCAS does not include an activities section or space for any other writing. The 47 lines is all you have. Some universities might accept information if there are particularly important extenuating circumstances that must be conveyed. This can be done via email, but typically they don’t want to see more than the UCAS statement and your school’s reference provides. Now, let’s take a look at some of my favorite UCAS personal statement examples with some analysis for why I think these are great. When I was ten, I saw a documentary on Chemistry that really fascinated me. Narrated by British theoretical physicist Jim Al-Khalili, it explained how the first elements were discovered and how Chemistry was born out of alchemy. I became fascinated with Chemistry and have remained so ever since. I love the subject because it has very theoretical components, for example quantum Chemistry, while also having huge practical applications. In this introduction, the student shows where his interest in Chemistry comes from. Adding some additional academic detail (in this case, the name of the scientist) helps guide the reader into more specific information on why this subject is interesting to him. This aspect of Chemistry is important to me. I have, for example, used machine learning to differentiate between approved and experimental drugs. On the first run, using drug molecules from the website Drug Bank, I calculated some molecular descriptors for them. I started with a simple logistic regression model and was shocked to find that it had apparently classified almost all molecules correctly. This result couldn’t be right; it took me nearly a month to find the error. I accidentally normalized the molecular-descriptor data individually, rather than as a combined data set, thereby encoding the label into the input. On a second run, after fixing the error, I used real machine learning libraries. Here I actually got some performance with my new algorithm, which I could compare to professional researchers’ papers. The highest accuracy I ever saw on my screen was 86 percent. The researchers’ result was 85 percent; thanks to more modern machine learning methods, I narrowly beat them. I have also studied Mathematics and Physics at A Level and have been able to dive into areas beyond the A Level syllabus such as complex integration in math and the Schrödinger equation in Physics. This paragraph lays out a clear case for this student’s aptitude for, and interest in, Chemistry. He explains in detail how he has explored his intended major, using academic terminology to show us he has studied the subject deeply. Knowing an admissions reader is looking for evidence that this student has a talent for Chemistry, this paragraph gives them the evidence they need to admit him. Additionally, I have worked on an undergraduate computer science course on MIT Opencourseware, but found that the content followed fixed rules and did not require creativity. At the time I was interested in neural networks and listened to lectures by professor Geoffrey Hinton who serendipitously mentioned his students testing his techniques on ‘Kaggle Competitions’. I quickly got interested and decided to compete on this platform. Kaggle allowed me to measure my machine learning skills against competitors with PhDs or who are professional data scientists at large corporations. With this kind of competition naturally I did not win any prizes, but I worked with the same tools and saw how others gradually perfected a script, something which has helped my A Level studies immensely. Introducing a new topic, the student again uses academic terminology to show how he has gone beyond the confines of his curriculum to explore the subject at a higher level. In this paragraph, he demonstrates that he has studied university-level Chemistry. Again, this helps the reader to see that this student is capable of studying a Chemistry degree. I have been keen to engage in activities beyond the classroom. For example, I have taken part in a range of extracurricular activities, including ballroom dancing, public speaking, trumpet, spoken Mandarin, and tennis, achieving a LAMDA distinction at level four for my public speaking. I have also participated in Kaggle competitions, as I’m extremely interested in machine learning. For example, I have used neural networks to determine the causes of Amazon deforestation from satellite pictures in the ‘Planet: Understanding the Amazon from Space’ competition. I believe that having worked on projects spanning several weeks or even months has allowed me to build a stamina that will be extremely useful when studying at university. This penultimate paragraph introduces the student’s extracurricular interests, summing them up in a sentence. Those activities that can demonsrate skills whcih are transferable to the study of Chemistry are given a bit more explanation. The student’s descriptions in each paragraph are very detailed, with lots of specific information about awards, classes and teachers. What I hope to gain from an undergraduate (and perhaps post-graduate) education in Chemistry is to deepen my knowledge of the subject and potentially have the ability to successfully launch a startup after university. I’m particularly interested in areas such as computational Chemistry and cheminformatics. However, I’m open to studying other areas in Chemistry, as it is a subject that truly captivates me. In the conclusion, the student touches on his future plans, using specific terminology which shows his knowledge of Chemistry. This also reveals that he aims to have a career in this field, which many admission readers find appealing as it demonstrates a level of commitment to the subject. This next statement has to accomplish a number of tasks, given the subject the student is applying for. As a vocational degree, applicants for veterinary medicine are committing to a career as well as a subject to study, so they need to give information that demonstrates they understand the reality of a career in this area. It also needs to explain their motivation for this interest, which quite often is demonstrated through work experience (something which is often a condition for entry into these programs). Finally, as this is a highly-academic subject to study at university, the author should include a good level of academic teminology and experiences in the statement. There is nothing more fascinating to me than experiencing animals in the wild, in their natural habitat where their behaviour is about the survival of their species. I was lucky enough to experience this when in Tanzania. While observing animals hunting, I became intrigued by their musculature and inspired to work alongside these animals to help them when they are sick, as a veterinarian. In an efficient way, the applicant explains her motivation to become a vet, then squeezes in a bit of information about her experience with animals. As a horse rider and owner for nearly ten years, I have sought opportunities to learn as much as I can about caring for the animal. I helped around the yard with grooming and exercise, bringing horses in and out from the fields, putting on rugs, and mucking out. I have also been working at a small animal vet clinic every other Saturday for over 2.5 years. There, my responsibilities include restocking and sterilising equipment, watching procedures, and helping in consultations. Exposure to different cases has expanded my knowledge of various aspects, such as assisting with an emergency caesarean procedure. Due to a lack of staff on a Saturday, I was put in charge of anaesthesia while the puppies were being revived. I took on this task without hesitation and recorded heart and respiration rate, capillary refill time, and gum colour every five minutes. Other placements following an equine vet, working on a polo farm, and volunteering at a swan sanctuary have also broadened my experience with different species and how each possesses various requirements. During pre-vet summer courses, I was also introduced to farm animals such as pigs, cows, sheep and chicken. I spend some time milking dairy cows and removing clustered dust from chicken feet, as well as tipping sheep in order to inspect their teats. In this paragraph, she synthesizes personal experience with an academic understanding of vet medicine. She demonstrates that she is committed to animals (helping in the yard, regular Saturday work, assistance with procedures), that she has gained a variety of experiences, and that she understands some of the conditions (caesareans, clustered dust) that vets have to deal with. Note that she also briefly discusses ‘pre-vet summer courses,’ adding credibility to her level of experience. I have focused on HL Biology and HL Chemistry for my IB Diploma. I was particularly excited to study cell biology and body systems because these subjects allowed me to comprehend how the body works and are applicable to animal body functions. Topics like DNA replication as well as cell transcription and translation have helped me form a fundamental understanding of genetics and protein synthesis, both important topics when looking into hereditary diseases in animals. Learning about chemical reactions made me consider the importance of pharmaceutical aspects of veterinary medicine, such as the production of effective medicine. Vaccines are essential and by learning about the chemical reactions, I f developed a more nuanced understanding about how they are made and work. Now the statement turns to academic matters, linking her IB subjects to the university studies she aspires to. She draws out one particular example that makes a clear link between school and university-level study. I have also written my Extended Essay discussing the consequences of breeding laws in the UK and South Australia in relation to the development of genetic abnormalities in pugs and German shepherds. This topic is important, as the growing brachycephalic aesthetic of pugs is causing them to suffer throughout their lifetime. Pedigree dogs, such as the German shepherd, have a very small gene pool and as a result, hereditary diseases can develop. This becomes an ethical discussion, because allowing German shepherds to suffer is not moral; however, as a breed, they aid the police and thus serve society. The IB Extended Essay (like an A Level EPQ or an Capstone project), is a great topic to discuss in a personal statement, as these activities are designed to allow students to explore subjects in greater detail. The first sentence here is a great example of what getting more specific looks like because it engages more directly with what the student is actually writing about in this particular paragraph then it extrapolates a more general point of advice from those specificities. By choosing to write her Extended Essay on a topic of relevance to veterinary medicine, she has given herself the opportunity to show the varied aspects of veterinary science. This paragraph proves to the reader that this student is capable and motivated to study veterinary medicine. I have learned that being a veterinarian requires diagnostic skills as well as excellent communication and leadership skills. I understand the importance and ethics of euthanasia decisions, and the sensitivity around discussing it withanimal owners. I have developed teamwork and leadership skills when playing varsity football and basketball for four years. My communication skills have expanded through being a Model U.N. and Global Issues Network member. This small paragraph on her extracurricular activities links them clearly to her intended area of study, both in terms of related content and necessary skills. From this, the reader gains the impression that this student has a wide range of relevant interests. When I attend university, I not only hope to become a veterinarian, but also a leader in the field. I would like to research different aspects of veterinary medicine, such as diseases. As a vet, I would like to help work towards the One Health goal; allowing the maintenance of public health security. This affects vets because we are the ones working closely with animals every day. In the conclusion, she ties things together and looks ahead to her career. By introducing the concept of ‘One Health’, she also shows once again her knowledge of the field she is applying to. Standing inside a wind tunnel is not something every 17 year old aspires to, but for me the opportunity to do so last year confirmed my long-held desire to become a mechanical engineer. This introduction is efficient and provides a clear direction for the personal statement. Though it might seem that it should be more detailed, for a student applying to study a course that requires limited extended writing, being this matter-of-fact works fine. I enjoy the challenge of using the laws of Physics, complemented with Mathematical backing, in the context of everyday life, which helps me to visualise and understand where different topics can be applied. I explored the field of aeronautics, specifically in my work experience with Emirates Aviation University. I explored how engineers apply basic concepts of air resistance and drag when I had the opportunity to experiment with the wind tunnel, which allowed me to identify how different wing shapes behave at diverse air pressures. My interest with robotics has led me to take up a year-long internship with MakersBuilders, where I had the chance to explore physics and maths on a different plane. During my internship I educated young teenagers on a more fundamental stage of building and programming, in particular when we worked on building a small robot and programmed the infra-red sensor in order to create self-sufficient movement. This exposure allowed me to improve my communication and interpersonal skills. In this paragraph the student adds evidence to the initial assertion, that he enjoys seeing how Physics relates to everyday life. The descriptions of the work experiences he has had not only show his commitment to the subject, but also enable him to bring in some academic content to demonstrate his understanding of engineering and aeronautics. I’m interested in the mechanics side of Maths such as circular motion and projectiles; even Pure Maths has allowed me to easily see patterns when working and solving problems in Computer Science. During my A Level Maths and Further Maths, I have particularly enjoyed working with partial fractions as they show how reverse methodology can be used to solve addition of fractions, which ranges from simple addition to complex kinematics. Pure Maths has also enabled me to better understand how 3D modelling works with the use of volumes of revolution, especially when I learned how to apply the calculations to basic objects like calculating the amount of water in a bottle or the volume of a pencil. This paragraph brings in the academic content at school, which is important when applying for a subject such as engineering. This is because the admissions reader needs to be reassured that the student has covered the necessary foundational content to be able to cope with Year 1 of this course. In my Drone Club I have been able to apply several methods of wing formation, such as the number of blades used during a UAS flight. Drones can be used for purposes such as in Air-sea Rescue or transporting food to low income countries. I have taken on the responsibility of leading and sharing my skills with others, particularly in the Drone Club where I gained the certification to fly drones. In coding club, I participated in the global Google Code competition related to complex, real-life coding, such as a program that allows phones to send commands to another device using Bluetooth. My Cambridge summer course on math and engineering included the origins of a few of the most important equations and ideologies from many mathematicians such as, E=mc2 from Einstein, I also got a head start at understanding matrices and their importance in kinematics. Last summer, I completed a course at UT Dallas on Artificial Intelligence and Machine Learning. The course was intuitive and allowed me to understand a different perspective of how robots and AI will replace humans to do complex and labour-intensive activities, customer service, driverless cars and technical support. In this section, he demonstrates his commitment to the subject through a detailed list of extracurricular activities, all linked to engineering and aeronautics. The detail he gives about each one links to the knowledge and skills needed to succeed in these subjects at university. I have represented Model UN as a delegate and enjoyed working with others to solve problems. For my Duke of Edinburgh Award, I partook in several activities such as trekking and playing the drums. I enjoy music and I have reached grade 3 for percussion. I have also participated in a range of charitable activities, which include assisting during Ramadan and undertaking fun-runs to raise money for cancer research. As with the introduction, this is an efficient use of language, sharing a range of activities, each of which has taught him useful skills. The conclusion that follows is similarly efficient and to-the-point. I believe that engineering is a discipline that will offer me a chance to make a tangible difference in the world, and I am certain I will enjoy the process of integrating technology with our everyday life. Applying for a joint honours course presents a particular challenge of making the case that you are interested in the first subject, the second subject and (often overlooked) the combination of the two. In this example, the applicant uses her own academic studies and personal experiences to make her case. I usually spend my summer breaks in Uttar Pradesh, India working at my grandparents’ NGO which produces bio-fertilizers for the poor. While working, I speak to many of the villagers in the nearby villages like Barokhar and Dharampur and have found out about the various initiatives the Government has taken to improve the production of wheat and rice. I understand the hardships they undergo and speaking to them has shown me the importance of Social Policy and the role the government plays in improving the lives of people and inspired me to pursue my university studies in this field. In the introduction, this applicant explains where her interlinking experiences come from: she has personal experiences that demonstrate how economics impacts the most vulnerable in society. In doing so, she shows the admissions reader that she has a deep interest in this combination and can move on to discussing each subject in turn. My interest in these areas has been driven by the experiences I had at high school and beyond. I started attending Model United Nations in the 9th grade and have been to many conferences, discussing problems like the water crisis and a lack of sustainability in underdeveloped countries. These topics overlapped with my study of economics and exciting classroom discussions on what was going on how different events would impact economies, for instance how fluctuations in oil prices will affect standards of living. Studying Economics has expanded my knowledge about how countries are run and how macroeconomic policies shape the everyday experiences of individuals. Unusually, this applicant does not go straight into her classroom experiences but instead uses one of her extracurricular activities (Model United Nations) in her first paragraph. For students applying for subjects that are not often taught at school (Social Policy in this example), this can be a good idea, as it allows you to bring in material that you have self-studied to explain why you are capable of studying each subject at university. Here, she uses MUN discussions to show she understands some topics in social policy that are impacting the world. By taking up history as a subject in Grade 11 and 12, I have seen the challenges that people went through in the past, and how different ideas gained momentum in different parts of the world such as the growth of communism in Russia and China and how it spread to different countries during the Cold War. I learned about the different roles that governments played in times of hardships such as that which President Roosevelt’s New Deal played during the Great Depression. From this, I gained analytical skills by scrutinizing how different social, political and economic forces have moulded societies in the past. In this paragraph, she then takes the nearest possible class to her interest in Social Policy and draws elements from it to add to her case for Social Policy. Taking some elements from her history classes enables her to add some content to this statement, before linking to the topic of economics. To explore my interest in Economics, I interned at Emirates National Bank of Dubai, one of the largest banks in the Middle East, and also at IBM. At Emirates NBD, I undertook a research project on Cash Management methods in competitor banks and had to present my findings at the end of the internship. I also interned at IBM where I had to analyze market trends and fluctuations in market opportunity in countries in the Middle East and Africa. I had to find relations between GDP and market opportunity and had to analyze how market opportunity could change over the next 5 years with changing geo-political situations. I have also attended Harvard University’s Youth Lead the Change leadership conference where I was taught how to apply leadership skills to solve global problems such as gender inequality and poverty. Economics is explored again through extracurriculars, with some detail added to the general statement about the activities undertaken on this work experience. Though the level of academics here is a little thin because this student’s high school did not offer any classes in Economics, she does as well as she can to bring in academic content. I have partaken in many extra-curricular activities which have helped me develop the skills necessary for this course. Being a part of the Press Club at school gave me an opportunity to hone my talent for the written word and gave me a platform to talk about global issues. Volunteering at a local library taught me how to be organized. I developed research and analytical skills by undertaking various research projects at school such as the sector-wide contribution of the Indian economy to the GDP in the previous year. As a member of the Business and Economic Awareness Council at school, I was instrumental in organizing many economics-based events such as the Business Fair and Innovation Mela. Being part of various Face to Faith conferences has provided me with an opportunity to interact with students in Sierra Leone, India and Korea and understand global perspectives on issues like malaria and human trafficking. The extracurricular activities are revisited here, with the first half of this paragraph showing how the applicant has some transferable skills from her activities that will help her with this course. She then revisits her interest in the course studies, before following up with a closing section that touches on her career goals: The prospect of pursuing these two subjects is one that I eagerly anticipate and I look forward to meeting the challenge of university. In the future, I wish to become an economist and work at a think tank where I will be able to apply what I have learnt in studying such an exciting course. This applicant is also a joint-honours applicant, and again is applying for a subject that she has not been able to study at school. Thus, bringing in her own interest and knowledge of both subjects is crucial here. At the age of four, I remember an argument with my mother: I wanted to wear a pink ballerina dress with heels, made for eight-year-olds, which despite my difficulty in staying upright I was determined to wear. My mother persistently engaged in debate with me about why it was not ok to wear this ensemble in winter. After two hours of patiently explaining to me and listening to my responses she convinced me that I should wear something different, the first time I remember listening to reason. It has always been a natural instinct for me to discuss everything, since in the course of my upbringing I was never given a simple yes or no answer. Thus, when I began studying philosophy, I understood fully my passion for argument and dialogue. This is an unusual approach to start a UCAS Personal Statement, but it does serve to show how this student approaches the world and why this combination of subjects might work for her. Though it could perhaps be drawn out more explicitly, here she is combining an artistic issue (her clothes) with a philosophical concern (her debate with her mother) to lead the reader into the case she is making for admission into this program. This was first sparked academically when I was introduced to religious ethics; having a fairly Christian background my view on religion was immature. I never thought too much of the subject as I believed it was just something my grandparents did. However, when opened up to the arguments about god and religion, I was inclined to argue every side. After research and discussion, I was able to form my own view on religion without having to pick a distinctive side to which theory I would support. This is what makes me want to study philosophy: it gives an individual personal revelation towards matters into which they may not have given too much thought to. There is some good content here that discusses the applicant’s interest in philosophy and her own motivation for this subject, though there is a lack of academic content here. Alongside this, taking IB Visual Arts HL has opened my artistic views through pushing me out of my comfort zone. Art being a very subjective course, I was forced to choose an opinion which only mattered to me, it had no analytical nor empirical rights or wrongs, it was just my taste in art. From studying the two subjects alongside each other, I found great value, acquiring a certain form of freedom in each individual with their dual focus on personalized opinion and taste in many areas, leading to self- improvement. In this section, she uses her IB Visual Arts class to explore how her interest in philosophy bleeds into her appreciation of art. Again, we are still awaiting the academic content, but the reader will by now be convinced that the student has a deep level of motivation for this subject. When we consider how rare this combination is, with very few courses for this combination available, the approach to take slightly longer to establish can work. For this reason, I find the work of Henry Moore fascinating. I am intrigued by his pieces, especially the essence of the ‘Reclining Nude’ model, as the empty holes inflicted on the abstract human body encouraged my enthusiasm for artistic interpretation. This has led me to contemplate the subtlety, complexity and merit of the role of an artist. Developing an art piece is just as complex and refined as writing a novel or developing a theory in Philosophy. For this reason, History of Art conjoins with Philosophy, as the philosophical approach towards an art piece is what adds context to the history as well as purpose behind it. Finally, we’re given the academic content. Cleverly, the content links both the History of Art and Philosophy together though a discussion of the work of Henry Moore. Finding examples that conjoin the subjects that make up a joint-honours application is a great idea and works well here. Studying Philosophy has allowed me to apply real life abstractions to my art, as well as to glean a deeper critical analysis of art in its various mediums. My IB Extended Essay examined the 1900s Fauve movement, which made a huge breakthrough in France and Hungary simultaneously. This was the first artistic movement which was truly daring and outgoing with its vivid colours and bold brush strokes. My interest expanded to learning about the Hungarian artists in this movement led by Henri Matisse. Bela Czobel was one of the few who travelled to France to study but returned to Hungary, more specifically Nagybanya, to bestow what he had learned. Again in this paragraph, the author connects the subjects. Students who are able to undertake a research project in their high school studies (such as the IB Extended Essay here, or the A Level Extended Project or AP Capstone) can describe these in their UCAS personal statements, as this level of research in an area of academic study can enliven and add depth to the writing, as is the case here. As an international student with a multicultural background, I believe I can adapt to challenging or unfamiliar surroundings with ease. I spent two summers working at a nursery in Hungary as a junior Assistant Teacher, where I demonstrated leadership and teamwork skills that I had previously developed through commitment to sports teams. I was a competitive swimmer for six years and have represented my school internationally as well as holding the school record for 100m backstroke. I was elected Deputy Head of my House, which further reflects my dedication, leadership, teamwork and diligence. As in the previous examples, this statement gives a good overview of the applicant’s extracurricular activities, with a mention of skills that will be beneficial to her studies at university. She then concludes with a brief final sentence: I hope to carry these skills with me into my university studies, allowing me to enrich my knowledge and combine my artistic and philosophical interests. A good range of UK universities now offer courses called ‘Liberal Arts’ (or similar titles such as ‘Flexible Combined Honours’), which allows students to study a broader topic of study–perhaps combining three or four subjects–than is typically available in the UK system. This presents a challenge in the personal statement, as within the 47 line / 4000 character limit, the applicant will have to show academic interest and knowledge in a range of subjects while also making the case to be admitted for this combined programme of study. As a child I disliked reading; however, when I was 8, there was one particular book that caught my attention: The Little Prince. From that moment onwards, my love for literature was ignited and I had entered into a whirlwind of fictional worlds. While studying and analysing the classics from The Great Gatsby to Candide, this has exposed me to a variety of novels. My French bilingualism allowed me to study, in great depth, different texts in their original language. This sparked a new passion of mine for poetry, and introduced me to the works of Arthur Rimbaud, who has greatly influenced me. Through both reading and analysing poetry I was able to decipher its meaning. Liberal Arts gives me the opportunity to continue to study a range of texts and authors from different periods in history, as well as related aspects of culture, economy and society. Here we have a slightly longer than usual opening paragraph, but given the nature of the course being applied for this works well. A personal story segueing from literature to modern languages to history and cultural studies shows that this student has a broad range of interests within the humanities and thus is well-suited to this course of study. Liberal Arts is a clear choice for me. Coming from the IB International Baccalaureate Diploma programme I have studied a wide range of subjects which has provided me with a breadth of knowledge. In Theatre, I have adapted classics such as Othello by Shakespeare, and playing the role of moreover acting as Desdemona forced me to compartmentalise her complex emotions behind the early-modern English text. Studying History has taught me a number of skills; understanding the reasons behind changes in society, evaluating sources, and considering conflicting interpretations. From my interdisciplinary education I am able to critically analyse the world around me. Through studying Theory of Knowledge, I have developed high quality analysis using key questions and a critical mindset by questioning how and why we think and why. By going beyond the common use of reason, I have been able to deepen greaten my understanding and apply my ways of knowing in all subjects; for example in science I was creative in constructing my experiment (imagination) and used qualitative data (sense perception). Students who are taking the IB Diploma, with its strictures to retain a broad curriculum, are well-suited to the UK’s Liberal Arts courses, as they have had practice seeing the links between subjects. In this paragraph, the applicant shows how she has done this, linking content from one subject to skills developed in another, and touching on the experience of IB Theory of Knowledge (an interdisciplinary class compulsory for all IB Diploma students) to show how she is able to see how different academic subjects overlap and share some common themes. Languages have always played an important role in my life. I was immersed into a French nursery even though my parents are not French speakers. I have always cherished the ability to speak another language; it is something I have never taken for granted, and it is how I individualise myself. Being bilingual has allowed me to engage with a different culture. As a result, I am more open minded and have a global outlook. This has fuelled my desire to travel, learn new languages and experience new cultures. This course would provide me with the opportunity to fulfil these desires. Having written my Extended Essay in French on the use of manipulative language used by a particular character from the French classic Dangerous Liaisons I have had to apply my skills of close contextual reading and analysing to sculpt this essay. These skills are perfectly applicable to the critical thinking that is demanded for the course. Within the humanities, this student has a particular background that makes her stand out, having become fluent in French while having no French background nor living in a French-speaking country. This is worth her exploring to develop her motivation for a broad course of study at university, which she does well here. Studying the Liberal Arts will allow me to further my knowledge in a variety of fields whilst living independently and meeting people from different backgrounds. The flexible skills I would achieve from obtaining a liberal arts degree I believe would make me more desirable for future employment. I would thrive in this environment due to my self discipline and determination. During my school holidays I have undertaken working in a hotel as a chambermaid and this has made me appreciate the service sector in society and has taught me to work cohesively with others in an unfamiliar environment. I also took part in a creative writing course held at Keats House, where I learnt about romanticism. My commitment to extracurricular activities such as varsity football and basketball has shown me the importance of sportsmanship and camaraderie, while GIN (Global Issue Networking) has informed me of the values of community and the importance for charitable organisations. The extracurricular paragraph here draws out a range of skills the student will apply to this course. Knowing that taking a broader range of subjects at a UK university requires excellent organizational skills, the student takes time to explain how she can meet these, perhaps going into slightly more detail than would be necessary for a single-honours application to spell out that she is capable of managing her time well. She then broadens this at the end by touching on some activities that have relevance for her studies. My academic and personal preferences have always led me to the Liberal Arts; I feel as though the International Baccalaureate, my passion and self-discipline have prepared me for higher education. From the academics, extracurriculars and social aspects, I intend to embrace the entire experience of university. In the final section, the candidate restates how she matches this course. Overall, you can see how the key factor in a UCAS statement is the academic evidence, with students linking their engagement with a subject to the course of study that they are applying to. Using the courtroom exercise analogy, the judge here should be completely convinced that the case has been made, and will therefore issue an offer of admittance to that university.
4e24fbab82c0c017	From Wikipedia, the free encyclopedia (Redirected from Wave motion) Jump to navigation Jump to search Different types of wave with varying rectifications In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.[1] There are two main types of waves: mechanical and electromagnetic. Mechanical waves propagate through a physical matter, whose substance is being deformed. Restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbours. When the molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave. Electromagnetic waves do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. A wave can be transverse, where a disturbance creates oscillations that are perpendicular to the propagation of energy transfer, or longitudinal: the oscillations are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse in free space. General features Surface waves in water showing water ripples Mathematical description of one-dimensional waves Wave equation Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t) • in the direction in space. E.g., let the positive direction be to the right, and the negative direction be to the left. • with constant amplitude • with constant velocity , where is • with constant waveform, or shape This wave can then be described by the two-dimensional functions (waveform traveling to the right) (waveform traveling to the left) or, more generally, by d'Alembert's formula:[4] representing two component waveforms and traveling through the medium in opposite directions. A generalized representation of this wave can be obtained[5] as the partial differential equation General solutions are based upon Duhamel's principle.[6] Wave forms Sine, square, triangle and sawtooth waveforms. In the case of a periodic function F with period λ, that is, F(x + λvt) = F(x vt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F(xv(t + T)) = F(x vt) provided vT = λ, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[8] Amplitude and modulation Amplitude modulation can be achieved through f(x,t) = 1.00sin(2pi/0.10(x-1.00t)) and g(x,t) = 1.00sin(2pi/0.11(x-1.00t))only the resultant is visible to improve clarity of waveform. where is the amplitude envelope of the wave, is the wavenumber and is the phase. If the group velocity (see below) is wavelength-independent, this equation can be simplified as:[12] Phase velocity and group velocity The red square moves with the phase velocity, while the green circles propagate with the group velocity Phase velocity is the rate at which the phase of the wave propagates in space: any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as A wave with the group and phase velocities going in different directions Group velocity is a property of waves that have a defined envelope, measuring propagation through space (i.e. phase velocity) of the overall shape of the waves' amplitudes—modulation or envelope of the wave. Sine waves Sinusoidal waves correspond to simple harmonic motion. Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (also called harmonic wave or sinusoid) with an amplitude described by the equation: • is the space coordinate • is the time coordinate • is the wavenumber • is the angular frequency • is the phase constant. The wavelength is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber , the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation The period is the time for one complete cycle of an oscillation of a wave. The frequency is the number of periods per unit time (per second) and is typically measured in hertz denoted as Hz. These are related by: The angular frequency represents the frequency in radians per second. It is related to the frequency or period by The wavelength of a sinusoidal waveform traveling at constant speed is given by:[14] where is called the phase speed (magnitude of the phase velocity) of the wave and is the wave's frequency. Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.[16] Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.[17] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.[18][19] The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. An arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases.[20][21] In particular, many media are linear, or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform.[22] When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition. Plane waves Standing waves Physical properties Transmission and media Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter. Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel. Mechanical waves Waves on strings Acoustic waves Acoustic or sound waves travel at speed given by Water waves Shallow water wave.gif Seismic waves Shock waves Formation of a shock wave by a plane. Electromagnetic waves Onde electromagnétique.png Quantum mechanical waves Schrödinger equation Dirac equation The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles. de Broglie waves Louis de Broglie postulated that all particles with momentum have a wavelength where the wavelength is determined by the wave vector k as: and the momentum by: In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.[28] Gaussian wave packets also are used to analyze water waves.[29] For example, a Gaussian wavefunction ψ might take the form:[30] at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as λ0 = 2π / k0. It is well known from the theory of Fourier analysis,[31] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.[32] Given the Gaussian: the Fourier transform is: The Gaussian in space therefore is made up of waves: Gravity waves Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one example. Gravitational waves Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016.[33] Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity. See also Waves in general Electromagnetic waves In fluids In quantum mechanics In relativity Other specific types of waves Related topics 1. ^ Pragnan Chakravorty, "What Is a Signal? [Lecture Notes]," IEEE Signal Processing Magazine, vol. 35, no. 5, pp. 175-177, Sept. 2018. https://doi.org/10.1109/MSP.2018.2832195 2. ^ Lev A. Ostrovsky & Alexander I. Potapov (2001). Modulated waves: theory and application. Johns Hopkins University Press. ISBN 0-8018-7325-8. 6. ^ Jalal M. Ihsan Shatah; Michael Struwe (2000). "The linear wave equation". Geometric wave equations. American Mathematical Society Bookstore. pp. 37 ff. ISBN 0-8218-2749-9. 13. ^ Stefano Longhi; Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals". In Hugo E. Hernández-Figueroa; Michel Zamboni-Rached; Erasmo Recami. Localized Waves. Wiley-Interscience. p. 329. ISBN 0-470-10885-1. 15. ^ Paul R Pinet (2009). op. cit. p. 242. ISBN 0-7637-5993-7. 16. ^ Mischa Schwartz; William R. Bennett & Seymour Stein (1995). Communication Systems and Techniques. John Wiley and Sons. p. 208. ISBN 978-0-7803-4715-1. 21. ^ Kimball A. Milton; Julian Seymour Schwinger (2006). Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Springer. p. 16. ISBN 3-540-29304-3. Thus, an arbitrary function f(r, t) can be synthesized by a proper superposition of the functions exp[i (k·r−ωt)]... 22. ^ Raymond A. Serway & John W. Jewett (2005). "§14.1 The Principle of Superposition". Principles of physics (4th ed.). Cengage Learning. p. 433. ISBN 0-534-49143-X. 23. ^ Newton, Isaac (1704). "Prop VII Theor V". Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of Curvilinear Figures. 1. London. p. 118. All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these... 24. ^ Anderson, John D. Jr. (January 2001) [1984], Fundamentals of Aerodynamics (3rd ed.), McGraw-Hill Science/Engineering/Math, ISBN 0-07-237335-0 25. ^ M. J. Lighthill; G. B. Whitham (1955). "On kinematic waves. II. A theory of traffic flow on long crowded roads". Proceedings of the Royal Society of London. Series A. 229: 281–345. Bibcode:1955RSPSA.229..281L. doi:10.1098/rspa.1955.0088. And: P. I. Richards (1956). "Shockwaves on the highway". Operations Research. 4 (1): 42–51. doi:10.1287/opre.4.1.42. 27. ^ Ming Chiang Li (1980). "Electron Interference". In L. Marton; Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0-12-014653-3. 30. ^ Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2nd ed.). Springer. p. 60. ISBN 3-540-67458-6. 31. ^ Siegmund Brandt; Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN 0-387-95141-5. 33. ^ "Gravitational waves detected for 1st time, 'opens a brand new window on the universe'". CBC. 11 February 2016. • Fleisch, D.; Kinnaman, L. (2015). A student's guide to waves. Cambridge, UK: Cambridge University Press. ISBN 978-1107643260. • Griffiths, G.; Schiesser, W. E. (2010). Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple. Academic Press. ISBN 9780123846532. External links • Interactive Visual Representation of Waves • Linear and nonlinear waves • Science Aid: Wave properties—Concise guide aimed at teens Retrieved from "https://en.wikipedia.org/w/index.php?title=Wave&oldid=865181822" This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Wave_motion This page is based on the copyrighted Wikipedia article "Wave"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA
bd74c764cfa0fda2	You are currently browsing the monthly archive for November 2009. The Schrödinger equation \displaystyle i \hbar \partial_t \|\psi \rangle = H \|\psi\rangle is the fundamental equation of motion for (non-relativistic) quantum mechanics, modeling both one-particle systems and {N}-particle systems for {N>1}. Remarkably, despite being a linear equation, solutions {\|\psi\rangle} to this equation can be governed by a non-linear equation in the large particle limit {N \rightarrow \infty}. In particular, when modeling a Bose-Einstein condensate with a suitably scaled interaction potential {V} in the large particle limit, the solution can be governed by the cubic nonlinear Schrödinger equation \displaystyle i \partial_t \phi = \Delta \phi + \lambda \|\phi\|^2 \phi. \ \ \ \ \ (1) I recently attended a talk by Natasa Pavlovic on the rigorous derivation of this type of limiting behaviour, which was initiated by the pioneering work of Hepp and Spohn, and has now attracted a vast recent literature. The rigorous details here are rather sophisticated; but the heuristic explanation of the phenomenon is fairly simple, and actually rather pretty in my opinion, involving the foundational quantum mechanics of {N}-particle systems. I am recording this heuristic derivation here, partly for my own benefit, but perhaps it will be of interest to some readers. This discussion will be purely formal, in the sense that (important) analytic issues such as differentiability, existence and uniqueness, etc. will be largely ignored. Read the rest of this entry » Read the rest of this entry » [A little bit of advertising on behalf of my maths dept. Unfortunately funding for this scholarship was secured only very recently, so the application deadline is extremely near, which is why I am publicising it here, in case someone here may know of a suitable applicant. – T.] Eligibility Requirements: • 12th grader applying to UCLA for admission in Fall of 2010. • Outstanding academic record and standardized test scores. • Evidence of exceptional background and promise in mathematics, such as: placing in the top 25% in the U.S.A. Mathematics Olympiad (USAMO) or comparable (International Mathematics Olympiad level) performance on a similar national competition. • Strong preference will be given to International Mathematics Olympiad medalists. In the course of the ongoing logic reading seminar at UCLA, I learned about the property of countable saturation. A model {M} of a language {L} is countably saturated if, every countable sequence {P_1(x), P_2(x), \ldots} of formulae in {L} (involving countably many constants in {M}) which is finitely satisfiable in {M} (i.e. any finite collection {P_1(x),\ldots,P_n(x)} in the sequence has a solution {x} in {M}), is automatically satisfiable in {M} (i.e. there is a solution {x} to all {P_n(x)} simultaneously). Equivalently, a model is countably saturated if the topology generated by the definable sets is countably compact. Update, Nov 19: I have learned that the above terminology is not quite accurate; countable saturation allows for an uncountable sequence of formulae, as long as the constants used remain finite. So, the discussion here involves a weaker property than countable saturation, which I do not know the official term for. If one chooses a special type of ultrafilter, namely a “countably incomplete” ultrafilter, one can recover the full strength of countable saturation, though it is not needed for the remarks here. Most models are not countably saturated. Consider for instance the standard natural numbers {{\Bbb N}} as a model for arithmetic. Then the sequence of formulae “{x > n}” for {n=1,2,3,\ldots} is finitely satisfiable in {{\Bbb N}}, but not satisfiable. However, if one takes a model {M} of {L} and passes to an ultrapower {M}, whose elements {x} consist of sequences {(x_n)_{n \in {\Bbb N}}} in {M}, modulo equivalence with respect to some fixed non-principal ultrafilter {p}, then it turns out that such models are automatically countably compact. Indeed, if {P_1(x), P_2(x), \ldots} are finitely satisfiable in {M}, then they are also finitely satisfiable in {M} (either by inspection, or by appeal to Los’s theorem and/or the transfer principle in non-standard analysis), so for each {n} there exists {x_n \in M} which satisfies {P_1,\ldots,P_n}. Letting {x = (x_n)_{n \in {\Bbb N}} \in M} be the ultralimit of the {x_n}, we see that {x} satisfies all of the {P_n} at once. In particular, non-standard models of mathematics, such as the non-standard model {{\Bbb N}} of the natural numbers, are automatically countably saturated. This has some cute consequences. For instance, suppose one has a non-standard metric space {X} (an ultralimit of standard metric spaces), and suppose one has a standard sequence {(x_n)_{n \in {\mathbb N}}} of elements of {X} which are standard-Cauchy, in the sense that for any standard {\varepsilon > 0} one has {d( x_n, x_m ) < \varepsilon} for all sufficiently large {n,m}. Then there exists a non-standard element {x \in X} such that {x_n} standard-converges to {x} in the sense that for every standard {\varepsilon > 0} one has {d(x_n, x) < \varepsilon} for all sufficiently large {n}. Indeed, from the standard-Cauchy hypothesis, one can find a standard {\varepsilon(n) > 0} for each standard {n} that goes to zero (in the standard sense), such that the formulae “{d(x_n,x) < \varepsilon(n)}” are finitely satisfiable, and hence satisfiable by countable saturation. Thus we see that non-standard metric spaces are automatically “standardly complete” in some sense. This leads to a non-standard structure theorem for Hilbert spaces, analogous to the orthogonal decomposition in Hilbert spaces: Theorem 1 (Non-standard structure theorem for Hilbert spaces) Let {H} be a non-standard Hilbert space, let {S} be a bounded (external) subset of {H}, and let {x \in H}. Then there exists a decomposition {x = x_S + x_{S^\perp}}, where {x_S \in H} is “almost standard-generated by {S}” in the sense that for every standard {\varepsilon > 0}, there exists a standard finite linear combination of elements of {S} which is within {\varepsilon} of {S}, and {x_{S^\perp} \in H} is “standard-orthogonal to {S}” in the sense that {\langle x_{S^\perp}, s\rangle = o(1)} for all {s \in S}. Proof: Let {d} be the infimum of all the (standard) distances from {x} to a standard linear combination of elements of {S}, then for every standard {n} one can find a standard linear combination {x_n} of elements of {S} which lie within {d+1/n} of {x}. From the parallelogram law we see that {x_n} is standard-Cauchy, and thus standard-converges to some limit {x_S \in H}, which is then almost standard-generated by {S} by construction. An application of Pythagoras then shows that {x_{S^\perp} := x-x_S} is standard-orthogonal to every element of {S}. \Box This is the non-standard analogue of a combinatorial structure theorem for Hilbert spaces (see e.g. Theorem 2.6 of my FOCS paper). There is an analogous non-standard structure theorem for {\sigma}-algebras (the counterpart of Theorem 3.6 from that paper) which I will not discuss here, but I will give just one sample corollary: Theorem 2 (Non-standard arithmetic regularity lemma) Let {G} be a non-standardly finite abelian group, and let {f: G \rightarrow [0,1]} be a function. Then one can split {f = f_{U^\perp} + f_U}, where {f_U: G \rightarrow [-1,1]} is standard-uniform in the sense that all Fourier coefficients are (uniformly) {o(1)}, and {f_{U^\perp}: G \rightarrow [0,1]} is standard-almost periodic in the sense that for every standard {\varepsilon > 0}, one can approximate {f_{U^\perp}} to error {\varepsilon} in {L^1(*G)} norm by a standard linear combination of characters (which is also bounded). This can be used for instance to give a non-standard proof of Roth’s theorem (which is not much different from the “finitary ergodic” proof of Roth’s theorem, given for instance in Section 10.5 of my book with Van Vu). There is also a non-standard version of the Szemerédi regularity lemma which can be used, among other things, to prove the hypergraph removal lemma (the proof then becomes rather close to the infinitary proof of this lemma in this paper of mine). More generally, the above structure theorem can be used as a substitute for various “energy increment arguments” in the combinatorial literature, though it does not seem that there is a significant saving in complexity in doing so unless one is performing quite a large number of these arguments. One can also cast density increment arguments in a nonstandard framework. Here is a typical example. Call a non-standard subset {X} of a non-standard finite set {Y} dense if one has {\|X\| \geq \varepsilon \|Y\|} for some standard {\varepsilon > 0}. Theorem 3 Suppose Szemerédi’s theorem (every set of integers of positive upper density contains an arithmetic progression of length {k}) fails for some {k}. Then there exists an unbounded non-standard integer {N}, a dense subset {A} of {[N] := \{1,\ldots,N\}} with no progressions of length {k}, and with the additional property that \displaystyle \frac{\|A \cap P\|}{\|P\|} \leq \frac{\|A \cap [N]\|}{N} + o(1) for any subprogression {P} of {[N]} of unbounded size (thus there is no sizeable density increment on any large progression). Proof: Let {B \subset {\Bbb N}} be a (standard) set of positive upper density which contains no progression of length {k}. Let {\delta := \limsup_{\|P\| \rightarrow \infty} \|B \cap P\|/\|P\|} be the asymptotic maximal density of {B} inside a long progression, thus {\delta > 0}. For any {n > 0}, one can then find a standard integer {N_n \geq n} and a standard subset {A_n} of {[N_n]} of density at least {\delta-1/n} such that {A_n} can be embedded (after a linear transformation) inside {B}, so in particular {A_n} has no progressions of length {k}. Applying the saturation property, one can then find an unbounded {N} and a set {A} of {[N]} of density at least {\delta-1/n} for every standard {n} (i.e. of density at least {\delta-o(1)}) with no progressions of length {k}. By construction, we also see that for any subprogression {P} of {[N]} of unbounded size, {A} hs density at most {\delta+1/n} for any standard {n}, thus has density at most {\delta+o(1)}, and the claim follows. \Box This can be used as the starting point for any density-increment based proof of Szemerédi’s theorem for a fixed {k}, e.g. Roth’s proof for {k=3}, Gowers’ proof for arbitrary {k}, or Szemerédi’s proof for arbitrary {k}. (It is likely that Szemerédi’s proof, in particular, simplifies a little bit when translated to the non-standard setting, though the savings are likely to be modest.) I’m also hoping that the recent results of Hrushovski on the noncommutative Freiman problem require only countable saturation, as this makes it more likely that they can be translated to a non-standard setting and thence to a purely finitary framework. Let {X} be a finite subset of a non-commutative group {G}. As mentioned previously on this blog (as well as in the current logic reading seminar), there is some interest in classifying those {X} which obey small doubling conditions such as {\|X \cdot X\| = O(\|X\|)} or {\|X \cdot X^{-1}\| = O(\|X\|)}. A full classification here has still not been established. However, I wanted to record here an elementary argument (based on Exercise 2.6.5 of my book with Van Vu, which in turn is based on this paper of Izabella Laba) that handles the case when {\|X \cdot X\|} is very close to {\|X\|}: Proposition 1 If {\|X^{-1} \cdot X\| < \frac{3}{2} \|X\|}, then {X \cdot X^{-1}} and {X^{-1} \cdot X} are both finite groups, which are conjugate to each other. In particular, {X} is contained in the right-coset (or left-coset) of a group of order less than {\frac{3}{2} \|X\|}. Remark 1 The constant {\frac{3}{2}} is completely sharp; consider the case when {X = \{e, x\}} where {e} is the identity and {x} is an element of order larger than {2}. This is a small example, but one can make it as large as one pleases by taking the direct product of {X} and {G} with any finite group. In the converse direction, we see that whenever {X} is contained in the right-coset {S \cdot x} (resp. left-coset {x \cdot S}) of a group of order less than {2\|X\|}, then {X \cdot X^{-1}} (resp. {X^{-1} \cdot X}) is necessarily equal to all of {S}, by the inclusion-exclusion principle (see the proof below for a related argument). Proof: We begin by showing that {S := X \cdot X^{-1}} is a group. As {S} is symmetric and contains the identity, it suffices to show that this set is closed under addition. Let {a, b \in S}. Then we can write {a=xy^{-1}} and {b=zw^{-1}} for {x,y,z,w \in X}. If {y} were equal to {z}, then {ab = xw^{-1} \in X \cdot X^{-1}} and we would be done. Of course, there is no reason why {y} should equal {z}; but we can use the hypothesis {\|X^{-1} \cdot X\| < \frac{3}{2}\|X\|} to boost this as follows. Observe that {x^{-1} \cdot X} and {y^{-1} \cdot X} both have cardinality {\|X\|} and lie inside {X^{-1} \cdot X}, which has cardinality strictly less than {\frac{3}{2} \|X\|}. By the inclusion-exclusion principle, this forces {x^{-1} \cdot X \cap y^{-1} \cdot X} to have cardinality greater than {\frac{1}{2}\|X\|}. In other words, there exist more than {\frac{1}{2}\|X\|} pairs {x',y' \in X} such that {x^{-1} x' = y^{-1} y'}, which implies that {a = x' (y')^{-1}}. Thus there are more than {\frac{1}{2}\|X\|} elements {y' \in X} such that {a = x' (y')^{-1}} for some {x'\in X} (since {x'} is uniquely determined by {y'}); similarly, there exists more than {\frac{1}{2}\|X\|} elements {z' \in X} such that {b = z' (w')^{-1}} for some {w' \in X}. Again by inclusion-exclusion, we can thus find {y'=z'} in {X} for which one has simultaneous representations {a = x' (y')^{-1}} and {b = y' (z')^{-1}}, and so {ab = x'(z')^{-1} \in X \cdot X^{-1}}, and the claim follows. In the course of the above argument we showed that every element of the group {S} has more than {\frac{1}{2}\|X\|} representations of the form {xy^{-1}} for {x,y \in X}. But there are only {\|X\|^2} pairs {(x,y)} available, and thus {\|S\| < 2\|X\|}. Now let {x} be any element of {X}. Since {X \cdot x^{-1} \subset S}, we have {X \subset S \cdot x}, and so {X^{-1} \cdot X \subset x^{-1} \cdot S \cdot x}. Conversely, every element of {x^{-1} \cdot S \cdot x} has exactly {\|S\|} representations of the form {z^{-1} w} where {z, w \in S \cdot x}. Since {X} occupies more than half of {S \cdot x}, we thus see from the inclusion-exclusion principle, there is thus at least one representation {z^{-1} w} for which {z, w} both lie in {X}. In other words, {x^{-1} \cdot S \cdot x = X^{-1} \cdot X}, and the claim follows. \Box To relate this to the classical doubling constants {\|X \cdot X\|/\|X\|}, we first make an easy observation: Lemma 2 If {\|X \cdot X\| < 2\|X\|}, then {X \cdot X^{-1} = X^{-1} \cdot X}. Again, this is sharp; consider {X} equal to {\{x,y\}} where {x,y} generate a free group. Proof: Suppose that {xy^{-1}} is an element of {X \cdot X^{-1}} for some {x,y \in X}. Then the sets {X \cdot x} and {X \cdot y} have cardinality {\|X\|} and lie in {X \cdot X}, so by the inclusion-exclusion principle, the two sets intersect. Thus there exist {z,w \in X} such that {zx=wy}, thus {xy^{-1}=z^{-1}w \in X^{-1} \cdot X}. This shows that {X \cdot X^{-1}} is contained in {X^{-1} \cdot X}. The converse inclusion is proven similarly. \Box Proposition 3 If {\|X \cdot X\| < \frac{3}{2} \|X\|}, then {S := X \cdot X^{-1}} is a finite group of order {\|X \cdot X\|}, and {X \subset S \cdot x = x \cdot S} for some {x} in the normaliser of {S}. The factor {\frac{3}{2}} is sharp, by the same example used to show sharpness of Proposition 1. However, there seems to be some room for further improvement if one weakens the conclusion a bit; see below the fold. Proof: Let {S = X^{-1} \cdot X = X \cdot X^{-1}} (the two sets being equal by Lemma 2). By the argument used to prove Lemma 2, every element of {S} has more than {\frac{1}{2}\|X\|} representations of the form {xy^{-1}} for {x,y \in X}. By the argument used to prove Proposition 1, this shows that {S} is a group; also, since there are only {\|X\|^2} pairs {(x,y)}, we also see that {\|S\| < 2\|X\|}. Pick any {x \in X}; then {x^{-1} \cdot X, X \cdot x^{-1} \subset S}, and so {X \subset x\cdot S, S \cdot x}. Because every element of {x \cdot S \cdot x} has {\|S\|} representations of the form {yz} with {y \in x \cdot S}, {z \in S \cdot x}, and {X} occupies more than half of {x \cdot S} and of {S \cdot x}, we conclude that each element of {x \cdot S \cdot x} lies in {X \cdot X}, and so {X \cdot X = x \cdot S \cdot x} and {\|S\| = \|X \cdot X\|}. The intersection of the groups {S} and {x \cdot S \cdot x^{-1}} contains {X \cdot x^{-1}}, which is more than half the size of {S}, and so we must have {S = x \cdot S \cdot x^{-1}}, i.e. {x} normalises {S}, and the proposition follows. \Box Because the arguments here are so elementary, they extend easily to the infinitary setting in which {X} is now an infinite set, but has finite measure with respect to some translation-invariant Kiesler measure {\mu}. We omit the details. (I am hoping that this observation may help simplify some of the theory in that setting.) Read the rest of this entry » Read the rest of this entry » Read the rest of this entry » As the previous discussion on displaying mathematics on the web has become quite lengthy, I am opening a fresh post to continue the topic. I’m leaving the previous thread open for those who wish to respond directly to some specific comments in that thread, but otherwise it would be preferable to start afresh on this thread to make it easier to follow the discussion. It’s not easy to summarise the discussion so far, but the comments have identified several existing formats for displaying (and marking up) mathematics on the web (mathMLjsMath, MathJaxOpenMath), as well as a surprisingly large number of tools for converting mathematics into web friendly formats (e.g. LaTeX2HTMLLaTeXMathML, LaTeX2WPWindows 7 Math Inputitex2MMLRitexGellmumathTeXWP-LaTeXTeX4htblahtexplastexTtHWebEQtechexplorer, etc.). Some of the formats are not widely supported by current software, and by current browsers in particular, but it seems that the situation will improve with the next generation of these browsers. It seems that the tools that already exist are enough to improvise a passable way of displaying mathematics in various formats online, though there are still significant issues with accessibility, browser support, and ease of use. Even if all these issues are resolved, though, I still feel that something is still missing. Currently, if I want to transfer some mathematical content from one location to another (e.g. from a LaTeX file to a blog, or from a wiki to a PDF, or from email to an online document, or whatever), or to input some new piece of mathematics, I have to think about exactly what format I need for the task at hand, and what conversion tool may be needed. In contrast, if one looks at non-mathematical content such as text, links, fonts, non-Latin alphabets, colours, tables, images, or even video, the formats here have been standardised, and one can manipulate this type of content in both online and offline formats more or less seamlessly (in principle, at least – there is still room for improvement), without the need for any particularly advanced technical expertise. It doesn’t look like we’re anywhere near that level currently with regards to mathematical content, though presumably things will improve when a single mathematics presentation standard, such as mathML, becomes universally adopted and supported in browsers, in operating systems, and in other various pieces of auxiliary software. Anyway, it has been a very interesting and educational discussion for me, and hopefully for others also; I look forward to any further thoughts that readers have on these topics. (Also, feel free to recapitulate some of the points from the previous thread; the discussion has been far too multifaceted for me to attempt a coherent summary by myself.)
e943cbf42dea25cb	A-Z Index \| Phone Book \| Careers Scaling the Nanowire December 23, 2008 By Karyn Hede The preeminent physicist-futurist Richard Feynman famously declared in a 1959 address to the American Physical Society that “there’s plenty of room at the bottom.” He then invited them to enter the strange new world of nanoscale materials, none of which had actually been invented, except in Feynman’s fantastical imagination. It took another generation of scientists before nanotechnology emerged, but Feynman’s assertion still rings true. There’s plenty of room at the nanoscale and scientists at Lawrence Berkeley National Laboratory (LBNL) in California are at the forefront in constructing new materials there. Paul Alivisatos, director of LBNL’s Materials Science Division, is a world leader in nanostructures and inventor of many technologies using quantum dots — special kinds of semiconductor nanocrystals. Quantum dots, which are one ten-millionth of an inch in diameter, fluoresce brightly, are exceedingly stable and don’t interfere with biological processes because they are made of inert minerals. Alivisatos and his colleagues have constructed dozens of variations in which the fluorescent color changes with the dot’s size. Today life-science researchers use quantum dots as markers, allowing them to visualize with extreme accuracy individual genes, proteins and other small molecules inside living cells and fulfilling a prediction Feynman made in his famous lecture. LBNL physicist Lin-Wang Wang likes to say that some day we will view the 21st Century as the “nanostructure” age, much as we associate Neolithic humans with the Stone Age and their descendents with the Bronze and Iron ages. Despite their obvious usefulness, however, the behavior of materials built from nanometer particles is still not completely understood or fully predictable. Part of the problem is that electron wavelengths also are on the nanometer scale. Electrons’ quantum mechanical properties — the consequence of their wave-like behaviors — are changed by the sizes and geometries of the quantum dots, a subject that still generates heated discussion among physicists. These size and geometric changes allow electrons in semiconductor nanostructures to generate new energetic and optical properties. Wang and his LBNL colleague Andrew Canning, a computational physicist who helped pioneer the application of parallel computing to material science, want to use computational methods to understand the emergent behaviors of novel materials, such as quantum dots, built from these exceedingly small blocks. “There are a lot of challenges and there are still many mysteries to be solved,” Wang says. “For example, we still don’t quite understand the dynamics of the electron inside a quantum dot or a quantum rod. There is a lot of surface area in a quantum structure, much more than the same material in bulk. So how the surface is coupled with the interior states and how this affects the nanostructure properties is not well understood.” The Rules are Different in Nanoscale The research team is not starting from scratch, of course. There are established equations that predict the behavior of the electron wave function in these materials. The devil lies in the size of the problem. “In terms of computation the nanostructure is challenging. For example, if you have a bulk material the crystal structure is a very small unit cell, just a few atoms, that repeats itself many, many times,” Wang says. “So computationally, you can treat bulk structures by calculating one unit cell — you only deal with a few atoms. With only a few atoms, you can represent the whole, much larger structure of the material. However, for a quantum dot or a quantum wire you have to treat the whole system together. These systems usually contain a few thousand to tens of thousands of atoms, and that makes the computation challenging.” To solve a problem containing thousands of atoms requires new algorithms that handle the physics differently without compromising accuracy and parallel computing on a massive scale. That’s where Canning’s expertise came in. “We know we need to solve the Schrödinger equation for these problems, but to do so fully is exceedingly computationally expensive,” Canning says. “What we did was make advances to approximate, solve the problem, and still get the physics right.” Canning collaborated with Steven Louie’s group at the University of California-Berkeley, to improve the Parallel Total Energy Code (Paratec), an ab initio, quantum- mechanical, total energy program. The program runs on Franklin, the Cray XT4 at LBNL’s National Energy Research Scientific Computing Center (NERSC). The massively parallel system has 9,660 compute nodes, but is due to receive an upgrade, increasing its processing capability to a theoretical peak of about 360 teraflops. “Paratec enables us to calculate thousand-atom nanosystems,” Canning says. “The calculation is fast and scales to the cube of the system, rather than exponentially, as a true solution of the many-body Schrödinger equation.” Besides massive parallelization of the codes, the researchers also developed many new algorithms for nanostructure calculations. For example, Wang devised a linear scaling method, called the folded spectrum method, for use on large-scale electronic structure calculations. The conventional methods in Paratec must calculate thousands of electron wave functions, but the Escan code uses the folded spectrum method to calculate only a few states near the nanostructure energy band gap. That means the computation scales linearly to the size of the problem — a critical requirement for efficient nanoscience computation. Wang and Canning recently worked with Osni Marques at LBNL and Jack Dongarra’s group at the University of Tennessee, Knoxville, to reinvestigate and significantly improve the Escan code by adding more advanced algorithms. Wang and his colleagues also have recently invented a linear scaling three-dimensional fragment (LS3DF) method, which can be hundreds of times faster than a conventional method in calculating the total energy of a given nanostructure. The code has run at 107 teraflops on 137,072 processors of Intrepid, Argonne National Laboratory’s IBM Blue Gene/P. The researchers have in essence designed a new algorithm to solve an existing physical problem with petascale computation. Wang says the LS3DF program is designed for materials science applications such as studying material defects, metal alloys and large organic molecules. Within a nanostructure, the physicists are interested mainly in the location and energy level of electrons in the system because that determines the properties of a nanomaterial. For example, Wang says, electrons within a quantum rod or dot can occupy a series of quantum energy states or levels as they orbit the atomic nucleus and interact with each other. The color emitted by the material typically depends on these energy states. Specifically, the scientists focus on two quantum energy levels: the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), which the Escan code can calculate. The energy difference between these two levels determines the material’s color. The color also changes with the quantum dot’s size, providing one way to engineer its properties. In principle, knowing the electronic properties of a given material lets the researchers predict how a new nanostructure will behave before actually spending the time and money to make it. It’s a potentially less expensive way to experiment with new nanomaterials, Wang says. Solar Cells on the Cheap That kind of predictive power led to a discovery that may contribute to energy independence for the United States. In collaboration with scientist Yong Zhang, a senior scientist at the National Renewable Energy Laboratory in Golden, Colorado, Wang and his colleagues predicted a new kind of solar cell that could be manufactured inexpensively and from environmentally friendly materials. The prediction is based on a new kind of nanostructure architecture that takes advantage of the unique electronic properties of materials constructed from nanometer-sized units. In order for a material to generate electricity efficiently from sunlight it must have an electron excitation potential around 2 electron volts (eV). Silicon is one such material, and most commercial solar cells use silicon that is cut into thin sheets from bulk material. These solar cells are expensive and it takes years to recoup the cost of manufacturing and installation. Wang and Zhang collaborated with Joshua Schrier, Denis Demchenko and Paul Alivisatos to propose using zinc oxide (the white stuff often found in sunscreens) and zinc sulfide (an abundant, easily produced mineral) in a novel solar cell. The two materials by themselves could never generate solar energy efficiently. The key is how they are combined. The researchers designed an architecture with a zinc sulfide nanostructure core surrounded by a thin shell of zinc oxide to form a nanowire. Using a new code developed by Wang and a team of LBNL computational scientists, the researchers simulated the electronic wave properties of the proposed solar cell. “We wanted to reduce the band gap,” Wang says. “By staggering the band energy alignment of the materials we calculated the overall band gap would be 2 eV, which produces a high efficiency limit of 23 percent.” The researchers published their finding in the journal Nanoletters and got immediate response. The publication was one of the top 10 most-viewed Nanoletters articles in 2007. Wang says several groups worldwide now are working on building versions of the nanowire and testing their efficiency. If they function as predicted, the group will have invented a safe, abundant, stable and environmentally benign solar cell that can be built immediately using existing manufacturing methods. Looking forward, the scientists are using these newly developed computational methods to help guide other new nanostructure materials. Even now, when the experimental group creates a new nanostructure, the properties are not always easy to predict. “When you do the experiment, you don’t really know what result you are getting,” Wang says. “How to explain experiments is sometimes challenging. That’s the first task of simulation, to explain experimental results. Then the second task is to get some guidance for experimental design — to figure out what kind of experiment to do. That’s theory guiding the experiment.” Both kinds of computation keep the group busy. Experimentalists at LBNL and collaborators at Washington University, NREL and worldwide are busy creating new nanomaterials: rod-shaped semiconductor nanocrystals that could be stacked to create tiny electronic devices; nanowires of various semiconductor materials; and quantum dots that can track tumors and help physicians diagnose and treat cancers more specifically. There appear to be few limits to the number of nanostructures that can be created, proving once again that Feynman was right — there’s lots of room in the nanosphere. About Computing Sciences at Berkeley Lab
8766483068017748	Superposition Principle Get Superposition Principle essential facts below. View Videos or join the Superposition Principle discussion. Add Superposition Principle to your PopFlock.com topic list for future reference or share this resource on social media. Superposition Principle Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths. Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation. In physics and systems theory, the superposition principle,[1] also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). The homogeneity and additivity properties together are called the superposition principle. A linear function is one that satisfies the properties of superposition. It is defined as for scalar a. This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour. The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. Relation to Fourier analysis and similar methods By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific, simple form, often the response becomes easier to compute. For example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses. As another common example, in Green's function analysis, the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses. Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves. Wave superposition Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave. Waves are usually described by variations in some parameter through space and time--for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point. In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In many cases (for example, in the classic wave equation), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at top.) Wave diffraction vs. wave interference With regard to wave superposition, Richard Feynman wrote:[2] No-one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do is, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used. Other authors elaborate:[3] The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically they are two limiting cases of superposition effects. Yet another source concurs:[4] Inasmuch as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is therefore a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront. Wave interference The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-cancelling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. In other cases, such as in a line array, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference. green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Interference of two waves.svg wave 1 wave 2 Two waves in phase Two waves 180° out of phase Departures from linearity In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles nonlinear optics and nonlinear acoustics. Quantum superposition In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "quantum superposition") of (possibly infinitely many) other wave functions of a certain type--stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[5] The projective nature of quantum-mechanical-state space makes an important difference: it does not permit superposition of the kind that is the topic of the present article. A quantum mechanical state is a ray in projective Hilbert space, not a vector. The sum of two rays is undefined. To obtain the relative phase, we must decompose or split the ray into components where the and the belongs to an orthonormal basis set. The equivalence class of allows a well-defined meaning to be given to the relative phases of the .[6] There are some likenesses between the superposition presented in the main on this page, and quantum superposition. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics." According to Dirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]."[7] Boundary value problems A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation with some boundary specification For example, in Laplace's equation with Dirichlet boundary conditions, F would be the Laplacian operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R. In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: while the boundary values superpose: Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary value problems. Additive state decomposition Consider a simple linear system : By superposition principle, the system can be decomposed into Superposition principle is only available for linear systems. However, the Additive state decomposition can be applied not only to linear systems but also nonlinear systems. Next, consider a nonlinear system where is a nonlinear function. By the additive state decomposition, the system can be 'additively' decomposed into This decomposition can help to simplify controller design. Other example applications • In electrical engineering, in a linear circuit, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. The use of Fourier analysis on this basis is particularly common. For another, related technique in circuit analysis, see Superposition theorem. • In physics, Maxwell's equations imply that the (possibly time-varying) distributions of charges and currents are related to the electric and magnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields which arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the heat equation. • In mechanical engineering, superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[8] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[9] • In hydrogeology, the superposition principle is applied to the drawdown of two or more water wells pumping in an ideal aquifer. • In process control, the superposition principle is used in model predictive control. • The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization. • In music, theorist Joseph Schillinger used a form of the superposition principle as one basis of his Theory of Rhythm in his Schillinger System of Musical Composition. According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by Leonhard Euler and then by Joseph Lagrange. Later it became accepted, largely through the work of Joseph Fourier.[10] See also 1. ^ The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London 2. ^ Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [1] 3. ^ N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [2] 4. ^ Tim Freegard, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [3] 5. ^ Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62 ISBN 978-0-486-66772-0 6. ^ Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185-195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623. 7. ^ Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK, p. 14. 8. ^ Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 ISBN 0-07-252036-1 9. ^ Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 ISBN 0-13-301458-4 10. ^ Brillouin, L. (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw-Hill, New York, p. 2. Further reading External links Music Scenes
efd0cd1e37c77e84	Home \| english \| Impressum \| Datenschutz \| Sitemap \| KIT I. Multiscale computational methods Bild QM/MM Multiscale computational methods To study chemical reactivity and spectroscopic properties of complex systems like proteins or organic semi-conductors, quantum mechanical (QM) methods are required. Due to the large size of these structures, a simple application of QM methods is impossible due to the high computational cost. Therefore, QM methods are conveniently combined with empirical force field methods (Molecular Mechanics: MM) in the so- called QM/MM methods. In such approaches, the active site of a protein consisting of several tens of atoms is treated with QM, while the remainder of the protein is handled with the computationally cheaper MM methods. This approach can be taken to treat even larger scales by combining the QM/MM methods with continuum approaches. Also, various QM methods with different accuracy can be applied in the active site, building onion-like computation shells of methods. Besides the development of such method-combinations, a large part of our effort concentrates on the development of fast and accurate approximations to Density Functional Theory (DFT) resulting in the DFTB suite of methods which are about three orders of magnitude faster than DFT, however, keeping a similar accuracy for a broad range of molecular properties. Advancement of DFTB II. Electron Transfer in proteins and organic materials Charge Transfer in Biomolecules Charge Transfer in Biomolecules Modeling of electron transfer (ET) events in proteins, DNA and organic materials is a particular challenge for computational chemistry due to the extensive size of the involved QM region on the one hand and to the prolonged time scales on the other. An interesting sub-class of ET processes is those which appear to be relatively fast – on the sub-nanosecond range – and for which the standard approximative schemes like Marcus' theory may no longer be valid. For such processes, we have developed a novel multi-scale methodology which intertwines efficient QM calculations with MM methods and allows for extended sampling of the processes. The electron (or electron hole) is propagated according to the Schrödinger equation within a semi-classical approach while the movement of atoms is described by classical Newton’s equations of motion. The recent applications are concerned with several topical ET systems. In double-stranded DNA, hole transfer occurs over long distances of up to 100 ns, which has consequences for radiatively induced damage and repair processes. The mechanism of this transfer is controversial to-date, and we are performing simulations designed to resolve this issue. Another application is the hole transfer in the DNA photolyase protein, which takes place in several consecutive steps. It represents a paradigmatic system that violates the assumptions of the classical theory of ET, while our simulations describe the transfer well. A new class of systems that we are starting to work is organic semi-conductors. This project is in an early stage of development, which was made necessary by the fundamental differences between the relevant materials and the biomolecular complexes studied so far. Charge Transfer in DNA Organic Electronics Charge Transfer in Proteins III. Photo-active proteins Photo-active Proteins Photo-active Proteins Many proteins make use of the sun light to drive important processes in the cell, like energy generation, or signal conversion. In the last years, we studied intensely the bacterial protein Bacteriorhodopsin (bR), which acts as light driven proton pump. It has been in the focus of several scientific fields for several decades, including various biological, chemical and biophysical approaches, which lead to a detailed understanding of the chemical events and structural changes that accompany the proton pumping process. Simulations, in particular those based on QM/MM methods complemented the picture by computing proton transfer pathways through the protein and energy barriers and relating spectroscopic measurements with the underlying structure.The channelrhodopsins (ChRs) are currently the most regarded members of the rhodopsin proteins and may be considered the most intriguing photo-active proteins discovered in the last decade. They are light-induced ion channels and have drawn significant attention from neuroscientists. They can be readily expressed in a variety of targets, e.g. mammalian neurons, to generate action potentials with light. The ChRs are the foundation of the new method of optogenetics. Despite of their advanced application, surprisingly little is known about their detailed structure and functional mechanism. Structure and Function of the Channelrhodopsines Proton Transfer in Bacteriorhodopsin Color Tuning in Rhodopsin and Cone Visual Pigments IV. Interaction of peptides with membranes Membrane Simulations Membranes are crucial parts of every cell and a majority of biochemical reactions in metabolism, signalling and transport involves membranes and membrane proteins. Obtaining structural data for anisotropic and complex membrane systems is difficult and computer simulations can help tremendously here. We study the physico-chemical properties of lipid bilayer biomembrane models using high-efficiency particle dynamics simulations that allow even very slow processes like phase transitions to be described. Both coarse-grained models for large scale processes and atomistic force field molecular dynamics simulations are employed. In collaboration with the group of Prof. Anne Ulrich we draw the connection between simulation results and NMR data to study the mechanism of antimicrobial, membrane penetrating peptides.
636b5754c9937af5	Duke University Press Signs French National License Agreement with ISTEX 1. here we consider some as say prof dr mircea orasanu and prof horia orasanu as followings This paper has considered path following control of planar snake robots by using virtual holonomic constraints. The equations of motion of the snake robot were derived using a Lagrangian framework. We then introduced virtual holonomic constraints that defined the geometry of a constraint manifold for the robot. We showed that the constraint manifold can be made positively invariant by a suitable choice of feedback, and we designed an input-output feedback linearizing control law to exponentially stabilize the constraint manifold for the system. We presented simulation and experimental results which validated the theoretical design. In particular, the robot successfully converged to and followed a desired straight path. As a topic of future work, we aim to prove the practical stability of the desired path with the proposed control approach. Furthermore, a formal proof for boundedness of the solutions of the dynamic compensator remains as a topic of future work. Moreover, application of the proposed control strategy for more complex paths such as curved paths, using different path following guidance laws, remains as a topic of future work. Complex systems are large interdisciplinary research topics that have been studied by means of a mixed basic theory that mainly derives from physics and computer simulation. Such systems are made of many interacting elementary units that are called “agents”. The way in which such a system manifests itself cannot be exclusively predicted only by the behavior of individual elements. Its manifestation is also induced by the manner in which the elements relate in order to influence global behavior. The most significant properties of complex systems are emergence, self-organization, adaptability, etc. [1–4]. Examples of complex systems can be found in human societies, brains, the Internet, ecosystems, biological evolution, stock markets, economies and many others [1, 2]. Particularly, polymers are examples of such complex systems. Their forms include a multitude of organizations starting from simple, linear chains of identical structural units and ending with very complex chains consisting of sequences of amino acids that form the building blocks of living fields. One of the most intriguing polymers in nature is DNA, which creates cells by means of a simple, but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems, such as organs, which, in turn, form even more complex systems, such as organisms. The study of complex systems can offer a glimpse into the realistic dynamics of polymers and solve certain difficult problems (protein folding) [1–4]. Correspondingly, theoretical models that describe the dynamics of complex systems are sophisticated [1–4]. However, the situation can be standardized taking into account that the complexity of interaction processes imposes various temporal resolution scales, while pattern evolution implies different freedom degrees [5]. In order to develop new theoretical models, we must admit that complex systems displaying chaotic behavior acquire self-similarity (space-time structures seem to appear) in association with strong fluctuations at all possible space-time scales [1–4]. Then, in the case of temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a collection of potential trajectories, while the concept of definite positions by that of probability density. One of the most interesting examples is the collision process in complex systems, a case in which the dynamics of the particles can be described by non-differentiable curves. Since non-differentiability appears as the universal property of complex systems, it is necessary to construct a non-differentiable physics. Thus, the complexity of the interaction processes is replaced by non-differentiability; accordingly, it is no longer necessary to use the whole classical “arsenal” of quantities from standard physics (differentiable physics). This topic was developed within scale relativity theory (SRT) [6,7] and non-standard scale relativity theory (NSSRT) [8–22]. In this case, we assume that the movements of complex system entities take place on continuous, but non-differentiable, curves (fractal curves), so that all physical phenomena involved in the dynamics depend not only on space-time coordinates, but also on space-time scale resolution. From such a perspective, physical quantities describing the dynamics of complex systems may be considered fractal functions [6,7]. Moreover, the entities of the complex system may be reduced to and identified with their own trajectories, so that the complex system will behave as a special fluid lacking interaction (via their geodesics in a non-differentiable (fractal) space). We have called such fluid a “fractal fluid” [8–22]. In the present paper, we shall introduce new concepts, like non-differentiable entropy, informational non-differentiable entropy, informational non-differentiable energy, etc., in the NSSRT approach (the scale relativity theory with an arbitrary constant fractal dimension). Based on a fractal potential, which is the “source” of the non-differentiability of trajectories of the complex system entities, we establish the relationships among non-differentiable entropy. The correlation fractal potential-non-differentiable entropy implies uncertainty relations in the hydrodynamic representation, while the correlation of informational non-differentiable entropy/informational non-differentiable energy implies specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. The constant value of the informational non-differentiable energy made explicit for the harmonic oscillator induces a quantification condition. We note that there exists a large class of complex systems that take smooth trajectories. However, the analysis of the dynamics of these classes is reducible to the above-mentioned statements by neglecting their fractality. Head angle tracking error in simulations. The head angle tracking error converges exponentially to zero. Joint angles tracking the reference joint angles in simulations. The joints of the robot track the sinusoidal motions (above). The joint tracking errors converge exponentially to zero (below). Figure 5 Figure 6 The motion of the center of mass in the x – y plane in simulations. The position of the CM of the robot (blue) converges to and follows the desired straight line path (the x-axis). Figure 7 Joint angles tracking the reference joint angles in simulations in the presence of measurement noise. The joints of the robot track the sinusoidal motions in the presence of measurement noise (above). The joint tracking errors converge exponentially . The experiment was carried out using the snake robot Wheeko [16]. The robot, which is shown in Figure Figure2,2, has 10 identical joint modules, i.e. N=11 links. Each joint module is equipped with a set of passive wheels which give the robot anisotropic ground friction properties during motion on flat surfaces. The wheels are able to slip sideways and thus do not introduce nonholonomic velocity constraints in the system. Each joint is driven by a Hitec servo motor (HS-5955TG; Hitec RCD USA, Inc., Poway, CA, USA), and the joint angles are measured using magnetic rotary encoders. The motion of the snake robot was measured using a camera-based motion capture system from OptiTrack of type Flex 13 (NaturalPoint, Inc., Corvallis, OR, USA). The system consists of 16 cameras which are sampled at 120 frames per second and which allow reflective markers to be tracked on a submillimetre level. During the experiment, reflective markers were mounted on the head link of the snake robot in order to measure the position (px,N,py,N) and orientation (θN) of the head. These measurements were combined with the measured joint angles (q1,…,qN−1) of the snake robot in order to measure the absolute link angles (1) and the position of the CM (px,py) of the robot. In order to obtain the derivatives of the reference head angle (46), we used the same technique as in the simulations, i.e. passing ΦN through a low-pass filter of the form (67). The parameters of the low-pass filter were set to ωn=π/2 and ψf=1. In the following, we elaborate on a few adjustments that were made in the implemented path following controller in order to comply with the particular properties and capabilities of the physical snake robot employed in the experiment. We conjecture that these adjustments only marginally affected the overall motion of the robot. The successful path following behaviour of the robot demonstrated below supports this claim. Since the experimental setup only provided measurements of the joint angles and the position and orientation of the head link, we chose to implement the joint controller in (57) as 𝜗i = −kpyi where i∈{1,…,10}. We conjecture that eliminating the joint angular velocity terms from (57) did not significantly change the dynamic behaviour of the system since the joint motion was relatively slow during the experiment. The main consequence of excluding the velocity terms from (57) is that we potentially introduce a steady-state error in the tracking of the joint angles. Consequently, since with the joint control law (68) the derivative terms in (63) are identically zero, they need not to be linearized in the head angle dynamics by the dynamic compensator. As the result, we implemented the dynamic compensator of the form where the controller gains were kp,θN = 20, kd,θN = 1, kp=10, and kd=5. We saturated the joint angle offset ϕo according to ϕo∈[−π/6,π/6], in order to keep the joint reference angles within reasonable bounds w.r.t the maximum allowable joint angles of the physical snake robot. Moreover, from Figure Figure2,2, it can be seen that the head link of the physical snake robot does not touch the ground since the ground contact points occur at the location of the joints. As a results, we implemented (69) with fθN ≡ 0. The solutions of the dynamic compensator (69) were obtained by numerical integration in LabVIEW which was used as the development environment. We chose the look-ahead distance of the path following controller as Δ=1.4 m. The initial values for the configuration variables of the snake robot were qi=0 rad, θN=−π/2 rad, px=0.3 m, and py=1.7 m, i.e. the snake robot was initially headed towards the desired path (the x-axis), and the initial distance from the CM to the desired path was 1.7 m. Furthermore, the parameters of the constraint functions for the joint angles, i.e. (45), were α=π/6, η=70πt/180, and δ=36π/180, while the ground friction coefficients were ct=1 and cn=10 (i.e identical to the simulation parameters).Figure 12 Comparison between experiments and simulations. Comparison of the convergence of the cross-track error during simulations (red) and experiments (blue). Figure 13 An image of the motion of the robot during the experiments. The robot converges to and follows the desired path. 2. Hallmarks of Non-Differentiability Let us assume that the motion of complex system entities takes place on fractal curves (continuous, but non-differentiable). A manifold that is compatible with such movement defines a fractal space. The fractal nature of space generates the breaking of differential time reflection invariance. In such a context, the usual definitions of the derivative of a given function with respect to time [6,7], are equivalent in the differentiable case. The passage from one to the other is performed via Δt → − Δt transformation (time reflection invariance at the infinitesimal level). In the non-differentiable case, (dQ+dt) and (dQ−dt) are defined as explicit functions of t and dt, The sign (+) corresponds to the forward process, while (−) corresponds to the backward process. Then, in space coordinates dX, we can write [6,7]: with v± the forward and backward mean speeds, and dξ± a measure of non-differentiability (a fluctuation induced by the fractal properties of trajectory) having the average: where the symbol 〈〉 defines the mean value. While the speed-concept is classically a single concept, if space is a fractal, then we must introduce two speeds (v+ and v−), instead of one. These “two-values” of the speed vector represent a specific consequence of non-differentiability that has no standard counterpart (according to differential physics). However, we cannot favor v+ as compared to v−. The only solution is to consider both the forward (dt > 0) and backward (dt < 0) processes. Then, it is necessary to introduce the complex speed [6,7]: If VD is differentiable and resolution scale (dt) speed independent, then VF is non-differentiable and resolution scale (dt) speed dependent. Using the notations dx± = d±x, Equation (6) becomes: This enables us to define the operator: Let us now assume that the fractal curve is immersed in a three-dimensional space and that X of components Xi (i = 1, 2, 3) is the position vector of a point on the curve. Let us also consider a function f(X, t) and the following series expansion up to the second order: Using notations, dXi±=d±Xi , the forward and backward average values of this relation take the form: We shall stipulate the following: the mean values of function f and its derivatives coincide with themselves, and the differentials d±Xi and dt are independent. Therefore, the averages of their products coincide with the product of averages. Thus, Equation (10) becomes: or more, using Equation (3), where the quantities ⟨d±xid±ξl⟩, ⟨d±ξid±xl⟩ are null based on the Relation (5) and also on the above property referring to a product mean. Since dξ± describes the fractal properties of the trajectory with the fractal dimension DF [23], it is natural to impose that (dξ±)DF is proportional with resolution scale dt [6,7], where D is a coefficient of proportionality (for details, see [6,7]). In Nottale’s theory [6,7], D is a coefficient associated with the transition fractal-non-fractal. Let us focus now on the mean ⟨dξi±dξl±⟩ , which has statistical significance [6,7]. This means that at any point on a fractal path, the local acceleration, ∂tVˆ , the non-linearly (convective) term, (Vˆ⋅∇)Vˆ, and the dissipative one, D(dt)2DF−1ΔVˆ, are in balance. Therefore, the complex system dynamics can be assimilated with a “rheological” fluid dynamics. Such a dynamics is described by the complex velocity field Vˆ, by the complex acceleration field dˆVˆdt, etc., as well as by the imaginary viscosity type coefficient iD(dt)2DF−1 For irrotational motions of the complex system entities: Vˆ can be chosen with the form: where ϕ = ln ψ is the velocity scalar potential. Substituting (22) in (20), we obtain: or more: Using the identities [7]: the Equation (23) becomes: This equation can be integrated up to an arbitrary phase factor, which may be set to zero by a suitable choice of phase of ψ and this yields: Relation (24) is a Schrödinger-type equation. For motions of complex system entities on Peano’s curves, DF = 2, Equation (24) takes the Nottale’s form [6,7]. Moreover, for motions of complex system entities on Peano’s curves at the Compton scale, D=h2m0 (for details, see [6,7]), with ħ the reduced Planck constant and m0 the rest mass of the complex system entities, Relation (24) becomes the standard Schrödinger equation. If ψ=ρe−−√iS , with ρ√ the amplitude and S the phase of ψ, the complex velocity field (22) takes the explicit form: Substituting (25) into (20) and separating the real and the imaginary parts, up to an arbitrary phase factor, which may be set to zero by a suitable choice of the phase of ψ, we obtain: with Q the specific fractal potential (specific non-differentiable potential): The specific fractal potential can simultaneously work with the standard potentials (for instance, an external scalar potential). The first Equation (26) represents the specific momentum conservation law, while the second Equation (26) exhibits the state density conservation law. Equations (26) and (27) define the fractal hydrodynamics model (FHM). The following conclusions are obvious: • Any entity of the complex system is in permanent interaction with the fractal medium through a specific fractal potential. • The fractal medium is identified with a non-relativistic fractal fluid described by the specific momentum and state density conservation laws (probability density 2. here we consider that can be appear some situations as say prof dr mircea orasanu and prof horia orasanu concerning as followed In keeping with the aims of the journal the editorial hand is used very lightly. This is an international unrefereed journal which aims to stimulate the sharing of ideas for no other reason than an interest in the ideas and love of discussion among its contributors and readers. If a contribution has some relevance to the broad areas of interest concerned, and contains some features of value it will be included; and these criteria are used very liberally. Please send any items for inclusion to the editor including an electronic copy on disc or E-mail. Word for Windows versions 6 and 7 preferred, but most word processing formats can be accommodated. Most items are welcome include papers, short contributions, letters, discussions, provocations, reactions, half-baked ideas, notices of relevant research groups, conferences or publications, reviews of books and papers, or even books or papers for review. Diagrams, figures and other inserted items are not always easy to accommodate and can use up a great deal of web space, so please use these economically in submissions. First, there is the continuing assumption (inherited from the conservatives) that heaping more assessments on schools and going ‘back-to-basics’ is the way to improve schooling. I expected a more thoughtful and progressive approach from Labour. Rather than any careful reflection on what should be the aims of education in a post-industrial society, there is a strong subscription to what I called the Technological Pragmatist ideology. This involves overriding utilitarian aims for education, and automatic support for anything related to industry, commerce, technology, narrow basic skills and skill certification in education. The subscription to these values is so strong and uncritical it overrides any concerns with rounded personal development (such as the arts in primary education), personal empowerment (such as critical mathematical literacy, democratic or related social skills), or creative student work (such as project or investigational work). The new government exhibits no curiosity or doubts about its aims and programme, or concern about its evidential foundation, and simply dictates its educational policy from the centre on the basis, recently established by the conservatives, that “It’s my ball, so I’ll make up the rules”. Second, teacher education has become more and more centrally regulated over the past 15 years. Of course public accountability is both desirable and necessary, but for 15 years the government has imposed a set of tight aims, procedures and regulations, and then as we conform to the required changes, criticised teacher education for being misconceived. Thus central policy has been a continuous cycle of ever changing goals and higher hoops to jump through, followed by criticism. Under the conservatives this was fuelled by the Industrial Trainers who in power and in New Right think-tanks critiqued teacher education for being Marxist instead of politically neutral, obsessed with irrelevant PC-isms (anti-sexism and anti-racism) rather than good old fashioned school subjects, and for being theory rather than practice driven. All these charges are patently untrue as anyone working in teacher education knows. The primary concerns are subject matter knowledge, subject pedagogy and practical teaching practice. Even if we didn’t think these are the most important things for future teachers (which all of us in teacher education do) strictly enforced government regulations upon which our accreditation depends would prevented us doing anything else anyway. To ensure that their heavy handed agenda was implemented conservative governments created the Teacher Training Agency and placed well known members of New Right think tanks in charge of all of its committees to conduct an ideologically driven assault on teacher education and its progressive values. So what did the New Labour Government do on taking over the running of teacher education? It sustained and endorsed the activities of the Teacher Training Agency in establishing a National Curriculum for Teacher Education, in conjunction with Department for Education and Employment (the government ministry of education). The primary school version of this, specifying the required legal basis for primary teacher education, does not mention history, geography, foreign languages, dance, drama, or music, even 2 FORMULATIONThe assessment of the student’s mathematical ability in mathematics concerns the following qualities: The ability to use, develop, and express mathematical knowledge. The assessment concerns the student’s ability to use and develop her/his mathematical “knowing” to interpret and deal with different kind of tasks and situations that can be found in school and society, for example the ability to discover patterns and relationships, suggest solutions, make (rough) estimations, reflect on and interpret her/his results and assess their reasonableness. Independence and creativity are important aspects to consider in assessment as well as clarity, thoroughness, and skill. An important aspect of knowing mathematics is the student’s ability to express her/his thoughts orally and in writing, with the help of the mathematical language of symbols and supported by concrete material and pictures. The ability to follow, understand and scrutinise reasoning in mathematics. The assessment concerns the student’s ability to be receptive of (“ta del av”) and use information in oral as well as written form, for example the ability to listen to, follow and scrutinise explanations and arguments given by others. Furthermore, attention is given to the student’s ability to independently and critically decide on mathematically founded descriptions and solutions to problems found in different contexts in school and society. The ability to reflect on the significance of mathematics in culture and society. The assessment concerns the student’s awareness (insight) and feeling for the value and limitations of mathematics as a tool and aid of assistance in other school-subjects, in everyday life and society and in communication between people. It also concerns the student’s knowledge about the significance of mathematics in a historical perspective. 6.1.2 Criteria (standards) for Pass with distinction The student uses mathematical concepts and methods in order to formulate and solve problems. (‘Formulate’ can be interpreted with the help of aims to strive for, indicating that ‘formulate’ means the same as ‘mathematise’ or perhaps ‘model’.) . Therefore, the geodesic equations reduce in this case to (6) \|xu\|2 u + u2 xuu•xu + v2xvv•xu = 0 (7) \|xv\|2 v + 2uvxuv•xv = 0 Isolating the v-terms in (7) gives (8) , where G= xv•xv. Leave a Reply WordPress.com Logo Google+ photo Twitter picture Facebook photo Connecting to %s
afe8840eeba19e0e	Day Six – the State of Superposition And God saw all that He had made, and behold it was very good, and it was evening and it was morning, the sixth day. (Gen. 1:31) The Biblical narrative of creation concludes with the above verse. Rabbi Shlomo Yitzchaki (a.k.a. Rashi) comments on this verse: Related image the sixth day: Scripture added a “hey” on the sixth [day], at the completion of the Creation, to tell us that He stipulated with them, [“you were created] on the condition that Israel accept the Five Books of the Torah.” [The numerical value of the “hey” is five.] (Tanchuma Bereishith 1). Another explanation for “the sixth day”: They [the works of creation] were all suspended until the “sixth day,” referring to the sixth day of Sivan, which was prepared for the giving of the Torah (Shab. 88a). [The letter “hey” is the definite article, alluding to the definitive sixth day, the sixth day of Sivan, when the Torah was given] What Rashi is saying here is that the world was suspended in the state of superposition of being created and not being created depending on the future choice to be made by Jewish people – to accept the Torah or not. Until that moment – the sixth day of Sivan – the world existed in a quantum-mechanical state of superposition of existence and not existence, as it were, just as a Schrödinger cat. In quantum mechanics, a system (say, a particle) is described by a wave function (or wavefunction), that describes the distribution of probabilities to find the system (e.g., particle) in a particular area of space. The wavefunction obeys the Schrödinger equation that predicts the evolution of the wavefunction in time. Why the evolution of the wavefunction is deterministic, the predictions based on this equation are not deterministic. All we can predict is the probability of finding a particle in a particular area of space. However, when we conduct an experiment, we always get a definitive result for the measured property. This paradox is called the Measurement Paradox, which different competing interpretations of quantum mechanics strive to explain. The Everett’s many-worlds interpretation of quantum mechanics postulates that a choice splits the universe into branches, where each of the possible outcomes is realized. In the orthodox Copenhagen interpretation of quantum mechanics, the choice causes the collapse of the wavefunction, whereby the plurality of possibilities is collapsed into a single actuality. Until such collapse, the system (e.g., particle) is in a state of superposition of all possible states. Thus, a Schrödinger cat exists in a state of superposition of being alive and dead at the same time, until its state is collapsed by the act of observation, whereby the observer “chooses,” albeit subconsciously, between two possible states. According to Rashi, until the Sinaitic revelation, the world existed in a superposition of two states – existence and non-existence. “[The works of creation] were all suspended until the “sixth day.” It was the choice of the Jewish people who accepted the Torah that collapsed the wavefunction and broad the world into existence. I don’t know if Rashi knew quantum mechanics, but his logic is clearly quantum-mechanical. This commentary is one of the most explicit examples of the quantum-mechanical concept of superposition and the collapsed the wavefunction as it is used in the Torah. It gives me a great pleasure to thank our son, Elie, who pointed out to me this fascinating parallel. May the Almighty send him, Eliyahu Chaim Yitzchak ben Leah, refuah shaleimah! About the Author Dr. Alexander Poltorak is Chairman and CEO of General Patent Corporation. He is also an Adjunct Professor of Physics at The City College of New York. In the past, he served as Assistant Professor of Physics at Touro College, Assistant Professor of Biomathematics at Cornell University Medical College, and Adjunct Professor of Law at the Globe Institute for Technology. He holds a Ph.D. in theoretical physics. Related Topics Related Posts
ddcc74c4a60ca425	This paper is devoted to the derivation of a pleasingly parallel Galerkin method for the time-independent N-body Schrödinger equation, and its time-dependent version modeling molecules subject to an external electric field (Bandrauk 1994; Bandrauk et al., J. Phys. B-Atom. Mol. Opt. Phys. 46(15), 153001, 2013; Cohen-Tannoudji et al. 1992). In this goal, we develop a Schwarz waveform relaxation (SWR) domain decomposition method (DDM) for the N-body Schrödinger equation. In order to optimize the efficiency and accuracy of the overall algorithm, (i) we use mollifiers to regularize the singular potentials and to approximate the Schrödinger Hamiltonian, (ii) we select appropriate orbitals, and (iii) we carefully derive and approximate the SWR transmission conditions. Some low-dimensional numerical experiments are presented to illustrate the methodology. Additional Metadata Keywords Domain decomposition method, Mollifiers, N-body Schrödinger equation, Parallel computing Persistent URL Journal Numerical Algorithms Lorin, E. (2018). Domain decomposition method for the N-body time-independent and time-dependent Schrödinger equations. Numerical Algorithms, 1–40. doi:10.1007/s11075-018-0566-3
cd01e278c6147268	Citation for this page in APA citation style. Close Mortimer Adler Rogers Albritton Alexander of Aphrodisias Samuel Alexander William Alston Louise Antony Thomas Aquinas David Armstrong Harald Atmanspacher Robert Audi Alexander Bain Mark Balaguer Jeffrey Barrett William Belsham Henri Bergson George Berkeley Isaiah Berlin Richard J. Bernstein Bernard Berofsky Robert Bishop Max Black Susanne Bobzien Emil du Bois-Reymond Hilary Bok Laurence BonJour George Boole Émile Boutroux Michael Burke Joseph Keim Campbell Rudolf Carnap Ernst Cassirer David Chalmers Roderick Chisholm Randolph Clarke Samuel Clarke Anthony Collins Antonella Corradini Diodorus Cronus Jonathan Dancy Donald Davidson Mario De Caro Daniel Dennett Jacques Derrida René Descartes Richard Double Fred Dretske John Dupré John Earman Laura Waddell Ekstrom Herbert Feigl John Martin Fischer Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. 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Jay Wallace Ted Warfield Roy Weatherford William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Michael Arbib Walter Baade Bernard Baars Gregory Bateson John S. Bell Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Donald Campbell Anthony Cashmore Eric Chaisson Jean-Pierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher Joseph Fourier Philipp Frank Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. 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Wilson H. Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Free Will Mental Causation James Symposium The Problem of Microscopic Reversibility Loschmidt's Paradox In 1874, Josef Loschmidt criticized his younger colleague Ludwig Boltzmann's 1866 attempt to derive from basic classical dynamics the increasing entropy required by the second law of thermodynamics. Increasing entropy is the intimate connection between time and the second law of thermodynamics that Arthur Stanley Eddington later called the Arrow of Time. (The fundamental arrow of time is the expansion of the universe, which makes room for all the other arrows.) Despite never seeing entropy decrease in an isolated system, attempts to "prove" that it always increases have been failures. Loschmidt's criticism was based on the simple idea that the laws of classical dynamics are time reversible. Consequently, if we just turned the time around, the time evolution of the system should lead to decreasing entropy. Of course we cannot turn time around, but a classical dynamical system will evolve in reverse if all the particles could have their velocities exactly reversed. Apart from the practical impossibility of doing this, Loschmidt had shown that systems could exist for which the entropy should decrease instead of increasing. This is called Loschmidt's "Reversibility Objection" (Umwiederkehreinwand) or "Loschmidt's paradox." We call it the problem of microscopic reversibility. We can visualize the free expansion of a gas that occurs when we rapidly withdraw a piston. Because this is a movie, we can reverse the movie to show what Loschmidt imagined would happen. But Boltzmann thought that even if the particles could all have their velocities reversed, minute errors in the collisions would likely prevent a perfect return to the original state. Forward Time Time Reversal To demonstrate the randomness in each collision, which Boltzmann described as "molecular disorder (molekular ungeordnet) we need a program that reverses the velocities of the gas particles, and adds randomness into the collisions. (This is a work in progress.) Information physics claims that microscopic reversibility is actually extremely unlikely and that the intrinsic path information in particles needed to reduce entropy is erased by matter-radiation interactions or by internal quantum transitions in the colliding atoms considered as a "quasi"-molecule. Microscopic time reversibility is one of the foundational assumptions of both classical mechanics and quantum mechanics. It is mistakenly thought to be the basis for the "detailed balancing" of chemical reactions in thermodynamic equilibrium. In fact microscopic reversibility is an assumption that is only statistically valid in the same limits as any "quantum to classical transition." This is the limit when the number of particles is large enough that we can average over quantum effects. Quantum events also approach classical behavior in the limit of large quantum numbers, which Niels Bohr called the "correspondence principle." It may seem presumptuous for an information philosopher to challenge such a fundamental principle of statistical mechanics and even quantum statistical physics as microscopic reversibility. What "detailed balancing" means is that in thermodynamic equilibrium, the number of forward reactions is exactly balanced by the number of reverse reactions. And this is correct. But microscopic reversibility, while still true when considering averages over time, should not be confused with the time reversibility of a specific individual collision between particles. We will examine the collision of two atoms and show that if their velocities are reversed at some time after the collision, it is highly improbable that they will retrace their paths. This does not mean that, given enough particle collisions, there will not be statistically many collisions that are essentially the same as the "reverse collisions" needed for detailed balancing in chemical reactions, for transport processes with the Boltzmann equation, and for the Onsager reciprocal relations in non-equilibrium conditions. The Origin of Irreversibility Our careful quantum analysis shows that time reversibility fails even in the most ideal conditions (the simplest case of two particles in collision), provided internal quantum structure or the quantum-mechanical interaction with radiation is taken into account. Albert Einstein was the first to see this, first in his 1909 extension of work on the photoelectric effect but especially in his 1916-17 work on the emission and absorption of radiation. This was the work in which Einstein showed that quantum theory implies ontological chance, which he famously disliked, ("God does not play dice!"). For Einstein, detailed balancing was not the result of microscopic reversibility, it was his starting assumption. Einstein's work is sometimes cited as proof of detailed balancing and microscopic reversibility. (Wikipedia, for example.) In fact, Einstein used Boltzmann's assumption of detailed balancing, along with the "Boltzmann principle" that the probability of states with energy E is reduced by the exponential "Boltzmann factor," f(E) ∝ e-E/kT, to derive his transition probabilities for emission and absorption of radiation. Einstein also derived Planck's radiation law and Bohr's second "quantum postulate" Em - En = hν. But Einstein distinctly denied any symmetry in the elementary processes of emission and absorption. As early as 1909, he noted that the elementary process of emission is not "invertible." There are outgoing spherical waves of radiation, but incoming spherical waves are never seen. In a deterministic universe, the path information needed to predict the future motions of all particles would be preserved. If information is a conserved quantity, the future and the past are all contained in the present. The information about future paths is precisely the same information that, if reversed, would predict microscopic reversibility of each and every collision. The introduction of ontological probabilities and statistics would deny such determinism. If the motions of particles have a chance element, such determinism can not exist. And this is exactly what Einstein did in his papers on the emission and absorption of radiation by matter. He found that quantum theory implies ontological chance. A "weakness in the theory," he called it. What we might call Einstein's "radiation asymmetry" was introduced with these words, The elementary process of the emission and absorption of radiation is asymmetric, because the process is directed, as Einstein had explicitly noted first in 1909, and we think he had seen as early as 1905. The apparent isotropy of the emission of radiation is only what Einstein called "pseudo-isotropy" (Pseudoisotropie), a consequence of time averages over large numbers of events. Einstein often substituted time averages for space averages, or averages over the possible states of a system in statistical mechanics. a quantum theory free from contradictions can only be obtained if the emission process, just as absorption, is assumed to be directional. In that case, for each elementary emission process Zm->Zn a momentum of magnitude (εm—εn)/c is transferred to the molecule. If the latter is isotropic, we shall have to assume that all directions of emission are equally probable. If the molecule is not isotropic, we arrive at the same statement if the orientation changes with time in accordance with the laws of chance. Moreover, such an assumption will also have to be made about the statistical laws for absorption, (B) and (B'). Otherwise the constants Bmn and Bnm would have to depend on the direction, and this can be avoided by making the assumption of isotropy or pseudo-isotropy (using time averages). Now the principle of microscopic reversibility is a fundamental assumption of statistical mechanics. It underlies the principle of "detailed balancing," which is critical to the understanding of chemical reactions. In thermodynamic equilibrium, the number of forward reactions is exactly balanced by the number of reverse reactions. But microscopic reversibility, while true in the sense of averages over time, should not be confused with the reversibility of individual collisions between molecules. The equations of classical dynamics are reversible in time. And the deterministic Schrödinger equation of motion in quantum mechanics is also time reversible. But the interactions of photons and material particles like electrons and atoms are distinctly not reversible! An explanation of microscopic irreversibility in atomic and molecular collisions would provide the needed justification for Ludwig Boltzmann's assumption of "molecular disorder" and strengthen his H-Theorem. This is what we hope to do. In quantum mechanics, microscopic time reversibility is assumed true by most scientists because the deterministic Schrödinger equation itself is time reversible. But the Schrödinger equation only describes the deterministic time evolution of the probabilities of various quantum events, which are themselves not deterministic and not reversible. When an actual event occurs, the probabilities of multiple possible events collapse to the actual occurrence of one event. In quantum mechanics, this is the irreversible collapse of the wave function that John von Neumann called "Process 1." The possibility of quantum transitions between closely spaced vibrational and rotational energy levels in the "quasi-molecule' introduces indeterminacy in the future paths of the separate atoms. The classical path information needed to ensure the deterministic dynamical behavior has been partially erased. The memory of the past needed to predict the "determined" future has been lost. Even assuming the practical impossibility of a perfect classical time reversal, in which we simply turn the two particles around, quantum physics would require two measurements to locate the two particles, followed by two state preparations to send them in the opposite direction. These could only be made within the precision of Heisenberg's uncertainty principle and so could not perfectly produce microscopic reversibility, which is thus only a classical idealization, like the idea of determinism.. Heisenberg indeterminacy puts calculable limits on the accuracy with which perfect reversed paths can be achieved. Let us assume this impossible task can be completed, and it sends the two particles into the reverse collision paths. But on the return path, there is only a finite probability that a "sum over histories" calculation will produce the same (or exactly reversed) quantum transitions between vibrational and rotational states that occurred in the first collision. Thus a quantum description of a two-particle collision establishes the microscopic irreversibility that Boltzmann sometimes described as his assumption of "molecular disorder." In his second (1877) derivation of the H-theorem, Boltzmann used a statistical approach and the molecular disorder assumption to get away from the time-reversibility assumptions of classical dynamics. We must develop a deep insight into Einstein's asymmetry between light and matter, one that was appreciated as early as the 1880's by Max Planck's great mentor Gustave Kirchhoff, but was not understood in quantum mechanical terms until Einstein's understanding of nonlocality and the relation between waves and particles in 1909.. It is still ignored in quantum statistical mechanics by those who mistakenly think that the time reversible Schrödinger equation means microscopic interactions are reversible. Maxwell and Boltzmann had shown that collisions between material particles, analyzed statistically, cause the distribution of positions and velocities to approach their equilibrium Maxwell-Boltzmann distribution. A bit later, Kirchhoff and Planck knew that an extreme non-equilibrium distribution of radiation, for example a monochromatic radiation field, will remain out of equilibrium indefinitely. But if that radiation interacts with even the tiniest amount of matter, a speck of carbon black was their example, all the wavelengths of the spectrum - the Kirchhoff law - soon appear. So we can say that the approach to equilibrium of a radiation field has the same origin of irreversibility as that of matter. Radiation without matter cannot equilibrate. Photons do not interact, except at the extremely high energies where they can convert to matter and anti-matter. Our new insight is that matter without radiation also cannot equilibrate in a way that escapes the reversibility and recurrence objections, as is taught in every textbook and review article on statistical mechanics to this day. It is thus the irreversible interaction of the two, light and matter, photons and electrons, that lies behind the increase of entropy in the universe. The second law of thermodynamics would not explain the increase of entropy except for the microscopic irreversibility that we have shown to be the case. Microscopic irreversibility not only explains the second law, it validates Boltzmann's brilliant assumption of "molecular disorder" to justify his statistical arguments. Zermelo's paradox was a later criticism of Ludwig Boltzmann's attempt to derive the increasing entropy required by the second law of thermodynamics. It also involves time. Assuming infinite available time, a finite universe with fixed matter, energy, and information will at some point return to any given earlier state. We now know that even a finite part of the universe cannot return to exactly the same state, because the surrounding universe will have aged and be in a different information state. This is the information philosophy solution to the problem of eternal recurrence, as seen by Arthur Stanley Eddington and H. Dieter Zeh. The Origin of Irreversibility (pdf) Microscopic Irreversibility, chapter 25 of Great Problems of Philosophy and Physics Solved?. For Teachers Chapter 5.7 - Recurrence Problem Chapter 5.9 - Universals Part Four - Freedom Part Six - Solutions Normal \| Teacher \| Scholar
a76f8497fb5e712e	My questions are about de Broglie-Bohm "pilot wave" interpretation of quantum mechanics (a.k.a. Bohmian mechanics). 1. Do quasiparticles have any meaning in Bohmian mechanics, or not? Specifically, is it possible to trace the motion of a quasiparticle (e.g. a phonon, or a hole) by watching Bohmian trajectories? 2. Bohmian mechanics provides some explanation of difficulties related to quantum measurement process. But imagine that, in some kind of "theory of everything", all known elementary particles (leptons, quarks, gluons etc.) are actually quasiparticles (real particles are always confined). Would the explanation, provided by Bohmian mechanics, survive in this case? • 2 $\begingroup$ I think that your inherent suggestion that the Bohmian mechanics suffers from an inconsistency because of quasiparticles is right. QM shows that it doesn't matter whether one says that something is a particle or quasiparticle: the Hilbert spaces and dynamics and probabilistic predictions are isomorphic. In the Bohmian mechanics, it always matters because particles' degrees of freedom are "beable" while other e.g. quasiparticles' degrees of freedom are not. For the same reason, quantum computers can't simulate the reality in Bohmian mechanics. It just doesn't work. $\endgroup$ – Luboš Motl Oct 6 '11 at 12:03 • 1 $\begingroup$ @Luboš Motl: Electron (a particle) in Hydrogen atom is always in a mixed state - it does not have an electron wave function $\psi (\vec{r}_e)$. Instead, it is a quasi-particle that may be in a pure state $\psi (\vec{r})$ where $\vec{r}=\vec{r}_e-\vec{r}_p$. As well, proton is also in a mixed state but the center of mass may be in a pure state like $exp\left ( i(\vec{P}\vec{R}-\frac{P^2}{2M_A\hbar}t)\right)$.There are no such real particles with $\mu$ and $M_A=m_e+m_p$; the corresponding variables describe quasi-particles. It is namely quasi-particle energy in atom that is quantized. $\endgroup$ – Vladimir Kalitvianski Oct 6 '11 at 13:13 • 5 $\begingroup$ The answer to your question, @Vladimir, is obviously No: every quantum system (whether it is a particle or quasiparticle) may always be found in a pure state. That's why any quantum system may exhibit 100% destructive interference etc., something that would indeed be impossible if an object were "forced" to be partly mixed, and this is one reason why any picture of the quantum phenomena that tries to be more classical than QM (including Bohmian mechanics) is inevitable inconsistent with the tests of interference and other empirically established features of the quantum world. $\endgroup$ – Luboš Motl Oct 6 '11 at 15:57 • 3 $\begingroup$ What do you mean by "meaning"? How can you say that quasiparticles have any meaning in standard quantum mechanics? They're just a very good way of describing an emergent phenomenon; they don't really exist in the usual sense of the word. $\endgroup$ – Peter Shor Oct 8 '11 at 21:27 • 5 $\begingroup$ Can I also say that after thinking it over, I suspect there may be a very good question lurking in here. Namely: do the trajectories of quasiparticles in Bohmian mechanics obey the same rules as the trajectories of particles in Bohmian mechanics? And if not, what rules do they obey? $\endgroup$ – Peter Shor Oct 10 '11 at 22:21 Nobody wrote an answer, so I'll give it a try 1. Indeed in Bohmian QM quasiparticles are not on the same ground as ordinary particles. For example consider phonons in a crystal lattice. From a Bohmian POV, the atom position observables have fundamental significance. But observables natural from the phonon POV, such as phonon number are not functions of the atom positions. Of course these observables are still functions of atom positions + momenta, so in principle they can be assigned values along a Bohmian trajectory. However, this approach has serious issues. For one thing the "phononic" observables are still not going to play a symmetric role with the atomic positions. For another, the phonon number will not be integer! This shows it hardly makes sense to talk about phonon trajectories 2. Actually, the elementary particles are "quasiparticles". They are excitations of the quantum fields. This indeed means the Bohmian approach runs into trouble, since in Bohmian QFT the fields become the fundamental observables, making it inconsistent with nonrelativistic Bohmian QM. Actually it's not the only problem of Bohmian QFT: it also fails to be Lorentz invariant In the simplest harmonic model, the occupation number of each mode is a linear function of its energy i.e. something quadratic in both positions and momenta 1. Given that Bohmian mechanics (better named de Broglie-Bohm or dBB theory) contains the Schrödinger equation, it contains quantum theory completely, and therefore all things which make sense in quantum theory remain meaningful. Of course, for quasiparticles nobody plans to construct trajectories. And it is also not possible to simply "watch" Bohmian trajectories. 1. dBB theory is usually presented as a theory for particles, but this is far too restrictive. The better way is to present it as a theory for the configuration space: It defines a trajectory for the configuration $q(t)$. So, the configuration may be, as well, a field $f(x)$ on the space. Then the "trajectory" is $f(x,t)$. The mathematics of dBB theory works if the Hamiltonian has the form $H=p^2 + V(q)$. This works nicely with relativistic field theories $L = (d_tf)^2-(d_xf)^2$ + interaction terms gives momentum $p(x)=d_t$ $f(x)$ and gives a Hamiltonian quadratic in $p(x)$. All the advantages provided by dBB theory do not depend on the question what is the particular configuration space. Thus, they would survive whatever the choice of Q is. All one has to care for is a way to obtain a choice of a configuration space so that $H=p^2 + V(q)$. For bosonic field theories no problem, for fermionic field theories proposals exist too. In the worst case, one can consider partial realizations (like defining trajectories only for bosons). This is less beautiful but preserves the major advantages (realism, causal explanation, no measurement problem). • $\begingroup$ It's just an elementary mistake to say that a "theory A containing the key equation of theory B" may do everything that B does. Theories are only equivalent if their objects may be mapped in a one-to-one way, the observable evolution laws are isomorphic in both, and therefore if A isn't "missing" anything from B, but it also has nothing extra on top of B. Bohmian mechanics has extra "beables" - classical particle positions etc. (which are sometimes said to be measured instead of the collapsed wave functions) - so it is clearly not equivalent to QM. $\endgroup$ – Luboš Motl May 16 '16 at 14:11 Your Answer
e4131e2681c56a61	Saturday, February 29, 2020 Netflix goes to Africa: Queen Sono Friday, February 28, 2020 The Hudson, an orange bouy, and the George Washington Bridge The world is changing, and increasingly uncertain Farhad Manjoo, Admit It: You Don’t Know What Will Happen Next, NYTimes, Feb 26, 2020. A projection of certainty is often a crucial part of commentary; nobody wants to listen to a wishy-washy pundit. But I worry that unwarranted certainty, and an under-appreciation of the unknown, might be our collective downfall, because it blinds us to a new dynamic governing humanity: The world is getting more complicated, and therefore less predictable. Yes, the future is always unknowable. But there’s reason to believe it’s becoming even more so, because when it comes to affairs involving masses of human beings — which is most things, from politics to markets to religion to art and entertainment — a range of forces is altering society in fundamental ways. These forces are easy to describe as Davos-type grand concepts: among others, the internet, smartphones, social networks, the globalization and interdependence of supply chains and manufacturing, the internationalization of culture, unprecedented levels of travel, urbanization and climate change. But their effects are not discrete. They overlap and intertwine in nonlinear ways, leaving chaos in their wake. In the last couple of decades, the world has become unmoored, crazier, somehow messier. The black swans are circling; chaos monkeys have been unleashed. And whether we’re talking about the election, the economy, or most any other corner of humanity, we in the pundit class would do well more often to strike a note of humility in the face of the expanding unknown. We ought to add a disclaimer to everything we say: “I could be wrong! We all could be wrong!” Thursday, February 27, 2020 Norman Rockwell gets political Hudson River, Verrazano-Narrows Bridge in the distance A guerilla attack on (bogus) copyright claims over musical melodies: Create them all and release them into the publid domain Two programmer-musicians wrote every possible MIDI melody in existence to a hard drive, copyrighted the whole thing, and then released it all to the public in an attempt to stop musicians from getting sued. Programmer, musician, and copyright attorney Damien Riehl, along with fellow musician/programmer Noah Rubin, sought to stop copyright lawsuits that they believe stifle the creative freedom of artists. Often in copyright cases for song melodies, if the artist being sued for infringement could have possibly had access to the music they're accused of copying—even if it was something they listened to once—they can be accused of "subconsciously" infringing on the original content. One of the most notorious examples of this is Tom Petty's claim that Sam Smith's “Stay With Me” sounded too close to Petty's “I Won’t Back Down." Smith eventually had to give Petty co-writing credits on his own chart-topping song, which entitled Petty to royalties. Defending a case like that in court can cost millions of dollars in legal fees, and the outcome is never assured. Riehl and Rubin hope that by releasing the melodies publicly, they'll prevent a lot of these cases from standing a chance in court. In a recent talk about the project, Riehl explained that to get their melody database, they algorithmically determined every melody contained within a single octave. Monday, February 24, 2020 Wynton Marsalis on the difference between African rhythm and jazz rhythm Ethan Iverson [EI] is interviewing Wynton Marsalis [WM] about his composition Congo Square, which combines the Lincoln Center Jazz Archestra with Odadaa!, a West African drum ensemble led by Yacub Addy. At various points in the interview Iverson plays a short clip for Marsalis. He did so just before this passage: EI: Now, what is that break? WM: Carlos Henriquez showed me that one. If you hear it in 4, it’s easy, but if you hear it in 6, it’s hard. But in 4 it is square, right on the beat, but maybe we “place” them a little bit. We have to adjust to the 6 Odadaa! is playing, especially since they are in the middle of a phrase. As conductor, I adjust to the bell. EI: That’s a mysterious moment; that’s why I like it so much. WM: The hardest thing is to get us to play with the bell pattern. EI: The up-and-down of the beat is not American. WM: No, it’s not, it’s more like a clave. And like Yacub told me: “In order for us all to play together, y’all will have to play with us.” For me, it was a blessing to have Carlos and Ali [Jackson], who spent a lot of time at night working it out; a real labor of love. They would sit up with me and go through rhythm patterns and say, “No, that’s not it.” Then, eventually, “This is it.” YES! to this: "The up-and-down of the beat is not American." I learned that from years of playing with the late Ade Knowles when I was living in Troy, New York. Early in his career Ade had toured as a drummer and percussionist with Gil Scott-Heron. I met him when he was an administrator at RPI (Rensselaer Polytechnic Institute) and I was on the faculty. "The Magic of the Bell" is a piece I wrote about a particularly magical rehearsal with Ade. Abstract words in prose fiction [#DH] Friday, February 21, 2020 Of telomeres, senescence, cancer, laboratory mice, evolutionary biology, and institutional failure: From the life of Bret Weinstein This is a long podcast (over two hours), and it takes awhile to get off the ground, but it's worth your attention. About the podcast: All of our Mice are Broken. On this episode of The Portal, Bret and Eric sit down alone with each other for the first time in public. There was no plan. There was however, a remarkable story of science at its both best and worst that had not been told in years. After an initial tussle, we dusted off the cobwebs and decided to reconstruct it raw and share it with you, our Portal audience, for the first time. I don't think it will be the last as we are now again looking for our old notes to tighten it up for the next telling. We hope you find it interesting, and that it inspires you younger and less established scientists to tell your stories using this new medium of long form podcasting. We hope the next place you hear this story will be in a biology department seminar room in perhaps Cambridge, Chicago, Princeton, the Bay Area or elsewhere. Until then, be well and have a listen to this initial and raw version. Louis Armstrong on the cover of Time Magazine Thursday, February 20, 2020 The egalitarian proclivities of Louis Armstrong M.H, Miller, Louis Armstrong, The King of Queens, NYTimes, 20 Feb 2020: Armstrong was born in New Orleans in 1901, dropped out of school as a child and was a successful touring musician in his early 20s. By 1929, he was living in Harlem, though as one of the most popular recording artists in the country, he traveled about 300 nights a year. In 1939, he met his fourth and final wife, Lucille Wilson, a dancer at Harlem’s Cotton Club. Lucille, who spent part of her childhood in Corona, decided it was time for her husband to settle down in a house, a real house, instead of living out of hotel rooms. (Even their wedding took place on the road, in St. Louis, at the home of the singer Velma Middleton.) One day, when Armstrong was away at a gig, she put a down payment of $8,000 (around $119,000 in today’s money) on 34-56 107th Street. She didn’t tell him she’d done this until eight months later, during which time she made the mortgage payments herself. [...] From the outside, the two-bedroom, 3,000-square-foot house looks just like any other on the block, which was deliberate. Armstrong often referred to himself as “a salary man” and felt at ease alongside the telephone operators, schoolteachers and janitors of Corona, a neighborhood that, in a testament to how much of his life was spent in jazz clubs, he referred to affectionately as “that good ol’ country life.” One of the earliest integrated areas of New York, Corona was mostly home to middle-class African-Americans and Italian immigrants when the Armstrongs moved in. The demographics would change in the coming decades — Latin Americans began replacing the Italians in the ’60s, and now make up most of the neighborhood — but not much else. There was never a mass wave of gentrification or development here, and Armstrong himself was so concerned with blending in with his working-class neighbors that when his wife decided to give the house a brick facade, Armstrong went door-to-door down the block asking the other residents if they wanted him to pay for their houses to receive the same upgrade. (A few of his neighbors took him upon the offer, which accounts for the scattered presence of brick homes on the street to this day.) [...] He played behind the Iron Curtain during the Cold War and in the Democratic Republic of Congo during decolonization in 1960, during which both sides of a civil war called a truce to watch him perform, then picked up fighting again once his plane took off. There are few American figures as legendary and beloved, and yet, as Harris told me, a common reaction people have upon entering his home is, “This reminds me of my grandmother’s house.” Certainly the living room recalls a ’60s vision of Modernism with a vaguely minimalist formality. Wednesday, February 19, 2020 Illustrated Japanese books from 1600-1912 Surveillance tech is deeply flawed [[Surprise! Surprise!]] Charlie Warzel, All This Dystopia, and for What?, NYTimes 20 Feb 2020. The above examples all represent a different, equally troubling brand of dystopia — one full of false positives, confusion and waste. In these examples the technology is no less invasive. Your face is still scanned in public, your online information is still leveraged against you to manipulate your behavior and your financial data is collected to compile a score that may determine if you can own a home or a car. Your privacy is still invaded, only now you’re left to wonder if the insights were accurate. As lawmakers ponder facial recognition bans and comprehensive privacy laws, they’d do well to consider this fundamental question: Setting aside even the ethical concerns, are the technologies that are slowly eroding our ability to live a private life actually delivering on their promises? Companies like NEC and others argue that outright bans on technology like facial recognition “stifle innovation.” Though I’m personally not convinced, there may be kernels of truth to that. But before giving these companies the benefit of the doubt, we should look deeper at the so-called innovation to see what we’re really gaining as a result of our larger privacy sacrifice. Friday, February 14, 2020 Romantic kissing is not universal In addition, cross-cultural ethnographic data was used to analyze the relationship between any presence of romantic kissing and a culture’s complexity of social stratification. The report finds that complex societies with distinct social classes (e.g. industrialized societies) have a much more frequent occurrence of this type of kissing than egalitarian societies (e.g. foragers). More at the link (H/t Tyler Cowen). Thursday, February 13, 2020 Conjunctions: transportation and graffiti [Jersey City] Bernie Sanders isn't a socialist The thing is, Bernie Sanders isn’t actually a socialist in any normal sense of the term. He doesn’t want to nationalize our major industries and replace markets with central planning; he has expressed admiration, not for Venezuela, but for Denmark. He’s basically what Europeans would call a social democrat — and social democracies like Denmark are, in fact, quite nice places to live, with societies that are, if anything, freer than our own. So why does Sanders call himself a socialist? I’d say that it’s mainly about personal branding, with a dash of glee at shocking the bourgeoisie. And this self-indulgence did no harm as long as he was just a senator from a very liberal state. But if Sanders becomes the Democratic presidential nominee, his misleading self-description will be a gift to the Trump campaign. So will his policy proposals. Single-payer health care is (a) a good idea in principle and (b) very unlikely to happen in practice, but by making Medicare for All the centerpiece of his campaign, Sanders would take the focus off the Trump administration’s determination to take away the social safety net we already have. Has "civilization" entered a phase of decadence? That's what Ross Douthat argues in his new book, The Decadent Society: How We Became the Victims of Our Own Success, which Damon Linker reviews in The Week, February 13, 2020. Decadence? By calling us "decadent," Douthat doesn't mean that we're succumbing to imminent decline and collapse. Following esteemed cultural critic Jacques Barzun, Douthat instead defines decadence as a time when art and life seem exhausted, when institutions creak, the sensations of "repetition and frustration" are endemic, "boredom and fatigue are great historical forces," and "people accept futility and the absurd as normal." Douthat goes on to refine the definition: Decadence refers to economic stagnation, institutional decay, and cultural and intellectual exhaustion at a high level of material prosperity and technological development. It describes a situation in which repetition is more the norm than innovation; in which sclerosis afflicts public institutions and private enterprises alike; in which intellectual life seems to go in circles; in which new developments in science, new exploratory projects, underdeliver compared with what people recently expected. And crucially, the stagnation and decay are often a direct consequence of previous development: the decadent society is, by definition, a victim of its own significant success. Douthat certainly isn't a favorite of mine, and I've got problems with the word "decadent", but that description is consistent with my own view, based on the theory of cultural ranks that David Hays and I developed, that we're exhausting the cultural resources we've inherited but have not yet managed to invent new modes of thinking, feeling, living, and exploring. Near the end Linker observes: Interestingly, one way to describe the populist insurgencies taking place around us is to say that they're a rebellion against the decadence of the post-Cold War world — the sense that history came to an end in 1989, with all significant ideological disputes resolved and politics reduced to the fine-tuning of liberal democratic government. Francis Fukuyama's own high-level punditry on the subject was actually far more ambivalent than it's usually credited with being. Although Fukuyama argued that liberal democracy triumphed over communism because it was more capable of fulfilling humanity's material and spiritual needs than any other political and economic system, he also worried with uncanny prescience that a world in which liberal democracy was the only available option could be marked by boredom, repetition, and sterility — and that the intolerable character of such decadence could inspire anti-liberal movements that aimed to restart history once again. Douthat's book can be read as a melancholy sequel to Fukuyama's The End of History and the Last Man that confirms the author's darkest predictions but without endorsing (or seriously wrestling with) any of the concrete efforts going on around us to overcome our own malaise by breaking away from decadent liberalism — whether it's Donald Trump's MAGA presidency, the Catholic conservatism of Poland's Law and Justice Party, Marion Maréchal's National Rally in France, the National Conservatism spearheaded by Yoram Hazony, or Viktor Orban's anti-liberal and pro-natalist populism in Hungary. Given that Douthat is a conservative who longs for renewal, rebirth, and revitalization — for an end to the decadence he thinks plagues us — it's surprising that he has so little to say about these efforts in the book. [...] Douthat sees a lot, and far more than most of our less profoundly discontented commentators. That makes him an excellent pundit — maybe the best of our moment. But in his new book he also avoids a forthright confrontation with the political correlates of his own moral, aesthetic, intellectual, and spiritual dissatisfactions. In its place we find idle speculations about alternative realities. Which may mean that, for all its strengths, Douthat's book about decadence is more than a little decadent itself. That is to say that Douthat is himself trapped in the same exhausted cultural forms. Who among us isn't? * * * * * See Douthat's recent NYTimes oped (Feb, 7, 2020): The Age of Decadence. Here's now it opens: Everyone knows that we live in a time of constant acceleration, of vertiginous change, of transformation or looming disaster everywhere you look. Partisans are girding for civil war, robots are coming for our jobs, and the news feels like a multicar pileup every time you fire up Twitter. Our pessimists see crises everywhere; our optimists insist that we’re just anxious because the world is changing faster than our primitive ape-brains can process. But what if the feeling of acceleration is an illusion, conjured by our expectations of perpetual progress and exaggerated by the distorting filter of the internet? What if we — or at least we in the developed world, in America and Europe and the Pacific Rim — really inhabit an era in which repetition is more the norm than invention; in which stalemate rather than revolution stamps our politics; in which sclerosis afflicts public institutions and private life alike; in which new developments in science, new exploratory projects, consistently underdeliver? What if the meltdown at the Iowa caucuses, an antique system undone by pseudo-innovation and incompetence, was much more emblematic of our age than any great catastrophe or breakthrough? The truth of the first decades of the 21st century, a truth that helped give us the Trump presidency but will still be an important truth when he is gone, is that we probably aren’t entering a 1930-style crisis for Western liberalism or hurtling forward toward transhumanism or extinction. Instead, we are aging, comfortable and stuck, cut off from the past and no longer optimistic about the future, spurning both memory and ambition while we await some saving innovation or revelation, growing old unhappily together in the light of tiny screens. The farther you get from that iPhone glow, the clearer it becomes: Our civilization has entered into decadence. Wednesday, February 12, 2020 Pandemics and cooperation between nation-states Thomas Bollyky and Samantha Kiernan, No Nation Can Fight Coronavirus on Its Own, Lawfare, February 12, 2020: "Infectious diseases were the first global problem that nation-states realized they could not solve without international cooperation." This came about in the mid-19th century: For most of human history, plagues, parasites and pests were a domestic affair. Quarantine was the principal means by which nations contained the microbes that were brought by invading armies and the passengers, both human and vermin, on trading ships and caravans. Those isolation measures proved ineffective, however, against the six pandemics of cholera that swept the United States, the Middle East, Russia and Europe in the 19th century. A terrifying disease that struck seemingly healthy people, cholera killed tens of thousands in the cities of Europe and the United States—and, very likely, many more in India, where the pandemics originated. The economic costs of uncoordinated quarantines hurt nations and merchants alike. In 1851, European states gathered for the first International Sanitary Conference to discuss cooperation on cholera, plague and yellow fever. That convention, and those that followed, led to the first treaties on international infectious disease control and—in 1902—the International Sanitary Bureau, which later became the Pan American Health Organization. These international initiatives were the early models for later agreements and agencies on other transnational concerns, such as pollution, the opium trade and unsafe labor practices. Microbes have continued to inspire episodes of cooperation among even bitter rivals. The WHO, the United Nation’s first specialized agency, was created in 1946 in response to the horrors of World War II. Its early days were devoted to international campaigns against the great scourges of that era, such as malaria, smallpox and tuberculosis. At the height of the Cold War, the smallpox immunization campaign motivated the United States and the Soviet Union to join forces in an effort that succeeded in eradicating the disease in 1980. In El Salvador, an international vaccination campaign against pediatric infections led to a pause in the country’s 14-year civil war for the sole purpose of immunizing children. And the current coronavirus epidemic? There is much we do not know yet about how easily the virus spreads or its severity. But there is reason to think that the scale of this coronavirus outbreak and the likelihood of epidemics of the virus occurring outside China may inspire more cooperation than even the five previous occasions that the WHO designated as international public health emergencies: the H1N1 influenza pandemic (2009), the re-emergence of polio in several nations (2014), the Ebola outbreak in West Africa (2014), the Zika virus outbreak (2016) and the Ebola virus outbreak in the Democratic Republic of Congo (2019). In a little over one month, the coronavirus has more than five times the number of laboratory-confirmed cases (43,114 as of Feb. 11) than the outbreak of SARS did in four months (8,096). The novel coronavirus has already spread to at least 26 countries, far more than the current outbreak of the Ebola virus in the Democratic Republic of Congo, its predecessor in West Africa in 2013-2015, or during the resurgence of polio in Afghanistan, Nigeria and Pakistan in 2014. The mortality rate for known cases of the novel coronavirus has been about 2-3 percent, deadlier than the Zika virus or the 2009 H1N1 swine flu. [...] Perhaps a pandemic of novel coronavirus, if it occurs, would be a sufficiently frightening antagonist to force international cooperation, even at a moment that otherwise has proved inhospitable to global governance. If so, this novel coronavirus will do what climate change, tariff threats and the prospect of nuclear proliferation on the Korean peninsula could not: force nations to work together. Bird in red and black (flooded by light) Rodney Brooks on AI and robotics As you may know, Rodney Brooks is a pioneering robotics researcher and entrepreneur (his company markets the Roomba) who once headed the AI lab at MIT. He has a blog where he's been commenting on AI. Here's a post where he has links to eight posts on the future of AI and robotics that he posted between August of 2017 and July of 2018, Future of Robotics and Artificial Intelligence. This post is from July, 2018, where he gives a capsule overview of the history of AI, Steps Toward Super Intelligence I, How We Got Here. He lists for main approaches, with approximate start dates: 1. Symbolic (1956) 2. Neural networks (1954, 1960, 1969, 1986, 2006, …) 3. Traditional robotics (1968) 4. Behavior-based robotics (1985) Neural networks, as you see, has a spotty history. The basic idea is relatively old (as work in AI goes). 1986 marks the advent of back-propagation along with multilayered networks while the 2006 dates marks some new techniques ("deep learning"), much more computing power, and huge sets of training data. I found this discussion particularly useful. He shows us the following photo: A Google program was able to generate this caption, “A group of young people playing a game of Frisbee”, and goes on to note: I think this is when people really started to take notice of Deep Learning. It seemed miraculous, even to AI researchers, and perhaps especially to researchers in symbolic AI, that a program could do this well. But I also think that people confused performance with competence (referring again to my seven deadly sins post). If a person had this level of performance, and could say this about that photo, then one would naturally expect that the person had enough competence in understanding the world, that they could probably answer each of the following questions: • what is the shape of a Frisbee? • roughly how far can a person throw a Frisbee? • can a person eat a Frisbee? • roughly how many people play Frisbee at once? • can a 3 month old person play Frisbee? • is today’s weather suitable for playing Frisbee? But the Deep Learning neural network that produced the caption above can not answer these questions. It certainly has no idea what a question is, and can only output words, not take them in, but it doesn’t even have any of the knowledge that would be needed to answer these questions buried anywhere inside what it has learned. Brooks' own work has been in the fourth approach, behavior-based robotics, where he is a pioneer. He remarks: ...I started to reflect on how well insects were able to navigate in the real world, and how they were doing so with very few neurons (certainly less that the number of artificial neurons in modern Deep Learning networks). In thinking about how this could be I realized that the evolutionary path that had lead to simple creatures probably had not started out by building a symbolic or three dimensional modeling system for the world. Rather it must have begun by very simple connections between perceptions and actions. In the behavior-based approach that this thinking has lead to, there are many parallel behaviors running all at once, trying to make sense of little slices of perception, and using them to drive simple actions in the world. Often behaviors propose conflicting commands for the robot’s actuators and there has to be a some sort of conflict resolution. But not wanting to get stuck going back to the need for a full model of the world, the conflict resolution mechanism is necessarily heuristic in nature. Just as one might guess, the sort of thing that evolution would produce. Behavior-based systems work because the demands of physics on a body embedded in the world force the ultimate conflict resolution between behaviors, and the interactions. Furthermore by being embedded in a physical world, as a system moves about it detects new physical constraints, or constraints from other agents in the world. Finally, Brooks has created a predictions scorecard in three areas, self-driving cars, AI and machine learning, and space industry. He first posted it on January 1, 2018 and has updated it on Jan. 1 of 2019 and again, Jan. 1 2020. The list contains (I would guess) over 50 specific items distributed over those categories with specific dates attached. It makes for very interesting reading. Spain is now the world's healthiest country Music versus algorithms Alexis Petridis reviews Ted Gioia's current book, Music: A Subversive History: In terms of scope, well, put it this way: it starts out talking about a bear’s thighbone that Neanderthal hunters apparently turned into a primitive flute somewhere between 43,000 and 82,000 years ago and ends up, 450 pages later, discussing K-pop and EDM. His central theory: music is a kind of magical, ungovernable force that connects us to ancient shamanistic rituals, it’s primarily fuelled by sex and violence – anyone horrified by the lyrics of drill or death metal should consider that the first instruments were made from body parts and would once have literally dripped with blood – and all attempts to reduce it to mathematical formulae or “quasi-science”, while useful, go against its intrinsic nature. He’s really not keen on Pythagoras, whose mathematical theories about tuning underpin “music as it is taught in every university and conservatory in the world today”. I didn’t agree with everything Gioia had to say, but something about that central theory stuck with me. For one thing, there is something magical and ungovernable about music: that weird tingling sensation you get when you hear something you love - a friend of mine calls it the Holy Shiver - is involuntary. It just happens. And we live in an era when music has never been more governed by mathematics. Algorithms are supposed to be able to predict everything, from what you want to hear next to whether or not a song’s going to be a hit: the digital strategist who developed the software behind the AI record label that’s just launched was also “involved in the development and marketing of stars such as Avicii, Logic, Mike Posner and Swedish House Mafia”. For a series of anecdotes illustrating music's power, see my working paper, Emotion & Magic in Musical Performance. Tuesday, February 11, 2020 This is your brain on art Monday, February 10, 2020 Fallen angel When I saw this at night I wondered what it was, perhaps some strange art project? When I came back the next day I realized that it was simply a decorative angel that had fallen down. Howard Rheingold on democracy and online media Howard Rheingold, Democracy is losing the online arms race, February 4, 2020. Opening paragraphs: Democracy is threatened by an arms race that the forces of deception are winning. While microtargeted computational propaganda, organized troll brigades, coordinated networks of bots, malware, scams, epidemic misinformation, miscreant communities such as 4chan and 8chan, and professionally crafted rivers of disinformation continue to evolve, infest, and pollute the public sphere, the potential educational antidotes – widespread training in critical thinking, media literacies, and crap detection – are moving at a leisurely pace, if at all. When I started writing about the potential for computer-mediated communication, decades before online communication became widely known as “social media,” my inquiries about where the largely benign online culture of the 1980s might go terribly wrong led me to the concept of the “public sphere,” most notably explicated by the German political philosopher Jurgen Habermas. “What is the most important critical uncertainty about mass adoption of computer mediated communication?” was the question I asked myself, and I decided that the most serious outcome of this emerging medium would have to do with whether citizens gain or lose liberty with the rising adoption of digital media and networks. It didn’t take a lot of seeking to find Habermas’ work when I started pursuing this question. Although Habermas’ prose is dense, the notion is simple: Democracies are not just about voting for leaders and policy-makers; democratic societies can only take root in populations that are educated enough and free enough to communicate about issues of concern and to form public opinion that influences policy. Five skillsets for online life: When I set out to write Net Smart: How to Thrive Online, I decided that five essential skillsets/bodies of lore/skills were necessary to thrive online – and by way of individual thriving, to enhance the value of the commons: literacies of attention, crap detection, participation, collaboration, and network awareness: · Attention because it is the foundation of thought and communication, and even a decade ago it was clear that computer and smartphone screens were capturing more and more of our attention. · Crap detection because we live in an age where it is possible to ask any question, any time, anywhere, and get a million answers in a couple seconds – but where it is now up to the consumer of information to determine whether the information is authentic or phony. · Participation because the birth and the health of the Web did not come about because and should not depend upon the decisions of five digital monopolies, but was built by millions of people who put their cultural creations and their inventions online, nurtured their own communities, invented search engines in their dorm rooms and the Web itself in a physics lab. · Collaboration because of the immense power of social production, virtual communities, collective intelligence, smart mobs afforded by access to tools and knowledge of how to use them. · Network awareness because we live in an age of social, political, and technological networks that affect our lives, whether we understand them or not. In an ideal world, the social and political malignancies of today’s online culture could be radically reduced, although not eliminated, if a significant enough portion of the online population was fluent or at least basically conversant in these literacies – in particular, while it seems impossible to stem the rising tide of crap at its sources, its impact could be significantly reduced if most of the online population was educated in crap detection. On attention: I confronted issues of attention in the classroom during my decade of teaching at UC Berkeley and Stanford – as does any instructor who faces a classroom of students who are looking at their laptops and phones in class. Because I was teaching social media issues and social media literacies, it seemed to me to be escaping the issue by simply banning screentime in class – so we made our attention one of our regular activities. I asked my co-teaching teams (I asked teams of three learners to take responsibility for driving conversation during one-third of our class time) to make up “attention probes” that tested our beliefs and behavior. When I researched attentional discipline for Net Smart, I found an abundance of evidence from millennia-old contemplative traditions to contemporary neuroscience for the plasticity of attention. Simply paying attention to one’s attention – the methodology at the root of mindfulness meditation – can be an important first step to control. It doesn’t seem that attention engineers, despite their wild success, have the overwhelming advantage in the arms race with attention education that surveillance capitalists and computational propagandists deploy with their big data, bots, and troll armies. The lopsided arms race is what leads me to conclude that education in crap detection, attention control, media literacy, and critical thinking are important, but are not sufficient. Regulation of the companies who wield these new and potentially destructive powers will also be necessary. There's more at the link. Saturday, February 8, 2020 Wild Child The hill was covered with strange grassy mounds about the size of molehills. The adults had no idea what they were — which was very exciting to me, realizing that there were things in the world that not even the adults understood. So I filled in the blanks for myself and decided they must be burial mounds for fairies. This was the magical landscape that inspired my book “The Wizards of Once.” For the wildwood in that book, I took particular inspiration from the ancient wood of Kingley Vale in Sussex. Its trees have gnarled, expressive faces, and roots that embed into the earth with an almost visceral power. The more you learn about trees, the more magical you realize they are. Did you know, for example, that trees can communicate with each other through their roots, even when they are many miles apart? Trees grow throughout children’s books. From “Peter Pan” to “A Monster Calls,” “The Lord of the Rings” to “Harry Potter,” trees are refuges, prisons and symbols of nature’s potency. They can be a friendly home, like the Hundred Acre Wood in “Winnie-the-Pooh,” or give a sense of menace, like the snowy forest in “The Lion, the Witch and the Wardrobe.” They can also be symbolic, like the cement-filled dying tree in “To Kill a Mockingbird.” The writers I loved when I was a child were similarly inspired by magical landscapes and nature: Ursula K. Le Guin, J.R.R. Tolkien, L. Frank Baum, Diana Wynne Jones, Lloyd Alexander, Robert Louis Stevenson, T.H. White — and so many others. Today, children have much less unsupervised access to the countryside. I worry that they may never know the magic of the wilderness, the power of trees and the thrilling excitement of exploring nature without an adult hovering behind them. And so I write books for children who will never know what the freedom of my childhood was like. Friday, February 7, 2020 “Undecidability, Uncomputability and the Unity of Physics. Part 1.” That's the title of a post by Tim Palmer at Backreaction. Here's the opening: Our three great theories of 20th Century physics – general relativity theory, quantum theory and chaos theory – seem incompatible with each other. The difficulty combining general relativity and quantum theory to a common theory of “quantum gravity” is legendary; some of our greatest minds have despaired – and still despair – over it. Superficially, the links between quantum theory and chaos appear to be a little stronger, since both are characterised by unpredictability (in measurement and prediction outcomes respectively). However, the Schrödinger equation is linear and the dynamical equations of chaos are nonlinear. Moreover, in the common interpretation of Bell’s inequality, a chaotic model of quantum physics, since it is deterministic, would be incompatible with Einstein’s notion of relativistic causality. Finally, although the dynamics of general relativity and chaos theory are both nonlinear and deterministic, it is difficult to even make sense of chaos in the space-time of general relativity. This is because the usual definition of chaos is based on the notion that nearby initial states can diverge exponentially in time. However, speaking of an exponential divergence in time depends on a choice of time-coordinate. If we logarithmically rescale the time coordinate, the defining feature of chaos disappears. Trouble is, in general relativity, the underlying physics must not depend on the space-time coordinates. So, do we simply have to accept that, “What God hath put asunder, let no man join together”? I don’t think so. A few weeks ago, the Foundational Questions Institute put out a call for essays on the topic of “Undecidability, Uncomputability and Unpredictability”. I have submitted an essay in which I argue that undecidability and uncomputability may provide a new framework for unifying these theories of 20th Century physics. I want to summarize my argument in this and a follow-on guest post. What interests me is simply the conjunction mentioned in the opening line, general relativity theory, quantum theory and chaos theory, which featured in a post in January, The third 20th-century revolution in physics [non-linear dynamics]. Here are two (of three) observations I made at the end of that post: • It seems to me that quantum mechanics and relativity are focused on explanatory principles whereas non-linear dynamics tends more toward description, description of a wide variety of phenomena. Moreover quantum mechanics and relativity are most strongly operative in different domains, the microscopic and macroscopic respectively. • Computation: many cases there are various computational paths from the initial state to the completion of the computation. As a simple example, when adding a group of numbers, the order of the numbers doesn't matter; the sum will be the same in each case. In the case of non-linear systems successive states in the computation 'mirror' successive states in the system being modeled so the temporal evolution of the computation is intrinsic to the model rather than extrinsic. Does that second observation imply that (something like) computation is inherent in the physical nature of the universe and is not merely an intellectual operation carried out by various artificial means? Life in the ocean depths, minerals too Wil S. Hylton, History's Largest Mining Operation Is About to Begin, The Atlantic, January/February 2020. As the title indicates, the article is primarily about mining the ocean depths, but this passage struck me: Until recently, marine biologists paid little attention to the deep sea. They believed its craggy knolls and bluffs were essentially barren. The traditional model of life on Earth relies on photosynthesis: plants on land and in shallow water harness sunlight to grow biomass, which is devoured by creatures small and large, up the food chain to Sunday dinner. By this account, every animal on the planet would depend on plants to capture solar energy. Since plants disappear a few hundred feet below sea level, and everything goes dark a little farther down, there was no reason to expect a thriving ecosystem in the deep. Maybe a light snow of organic debris would trickle from the surface, but it would be enough to sustain only a few wayward aquatic drifters. That theory capsized in 1977, when a pair of oceanographers began poking around the Pacific in a submersible vehicle. While exploring a range of underwater mountains near the Galápagos Islands, they spotted a hydrothermal vent about 8,000 feet deep. No one had ever seen an underwater hot spring before, though geologists suspected they might exist. As the oceanographers drew close to the vent, they made an even more startling discovery: A large congregation of animals was camped around the vent opening. These were not the feeble scavengers that one expected so far down. They were giant clams, purple octopuses, white crabs, and 10-foot tube worms, whose food chain began not with plants but with organic chemicals floating in the warm vent water. For biologists, this was more than curious. It shook the foundation of their field. If a complex ecosystem could emerge in a landscape devoid of plants, evolution must be more than a heliological affair. Life could appear in perfect darkness, in blistering heat and a broth of noxious compounds—an environment that would extinguish every known creature on Earth. “That was the discovery event,” an evolutionary biologist named Timothy Shank told me. “It changed our view about the boundaries of life. Now we know that the methane lakes on one of Jupiter’s moons are probably laden with species, and there is no doubt life on other planetary bodies.” As for mining: Deepwater plains are also home to the polymetallic nodules that explorers first discovered a century and a half ago. Mineral companies believe that nodules will be easier to mine than other seabed deposits. To remove the metal from a hydrothermal vent or an underwater mountain, they will have to shatter rock in a manner similar to land-based extraction. Nodules are isolated chunks of rocks on the seabed that typically range from the size of a golf ball to that of a grapefruit, so they can be lifted from the sediment with relative ease. Nodules also contain a distinct combination of minerals. While vents and ridges are flecked with precious metal, such as silver and gold, the primary metals in nodules are copper, manganese, nickel, and cobalt—crucial materials in modern batteries. As iPhones and laptops and electric vehicles spike demand for those metals, many people believe that nodules are the best way to migrate from fossil fuels to battery power. The ISA has issued more mining licenses for nodules than for any other seabed deposit. Most of these licenses authorize contractors to exploit a single deepwater plain. Known as the Clarion-Clipperton Zone, or CCZ, it extends across 1.7 million square miles between Hawaii and Mexico—wider than the continental United States. When the Mining Code is approved, more than a dozen companies will accelerate their explorations in the CCZ to industrial-scale extraction. Their ships and robots will use vacuum hoses to suck nodules and sediment from the seafloor, extracting the metal and dumping the rest into the water. How many ecosystems will be covered by that sediment is impossible to predict. Ocean currents fluctuate regularly in speed and direction, so identical plumes of slurry will travel different distances, in different directions, on different days. The impact of a sediment plume also depends on how it is released. Slurry that is dumped near the surface will drift farther than slurry pumped back to the bottom. The circulating draft of the Mining Code does not specify a depth of discharge. The ISA has adopted an estimate that sediment dumped near the surface will travel no more than 62 miles from the point of release, but many experts believe the slurry could travel farther. A recent survey of academic research compiled by Greenpeace concluded that mining waste “could travel hundreds or even thousands of kilometers.
b00c414557c5233a	Random Walking – CUPS Press Office Article July 16, 2021 CUPS papers distilled Random Walking – When having no clue where to go still makes you reach your destination In empirical sciences, theories can be sorted into three categories with ascending gain of knowledge: empirical, semi-empirical and ab initio. Their difference can best be explained by an example: In astronomy the movement of all planets was known since ancient times. By pure observation astronomers could predict where in the sky a certain planet would be at a given time. They had the knowledge of how they moved but actually no clue why they did so. Their knowledge was purely empirical, meaning purely based on observation. Kepler became the first to develop a model by postulating that the sun is the center of the planetary system and the planet’s movement is controlled by her. Since Kepler could not explain why the planets would be moved by the sun, he had to introduce free parameters which he varied until the prediction of the model matched the observations. This is a so-called semi-empirical model. But it was not until Newton who with his theory of gravity could predict the planets movements without any free parameters or assumptions but purely by an ab initio, latin for “from the beginning”, theory based on fundamental principles of nature, namely gravity. As scientists are quite curious creatures, they always want to know not only how things work but also why they work this way. Therefore, developing ab initio theories is the holy grail in every discipline. Luckily, in quantum mechanics the process of finding ab initio theories has been strongly formalized. This means that if we want to know the property of a system, for example its velocity, we just have to kindly ask the system for it. This is done by applying a tool, the so-called operator, belonging to the property of interest on the function describing the system’s current state. The result of this operation is the property we are interested in. Think of a pot of water. We want to know its temperature? We use a tool to measure temperature, a thermometer. We want to know its weight? We use the tool to measure its weight, a scale. An operator is a mathematical tool which transforms mathematical functions and provides us with the functions property connected to the operator. Think of the integral sign which is an operator too. The integral is just the operator of the area under a function and the x-axis. The problem is, how do we know the above mentioned function describing the system’s state? Fortunately, smart people developed a generic way to answer this problem too: We have to solve the so-called Schrödinger equation. Writing down this equation is comparably easy, we just need to know the potentials of all forces acting on the system and we can solve it. Well, if we can solve it. It can be shown that analytical solutions, that means solutions which can be expressed by a closed mathematical expression, only exist for very simple systems, if at all. For everything else numerical approaches have to be applied. While they still converge towards the exact solution, this takes a lot of computational time. The higher the complexity of the system, the more time it takes. So for complex systems even the exact numerical approach quickly becomes impractical to use. One way out of this misery is simplification. With clever assumptions about the system, based on its observation one can drastically reduce the complexity of the calculations. With this approach, we are able to, within reasonable time, find solutions for the problem which are not exact, but exact within a certain error range. Another way to find a solution for these complex problems is getting help from one of nature’s most powerful and mysterious principles: chance. The problem of the numerical exact solving approach is that it has to walk through an immensely huge multidimensional space, spanned by the combinations of all possible interactions between all involved particles. Think billions of trillions times billions of trillions. By using a technique called Random Walking the time to explore this space can be significantly reduced. Again, let’s take an example: Imagine we want to know how many trees grow in a forest. The exact solution would be dividing the forest into a grid of e.g., 1 square foot, and counting how many trees are in each square. A random walk would start in the forest center. Then we randomly choose a direction and a distance to walk before counting the trees in the resulting square. If we repeat this just long enough we eventually will have visited every square, therefore knowing the exact number, meaning the random walk converges towards the exact result. By having many people starting together and doing individual random walks stopping when their results deviation is below a certain threshold a quite accurate approximation can be obtained in little time. Columbia postdoc Benjamin Rudshteyn and his colleagues developed a very efficient algorithm based on this method specifically tailored for calculating molecules containing transition metals such as copper, niobium or gold. While being ubiquitous in biology and chemistry, and playing a central role in important fields such as the development of new drugs or high-temperature superconductors, these molecules are difficult to treat both experimentally and theoretically due to their complex electronic structures. They tested their method by calculating for a collection of 34 tetrahedral, square planar, and octahedral 3D metal-containing complexes the energy needed to dissociate a part of the molecule from the rest. For this, precise knowledge of the energy states of both the initial molecule and the products is needed. By comparing their results with precise experimental data and results of conventional theoretical methods they could show that their method results in at least two times increased accuracy as well as increased robustness, meaning little variation in the statistical uncertainty between the different complexes. Figure 1: Illustration of the three types of geometry of the datasets molecules: Octahedral (a), square planar (b) and tetrahedral (c), with the transition metal being the central sphere. In (d) the dissociation of part of the molecule is shown While still requiring the computational power of modern supercomputers, their findings push the boundaries of the size of transition metal containing molecules for which reliable theoretical data can be produced. These results can then be used as an input to train methods using approximations to further reduce the computational time needed for the calculations. Dr. Benjamin Rudshteyn is currently a postdoc in the Friesner Group of Theoretical Chemistry, lead by Prof. Dr. Richard A. Friesner, in the Department of Biological Sciences at Columbia University.
2dbf816d280f7388	It is susceptible to attack by many insect-pests, and more severe It is susceptible to THZ1 price attack by many insect-pests, and more severely affected by the fruit and shoot borer (FSB). These insects effectively damage (60–70%) the crop even following the average 4.6 kg of insecticides and pesticides per hectare [2]. Therefore, to control the indiscriminate use of insecticides, the transgenic approach is being opted that is eco-friendly and shows promise to control the FSB infecting brinjal. The use of insecticidal proteins from the bacterium Bacillus thuringiensis (Bt) in the improvement of crop productivity via transgenic crop (Bt crop) MGCD0103 manufacturer is being promoted in most cases. However, the potential risk associated with the impact of transgenic crops on non-target microorganisms and flora and fauna in the environment, is still a matter of concern. Bt crops have the potential to alter the microbial community dynamics in the soil agro-ecosystem owing to the release of toxic Cry proteins into the soil via root exudates [3], and through decomposition of the crop residues [4]. The available reports, however, are not consistent regarding the nature of interaction of transgenic crops with the native microbial community. Icoz and Stotzky [5] presented a comprehensive analysis of the fate and effect of Bt crops in soil ecosystem and emphasized LY2109761 research buy for the risk assessment studies of transgenic crops. Phylogenetically, actinomycetes are the member of taxa under high G + C sub-division of the Gram positive bacteria [6]. Apart from bacteria and fungi, actinomycetes are an important microbial group known to be actively involved in degradation of complex organic materials in soils and contribute to the biogeochemical cycle [7]. The presence of Micromonospora in soils contributes to the production of secondary metabolite (antibiotics) like anthraquinones [8], and Arthrobacter globiformis degrades substituted phenyl urea in soil [9]. Nakamurella group are known for the production of catalase and storing polysaccharides [10]. Thermomonospora, common to decaying organic Branched chain aminotransferase matter, are known for plant cell degradation [11]. Frankia is widely known for N2 fixation [12], Sphaerisporangium album in starch hydrolysis and nitrate reduction in soils [13], Agromyces sp. degrades organophosphate compounds via phosphonoacetate metabolism through catabolite repression by glucose [14]. Janibacter in rhizospheric soils, are widely known to degrade 1, 1-dichloro-2, 2- bis (4-chlorophenyl) ethylene (DDE) [15], while Streptomyces for the production of chitinase as well as antibiotics [16]. These studies suggest that most of the representative genera of actinomycetes in the soil, contribute to maintenance of the soil fertility. Most studies on transgenic crops have been carried out on cotton, corn, tomato, papaya, rice, etc., with emphasis on protozoal, bacterial and fungal communities [5]. VanSaun2, Lynn M Matrisian2, D Lee Gorden2 1 Department of Surg VanSaun2, Lynn M. Matrisian2, D. Lee Gorden2 1 Department of Surgery, St. Mary’s Hospital, College of Medicine, The Catholic University of Korea, Seoul, Korea Republic, 2 Department of Cancer Biology, Vanderbilt University, Nashville, Tennessee, USA Purpose: Pro-inflammatory processes of the early postoperative states may induce peritoneal metastases in patients with advanced diseases. To identify that wound 4-Hydroxytamoxifen cost healing response after an abdominal incision leads to increased MMP-9 activity locally, therefore providing a favorable environment for peritoneal metastasis. Increased MMP9 in a post-operative injury setting increases the number and severity of peritoneal metastasis when compared to mice without wounds. Methods: Eighteen C57bl/6 J male mice were obtained at 8 weeks of age. Metastatic tumors were initiated using a peritoneal injection model with syngeneic MC38 murine colon cancer cells. Peritoneal EPZ5676 cost injections were performed into the intraperitoneum at right lower quadrant area via 25G syringe. A 1.5 cm upper midline incision was made in the abdominal wall to recapitulate the postoperative wound model. The abdominal wall was closed by a continuous 4-0 prolene suture with 5 stitches. Mice were sacrificed at various time points. And we observed the rate of the peritoneal metastasis from each group. Results: By making incision into the abdominal wall, we induced inflammation of the mouse and observed the incidence of the peritoneal metastasis was increased(Fig.1). Early stage of wound healing process increases pro-inflammatory cytokines and number of inflammatory cells in the peritoneum, and this leads to increase pro-MMP9 proteins. And the inflammatory process which initiated by the wound, in turn, increased the proliferation of the mesothelial cells and provoked expression of the inflammatory cells and increased parietal peritoneal metastasis. Conclusion: stage of wound healing process increases pro-inflammatory cytokines and number of inflammatory cells in the peritoneum, and this leads to increase pro-MMP9 proteins. So the increased pro-MMP9 proteins play a key role on the growth and progressions of cancer cells in peritoneal Cobimetinib concentration metastasis. Figure 1. Poster No. 87 Cytokine-Mediated Activation of Gr-1 + Inflammatory Cells and Macrophages in Squamous Cell Selleck YM155 Carcinoma towards a Tumor-Supporting Pro-Invasive and Pro-Angiogenic Phenotype Nina Linde 1 , Dennis Dauscher1, Margareta M. Mueller1 1 Group Tumor and Microenvironment, German Cancer Research Center, Heidelberg, Germany Inflammatory cells have been widely accepted to contribute to tumor formation and progression. In a HaCaT model for human squamous cell carcinoma (SCC) of the skin, we have observed that infiltration of inflammatory cells does not only promote tumorigenesis but is indispensable for persisten angiogenesis and the development of malignant tumors. Recent studies suggest that BRCA proteins are required for protec Recent studies suggest that BRCA proteins are required for protecting the genome from damage [12]. Mutations in BRCA genes have been established to predispose women to breast and ovarian cancer, the end point of BRCA protein dysfunction. Mutations in both genes are spread throughout the entire gene. More than 600 different mutations have been identified in BRCAl gene and 450 mutations in BRCA. The majorities of mutations, known to be disease-causing, results in a truncated protein due to frame shift, nonsense, or splice site alternations. Nonsense mutations occur when the nucleotide substitution produces a stop codon (TGA, TAA, or TAG) and translation of the protein is terminated at this point. Frame shift mutations occur when one or more nucleotides are either inserted or deleted, resulting in missing or non-functional protein. Splice site mutations cause abnormal inclusion or exclusion of DNA in the coding sequence, resulting in an abnormal protein. Other kind of mutations results from a single nucleotide substitution is missense mutations in which the substitution changes a single amino acid but does not affect the remainder of the protein translation [13, 14]. Studies of BRCAl mutation occurrence suggested that nearly half of the families at high risk for breast cancer carried BRCAl mutation [15]. However, other analysis suggest that the actual incidence of BRCAl in high risk families (>3 cases of breast and/or ovarian SC79 cost cancer) might be as low as 12.8% to 16% [4]. Substantial variation in the prevalence of BRCA1 mutations in high risk families in various countries has been observed which are more common than BRCA2 mutations [16, 17]. The main objectives of the present work were to identify germline mutations in BRCA1 (exons 2, 8, 13, 22) and BRCA2 (exon 9) genes for the early detection of presymptomatic mutation carriers in Egyptian healthy females who were first degree relatives of affected women from families with and without family history of breast cancer. Subjects and Methods Patients and families Sixty breast cancer patients (index patients), Selleckchem Fludarabine derived from 60 families, considered being at high risk, due to medicinal examination and they were grid 3 patients, were selected for molecular genetic testing of BRCA1 and BRCA2 genes. They were referred to the Clinical Oncology Unit in Medical Research learn more Institute, Alexandria University, for chemotherapy as part of their curative treatment after mastectomy. Selected index patients were preferred to be at early onset age at diagnosis, possessing a positive family history and bilateral breast cancer. The study also included one hundred and twenty healthy first degree female relatives of index patients either sisters and/or daughters for early detection of mutation carriers. The decision to undergo genetic testing was taken after the participants were informed about benefits and importance of genetic testing. CrossRef 31 Tang CG, Chen YH, Xu B, Ye XL, Wang ZG: Well-width d CrossRef 31. Tang CG, Chen YH, Xu B, Ye XL, Wang ZG: Well-width dependence of in-plane optical anisotropy in (001) GaAs/AlGaAs quantum wells induced by in-plane uniaxial strain and interface asymmetry . J Appl Phys 2009,105(10):103108.CrossRef 32. Tang CG, Chen YH, Ye XL, Wang ZG, Zhang WF: Strain-induced in-plane optical anisotropy in (001) GaAs/AlGaAs superlattice studied by reflectance difference spectroscopy . J Appl Phys 2006,100(11):113122.CrossRef 33. Krebs O, Voisin P: Giant optical anisotropy of semiconductor heterostructures with no common atom and the quantum-confined Pockels effect . Phys Rev Lett 1996, 77:1829.CrossRef 34. Yu J, Chen Y, Cheng S, Lai Y: Spectra of circular and linear photogalvanic effect at BIX 1294 in vivo inter-band GDC-0449 mw excitation in In 0.15 Ga 0.85 As/Al 0.3 Ga 0.7 As multiple quantum wells . Phys E: Low-dimensional Systems and Nanostructures 2013,49(0):92–96. 35. Takagi T: Refractive index of Ga 1-x In x As prepared by vapor-phase epitaxy . Japanese J Appl Phys 1978, 17:1813–1817.CrossRef 36. Park YS, Reynolds DSC: Exciton structure in photoconductivity of CdS, CdSe, and CdS: Se single crystals . Phys Rev 1963, 132:2450–2457.CrossRef 37. Ohno Y, Terauchi R, Adachi T, Matsukura F, Ohno H: Spin relaxation CX-5461 nmr in GaAs(110) quantum wells . Phys Rev Lett 83:4196–4199. 38. Damen TC, Via L, Cunningham JE, Shah J, Sham LJ: Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells . Phys Rev Lett 1991, 67:3432–3435.CrossRef 39. Roussignol P, Rolland P, Ferreira R, Delalande C, Bastard G, Vinattieri A, Martinez-Pastor Protein kinase N1 J, Carraresi L, Colocci M, Palmier JF, Etienne B: Hole polarization and slow hole-spin relaxation in an n-doped quantum-well structure . Phys Rev B 1992, 46:7292–7295.CrossRef 40. Mattana R, George J-M, Jaffrès H, Nguyen Van Dau F, Fert A, Lépine B, Guivarc’h A, Jézéquel G: Electrical detection of spin accumulation in a p-type GaAs quantum well . Phys Rev Lett 2003, 90:166601.CrossRef 41. Bulaev DV, Loss D: Spin relaxation and decoherence of holes in quantum dots . Phys Rev Lett 2005, 95:076805.CrossRef 42. Gvozdic DM, Ekenberg U: Superefficient electric-field-induced spin-orbit splitting in strained p-type quantum wells . Europhys Lett 2006, 73:927.CrossRef 43. Chao CY, Chuang SL: Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells . Physical Review B 1992,46(7):4110.CrossRef 44. Foreman BA: Analytical envelope-function theory of interface band mixing . Phys Rev Lett 1998,81(2):425.CrossRef 45. Muraki K, Fukatsu S, Shiraki Y, Ito R: Surface segregation of in atoms during molecular-beam epitaxy and its influence on the energy-levels in InGaAs/GaAs quantum-wells . Appl Phys Lett 1992,61(5):557–559.CrossRef 46. Chen YH, Wang ZG, Yang ZY: A new interface anisotropic potential of zinc-blende semiconductor interface induced by lattice mismatch . Chinese Phys Lett 1999,16(1):56–58.CrossRef 47. The inset in (e) shows the corresponding selected area diffractio The inset in (e) shows the corresponding selected area diffraction pattern with a zone axis of [1–30]. The second processing parameter we investigated was the vapor pressure. Figure 3a,b,c show our SEM studies for 100, 300, and 500 Torr, respectively. It turns out that CoSi Selleckchem U0126 nanowires grew particularly well at the reaction pressure of 500 Torr. In this experiment, the higher the vapor pressure, the longer the nanowires grown. Additionally, with the increasing vapor pressure, the number of nanoparticles reduces, Tariquidar but the size of the nanoparticles increases. Figure 3 SEM images of CoSi nanowires. At vapor pressures AZD8931 research buy of (a) 100, (b) 300, and (c) 500 Torr, respectively. For the synthesis of cobalt silicide nanowires, the third and final processing parameter we studied was the gas flow rate. We conducted experiments at the gas flow rate of 200, 250, 300, and 350 sccm, obtaining the corresponding results shown in Figure 4a,b,c,d, respectively. It can be found in the SEM images of Figure 4 that at 850°C ~ 880°C, the number of CoSi nanowires reduced with the increasing gas flow rate; thus, more CoSi nanowires appeared as the gas flow rate was lower. Figure 4 SEM images of CoSi nanowires. At gas flow rates of (a) 200, (b) 250, (c) 300, and (d) 350 sccm, respectively. The growth mechanism of the cobalt silicide nanowires in this work is of interest. Figure 5 is the schematic illustration of the growth mechanism, showing the proposed growth steps of CoSi nanowires with a SiOx outer layer. When the system temperature did not reach the reaction temperature, CoCl2 reacted with H2 (g) to form Co following step (1) of Figure 5: Figure 5 The schematic illustration of the growth mechanism. (1) CoCl2(g) + H2(g) → Co(s) + 2HCl(g), (2) 2CoCl2(g) + 3Si(s) → 2CoSi(s) + SiCl4(g), (3) SiCl4(g) + 2H2(g) → Si(g) + 4HCl(g), (4) 2Si(g) + O2(g) → 2SiO(g), and (5) Co(solid or vapor) + 2SiO(g) → CoSi(s) + SiO2(s). The Co atoms agglomerated to PTK6 form Co nanoparticles on the silicon substrate. When the system temperature reached the reaction temperatures, 850°C ~ 880°C, CoCl2 reacted with the silicon substrate to form a CoSi thin film and SiCl4 based on step (2) of Figure 5: The SiCl4 product then reacted with H2(g) to form Si(g) following step (3) of Figure 5: The Si here reacted with either residual oxygen or the exposed SiO2 surface to form SiO vapor from step (4) of Figure 5[30]: The SiO vapor reacted with Co nanoparticles via vapor-liquid–solid mechanism. We will comment not only on the strengths but also on the technic We will comment not only on the strengths but also on the technical pitfalls and the current limitations of the technique, discussing the performance of DFT and the foreseeable achievements in the near future. Theoretical background To appreciate the special place of DFT in the modern arsenal of quantum chemical methods, it is useful first to have a look into VX-689 nmr the more traditional wavefunction-based approaches. These attempt to provide approximate solutions to the Schrödinger equation, the fundamental equation of quantum mechanics that describes any given chemical system. The most fundamental of these approaches originates from the pioneering work of Hartree and Fock in the 1920s (Szabo and Ostlund 1989). The HF method assumes that the exact N-body wavefunction of the C59 wnt chemical structure system BIBF 1120 solubility dmso can be approximated by a single Slater determinant of N spin-orbitals. By invoking the variational principle, one can derive a set of N-coupled equations for the N spin orbitals. Solution of these equations yields the Hartree–Fock wavefunction and energy of the system, which are upper-bound approximations of the exact ones. The main shortcoming of the HF method is that it treats electrons as if they were moving independently of each other; in other words, it neglects electron correlation. For this reason, the efficiency and simplicity of the HF method are offset by poor performance for systems of relevance to bioinorganic chemistry. Thus, HF is now principally used merely as a starting acetylcholine point for more elaborate “post-HF” ab initio quantum chemical approaches, such as coupled cluster or configuration interaction methods, which provide different ways of recovering the correlation missing from HF and approximating the exact wavefunction. Unfortunately, post-HF methods usually present difficulties in their application to bioinorganic and biological systems, and their cost is currently still prohibitive for molecules containing more than about 20 atoms. Density functional theory attempts to address both the inaccuracy of HF and the high computational demands of post-HF methods by replacing the many-body electronic wavefunction with the electronic density as the basic quantity (Koch and Holthausen 2000; Parr and Yang 1989). Whereas the wavefunction of an N electron system is dependent on 3N variables (three spatial variables for each of the N electrons), the density is a function of only three variables and is a simpler quantity to deal with both conceptually and practically, while electron correlation is included in an indirect way from the outset. Modern DFT rests on two theorems by Hohenberg and Kohn (1964). The first theorem states that the ground-state electron density uniquely determines the electronic wavefunction and hence all ground-state properties of an electronic system. The extracted ΦB values of these samples are presented in the Fig The extracted ΦB values of these samples are presented in the Figure 4. The highest ΦB value attained by the ABT-737 nmr sample annealed in O2 ambient (3.72 eV) was higher than that of metal-organic selleck inhibitor decomposed CeO2 (1.13 eV) spin-coated on n-type GaN substrate [20]. No ΦB value has been extracted for the sample annealed in N2 ambient due to the low E B and high J of this sample, wherein the gate oxide breaks down prior to the FN tunneling mechanism. Figure 7 Experimental data fitted well with FN tunneling model. Experimental data (symbol) of samples annealed in O2, Ar (HJQ and KYC, unpublished work), and FG ambient fitted well with FN tunneling model (line). Table 1 compares the computed ΔE c values from the XPS characterization with the ΦB value extracted from the FN tunneling model. From this table, it is distinguished that the E B of the sample annealed in O2 ambient is dominated by the breakdown of IL as PI3K Inhibitor Library datasheet the obtained value of ΦB from the FN tunneling model is comparable with the value of ΔE c(IL/GaN) computed from the XPS measurement. For samples annealed in Ar and FG ambient, the acquisition of ΦB value that is comparable to the ΔE c(Y2O3/GaN) indicates that the E B of these samples is actually dominated by the breakdown of bulk Y2O3. Since the leakage current of the sample annealed in N2 ambient is not governed by FN tunneling mechanism, a conclusion in determining whether the E B of this sample is dominated by the breakdown of IL, Y2O3, or a combination of both cannot be deduced. Based on the obtained values of ΔE c(Y2O3/GaN), ΔE c(IL/GaN), and ΔE c(Y2O3/IL), the E B of this sample is unlikely to be dominated by IL due to the acquisition of a negative ΔE c(IL/GaN) value for this sample. Thus, the E B of this sample is most plausible to be dominated by either Y2O3 or a combination of Y2O3 and IL. However, the attainment of ΔE c(Y2O3/IL) value which is larger than that of ΔE c(Y2O3/GaN) value obtained for the samples annealed in Ar and FG ambient eliminates the latter possibility. The reason behind Methisazone it is if the E B of the sample annealed in N2 ambient is dominated by the combination of Y2O3 and IL, this sample should be able to sustain a higher E B and a lower J than the samples annealed in Ar and FG ambient. Therefore, the E B of the sample annealed in N2 ambient is most likely dominated by the breakdown of bulk Y2O3. Table 1 Comparison of the obtained Δ E c and Φ B values XPS: conduction band offset J-E Y 2 O 3 /GaN IL/GaN Y 2 O 3 /IL Barrier height O2 3.00 3.77 0.77 3.72 Ar 1.55 1.40 0.15 1.58 FG 0.99 0.68 0.31 0.92 N2 0.70 −2.03 2.73 a aNot influenced by FN tunneling. Therefore, barrier height is not extracted from the FN tunneling model. A 100 μL drop of MSgg was mounted on top of the biofilm and NO microprofiles SBI-0206965 research buy were measured immediately with an NO microsensor as described previously [43]. For each experimental treatment, MSgg was supplied either with or without 300 μM of the NO donor SNAP. SNAP was mixed Similar observations were made for the total score of these quest Similar observations were made for the total score of these questionnaires (Fig. 3). Doramapimod ic50 TH-302 price Patients with a fracture on the right side had significantly higher scores immediately after the fracture for the IOF physical function domain [right vs left, median (interquartile range, IQR): 89 (75, 96) vs 71 (61, 86), P = 0.002]. A fracture on the dominant side was associated with higher scores than a fracture on the non-dominant side with regard to physical function [89 (75, 96) vs 70 (59, 82), P < 0.001] and overall score [67 (54, 79) vs 56 (47, 67), P = 0.016]. The latter is shown in Fig. 4. Patients undergoing surgical treatment had lower scores of Qualeffo-41, indicating better quality of life, on general health (P = 0.013) and mental health selleck compound (P = 0.004) than patients with non-surgical treatment. Patients using analgesics had a higher scores of the IOF-wrist fracture questionnaire on pain (P = 0.009), on physical function (P = 0.001) and a higher overall score (P = 0.002) than patients not using analgesics. Table 5 Comparison of IOF-wrist domain and EQ-5D scores over time IOF-wrist EQ-5D Pain Upper limb symptoms Physical function General health Overall score Overall score Baseline 50 (25, 50) 25 (8, 42) 75 (61, 93) 75 (50, 75) 60 (50, 73) 0.59 (0.26, 0.72) 104 104 105 92 105 104 6 weeks 25 (25, 50) 29 (8,42) 57 (36, 79) 50 (25, 75) 48 (31, 65) 0.66 (0.59, 0.78) 0.002 0.688 <0.001 0.001 <0.001 <0.001 17-DMAG (Alvespimycin) HCl 98 98 98 95 98 97 3 months 25 (25, 50) 25 (8, 42) 25 (11, 46) 25 (0, 50) 25 (13, 46) 0.76 (0.66, 0.88) <0.001 0.007 <0.001 <0.001 <0.001 <0.001 89 89 89 88 89 85 6 months 25 (0, 50) 17 (8, 33) 14 (0, 33) 25 (0, 50) 15 (4, 34) 0.78 (0.69, 1.00) <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 87 87 87 87 87 86 12 months 0 (0, 25) 8 (0, 25) 4 (0, 29) 0 (0, 25) 8 (2, 27) 0.80 (0.69, 1.00) <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 87 87 87 86 87 85 Data presented as: median score (IQR) p value for difference between time point score and baseline score No. of subjects Fig. 2 IOF-wrist fracture median domain scores by time point Fig. 3 IOF-wrist fracture and Qualeffo-41 (spine) median overall scores by time point Fig. 4 IOF-wrist fracture median overall score by side of fracture and by time point Utility data could be calculated from the EQ-5D results. Immediately after the fracture, the utility was 0.59, increasing to 0.76 after 3 months and to 0.80 after 1 year. Assuming that the quality of life and the utility after 1 year are similar to that before the fracture, the utility loss due to the distal radius fracture is more than 0.20 in the first weeks. Most of the utility loss was regained after 3 months. Discussion The results from this study show that the IOF-wrist fracture questionnaire has an adequate repeatability, since the kappa statistic was moderate to good for most questions and quite similar to data obtained with Qualeffo-41 [10]. The concentration of RNA was adjusted to 100 ng/μl, and the sampl The concentration of RNA was adjusted to 100 ng/μl, and the samples were stored at −70°C. cDNA templates were synthesized from 50 ng RNA with PrimeScript™ 1st strand cDNA Synthesis Kit (TaKaRa) and gene-specific primers at 42°C for 15 m, 85°C for 5 s. Real-time PCR was performed with the cDNA and SYBR Premix Ex Taq (TaKaRa) using a StepOne Real-Time PCR System (Applied Biosystems). The quantity of cDNA measured by real-time PCR was normalised to the abundance of 16S cDNA. Real-time RT-PCR was repeated three times in triplicate parallel experiments. Statistical analysis The paired t test was used for statistical comparisons between groups. The level of statistical significance was set at a P value of ≤ 0.05. Results AI-2 inhibits biofilm formation FRAX597 order in a concentration-dependent manner under static conditions Previous studies showed that biofilm formation was influenced by the LuxS/AI-2 system both in Gram-positive and Gram-negative bacteria [32, 34]. The genome of S. aureus encodes a typical luxS gene, which plays a role in the regulation of capsular polysaccharide synthesis and virulence [43]. In this study, to investigate whether LuxS/AI-2 system regulates Protein Tyrosine Kinase inhibitor biofilm formation in S. aureus, we monitored the biofilm formation of S. aureus WT NCT-501 concentration strain RN6390B and the isogenic derivative ΔluxS strain using a microtitre plate assay. As shown in Figure 1A, the WT strain formed almost no biofilm after 4 h incubation at 37°C. However, the ΔluxS strain formed strong biofilms as measured by quantitative spectrophotometric analysis based on OD560 after crystal violet staining (Figure 1A). This discrepancy could be complemented by introducing a plasmid that contains the luxS gene (Figure 1B). Figure 1 Biofilm formation under static conditions and chemical complementation by DPD of different concentrations. Biofilm growth of S. aureus WT (RN6390B), ΔluxS and ΔluxS complemented with different concentrations of chemically synthesized DPD in 24-well plates for 4 h under aerobic conditions (A1: 0.39 nM, A2: 3.9 nM, A3: 39 nM, A4: 390 nM). The cells that adhered to the plate after staining with crystal violet were measured by OD560 . The effects of LuxS could be attributed to its central metabolic function or the AI-2-mediated next QS regulation, which has been reported to influence biofilm formation in some strains [32–34]. To determine if AI-2, as a QS signal, regulates biofilm formation in S. aureus, the chemically synthesized pre-AI-2 molecule DPD at concentrations from 0.39 nM to 390 nM was used to complement the ΔluxS strain. The resulting data suggested that exogenous AI-2 could decrease biofilm formation of the ΔluxS strain and the effective concentration for complementation was from 3.9 nM to 39 nM DPD (Figure 1A). As expected, these concentrations were within the range that has been reported [51]. The phenomenon that the higher concentration of AI-2 does not take effect on biofilm formation is very interesting, which has also been found in other species [51].
860c8867feebf994	Nonextensive approach to decoherence in quantum mechanics Nonextensive approach to decoherence in quantum mechanics 22 January 2001 Physics Letters A 279 (2001) 56–60 Nonextensive approach to decoherence in quantum mechanics A. Vidiella-... 99KB Sizes 2 Downloads 106 Views 22 January 2001 Physics Letters A 279 (2001) 56–60 Nonextensive approach to decoherence in quantum mechanics A. Vidiella-Barranco ∗ , H. Moya-Cessa INAOE, Coordinación de Optica, Apdo. Postal 51 y 216, 72000 Puebla, Pue., Mexico Received 22 May 2000; received in revised form 13 October 2000; accepted 12 December 2000 Communicated by A.R. Bishop Abstract We propose a nonextensive generalization (q-parametrized) of the von Neumann equation for the density operator. Our model naturally leads to the phenomenon of decoherence, and unitary evolution is recovered in the limit of q → 1. The resulting evolution yields a nonexponential decay for quantum coherences, fact that might be attributed to nonextensivity. We discuss, as an example, the loss of coherence observed in trapped ions.  2001 Elsevier Science B.V. All rights reserved. PACS: 05.30.Ch; 03.65.Bz; 42.50.Lc In the past decade, there have been substantial advances regarding a nonextensive, q-parametrized generalization of Gibbs–Boltzmann statistical mechanics [1,2]. The q-parametrization is based on an approximate expression for the exponential, or the qexponential, 1/(1−q) , eq (x) = 1 + (1 − q)x (1) being the ordinary exponential recovered in the limit of q → 1 (e1 (x) = ex ). Thus we have a nonextensive q-exponential, i.e., eq (x)eq (y) = eq (x + y + (1 −q)xy). Such a parametrization is the basis of Tsallis statistics, in which it is used a q-parametrized natural logarithm for the definition of a generalized entropy. We recall that in Tsallis’ statistics the entropy follows the pseudo-additive rule [1,2] Sq (A + B) = Sq (A) + Sq (B) + (1 − q)Sq (A)Sq (B), where A and B * Corresponding author. Permanent address: Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, 13083-970 Campinas, SP, Brazil. E-mail addresses: [email protected] (A. Vidiella-Barranco), [email protected] (H. Moya-Cessa). represent two independent systems, i.e., pij (A + B) = pi (A)pj (B). Hence for q < 1 (q > 1) we have the supper-additive (sub-additive) regime. The parameter q thus characterizes the degree of extensivity of the physical system considered. Tsallis’ entropy seems to be privileged, in the sense that it has a well defined concavity [1,3]. Moreover, it has also been shown that most of the formal structure of the standard statistical mechanics (and thermodynamics) is retained within the nonextensive statistics, e.g., the H-theorem [4], and the Onsager reciprocity theorem [5]. Nonextensive effects are usual in many systems exhibiting, for instance, long-range interactions, and the nonextensive formalism has been successfully applied not only to several interesting physical problems, but also outside physics [2]. We may cite for instance, applications to statistical mechanics [6], Lévy anomalous superdiffusion [7], low-dimensional dissipative systems [8], and others [2]. This means that departures from exponential (or logarithmic) behavior not so rarely occur in nature, and that the parametrization given by Tsallis’ statistics seems to be adequate for treating some of them. Discussions concerning the 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 8 2 0 - 3 A. Vidiella-Barranco, H. Moya-Cessa / Physics Letters A 279 (2001) 56–60 quantification of quantum entanglement [9,10] as well as the implications on local realism [11] and quantum measurements [12] may be also found, within the nonextensive formalism. Quantum entanglement has itself a nonlocal nature, and this may have somehow a connection with the general idea of nonextensivity. In fact, it has been shown that entanglement may be enhanced in the supper-additive regime (for q < 1) [9,10]. Thus, although nonextensive ideas open up new interesting possibilities, their application to a field such as foundations of quantum mechanics has not been so frequent. For instance, one could mention works proposing a nonextensive generalization of Schrödinger equation [13]. In this Letter we address, within a nonextensive approach, fundamental aspects of quantum theory which make necessary the use of the density operator representation. In quantum mechanics, the issue of decoherence, or the transformation of pure states into statistical mixtures has been attracting a lot attention, recently, especially due to the potential applications, such as quantum computing and quantum cryptography [14], that might arise in highly controlled purely quantum mechanical systems, e.g., in trapped ion systems [15]. That extraordinary control is also allowing to address fundamental problems in quantum mechanics, such as the Schrödinger cat problem [16], and the question of the origin of decoherence itself. Despite of the progresses, it does not exist yet a proper theory handling the question of the loss of coherence in quantum mechanics, although there are several propositions, involving either some coupling with an environment [17,18], or spontaneous (intrinsic) mechanisms [18]. Indeed dissipative environments, in which it is verified loss of energy, tend to cause decoherence. In spite of the destructive action of dissipation, though, it has been worked out a scheme to recover quantum coherences in cavity fields [19], even if its environment is at T 6= 0, and after the system’s energy, and coherences, have substantially decayed. Regarding the intrinsic decoherence, several models have been already presented [18], and decoherence has been attributed, for instance, either to stochastic jumps in the wave function [20], or gravitational effects [21]. These models may contain one [20] or even two new parameters [22]. It is normally proposed some kind of modification of the von Neumann equation for the density operator ρ, ˆ i d ρˆ = − Hˆ , ρˆ . (2) h¯ dt A typical model yielding intrinsic decoherence, such as Milburn’s [20] (see also Ref. [23]), gives the following modified equation for the evolution of ρ: ˆ τ i d ρˆ ≈ − Hˆ , ρˆ − 2 Hˆ , Hˆ , ρˆ , (3) h¯ dt 2h¯ where τ is a fundamental time step. The second term in the right-hand side (double commutator), leads to an exponential decay of coherences without energy loss [20,24]. Nevertheless, other models concerning decoherence [21], as well as quantum measurements [25], predict nonexponential decay of coherences, going as exp(−γ t 2 ) rather than as exp(−γ t). In [21], decoherence is attributed to a whormhole– matter coupling of nonlocal nature. This suggests that decoherence might not be appropriately described by a Markovian stochastic process [20]. Moreover, experimental evidence of nonexponential decay of quantum coherences has been recently reported, in a experiments involving trapped ions [26], as well as in quantum tunneling of cold atoms in an accelerating optical potential [27]. In Ref. [23] it is also considered the possibility of having nonexponential decay within a model for non-dissipative decoherence. Here we propose a novel model to treat decoherence, within a nonextensive realm. The motivation for that is related to the fact that physical systems are in general not isolated, and the control of unwanted interactions and fluctuations in real experiments is not an easy task. In particular, memory effects and long range interactions may be present in certain systems (non-Markovian character), and Tsallis statistics is considered to be relevant in such cases. Bearing this in mind, we were able to write down a q-parametrized (nonextensive) evolution equation for the density operator, which naturally leads to decoherence. As a first step we may express von Neumann’s ˆ beequation (Eq. (2)) in terms of the superoperator L, ˆ ˆ ˆ ing i h¯ Lρˆ = H ρˆ − ρˆ H , so that we may write its formal solution in a simple form ˆ ρ(0). ρ(t) ˆ = exp Lt ˆ (4) Now, we substitute the exponential superoperator ˆ by a q-parametrized exponential as defined exp(Lt) in Eq. (1): ˆ 1/(1−q)ρ(0). ˆ ρ(t) ˆ = 1 + (1 − q)Lt The standard unitary evolution given by von Neumann’s equation is recovered in the limit of q → 1. Hence, departing from Eq. (5), we may write a generalized (q-parametrized) von Neumann equation, or Lˆ d ρˆ = ρ(t). ˆ ˆ dt 1 + (1 − q)Lt Here we consider the case in which the parameter q is very close to one. This represents a small deviation from the unitary evolution given by (2), but yields loss of coherence. For \|1 − q\|Ωt 1, where Ω is the characteristic frequency of the physical system considered, we may write d ρˆ ˆ ≈ L − (1 − q)t Lˆ 2 ρ(t). ˆ (7) dt It may be shown, from Eq. (7), that there is decay of the nondiagonal elements in the basis formed by the eigenstates of Hˆ , characterizing decoherence, while the diagonal elements are not modified. Therefore the density operator remains normalized at all times, or Tr ρ(t) ˆ = Tr ρ(0) ˆ = 1, property which must hold in any proper model for decoherence. We shall remark that in order to have a physically acceptable dynamics, the extensivity parameter should be greater than one (q > 1), i.e., in the sub-additive regime. We may also rewrite Eq. (7) as i d ρˆ ≈ − Hˆ , ρˆ + g(t) Hˆ , Hˆ , ρˆ , h¯ dt where g(t) = (1 − q)t/h¯ 2 . Eq. (8) is similar to Eq. (3), apart from the time-dependent factor g(t) multiplying the double commutator. Note that Eq. (8) is only valid for short times, or \|1 − q\|Ωt 1. We have therefore constructed a novel model, based on a nonextensive generalization of von Neumann’s equation, which predicts loss of coherence, although with a time-dependence for the nondiagonal elements in the energy basis different from most that can be found in the literature [18]. This particular timedependence will lead to a nonexponential decay of quantum coherences, as we are going to show. Now we would like to apply our model to a specific system in which experimental results are available, or a single trapped ion interacting with laser fields. Quantum state engineering of motional states of a massive oscillator has been achieved [28], and the loss of coherence in that system has been already observed [26,28]. In the experiment described in Ref. [28], two Raman lasers induce transitions between two internal electronic states (\|ei and \|gi) of a single 9 Be+ trapped ion, as well as amongst center-of-mass vibrational states \|ni. In the interaction picture, and under the rotating wave approximation, the effective Hamiltonian (in one dimension) may be written as [28] † † I = h¯ Ω σ+ eiη(aˆ x +aˆ x )−iδt + σ− e−iη(aˆ x +aˆ x )+iδt , Hˆ int (9) Ω being the coupling constant, δ the detuning of the frequency difference of the two laser beams with respect to ω0 , which is the energy difference between \|ei and \|gi (E ¯ 0 ). The Lamb–Dicke parameter √ e − Eg = hω is η = k h¯ /(2mωx ), where k is the magnitude of the difference between the wavevectors of the two laser beams, ωx is the vibrational frequency of the ion center of mass having a mass m, and aˆ x (aˆ x† ), are annihilation and creation operators of vibrational excitations. The ion is Raman cooled to the (vibrational) vacuum state \|0i, and from that Fock (as well as coherent and squeezed) states may be prepared by applying a sequence of laser pulses [28]. If the atom is initially in the \|g, ni state, and the Raman lasers are tuned to the first blue sideband (δ = ωx ), the evolution according to the Hamiltonian in Eq. (9) (for small η) will be such that the system will perform Rabi oscillations between the states \|g, ni and \|e, n + 1i. If the excitation distribution of the initial vibrational state of the ion is Pn , occupation probability of the ground state Pg (t) will be given by X 1 −γn t Pn cos(2Ωn t) e . Pg (n, t) = 1 + (10) 2 n Here the Rabi frequency Ωn is e−η /2 ηL1n η2 , Ωn = Ω √ n+1 2 where Ω/2π = 500 kHz, L1n are Laguerre generalized polynomials, and γn = γ0 (n + 1)0.7 , with γ0 = 11.9 kHz. The experimental results are fitted using Eq. (10) [28]. The Rabi oscillations are damped, and an empirical damping factor, γn , is introduced in order to fit the experimental data [28]. There have been attempts to derive such an unusual dependence on n of γn , by taking into account the fluctuations in laser fields and the trap parameters [29]. Those models are somehow connected to a evolution of the type given by Eq. (3) [20,24], which predicts loss of coherence without loss of energy. In fact, at a time-scale in which the ion energy remains almost constant, decoherence is considerably large [15,28]. Moreover, the actual causes of loss of coherence in experiments with trapped ions are still not identified [15,30]. Our approach differs from previous ones because we start from a different dynamics which leads to a peculiar nonexponential time-dependence. By employing a similar methodology to the one discussed in reference [24], we may calculate, from Eq. (8), the probability of the atom to occupy the ground state, obtaining q Pg (n, t) = X 1 −γn,q t 2 1+ cos(2Ωn t) e , 2 n where γn,q = (q − 1)Ωn2/2, i.e., the damping factor arising in our model depends on the Rabi frequency Ωn and on the parameter q in a simple way. It is also verified a nonexponential decay which goes as exp(−γ t 2 ) rather than an exponential one. Now we may proceed with a graphical comparison between the curves plotted from expressions (10) and (12). This is shown in Fig. 1, for two distinct cases. In Fig. 1(a) we have plotted both the probability Pg in Eq. (12) (dashed line) and the one in Eq. (10) (full line), as a function of time, for an initial state \|g, 0i (Pn = δn,0 ), a Lamb–Dicke parameter η = 0.202, and q = 1.001. We note a clear departure from exponential behavior in the curve arising from our model, although they might coincide for some time-intervals. We remark that Eq. (12) is valid for times such that \|1−q\|Ωt 1. Here \|1 − q\|Ωtmax ≈ 0.17. Right below, in Fig. 1(b), there is a similar plot, but using a different initial condition for the distribution of excitations of the ion state, e.g., a coherent state \|αi, with n = α 2 = 3 (Pn = exp(−α 2 )α 2n /n!) instead of the vacuum state \|0i. In the coherent state case both curves are even closer. In order to better appreciate the effect of nonextensivity, we have plotted Pg with q = 1 (dotted curves), which represent the unitary evolution. In general, our results are in reasonable agreement with the experimental data, as it may be seen in Fig. 1 and in Ref. [28]. This Fig. 1. Probability of having the atom in the ground state Pg as a function of time: (a) for an initial (vibrational) vacuum state and (b) for an initial (vibrational) coherent state with n = 3. In either case, η = 0.202 and Ω/2π = 500 kHz. The solid lines are calculated from from Eq. (10), and the dashed lines from our model, Eq. (12), with an extensivity parameter q = 1.001. The dotted curves represent the unitary evolution (q = 1). means that a nonexponential decay may also account for the loss of coherence in the trapped ions system. In our model, the effects leading to decoherence are embodied within the nonextensive parameter q, rather than into a “fundamental time step” τ , as it occurs in Milburn’s model. Nonextensivity may be especially relevant when nonlocal effects, for instance, longrange interactions, or memory effects are involved [2]. In the specific example treated here, the ion is not completely isolated from its surroundings; electric fields generated in the trap electrodes couple to the ion charge, and this is considered to be a genuine source of decoherence to the vibrational motion of the ion [15,30]. In conclusion, we have presented a novel approach for treating the loss of coherence in quantum mechan- ics, based on a nonextensive formalism. In our model, decoherence depends on a single parameter, q, related to the nonextensive properties of the physical system considered. We obtain, with such a parametrization, an evolution equation which is a (nonextensive) generalization of von Neumann’s equation for the density operator, and that leads, in general, to a nonexponential decay of quantum coherences. We have applied our model to a concrete physical problem, that is, the decoherence occurring in the quantum state of a single trapped ion undergoing harmonic oscillations. This is a well know case in which decoherence may be readily tracked down, yet it does not exist an easy way to consider all the possible effects leading to decoherence. We have found that for values of the parameter q rather close to one (in the sub-additive regime), our model is in reasonable agreement with the available experimental results, without the need of introducing any supplementary assumptions. Acknowledgements One of us, H.M.-C., thanks R. Bonifacio and P. Tombesi for useful comments. A.V.-B. would like to thank the hospitality at INAOE, México. This work was partially supported by CONACyT (Consejo Nacional de Ciencia y Tecnología, México) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil). C. Tsallis, J. Stat. Phys. 52 (1988) 479. C. Tsallis, Braz. J. Phys. 29 (1999) 1. R.S. Mendes, Physica A 242 (1999) 299. A.M. Mariz, Phys. Lett. A 165 (1992) 409; J.D. Ramshaw, Phys. Lett. A 175 (1993) 169. [5] M.O. Caceres, Phys. Lett. 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dd263b9f5ae698ec	logo Insalogo Insa Nanophysics - from wave propagation to photonics and nanotechnologies Programme (detailed contents): Wave phenomena, interferences and diffraction. Particle phenomena. The dual nature of waves and particles, application to electron microscopy. Quantum physics laws. Quantum effects and applications. Scanning tunnelling microscope, quantum wells and quantum dots, application to radioactivity, harmonic oscillator, vibrations of molecules and Infrared spectroscopy, kinetic momentum and its application to molecular rotation spectroscopy, spin and its applications in NMR and MRI. Atomic and molecular orbitals. X-rays . Lasers. Crystalline solids, concept of energy bands, application in semiconductor electronic devices. Based on an epistemological approach, the lectures lead the students progressively towards modern physics which is at the core of nanotechnologies. Main difficulties for students: v Link formalism to applications (devices or techniques based on quantum physics or wave propagation) v Solve differential equations involved in the resolution of the Schrödinger equation v The fundamentals of wave propagation and quantum physics that are necessary for the understanding of modern electronic devices and analytical techniques v The principle of analytical techniques commonly used in laboratories and the molecular mechanisms of quantum physics The student will be able to: v Formulate in his own words some nano-scale mechanisms and give concrete examples of micro and nano-devices together with well-known analytical methods using these mechanisms v Master elementary phenomena of nano-scale physics v Select the best method for a specific characterisation on the basis of the acquired concepts. v Carry out some nano-scale characterisation methods v Link mathematical formalism of quantum physics to real applications v Grasp the necessary approximations that are required in quantum physics v Bring together these different concepts to assimilate them, extract them from their context in order to apply them to real situations. context in order to apply them to real situations. Needed prerequisite 1st year Mechanics, Electrostatics, Optics and Mathematics Form of assessment Additional information wave phenomena interferences and diffraction the dual nature of waves and particles electron microscopy quantum physics laws Tunnel effect and scanning tunnelling microscope quantum wells quantum dots harmonic oscillator kinetic momentum infrared spectroscopy electron spin energy bands in solid state materials
078a5f7d90b2514f	Saturday, February 27, 2021 TGD based explanation for the asymmetry between anti-u and anti-d sea quarks in proton I encountered in FB a highly interesting popular article "Decades-Long Experiment Finds Strange Mix of Antimatter in The Heart of Every Proton" (see this). The popular article tells about the article "The asymmetry of antimatter in the proton" of Dove et al published in Nature (see this). This article is behind the paywall but the same issue of Nature has an additional article "Antimatter in the proton is more down than up" (see this) explaining the finding. What is found is an asymmetry for u and antiquarks in the sense that there are slightly more d-type antiquarks (anti-d) than u type antiquarks (anti-u) in quark sea. This asymmetry does not seem to depend on the longitudinal momentum fraction of the antiquark: the ratio of anti-down and anti-up distribution functions is smaller than one and constant. A model assuming that proton is part of time in a state consisting of neutron and virtual pion seems to fit at qualitative level into the picture. Unfortunately, the old-fashioned strong interaction theory based on nuclei and pions does not converge by the quite too large value of proton pion coupling constant. I looked at the situation in more detail and developed a simple TGD based model based on the already existing picture developed by taking seriously the so called X boson as 17.5 MeV particle and the empirical evidence for scaled down variants of pion predicted by TGD (see this). What TGD can give is the replacement of virtual mesons with real on mass shell mesons but with p-adically scaled down mass and a concrete topologicaldescription of strong interactions at the hadronic and nuclear level in terms of reconnections of flux tubes. 1. Basic data about quark and nucleon masses To get a quantitative grasp about the situation, one can first see what is known about masses of u and d quarks. 1. One estimate for u and d quark masses (one must taken the proposals very cautiously) can be found here. The mass ranges are for u 1.7-3.3 MeV and and for d 4.1-5.8 MeV. 2. In the first approximation n-p mass difference 1.3 MeV would be just d-u mass difference varying in the range 1.2 MeV-4.1 MeV and has correct sign and correct order of magnitude. 4.1 MeV for d and 3.3 MeV for u would produce the n-p mass difference correctly. 3. Coulomb interactions give a contribution which is vanishing for p and and negative for neutron Ec(n) =-α× ℏ/3Re, Re is proton's electromagnetic scale. This contribution reduces neutron mass. If Rem is taken to be proton Compton radius this gives about Ec≈ - 3.2 MeV. This would predict mass n-p difference in the range -1.1-0.9 MeV. This favors maximal n-p mass difference 4.1 MeV and m(u)= 1.7 MeV and md =5.8 MeV: d-u mass difference would be 4.1 MeV roughly 4 times electron mass. 2. TGD based picture about hadronic an nuclear interactions Consider first the TGD inspired topological model for hadronic an nuclear interactions. 1. The notion of magnetic body (MB) assignable to color and em and electroweak interactions is essential. Interactions are described by virtual particle exchanges in quantum field framework. In TGD they are described by reconnections of U-shaped flux tubes which are like tentacles. In interaction these tentacles reconnect and give rise to a pair of flux tubes connecting the particles. The flux tubes would carry monopole flux so that single flux tube cannot be split. These flux tube pairs serve also as correlates of entanglement replacing wormholes as their correlates in ER-EPR picture. This picture looks rather biological and was developed first as a model of bio-catalysis. The picture should apply quite generally to short range interactions at least. 2. The U-shaped flux tubes of color MB replace virtual pion and and rho meson exchanges in the old fashioned picture about strong interactions. They represent in TGD framework real particles but with p-adically scaled down mass. For instance, pions are predicted to have scaled down variants with masses different by a negative power of 2 from pion mass. Same is true for rho. Now the masses would be below MeV range, which is the energy scale of nuclear strong interactions. Also nuclear strong interactions would occur in this manner. The fact that flux tubes have much longer length than nuclear size would explain the mysterious finding that in nuclear decay the fragments manage to generate their angular momenta after the reaction: the flux tubes would make possible the exchange of angular momentum required by angular momentum conservation. 3. A model for the anti-quark asymmetry Consider now a model anti-quark asymmetry for sea quarks. 1. Quarks and antiquarks would appear at these flux tubes. The natural first guess is meson like states are in question. The generation of u-anti-d type pion or rho would transform proton to neutron if the valence u transforms to valence d and W boson with scaled down mass. Note that the scaling down would make weak interaction stronger since weak boson exchange amplitude is proportional to 1/mW2). This would give the analog of neuron plus charge virtual pion. Taking two sea quarks would lead to trouble with the too large Coulomb interaction energy about -10 MeV of negatively charged sea with positively charged valence part of proton if the sea is of the same size as proton. 2. Does the scaled down W decay to u-anti-d forming a scaled down meson? Or should one regard u-anti-d as a scaled down W having also the spin zero state analogous to pion since it is massive? 3. Here comes a connection with old-fashioned and long ago forgotten hadron physics. Thepartially conserved axial current hypothesis (PCAC) gives a connection between strong and weak interactions forgotten when QCD emerged as the final theory. PCAC says that the divergence of axial weak currents associated with weak bosons are proportional to pions. Are the two pictures more or less equivalent? Virtual pion exchange could be regarded as a weak interaction! Also conserved vector current hypothesis (CVC) is part of this picture. This is not new: I have developed this picture earlier in an attempt to understand what the reported X boson with 17.5 MeV mass is in the TGD framework. Scaled down pion would be in question (see this). 4. What about masses? Since the flux loop would have considerably greater size than proton, the mass scale of udbar state would be smaller than say MeV, and the contribution to mass of proton would be small. 5. Why the asymmetry for anti-quarks of sea? The generation u-anti-d loop would increase the charge of the core region by two 2 units and transform it to Δ. This looksneither plausible nor probable. Proton would be a superposition consisting mostly of the proton of good old QCD and neutron plus flux loop with quantum numbers of a scaled down pion. 6. Also the presence of scaled down ρ meson loops can be considered. Their presence would turn the spin of the core part of the proton opposite for some fraction of time. One can wonder whether this could relate to the spin puzzle of proton. For the TGD based model of X boson see the article "X boson as evidence for nuclear string model". See the article a The asymmetry of antimatter in proton from TGD point of view. For a summary of earlier postings see Latest progress in TGD. Articles and other material related to TGD. Thursday, February 25, 2021 Is analysis as a cognitive process analogous to particle reaction? The latest work with number theoretical aspects of M8-H duality (see this) was related to the question whether it allows and in fact implies Fourier analysis in number theoretical universal manner at the level of H= M4× CP2. The problem is that at the level of M8 analogous to momentum space polynomials define the space-time surfaces and number theoretically periodic functions are not a natural or even possible basis. At the level of H the space-time surfaces can be regarded orbits of 3-surfaces representing particles and are dynamical so that Fourier analysis is natural. That this is the case, is suggested by the fact that M8-H duality maps normal spaces of space-time surface to points of CP2. Normal space is essentially the velocity space for surface deformations normal to the surface, which define the dynamical degrees of freedom by general coordinate invariance. Therefore dynamics enters to the picture. It turns out that the conjecture finds support. Consider now the topic of this posting. Number theoretic vision about TGD is forced by the need to to describe the correlates of cognition. It is not totally surprising that these considerations lead to new insights related to the notion of cognitive measurement involving a cascade of measurements in the group algebra of Galois group as a possible model for analysis as a cognitive process (see this, this and this). 1. The dimension n of the extension of rationals as the degree of the polynomial P=Pn1∘ Pn2∘ ... is the product of degrees of degrees ni: n=∏ini and one has a hierarchy of Galois groups Gi associated with Pni∘.... Gi+1 is a normal subgroup of Gi so that the coset space Hi=Gi/Gi+1 is a group of order ni. The groups Hi are simple and do not have this kind of decomposition: simple finite groups appearing as building bricks of finite groups are classified. Simple groups are primes for finite groups. 2. The wave function in group algebra L(G) of Galois group G of P has a representation as an entangled state in the product of simple group algebras L(Hi). Since the Galois groups act on the space-time surfaces in M8 they do so also in H. One obtains wave functions in the space of space-time surfaces. G has decomposition to a product (not Cartesian in general) of simple groups. In the same manner, L(G) has a representation of entangled states assignable to L(Hi) (see this and this). This picture leads to a model of cognitive processes as cascades of "small state function reductions" (SSFRs) analogous to "weak" measurements. 1. Cognitive measurement would reduce the entanglement between L(H1) and L(H2), the between L(H2) and L(H3) and so on. The outcome would be an unentangled product of wave functions in L(Hi) in the product L(H1)× L(H2)× .... This cascade of cognitive measurements has an interpretation as a quantum correlate for analysis as factorization of a Galois group to its prime factors defined by simple Galois groups. Similar interpretation applies in M4 degrees of freedom. 2. This decomposition could correspond to a replacement of P with a product ∏i Pi of polynomials with degrees n= n1n2..., which is irreducible and defines a union of separate surfaces without any correlations. This process is very much analogous to analysis. 3. The analysis cannot occur for simple Galois groups associated with extensions having no decomposition to simpler extensions. They could be regarded as correlates for irreducible primal ideas. In Eastern philosophies the notion of state empty of thoughts could correspond to these cognitive states in which SSFRs cannot occur. 4. An analogous process should make sense also in the gravitational sector and would mean the splitting of K=nA appearing as a factor ngr=Kp to prime factors so that the sizes of CDs involved with the resulting structure would be reduced. Note that ep(1/K) is the root of e defining the trascdental infinite-D extension rationals which has finite dimension Kp for p-adic number field Qp. This process would reduce to a simultaneous measurement cascade in hyperbolic and trigonometric Abelian extensions. The IR cutoffs having interpretation as coherence lengths would decrease in the process as expected. Nature would be performing ordinary prime factorization in the gravitational degrees of freedom. This cognitive process would also have a geometric description. 1. For the algebraic EQs, the geometric description would be as a decay of n-sheeted 4-surface with respect to M4 to a union of ni-sheeted 4-surfaces by SSFRs. This would take place for flux tubes mediating all kinds of interactions. In gravitational degrees of freedom, that is for trascendental EQs, the states with ngr=Kp having bundles of Kp flux tubes would deca to flux tubes bundles of ngr,i=Kip, where Ki is a prime dividing K. The quantity log(K) would be conserved in the process and is analogous to the corresponding conserved quantity in arithmetic quantum field theories and relates to the notion of infinite prime inspired by TGD \citeallbvisionc. 2. This picture leads to ask whether one could speak of cognitive analogs of particle reactions representing interactions of "thought bubbles" i.e. space-time surfaces as correlates of cognition. The incoming and outgoing states would correspond to a Cartesian product of simple subgroups: G=∏×i Hi. In this composition the order of factors does not matter and the situation is analogous to a many particle system without interactions. The non-commutativity in general case leads to ask whether quantum groups might provide a natural description of the situation. See the article Is M8-H duality consistent with Fourier analysis at the level of M4× CP2? or the chapter Breakthrough in understanding of M8-H duality. For a summary of earlier postings see Latest progress in TGD. Articles and other material related to TGD.   Is M8-H duality consistent with Fourier analysis at the level of M4× CP2? M8-H duality predicts that space-time surfaces as algebraic surfaces in complexified M8 (complexified octonions) determined by polynomials can be mapped to H=M4× CP2. The proposal (see this) is that the strong form of M8-H duality in M4 degrees of freedom is realized by the inversion map pk∈ M4→ ℏeff×pk/p2. This conforms with the Uncertainty Principle. However, the polynomials do not involve periodic functions typically associated with the minimal space-time surfaces in H. Since M8 is analogous to momentum space, the periodicity is not needed. In contrast to this, the representation of the space-time surfaces in H obey dynamics and the H-images of X4⊂ M8 should involve periodic functions and Fourier analysis for CP2 coordinates as functions of M4 coordinates. Neper number, and therefore trigonometric and exponential functions are p-adically very special. In particular, ep is a p-adic number so that roots of e define finite-D extensions of p-adic numbers. As a consequence, Fourier analysis extended to allow exponential functions required in the case of Minkowskian signatures is a number theoretically universal concept making sense also for p-adic number fields. The map of the tangent space of the space-time surface X4⊂ M8 to CP2 involves the analog velocity missing at the level of M8 and brings in the dynamics of minimal surfaces. Therefore the expectation is that the expansion of CP2 coordinates as exponential and trigonometric functions of M4 coordinates emerges naturally. The possible physical interpretation of this picture is considered. The proposal is that the dimension of extension of rationals (EQ) resp. the dimension of the transcendental extension defined by roots of Neper number correspond to relatively small values of heff assignable to gauge interactions resp. to very large value of gravitational Planck constant ℏgr originally introduced by Nottale. Also the connections with the quantum model for cognitive processes as cascades of cognitive measurements in the group algebra of Galois group (see this and this) and its counterpart for the transcendental extension defined by the root of e are considered. The geometrical picture suggests the interpretation of cognitive process as an analog of particle reaction emerges. For a summary of earlier postings see Latest progress in TGD. Articles and other material related to TGD. Wednesday, February 17, 2021 Can one regard leptons as effectively local 3-quark composites? The idea about leptons as local composites of 3 quarks (see this) is strongly suggested by the mathematical structure of TGD. Later it was realized that it is enough that leptons look like local composites in scales longer than CP2 scale defining the scale of the partonic 2-surface assignable to the particle. The proposal has profound consequences. One might say that SUSY in the TGD sense has been below our nose for more than a century. The proposal could also solve matter-antimatter asymmetry since the twistor-lift of TGD predicts the analog of Kähler structure for Minkowski space and a small CP breaking, which could make possible a cosmological evolution in which quarks prefer to form baryons and antiquarks to form leptons. The basic objection is that the leptonic analog of Δ might emerge. One must explain why this state is at least experimentally absent and also develop a detailed model. In the article Can one regard leptons as effectively local 3-quark composites? the construction of leptons as effectively local 3 quark states allowing effective description in terms of the modes of leptonic spinor field in H=M4× CP2 having H-chirality opposite to quark spinors is discussed in detail. See the article Can one regard leptons as effectively local 3-quark composites? and the chapter The Recent View about SUSY in TGD Universe. For a summary of earlier postings see Latest progress in TGD. Articles and other material related to TGD.   Monday, February 08, 2021 Quantum asymmetry between space and time I received a link to a popular article about a test the proposal of Joan Vaccaro that if time reversal symmetry T were exact, our Universe would be radically different (thanks for Reza Rastmanesh and Gary Ehlenberg) For instance, wave functions would be wave packets in 4-D sense and conservation laws would be lost. Breaking of T would however come in rescue and give rise to the world in which we live. This proposal does not make sense in standard quantum theory but JV proposes a modification of path integral for single particle wave mechanism leading to the result. I found that I could not understand anything about the popular article. I however found the article "Quantum asymmetry between space and time" by Joan Vaccaro. I tried to get grasp about the formula jungle of the article but became only confused. I expected clear logical arguments but I found none. My comments are based mostly on the abstract of the article. Asymmetry between time and space My comments: 1. One might argue like JV does if one does not keep in mind that Lorentz invariance allows to distinguish between timelike and space-like directions and base the notion of causality on their properties. In Euclidian geometry there would be no such special choice of time coordinate. But also now field equations would define slicing of space-time to 3-D slices since initial values at them would fix the time solution. Now however the slices could be chosen in very manner manners - for instance 3-spheres rather than hyperplanes as in Minkowski space. 2. JV argues that there is an asymmetry in the quantum description of space and time in conventional quantum theory. The spatial coordinates of particle are treated as operators but time is not represented as an operator. The first mis-understanding is that the position of operator of particle is not space-time coordinate but specifies position of point-like particle in space time. The second mis-understanding is to think that the configuration space of the particle would be 4-D space-time. The configuration space of particle in non-relativistic wave mechanics is 3-space and time is the evolution parameter for unitary time evolution, not space-time coordinate. In the relativistic picture it could correspond to proper time along a world line. In quantum field theory (QFT) the spatial and temporal coordinates are in completely symmetric position. Wave mechanics is an approximation in which one considers only singlenon-relativistic particle. One should start from QFT or some more advanced to see whether the idea makes sense. 3. JV identifies subjective and geometric time as practically all colleagues do. In geometric time time evolution is determined by field equations and conservation laws. In TGD zero energy ontology (ZEO) does not identify these times and resolves the problems caused by identification of these two times. The counterpart of time evolution with respect to subjective time is sequence of small state function reductions. 4. The asymmetry between the two time directions appears in two manners. 1. There is the thermo-dynamical arrow of time usually assumed to be the same always. In TGD both arrows are possible and the arrow changes in "big" (ordinary) state function reduction (BSFR). Subjective time correlates with geometric time but is not identical with it is closely related to the thermo-dynamical breaking of time reversal. 2. The field equations (geometric time) have slightly broken time reflection symmetry T. This breaking is quite different from the generation of the thermo-dynamical arrow of time. Could we give up the conservation laws and unitary time evolution and could the breaking of time reversal symmetry bring them back? In this case, equations of motion and conservation laws are undefined or inapplicable. However, if T symmetry is violated, then the same sum over paths formalism yields states that are localized in space and distributed without bound over time, creating an asymmetry between time and space. Moreover, the states satisfy an equation of motion (the Schrdinger equation) and conservation laws apply. This suggests that the timespace asymmetry is not elemental as currently presumed, and that T violation may have a deep connection with time evolution. My comments: 1. JV is ready to give up symmetries and conservation laws altogether in the new definition of path integral but of course brings them implicitly in by choice of Hamiltonian and by using the basic concepts like momentum and energy which are lost if one does not have Poincare symmetry. What remains is an attempt to repair the horrible damage done. The hope is that the tiny breaking of T invariance would be capable of this feat. 2. Author uses a lot of formulas to show that T breaking can save the world. There are however ad hoc assumptions such as coarse graining and assumptions about the difference between Hamiltonian and time reversed Hamiltonian argued to lead to the basic formulas of standard quantum theory. The proposed formulas are based on single particle wave mechanics and do not generalize to the level of QFT. If one is really ready to throw away the basic conservation laws and therefore corresponding symmetries also the basic starting point formulas become non-sensible. Holistic mathematical thinking would help enormously the recent day theoretical physicists but it is given the label "philosophical" having the same emotional effect as "homeopathic" to the average colleague. What my colleague called formula disease has been the basic problem of theoretical physics for more than half century. 3. This modification of path integral formula looks rather implausible to me. 1. Giving up the arrow of time in the sum over paths formalism breaks the interpretation as a representation for Hamiltonian time evolution (path integral is mathematically actually not well-defined and is meant to represent just Schroedinger equation). If there were no asymmetry between time and space, quantum states would be wave packets in 4-D sense rather in 3-D sense. This is of course complete nonsense and in conflict with conservation laws: an entire galaxy could appear from nothing and disappear. Author notices this but does not seem to worry about consequences. By the way, in TGD inspired theory of consciousnessthe mental image of the galaxy can do this but this does not mean the disappearance of the galaxy! The use of the wave mechanics which is not Lorentz invariant, hides the loss of Lorentz invariance implied by the formalism whereas ordinary Schrödinger equation as non-relativistic approximation of Dirac equation does not break Lorentz invariance in non-relativistic approximation. 2. It is optimistically assumed that the tiny breaking of T symmetry could change the situation completely so that the predictions of the standard quantum theory would emerge. Somehow the extremely small breaking of T would bring back arrow of time and save conservation laws and unitary time evolution and we could be confident that our galaxy exists also to-morrow. Why the contributions to modified path integral for which time arrow is not fixed would magically interfere to zero by a tiny breaking of T invariance? The proposal seems to be in conflict with elementary particle physics. The view is that neutral kaon is a superposition of a state and its T mirror image and this means tiny T breaking. Neutrinos also break T symmetry slightly. In this framework all other particles would represent complete T breaking. Usually the interpretation is just the opposite. This does not make sense to me. 3. The test for proposal would be based on the idea that neutrino from nuclear reaction defined flux diminishing as 1/r2, r the distance from the reactor. This should somehow cause an effect on clocks proportional to 1/r2 due to the incomplete destructive interference of the contributions breaking the ordinary picture. I do not understand the details for how this was thought to take place. The small T violation of neutrinos would affect the maximal T violation in environment and somehow affect the local physics and be visible as a modification of clock time - maybe by modification of the Hamiltonian modelling the clock. This is really confusing since just the small T violation is assumed to induce the selection of the arrow of time meaning maximal T violation! To sum up, I am confused. For a summary of earlier postings see Latest progress in TGD. Articles and other material related to TGD. A test for the quantum character of gravity as a test for quantum TGD The following represents comments to a popular article describing a proposed experiment possibly conforming the quantum character of gravity. Thanks for the link to Gary Ehlenberg. The proposal in idealized form is following. 1. One has two diamonds which are originally in superposition of two opposite spin directions. 2. One applies magnetic field with gradient. We know that in external magnetic field which is not constant, particle with spin is deflected to a direction which is opposite for opposite sping directions. If one has a superposition of spins then one would obtain superposition of deflected spin states with different spatial positions. The diamond can be said to be at two positions simultaneously. This is the key effect and leads to the question about whether and how it could be described in general relativity. Consider now the situation in various approaches to quantum gravity. 1. The situation is problematic from the point of view of classical GRT since the gravitational fields created by the different spin components of superposition would have different distribution of mass. Classical GRT space-time determined by Einstein's equations would not be unique. In order to avoidthis one could argue that state function reduction (SFR) occurs immediatelyso that the spin is well-defined and space-time is unique. This looks very ugly but to my surprise Dyson takes this possibility seriously. 2. In quantum gravity based on GRTview it seems that one should allow superposition of space-times or at least of 3-spaces. The superposition of 3-geometries would conform with Wheelerian superspace view but leads to the problems with time and conservationlaws quite generally. Wheeler's superspace did not work mathematically. 3. In the description in terms of quantum gravitons one would give up the GRT picture altogether and string model view is something akin to this. Unfortunately, the attemptsto formulate quantum gravitation as a field theory in empty Minkowski space using Einstein's action fail. The divergences are horrible and renormalization is not possible. String models in turn try to get space-time by spontaneous compactification but this led to the landscape misery. TGD based picture avoids the problems of these approaches. 1. In zero energy ontology (ZEO) the quantum states are superpositions of space-time surfaces : not superpositions of general 4-geometries making even not sense in standard QM or of 3-surfaces as they would be in generalization of Wheeler's superspace approach. The space-time surfaces would be preferred extremals of the action defining the theory and one obtains holography from general coordinate invariance: 3-surface determines space-time surface just as the point of Bohr orbit defines the Bohr orbit. 2. In this picture superposition of states with position of particle depending on spin direction corresponds to superposition of different space-time surfaces. Also the entanglement between various components of the state involving two spin superposed states of the two diamonds involved, is possible and generated in the time evolution. Entanglementwould have as a correlate magnetic flux tubes connecting the diamonds. One would have superposition of space-time surfaces corresponding to different flux tube lengths connecting spin states at various positions. Thereisno reason to expect that entanglement would not be generated. The expected positive outcome of the experiment could be seen as a proof of not only quantum character of gravitations but evidences for the superposition of space-time surfaces and ZEO. 3. Diamonds are rather large objects and the small value of Planck constant is the trouble in the standard approach. It might be possible to overcome this problem in TGD. TGD predicts that dark matter corresponds to phases of ordinary matter with arbitrarily large values of h_eff at MB glux tubes would have macroscopic quantum coherence in the scale of flux tube and could induce coherence of ordinary matter (diamonds) at their end as controllers of the ordinary matter. This would not be quantum coherence but would make visible the quantum coherence at the level of MB. For a summary of earlier postings see Latest progress in TGD. Articles and other material related to TGD.
4bd2502e6d4f0b31	Free particle wave equations are derived from kinematic laws using the quantum mechanical substitutions For example Any relativistic wave equation (e.g. Klein-Gordon or Dirac equations) has both positive and negative energy solutions The negative energy solutions correspond to the motion of antiparticles; antiparticles must exist if quantum mechanics and special relativity are correct. Time Reversal (tÆ-t) A wavefunction cannot be in an eigenstate of T, because T is an antiunitary operator which changes functions to their complex conjugate. e.g. if y = ei(px-Et) then (using Tp =-p) Ty = ei(-px+Et) = y* One can easily see that the eigenvalue equation Ty = ly has no solutions, even for a complex eigenvalue l= reiq. This does not mean T may not be a symmetry of the Hamiltonian. It does mean that wavefunctions cannot be eigenstates of T. Probabilities and expectation values are unchanged if T is a symmetry. e.g. For a free particle wave function T(yy) = y y = yy This may not be true for interactions if complex matrices are involved. T symmetry, as well as CP, is violated at the 0.2% level in neutral kaon decays. Time reversal can be tested in several ways, for example, consider a two body reaction involving initial state spinless particles a & b, and final state spinless particles c & d: a(pa) + b(pb) Þ c(pc) + d(pd) Under time reversal we get c(-pc) + d(-pd) Þ a(-pa) + b(-pb) and applying parity we have c(pc) + d(pd) Þ a(pa) + b(pb) So PT symmetry would imply that the rate for the reaction a+b Þc+d should be the same as the rate for the reaction c+d Þa+b. This is the principal of detailed balance. Symmetries and Conservation Laws A system is said to have a symmetry if the laws of physics for that system are invariant under some transformation. For example, if the laws of physics are independent of location in space, then if y(x) is a solution of the Schrödinger equation i dy/dt = Hy & i dy/dt = y*H then y (x+d) is also a solution, where d is a displacement in x. Consider an infinitesmal transformation, dÞ0. (This is quite general, since space translation is a continuous operation, so any arbitrary finite translation, D, can be made in infinitesmal steps.) But remembering that the momentum operator is This must also be a solution of the Schrödinger equation Something (e.g. F) is conserved if its expectation value is independent of time. Assume the operator F does not depend on t, then In this case, we have assumed space invariance and the momentum operator has appeared, so let's check So the assumption that physics is invariant under space transformations requires that momentum be invariant! In general such continuous symmetries give additive conservation laws. In general, if a operator commutes with the Hamiltonian, the corresponding observable is a conserved quantity. e.g. Energy, momentum, and angular momentum conservation correspond to invariance under time and space translations, and space rotations. Since these translations can be described by the Lorentz group, we can say that energy-momentum conservation are due to the Lorentz group symmetry of the universe. Note, however, that we often write down Hamiltonians which do not conserve momentum or energy. e.g. When there is an external potential, or friction. In these cases momentum and energy may not be conserved, but this is because our wavefunction does not include the whole system. "Charge" Conservation Conservation of "charges" such as electric charge, baryon number, lepton number, strangeness, … correspond to invariance under various continuous symmetry transformations. (Note: A continuous symmetry may correspond to a charge which happens to be discretely quantized. Such quantization is due to the charges of the fundamental particles; it is not an intrinsic consequence of charge conservation or the corresponding symmetry.) For example, consider electric charge conservation. Let Q be the electric charge operator with eigenvalues q, i.e. Electric charge conservation is equivalent to saying that there is a symmetry under continuous transformations of the form where e is an arbitrary parameter. This is known as a U(1) gauge transformation, where U(1) is the name of the mathematical group describing the characteristics of the transformation. This transformation is just the finite version of an infinitesimal transformation: Transforming our normal free particle Hamiltonian we have This is known as a global U(1) gauge transformation if e is the same at all points in space & time. We can easily see that the free particle Schrödinger equation is invariant under the U(1) gauge transformation so electric charge is a conserved quantity in exact analogy with our previous discussion of momentum. Saying that there is a global symmetry is equivalent saying there is a conserved quantity. Things become more interesting if we ask if there is invariance under local U(1) gauge transformations, where we allow e to vary locally in space and time, i.e. e= e(x,t) is an arbitrary function. Then we have So local U(1) gauge invariance does not hold for our free particle wave equation. But what if we insist? The only way to make U(1) gauge invariance true is to change the Hamiltonian. In fact, we must add vector and scalar fields to the Hamiltonian so that the new Hamiltonian is where A(x,t) and V(x,t) transform under U(1) gauge transformations such that This Hamiltonian has exactly the form of the Hamiltonian of an electrically charged particle in a vector and scalar electromagnetic potential. i.e. Requiring local U(1) gauge invariance requires that the particle interact with a field. In this case all of electromagnetism can be summed up by saying there is a local U(1) gauge invariance of the universe.
57cc34a6b943b908	(Roughly) Daily Posts Tagged ‘simulation “Reality is broken”… Written by LW October 23, 2017 at 1:01 am “Acedia est tristitia vocem amputans”… Don’t worry, be happy at: “Before Sloth Meant Laziness, It Was the Spiritual Sin of Acedia.” * “Acedia is a sadness that silences the voice” a saying of Gregory of Nyssa, quoted by Aquinas As we work on our attitudes, we might send provocative birthday greetings to Jean Baudrillard; he was born on this date in 1929. A sociologist, philosopher, cultural theorist, political commentator, and photographer, he is best known for his analyses of media, contemporary culture, and technological communication, as well as his formulation of concepts such as simulation and hyperreality. He wrote widely– touching subjects including consumerism, gender relations, economics, social history, art, Western foreign policy, and popular culture– and is perhaps best known for Simulacra and Simulation (1981). Part of a generation of French thinkers that included Gilles Deleuze, Jean-François Lyotard, Michel Foucault, Jacques Derrida, and Jacques Lacan, with all of whom Baudrillard shared an interest in semiotics, he is often seen as a central to the post-structuralist philosophical school. “Animals have no unconscious, because they have a territory. Men have only had an unconscious since they lost a territory.” … Jean Baudrillard, Simulacra and Simulation As we get in touch with our inner omnivore, we might send passionate birthday greetings to The Maid of Orléans, Joan of Arc; she was born on this date in 1412. Joan entered history in spectacular fashion during the spring of 1429: following what she maintained was the command of God, Joan led the French Dauphin’s armies in a series of stunning military victories over the English, effectively reversing the course of the Hundred Years’ War. But she was captured in 1430 by the Burgundians, a faction (led by the Duke of Burgundy) allied with the English. The French King, Charles VII, declined to ransom her from the Burgundians who then “sold” her to the English. In December of that year, she was transferred to Rouen, the military headquarters and administrative capital in France of King Henry VI of England, and placed on trial for heresy before a Church court headed by a Bishop loyal to the English. Joan was convicted and executed in May of 1431. She was exonerated in 1456 when the verdict was reversed on appeal by the Inquisitor-General. She became a French national heroine, and in 1920 was canonized a saint of the Roman Catholic Church. “Boulainvilliers tells of her birth in Domrémy, and it is he who gives us an exact date, which may be the true one, saying that she was born on the night of Epiphany, 6 January” – Pernoud’s Joan of Arc By Herself and Her Witnesses, p. 98 Written by LW January 6, 2015 at 1:01 am Playing at life… For years both scientists and science fiction writers have suggested that a sufficiently advanced civilisation could– and thus would– create a simulated universe. And since simulations beget simulations (within simulations, within simulations, etc., etc.), there would ultimately be many more simulated universes than real ones… meaning that it would be more likely than not that any one universe– say, ours– is artificial. Silas Beane, working with a team at the University of Bonn says he have evidence this may be true. The Physics arXiv Blog at Technology Review explains: What they find is interesting. They say that the lattice spacing imposes a fundamental limit on the energy that particles can have. That’s because nothing can exist that is smaller than the lattice itself. But Beane and co calculate that the lattice spacing imposes some additional features on the spectrum. “The most striking feature…is that the angular distribution of the highest energy components would exhibit cubic symmetry in the rest frame of the lattice, deviating signiﬁcantly from isotropy,” they say. In other words, the cosmic rays would travel preferentially along the axes of the lattice, so we wouldn’t see them equally in all directions. That’s a measurement we could do now with current technology. Finding the effect would be equivalent to being able to to ‘see’ the orientation of lattice on which our universe is simulated… Read the whole mind-blowing story (including some comforting caveats) at “The Measurement That Would Reveal The Universe As A Computer Simulation” (and find Beane’s paper here). As we wonder if there’s a “reset” button, we might spare a thought for the extraordinary Erwin Rudolf Josef Alexander Schrödinger; he died on this date in 1961. A physicist best remembered in his field for his contributions to the development of quantum mechanics (e.g., the Schrödinger equation), and more generally for his “Schrödinger’s cat thought experiment– a critique of the Copenhagen interpretation of quantum mechanics– he also wrote on philosophy and theoretical biology. Indeed, both James Watson, and independently, Francis Crick, co-discoverers of the structure of DNA, credited Schrödinger’s What is Life? (1944), with its theoretical description of how the storage of genetic information might work, as an inspiration. – from Science and Humanism, 1951 Written by LW January 4, 2013 at 1:01 am From the Not-Sure-I-Really-Want-To-Know Department… As readers know, some physicists believe that the universe as we know it is actually a giant hologram, giving us the illusion of three-dimensions, while in fact all the action is occurring on a two-dimensional boundary region (see here, here, and here)… shadows on the walls of a cave, indeed. But lest one mistake that for the frontier of freakiness, others (c.f., e.g., here and here) believe that the existence we experience is nothing more (or less) than a Matrix-like simulation… A common theme of science fiction movies and books is the idea that we’re all living in a simulated universe—that nothing is actually real. This is no trivial pursuit: some of the greatest minds in history, from Plato, to Descartes, have pondered the possibility. Though, none were able to offer proof that such an idea is even possible. Now, a team of physicists working at the University of Bonn have come up with a possible means for providing us with the evidence we are looking for; namely, a measurable way to show that our universe is indeed simulated. They have written a paper describing their idea and have uploaded it to the preprint server arXiv… Phys.Org has the whole story at “Is it real? Physicists propose method to determine if the universe is a simulation“; the paper mentioned above can be downloaded here. As we reach for the “reset” button, we might send carefully-calculated birthday greetings to Paul Isaac Bernays; he was born on this date in 1888. A close associate of David Hilbert (of “Hilbert’s Hotel” fame), Bernays was one the foremost philosophers of mathematics of the Twentieth Century, who made important contributions to mathematical logic and axiomatic set theory. Bernays is perhaps best remembered for his revision and improvement of the (early, incomplete) set theory advanced by John von Neumann in the 1920s; Bernays’s work, with some subsequent modifications by Kurt Gödel, is now known as the Von Neumann–Bernays–Gödel set theory. Lest, per the simulation speculation above suggest that cosmology has a hammerlock on weirdness: Set theory is used, among other purposes, to describe the symmetries inherent in families of elementary particles and in crystals. Materials such as a liquid or a gas in equilibrium, made of uniformly distributed particles, exhibit perfect spatial symmetry—they look the same everywhere and in every direction… a condition that “breaks” at very low temperature, when the particles form crystals (which have some symmetry, but less)… Now Nobel Laureate Frank Wilczek has suggested that there may exist “Time Crystals“– whose structure would repeat periodically, as with an ordinary crystal, but in time rather than in space… a kind of “perpetual motion ‘machine'” (weirder yet, one that doesn’t violate the laws of thermodynamics). Paul Bernays Written by LW October 17, 2012 at 1:01 am Special Summer Cheesecake Edition… Y’all be good… %d bloggers like this:
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Dieter Zeh Ernst Zermelo Wojciech Zurek Fritz Zwicky Free Will Mental Causation James Symposium Stuart Kauffman Stuart Kauffman is a medical doctor, a theoretical biologist, a MacArthur fellow, and a strong defender of the idea that the self-organization of complex adaptive systems must be added to Darwinian evolutionary theory to account for the emergence of life. Kauffman was an early faculty member in residence at the Santa Fe Institute, which is well known for its contributions to complexity theory, chaos theory, and information theory. His strong views on the need to augment natural selection have been controversial among systems biologists, but Kauffman said that at the outset about his work, which he called a "heretical possibility." The origin of life, rather than having been vastly improbable, is instead an expected collective property of complex systems of catalytic polymers and the molecules on which they act. Life, in a deep sense, crystallized as a collective self-reproducing metabolism in a space of possible organic reactions. If this is true, then the routes to life are many and its origin is profound yet simple. This view is indeed heretical. Most students of the origin of life hold that life must be based on the self-templating character of RNA or RNA-like molecules. Because of such self-templating, any RNA molecule would specify its base pair complement; hence a "nude gene" might reproduce itself. After that, according to most thinkers, these simplest replicating molecules built up around themselves the complex set of RNA, DNA, and protein molecules which constituted a self-reproducing system coordinating a metabolic flow and capable of evolving. Chapter 7 unfolds this new view, which is based on the discovery of an expected phase transition from a collection of polymers which do not reproduce themselves to a slightly more complex collection of polymers which do jointly catalyze their own reproduction. In this theory of the origin of life, it is not necessary that any molecule reproduce itself. Rather, a collection of molecules has the property that the last step in the formation of each molecule is catalyzed by some molecule in the system. The phase transition occurs when some critical complexity level of molecular diversity is surpassed. At that critical level, the ratio of reactions among the polymers to the number of polymers in the system passes a critical value, and a connected web of catalyzed reactions linking the polymers arises and spans the molecular species in the system. This web constitutes the crystallization of catalytic closure such that the system of polymers becomes collectively self-reproducing. While heretical, this new body of theory is robust in the sense that the conclusions hold for a wide variety of assumptions about prebiotic chemistry, about the kinds of polymers involved, and about the capacities of those polymers to catalyze reactions transforming either themselves or other, very similar polymers. It is also robust in leading to a fundamental new conclusion: Molecular systems, in principle, can both reproduce and evolve without having a genome in the familiar sense of a template-replicating molecular species. It is no small conclusion that heritable variation, and hence adaptive evolution, can occur in a self-reproducing molecular system lacking a genome. Since Darwin's theory of evolution, Mendel's discovery of the "atoms" of heredity, and Weismann 's theory of the germ plasm, biologists have argued that evolution requires a genome. False, I claim. Also, this new body of theory is fully testable. If correct, sufficiently complex systems of RNA or protein polymers should be collectively autocatalytic. In Chapter 8 these new concepts are extended to the crystallization of a connected metabolism. I strongly suspect that, rather than having formed piecemeal, a connected metabolism, like a self-reproducing set of catalytic polymers, emerged spontaneously as a phase transition when a sufficient number of potentially catalytic polymers were mixed with a sufficiently complex set of organic molecules. In this condition, a critical ratio of number of catalyzed reactions to number of molecular species present is surpassed, and a connected web of catalyzed transformations arises. Life began whole and integrated, not disconnected and disorganized, Kauffman's ideas about autocatalytic systems are shared by Terrence Deacon. Kauffman thought that he might even discover "laws" of self-organization. In his 1995 book, At Home in the Universe, he identified the discovery of such laws as showing that human life followed directly from these pre-existing laws, which would replace the arbitrary and purposeless system of Darwinian natural selection. Humans (and even paradise?) would be implicit in those laws. Random variation, selection sifting. Here is the core, the root. Here lies the brooding sense of accident, of historical contingency, of design by elimination. At least physics, cold in its calculus, implied a deep order, an inevitability. Biology has come to seem a science of the accidental, the ad hoc, and we just one of the fruits of this ad hocery. Were the tape played over, we like to say, the forms of organisms would surely differ dramatically. We humans, a trumped-up, tricked-out, horn-blowing, self-important presence on the globe, need never have occurred. So much for our pretensions; we are lucky to have our hour. So much, too, for paradise. Where, then, does this order come from, this teeming life I see from my window: urgent spider making her living with her pre-nylon web, coyote crafty across the ridgetop, muddy Rio Grande aswarm with nosee- ems (an invisible insect peculiar to early evenings)? Since Darwin, we turn to a single, singular force, Natural Selection, which we might as well capitalize as though it were the new deity. Random variation, selection-sifting. Without it, we reason, there would be nothing but incoherent disorder. I shall argue in this book that this ldea is wrong. For, as we shall see, the emerging sciences of complexity begin to suggest that the order is not all accidental, that vast veins of spontaneous order lie at hand. Laws of complexity spontaneously generate much of the order of the natural world. It is only then that selection comes into play, further molding and refining. Such veins of spontaneous order have not been entirely unknown, yet they are just beginning to emerge as powerful new clues to the origins and evolution of life. We have all known that simple physical systems exhibit spontaneous order: an oil droplet in water forms a sphere; snowflakes exhibit their evanescent sixfold symmetry. What is new is that the range of spontaneous order is enormously greater than we have supposed. Profound order is being discovered in large, complex, and apparently random systems. I believe that this emergent order underlies not only the origin of life itself, but much of the order seen in organisms today. So, too, do many of my colleagues, who are starting to find overlapping evidence of such emergent order in all different kinds of complex systems. The existence of spontaneous order is a stunning challenge to our settled ideas in biology since Darwin. Most biologists have believed for over a century that selection is the sole source of order in biology, that selection alone is the "tinkerer" that crafts the forms. But if the forms selection chooses among were generated by laws of complexity, then selection has always had a handmaiden. It is not, after all, the sole source of order, and organisms are not just tinkered-together contraptions, but expressions of deeper natural laws. If all this is true, what a revision of the Darwinian worldview will lie before us! Not we the accidental, but we the expected. Information philosophy completely agrees with the idea of replacing the idea of God with the cosmic creative process. Kauffman's plaintive "so much for paradise" lament above is extended in his latest book, Reinventing the Sacred, to the thesis that his "new view of God" would replace a "creator" with the "ceaseless creativity of the universe, biosphere, and human culture and history." Creativity is emergent and unexplainable by reduction to the "causally closed" world of natural law. The augmentation of Darwinian natural selection with "partially lawless" self-organization of Kauffman's first two books is extended to finding a new "place for our spirituality." Since humans invented God, he says, we can reinvent "God as the natural creativity of the universe." This is an eminently worthwhile project. Its success will depend on the appeal of the arguments and explanations to secularists wanting substitute reasons beyond the humanism that Kauffman calls "too thin" (p.xii). For him, the new sacred will provide explanations for mind, consciousness, agency, meaning, purpose, values, and life itself. Can Kauffman's explanations sway those with a belief in a God that promises an escape from death, an afterlife in paradise? Probably not, but short of that, deeper explanations for great problems in biology, psychology, and philosophy make Kauffman's project worth examining very closely. Like many emergentists, Kauffman attacks the reductionist views epitomized by the great physicist Steven Weinberg's two famous dicta: "The explanatory arrows always point downward" to physics, and "The more we comprehend the universe, the more pointless it seems." In brief, reductionism is the view that society is to be explained in terms of people, people in terms of organs, organs by cells, cells by biochemistry, biochemistry by chemistry, and chemistry by physics. To put it even more crudely, it is the view that in the end, all of reality is nothing but whatever is "down there" at the current base of physics: quarks or the famous strings of string theory, plus the interactions among these entities. Physics is held to be the basic science in terms of which all other sciences will ultimately be understood. As Weinberg puts it, all explanations of higher-level entities point down to physics. And in physics there are only happenings, only facts. What does Kauffman put in place of reductionism, which he calls the "Galilean ideal?" Beyond the complexity and self-organization of complex adaptive auto-catalytic systems of his earlier work, he adds a "partially lawless" element that he sees in a non-standard interpretation of quantum mechanics called decoherence. His explanation of mind and consciousness depends on his idea of a "poised state" must conscious mind be classical, rather than quantum or a mixture of quantum and classical? Could consciousness be a very special poised state between quantum coherence and decoherence to classicity by which "immaterial, nonobjective" mind "acts" on matter? Most physicists say this is impossible. As I will show in the next chapter, recent theories and experiments suggest otherwise. I am hardly the first person to assert that consciousness may be related to quantum phenomena. In 1989, the physicist Roger Penrose, in The Emperor's New Mind, proposed that consciousness is related to quantum gravity, the still missing union of general relativity and quantum mechanics. Here I will take a different tack and suggest that consciousness is associated with a poised state between quantum "coherent" behavior and what is called "decoherence" of quantum possibilities to "classical" actual events. I will propose that this is how the immaterial—not objectively real—mind has consequences for the actual classical physical world. I warn you that this hypothesis is highly controversial—the most scientifically improbable thing I say in this book. Yet as we will see, there appear to be grounds to investigate it seriously... I will make use of decoherence to classical behavior as the means by which a quantum coherent conscious mind of pure possibilities can have actual classical consequences in the physical world. This will be how mind has consequences for matter. Note that I do not say "how mind acts on matter," because I am proposing that the consequences in the classical world of the quantum mind are due to decoherence, which is not itself causal in any normal classical sense. Thus I will circumvent the worry about how the immaterial mind has causal effects on matter by asserting that the quantum coherent mind decoheres to have consequences for the classical world, but does not act causally on the material world. As we will see, this appears to circumvent the very old problem of mental causation and provide a possible, if still scientifically unlikely, solution to how the mind "acts" on matter... The cornerstone of my theory is that the conscious mind is a persistently poised quantum coherent-decoherent system, forever propagating quantum coherent behavior, yet forever also decohering to classical behavior. I describe the requirements for this theory in more detail below. Here, mind—consciousness, res cogitans—is identical with quantum coherent immaterial possibilities, or with partially coherent quantum behavior, yet via decoherence, the quantum coherent mind has consequences that approach classical behavior so very closely that mind can have consequences that create actual physical events by the emergence of classicity. Thus, res cogitans has consequences for res extensa! Immaterial mind has consequences for matter. More, in the poised quantum coherent-decoherent biomolecular system I will posit, quantum coherent, or partially coherent, possibilities themselves continue to propagate, but remain acausal. This will become mental processes begetting mental processes. Information philosophy agrees that mind is immaterial, but it is also "objectively real" Some physicists will object to my use of the word immaterial with reject to quantum phenomenon. They would want to say instead, "not objectively real," like a rock. I am happy to accept this language, and will use immaterial to mean "not objectively real." In a recent paper on arXiv, Kauffman speculated about solutions to the problem of free will, the mind-body problem, and suggested a new interpretation of quantum mechanics. He calls the "causal closure" of classical physics (basically reduction ) the source of the idea that we are machines and our minds are epiphenomenal. He proposes a new dualism of ontologically real actuals (he uses Descartes' term res extensa) and ontologically real possibles (he calls res potentia, which could come from Aristotle or Heisenberg). The Schrödinger equation describes the evolution of these possibilities. He puzzles over nonlocality. And he presents his "poised realm," which hovers reversibly between quantum coherent and "classical" worlds. He then proposes a "quantum mind" in which decoherence produces an "acausal loss of phase information from the open quantum system to the universe." This has "acausal consequences for the classical 'meat' of the brain," he says. Kauffman suggests that he can "decircularize" the "Strong Free Will Theorem of John Conway and Simon Kochen. He first states the two part standard argument against free will. If the mind is determined, "classical physics holds we have no free will at all." If we try to use quantum indeterminism to achieve an ontologically free will, it is merely random...So a random quantum event occurs in my brain,...but I am not “responsible”, the quantum event was random. So even if measurement is real, and ontologically indeterminate, so underlies a “free will”, that will cannot be responsible. Kauffman proposes "a broad new formulation of quantum mechanics in terms a new triad: Actuals, Possibles and Mind - conscious observation acausally mediating measurement, and doing. Here new actuals create new possibles which are available via mind to be measured to create new actuals to create new "adjacent possibles" in a persistent becoming of the universe. I want us to consider a totally new view of quantum mechanics and reality, consisting of ontologically real actuals that obey the law of the excluded middle, ontologically real possibles that do not obey the law of the excluded middle in quantum behavior before measurement, and mind measuring and responsible free willed. In this view, measurement creates new actuals that acausally and outside of space and inside time consistent with Special Relativity, can instantaneously and acausally alter what is now possible, hence account for instantaneous changes in wave functions upon measurement and non-locality. If new "actuals" are created acausally by quantum mind "measurements," the outcomes are statistical, that is random. We need the randomness to create the possibles from which the outcome/choice/selection is adequately determined In turn new possibles are available for mind to measure and do, creating new actuals, in a persistent cycle of quantum enigma free willed and conscious becoming in a radically participatory universe with a non-epiphenomenal “cosmic” consciousness and doings wherever measurements happen. Normal \| Teacher \| Scholar
1732d385a7771ca0	#12: The Stoic Formula for a Good Life Take a look at the following formula: It is not the formula for having a happy and meaningful life; it is instead the time-dependent Schrödinger equation. It describes how the quantum state of a physical system changes over time. If you want to have a good life, you do not need to master this formula, which is a good thing since so many of us have forgotten how to do partial differential equations. Now take a look at this formula: X = the number of days you have left to live It is a formula that anyone can use, even someone with the most rudimentary mathematical ability. It also comes as close as anything to being the Stoic formula for having a happy, meaningful life. For most people, most of the time, the exact value of X is unknown. But one thing that is certain is that X is finite: none of us will live forever. Some people live in denial of this fact. Many more resent it: they wish they could live forever. But from a Stoic point of view, if we wish to have a happy, meaningful life, it is very important that we keep in mind that our own personal X is a finite number. Here’s why. If you have X days left to live, today will represent 1/X of your life. Thus, if X = 2, today represents 1/2 of your life, and if X = 100, today represents 1/100 of your life. But suppose X were infinite. Then the value of today would be 1/∞, which is equal to 0. What this means is that if you go through life believing that you are immortal, your days will have zero value. You can waste a day and not lose anything by doing so. You will end up like Bill Murray in the movie Groundhog Day—which is, by the way, on my list of the top five “meaning of life” movies. Or you will end up like many young people who have not yet acknowledged their own mortality. If you believe that you have finitely many days to live, though, they will be things of value, and the fewer of them you have, the more valuable they will be. You will realize that life is simply too short for the kind of nonsense that many people allow themselves to get distracted by. Time you spend on petty quarrels is precious time wasted. So is time spent worrying about things you cannot control. I have known bedridden 90-year-olds who were vastly more appreciative of life and therefore more fully alive than many 20-year-olds. When a 90-year-old opens her eyes in the morning and sees the ceiling, it is cause for celebration: another day of life has been granted, another chance to get it right! A 20-year-old, by way of contrast, might wake up angry about something, only to spend the rest of the day complaining about it. She might complain about her clothes, her hair, her job, her boyfriend—you name it. She might vent these complaints because she thinks it is the duty of other people to remove the obstacles to her happiness. Or maybe all she wants from other people is their sympathy. Either way, it is a recipe for a miserable existence. The illusion of immortality comes with a hefty price tag. The Stoics spent time contemplating their own death. They didn’t fear death; nor did they fixate on it. But they did periodically entertain flickering thoughts about it. By doing so, they put their daily life into perspective: yes, today may not be going as planned, but it is nevertheless a precious thing. It is one of the X days that I have left to live, so I must not waste it. Indeed, I must do my best to savor it. Yes, the person you are talking to may be annoying and even offensive. Yes, your boss may be overly demanding. Yes, it may be raining and you have forgotten your umbrella. Yes, yes, yes. And yet, here you are, alive and breathing, in a world that is profoundly beautiful, if only you make it your business to look for its beauty.
1b0d6f163d626830	Call For Abstract Session 1Organic and Inorganic Chemistry The word "organic" means something very different in chemistry than it does when you're talking about produce and food. Organic compounds and inorganic compounds form the basis of chemistry. The primary difference between organic compounds and inorganic compounds is that organic compounds always contain carbon while most inorganic compounds do not contain carbon. Also, nearly all organic compounds contain carbon-hydrogen or C-H bonds. Organic and inorganic chemistry are two of the main disciplines of chemistry. An organic chemist studies organic molecules and reactions, while an inorganic chemistry focuses on inorganic reactions. Session 2Analytical Chemistry Analytical chemistry is often described as the area of chemistry responsible for characterizing the composition of matter, both qualitatively (Is there any lead in this sample?) and quantitatively (How much lead is in this sample?). Analytical chemistry is not a separate branch of chemistry, but simply the application of chemical knowledge. Session 3Green Chemistry Session 4Biochemistry Biochemistry is the branch of science that explores the chemical processes within and related to living organisms. It is a laboratory based science that brings together biology and chemistry. By using chemical knowledge and techniques, biochemists can understand and solve biological problems. Biochemistry allows us to understand how chemical processes such as respiration, produces life functions in all living organisms. Session 5Environmental Chemistry Session 6Medicinal Chemistry Medicinal chemistry deals with the design, optimization and development of chemical compounds for use as drugs. It is inherently a multidisciplinary topic — beginning with the synthesis of potential drugs followed by studies investigating their interactions with biological targets to understand the medicinal effects of the drug, its metabolism and side-effects. Session 7Materials Chemistry Materials chemistry provides the link between atomic, molecular and supramolecular behaviour and the useful properties of a material. It lies at the core of many chemical-using industries. A wide range of materials, includes organic materials and polymers, nanomaterials and nanoporous materials. Session 8Petro Chemistry Petrochemistry is the branch of chemistry defines refining and processing of chemistry concerned with crude oil and fossil fuel. Examples of petrochemicals includes: ammonia, acetylene, benzene, and polystyrene. Petrochemistry covers the areas of good style of materials like plastics, explosives, fertilizers, and artificial fibers. Session 9Nuclear Chemistry Nuclear chemistry is the subdiscipline of chemistry that is concerned with changes in the nucleus of elements. These changes are the source of radioactivity and nuclear power. Since radioactivity is associated with nuclear power generation, the concomitant disposal of radioactive waste, and some medical procedures, everyone should have a fundamental understanding of radioactivity and nuclear. Session 10Physical and Theoretical Chemistry Physical Chemistry is the application of physical principles and measurements to understand the properties of matter, as well as for the development of new technologies for the environment, energy and medicine. Theoretical and computational tools are to provide atomic-level understanding for applications such as: nanodevices for bio-detection and receptors, interfacial chemistry of catalysis and implant Session 11Biological Chemistry Biological chemistry, is the study of chemical processes within and relating to living organisms. Biological chemistry is closely related to molecular biology, the study of the molecular mechanisms by which genetic information encoded in DNA is able to result in the processes of life. Session 12Geochemistry Geochemistry is the branch of Earth Science that applies chemical principles to deepen an understanding of the Earth system and systems of other planets. Geochemists consider Earth composed of discrete spheres — rocks, fluids, gases and biology — that exchange matter and energy over a range of time scales. Session 13Quantum Chemistry Quantum chemistry, a sub-discipline of chemistry that focuses on the properties and behavior of subatomic particles, especially electrons. Quantum chemistry applies quantum mechanics to the theoretical study of chemical systems. It aims, in principle, to solve the Schrödinger equation for the system under scrutiny; however, its complexity for all but the simplest of atoms or molecules requires simplifying assumptions and approximations, creating a trade-off between accuracy and computational cost. Session 14Polymer Chemistry Polymer chemistry is a chemistry subdiscipline that deals with the structures, chemical synthesis and properties of polymers, primarily synthetic polymers such as plastics and elastomers. Polymer chemistry is related to the broader field of polymer science, which also encompasses polymer physics and polymer engineering. Session 15Clinical Chemistry Clinical Chemistry also known as chemical pathology, clinical biochemistry or medical biochemistry is the area of chemistry that is generally concerned with analysis of bodily fluids for diagnostic and therapeutic purposes. It is an applied form of biochemistry. The discipline originated with the use of simple chemical reaction tests for various components of blood and urine. Session 16Electrochemistry Session 17Synthetic Chemistry Synthetic Chemistry is the study of the connection between structure and reactivity of organic molecules. Synthetic chemistry is most frequently used in the preparation of mono-functional and di-functional compounds from the smaller entities. It is widely used for the production of organic compounds that are having commercial interest. Session 18Natural Product Chemistry Natural Products Chemistry deals with chemical compounds found in nature that usually has a pharmacological or biological activity for use in pharmaceutical drug discovery and drug design. Natural Products Chemistry is related to the study of chemistry and biochemistry of naturally occurring compounds or the biology of living systems from which they are obtained. Session 19Agricultural and Food Chemistry Agricultural and food chemistry deals with the chemistry and biochemistry of agriculture and food with a focus on original research representing complete studies, rather than incremental studies it covers the chemistry of pesticides, veterinary drugs fertilizers, and other agrochemicals, together with their metabolism, toxicology, and environmental fate. Session 20Nanochemistry Session 21Industrial Chemistry Industrial chemistry is concerned with using chemical and physical processes to transform raw materials into products that are beneficial to humanity. This includes the manufacture of basic chemicals to produce products for various industries.
47ae4ae6021efb4e	Vector Space Get Vector Space essential facts below. View Videos or join the Vector Space discussion. Add Vector Space to your Like2do.com topic list for future reference or share this resource on social media. Vector Space Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs. Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations. They offer a framework for Fourier expansion, which is employed in image compression routines, and they provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. Introduction and definition The concept of vector space will first be explained by describing two particular examples: First example: arrows in the plane The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead. The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently, 2w is the sum w + w. Moreover, (-1)v = -v has the opposite direction and the same length as v (blue vector pointing down in the right image). Vector addition: the sum v + w (black) of the vectors v (blue) and w (red) is shown. Scalar multiplication: the multiples -v and 2w are shown. Second example: ordered pairs of numbers A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is defined as follows: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) a (x, y) = (ax, ay). The first example above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points. In this article, vectors are represented in boldface to distinguish them from scalars.[nb 1] A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. • The first operation, called vector addition or simply addition + : V × V -> V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (Note that the resultant vector is also an element of the set V ). • The second operation, called scalar multiplication · : F × V -> V, takes any scalar a and any vector v and gives another vector av. (Similarly, the vector av is an element of the set V ). Elements of V are commonly called vectors. Elements of F are commonly called scalars. In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively. To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[1] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F. Axiom Meaning Associativity of addition u + (v + w) = (u + v) + w Commutativity of addition u + v = v + u Identity element of addition There exists an element 0 ? V, called the zero vector, such that v + 0 = v for all v ? V. Inverse elements of addition For every v ? V, there exists an element -v ? V, called the additive inverse of v, such that v + (-v) = 0. Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v [nb 2] Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F. Distributivity of scalar multiplication with respect to vector addition a(u + v) = au + av Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands: (xv, yv) + (xw, yw) = (xw, yw) + (xv, yv). Likewise, in the geometric example of vectors as arrows, v + w = w + v since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space. Subtraction of two vectors and division by a (non-zero) scalar can be defined as v - w = v + (-w), v/a = (1/a)v. When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers C, the vector space is called a complex vector space. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed field F. The notion is then known as an F-vector spaces or a vector space over F. A field is, essentially, a set of numbers possessing addition, subtraction, multiplication and division operations.[nb 3] For example, rational numbers form a field. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general vector spaces, no notion of nearness, angles or distances. To deal with such matters, particular types of vector spaces are introduced; see below. Alternative formulations and elementary consequences Vector addition and scalar multiplication are operations, satisfying the closure property: u + v and av are in V for all a in F, and u, v in V. Some older sources mention these properties as separate axioms.[2] In the parlance of abstract algebra, the first four axioms are equivalent to requiring the set of vectors to be an abelian group under addition. The remaining axioms give this group an F-module structure. In other words, there is a ring homomorphism f from the field F into the endomorphism ring of the group of vectors. Then scalar multiplication av is defined as (f(a))(v).[3] There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to the additive group of vectors: for example the zero vector 0 of V and the additive inverse -v of any vector v are unique. Other properties follow from the distributive law, for example av equals 0 if and only if a equals 0 or v equals 0. Vector spaces stem from affine geometry via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutions to an equation of two variables with points on a plane curve.[4] In 1804, to achieve geometric solutions without using coordinates, Bolzano introduced certain operations on points, lines and planes, which are predecessors of vectors.[5] His work was then used in the conception of barycentric coordinates by Möbius in 1827.[6] In 1828 C. V. Mourey suggested the existence of an algebra surpassing not only ordinary algebra but also two-dimensional algebra created by him searching a geometrical interpretation of complex numbers.[7] The definition of vectors was founded on Bellavitis' notion of the bipoint, an oriented segment of which one end is the origin and the other a target, then further elaborated with the presentation of complex numbers by Argand and Hamilton and the introduction of quaternions and biquaternions by the latter.[8] They are elements in R2, R4, and R8; their treatment as linear combinations can be traced back to Laguerre in 1867, who also defined systems of linear equations. In 1857, Cayley introduced matrix notation, which allows for a harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.[9] In his work, the concepts of linear independence and dimension, as well as scalar products, are present. In fact, Grassmann's 1844 work exceeds the framework of vector spaces, since his consideration of multiplication led him to what are today called algebras. Peano was the first to give the modern definition of vector spaces and linear maps in 1888.[10] An important development of vector spaces is due to the construction of function spaces by Lebesgue. This was later formalized by Banach and Hilbert, around 1920.[11] At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.[12] Vector spaces, including infinite-dimensional ones, then became a firmly established notion, and many mathematical branches started making use of this concept. Coordinate spaces The simplest example of a vector space over a field F is the field itself, equipped with its standard addition and multiplication. More generally, a vector space can be composed of n-tuples (sequences of length n) of elements of F, such as (a1, a2, ..., an), where each ai is an element of F.[13] A vector space composed of all the n-tuples of a field F is known as a coordinate space, usually denoted Fn. The case n = 1 is the above-mentioned simplest example, in which the field F is also regarded as a vector space over itself. The case F = R and n = 2 was discussed in the introduction above. Complex numbers and other field extensions The set of complex numbers C, i.e., numbers that can be written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: (x + iy) + (a + ib) = (x + a) + i(y + b) and c ? (x + iy) = (c ? x) + i(c ? y) for real numbers x, y, a, b and c. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same (i.e., it is isomorphic) to the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number x + i y as representing the ordered pair (x, y) in the complex plane then we see that the rules for sum and scalar product correspond exactly to those in the earlier example. More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field F containing a smaller field E is an E-vector space, by the given multiplication and addition operations of F.[14] For example, the complex numbers are a vector space over R, and the field extension is a vector space over Q. Function spaces Functions from any fixed set ? to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function (f + g) given by (f + g)(w) = f(w) + g(w), and similarly for multiplication. Such function spaces occur in many geometric situations, when ? is the real line or an interval, or other subsets of R. Many notions in topology and analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.[15] Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of functional analysis, see below. Algebraic constraints also yield vector spaces: the vector space F[x] is given by polynomial functions: f(x) = r0 + r1x + ... + rn-1xn-1 + rnxn, where the coefficients r0, ..., rn are in F.[16] Linear equations Systems of homogeneous linear equations are closely tied to vector spaces.[17] For example, the solutions of a + 3b + c = 0 4a + 2b + 2c = 0 are given by triples with arbitrary a, b = a/2, and c = -5a/2. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely Ax = 0, where A = is the matrix containing the coefficients of the given equations, x is the vector (a, b, c), Ax denotes the matrix product, and 0 = (0, 0) is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example, f(x) + 2f?(x) + f(x) = 0 yields f(x) = a e-x + bx e-x, where a and b are arbitrary constants, and ex is the natural exponential function. Basis and dimension A vector v in R2 (blue) expressed in terms of different bases: using the standard basis of R2v = xe1 + ye2 (black), and using a different, non-orthogonal basis: v = f1 + f2 (red). Bases allow one to represent vectors by a sequence of scalars called coordinates or components. A basis is a (finite or infinite) set B = {bi}i ? I of vectors bi, for convenience often indexed by some index set I, that spans the whole space and is linearly independent. "Spanning the whole space" means that any vector v can be expressed as a finite sum (called a linear combination) of the basis elements: where the ak are scalars, called the coordinates (or the components) of the vector v with respect to the basis B, and bik (k = 1, ..., n) elements of B. Linear independence means that the coordinates ak are uniquely determined for any vector in the vector space. For example, the coordinate vectors e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0), to en = (0, 0, ..., 0, 1), form a basis of Fn, called the standard basis, since any vector (x1, x2, ..., xn) can be uniquely expressed as a linear combination of these vectors: (x1, x2, ..., xn) = x1(1, 0, ..., 0) + x2(0, 1, 0, ..., 0) + ... + xn(0, ..., 0, 1) = x1e1 + x2e2 + ... + xnen. The corresponding coordinates x1, x2, ..., xn are just the Cartesian coordinates of the vector. Every vector space has a basis. This follows from Zorn's lemma, an equivalent formulation of the Axiom of Choice.[18] Given the other axioms of Zermelo-Fraenkel set theory, the existence of bases is equivalent to the axiom of choice.[19] The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. Dimension theorem for vector spaces).[20] It is called the dimension of the vector space, denoted by dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.[21] The dimension of the coordinate space Fn is n, by the basis exhibited above. The dimension of the polynomial ring F[x] introduced above is countably infinite, a basis is given by 1, x, x2, ... A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.[nb 4] Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation.[22] For example, the solution space for the above equation is generated by e-x and xe-x. These two functions are linearly independent over R, so the dimension of this space is two, as is the degree of the equation. A field extension over the rationals Q can be thought of as a vector space over Q (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of Q, and otherwise ignoring the field multiplication). The dimension (or degree) of the field extension Q(?) over Q depends on ?. If ? satisfies some polynomial equation with rational coefficients qn, ..., q0 (in other words, if ? is algebraic), the dimension is finite. More precisely, it equals the degree of the minimal polynomial having ? as a root.[23] For example, the complex numbers C are a two-dimensional real vector space, generated by 1 and the imaginary unit i. The latter satisfies i2 + 1 = 0, an equation of degree two. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C). If ? is not algebraic, the dimension of Q(?) over Q is infinite. For instance, for ? = ? there is no such equation, in other words ? is transcendental.[24] Linear maps and matrices The relation of two vector spaces can be expressed by linear map or linear transformation. They are functions that reflect the vector space structure--i.e., they preserve sums and scalar multiplication: f(x + y) = f(x) + f(y) and f(a · x) = a · f(x) for all x and y in V, all a in F.[25] An isomorphism is a linear map f : V -> W such that there exists an inverse map g : W -> V, which is a map such that the two possible compositions f ? g : W -> W and g ? f : V -> V are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective).[26] If there exists an isomorphism between V and W, the two spaces are said to be isomorphic; they are then essentially identical as vector spaces, since all identities holding in V are, via f, transported to similar ones in W, and vice versa via g. Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces. For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow v departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the x- and y-component of the arrow, as shown in the image at the right. Conversely, given a pair (x, y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v. Linear maps V -> W between two vector spaces form a vector space HomF(V, W), also denoted L(V, W).[27] The space of linear maps from V to F is called the dual vector space, denoted V.[28] Via the injective natural map V -> V, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.[29] Once a basis of V is chosen, linear maps f : V -> W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.[30] If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.[31] Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional F-vector space V is isomorphic to Fn. There is, however, no "canonical" or preferred isomorphism; actually an isomorphism ? : Fn -> V is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via ?. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context, see below. A typical matrix Matrices are a useful notion to encode linear maps.[32] They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix A gives rise to a linear map from Fn to Fm, by the following , where denotes summation, or, using the matrix multiplication of the matrix A with the coordinate vector x: x ? Ax. Moreover, after choosing bases of V and W, any linear map f : V -> W is uniquely represented by a matrix via this assignment.[33] The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r1, r2, and r3. The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.[34] The linear transformation of Rn corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive. Eigenvalues and eigenvectors Endomorphisms, linear maps f : V -> V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying ?v = f(v), where ? is a scalar, is called an eigenvector of f with eigenvalue ?.[nb 5][35] Equivalently, v is an element of the kernel of the difference f - ? · Id (where Id is the identity map V -> V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue ? is equivalent to det(f - ? · Id) = 0. By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in ?, called the characteristic polynomial of f.[36] If the field F is large enough to contain a zero of this polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.[nb 6] The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below. Basic constructions In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object X by specifying the linear maps from X to any other vector space. Subspaces and quotient spaces A line passing through the origin (blue, thick) in R3 is a linear subspace. It is the intersection of two planes (green and yellow). A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a linear subspace of V, or simply a subspace of V, when the ambient space is unambiguously a vector space.[37][nb 7] Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing the set S. Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S.[38] A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension n, a vector hyperplane is thus a subspace of dimension n - 1. The counterpart to subspaces are quotient vector spaces.[39] Given any subspace W ? V, the quotient space V/W ("V modulo W") is defined as follows: as a set, it consists of v + W = {v + w : w ? W}, where v is an arbitrary vector in V. The sum of two such elements v1 + W and v2 + W is (v1 + v2) + W, and scalar multiplication is given by a · (v + W) = (a · v) + W. The key point in this definition is that v1 + W = v2 + W if and only if the difference of v1 and v2 lies in W.[nb 8] This way, the quotient space "forgets" information that is contained in the subspace W. The kernel ker(f) of a linear map f : V -> W consists of vectors v that are mapped to 0 in W.[40] Both kernel and image im(f) = {f(v) : v ? V} are subspaces of V and W, respectively.[41] The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field F) is an abelian category, i.e. a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups.[42] Because of this, many statements such as the first isomorphism theorem (also called rank-nullity theorem in matrix-related terms) V / ker(f) ? im(f). and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups. An important example is the kernel of a linear map x ? Ax for some fixed matrix A, as above. The kernel of this map is the subspace of vectors x such that Ax = 0, which is precisely the set of solutions to the system of homogeneous linear equations belonging to A. This concept also extends to linear differential equations , where the coefficients ai are functions in x, too. In the corresponding map the derivatives of the function f appear linearly (as opposed to f(x)2, for example). Since differentiation is a linear procedure (i.e., (f + g)? = f? + g ? and (c·f)? = c·f? for a constant c) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D(f) = 0 form a vector space (over R or C). Direct product and direct sum The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. The direct product of a family of vector spaces Vi consists of the set of all tuples (vi)i ? I, which specify for each index i in some index set I an element vi of Vi.[43] Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum (also called coproduct and denoted ), where only tuples with finitely many nonzero vectors are allowed. If the index set I is finite, the two constructions agree, but in general they are different. Tensor product The tensor product V ?FW, or simply V ? W, of two vector spaces V and W is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map g : V × W -> X is called bilinear if g is linear in both variables v and w. That is to say, for fixed w the map v ? g(v, w) is linear in the sense above and likewise for fixed v. The tensor product is a particular vector space that is a universal recipient of bilinear maps g, as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors v1 ? w1 + v2 ? w2 + ... + vn ? wn, subject to the rules a · (v ? w) = (a · v) ? w = v ? (a · w), where a is a scalar, (v1 + v2) ? w = v1 ? w + v2 ? w, and v ? (w1 + w2) = v ? w1 + v ? w2.[44] Commutative diagram depicting the universal property of the tensor product. These rules ensure that the map f from the V × W to V ? W that maps a tuple (v, w) to v ? w is bilinear. The universality states that given any vector space X and any bilinear map g : V × W -> X, there exists a unique map u, shown in the diagram with a dotted arrow, whose composition with f equals g: u(v ? w) = g(v, w).[45] This is called the universal property of the tensor product, an instance of the method--much used in advanced abstract algebra--to indirectly define objects by specifying maps from or to this object. Vector spaces with additional structure From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question--crucial to analysis--whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures. A vector space may be given a partial order [46] For example, n-dimensional real space Rn can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions f = f+ - f-, where f+ denotes the positive part of f and f- the negative part.[47] Normed vector spaces and inner product spaces "Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted and , respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm . Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.[48] Coordinate space Fn can be equipped with the standard dot product: In R2, this reflects the common notion of the angle between two vectors x and y, by the law of cosines: Because of this, two vectors satisfying are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R4 endowed with the Lorentz product In contrast to the standard dot product, it is not positive definite: also takes negative values, for example for . Singling out the fourth coordinate--corresponding to time, as opposed to three space-dimensions--makes it useful for the mathematical treatment of special relativity. Topological vector spaces Convergence questions are treated by considering vector spaces V carrying a compatible topology, a structure that allows one to talk about elements being close to each other.[50][51] Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in V, and a in F vary by a bounded amount, then so do x + y and ax.[nb 9] To make sense of specifying the amount a scalar changes, the field F also has to carry a topology in this context; a common choice are the reals or the complex numbers. In such topological vector spaces one can consider series of vectors. The infinite sum denotes the limit of the corresponding finite partial sums of the sequence (fi)i?N of elements of V. For example, the fi could be (real or complex) functions belonging to some function space V, in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples. Unit "spheres" in R2 consist of plane vectors of norm 1. Depicted are the unit spheres in different p-norms, for p = 1, 2, and ?. The bigger diamond depicts points of 1-norm equal to 2. A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem.[52] In contrast, the space of all continuous functions on [0,1] with the same topology is complete.[53] A norm gives rise to a topology by defining that a sequence of vectors vn converges to v if and only if Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study--a key piece of functional analysis--focusses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.[54] The image at the right shows the equivalence of the 1-norm and ?-norm on R2: as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data. From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) V -> W, maps between topological vector spaces are required to be continuous.[55] In particular, the (topological) dual space V consists of continuous functionals V -> R (or to C). The fundamental Hahn-Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[56] Banach spaces Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.[57] A first example is the vector space l p consisting of infinite vectors with real entries x = (x1, x2, ...) whose p-norm (1 p given by for p < ? and is finite. The topologies on the infinite-dimensional space l p are inequivalent for different p. E.g. the sequence of vectors xn = (2-n, 2-n, ..., 2-n, 0, 0, ...), i.e. the first 2n components are 2-n, the following ones are 0, converges to the zero vector for p = ?, but does not for p = 1: , but More generally than sequences of real numbers, functions f -> R are endowed with a norm that replaces the above sum by the Lebesgue integral The space of integrable functions on a given domain ? (for example an interval) satisfying \|\|p < ?, and equipped with this norm are called Lebesgue spaces, denoted Lp(?).[nb 10] These spaces are complete.[58] (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.[nb 11]) Concretely this means that for any sequence of Lebesgue-integrable functions f1, f2, ... with \|\|p < ?, satisfying the condition there exists a function f(x) belonging to the vector space Lp(?) such that Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.[59] Hilbert spaces The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red). Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.[60] The Hilbert space L2(?), with inner product given by where denotes the complex conjugate of g(x),[61][nb 12] is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions fn with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions f by polynomials.[62] By the Stone-Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial.[63] A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space H, in the sense that the closure of their span (i.e., finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of H, its cardinality is known as the Hilbert space dimension.[nb 13] Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but together with the Gram-Schmidt process, it enables one to construct a basis of orthogonal vectors.[64] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space. The solutions to various differential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular physical properties are used as basis functions, often orthogonal.[65] As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in time by means of a partial differential equation, whose solutions are called wavefunctions.[66] Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.[67] Algebras over fields A hyperbola, given by the equation x ? y = 1. The coordinate ring of functions on this hyperbola is given by R[x, y] / (x · y - 1), an infinite-dimensional vector space over R. General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field.[68] Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone-Weierstrass theorem mentioned above, for example, relies on Banach algebras which are both Banach spaces and algebras. Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic geometry, because they are rings of functions of algebraic geometric objects.[69] Another crucial example are Lie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ([x, y] denotes the product of x and y): Examples include the vector space of n-by-n matrices, with [x, y] = xy - yx, the commutator of two matrices, and R3, endowed with the cross product. The tensor algebra T(V) is a formal way of adding products to any vector space V to obtain an algebra.[71] As a vector space, it is spanned by symbols, called simple tensors v1 ? v2 ? ... ? vn, where the degree n varies. The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product ?, much the same way as with the tensor product of two vector spaces introduced above. In general, there are no relations between v1 ? v2 and v2 ? v1. Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing v1 ? v2 = - v2 ? v1 yields the exterior algebra.[72] When a field, F is explicitly stated, a common term used is F-algebra. Vector spaces have many applications as they occur frequently in common circumstances, namely wherever functions with values in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods.[73]Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as group theory.[74] A distribution (or generalized function) is a linear map assigning a number to each "test" function, typically a smooth function with compact support, in a continuous way: in the above terminology the space of distributions is the (continuous) dual of the test function space.[75] The latter space is endowed with a topology that takes into account not only f itself, but also all its higher derivatives. A standard example is the result of integrating a test function f over some domain ?: When ? = {p}, the set consisting of a single point, this reduces to the Dirac distribution, denoted by ?, which associates to a test function f its value at the p(f) = f(p). Distributions are a powerful instrument to solve differential equations. Since all standard analytic notions such as derivatives are linear, they extend naturally to the space of distributions. Therefore, the equation in question can be transferred to a distribution space, which is bigger than the underlying function space, so that more flexible methods are available for solving the equation. For example, Green's functions and fundamental solutions are usually distributions rather than proper functions, and can then be used to find solutions of the equation with prescribed boundary conditions. The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (e.g., using the Lax-Milgram theorem, a consequence of the Riesz representation theorem).[76] Fourier analysis The heat equation describes the dissipation of physical properties over time, such as the decline of the temperature of a hot body placed in a colder environment (yellow depicts colder regions than red). Resolving a periodic function into a sum of trigonometric functions forms a Fourier series, a technique much used in physics and engineering.[nb 14][77] The underlying vector space is usually the Hilbert space L2(0, 2?), for which the functions sin mx and cos mx (m an integer) form an orthogonal basis.[78] The Fourier expansion of an L2 function f is The coefficients am and bm are called Fourier coefficients of f, and are calculated by the formulas[79] In physical terms the function is represented as a superposition of sine waves and the coefficients give information about the function's frequency spectrum.[80] A complex-number form of Fourier series is also commonly used.[79] The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality.[81] Applied to the group R, it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice encoding positions of atoms in crystals.[82] Fourier series are used to solve boundary value problems in partial differential equations.[83] In 1822, Fourier first used this technique to solve the heat equation.[84] A discrete version of the Fourier series can be used in sampling applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier series is finite and its value is equal to the sampled values at all points.[85] The set of coefficients is known as the discrete Fourier transform (DFT) of the given sample sequence. The DFT is one of the key tools of digital signal processing, a field whose applications include radar, speech encoding, image compression.[86] The JPEG image format is an application of the closely related discrete cosine transform.[87] The fast Fourier transform is an algorithm for rapidly computing the discrete Fourier transform.[88] It is used not only for calculating the Fourier coefficients but, using the convolution theorem, also for computing the convolution of two finite sequences.[89] They in turn are applied in digital filters[90] and as a rapid multiplication algorithm for polynomials and large integers (Schönhage-Strassen algorithm).[91][92] Differential geometry The tangent space to the 2-sphere at some point is the infinite plane touching the sphere in this point. The tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point.[nb 15] Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the generalization to higher-dimensional differentiable manifolds.[93] Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.[94] Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.[95][96] The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify compact Lie groups.[97] Vector bundles A Möbius strip. Locally, it looks like U × R. A vector bundle is a family of vector spaces parametrized continuously by a topological space X.[93] More precisely, a vector bundle over X is a topological space E equipped with a continuous map ? : E -> X such that for every x in X, the fiber ?-1(x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space V, the projection X × V -> X makes the product X × V into a "trivial" vector bundle. Vector bundles over X are required to be locally a product of X and some (fixed) vector space V: for every x in X, there is a neighborhood U of x such that the restriction of ? to ?-1(U) is isomorphic[nb 16] to the trivial bundle U × V -> U. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space X) be "twisted" in the large (i.e., the bundle need not be (globally isomorphic to) the trivial bundle X × V). For example, the Möbius strip can be seen as a line bundle over the circle S1 (by identifying open intervals with the real line). It is, however, different from the cylinder S1 × R, because the latter is orientable whereas the former is not.[98] Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S1 is globally isomorphic to S1 × R, since there is a global nonzero vector field on S1.[nb 17] In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S2 which is everywhere nonzero.[99]K-theory studies the isomorphism classes of all vector bundles over some topological space.[100] In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions O.[] The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms. Modules are to rings what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F, yield modules.[101] The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (i.e., abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field with the elements being called vectors. Some authors use the term vector space to mean modules over a division ring.[102] The algebro-geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles. Affine and projective spaces An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red). Roughly, affine spaces are vector spaces whose origins are not specified.[103] More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map V × V -> V, (v, a) ? a + v. If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a fixed vector x ? W; this space is denoted by x + V (it is a coset of V in W) and consists of all vectors of the form x + v for v ? V. An important example is the space of solutions of a system of inhomogeneous linear equations Ax = b generalizing the homogeneous case b = 0 above.[104] The space of solutions is the affine subspace x + V where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the nullspace of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting at infinity.[105]Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively. See also 1. ^ It is also common, especially in physics, to denote vectors with an arrow on top: . 2. ^ This axiom and the next refer to two different operations: scalar multiplication: bv; and field multiplication: ab. They do not assert the associativity of either operation. More formally, scalar multiplication is a group action of the multiplicative group of the field F on the vector space V. 3. ^ Some authors (such as Brown 1991) restrict attention to the fields R or C, but most of the theory is unchanged for an arbitrary field. 4. ^ The indicator functions of intervals (of which there are infinitely many) are linearly independent, for example. 5. ^ The nomenclature derives from German "eigen", which means own or proper. 6. ^ Roman 2005, ch. 8, p. 140. See also Jordan-Chevalley decomposition. 7. ^ This is typically the case when a vector space is also considered as an affine space. In this case, a linear subspace contains the zero vector, while an affine subspace does not necessarily contain it. 8. ^ Some authors (such as Roman 2005) choose to start with this equivalence relation and derive the concrete shape of V/W from this. 9. ^ This requirement implies that the topology gives rise to a uniform structure, Bourbaki 1989, ch. II 10. ^ The triangle inequality for \|-\|p is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a seminorm. 11. ^ "Many functions in L2 of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the L2 norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", Dudley 1989, §5.3, p. 125 12. ^ For p ?2, Lp(?) is not a Hilbert space. 13. ^ A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra above. For distinction, the latter is then called a Hamel basis. 14. ^ Although the Fourier series is periodic, the technique can be applied to any L2 function on an interval by considering the function to be continued periodically outside the interval. See Kreyszig 1988, p. 601 15. ^ That is to say (BSE-3 2001), the plane passing through the point of contact P such that the distance from a point P1 on the surface to the plane is infinitesimally small compared to the distance from P1 to P in the limit as P1 approaches P along the surface. 16. ^ That is, there is a homeomorphism from ?-1(U) to V × U which restricts to linear isomorphisms between fibers. 17. ^ A line bundle, such as the tangent bundle of S1 is trivial if and only if there is a section that vanishes nowhere, see Husemoller 1994, Corollary 8.3. The sections of the tangent bundle are just vector fields. 1. ^ Roman 2005, ch. 1, p. 27 2. ^ van der Waerden 1993, Ch. 19 3. ^ Bourbaki 1998, §II.1.1. Bourbaki calls the group homomorphisms f(a) homotheties. 4. ^ Bourbaki 1969, ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78-91 5. ^ Bolzano 1804 6. ^ Möbius 1827 7. ^ Crowe, Michel J. (1994), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Dover, p. 11 and 16, ISBN 0-486-67910-1 8. ^ Hamilton 1853 9. ^ Grassmann 2000 10. ^ Peano 1888, ch. IX 11. ^ Banach 1922 12. ^ Dorier 1995, Moore 1995 13. ^ Lang 1987, ch. I.1 14. ^ Lang 2002, ch. V.1 15. ^ e.g. Lang 1993, ch. XII.3., p. 335 16. ^ Lang 1987, ch. IX.1 17. ^ Lang 1987, ch. VI.3. 18. ^ Roman 2005, Theorem 1.9, p. 43 19. ^ Blass 1984 20. ^ Halpern 1966, pp. 670-673 21. ^ Artin 1991, Theorem 3.3.13 22. ^ Braun 1993, Th. 3.4.5, p. 291 23. ^ Stewart 1975, Proposition 4.3, p. 52 24. ^ Stewart 1975, Theorem 6.5, p. 74 25. ^ Roman 2005, ch. 2, p. 45 26. ^ Lang 1987, ch. IV.4, Corollary, p. 106 27. ^ Lang 1987, Example IV.2.6 28. ^ Lang 1987, ch. VI.6 29. ^ Halmos 1974, p. 28, Ex. 9 30. ^ Lang 1987, Theorem IV.2.1, p. 95 31. ^ Roman 2005, Th. 2.5 and 2.6, p. 49 32. ^ Lang 1987, ch. V.1 33. ^ Lang 1987, ch. V.3., Corollary, p. 106 34. ^ Lang 1987, Theorem VII.9.8, p. 198 35. ^ Roman 2005, ch. 8, p. 135-156 36. ^ Lang 1987, ch. IX.4 37. ^ Roman 2005, ch. 1, p. 29 38. ^ Roman 2005, ch. 1, p. 35 39. ^ Roman 2005, ch. 3, p. 64 40. ^ Lang 1987, ch. IV.3. 41. ^ Roman 2005, ch. 2, p. 48 42. ^ Mac Lane 1998 43. ^ Roman 2005, ch. 1, pp. 31-32 44. ^ Lang 2002, ch. XVI.1 45. ^ Roman 2005, Th. 14.3. See also Yoneda lemma. 46. ^ Schaefer & Wolff 1999, pp. 204-205 47. ^ Bourbaki 2004, ch. 2, p. 48 48. ^ Roman 2005, ch. 9 49. ^ Naber 2003, ch. 1.2 50. ^ Treves 1967 51. ^ Bourbaki 1987 52. ^ Kreyszig 1989, §4.11-5 53. ^ Kreyszig 1989, §1.5-5 54. ^ Choquet 1966, Proposition III.7.2 55. ^ Treves 1967, p. 34-36 56. ^ Lang 1983, Cor. 4.1.2, p. 69 57. ^ Treves 1967, ch. 11 58. ^ Treves 1967, Theorem 11.2, p. 102 59. ^ Evans 1998, ch. 5 60. ^ Treves 1967, ch. 12 61. ^ Dennery 1996, p.190 62. ^ Lang 1993, Th. XIII.6, p. 349 63. ^ Lang 1993, Th. III.1.1 64. ^ Choquet 1966, Lemma III.16.11 65. ^ Kreyszig 1999, Chapter 11 66. ^ Griffiths 1995, Chapter 1 67. ^ Lang 1993, ch. XVII.3 68. ^ Lang 2002, ch. III.1, p. 121 69. ^ Eisenbud 1995, ch. 1.6 70. ^ Varadarajan 1974 71. ^ Lang 2002, ch. XVI.7 72. ^ Lang 2002, ch. XVI.8 73. ^ Luenberger 1997, §7.13 74. ^ See representation theory and group representation. 75. ^ Lang 1993, Ch. XI.1 76. ^ Evans 1998, Th. 6.2.1 77. ^ Folland 1992, p. 349 ff 78. ^ Gasquet & Witomski 1999, p. 150 79. ^ a b Gasquet & Witomski 1999, §4.5 80. ^ Gasquet & Witomski 1999, p. 57 81. ^ Loomis 1953, Ch. VII 82. ^ Ashcroft & Mermin 1976, Ch. 5 83. ^ Kreyszig 1988, p. 667 84. ^ Fourier 1822 85. ^ Gasquet & Witomski 1999, p. 67 86. ^ Ifeachor & Jervis 2002, pp. 3-4, 11 87. ^ Wallace Feb 1992 88. ^ Ifeachor & Jervis 2002, p. 132 89. ^ Gasquet & Witomski 1999, §10.2 90. ^ Ifeachor & Jervis 2002, pp. 307-310 91. ^ Gasquet & Witomski 1999, §10.3 92. ^ Schönhage & Strassen 1971 93. ^ a b Spivak 1999, ch. 3 94. ^ Jost 2005. See also Lorentzian manifold. 95. ^ Misner, Thorne & Wheeler 1973, ch. 1.8.7, p. 222 and ch. 2.13.5, p. 325 96. ^ Jost 2005, ch. 3.1 97. ^ Varadarajan 1974, ch. 4.3, Theorem 4.3.27 98. ^ Kreyszig 1991, §34, p. 108 99. ^ Eisenberg & Guy 1979 100. ^ Atiyah 1989 101. ^ Artin 1991, ch. 12 102. ^ Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007. 103. ^ Meyer 2000, Example 5.13.5, p. 436 104. ^ Meyer 2000, Exercise 5.13.15-17, p. 442 105. ^ Coxeter 1987 Historical references Further references External links Top US Cities
4dbd21bc0ed7e88d	Recent Publications 1. Title: Nuclear quantum effects on the structural properties of solids Author(s): Scivetti I., Hughes D., Gidopoulos N., Caro A., Kohanoff J., AIP Conference Proceedings, 963, pp. 212-223 (SEP 25 2007) 2. Title: Can Marcus theory be applied to redox processes in ionic liquids? A comparative simulation study of dimethylimidazolium liquids and acetonitrile Author(s): Lynden-Bell R.M. Journal Of Physical Chemistry B, 111, No. 36, pp. 10800-10806 (SEP 13 2007) doi: 10.1021/jp074298s Simulations of a model system of charged spherical ions in the ionic liquids dimethylimidazolium chloride, [dmim][Cl], dimethylimidazolium hexafluorophosphate, [dmim][PF6], and the polar liquid acetonitrile, MeCN, are used to investigate the applicability of Marcus theory to electrochemical half-cell redox processes in these liquids. The free energy curves for solvent fluctuations are found to be approximately parabolic and the Marcus solvent reorganization free energies and activation free energies are determined for six possible redox processes in each solvent. The similarities between the different types of solvent are striking and are attributed to the essentially long-range nature of the relevant interactions and the effectiveness of the screening of the ion potential. Nevertheless, molecular effects are seen in the variation of solvent screening potential with distances up to 2 nm. 3. Title: Vibrational frequencies of CO on Pt(111) in electric field: A periodic DFT study Author(s): Lozovoi A.Y., Alavi A. Journal Of Electroanalytical Chemistry, 607, No. 1-2, pp. 140-146 (SEP 1 2007) doi: 10.1016/j.jelechem.2007.02.030 The stretching frequency of CO adsorbed on Pt(1 1 1), vc-0, changes if an external electric field F is applied (the vibrational Stark effect). The slope dv(c-o)/dF measured in electrochemical cells, is a factor of two larger than that obtained in UHV experiments. The reasons for this discrepancy are still being debated, whereas existing DFT calculations "support" the electrochemical value. This study, on the contrary, produces a slope close to that found in UHV. The difference between our ab initio calculations and previous studies is that the latter were performed with finite clusters, whereas here the electric field is applied to a periodic supercell. This highlights the importance of the computational setup, and in particular the boundary conditions, for calculations of charged systems. (c) 2007 Elsevier B.V. All rights reserved. 4. Title: Structure-related effects on the domain wall migration in atomic point contacts Author(s): Stamenova M., Sahoo S., Todorov T.N., Sanvito S. Journal Of Magnetism And Magnetic Materials, 316, No. 2, pp. E934-E936 (SEP 2007) doi: 10.1016/j.jmmm.2007.03.147 We investigate the interplay between magnetic and structural dynamics in magnetic point contacts under current-carrying conditions. In particular we address the dependence of the results upon the specific parameterization of the elastic properties of the material. Our analysis shows a strong influence of the structural relaxation on the energy barrier for domain wall migration, but a negligible dependence of the mechanical properties on the magnetic state. These results are stable against the various material specific choices of parameters describing different transition metals. (c) 2007 Elsevier B.V. All rights reserved. 5. Title: Nonequilibrium statistical mechanics of classical nuclei interacting with the quantum electron gas Author(s): Wang Y., Kantorovich L. Physical Review B, 76, No. 14, Art. No. 144304 (OCT 19 2007) doi: 10.1103/PhysRevB.76.144304 Kinetic equations governing time evolution of positions and momenta of atoms in extended systems are derived using quantum-classical ensembles within the nonequilibrium statistical operator method (NESOM). Ions are treated classically, while their electrons quantum mechanically; however, the statistical operator is not factorized in any way and no simplifying assumptions are made concerning the electronic subsystem. Using this method, we derive kinetic equations of motion for the classical degrees of freedom (atoms) which account fully for the interaction and energy exchange with the quantum variables (electrons). Our equations, alongside the usual Newton-like terms normally associated with the Ehrenfest dynamics, contain additional terms, proportional to the atoms velocities, which can be associated with the electronic friction. Although previously electronic friction was introduced into molecular dynamics equations of atoms only using model treatments, we show that this result is general and model independent and thus must be expected in treating any system as an additional force acting on slow degrees of freedom due to coupling with the fast ones. Possible ways of calculating the friction forces, which are shown to be given via complicated nonequilibrium correlation functions, are discussed. In particular, we demonstrate that the correlation functions are directly related to the thermodynamic Matsubara Green\'s functions, and this relationship allows for the diagrammatic methods to be used in treating electron-electron interaction perturbatively when calculating the correlation functions. 6. Title: Competition of charge density waves and superconductivity in Sulfur Author(s): Degtyareva O., Magnitskaya M.V., Kohanoff J., Profeta G., Scandolo S., McMahon M.I., Gregoryanz E., Physical Review Letters, 99, Art. No. 155505 (OCT 15 2007) doi: 10.1103/PhysRevLett.99.155505 Full Text A one-dimensional charge-density wave (CDW) instability is shown to be responsible for the formation of the incommensurate modulation of the atomic lattice in the high-pressure phase of sulfur. The coexistence of, and competition between, the CDW and the superconducting state leads to the previously observed increase of Tc up to 17 K, which we attribute to the suppression of the CDW instability, the same phenomenology found in doped layered dichalcogenides. 7. Title: Simulations of ionic liquids, solutions, and surfaces Author(s): Lynden-Bell R.M., Del Popolo M.G., Youngs T.G.A., Kohanoff J., Hanke C.G., Harper J.B., Pinilla C.C. Accounts of Chemical Research, 40, No. 11, pp. 1138-1145 (OCT 2007) doi: 10.1021/ar700065s We have been using atomistic simulation for the last 10 years to study properties of imidazolium-based ionic liquids. Studies of dissolved molecules show the importance of electrostatic interactions in both aromatic and hydrogen-bonding solutes. However, the local structure strongly depends upon ionion and solutesolvent interactions. We find interesting local alignments of cations at the gasliquid and solidliquid interfaces, which give a potential drop through the surface. If the solid interface is charged, this charge is strongly screened over distances of a few nanometres and this screening decays on a fast time scale. We have studied the sensitivity of the liquid structure to force-field parameters and show that results from ab initio simulations can be used in the development of force fields. 8. Title: Clusters, liquids, and crystals of dialkyimidazolium salts. A combined perspective from a initio and classical computer simulations Author(s): Del Popolo M.G., Kohanoff J., Lynden-Bell R.M., Pinilla C Accounts of Chemical Research, 40, No. 11, pp. 1156-1164 (NOV 2007) doi: 10.1021/ar700069c We summarize results obtained by a combination of ab initio and classical computer simulations of dialkylimidazolium ionic liquids in different states of aggregation, from crystals to liquids and clusters. Unusual features arising from the competition between electrostatic, dispersion, and hydrogen-bonding interactions are identified at the origin of observed structural patterns. We also discuss the way Bronsted acids interact with ionic liquids leading to the formation of hydrogen-bonded anions. 9. Title: Dynamical simulation of inelastic quantum transport 10. Title: Polarization relaxation in an ionic liquid confined between electrified walls Author(s): Pinilla C., Del Popolo M.G., Kohanoff J., Lynden-Bell R.M. Journal Of Physical Chemistry B, 111, No. 18, pp. 4877-4884 (MAY 10 2007) doi: 10.1021/jp067184+ The response of a room temperature molten salt to an external electric field when it is confined to a nanoslit is studied by molecular dynamics simulations. The fluid is confined between two parallel and oppositely charged walls, emulating two electrified solid-liquid interfaces. Attention is focused on structural, electrostatic, and dynamical properties, which are compared with those of the nonpolarized fluid. It is found that the relaxation of the electrostatic potential, after switching the electric field off, occurs in two stages. A first, subpicosecond process accounts for 80% of the decay and is followed by a second subdiffusive process with a time constant of 8 ps. Diffusion is not involved in the relaxation, which is mostly driven by small anion translations. The relaxation of the polarization in the confined system is discussed in terms of the spectrum of charge density fluctuations in the bulk. 11. Title: Neutral and charged 1-butyl-3-methylimidazolium triflate clusters: Equilibrium concentration in the vapor phase and thermal properties of nanometric droplets Author(s): Ballone P., Pinilla C., Kohanoff J., Del Popolo M.G. Journal Of Physical Chemistry B, 111, No. 18, pp. 4938-4950 (MAY 10 2007) doi: 10.1021/jp067484r Ground state energy, structure, and harmonic vibrational modes of 1-butyl-3-methylimidazolium triflate ([bmim][Tf]) clusters have been computed using an all-atom empirical potential model. Neutral and charged species have been considered up to a size (30 [bmim][Tf] pairs) well into the nanometric range. Free energy computations and thermodynamic modeling have been used to predict the equilibrium composition of the vapor phase as a function of temperature and density. The results point to a nonnegligible concentration of very small charged species at pressures (P similar to 0.01 Pa) and temperatures (T >= 600 K) at the boundary of the stability range of [bmim][Tf]. Thermal properties of nanometric neutral droplets have been investigated in the 0 <= T <= 700 K range. A near-continuous transition between a liquidlike phase at high T and a solidlike phase at low T takes place at T similar to 190 K in close correspondence with the bulk glass point T-g similar to 200 K. Solidification is accompanied by a transition in the dielectric properties of the droplet, giving rise to a small permanent dipole embedded into the solid cluster. The simulation results highlight the molecular precursors of several macroscopic properties and phenomena and point to the close competition of Coulomb and dispersion forces as their common origin. 12. Title: Quantum annealing of an ising spin-glass by Green's function Monte Carlo Author(s): Stella L., Santoro G.E., Physical Review E, 75, No. 3, Art. No. 036703 (March 2007) doi: 10.1103/PhysRevE.75.036703 We present an implementation of quantum annealing (QA) via lattice Green's function Monte Carlo (GFMC), focusing on its application to the Ising spin glass in transverse field. In particular, we study whether or not such a method is more effective than the path-integral Monte Carlo- (PIMC) based QA, as well as classical simulated annealing (CA), previously tested on the same optimization problem. We identify the issue of importance sampling, i.e., the necessity of possessing reasonably good (variational) trial wave functions, as the key point of the algorithm. We performed GFMC-QA runs using such a Boltzmann-type trial wave function, finding results for the residual energies that are qualitatively similar to those of CA (but at a much larger computational cost), and definitely worse than PIMC-QA. We conclude that, at present, without a serious effort in constructing reliable importance sampling variational wave functions for a quantum glass, GFMC-QA is not a true competitor of PIMC-QA. 13. Title: Monte Carlo study of thermodynamic properties and clustering in the bcc Fe-Cr system Author(s): Lavrentiev M.Y., Drautz R., Nguyen-Manh D., Klaver T.P.C., Dudarev S.L. Physical Review B, 75, No. 1, Art. No. 014208 (JAN 2007) doi: 10.1103/PhysRevB.75.014208 Iron-chromium alloys are characterized by a complex phase diagram, by the small negative enthalpy of mixing at low Cr concentrations in the bcc alpha-phase of Fe, and by the inversion of the short-range order parameter. We present Monte Carlo simulations of the binary Fe-Cr alloy based on the cluster expansion approximation for the enthalpy of the system. The set of cluster expansion coefficients is validated against density functional calculations of energies of small clusters of chromium in bcc structure. We show that in the limit of small Cr concentration the enthalpy of mixing remains negative up to fairly high temperatures, and individual Cr atoms remain well separated from each other. Clustering of Cr atoms begins at concentrations exceeding approximately 10% at 800 K and 20% at 1400 K, with Cr-Fe interfaces being parallel to the [110] planes. Calculations show that the first and the second short-range order parameters change sign at approximately 10.5% Cr, in agreement with experimental observations. Semi-grand-canonical ensemble simulations used together with experimental data on vibrational entropy of mixing give an estimate for the temperature of the top of the alpha-alpha(') miscibility gap. We find that the complex ordering reactions occurring in Fe-Cr, as well as the thermodynamic properties of the alloy, can be reasonably well described using a few concentration-independent cluster expansion coefficients. 14. Title: Lattice melting and rotation in perpetually pulsating equilibria Author(s): Pichon C., Lynden-Bell D., Pichon J., Lynden-Bell R. Physical Review E, 75, No. 1, Art. No. 011125 (JAN 2007) doi: 10.1103/PhysRevE.75.011125 Systems whose potential energies consists of pieces that scale as r(-2) together with pieces that scale as r(2), show no violent relaxation to Virial equilibrium but may pulsate at considerable amplitude forever. Despite this pulsation these systems form lattices when the nonpulsational "energy" is low, and these disintegrate as that energy is increased. The "specific heats" show the expected halving as the "solid" is gradually replaced by the "fluid" of independent particles. The forms of the lattices are described here for N <= 18 and they become hexagonal close packed for large N. In the larger N limit, a shell structure is formed. Their large N behavior is analogous to a gamma=5/3 polytropic fluid with a quasigravity such that every element of fluid attracts every other in proportion to their separation. For such a fluid, we study the "rotating pulsating equilibria" and their relaxation back to uniform but pulsating rotation. We also compare the rotating pulsating fluid to its discrete counterpart, and study the rate at which the rotating crystal redistributes angular momentum and mixes as a function of extra heat content. 15. Title: Phase diagrams and surface properties of modified water models Author(s): Alejandre J., Lynden-Bell R.M. Molecular Physics, 105, No. 23, pp. 3029-3033 (DEC 2007) doi: 10.1080/00268970701733405 The surface properties and phase diagrams are examined for a number of modified water models. In the 'bent' family of models where the bond angle is decreased and the network structure is lost the surface tension is lower than in SPC/E water and the critical temperature is lower. In the 'hybrid' family of models which are hybrids between SPC/E water and a Lennard-Jones liquid the surface tension and the critical temperature are higher that in SPC/E water. These properties correlate well with the varying ability of the liquids to dissolve hard spheres. The surface potential, on the other hand, is slightly smaller in magnitude in the hybrid models than in SPC/E water because there is slightly less alignment of the dipoles in the surface layer. The degree of molecular alignment in the surface and the consequent surface potential drop is much lower in magnitude in the bent models than in SPC/E water. 16. Title: Does Marcus theory apply to redox processes in ionic liquids? A simulation study Author(s): Lynden-Bell R.M. Electrochemistry Communications, 9, No. 8, pp. 1857-1861 (AUG 2007) doi: 10.1016/j.elecom.2007.04.010 Simulation studies on a model system of a spherical ion with various charges in two imidazolium ionic liquids and in acetonitrile are compared. The average vertical ionisation potentials as a function of the charge on the ion are similar for all three systems. The Landau free energies of each system as a function of the vertical ionisation potential are computed and are close to being parabolic. Results are shown for the solvent reorganisation energies and for the activation free energies. The similarities of all these quantities are interpreted in terms of continuum models. However, the dynamics are likely to be very different in a polar liquid and in an ionic liquid as in the former case screening occurs by reorientation of molecules and in the latter case it occurs by translation of ions. (c) 2007 Elsevier B.V. All rights reserved. 17. Title: Local and semilocal density functional computations for crystals of 1-alkyl-3-methyl-imidazolium salts Author(s): Del Popolo M.G., Pinilla C., Ballone P. Journal Of Chemical Physics, 126, No. 14, Art. No. 144705 (APR 14 2007) doi: 10.1063/1.2715571 The accuracy and reliability of popular density functional approximations for the compounds giving origin to room temperature ionic liquids have been assessed by computing the T=0 K crystal structure of several 1-alkyl-3-methyl-imidazolium salts. Two prototypical exchange-correlation approximations have been considered, i.e., the local density approximation (LDA) and one gradient corrected scheme [PBE-GGA, Phys. Rev. Lett. 77, 3865 (1996)]. Comparison with low-temperature x-ray diffraction data shows that the equilibrium volume predicted by either approximations is affected by large errors, nearly equal in magnitude (similar to 10%), and of opposite sign. In both cases the error can be traced to a poor description of the intermolecular interactions, while the intramolecular structure is fairly well reproduced by LDA and PBE-GGA. The PBE-GGA optimization of atomic positions within the experimental unit cell provides results in good agreement with the x-ray structure. The correct system volume can also be restored by supplementing PBE-GGA with empirical dispersion terms reproducing the r(-6) attractive tail of the van der Waals interactions. (c) 2007 American Institute of Physics. 18. Title: The Computational Modeling of Alloys at the Atomic Scale: From Ab Initio and Thermodynamics to Radiation-Induced Heterogeneous Precipitation Author(s): Caro A., Caro M., Klaver T.P.C. Sadigh B., Lopasso E.M., Srinivasan S.G. JOM Journal of the Minerals, Metals and Materials Society, 59, No. 4, pp. 52-57 (APR 2007) doi: 10.1007/s11837-007-0055-y Full Text This paper describes a strategy to simulate radiation damage in FeCr alloys wherein magnetism introduces an anomaly in the heat of formation of the solid solution. This has implications for the precipitation of excess chromium in the α′ phase in the presence of eterogeneities. These complexities pose many challenges for atomistic (empirical) methods. To address such issues the authors have developed a modified manybody potential by rigorously fitting thermodynamic properties including free energy. Multi-million atom displacement Monte Carlo simulations in the transmutation ensemble, using the new potential, predict that thermodynamically grain boundaries, dislocations, and free surfaces are not preferential sites for α′precipitation. 19. Title: Robust nonadiabatic molecular dynamics for metals and insulators Author(s): Stella L., Meister M., Fisher A. J., Horsfield A. P. Journal of Physical Chemistry, 127, No. 21, Art. No. 214104 (4 December 2007) doi: 10.1063/1.2801537 We present a new formulation of the correlated electron-ion dynamics (CEID) scheme, which systematically improves Ehrenfest dynamics by including quantum fluctuations around the mean-field atomic trajectories. We show that the method can simulate models of nonadiabatic electronic transitions and test it against exact integration of the time-dependent Schrödinger equation. Unlike previous formulations of CEID, the accuracy of this scheme depends on a single tunable parameter which sets the level of atomic fluctuations included. The convergence to the exact dynamics by increasing the tunable parameter is demonstrated for a model two level system. This algorithm provides a smooth description of the nonadiabatic electronic transitions which satisfies the kinematic constraints (energy and momentum conservation) and preserves quantum coherence. The applicability of this algorithm to more complex atomic systems is discussed. 20. Title: Magnetocaloric effect in metamagnetic systems Author(s): Triguero C., Porta M., Planes A. Physical Review B, 76, No. 9, pp. 094415- (24 September 2007) doi: 10.1103/PhysRevB.76.094415 We propose a model to account for the magnetocaloric effect associated with field-driven first-order magnetic transitions. The model is based on an Ising model for the magnetic degrees of freedom which are coupled to lattice vibrations. The coupling is introduced within the framework of a bond proportion model, which assumes that nearest-neighbor pairs of antiparallel and parallel spins have different stiffness. The model is solved by means of a variational mean field technique and it is found that, for strong enough coupling, it displays a field-induced first-order magnetic transition. The magnetocaloric properties are obtained and results are compared with experimental results reported for systems undergoing metamagnetic transitions. 21. Title: Evolucao estrutural de nanotubos de oxidos de vanadio: efeito da temperatura Author(s): Ferreira O.P., Alves O.L., Souza Filho A.G., Santos E.J.G., Mendes Filho J. Chemical Physics Letters, 437, No. 1-3, pp. 79-82 (22 March 2007) doi: 10.1016/j.cplett.2007.01.071 The interaction of 2,3,7,8-tetrachlorinated dibenzo-p-dioxin (dioxin) with carbon nanotubes is analyzed by ab initio methods. The structural and electronic properties of the dioxin interacting with the pristine, defective and B-, N-, and Si-doped SWNTs are considered by studying different geometries. We have found that the dioxin interacts with a pure carbon nanotube although it depends on the geometry of the molecule approximation, increasing if the tube is defective. 22. Title: Atomistic study of ordinary 1/2 < 110] screw dislocations in single-phase and lamellar gamma-TiAl Author(s): Katzarov I.H., Cawkwell M.J., Paxton A.T., Finnis M.W. Philosophical Magazine, 87, No. 12, pp. 1795-1809 (2007) doi: 10.1080/14786430601080252 Full Text Computer simulation of the core structure and glide of ordinary 1/2 [110] screw dislocations in single-phase L1(0) TiAl and in two lamellae forming a twin gamma/gamma-interface has been performed using recently constructed Bond-Order Potentials (BOPs). BOPs represent a semi-empirical, numerically efficient scheme that works within the orthogonal tight-binding approximation and is able to capture the directionality of bonding. We have studied dislocation glide in perfect L10 TiAl and along a twin interface, transmission of an ordinary screw dislocation between lamellae, and the core structure, mobility and detachment of an interfacial 1/2 [110] screw dislocation from a twin boundary under applied shear stresses in directions parallel and perpendicular to a (111) plane. Our results show that the glide of ordinary 1/2 [110] straight screw dislocations under applied stresses in L1(0) TiAl is characterized by zigzag movement on two conjugated {111} planes. The non-planar core of the 1/2 (110] screw dislocation is distorted asymmetrically when the elastic centre of the dislocation is close to a twin gamma/gamma-interface and the dislocation moves on one of the (111) planes, depending on the magnitude of the corresponding Schmid factor. Ordinary dislocations become ordinary interfacial dislocations when they reach the interface. With increasing applied stress they can glide into the adjacent lamella, leaving no remnant interfacial dislocation. 23. Title: Interstitials in FeCr alloys studied by density functional theory Author(s): Klaver T.P.C., Olsson P., Finnis M.W. Physical Review B, 76, pp. 214110- (2007) doi: 10.1103/PhysRevB.76.214110 Density functional theory calculations have been used to study relaxed interstitial configurations in FeCr alloys. The ionic and electronic ground states of 69 interstitial structures have been determined. Interstitials were placed in alloys with up to 14 at. % Cr. Cr atoms were either monatomically dispersed or clustered together within a periodically repeated supercell consisting of 4 4 4 cubes of bcc unit cells. The distance between the interstitials and Cr atoms was varied within the supercells. It is shown that Cr atoms beyond third-nearest-neighbor distance from the interstitial can still have an interaction with it of up to 0.9 eV. The multibody nature of the Cr-Cr interactions causes the Cr-interstitial interaction to be strongly concentration dependent. The Cr-Cr interaction in defect-free alloys is also dependent on the overall Cr concentration. The effective Cr-Cr repulsion is weaker in alloys than in an environment of pure Fe. Apart from the Cr concentration, the Cr-interstitial interaction also depends on the dispersion level of Cr atoms beyond third-nearestneighbor distance from the interstitial. The formation energy differences between dumbbell interstitials with different orientations are independent of the Cr concentration. We show that the long-range influence of Cr atoms on the interstitial is not due to the interstitial strain field protruding into Cr-rich parts of the supercells. The Fermi-level and band energies were found not to be the sole governing parameter in determining the formation energies. Implications for the construction of empirical potentials are discussed. 24. Title: Glucose solvation by the ionic liquid 1,3-dimethylimidazolium chloride: A simulation study Author(s): Youngs T.G.A., Hardacre C., Holbrey J.D., J. Phys. Chem. B, 111, pp. 13765-13774 (2007) doi: 10.1021/Jp076728k 25. Title: Structure and solvation in ionic liquids Author(s): Hardacre C., Holbrey J.D., Nieuwenhuyzen M., Youngs T.G.A., Acc. Chem. Res., 40, pp. 1146-1155 (2007) doi: 10.1021/ar700068x 26. Title: Macroscopic limit of time-dependent density-functional theory for adiabatic local approximations of the exchange-correlation kernel Author(s): Gruening M., Gonze X. Physical Review B, 76, pp. 035126- (2007)
e3d95e0b46b38673	Blog Archives No, seriously, why can’t orbitals be observed? The concept of electronic orbital has become such a useful and engraved tool in understanding chemical structure and reactivity that it has almost become one of those things whose original meaning has been lost and replaced for a utilitarian concept, one which is not bad in itself but that may lead to some wrong conclusions when certain fundamental facts are overlooked. Last week a wrote -what I thought was- a humorous post on this topic because a couple of weeks ago a viewpoint in JPC-A was published by Pham and Gordon on the possibility of observing molecular orbitals through microscopy methods, which elicited a ‘seriously? again?‘ reaction from me, since I distinctly remember the Nature article by Zuo from the year 2000 when I just had entered graduate school. The article is titled “direct observation of d-orbital holes.” We discussed this paper in class and the discussion it prompted was very interesting at various levels: for starters, the allegedly observed d-orbital was strikingly similar to a dz2, which we had learned in class (thanks, prof. Carlos Amador!) that is actually a linear combination of d(z2-x2) and d(z2-y2) orbitals, a mathematical -lets say- trick to conform to spectroscopic observations. Pham and Gordon are pretty clear in their first paragraph: “The wave function amplitude ΨΨ is interpreted as the probability density. All observable atomic or molecular properties are determined by the probability and a corresponding quantum mechanical operator, not by the wave function itself. Wave functions, even exact wave functions, are not observables.” There is even another problem, about which I wrote a post long time ago: orbitals are non-unique, this means that I could get a set of orbitals by solving the Schrödinger equation for any given molecule and then perform a unit transformation on them (such as renormalizing them, re-orthonormalizing them to get a localized version, or even hybridizing them) and the electronic density derived from them would be the same! In quantum mechanical terms this means that the probability density associated with the wave function internal product, ΨΨ, is not changed upon unit transformations; why then would a specific version be “observed” under a microscope? As Pham and Gordon state more eloquently it has to do with the Density of States (DOS) rather than with the orbitals. Furthermore, an orbital, or more precisely a spinorbital, is conveniently (in math terms) separated into a radial, an angular and a spin component R(r)Ylm(θ,φ)σ(α,β) with the angular part given by the spherical harmonic functions Ylm(θ,φ), which in turn -when plotted in spherical coordinates- create the famous lobes we all chemists know and love. Zuo’s observation claim was based on the resemblance of the observed density to the angular part of an atomic orbital. Another thing, orbitals have phases, no experimental observation claims to have resolved those. Now, I may be entering a dangerous comparison but, can you observe a 2? If you say you just did, well, that “2” is just a symbol used to represent a quantity: two, the cardinality of a set containing two elements. You might as well depict such quantity as “II” or “⋅⋅” but still cannot observe “a two”. (If any mathematician is reading this, please, be gentle.) I know a number and a function are different, sorry if I’m just rambling here and overextending a metaphor. Pretending to having observed an orbital through direct experimental methods is to neglect the Born interpretation of the wave function, Heisenberg’s uncertainty principle and even Schrödinger’s cat! (I know, I know, Schrödinger came up with this gedankenexperiment in order to refute the Copenhagen interpretation of quantum mechanics, but it seems like after all the cat is still not out of the box!) So, the take home message from the viewpoint in JPC is that molecular properties are defined by the expected values of a given wave function for a specific quantum mechanical operator of the property under investigation and not from the wave function itself. Wave functions are not observables and although some imaging techniques seem to accomplish a formidable task the physical impossibility hints to a misinterpretation of facts. I think I’ll write more about this in a future post but for now, my take home message is to keep in mind that orbitals are wave functions and therefore are not more observable (as in imaging) than a partition function is in statistical mechanics. %d bloggers like this:
7ae8f1586e6c6ad4	Quantum mechanics/Wavepackets From Wikiversity Jump to navigation Jump to search created by w:User:Farhan babra In quantum mechanics, when a particle is not subjected to any external force (i.e., it exists in a region of constant potential, which can always be chosen to be zero, because one has the freedom of assigning zero potential to all points in space), the free Schrödinger equation describing it is given by which, in one dimension, has the general solution of the form This represents plane waves. These plane wave solutions are not normalisable because the resulting probability density is uniform everywhere. The Hamiltonian commutes with the momentum operator , whose eigenfunctions are . There exist two methods of constructing normalisable solutions (wave functions). First, one may restrict the wave functions within a region (see Particle in a box). Another way is to take linear combinations of plane waves with several k (momenta), called wave packets, where is the wave function of the same particle in the momentum basis. The can be normalised by choosing appropriate s . and constitute a fourier transform pair for t=0 (we are given the wave packet at some t, which can be taken to be zero, and asked to determine the wave function at a later time). One may check that both are normalised by the application of Plancherel theorem. If we require to localise the particle in some finite region, we need some of the coefficients to be non-zero, i.e, a superposition of waves with different k's. Therefore, the more precise the location of particle, the more plane waves needed with a wide range of momenta, the less precise the momentum of the particle---the Heisenberg Uncertainty inequality. Evolution of wave packet[edit] Each energy eigenfunction evolves by acquiring a phase factor , where corresponds to the energy eigenvalue associated with k. can have a more complex form if the particle encounters a different potential, potential barrier, for example. Ascribing this time factor to each constituent eigenfunction, yields the travelling wave packet. Suppose the wave packet is peaked around at some particular k; expanding around its vicinity and retaining up to two terms, yields a moving wave packet (the envelope of plane waves) with a group velocity By contrast, each plane wave travels with the corresponding phase velocity . The group velocity, i.e., the apparent velocity of wave packet with which it travels, is twice the phase velocity, the velocity of the constituents.
2479db737bde1a51	10 The Aharonov–Bohm effect Consider once more the two-slit experiment with electrons. What happens if we place a thin, long solenoid right behind (or right in front of) the slit plate, between the two slits. If no current is flowing in the solenoid, we observe the usual interference pattern. But if a current is flowing, the interference pattern is shifted sideways (right or left, depending on the direction of the current). This effect could easily have been predicted in the late 1920’s, by which time all the necessary physics was in place. Yet by many accounts it was predicted three decades later, in 1959, by Yakir Aharonov and David Bohm.[1] (Actually it was first predicted two decades later,[2] but it made a splash only after the publication of the paper by Aharonov and Bohm. This was followed by an actual demonstration of the effect in less than a year.) Why did it take that long for this effect to be predicted and taken note of? Take another look at the rectangle on the right side of Figure 2.17.1. The flux of the magnetic field through this rectangle, we observed, determines the difference between the actions of the paths A→B→C and A→D→C. Think of these paths as leading from the electron gun G through either L or R to a detector D at the screen: G→L→D and G→R→D. As we glean from Equation 2.17.1, the action for each path depends on the vector potential along the path. The vector potential A (as well as the magnetic field B) associated with a current-bearing solenoid are shown in Figure 3.10.1. If this solenoid is introduced into the setup of the two-slit experiment, the action for the left path (G→L→D) is increased, while the action for the right path (G→R→D) is reduced. Stepping up the current in the solenoid increases the difference between these actions. Since this difference determines the positions of the maxima and minima at the screen, the interference pattern is shifted to the left as a result. A solenoid, the current J flowing through it, and the resulting fields A and B. There are several reasons why it took that long to predict this remarkable effect. For one, classical electrodynamics can be formulated exclusively in terms of the electric field E and the magnetic field B. For another, while the four components of V and A uniquely determine the six components of E and B, they themselves are not not unique. E and B are invariant under the substitutions (3.10.1)xxxV → V−∂tα,xxxAx → Ax+∂xα,xxxAy → Ay+∂yα,xxxAz → Az+∂zα, where α is a function of the spacetime coordinates t,x,y,z. On the other hand, it was well known that the Schrödinger equation could accommodate electromagnetic effects only in terms of the potentials V and A, and that the actions associated with loops are also invariant under these substitutions. (Since these substitution change the actions for G→L→D and G→R→D by equal amounts, the difference between the two actions, which equals the action associated with the loop G→L→D→R→G, remains unchanged.) Yet physicists were unable to disabuse themselves of the notion that E and B are physically real, while the potentials “have no physical meaning and are introduced solely for the purpose of mathematical simplification of the equations,” as Fritz Rohrlich wrote.[3] The general idea at the time was that the electromagnetic field is a physical entity in its own right, that it is locally acted upon by charges, that it locally acts on charges, and that it mediates the action of charges on charges by locally acting on itself. At the heart of this notion was the so-called principle of local action, felicitously articulated by DeWitt and Graham[4] in an American Journal of Physics resource letter: physicists are, at bottom, a naive breed, forever trying to come to terms with the “world out there” by methods which, however imaginative and refined, involve in essence the same element of contact as a well-placed kick. With the notable exception of Roger Boscovich, a Croatian physicist and philosopher who flourished in the 18th Century, it does not seem to have occurred to anyone that local action is as unintelligible as the apparent ability of material objects to act where they are not, which the principle of local action has supposedly done away with. (The impression that local action is intelligible rests on such familiar experiences as pulling a rope or pushing a stalled car. As soon as we take a microscopic look at what happens between the pushing hands and the car, we discover that this seemingly local action involves net interatomic or intermolecular repulsive forces acting at a distance.) If it is believed that electromagnetic effects on charges are produced via a continuous sequence of local cause-effect relations, then something like the Aharonov–Bohm effect cannot be foreseen, for the values of E and B along the alternative electron paths can be made arbitrarily small by making the solenoid sufficiently long. And even if A were a physical entity and as such responsible for the Aharonov–Bohm effect, it could not produce it locally since the local values of A are arbitrary. → Next 1. [↑] Aharonov, Y., and Bohm, D. (1959). Significance of electromagnetic potentials in quantum theory. Physical Review 115, 485–491. 2. [↑] Ehrenberg, W., and Siday, R.E. (1949). The Refractive Index in Electron Optics and the Principles of Dynamics. Proceedings of the Physical Society B 62, 8–21. 3. [↑] Rohrlich, F (1965). Classical Charged Particles, Addison–Wesley, pp. 65–66. 4. [↑] DeWitt, B.S., and Graham, R.N. (1971). Resource letter IQM-1 on the interpretation of quantum mechanics, American Journal of Physics 39, 724–738.
08841cf695230a75	Open systems consisting of mutually interacting objects have a clustering tendency and all they, under certain conditions, show definite “shapes” or aggregate patterns, known by emergence phenomena. This occurs, according to Baas, “(…) when something happens in the course of the interaction among a system’s parts that produce a feature of the system that wasn’t present when the individual parts were considered separately”. These systems may be constituted, inter alia, by a bunch of nucleons in an atomic nucleus, a cloud of electrons around an atom, molecules of a substance, planets, stars, galaxies, a cluster of bacteria, a school of fish, a herd of cattle and human society. Natural phenomena must obey two basic principles: the second law of thermodynamics and the law concerning energy constraints. It will be shown several examples of systems involving inanimate and living objects in which the system “looks for paths” related to Hamilton’s variational principle or usually known by the principle of least action13:228, which leads the system, to minimal state of energy expenditure. Recently 33 showed that data processing networks minimizes energy expenditure through clustering process. It will be shown that clustering phenomena and the well-known “herding effects” in a collection of living objects is derived from this variational principle. For this kind of phenomenon it will be shown that this energy saving principle comes before any other tenet commanding living beings, or in other words, the clustering and the herding effects can be explained by the action of least energy consumption principle. In other words, living aggregations that have been viewed, classically, as an evolutionary advantageous process, in which participants share the benefits of protection, information and mate choice, comes as a fortuitous consequence and cannot be considered the key factor or reason of these effects, being preceded before by energy constraints. A question raised by 29: are all emergent properties of animal aggregations functional or are some simple patterns? Could certain class of human behavior dynamics be described by physics concepts like the Reynolds number of hydrodynamics? To what extent the emerging results of complex systems are universal and could be extended ubiquitously to all? There are some succeed results in this sense: the conceptual framework provided by scaling and universality in animate and inanimate systems, for example, explain the aggregation phenomenon described by a simple physical model6,40,41 and seems to be applicable on DNA sequences, heartbeat intervals, avalanche-like lung inflation, urban growth and company growth. The purpose of this paper is: • Offers an explanation of physical clustering behavior, driven by fundamental principles, common to all open systems, based on system composed by mutually interacting inanimate or living members. • Analyze some elected behavior of living systems associated with a non-rational behavior, commonly known as “herding effect”. • Discussing also the possibility of addressing part of social dynamics from the exact sciences viewpoint, given its resemblance with the behavior of fluids and thereafter, subject to formal description through mathematical equations. Collective ubiquitous behavior Starting with a microscope world, the “shape” of the nucleus is nothing but the density distribution of the particles—protons, neutrons, hadrons, quarks and others—governed by a variational principle, through “paths” which follows the Hamilton’s principle of least action and minimize the energy state of the system. One of most successful model to describe the atomic and nuclear “shapes” on all except the lightest atoms and nuclei, has over 85 years of age, known by Hartree central-field approximation15. It was originally conceived for atomic systems and assumes that each of the atomic electrons moves in a “mean” spherically (for isotropic property of space) attractive potential field due to the nucleus (positive charge) and all the other electrons (negative charge). Owing to the analytical impossibility of treating the “three or more body problem”, the Schrödinger equation is solved for each electron in its own central field, neglecting the presence of the other electrons, and the resulting wave functions made consistent with the fields from which they are calculated. The important fact is that the Hartree method is in complete agreement with the Hamilton’s variational principle of least action or, in other words, it minimizes the energy state of the system35:284. Nuclear aggregation can also be described successfully through central Hartree “mean potential” generated by attractive nuclear force acting mutually between the nucleons, coming out the “shell model” nuclear theory. Some living objects have mutual attraction force acting between them, so the “mean potential” concept can be applied to describe the tendency of a given kind of animals to the aggregation phenomenon. In particular the human being has the known “bio-social attraction”5,36, giving rise to the equivalent aggregative “mean potential”, leading to human tendency to clustering, driven by physical reasons: the Hamilton’s variational principle of least action. This enlightens the issue raised by Parrish in the introduction in the sense that, although clustering members share the benefits of protection, mate choice, and centralized information against the costs of limiting resources, it cannot be seen as a selective evolutionary criterion, since aggregation phenomenon happens regardless of these advantages. Furthermore, in human societies optimal group size is often exceeded, with regrettable environmental consequence: pollution, epidemics and social degradation27,29. The problem of animal clustering was also treated by Flierl et al.10, using the “individual-based model” computer simulation (cellular automata model) to study the environmental and social forces leading to grouping. They describe social behavior at the level of the individual animal in the Lagrangian framework in terms of their position, velocity, acceleration, internal states and environmental variables (see 14 for a review). Therefore, the aggregation phenomenon in living and inanimate systems is explained by variational principle of least action and is usually followed by another “non-thinking” or irrational behavior driven by energy saving laws, the herding effect. The herding effect “Let them alone: they are blind leaders of the blind. And if the blind lead the blind, both shall fall into the ditch.” [Matthew 15:14] In 4. Unanimity is always stupid” (Rodrigues31, Brazilian journalist and novelist) “Think, really think, it was so laborious, as hard as carrying heavy chests to the attic.” (Yalon42, writer and psychiatrist professor at Stanford University) 1. It’s known fact that when a large numbers of lemmings migrate, a lot of them inevitably drown while crossing rivers and lakes. 2. In the town of Gevas, Turkey, 450 sheep jump to their deaths (, 7/8/2005). 3. In Falmouth, Cornwall, England, 26 dolphins died after beaching (, June 11, 2008). 4. Thousands of jumb squid have beached themselves on central California shores this week (, December 15, 2012). 5. “…Touched by the wand of speculation, frenzy runs through the entire nation…” (The Gazette of August 13, 1791, on the financial crash of 1791, in 38). Imitative behavior transmission among animals is relatively common in primate, vertebrate and invertebrate organisms. All the facts mentioned above have in common possibly the same explanation known by herding effect, based on allelomimesis phenomenon, or in other words, “doing what your neighbors do”29. One of underlying cause of this phenomenon may be a mere extension of the non-rational principle which drives the clustering behavior, i.e., the variational principle of least action or energy expenditure. Saving energy is one of the driving forces which may explain these natural phenomena of living systems, which submitted to internal/external changes will adapt to a new situation with minimum energy expenditure, looking for a most “comfortable” state, i.e., a state of least effort44. Considering specifically the human beings the neural processing of information is metabolically expensive. The metabolism of the brain consumes about 20% of the total metabolic energy consumption, although the brain is only 2% of whole body weight1,23,32. The number of decisions that an individual makes on a daily since his awakening is immense2,4: choice of stores, products, schools, political candidates, technologies, how many children to have, drug, alcohol and cigarette use, religions, fads, fashion and custom – your brain is continuously requested from small tasks till presumed sophisticated decisions needed, for example, the investment decisions in the capital market. A traditional paradigm on this market is one of rational utility maximizing agents: investment decisions reflect agent’s rationally formed and decisions are made under all available information in an efficient way. However, what happens is that managers simply mimic the investment decisions of other managers, ignoring substantial private information, which may lead the market to catastrophic situations. There is a huge bibliography on this subject: see for example 3,8,11,34,37. On the modern influence of the WEB sites in the herding effect see 26. There is no doubt about the importance of herding effect on the mass psychology and its significance for political leaders and product marketing. From a simple choice of a toothbrush till to convince people about a political ideology are issues deeply driven by this effect. Fluid dynamic approach Nuclei in the fundamental state have in general “spherical” shape, for isotropic symmetry of space, mentioned above. However, when the matter density reaches about, case the inner crust of a neutron star, the nucleus “looks” for a more “comfortable” state or energetically most favorable nucleus shape and changes from spherical to cylinder, slab, cylindrical hole and spherical hole28. One of the most successful models to describe the nuclear binding energy and a class of collective phenomena like the nuclear fission, nuclear vibration and nuclear rotation have been derived from the hydrodynamic theory, the Liquid-Drop Model, for almost 80 years30,39,43. In this model, the core is treated as a deformable drop of nuclear “liquid”, in interaction with few extra nucleons in an unfilled shell, in order to obtain other nuclear nonspherical emergent shapes, through dynamical “paths” which minimize the energy of the system. In the same way, that is, obeying the energy constraints, groups of atoms, through covalent or ionic coupling, can develop from simple bounded atoms to complex chain of molecules. One theory sustained by most scientists about the emergence of life on earth considers the origin of cells from inanimate matter through a spontaneous and gradual increase of molecular complexity. This process starts with a closed spherical molecular aggregation (micelles), like the microscope process described before. Micelles formation is an example of self-organization which is obtained by addition of some liquid soap in water at a concentration higher than the critical micelle concentration. In this soap a spherical micelle aggregates spontaneously appears. This process takes place with a negative free energy change, or looking for a more “comfortable” energy state and by an increase of entropy. This occurs by the fact that amphiphilic molecules tend to aggregate in order to decrease the unfavorable contacts with water molecules—thus, upon aggregation, water molecules are set “free”. Micelles are the spontaneous self-aggregation of membranogenic surfactants into a vesicle, with an interior water pool that can host water-soluble molecules. If this self-aggregation takes place also in the presence of hydrophobic molecules, and/or ionic molecules, these can organize themselves into the bilayer or on the surface of the vesicle.”24 The remarkable fact on this self-organizing phenomenon is that, despite the apparent second thermodynamic law violation—the system going spontaneously from a less ordered to a more ordered state, it occurs with an increase of total entropy24. Analogously, another phenomenon which apparently violates the entropy principle is one that a pot of water is placed on the stove flame, an emergence occurs before the water reaches a turbulent or boiling state—a toroidal conformation of the fluid, provided by the convection water flowing, showing a “doughnut” pattern shape. The most of natural phenomena cannot be described mathematically due to a non-linearity coming from the mutually many bodies interaction, even for the so called Newtonian problems. A paradigmatic approach for analyze complex systems is through the fluid dynamics theory, since it contains the main ubiquitous aspects of emergent phenomena concerning these systems. According to Hofstadter: “Fluid concepts are necessarily, I believe, to understand emergent aspects of a complex system. I suspect that conceptual fluidity can only come out of a seething mass of subcognitive activities, just as the less abstract fluidity of real physical liquids is necessarily an emergent phenomenon, a statistical outcome of vast swarms of molecules jouncing incoherently one against another.”7 Concerning human beings, the behavior of pedestrian crowds is similar to that of gases or fluids16,17,18, and footsteps of pedestrians in snow look similar to streamlines of fluids. At borderlines between opposite directions of walking one can observe “viscous fingering”19. The process by which organisms form groups and how social forces interact with environmental variability was studied by Flierl10, utilizing a model derived from fluid dynamics specifically considering a system embedded in a turbulent flow fields. The origin of the word turbulence is generally believed to have been coined by Leonardo da Vinci which conceived the chaotic flow of a waterfall, as analogous to a crowd of agitated people (turba), which derives turbolentia, the Latin origin of turbulence25. According to Kauffman20,21,22; in 12, the two key variables that drives the dynamics of a complex system are the number of agents (N) and the density of their connection (ρ). The behavior of the system will change through a sequence of critical thresholds, from stable (lamellar) to turbulent behavior by increasing N and/or ?. However, if the complex system is seen from the perspective of fluid dynamics, two other variables become important to account the dynamic stability: the velocity (v) of the agents (N) and the “viscosity” (µ) pervading them, through the Reynolds number R9: R = ρ.v.L/µ For an ordinary fluid, v means the velocity, ρ the density, µ the viscosity and L a parameter concerning some kind of size measure of the system. Thus, the numerator represents the inertial properties, while the denominator is connected to the fluid viscosity or motion braking. The R number therefore measures the relationship between these two forces in the fluid whose flow is laminar if R is small, in which case the numerator is small (low speed, density or size) or large denominator (large viscosity µ). Otherwise, if R is large the fluid can flow in chaotic or turbulent regime. This concept may be transferred qualitatively to other complex systems, for example, the financial system. Globalization has led to a sharp increase of participants thus increasing the “density” of transactions in these interconnected systems. On the other hand, information processing technology has also made possible the considerably increase in the “speed” v of these transactions without any increase in the “viscosity” in that environment. Viscosity µ means, in this case, a set of restrictions and regulations which “put the brakes on” the financial “flows”. The USA is confessedly a country in which this variable is lower than other countries for historical reasons, since they are the emblematic “proxy” of the liberal economic system. As a consequence, turbulent episodes have occurred with increasing frequency triggered by disproportionate facts showing a strong non-linearity and instability in these systems. Saving energy seems to be a fundamental principle that drives much of the dynamic behavior of living and inanimate systems. Clustering effect for both living and inanimate systems and herding behavior for living systems may be understand as a consequence of physical sciences: the variational principle of least action. Some non-living concepts due to fluid dynamics may be employed, at least in a qualitatively way, to understand the turbulence behavior of financial systems crash.
6b5ee9abbe4679c5	In my previous post, I briefly discussed the work of the four Fields medalists of 2010 (Lindenstrauss, Ngo, Smirnov, and Villani). In this post I will discuss the work of Dan Spielman (winner of the Nevanlinna prize), Yves Meyer (winner of the Gauss prize), and Louis Nirenberg (winner of the Chern medal). Again by chance, the work of all three of the recipients overlaps to some extent with my own areas of expertise, so I will be able to discuss a sample contribution for each of them. Again, my choice of contribution is somewhat idiosyncratic and is not intended to represent the “best” work of each of the awardees. — 1. Dan Spielman — Dan Spielman works in numerical analysis (and in particular, numerical linear algebra) and theoretical computer science. Here I want to talk about one of his key contributions, namely his pioneering work with Teng on smoothed analysis. This is about an idea as much as it is about a collection of rigorous results, though Spielman and Teng certainly did buttress their ideas with serious new theorems. Prior to this work, there were two basic ways that one analysed the performance (which could mean run-time, accuracy, or some other desirable quality) of a given algorithm. Firstly, one could perform a worst-case analysis, in which one assumed that the input was chosen in such an “adversarial” fashion that the performance was as poor as possible. Such an analysis would be suitable for applications such as certain aspects of cryptography, in which the input really was chosen by an adversary, or in high-stakes situations in which there was zero tolerance for any error whatsoever; it is also useful as a “default” analysis for when no realistic input model is available. At the other extreme, one could perform an average-case analysis, in which the input was chosen in a completely random fashion (e.g. a random string of zeroes and ones, or a random vector whose entries were all distributed according to a Gaussian distribution). While such input models were usually not too realistic (except in situations where the signal-to-noise ratio was very low), they were usually fairly simple to analyse (using tools such as concentration of measure). In many situations, the worst-case analysis is too conservative, and the average-case analysis is too optimistic or unrealistic. For instance, when using the popular simplex method to solve linear programming problems, the worst-case run-time can be exponentially large in the size of the problem, whereas the average-case run-time (in which one is fed a randomly chosen linear program as input) is polynomial. However, the typical linear program that one encounters in practice has enough structure to it that it does not resemble a randomly chosen program at all, and so it is not clear that the average-case bound is appropriate for the type of inputs one has in practice. At the other extreme, the exponentially bad worst-case inputs were so rare that they never seemed to come up in practice either. To obtain a better input model, Spielman and Teng considered a smoothed-case model, in which the input was the sum of a deterministic (and possibly worst-case) input, and a small noise perturbation, which they took to be Gaussian to simplify their analysis. This reflected the presence of measurement error, roundoff error, and similar sources of noise in real-life applications of numerical algorithms. Remarkably, they were able to analyse the run-time of the simplex method for this model, concluding (after a lengthy technical argument) that under reasonable choices of parameters, the run time was polynomial time in the length, thus explaining the empirically observed phenomenon that the simplex method tended to run a lot better in practice than its worst-case analysis would predict, even if one started with extremely ill-conditioned inputs, provided that there was a bit of noise in the system. One of the ingredients in their analysis was a quantitative bound on the condition number of an arbitrary matrix when it is perturbed by a random gaussian perturbation; the point being that random perturbation can often make an ill-conditioned matrix better behaved. (This is perhaps analogous in some ways to the empirical experience that some pieces of machinery work better after being kicked.) Recently, new tools from additive combinatorics (in particular, inverse Littlewood-Offord theory) have enabled Rudelson-Vershynin, Vu and myself, and others to generalise this bound to other noise models, such as random Bernoulli perturbations, which are a simple model for modeling digital roundoff error. — 2. Yves Meyer — Yves Meyer has worked in many fields over the years, from number theory to harmonic analysis to PDE to signal processing. As the Gauss prize is concerned with impact on fields outside of mathematics, Meyer’s major contributions to the theoretical foundations of wavelets, which are now a basic tool in signal processing, were undoubtedly a major consideration in awarding this prize. But I would like to focus here on another of Yves’ contributions, namely the Coifman-Meyer theory of paraproducts developed with Raphy Coifman, which is a cornerstone of the para-differential calculus that has turned out to be an indispensable tool in the modern theory of nonlinear PDE. Nonlinear differential equations, by definition, tend to involve a combination of differential operators and nonlinear operators. The simplest example of the latter is a pointwise product {uv} of two fields {u} and {v}. One is thus often led to study expressions such as {D(uv)} or {D^{-1}(uv)} for various differential operators {D}. For first order operators {D}, we can handle derivatives using the product rule (or Leibniz rule) from freshman calculus: \displaystyle D(uv) = (Du)v + u(Dv). We can then iterate this to handle higher order derivatives. For instance, we have \displaystyle D^2(uv) = (D^2 u) v + 2 (Du) (Dv) + u (D^2 v), \displaystyle D^3(uv) = (D^3 u) v + 3 (D^2u) (Dv) + 3 (Du) (D^2 v) + u (D^3 v), and so forth, assuming of course that all functions involved are sufficiently regular so that all expressions make sense. For inverse derivative expressions such as {D^{-1}(uv)}, no such simple formula exists, although one does have the very important integration by parts formula as a substitute. And for fractional derivatives such as {\|D\|^\alpha(uv)} with {\alpha > 0} a non-integer, there is also no closed formula of the above form. Note how the derivatives on the product {uv} get distributed to the individual factors {u,v}, with {u} absorbing all the derivatives in one term, {v} absorbing all the derivatives in another, and the derivatives being shared between {u} and {v} in other terms. Within the usual confines of differential calculus; we cannot pick and choose which of the terms we like to keep, and which ones to discard; we must treat every single one of the terms that arise from the Leibniz expansion. This can cause difficulty when trying to control the product of two functions of unequal regularity – a situation that occurs very frequently in nonlinear PDE. For instance, if {u} is {C^1} (once continuously differentiable), and {v} is {C^3} (three times continuously differentiable), then the product {uv} is merely {C^1} rather than {C^3}; intuitively, we cannot “prevent” the three derivatives in the expression {D^3(uv)} from making their way to the {u} factor, which is not prepared to absorb all of them. However, it turns out that if we split the product {uv} into paraproducts such as the high-low paraproduct {\pi_{hl}(u,v)} and the low-high paraproduct, we can effectively separate these terms from each other, allowing for a much more flexible analysis of the situation. The concept of a paraproduct can be motivated by using the Fourier transform. For simplicity let us work in one dimension, with {D} being the usual differential operator {D = \frac{d}{dx}}. Using a Fourier expansion (and assuming as much regularity and integrability as is needed to justify the formal manipulations) of {u} into components of different frequencies, we have \displaystyle u(x) = \int_{\bf R} \hat u(\xi) e^{i x \xi}\ d\xi (here we use the usual PDE normalisation in which we try to hide the {2\pi} factor) and similarly \displaystyle v(x) = \int_{\bf R} \hat v(\eta) e^{i x \eta}\ d\eta and thus \displaystyle uv(x) = \int_{\bf R} \int_{\bf R} \hat u(\xi) \hat v(\eta) e^{ix(\xi+\eta)}\ d\xi d\eta \ \ \ \ \ (1) and thus, by differentiating under the integral sign \displaystyle D^k(uv)(x) = i^k \int_{\bf R} \int_{\bf R} (\xi+\eta)^k \hat u(\xi) \hat v(\eta) e^{ix(\xi+\eta)}\ d\xi d\eta. In contrast, we have \displaystyle (D^j u) (D^{k-j} v)(x) = i^k \int_{\bf R} \int_{\bf R} \xi^j \eta^{k-j} \hat u(\xi) \hat v(\eta) e^{ix(\xi+\eta)}\ d\xi d\eta; thus, the iterated Leibnitz rule just becomes the binomial formula \displaystyle (\xi+\eta)^2 = \xi^2 + 2\xi\eta + \eta^2, \quad (\xi+\eta)^3 = \xi^3 + 3 \xi^2 \eta + 3 \xi \eta^2 + \eta^3, \ldots after taking Fourier transforms. Now, it is certainly true that when dealing with an expression such as {(\xi+\eta)^2}, all three terms {\xi^2, 2\xi\eta, \eta^2} need to be present. But observe that when dealing with a “high-low” frequency interaction, in which {\xi} is much larger in magnitude than {\eta}, then the first term dominates: {(\xi+\eta)^2 \sim \xi^2}. Conversely, with a “low-high” frequency interaction, in which {\eta} is much larger in magnitude than {\xi}, we have {(\xi+\eta)^2 \sim \eta^2}. (There are also “high-high” interactions, in which {\xi} and {\eta} are comparable in magnitude, and {(\xi+\eta)^2} can be significantly smaller than either {\xi^2} or {\eta^2}, but for simplicity of discussion let us ignore this case.) It then becomes natural to try to decompose the product {uv} into “high-low” and “low-high” pieces (plus a “high-high” error), for instance by inserting suitable cutoff functions {m_{hl}(\xi,\eta)} or {m_{lh}(\xi,\eta)} in (1) to the regions {\|\xi\| \gg \|\eta\|} or {\|\xi\| \ll \|\eta\|} to create the paraproducts \displaystyle \pi_{hl}(u,v)(x) = \int_{\bf R} \int_{\bf R} m_{hl}(\xi,\eta) \hat u(\xi) \hat v(\eta) e^{ix(\xi+\eta)}\ d\xi d\eta \displaystyle \pi_{lh}(u,v)(x) = \int_{\bf R} \int_{\bf R} m_{lh}(\xi,\eta) \hat u(\xi) \hat v(\eta) e^{ix(\xi+\eta)}\ d\xi d\eta Such paraproducts were first introduced by Calderón, and more explicitly by Bony. Heuristically, {\pi_{hl}(u,v)} is the “high-low” portion of the product {uv}, in which the high frequency components of {u} are “allowed” to interact with the low frequency components of {v}, but no other frequency interactions are permitted, and similarly for {\pi_{lh}(u,v)}. The para-differential calculus of Bony, Coifman, and Meyer then allows one to manipulate these paraproducts in ways that are very similar to ordinary pointwise products, except that they behave better with respect to the Leibniz rule or with more exotic differential or integral operators. For instance, one has \displaystyle D^k \pi_{hl}(u,v) \approx \pi_{hl}(D^k u, v) \displaystyle D^k \pi_{lh}(u,v) \approx \pi_{lh}(u, D^k v) for differential operators {D^k} (and more generally for pseudodifferential operators such as {\|D\|^\alpha}, or integral operators such as {D^{-1}}), where we use the {\approx} symbol loosely to denote “up to lower order terms”. Furthermore, many of the basic estimates of the pointwise product, in particular Hölder’s inequality, have analogues for paraproducts; this is a special case of what is now known as the Coifman-Meyer theorem, which is fundamental in this subject, and is proven using Littlewood-Paley theory. The same theory in fact gives some estimates for paraproducts beyond what are available for products. For instance, if {u} is in {C^1} and {v} is in {C^3}, then the paraproduct {\pi_{lh}(u,v)} is “almost” in {C^3} (modulo some technical logarithmic divergences which I will not elaborate on here), in contrast to the full product {uv} which is merely in {C^1}. Paraproducts also allow one to extend the classical product and chain rules to fractional derivative operators, leading to the fractional Leibniz rule \displaystyle \|D\|^\alpha(uv) \approx (\|D\|^\alpha u) v + u (\|D\|^\alpha v) and fractional chain rule \displaystyle \|D\|^\alpha(F(u)) \approx (\|D\|^\alpha u) F'(u) which are both very useful in nonlinear PDE (see e.g. this book of Taylor for a thorough treatment). See also this brief Notices article on paraproducts by Benyi, Maldonado, and Naibo. — 3. Louis Nirenberg — Louis Nirenberg has made an amazing number of contributions to analysis, PDE, and geometry (e.g. John-Nirenberg inequality, Nirenberg-Treves conjecture (recently solved by Dencker), Newlander-Nirenberg theorem, Gagliardo-Nirenberg inequality, Caffarelli-Kohn-Nirenberg theorem, etc.), while also being one of the nicest people I know. I will mention only two results of his here, one of them very briefly. Among other things, Nirenberg and Kohn introduced the pseudo-differential calculus which, like the para-differential calculus mentioned in the previous section, is an extension of differential calculus, but this time focused more on generalisation to variable coefficient or fractional operators, rather than in generalising the Leibniz or chain rules. This calculus sits at the intersection of harmonic analysis, PDE, von Neumann algebras, microlocal analysis, and semiclassical physics, and also happens to be closely related to Meyer’s work on wavelets; it quantifies the positive aspects of the Heisenberg uncertainty principle, in that one can observe position and momentum simultaneously so long as the uncertainty relation is respected. But I will not discuss this topic further today. Instead, I would like to focus here instead on a gem of an argument of Gidas, Ni, and Nirenberg, which is a brilliant application of Alexandrov’s method of moving planes, combined with the ubiquitous maximum principle. This concerns solutions to the ground state equation \displaystyle \Delta Q + Q^p = Q \ \ \ \ \ (2) where {Q: {\bf R}^n \rightarrow {\bf R}^+} is a smooth positive function that decays exponentially at infinity, {p > 1} is an exponent, and {\Delta := \sum_{j=1}^n \frac{\partial^2}{\partial x_j^2}} is the Laplacian. This equation shows up in a number of contexts, including the nonlinear Schrödinger equation and also, by coincidence, in connection with the best constants in the Gagliardo-Nirenberg inequality. The existence of ground states {Q} can be proven by the variational principle. But one can say much more: Lemma 1 (Gidas-Ni-Nirenberg) All ground states {Q} are radially symmetric with respect to some origin. To show this radial symmetry, a small amount of Euclidean geometry shows that it is enough to show that there is a lot of reflection symmetry: Lemma 2 (Gidas-Ni-Nirenberg, again) If {Q} is a ground state and {\omega \in S^{n-1}} is a unit vector, then there exists a hyperplane orthogonal to {\omega} with respect to which {Q} is symmetric. To prove this lemma, we use the moving planes method, sliding in a plane orthogonal to {\omega} from infinity. More precisely, for each {t \in {\bf R}}, let {\Pi_t} be the hyperplane {\{ x: x \cdot \omega = t \}}, let {H_t} be the associated half-space {\{ x: x \cdot \omega \leq t\}}, and let {Q_t: H_t \rightarrow {\bf R}} be the function {Q_t(x) := Q(x) - Q(r_t(x))}, where \displaystyle r_t(x) := x + 2 (t - x \cdot \omega) \omega is the reflection through {\Pi_t}; thus {Q_t} is the difference between {Q} and its reflection in {\Pi_t}. In particular, {Q_t} vanishes on the boundary {\Pi_t} of the half-space {H_t}. Intuitively, the argument proceeds as follows. It is plausible that {Q_t} is going to be positive in the interior of {H_t} for large positive {t}, but negative in the interior of {H_t} for large negative {t}. Now imagine sliding {t} down from {+\infty} to {-\infty} until one reaches the first point {t = t_0} where {Q_{t_0}} ceases to be positive in the interior of {H_{t_0}}, then it attains its minimum at some zero in the interior of {H_{t_0}}. But by playing around with (2) (using the Lipschitz nature of the map {Q \mapsto Q^p} when {Q} is bounded) we know that {Q_{t_0}} obeys an elliptic constraint of the form {\Delta Q_{t_0} = O( \|Q_{t_0}\| )}. Applying the maximum principle, we can then conclude that {Q_{t_0}} vanishes identically in {H_{t_0}}, which gives the desired reflectoin symmetry. (Now it turns out that there are some technical issues in making the above sketch precise, mainly because of the non-compact nature of the half-space {H_t}, but these can be fixed with a little bit of fiddling; see for instance Appendix B of my PDE textbook.)
a13bacd0ecc19f75	Abraham Meets Abraham from a Parallel Universe And he [Abraham] lifted up his eyes and looked, and, lo, three men stood over against him… (Genesis 18:2) On this blog, we often discuss the collapse of the wavefunction as the result of a measurement. This phenomenon is called the “measurement problem.” There are several reasons, why the collapse of the wavefunction – part and parcel of the Copenhagen interpretation of quantum mechanics – is called a problem. Firstly, it does not follow from the Schrödinger equation, the main equation of quantum mechanics that describes the evolution of the wavefunction in time, and is added ad hoc. Secondly, nobody knows how the collapse happens or how long it takes to collapse the wavefunction. This is not to mention that any notion that the collapse of the wavefunction is caused by human consciousness, [...]
36ca0dd32e5c4b3f	Hartree–Fock method (Redirected from Hartree-Fock theory) Brief historyEdit Early semi-empirical methodsEdit The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B. Lindsay, and himself) set in the old quantum theory of Bohr. In the Bohr model of the atom, the energy of a state with principal quantum number n is given in atomic units as . It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the quantum defect d as an empirical parameter, the energy levels of a generic atom were well approximated by the formula , in the sense that one could reproduce fairly well the observed transitions levels observed in the X-ray region (for example, see the empirical discussion and derivation in Moseley's law). The existence of a non-zero quantum defect was attributed to electron–electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial screening of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data. Hartree methodEdit In 1927, D. R. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions.[3] Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., ab initio. His first proposed method of solution became known as the Hartree method, or Hartree product. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions.[4][5] In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function.[6][7] The Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of quantum statistics. A solution to the lack of anti-symmetry in the Hartree method came when it was shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. Fock's original method relied heavily on group theory and was too abstract for contemporary physicists to understand and implement. In 1935, Hartree reformulated the method to be more suitable for the purposes of calculation.[8] The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the central field approximation, to impose the condition that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, calculating a solution by hand using the Hartree–Fock equations for a medium-sized atom was laborious; small molecules required computational resources far beyond what was available before 1950. Hartree–Fock algorithmEdit • The mean-field approximation is implied. Effects arising from deviations from this assumption are neglected. These effects are often collectively used as a definition of the term electron correlation. However, the label "electron correlation" strictly spoken encompasses both Coulomb correlation and Fermi correlation, and the latter is an effect of electron exchange, which is fully accounted for in the Hartree–Fock method.[9][10] Stated in this terminology, the method only neglects the Coulomb correlation. However, this is an important flaw, accounting for (among others) Hartree–Fock's inability to capture London dispersion.[11] Variational optimization of orbitalsEdit Algorithmic flowchart illustrating the Hartree–Fock method The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as spin-orbitals. For an atomic orbital calculation, these are typically the orbitals for a hydrogen-like atom (an atom with only one electron, but the appropriate nuclear charge). For a molecular orbital or crystalline calculation, the initial approximate one-electron wave functions are typically a linear combination of atomic orbitals (LCAO). Since the Fock operator depends on the orbitals used to construct the corresponding Fock matrix, the eigenfunctions of the Fock operator are in turn new orbitals, which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals is calculated. The Hartree–Fock electronic wave function is then the Slater determinant constructed from these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed. Mathematical formulationEdit The Fock operatorEdit Because the electron–electron repulsion term of the molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation (outlined under Hartree–Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear–nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below, for closed-shell atoms or molecules (with two electrons in each spatial orbital).[12] The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature. is the one-electron Fock operator generated by the orbitals , and is the one-electron core Hamiltonian. Also is the Coulomb operator, defining the electron–electron repulsion energy due to each of the two electrons in the j-th orbital.[12] Finally, is the exchange operator, defining the electron exchange energy due to the antisymmetry of the total N-electron wave function.[12] This "exchange energy" operator is simply an artifact of the Slater determinant. Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation Linear combination of atomic orbitalsEdit Numerical stabilityEdit Weaknesses, extensions, and alternativesEdit Software packagesEdit See alsoEdit 1. ^ Froese Fischer, Charlotte (1987). "General Hartree-Fock program". Computer Physics Communications. 43 (3): 355–365. Bibcode:1987CoPhC..43..355F. doi:10.1016/0010-4655(87)90053-1. 3. ^ Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field". Math. Proc. Camb. Philos. Soc. 24 (1): 111. doi:10.1017/S0305004100011920. 4. ^ Slater, J. C. (1928). "The Self Consistent Field and the Structure of Atoms". Phys. Rev. 32 (3): 339. doi:10.1103/PhysRev.32.339. 5. ^ Gaunt, J. A. (1928). "A Theory of Hartree's Atomic Fields". Math. Proc. Camb. Philos. Soc. 24 (2): 328. doi:10.1017/S0305004100015851. 6. ^ Slater, J. C. (1930). "Note on Hartree's Method". Phys. Rev. 35 (2): 210. doi:10.1103/PhysRev.35.210.2. 7. ^ Fock, V. A. (1930). "Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems". Z. Phys. (in German). 61 (1): 126. doi:10.1007/BF01340294. Fock, V. A. (1930). "„Selfconsistent field" mit Austausch für Natrium". Z. Phys. (in German). 62 (11): 795. doi:10.1007/BF01330439. 8. ^ Hartree, D. R.; Hartree, W. (1935). "Self-consistent field, with exchange, for beryllium". Proc. Royal Soc. Lond. A. 150 (869): 9. doi:10.1098/rspa.1935.0085. External linksEdit
e73e4273be8c02cb	Specific information • Ikariam uses "s23" in its URL as the server designation for this world, no matter which community / language version you use. • Ikariam: • Ikariam uses a few specialty servers for the following reasons: Statistical information • The Psi world makes up: • Psi is active in 10 communities. • These communities / servers represent: • 31.25 % of the 32 grand-total number of communities. • 4.5249 % of the 221 total Greek alphabet servers. • 2.5253 % of the 396 player accessible servers, excluding the specialty (Speed, Test and War) servers. • 2.331 % of the 429 total player accessible servers. • 2.3095 % of the 433 grand total number of all of the world servers. • Each individual community that has the Psi server is 10 % of the group of 10 . Special Attributes Psi Psi (s23) is a Special Server in the Ar AR, Cz CZ, Hu HU and Pt PT communities. 1. Ar AR 2. Cz CZ 3. Hu HU and Pt PT General information Psi (uppercase Ψ, lowercase ψ) is the 23rd letter of the modern Greek alphabet and has a numeric value of 700. In both Classical and Modern Greek, the letter indicates the combination /ps/ (like in English "lapse"). In Greek, this consonant cluster can occur in the syllable initial position, as in the Greek word "ψάρι" [psári (=fish)]. However, in some languages (including English) this combination is not possible at the beginning of a syllable. In Latin, Greek words beginning with psi are transcribed by ps-, but seem to have been pronounced simply as s-, with a quiescent p. This pronunciation has affected that of Greek loanwords beginning with the letter in several languages. In English, for example, psychology is pronounced with a silent p, and the name of the letter is often pronounced [saɪ] ("sigh"), although this is not always the case. Any person who has learned Greek is aware that the correct pronunciation is actually "see". The letter was adopted into the Old Italic alphabet, and its shape is continued into the Algiz rune of the Elder Futhark. The letter may have originated from the practice of writing the sigma over the pi, eventually making the combination into a single letter.[citation needed] Psi was also adopted into the early Cyrillic alphabet. See psi (Cyrillic) (Ѱ, ѱ). This may have also been since the letter, both in lower case and uppercase form, resembles the trident wielded by Poseidon, the Greek god of water/ocean. The letter psi is commonly used in physics for representing a wavefunction in quantum mechanics, particularly with the Schrödinger equation and bra-ket notation: . It is also used to represent the (generalized) positional states of a qubit in a quantum computer. Other World ( Active type ) servers All items (16)
250f67ce9aed52d4	Tuesday, August 31, 2010 Why complex numbers are fundamental in physics History of complex numbers in mathematics Cardano was able to additively shift "x" by "a/3" ("a" is the quadratic coefficient of the original equation) to get rid of the quadratic coefficient. Without a loss of generality, he was therefore solving the equations of the type x3 + bx + c = 0 that only depends on two numbers, "b, c". Cardano was aware of one of the three solutions to the equation; it was co-co-communicated to him by Tartaglia (The Stammerer), also known as Niccolo Fontana. It is equal to x1 = cbrt[-c/2 + sqrt(c2/4+b3/27)] + + cbrt[-c/2 - sqrt(c2/4+b3/27)] Here, cbrt is the cubic root. You can check it is a solution if you substitute it to the original equation. Now, using the modern technologies, it is possible to divide the cubic polynomial by "(x - x_1)" to obtain a quadratic polynomial which produces the remaining two solutions once it is solved. Let's assume that the cubic polynomial has 3 real solutions. The shocking revelation came in 1572 when Rafael Bambelli was able to find real solutions using the complex numbers as tools in the intermediate calculations. This is an event that shows that the new tool was bringing you something useful: it wasn't just a piece of unnecessary garbage for which the costs are equal the expenses and that should be cut away by Occam's razor: it actually helps you to solve your old problems. Consider the equation x3 - 15x - 4 = 0. Just to be sure where we're going, compute the three roots by Mathematica or anything else. They're equal to x1,2,3 = {4, -2-sqrt(3), -2+sqrt(3)} The coefficient "b=-15" is too big and negative, so the square root in Cardano's formula is the square root of "(-15)^3/27 + 4^2/4" which is a square root of "-125+4" or "-121". You can't do anything about that: it is negative. The argument could have been positive for other cubic polynomials if the coefficient "b" were positive or closer to zero, instead of "-15", but with "-15", it's just negative. Bombelli realized the bombshell that one can simply work with the "sqrt(-121)" as if it were an actual number; we don't have to give up once we encounter the first unusual expression. Note that it is being added to a real number and a cube root is computed out of it. Using the modern language, "sqrt(-121)" is "11i" or "-11i". The cube roots are general complex numbers but if you add two of them, the imaginary parts cancel. Only the real parts survive. Bombelli was able to indirectly do this calculation and show that x1 = cbrt(2+11i) + cbrt(2-11i) = (2+i) + (2-i) = 4 which matches the simplest root. That was fascinating! Please feel free to verify that (2+i)^3 is equal to "8+12i-6-i = 2+11i" and imagine that the historical characters would write "sqrt(-1)" instead of "i". By the way, it is trivial to calculate the other two roots "x_2, x_3" if you simply multiply the two cubic roots, cbrt, which were equal to "(2+-i)", by the two opposite non-real cubic roots of unity, "exp(+-2.pi.i/3) = -1/2+-i.sqrt(3)/2". When additions to these insights were made by John Wallis in 1673 and later by Euler, Cauchy, Gauss, and others, complex numbers took pretty much their modern form and mathematicians have already known more about them than the average TRF readers - sorry. ;-) Fundamental theorem of algebra Complex numbers have many cool properties. For example, every N-th order algebraic (polynomial) equation with real (or complex) coefficients has exactly "n" complex solutions (some of them may coincide, producing multiple roots). How do you prove this statement? Using powerful modern TRF techniques, it's trivial. At a sufficiently big circle in the complex plane, the N-th order polynomial qualitatively behaves like a multiple of "x^N". In particular, the complex phase of the value of this polynomial "winds" around the zero in the complex plane N times. Or the logarithm of the polynomial jumps by 2.pi.i.N, if you wish. You may divide the big circle into an arbitrarily fine grid and the N units of winding have to come from some particular "little squares" in the grid: the jump of the logarithm over the circle is the sum of jumps of the logarithm over the round trips around the little squares that constitute the big circle. The little squares around which the winding is nonzero have to have the polynomial equal to zero inside (otherwise the polynomial would be pretty much constant and nonzero inside, which would mean no winding) - so the roots are located in these grids. If the winding around a small square is greater than one, there is a multiple root over there. In this way, you can easily find the roots and their number is equal to the degree of the polynomial. Fine. People have learned lots of things about the calculus - and functions of complex variables. They were mathematically interesting, to say the least. Complex numbers are really "new" because they can't be reduced to real diagonal matrices. That wouldn't be true e.g. for "U-complex" numbers "a+bU" where "U^2=+1": you could represent "U" by "sigma_3", the Pauli matrix, which is both real and diagonal. Complex numbers have unified geometry and algebra. The exponential of an imaginary number produces sines and cosines - and knows everything about the angles and rotations (multiplication by a complex constant is a rotation together with magnification). The behavior of many functions in the complex plane - e.g. the Riemann zeta function - has been linked to number theory (distribution of primes) and other previously separate mathematical disciplines. There's no doubt that complex numbers are essential in mathematics. Going to physics In classical physics, complex numbers would be used as bookkeeping devices to remember the two coordinates of a two-dimensional vector; the complex numbers also knew something about the length of two-dimensional vectors. But this usage of the complex numbers was not really fundamental. In particular, the multiplication of two complex numbers never directly entered physics. This totally changed when quantum mechanics was born. The waves in quantum mechanics had to be complex, "exp(ikx)", for the waves to remember the momentum as well as the direction of motion. And when you multiply operators or state vectors, you actually have to multiply complex numbers (the matrix elements) according to the rules of complex multiplication. Now, we need to emphasize that it doesn't matter whether you write the number as "exp(ikx)", "cos(kx)+i.sin(kx)", "cos(kx)+j.sin(kx)", or "(cos kx, sin kx)" with an extra structure defining the product of two 2-component vectors. It doesn't matter whether you call the complex numbers "complex numbers", "Bambelli's spaghetti", "Euler's toilets", or "Feynman's silly arrows". All these things are mathematically equivalent. What matters is that they have two inseparable components and a specific rule how to multiply them. The commutator of "x" and "p" equals "xp-px" which is, for two Hermitean (real-eigenvalue-boasting) operators, an anti-Hermitean operator i.e. "i" times a Hermitean operator (because its Hermitean conjugate is "px-xp", the opposite thing). You can't do anything about it: if it is a c-number, it has to be a pure imaginary c-number that we call "i.hbar". The uncertainty principle forces the complex numbers upon us. So the imaginary unit is not a "trick" that randomly appeared in one application of some bizarre quantum mechanics problem - and something that you may humiliate. The imaginary unit is guaranteed to occur in any system that reduces to classical physics in a limit but is not a case of classical physics exactly. Completely universally, the commutator of Hermitean operators - that are "deduced" from real classical observables - have commutators that involve an "i". That means that their definitions in any representation that you may find have to include some "i" factors as well. Once "i" enters some fundamental formulae of physics, including Schrödinger's (or Heisenberg's) equation, it's clear that it penetrates to pretty much all of physics. In particular: In quantum mechanics, probabilities are the only thing we can compute about the outcomes of any experiments or phenomena. And the last steps of such calculations always include the squaring of absolute values of complex probability amplitudes. Complex numbers are fundamental for all predictions in modern science. Thermal quantum mechanics One of the places where imaginary quantities occur is the calcuation of thermal physics. In classical (or quantum) physics, you may calculate the probability that a particle occupies an energy-E state at a thermal equilibrium. Because the physical system can probe all the states with the same energy (and other conserved quantities), the probability can only depend on the energy (and other conserved quantities). By maximizing the total number of microstates (and entropy) and by using Stirling's approximation etc., you may derive that the probabilities go like "exp(-E/kT)" for the energy-E states. Here, "T" is called the temperature and Boltzmann's constant "k" is only inserted because people began to use stupidly different units for temperature than they used for energy. This exponential gives rise to the Maxwell-Boltzmann and other distributions in thermodynamics. The exponential had to occur here because it converts addition to multiplication. If you consider two independent subsystems of a physical system (see Locality and additivity of energy), their total energy "E" is just the sum "E1+E2". And the value of "exp(-E/kT)" is simply the product of "exp(-E1/kT)" and "exp(-E2/kT)". This product is exactly what you want because the probability of two independent conditions is the product of the two separate probabilities. The exponential has to be everywhere in thermodynamics. Fine. When you do the analogous reasoning in quantum thermodynamics, you will still find that the exponential matters. But the classical energy "E" in the exponent will be replaced by the Hamiltonian "H", of course: it's the quantum counterpart of the classical energy. The operator "exp(-H/kT)" will be the right density matrix (after you normalize it) that contains all the information about the temperature-T equilibrium. There is one more place where the Hamiltonian occurs in the exponent: the evolution operator "exp(H.t/i.hbar)". The evolution operator is also an exponential because you may get it as a composition of the evolution by infinitesimal intervals of time. Each of these infinitesimal evolutions may be calculated from Schrödinger's equation and [1 + H.t/(i.hbar.N)]N = exp(H.t/i.hbar) in the large "N" limit: we divided the interval "t" to "N" equal parts. If you don't want to use any infinitesimal numbers, note that the derivative of the exponential is an exponential again, so it is the right operator that solves the Schrödinger-like equation. So fine, the exponentials of multiples of the Hamiltonian appear both in the thermal density matrix as well as in the evolution operator. The main "qualitative" difference is that there is an "i" in the evolution operator. In the evolution operator, the coefficient in front of "H" is imaginary while it is real in the thermal density matrix. But you may erase this difference if you consider an imaginary temperature or, on the contrary, you consider the evolution operator by an imaginary time "t = i.hbar/k.T". Because the evolution may be calculated in many other ways and additional tools are available, it's the latter perspective that is more useful. The evolution by an imaginary time calculates thermal properties of the system. Now, is it a trick that you should dismiss as an irrelevant curiosity? Again, it's not. This map between thermal properties and imaginary evolution applies to the thermodynamics of all quantum systems. And because everything in our world is quantum at the fundamental level, this evolution by imaginary time is directly relevant for the thermodynamics of anything and everything in this world. Any trash talk about this map is a sign of ignorance. Can we actually wait for an imaginary time? As Gordon asked, can such imaginary waiting be helpful to explain why we're late for a date with a woman (or a man, to be really politically correct if a bit disgusting)? Well, when people were just animals, Nature told us to behave and to live our lives in the real time only. However, theoretical physicists have no problem to live their lives in the imaginary or complex time, too. At least they can calculate what will happen in their lives. The results satisfy most of the physical consistency conditions you expect except for the reality conditions and the preservation of the total probabilities. ;-) Frankly speaking, you don't want to live in the imaginary time but you should certainly be keen on calculating with the imaginary time! Analytic continuation The thermal-evolution map was an example showing that it is damn useful to extrapolate real arguments into complex values if you want to learn important things. However, thermodynamics is not the only application where this powerful weapon shows its muscles. More precisely, you surely don't have to be at equilibrium to see that the continuations of quantities to complex values will bring you important insights that can't be obtained by inequivalent yet equally general methods. The continuation into imaginary values of time is linked to thermodynamics, the Wick rotation, or the Hartle-Hawking wave function. Each of these three applications - and a few others - would deserve a similar discussion to the case of the "thermodynamics as imaginary evolution in time". I don't want to describe all of conceptual physics in this text, so let me keep the thermodynamic comments as the only representative. Continuation in energy and momentum However, it's equally if not more important to analytically continue in quantities such as the energy. Let us immediately say that special relativity downgrades energy to the time component of a more comprehensive vector in spacetime, the energy-momentum vector. So once we will realize that it's important to analytically continue various objects to complex energies, relativity makes it equally important to continue analogous objects to complex values of the momentum - and various functions of momenta such as "k^2". Fine. So we are left with the question: Why should we ever analytically continue things into the complex values of the energy? A typical laymen who doesn't like maths too much thinks that this is a contrived, unnatural operation. Why would he do it? A person who likes to compute things with the complex numbers asks whether we can calculate it. The answer is Yes, we can. ;-) And when we do it, we inevitably obtain some crucial information about the physical system. A way to see why such things are useful is to imagine that the Fourier transform of a step function, "theta(t)" (zero for negative "t", one for positive "t"), is something like "1/(E-i.epsilon)". If you add some decreasing "exp(-ct)" factor to the step function, you may replace the infinitesimal "epsilon" by a finite constant. Anyway, if you perturb the system at "t=0", various responses will only exist for positive values of "t". Many of them may exponentially decrease - like in oscillators with friction. All the information about the response at a finite time can be obtained by continuing the Fourier transform of various functions into complex values of the energy. Because many physical processes will depend "nicely" or "analytically" on the energy, the continuation will nicely work. You will find out that in the complex plane, there can be non-analyticities - such as poles - and one can show that these singular points or cuts always have a physical meaning. For example, they are identified with possible bound states, their continua, or resonances (metastable states). The information about all possible resonances etc. is encoded in the continuation of various "spectral functions" - calculable from the evolution - to complex values of the energy. Unitarity (preservation of the total probabilities) can be shown to restrict the character of discontinuities at the poles and branch cuts. Some properties of these non-analyticities are also related to the locality and other things. There are many links over here for many chapters of a book. However, I want to emphasize the universal, "philosophical" message. These are not just "tricks" that happen to work as a solution to one particular, contrived problem. These are absolutely universal - and therefore fundamental - roles that the complex values of time or energy play in quantum physics. Regardless of the physical system you consider (and its Hamiltonian), its thermal behavior will be encoded in its evolution over an imaginary time. If Hartle and Hawking are right, then regardless of the physical system, as long as it includes quantum gravity, the initial conditions of its cosmological evolution are encoded in the dynamics of the Euclidean spacetime (which contains an imaginary time instead of the real time from the Minkowski spacetime). Regardless of the physical system, the poles of various scattering amplitudes etc. (as functions of complexified energy-momentum vectors) tell you about the spectrum of states - including bound states and resonances. Before one studies physics, we don't have any intuition for such things. That's why it's so important to develop an intuition for them. These things are very real and very important. Everyone who thinks it should be taboo - and it should be humiliated - if someone extrapolates quantities into complex values of the (originally real) physical arguments is mistaken and is automatically avoiding a proper understanding of a big portion of the wisdom about the real world. Most complex numbers are not "real" numbers in the technical sense. ;-) But their importance for the functioning of the "real" world and for the unified explanation of various features of the reality is damn "real". And that's the memo. 1. Learn Geometric Algebra and then you won't need complex numbers anymore (for physics) Complex numbers are nothing more than a subalgebra of GA/Clifford algebra. Nothing special about them at all. 2. Holy cow, gezinorgiva. There is everything fundamental and special about the complex numbers as you would know if you have read at least my modest essay about them. The complex numbers may be a subset of many other sets but the complex numbers are much more fundamental than any of these sets. The nearest college or high school is recommended. 3. There is an interesting article related to the topic of this post by C.N. Yang in the book "Schrodinger, Centenary celebration of a polymath" E. C.W. Kilmister, entitled "Square root of minus one, complex phases and Erwin Schrodinger". There Yang quotes Dirac as saying that as a young man he thought that non commutativity was the most revolutionary and essentially new feature of quantum mechanics, but as he got older he got to think that that was the entrance of complex numbers in physics in a fundamental way (as opposite to as auxiliary tools as in circuit theory). He describes Schrodinger struggles to come to terms with that, after unsuccessfully trying to get rid of "i". Also included is the role of previous work by Schrodinger in Weyl's seminal gauge theory ideas in his discovering of quantum mechanics. 4. This comment has been removed by the author. 5. Noncommutativity implies complex numbers; the Pauli spin matrices sigma_x sigma_y sigma_z multiply to give i. 6. Carl, please... Since you're gonna be a student again, you will have to learn how to think properly again. Your statement is illogical at every conceivable level. First, the "product" of the three Pauli matrices has nothing directly to do with noncommutativity. The latter is a property of two matrices, not three matrices. The product is not a commutator (although it's related to it). Second, the fact that the product includes an "i" is clearly a consequence of the fact that in the conventional basis, one of the Pauli matrices - namely sigma_{y} - is pure imaginary. This imaginary value of sigma_{y} is the reason, not a consequence, of the product's being imaginary. Third, it's easy to see that noncommutativity doesn't imply any complex numbers in general. The generic real - non-complex - matrices (e.g. the non-diagonal ones) are noncommutative but their commutator is always a real matrix. Noncommutativity by itself is completely independent of complexity of the numbers. And indeed, complex numbers themselves are commutative, not non-commutative. The only way to link noncommutativity and complex numbers is to compute the eigenvalues of the commutator of two Hermitean operators. Because their commutator is anti-Hermitean, its eigenvalues are pure imaginary. For example, the commutator can be an imaginary c-number, e.g. in xp-px. For more general operators, the eigenvalues are typically computed from a characteristic equation that will contain (x^2+r^2) factors, producing ir and -ir as eigenvalues. 7. It's actually impossible to avoid the existence of complex numbers even in real analysis—or at least to avoid their effects. Consider the Taylor series of the function f(x)=1/(1-x^2) centered around x=0. The series is given by f(x)=1+x^2+x^4+x^6+... . It can be seen that this Taylor series is divergent for \|x\|>1 and so the Taylor series will fail for large x. This isn't very surprising as it can be seen that f(x) has obvious singularities at x=-1,+1 and so the Taylor series could not possibly extend beyond these points. However, more interesting is the same approach to the function g(x)=1/(1+x^2). This function is perfectly well behaved, having no singularities of any order in the real number. Yet its Taylor series g(x)=1-x^2+x^4-x^6+... is divergent for \|x\|>1, despite there seemingly being no corresponding singularity as in the previous case. Analysis in the reals leads to the idea of a radius of convergence, but gives no clear idea where this comes from. In fact using complex numbers the reason becomes clear. g(x) has singularities at x=-i,+i. Despite these existing only in the complex plane, their effects can be felt for the real function. In fact the radius of converge of a Taylor series is the distance from the central point to the nearest singularity—be it in the real or complex plane(See the book "Visual Complex Analysis" for more). Complex numbers become fundamental and indeed in some sense unavoidable the moment we introduce multiplication and division into our algebra. This is because these operations—and most (all?) elementary operations—hold for complex numbers in general and not just for the real numbers. Once we write expressions like (x^2+7)/(x^4-3), while we may mean for x to be a purely real number, the complex numbers will work in this equation just as well, and indeed more importantly, will continue to work as we perform all elementary algebraic operations on the expression; "BOMDAS" operations, radixes, even taking exponents and logs. This should not come as too much of a surprise, and we could have started—like the Pythagoreans—by meaning for the expression to be restricted to rational numbers and even disregarding irrational numbers entirely. But irrational numbers will work in the original expression and in through all our rational manipulations. What we claim holds for a subset of numbers holds for the large set too. And the existence of this larger set has concrete implications for expressions on the subset. So, when we write down any equation at all, we must be careful. We may mean for it to hold for some restricted class of numbers, but there may be much wider implications. While I am not a physicist, I suspect a similar situation arise. We may want or expect the quantities we measure to expressible in purely real numbers; but the universe may have other ideas. 8. Sorry, getting old. I meant "Clifford or geometric algebra" rather than "noncommutative". I was continuing the comment by gezinoriva. And the "i" is not "clearly a consequence" of a basis choice. The product sigma_x sigma_y sigma_z is an element of the Clifford algebra that commutes with everything in the algebra and squares to -1. That's what makes it's interpretation "i" and this does not depend on basis choice. 9. Dear Carl, it's completely unclear to me why you think that you have "explained" complex numbers. A number that squares to minus one is the defining property of the imaginary unit "i". You just "found" one (complicated) application - among thousands - where the imaginary unit and/or complex numbers emerge. The condition that a quantity squares to a negative number appears at thousands of other places, too. For example, it's the coefficient in the exponent of oscillating functions - that are eigenvectors under differentiation. Why do you think that Clifford algebras are special? 10. Dear Lumos; The Clifford algebras are special as they are related to the geometry of space-time. For example, (ignoring the choice of basis and only looking at algebraic relations) Dirac's gamma matrices are a Clifford algebra. Generalizing to higher dimension people expect that the generalization of the gamma matrices will also be a Clifford algebra. On this subject I sort of follow David Hestenes; his work geometrizes the quantum wave functions, but I prefer to geometrize the pure density matrices. But other than that, his work explains some of the justification. See his papers at geocalc.clas.asu.edu My concentration on this subject is due to my belief that geometry is more fundamental than symmetry. (This belief is a "working belief" only, that is, what I really believe is that it's more useful for me to assume this belief than to assume the default which almost everyone else assumes.) A beautiful example of putting geometry ahead of symmetry are Hestenes' description of point groups in geometric / Clifford algebra. I'm sure you'll enjoy these: Point Groups and Space Groups in GA and Crystallographic Space Groups 11. Apologies, Carl, but what you write is a crackpottery that makes no sense. Clifford algebras are related to the geometry of spacetime? So is the Hartle-Hawking wave function, black holes, wormholes quintic hypersurface, conifold, flop transition, and thousands of other things I can enumerate. In one half of them, complex numbers play an important role. Also, what the hell do you misunderstand about the generalization of gamma matrices to higher dimensions - which are still just ordinary gamma matrices - that you describe them in this mysterious way? You just don't know what you're talking about. 12. Mathematics is an infinite subject and uses complex numbers in an infinite number of ways. Who cares. What's important is when they appear in the definition of space itself, before QM or SR or GR. In the traditional physics approach, the Pauli spin matrices are just useful matrices for describing spin-1/2. But they can be given a completely geometric meaning and i falls out as the product. See equation (1.8) of Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics Regarding the relationship between higher dimensions and gamma matrices see the wikipedia article Higher dimensional gamma matrices It defines the higher dimensional gamma matrices as matrices that satisfy the Clifford algebra relations. But this is well known to string theorists, why are you asking? I must be misunderstanding you. 13. Dear Carl, your comment is a constant stream of nonsense. First, in physics, one can't define space without relativity or whatever replaces it. You either have a space of relativistic physics, or space of non-relativistic physics, but you need some space and its detailed physical properties always matter because they define mathematically inequivalent structures. So it's not possible to define "space before anything else" such as relativity: space is inseparably linked to its physical properties. In particular, space of Newtonian physics is simply incorrect for physics when looked at with some precision - e.g. in the presence of gravity or high speeds. Second, the examples I wrote were also linked to space - and they were arguably linked to space much more tightly than your Clifford algebra example. So it is nonsensical for you to return to the thesis that your example is more "space-related" or more fundamental than mine. I have irrevocably shown you that it's not. Third, it's just one problem with your statements that the Clifford algebra is not "the most essential thing" for space. Another problem is the fact that space itself is not more fundamental than many other notions in physics. Space itself is just one important concept in physics - and there are many others, equally important ones, and they're also linked to complex numbers. All of them can be fundamental in some descriptions, all of them - including space - may be emergent. It's just irrational to worship the concept of space as something special. So even your broader assumption that what is more tightly linked to space has to be more fundamental is a symptom of your naivite - or a quasi-religious bias. Fourth, it was you, not me, who claimed that he has some problems with totally elementary things such as Dirac matrices in higher dimensions. So why the fuck are you now reverting your statement? Previously, you wrote "Generalizing to higher dimension people expect that the generalization of the gamma matrices will also be a Clifford algebra." I have personally learned Dirac matrices for all possible dimensions at the very first moments when I encountered the Dirac matrices, I have always taught them in this way as well, and that's how it should be because the basic properties and algorithms of Dirac matrices naturally work in any dimension - and only by doing the algorithm in an arbitrary dimension, one really understands what the algorithm and the properties of the matrices are. So why are you creating a non-existent controversy about the Dirac matrices in higher dimensions? It's a rudimentary piece of maths. Moreover, in your newest comment, you directly contradicted your previous comment when you claimed that it was me, and not you, who claimed that there was a mystery with higher-dimensional matrices. There are about 5 completely fundamental gaps in your logic. One of them would be enough for me to think that the author of a comment isn't able to go beyond a sloppy thinking. But your reasoning is just defective at every conceivable level. I just don't know how to interact with this garbage. You're just flooding this blog with complete junk. 15. Dear Lubos, i don't agree that i has to be represented as a c-number. In fact i think many of the posters have been trying to say (poorly) the following: i can be definitely defined algebraically as a c-number can be wrote in the representation of a conmutative subalgebra of SO(2) defined by the isomorphism: a + ib <=> ( ( a , b) , ( -b , a) ) (sorry, i had to write a matrix as a list of rows, i hope its clear) 16. Dear seňor karra, of course, I realize this isomorphism with the matrices. But it's just a convention whether you express the "number that squares to minus one" as a matrix or as a new letter. It's mathematically the same thing. The important thing is that you introduce a new object with new rules. In particular, in "your" case, you must guarantee that the matrices you call "complex numbers" are not general matrices but just combinations of the "1" and "i" matrices. In the case of a letter "i", you must introduce its multiplication rules. 17. @Lumo: Clifford algebra is the generalization of complex numbers and quaternions to arbitrary dimensions. Just google it. Therefore it should be no controversy here. Clifford algebra (or geometric algebra) has been very successful in reformulating every theory of physics into the same mathematical language. That has, among other tings, emphasized the similarities and differences between the theories of physics in a totally new way. One elegant feature of this reformulation is to reduce Maxwell's equations into one single equation. The reason why Clifford algebra has lately been renamed "geometric algebra" is that quantities of the algebra are given geometric interpretations, and the Clifford product are effective in manipulating these geometric quantities directly. Together with the extension of the algebra to a calculus this formalism has the power to effectively model many geometries such as projective, conformal, and differential geometry. In the geometric algebra over three dimensions most quantities are interpreted as lines, planes and volumes. Plane and volume segments of unit size are represented with algebraic objects that square to minus one. In the reformulation of quantum mechanics with geometric algebra (describes geometry of the three dimensions of physical space), the unit imaginary from the standard treatment is identified with several different quantities in the algebra. In some situations, as in the Schrödinger equation, the unit imaginary times h bar is identified with the spin of the particle by the geometric algebra reformulation. This, together with other results of the reformulation suggests that spin is an intrinsic part of every aspect of quantum mechanics, and that spin may be the only cause of quantum effects. If you have the time and interest I strongly suggest reading a little about geometric algebra. Geometric algebra is not on a collision course with complex numbers. In fact, geometric algebra embrace, generalize and deploy them to a much larger extent than before. 18. Dear Hugo, the very assertion that "the Clifford algebra is the generalization of complex numbers to any dimension" is largely vacuous. Complex numbers play lots of roles and they're unique in the most important roles. One may hide his head into the sand and forget about some important properties of the complex numbers - e.g. the fact that every algebraic equation of N-th degree has N solutions, not necessarily different, in the complex realm (something that makes C really unique) - but if he does forget them, he's really throwing the baby out with the bath water. Of course that if you forget about some conditions, you may take the remaining conditions (the subset) and find new solutions besides C, "generalizations". But it's surely morally invalid to say that the Clifford algebra is "the" generalization. It's at most "a" generalization in some particular direction - one that isn't extremely important. 19. Ok, that's a semi-important point for the physicist; Clifford algebra is _a_ generalization of complex numbers and quaternions. It is puzzling that all you managed to extract from my comment was that I should have written "a" in stead of "the". My comment was about the role of Clifford algebra in physics. When you state that Clifford algebra is not important you should consider explaining why, if you don't want to be regarded as ignorant and "not important" yourself. 20. Dear Huge, your "the" instead of "a" was a very important mistake, one that summarizes your whole misunderstanding of the importance of complex numbers. This article was about the importance of complex numbers in physics and the branches of mathematics that are used in physics. This importance can't be understated. Clifford algebras simply came nowhere close to it. They're many orders of magnitude less important than complex numbers. There may exist mathematical fundamentalists and masturbators who would like if physics were all about Clifford algebras but the fact still is that physics is not about them. They're a generalization of complex numbers that isn't too natural from a physics viewpoint. After all, even quaternions themselves have an extremely limited role in physics, too. The relative unimportance of Clifford algebras in physics may be interpreted in many different ways. For example, it is pretty much guaranteed that a big portion of top physicists don't even know what a Clifford algebra actually is. Mostly those who were trained as mathematicians do know it. Others who "vaguely know" will tell you that it's an algebra of gamma matrices for spinors, or something like that, but they won't tell you why you would talk about them with such a religious fervor because the relevant maths behind gamma matrices is about representations of Lie groups and Lie algebras, not new kinds of algebras. Moreover, many of them will rightfully tell you that the overemphasis of Clifford algebras means an irrational preference for spinor representations (and pseudo/orthogonal groups) over other reps and other groups (including exceptional ones). It's just a wrong way of thinking to consider the concept of Clifford algebras fundamental. Physicists don't do it because it's just not terribly useful to talk in this way but even sensible mathematicians shouldn't be thinking in this way. 21. "Huge" should have been "Hugo". One more comment. People who believe that Clifford algebras are important and start to study physics are often distracted by superficial similarities that hide big physics differences. For example, Lie superalgebras are very important in physics (although less than complex numbers, of course), generalizing ordinary Lie algebras in a way that must be allowed in physics and is used in Nature. However, people with the idea that Clifford algebras are fundamental often try to imagine that superalgebras are just a special case etc. See e.g. this question on Physics Stack Exchange. The answer is, of course, that superalgebras don't have to be Clifford algebras. They may be more complicated etc. Moreover, the analogy between the algebra of Dirac matrices on one hand and Grassmann numbers on the other hand is just superficial. In physics, it's pretty important we distinguish them. The gamma matrices may anticommute but they're still matrices of Grassmann-even numbers which are different objects than Grassmann-odd numbers. When we associate fields to points in spacetime, the difference between Grassmann-odd and Grassmann-even objects is just huge, despite the same "anticommutator". When talking about objects such as spinors, the fundamental math terms are groups, Lie groups, Lie algebras, and their representations. For fields, one also adds bundles, fibers, and so on, perhaps, although the language is only used by "mathematical" physicists. But Clifford algebras are at most a name given by one particular anticommutator that appears once when we learn about spinors etc. and it never appears again. It doesn't bring a big branch of maths that should be studied for a long time. It's just a name for one equation among thousands of equations. It's not being manipulated with in numerous ways like we manipulate complex numbers or Lie algebras. The Clifford algebras are the kind of objects invented by mathematicians who predetermined that a particular generalization should be ever more important except that the subsequent research showed the assumption invalid and some people are unwilling to see this fact. 22. A number whose square is less than or equal to zero is termed as an imaginary number. Let's take an example, √-5 is an imaginary number and its square is -5. An imaginary number can be written as a real number but multiplied by the imaginary unit.in a+bi complex number i is called the imaginary unit,in given expression "a" is the real part and b is the imaginary part of the complex number. The complex number can be identified with the point (a, b). one-to-one correspondence 23. is the fundamental reason to explain the absolute asymmetry between left-right handed rotation frames in the non-euclidean geometry generated by the double torsion given by complex numbers and its comjugate complexes,or best the quaternions,through of anticommutativity to 4-dimensions that connect space and time into spacetime continuos.the biquaternions calcule the motion to curve manifolds to 4-dimensions. 24. I stumbled across this point while Googling Dirac's famous comment that it took people many years to become comfortable with complex numbers, so it was likely it would take another 100 years before they are comfortable with spinors. It is not quite what I was looking for, but it is certainly a good article. But I must admit that having more of a mathematician's inclination than a physicist's, I don't see what the fuss is all about. Mathematicians accept imaginary numbers as 'real' for a number of reasons, and our insight into their reality deepens over time, so that now I would say the reason we accept them as real and interesting can be summarized in a way that appears at first glance very different from the valid historical reasons Lubos gives: that the complex numbers are a perfectly valid algebraic extension of the reals, an extension with unique properties (among all other algebraic extensions) that explain many of the historical reasons for our interest in complex numbers. But if the reader finds that too obscure, there is always the matrix representation of complex numbers, one of the discoveries that put to rest many of the historical doubts about the 'reality' of complex numbers: represent a complex number a+bi as a matrix a00=a, a01=-1, a10=b a11=a. Such matrices are certainly real; their simplicity and symmetry suggest they should be both significant and easy to study. Lubos's post lists many of the reasons that suggestion has been amply justified over the years. 25. for more about imaginary number please read a paper namely 'complex number theory without imaginary number' at web http://www.jourlib.org/search?kw=Deepak%20Bhalchandra%20Gode&searchField=authors 26. Sorry, this "paper" at most tries to propose a new notation to write complex numbers and as far as I can say, it is a completely incoherent notation unmasking the stupidity of its author. 27. no need to apologies Brother, did you really got what is written in this
b5e0f0e85866e6e1	The original question was: I’ve heard somewhere that they’re also trying to build computers using molecules, like DNA. In general would it work to try and simulate a factoring algorithm using real world things, and then let the physics of the interactions stand-in for the computer calculation? Since the real-world does all kinds of crazy calculations in no time. Physicist: The amount of time that goes into, say, calculating how two electrons bounce off of each other is very humbling. It’s terribly frustrating that the universe has no hang ups about doing it so fast. In some sense, basic physical laws are the basis of how all calculations are done. It’s just that modern computers are “Turing machines“, a very small set of all the possible kinds of computational devices. Basically, if your calculating machine consists of the manipulation of symbols (which in turn can always be reduced to the manipulation of 1’s and 0’s), then you’re talking about Turing machines. In the earlier epoch of computer science there was a strong case for analog computers over digital computers. Analog computers use the properties of circuit elements like capacitors, inductors, or even just the layout of wiring, to do mathematical operations. In their heyday they were faster than digital computers. However, they’re difficult to design, not nearly as versatile, and they’re no longer faster. Nordsieck's Differential Analyzer was an analog computer used for solving differential equations. Any physical phenomena that represents information in definite, discrete states is doing the same thing a digital computer does, it’s just a question of speed. To see other kinds of computation it’s necessary to move into non-digital kinds of information. One beautiful example is the gravity powered square root finder. Newtonian physics used to find the square root of numbers. Put a marble next to a number, N, (white dots) on the slope, and the marble will land on the ground at a distance proportional to √N. When you put a marble on a ramp the horizontal distance it will travel before hitting the ground is proportional to the square root of how far up the ramp it started. Another mechanical calculator, the planimeter, can find the area of any shape just by tracing along the edge. Admittedly, a computer could do both calculations a heck of a lot faster, but they’re still descent enough examples. Despite the power of digital computers, it doesn’t take much looking around to find problems that can’t be efficiently done on them, but that can be done using more “natural” devices. For example, solutions to “harmonic functions with Dirichlet boundary conditions” (soap films) can be fiendishly difficult to calculate in general. The huge range of possible shapes that the solutions can take mean that often even the most reasonable computer program (capable of running in any reasonable time) will fail to find all the solutions. Part of Richard Courant's face demonstrating a fancy math calculation using soapy water and wires. So, rather than burning through miles of chalkboards and a swimming pools of coffee, you can bend wires to fit the boundary conditions, dip them in soapy water, and see what you get. One of the advantages, not generally mentioned in the literature, is that playing with bubbles is fun. Today we’re seeing the advent of a new type of computer, the quantum computer, which is kinda-digital/kinda-analog. Using quantum mechanical properties like super-position and entanglement, quantum computers can (or would, if we can get them off the ground) solve problems that would take even very powerful normal computers a tremendously long time to solve, like integer factorization. “Long time” here means that the heat death of the universe becomes a concern. Long time. Aside from actual computers, you can think of the universe itself, in a… sideways, philosophical sense, as doing simulations of itself that we can use to understand it. For example, one of the more common questions we get are along the lines of “how do scientists calculate the probability/energy of such-and-such chemical/nuclear reaction”. There are certainly methods to do the calculations (have Schrödinger equation, will travel), but really, if you want to get it right (and often save time), the best way to do the calculation is to let nature do it. That is, the best way to calculate atomic spectra, or how hot fire is, or how stable an isotope is, or whatever, is to go to the lab and just measure it. Even cooler, a lot of optimization problems can be solved by looking at the biological world. Evolution is, ideally, a process of optimization (though not always). During the early development of sonar and radar there were (still are) a number of questions about what kind of “ping” would return the greatest amount of information about the target. After a hell of a lot of effort it was found that the researchers had managed to re-create the sonar ping of several bat species. Bats are still studied as the results of what the universe has already “calculated” about optimal sonar techniques. You can usually find a solution through direct computation, but sometimes looking around works just as well. This entry was posted in -- By the Physicist, Computer Science, Engineering, Entropy/Information, Physics. Bookmark the permalink. 6 Responses to Q: Since the real-world does all kinds of crazy calculations in no time, can we use physics to calculate stuff? 1. Ron says: This is kind of like the earth being created as a computer to calculate the ultimate question, of which, the answer is 42. To bad the Vogons destroyed it in order to build a hyperspace express route just before the solution was found. 2. Steve says: One of my favourite articles so far. Thanks for being awesome! 3. odin says: Great read as usual. 4. The Physicist The Physicist says: I do hereby blush. 5. Will says: When the king of the Norse Pantheon is giving you praise you know you’ve done something right eh? 6. Pingback: Luonto laskee tai analogisia laskukoneita \| Markuksen Sielunmaisema Leave a Reply
771458a15d748820	Take the 2-minute tour × All of us have probably been exposed to questions such as: "What are the applications of group theory...". This is not the subject of this MO question. Here is a little newspaper article that I found inspiring: Madam, – In response to Marc Morgan’s question, “Does mathematics have any practical value?” (November 8th), I wish to respond as follows. Apart from its direct applications to electrical circuits and machinery, electronics (including circuit design and computer hardware), computer software (including cryptography for internet transaction security, business software, anti-virus software and games), telephones, mobile phones, fax machines, radio and television broadcasting systems, antenna design, computer game consoles, hand-held devices such as iPods, architecture and construction, automobile design and fabrication, space travel, GPS systems, radar, X-ray machines, medical scanners, particle research, meteorology, satellites, all of physics and much of chemistry, the answer is probably “No”. – Yours, etc, The Irish Times - Wednesday, November 10, 2010 The above article article seems to provide an ideal source of solutions to a perennial problem: How to tell something interesting about math to non-mathematicians, without losing your audience? However, I am embarrassed to admit that I have no idea what kind of math gets used for antenna designs, computer game consoles, GPS systems, etc. I would like to have a list applications of math, from the point of view of the applications. To make sure that each answer is sufficiently structured and developed, I shall impose some restrictions on their format. Each answer should contain the following three parts, roughly of the same size: • Start by the description of a practical problem that any layman can understand. • Then I would appreciate to have an explanation of why it is difficult to solve without mathematical tools. • Finally, there should be a little explanation of the kind of math that gets used in the solution. ♦♦ My ultimate goal is to have a nice collection of examples of applications of mathematics, for the purpose of casual discussions with non-mathematicians. ♦♦ As usual with community-wiki questions: one answer per post. share\|improve this question Related: mathoverflow.net/questions/2556/… – Qiaochu Yuan Feb 24 '11 at 18:55 I think you'd need a list of topics that you'd like to know more about, lest this develops in a whole new encyclopedia. I just picked three examples from the newpaper piece you cited. – Tim van Beek Feb 25 '11 at 3:05 @Tim: The article I cited provides such a list. I'd already be quite happy if an example could be provided for each of the items in there. – André Henriques Feb 25 '11 at 10:51 @André: Ok, is there an item on the list that has not been addressed and that your are particularly interested in? I feel like I could write books about every one :-) – Tim van Beek Feb 27 '11 at 10:49 @Tim: Actually yes: antenna design. – André Henriques Feb 27 '11 at 15:02 10 Answers 10 up vote 48 down vote accepted Sending a man to the Moon (and back). Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." (Quoted from Hilbert-Courant by Constance Reid, Springer, 1986, p. 92). The task obviously required solving plenty of scientific and technological problems. But the key breakthrough that made it all possible was Richard Arenstorf's discovery of a stable 8-shaped orbit between the Earth and the Moon. This involved the development of a numerical algorithm for solving the restricted three-body problem which is just a special non-linear second order ODE (see also my answer to the previous MO question). Another orbit, also mapped by Arenstorf, was later used in the dramatic rescue of the Apollo 13 crew. share\|improve this answer One typical way that GPS is invoked as an application of mathematics is through the use of general relativity. Most people have a rough idea of what the GPS system does: there are some (27) satellites flying in the sky, and a GPS device on the surface of the earth determines its position by radio communication with the satellites. It is also pretty clear that this is a hard problem to solve, with or without mathematics. The basic idea is that if your GPS device measures its distance between 3 different satellites, then it knows that it lies on three level sets which must intersect at a point. This is the standard idea of triangulation. Of course measuring distance is hard to do, and relativity comes into play in many different, nontrivial ways, but there is one way in particular that is interesting and easy to explain. If one uses the euclidean metric to determine the distance (so, straight lines) from the GPS to the satellite, then it will be impossible to determine the location on the earth to a high degree of accuracy. So instead the GPS system uses the kerr metric, that is the lorentz metric that models spacetime outside of a spherically symmetric, rotating body. Naturally this metric gives a different, more accurate distance between the observer on earth and the satellite. The thing that is surprising to people is that the switch from euclidean to kerr is required to get really accurate gps readings. In other words, without relativity you might not be able to use that iphone app to find your car in the grocery store parking lot. People are often surprised and interested to learn that the differences between relativity and newtonian gravity really are observable. Other standard examples are the precession of the perihelion of mercury (which was a famous unsolved problem before the introduction of GR) and the demonstration that light rays do not travel along straight lines by photographing the sun during an eclipse. This last observation demonstrated, for instance, that the metric on the universe is not the trivial flat one. share\|improve this answer Another place where relativity is relevant to GPS is time dilation. Fundamentally GPS computes distances by calculating time differences, so every satellite contains an atomic clock. Before they launch them they calibrate the clocks, but they have to be detuned by an amount that takes into account both the SR and GR time dilation effects in order to be accurate when they get into orbit. – hobbs Feb 9 '14 at 3:10 A particularly striking application to physics and chemistry is explained in Singer's book Linearity, symmetry, and prediction in the hydrogen atom. The practical problem, in the large, is easy to state: what is the stuff around us made of, and why does it react with other stuff the way it does? More precisely, what explains the structure of the periodic table? There is no a priori reason that the elements ought to naturally arrange themselves in rows of size $2, 8, 8, 18, 18, ...$ with repeating chemical properties. This periodic structure profoundly shapes the nature of the world around us and so ought to be well worth trying to understand on a deeper level. Physically, the answer has to do with the way that electrons arrange themselves around a nucleus, one of the classic examples of the breakdown of classical mechanics. The Bohr model posits that electrons are arranged in discrete orbitals $n = 1, 2, 3, ... $ with energy levels proportional to $- \frac{1}{n^2}$ such that the $n^{th}$ energy level admits at most $2n^2$ electrons. This behavior $- \frac{1}{n^2}$ can be empirically deduced by an examination of atomic spectra but the Bohr model still does not provide a conceptual explanation of it. That explanation comes from full-blown quantum mechanics, which already requires a fair amount of nontrivial mathematics. For our purposes quantum mechanics will be described by a Hilbert space $K = L^2(X)$ where $X$ is the classical phase space (e.g. $\mathbb{R}^3$) and a self-adjoint operator $H : K \to K$, the Hamiltonian, which will describe the evolution of states via the Schrödinger equation. The simplest case is that of an electron orbiting a single proton, in which case one can explicitly write down the potential. In this case the Schrödinger equation can be solved fairly explicitly and the answer tells you what electron orbitals look like, but it turns out that one can do much better: it is possible to predict the solutions and their properties using representation theory. To start with, the Coulomb potential has a spherical symmetry, so this endows $K$ with the structure of a unitary representation of $\text{SO}(3)$. By identifying two wave functions together if they lie in the same representation we can hope to have a physical classification of the possible states of an electron; the idea is that physical quantities we care about should be invariant under physical symmetries (e.g. mass, energy, charge). The action of $\text{SO}(3)$ breaks up the space of possible states based on their angular momentum (Noether's theorem). The corresponding representations have dimensions $1, 3, 5, 7, ...$ and indeed we find that we can decompose the number of elements in each row of the periodic table as $$2 = 1 + 1$$ $$8 = 1 + 1 + 3 + 3$$ $$18 = 1 + 1 + 3 + 3 + 5 + 5$$ corresponding to the possible angular momentum values allowed at each energy level. Of course these symmetry considerations apply to every spherically symmetric system so the $\text{SO}(3)$ symmetry cannot tell us anything more specific. But it turns out there is even more symmetry to exploit. First of all, remarkably enough the $\text{SO}(3)$ symmetry extends to an $\text{SO}(4)$ symmetry. (I do not really know a conceptual explanation of this, unfortunately; I have a half-baked one which I'm not sure is valid.) The irreducible representations of $\text{SO}(4)$ occurring here are precisely the ones of dimensions $1, 1 + 3, 1 + 3 + 5, ...$ and they break up into irreducible $\text{SO}(3)$ representations in exactly the right way to account for the above pattern up to a factor of $2$. Second of all, the factor of $2$ is accounted for by an additional action of $\text{SU}(2)$ coming from electron spin (the thing that makes MRI machines work). So representation theory provides a strikingly elegant answer to the question of how the periodic table is arranged (if one accepts that a single proton is a good approximation to a general atomic nucleus). Of course there is much more to say here about the relation between representation theory and physics and chemistry, but I am not the one to ask... share\|improve this answer My half-baked explanation for the SO(4) symmetry is this: the wave functions we are interested are localized near the origin, so their Fourier transforms are smeared out in momentum space. Smeared-out functions on R^3 look like functions on S^3, and the proton in R^3 is completely smeared out in S^3, so... there is an SO(4) symmetry in momentum space. Or something like that. – Qiaochu Yuan Feb 25 '11 at 1:10 You could mention almost any physical theory (e.g. General Relativity, anything with PDEs, etc.) - almost all involve some degree of nontrival mathematics. Why QM in particular? I actually think it's a particularly bad example! You say: "...but the Bohr model still does not provide a conceptual explanation of it." But I would say the same applies to Quantum Mechanics! Maybe it gives correct formulae and predictions for (currently known) experimental data, but it's TOTALLY crazy and strange. Maybe in 100-200 years from now, it will have been replaced by something totally different! – Zen Harper Feb 25 '11 at 5:34 ...just to make it clear, I'm obviously not suggesting that QM is not useful or not mathematical! I just think there are many other things from mathematical physics which are much less crazy and counterintuitive, and so would be much better examples for conversation with nonmathematicians. To me, QM looks like a totally incorrect theory which only gives the correct formulae by chance. A proper explanation is still waiting to be found. – Zen Harper Feb 25 '11 at 5:40 @Qiaochu: That's fine as a basic example of the use of representation theory. But note that much more subtle aspects of how the Periodic Table is organized, which for a long time were only experimentally observed, have just recently been described mathematically: by detailed studies of asymptotic properties of the Schrödinger equation combined with some rep theoretical aspects, see <a href="mpip-mainz.mpg.de/theory/events/namet2010/…;. – Thomas Sauvaget Feb 25 '11 at 10:07 @Qiaochu: Sorry the link should be mpip-mainz.mpg.de/theory/events/namet2010/… Also, the SO(4) symmetry is a property of the classical 2-body problem, which is thus inherited by the quantum one: for a natural conceptual explanation you can have a look at this wikipedia section en.wikipedia.org/wiki/… (and that whole wikipedia page for more details on the corresponding additional conserved quantity). – Thomas Sauvaget Feb 25 '11 at 10:16 Here are some examples to quote my favorite one: In 1998, mathematics was suddenly in the news. Thomas Hales of the University of Pittsburgh, Pennsylvania, had proved the Kepler conjecture, showing that the way greengrocers stack oranges is the most efficient way to pack spheres. A problem that had been open since 1611 was finally solved! On the television a greengrocer said: “I think that it's a waste of time and taxpayers' money.” I have been mentally arguing with that greengrocer ever since: today the mathematics of sphere packing enables modern communication, being at the heart of the study of channel coding and error-correction codes. In 1611, Johannes Kepler suggested that the greengrocer's stacking was the most efficient, but he was not able to give a proof. It turned out to be a very difficult problem. Even the simpler question of the best way to pack circles was only proved in 1940 by László Fejes Tóth. Also in the seventeenth century, Isaac Newton and David Gregory argued over the kissing problem: how many spheres can touch a given sphere with no overlaps? In two dimensions it is easy to prove that the answer is 6. Newton thought that 12 was the maximum in 3 dimensions. It is, but only in 1953 did Kurt Schütte and Bartel van der Waerden give a proof. The kissing number in 4 dimensions was proved to be 24 by Oleg Musin in 2003. In 5 dimensions we can say only that it lies between 40 and 44. Yet we do know that the answer in 8 dimensions is 240, proved back in 1979 by Andrew Odlyzko of the University of Minnesota, Minneapolis. The same paper had an even stranger result: the answer in 24 dimensions is 196,560. These proofs are simpler than the result for three dimensions, and relate to two incredibly dense packings of spheres, called the E8 lattice in 8-dimensions and the Leech lattice in 24 dimensions. This is all quite magical, but is it useful? In the 1960s an engineer called Gordon Lang believed so. Lang was designing the systems for modems and was busy harvesting all the mathematics he could find. He needed to send a signal over a noisy channel, such as a phone line. The natural way is to choose a collection of tones for signals. But the sound received may not be the same as the one sent. To solve this, he described the sounds by a list of numbers. It was then simple to find which of the signals that might have been sent was closest to the signal received. The signals can then be considered as spheres, with wiggle room for noise. To maximize the information that can be sent, these 'spheres' must be packed as tightly as possible. In the 1970s, Lang developed a modem with 8-dimensional signals, using E8 packing. This helped to open up the Internet, as data could be sent over the phone, instead of relying on specifically designed cables. Not everyone was thrilled. Donald Coxeter, who had helped Lang understand the mathematics, said he was “appalled that his beautiful theories had been sullied in this way”. share\|improve this answer computerized tomography In computerized tomograhy one meaures X-ray images of a body from different angles. Each X-ray image roughly corrsponds to a projection of the density distribution along a certain direction. To obtain the full density distribution inside the body one has to invert the Radon transform. This is an interesting problem from integral geometry which is also challenging concerning the numerical implementation, since the inverse is known the be discontinuous and hence, regularization techniques has to be employed. Another interesting aspect of this story is that the mathematical problem of the inversion of the Radon transform was done aound 1917 (by Johan Radon itself) while this was totally unknown to the inventors of computerized tomography as it is used today. share\|improve this answer example: Information send over the internet needs to be secure such that only the sender and the recipient can understand and use it. Example: Man in the middle attack on a bank transaction: You send an order to your bank to pay 100 Dollars to Mr. X. I intercept this transmission and change the order to your bank to make them send me 100 000 Dollars instead. Since every information send over the internet passes through a lot of different computers (gateways), all I need to intercept your message is access to one of those computers. Thousands of network administrators do have such an access (this is grossly simplified of course). In order to secure the information, me and my bank need to know an algorithm for cryptography. Commonly used are algorithms using public/private key pairs. These consist of functions such that 1. the bank publishes a public key k, 2. I can apply a function f mapping the information inf I would like to send, using the public key k, to an encrypted message f(inf, k). 3. The whole punchline is that the inverse function can only be computed by knowing the private key, which only my bank knows. So only my bank can compute the information inf knowing f(inf, k). Commonly used algorithms are based on the assumption that there is no efficient algorithm to factorize large numbers, i.e. compute the prime factors of a given large number. The validity has not been proven. So you can a) get famous by proving this assumption, b) get either famous (and provoke a collapse of internet banking) or insanely rich by finding an algorithm that computes prime factors efficiently, c) get famous and rich by finding an efficient algorithm for public/private key encryption that is efficient and provable safe. share\|improve this answer I don't want to be the person who makes most of the current coding algorithms useless ... this seems to be quite dangerous. – Martin Brandenburg Feb 25 '11 at 9:15 It would be dangerous if someone developed a cracking algorithm to use this for his own good instead of publishing it. – Tim van Beek Feb 27 '11 at 10:48 computer game consoles Many computer games today display some sort of 3D real time graphics. There are two important aspects: a) the display of a 2D projection of a 3D object model needs involved algorithms that calculate what is visible from the viewpoint of the observer, how objects look like from that perspective and calculate shading and light effects on colors. These algorithms need concepts from 3D geometry (linear algebra, vectors, areas, projection operators etc.). Many computer science departments have classes for the involved mathematics. b) the animation effects of many games are calculated by numerical solutions of partial differential equations describing the physical motion of solid bodies and fluids. In order to animate fluids like water, for example, computer games use finite element approximations to the Navier-Stokes equations. Needless to say, this is a very active area of current research. (Computer consoles are actually used in research involving computational fluid dynamics because they are cheap, easy to program and very powerful.) The last part is also true for automobile design and fabrication: Car companies need to test new car designs for example for mechanical problems: Are there any parts that will make noise once you drive faster than 50 km/h? This is tested with software that simulates the mechanical parts of the proposed car design using finite element approximations to equations of solid state mechanics. The same technique is also used to simulate crash tests. The design of the car body is done via CAD (computer aided design) software. This software uses approximation and interpolation algorithms to calculate external surfaces that are as smooth as possible while satisfying boundary conditions that are specified by the designer. These approximations are done e.g. by spline interpolation. Numerical approximations of computational fluid dynamics are also used to simulate tests in the wind tunnel. This actually saves a lot of money. (It is also the reason why modern cars all kinda look alike.) share\|improve this answer circuit design and computer hardware Example: a robotic arm has to create a complicated electronic circuit by putting a conducting material on a non-conducting base. The robotic arm has to traverse the whole graph that makes up the circuit at least once. In order to reduce the time the roboter needs to create one electronic circuit, the way it has to traverse needs to be minimized, that is one needs a good approximate solution to the traveling salesman problem. I know of examples where improved heuristic approximation algorithms have increased the output by several percents (the student doing the math thesis on this was rewarded by the company producing these circuits with several hundred thousands of dollars. No, it wasn't me.). share\|improve this answer (Dredged up from the murky past...) Designing control systems usually involves building a logic circuit that has several inputs and one or two outputs. Sometimes states are involved (sequencing of traffic lights, coin collectors for vending machines), sometimes not. In designing such control logic, many equations get written down which represent things like "If these three switches are off and these others are on, flip this switch over here". Once one has the equations written down, (often as a Boolean function, a map from {0,1}^n to {0,1}) one has to build the circuit implementing these equations. Often times, the medium for implementation is a gate array, which may be a field of NAND logic gates that can be wired together, or a programmable logic device, which is like two or more gate arrays, some with ANDS, some with ORS, some NOT gates, flip-flops which are like little memory stores, and so on. The major question is: are there enough gates on the device to build all the logic represented by the equations? To this end, computer programs called logic minimizers are used. They have certain definite rules (related to manipulation terms in Boolean logic) and certain heuristics (guidelines and methods for following the guidelines) to follow in order to minimize the number of, say, AND and OR gates used in representing the equations. The mathematics of representing any Boolean function as a series of AND and OR gates, and finding equivalent representations, has been developed and used since George Boole set down the algebraic form of what is now called Boolean Logic. Computer Science, abstract algebra, clone theory, all have played and continue to play an essential role in solving instances of this kind of problem. The fact that it is not completely solved is related to one of the Millenium Prize problems (P-NP) . Gerhard "Ask Me About PLD Chips" Paseman, 2011.02.24 share\|improve this answer The error-correction required for cell phones and 3G and 4G devices to work is mathematics! share\|improve this answer Your Answer
1f159318e16e1c47	Max Planck Max Planck Max planck.jpg Max Karl Ernst Ludwig Planck April 23, 1858 Kiel, Germany Died October 4, 1947 Göttingen, Germany Residence Flag of Germany.svg Germany Nationality Flag of Germany.svg German Field Physicist Institutions University of Kiel Humboldt-Universität zu Berlin Georg-August-Universität Göttingen Alma mater Ludwig-Maximilians-Universität München Academic advisor Philipp von Jolly Notable students Gustav Ludwig Hertz Nobel.svg Erich Kretschmann Walther Meißner Walter Schottky Max von Laue Nobel.svg Max Abraham Moritz Schlick Walther Bothe Nobel.svg Known for Planck's constant, quantum theory Notable prizes Nobel.svg Nobel Prize in Physics (1918) He was the father of Erwin Planck. Max Karl Ernst Ludwig Planck (April 23, 1858 – October 4, 1947) was a German physicist who is widely regarded as one of the most significant scientists in history. He developed a simple but revolutionary concept that was to become the foundation of a new way of looking at the world, called quantum theory. In 1900, to solve a vexing problem concerning the radiation emitted by a glowing body, he introduced the radical view that energy is transmitted not in the form of an unbroken (infinitely subdivisible) continuum, but in discrete, particle-like units. He called each such unit a quantum (the plural form being quanta). This concept was not immediately accepted by physicists, but it ultimately changed the very foundations of physics. Planck himself did not quite believe in the reality of this concept—he considered it a mathematical construct. In 1905, Albert Einstein used that concept to explain the photoelectric effect, and in 1913, Niels Bohr used the same idea to explain the structures of atoms. From then on, Planck's idea became central to all of physics. He received the Nobel Prize in 1918, and both Einstein and Bohr received the prize a few years later. Planck was also a deeply religious man who believed that religion and science were mutually compatible, both leading to a larger, universal truth. By basing his convictions on seeking the higher truth, not on doctrine, he was able to stay open-minded when it came to formulating scientific concepts and being tolerant toward alternative belief systems. Life and work Early childhood Planck came from a traditional, intellectual family. His paternal great-grandfather and grandfather were both theology professors in Göttingen, his father was a law professor in Kiel and Munich, and his paternal uncle was a judge. Planck was born in Kiel to Johann Julius Wilhelm Planck and his second wife, Emma Patzig. He was the sixth child in the family, including two siblings from his father's first marriage. Among his earliest memories was the marching of Prussian and Austrian troops into Kiel during the Danish-Prussian War in 1864. In 1867, the family moved to Munich, and Planck enrolled in the Maximilians gymnasium. There he came under the tutelage of Hermann Müller, a mathematician who took an interest in the youth and taught him astronomy and mechanics as well as mathematics. It was from Müller that Planck first learned the principle of conservation of energy. Planck graduated early, at age 16. This is how Planck first came in contact with the field of physics. Planck was extremely gifted when it came to music: He took singing lessons and played the piano, organ, and cello, and composed songs and operas. However, instead of music, he chose to study physics. Munich physics professor Philipp von Jolly advised him against going into physics, saying, "in this field, almost everything is already discovered, and all that remains is to fill a few holes." Planck replied that he did not wish to discover new things, only to understand the known fundamentals of the field. In 1874, he began his studies at the University of Munich. Under Jolly's supervision, Planck performed the only experiments of his scientific career: Studying the diffusion of hydrogen through heated platinum. He soon transferred to theoretical physics. In 1877, he went to Berlin for a year of study with the famous physicists Hermann von Helmholtz and Gustav Kirchhoff, and the mathematician Karl Weierstrass. He wrote that Helmholtz was never quite prepared (with his lectures), spoke slowly, miscalculated endlessly, and bored his listeners, while Kirchhoff spoke in carefully prepared lectures, which were, however, dry and monotonous. Nonetheless, he soon became close friends with Helmholtz. While there, he mostly undertook a program of self-study of Rudolf Clausius's writings, which led him to choose heat theory as his field. In October 1878, Planck passed his qualifying exams and in February 1879, defended his dissertation, Über den zweiten Hauptsatz der mechanischen Wärmetheorie (On the second fundamental theorem of the mechanical theory of heat). He briefly taught mathematics and physics at his former school in Munich. In June 1880, he presented his habilitation thesis, Gleichgewichtszustände isotroper Körper in verschiedenen Temperaturen (Equilibrium states of isotropic bodies at different temperatures). Academic career With the completion of his habilitation thesis, Planck became an unpaid private lecturer in Munich, waiting until he was offered an academic position. Although he was initially ignored by the academic community, he furthered his work on the field of heat theory and discovered one after the other the same thermodynamical formalism as Josiah Willard Gibbs without realizing it. Clausius's ideas on entropy occupied a central role in his work. In April 1885, the University of Kiel appointed Planck an associate professor of theoretical physics. Further work on entropy and its treatment, especially as applied in physical chemistry, followed. He proposed a thermodynamic basis for Arrhenius's theory of electrolytic dissociation. Within four years, he was named the successor to Kirchhoff's position at the University of Berlin—presumably thanks to Helmholtz's intercession—and by 1892 became a full professor. In 1907, Planck was offered Boltzmann's position in Vienna, but turned it down to stay in Berlin. During 1909, he was the Ernest Kempton Adams Lecturer in Theoretical Physics at Columbia University in New York City. He retired from Berlin on January 10, 1926, and was succeeded by Erwin Schrödinger. In March 1887, Planck married Marie Merck (1861-1909), sister of a school fellow, and moved with her into a sublet apartment in Kiel. They had four children: Karl (1888-1916), the twins Emma (1889-1919) and Grete (1889-1917), and Erwin (1893-1945). After the appointment to Berlin, the Planck family lived in a villa in Berlin-Grunewald, Wangenheimstraße 21. Several other professors of Berlin University lived nearby, among them the famous theologian Adolf von Harnack, who became a close friend of Planck. Soon the Planck home became a social and cultural center. Numerous well-known scientists—such as Albert Einstein, Otto Hahn, and Lise Meitner—were frequent visitors. The tradition of jointly playing music had already been established in the home of Helmholtz. After several happy years, the Planck family was struck by a series of disasters: In July 1909, Marie Planck died, possibly from tuberculosis. In March 1911, Planck married his second wife, Marga von Hoesslin (1882-1948); in December his third son, Herrmann, was born. During the First World War, Planck's son Erwin was taken prisoner by the French in 1914, and his son Karl was killed in action at Verdun in 1916. His daughter Grete died in 1917 while giving birth to her first child; her sister lost her life two years later under the same circumstances, after marrying Grete's widower. Both granddaughters survived and were named after their mothers. Planck endured all these losses with stoic submission to fate. During World War II, Planck's house in Berlin was completely destroyed by bombs in 1944, and his youngest son, Erwin, was implicated in the attempt made on Hitler's life on July 20, 1944. Consequently, Erwin died a horrible death at the hands of the Gestapo in 1945. Professor at Berlin University In Berlin, Planck joined the local Physical Society. He later wrote about this time: "In those days I was essentially the only theoretical physicist there, whence things were not so easy for me, because I started mentioning entropy, but this was not quite fashionable, since it was regarded as a mathematical spook." Thanks to his initiative, the various local Physical Societies of Germany merged in 1898 to form the German Physical Society (Deutsche Physikalische Gesellschaft, DPG), and Planck was its president from 1905 to 1909. Planck started a six semester course of lectures on theoretical physics. Lise Meitner described the lectures as "dry, somewhat impersonal." An English participant, James R. Partington, wrote, "using no notes, never making mistakes, never faltering; the best lecturer I ever heard." He continues: "There were always many standing around the room. As the lecture-room was well heated and rather close, some of the listeners would from time to time drop to the floor, but this did not disturb the lecture." Planck did not establish an actual "school," the number of his graduate students was only about 20 altogether. Among his students were the following individuals. The year in which each individual achieved the highest degree is indicated after the person's name (outside the parentheses); the individual's year of birth and year of death are given within parentheses. Max Abraham 1897 (1875-1922) Moritz Schlick 1904 (1882-1936) Walther Meißner 1906 (1882-1974) Max von Laue 1906 (1879-1960) Fritz Reiche 1907 (1883-1960) Walter Schottky 1912 (1886-1976) Walther Bothe 1914 (1891-1957) Black-body radiation In 1894, Planck had been commissioned by electricity companies to discover how to generate the greatest luminosity from light bulbs with the minimum energy. To approach that question, he turned his attention to the problem of black-body radiation. In physics, a black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. Black bodies below around 700 K (430 °C) produce very little radiation at visible wavelengths and appear black (hence the name). Above this temperature, however, they produce radiation at visible wavelengths, starting at red and going through orange, yellow, and white before ending up at blue, as the temperature is raised. The light emitted by a black body is called black-body radiation (or cavity radiation). The amount and wavelength (color) of electromagnetic radiation emitted by a black body is directly related to its temperature. The problem, stated by Kirchhoff in 1859, was: How does the intensity of the electromagnetic radiation emitted by a black body depend on the frequency of the radiation (correlated with the color of the light) and the temperature of the body? This question had been explored experimentally, but the Rayleigh-Jeans law, derived from classical physics, failed to explain the observed behavior at high frequencies, where it predicted a divergence of the energy density toward infinity (the "ultraviolet catastrophe"). Wilhelm Wien proposed Wien's law, which correctly predicted the behavior at high frequencies but failed at low frequencies. By interpolating between the laws of Wien and Rayleigh-Jeans, Planck formulated the now-famous Planck's law of black-body radiation, which described the experimentally observed black-body spectrum very well. It was first proposed in a meeting of the DPG on October 19, 1900, and published in 1901. By December 14, 1900, Planck was already able to present a theoretical derivation of the law, but this required him to use ideas from statistical mechanics, as introduced by Boltzmann. So far, he had held a strong aversion to any statistical interpretation of the second law of thermodynamics, which he regarded as having an axiomatic nature. Compelled to use statistics, he noted: "… an act of despair … I was ready to sacrifice any of my previous convictions about physics …" The central assumption behind his derivation was the supposition that electromagnetic energy could be emitted only in quantized form. In other words, the energy could only be a multiple of an elementary unit. Mathematically, this was expressed as: E = h \nu where h is a constant that came to be called Planck's constant (or Planck's action quantum), first introduced in 1899, and \nu is the frequency of the radiation. Planck's work on quantum theory, as it came to be known, was published in the journal Annalen der Physik. His work is summarized in two books Thermodynamik (Thermodynamics) (1897) and Theorie der Wärmestrahlung (theory of heat radiation) (1906). At first, Planck considered that quantization was only "a purely formal assumption … actually I did not think much about it…" This assumption, incompatible with classical physics, is now regarded as the birth of quantum physics and the greatest intellectual accomplishment of Planck's career. (However, in a theoretical paper published in 1877, Ludwig Boltzmann had already been discussing the possibility that the energy states of a physical system could be discrete.) In recognition of this accomplishment, Planck was awarded the Nobel prize for physics in 1918. The discovery of Planck's constant enabled him to define a new universal set of physical units—such as Planck length and Planck mass—all based on fundamental physical constants. Subsequently, Planck tried to integrate the concept of energy quanta with classical physics, but to no avail. "My unavailing attempts to somehow reintegrate the action quantum into classical theory extended over several years and caused me much trouble." Even several years later, other physicists—including Lord Rayleigh, James Jeans, and Hendrik Lorentz—set Planck's constant to zero, in an attempt to align with classical physics, but Planck knew well that this constant had a precise, nonzero value. "I am unable to understand Jeans' stubbornness—he is an example of a theoretician as should never be existing, the same as Hegel was for philosophy. So much the worse for the facts, if they are wrong." Max Born wrote about Planck: "He was by nature and by the tradition of his family conservative, averse to revolutionary novelties and skeptical towards speculations. But his belief in the imperative power of logical thinking based on facts was so strong that he did not hesitate to express a claim contradicting to all tradition, because he had convinced himself that no other resort was possible." Einstein and the theory of relativity To explain the photoelectric effect (discovered by Philipp Lenard in 1902), Einstein proposed that light consists of quanta, which he called photons. Planck, however, initially rejected this theory, as he was unwilling to completely discard Maxwell's theory of electrodynamics. Planck wrote, "The theory of light would be thrown back not by decades, but by centuries, into the age when Christian Huygens dared to fight against the mighty emission theory of Isaac Newton …" In 1910, Einstein pointed out the anomalous behavior of specific heat at low temperatures as another example of a phenomenon that defies explanation by classical physics. To resolve the increasing number of contradictions, Planck and Walther Nernst organized the First Solvay Conference in Brussels in 1911. At this meeting, Einstein was finally able to convince Planck. Meanwhile, Planck had been appointed dean of Berlin University. Thereby, it was possible for him to call Einstein to Berlin and establish a new professorship for him in 1914. Soon the two scientists became close friends and met frequently to play music together. World War I and the Weimar Republic At the onset of the First World War Planck was not immune to the general excitement of the public: "… besides of much horrible also much unexpectedly great and beautiful: The swift solution of the most difficult issues of domestic policy through arrangement of all parties… the higher esteem for all that is brave and truthful…" He refrained from the extremes of nationalism. For instance, he voted successfully for a scientific paper from Italy to receive a prize from the Prussian Academy of Sciences in 1915, (Planck was one of its four permanent presidents), although at that time Italy was about to join the Allies. Nevertheless, the infamous "Manifesto of the 93 intellectuals," a polemic pamphlet of war propaganda, was also signed by Planck. Einstein, on the other hand, retained a strictly pacifist attitude, which almost led to his imprisonment, from which he was saved only by his Swiss citizenship. But already in 1915, Planck revoked parts of the Manifesto, (after several meetings with Dutch physicist Lorentz), and in 1916, he signed a declaration against the German policy of annexation. In the turbulent post-war years, Planck, by now the highest authority of German physics, issued the slogan "persevere and continue working" to his colleagues. In October 1920, he and Fritz Haber established the Notgemeinschaft der Deutschen Wissenschaft (Emergency Organization of German Science), which aimed at providing support for the destitute scientific research. They obtained a considerable portion of their funds from abroad. In this time, Planck held leading positions also at Berlin University, the Prussian Academy of Sciences, the German Physical Society, and the Kaiser Wilhelm Gesellschaft (KWG, which in 1948 became the Max Planck Gesellschaft). Under such circumstances, he himself could hardly conduct any more research. He became a member of the Deutsche Volks-Partei (German People's Party), the party of peace Nobel prize laureate Gustav Stresemann, which aspired to liberal aims for domestic policy and rather revisionist aims for international politics. He disagreed with the introduction of universal suffrage and expressed later the view that the Nazi dictatorship was the result of "the ascent of the rule of the crowds." Quantum mechanics At the end of the 1920s, Bohr, Werner Heisenberg, and Wolfgang Pauli had worked out the Copenhagen interpretation of quantum mechanics. It was, however, rejected by Planck, as well as Schrödinger and Laue. Even Einstein had rejected Bohr's interpretation. Planck called Heisenberg's matrix mechanics "disgusting," but he gave the Schrödinger equation a warmer reception. He expected that wave mechanics would soon render quantum theory—his own brainchild—unnecessary. Nonetheless, scientific progress ignored Planck's concerns. He experienced the truth of his own earlier concept, after his struggle with the older views. He wrote, "A new scientific truth does not establish itself by its enemies being convinced and expressing their change of opinion, but rather by its enemies gradually dying out and the younger generation being taught the truth from the beginning." Nazi dictatorship and World War II When the Nazis seized power in 1933, Planck was 74. He witnessed many Jewish friends and colleagues expelled from their positions and humiliated, and hundreds of scientists emigrated from Germany. Again he tried the "persevere and continue working" slogan and asked scientists who were considering emigration to stay in Germany. He hoped the crisis would abate soon and the political situation would improve again. There was also a deeper argument against emigration: Emigrating non-Jewish scientists would need to look for academic positions abroad, but these positions better served Jewish scientists, who had no chance of continuing to work in Germany. Hahn asked Planck to gather well-known German professors, to issue a public proclamation against the treatment of Jewish professors. Planck, however, replied, "If you are able to gather today 30 such gentlemen, then tomorrow 150 others will come and speak against it, because they are eager to take over the positions of the others." Although, in a slightly different translation, Hahn remembers Planck saying: "If you bring together 30 such men today, then tomorrow 150 will come to denounce them because they want to take their places." Under Planck's leadership, the KWG avoided open conflict with the Nazi regime. One exception was Fritz Haber. Planck tried to discuss the issue with Adolf Hitler but was unsuccessful. In the following year, 1934, Haber died in exile. As the political climate in Germany gradually became more hostile, Johannes Stark, prominent exponent of Deutsche Physik ("German Physics," also called "Aryan Physics") attacked Planck, Arnold Sommerfeld, and Heisenberg for continuing to teach the theories of Einstein, calling them "white Jews." The "Hauptamt Wissenschaft" (Nazi government office for science) started an investigation of Planck's ancestry, but all they could find out was that he was "1/16 Jewish." In 1938, Planck celebrated his 80th birthday. The DPG held an official celebration, during which the Max Planck medal (founded as the highest medal by the DPG in 1928) was awarded to French physicist Louis de Broglie. At the end of 1938, the Prussian Academy lost its remaining independence and was taken over by Nazis (Gleichschaltung). Planck protested by resigning his presidency. He continued to travel frequently, giving numerous public talks, such as his famous talk on "Religion and Science." Five years later, he was still sufficiently fit to climb 3,000-meter peaks in the Alps. During the Second World War, the increasing number of Allied bombing campaigns against Berlin forced Planck and his wife to leave the city temporarily and live in the countryside. In 1942, he wrote: "In me an ardent desire has grown to persevere this crisis and live long enough to be able to witness the turning point, the beginning of a new rise." In February 1944, his home in Berlin was completely destroyed by an air raid, annihilating all his scientific records and correspondence. Finally, he was in a dangerous situation in his rural retreat during the rapid advance of Allied armies from both sides. After the end of the war, Planck, his second wife, and their son Herrmann moved to Göttingen, where he died on October 4, 1947. Max Planck commemorated on the German 2 Mark Coin Religious views Max Planck was a devoted Christian from early life to death. As a scientist, however, he was very tolerant toward other religions and alternate views, and was discontent with the church organization's demands for unquestioning belief. He noted that "natural laws … are the same for men of all races and nations." Planck regarded the search for universal truth as the loftiest goal of all scientific activity. Perhaps foreseeing the central role it now plays in current thinking, Planck made great note of the fact that the quantum of action retained its significance in relativity because of the relativistic invariance of the Principle of Least Action. Max Planck's view of God can be regarded as pantheistic, with an almighty, all-knowing, benevolent but unintelligible God who permeates everything, manifest by symbols, including physical laws. His view may have been motivated by an opposition—like that of Einstein and Schrödinger—to the positivist, statistical, subjective universe of scientists such as Bohr, Heisenberg, and others. Planck was interested in truth and the Universe beyond observation, and he objected to atheism as an obsession with symbols.[1] Planck was the very first scientist to contradict the physics established by Newton. This is why all physics before Planck is called "classical physics," while all physics after him is called "quantum physics." In the classical world, energy is continuous; in the quantum world, it is discrete. On this simple insight of Planck's was constructed all of the new physics of the twentieth century. Unlike religion with its great leaps, science proceeds by baby steps. The small step taken by Planck was the first of the many needed to reach the current "internal wave and external particle" view of modern physics a century later. Honors and medals • "Pour le Mérite" for Science and Arts 1915 (in 1930 he became chancellor of this order) • Nobel Prize in Physics 1918 (awarded 1919) • Lorentz Medal 1927 • Adlerschild des Deutschen Reiches (1928) • Max Planck medal (1929, together with Einstein) • Planck received honorary doctorates from the universities of Frankfurt, Munich (TH), Rostock, Berlin(TH), Graz, Athens, Cambridge, London, and Glasgow • The asteroid 1069 was given the name "Stella Planckia" (1938) Planck units • Planck time • Planck length • Planck temperature • Planck current • Planck power • Planck density • Planck mass 1., The Religious Affiliation of Physicist Max Planck. Retrieved July 16, 2007. Selected publications by Planck • Gamow, George. 1966. Thirty Years That Shook Physics: The Story of Quantum Theory. Garden City, NY: Doubleday. • Heilbron, J. L. 2000. The Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science. Cambridge, MA: Harvard University Press. ISBN 0-674-00439-6 • Rosenthal-Schneider, Ilse. 1980. Reality and Scientific Truth: Discussions with Einstein, von Laue, and Planck. Wayne State University. ISBN 0-8143-1650-6 External links All links retrieved May 6, 2015.
38010dbae435b0ca	Baylor University Department of Physics College of Arts and Sciences Baylor > Physics > News Physics News News Categories • Baylor • Colloquium • Faculty Meetings • Graduate • Outreach • Research Seminars • Social Events • SPS Top News • Error fix for long-lived qubits brings quantum computers nearer • 2 Newfound Alien Planets May Be Capable of Supporting Life • Three's Company: Newly Discovered Planet Orbits a Trio of Stars • A Fifth Force: Fact or Fiction? • Neutrinos hint at why antimatter didn’t blow up the universe • Relativistic codes reveal a clumpy universe • Quantum computer makes first high-energy physics simulation • Scientists have detected gravitational waves for the second time • Surprise! The Universe Is Expanding Faster Than Scientists Thought • Building Blocks of Life Found in Comet's Atmosphere • Quantum cats here and there • Silicon quantum computers take shape in Australia • New Support for Alternative Quantum View • Dark matter does not include certain axion-like particles • Scientists discover new form of light • How light is detected affects the atom that emits it • AI learns and recreates Nobel-winning physics experiment • Scientists Talk Privately About Creating a Synthetic Human Genome • Boiling Water May Be Cause of Martian Streaks • Physicists Abuzz About Possible New Particle as CERN Revs Up • Gravitational lens reveals hiding dwarf dark galaxy • Leonardo Da Vinci's Living Relatives Found • Stephen Hawking: We Probably Won't Find Aliens Anytime Soon • Measurement of Universe's expansion rate creates cosmological puzzle • 'Bizarre' Group of Distant Black Holes are Mysteriously Aligned • Isaac Newton: handwritten recipe reveals fascination with alchemy • Surprise! Gigantic Black Hole Found in Cosmic Backwater • New Bizarre State of Matter Seems to Split Fundamental Particles • Researchers made the smallest diode using a DNA molecule • Astronomers Discover Colossal 'Super Spiral' Galaxies • How a distant planet could have killed the dinosaurs • Search for alien signals expands to 20,000 star systems • Video: ESA Euronews - Is there life on the Red Planet? • New Tetraquark Particle Sparks Doubts • 7 Theories on the Origin of Life • NIST Creates Fundamentally Accurate Quantum Thermometer • DNA data storage could last thousands of years • Hints of new LHC particle get slightly stronger • Snake walk: The physics of slithering • Astronomers discover a new galaxy far, far(ther) away • Space experts warn Congress that NASA’s “Journey to Mars” is illusory • Never-Seen-Before Tetraquark Particle Possibly Spotted in Atom Smasher • ET search: Look for the aliens looking for Earth • Physicists create first photonic Maxwell's demon • Black holes banish matter into cosmic voids • How NASA's new telescope could unlock some mysteries of the universe • 5D Black Holes Could Break Relativity • Reactor data hint at existence of fourth neutrino • Single-particle ‘spooky action at a distance’ finally demonstrated • NASA and 'Star Trek' Combine Forces to Make Replicators a Reality • 'Five-dimensional' glass discs can store data for up to 13.8 billion years • The Hubble Space Telescope Just Snapped Photos of the Biggest Black Hole We've Ever Observed • LIGO Discovers the Merger of Two Black Holes • Einstein's gravitational waves 'seen' from black holes • Earth-like Planets Have Earth-like Interiors • Physicists find signs of four-neutron nucleus • Have Gravitational Waves Finally Been Spotted? • Astronomers build Earth-sized telescope to see Milky Way black hole • The telescope gets its first major upgrade in centuries • New Theory of Secondary Inflation Expands Options for Avoiding an Excess of Dark Matter • Stephen Hawking: Black Holes Have 'Hair • the Riemann Hypothesis has finally been SOLVED by a Nigerian professor • Physicists propose the first scheme to teleport the memory of an organism • Scientists struggle to stay grounded after possible gravitational wave signal • NASA's Kepler Comes Roaring Back with 100 New Exoplanet Finds • Physicists figure out how to retrieve information from a black hole • New study asks: Why didn't the universe collapse? • Physicists in Europe Find Tantalizing Hints of a Mysterious New Particl • German physicists see landmark in nuclear fusion quest • Scientists detect the magnetic field that powers our galaxy’s supermassive black hole • Controversial experiment sees no evidence that the universe is a hologram • How to encrypt a message in the afterglow of the big bang • LISA Pathfinder Heads to Space • Scientists Create New Kind Of Diamond At Room Temperature • Japanese scientists create touchable holograms • Positrons Are Plentiful In Ultra-Intense Laser Blasts • Scientists Link Moon’s Tilt and Earth’s Gold • NASA's New 'Star Trek' Tech Is Designed to Detect Alien Life • A Century Ago, Einstein’s Theory of Relativity Changed Everything • Is Earth Growing a Hairy Dark Matter 'Beard'? • Scientists caught a new planet forming for the first time ever • Experiment records extreme quantum weirdness • Scientists look into hydrogen atom, find old recipe for pi • Strong forces make antimatter stick Tiny 'Atomic Memory' Device Could Store All Books Ever Written A new "atomic memory" device that encodes data atom by atom can store hundreds of times more data than current hard disks can, a new study finds. "You would need just the area of a postage stamp to write out all books ever written," said study senior author Sander Otte, a physicist at the Delft University of Technology's Kavli Institute of Nanoscience in the Netherlands. In fact, the researchers estimated that if they created a cube 100 microns wide — about the same diameter as the average human hair — made of sheets of atomic memory separated from one another by 5 nanometers, or billionths of a meter, the cube could easily store the contents of the entire U.S. Library of Congress. Error fix for long-lived qubits brings quantum computers nearer Useful quantum computers are one step closer, thanks to the latest demonstration of a technique designed to stop them making mistakes. Quantum computers store information as quantum bits, or qubits. Unlike binary bits, which store a 0 or a 1, qubits can hold a mixture of both states at the same time, boosting their computing potential for certain types of problems. But qubits are fragile – their quantum nature means they can’t hold data for long before errors creep in. So researchers wanting to build large-scale computers invented quantum error correction (QEC). This technique encodes a bit of quantum information using many physical qubits. The code is designed to provide wiggle-room, making it possible to recover from errors. A similar concept is used to handle errors in binary bits on hard drives and DVDs, but things are more difficult in the quantum realm. The rules of quantum mechanics mean you can’t directly read the state of a qubit without destroying it – it’s like opening the box to take a look at Schrödinger’s cat. That means we need more sophisticated codes for quantum computers than for DVDs. 2 Newfound Alien Planets May Be Capable of Supporting Life New Science Experiment Will Tell Us If The Universe Is a Hologram Do we live in a two-dimensional hologram? Scientists may discover that we live in a Matrix-like illusion during an experiment from the U.S. Department of Energy’s Fermi National Accelerator Laboratory that will look for “holographic noise” and will collect data over the next years. Although our world appears three-dimensional to us, this could be entirely illusive. Scientists are theorizing that our universe may operate much like a TV screen; just like a TV screen has pixels which create seamless images, spacetime could be contained in two-dimensional packets that make reality appear three-dimensional to the human eye. Like a hologram, a three-dimensional image would be coded onto a two-dimensional medium. If this theory is correct, these “pixels” of spacetime would be ten trillion trillion times smaller than an atom. “We want to find out whether space-time is a quantum system just like matter is,” said Craig Hogan, director of Fermilab’s Center for Particle Astrophysics. “If we see something, it will completely change ideas about space we’ve used for thousands of years.” If these packets of spacetime exist, then they are expected to be governed by the Heisenberg uncertainty principle. According to this principle, it is impossible to simultaneously ascertain the exact speed and exact location of a subatomic particle. If spacetime consists of two-dimensional fragments, then all of space would potentially be governed by this principle. If this were the case, then space, like matter, would have perpetual quantum vibrations regardless of its energy level. Three's Company: Newly Discovered Planet Orbits a Trio of Stars Three's Company: Newly Discovered Planet Orbits a Trio of Stars A new planet, HD 131399Ab, in a triple star system was recently discovered about 340 light-years from Earth in the constellation of Centaurus. Credit: ESO/L. Calçada A newly discovered planet has been spotted orbiting three stars at once, in a highly exotic celestial arrangement. "The planet is orbiting star A — the lonely star in this scenario," Kevin Wagner, a first-year doctoral student at the University of Arizona, told The planet and star A are then orbited by a pair of stars that the scientists call "star B" and "star C. (Check below to see a video of the system's orbital dance). The strange new world, HD 131399Ab, lies 340 light-years from Earth, in the constellation Centaurus. For about half of its orbit through the system, all three stars are visible in the sky. A Fifth Force: Fact or Fiction? Science and the internet have an uneasy relationship: Science tends to move forward through a careful and tedious evaluation of data and theory, and the process can take years to complete. In contrast, the internet community generally has the attention span of Dory, the absent-minded fish of "Finding Nemo"(and now "Finding Dory") — a meme here, a celebrity picture there — oh, look … a funny cat video. Thus people who are interested in serious science should be extremely cautious when they read an online story that purports to be a paradigm-shifting scientific discovery. A recent example is one suggesting that a new force of nature might have been discovered. If true, that would mean that we have to rewrite the textbooks. So what has been claimed? In an article submitted on April 7, 2015, to the arXiv repository of physics papers, a group of Hungarian researchers reported on a study in which they focused an intense beam of protons (particles found in the center of atoms) on thin lithium targets. The collisions created excited nuclei of beryllium-8, which decayed into ordinary beryllium-8 and pairs of electron-positron particles. (The positron is the antimatter equivalent of the electron.) They claimed that their data could not be explained by known physical phenomena in the Standard Model, the reigning model governing particle physics. But, they purported, they could explain the data if a new particle existed with a mass of approximately 17 million electron volts, which is 32.7 times heavier than an electron and just shy of 2 percent the mass of a proton. The particles that emerge at this energy range, which is relatively low by modern standards, have been well studied. And so it would be very surprising if a new particle were discovered in this energy regime. However, the measurement survived peer review and was published on Jan. 26, 2016, in the journal Physical Review Letters, which is one of the most prestigious physics journals in the world. In this publication, the researchers, and this research, cleared an impressive hurdle. [What's That? Your Physics Questions Answered] Their measurement received little attention until a group of theoretical physicists from the University of California, Irvine (UCI), turned their attention to it. As theorists commonly do with a controversial physics measurement, the team compared it with the body of work that has been assembled over the last century or so, to see if the new data are consistent or inconsistent with the existing body of knowledge. In this case, they looked at about a dozen published studies. What they found is that though the measurement didn't conflict with any past studies, it seemed to be something never before observed — and something that couldn't be explained by the Standard Model. Neutrinos hint at why antimatter didn’t blow up the universe Relativistic codes reveal a clumpy universe A new set of codes that, for the first time, are able to apply Einstein's complete general theory of relativity to simulate how our universe evolved, have been independently developed by two international teams of physicists. They pave the way for cosmologists to confirm whether our interpretations of observations of large-scale structure and cosmic expansion are telling us the true story. The impetus to develop codes designed to apply general relativity to cosmology stems from the limitations of traditional numerical simulations of the universe. Currently, such models invoke Newtonian gravity and assume a homogenous universe when describing cosmic expansion, for reasons of simplicity and computing power. On the largest scales the universe is homogenous and isotropic, meaning that matter is distributed evenly in all directions; but on smaller scales the universe is clearly inhomogeneous, with matter clumped into chains of galaxies and filaments of dark matter assembled around vast voids. Quantum computer makes first high-energy physics simulation Physicists have performed the first full simulation of a high-energy physics experiment — the creation of pairs of particles and their antiparticles — on a quantum computer1. If the team can scale it up, the technique promises access to calculations that would be too complex for an ordinary computer to deal with. To understand exactly what their theories predict, physicists routinely do computer simulations. They then compare the outcomes of the simulations with actual experimental data to test their theories. In some situations, however, the calculations are too hard to allow predictions from first principles. This is particularly true for phenomena that involve the strong nuclear force, which governs how quarks bind together into protons and neutrons and how these particles form atomic nuclei, says Christine Muschik, a theoretical physicist at the University of Innsbruck in Austria and a member of the simulation team. Many researchers hope that future quantum computers will help to solve this problem. These machines, which are still in the earliest stages of development, exploit the physics of objects that can be in multiple states at once, encoding information in ‘qubits’, rather than in the on/off state of classical bits. A computer made of a handful of qubits can perform many calculations simultaneously, and can complete certain tasks exponentially faster than an ordinary computer. Coaxing qubits Esteban Martinez, an experimental physicist at the University of Innsbruck, and his colleagues completed a proof of concept for a simulation of a high-energy physics experiment in which energy is converted into matter, creating an electron and its antiparticle, a positron. The team used a tried-and-tested type of quantum computer in which an electromagnetic field traps four ions in a row, each one encoding a qubit, in a vacuum. They manipulated the ions’ spins — their magnetic orientations — using laser beams. This coaxed the ions to perform logic operations, the basic steps in any computer calculation. After sequences of about 100 steps, each lasting a few milliseconds, the team looked at the state of the ions using a digital camera. Each of the four ions represented a location, two for particles and two for antiparticles, and the orientation of the ion revealed whether or not a particle or an antiparticle had been created at that location. The team’s quantum calculations confirmed the predictions of a simplified version of quantum electrodynamics, the established theory of the electromagnetic force. “The stronger the field, the faster we can create particles and antiparticles,” Martinez says. He and his collaborators describe their results on 22 June in Nature1. Four qubits constitute a rudimentary quantum computer; the fabled applications of future quantum computers, such as for breaking down huge numbers into prime factors, will require hundreds of qubits and complex error-correction codes. But for physical simulations, which can tolerate small margins of error, 30 to 40 qubits could already be useful, Martinez says. John Chiaverini, a physicist who works on quantum computing at the Massachusetts Institute of Technology in Cambridge, says that the experiment might be difficult to scale up without significant modifications. The linear arrangement of ions in the trap, he says, is “particularly limiting for attacking problems of a reasonable scale”. Muschik says that her team is already making plans to use two-dimensional configurations of ions. Are we there yet? “We are not yet there where we can answer questions we can’t answer with classical computers,” Martinez says, “but this is a first step in that direction.” Quantum computers are not strictly necessary for understanding the electromagnetic force. However, the researchers hope to scale up their techniques so that they can simulate the strong nuclear force. This may take years, Muschik says, and will require not only breakthroughs in hardware, but also the development of new quantum algorithms. These scaled-up quantum computers could help in understanding what happens during the high-speed collision of two atomic nuclei, for instance. Faced with such a problem, classical computer simulations just fall apart, says Andreas Kronfeld, a theoretical physicist who works on simulations of the strong nuclear force at the Fermi National Accelerator Laboratory (Fermilab) near Chicago, Illinois. Another example, he says, is understanding neutron stars. Researchers think that these compact celestial objects consist of densely packed neutrons, but they’re not sure. They also don’t know the state of matter in which those neutrons would exist. E.T. Phones Earth? 1,500 Years Until Contact, Experts Estimate It's December 15, 2015, and an auditorium in Geneva is packed with physicists. The air is filled with tension and excitement because everybody knows that something important is about to be announced. The CERN Large Hadron Collider (LHC) has recently restarted operations at the highest energies ever achieved in a laboratory experiment, and the first new results from two enormous, complex detectors known as ATLAS and CMS are being presented. This announcement has been organized hastily because both detectors have picked up something completely unexpected. Rumors have been circulating for days about what it might be, but nobody knows for sure what is really going on, and the speculations are wild. Scientists have detected gravitational waves for the second time Scientists with the LIGO collaboration claim they have once again detected gravitational waves — the ripples in space-time produced by objects moving throughout the Universe. It’s the second time these researchers have picked up gravitational wave signals, after becoming the first team in history to do so earlier this year. “A black hole has no hair.” That mysterious, koan-like statement by the theorist and legendary phrasemaker John Archibald Wheeler of Princeton has stood for half a century as one of the brute pillars of modern physics. It describes the ability of nature, according to classical gravitational equations, to obliterate most of the attributes and properties of anything that falls into a black hole, playing havoc with science’s ability to predict the future and tearing at our understanding of how the universe works. Now it seems that statement might be wrong. Recently Stephen Hawking, who has spent his entire career battling a form of Lou Gehrig’s disease, wheeled across the stage in Harvard’s hoary, wood-paneled Sanders Theater to do battle with the black hole. It is one of the most fearsome demons ever conjured by science, and one partly of his own making: a cosmic pit so deep and dense and endless that it was long thought that nothing — not even light, not even a thought — could ever escape. But Dr. Hawking was there to tell us not to be so afraid. In a paper to be published this week in Physical Review Letters, Dr. Hawking and his colleagues Andrew Strominger of Harvard and Malcolm Perry of Cambridge University in England say they have found a clue pointing the way out of black holes. “They are not the eternal prisons they were once thought,” Dr. Hawking said in his famous robot voice, now processed through a synthesizer. “If you feel you are trapped in a black hole, don’t give up. There is a way out.” Black holes are the most ominous prediction of Einstein’s general theory of relativity: Too much matter or energy concentrated in one place would cause space to give way, swallowing everything inside like a magician’s cloak. An eternal prison was the only metaphor scientists had for these monsters until 40 years ago, when Dr. Hawking turned black holes upside down — or perhaps inside out. His equations showed that black holes would not last forever. Over time, they would “leak” and then explode in a fountain of radiation and particles. Ever since, the burning question in physics has been: When the black hole finally goes, does it give up the secrets of everything that fell in? Dr. Hawking’s calculation was, and remains, hailed as a breakthrough in understanding the connection between gravity and quantum mechanics, between the fabric of space and the subatomic particles that live inside it — the large and the small in the universe. But there was a hitch. By Dr. Hawking’s estimation, the radiation coming out of the black hole as it fell apart would be random. As a result, most of the “information” about what had fallen in — all of the attributes and properties of the things sucked in, whether elephants or donkeys, Volkswagens or Cadillacs — would be erased. In a riposte to Einstein’s famous remark that God does not play dice, Dr. Hawking said in 1976, “God not only plays dice with the universe, but sometimes throws them where they can’t be seen.” But his calculation violated a tenet of modern physics: that it is always possible in theory to reverse time, run the proverbial film backward and reconstruct what happened in, say, the collision of two cars or the collapse of a dead star into a black hole. The universe, like a kind of supercomputer, is supposed to be able to keep track of whether one car was a green pickup truck and the other was a red Porsche, or whether one was made of matter and the other antimatter. These things may be destroyed, but their “information” — their essential physical attributes — should live forever. In fact, the information seemed to be lost in the black hole, according to Dr. Hawking, as if part of the universe’s memory chip had been erased. According to this theorem, only information about the mass, charge and angular momentum of what went in would survive. Nothing about whether it was antimatter or matter, male or female, sweet or sour. A war of words and ideas ensued. The information paradox, as it is known, was no abstruse debate, as Dr. Hawking pointed out from the stage of the Sanders Theater in April. Rather, it challenged foundational beliefs about what reality is and how it works. If the rules break down in black holes, they may be lost in other places as well, he warned. If foundational information disappears into a gaping maw, the notion of a “past” itself may be in jeopardy — we couldn’t even be sure of our own histories. Our memories could be illusions. “It’s the past that tells us who we are. Without it we lose our identity,” he said. Fortunately for historians, Dr. Hawking conceded defeat in the black hole information debate 10 years ago, admitting that advances in string theory, the so-called theory of everything, had left no room in the universe for information loss. At least in principle, then, he agreed, information is always preserved — even in the smoke and ashes when you, say, burn a book. With the right calculations, you should be able reconstruct the patterns of ink, the text. Dr. Hawking paid off a bet with John Preskill, a Caltech physicist, with a baseball encyclopedia, from which information can be easily retrieved. But neither Dr. Hawking nor anybody else was able to come up with a convincing explanation for how that happens and how all this “information” escapes from the deadly erasing clutches of a black hole. Indeed, a group of physicists four years ago tried to figure it out and suggested controversially that there might be a firewall of energy just inside a black hole that stops anything from getting out or even into a black hole. The new results do not address that issue. But they do undermine the famous notion that black holes have “no hair” — that they are shorn of the essential properties of the things they have consumed. About four years ago, Dr. Strominger started noodling around with theoretical studies about gravity dating to the early 1960s. Interpreted in a modern light, the papers — published in 1962 by Hermann Bondi, M. G. J. van der Burg, A. W. K. Metzner and Rainer Sachs, and in 1965 by Steven Weinberg, later a recipient of the Nobel Prize — suggested that gravity was not as ruthless as Dr. Wheeler had said. Looked at from the right vantage point, black holes might not be not be bald at all. The right vantage point is not from a great distance in space — the normal assumption in theoretical calculations — but from a far distance in time, the far future, technically known as “null infinity.” “Null infinity is where light rays go if they are not trapped in a black hole,” Dr. Strominger tried to explain over coffee in Harvard Square recently.From this point of view, you can think of light rays on the surface of a black hole as a bundle of straws all pointing outward, trying to fly away at the speed of, of course, light. Because of the black hole’s immense gravity, they are stuck. But the individual straws can slide inward or outward along their futile tracks, slightly advancing or falling back, under the influence of incoming material. When a particle falls into a black hole, it slides the straws of light back and forth, a process called a supertranslation. “One often hears that black holes have no hair,” Dr. Strominger and a postdoctoral researcher, Alexander Zhiboedov, wrote in a 2014 paper. Not true: “Black holes have a lush infinite head of supertranslation hair.” Enter Dr. Hawking. For years, he and Dr. Strominger and a few others had gotten together to work in seclusion at a Texas ranch owned by the oilman and fracking pioneer George P. Mitchell. Because Dr. Hawking was discouraged from flying, in April 2014 the retreat was in Hereford, Britain. It was there that Dr. Hawking first heard about soft hair — and was very excited. He, Dr. Strominger and Dr. Perry began working together. In Stockholm that fall, he made a splash when he announced that a resolution to the information paradox was at hand — somewhat to the surprise of Dr. Strominger and Dr. Perry, who has been trying to maintain an understated stance. Although information gets hopelessly scrambled, Dr. Hawking declared, it “can be recovered in principle, but it is lost for all practical purposes.” In January, Dr. Hawking, Dr. Strominger and Dr. Perry posted a paper online titled “Soft Hair on Black Holes,” laying out the basic principles of their idea. In the paper, they are at pains to admit that knocking the pins out from under the no-hair theorem is a far cry from solving the information paradox. But it is progress. Their work suggests that science has been missing something fundamental about how black holes evaporate, Dr. Strominger said. And now they can sharpen their questions. “I hope we have the tiger by the tail,” he said. Whether or not soft hair is enough to resolve the information paradox, nobody really knows. Reaction from other physicists has been reserved. Surprise! The Universe Is Expanding Faster Than Scientists Thought The universe is expanding 5 to 9 percent faster than astronomers had thought, a new study suggests. "This surprising finding may be an important clue to understanding those mysterious parts of the universe that make up 95 percent of everything and don't emit light, such as dark energy, dark matter and dark radiation," study leader Adam Riess, an astrophysicist at the Space Telescope Science Institute and Johns Hopkins University in Baltimore, said in a statement. Riess — who shared the 2011 Nobel Prize in physics for the discovery that the universe's expansion is accelerating — and his colleagues used NASA's Hubble Space Telescope to study 2,400 Cepheid stars and 300 Type Ia supernovas. Building Blocks of Life Found in Comet's Atmosphere For the first time, scientists have directly detected a crucial amino acid and a rich selection of organic molecules in the dusty atmosphere of a comet, further bolstering the hypothesis that these icy objects delivered some of life's ingredients to Earth. The amino acid glycine, along with some of its precursor organic molecules and the essential element phosphorus, were spotted in the cloud of gas and dust surrounding Comet 67P/Churyumov-Gerasimenko by the Rosetta spacecraft, which has been orbiting the comet since 2014. While glycine had previously been extracted from cometary dust samples that were brought to Earth by NASA's Stardust mission, this is the first time that the compound has been detected in space, naturally vaporized. The discovery of those building blocks around a comet supports the idea that comets could have played an essential role in the development of life on early Earth, researchers said. Quantum cats here and there The story of Schrödinger's cat being hidden away in a box and being both dead and alive is often invoked to illustrate the how peculiar the quantum world can be. On a twist of the dead/alive behavior, Wang et al. now show that the cat can be in two separate locations at the same time. Constructing their cat from coherent microwave photons, they show that the state of the “electromagnetic cat” can be shared by two separated cavities. Going beyond common-sense absurdities of the classical world, the ability to share quantum states in different locations could be a powerful resource for quantum information processing. Planet 1,200 Light-years Away Is A Good Prospect For Habitability The planet, which is about 1,200 light-years from Earth in the direction of the constellation Lyra, is approximately 40 percent larger than Earth. At that size, Kepler-62f is within the range of planets that are likely to be rocky and possibly could have oceans, said Aomawa Shields, the study's lead author and a National Science Foundation astronomy and astrophysics postdoctoral fellow in UCLA's department of physics and astronomy. NASA's Kepler mission discovered the planetary system that includes Kepler-62f in 2013, and it identified Kepler-62f as the outermost of five planets orbiting a star that is smaller and cooler than the sun. But the mission didn't produce information about Kepler-62f's composition or atmosphere or the shape of its orbit. "We found there are multiple atmospheric compositions that allow it to be warm enough to have surface liquid water," said Shields, a University of California President's Postdoctoral Program Fellow. "This makes it a strong candidate for a habitable planet." Has a Hungarian Physics Lab Found a Fifth Force of Nature? A laboratory experiment in Hungary has spotted an anomaly in radioactive decay that could be the signature of a previously unknown fifth fundamental force of nature, physicists say—if the finding holds up. Attila Krasznahorkay at the Hungarian Academy of Sciences’s Institute for Nuclear Research in Debrecen, Hungary, and his colleagues reported their surprising result in 2015 on the arXiv preprint server, and this January in the journal Physical Review Letters. But the report – which posited the existence of a new, light boson only 34 times heavier than the electron—was largely overlooked. Then, on April 25, a group of US theoretical physicists brought the finding to wider attention by publishing its own analysis of the result on arXiv. The theorists showed that the data didn’t conflict with any previous experiments—and concluded that it could be evidence for a fifth fundamental force. “We brought it out from relative obscurity,” says Jonathan Feng, at the University of California, Irvine, the lead author of the arXiv report. Four days later, two of Feng's colleagues discussed the finding at a workshop at the SLAC National Accelerator Laboratory in Menlo Park, California. Researchers there were sceptical but excited about the idea, says Bogdan Wojtsekhowski, a physicist at the Thomas Jefferson National Accelerator Facility in Newport News, Virginia. “Many participants in the workshop are thinking about different ways to check it,” he says. Groups in Europe and the United States say that they should be able to confirm or rebut the Hungarian experimental results within about a year. Silicon quantum computers take shape in Australia Silicon is at the heart of the multibillion-dollar computing industry. Now, efforts to harness the element to build a quantum processor are taking off, thanks to elegant designs from an Australian collaboration. In July, the Centre for Quantum Computation and Communication Technology, which is based at the University of New South Wales (UNSW) in Sydney, will receive the first instalment of a Aus$46-million (US$33-million) investment. The money comes from government and industry sources whose goal is to create a practical quantum computer. At an innovation forum in London on 6 May, hosted by Nature and start-up accelerator Entrepreneur First, two physicists from a group at the UNSW pitched a plan to reach that goal. Their audience was a panel of entrepreneurs and scientists, who critiqued ideas for commercializing a range of quantum technologies, including sensors, computer security and a quantum internet as well as quantum computers. So far, the UNSW team has demonstrated a system with quantum bits, or qubits, only in a single atom. Useful computations will require linking qubits in multiple atoms. But the team’s silicon qubits hold their quantum state nearly a million times longer than do systems made from superconducting circuits, a leading alternative, UNSW physicist Guilherme Tosi told participants at the event. This helps the silicon qubits to perform operations with one-sixth of the errors of superconducting circuits. If the team can pull off this low error rate in a larger system, it would be “quite amazing”, said Hartmut Neven, director of engineering at Google and a member of the panel. But he cautioned that in terms of performance, the system is far behind others. The team is aiming for ten qubits in five years, but both Google and IBM are already approaching this with superconducting systems. And in five years, Google plans to have ramped up to hundreds of qubits. New Support for Alternative Quantum View Of the many counterintuitive features of quantum mechanics, perhaps the most challenging to our notions of common sense is that particles do not have locations until they are observed. This is exactly what the standard view of quantum mechanics, often called the Copenhagen interpretation, asks us to believe. Instead of the clear-cut positions and movements of Newtonian physics, we have a cloud of probabilities described by a mathematical structure known as a wave function. The wave function, meanwhile, evolves over time, its evolution governed by precise rules codified in something called the Schrödinger equation. The mathematics are clear enough; the actual whereabouts of particles, less so. Until a particle is observed, an act that causes the wave function to “collapse,” we can say nothing about its location. Albert Einstein, among others, objected to this idea. As his biographer Abraham Pais wrote: “We often discussed his notions on objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it.” But there’s another view — one that’s been around for almost a century — in which particles really do have precise positions at all times. This alternative view, known as pilot-wave theory or Bohmian mechanics, never became as popular as the Copenhagen view, in part because Bohmian mechanics implies that the world must be strange in other ways. In particular, a 1992 study claimed to crystalize certain bizarre consequences of Bohmian mechanics and in doing so deal it a fatal conceptual blow. The authors of that paper concluded that a particle following the laws of Bohmian mechanics would end up taking a trajectory that was so unphysical — even by the warped standards of quantum theory — that they described it as “surreal.” Nearly a quarter-century later, a group of scientists has carried out an experiment in a Toronto laboratory that aims to test this idea. And if their results, first reported earlier this year, hold up to scrutiny, the Bohmian view of quantum mechanics — less fuzzy but in some ways more strange than the traditional view — may be poised for a comeback. Dark matter does not include certain axion-like particles Scientists believe 80 percent of the universe is made up of dark matter. What exactly constitutes dark matter? Scientists still aren't sure. A new study, published this week in the journal Physical Review Letters, grows the list of particles not found in dark matter. Astronomers have previously hypothesized that axion-like particles, or ALPs, might make up dark matter. Given their diminutive size -- registering at a billionth the mass of a single electron -- it was a logical guess. But when researchers at Stockholm University used NASA's gamma-ray telescope on the Fermi satellite to look for ALPs in the Perseus galaxy cluster, they came up empty-handed. ALPs can be briefly transformed into light-emitting matter when they travel through intense electromagnetic fields. Likewise, light particles like gamma radiation can briefly transform into ALPs. No such transformations, however, were detected near the center of the Perseus cluster. While the research didn't offer any revelations on the makeup of dark matter, scientists believe they can now exclude certain types of ALPs in the ongoing search for the elusive matter. "The ALPs we have been able to exclude could explain a certain amount of dark matter," Manuel Meyer, a physicist at Stockholm University, said in a news release. "What is particularly interesting is that with our analysis we are reaching a sensitivity that we thought could only be obtained with dedicated future experiments on Earth." So, the hunt for dark matter details continues. Scientists discover new form of light Researchers in Ireland have discovered a new form of light. Their discovery is expected to reshape scientists' understanding of light's basic nature. Angular momentum describes the rotation of a light beam around its axis. Until now, researchers believed the angular momentum was always a multiple of Planck's constant -- a constant ratio that describes the relationship between photon energy and frequency, and also sets the scale for quantum mechanics. The newly discovered form of light, however, features photons with an angular momentum of just half the value of Planck's constant. The difference sounds small, but researchers say the significance of the discovery is great. "For a beam of light, although traveling in a straight line it can also be rotating around its own axis," John Donegan, a professor at Trinity College Dublin's School of Physics, explained in a news release. "So when light from the mirror hits your eye in the morning, every photon twists your eye a little, one way or another." "Our discovery will have real impacts for the study of light waves in areas such as secure optical communications," Donegan added. Researchers made their discovery after passing light through special crystals to create a light beam with a hollow, screw-like structure. Using quantum mechanics, the physicists theorized that the beam's twisting photons were being slowed to a half-integer of Planck's constant. The team of researchers then designed a device to measure the beam's angular momentum as it passed through the crystal. As they had predicted, they registered a shift in the flow of photons caused by quantum effects. The researchers described their discovery in a paper published this week in the journal Science Advances. "What I think is so exciting about this result is that even this fundamental property of light, that physicists have always thought was fixed, can be changed," concluded Paul Eastham, assistant professor of physics at Trinity. How light is detected affects the atom that emits it When the photon enters your eye, something similar happens but in reverse. The photon is absorbed by a molecule in the retina and its energy kicks that molecule into an excited state. Light is both a particle and a wave, and this duality is fundamental to the physics that rule the Lilliputian world of atoms and molecules. Yet it would seem that in this case the wave nature of light can be safely ignored. Kater Murch, assistant professor of physics in Arts and Sciences at Washington University in St. Louis, might give you an argument about that. His lab is one of the first in the world to look at spontaneous emission with an instrument sensitive to the wave rather than the particle nature of light, work described in the May 20th issue of Nature Communications. His experimental instrument consists of an artificial atom (actually a superconducting circuit with two states, or energy levels) and an interferometer, in which the electromagnetic wave of the emitted light interferes with a reference wave of the same frequency. This manner of detection turns everything upside down, he said. All that a photon detector can tell you about spontaneous emission is whether an atom is in its excited state or its ground state. But the interferometer catches the atom diffusing through a quantum "state space" made up of all the possible combinations, or superpositions, of its two energy states. This is actually trickier than it sounds because the scientists are tracking a very faint signal (the electromagnetic field associated with one photon), and most of what they see in the interference pattern is quantum noise. But the noise carries complementary information about the state of the artificial atom that allows them to chart its evolution. When viewed in this way, the artificial atom can move from a lower energy state to a higher energy one even as its follows the inevitable downward trajectory to the ground state. "You'd never see that if you were detecting photons," Murch said. So different detectors see spontaneous emission very differently. "By looking at the wave nature of light, we are able see this lovely diffusive evolution between the states," Murch said. But it gets stranger. The fact that an atom's average excitation can increase even when it decays is a sign that how we look at light might give us some control over the atoms that emitted the light, Murch said. This might sound like a reversal of cause and effect, with the effect pushing on the cause. It is possible only because of one of the weirdest of all the quantum effects: When an atom emits light, quantum physics requires the light and the atom to become connected, or entangled, so that measuring a property of one instantly reveals the value of that property for the other, no matter how far away it is. Or put another way, every measurement of an entangled object perturbs its entangled partner. It is this quantum back-action, Murch said, that could potentially allow a light detector to control the light emitter. "Quantum control has been a dream for many years," Murch said. "One day, we may use it to enhance fluorescence imaging by detecting the light in a way that creates superpositions in the emitters. "That's very long term, but that's the idea," he said. AI learns and recreates Nobel-winning physics experiment The Australian National University team cooled a bit of gas down to 1 microkelvin — that’s a thousandth of a degree above absolute zero — then handed over control to the AI. It then had to figure out how to apply its lasers and control other parameters to best cool the atoms down to a few hundred nanokelvin, and over dozens of repetitions, it found more and more efficient ways to do so. “It did things a person wouldn’t guess, such as changing one laser’s power up and down, and compensating with another,” said ANU’s Paul Wigley, co-lead researcher, in a news release. “I didn’t expect the machine could learn to do the experiment itself, from scratch, in under an hour. It may be able to come up with complicated ways humans haven’t thought of to get experiments colder and make measurements more precise.” Bose-Einstein condensates have strange and wonderful properties, and their extreme sensitivity to fluctuations in energy make them useful for other experiments and measurements. But that same sensitivity makes the process of creating and maintaining them difficult. The AI monitors many parameters at once and can adjust the process quickly and in ways that humans might not understand, but which are nevertheless effective. The result: condensates can be created faster, under more conditions, and in greater quantities. Not to mention the AI doesn’t eat, sleep, or take vacations. “It’s cheaper than taking a physicist everywhere with you,” said the other co-lead researcher, Michael Hush, of the University of New South Wales. “You could make a working device to measure gravity that you could take in the back of a car, and the artificial intelligence would recalibrate and fix itself no matter what.” This AI is extremely specific in its design, of course, and can’t be applied as-is to other problems; for more flexible automation, physicists will still have to rely on the general-purpose research units called “graduate students.” Scientists Talk Privately About Creating a Synthetic Human Genome “Would it be O.K., for example, to sequence and then synthesize Einstein’s genome?” Drew Endy, a bioengineer at Stanford, and Laurie Zoloth, a bioethicist at Northwestern University, wrote in an essay criticizing the proposed project. “If so how many Einstein genomes should be made and installed in cells, and who would get to make them?” Continue reading the main story Scientists Seek Moratorium on Edits to Human Genome That Could Be Inherited DEC. 3, 2015 British Researcher Gets Permission to Edit Genes of Human Embryos FEB. 1, 2016 George Church, a professor of genetics at Harvard Medical School and an organizer of the proposed project, said there had been a misunderstanding. The project was not aimed at creating people, just cells, and would not be restricted to human genomes, he said. Rather it would aim to improve the ability to synthesize DNA in general, which could be applied to various animals, plants and microbes. He said the meeting was closed to the news media, and people were asked not to tweet because the project organizers, in an attempt to be transparent, had submitted a paper to a scientific journal. They were therefore not supposed to discuss the idea publicly before publication. He and other organizers said ethical aspects have been amply discussed since the beginning. The project was initially called HGP2: The Human Genome Synthesis Project, with HGP referring to the Human Genome Project. An invitation to the meeting at Harvard said that the primary goal “would be to synthesize a complete human genome in a cell line within a period of 10 years.” Boiling Water May Be Cause of Martian Streaks The results of Earth-bound lab experiments appear to back up the theory that dark lines on Martian slopes are created by water — though in an otherworldly manner, scientists said Monday.A team from France, Britain and the United States constructed models and simulated Mars conditions to follow up on a 2015 study which proffered “the strongest evidence yet” for liquid water — a prerequisite for life — on the Red Planet. That finding had left many scientists scratching their heads as the low pressure of Mars’ atmosphere means that water does not survive long in liquid form. It either boils or freezes. An international team of astronomers has discovered three Earth-like exoplanets orbiting an ultra-cool dwarf star—the smallest and dimmest stars in the Galaxy—now known as TRAPPIST-1. The discovery, made with the TRAPPIST telescope at ESO's La Silla Observatory, is significant not only because the three planets have similar properties to Earth, suggesting they could harbor life, but also because they are relatively close (just 40 light years away) and they are the first planets ever discovered orbiting such a dim star. A research paper detailing the teams findings was published today in the journal Nature. "What is super exciting is that for the first time, we have extrasolar worlds similar in size and temperature to Earth—planets that could thus, in theory, harbor liquid water and host life on at least a part of their surfaces—for which the atmospheric composition can be studied in detail with current technology," lead researcher Michaël Gillon of the University of Liège in Belgium said in an email to Popular Mechanics. The real reasons nothing can go faster than the speed of light We are told that nothing can travel faster than light. This is how we know it is true Physicists Abuzz About Possible New Particle as CERN Revs Up Alien life: A new paper shows that the discoveries of exoplanets, plus a revised Drake's equation, produces a new, empirically valid probability of whether any other advanced civilizations have ever existed. Astronomers revised the half-century old Drake equation, which attempts to calculate the probability of the existence of advanced alien civilizations, to determine whether any such civilizations have existed at any point in the history of the universe. They found that the chances that a human civilization evolved on Earth and nowhere else in the universe are less than about one in 10 billion trillion. ne of Stephen Hawking's most brilliant and disturbing theories may have been confirmed by a scientist who created a sound “black hole” in his laboratory, potentially paving the way for a Nobel Prize. Research by Professor Hawking, a cosmologist at Cambridge University, disputes the notion that black holes are a gravitational sinkhole, pulling in matter and never allowing anything to escape, even light. His model, developed in the 1970s, instead suggested that black holes could actually emit tiny particles, allowing energy to escape. If true, it would mean some black holes could simply evaporate completely with profound implications for our understanding of the universe.But such is the weakness of the emitted particle combined with the remoteness of even the nearest of black holes, his mathematical discovery has yet to be verified by observation. Instead Jeff Steinhauer, professor of physics at the Technion university in Haifa, created something analagous to a “black hole” for sound in his laboratory. In a paper published on the physics website arXiv, and reported by The Times, he described how he cooled helium to close to absolute zero before manipulating it in such a way that sound could not cross it, like a black hole's event horizon. He said he found evidence that phonons – the sound equivalent of light's photons - were leaking out, rather as Prof Hawking had predicted for black holes.The results have yet to be replicated elsewhere and scientists say they will want to check the effect is not caused by another factor. If confirmed, it would strengthen Prof Hawking's case for science's greatest prize. Although his theory has a lot of support, Nobel Prizes for Physics are not awarded without experimental proof. Earlier this year, Prof Hawking used the BBC's Reith Lecture to make the case that his work was close to being proven, both in the laboratory and from echoes of the very earliest moments of our universe.“I am resigned to the fact that I won’t see proof of Hawking radiation directly. “There are solid state analogues of black holes and other effects, that the Nobel committee might accept as proof,” he said. “But there’s another kind of Hawking radiation, coming from the cosmological event horizon of the early inflationary universe. “I am now studying whether one might detect Hawking radiation in primordial gravitational waves . . . so I might get a Nobel prize after all.” Gravitational lens reveals hiding dwarf dark galaxy Originally, scientists were simply trying to capture an image of the gravitational lens SDP.81 using the Atacama Large Millimeter Array. Their efforts were part of a 2014 survey aimed at testing ALMA's new, high-resolution capabilities. More than a year later, however, the image revealed a surprise -- a dwarf dark galaxy hiding in the halo of a larger galaxy, positioned some 4 billion light-years from Earth.A gravitational lens, or gravitational lensing, is a phenomenon whereby the gravity of a closer galaxy bends the light of a more distant galaxy, creating a magnifying lens-like effect. The phenomenon is often used to study galaxies that would otherwise be too far away to see. Astronomers initially assumed SDP.81 revealed the light of two galaxies -- that of a more distant galaxy, 12 billion light-years away, and that of the a closer galaxy, 4 billion light-years away. But new analysis of the image by researchers at Stanford University has revealed evidence of a dwarf dark galaxy. "We can find these invisible objects in the same way that you can see rain droplets on a window. You know they are there because they distort the image of the background objects," astronomer Yashar Hezaveh explained in a news release. The gravitational influence of dark matter distorted the light bending through the gravitational lens. Hezaveh and his colleagues recruited the power of several supercomputers to scan the radio telescope data for anomalies within the halo of SDP.81. They succeeded in identifying a unique clump of distortion, less than one-thousandth the mass of the Milky Way. The work may pave the way for the discovery of more collections of dark matter and also solve a discrepancy that's long plagued cosmologists and astronomers. Leonardo Da Vinci's Living Relatives Found Leonardo da Vinci lives on, according to two Italian researchers who have tracked down the living relatives of the Renaissance genius. It was believed that no traces were left of the painter, engineer, mathematician, philosopher and naturalist. The remains of Leonardo, who died in 1519 in Amboise, France, were dispersed in the 16th century during religious wars. But according to historian Agnese Sabato and art historian Alessandro Vezzosi, director of the Museo Ideale in the Tuscan town of Vinci, where the artist was born in 1452, Da Vinci's family did not go extinct. Stephen Hawking: We Probably Won't Find Aliens Anytime Soon Will humanity find intelligent alien life anytime soon? Probably not, according to theoretical physicist Stephen Hawking. Hawking made the prediction yesterday (April 12) during the Breakthrough Starshot announcement in New York City. At the news conference, Hawking, along with Russian billionaire investor Yuri Milner and a group of scientists, detailed a new project that aims to send a multitude of tiny, wafer-size spaceships into space to the neighboring star system Alpha Centauri. If these tiny spaceships travel at 20 percent the speed of light, they'll be able to reach Alpha Centauri in just 20 years, Milner said. Once there, the spacecraft will be able to do a 1-hour flyby of Alpha Centauri and collect data that's impossible to gather from Earth, such as taking close-up photos of the star system, probing space dust molecules and measuring magnetic fields, said Avi Loeb, chairman of the Breakthrough Starshot Advisory Committee and a professor of science at Harvard University. Measurement of Universe's expansion rate creates cosmological puzzle The most precise measurement ever made of the current rate of expansion of the Universe has produced a value that appears incompatible with measurements of radiation left over from the Big Bang1. If the findings are confirmed by independent techniques, the laws of cosmology might have to be rewritten.This might even mean that dark energy — the unknown force that is thought to be responsible for the observed acceleration of the expansion of the Universe — has increased in strength since the dawn of time. “I think that there is something in the standard cosmological model that we don't understand,” says astrophysicist Adam Riess, a physicist at Johns Hopkins University in Baltimore, Maryland, who co-discovered dark energy in 1998 and led the latest study. Kevork Abazajian, a cosmologist at the University of California, Irvine, who was not involved in the study, says that the results have the potential of “becoming transformational in cosmology”. Stephen Hawking Helps Launch Project 'Starshot' for Interstellar Space Exploration 'Bizarre' Group of Distant Black Holes are Mysteriously Aligned A highly sensitive radio telescope has seen something peculiar in the depths of our cosmos: A group of supermassive black holes are mysteriously aligned, as if captured in a synchronized dance.These black holes, which occupy the centers of galaxies in a region of space called ELAIS-N1, appear to have no relation to one another, separated by millions of light-years. But after studying the radio waves generated by the twin jets blasting from the black holes’ poles, astronomers using data from the Giant Metrewave Radio Telescope (GMRT) in India realized that all the jets were pointed in the same direction, like arrows on compasses all pointing “north.”This is the first time a group of supermassive black holes in galactic cores have been seen to share this bizarre relationship and, at first glance, the occurrence should be impossible. What we are witnessing is a cluster of galaxies, that all have central supermassive black holes that have their axes of rotation pointed in the same direction. “Since these black holes don’t know about each other, or have any way of exchanging information or influencing each other directly over such vast scales, this spin alignment must have occurred during the formation of the galaxies in the early universe,” said Andrew Russ Taylor, director of the Inter-University Institute for Data Intensive Astronomy in Cape Town, South Africa. Taylor is lead author of the study published in the journal Monthly Notices of the Royal Astronomical Society. Isaac Newton: handwritten recipe reveals fascination with alchemy A 17th-century recipe written by Isaac Newton is now going online, revealing more about the physicist’s relationship with the ancient science of alchemy. Calling for ingredients such as "one part Fiery Dragon" and "at least seven Eagles of mercury," the handwritten recipe describes how to make "sophick mercury," seen at the time as an essential element in creating the "philosopher’s stone," a fabled substance with the power to turn base metals, like lead, into gold. The manuscript, which is written in Latin and English, was acquired in February by Philadelphia-based nonprofit the Chemical Heritage Foundation, National Geographic reports. The foundation is now working to upload digital images and transcriptions of the text to an online database. Scientists may not have been able to spot the proposed ninth planet in our solar system, or even confirm that it exists, but that hasn't stopped them from imagining how it looks. Astrophysicists from the University of Bern recently showed off a new model of the possible evolution of Planet Nine, a planet hypothesized to explain the movement bodies at our solar system's edge. Published in the Journal Astronomy and Astrophysics, the model shows the possible size, temperature, and brightness of the mysterious planet. New research suggests that the sugar ribose -- the "R" in RNA -- is probably found in comets and asteroids that zip through the solar system and may be more abundant throughout the universe than was previously thought. The finding has implications not just for the study of the origins of life on Earth, but also for understanding how much life there might be beyond our planet. Scientists already knew that several of the molecules necessary for life including amino acids, nucleobases and others can be made from the interaction of cometary ices and space radiation. But ribose, which makes up the backbone of the RNA molecule, had been elusive -- until now. The new work, published Thursday in Science, fills in another piece of the puzzle, said Andrew Mattioda, an astrochemist at NASA Ames Research Center, who was not involved with the study. Surprise! Gigantic Black Hole Found in Cosmic Backwater One of the biggest black holes ever found sits in a cosmic backwater, like a towering skyscraper in a small town. Astronomers have spotted a supermassive black hole containing 17 billion times the mass of the sun — only slightly smaller than the heftiest known black hole, which weighs in at a maximum of 21 billion solar masses — at the center of the galaxy NGC 1600. That's a surprise, because NGC 1600, which lies 200 million light-years from Earth in the constellation Eridanus, belongs to an average-size galaxy group, and the monster black holes discovered to date tend to be found in dense clusters of galaxies. So researchers may have to rethink their ideas about where gigantic black holes reside, and how many of them might populate the universe, study team members said. New Bizarre State of Matter Seems to Split Fundamental Particles A bizarre new state of matter has been discovered — one in which electrons that usually are indivisible seem to break apart. The new state of matter, which had been predicted but never spotted in real life before, forms when the electrons in an exotic material enter into a type of "quantum dance," in which the spins of the electrons interact in a particular way, said Arnab Banerjee, a physicist at Oak Ridge National Laboratory in Tennessee. The findings could pave the way for better quantum computers, Banerjee said. Is Mysterious 'Planet Nine' Tugging on NASA Saturn Probe? Researchers made the smallest diode using a DNA molecule The study could lead to nanoscale electronic components and devices. A team of researchers from the University of Georgia and Ben-Gurion University has developed an electronic component so tiny, you can't even see it under an ordinary microscope. See, the team used a single DNA molecule to create a diode, a component that conducts electricity mostly in one direction. Further, the DNA molecule they designed for the study only has 11 base pairs. That makes it a pretty short helix, considering a human genome has approximately 3 billion pairs. To allow a current to flow through the DNA, the team inserted a molecule called "coralyne" into the helix. What the team came up with was a diode, because the current was 15 times stronger for negative voltages than for positive. The study's lead author Bingqian Xu decided to experiment on DNA to create minuscule components, since we can't exactly use silicon for parts that size. Astronomers Discover Colossal 'Super Spiral' Galaxies How a distant planet could have killed the dinosaurs A new paper revamps the age-old theory of Planet X – a distant planet with an gravitational pull that dislodges stray comets and sends them toward Earth.A theoretical giant planet orbiting at the far edges of our solar system may be redirecting stray comets and asteroids into the inner solar system, one or some of which could have caused the dinosaur extinction event on Earth. The theory might seem like science fiction, but a study by astrophysicist Daniel Whitmire offers scientific data to back up the claim. Dr. Whitmire, who now teaches math at the University of Arkansas, says the mysterious planet has been causing extinction events at regular intervals – every 27 million years – on Earth. Sound familiar? The new paper, published in the Monthly Notices of the Royal Astronomical Society, is a revision of a previous study Whitmore and his partner John Matese proposed in 1985. Search for alien signals expands to 20,000 star systems The search for radio signals from alien worlds is expanding to 20,000 star systems that were previously considered poor targets for intelligent extraterrestrial life, US researchers said Wednesday. The ExoMars spacecraft has blasted off from the Baikonur Cosmodrome in Kazakhstan to search for signs of life on the Red Planet. It's a mission that presents incredible scientific and engineering challenges - as it looks to unravel some of the mysteries of our Solar System. Watch this European Space Agency video to understand this mission. New Tetraquark Particle Sparks Doubts Exotic particles can be incredibly ephemeral, sticking around for tiny fractions of a second before decaying. The recent discovery of a new type of particle called a tetraquark may turn out to be equally short-lived, according to a new study casting doubt on the finding, although the issue is not yet settled. The new tetraquark — an arrangement of four quarks, the fundamental particles that build up the protons and neutrons inside atoms — was first announced in late February by physicists taking part in the DZero experiment at the Tevatron collider at the Fermi National Accelerator Laboratory (Fermilab) in Illinois. The finding represented a surprising configuration of quarks of four different flavors that was not predicted and could help elucidate the maddeningly complex rules that govern these particles. But now scientists at the Large Hadron Collider (LHC) — the world's largest particle accelerator, buried beneath Switzerland and France — say they have tried and failed to find confirming evidence for the particle in their own data. "We don't see any of these tetraquarks at all," says Sheldon Stone, a Syracuse University physicist who led the analysis for the Large Hadron Collider Beauty (LHCb) experiment. "We contradict their result." 7 Theories on the Origin of Life Life on Earth began more than 3 billion years ago, evolving from the most basic of microbes into a dazzling array of complexity over time. But how did the first organisms on the only known home to life in the universe develop from the primordial soup? One theory involved a “shocking” start. Another idea is utterly chilling. And one theory is out of this world! This article reveals the different scientific theories on the origins of life on Earth. NIST Creates Fundamentally Accurate Quantum Thermometer While the method is not yet ready for commercialization, it reveals how an object’s thermal energy—its heat—can be determined precisely by observing its physical properties at the quantum scale. While the initial demonstration has an absolute accuracy only within a few percentage points, the NIST approach works over a wide temperature range encompassing cryogenic and room temperatures. It is also accomplished with a small, nanofabricated photonic device, which opens up possible applications that are not practical with conventional temperature standards. DNA data storage could last thousands of years Researchers in Switzerland have developed a method for writing vast amounts of information in DNA and storing it inside a synthetic fossil, potentially for thousands of years. In past centuries, books and scrolls preserved the knowledge of our ancestors, even though they were prone to damage and disintegration. In the digital era, most of humanity's collective knowledge is stored on servers and hard drives. But these have a limited lifespan and need constant maintenance. Scientists from ETH Zurich have taken inspiration from the natural world in a bid to devise a storage medium that could last for potentially thousands of years. They say that genetic material found in fossils hundreds of thousands of years old can be isolated and analyzed as it has been protected from environmental stresse Hints of new LHC particle get slightly stronger That excess of photons seen by the CMS experiment has now become slightly more significant, owing to a fresh analysis reported on 17 March at a conference in La Thuile, Italy. But to the disappointment of many, the significance seen by ATLAS actually went down a bit, as a result of a more conservative interpretation of the data. Snake walk: The physics of slithering Innumerable critters have evolved superb ways to scuttle and slither - or even burrow and "swim" - across the most unhelpful of terrains: those that flow. If you've ever tried to walk up a sand dune, then you are familiar with the problem: unstable ground makes a mission out of locomotion. Now, imagine doing it on your belly. This is why a team of physicists is playing with snakes in a custom-built sand pit. The way they move is a marvel. (The snakes, not the physicists.) If you've ever tried to walk up a sand dune, then you are familiar with the problem: unstable ground makes a mission out of locomotion. Now, imagine doing it on your belly. Ms Perrin Schiebel is studying for a PhD in physics at the Georgia Institute of Technology in Atlanta, US. She has spent many months putting 10 of these snakes through their slippery paces in a sand-filled aquarium. "One of the things that's really interesting about snakes is that their entire body is, in this type of locomotion, in sliding contact with the ground," Ms Schiebel explains. Astronomers discover a new galaxy far, far(ther) away A team of astronomers say they have discovered a hot, star-popping galaxy that – at 13.4 billion light years away – is much farther than any galaxy previously identified, both in time and distance. Using a technique that has raised some skepticism among rival astronomers, they say they’ve identified a galaxy from a time when the universe was only about 400 million years old. That’s a time period commonly believed to be impossible to observe with today’s technology. The discovery far surpasses previous records for distance and time, and may be farthest that can be seen until a new space telescope is launched, the astronomers report in a paper published Thursday in Astrophysical Journal. For the last half-decade, NASA has resolutely declared that it has embarked on a Journey to Mars. Virtually every agency achievement has, in one way or another, been characterized as furthering this ambition. Even last summer when the New Horizons spacecraft flew by Pluto, NASA Administrator Charles Bolden said it represented “one more step” on the Journey to Mars. But as the end of President Obama’s second term in office nears, Congress has begun to assess NASA’s Mars ambitions. On Wednesday during a House space subcommittee hearing, legislators signaled that they were not entirely pleased with those plans. Comments from lawmakers, and the three witnesses called to the hearing, indicate NASA’s Journey to Mars may receive some pushback in the next year or two. Some of the most critical testimony came from John Sommerer, a space scientist who spent more than a year as chairman of a National Research Council technical panel reviewing NASA’s human spaceflight activities. That panel’s work, summarized in a 2014 report titled Pathways to Exploration, considered possible pathways to Mars. Never-Seen-Before Tetraquark Particle Possibly Spotted in Atom Smasher Evidence for a never-before-seen particle containing four types of quark has shown up in data from the Tevatron collider at the Fermi National Accelerator Laboratory (Fermilab) in Illinois. The new particle, a class of "tetraquark," is made of a bottom quark, a strange quark, an up quark and a down quark. The discovery could help elucidate the complex rules that govern quarks — the tiny fundamental particles that make up the protons and neutrons inside all the atoms in the universe. Protons and neutrons each contain three quarks, which is by far the most stable grouping. Pairs of quarks, called mesons, also commonly appear, but larger conglomerations of quarks are extremely rare. Scientists at the Large Hadron Collider (LHC) in Switzerland last year saw the first signs of a pentaquark—a grouping of five quarks—which had long been predicted but never seen. The first tetraquark was found in 2003 at the Belle experiment in Japan, and since then physicists have encountered a half dozen different arrangements. But the new one, if confirmed, would be special. “What’s unique in this case is that we basically have four quarks, which are all different—bottom, up, strange and down,” says Dmitri Denisov, co-spokesperson for the DZero experiment. “In all previous configurations usually two quarks are the same. Is this telling us something? I hope yes.” The unusual arrangement, dubbed X(5568) in a paper submitted toPhysical Review Letters, could reflect some deeper rule about how the different types, or “flavors,” of quarks bind together—a process enabled by the strongest force in nature, called, appropriately, the strong force. Physicists have a theory—called quantum chromodynamics—that describes how the strong force works, but it is incredibly unwieldy and difficult to make predictions with. “While we understand many features of the strong force, we don’t understand everything, especially how the strong force acts on large distances,” Denisov says. “And on a fundamental level we still don’t have a very good model of how quarks interact when there are quite a few of them joined together.” ET search: Look for the aliens looking for Earth By watching how the light dims as a planet orbits in front of its parent star, NASA’s Kepler spacecraft has discovered more than 1,000 worlds since its launch in 2009. Now, astronomers are flipping that idea on its head in the hope of finding and talking to alien civilizations. Scientists searching for extraterrestrial intelligence should target exoplanets from which Earth can be seen passing in front of the Sun, says René Heller, an astronomer at the Max Planck Institute for Solar System Research in Göttingen, Germany. By studying these eclipses, known as transits, civilizations on those planets could see that Earth has an atmosphere that has been chemically altered by life. “They have a higher motivation to contact us, because they have a better means to identify us as an inhabited planet,” Heller says. About 10,000 stars that could harbour such planets should exist within about 1,000 parsecs (3,260 light years) of Earth, Heller and Ralph Pudritz, an astronomer at McMaster University in Hamilton, Canada, report in the April issue of Astrobiology1. They argue that future searches for signals from aliens, such as the US$100-million Breakthrough Listen project, should focus on these stars, which fall in a band of space formed by projecting the plane of the Solar System out into the cosmos. Breakthrough Listen currently has no plans to search this region; it is targeting both the centre and the plane of our galaxy, which is not the same as the plane of the Solar System, as well as stars and galaxies across other parts of the sky. Physicists create first photonic Maxwell's demon Maxwell's demon, a hypothetical being that appears to violate the second law of thermodynamics, has been widely studied since it was first proposed in 1867 by James Clerk Maxwell. But most of these studies have been theoretical, with only a handful of experiments having actually realized Maxwell's demon. Now in a new paper, physicists have reported what they believe is the first photonic implementation of Maxwell's demon, by showing that measurements made on two light beams can be used to create an energy imbalance between the beams, from which work can be extracted. One of the interesting things about this experiment is that the extracted work can then be used to charge a battery, providing direct evidence of the "demon's" activity. The physicists, Mihai D. Vidrighin, et al., carried out the experiment at the University of Oxford and published a paper on their results in a recent issue of Physical Review Letters. Black holes banish matter into cosmic voids In recent decades, astronomers have cultivated a picture of the universe dominated by unseen matter, in which – on the largest scales – galaxies and everything they contain are concentrated into honeycomb-like filaments stretching around the edge of enormous voids. Until a recent study, the voids were thought to be almost empty. Now astronomers in Austria, Germany and the United States say these dark areas in space could contain as much as 20% of the ordinary matter of our cosmos. They also say that galaxies make up only 1/500th of the volume of the universe. The team, led by Dr. Markus Haider of the Institute of Astro- and Particle Physics at the University of Innsbruck in Austria, published these results in a new paper in Monthly Notices of the Royal Astronomical Society on February 24, 2016. How NASA's new telescope could unlock some mysteries of the universe One of the most highly anticipated astronomical missions of the next decade is now officially in the works at the NASA. The Wide Field Infrared Survey Telescope, or WFIRST, has been under study for years and it was formally decided Wednesday that the project would be moving forward. “WFIRST has the potential to open our eyes to the wonders of the universe, much the same way Hubble [Space Telescope] has,” said NASA Science Mission Directorate associate administrator John Grunsfeld in an agency release. “This mission uniquely combines the ability to discover and characterize planets beyond our own solar system with the sensitivity and optics to look wide and deep into the universe in a quest to unravel the mysteries of dark energy and dark matter.” 5D Black Holes Could Break Relativity Ring-shaped, five-dimensional black holes could break Einstein's theory of general relativity, new research suggests. There's a catch, of course. These 5D "black rings" don't exist, as far as anyone can tell. Instead, the new theoretical model may point out one reason why we live in a four-dimensional universe: Any other option could be a hot mess. "Here we may have a first glimpse that four space-time dimensions is a very, very good choice, because otherwise, something pretty bad happens in the universe," said Ulrich Sperhake, a theoretical physicist at the University of Cambridge in England. Reactor data hint at existence of fourth neutrino Neutrinos, electrically neutral particles that sense only gravity and the weak nuclear force, interact so feebly with matter that 100 trillion zip unimpeded through your body every second. They come in three known types: electron, muon and tau. The Daya Bay results suggest the possibility that a fourth, even more ghostly type of neutrino exists — one more than physicists’ standard theory allows. Dubbed the sterile neutrino, this phantom particle would carry no charge of any kind and would be impervious to all forces other than gravity. Only when shedding its invisibility cloak by transforming into an electron, muon or tau neutrino could the sterile neutrino be detected. Definitive evidence “would open up a whole new avenue of research,” says particle physicist Stephen Parke of the Fermi National Accelerator Laboratory in Batavia, Ill. Possible evidence for the sterile particle comes from a mismatch between theory and experiment. If a nuclear reactor produces a beam of just one type of neutrino, theory predicts that some should change their identity as they travel to a far-off detector (SN Online: 10/6/15). Analyzing more than 300,000 electron antineutrinos (the antimatter counterpart of the electron neutrino) collected from the Daya Bay nuclear reactors during 217 days of operation, researchers found 6 percent fewer of the particles than predicted by the standard particle physics model. Particle physicist Kam-Biu Luk of the University of California, Berkeley and the Lawrence Berkeley National Laboratory and colleagues report the findings in the Feb. 12 Physical Review Letters. One explanation for the deficit is that some of the electron antineutrinos have transformed into an undetectable, lightweight sterile neutrino, about one-millionth the mass of an electron, says Luk. Other nuclear reactor studies, including an experiment at the Bugey reactor in Saint-Vulbas, France, have seen similar electron antineutrino deficits, he notes. Studies with muon antineutrino beams at some particle accelerators have seen an excess of electron antineutrinos, which might be attributed to a different kind of sleight-of-hand by the unseen sterile neutrinos. The Daya Bay result provides the most precise measure yet of the energies of antielectron neutrinos at a nuclear reactor. Even so, the statistical significance of the deficit is not high enough to rate the finding a discovery. The result is a “three-sigma” finding, meaning that there’s about a 0.3 percent probability that such a paucity of electron antineutrinos would have occurred if no sterile neutrino exists. Physicists generally want a discrepancy to have a significance of five-sigma, or a 0.00003 percent chance of being a fluke, before they will label it a discovery. Besides the hint of sterile neutrinos, the Daya Bay results reveal a second strange feature — an excess of electron antineutrinos (compared with theoretical predictions) at an energy of around 5 million electron volts. That could be a sign of completely new physics awaiting discovery (or simply that scientists don’t have a detailed enough grasp of the output of nuclear reactors). A revised understanding of that feature might even do away with the need for a lightweight sterile neutrino to explain the overall deficit in electron antineutrinos. But if definitive evidence for a light sterile neutrino is eventually found, it “would turn the theory community on its head,” says Parke, and could have a bigger impact than the discovery of the Higgs boson, the Nobel-winning finding that explains why elementary particles have mass. “Finding a sterile neutrino is extremely important because it would be the first discovery of a particle which cannot be accommodated in the framework of the so-called standard model,” says particle physicist Carlo Giunti of the University of Turin in Italy. One of the earliest experiments that suggested the presence of sterile neutrinos was the Liquid Scintillator Neutrino Detector, which operated at the Los Alamos National Laboratory in New Mexico from 1993 to 1998. The LSND found that muon antineutrinos beamed into 167 tons of mineral oil had morphed into electron antineutrinos in a way that seemed to require a fourth type of neutrino to exist. A follow-up experiment at Fermilab, called MiniBooNE, ran from 2002 to 2012, with equivocal results. Another Fermilab experiment, MicroBooNE, began operation last October. MicroBooNE is the first of three liquid argon detectors, spaced at different distances near neutrino sources at Fermilab, that will track with unprecedented precision the transformation of neutrinos from one type to another. Located 470 meters from Fermilab’s Booster Neutrino Beamline, MicroBooNE is the middle of the trio, to be joined in 2018 by ICARUS, the farthest detector, at a distance of about 600 meters from the beamline, and the Short-Baseline Near Detector, placed just 100 meters from the source. First results from the trio are expected in 2021, says experimental particle physicist Peter Wilson of Fermilab. The detectors will also serve as a prototype for the Deep Underground Neutrino Experiment, a large-scale experiment that will send Fermilab-generated neutrinos on a 1,300-kilometer journey to the Sanford Underground Research Facility near Lead, S.D. In the meantime, the Daya Bay collaboration has teamed up with another Fermilab experiment, the Main Injector Neutrino Oscillation Search, to continue to seek signs of the sterile neutrinos. Although data from accelerator and reactor experiments do not yet paint a consistent picture, “we will soon know better whether a light sterile neutrino is waiting for us to unveil,” says Luk. If a light sterile neutrino exists, it might have siblings about 1,000 times heavier. These particles could contribute to the as-yet-unidentified dark matter, the invisible gravitational glue that keeps galaxies from flying apart and shapes the large-scale structure of the universe. Fingerprints of this particle will be sought with an experiment called KATRIN, which examines the radioactive decay of tritium, a heavy isotope of hydrogen, at the Karlsruhe Institute of Technology in Germany. Sterile neutrinos that are even more massive, more than a trillion times heavier than the electron, could explain an ever bigger cosmic mystery — the mismatch between the amounts of matter and antimatter in the cosmos. Possessing an energy at least a million times greater than can be produced at the Large Hadron Collider, the world’s most powerful particle accelerator, a superheavy sterile neutrino in the early universe would have made a smidgen more matter than antimatter. Over time, the tiny imbalance, reproduced in countless nuclear reactions, would have generated the matter-dominated universe seen today (SN: 1/26/13, p. 18). “For cosmology, the [lightweight] sterile neutrino that we are talking about cannot solve the problem of the matter-antimatter asymmetry, but it is likely that the sterile neutrino is connected with other new particles that can solve the problem,” says Giunti. Scientists see another, more practical, benefit for studying neutrinos. By recording the antineutrino output of nuclear reactors, detectors can discern the relative amounts of plutonium and uranium, the raw materials for making nuclear weapons. Gram for gram, fissioned plutonium and uranium have distinctive fingerprints in both the energy and rate of antineutrinos they produce, says physicist Adam Bernstein of the Lawrence Livermore National Laboratory in California. Closeup monitoring of reactors, from a distance of 10 to 500 meters, has already been demonstrated; detectors capable of monitoring weapons activity from several hundred kilometers away is possible but will require additional research and funding, Bernstein says. Single-particle ‘spooky action at a distance’ finally demonstrated For the first time, researchers have demonstrated what Albert Einstein called "spooky action at a distance" using a single particle. And not only is it a huge deal for our understanding of quantum mechanics, it also proves that the physics genius got something wrong. Spooky action at a distance, or quantum entanglement, in a single particle is a strange form of entanglement that could greatly help to improve quantum computing and communications. Unlike regular quantum entanglement, which involves two particles being defined only by being opposites of each other, single particles that are entangled have a wave function that's spread over huge distances, but are never actually in more than one place While it isn't quite the magical wave of hand envisioned in Star Trek, 3D printers are still pretty close to the replicators seen from The Next Generation on, able to fabricate body parts, cars, and even food from raw materials. It's no wonder NASA wants them to build tools, rocket engines, and even housing on Mars. But now, NASA has launched a challenge meant to bring kids into the mix: it wants 3D printed food implements. It's not quite the same as, say, a 3D printed pizza. But what NASA wants is, essentially, kitchenware and food growing aides. In the Star Trek Replicator competition, it wants "non-edible, food-related item for astronauts to 3D print in the year 2050," which is around when we'll supposedly be on Mars. Ish. The design guidelines are fairly open ended, but there are a few ground rules. It must not be bigger than 6 inches cubed (6"x6"x6".) The K-12 student designing it must designate where it will be 3D printed, and why it's suited for that environment. It must advance long term space exploration. And it must involve only a single material (so there's no printing, say, a metal alloy and plastic into the same object.) The contest kicks off today, with entries due by May 1. Four finalists will win a 3D printer for their school and a PancakeBot for themselves, while the grand prize is a tour of the Intrepid Sea, Air, & Space Museum in New York with an astronaut, a bunch of Star Trek swag, and more. There are multiple eligible age groups, so if you have a school aged child, this could be just the contest for them. More information is available here. Photographs fade, books rot, and even hard drives eventually fester. When you take the long view, preserving humanity's collective culture isn't a marathon, it's a relay — with successive generations passing on information from one slowly-failing storage medium to the next. However, this could change. Scientists from the University of Southampton in the UK have created a new data format that encodes information in tiny nanostructures in glass. A standard-sized disc can store around 360 terabytes of data, with an estimated lifespan of up to 13.8 billion years even at temperatures of 190°C. That's as old as the Universe, and more than three times the age of the Earth. The method is called five-dimensional data storage, and was first demonstrated in a paper in 2013. Since then, the scientists behind it say they've more or less perfected their technique, and are now looking to move the technology forward and perhaps even commercialize it. "We can encode anything," Aabid Patel, a postgraduate student involved in the research tells The Verge. "We’re not limited to anything — just give us the file and we can print it [onto a disc]." In order to demonstrate the format's virtues, the team from the University of Southampton have created copies of the King James Bible, Isaac Newton's Opticks (the foundational text of the study of light and lenses), and the United Nations' Universal Declaration of Human Rights, which was presented to the UN earlier this month. A new paper on the format will be given tomorrow by the team's lead researcher, Professor Peter Kazinsky, at the Society for Optical Engineering Conference in San Francisco. To understand why these discs can store so much information for such a long time, it's best to compare them to a regular CD. Data is read from a normal CD by shining a laser at a tiny line with bumps in it. Whenever the laser hits a bump. it's reflected back and recorded as a 1; whenever there's no bump, it's recorded as a 0. These are just two "dimensions" of information — on or off — but from them, CDs can store anything: music, books, images, videos, or software. But because this bumpy line is stored on the surface of the CD, it's vulnerable. It can be eroded either by physical scratches and scuffs, or by exposure to oxygen, heat, and humidity. 5D discs, by comparison, store information within their interior using tiny physical structures known as "nanogratings." Much like those bumpy lines in the CDs, these change how light is reflected, but instead of doing so in just two "dimensions," the reflected light encodes five — hence the name. The changes to the light can be read to obtain pieces of information about the nanograting's orientation, the strength of the light it refracts, and its location in space on the x, y, and z axes. These extra dimensions are why 5D discs can store data so densely compared to regular optical discs. A Blu-ray disc can hold up to 128GBs of data (the same as the biggest iPhone), while a 5D disc of the same size could store nearly 3,000 times that: 360 terabytes of information. These discs can potentially last for so long because glass is a tough material which needs a lot of heat to melt or warp it, and it's chemically stable too. (Think about all those science experiments that use glass beakers to contain reactive materials without anything bad happening to them.) This makes the 5D discs safe up to temperatures of 1,000°C, say the researchers. A new photograph of galaxy NGC 4889 may look peaceful from such a great distance, but it’s actually home to one of the biggest black holes that astronomers have ever identified. The Hubble Space Telescope allowed scientists to capture photos of the galaxy, located in the Coma Cluster about 300 million light-years away. The supermassive black hole hidden away in NGC 4889 breaks all kinds of records, even though it is currently classified as dormant. So how big is it, exactly? Well, according to our best estimates, the supermassive black hole is roughly 21 billion times the size of the Sun, and its event horizon (an area so dense and powerful that light can’t escape its gravity) measures 130 billion kilometers in diameter. That’s about 15 times the diameter of Neptune’s orbit around the Sun, according to scientists at the Hubble Space Telescope. At one point, the black hole was fueling itself on a process called hot accretion. Space stuff like gases, dust, and galactic debris fell towards the black hole and created an accretion disk. Then that spinning disk of space junk, accelerated by the strong gravitational pull of the largest known black hole, emitted huge jets of energy out into the galaxy. LIGO Discovers the Merger of Two Black Holes Big news: the Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected its first gravitational-wave signal! Not only is the detection of this signal a major technical accomplishment and an exciting confirmation of general relativity, but it also has huge implications for black-hole astrophysics. What did LIGO see? LIGO is designed to detect the ripples in space-time created by two massive objects orbiting each other. These waves can reach observable amplitudes when a binary system consisting of two especially massive objects — i.e., black holes or neutron stars — reach the end of their inspiral and merge. LIGO has been unsuccessfully searching for gravitational waves since its initial operations in 2002, but a recent upgrade in its design has significantly increased its sensitivity and observational range. The first official observing run of Advanced LIGO began 18 September 2015, but the instruments were up and running in “engineering mode” several weeks before that. And it was in this time frame — before official observing even began! — that LIGO spotted its first gravitational wave signal: GW150914. One of LIGO’s two detection sites, located near Hanford in eastern Washington. [LIGO] The signal, detected on 14 September, 2015, provides astronomers with a remarkable amount of information about the merger that caused it. From the detection, the LIGO team has extracted the masses of the two black holes that merged, 36+5-4 and 29+4-4 solar masses, as well as the mass of the final black hole formed by the merger, ~62 solar masses. The team also determined that the merger happened roughly a billion light-years away (at a redshift of z~0.1), and the direction of the signal was localized to an area of ~600 square degrees (roughly 1% of the sky). Why is this detection a big deal? This is the first direct detection of gravitational waves, providing spectacular further confirmation of Einstein’s general theory of relativity. But the implications of GW150914 go far beyond this confirmation. This detection is a huge deal for astrophysics because it’s the first direct evidence we’ve had that: “Heavy” stellar-mass black holes exist. We’ve reliably measured black holes of masses up to 10–20 solar masses in X-ray binaries (binary systems in which a single neutron star or black hole accretes matter from a donor star). But this is the first proof we’ve found that stellar-mass black holes of >25 solar masses can form in nature. Binaries consisting of two black holes can form in nature. As we’ll discuss shortly, there are two theorized mechanisms for the formation of these black-hole binaries. Until now, however, there was no guarantee that either of those mechanisms worked! These black-hole binaries can inspiral and merge within the age of the universe. The formation of a black-hole binary is no guarantee that it will merge on a reasonable timescale: if the binary forms with enough separation, it could take longer than the age of the universe to merge. This detection proves that black-hole binaries can form with small enough separation to merge on observable timescales. What can we learn from GW150914? One of the key questions we’d like to answer is: how do binary black holes form? Two primary mechanisms have been proposed: A binary star system contains two stars that are each massive enough to individually collapse into a black hole. If the binary isn’t disrupted during the two collapse events, this forms an isolated black-hole binary. Single black holes form in dense cluster environments and then — because they are the most massive objects — sink to the center of the cluster. There they form pairs through dynamical interactions. Now that we’re able to observe black-hole binaries through gravitational-wave detections, one way we could distinguish between the two formation mechanisms is from spin measurements. If we discover a clear preference for the misalignment of the two black holes’ spins, this would favor formation in clusters, where there’s no reason for the original spins to be aligned. The current, single detection is not enough to provide constraints, but if we can compile a large enough sample of events, we can start to present a statistical case favoring one channel over the other. What does GW150914 mean for the future of gravitational-wave detection? The fact that Advanced LIGO detected an event even before the start of its first official observing run is certainly promising! The LIGO team estimates that the volume the detectors can probe will still increase by at least a factor of ~10 as the observing runs become more sensitive and of longer duration. In addition, LIGO is not alone in the gravitational-wave game. LIGO’s counterpart in Europe, Virgo, is also undergoing design upgrades to increase its sensitivity. Within this year, Virgo should be able to take data simultaneously with LIGO, allowing for better localization of sources. And the launch of (e)LISA, ESA’s planned space-based interferometer, will grant us access to a new frequency range, opening a further window to the gravitational-wave sky. The detection of GW150914 marks the dawn of a new field: observational gravitational-wave astronomy. This detection alone confirms much that was purely theory before now — and given that instrument upgrades are still underway, the future of gravitational-wave detection looks incredibly promising. This awesome video (produced by SXS lensing) shows an actual simulation of the black-hole merger GW150914. Time is slowed by a factor of 100, compared to the actual merger. The two black holes — of 29 and 36 solar masses — warp the space-time around them, causing the distorted view. Einstein's gravitational waves 'seen' from black holes The international team says the first detection of these gravitational waves will usher in a new era for astronomy. It is the culmination of decades of searching and could ultimately offer a window on the Big Bang. The research, by the Ligo Collaboration, has been published today in the journal Physical Review Letters. The collaboration operates a number of labs around the world that fire lasers through long tunnels, trying to sense ripples in the fabric of space-time. The signals they detect are incredibly subtle and disturb the machines, known as interferometers, by just fractions of the width of an atom. But this black hole merger was picked up almost simultaneously by two widely separated Ligo facilities in the US. The merger radiated three times the mass of the sun in pure gravitational energy. "We have detected gravitational waves," Prof David Reitze, executive director of the Ligo project, told journalists at a news conference in Washington DC. "It's the first time the Universe has spoken to us through gravitational waves. Up until now, we've been deaf." Prof Karsten Danzmann, from the Max Planck Institute for Gravitational Physics and Leibniz University in Hannover, Germany, is a European leader on the collaboration. He said the detection was one of the most important developments in science since the discovery of the Higgs particle, and on a par with the determination of the structure of DNA. "There is a Nobel Prize in it - there is no doubt," he told the BBC. "It is the first ever direct detection of gravitational waves; it's the first ever direct detection of black holes and it is a confirmation of General Relativity because the property of these black holes agrees exactly with what Einstein predicted almost exactly 100 years ago." Gravitational waves are prediction of the Theory of General Relativity Their existence has been inferred by science but only now directly detected They are ripples in the fabric of space and time produced by violent events Accelerating masses will produce waves that propagate at the speed of light Detectable sources ought to include merging black holes and neutron stars Detecting the waves opens up the Universe to completely new investigations That view was reinforced by Prof Stephen Hawking, who is an expert on black holes. Speaking exclusively to BBC News, he said he believed that the detection marked a key moment in scientific history. "Apart from testing (Albert Einstein's theory of) General Relativity, we could hope to see black holes through the history of the Universe. We may even see relics of the very early Universe during the Big Bang at some of the most extreme energies possible." Team member Prof Gabriela González, from Louisiana State University, said: "We have discovered gravitational waves from the merger of black holes. It's been a very long road, but this is just the beginning. "Now that we have the detectors to see these systems, now that we know binary black holes are out there - we'll begin listening to the Universe." The Ligo laser interferometers in Hanford, in Washington, and Livingston, in Louisiana, were only recently refurbished and had just come back online when they sensed the signal from the collision. This occurred at 10.51 GMT on 14 September last year. On a graph, the data looks like a symmetrical, wiggly line that gradually increases in height and then suddenly fades away. "We found a beautiful signature of the merger of two black holes and it agrees exactly - fantastically - with the numerical solutions to Einstein equations... it looked too beautiful to be true," said Prof Danzmann. Prof Sheila Rowan, who is one of the lead UK researchers involved in the project, said that the first detection of gravitational waves was just the start of a "terrifically exciting" journey. "The fact that we are sitting here on Earth feeling the actual fabric of the Universe stretch and compress slightly due to the merger of black holes that occurred just over a billion years ago - I think that's phenomenal. It's amazing that when we first turned on our detectors, the Universe was ready and waiting to say 'hello'," the Glasgow University scientist told the BBC. Being able to detect gravitational waves enables astronomers finally to probe what they call "dark" Universe - the majority part of the cosmos that is invisible to the light telescopes in use today. "Gravitational waves go through everything. They are hardly affected by what they pass through, and that means that they are perfect messengers," said Prof Bernard Schutz, from Cardiff University, UK. "The information carried on the gravitational wave is exactly the same as when the system sent it out; and that is unusual in astronomy. We can't see light from whole regions of our own galaxy because of the dust that is in the way, and we can't see the early part of the Big Bang because the Universe was opaque to light earlier than a certain time. "With gravitational waves, we do expect eventually to see the Big Bang itself," he told the BBC. In addition, the study of gravitational waves may ultimately help scientists in their quest to solve some of the biggest problems in physics, such as the unification of forces, linking quantum theory with gravity. At the moment, General Relativity describes the cosmos on the largest scales tremendously well, but it is to quantum ideas that we resort when talking about the smallest interactions. Being able to study places in the Universe where gravity is really extreme, such as at black holes, may open a path to new, more complete thinking on these issues. The separate paths bounce back and forth between damped mirrors Eventually, the two light parts are recombined and sent to a detector Gravitational waves passing through the lab should disturb the set-up Theory holds they should very subtly stretch and squeeze its space. This ought to show itself as a change in the lengths of the light arms (green). The photodetector captures this signal in the recombined beam Scientists have sought experimental evidence for gravitational waves for more than 40 years. Einstein himself actually thought a detection might be beyond the reach of technology. His theory of General Relativity suggests that objects such as stars and planets can warp space around them - in the same way that a billiard ball creates a dip when placed on a thin, stretched, rubber sheet. Gravity is a consequence of that distortion - objects will be attracted to the warped space in the same way that a pea will fall in to the dip created by the billiard ball. Inspirational moment. Einstein predicted that if the gravity in an area was changed suddenly - by an exploding star, say - waves of gravitational energy would ripple across the Universe at light-speed, stretching and squeezing space as they travelled. Although a fantastically small effect, modern technology has now risen to the challenge. Much of the R&D work for the Washington and Louisiana machines was done at Europe's smaller GEO600 interferometer in Hannover. "I think it's phenomenal to be able to build an instrument capable of measuring [gravitational waves]," said Prof Rowan. "It is hugely exciting for a whole generation of young people coming along, because these kinds of observations and this real pushing back of the frontiers is really what inspires a lot of young people to get into science and engineering." Earth-like Planets Have Earth-like Interiors Physicists find signs of four-neutron nucleus The suspected discovery of an atomic nucleus with four neutrons but no protons has physicists scratching their heads. If confirmed by further experiments, this “tetraneutron” would be the first example of an uncharged nucleus, something that many theorists say should not exist. “It would be something of a sensation,” says Peter Schuck, a nuclear theorist at the National Center for Scientific Research in France who was not involved in the work. Details on the tetraneutron appear in the Feb. 5 Physical Review Letters. Have Gravitational Waves Finally Been Spotted? Astronomers may finally have found elusive gravitational waves, the mysterious ripples in the fabric of spacetime whose existence was first predicted by Albert Einstein in 1916, in his famous theory of general relativity. Scientists are holding a news conference Thursday (Feb. 11) at 10:30 a.m. EST (1530 GMT) at the National Press Club in Washington, D.C., to discuss the search for gravitational waves, which zoom through space at the speed of light. A media advisory describing the news conference is brief and somewhat vague, promising merely a "status report" on the ongoing hunt by the scientists using the Laser Interferometer Gravitational-Wave Observatory, or LIGO. But there's reason to suspect that researchers will announce a big discovery at the Thursday event. Astronomers build Earth-sized telescope to see Milky Way black hole An Earth-sized telescope will allow astronomers to glimpse the black hole at the centre of the Milky Way. Scientists across the globe are currently linking up telescopes across the globe to form the Event Horizon Telescope which will be the first instrument ever to take detailed pictures of a black hole. Even though the Milky Way’s black hole, known as Sagittarius A* (pronounced ‘Sagittarius A-star’), is four million times more massive than the sun, it is tiny to the eyes of astronomers. It is the equivalent of standing in New York and reading the date on a penny in Germany or seeing a grapefruit on the Moon for someone standing on Earth. But if successful, it will prove for the first time that black holes have ‘event horizons’ – an edge from which nothing can escape, not even light. "The goals of the EHT are to test Einstein's theory of general relativity, understand how black holes eat and generate relativistic outflows, and to prove the existence of the event horizon, or 'edge,' of a black hole," says Dan Marrone. The telescope gets its first major upgrade in centuries The general design of a telescope has remained more or less the same since the techology was first invented in the 17th century. Like an eye, the telescope collects light, and that light is then reflected to form an image. If you want to use one to see a really long way - into the depths of space, say - you'll need a really big one. “We can only scale the size and weight of telescopes so much before it becomes impractical to launch them into orbit and beyond,” says Danielle Wuchenich, senior research scientist at Lockheed Martin’s Advanced Technology Center in California. “Besides, the way our eye works is not the only way to process images from the world around us.” Lockheed Martin is now working on a new technology that promises to drastically reduce the size of telescope needed to see long distances. Its new system, SPIDER, (or 'Segmented Planar Imaging Detector for Electro-optical Reconnaissance', to give it its full title) does away with the large lenses or mirrors found in traditional refracting and reflecting telescopes, and replaces them with hundreds or thousands of tiny lenses. Dr Alan Duncan at Lockheed Martin explains: "SPIDER is a new way of collecting light to form images ... We collect the light, couple it into the silicon chip, move it around and combine it in a way that we can measure it with just ordinary detectors like you would have in your cellphone camera. And then (we) take all that data that's collected by those detectors, process it in a computer and form an image." Standard cosmology -- that is, the Big Bang Theory with its early period of exponential growth known as inflation -- is the prevailing scientific model for our universe It suggest that he entirety of space and time ballooned out from a very hot, very dense point into a homogeneous and ever-expanding vastness. This theory accounts for many of the physical phenomena we observe. But what if that's not all there was to it? A new theory from physicists at the U.S. Department of Energy's Brookhaven National Laboratory, Fermi National Accelerator Laboratory, and Stony Brook University, which will publish online on January 18 in Physical Review Letters, suggests a shorter secondary inflationary period that could account for the amount of dark matter estimated to exist throughout the cosmos. "In general, a fundamental theory of nature can explain certain phenomena, but it may not always end up giving you the right amount of dark matter," said Hooman Davoudiasl, group leader in the High-Energy Theory Group at Brookhaven National Laboratory and an author on the paper. "If you come up with too little dark matter, you can suggest another source, but having too much is a problem." Measuring the amount of dark matter in the universe is no easy task. It is dark after all, so it doesn't interact in any significant way with ordinary matter. Nonetheless, gravitational effects of dark matter give scientists a good idea of how much of it is out there. The best estimates indicate that it makes up about a quarter of the mass-energy budget of the universe, while ordinary matter -- which makes up the stars, our planet, and us -- comprises just 5 percent. Dark matter is the dominant form of substance in the universe, which leads physicists to devise theories and experiments to explore its properties and understand how it originated. Some theories that elegantly explain perplexing oddities in physics -- for example, the inordinate weakness of gravity compared to other fundamental interactions such as the electromagnetic, strong nuclear, and weak nuclear forces -- cannot be fully accepted because they predict more dark matter than empirical observations can support. In standard cosmology, the exponential expansion of the universe called cosmic inflation began perhaps as early as 10-35 seconds after the beginning of time -- that's a decimal point followed by 34 zeros before a 1. This explosive expansion of the entirety of space lasted mere fractions of a fraction of a second, eventually leading to a hot universe, followed by a cooling period that has continued until the present day. Then, when the universe was just seconds to minutes old -- that is, cool enough -- the formation of the lighter elements began. Between those milestones, there may have been other inflationary interludes, said Davoudiasl. "They wouldn't have been as grand or as violent as the initial one, but they could account for a dilution of dark matter," he said. "At this point, the abundance of dark matter is now baked in the cake," said Davoudiasl. "Remember, dark matter interacts very weakly. So, a significant annihilation rate cannot persist at lower temperatures. Self-annihilation of dark matter becomes inefficient quite early, and the amount of dark matter particles is frozen." "It's definitely not the standard cosmology, but you have to accept that the universe may not be governed by things in the standard way that we thought," he said. "But we didn't need to construct something complicated. We show how a simple model can achieve this short amount of inflation in the early universe and account for the amount of dark matter we believe is out there." "If this secondary inflationary period happened, it could be characterized by energies within the reach of experiments at accelerators such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider," he said. Only time will tell if signs of a hidden sector show up in collisions within these colliders, or in other experimental facilities. Stephen Hawking: Black Holes Have 'Hair Black holes may sport a luxurious head of "hair" made up of ghostly, zero-energy particles, says a new hypothesis proposed by Stephen Hawking and other physicists. The new paper, which was published online Jan. 5 in the preprint journal arXiv, proposes that at least some of the information devoured by a black hole is stored in these electric hairs. Still, the new proposal doesn't prove that all the information that enters a black hole is preserved. the Riemann Hypothesis has finally been SOLVED by a Nigerian professor Riemann Hypothesis is considered one of the hardest maths problem Devised in 1859, it has been resolved by professor Dr Opeyemi Enoch He has been given $1million (£658,000) for the work into prime numbers Riemann Hypothesis was one of the seven Millennium Problems in Mathematics set by the Clay Mathematics Institute in 2000 Read more: Physicists propose the first scheme to teleport the memory of an organism They also proposed a scheme to create a Schrodinger's cat state in which a microorganism can be in two places at the same time. This is an important step towards potentially teleporting an organism in future. Scientists struggle to stay grounded after possible gravitational wave signal Not for the first time, the world of physics is abuzz with rumours that gravitational waves have been detected by scientists in the US. — Lawrence M. Krauss (@LKrauss1) January 11, 2016"> Krauss said he was 60% confident that the rumour was true, but said he would have to see the scientists’ data before drawing any conclusions about whether the signal was genuine or not. Researchers on a large collaboration like Ligo will have any such paper internally vetted before sending it for publication and calling a press conference. In 2014, researchers on another US experiment, called BICEP2, called a press conference to announce the discovery of gravitational waves, but others have since pointed out that the signal could be due entirely to space dust. Speaking about the LIGO team, Krauss said: “They will be extremely cautious. There’s no reason for them to make a claim they are not certain of.” If gravitational waves have been discovered, astronomers could use them to observe the cosmos in a way that has been impossible to date. “We would have a new window on the universe,” Krauss said. “Gravitational waves are generated in the most exotic, strange locations in nature, such as at the edge of black holes at the beginning of time. We are pretty certain they exist, but we’ve not been able to use them to probe the universe.” NASA's Kepler Comes Roaring Back with 100 New Exoplanet Finds Physicists figure out how to retrieve information from a black hole Black holes earn their name because their gravity is so strong not even light can escape from them. Oddly, though, physicists have come up with a bit of theoretical sleight of hand to retrieve a speck of information that's been dropped into a black hole. The calculation touches on one of the biggest mysteries in physics: how all of the information trapped in a black hole leaks out as the black hole "evaporates." Many theorists think that must happen, but they don't know how. Unfortunately for them, the new scheme may do more to underscore the difficulty of the larger "black hole information problem" than to solve it. "Maybe others will be able to go further with this, but it's not obvious to me that it will help," says Don Page, a theorist at the University of Alberta in Edmonton, Canada, who was not involved in the work. You can shred your tax returns, but you shouldn't be able to destroy information by tossing it into a black hole. That's because, even though quantum mechanics deals in probabilities—such as the likelihood of an electron being in one location or another—the quantum waves that give those probabilities must still evolve predictably, so that if you know a wave's shape at one moment you can predict it exactly at any future time. Without such "unitarity" quantum theory would produce nonsensical results such as probabilities that don't add up to 100%. But suppose you toss some quantum particles into a black hole. At first blush, the particles and the information they encode is lost. That's a problem, as now part of the quantum state describing the combined black hole-particles system has been obliterated, making it impossible to predict its exact evolution and violating unitarity. Now, Aidan Chatwin-Davies, Adam Jermyn, and Sean Carroll of the California Institute of Technology in Pasadena have found an explicit way to retrieve information from one quantum particle lost in a black hole, using Hawking radiation and the weird concept of quantum teleportation. New study asks: Why didn't the universe collapse? he models that best describe the Big Bang and birth of the universe have one glaring problem. Most of them predict a collapse almost immediately after inflation. There was nothing, then there was something. And then there was nothing again. As we know from living and breathing and looking up at a sky action-packed with cosmic activity, there's definitely something more than nothing out there. So why is there still something? Why did the universe's tendency to expand overcome its tendency to collapse? A new study published in the Physical Review Letters is just the latest to try to inch closer to a place where physicists might be able to answer those questions. In this particular paper, researchers try to work out the details of the relationship between Higgs boson particles and gravity -- a relationship scientists believe kept an early, unstable universe from collapsing. Their latest calculations confirm that the stronger the bond between Higgs fields and gravity, the greater the chance of instability and a transition to a negative energy vacuum state, a place with little energy only a few particles popping in and out of existence. A coupling strength above one would have certainly spelled doom for the early universe, scientists at the University of Copenhagen determined. The new math helps narrow the likely coupling range to between 0.1 and 1. Physicists in Europe Find Tantalizing Hints of a Mysterious New Particl Two teams of physicists working independently at the Large Hadron Collider at CERN, the European Organization for Nuclear Research, reported on Tuesday that they had seen traces of what could be a new fundamental particle of nature. One possibility, out of a gaggle of wild and not-so-wild ideas springing to life's the day went on, is that the particle — assuming it is real — is a heavier version of the Higgs boson, a particle that explains why other particles have mass. Another is that it is a graviton, the supposed quantum carrier of gravity, whose discovery could imply the existence of extra dimensions of space-time. At the end of a long chain of “ifs” could be a revolution, the first clues to a theory of nature that goes beyond the so-called Standard Model, which has ruled physics for the last quarter-century. It is, however, far too soon to shout “whale ahoy,” physicists both inside and outside CERN said, noting that the history of particle physics is rife with statistical flukes and anomalies that disappeared when more data was compiled. A coincidence is the most probable explanation for the surprising bumps in data from the collider, physicists from the experiments cautioned, saying that a lot more data was needed and would in fact soon be available. “I don’t think there is anyone around who thinks this is conclusive,” said Kyle Cranmer, a physicist from New York University who works on one of the CERN teams, known as Atlas. “But it would be huge if true,” he said, noting that many theorists had put their other work aside to study the new result. German physicists see landmark in nuclear fusion quest Nuclear fusion entails fusing atoms together to generate energy -- a process similar to that in the Sun -- as opposed to nuclear fission, where atoms are split, which entails worries over safety and long-term waste. After spending a billion euros ($1.1 billion) and nine years' construction work, physicists working on a German project called the "stellarator" said they had briefly generated a super-heated helium plasma inside a vessel -- a key point in the experimental process. Scientists detect the magnetic field that powers our galaxy’s supermassive black hole The Milky Way, like most galaxies, has a supermassive black hole sitting right in its center. Now, for the first time, scientists have detected a magnetic field just outside the event horizon — or outer boundary — of that black hole. Why do we care? Because that magnetic field is probably what makes our neighborhood black hole so powerful. Controversial experiment sees no evidence that the universe is a hologram It's a classic underdog story: Working in a disused tunnel with a couple of lasers and a few mirrors, a plucky band of physicists dreamed up a way to test one of the wildest ideas in theoretical physics—a notion from the nearly inscrutable realm of "string theory" that our universe may be like an enormous hologram. However, science doesn't indulge sentimental favorites. After years of probing the fabric of spacetime for a signal of the "holographic principle," researchers at Fermi National Accelerator Laboratory (Fermilab) in Batavia, Illinois, have come up empty, as they will report tomorrow at the lab. The null result won't surprise many people, as some of the inventors of the principle had complained that the experiment, the $2.5 million Fermilab Holometer, couldn't test it. But Yanbei Chen, a theorist at the California Institute of Technology in Pasadena, says the experiment and its inventor, Fermilab theorist Craig Hogan, deserve some credit for trying. "At least he's making some effort to make an experimental test," Chen says. "I think we should do more of this, and if the string theorists complain that this is not testing what they're doing, well, they can come up with their own tests." The holographic principle springs from the theoretical study of black holes, spherical regions where gravity is so intense that not even light can escape. Theorists realized that a black hole has an amount of disorder, or entropy, that is proportional to its surface area. As entropy is related to information content, some theorists suggested that an information-area connection might be extended to any properly defined volume of space and time, or spacetime. Thus, crudely speaking, the maximum amount of information contained in a 3D region of space would be proportional its 2D surface area. The universe would then work a bit like a hologram, in which a 2D pattern captures a 3D image. How to encrypt a message in the afterglow of the big bang If you’ve got a secret to keep safe, look to the skies. Physicists have proposed using the afterglow of the big bang to make encryption keys. The security of many encryption methods relies on generating large random numbers to act as keys to encrypt or decipher information. Computers can spawn these keys with certain algorithms, but they aren’t truly random, so another computer armed with the same algorithm could potentially duplicate the key. An alternative is to rely on physical randomness, like the thermal noise on a chip or the timing of a user’s keystrokes. Now Jeffrey Lee and Gerald Cleaver at Baylor University in Waco, Texas, have taken that to the ultimate extreme by looking at the cosmic microwave background (CMB), the thermal radiation left over from the big bang. LISA Pathfinder Heads to Space A trailblazing mission took to the skies early this morning as a Vega rocket carrying the LISA Pathfinder lit up the night over Kourou, French Guiana. Originally known as the Small Missions for Advanced Research in Technology (SMART 2) and a forerunner to the full-fledged Evolved Laser Interferometer Space Antenna (eLISA) project, LISA Pathfinder will test the technologies key to conducting long-baseline laser interferometry in space. Coming almost exactly 100 years after Einstein proposed his theory of general relativity, this mission will prove vital in the hunt for one of the theory’s more bizarre predictions: gravitational waves. The equations of general relativity say that accelerating massive objects, such as exploding stars or a pair of whirling black holes, ought to send ripples through spacetime. There’s solid indirect evidence that gravitational waves exist, but direct detection has eluded scientists so far. LISA Pathfinder paves the way for eLISA, which will take that hunt into space. Slated for launch in 2034, eLISA will use three free-flying spacecraft to create a triangular baseline a million kilometers on a side — a feat impossible on Earth. Lasers will measure the position of two masses suspended at the end of each arm, and then researchers will analyze the data to look for the very slight jiggling induced by gravitational waves passing by. The unique setup and location will give eLISA an unprecedented sensitivity . Scientists Create New Kind Of Diamond At Room Temperature Researchers have created a new phase of solid carbon with qualities previously thought to be impossible that can be used to create diamonds at room temperature and the same atmospheric pressure as the ambient air. Scientists at North Carolina State University call it Q-carbon and say it is distinct from the other known solid forms of carbon – graphite and diamond. “The only place it may be found in the natural world would be possibly in the core of some planets,” says NC State’s Jay Narayan, lead author of three papers on the findings, including one published today in Journal of Applied Physics. Q-carbon is ferromagnetic, which he says was thought to be impossible,and is also harder than diamond and can glow when exposed to even a small amount of energy. Japanese scientists create touchable holograms A group of Japanese scientists have created touchable holograms, three dimensional virtual objects that can be manipulated by human hand. Using femtosecond laser technology the researchers developed 'Fairy Lights, a system that can fire high frequency laser pulses that last one millionth of one billionth of a second. The pulses respond to human touch, so that - when interrupted - the hologram's pixels can be manipulated in mid-air. Positrons Are Plentiful In Ultra-Intense Laser Blasts In a series of experiments described recently in the online journal Scientific Reports published by Nature, the researchers used UT’s Texas Petawatt Laser to make large number of positrons by blasting tiny gold and platinum targets. Although the positrons were annihilated in a fraction of a microsecond, the experiments have implications for new realms of physics and astrophysics research, medical therapy and perhaps even space travel, said Rice physicist Edison Liang, lead author of the study. “There are many futuristic technologies related to antimatter that people have been dreaming about for the last 50 years,” said Liang, the Andrew Hays Buchanan Professor of Astrophysics. “One is that antimatter is the most efficient form of energy storage. When antimatter annihilates with matter, it becomes pure energy. Nothing is left behind, unlike in fusion or fission or chemical-based reactions.” Scientists Link Moon’s Tilt and Earth’s Gold By now, with the moon a quarter million miles from Earth, the sun’s gravity should have tipped the moon’s orbit to lie in the same plane as the orbits of the planets. But it has not. The moon’s orbit is about 5 degrees askew. “That the lunar inclination is as small as it is gives us some confidence that the basic idea of lunar formation from an equatorial disk of debris orbiting the proto-Earth is a good one,” said Kaveh Pahlevan, a planetary scientist at the Observatory of the Côte d’Azur in Nice, France. “But the story must have a twist.” A Century Ago, Einstein’s Theory of Relativity Changed Everything Is Earth Growing a Hairy Dark Matter 'Beard'? Dark matter is thought to be everywhere, literally, but we can’t see it; we can only detect its gravitational presence over large cosmic scales. Now, theoretical physicists are theorizing what configuration the dark stuff may take around Earth. And it’s becoming a bit of a hairy subject. If we are to take the findings of a recent computer simulation to heart, it looks as if the planets in our solar system are growing rather trendy dark matter “beards,” an idea that not only reveals previously unknown interplanetary fashion trend, it could also provide a guide as to where to seek out direct evidence of the invisible matter that is thought to make up 85 percent of the mass of the entire universe. Scientists caught a new planet forming for the first time ever When a new star is born, it creates a disk full of gas and dust — the stuff of planetary formation. But it's hard to catch alien stars in the process of planetary baby-making, because the same dust that creates planets helps obscure these distant solar systems from our sight. We've found young planets and old ones alike, but none of them have actually been in the process of forming — until now. This new study was led by University of Arizona graduate student Stephanie Sallum and Kate Follette, a former fellow graduate student who has since moved on to postdoctoral research at Stanford University. The two women were working on separate PhD projects, but had decided to focus on the same star — LkCa15, located 450 light years from Earth. "The reason we selected this system is because it’s built around a very young star that has material left over from the star-formation process," Follette said in a statement. "It’s like a big doughnut. This system is special because it’s one of a handful of disks that has a solar-system size gap in it. And one of the ways to create that gap is to have planets forming in there." The women and their colleagues set high-powered telescopes to look at the system and used a new technique to look for protoplanets. They searched for the light emitted by hydrogen as the gas falls toward a newly forming planet. That process is hot — roughly 17,500 degrees Fahrenheit — and it produces a signature red glow. Experiment records extreme quantum weirdness An experiment in Singapore has pushed quantum weirdness close to its absolute limit. Researchers from the Centre for Quantum Technologies (CQT) at the National University of Singapore and the University of Seville in Spain have reported the most extreme 'entanglement' between pairs of photons ever seen in the lab. The result was published 30 October 2015 in Physical Review Letters. The achievement is evidence for the validity of quantum physics and will bolster confidence in schemes for quantum cryptography and quantum computing designed to exploit this phenomenon. "For some quantum technologies to work as we intend, we need to be confident that quantum physics is complete," says Poh Hou Shun, who carried out the experiment at CQT. "Our new result increases that confidence," he says. The researchers looked at 33.2 million optimized photon pairs. Each pair was split up and the photons measured separately, then the correlation between the results quantified. In such a Bell test, the strength of the correlation says whether or not the photons were entangled. The measures involved are complex, but can be reduced to a simple number. Any value bigger than 2 is evidence for quantum effects at work. But there is also an upper limit. Quantum physics predicts the correlation measure cannot get any bigger than 2sqrt(2) ~2.82843. In the experiment at CQT, they measure 2.82759 +/- 0.00051 - within 0.03% of the limit. If the peak value were the top of Everest, this would be only 2.6 metres below the summit. Scientists look into hydrogen atom, find old recipe for pi Published in 1655 by the English mathematician John Wallis, the Wallis product is an infinite series of fractions that, when multiplied, equal pi divided by 2. It has not appeared in physics at all until now, when University of Rochester scientists Carl Hagen and Tamar Friedmann collaborated on a problem set that Dr. Hagen had developed for his quantum mechanics class. Instead of using Niels Bohr’s near-century-old calculations for the energy states of hydrogen, Hagen had his students use a method called the variational principle, just to see what might happen. Ultimately, the calculations demanded mathematical expertise, which came in the form of Dr. Friedmann, who is both a mathematician and a physicist. “One of the things that I’m able to do is talk to both mathematicians and physicists, and that basically requires translating between two languages,” says Friedmann, who studied mathematics as an undergraduate student at Princeton, where she also earned a PhD. in Theoretical and Mathematical Physics. Friedmann says that asking new questions in math from physics and seeking to understand the physical systems from a mathematical standpoint enriched her understanding of problems in both disciplines. Friedmann tends to take on problems that might not even have an answer; she thinks that’s the type of approach that can lead to new discoveries. “And when they happen,” Friedmann says, “it’s really amazing.” Strong forces make antimatter stick Antimatter is a shadowy mirror image of the ordinary matter we are familiar with. For the first time, scientists have measured the forces that make certain antimatter particles stick together. The findings, published in Nature, may yield clues to what led to the scarcity of antimatter in the cosmos today. The forces between antimatter particles - in this case antiprotons - had not been measured before. If antiprotons were found to behave in a different way to their "mirror images" (the ordinary proton particles that are found in atoms) it might provide a potential explanation for what is known as "matter/antimatter asymmetry". News Archives
d7839ea37cddf558	Viewpoint: Light Bends Itself into an Arc Zhigang Chen, Department of Physics and Astronomy, San Francisco State University, San Francisco, CA 94132, USA Published April 16, 2012 \| Physics 5, 44 (2012) \| DOI: 10.1103/Physics.5.44 Nondiffracting Accelerating Wave Packets of Maxwell’s Equations Ido Kaminer, Rivka Bekenstein, Jonathan Nemirovsky, and Mordechai Segev Published April 16, 2012 \| PDF (free) +Enlarge image Figure 1 (Left) Ref. [1]. (Right) Courtesy D. N. Christodoulides Figure 1 Kaminer et al. showed that shape-preserving beams of light that travel along a circular trajectory emerge as solutions to Maxwell’s equations. (Left) Calculated propagation of a self-bending beam. This solution assumes the wave’s electric field is polarized in the transverse direction (TE polarization). (Right) Illustration of a nondiffracting beam bending around an obstacle. Apart from the broadening effects of diffraction, light beams tend to propagate along a straight path. Mirrors, lenses, and light guides are all ways to force light to take a more circuitous path, but an alternative that many researchers are exploring is to prepare light beams that can bend themselves along a curved path, even in vacuum. In a paper in Physical Review Letters, Ido Kaminer and colleagues at Technion, Israel, report on wave solutions to Maxwell’s equations that are both nondiffracting and capable of following a much tighter circular trajectory than was previously thought possible [1]. Apart from fundamental scientific interest, such wave solutions may lead to the possibility of synthesizing shape-preserving optical beams that make curved left- or right-handed turns on their own. The equations describing these light waves could also be generalized to describe similar behavior in sound and water waves. The idea for making specially shaped light waves that could bend without dispersion actually emerged from quantum physics. In 1979, Berry and Balazs realized that the force-free Schrödinger equation could give rise to solutions in the form of nonspreading “Airy” wave packets [2] that freely accelerate even in the absence of any external potential. This early work remained dormant in the literature for decades, until Christodoulides and co-workers demonstrated the optical analog of Airy wave packets: specially shaped beams of light that did not diffract over long distances but could bend (or, self-accelerate) sideways [3] (see 28 November 2007 Focus story ). Such self-accelerating Airy beams have since attracted a great deal of interest due to their unique properties, and they provide the basis for a number of proposed applications including optical micromanipulation [4], plasma guidance and light bullet generation [5], and routing surface plasmon polaritons [6] (see 6 September 2011 Viewpoint ). A typical simplification when solving Maxwell’s wave equations is to assume the light waves are paraxial, meaning the angle between the wave vectors that constitute a wavepacket and the optical axis is small enough that the wave does not deviate too much from its propagation direction. Under this paraxial approximation, the resulting time-independent scalar Helmholtz equation takes the same form of the Schrödinger equation, and it was this relationship that led to the proposal that finite-power optical Airy beams could be attainable in experiments [3]. (Unlike other nondiffracting beams such as the well-known Bessel beams [7], Airy beams have a unique spatial phase structure, do not rely on simple conical superposition of plane waves, and can self-accelerate.) At small angles, Airy beams follow parabolic trajectories similar to ballistic projectiles moving under the force of gravity [8], but at large angles, beyond the paraxial approximation, they cannot maintain their shape-preserving property as they propagate. Therefore, it is important to identify mechanisms that could allow self-accelerating beams to propagate in a true diffraction-free manner even for large trajectory bending. Several studies have searched for accelerating beams beyond the paraxial regime. In one study [9], nonparaxial Airy beams were sought as exact solutions of Maxwell’s wave equation, but these beams tend to break up and decay as they propagate because parts of them consist of evanescent waves. In yet another study [10], the so-called caustic method was used to effectively stretch the paraxial Airy beams to the nonparaxial regime, but these “caustic-designed” accelerating beams don’t preserve their shape like nondiffracting paraxial Airy beams. Thus, a natural question arose: Could a beam accelerate at large nonparaxial angles but still hold its shape? Since beam propagation is governed by Maxwell’s equations, it was equivalent to asking: Were there any solutions to Maxwell’s equations that allowed nondiffracting, self-accelerating beams? In their new work, Kaminer et al. report they have found shape-preserving nonparaxial accelerating beams (NABs) as a complete set of general solutions to the full Maxwell’s equations. Differently from the paraxial Airy beams that accelerate along a parabolic trajectory, these nonparaxial beams accelerate in a circular trajectory. To find the solutions for the NABs, Kaminer et al. started with the scalar Maxwell’s wave equation for a given polarization, such as TE (transverse electric) polarization, where the electric field is perpendicular to the direction of the wave. Since the equation exhibits full symmetry between the x and z coordinates, the solutions of shape-preserving beams must have circular symmetry. The authors therefore transformed the equation into polar coordinates and looked for shape-preserving solutions where the field amplitude didn’t vary with angle. In polar coordinates, the solution to the wave equation is a Bessel function; transforming back to Cartesian coordinates, the solution must be separated into forward and backward propagating waves in Fourier space. However, only the forward propagating part forms the desired accelerating beam, so the Kaminer et al. solutions are properly called “half Bessel wave packets.” The authors found a solution for TM (transverse magnetic) polarization through a similar procedure. For both TE and TM polarizations, the beams preserve their shape while the quarter-circle bending could occur after a propagation distance of just 35μm. In addition, the authors studied the properties of these Bessel-like accelerating beams and found that the Poynting vector of the main lobe can turn by more than 90 [1]. The left part of Fig. 1 shows a typical solution of the shape-preserving beam, which can sweep out a quarter circle. It is important to note that this one-dimensional beam propagates initially along the longitudinal z direction while its curved trajectory in the x-z plane is determined by a Bessel function. This is quite different from the traditional nondiffracting Bessel beam, which propagates in a straight line while its two-dimensional transverse pattern follows a Bessel function [7]. As the authors point out, the nonparaxial shape-preserving accelerating beams found in their work originates from the full vector solutions of Maxwell’s equation. Moreover, in their scalar form, these beams are the exact solutions for nondispersive accelerating wave packets of the most common wave equation describing time-harmonic waves. As such, this work has profound implications for other linear wave systems in nature, ranging from sound waves and surface waves in fluids to many kinds of classical waves. Furthermore, based on previous successful demonstrations of self-accelerating Airy beams [3–6], one would expect that the nonparaxial Bessel-like accelerating beams proposed in this study could be readily realized in experiment. Apart from many exciting opportunities for these beams in various applications, such as beams that self-bend around an obstacle (Fig. 1, right), one might expect one day light could really travel around a circle by itself, bringing the search for an “optical boomerang” into reality. 1. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012). 2. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979). 3. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007); G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007). 4. J. Baumgartl, M. Mazilu, and K. Dholakia, Nature Photon. 2, 675 (2008). 5. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324,229 (2009); A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nature Photon. 4,103 (2010). 6. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, Opt. Lett. 36,3191 (2011); A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107,116802 (2011); L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, 107,126804 (2011). 7. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58,1499 (1987). 8. G. A. Siviloglou J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33,207 (2008); Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, 35,2260 (2010). 9. A. V. Novitsky and D. V. Novitsky, Opt. Lett. 34, 3430 (2009). 10. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, Opt. Express 19,16455 (2011). About the Author: Zhigang Chen Zhigang Chen Zhigang Chen received his Ph.D. in physics from Bryn Mawr College in 1995. He was a postdoctoral research associate and then a senior research staff member at Princeton University. He has been on the faculty in the Department of Physics and Astronomy at San Francisco State University since 1998. Subject Areas New in Physics
7d3770ce300c6590	Electron : definition of Electron and synonyms of Electron (English) definitions - Electron electron (n.) 1.an elementary particle with negative charge Advertizing ▼ Merriam Webster 1. Amber; also, the alloy of gold and silver, called electrum. [archaic] Advertizing ▼ definition (more) definition of Wikipedia synonyms - Electron electron (n.) atom, negatron, particle see also - Electron electron (n.) -Bacterial Electron Transport Chain Complex Proteins • Bacterial Electron Transport Complex I • Bacterial Electron Transport Complex II • Bacterial Electron Transport Complex III • Bacterial Electron Transport Complex IV • Cryo-electron Microscopy • Electron Beam Computed Tomography • Electron Beam Tomography • Electron Cryomicroscopy • Electron Diffraction Microscopy • Electron Energy-Loss Spectroscopy • Electron Microscopy • Electron Microscopy, Scanning Transmission • Electron Microscopy, Transmission • Electron Nuclear Double Resonance • Electron Paramagnetic Resonance • Electron Probe Microanalysis • Electron Scanning Microscopy • Electron Spectroscopic Imaging • Electron Spin Resonance • Electron Spin Resonance Spectroscopy • Electron Transfer Flavoprotein • Electron Transfer Flavoprotein Alpha Subunit Deficiency • Electron Transfer Flavoprotein Beta Subunit Deficiency • Electron Transfer Flavoprotein Deficiency • Electron Transfer Flavoprotein Dehydrogenase Deficiency • Electron Transport • Electron Transport Chain Complex Proteins • Electron Transport Chain Deficiencies, Mitochondrial • Electron Transport Complex I • Electron Transport Complex II • Electron Transport Complex III • Electron Transport Complex IV • Electron-Transferring Flavoproteins • Microanalysis, Electron Probe • Microscopy, Electron • Microscopy, Electron Diffraction • Microscopy, Electron, Scanning • Microscopy, Electron, Scanning Transmission • Microscopy, Electron, Transmission • Microscopy, Electron, X-Ray Microanalysis • Mitochondrial Electron Transport Chain Complex Proteins • Mitochondrial Electron Transport Chain Deficiencies • Mitochondrial Electron Transport Complex I • Mitochondrial Electron Transport Complex II • Mitochondrial Electron Transport Complex III • Mitochondrial Electron Transport Complex IV • Scanning Electron Microscopy • Spectroscopy, Electron Energy-Loss • Transmission Electron Microscopy • X-Ray Microanalysis, Electron Microscopic • X-Ray Microanalysis, Electron Probe • electron accelerator • electron beam • electron dot structure • electron gun • electron lens • electron microscope • electron microscopic • electron microscopy • electron multiplier • electron multiplier tube • electron optics • electron orbit • electron paramagnetic resonance • electron radiation • electron shell • electron spin resonance • electron tube • electron volt • free electron • unbound electron • valence electron analogical dictionary MESH root[Thème] electron [MeSH] lepton (particule) (fr)[Classe] atome (fr)[DomainDescrip.] electron (n.) A glass tube containing a glowing green electron beam Composition Elementary particle[2] Statistics Fermionic Generation First Interactions Gravity, Electromagnetic, Weak Symbol e , β Antiparticle Positron (also called antielectron) Theorized Richard Laming (1838–1851),[3] G. Johnstone Stoney (1874) and others.[4][5] Discovered J. J. Thomson (1897)[6] Mass 9.10938291(40)×10−31 kg[7] 5.4857990946(22)×10−4 u[7] [1,822.8884845(14)]−1 u[note 1] 0.510998928(11) MeV/c2[7] Electric charge −1 e[note 2] −1.602176565(35)×10−19 C[7] −4.80320451(10)×10−10 esu Magnetic moment −1.00115965218076(27) μB[7] Spin 12 The electron (symbol: e ) is a subatomic particle with a negative elementary electric charge.[8] It has no known components or substructure; in other words, it is generally thought to be an elementary particle.[2] An electron has a mass that is approximately 1/1836 that of the proton.[9] The intrinsic angular momentum (spin) of the electron is a half-integer value in units of ħ, which means that it is a fermion. The antiparticle of the electron is called the positron; it is identical to the electron except that it carries electrical and other charges of the opposite sign. When an electron collides with a positron, both particles may be totally annihilated, producing gamma ray photons. Electrons, which belong to the first generation of the lepton particle family,[10] participate in gravitational, electromagnetic and weak interactions.[11] Electrons, like all matter, have quantum mechanical properties of both particles and waves, so they can collide with other particles and can be diffracted like light. However, this duality is best demonstrated in experiments with electrons, due to their tiny mass. Since an electron is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle.[10] In 1894, Stoney coined the term electron to describe these elementary charges, saying, "... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron".[21] The word electron is a combination of the word electric and the suffix -on, with the latter now used to designate a subatomic particle, such as a proton or neutron.[22][23] A round glass vacuum tube with a glowing circular beam inside A beam of electrons deflected in a circle by a magnetic field[24] The German physicist Johann Wilhelm Hittorf undertook the study of electrical conductivity in rarefied gases. In 1869, he discovered a glow emitted from the cathode that increased in size with decrease in gas pressure. In 1876, the German physicist Eugen Goldstein showed that the rays from this glow cast a shadow, and he dubbed the rays cathode rays.[25] During the 1870s, the English chemist and physicist Sir William Crookes developed the first cathode ray tube to have a high vacuum inside.[26] He then showed that the luminescence rays appearing within the tube carried energy and moved from the cathode to the anode. Furthermore, by applying a magnetic field, he was able to deflect the rays, thereby demonstrating that the beam behaved as though it were negatively charged.[27][28] In 1879, he proposed that these properties could be explained by what he termed 'radiant matter'. He suggested that this was a fourth state of matter, consisting of negatively charged molecules that were being projected with high velocity from the cathode.[29] In 1896, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson,[12] performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.[6] Thomson made good estimates of both the charge e and the mass m, finding that cathode ray particles, which he called "corpuscles," had perhaps one thousandth of the mass of the least massive ion known: hydrogen.[6][13] He showed that their charge to mass ratio, e/m, was independent of cathode material. He further showed that the negatively charged particles produced by radioactive materials, by heated materials and by illuminated materials were universal.[6][31] The name electron was again proposed for these particles by the Irish physicist George F. Fitzgerald, and the name has since gained universal acceptance.[27] Robert Millikan While studying naturally fluorescing minerals in 1896, the French physicist Henri Becquerel discovered that they emitted radiation without any exposure to an external energy source. These radioactive materials became the subject of much interest by scientists, including the New Zealand physicist Ernest Rutherford who discovered they emitted particles. He designated these particles alpha and beta, on the basis of their ability to penetrate matter.[32] In 1900, Becquerel showed that the beta rays emitted by radium could be deflected by an electric field, and that their mass-to-charge ratio was the same as for cathode rays.[33] This evidence strengthened the view that electrons existed as components of atoms.[34][35] The electron's charge was more carefully measured by the American physicists Robert Millikan and Harvey Fletcher in their oil-drop experiment of 1909, the results of which were published in 1911. This experiment used an electric field to prevent a charged droplet of oil from falling as a result of gravity. This device could measure the electric charge from as few as 1–150 ions with an error margin of less than 0.3%. Comparable experiments had been done earlier by Thomson's team,[6] using clouds of charged water droplets generated by electrolysis,[12] and in 1911 by Abram Ioffe, who independently obtained the same result as Millikan using charged microparticles of metals, then published his results in 1913.[36] However, oil drops were more stable than water drops because of their slower evaporation rate, and thus more suited to precise experimentation over longer periods of time.[37] Around the beginning of the twentieth century, it was found that under certain conditions a fast moving charged particle caused a condensation of supersaturated water vapor along its path. In 1911, Charles Wilson used this principle to devise his cloud chamber, allowing the tracks of charged particles, such as quickly-moving electrons, to be photographed.[38] Atomic theory By 1914, experiments by physicists Ernest Rutherford, Henry Moseley, James Franck and Gustav Hertz had largely established the structure of an atom as a dense nucleus of positive charge surrounded by lower-mass electrons.[39] In 1913, Danish physicist Niels Bohr postulated that electrons resided in quantized energy states, with the energy determined by the angular momentum of the electron's orbits about the nucleus. The electrons could move between these states, or orbits, by the emission or absorption of photons at specific frequencies. By means of these quantized orbits, he accurately explained the spectral lines of the hydrogen atom.[40] However, Bohr's model failed to account for the relative intensities of the spectral lines and it was unsuccessful in explaining the spectra of more complex atoms.[39] Chemical bonds between atoms were explained by Gilbert Newton Lewis, who in 1916 proposed that a covalent bond between two atoms is maintained by a pair of electrons shared between them.[41] Later, in 1923, Walter Heitler and Fritz London gave the full explanation of the electron-pair formation and chemical bonding in terms of quantum mechanics.[42] In 1919, the American chemist Irving Langmuir elaborated on the Lewis' static model of the atom and suggested that all electrons were distributed in successive "concentric (nearly) spherical shells, all of equal thickness".[43] The shells were, in turn, divided by him in a number of cells each containing one pair of electrons. With this model Langmuir was able to qualitatively explain the chemical properties of all elements in the periodic table,[42] which were known to largely repeat themselves according to the periodic law.[44] Quantum mechanics In his 1924 dissertation Recherches sur la théorie des quanta (Research on Quantum Theory), French physicist Louis de Broglie hypothesized that all matter possesses a De Broglie wave similar to light.[48] That is, under the appropriate conditions, electrons and other matter would show properties of either particles or waves. The corpuscular properties of a particle are demonstrated when it is shown to have a localized position in space along its trajectory at any given moment.[49] Wave-like nature is observed, for example, when a beam of light is passed through parallel slits and creates interference patterns. In 1927, the interference effect was demonstrated with a beam of electrons by English physicist George Paget Thomson with a thin metal film and by American physicists Clinton Davisson and Lester Germer using a crystal of nickel.[50] A symmetrical blue cloud that decreases in intensity from the center outward The success of de Broglie's prediction led to the publication, by Erwin Schrödinger in 1926, of the Schrödinger equation that successfully describes how electron waves propagated.[51] Rather than yielding a solution that determines the location of an electron over time, this wave equation can be used to predict the probability of finding an electron near a position. This approach was later called quantum mechanics, which provided an extremely close derivation to the energy states of an electron in a hydrogen atom.[52] Once spin and the interaction between multiple electrons were considered, quantum mechanics allowed the configuration of electrons in atoms with higher atomic numbers than hydrogen to be successfully predicted.[53] In 1928, building on Wolfgang Pauli's work, Paul Dirac produced a model of the electron - the Dirac equation, consistent with relativity theory, by applying relativistic and symmetry considerations to the hamiltonian formulation of the quantum mechanics of the electro-magnetic field.[54] In order to resolve some problems within his relativistic equation, in 1930 Dirac developed a model of the vacuum as an infinite sea of particles having negative energy, which was dubbed the Dirac sea. This led him to predict the existence of a positron, the antimatter counterpart of the electron.[55] This particle was discovered in 1932 by Carl D. Anderson, who proposed calling standard electrons negatrons, and using electron as a generic term to describe both the positively and negatively charged variants. Particle accelerators With a beam energy of 1.5 GeV, the first high-energy particle collider was ADONE, which began operations in 1968.[59] This device accelerated electrons and positrons in opposite directions, effectively doubling the energy of their collision when compared to striking a static target with an electron.[60] The Large Electron-Positron Collider (LEP) at CERN, which was operational from 1989 to 2000, achieved collision energies of 209 GeV and made important measurements for the Standard Model of particle physics.[61][62] Fundamental properties The electron has no known substructure.[2][71] Hence, it is defined or assumed to be a point particle with a point charge and no spatial extent.[10] Observation of a single electron in a Penning trap shows the upper limit of the particle's radius is 10−22 meters.[72] There is a physical constant called the "classical electron radius", with the much larger value of 2.8179×10−15 m. However, the terminology comes from a simplistic calculation that ignores the effects of quantum mechanics; in reality, the so-called classical electron radius has little to do with the true fundamental structure of the electron.[73][note 5] There are elementary particles that spontaneously decay into less massive particles. An example is the muon, which decays into an electron, a neutrino and an antineutrino, with a mean lifetime of 2.2×10−6 seconds. However, the electron is thought to be stable on theoretical grounds: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation.[74] The experimental lower bound for the electron's mean lifetime is 4.6×1026 years, at a 90% confidence level.[75] Quantum properties Virtual particles While an electron–positron virtual pair is in existence, the coulomb force from the ambient electric field surrounding an electron causes a created positron to be attracted to the original electron, while a created electron experiences a repulsion. This causes what is called vacuum polarization. In effect, the vacuum behaves like a medium having a dielectric permittivity more than unity. Thus the effective charge of an electron is actually smaller than its true value, and the charge decreases with increasing distance from the electron.[79][80] This polarization was confirmed experimentally in 1997 using the Japanese TRISTAN particle accelerator.[81] Virtual particles cause a comparable shielding effect for the mass of the electron.[82] The interaction with virtual particles also explains the small (about 0.1%) deviation of the intrinsic magnetic moment of the electron from the Bohr magneton (the anomalous magnetic moment).[69][83] The extraordinarily precise agreement of this predicted difference with the experimentally determined value is viewed as one of the great achievements of quantum electrodynamics.[84] In classical physics, the angular momentum and magnetic moment of an object depend upon its physical dimensions. Hence, the concept of a dimensionless electron possessing these properties might seem inconsistent. The apparent paradox can be explained by the formation of virtual photons in the electric field generated by the electron. These photons cause the electron to shift about in a jittery fashion (known as zitterbewegung),[85] which results in a net circular motion with precession. This motion produces both the spin and the magnetic moment of the electron.[10][86] In atoms, this creation of virtual photons explains the Lamb shift observed in spectral lines.[79] An electron generates an electric field that exerts an attractive force on a particle with a positive charge, such as the proton, and a repulsive force on a particle with a negative charge. The strength of this force is determined by Coulomb's inverse square law.[87] When an electron is in motion, it generates a magnetic field.[88] The Ampère-Maxwell law relates the magnetic field to the mass motion of electrons (the current) with respect to an observer. It is this property of induction which supplies the magnetic field that drives an electric motor.[89] The electromagnetic field of an arbitrary moving charged particle is expressed by the Liénard–Wiechert potentials, which are valid even when the particle's speed is close to that of light (relativistic). A graph with arcs showing the motion of charged particles Atoms and molecules Four bolts of lightning strike the ground Electrons inside conducting solids, which are quasi-particles themselves, when tightly confined at temperatures close to absolute zero, behave as though they had split into two other quasiparticles: spinons and holons.[122][123] The former carries spin and magnetic moment, while the latter electrical charge. Motion and energy The plot starts at zero and curves sharply upward toward the right \displaystyle K_{\mathrm{e}} = (\gamma - 1)m_{\mathrm{e}} c^2, γ + γe+ + e np + e + ν A branching tree representing the particle production + ν + ν + ν Plasma applications Particle beams The electron microscope directs a focused beam of electrons at a specimen. As the beam interacts with the material, some electrons change their properties, such as movement direction, angle, relative phase and energy. By recording these changes in the electron beam, microscopists can produce atomically resolved image of the material.[165] In blue light, conventional optical microscopes have a diffraction-limited resolution of about 200 nm.[166] By comparison, electron microscopes are limited by the de Broglie wavelength of the electron. This wavelength, for example, is equal to 0.0037 nm for electrons accelerated across a 100,000-volt potential.[167] The Transmission Electron Aberration-corrected Microscope is capable of sub-0.05 nm resolution, which is more than enough to resolve individual atoms.[168] This capability makes the electron microscope a useful laboratory instrument for high resolution imaging. However, electron microscopes are expensive instruments that are costly to maintain. Other applications Electrons are at the heart of cathode ray tubes, which have been used extensively as display devices in laboratory instruments, computer monitors and television sets.[173] In a photomultiplier tube, every photon striking the photocathode initiates an avalanche of electrons that produces a detectable current pulse.[174] Vacuum tubes use the flow of electrons to manipulate electrical signals, and they played a critical role in the development of electronics technology. 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External links All translations of Electron sensagent's content • definitions • synonyms • antonyms • encyclopedia Dictionary and translator for handheld ⇨ New : sensagent is now available on your handheld Advertising ▼ sensagent's office Shortkey or widget. Free. Windows Shortkey: sensagent. Free. Vista Widget : sensagent. Free. Webmaster Solution Try here or get the code Business solution Improve your site content Add new content to your site from Sensagent by XML. Crawl products or adds Get XML access to reach the best products. Index images and define metadata Get XML access to fix the meaning of your metadata. Please, email us to describe your idea. The English word games are: ○ Anagrams ○ Wildcard, crossword ○ Lettris ○ Boggle. English dictionary Main references Most English definitions are provided by WordNet . English Encyclopedia is licensed by Wikipedia (GNU). The SensagentBox are offered by sensAgent. 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22cd150a4a8f9d4c	« · » Section 8.2: The Quantum-mechanical Free-particle Solution Please wait for the animation to completely load. In order to tackle the free-particle problem, we begin with the Schrödinger equation in one dimension with V(x) = 0, [−(ħ2/2m)∂2Ψ(x,t)/∂x2 = (∂/∂t)Ψ(x,t) . (8.2) We can simplify the analysis somewhat by performing a separation of variables and therefore considering the time-independent Schrödinger equation: −(ħ2/2m)(d2/dx2) ψ(x) = E ψ(x) , (8.3) which we can rewrite as: [(d2/dx2) + k2] ψ(x) = 0 , (8.4) where k2 ≡ 2mE/ħ2. We find the solutions to the equation above are of the form ψ(x) = Aexp(ikx) where we allow k to take both positive and negative values.2 Unlike a bound-state problem, such as the infinite square well, there are no boundary conditions to restrict the k and therefore E values. However, each plane wave has a definite k value and therefore a definite momentum (and also a definite energy) since pψ(x) = −(d/dx) ψ(x) = ħkψ(x), again with k taking on both positive and negative values (so that pψ(x) = ± ħ\|k\|ψ(x)). The time dependence is now straightforward from the Schrödinger equation: (∂/∂t) Ψ(x,t) = EΨ(x,t) , (8.6) or by acting the time-evolution operator, UT(t) = eiHt/ħ, on ψ(x, t). Both procedures yield ψ(x,t) = AeikxiEt/ħ (again k can take positive and negative values) and since E = p2/2m = ħ2k2/2m, we also have that ψ(k > 0)(x,t) = Aexp(ikxiħk2t/2m) or ψ(k < 0)(x,t) = Aexp(-i\|k\|xiħk2t/2m), (8.7) where ħk2/2m ≡ ω. These solutions describe both right-moving (k > 0) and left-moving (k < 0) plane waves. Recall that solutions to the classical wave equation are in the form of f(kx −/+ ωt) for a wave moving to the right (−) or left (+). These quantum-mechanical plane waves, however, are complex functions and can be written in the form f(±\|k\|x − ωt). RestartIn the animation, ħ = 2m = 1. A right-moving plane wave is represented in terms of its amplitude and phase (as color) and also its real, cos(kx - ħk2t/2m), and imaginary, sin(kx ħk2t/2m), parts. What is the velocity of this wave? If this were a classical free particle with non-relativistic velocity, E = mv2/2 = p2/2m and vclassical = p/m as expected. But what about our solution? The velocity of our wave is ω/k which gives: ħk/2m = p/2m, half of the expected (classical) velocity! This velocity is the phase velocity. If instead we consider the group velocity, vg = ∂ω/∂k, (8.8) )we find that vg = ∂(ħk2/2m)/∂k = ħk/m, (8.9) the expected (classical) velocity. Consider the right-moving wave, ψ(x,t) = Aexp(ikxiħk2t/2m) , which has a definite momentum, p = ħk. We notice that the amplitude of the wave function, A, is a finite constant over all space. However, we also find that ∫ ψψ dx = ∞ [integral from −∞ to +∞] even though ψψ = \|A\|2 is finite. While the plane wave is a definite-momentum solution to the Schrödinger equation, it is not a localized solution. In this case then, we must discard, or somehow modify, these solutions if we wish a localized and normalized free-particle description. 2The most general solution to the differential equation is ψ(x) = Aeikx + Beikx (8.5) with k values positive. 3We can however, box normalize the wave function. In box normalization, we normalize the wave function such that over a finite region of space the wave function is normalized. The OSP Network: Open Source Physics - Tracker - EJS Modeling Physlet Physics Physlet Quantum Physics
01069018e05611c8	JETP Letters , 94:243 \| Cite as On a two-dimensional Schrödinger equation with a magnetic field with an additional quadratic integral of motion • V. G. MarikhinEmail author The problem of commuting quadratic quantum operators with a magnetic field has been considered. It has been shown that any such pair can be reduced to the canonical form, which makes it possible to construct an almost complete classification of the solutions of equations that are necessary and sufficient for a pair of operators to commute with each other. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables; this change is similar to that in the theory of integrable tops. As an example, this procedure has been considered for the two-dimensional Schrödinger equation with the magnetic field; this equation has an additional quantum integral of motion. Magnetic Field Canonical Form JETP Letter Classical Case Static Magnetic Field 1. 1. E. V. Ferapontov and A. P. Veselov, J. Math. Phys. 42, 590 (2001).MathSciNetADSzbMATHCrossRefGoogle Scholar 2. 2. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Dokl. Akad. Nauk SSSR 229, 15 (1976).MathSciNetGoogle Scholar 3. 3. S. P. Novikov and A. P. Veselov, Am. Math. Soc. Transl. 179, 109 (1997).MathSciNetGoogle Scholar 4. 4. J. Berube and P. Winternitz, J. Math. Phys. 45, 1959 (2004).MathSciNetADSzbMATHCrossRefGoogle Scholar 5. 5. V. G. Marikhin and V. V. Sokolov, Theor. Math. Phys. 149, 1425 (2006).MathSciNetzbMATHCrossRefGoogle Scholar 6. 6. L. P. Eisenhart, Ann. Math. 35, 284 (1934).MathSciNetCrossRefGoogle Scholar 7. 7. V. G. Marikhin and V. V. Sokolov, Reg. Chaot. Dynamics 10, 59 (2005).MathSciNetzbMATHCrossRefADSGoogle Scholar Copyright information © Pleiades Publishing, Ltd. 2011 Authors and Affiliations 1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow regionRussia Personalised recommendations
f46492aa399a7fe4	First published on Sentient Developments Date: May 2009 May 1, 2009 David Pearce is guest blogging this week A World Without Suffering? the lion lies down with the lamb In November 2005, at the Society for Neuroscience Congress, the Dalai Lama observed: "If it was possible to become free of negative emotions by a riskless implementation of an electrode - without impairing intelligence and the critical mind - I would be the first patient." Smart neurostimulation, long-acting mood-enhancers, genetically re-engineering our hedonic "set-point" (etc) aren't therapeutic strategies associated with Buddhist tradition. Yet if we are morally serious about securing the well-being of all sentient life, then we have to exploit advanced technology to the fullest possible extent. Nothing else will work (short of some exotic metaphysics that is hard to reconcile with the scientific world-picture). Non-biological strategies to enrich psychological well-being have been tried on a personal level over thousands of years - and proved inadequate at best. This is because they don't subvert the brutally efficient negative feedback mechanisms of the hedonic treadmill - a legacy of millions of years of natural selection. Nor is the well-being of all sentient life feasible in a Darwinian ecosystem where the welfare of some creatures depends on eating or exploiting others. The lion can lie down with the lamb; but only after both have been genetically tweaked. Any solution to the problem of suffering ultimately has to be global. 1. Abstain from eating meat I know many readers of Sentient Developments are Buddhists. Not all of them will agree with the above analysis. Some readers may suspect that I'm just trying to cloak my techno-utopianism in the mantle of venerable Buddhist wisdom. (Heaven forbid!) In fact the abolitionist project is just a blueprint for implementing the aspiration of Gautama Buddha two and a half millennia ago: "May all that have life be delivered from suffering". I hope other researchers will devise (much) better blueprints; and the project will one day be institutionalized, internationalized, properly funded, transformed into a field of rigorous academic scholarship, and eventually government-led. I've glossed over a lot of potential pitfalls and technical challenges. Here I'll just say I think they are a price worth paying for a cruelty-free world. Many thanks to George for inviting me to guest-blog this week. And many thanks to Sentient Developments readers for their critical feedback. It's much appreciated. * * * April 30, 2009 David Pearce is guest blogging this week. Guest blogger David Pearce answers your questions (part 3) Michael Kirkland alleges that the abolitionist project violates the Golden Rule. Maybe. But recall J.S. Mill: “In the golden rule of Jesus of Nazareth, we read the complete sprit of the ethics of utility.” The abolitionist project, and indeed superhappiness on a cosmic scale, follows straightforwardly from application of the principle of utility in a postgenomic era. Yes, there are problems in interpreting an ethic of reciprocity in the case of non-human animals, small children, and the severely mentally handicapped. But the pleasure-pain axis seems to be common to the vertebrate line and beyond. Michael also believes that my ideas about predators are "monstrous" and "genocidal". OK, here is a thought-experiment. Imagine if there were a Predator species that treated humans in the way cats treat mice. Let's assume that the Predator is sleek, beautiful, elegant, and is endowed with all the virtues we ascribe to cats. In common with cats, the Predator simply doesn't understand the implications of what it is doing when tormenting and killing us in virtue of its defective theory of mind. What are our policy options? We might decide to wipe out the race of Predators altogether - "genocide" so to speak - from the conviction that a race of baby-killers was no more valuable than the smallpox virus. Or alternatively, we might be more forgiving and tweak its genome, reprogramming the Predator so it no longer preyed on us (or anyone else). However, perhaps one group of traditionally-minded humans decide to protest against adopting even the second, "humane" option. Tweaking its genome so the Predator no longer preys on humans will destroy a vital part of its species essence, the ecological naturalists claim. Curbing its killer instincts would be "unnatural" and have dangerously unpredictable effects on a finely-balanced global ecosystem (etc). Perhaps a few extremists even favour a "rewilding" option in which the Predator should be reintroduced to human habitats where it is now extinct. Should we take such an ethic seriously? I hope not. Granted, the above analogy sounds fantastical. But the parallel isn't wholly far-fetched. If your child in Africa has just been mauled to death by a lion, you'll be less enthusiastic about the noble King Of The Beasts than a Western armchair wildlife enthusiast. A living world based around creatures eating each is barbaric; and later this century it's going to become optional. One reason we're not unduly troubled by the cruelties of Darwinian life is that our wildlife "documentaries" give a Disneyfied view of Nature, with their tasteful soundtracks and soothing commentary to match. Unlike on TV news, the commentator never says: "some of the pictures are too shocking to be shown here". But the reality is often grisly in ways that exceed our imagination. In response to Leafy: Are cats that don't kill really "post-felines" - not really cats at all? Maybe not, but only in a benign sense. A similar relationship may hold between humans and the (post-)humans we are destined to become. Carl worries that I might hold a "[David] Chalmers-style 'supernatural' account of consciousness". We differ, but like Chalmers, I promise I am a scientific naturalist. The behaviour of the stuff of the world is exhaustively described by the universal Schrödinger equation (or its relativistic generalization). This rules out dualism (casual closure) or epiphenomenalism (epiphenomenal qualia would lack the causal efficacy to inspire talk about their own existence). But theoretical physics is completely silent on the intrinsic nature of the stuff of the world; physics describes only its formal structure. There is still the question of what "breathes fire into the equations and makes a universe for them to describe", in Hawking's immortal phase; or alternatively, in John Wheeler's metaphor, "What makes the Universe fly?" Of the remaining options, monistic idealism is desperately implausible and monistic materialism is demonstrably false (i.e. one isn't a zombie.) So IMO the naturalist must choose the desperately implausible option. See Galen Strawson [Consciousness and Its Place in Nature: Does physicalism entail panpsychism? (2006)] for a defence of an ontology he'd hitherto dismissed as "crazy"; and Oxford philosopher Michael Lockwood [Mind, Brain and the Quantum (1991)] on how one's own mind/brain gives one privileged access to the intrinsic nature of the "fire in the equations". I think two distinct problems of consciousness need to be distinguished here. A solution to the second presupposes a solution to the first. First, why does consciousness exist in the natural world at all? This question confronts the materialist with the insoluble "explanatory gap". But for the monistic idealist who thinks fields of microqualia are all that exist, this question is strictly equivalent to explaining why anything at all exists. Mysteries should not be multiplied beyond necessity. Secondly, why is it that, say, an ant colony or the population of China or (I'd argue) a digital computer - with its classical serial architecture and "von Neumann bottleneck" - don't support a unitary consciousness beyond the aggregate consciousness of its individual constituents, whereas a hundred billion (apparently) discrete but functionally interconnected nerve cells of a waking/dreaming vertebrate CNS can generate a unitary experiential field? I'd argue that it's the functionally unique valence properties of the carbon atom that generate the macromolecular structures needed for unitary conscious mind from the primordial quantum mind-dust. No, I don't buy the Hameroff Penrose "Orch-OR" model of consciousness. But I do think Hameroff is right insofar as the mechanism of general anaesthesia offers a clue to the generation of unitary conscious mind via quantum coherence. The challenge is to show a) how ultra-rapid thermally-induced decoherence can be avoided in an environment as warm and noisy as the mind/brain; and 2) how to "read off" the values of our qualia from the values of the solutions of the quantum mechanical formalism. Proverbially, the dominant technology of an age supplies it root metaphor of mind. Our dominant technology is the digital computer. IMO our next root metaphor is going to be the quantum computer - our new root metaphor both of conscious mind and the multiverse itself. But if your heart sinks when you read the title of any book - or blog post - containing the words "quantum" and "consciousness", then I promise mine does too. * * * April 29, 2009 David Pearce is guest blogging this week Guest blogger David Pearce answers your questions (part 2) Gary Francione and Peter Singer? Despite their different perspectives, I admire them both. As an ethical utilitarian rather than a rights theorist, I'm probably closer to Peter Singer. But IMO a utilitarian ethic dictates that factory-farmed animals don't just need "liberating", they need to be cared for. Non-human animals in the wild simply aren't smart enough adequately to look after themselves in times of drought or famine or pestilence, for instance, any more than are human toddlers and infants, and any more than were adult members of Homo sapiens before the advent of modern scientific medicine, general anaesthesia, and painkilling drugs. [Actually, until humanity conquers ageing and masters the technologies needed reliably to modulate mood and emotion, this control will be woefully incomplete.] * * * April 28, 2009 David Pearce is guest blogging this week Guest blogger David Pearce answers your questions In yesterday's post I promised I'd try to respond to comments and questions. Here is a start - but alas I'm only skimming the surface of some of the issues raised. t.theodorus ibrahim asks: "what are the current prospects of a merger or synergy between transhumanist concerns of the rights of natural humans within a world dominated by greater intelligences and the existing unequal power relations between animals - especially between human and non-human animals? Will the human reich give way to a transhuman reich?!" The worry that superintelligent posthumans might treat natural humans in ways analogous to how humans treat non-human animals is ultimately (I'd argue) misplaced. But reflection on such a disturbing possibility might (conceivably) encourage humans to behave less badly to our fellow creatures. Parallels with the Third Reich are best used sparingly - the term "Nazi" is too often flung around in overheated rhetoric. Yet when a Jewish Nobel laureate (Isaac Bashevis Singer, The Letter Writer) writes: "In relation to them, all people are Nazis; for the animals it is an eternal Treblinka", he is not using his words lightly. whether we're talking about vivisection to satisfy scientific curiosity or medical research, or industrialized mass killing of "inferior" beings, the parallels are real - though like all analogies when pressed too far, they eventually break down. One reason I'm optimistic some kind of "paradise engineering" is more likely than dystopian scenarios is that the god-like technical powers of our descendants will most likely be matched by a god-like capacity for empathetic understanding - a far richer and deeper understanding than our own faltering efforts to understand the minds of sentient beings different from "us". Posthuman ethics will presumably be more sophisticated than human ethics - and certainly less anthropocentric. Also, it's worth noting that a commitment to the well-being of all-sentience is enshrined in the Transhumanist/H+ Declaration - a useful reminder of what, I hope, unites us. Spaceweaver, if I understand correctly, argues that perpetual happiness would lead to stasis. Lifelong bliss would bring the evolutionary development of life to a premature end: discontent is the motor of progress. I'd agree this sort of scenario can't be excluded. Perhaps we'll get stuck in a sub-optimal rut: some kind of fool's paradise or Brave New World. But IMO there are grounds for believing that our immediate descendants (and perhaps our elderly selves?) will be both happier and better motivated. Enhancing dopamine function, for instance, boosts exploratory behaviour, making personal stasis less likely. By contrast, it's depressives who get "stuck in a rut". Moreover if we recalibrate the hedonic treadmill rather than dismantle it altogether, then we can retain the functional analogues of discontent minus its nasty "raw feels". See "Happiness, Hypermotivation and the Meaning Of Life." FrF asks: "Could it be that transhumanists who are, for whatever reason, dissatisfied with their current lives overcompensate for their suffering by proposing grand schemes that could potentially bulldoze over all other people?" For all our good intentions, yes. The history of utopian thought is littered with good ideas that went horribly wrong. However, technologies to retard and eventually abolish ageing, or tools to amplify intelligence, are potentially empowering - liberating, not "bulldozing". Transhumanists/H+-ers aren't trying to force anyone to be eternally youthful, hyperintelligent or even - God forbid - perpetually happy. Genuine dilemmas do lie ahead: for instance, is a commitment to abolishing aging really consistent with a libertarian commitment to unlimited procreative freedom? But I think one reason many people are suspicious of future radical mood-enhancement technologies, for instance, is the curious fear that someone, somewhere is going to force them to be happy against their will. It's also worth stressing that radically enriching hedonic tone is consistent with preserving most of your existing preference architecture intact: indeed in one sense, raising our "hedonic set-point" is a radically conservative strategy to improve our quality of life. Uniform bliss would be the truly revolutionary option: and uniform lifelong bliss probably isn't sociologically viable for an entire civilisation in the foreseeable future. However, stressing a future of hedonic gradients and the possibility of preference architecture conservation is arguably to underplay the cognitive significance of the hedonic transition (IMO) in prospect. Just as the world of the clinically depressed today is unimaginably different from the world of the happy, so I reckon the world of our superhappy successors will be unimaginably more wonderful than contemporary life at its best. In what ways? Well, even if I knew, I wouldn't have the conceptual equipment to say. Visigoth rightly points out that any long-acting oxytocin-therapy [or something similar] would leave us vulnerable to being "suckered". Could genetically non-identical organisms ever wholly trust each other - or ever be wholly trustworthy - without compromising the inclusive fitness of their genes? Maybe not. I discuss the nature of selection pressure in a hypothetical post-Darwinian world here: "The Reproductive Revolution: Selection Pressure in a Post-Darwinian World". Needless to say, any conclusions drawn are only tentative. Away from genetics, it's worth asking what does it mean to be "suckered" when life no longer resembles a zero-sum game and pleasure is no longer a finite resource - a world where everyone has maximal control over their own reward circuity. Right now this scenario sounds fantastical; but IMO the prospect is neither as fantastical (nor as remote) as it sounds. See the Affective Neuroscience and Biopsychology Lab for how close we're coming to discovering the anatomical location and molecular signature of pure bliss. More tomorrow.... * * * April 27, 2009 David Pearce is guest blogging this week David Pearce * * * H+ Interview (Sept. 2009) Cronopis Interview (Dec. 2007) Vanity Fair Interview (April 2007) Nanoaging Interview (Dec. 2005) The Neofiles Interview (Dec. 2003) DP Drug Regimen [August 2005] The End of Suffering? [Philosophy Now, July-August 2006] The Hedonistic Imperative Future Opioids Utopian Surgery? The End of Suffering Wirehead Hedonism The Good Drug Guide Paradise Engineering The Abolitionist Project Nanotechnology Hotlinks Reprogramming Predators The Reproductive Revolution MDMA: Utopian Pharmacology Critique of Huxley's Brave New World DP (Wikipedia) + DP Friends (Facebook) E-mail Dave
6a4af689f4dc0acd	Laurent Jaeken, 1, Vladimir Vasilievich Matveev2 1 Laboratory of Biochemistry, Karel de Grote University College, Department of Applied Engineering, Salesianenlaan 30, B-2660, Antwerp, Belgium 2 Laboratory of Cell Physiology, Institute of Cytology, Russian Academy of Sciences, Tikhoretsky Ave 4, St. Petersburg 194064, Russia Article Metrics CrossRef Citations: Total Statistics: Full-Text HTML Views: 937 Abstract HTML Views: 714 PDF Downloads: 141 Total Views/Downloads: 1792 Unique Statistics: Full-Text HTML Views: 574 Abstract HTML Views: 460 PDF Downloads: 116 Total Views/Downloads: 1150 © Jaeken and Matveev; Licensee Bentham Open. Address correspondence to this author at the Laboratory of Biochemistry, Karel de Grote University College, Department of Applied Engineering, Salesianenlaan 30, B-2660, Antwerp, Belgium; Tel:+1323 663 30 31;Fax: +1323 293 34 08; Emails:;; Observations of coherent cellular behavior cannot be integrated into widely accepted membrane (pump) theory (MT) and its steady state energetics because of the thermal noise of assumed ordinary cell water and freely soluble cytoplasmic K+. However, Ling disproved MT and proposed an alternative based on coherence, showing that rest (R) and action (A) are two different phases of protoplasm with different energy levels. The R-state is a coherent metastable low-entropy state as water and K+ are bound to unfolded proteins. The A-state is the higher-entropy state because water and K+ are free. The R-to-A phase transition is regarded as a mechanism to release energy for biological work, replacing the classical concept of high-energy bonds. Subsequent inactivation during the endergonic A-to-R phase transition needs an input of metabolic energy to restore the low entropy R-state. Matveev’s native aggregation hypothesis allows to integrate the energetic details of globular proteins into this view. Keywords:: Entropy, coherence, polarized water, unfolded proteins, native aggregation, association-induction hypothesis. Among physicists there have always been people who are interested in basic physical approaches to the phenomena of life. Unlike mathematics, physics always requires models or mechanisms without which the tools of the discipline are not applicable. Cell physiology has suggested two opposing models of the living cell to physicists: (i) a cell is a membrane-delineated bubble with an electrolyte solution inside; (ii) a cell is a different phase relative to the surrounding fluid. The main difference between these models is in the physical state of intracellular water and K+ (the main cation of the cell). For the first model, water and K+ are free in the cell. Membrane (pump) theory (MT) was founded on this basis [1-7]. It describes a large set of interrelated physiological phenomena (solute distribution, transport across membranes, cell potentials, osmotic behavior, aspects of motility, etc.). One of its basic concepts, that of the Na+/K+ pump using a continuous energy supply from the hydrolysis of the high-energy bond of ATP [8], is intimately linked with that of steady state energetics; the metaphor that life can be compared to a burning candle. In contrast, the idea that a cell is a different phase forms the basis for a number of bulk-phase theories [9-15] describing the same physiological phenomena in a completely different way. According to the latter, water and K+ are bound in the cell and the protoplasm resembles a drop of gel. Of course, these fundamentally different models require very different physical approaches. MT failed to offer key ideas for creating a general physical approach to the cell. Instead, various specific questions were subjected to physical analysis such as the explanations for membrane potential, muscle contraction, protein conformation, etc [16]. However, these fragments of knowledge cannot be assembled into a holistic theory of the cell. The separate topics are like the reflections of a whole cell in a broken mirror of physics. The fragments were obtained by different authors using different physical approaches, often contradicting each other. On the other hand, Schrödinger [17] proposed a unified view of the cell, but his ideas had little effect on cell physiology and were discussed mainly among physicists. In our opinion, this is because MT has the widest following among physiologists and the broader interdisciplinary area of biology. Unfortunately, MT proved to be incompatible with the scale of Schrödinger's ideas. We would like to show that Schrödinger's ideas are fully compatible with the bulk-phase model of the cell. But this will entrain another model for cell energetics. Ling's Association-Induction Hypothesis (AIH) [11, 12, 18-26] is the supreme achievement of the bulk-phase approach to physiological phenomena in the living cell. According to Ling, the cell is a holistic system because water and K+ are adsorbed to a matrix of cell proteins making that these three physiologically relevant components act as a single whole. Owing to a network of proteins with natively unfolded conformation, which strongly binds, orients and polarizes water, the cell acquires the capacity for coherent behavior. Since water is bound, limited in motion and ordered in space, the entropy of the cell is lower than in a cell with free water. The entropic contribution of water is very important because water makes up most of the mass of the cell (its intracellular concentration is about 44 M). The bound state of K+ is another contribution to the reduction of entropy. Furthermore, the presence of a network of interconnected protein molecules, called ‘cell matrix’, reduces the entropy of the protein component of the system. All these features are inherent to the cell during the resting state (R-state): relaxed myofibrils, resting potential, inactive secretion processes, unfertilized egg, dormant spores and cysts, etc. So, the cell during the R-state is a protein-water-K+ complex having cooperative properties. This complex appears as a single coherent whole. During the protoplasmic phase transition from the R-state to an active state (A-state), the complex is transformed and water and K+ become free. The entropy of the system increases. The free energy of the system is reduced (because of the entropic contribution). The released energy is utilized for biological work: myofibril contraction, action potential, secretion, fertilization, germination, etc. Thus, in this view, entropy plays a key role in cell energetics, which makes Ling's cell model qualitatively compatible with Schrödinger's view. The physical concept of phase transition is not sufficiently detailed to serve as an adequate account of cell physiology. However, the recently published [15] native aggregation hypothesis (NAH) provides specific physiological and biochemical details. During the R-to-A phase transition, natively unfolded proteins form temporary secondary structures that interact only with secondary structures complementary to them. Natively unfolded and globular proteins temporarily form new active functional systems. The energetic aspects of this process are the immediate objective of this article. We believe that a mechanism of energy transformation in the living cell (its release and utilization) must include a change in the sorption properties of proteins as a key event. Water and K+ ions are the main adsorbents in this mechanism. ATP is seen as the regulator of the sorption properties of proteins. The source of energy for biological work is not ATP alone but supra-molecular ATP-protein-water-K+ complexes with an ordered structure and coherent properties. Our larger goal is to attract the attention of physicists, physical chemists, cell biologists and biochemists to the problems discussed. We hope to offer a new synthesis of ideas from different scientific disciplines. This is the time to join forces. 1. The Origin of Generally Accepted Membrane Theory (MT) In 1930 a crucial experiment by the famous physiologist A.V. Hill [2, 3] seemed to lay down the basis of ‘membrane (pump) theory’ (MT). This theory holds (a) that cells are delineated by an intact plasma membrane separating the external solution of ions and other solutes from the cytoplasmic solution and (b) that the plasma membrane exerts some pumping activity, thought to be responsible for the observed concentration gradients of ions, which are not in thermodynamic equilibrium when calculated using the law of Nernst. (c) This non-equilibrium situation leads to steady state cell energetics, in which metabolism-dependent pumping unceasingly compensates for passive leaks. Before 1930, ideas in the realm of MT were rivaled by bulk-phase theories, in which the colloid phase of the cytoplasm was thought to be responsible for concentration differences between a cell’s interior and its surroundings. In order to stop the endless debate, Hill [2] determined the distribution coefficient of a neutral molecule, urea, between the cytoplasm of resting muscle cells and the Ringer solution in which they were bathed. He found it to be close to one. Hence he concluded that intracellular water has similar properties to the water outside the cell. He also concluded that in order to explain the measured osmotic pressure [3], the most abundant intracellular ion (K+) must be in solution. These inferences seemed to oppose the possibility of a colloidal cytoplasm. They implied that the aqueous environment inside a cell must be a solution and hence the law of Nernst should be applied. This led to the conclusions that membrane pumps must exist and that the cell exhibits steady state energetics. Hill’s experiment caused most proponents of bulk phase theories to adopt MT instead. In 1957 Skou [27: review] discovered that the plasma membrane protein Na+/K+-ATPase shared some key properties with the physiological process of active Na+ and K+ transport, suggesting that this membrane-located enzyme is in fact the Na+/K+ pump. But it took much additional research before general agreement was reached. Hoffman [28] confirmed that ATP was indeed the direct metabolic intermediate. From the then-popular view of Lipmann [8], it followed that hydrolysis of the high-energy bond of ATP would act as an instantaneous injector of energy into the pump. Reconstitution of the purified Na+/K+-ATPase into phospholipid vesicles by Hilden et al., [29] in 1974 is generally considered as definite proof of the pumping activity of this enzyme. A year later, Glynn and Karlish [30] wrote a much cited review of all the work done up to then, unfortunately without mentioning an important alternative explanation, the ‘association-induction hypothesis’ (AIH), worked out by Ling during the preceding decade [11] and presented at an international conference two years earlier [31, 32]. While Ling’s alternative has remained virtually unknown – but to date also un-refuted – MT quickly developed into an unquestioned paradigm, appearing as the only explanation for a large number of interconnected physiological phenomena. It seemed to explain non-equilibrium gradients and the transport of ions and other solutes by the direct action of membrane-located pumps and indirectly by coupled transport mechanisms [33] (for another explanation see [12, 18, 19]; for trans-epithelial transport see [18, 34]). By showing how concentrations of low molecular mass solutes could be kept in steady state, an explanation for osmotic balance and hence cell volume was given [27] (for another explanation see [11, 14, 18, 19]). The electric imbalance of pumping 3 Na+ outwards in exchange for 2 K+ inwards [35] was taken as an explanation for the existence and the observed value (-70 mV in neurons) of the resting potential and seemed to elucidate the property of excitability of some cell types [7] (for other explanations see [11, 14, 18, 19, 36-39]). Respiration-dependent proton transport in mitochondria and bacteria and light-driven proton transport between internal and external bulk phases were considered to energize membrane-dependent ATP synthesis in these organelles and in bacteria by means of chemiosmosis [40] (for other explanations see [18, 41-44]). Clearly, the idea of membrane-located ion-pumps became central to a large and very central part of classical cell physiology, despite many criticisms from many different fronts. Two recent reviews cover the present status and historical development of this issue: one by Ling [25] discussing the pro’s and contra’s of both MT and its alternative (AIH); the other by Glynn [45], who unfortunately continues to discuss only MT. 2. The Problem with MT Three kinds of observations opposing MT and its steady state energetics are selected here from a much broader context (see [25] for a historical review). They are: (a) the rapidly increasing number of observations supporting the coherent behavior of life; (b) the abundant evidence that cell water is largely structured; and (c) the abundant evidence that cellular K+ is mostly in a bound (adsorbed) state. These three kinds of observations are directly interlinked because the adsorption properties of (some) proteins for water and K+ ensure coherent behavior and a high degree of order down to the molecular level. The disorder due to free water (about 44 M) and free K+ (about 0.15 M), as MT protagonists contend, would not support most of the coherent processes. There is an additional link between the state of water and that of K+, already given in the discussion of the experiments of Hill. The measured osmotic pressure of the muscle cells in Hill’s experiment can be explained in two ways. When water is free, this implies that there is no colloid and hence no colloid osmotic pressure. Then there is just the normal osmotic pressure of the proteins and this is not sufficient. Therefore, the main intracellular solute K+ (about 0,150 M) must also be osmotically active, i.e. it must be free. In contrast, when water is structured, there must be a colloid phase and colloids have an osmotic pressure which is much higher than expected on behalf of their molar activity (sometimes called colloid osmotic pressure), and which for cells almost on its own explains the measured osmotic pressure [12, 18, 19]. Hence, K+ should be osmotically inactive by being bound. Observations during the past 50 years culminated in favor of the second possibility. But how then should the findings of Hill be explained? In 1973, Ling et al., [31] confirmed after the experiment of Hill, but also performed a large number of similar experiments using many solutes other than urea. They found that distribution coefficients for substances as small as urea are indeed close to one, but for larger neutral molecules they are definitely below unity: the larger the molecule, the lower the coefficient. It became clear that Hill’s choice of urea had been quite unfortunate (and is an example of the serious errors to which the most reputable scientists can be prone). The much more elaborate studies of Ling et al., [32] (many other substances were tested later [11, 18-20]) led to the opposite conclusions: (a) intracellular water cannot be ordinary water and must be structured, (b) intracellular K+ cannot be in solution but must be largely bound, and (c) cytoplasm must be colloid-like, possessing solvencies that are quite different from those in ordinary water. Among the solutes studied were sodium salts. The distribution coefficient for Na+ was shown to explain its gradient fully without the need for pumping [11, 18-20, 25]. The structured and polarized state of almost all cellular water and the physical state of the intracellular K+ as largely adsorbed were both demonstrated using a wide variety of modern techniques. A brief overview of some major pieces of evidence will be given in the next two sections. 3. The Polarized State of Cellular Water Early ideas about bound water failed because of the lack of a physical theory describing such water at the molecular level. In 1965, Ling [46] developed such a theory and made it part of his AIH. Briefly, it states that a surface, for instance a polymer matrix, is able to orient and electrically polarize the dynamic structure of water when it bears an appropriately spaced checkerboard pattern of sufficiently strong positive (P) charges (these may be partial positive charges such as on –NH groups) in regular alternation with sufficiently strong negative (N) charges (these may be partial negative charges such as on –CO groups) together forming a so-called ‘NP-NP- system’; or a comparable pattern of alternating (partial) positive (P) and neutral (0) sites forming a ‘P0-P0- system’, or alternating (partial) negative (N) and neutral (0) sites forming a ‘N0-N0-system’. The water molecules in close contact with the polymer will be spatially oriented by the polymer charges, but in addition, the stronger dipoles of the polymer will induce a greater dipole moment in the water molecules, so that the latter also become electrically polarized. In consequence, the dipole moment of the adsorbed water is higher than that of bulk water. The result is that further layers of water are induced one after the other. Ling [46] called this Polarized Multi-Layer water or PML-water. It exhibits long-range dynamic structuring. The higher total dipole moment stabilizes the multi-layers because it leads to stronger hydrogen bonds and this makes it a poor solvent for most solutes that are highly soluble in normal (bulk) water. Translational as well as rotational freedom of the water molecules is reduced and the residence time of each water molecule at a specific location is increased. Together, this makes all the physicochemical properties of PML-water differ from those of normal water. And this is the way to test the presence of PML-water in cells. The aberrant properties of PML-water were first studied in inanimate model systems of suitable polymers and then compared with the properties of cell water [11, 18-21]. The results obtained from this approach as well as an updated theoretical background were recently reviewed [20]. An important question concerns the thickness of the oriented-polarized multilayers. Giguère and Harvey [47] studied a water film between two parallel polished AgCl plates at a distance of 10 µm and unexpectedly found that it did not freeze even at -176°C. The infrared absorption spectrum was indistinguishable from that at 31°C. This means that about 30,000 layers of water molecules are involved. The experiment was highly reproducible, indicating that the situation represents a true equilibrium state and not just a metastable equilibrium as in the case of supercooling. More recently, Ise et al., ([48], see also [49]) obtained striking micro-cinematography of latex particles suspended in both ordered and disordered water, revealing long stochastic trajectories for the latter, as might be expected from Brownian motion, and a tetragonal quasi-crystalline pattern of particles almost free of movement for the former. This is remarkable because the individual particles are situated several micrometers from each other, yet are ordered in a tetragonal pattern. Moreover, experiments by Zheng and Pollack ([50], see also [49]) generated water polarization up to hundreds of micrometers away from the surfaces of hydrophilic gels and of muscle preparations. All three of these sets of observations support Ling’s conclusion [20] that an idealized system may enhance water-to-water adsorption energy ad infinitum. Applying his theory of NP-NP-systems to cells, Ling [46] deduced that there is but one molecular structure present in sufficient abundance to explain the high degree of polarization of all or almost all cell water. It consists of the backbone NH-CO groups of the peptide bonds of proteins in a fully-extended conformation. In the absence of disturbing side-groups (see e.g. gelatin), the backbone of a polypeptide gives an almost perfect fit, sufficient to bridge distances between neighboring proteins in a cell. Ling [12, 18-21] then demonstrated that free NH-CO groups of unfolded proteins are indeed able to bind water strongly, and to orient and polarize it with formation of multilayers. Although extended proteins are essentially linear polymers and not idealized surfaces, Ling argued that there are ‘generous margins for effective polarization-orientation of the entire cell water by a relatively small amount of intracellular proteins that assume the fully-extended conformation of an NP-NP-NP system’ [20, p. 127]. Orientation and polarization of water by an unfolded protein matrix are characterized by very low entropy. The question, however, is: from where does the energy to unfold these proteins come? Ling [18-26] found that it came from the stable adsorption of ATP to ATP-binding sites on key proteins, which together with other proteins form an auto-cooperatively linked protein-water-ion ensemble of fully extended proteins, PML-water and adsorbed K+ (see later), which he defined as the cell-matrix. This protein-water-ion complex occurs in cells or parts of cells when they are in the resting (inactive) state (R-state). Relaxed myofibrils are a well-defined example. This state, although low-entropic, is stable in the sense that it can be maintained over long periods without a permanent supply of energy [51, 52], even years in the case of Artemia cysts [52]. However, its low entropy makes clear that it is nevertheless metastable. Its remarkable stability is due to the stable binding (adsorption) of ATP, exerting a strong inductive effect leading to polypeptide unfolding, which is transmitted auto-cooperatively through the whole linked cell-matrix system. Actually the propagated inductive effect has three main effects. It results in protein unfolding, water binding and polarization and K+ adsorption [11, 53]. This concept clarifies the words ‘association’ and ‘induction’ in the name of this theory: ‘association-induction hypothesis’ [11, 18-26, 46]. At the moment when Ling introduced his cell-matrix concept [31], the idea of in vivo unfolded proteins would have met with much skepticism. Common perception at the time held that proteins exist in a folded conformation in vivo, mainly because they were isolated as such from living material and continued functioning in vitro. Nowadays, the existence of unfolded or partially unfolded proteins, usually called ‘intrinsically unstructured’ proteins (or IDPs) [54-58], starts to be recognized. According to estimations derived from primary structure data bases 40% of all human proteins contain at least one intrinsically disordered segment of 30 amino acids or more and 25% are likely disordered over their entire length. The other proteins are thought to be globular in vivo [57]. Even if the methods used to arrive at these figures may be far from accurate, the occurrence of both IDPs and globular proteins can no longer be dismissed. These findings shake classical ideas about the relation between a protein’s 3D-structure and its function to its foundations [58]. It should be noted that the ideas associated with IDPs may not exactly correspond to Ling’s view of fully extended cell-matrix proteins, but there may be a large overlap. Main difference is that to Ling’s view of fully extended proteins a whole basically complete alternative physiology is attached. If explanations for the occurrence of IDP’s have to be found, the AIH may give some important answers. Definitely, the modern field of protein biophysics would benefit a lot by taking knowledge of this broad alternative. Pollack [14] also proposed a very extensive bulk-phase theory: his ‘gel-sol-hypothesis’ (GSH). It is built upon the fundamental aspects of Ling’s AIH. It takes over most of the concepts of the latter, adds additional evidence for it and explains many physiological processes within this broad context. AIH and GSH are both treated in his very readable book ‘Cells, gels and the engines of life’ [14]. In this book Pollack further explores the cell from the point of view of colloid science. Many details of physiological situations are given whereby controlled unfolding of specific proteins is linked to function and explains the existing data. To the list of cellular processes already described in Ling’s AIH (the role of ATP, ion distribution, solute transport, permeability, cell volume control, cell potentials, oxidative phosphorylation, epithelial transport, muscle contraction, control of protein synthesis, growth, differentiation and its malfunction), Pollack adds the processes of exocytosis, amoeboid movement, ciliary beating, mitotic chromosome displacement and cytokinesis. He also remarks that there may be another additional mechanism to obtain multi-layer water orientation and polarization. Some domains of the hydrophilic surfaces of such cytoskeletal proteins as actin and tubulin exhibit a checkerboard pattern of charges at the correct distance (as defined by Ling) to achieve water polarization, even in the folded state. This means that protein unfolding is not always required, although it certainly occurs in many cases. In Ling’s article from 1965 [46] and in certain of his other publications [18, 19, 21], some figures illustrate the orientation of water molecules when bound to an appropriate checkerboard pattern of complementary charges. These figures schematically and in a static way depict the situation in two dimensions. Water, however, has a three-dimensional structure. Chaplin [59, 60] built models to show how polarized water would appear in three dimensions. He described two main conformations of water-clusters, each consisting of 280 strongly bound water molecules operating as if it were a single macromolecule. With this model he could explain almost all properties of water, including the many anomalies. The two conformations were found to be easily transformed into each other without breaking intra- or intermolecular hydrogen bonds. One conformation has high density (HDW, high density water) and is non-polarized; it predominates in normal water. The other conformation has low density (LDW), is polarized and is abundantly present in cells. K+ appears to fit well into the central cavity of LDW, while Na+ does not. The latter ion has a much higher solubility in HDW, in agreement with the observations of Ling [18, 19]. On the basis of this finding, Chaplin proposes a third mechanism for water polarization, whereby K+ ions adsorbed to protein carboxyl groups act as nucleation sites for the formation of an LDW cluster with high polarization, from which water polarization spreads further. This idea agrees with the extensive studies of Wiggins [61] on the properties of HDW and LDW. She showed that both types of water can be obtained in the small pores (averaging 2 nm diameter) present in cellulose acetate films. Certain ions (K+, NH4+, Cl-, HCO3-, HSO4-, H2PO4-) appear to fit well into the central cavity of a LDW cluster and exhibit reasonable solubility. Others (H+, Li+, Na+, Ca2+, Mg2+) do not fit, but fit and show good solubility in HDW. Surprisingly, it was also found that at high concentrations the first group induces the conversion of LDW into HDW, while the second group at high concentrations induces the conversion of HDW into LDW. These properties make it possible to generate a cycle. In a remarkable experiment, Wiggins and MacClement [62] succeeded in synthesizing ATP from ADP and KH2PO4 (5 mM) in the presence of 100 mM KCl, with the pores of cellulose acetate forming the micro-domains of water and with no additional energy input. A major difference from the enzyme-catalyzed reaction is that it requires a few days, but one can learn from the mechanism. It is quite possible that hydrolytic and biosynthetic enzymes, including ATPases and ATP synthase, evolved as catalysts of this older HDW-LDW cycle. Proposals are given as to how the two states of water may contribute to the mechanisms of cation pumps and other enzymes. This is a real opportunity for the study of these enzymes, including Na+/K+-ATPase. The three mechanisms for water polarization (Ling, Pollack and Chaplin) do not exclude each other. Their relative contributions, however, remain to be determined. These three mechanisms and the more general idea of water orientation and polarization are also in full agreement with Kaivarainen’s new fundamental physical theory of phases, called ‘new hierarchical physical theory for the solid and liquid phases’, which is derived from basic principles [63]. Kaivarainen gives many applications of his theory to cell biology. 4. The Adsorbed State of Cellular K+ After the experiment of Hill [2, 3] had set the stage in favor of MT, some other experiments seemed to prove that cellular K+ is free. However, it was demonstrated later that these experiments or their interpretations were also problematic [12, 18, 19, 22, 64]. In contrast, a large number of different approaches all point to bound K+. Some of these studies do not allow us to decide where and on which kind of binding sites adsorption occurs. Others, however, give this information quite specifically. Indications for bound K+ as such comprise: (a) the study of distribution coefficients by Ling et al. [32] discussed above, indicating that Hill [2, 3] found an exception by only studying urea; (b) the fact that resting muscle cells without an intact cell membrane (EMOC preparations: see further) are able to sustain normal intracellular K+ concentrations for at least two days [51]; (c) the demonstration that Artemia cysts survive for four years without energy consumption [52]; (d) reinvestigations of measurements of ionic conductivity in various cell types and of K+ mobility in frog muscle cells, giving much lower values than for free K+ and providing convincing explanations why some earlier studies gave high values [12, 18, 19, 22]; (e) proof that use of a K+ selective electrode readily injures the cell, resulting in much too high activity measurements, so that more confidence can be given to those experimental set-ups giving low values [18, 19, 22, 64]; (f) X-ray absorption-edge fine structure determination of red blood cells giving a result that differs markedly from control measurements on a KCl solution of the same concentration [65]; (g) 39K+ NMR spectra of neurons, muscle cells and E. coli, which produced relaxation times much shorter than those of 0.1 M to 0.4 M KCl solutions and comparable to those of K+ adsorbed on a Dowex-50 cation-exchange resin [66, 67, 12, 18, 19, 22]; (h) demonstration that the relationship between internal and external 42K+ concentration follows Langmuir’s adsorption isotherm, whereby real proof of adsorption was given by studying the influence of inhibitors that compete in different ways for the same adsorption sites (unlabeled Cs+ and unlabeled K+) [19, 68]. In 1952, Ling had already deduced that K+ might be adsorbed to the β- and γ-carboxyl groups of proteins [11, 12, 18, 19, 22]. Kellermayer et al., [69] showed that K+ remained associated with cytoskeletal proteins for several minutes after treatment of living cells with the non-ionic detergent Brij 68, whereas if it were freely soluble in the cytosol its liberation would have been detected in seconds. In addition, a stoichiometric relationship between protein carboxyl group concentration and K+ concentration was demonstrated [19]. The most convincing results, however, came from the extensive electron microscopic investigations of frog Sartorius muscles mainly by Edelmann [70-72], using most delicate preparation techniques and advanced methods of analysis. Freeze-dried embedded and frozen hydrated thick sections of chemically unfixed specimens were used without staining, so that the location of K+ (by far the most abundant metal) could be directly observed. Alternatively, specimens in which K+ was partially substituted by one its more electron dense competitors, Rb+, Cs+ or Tl+, were used. In all cases K+ or its analogue was preferentially located in the A-band, where the heads of myosin are exclusively present, with a fainter electron density at the Z-line. Quantification was obtained with electron probe X-ray microanalysis (see also [73]) and laser microprobe mass-spectroscopic analysis. Autoradiography of single air-dried [19] or frozen fully hydrated muscle fibers [72] using 86Rb and 134Cs revealed the same specific localization. The presence in the cell of PML-water and adsorbed K+, for which abundant arguments have been given in this and the foregoing sections, are a necessary condition for reaching coherence. Coherence needs nearest neighbor interactions of the correct extent. The law of Nernst, as applied by MT, is based on the independence of solute molecules, i.e. the absence of interactions, so that stochastic behavior ensues. Arguments for coherent cellular behavior, which are the subject of the next section, are in fact supplementary arguments for a high level of order, also with respect to water and the main cation, K+. 5. The Coherent Behavior of Life In his book ‘What is life?’ Schrödinger [17] argued that concentrations of many important cellular components are too low to apply statistical laws in a meaningful way. This is the case for DNA and some regulatory substances and even for ions in small cells and subcellular compartments. In particular, Schrödinger envisioned the laws of diffusion, which are also part of the law of Nernst. To give an example: in liver cells an average mitochondrial matrix at neutral pH contains six free protons during state IV and only one during state III (fully activated and contracted). How one can make a bulk gradient with that [74]? In order to alleviate this undeniable difficulty, Schrödinger proposed that cellular processes may make use of coherent mechanisms, analogous to those found in physical systems close to absolute zero but applicable at body temperature. His argument was that such mechanisms are much more precise and able to operate below the level of thermal noise, exactly what would be needed to explain life. His proposal was largely neglected, mainly because at that time all data, in particular the findings of Hill [2, 3], were pointing towards the laws of diffusion and of Nernst as providing exact explanations for cell physiology. As a result, investigations into life’s coherence originated at the very margins of biological science. Several independent approaches to the question of the coherence of life will be reviewed. 1. Ling’s AIH [11, 53] was probably the first broad physiological theory in which coherence figured as a central theme: actually as the primary mechanism of all actions in living cells. In the R-state the cell-matrix is seen as a metastable high-energy low-entropy polarized coherent whole, a cooperatively-linked protein-water-ion complex. It is a coherent static system extremely sensitive to all kinds of very weak stimuli. When the energy delivered by a stimulus and mediated by Ca2+ (or another mediator) reaches the activation threshold, the system as a whole undergoes a conformational change, leading to the A-state, which is less polarized and has lower energy and higher entropy. The energy difference between the R- and A-states, attributable to an increase in entropy, is used for biological work: mechanical, electrical, chemical, transport work and coherent emissions (see later). The higher entropy is due to folding of the cell-matrix proteins, whereby adsorbed layers of water are depolarized and dissociated together with K+. This is just what Schrödinger imagined: life processes energized by negative entropy. The conformational change itself is also a coherent process, but of a dynamic type. This can be understood from the all-or-none character of the folding of a random coil into an α-helix (or other type of protein secondary structure), and, linked to it, the all-or-none character of long-range water depolarization. Biological work is performed at the onset of this conformational change, because the suddenly-formed secondary structures bring about interactions with other activated molecules, in which interaction-capable secondary structures appear as well (see also [15]). The formation of actomyosin is a good example. At the end of this dynamic process, after work is done, the cell-matrix ends up in the A-state, which is much less coherent; perhaps even not far from an irreversible death state. In muscle cells the A-state is called rigor mortis. In a healthy living cell, however, metabolism synthesizes new ATP, which binds to the ATP-binding proteins of the cell-matrix and restores the high-energy low-entropy R-state. Proteins again unfold, re-adsorb and polarize cell water, and re-adsorb K+. Metabolism thus brings about the reverse conformational changes, also a dynamic coherent process. Metabolism is the factor that prevents the death state from succeeding the A-state. It is responsible for reversion to the R-state following power output. This R-to-A-to-R cycle and its energetics was already developed by Ling in his first book (1962, [11]). It is presented graphically by the red line in Fig. (1) (the green line will be explained at the end of this article). Small note: according to Ling [18], liberation of rather large amounts of K+ in local myosin-containing compartments of muscle cells brings about strong local unidirectional osmotic fluxes of water towards that region, and this is the major force that brings about contraction, a mechanism that explains the concomitant thickening of muscles. Apart from this interesting proposal, a very penetrating controversy exists with respect to the energetics of muscle contraction. Formation of firm protein-protein (actin-myosin) linkages, when considered on its own, may at first sight imply a decrease in entropy. If the assumptions of MT were correct – i.e. if the physical state of cellular water and K+ during the R- and A-states remained the same - the classical conclusion that muscle contraction is a ‘disorder-to-order transition’ [75] appears logical. But all the evidence favors Ling’s alternative view, with desorption and depolarization of water and desorption of K+ during the R-to-A transition. It then follows that the small decrease in entropy due to actomyosin formation is largely offset by the large increase in entropy due to liberation and depolarization of water and K+ desorption. Therefore, exactly the opposite is true: muscle contraction is an ‘order-to-disorder transition’. The release of rigor by addition of ATP [76] is the ultimate corroboration of Ling’s and Schrödinger’s view. The survival of Artemia cysts for long periods, with measured zero-energy consumption [52], is also consonant with this. The above controversy illustrates the problem with MT not involving a holistic physical approach to the study of life, and underlines the need of a new model for cell energetics. According to Ling [11, 12, 18, 23, 24, 53] the mechanism that brings about the R-to-A and A-to-R conformational changes is based on the induction effect of Lewis: the basis of chemistry, hence also of biochemistry. This mechanism includes the inductive effects exerted by so-called ‘cardinal adsorbents’ at the site of induction and their long-range propagation and spreading throughout the cooperatively-linked proteins of the cell-matrix. Cardinal adsorbents are controlling substances binding to controlling binding sites (cardinal adsorption sites) on the cell-matrix. They include: all kinds of external and internal stimuli, hormones, neural transmitters, growth factors, second messengers, nucleotides and other allosteric modulators, drugs, etc.. For instance, in the case of myofibrils, Ca2+ is the activator and ATP restores the R-state. The sites for binding water, K+, etc., are called ‘regular adsorption sites’, while the binding sites for ATP or Ca2+ are cardinal adsorption sites, because of their controlling function and their one-on-many effect. Indeed, the inductive (electron withdrawing) effect of the binding of one ATP molecule to one corresponding ATP binding cardinal adsorption site is propagated and spread through the whole system of auto-cooperatively interconnected proteins, reaching not only all these proteins but also all associated PML-water and all adsorbed K+. Temporary chemical modifications of proteins such as phosphorylation can play a similar role to cardinal adsorbents. The essential feature of cooperativity, in contrast to non-cooperativity, is energetic interaction between near-neighbor sites. It is described quantitatively in the theoretically-derived ‘Yang and Ling cooperative adsorption isotherm’ [11, 14, 18, 19, 23, 24, 53]. This equation allows of auto-, hetero- and non-cooperative adsorption phenomena to be precisely verified experimentally, and has been applied as such by Ling’s group to many of their own experiments as well as to many existing data in the literature. It is analogous to (but more general than) the empirical equation developed by Hill describing the oxy-hemoglobin dissociation curve. Ling also showed that the polypeptide backbone possesses the unique property of high polarizability and partial resonance, allowing it to act as a high-way for the propagation of inductive effects, resulting in water adsorption and polarization or desorption and depolarization. Propagated inductive effects also reach side chains resulting in K+ adsorption or desorption from β- and γ-carboxyl-groups and some comparable effects on positively charged side groups. On the basis of the inductive effects, all cardinal adsorbents were divided into two groups (eventually three): those generating an electron-withdrawing inductive effect (e.g. ATP) and those generating an electron-donating inductive effect (e.g. Ca2+). The third possibility, an electron-neutral effect, is probably not very relevant to functional processes. 2. In the late 1960s, Fröhlich [77, 78], a pioneer of the study of coherent behavior in physical systems near absolute zero, started exploring the eventuality of coherent behavior in living systems at physiological temperature. As a physicist he approached the problem from a completely different point of view, namely from electrodynamics. Living organisms are mostly made of dipolar (dielectric) molecules in a phase system, which is a quasi solid state. This implies that electric and visco-elastic (mechanical) forces may be in permanent interaction. The function of metabolism is to ‘pump’ the living system into an excited state. It should be noted that the latter idea is consistent with Ling’s R-state. Fröhlich uses the word ‘pump’ in a way that differs from its meaning in MT, because the excited state obtained by this pumping can be metastable. He uses pumping in the way Ling sees the function of ATP in restoring the high-energy R-state. The ensuing excited state is characterized by collective modes of vibration of a coherent nature. This holds that the giant dipolar structures of proteins, nucleic acids and some larger ensembles such as membranes, with their enormous electric fields (107 V/m), start to vibrate at characteristic frequencies common to the different components belonging to the system. These collective oscillations originate through the coupling of electric dipole displacements and mechanical displacements. Resonances may lead to emission of both electromechanical oscillations (phonons or sound waves) and electromagnetic radiations (photons), which may exhibit coherence as in a laser. These are propagated over a rather long distance, so they can eventually be measured outside the organism. This is the way to obtain information about the existence of these coherent collective modes. Alternatively, organisms can be subjected to extremely low intensities of the appropriate frequency, and this will lead to excitation of the corresponding cell function by resonance. Both approaches to these so-called ‘coherent excitations’ have been experimentally tested with positive results [79-82]. The group of Cosic [83, 84] further elaborated this idea in their ‘Resonance Recognition Model’. They used a calculation to transform an amino acid sequence into a frequency pattern. When this was done for proteins exerting similar functions, a typical common frequency was always found, indicative of that function, but also found in the frequency spectrum of their targets. It follows that it is characteristic of their reciprocal attraction and recognition by resonance. The frequency range for protein interactions is 1013 to 1015Hz (equivalent to the near-infrared, visible and near-UV spectra). Common frequencies were calculated and also measured, and the measurements were found to match the calculations. The authors concluded that the specificity of protein interactions is based on resonant electromagnetic energy transfer at a frequency specific for each kind of interaction or function. Fröhlich described three kinds of coherent excitations [80-82]. The first is static (metastable), low-entropic and highly polarized and became known as the ‘Fröhlich state’. Jaeken [85] argued that the R-state as described in physiological and molecular detail by Ling and Pollack (gel-state) conforms to the more general physical description of a Fröhlich state, formulated in terms of frequencies. Both descriptions evoke a low entropy, metastable (static), highly polarized, long-range coherent, excited (high-energy) state obtained through input of metabolic energy (ATP). The Fröhlich state is characterized by condensation of all modes into a single frequency, exhibiting coherence. Clearly this is due to the auto-cooperative linkage between all parts in Ling’s description. Second, Fröhlich also described a dynamic type of coherence, called a ‘Fröhlich wave’. The R-to-A (gel-sol) and A-to-R (sol-gel) phase transitions described in physiological and molecular terms respectively by Ling and by Pollack were both shown to conform [85] in physical terms to a Fröhlich wave. Third, Fröhlich envisioned the existence of coherent cyclic processes, known as ‘limit cycles’ [80-82, 85, 86]. The rhythmic succession of action states occurring for instance during prolonged muscle activity – if understood according to Ling’s AIH or Pollack’s GSH (but not according to MT) – exemplifies such limit cycles [85]. 3. In the early 1970s a third independent approach was taken by Davydov [87-90, also 91 for review]. He applied quantum mechanics to protein α-helices and showed theoretically that the latter might be able to conduct non-dispersed one-dimensional elastic lattice pulses along their length, which became known as ‘Davydov-solitons’. The concept of ‘solitons’ is used in non-linear physics to describe the loss-less transmission of waves of any kind. In the case of Davydov-solitons, dispersion of mechanical pulses along an α-helix (in an apolar environment) is exactly counteracted electronically by enhanced inductive effects on the longitudinal amide-2 hydrogen bonds, so that coherence is reached. Similarly, Davydov also described ‘electro-solitons’ and ‘protonic solitons’ (loss-less displacement of an electron charge respectively a protonic charge along an α-helix) [89] (see [91] for an overview). One or more of these types of solitons are believed to travel along microfilaments and microtubules, providing them with the ability to transmit and process data [92, 93]. Indications for this were recently supported by direct evidence [94]. A much more complex type of soliton is the action potential [91, 95], a well-studied example of a rest-to-action-to-rest (R-to-A-to-R) (gel-sol-gel) double phase transition [14, 18, 19, 36, 39, 95] in agreement with Ling’s AIH, Pollack’s GSH and the Fröhlich wave concept [80-82]. These studies show that an action potential is much more than the outdated concept of temporary permeability changes, as MT proponents believe (see Ling [12, 18, 19, 36]). 4. The group of Del Giudice [86, 96-99] followed a fourth path, starting from quantum field dynamics. The phenomenon, which in their theory is described as a ‘Bose condensation’ and its reverse, here called ‘Bose decondensation’, was shown to correspond to the description of Fröhlich waves [80-82, 86, 96,] and to R-to-A (gel-sol) and A-to-R (sol-gel) phase transitions according to AIH and GSH [85]. Actually, the four independent approaches described above could all be unified using Ling’s physiological paradigm as a basis [85]. This unification was achieved by interpreting the four coherent solutions of a non-linear Schrödinger equation, derived by Tuszinsky et al., [100], who had already united the views of Fröhlich and Davydov. 5. A fifth independent approach came from quantum non-linear optics [101-106]. Using specifically developed optic instrumentation, Popp [101] obtained quantitative measurements of the extremely weak photon emission (not to be confused with bioluminescence) of all living organisms in the visible and UV region, first observed by Gurwitsch and Gurwitsch in 1922 [102]. He proved the coherent (laser) properties of these so-called ‘biophotons’, defined as coherent photons produced by living organisms [103, 104] but absent at death. Biophotons may be produced by DNA [98, 103-106] and by microtubules, whereby excited water plays a particular role [98, 99, 106]. Their coherent nature allows holograms to be generated, with all the remarkable properties of the latter including distributedness and extremely high information density, and offering the possibility of constructive (within cells) and destructive (between cells) interference [92, 104]. Popp proposes that biophotons are ideally suited to coordinating the behavior of otherwise independent molecules located at distant places in cells, tissues, organs and even the whole body; and they could do this at the speed of light. Popp [88] concluded in a general way: ‘biophotons may well provide the necessary activation energy for triggering all biochemical reactions in a cell at the right time at the right place’. In that respect, they are responsible for the common observation that a body acts as a single unit, despite it being composed of innumerable individual molecules. Precisely this unity of behavior constitutes the problem with which Schrödinger struggled. Of course, the coherent nature of biophotons would be dispersed if they had to travel through a non-coherent cytoplasm, as assumed by MT. 6. Ingber [107, 108] demonstrated the tensegrity properties of the cytoskeleton. The latter consists of a three-dimensional construction of compression-resisting struts (microtubules) interconnected in an indirect way by means of pre-stressed microfilaments and intermediary filaments, which have more elastic properties, or which by sliding past each other acquire properties equivalent to elasticity. In mechanics such systems are called ‘tensegrity structures’. They exhibit tensegrity properties, which means that they behave as a unified whole in response to external mechanical forces, which become distributed over the entire network and result in global rearrangements. In this way a cell also achieves mechanical coherence [109]. The observed pre-stress [107] means that the cytoskeleton – also when at rest – stores mechanical energy and hence the R-state is also a high-energy structure from the mechanical perspective, complementing Ling’s view. This property makes it very sensitive to weak mechanical influences. Because cells are connected by integrins and junctional complexes to neighboring cells and to extracellular matrices, which also exhibit tensegrity properties, entire tissues and even organs exhibit tensegrity. The tensegrity concept allows us to explain how mechanical stimuli influence cellular information processing in the cytoplasm and in the nucleus through connections with the nuclear skeleton [108]. Some of its effects at the transcriptional level participate in morphogenesis, both normal and anomalous as in the case of cancer [108]. Because both the cytoskeleton and the extracellular matrix (i.e. collagen) exhibit piezoelectric properties [110], mechanical coherence goes along with electrical coherence [109]. And because many proteins, multi-protein systems and cell organelles are connected to the cytoskeleton [111] and can be directly influenced by data processing along the cytoskeletal network [92-94], chemical coherence may also become aligned with both mechanical and electrical coherence [109]. 7. Finally, Ho [49], pursuing the idea that coherence ought to be reflected in its structural organization down to the atomic and molecular level, went to the polarizing microscope and indeed directly observed the structural coherence of entire living cells, even organs in living organisms, proving that Schrödinger’s early proposal was fully justified. Ho [49] and Abbott [112] reviewed the indications that living organisms are indeed quantum coherent and can be regarded as macroscopic quantum systems. Recently, the existence of quantum coherence was directly measured in several in vitro experiments on parts of the photosynthetic apparatus [113-116]. Fig. (1). Energy levels of natively unfolded proteins (NUN) (red line) and natively folded proteins (NF) (green line) in the resting-state (R), action-state (A) and during R-to-A and A-to-R phase transitions. The energy level of NUN-proteins relative to that of NF-proteins during the A-state is approximately. For both groups it is somewhere between the levels of the fully unfolded state (see: NUN-proteins during the R-state) and fully folded state (see: NF-proteins during the R-state). The exact values depend on the kind of protein, but the energy difference of the R-to-A transition is always greater for NUN-proteins than for NF-proteins. During the R-state, NUN-proteins are inactive. They are unfolded, polarize cell water and adsorb K+, creating a high-energy low-entropy metastable state. A sufficiently strong stimulus is needed for them to leave their high-energy basin (threshold). NF-proteins are inactive as well during the R-state because they are tightly folded, hiding their reactive centers inside. Slight melting to an intermediate state is needed to expose these centers and make them active. The frame representing glycolysis and respiration points towards the processes that convert fuel, oxygen, ADP and Pi into CO2 , H2O, ATP and radiation (both coherent and waste heat), but is not intended to lie on the energy axis. Energy of entropic origin released during the R-to-A transition of NUN- proteins is used for activation of NF-proteins and for biological work (physiological functions). K+, Pi and ADP liberated during the R-to-A transition of NUN-proteins is thought to activate major rate controlling enzymes of glycolysis and respiration, in agreement with known allosteric mechanisms. So formed new ATP adsorbs to NUN-proteins and causes their A-to-R transition. Subsequent lowering of K+ and ATP activity deactivates glycolysis and respiration. A-to-R deactivation of NF-proteins either needs a small amount of energy to leave their A-state high-energy basin or is a spontaneous process, probably depending on the case. Therefore, both possibilities are drawn with dotted lines. In view of all these lines of evidence, direct and indirect, and the strong theoretical foundations from so many different independent approaches, which all point in the same direction, the coherent behavior of life can no longer be dismissed. Moreover, applications are quickly being made, particularly in diagnostic and therapeutic medicine, as there are increasingly strong indications that the lack of a sufficient level of coherence is the primary cause of cancer [117]. Genetic defects appear to originate later and secondarily as a result of insufficient coherence, which in a non-specific and progressive way destabilizes molecular and gene regulatory networks [117]. Despite all this evidence for coherence, together with the evidence for adsorption and polarization of water and adsorption of K+ to cell-matrix proteins , there is a major persistent obstruction to the uptake of these data by general textbooks and university courses. This obstruction is the persistence of MT. Therefore some additional evidence against an important aspect of MT, the concept of the Na+/K+ pump, must properly be given. 6. Evidence Against the Concept of Pumps The lines of evidence discussed below are not against the existence of Na+/K+-ATPase, nor against the main steps of its working mechanism as it is understood at present [27], but against the way of thinking about this enzyme/transporter as the pump responsible for the steady state concentrations of Na+ and K+ in cytosol. Evidence against the pump hypothesis properly includes the following. • Full inhibition of respiration and glycolysis in cells kept at 0°C, blocking the energy supply to the Na+-pump, did not result in a reduction of Na+ efflux. This early experiment of Ling [11, see also: 12, 18, 19, 22, 25] was later confirmed with cells at 17-23°C by Keynes and Maisel [118] - although Keynes later adopted MT - and by Conway et al. [119]. Extensive controls for remnant energy sources (creatine phosphate, ATP, ADP, lactate) and heat output measurements were undertaken (discussed by Ling [11, 12, 18, 19, 25]), confirming that the observed Na+ efflux was independent of a continuous energy supply. All authors agreed on the data. Taking the measured Na+ efflux rate, a resting potential of -90 mV and the method of calculation customary in MT, the conclusion – quoting Ling [11, 12, 18, 19, 25] – was: ‘the minimum energy needed by the Na pump at the plasma membrane alone was from 1542% to 3050% of the maximum available energy’. Comparable work done by Minkoff and Damadian [120] on E. coli led to a similar conclusion. • (2) The general consensus [27] is that in neurons, without inhibition of metabolism, the Na+/K+ pump consumes about 60% of the cell’s energy. However, there are many more pumps and they are located not only in plasma membrane, but also in the much larger area of intracellular membranes. Energy is also needed for cytoskeletal movement, for anabolism, regulation, etc. So, energy shortage remains a problem. If one considers that the R-state of a cell in the AIH perspective is a (metastable) physical equilibrium not consuming energy (see point 4 below for experimental proof), the problem of energy shortage is entirely taken away. • (3) Investigators in several laboratories (summarized in [12, 18, 19, 25]) observed Na+ exclusion and K+ accumulation in red cell ghost preparations upon addition of ATP, but only when these were prepared according to the method of Bodemann-Passow, which yields ghosts that are not pure membrane but are filled with cell-matrix (mainly hemoglobin, some actin and minor components). In contrast, when cytoplasm-free red cell ghosts consisting of pure membrane are prepared using methods with more thorough washings (discussed in [12, 18, 19, 25]), pumping activity was never observed. Obviously, this indicates that something in the cytoplasm is necessary for establishing ATP-dependent gradients [11, 12, 18, 19, 23-26]. • (4) In 1978 Ling [51] conducted a long series of experiments with so-called ‘EMOC preparations’ (EMOC = effectively membrane-(pump)-less open-ended muscle cells). Isolated resting frog Sartorius muscles were cut in the transverse direction and the membrane-less open-ended part was inserted into a Ringer solution, while the upper closed and membrane-delineated part was kept in the air. At the open-ended part in contact with the Ringer there are no membrane-bound pumps because there is no membrane, while in the upper part pumps present in the membrane cannot work because they are not in contact with the external Ringer solution (this was controlled). The intracellular concentrations of Na+ and K+ in the upper part of the cell remained unchanged for at least two days, despite the huge leak at the open end. Radioactively labeled 40K+ in the Ringer revealed that during this long period there was indeed transport of the label from the Ringer into the upper part of the cells, proving the existence of a K+ influx in the absence of effective pumps. The conclusions are surprising but straightforward. Since pumps must be ineffective in these EMOC-preparations, the constancy of the total concentrations of Na+ and K+ in the upper part of the cell over two or more days cannot be due to a steady state, as MT proposes. There is only one alternative: they must be due to a true physical equilibrium. This also means that neither pumps nor an intact membrane are necessary to maintain this situation. Classical cell energetics clearly needs to be replaced by a completely different mechanism consistent with these observations (or at least using this possibility as an additional control for experiments on "membrane” transport). From previous sections, a sound explanation of the EMOC experiments as conducted by Ling can be readily inferred. According to Ling, the intracellular Na+ concentration remains low despite direct open contact with a high Na+ concentration Ringer because Na+ is very poorly soluble in intracellular water that is polarized by the cytoplasmic colloid (interconnected cell-matrix); and the intracellular high K+ concentration remains high despite open contact with a low K+ concentration Ringer because K+ is largely adsorbed to the β- and γ-carboxyl-groups of the proteins of the same cytoplasmic colloid. He gathered substantial evidence for these two proposals (see earlier) [11, 12, 18-26, 51]. Together, these two effects (adsorbed water and K+) generate a physical equilibrium that has no need of a special membrane or of ion pumps located in it. Of course, an intact cell has a plasma membrane and hence these ions must pass through this membrane, but the function of proteins such as Na+/K+-ATPase must be different from that assumed by MT (see later). These EMOC experiments seem impossible to reconcile with a steady state theory and explain why the problems described above under points 1 to 4 exist. Instead of applying the Nernst equation to the total ion concentrations, as MT does, Ling [11, 12, 18, 19, 25] developed a new set of formulae in which distribution coefficients, diffusion effects and adsorption coefficients figure and total concentration is distinguished from activity (the free fraction). In contrast, MT does not even have a formula describing observed concentrations and is often careless about the concepts of concentration and activity (instead of explaining their physical nature). • (5). When authors (without exception) tried to prove the pumping function of Na+/K+-ATPase experimentally they were biased by the belief that this is the real function of this enzyme. It was like asking the Pope to prove the existence of God. Sorption processes were not considered as a real alternative to the ion-pump in any study devoted to the transport function of Na+/K+-ATPase (see [45] for review). Nevertheless, there has been abundant direct evidence for a bound state of K+ in living cells at least since the early 1980s (see [70] for review). • (6) It is always necessary to get important evidence from the opposite approach: namely, proofs indicating that all other explanations can be dismissed (as Ling does for MT). However, a large number of experiments together with a firm theoretical background point towards a serious alternative explanation for the actual Na+ and K+ distributions, which to date has remained effectively unchallenged. Each challenge was followed by a rebuttal [20-26]. • (7) Using EMOC preparations Ling [12, 18, 19, 25] made another crucial observation. In older EMOC preparations the cell-matrix at the open end gradually started to degrade. In the degraded parts, the Na+ concentration was higher and the K+ concentration lower than the normal cellular concentrations found in the non-degraded parts further away from the open end. Ling was able to observe a good correlation between local ion concentrations and local ATP concentration. It was found that a minimal ATP concentration is needed to keep water sufficiently polarized to maintain a low Na+ concentration and to keep K+ sufficiently adsorbed to maintain a high K+ concentration [12, 18, 19]. This relationship was beautifully addressed by Kellermayer’s group [121-123]. Red blood cells have a very narrowly-defined function. This probably explains why intracellular erythrocyte Na+ and K+ concentrations vary widely among different species and even within members of the same species, although the concentrations of these ions in the blood are very similar. In human red blood cells, Na+ and K+ concentrations are normal as in other cells. But dog erythrocytes contain 10 times more Na+ than K+. If MT is correct, the cause of this aberration has to be found in the working of the Na+/K+ pump. If AIH is correct, the cause should be an abnormally low ATP concentration. Although a relatively good correlation exists between the activity and/or the amount of Na+/K+-ATPase and the steady state levels of Na+ and K+ [122], the uptake rate of Rb+ (a widely used K+ analogue) is much higher than its leakage rate, irrespective of the kind of red blood cells [123]. This compromises any tight coupling between pump activity and leakage rate and therefore the whole steady state idea. In contrast, in the different kinds of red blood cells studied, there is a clear relationship between ATP concentration, Na+ and K+ content and the intracellular K+/Na+ concentration ratio. All this makes it clear that ATP is the crucial factor for maintaining the physical equilibrium, but as the experiments under points (1) and (7) demonstrate, ATP does not need to turn over by means of hydrolysis in order to keep the ion concentrations constant. Thus, the manner in which ATP energizes cellular systems has to be re-addressed, though MT has kept Lipmann’s idea of ‘hydrolysis of ATP’s high-energy bond’ on the stage for more than 70 years. 7. The Mechanism of Energy Delivery by ATP A central topic associated with the working of the Na+/K+ pump is how ATP energizes this system and other systems as well. The view offered by MT differs in a most fundamental way from that of Ling in his AIH. According to MT, energy is derived from the hydrolysis of ATP. MT followed the idea of Lipmann [8] that the phosphate-anhydride bond of ATP is a high-energy bond able to inject its binding energy into the system at the moment of hydrolysis. This would mean that the initial stages of strong actin-myosin binding constitute a higher energy state than dissociated actin and myosin in the relaxed state (see: the discussion on muscle contraction given before [75]). This is exactly opposite to the view of Ling [12, 18-26, 51]. According to Ling’s AIH, two properties of ATP are responsible for its mechanism as an energizer of cellular processes: (1) it can bind to cardinal binding sites on some proteins of the cell-matrix, whereby the adsorption energy lifts the system into a metastable high-energy low-entropy R-state (the R-state is high-energetic); (2) it can also be eliminated from these binding sites by the ATPase activity of these proteins. The latter property is important for the R-to-A transition involved in power output. Concerning the first property, the free energy of adsorption is huge: -39.7 kJ/mole for adsorption on (A-state) actomyosin [124] and -46.0 kJ/mol for adsorption on G-actin [125]. The equilibrium on G-actin adsorption constant in the case of (A-state) actomyosin is 9.84x106 [126], meaning that only 1 ATP molecule out of 9.84 million remains free. This also means that ATP does not have the opportunity to travel a long distance in the in vivo situation of macromolecular crowding. ATP generating enzymes must therefore be located in close vicinity to systems utilizing it. The role of ATP adsorption in restoring the R-state is well-established, most obviously in the case of unlocking rigor of actomyosin [76]. The stability of the resulting ATP-cell-matrix ensemble was proved by the experiments of Ling on EMOC preparations [51] and those of Clegg [52] on Artemia cysts. However, some discussion may remain about the exact moment of enzyme-bound ATP hydrolysis: before or after initiation of the R-to-A transition. Concerning the second property of ATP, most proteins that can adsorb ATP also possess ATPase activity. But the moment of hydrolysis may not be regarded as ‘the’ energy injection. First of all, this would neglect the high adsorption energies mentioned above. Secondly, enzyme-bound ATP hydrolysis is often close to equilibrium [127, 128]. In classic views the latter observations are not denied, but lead to possibly unrecognized confusion (see the controversy above about actomyosin formation, which is sometimes regarded as a mysterious ‘Maxwell’s demon’ [129]); a consequence of not having a holistic approach to cell physiology and of having ignored Ling’s AIH for so many years. Adsorption constants for ADP and Pi are much lower than for ATP. This allows ADP and Pi to dissociate, which is actually exergonic. So, energy stored in the high-energy low-entropy R-state can be tapped and transformed into work during the R-to-A transition in a stepwise manner and concomitantly with Pi and ADP liberation. Ling’s mechanism for ATP, which is intimately linked to the high-energy low-entropy status of the R-state, corroborates and adds detail to Schrödinger’s idea: ‘Life feeds on negative entropy’. After the above discussion of the properties of ATP, current models of the Na+/K+-ATPase are surrounded with questions. Why is nothing said about the adsorption energy of ATP? It is likely that the high energy state is achieved at the moment of ATP binding and not during the next step, when a phosphate group is transferred from ATP to an aspartate residue on the protein. How the reaction step, not really a hydrolysis but a transferase activity, could be an ‘energy injection’? Why are the effects of structured water at the membrane surface not taken into account? Given the data of Wiggins [61, 62] on ATP synthesis by means of changing the water structure, and given the idea that changes of water structure are likely to produce large differences in free energy on entropic grounds, this question should elicit immediate experimental verification. Why is it not seen that changes in binding preferences (affinities) for one kind of bound ion in favor of another (for instance Na+ instead of K+ or the reverse) is a quantitatively well described process in terms in Ling’s AIH, which states that these (observed) affinity changes are brought about by inductive effects exerted by cardinal adsorbents? The two affinity changes that occur during the transport cycle of the enzyme could be explained by changes from an electron withdrawing effect to an electron donating effect; and this has already been demonstrated. Indeed, under experimental conditions, Na+/K+ ATPase can be made to synthesize ATP. It is usually said that a sufficiently large gradient of Na+ and K+ is required to achieve this. However, Post et al. [130] succeeded in synthesizing ATP with a purified protein without an ion gradient. They concluded that ‘binding of sodium ion to a low-affinity site on the phosphoenzyme formed from inorganic phosphate is sufficient to induce a conformational change in the active center which permits transfer of the phosphate group to adenosine diphosphate’ [130]. The step of phosphorylation of the enzyme by Pi requires the presence of Mg2+ and K+, while the subsequent step of ATP synthesis from ADP and the phosphoenzyme is inhibited by Mg2+ and stimulated by Na+. By just reversing causes and effects one obtains the process in the ATPase direction. These findings, although they were not inspired by the AIH, completely shift attention from MT (based on ionic gradients) to the AIH (based on adsorption and its inductive effects) [18]. Why are the properties of ATP in a colloid environment not taken into account? ATP is highly soluble in ordinary water, as every biochemist discovers when he prepares an assay mixture. But oddly enough, the broader physical properties of ATP, which determine its behavior in colloidal ‘systems’, are recognized by neither chemists nor biologists. No one knows how ATP is distributed among the components of some commonly-used colloidal systems (cells, cell ghosts, vesicles reconstituted from purified Na+/K+-ATPase). All that is known at present is that ATP is amphipatic: log P = 1.64, where P is the partition coefficient of ATP in an octanol/water system [131] (for comparison, hexanol log P = 1.46 [131]). ATP’s amphipaticity indicates that it does not just swim in a solution, but must be distributed among the components of any colloidal system according to local hydrophilic respectively hydrophobic properties. Cells, cell ghosts and reconstituted vesicles with built-in-Na+/K+-ATPase all contain lipids and enzyme molecules that also have hydrophobic domains (see [132] for review). Since it is not known how the amphipaticity of ATP affects the properties of the colloidal systems studied, the data in favor of the pumping function of Na+/K+-ATPase cannot be considered as proof; nor can those on reconstituted Na+/K+ vesicles, commonly regarded as strong evidence. Just recall the experiments of Wiggins and MacClement [62] on ATP synthesis with cellulose acetate films. What then is the function of Na+/K+-ATPase? The earlier discussion about the experiments with red cell ghosts (section 6: point 3) clearly indicates that Na+ exclusion and K+ accumulation depend on something located in cytoplasm, which depends on ATP. In the case of cell-matrix-filled red cell ghosts a good guess can be made. Ling showed that hemoglobin together with actin (both adsorb ATP) and a yet to be determined minor cell-matrix component are responsible for the amounts of Na+ and K+ in the red cell ghosts [12, 18, 19, 24-26]. The results obtained by the group of Keller-mayer on entire red blood cells [121-123] and Ling’s experiments with EMOC preparations [51] confirm that the cell-matrix is responsible for the amounts of both ions, and that their concentrations do not depend on the working of membrane-located Na+/K+-ATPase. However, in intact cells, Na+ and K+ ions have to pass through the plasma membrane and this suggests that Na+/K+-ATPase is responsible for the flux through the membrane, though not for the intracellular concentrations. It may even be a regulator of the flux, as indicated by the more recent insight that Na+/K+-ATPase has a receptor function [14, 133]. This function agrees with observations of its connectivity to the cytoskeleton [134, 135]. Its function is definitely not to maintain steady state energetics, as differences in Rb+ uptake and release rates in different types of red blood cells demonstrate [123]. In the latter study the conclusion was reached that Na+/K+-ATPase is involved in the uptake of Rb+ (K+), which can be inhibited by ouabain, but that the Rb+ (K+) distribution/content is not affected either by this enzyme or by ouabain. This indicates that the Rb+ (K+) distribution/content may be primarily defined by adsorption. Na+/K+-ATPase is therefore not an important player with respect to cell energetics, but cytoplasmic K+-adsorbing and water polarizing ATPases belonging to Ling’s cell-matrix are. Cell energetics is determined by the cell-matrix. Therefore, it may be important to clarify Ling’s concept of the ‘cell-matrix’ further. The results with cell-matrix-filled red cell ghosts and with EMOC preparations indeed suggest that something comparable to a pumping function (in the sense of Fröhlich) is connected to the cell-matrix. But there are four main differences from how MT sees pumping: (a) the activity is located in the cytoplasm with only a minor contribution from the plasma membrane; (b) the activity is only needed for a very short period, at the moment of the A-to-R transition, not all the time; (c) during the R-state a physical equilibrium is established with regard to ion and other solute distributions, instead of a ceaselessly energy-consuming steady state; (d) the energy stems from ATP adsorption and not from ATP hydrolysis [136]. Ling’s [12, 18-26] definition of the cell-matrix as given earlier is a functional one. He is not primarily concerned with its morphology or composition. Certainly many proteins belonging to the cytoskeleton participate in the functions of the cell-matrix [18, 111] and many are ATPases or GTPases, but monomeric actin and tubulin, which may be only loosely interconnected, also polarize water and adsorb K+ [18], while they may act cooperatively via interspersed polarized water. Some membrane proteins and membrane surfaces also qualify for his definition [18]. So, there is no problem in seeing Na+/K+-ATPase as part of the cell-matrix, but this implies reckoning with the colloid context and the properties of associated water, and being open-minded with respect to membrane models [18]. In more recent work, Ling has coined the term ‘nano-protoplasm’ [24, 26] (Matveev [132] used the term ‘physiological atom’: the minimal structure that preserves the characteristic properties) in order to place more emphasis on the fact that the structural and functional units of life do not necessarily take up the entire cytoplasm or even large parts of it. They can be rather small. In a red blood cell, hemoglobin together with actin and the yet to be characterized minor protein may conform to the concept of cell-matrix and a minimal amount of it can yet be considered as nano-protoplasm. Fundamental to the definition of cell-matrix and nano-protoplasm is that its components – proteins, water and ions – possess ‘the ability to exist as coherent assemblies in either one of two alternative states, the resting and active living (or dead) state’ [24]. This definition means that at a given moment certain parts of the cell may be active while others are inactive. Because during the period of activity new ATP is generated, and because in most cells many parts are active at a given moment, a cell may be continuously busy with ATP production. This situation may give a false impression of something like a steady state, fueling the metaphor of the burning candle. It may obscure the fact that some parts are literally inactive and hence not consuming energy at that moment. In that regard, studies of myofibrils or Artemia cysts are very instructive because a complete R-state can be obtained. It should be realized that the R-state is difficult to study, because without specific measures important properties of the cell-matrix are lost by most in vitro preparatory methods and by the fixation methods used for light and electron microscopy. This makes correct interpretation of many experiments and also of many protein X-ray diffraction analyses very difficult. This has certainly helped to keep MT on the road. Perhaps MT is a good theory of a cell with a destroyed cell-matrix, capturing some aspects of the A-state but worthless for the R-state. To conclude, Ling’s proposal [12, 18-26, 46] that it is not the hydrolysis of ATP that energizes cellular processes, but its adsorption to certain cell-matrix proteins, establishing a long-term (meta)-stable R-state without further energy consumption once established, is amply proven by his EMOC preparations. The red line on Fig. (1) depicts his view of cell energetics. The question to be addressed now is whether or not his energy model is complete? According to Ling it is. However, in his AIH almost nothing is said about globular proteins, although he agrees that they occur in vivo. Does their existence makes a difference for cell energetics? We think it does, although only in a secondary way. In our view thermodynamics of globular proteins can be described by the green line in Fig. (1); and this line has to be added to the view presented by Ling. Our view is based on Matveev’s recent ‘native aggregation hypothesis’ (NAH). 8. Matveev’s Native Aggregation Hypothesis (NAH) Three aspects of Matveev’s NAH [15] are of prime importance for this article. First, there is the work done by the school of Nasonov, to which he belongs, on some physical properties of the R-state and the A-state, demonstrating that R-to-A and A-to-R are indeed phase transitions. In this work the meaning of R- and A-states and their transitions is the same as in Ling’s AIH (and incompatible with MT). Secondly, there is his idea of native aggregation as the universal mechanism for all kinds of physiological processes in living cells. Thirdly, there is his division of native proteins with respect to their in vivo conformation during the R-state into two well-defined groups from both a structural and an energetic point of view, the first group yielding the red line and the second group yielding the green line of Fig. (1). Addition of the second group allows the general outlines of an energy model, which fundamentally differs from the steady state model of MT, fully adopts Ling’s entropic model, but adds some aspects linked to globular proteins. Nasonov, like Ling, belonged to the small group of scientists who continued to believe in the bulk-phase nature of cells. Like Ling, he also studied the R-state of a living cell. This is a much-neglected field of inquiry, probably because nothing happens during the R-state and hence there is not very much to measure. The only way to learn about it is by comparison to the A-state. In this comparison some general physical parameters can be followed. Much of the work done appeared in the Russian language and as such it is largely unknown in the West. In his review of 2010 (in English), Matveev [15], a follower of Nasonov and also of Ling, summarized some main results of Nasonov’s school, adding his own results and making the link with the work of Ling. The school of Nasonov [10] studied the remarkable phenomenon that very different physical (pH, hydrostatic pressure, mechanical action) and chemical stimuli, when their intensity exceeds a certain threshold, are able to turn a resting cell (R-state) into an active one (A-state). It was found that all these different stimuli produce a similar set of general non-specific simultaneous changes in the physicochemical properties (viscosity, turbidity, hydrophobicity, structure) of very different types of living cells. Matveev [132] calls it ‘a universal reaction of the living cell’ or ‘protoreaction’. Activation of the cell causes a biphasic transition. Macroscopic viscosity first decreases and then increases above the resting level; cytoplasm first becomes more transparent, then more turbid, exceeding the resting level; hydrophobicity of cytoplasm as studied by means of vital stains first decreases and then rises above the resting level; etc. The first step can be considered still part of the R-state, being a phenomenon below the activation threshold (as is the case for generator potentials in neurons). The second step is initiated when the cell’s threshold is exceeded. It was called the ‘phase of excitation’ and corresponds to the R-to-A transition discussed in this article. At the structural level, when a living cell is viewed in the light microscope, extensive optically empty regions look quite homogeneous while additional structures appear upon excitation, which are also seen after fixation. There is therefore some similarity between the excited state and the death (fixed) state. Both Ling [12] and Matveev [15] consider the death state to be an irreversible ‘over-activated’ state, whereas the physiological active state is always reversible. A pathological state, such as in cancerous cells, is in between: an over-activated yet reversible state (see also [104, 117]). In order to see the rather structure-less R-state in the electron microscope, special precautions are needed. A remarkably homogeneous and very fine gel-like structure can then be observed, described by Porter et al., [137] as the ‘microtrabecular lattice’. In contrast, procedures used in classical electron microscopy reveal some kind of over-activated death state [15, 71, 137] characterized by much coarser structures. All these properties of the activated situation in contrast to the resting situation are very reminiscent of what happens during in vitro aggregation of isolated (globular) proteins [138-140]. Hence, Nasonov formulated what he called ‘the denaturational theory of cell excitation and damage’, which proposes that the cause of all the observed changes in physicochemical properties during cell activation has to be sought in stages of protein denaturation occurring in vivo. Misunderstandings may arise because of the term ‘denaturation of proteins’. In many textbooks the word ‘denaturation’ is applied to globular proteins. In that case, denaturation proceeds by consecutive steps of unfolding until a completely unfolded random coil is obtained. This process is endothermic. Heat-denaturation is one of the methods for achieving it. According to Ling’s AIH the cell-matrix during the R-state consists of a protein-water-ion complex with ‘fully-extended’ proteins. The structure of Porter’s microtrabecular lattice [137] may be compatible with this notion. Matveev [15] gives a new quite literal definition of ‘denaturation’, which allows the term also to be applied to the latter group of proteins. This new definition is intimately linked with the not yet widely-known concept of the R-state. The definition is derived from the idea that the ‘native’ or ‘natural’ state of a protein is how it occurs in the in vivo R-state. For a natively tightly-folded globular protein the act of denaturation then corresponds to the stages of unfolding. However, for a natively fully-extended protein, the act of denaturation corresponds to the process of folding. Matveev [15] expresses the latter idea as follows: ‘In the native state the key cell proteins are inert, non-reaction-capable; they do not interact with each other or with other biopolymers’. That all proteins may be inactivated during the R-state is proved by the observations of Clegg on Artemia cysts [52]. In these cysts even glycolysis and respiration must have come to a halt: zero activity. And now the new definition: ‘Loss of the state of inertia is denaturation’ [15]. When a fully-extended protein denatures (during R-to-A transition) temporary secondary structures appear and these structures are ‘reaction-capable’ in a specific way. Specificity suddenly appears on stage. On the other hand, a tightly-folded globular protein is inert and non-reactive because its reactive structures, since they possess hydrophobic regions, are hidden in the interior of the globule. Upon activation they undergo partial melting producing an intermediate state, whereby the reactive structures become exposed, ready for specific interaction. So, when a tightly-folded globular protein partially denatures during R-to-A transition, this happens by slight melting, just enough to expose their reactive secondary structures and make them ‘reaction-capable’. To begin with, the above description indicates that with respect to the process of denaturation and activation (the loss of the inert R-state), proteins fall into two well-defined groups: the natively fully-unfolded proteins described by Ling and hereafter called ‘NUN-proteins’; and the natively tightly-folded proteins, corresponding to the more classical picture of a protein, at least a few decades ago, hereafter called ‘NF-proteins’. This division is of prime importance for reaching an understanding of cell energetics because: (1) although the classical way of thinking about in vivo proteins has evolved quickly, it nevertheless remains strongly biased by MT and is incompatible with coherence and the in vivo state of water and K+, so wrong ideas about cell energetics are likely to persist for some time ; (2) Ling’s view takes away these problems with coherence and the in vivo state of water and K+, but in his cell-matrix concept he only describes NUN-proteins, devoting only a few sentences to globular proteins. For instance, in his 1992 book ([19], p. 72), he makes a clear distinction: ‘In globular native proteins, hydration is limited to polar side chains. … In fibrous proteins, some of the backbone NH-CO groups are not locked in intra-macromolecular H-bonds, and are thus free to sorb water. … The polar side chains, as a rule, sorb relatively small amounts of water’, whereas a few pages further (p. 100) he states about NUN-proteins:’ ‘Both water sorption in fully-extended proteins and that in living cells follow Bradley’s multilayer-adsorption’. This reveals why he does not discuss NF-proteins very much. Indeed, in order to explain water-sorption by the entire cell, NF-proteins are rather irrelevant. He calculated that if all proteins were globular in the native state, only around 6.25% of cell water would be influenced. However, it will be argued below that NF-proteins do have a contribution in cellular energetic, although it is a secondary one. Energetically the existence of two groups of proteins during the R-state, NUN- and NF-proteins, also follows from caloric measurements of protein unfolding. These studies reveal that the fully-folded state has a stability of between +20 to +80 kJ/mole [54] compared to the fully-unfolded state and so the unfolded state needs an external energy input in order to exist. In view of the AIH, the above energetic values make it clear that proteins that are closer to the lower value of +20 kJ/mole could exist in a largely unfolded state with an energy input derived from the adsorption of ATP, which is in the order of -45 kJ/mol [45, 124, 125]. These are NUN-proteins, characterized by a high-energy level. Their activation leads to folding, which is an exothermic process and allows power output. Owing to the adsorption and polarization of the surrounding water, the inactive ensemble of ATP-protein-water-ion has very low entropy. So, the power output is driven by entropy. Evidently, for proteins closer to the higher value of +80 kJ/mol, a fully-folded globular in vivo state is more likely. This state is stable by itself because it occupies a minimum energy well. Their active intermediate state, however, remains at a higher energy level and hence needs energy input (activation energy). These are the more classic globular or NF-proteins. Modern protein biophysics supports the existence of rather stable and reproducible intermediate states separated by a threshold from other states, and situated between the two extremes of the tightly-folded globule and the fully unfolded protein [55]. Dunker et al. [141] proposed the concept of the ‘molten globule’ and Uversky [142] of the ‘premolten globule’. Since the field is currently developing, preference is given here to the more general term ‘active intermediate’ state(s)’, particularly because the NAH deals with any kind of physiological process and not all cases need be similar. NAH holds that globular proteins are inactive in their most compact fully folded state hiding their reactive sites (the native state) and are activated by transition to a more loosened intermediate state, whereby the reactive sites are exposed to the surroundings and become reaction-capable. Activation of globular proteins by loosening their tightly folded state is evidently endergonic. It can be concluded that since the existence of globular proteins in vivo cannot be denied, neither does the endergonic nature of their activation, the green line in Fig. (1) has to be added. What counts in favor of the concept of native aggregation is that (1) a given protein can exist in an inactive inert state and in one or more active functional states; (2) in the active state(s) interaction-capable secondary structures appear and can be exposed to other activated biomolecules (not necessarily only proteins, actually any kind of ligand) so that physiological activity can take place temporarily. The rationale of the NAH lies in the facts that (a) the specificity required for each individual kind of physiological process during the A-state does not exist during the R-state but appears at the beginning of the R-to-A transition through the mechanism of protein denaturation (new definition); (b) despite the many kinds of different stimuli and the huge variety of specific physiological responses, the common mechanism of protein denaturation (new definition) always goes along with the same non-specific or universal changes of all physicochemical parameters associated with a protein-water-ion phase transition; and (c) that the formation of functional aggregates is not the result of a random Brownian process (as described in the literature on IDPs [56]), but is an organized event in agreement with the literature on coherence (see before) and with Cosic’s resonance recognition model [83, 84]. For a further more detailed treatment of these concepts including a number of clarifying examples reference [15] can be consulted. Summarizing in Matveev’s words [15]: ‘Two extreme protein states can be identified: the completely folded (the globular protein) and the completely unfolded states. Between these inactive (native) states, numerous intermediate, active forms can exist; it is these forms that produce native aggregation.’ 9. A New Proposal (Model) for Cell Energetics This new proposal, based on Ling’s AIH and extended by Matveev’s NAH, is depicted in Fig. (1). The red line is for NUN-proteins (see before: section 4 point 1) and the green line for NF-proteins. Below a complete overview for each stage from R over to A and back to R is given. The R-state Considering the entire living cell, the R-state is a metastable high-energy low-entropy state, stable over long periods of time, as EMOC preparations [51] and Clegg’s work on Artemia cysts [52] made clear. The term ‘metastable’ is used to indicate the stability of the high-energy character of the R-state, but also its potential to fall into lower-energy states, because it is very far from thermodynamic equilibrium (see [143] for details). However, as Ling [51] pointed out with his EMOC preparations, it is nevertheless a physical equilibrium, in need of neither pumping activity nor a continuous energy supply, as MT erroneously believed. The R-state can exist in a high energy basin, so a small amount of external energy is needed to leave this basin. This small amount can be regarded as a kind of activation energy, delivered for instance by the adsorption energy of activating cardinal adsorbents such as Ca2+ (or another activator). The amount of energy introduced by the external stimulus must be sufficient to reach the threshold; if not, the R-state persists. In the late 1960s, Prigogine [144] developed the thermodynamics of non-linear systems far from equilibrium and characterized by dynamic behavior. He discussed living systems from his new point of view. During activity (R-to-A and A-to-R transitions), living systems are indeed dynamic, so his thermodynamics could be applied. If MT were correct, these thermodynamics would also apply to the R-state, since MT regards the R-state as a steady state in need of continuous pumping activity, i.e. as a dynamic system. However, as explained above, Ling demonstrated beyond doubt that MT is wrong and that the R-state is a metastable static state, in physical equilibrium but far from thermodynamic equilibrium. This means that the thermodynamics of Prigogine cannot be used to describe the R-state. Only very recently, Prokhorenko and Matveev [143] derived the thermodynamics of static (metastable) non-linear systems far from equilibrium. They were developed from first principles and immediately applied to Ling’s description of a cell’s R-state. Ling’s R-state has therefore been placed on a firm thermodynamic foundation. Proteins making up the R-state must now be dissected into the two groups described above. NUN-proteins making up Ling’s cell-matrix are responsible for the fact that the R-state is a metastable high-energy low-entropy state. The stability of their association with bound ATP is responsible for the static character of this state, whereas the process of ATP binding during an A-to-R transition is responsible for lifting the system into the high-energy low-entropy basin. NF-proteins are stable on their own during the R-state, since they exist in the fully folded conformation, which for proteins represents the minimum energy basin. Summation of the energy levels of the two groups gives the high-energy level of the entire system in the R-state. It equals the energy level of NUN-proteins minus the energy level of NF-proteins. The result of this summation in all cases remains in favor of a net high-energy level. R-to-A transition R-to-A transition is initiated by an external stimulus or internal signal of sufficient strength to overcome the thresh-old. It is relayed to the appropriate intracellular site mostly indirectly, for instance mediated by Ca2+. The latter in turn acts on NUN-proteins that bear cardinal adsorption sites for activation (for instance Ca2+-binding sites). The strong inductive effect exerted by this binding is auto-cooperatively transmitted through the cooperatively-linked cell-matrix (NUN-protein system) and unleashes non-specific effects associated with the R-to-A phase transition, together with a whole range of specific functional effects, which are dependent on specific interactions among specific proteins (see NAH). Again, the two groups of proteins have to be distinguished. Folding of NUN-proteins is exergonic, whereby dissociation and depolarization of associated water and desorption of bound K+ are responsible for a large increase in entropy. Power output is generated from the stored low entropy. The slight unfolding of NF-proteins into an active intermediate conformation, however, is endergonic. It is proposed in this new model that the necessary energy is transferred from the exergonic reaction of NUN-proteins to NF-proteins. Mechanisms for such energy transfer will be given at the end of this article. The net exergonic nature of the R-to-A transition for the entire system is equal to the energy liberated by NUN-proteins minus the energy uptake by NF-proteins and always has a net exergonic character, so that power output is guaranteed. Native aggregation is a broad concept dealing with the temporary formation of active functional complexes of proteins with any kind of ligand (other proteins, polynucleotides, substrates, ions, etc.) at the onset of R-to-A transition [15]. Formation of an aggregate of two or more NUN-proteins (for instance, actomyosin formation) is exothermic, of a NUN-protein with a NF-protein is net exothermic. Interaction between two NF-proteins is not excluded, but energetically it is only possible after the activation energy for both proteins is delivered in some way, whereby ultimately this energy must come from the R-to-A transition of a NUN-protein system or directly from an external stimulus. The A-state In the A-state NUN-proteins and NF-proteins are in a quite different situation. Actin and myosin can be taken as examples of cell-matrix (NUN-) proteins [12, 14, 18, 70, 73, 134]. They possess unfolded regions [14, 75], polarize water [14, 18, 73], adsorb K+ [68-73], possess cardinal adsorption sites for ATP and exhibit ATPase activity [12, 14]. It was shown that upon activation of myofibrils weak binding between the S1-head of myosin and actin is followed by the release of Pi , allowing strong binding, which in turn leads to the power stroke and only then to ADP-release [145]. Together, these consecutive steps represent the R-to-A transformation, which is an exergonic dynamic process. After dissociation of ADP the system settles into its stable (static) low-energy basin, the A-state (rigor) with myosin heads free of nucleotides. Myofibrils will remain in this rigor state unless ATP is added [76, 145]. This fact illustrates the static and stable nature of the A-state as far as it concerns NUN-proteins. In contrast, during the A-state NF-proteins may remain active as they occur in a high-energy functional state, described above as an active intermediate state. The energy to lift NF-proteins into their active intermediate state can be considered as a kind of activation energy and should be delivered either directly from the stimulus or from the R-to-A transition of NUN-proteins. Mechanisms of energy transfer are described below. Summing up, the A-state has both a static low-energy component of NUN-proteins and a dynamic (relatively) high-energy component of NF-proteins. The high-energy level of NF-proteins during the A-state is, however, lower than the high-energy level of NUN-proteins during the R-state, so that the R-to-A transition is invariably exergonic. A-to-R transition Glycolysis and respiration generate new ATP. Some ATP is needed to activate these pathways (for instance hexokinase and phosphofructokinase reactions) before a larger amount of ATP can be generated. Therefore a cell always needs a minimal amount of ATP (see also EMOC experiments). In a healthy cell the minimal requirement for ATP remains sufficient during the A-state so that metabolic activity generates more ATP, lowering the ADP/ATP ratio. This new ATP quickly adsorbs to NUN-proteins of the cell-matrix because of the very high adsorption coefficients for these proteins (for actomyosin: 9.8x106 [46]). The adsorption energy of ATP (near 45kJ/mol) lifts NUN-proteins of the cell-matrix back into their inactive (meta)stable high-energy low-entropy basin of the R-state. As stated earlier, Ling made it unequivocally clear that the adsorption energy and not the hydrolysis energy of ATP is responsible for this A-to-R transition [12, 18-26, 46]. NF-proteins may return to their low-energy inactive fully folded state by means of several mechanisms discussed below. Their inactivation per se is exergonic, but eventually a small amount of energy is needed to reach the border of the high-energy basin they occupy. Whether or not this small amount is needed may depend on the particular case; therefore, two dotted green lines are drawn for this stage on Fig. (1). It depends whether the activated state is a real energy basin or a transient that needs only activation energy to start it, in which case it then flows exothermically downwards. Energy Transfer between NUN- and NF-proteins While the need to activate NF-proteins during R-to-A transition is self-evident, as explained above, the source of the energy is rather speculative in the sense that there are many possibilities and there might be different mechanisms for different cases. Activation energy for NF-proteins may be delivered either (a) rather directly from the stimulus itself or (b) indirectly from the stimulus through the intermediary of a NUN-system. In the latter case, the question is: how is energy transferred from a NUN-system to a NF-system? First, this energy transfer may occur when direct conformational contact is possible between proteins of the two groups. Many globular (NF) proteins are thought to be associated with elements of the cell-matrix (NUN-proteins) [111], enabling such direct transfer. Molecules of some proteins have both NF-segment(s) and NUN-segment(s) at the time [58]. Alternatively, energy could be tapped from the immediate surroundings. Indeed the most penetrating aspects of the R-to-A phase transition are due to the NUN-system. The local physicochemical environment is changed so profoundly and in an exothermic entropy-generating way that a nearby inactive NF-protein has plenty of energy in its immediate vicinity to be activated. The cooperative transformation of Ling's resting protein-water-K+ complex indeed results in a change of the physical conditions in protein microenvironments. The region that was previously in its R-state becomes more permeable. Diffusive fluxes of substances appear and with them come gradients of thermodynamic potentials: the higher the degree of cooperativity, the greater those potential gradients. In the case of ions, electric diffusion potentials are generated. These potentials can be energy sources for any kind of biological work including melting of some globular proteins. Proteins have side groups that vary greatly in polarity. This means that proteins are surface-active compounds. Living cells, as colloidal systems, exhibit many local phases and interphase boundaries. Interphase boundaries are surfaces for the adsorption of various substances, including proteins. Phase transitions fundamentally change the distribution of substances in the A-state as compared to the R-state. This may lead to changes in adsorption/desorption. The energy associated with these adsorption/desorption processes may easily destabilize a protein’s R-state and lead to melting of NF-proteins or to folding of (a new portion of) a NUN-protein. Energy transfer over longer distances is also possible, for instance by shuttling of allosteric modulators, second messengers, or ions. Some other mechanisms of energy transfer over longer distances are often of a coherent type: short or long distance transmission of energetic pulses along auto-cooperatively linked protein systems such as microfilaments or microtubules. Davydov-solitons and electro-solitons have been suggested for energy transfer along these transmission lines [92-94]. Biophotons may constitute another mechanism for long-distance transmission and coordination [103-105]. At a more local level Cosic’s resonance recognition model [83, 84] may be of particular importance. Glycolysis, Respiration and the K+-shuttle The energy model depicted on Fig. (1) is finalized with a proposal of (a) the likely mechanism of activation of the ATP regenerating system as the R-to-A transition proceeds; and (b) the mechanism of its de-activation as the R-state is reinstalled. A causative time sequence of events is proposed as the most obvious mechanism. In this proposal the R-to-A transition of for instance actin and myosin starts earlier than the R-to-A transition for ATP synthesis, because the latter has to wait upon liberation of ADP from actin [146] and myosin [145, 146]. Similarly, subsequent adsorption of newly formed ATP to actin and myosin [124, 125, 145, 146] restoring the R-state of the latter proteins precedes the inactivation of the ATP synthesizing machinery, because the latter has to wait the lowering of the ATP/ADP ratio caused by the ATP adsorption process (see Fig. 1). This proposal might look strange when one has the impression that ATP and ADP are distributed rather homogeneously in ‘cytosol’ and that ATP synthesis and consumption take place continuously, be it at a physiologically adaptable speed. However, the co-existence of R- and A-states simultaneously in different parts of the cell together with the amphipatic properties of nucleotides necessitate to leave the idea that free ATP is present all over the ‘cytosol’ and is synthesized permanently. The high value of the adsorption coefficient of ATP to certain proteins makes its free concentration in cellular regions, which are in the R-state almost nihil; and glycolysis is likely to be halted in that region. At the end of R-to-A transition the sudden local liberation of ADP must lead to a local quantitative conversion into new ATP, which then adsorbs fast and locally. In this story K+ may play a primary role. The key regulatory function of K+ during the R-to-A-to-R cycle is not generally realized. In a recent extensive review of the role of monovalent ions in enzyme function, Page and Di Cera [147] wrote: ‘Because the concentration of Na+ and K+ is tightly controlled in vivo, M+ ‘s do not function as regulators of enzyme activity’. Contrarily, according to Ling’s AIH, K+ activity is quite different between the R- and A-states, so K+ may effectively exert a regulatory function of prime importance. Ling [148] proposed such regulation in an un-cited short paper. K+ activity is very low during the R-state (about 2.5 mM) owing to adsorption to NUN-proteins of the cell-matrix. This bound K+ is then suddenly liberated at the onset of the R-to-A transition yielding a high K+ activity (about 150mM in neurons) during the A-state. Some of the liberated K+ ions may subsequently bind to some other proteins. Ling [148] mentions that two metabolic pathways in particular are activated by high K+ activity: glycolysis and respiration. Pyruvate kinase [149, 150] absolutely requires high K+ activity and so does phosphofructokinase [151]. Mitochondrial activation is dependent on a K+ influx [152-155], which according to the AIH is initiated at the onset of the R-to-A transition. At that moment, K+ may literally turn on these pathways as a main switch; and next, an increasing ADP/ATP ratio, associated with the power output of the R-to-A transition, further modulates the pathway flux in an allosteric way. In addition, increased ADP and Pi activity have a substrate effect on these pathways. As a result new ATP is quickly generated and binds to nearby NUN-proteins. This reinstalls the R-state of this group, whereby K+ is re-adsorbed. K+ activity drops again to a low value, causing its dissociation from regulatory glycolytic enzymes and leaving mitochondria. Glycolysis and respiration are again inactivated. If a cell is at complete rest, low K+ activity in the entire cell may lead to complete shutdown of glycolysis, respiration and other metabolic pathways. The total ATP concentration in the cell is high, but it is adsorbed to NUN-proteins in a stable manner. This at least follows from Clegg’s observations on Artemia cysts [52]. It is therefore time to realize that great changes in K+ activity during R-to-A and A-to-R transitions may operate as a real ON/OFF switch for ATP-generating pathways. Considering the entire R-to-A-to-R cycle, the effects of K+ may give rise to an automatic causative time sequence mechanism enabling the cycle to proceed and limiting the duration of the A-state, possibly also in the case of ‘limit cycles’, where cytoplasmic K+ oscillations may be expected. Mitochondrial K+ cycling has been described [155], but unfortunately it was not interpreted from Ling’s point of view, as has been done here. Schrödinger’s idea that ‘life feeds on negative entropy’ [17] could not find its place in MT because, according to MT, the media on both sides of the plasma membrane do not differ in entropy (ions and water are thought to be free on both sides). MT has therefore been the greatest obstacle to the spread of physical ideas and methods in cell physiology. As a result, there is a huge gap between cell physiology and both physics and chemistry (especially colloid chemistry). It is clear that the living cell is a multiphase system with many interfaces including the surfaces of proteins, protein complexes, and different supra-molecular structures. For most biologists the term ‘colloid’ is associated with the early 20th century. However, during the past two decades, colloid chemistry has developed a great deal. The knowledge obtained must now be used to solve problems of cell physiology. Biologists, who are working with colloidal systems (cells and cell models) but possibly not realizing it, may draw erroneous conclusions if they lack knowledge of physical and colloid chemistry. What seems to have been proven may be a serious mistake. It is necessary to combine the efforts of scientists from various specialties. Ling’s AIH is a good springboard for physicists who are interested in biology. But even more, everybody working in biosciences should have knowledge of Ling’s AIH, Pollack’s GSH and the coherent behavior of cells. An alternative view of such broad extent and importance including a different view of cellular energetic, expanded slightly by the present study, should not be missing from general textbooks and basic university courses. We are indebted to Paul Agutter and Ludwig Edelmann for valuable comments and to Paul Agutter for stylistic improvements. 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8d1a0baa429f732a	JMPJournal of Modern Physics2153-1196Scientific Research Publishing10.4236/jmp.2012.37075JMP-21118ArticlesPhysics&Mathematics Theory of Zero-Resistance States Generated by Radiation in GaAs/AlGaAs higejiFujita1KeiIto2AkiraSuzuki3Department of Physics, Faculty of Science, Tokyo University of Science, Shinjuku-ku, Tokyo, JapanResearch Division, National Center for University Entrance Examinations, Meguro-ku, Tokyo, JapanDepartment of Physics, University at Bu_alo, SUNY, Bu_alo, NY, USA;170720120307546552May 5, 2012June 10, 2012 July 1, 2012© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). Mani observed zero-registance states similar to those quantum-Hall-effect states in GaAs/AlGaAs but without the Hall resistance plateaus upon the application of radiations [R. G. Mani, Physica E 22, 1 (2004)]. An interpretation is presented. The applied radiation excites “holes”. The condensed composite (c)-bosons formed in the excited channel create a superconducting state with an energy gap. The supercondensate suppresses the non-condensed c-bosons at the higher energy, but it cannot suppress the c-fermions in the base channel, and the small normal current accompanied by the Hall field yeilds a B-linear Hall resistivity. Superconducting (Zero-Resistance) States; Composite-Boson (Fermion); <i>B</i>-Linear Hall Resistivity; Phonon (Fluxon) Exchange 1. Introduction In 2002 Mani et al. [1] observed a set of zero-resistance (superconducting) states in GaAs/AlGaAs heterojunction subjected to radiations at the low temperatures (~1.5 K) and the relatively low magnetic fields (~0.2 T). Figure 1 represents the data obtained by Mani [2,3] for the Hall () and diagonal () resistances in GaAs/AlGaAs at 50 GHz and 0.5 K. The resistance rises exponentially and symmetrically on both sides of the fields centered at with ω = radiation frequency, m = effective mass, e = electron charge, indicating the superconducting state with an energy gap in the elementary excitation spectrum. The phenomenon is similar to that observed in the same system in the traditional quantum Hall effect (QHE) regime (T ~ 1.5 K, b ~ 10 T) [4] with the main difference that the superconducting states are not accompanied by the Hall resistivity plateaus for the system subjected to radiation, see Figures 1(a) and (b). We call the temperature below which the superconducting state appears the critical temperature. The critical temperature (1.3 K) observed at is considerably higher than the traditional QHE critical temperature (0.5 K). Zudov et al. [5] reported similar magnetotransport properties for the system subjected to radiations with slightly different experimental conditions. They suggested that the principal resistivity minima occur at rather than (Mani’s case). They also noted a noticeable side resistivity minimum besides the principal set of the minima. In finer analysis Mani et al. [6] observed, see Figure 2, that (a) the deviation in the Hall resistance correlates with the resistance such that nearly vanishes when, and (b) is negative, and it is antisymmetric with respect to small B-fields: The property (b) means that there is a current due to “hole”-like particles having the charge of the opposite sign to that of the majority (“electron”-like) current carrier. In other words the applied radiation generates the “holes”. This may be checked by applying the circularly polarized lasers, which can excite “electrons” or “holes” selectively, depending on the sense of the circular polarization. The small slope means that the “hole” density is considerable higher than the “electron” Mani et al. [1] suggested for the cause of the spectral gap the electron pairing due to the excitons induced by radiation. Other mechanisms were proposed by several theoretical groups [7-11]. But none of them are conclusive. In particular no explanation is given to the question why the superconducting state can occur without the Hall resistivity plateau. Earlier Fujita et al. [12] developed a microscopic theory of the QHE based on the electron (fluxon)-phonon interaction. In this theory the composite (c-)particles (bosons or fermions) having a conduction electron and a number of flux quanta (fluxons) are bound by the phonon-exchange attraction. The composite moves as a boson (fermion) according to whether it contains the odd (even) number of fluxons in it. At the Landau Level (LL) occupation number (filling factor), odd, the c-bosons, each with fluxons, are generated, and condense below the critical temperature. The Hall resistivity plateau is due to the Meissner effect, as explained below. In the present work we shall extend our theory to Mani’s phenomena. We show that the most prominent zero-resistance states observed by Mani et al. [1] and Du et al. [3] represents the integer QHE with the state corresponding to the superconducting state at in Mani’s case (the state in Du’s case). If we write, the Mani-series 4/(4j + 1) can be recovered by setting,. Our model explains why the superconducting state under radiation does not accompany the Hall resistivity plateau. We predict that the fractional QHE should exist for. The aformentioned side dip observed by Zudov et al. [5] should correspond to the state at. This dip is missing in Mani et al.’s experiments. This is because the experimental temperature K is so low that the dip is overshadowed by the bosonic state at. 2. Theory of the Quantum Hall Effect If the magnetic field is applied slowly, the classical electron can continuously change from the straight line motion at zero field to the curved motion at a finite B. Quantum mechanically, the change from the momentum state to the Landau state requires a perturbation. We choose for this perturebation the phonon exchange between the electron and the fluxon. Consider the cparticle with a few fluxons. If the B-field is applied slowly the energy of the electron does not change but the cyclotron motion always acts so as to reduce the magnetic fields. Hence the total energy of the c-particle is less than the electron energy plus the unperturbed field energy. In other words the c-particle is stable against the break-up, and it is in a bound (negative energy) state. In our theory a c-particle is simply a dressed electron carrying fluxons. The c-particle moves as a boson (fermion) depending on the odd (even) number of fluxons in it. At the Landau level (LL) occupation number, odd, the c-bosons with fluxons are generated, and condense below certain critical temperature. The Hall resistivity plateau is due to the Meissner effect, see below. GaAs forms a zinc blende lattice. We assume that the interface is in the plane (001). The ions form a square lattice with the sides directed in [110] and [10]. The “electron” (wave packet) will then move isotropically with an effective mass. The ions also form a square lattice at a different height in [001]. The “holes”, each having a positive charge, will move similarly with an effective mass. A longitudinal phonon moving in [110] or in [10] can generate a charge (current) density variations, establishing an interaction between the phonon and the electron (phonon). If one phonon exchange is considered between the electron and the fluxon, a second-order perturbation calculation establishes an effective electron-fluxon interaction [14]: where is the phonon momentum (energy); the interaction strength between the electron (fluxon) and the phonon; the Landau quantum number is omitted; the bold denotes the two dimensional (2D) guiding center momentum and the italic the magnitude. If the energies () of the final and initial electron states are equal as in the degenerate LL, the effective interaction is attractive, i.e., . Following Bardeen, Cooper and Schrieffer (BCS) [13], we start with a Hamiltonian h with the phonon variables eliminated: where is the number operator for the “electron” (1) [“hole” (2), fluxon (3)] at momentum and spin with the energy. We represent the “electron” (“hole”) number by , where are annihilation (creation) operators satisfying the Fermi anticommutation rules: We represent the fluxon number by, with, satisfying the anticommutation rules. ,. The prime on the summation means the restriction:, = Debye frequency. If the fluxons are replaced by the conduction electrons (“electrons”, “holes”) our Hamiltonian is reduced to the original BCS Hamiltonian, Equation (2.14) of Ref. [13]. The “electron” and “hole” are generated, depending on the energy contour curvature sign [14]. For example only “electrons” (“holes”), are generated for a circular Fermi surface with the negative (positive) curvature whose inside (outside) is filled with electrons. Since the phonon has no charge, the phonon exchange cannot change the net charge. The pairing interaction terms in Equation (2) conserve the charge. The term, where, A = sample area, is the pairing strength, generates the transition in the “electron” states. Similary, the exchange of a phonon generates a transition in the “hole” states, represented by. The phonon exchange can also pair-create and pair-annihilate “electron” (“hole”)- fluxon composites, represented by, . At 0 K the system can have equal numbers of c-bosons, “electrons” (“holes”) composites, generated by. The c-bosons, each with one fluxon, will be called the fundamental (f) c-bosons. Their energies are obtained from [14] where is the reduced wavefunction for the fcboson; we neglected the fluxon energy. The denotes the strength after the ladder diagram binding, see below. For small, we obtain where is the Fermi velocity and the density of states per spin. Note that the energy depends linearly on the momentum. The system of free fc-bosons undergoes a Bose-Einstein condensation (BEC) in 2D at the critical temperature [13] The interboson distance calculated from this expression is. The boson size calculated from Equation (4), using the uncertainty relation and, is, which is a few times smaller than. Hence, the bosons do not overlap in space, and the model of free bosons is justified. For GaAs/AlGaAs, , me = electron mass. For the 2D electron density 1011 cm−2, we have cm∙s−1. Not all electrons are bound with fluxons since the simultaneous generations of ±fc-bosons is required. The minority carrier (“hole”) density controls the fc-boson density. For cm−2, K, which is reasonable. In the presence of Bose condensate below the unfluxed electron carries the energy, where the quasi-electron energy gap is the solution of Note that the gap depends on. At, there is no condensate and hence vanishes. Now the moving fc-boson below has the energy obtained from where replaced in Equation (3). We obtain where is determined from The energy difference: represents the T-dependent energy gap. The energy is negative. Otherwise, the fc-boson should break up. This limits to be at 0 K. The declines to zero as the temperature approaches from below. The fc-boson, having the linear dispersion (12), can move in all directions in the plane with the constant speed. The supercurrent is generated by the fc-bosons condensed monochromatically at the momentum directed along the sample length. The supercurrent density (magnitude), calculated by the rule:, is The induced Hall field (magnitude) equals. The magnetic flux is quantized, fluxon density. Hence we obtain If, , we obtain in agreement with the plateau value observed. The model can be extended to the integer QHE at. The field magnitude is less. The LL degeneracy is linear in, and hence the lowest LL’s must be considered. The fc-boson density per LL is the electron density over and the fluxon density is the boson density over: At there are c-bosons, each with two fluxons. The c-fermions have a Fermi energy. The c-fermions have effective masses. The Hall resistivity has a b-linear behavior while the resistivity is finite. Let us now take a general case, odd. Assume that there are sets of c-fermions with fluxons, which occupy the lowest LL’s. The cfermions subject to the available B-field form c-bosons with fluxons. In this configuration the c-boson density and the fluxon density are given by Equations (18). Using Equations (17) and (18) and assuming the fractional charge [15,16] we obtain as observed. In our theory the integer denotes the number of fluxons in the c-boson and the integer the number of the LL’s occupied by the parental c-fermions, each with fluxons. Our Hamiltonian in Equation (6) can generate and stabilize the c-particles with an arbitrary number of fluxons. For example a c-fermion with two fluxons is generated by two sets of the ladder diagram bindings, each between the electron and the fluxon. The ladder diagram binding arises as follows. Consider a hydrogen atom. The Hamiltonian contains kinetic energies of the electron and the proton, and the attractive Coulomb interaction. If we regard the Coulomb interaction as a perturbation and use a perturbation theory, we can represent the interaction process by an infinite set of ladder diagrams, each ladder step connecting the electron and the proton. The energy eigenvalues of this system is not obtained by using the perturbation theory but they are obtained by solving the Schrödinger equation directly. This example indicates that a two-body bound state is represented by an infinite set of ladder diagrams and that the binding energy (the negative of the ground-state energy) is calculated by a non-perturbative method. Jain introduced the effective magnetic field [17-19] relative to the standard field for the composite (c-) fermion at the even-denominator fraction. We extend this to the bosonic (odd-denominator) fraction. This means that the c-particle moves field-free at the exact fraction. The c-particle is viewd as the quasiparticle containing an electron circulating around fluxons. The jumping of the guiding centers (the CM of the c-particle) can occur as if they are subject to no B-field at the exact fraction. The excess (or deficit) of the magnetic field is simply the effective magnetic field B. The plateau in is formed due to the Meissner effect. Consider the case of zero temperature near. Only the energy matters. The fc-bosons are condensed with the ground-state energy, and hence the system energy at is, where is the number of –fc-bosons (or fc-bosons). The factor 2 arises since there are ±fcbosons. Away from we must add the magnetic field energy, so that When the field is reduced, the system tries to keep the same number by sucking in the flux lines. Thus the magnetic field becomes inhomogeneous outside the sample, generating the magnetic field energy . If the field is raised, the system tries to keep the same number by expeling out the flux lines. The inhomogeneous fields outside raise the field energy as well. There is a critical field. Beyond this value, the superconducting state is destroyed, generating a symmetric exponential rise in. In our discussion of the Hall resistivity plateau we used the fact that the ground-state energy of the fc-boson is negative, that is, the c-boson is bound. Only then the critical field can be defined. Here the phonon exchange attraction played an important role. The repulsive Coulomb interaction, which is the departure point of the prevalent theories [12], cannot generate a bound state. In the presence of the supercondensate the noncondensed c-boson has an energy gap. Hence the noncondensed c-boson density has the activation energy type exponential temperature-dependence: In the prevalent theories the energy gap for the fractional QHE is identified as the sum of the creation energies of a quasi-electron and a quasi-hole [20-23]. With this view it is difficult to explain why the activationenergy type temperature dependence shows up in the steady-state quantum transport. Some authors argue that the energy gap for the integer QHE is due to the LL separation =. But the separation is much greater than the observed. Besides from this view one cannot obtain the activation-type energy dependence. The BEC occurs at each LL, and therefore the c-boson density is less for high, see Equation (18), and the strengths become weaker as increases. 3. Quantum Hall Effect under Radiation The experiments by Mani et al. [6] indicate that the applied radiation excites a large number of “holes” in the system. Using these “holes” and the preexisting “electrons” the phonon exchange can pair-create ±c-bosons, which condense below in the excited channel. The c-bosons condensed with the momentum along the sample length are responsible for the supercurrent. In the presence of the condensed c-bosons, the non-condensed c-bosons have an energy gap, and therefore they are absent at 0 K. The c-fermions in the base channel have the energies but their energy spectra have no gap. Hence they are not completely suppressed at the lowest temperatures. They contribute a small normal current. They are subject to the Lorentz force:, and they generate a Hall field proportional to the field. Figure 2 show that the deviation in the Hall resistance, , is closely correlated to the resistance. We shall demonstrate this behavior based on the two channel (carriers) model. In the neighbarhood of the principal QHE at the carriers in the base and excited channels are respectively c-fermions and c-bosons condensed. The currents are additive. We write down the total current density as the sum of the fermionic current density and the bosonic: where and are the drift velocities of the fermions and bosons. The Hall fields are additive, too. Hence we have We note that the Hall effect condition () applies for the fermions and bosons. We therefore obtain Far away from the midpoint of the zero-resistance stretch the c-bosons are absent and hence the Hall resistivity becomes after the cancellation of. At the midpoint the c-bosons are dominant. Then, the Hall resistivity is approximately since where we used the flux quantization [], and the fact that the flux density equals the c-boson density. The Hall resistivity is not exactly equal to since the c-fermion current density is much smaller than the supercurrent density, but it does not vanish. In the horizontal stretch the system is superconducting, and hence the supercurrent dominates the normal current:. The deviation is, using Equation (26), If the field is raised (or lowered) a little from the midpoint, is a constant () due to the Meissner effect. If the field is raised high enough, the superconducting state is destroyed and the normal current sets in, generating a finite resistance and a vanishing. Hence the deviation and the diagonal resistance are closely correlated. In Figure 2 we observe that in the range where the Shubnikov-de Haas (ShdH) oscilations are observed for the resistance without radiation, the signature of oscillations also appear for the resistance with radiation. The ShdH oscillation arise only for the fermion carriers. The fermionic currents cannot be suppressed by the supercurrents. This ShdH signature in should remain. Hence our two-channel model is supported. Mani et al.’s experiments, Figure 2 of Ref. [1], show that the strength of the superconducting state does not change much for the radiation frequency in the range (47, 110) GHz. This feature may come as follows. The 2D density of states for the conduction electrons associated with the circular Fermi surface is independent of the electron energy, and hence the number of the excited electrons is roughly independent of the radiation energy (frequency). The “hole”-like excitations are absent with no radiation. We suspect that the “hole”-band edge is a distance away from the system’s Fermi level. This means that if the radiation energy is less than, the radiation can generate no superconducting state. This feature can be checked by applying radiation of frequencies lower than 47 GHz. Mani’s experiments on the simultaneous radiation excitations suggest that the critical frequency is between 15 and 47 GHz. In summary the QHE under radiation is the QHE at the upper channel. The condensed c-bosons generate a superconducting state with a gap in the c-boson energy spectrum. The supercondensate changes the c-fermion energy from to in the base channel. This energy spectrum has no gap, and hence the cfermions cannot be suppressed completely at the lowest temperatures, and generate a finite resistive current accompanied by the Hall field. This explains the b-linear Hall resistivity. Our microscopic theory can be tested experimentally by examining 1) the “hole”-like excitations by a circularly polarized laser; 2) the bosonic state at and; 3) the “hole” band edge. REFERENCESNOTESReferencesR. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson and V. Umansky, “Zero-Resistance States Induced by Electromagnetic-Wave Excitation in GaAs/AlGaAs Heterostructures,” Nature, Vol. 420, 2004, pp. 646-650. doi:10.1038/nature01277 R. G. Mani, “Zero-Resistance States Induced by Electromagnetic-Wave Excitation in GaAs/AlGaAs Hetero- structures,” Physica E, Vol. 22, 2004, pp. 1-6. doi:10.1016/j.physe.2003.11.204 R. R. Du, M. A. Zudov, C. L. Yang, Z. Q. Yuan, L. N. Pfeiffer and K. W. West, “Oscillatory and Vanishing Resistance States in Microwave Irradiated 2D Electron Systems,” In: Y. Wang, L. Engel and N. Bonesteel, Eds., High Magnetic Fields in Semiconductor Physics, World Scientific, Singapore, 2005, pp. 11-18. doi:10.1142/9789812701923_0001 D. C. Tsui, H. L. 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8ad2145aafd1a71b	Main The Beginning and the End of Everything: From the Big Bang to the End of the Universe Book cover The Beginning and the End of Everything: From the Big Bang to the End of the Universe 'Prepare to have your mind blown! A brilliantly written overview of the past, present and future of modern cosmology.' - DALLAS CAMPBELL, author of Ad Astra The Beginning and the End of Everything is the whole story as we currently understand it - from nothing, to the birth of our universe, to its ultimate fate. Authoritative and engaging, Paul Parsons takes us on a rollercoaster ride through billions of light years to tell the story of the Big Bang, from birth to death. 13.8 billion years ago, something incredible happened. Matter, energy, space and time all suddenly burst into existence in a cataclysmic event that's come to be known as the Big Bang. It was the birth of our universe. What started life smaller than the tiniest subatomic particle is now unimaginably vast and plays home to trillions of galaxies. The formulation of the Big Bang theory is a story that combines some of the most far-reaching concepts in fundamental physics with equally profound observations of the cosmos. From our realization that we are on a planet orbiting a star in one of many galaxies, to the discovery that our universe is expanding, to the groundbreaking theories of Einstein that laid the groundwork for the Big Bang cosmology of today - as each new discovery deepens our understanding of the origins of our universe, a clearer picture is forming of how it will all end. Will we ultimately burn out or fade away? Could the end simply signal a new beginning, as the universe rebounds into a fresh expanding phase? And was our Big Bang just one of many, making our cosmos only a small part of a sprawling multiverse of parallel universes? Categories: Physics\\Astronomy Year: 2018 Language: english Pages: 288 ISBN 10: 1782439560 ISBN 13: 978-1782439561 File: EPUB, 3.45 MB You may be interested in Физика горения газов Year: 1965 Language: russian File: PDF, 25.62 MB Mutual Causality in Buddhism and General Systems Theory Year: 1991 Language: english File: PDF, 5.20 MB The Theory Within the Theory ‘Gravity is a contributing factor in nearly 73 per cent of all accidents involving falling objects.’ Physics, in its most concentrated essence, is the study of the four fundamental forces of nature, namely: gravity, the force of attraction between objects with mass; electromagnetism, the interaction between electric charges and magnetic fields; and the strong and weak nuclear forces, which preside within the nuclei of atoms, binding the particles in the atomic nucleus together and governing subatomic phenomena such as radioactivity. Our story of the origin, evolution and ultimate demise of the universe is, for the most part, governed by just one of these forces: gravity. The nuclear forces, as the names suggests, aren’t anywhere near long-ranged enough to influence the large-scale behaviour of the universe, being confined within the cores of atoms. And the electromagnetic force, while it has a longer reach, is still not able to exert its influence across the gulf of space (its radiation, on the other hand, is a different matter – the light from distant stars and galaxies that paints the sky on a clear evening is nothing but a tangle of electromagnetic waves). Gravity, however, acts over much greater distances than the other forces. It is created by any concentration of mass or energy. Gravity keeps us stuck to the surface of the Earth; it locks the moon in orbit around our planet, and our planet in orbit around the sun. The influence of gravity doesn’t end there. It holds the sun and our solar system, and all of the other 250 billion stars in our Milky Way galaxy, wheeling in their gigantic circuits around the galactic centre. And it governs how the galaxies move relative to one another in the vastness of space. Gravity is the glue that binds our universe together. Our best working model of gravity is the general theory of relativity, devised in the early twentieth century by Albert Einstein. The theory was a triumph of the human intellect, offering bold new insights into the nature of space, time and the universe at large. But it was also one of staggering mathematical elegance and beauty. As we’ll see, it was to form the backbone of the Big Bang model describing our universe. The Ancient Greeks were the first people to invest any serious thought into the question of why heavy things fall, when the philosopher Aristotle mused on the subject in the fourth century BCE. Unfortunately, his ideas fell somewhat wide of the mark, supposing that ‘Earthy’ things, like rocks and bits of wood, move down towards where they seemingly belong – i.e., the Earth – whereas ‘fiery’ things, like, well, fire, move upwards towards where they belong: namely the sphere of fire that was posited to surround the Earth in the Ancient Greek view of the universe. In the sixth century CE, the Indian mathematician and astronomer Varahamihira was possibly the first person to imagine the concept of a universal force that keeps objects stuck to the ground and is responsible for holding the planets on their orbits around the Earth. The latter picture is, of course, incorrect – we know today that the planets orbit the sun and not the Earth, but both the Ancient Greek and Indian philosophers believed in a geocentric universe. It wasn’t until the late sixteenth century that thinkers began to get a handle on the true nature of gravity and its place in the universe. It was at this time that Galileo put forward his law of freefall, which says that two objects with different masses released simultaneously from the same height in a gravitational field should both fall at the same rate and hit the ground at exactly the same moment. The story goes that Galileo dropped two balls of different mass from the top of the Leaning Tower of Pisa and demonstrated that they reached the bottom simultaneously. Most historians of science now agree that this particular experiment probably never happened – though Galileo did conduct smaller-scale studies, measuring the speeds that different-mass balls rolled down sloping surfaces. Whatever the methodology, his conclusion was spot on – objects fall at the same rate, regardless of their mass. This has been verified many times since, perhaps most strikingly in a demonstration performed in 1971 by Apollo 15 astronaut David Scott, who dropped a hammer and a feather on the surface of the moon, observing them to glide in lockstep down into the lunar regolith. Attempting the same experiment on Earth, or any other planet with a significant atmosphere, doesn’t work because air resistance greatly slows the descent of the feather. Galileo’s law was basically a statement that gravity and acceleration are equivalent. Enter one of the greatest scientists to ever walk the Earth: Isaac Newton. Born on Christmas Day 1642 in Woolsthorpe-by-Colsterworth, a small village in Lincolnshire, England, Newton excelled at school and in 1661 was admitted to Trinity College, Cambridge, to study mathematics and ‘natural philosophy’ (a now largely defunct term meaning physics). By 1669, he had progressed to become the university’s Lucasian Professor of Mathematics – a chair that would later be held by Stephen Hawking. Newton’s scientific discoveries are manifold and include demonstrating that white light breaks down into a rainbow spectrum of different colours (essential for spectroscopy, as we saw in the last chapter), contributing to the development of calculus (a powerful mathematical technique for analysing quantities and their rates of change), and for building the world’s first ever reflecting telescope (which focuses light using a curved mirror, rather than a lens). A highly apocryphal story also has it that Newton invented the cat flap, so that his moggy could come and go without disrupting the great man’s experiments with light. Without doubt Newton’s greatest achievement was his book Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), published in 1687. Usually just referred to as Principia, it was written during an intense period of activity between 1684 and 1686, prompted by conversations with the astronomer Edmond Halley (discoverer of Halley’s Comet). Early in 1684, Halley had recounted to Newton his previous discussions with the British scientist Robert Hooke, in which Hooke claimed to have constructed a mathematical model that correctly explained the observed motion of the planets, by postulating that they orbited around the sun under the action of an inverse square law force. That means a force that gets weaker in proportion to the distance from the source squared. So, for example, if you doubled your distance from the sun then an inverse square law would make the gravitational force exerted on you diminish to 1/2 squared – that is, 1/4 – of what it was. Newton replied that he had already done this calculation himself, but claimed to have mislaid the manuscript. It took him some months to locate it, but in November that year Newton delivered to Halley a paper setting out a full mathematical derivation of how an inverse square law force leads directly to Kepler’s three laws of planetary motion. German mathematician Johannes Kepler had formulated these laws from observational data gathered by the Danish master astronomer Tycho Brahe. They are a set of three mathematical statements that provide an exceptionally accurate model for the motion of the planets in our solar system. However, the laws are purely empirical (contrived to explain the observations), with no underlying justification in physics. Published in the early seventeenth century, they state that: 1. The planets orbit in oval-shaped ellipses with the sun at one of its two focal points. A circle is just a special case where the two focal points coincide at the centre. You can draw a circle by banging a nail into a board. Slip a loop of string around both the nail and a pencil – move the pencil while keeping the string taut and the result is a circle. But if you now bang two nails into the board and repeat, the result is an ellipse with the two nails as the focal points. 2. A line drawn from the sun to a planet sweeps out equal areas in equal time. This law means planets must move faster when their orbits take them closer to the sun. 3. The square of the time taken to complete one orbit is proportional to the cube of the semi-major axis of the ellipse (half the length of the ellipse’s longest axis). So if, say, the semi-major axis is quadrupled then the time to complete one orbit increases by a factor of 8 (4 cubed is 64, square rooted gives 8). In our solar system, the planet Jupiter orbits 5.2 times further from the sun than the Earth – and cubing and then square-rooting this number gives Jupiter’s 11.86-year orbital period around the sun exactly. As the sceptics among you may have already spotted, it’s unclear whether Newton really had completed his inverse square law calculation earlier, or whether he essentially stole Hooke’s idea and then spent the intervening months working through the maths. Newton became embroiled in bitter disputes regarding the priority of a number of scientific discoveries throughout his professional life – most notably with the German mathematician Gottfried Leibniz over the development of calculus, which Newton viewed as plagiarism on the part of Leibniz (in reality, they both probably got there independently). Not content with merely disliking Leibniz, Newton actively despised the man – to the point that he would spend his time and energy writing anonymous reviews rubbishing his rival’s work. Indeed, it’s fair to say that Newton’s prodigious scientific talent was matched only by his reputation as an irascible and rather unpleasant human being. During his later life he became Warden and Master of the Royal Mint at the Tower of London, where he took pride in sending counterfeiters to their deaths on the gallows. So, it’s not inconceivable that Newton may have appropriated Hooke’s insight, if he thought it might have served his own ends. Then again, it does beg the question why, if Hooke really had done the calculation first, he didn’t publish it himself. We’ll probably never know the truth. What we do know is that Halley presented Newton’s paper to London’s Royal Society, where the proof that Kepler’s laws could all be condensed into the action of a single force of nature was very well received. Halley asked Newton for more of his work and the result, two years later, was Principia. Halley was so impressed that he even footed the printing costs for the three-volume work himself (the Royal Society had already blown its budget for that year publishing a tome entitled The History of Fish by Cambridge naturalist Francis Willughby). Principia appeared in print on 5 July 1687. Its rigorous and systematic approach to modelling the real world mathematically represented the birth of modern theoretical physics. Taking pride of place in the book was Newton’s universal law of gravitation. Newton summarized the law with a mathematical formula quantifying the force of gravitational attraction between two massive bodies. As his earlier calculations had suggested, the force was an inverse square law. But Newton added extra details, postulating that the force also grows in direct proportion to the masses of the two bodies (so if you double the mass of either body then the resulting force doubles too), all multiplied by a constant of proportionality known as the universal gravitational constant. Usually denoted by the letter G, this constant is a number that takes the value 6.67 divided by 100 billion, making it and the strength of the gravitational force exceptionally small. And this agrees with our experience – it takes the entire mass of the Earth to produce enough gravity to keep you stuck to the planet’s surface. The gravitational force between you and, say, this book is utterly negligible. Newton’s theory transcends Kepler’s laws. These actually describe planetary systems with a central star that’s very much heavier than its attending retinue of planets. That means the gravity of the planets is insufficient to make the star move very much, and so it can be thought of as stationary. If, on the other hand, you look at something like a binary star system, which comprises two stars of roughly equal mass, then each body has enough gravity to influence the other. Then, rather than one star simply orbiting the other, they will both orbit around their common centre of mass (imagine both stars on a see-saw – the centre of mass is the point between them where you’d need to put the pivot to make the see-saw balance). Newtonian gravity explains such systems very well. It can also deal with many more bodies than just two – although doing the calculations on paper in this case gets tricky. Instead, astrophysicists today use powerful computers to crunch out the solutions numerically; these often describe vast swarms of objects, such as diffuse clouds of matter condensing down to form galaxies. It should also be added that no apples were harmed in the making of this theory, although the classic story of Newton’s inspiration for the theory of gravity coming to him in a flash after an apple fell on his head probably does contain a grain of truth. Woolsthorpe Manor in Lincolnshire, where Newton grew up, is now owned by the National Trust (a British national-heritage organization) – and it still has an apple tree growing in its garden to this day. Newton certainly returned there for a time in 1666, when the Plague had forced him to temporarily leave Cambridge. In 1726, just a year before his death, Newton recounted to his friend William Stukeley – who would later pen a biography of Newton – how he had watched apples fall from the tree at Woolsthorpe, and wondered why they should always move perpendicular to the ground. He concluded, correctly, that matter must produce an attractive force and that the influence of this force could extend a great distance (the latter point coming to him while watching the moon hanging in the night sky one evening). There is no mention, however, of any apples ever landing on Newton’s head – which is probably for the best, since the variety he observed at Woolsthorpe, ‘Flower of Kent’, is a large-size cooking apple. It seems most likely that his being struck by one was an embellishment added later in the many retellings of the story. As well as gravity, Principia offered the first exposition of Newton’s laws of motion – three principles of physics governing the behaviour of moving objects. Newton’s first law states that objects remain stationary, or continue moving at constant speed, unless acted on by a force. The second law says that an object acted on by a force will accelerate. Based on experiments he had conducted, Newton was able to specify a formula – the famous equation – giving the magnitude of the force (F = ma) as the acceleration (that is, the rate at which the object’s speed is changing – represented in the equation by a) multiplied by its mass (m). Newton’s third law of motion says that for every force there’s an equal force pushing in the opposite direction. So, for example, when you sit down on a chair, the chair pushes upwards to support your weight. Newton’s laws of motion and gravity were revolutionary for their time, permitting the behaviour of bodies moving under the action of forces to be accurately predicted. Perhaps the most remarkable thing about them is their universality. The laws apply equally well to an apple falling from a tree here on Earth as they do to a spacecraft cruising the outer limits of the solar system. They held sway for over 200 years and are still used today to calculate the motion of objects moving at the slow speeds and in the relatively weak gravitational fields found on Earth and in our immediate neighbourhood of space. But, at the start of the twentieth century, a then little-known German physicist was about to recast our picture of motion and gravity for ever. His name was Albert Einstein. Born in Ulm, Germany, on 14 March 1879, Einstein was a reserved, bookish child who showed an interest and aptitude for science from a young age. He loathed authority, which made him unpopular with his school teachers; and that, combined with his pacifist nature, led him to leave Germany before his seventeenth birthday in order to avoid national service. Einstein travelled to Switzerland, where he eventually passed the entrance exam for Zurich Polytechnic in 1896 and enrolled to study maths and physics. Upon graduating in 1900, his intention was to continue in theoretical-physics research, but he struggled to secure a job in academia – largely because his contrarian nature had made it impossible to obtain a decent reference from his Zurich professors. So, in 1902, he took a job as a clerk at the Swiss patent office in Bern. He enjoyed the diversity of the work, the regular income and the fact that he could still pursue his own theoretical-physics research in his spare time. It was here that Einstein laid the foundations for his theory of relativity – which would dramatically change the laws of motion, and later replace Newton’s law of universal gravitation. Einstein’s suspicions that all was not well with the laws of motion as they stood were first aroused in a thought experiment conceived at the age of just sixteen. He wondered what would happen if you could travel at the speed of light and pull alongside a light beam. Einstein wanted to know whether it would be possible to make the beam appear stationary – but the thought experiment would soon reveal a deep inconsistency in the laws of physics as they stood. Pulling alongside a light beam, or anything else, in order to make it look stationary is an example of what’s called relative motion. If you’re in a car driving at 70 mph following a car doing 50 mph, then you’ll catch up with the car in front – and your relative speed of approach is 20 mph. That’s the speed that the car in front is seen to move relative to your own vehicle. If you were to pull alongside and slow down to 50 mph then the relative speed drops to zero and the other car appears stationary relative to your vehicle. Einstein, hypothetically speaking, wanted to do the same with a beam of light – to make it appear motionless. But here’s the problem. Back in 1861, the Scottish physicist James Clerk Maxwell had produced a unified theory of the electromagnetic force of nature – tying together electricity and magnetism as just different aspects of the same thing. The theory predicted that beams of light, as well as radio waves, X-rays and other types of radiation, are just electromagnetic waves – consisting of electric and magnetic fields vibrating at right angles to one another and moving through space. The speed at which these waves move dropped neatly out of Maxwell’s equations – 300,000 kilometres per second, or the speed of light. But this figure appeared in the theory as a fundamental constant of nature. That means it’s on the same footing as the three dimensions of space, or the value of π (the ratio of the diameter to the circumference of a circle, equal to 3.14159). Getting in your car and driving very fast has absolutely no effect on the value of π – measure the circumference and the diameter of a circle (though you might want to get someone else to take the wheel while you do this) and you’ll find their ratio will still take the same value, π, whether you drive at 10,100 or 1,000 mph. And in just the same way, the speed you drive at has no effect whatsoever on the speed of light. If you tried to chase after a light beam, the beam will continue to rush away from you at light-speed, regardless of your own speed. It’s as if you were trying to catch up with the car in the previous example but no matter how fast you travelled, it always moved away from you at the same 50 mph. Clearly, something had to give – either Maxwell was wrong, or simple addition and subtraction of speeds was not the correct way to calculate the relative speed between two moving bodies. Impressed with the new insights it offered, Einstein believed Maxwell’s new theory to be correct and so set about deriving a new way to calculate relative motion, under the assumption that the speed of a beam of light can never change. In this view of the world, it doesn’t matter how fast you travel – the relative speed between you and a light beam will not waiver from 300,000 kilometres per second. The result, published in 1905, became known as the special theory of relativity (‘special’ to distinguish it from the later ‘general’ theory, which we’ll come to shortly). At speeds very much less than light, Einstein’s new laws reduced to Newtonian theory, but as objects got faster and faster and ultimately approached light-speed their predictions grew wildly different, throwing up some effects that were at best counterintuitive – and often downright bizarre. First of all, Einstein found that in order to keep the speed of light constant, time in a moving frame of reference must slow down relative to time measured by a stationary observer. Imagine a beam of light moving inside a space rocket. An astronaut inside the rocket and a stationary external observer (looking in through the window) both have clocks to measure the time it takes the beam to travel from one side of the rocket to the other. When the rocket is stationary, both the observer and the astronaut agree how long it takes the light beam to cross the width of the rocket. Now the rocket fires up its engines, jets off at close to light-speed and the experiment is repeated. To the astronaut, little has changed and they record the same time for the light beam to cross the width of the rocket that they saw before. But for the external observer (let’s ignore for now the practical difficulties of timing a light beam through the window of a fast-moving rocket), something interesting happens. They see the light beam travel a greater distance, because in addition to crossing the width of the rocket it will also move forwards with the rocket’s motion. Since the light-beam must be moving at the same speed in both frames of reference (from Maxwell’s theory), the only way it can cover a greater distance is by taking longer to make the journey. That is, if the observer compares the ticking of their clock to the ticking of the astronaut’s clock (assuming they can see that through the window of the rocket as well), they’ll find that the moving clock is actually ticking slower – less time is passing in the astronaut’s frame of reference. Physicists call this phenomenon time dilation. Putting in some numbers, a clock moving at 86 per cent light-speed runs at roughly half the speed of one that’s stationary. That means that an astronaut travelling at this speed for twenty years, as measured by clocks back on Earth, ages only ten years. When they return to Earth they are only ten years older than when they left, while their former contemporaries have aged twenty years. From the astronaut’s perspective, they’ve effectively travelled forwards in time. [image: f0044] In the top diagram, as seen by an astronaut inside the rocket, a light beam simply travels from one side of the rocket to the other. In the lower diagram, as seen by an external observer, the light travels further because of the rocket’s motion. To keep the speed of light the same for both observers, less time elapses in the rocket’s frame of reference. This is time dilation. At faster speeds, this time difference becomes even greater. For example, in a rocket moving at 99.9 per cent light-speed the astronaut experiences less than a twentieth of the time that elapses on Earth. Travel at this speed for ten years (in their frame of reference) and they will return over 220 years in the future – by which time, all of their friends and family have passed away, and the Earth will be a radically different place. Note that this kind of time travel, where the astronaut moves solely into the future, is very different to time travel into the past. As anyone who’s seen the Back to the Future trilogy knows, this can invoke all manner of troublesome paradoxes – for instance, preventing your parents meeting and thus invalidating your own existence. For this reason, many physicists suspect backwards time travel to be impossible. Future-directed time travel, however, suffers from no such problems. Indeed, time dilation has been verified experimentally on many occasions – for example, in the decay times of high-speed subatomic particles. Some particles are known to be unstable and break apart over well-established timescales. However, when these particles are travelling at close to light-speed their decay times appear longer than they should be – in direct agreement with time dilation. In another experiment, conducted in 1971, super-accurate atomic clocks were placed aboard jet aircraft making round-the-world flights. An atomic clock uses the rapid vibrations of microwaves given off by atoms of the chemical element caesium (chosen for its long-term stability) to tick off almost vanishingly small intervals of time – enabling the clock to keep time to an accuracy of one second in 300 years. When the flying clocks were later compared with a reference clock on the ground, small time differences were found to have crept in that agreed with relativity’s predictions. As well as time dilation, Einstein also showed that the size of a moving object shrinks in its direction of motion, as measured by a stationary observer – a phenomenon known as ‘length contraction’. The shrinking factor is the same as the time dilation factor, so a rocket moving at 86 per cent light-speed appears roughly half as long as it does when stationary. You can see why this might be so if you imagine another light-beam inside the rocket; this one travelling from the back all the way to the front (rather than travelling across the width of the rocket, as in the previous example). The stationary external observer sees the beam travel the length of the rocket, while at the same time being carried forward some extra distance by the rocket’s motion. This extra distance travelled would make the beam appear to travel faster than light in the observer’s frame of reference. The only way the speed of light can be the same for both the astronaut and the observer is for the observer to see the rocket get shorter – and this is length contraction. [image: f0046] In the diagram at the top, as seen by an astronaut inside the rocket, a light beam travels from the back of the rocket to the front. In the diagram below, as seen by an external observer, the light travels further because of the rocket’s motion. To keep the speed of light the same for both observers, the rocket must get shorter as seen by the external observer. This phenomenon is known as length contraction. Special relativity even disrupts the notion of whether or not two events occur simultaneously – making this depend sensitively on the observer’s motion. Let’s go back to our space rocket and imagine that the astronaut switches on a light bulb right in the centre of the spacecraft, equidistant from each end. In the astronaut’s frame of reference the light reaches both ends of the rocket at the same instant – i.e., simultaneously. But from the point of view of the external observer, the back of the rocket is now rushing towards the light bulb while the front is moving away – meaning that they see the light reach the back first. Thanks to Einstein, scheduling meetings at close to light-speed requires not just a diary but also a degree in theoretical physics. In fact, the entire concept of space and time as distinct entities disappears in relativity – replaced by a unified four-dimensional fabric known as spacetime, which forms the stage on which the rest of physics plays out. This approach led to what must be the only equation in physics to have achieved household renown. In ordinary three-dimensional space, a moving object with mass has a property called kinetic energy – literally its energy of motion. But in the spacetime of relativity theory, Einstein found that massive objects also possess energy simply by virtue of their motion through time. This energy is equal to the object’s mass multiplied by the speed of light squared, or E = mc2. It means that mass and energy are equivalent, linked by the speed of light. When some of an object’s mass is destroyed, energy is released – and this equation tells you exactly how much. It would later turn out to be the founding principle upon which nuclear power is based (certain processes – called fission and fusion – can decrease the mass of the atomic nucleus, liberating extraordinary amounts of energy). Einstein’s study of how energy behaves in the theory of relativity also led to another startling revelation. He found that as a moving object accelerates, the energy needed to make it travel incrementally faster increases – to the point that the energy required to reach the speed of light is infinite. In other words, it cannot happen. This is the reason why the special theory of relativity is often cited as demanding that nothing can travel faster than light-speed. In fact, this isn’t strictly true. All relativity actually says is that nothing can cross the light barrier – be that from below or above. If you could somehow create an object that was already travelling faster than light, then it would take an infinite amount of energy to slow it below the light barrier – which is just as impossible as something moving slower than light gaining infinite energy to go faster. Subatomic particles that are born moving faster than light have been hypothesized – they’re known as tachyons – but as yet there’s no experimental evidence for their existence. For you and me, and everything else in the observable universe, light-speed is as fast as it gets. It’s rare for a theory to come along that’s as utterly revolutionary, and at times bewildering, as the special theory of relativity. The theory has since been tested to destruction – particle accelerators, such as the famous Large Hadron Collider at CERN, routinely push subatomic particles within the tiniest fraction of light-speed, and measurements of their behaviour, particularly their decay times (which, as we saw earlier, are made to appear longer because of time dilation), confirm the predictions of special relativity time and again. But as if special relativity wasn’t strange enough, Einstein was about to make things a whole lot stranger. Despite its successes, the special theory applies only in the absence of any gravitational fields: wishing to start simple, Einstein just hadn’t built gravity into the model. Indeed, this is why it’s called the ‘special’ theory – it specifically addresses the special case of motion in the absence of gravity. What Einstein sought now was a ‘general’ theory that added gravity into the mix. He knew from the outset that Newton’s theory of gravity must be at odds with relativity. In the Newtonian view the gravitational force travels across space instantaneously – meaning that if, for whatever reason, the sun was to mysteriously disappear then the effect of its gravity on the Earth would switch off immediately, and our planet would fly off into space. Einstein, however, had already shown that in special relativity nothing can travel faster than light-speed – gravity included. The Earth orbits 8.3 light-minutes from the sun, meaning that, if our star really did suddenly vanish, it would take 8.3 minutes for the absence of gravity to reach us. Until that time, we would keep orbiting as normal. If special relativity was correct – and the evidence suggested that it was – then Newtonian gravity had to be wrong. Einstein was also aware of Galileo’s law of freefall, which we met earlier in this chapter and which says that objects of different masses accelerate towards the ground at the same rate in a gravitational field – equivalent to saying that gravity and acceleration are the same thing. Einstein cemented this concept in a famous thought experiment of 1907, which he later described as ‘the happiest thought of my life’. In it, he realized that a person inside a sealed elevator with no windows would be unable to tell whether the elevator was accelerating upwards or was simply stationary in a gravitational field. Imagine lying on the floor of an elevator on Earth – the force of gravity keeps you stuck to the floor. Now imagine the same elevator out in space, far from any gravitational fields. When the elevator is stationary, you will be weightless. But if it now accelerates upwards fast enough you’ll find yourself pressed into the floor (just as you’re pressed into your seat when a car accelerates rapidly) and – with no windows through which to tell otherwise – you’ll be unable to distinguish whether what you’re feeling is due to genuine acceleration of the elevator, or the presence of a gravitational field. Taken further, it became clear to Einstein that no conceivable scientific experiment conducted inside the elevator, from the swinging of a pendulum to the growth of biological cells to the collision of subatomic particles, can tell between these two possibilities. Einstein elevated Galileo’s observation to become a central tenet of general relativity – which he called the equivalence principle. It was with this in mind that it dawned on Einstein what form his theory of gravity had to take. Going back to the imaginary elevator, he pictured shining a beam of light from one wall across to the opposite side. If we assume for a moment that the elevator really is accelerating upwards, then the point at which the light beam strikes the far wall will be slightly lower than the point at which it left – because in the time it takes the beam to cross the elevator’s width, the elevator itself has accelerated upwards a little way. To an observer inside the elevator, this means that the light-beam appears to curve downwards. But, remember, the equivalence principle says that no experiment can tell whether the lift is actually accelerating, or is just sitting in a gravitational field. And that means light must therefore also follow a curved trajectory in a gravitational field. For Einstein the implication was clear: the correct theory of gravity would involve taking the flat spacetime of special relativity and curving it according to the matter that it contains. Now the challenging work could really begin. As a mathematical framework for his theory, Einstein drew upon the work of a nineteenth-century German mathematician called Bernhard Riemann, who had pioneered a field of study known as differential geometry. This is a way of representing curved surfaces mathematically, a bit like the way latitude and longitude can be used to chart our position as we travel across the curved two-dimensional surface of the Earth. Only, Riemann’s mathematics could be applied to arbitrarily curved surfaces in not just two, but any number of dimensions – crucial if the theory was to describe the convoluted four-dimensional spacetime of general relativity. With differential geometry on-side, the question now was how to link its complex architecture to the contents of spacetime – to figure out exactly how mass and energy induce the required curvature. Solving this conundrum took Einstein until November 1915 when, close to exhaustion, and with fellow German mathematician David Hilbert hot on his heels, Einstein finally struck gold and arrived at the keystone of general relativity – the field equation which links matter to the geometry of space and time. On the left-hand side of the equation are the mathematical quantities from Riemann’s theory of differential geometry, describing spacetime curvature. And sitting on the right are the quantities specifying its contents. Whereas mass is the sole source of gravity in Newton’s theory, the right-hand side of Einstein’s field equation also factors in the energy present in space (for example, radiation) as well as properties of matter such as momentum and pressure. The fact that energy creates gravity in general relativity should come as no surprise given Einstein’s earlier discovery, from the special theory, that mass and energy are just different aspects of the same thing (because E = mc2). Einstein’s field equation empowered physicists to determine how spacetime becomes curved by mass and energy, and to then compute the motion of objects across that curved platform. For instance, the sun creates a giant concave depression in space, around the inside of which the planets circle like rolling marbles. The heavier the object, the deeper the indent in space it makes – and the harder it is to escape from the resulting gravitational field. In the extreme limit are black holes – objects so dense that the indentation they create in space is elongated into a deep funnel shape that not even light can escape from. As we’ll see in the next chapter, general relativity – as encapsulated by Einstein’s field equation – later permitted astrophysicists to formulate the first mathematical models describing the evolution of the universe. These models would explain beautifully Hubble and Humason’s discovery that the universe is expanding, and would later lead cosmologists to the idea that the universe may have had a beginning – the Big Bang. But we’re getting ahead of ourselves. In the winter of 1915, how could Einstein be sure that his version of the field equation was actually correct? One observation of the planets had long been baffling astronomers. In 1859, French mathematician Urbain Le Verrier had noticed that the elliptical orbit of the planet Mercury, the innermost world in our solar system, seems to be rotating around the sun, so that the planet itself traces out a rosette shape over time. The effect was tiny – Mercury’s orbit was rotating by just 0.012 degrees per century – but rotating it was, and nobody knew why. Initially it was thought that this behaviour could be caused by the gravity of another small planet, orbiting between Mercury and the sun – but all the astronomical surveys carried out to try to detect this new world had drawn a blank. Instead, it was Einstein’s new theory that held the answer. When general relativity was applied to the problem, it predicted that Mercury’s orbit should rotate at exactly the rate Le Verrier had observed. This was the first real experimental test of general relativity – something it could explain that Newton’s theory of gravity could not. And it passed with flying colours. [image: f0052] Mercury’s elliptical orbit around the sun gradually rotates, causing the planet to trace out a spiral pattern. General relativity explains the effect perfectly. But there was an even greater experimental triumph to come. Since space is curved in Einstein’s theory, light rays should also follow curved paths as they trace out its contours. But how to test this prediction? The most massive object near Earth is our star, the sun, which weighs in at a whopping 330,000 times the mass of our planet. Despite this enormous concentration of mass, a light ray scraping past the sun would only be deflected by a minuscule amount – just 0.0005 of a degree. That could just about be measured, were it not for the fact that the light from the star would be lost in the sun’s glare. English astronomer Arthur Eddington came up with an idea to get round this problem, by carrying out the observation during a solar eclipse – when the sun’s glare is blocked out by the silhouette of the moon. In 1919, Eddington led an expedition to the island of Príncipe, off the coast of Gabon, West Africa, where he took advantage of a solar eclipse visible there to photograph stars very close to the sun’s position in the sky. When he compared the photographs to the stars’ usual positions, he found that they had indeed moved ever so slightly during the eclipse – and by an amount that tallied with general relativity’s predictions. Following the announcement of Eddington’s findings, Einstein became an almost overnight celebrity. The press loved the fact that they had not just a brilliant scientist to cover but also a witty rebel who seldom failed to deliver a quote and (although he would never admit it) revelled in the attention. And the public were enrapt. At the height of ‘Einstein-mania’ in the early 1920s, he embarked upon a lecturing tour of the US, culminating in a meeting at the White House with President Warren G. Harding. The tour, meanwhile, was a sell-out – despite the fact that the lectures were all delivered in German. [image: f0054] How the bending of starlight around the sun shifts the observed positions of stars. Today, general relativity is tested every day in satnav devices. One consequence of the theory is that time runs more slowly in a gravitational field – and this effect must be corrected for, using Einstein’s equations, if the devices are to give accurate estimates of position. Civilian satnav receivers first became available in the early 2000s. More recently, astronomers verified the general theory’s final untested prediction when they made the first confirmed detection of gravitational waves. Just as casting a stone into a still lake sends ripples spreading out in giant circles towards the banks, so it is that violent gravitational phenomena – like exploding stars, or colliding black holes – create ripples in the flexible fabric of spacetime that travel out across the universe at the speed of light. The waves manifest themselves as small distortions in the distance between two points in space. They are tiny – two points a metre apart will typically see their separation distance fluctuate by just one billion-billionth of a centimetre (about one hundred-thousandth the size of the proton particles found inside the nuclei of atoms) when a gravitational wave passes. That’s a distance so small that the tiniest vibration from a passing car could drown out the signal, which is why detecting them has proven to be such a challenge. However, in 2016 – exactly 100 years after it was realized that the waves are a consequence of Einstein’s field equation – an international collaboration of scientists announced that they had made the first detection. The scientists used devices called interferometers, which bounce laser beams up and down 4-kilometre-long tunnels hundreds of times to amplify the size of the distortion that a passing gravitational wave creates. There are two such detectors in the United States, separated by 3,000 kilometres, and a third, smaller device in Italy – using multiple detectors rules out the likelihood of a false alarm. The historic first detection was made on 14 September 2015, and is believed to have comprised gravitational waves emanating from the merger of two black holes, each about thirty times the mass of the sun. It was located in the southern hemisphere of the sky (positional resolution is not yet good enough to be any more specific than that). At the time of writing, another five confirmed detections have been made. As we’ll see in Chapter 14, the detection of gravitational waves isn’t just the final confirmation of general relativity; it also begins a whole new era in the study of the universe. But we’re getting ahead of ourselves. As we saw in Chapter 1, in the early twentieth century baffling new telescopic observations of distant galaxies had emerged that were pushing established theory to the limits and beyond. The general theory of relativity – Einstein’s bold new picture of gravity – was about to run headlong into this cosmic conundrum. And the result would be nothing short of a scientific revolution in our understanding of the origin and evolution of the universe at large. From Out of Nowhere ‘In the beginning there was nothing, which exploded.’ We’ve seen how astronomical observations were used to infer the existence of the Big Bang. And we’ve seen how scientists devised ingenious theories to explain the universe’s 13.8-billion-year history. But one question remains. Namely: where did the Big Bang actually come from? Yes, it all came from a singularity that, in the framework provided by general relativity at least, was infinitely dense. And this singularity then exploded, chucking out particles, photons of radiation and the very fabric of spacetime itself, which all went on to form the universe today that we know and love. But just how did the singularity get there? And what possessed it to explode? The most elegant explanation, and probably the one that’s gained the most traction with cosmologists, is that the universe literally sprang from nothing. And by nothing, I really do mean nothing. If I’ve neglected to do the grocery shopping one week then there’ll be nothing in the fridge, but there’s still a fridge, there’s still the space inside the fridge (even if it is empty) and there’s still time there, threading fridges past, present and future together into a logical sequence of cause and effect. The precursor state to the Big Bang, on the other hand, had no matter, energy, space or time. There was nothing. It’s certainly plausible that our universe could have arisen from such desolate beginnings. If we believe that the universe is all there is and that there’s nothing beyond it then it makes the most sense for it to have also appeared from nothing. After all, a bubble in a pan of boiling water, and nothing but water, is most likely to have formed from . . . water. It also means that the universe’s overall energy should sum to zero. This is because of the law of conservation of energy, a fundamental principle in physics that says that mass and energy (remember, the two are one and the same thanks to Einstein – see Chapter 2) can be neither created nor destroyed. In other words, what goes in must come out – if the universe was made from nothing then the mass and energy of all its components today must still sum to precisely nothing. We’ve seen that pairs of subatomic particles can blink into existence and then vanish again such that the interval for which the pair exists, combined with their total energy, satisfies the Heisenberg uncertainty principle (see Chapter 6). This principle just says that the greater the energy of the particles, the shorter the time that they can live for, before recombining and annihilating. Borrowing the amount of energy needed to create all the matter in the universe therefore results in a universe that would be gone almost as soon as it appeared. However, if the total energy of the universe could be made zero, then it’s a very different story. If no energy at all was being borrowed, then the universe could, in principle, pop into existence from nothing and yet exist for ever. One of the first people to have realized that this is actually possible was Albert Einstein, back in the 1940s. The story goes that he was out walking in Princeton, New Jersey, with his friend and colleague George Gamow (of microwave-background and primordial-nucleosynthesis fame – see Chapter 4). Gamow was recounting how one of his colleagues, the German physicist Pascual Jordan, had calculated that it’s possible to create a star from nothing because its mass energy can be exactly balanced by its gravitational energy. The gravitational energy of a star, or any other object, in fact, is the energy required to assemble it assuming all of its constituent parts started out an infinite distance apart. It’s actually negative. That’s because the energy required to form the star must be equal and opposite to the energy needed to take it apart again – to take all of its constituent pieces and move them an infinite distance away from each other. And the energy required to disassemble a star is clearly positive – as the world’s space agencies regularly demonstrate, getting away from a gravitational field takes a considerable amount of energy, in the form of rocket fuel. Jordan had shown that this gravitational energy wasn’t just negative but could be equal in size to the star’s mass energy – so that, on balance, the total energy of the star was absolutely zero. When Einstein heard this, he realized straight away that the same reasoning could apply to the whole universe – the total mass and energy of its constituent matter and radiation could be balanced against its gravitational energy, making its total energy sum to zero. Einstein stopped in his tracks. He and Gamow were crossing a road at the time, forcing several cars to stop in order to avoid running them down. During the early 1970s, physicist Edward Tryon, of the City University of New York, embellished the idea using quantum theory to suggest that quantum uncertainty could have brought the universe into existence (taking the zero overall energy that the universe started out with and splitting it into positive mass and energy on one side and equal but opposite gravitational energy on the other) in the same way that quantum uncertainty allows particle-antiparticle pairs to appear as quantum fluctuations in the vacuum of empty space (see here). The idea was then developed further by the cosmologists who devised the inflationary universe theory, the notion that shortly after the Big Bang the universe underwent an exponential growth spurt (see Chapter 8). Alan Guth, the main originator of the theory of inflation, dubbed it the ‘free lunch universe’. In their version of events, the universe appeared from nothing as a quantum fluctuation, which should by rights have re-collapsed under its own enormous gravity and quickly disappeared again. However, that’s where inflation stepped in, swiftly expanding the universe out of the quantum realm before it had chance to re-collapse. Initially, the universe’s gravitational energy was balanced by the energy in the field of matter that drove inflation. But when inflation ended, this field decayed into the matter and energy that fills the universe today. In some sense, it’s this reheating of the universe after inflation that should be thought of as the actual ‘Big Bang’ – this is the point in cosmic history at which the current steadily expanding phase of the universe began, and where the stuff of the modern cosmos was actually forged. But that still left a question. It’s one thing to balance the energy budget, and prove that the universe could have been formed from nothing without violating any fundamental principles of physics. But it doesn’t explain how the infant universe actually came to be in an ocean of utter nothing. What was the theory, the actual physical laws, presiding over its creation? Any such theory will almost certainly make predictions about what the adult universe today should look like. And that’ll enable scientists to test it. Such an explanation can only lie in the domain of quantum gravity, the theory governing the ultra-small-scale behaviour of the gravitational force and its interaction with the other forces of nature. As we’ve already noted, during the Planck era, the universe was the size of a subatomic particle and so gravity at this time must have had to play by quantum rules. In fact, gravity in the early universe was entwined with the other forces into what’s known as the Theory of Everything, a quantum description of all four forces of nature wrapped up together, before gravity broke away when the temperature had dropped sufficiently at the end of the Planck era. But that’s a problem. Because quantum theory and gravity seem to be a marriage made in hell, with general relativity, our best theory of gravity, proving incompatible with the techniques that have led to the quantization of other forces, such as electromagnetism (see Chapter 6). Numerous approaches have been tried. We discussed string theory earlier, which treats particles as tiny vibrating loops of energy. Different particles can be thought of as different ‘notes’ played on these subatomic strings. Another is loop quantum gravity, in which space and time on tiny scales have a structure resembling chainmail armour, made of interlocking loops of energy knitted together. String theory is one quantum model for gravity that will, if it works, provide a framework that naturally unifies all the four forces together into one. Other approaches don’t necessarily offer that as a given. And physicists working in these areas have, perhaps wisely, opted to walk before they can run, focussing just on quantum gravity itself before trying to unify it with the other forces. One such idea has shown particular promise in explaining the first moments of the universe – and has accordingly become known as quantum cosmology. It was developed in the late 1960s by US theoretical physicists John Archibald Wheeler and Bryce DeWitt. They managed to construct a Schrödinger equation (see Chapter 6) that describes not just subatomic particles but the entire universe. Like the quantum descriptions of the other forces of nature, quantum gravity is ruled by randomness, dealing in probabilities rather than absolute, deterministic predictions. The solution to Wheeler and DeWitt’s equation is a wave function, like the wave function of a particle but giving probabilities for the state of the universe – that is, its gravitational field, as governed by the curvature of space and time. In the case of a simple subatomic particle of matter, the wave function might describe the particle’s location in space, with the peaks of the wave indicating where the particle is most likely to be found. That’s simple enough. But when you’re dealing not with a single particle but an entire universe, the wave function instantly becomes mind-bogglingly more complex. And so cosmologists make a simplifying assumption to keep their models tractable. Known as the mini-superspace approximation, it amounts to restricting the wave function to just a few numbers that describe the universe’s key properties – such as its size, and the state of the dominant form of matter or energy filling it. General relativity, in its most general form, gives answers that diverge to infinity when combined with quantum theory. But Wheeler and DeWitt found that the solution to Einstein’s equations for a homogeneous and isotropic universe, under the mini-superspace approximation, suffered no such problems – it could be quantized and gave sensible answers. Probabilities are calculated in the theory using the same method that US physicist Richard Feynman originally devised for the quantum theory of electromagnetism, and is known as the path integral approach. It means that if you want to calculate the probability of the mini-superspace describing the universe evolving from one state into another then you add up the probabilities for each of the different ways in which this can happen. It’s a little bit like rolling a pair of six-sided dice and wanting to know the probability that you’re going to roll a total of, say, 7. This can come about in a number of different ways, each of which occurs with a probability of 1 / 36 (1 / (6 × 6)). I could have 1 on the first die and 6 on the second, 2 on the first die and 5 on the second, 3 on the first and 4 on the second, and then all of these possibilities again with the order of the dice reversed – so, 6 on the first and 1 on the second, 5 on the first and 2 on the second, and 4 on the first and 3 on the second. So in all that’s six possible paths that give a total dice roll of 7, each occurring with a probability of 1 / 36, summing to a total probability of 1 / 6 (just 6 / 36, cancelling one factor of 6). Compare that to, say, the probability of rolling 12, which can happen only one way (two 6s) and so has a probability of just 1 / 36. In just the same way, the probability of the mini-superspace in quantum cosmology evolving from an initial state to a final state could be calculated by adding up the probabilities of all the different paths connecting the two. The trouble was, when it came to the birth of the universe, no one was really sure what the initial state should be. The final state was easy – that’s just the universe at the end of the Planck era, which undergoes inflation and then forms galaxies and clusters. But no one had the foggiest what the initial state should look like. In 1983, Hawking and his colleague American physicist James Hartle came up with a novel idea: they suggested that there was no initial state. This was called the no boundary proposal and it said that as you headed back to the last moment before the Big Bang, time morphed into a fourth spatial coordinate. The universe was then like the surface of a four-dimensional sphere, with space curved around on itself so that it had no boundaries. This meant that in the heart of the Big Bang, during the Planck era, there simply was no time and no concept of before and after. The conversion of time into a fourth spatial dimension in this way is a common feature of other quantum gravity models, and has shown promise in fields ranging from attempts to explain the nature of the cosmological constant to the physics of black holes. In Chapter 3, we drew an analogy between the early universe and the surface of the Earth, in that asking what came before the Big Bang makes about as much sense as trying to travel north of the North Pole. The analogy holds good here. If latitude is like time, then there are no latitudes north of the North Pole, while the Earth’s surface is an excellent example of a boundary-free space (albeit a two- rather than four-dimensional one) – you can travel any distance in any direction and never encounter an edge. The physical interpretation of the Hartle-Hawking initial state was that it corresponded to a universe being created from nothing, exactly as Einstein and Tryon had suggested. ‘Nothing’ in this sense was embodied by the fact that there really was nothing, no space or time, prior to the Planck era – just as there’s no Earth north of latitude +90. [image: f0188] The Hartle-Hawking initial state for the universe. The outer profile of the cone shape represents time which, at the Big Bang (the bottom of the diagram), curves around smoothly on itself to become like an extra dimension of space. The consequence of this is that the universe had no initial boundary in time which, if the theory is correct, means that it arose from nothing. The universe that emerges from the Hartle-Hawking state is actually closed, in the sense that its density exceeds the critical value needed to make it ultimately re-collapse in the far future. As we’ve seen, this means that it’s finite in extent and has no boundaries (just like the surface of an ordinary sphere). In contrast, the picture of the universe that’s been painted by studies of the cosmic microwave background radiation suggests that our universe has pretty much exactly the critical density, meaning that space is flat and infinite, and so will continue to expand for ever. But this is fine. One of the things that cosmological inflation does is to take a universe of arbitrary curvature and make it look as close as you like to flat – just as the surface of the Earth looks flat to us on its surface because it’s so much bigger than we are. This is exactly the resolution of the flatness problem that we explored in Chapter 8. Hartle and Hawking’s model is just one theory, but it demonstrated that a quantum treatment of the gravitational field really does have the power to explain how the universe could have appeared from nothing. Other theories have since been put forward, both within the framework of quantum cosmology, and also using other proposed quantum gravity approaches, such as string theory. In what would turn out to be the final piece of work before his death, Hawking investigated an alternative model based on string theory. Working with Belgian physicist Thomas Hertog, Hawking used an element of string theory known as the holographic principle, which says that the contents of any three-dimensional volume of space can be encoded on a two-dimensional surface forming the boundary of the space, to put forward a new theory as to how the universe began. The principle was partly inspired by Hawking’s earlier work on the physics of black holes, in which he’d found that the entropy of a black hole – the degree of disorder or ‘messiness’ associated with it (see Chapter 3) – increases not in proportion to the hole’s volume but in proportion to the area of its outer surface, the event horizon. Hawking and Hertog realized that just such a holographic boundary could constitute the initial state from which the universe was born. The earlier no-boundary model of Hartle and Hawking had applied quantum principles to a mini-superspace description of the universe, which was ultimately based on general relativity. This simplified scheme doesn’t seem to suffer from the issues that have plagued other attempts to usher general relativity into the quantum world. Nevertheless, Hawking and Hertog viewed the dependence on the quantum-unfriendly general relativity as a weakness. Their new holographic model had instead used string theory, a manifestly quantum approach to gravity, to chart the evolution of the universe away from its initial state, and the researchers viewed this as a far more reliable approach. However, the model was published only in May 2018 (in the Journal of High Energy Physics), and it thus remains to be seen whether this holographic description of the birth of the universe from nothing will enjoy the same longevity as its predecessor. The universe during the Planck era would have been a chaotic maelstrom, a writhing mass of space and time created by quantum randomness, with a foamy structure resembling the surface of a boiling pan of water. And this is exactly the chaotic state of the universe that would have served as the starting point for eternal inflation (see Chapter 8). The theory holds that random quantum fluctuations mean that somewhere the conditions will always be right for inflation to begin. And because inflation causes space to expand exponentially, these regions quickly come to dominate the universe. Eventually inflation ends in some regions, which then reheat and settle down to expand more sedately, just like the universe that we find ourselves in today. The remainder – indeed, the overwhelming majority – of the universe continues to inflate, and will still be inflating today. All it takes is for one tiny speck of space to continue inflating and that region will quickly dominate the volume of the universe, meaning that the process must continue for ever. And this is what’s meant by eternal inflation. Hertog speculates that it may be possible to test the holographic model by looking for gravitational waves created during eternal inflation. Just as inflation amplified tiny quantum fluctuations in the density of matter to create the seeds from which large-scale structures in the universe later grew, so it also amplified gravitational fluctuations in the curvature of space, which should now persist as ripples in space and time – exactly the gravitational waves that we met in Chapter 2, and which were discovered by ground-based detectors in 2014. In their paper, Hawking and Hertog calculated that the holographic model constrains the range of possible universes that can emerge from eternal inflation, and this should produce a measurably distinct gravitational-wave signature compared to the no-boundary proposal. The gravitational waves produced by inflation are too large to be picked up by current Earth-based detectors, but future space-based gravitational-wave observatories may well be up to the task. Gravitational-wave astronomy promises to revolutionize our view of the universe, and we’ll look at this in more detail in Chapter 14. If most of the universe really is still inflating today, as eternal inflation supposes, then there will be regions – far away from the space of our observable cosmos – where, right now, inflation is only just ending. Let that sink in for a moment. Somewhere out there, an almost unimaginable distance away, new universes are being born as you read this. Russian-American inflationary cosmologist Andrei Linde imagines the process creating an ever-growing cosmic tree-like structure, consisting of very many vast, inflating domains interspersed with patches of space that resemble our own observable neighbourhood – a construct he calls the eternally existing self-reproducing inflationary universe. It’s a sobering thought that our universe may, in fact, be just one of many – a dot in a sprawling multiverse of parallel worlds. And, as we’ll see in the next chapter, it’s a theory that offers an intriguing explanation for why our universe takes the form it does – and why we are here to worry about it all. The Long Dark Eternity ‘In 100 billion years, the universe will be a very strange place.’ If the fire and brimstone of a Big Crunch seems all too apocalyptic, you might prefer the somewhat gentler alternative – a scenario called the Heat Death. Here, the universe never really dies but gradually fades to black as the expansion of space marches on, and on, and on – for ever. Based on the current astronomical evidence, this seems the most likely fate for our universe. We saw that, in the absence of dark energy, the universe will expand without end if its matter and energy content is less than or equal to the so-called critical density (see Chapter 3). In this case there’s insufficient gravity to make the universe fall back in on itself in a Big Crunch. Studies of the CMB radiation using data gathered by the ESA’s Planck spacecraft suggest that our universe weighs in at around 0.999 of the critical density, implying that it will indeed expand for eternity (see here). And that’s without even considering the effect of dark energy, which accelerates the expansion of space through its repulsive gravity. Even if the density of the universe exceeded the critical value, the extra kick from dark energy could still conspire to make it eternally expanding. As we touched on in the previous chapter, it seems the only thing that can avert a Heat Death is a radical change in the nature of dark energy – for example, if it switched polarity to become gravitationally attractive at some point in the future. This can’t be ruled out, given how little we presently understand about dark energy and how it really works. But for this chapter, we’re going to assume that its gravity remains repulsive and that the universe is destined for a future of never-ending, relentless expansion. One of the first comprehensive studies into the future of an ever-expanding universe was carried out by American cosmologists Fred Adams and Gregory Laughlin, who summarized their findings in the 1999 book The Five Ages of the Universe. As the title suggests, they partitioned cosmic history into five distinct epochs. The first is known as the primordial age. We’ve already encountered this cosmic time period – it started at the Big Bang and lasted until the universe was a few hundred million years old. This is the era in which the universe was born. The forces of nature separated into distinct entities. Inflation happened, planting the density fluctuations from which galaxies later grew. Nucleosynthesis took place, converting the soup of particles emerging from the Big Bang into the first chemical elements. And, finally, the universe became transparent to radiation, allowing the microwave background radiation to stream freely out through space. The primordial age ended when the first stars in the universe began to switch on. These were the Population III stars (see Chapter 9), which first lit up around 180 million years after the Big Bang. Their appearance marked the beginning of Adams and Laughlin’s second cosmic age, the stelliferous era – the age of stars. This is the age in which we currently find ourselves, at a time around about 13.8 billion years after the Big Bang. The universe at this time is brightly lit by galaxies full of stars, each a blazing beacon, burning chemical elements by nuclear fusion reactions in their interiors. Stars don’t last for ever. Eventually they run out of fuel, the nuclear reactions within them switch off and they die. New generations of stars are continually forming – the Hubble Space Telescope, for example, has returned stunning images of giant hydrogen gas clouds in which new generations of stars are condensing and just beginning to shine. But this process cannot continue indefinitely. The raw materials for stars – the elements hydrogen and helium – eventually become depleted throughout the universe to the point that new stars can no longer form, and at this point the stelliferous era comes to an end. It’s estimated that this will happen when the universe is some 100,000 billion years old. In other words, not until the universe is more than 7,000 times older than it already is. Much will happen during the stelliferous era. In 4 billion years’ time, the Milky Way will collide with the Andromeda galaxy. The likely outcome of this is that both galaxies will merge into one, though the intricate spiral structure of each will be completely disrupted, leaving an irregularly shaped, amorphous body of stars. As we discussed in the previous chapter, the sun and solar system will probably survive the collision because direct encounters between stars will be rare (see here). All the while, space is expanding and sweeping the galaxies in the night sky further away from us. The further away they get, the faster they recede, and the more redshifted their light becomes, making them harder to see – until, when their rate of recession equals the speed of light, they ultimately fade from view. By the time the universe is 150 billion years old, still well within the stelliferous era, the galaxies beyond our Local Group (see Chapter 9) will have reached the very edge of the observable universe. We will still be able to see them as they fade, but they will be causally unreachable – it will be impossible by this time for signals to travel between them. Earth will be long gone by this time, as we saw in the previous chapter, devoured by the expanding sun during the death of the solar system many billions of years earlier. Humanity’s descendants may have settled on another world, or could inhabit colonies in space. Their astronomers will see the galaxies disappear from view one by one, leaving the Milky Way ultimately alone in the cosmos. Other changes are afoot. Before the universe celebrates its 1,000-billionth birthday, the gravity of the Local Group will have caused its members to all merge into one gigantic supergalaxy. And by 2,000 billion years, this will be the only galaxy visible within our observable universe – all others will be redshifted from view and no longer visible to us. Over the thousands of billions of years that follow, the stars, one by one, burn out and die, and are not replaced. The biggest stars literally live fast and die young. Those weighing upwards of sixty times the mass of the sun burn ferociously and vanish in just a few million years – a blink of the cosmic eye, comparable to the time that’s elapsed today since the first early humans emerged here on Earth. By comparison, stars like our own sun typically live for some 10 billion years. And the very lightest, those around a tenth the mass of the sun, shine with a feeble light but as a trade-off live for thousands of billions of years. This all means that from around 800 billion years after the Big Bang, the brightness of our galaxy – and all those other galaxies that are no longer visible to us – begins to dwindle away. When the universe has reached the ripe old age of 100,000 billion years, the last geriatrics in the last generation of stars die. With that, the stelliferous era ends and so begins a new and bleak cosmic age known as the degenerate era. Nothing to do with cosmic moral standards, degeneracy here refers to a very dense state of matter, in which subatomic particles are packed together extremely tightly. Particles such as electrons, protons and neutrons all obey a law of quantum theory called the Pauli exclusion principle, after the Austrian-born physicist Wolfgang Pauli who formulated it as a principle of quantum theory in 1925. It says that the particles don’t like to be in the same place with the same energy at the same time – to the point that if you try to force them into such a state, they experience a physical force pushing them apart. However, if you can overcome this force and squeeze the particles into the same state then the resulting matter is said to be degenerate. The corpses of stars are very often made from degenerate matter. Ordinarily, a star supports its own weight by the thermal pressure generated from the heat of nuclear reactions in its interior. Think of the way a piston expands when the gas inside is heated. In a star, the same tendency of hot gases to expand counters the inward pull of gravity and stops the star collapsing under its own weight. But when a star runs out of nuclear fuel and stops generating its own heat, then the thermal pressure drops and collapse is inevitable. The Indian mathematician and astrophysicist Subrahmanyan Chandrasekhar showed in the 1930s that for stars not exceeding about 1.4 times the mass of the sun, the degeneracy force between electrons is sufficient to halt the collapse and support the star against its own gravity. The resulting objects, supported entirely by electron degeneracy pressure, are known as white dwarf stars. They are superdense, packing approximately the mass of the sun into a sphere about the size of the Earth. And, as we saw in Chapter 12, our sun will end its days as one of these objects. But when the collapsing star is heavier than 1.4 solar masses, electron degeneracy pressure isn’t up to the task. Instead, electrons are forced into the same quantum state as their neighbours and the star continues to collapse. Protons and electrons are then squeezed together to form neutrons. If the star weighs less than about three times the mass of the sun, then neutron degeneracy pressure can step in to halt the collapse. The resulting object is called a neutron star. It squashes matter equivalent to several times the mass of the sun down into a sphere just 10 kilometres across. These objects are so dense that a litre of neutron star material weighs about the same as a mountain. Degenerate stellar objects are born extremely hot, with temperatures of tens of millions of degrees C. Although with no way to generate their own heat, they gradually cool, radiating their energy into space until they fade from view. There’s a limit to the amount of mass that can be supported by degenerate neutron pressure, and anything weighing more than three solar masses is doomed to collapse all the way down to a black hole – a spacetime singularity, a point of infinite density, surrounded by an event horizon from which nothing falling in can ever escape. In the degenerate era, most of the ordinary matter in the universe – the stuff that was formerly in stars – ends up in a degenerate state inside white dwarfs and neutron stars. But the universe isn’t done yet. Grand unified theories – which say that electromagnetism, and the strong and weak nuclear forces, were once different aspects of the same fundamental force in physics – assert that protons and neutrons will ultimately decay, breaking apart into lighter fundamental particles. This is thought to occur when the universe reaches between 10,000 billion billion billion and 1 billion billion billion billion years old. And that’s not the only pernicious force at work. As degenerate stellar remnants degrade and break down, their material is also being slowly eaten away by the universe’s population of black holes. They skulk through space, hoovering up anything that they happen upon until every last scrap is locked away in the dark hearts of these cosmic garbage collectors. Adams and Laughlin suggest this process would be complete by the time the universe is around 10,000 billion billion billion billion years old. By this time, the universe is completely and totally dark. Not only will all galaxies have been redshifted beyond the bounds of the observable universe, but all of the stars in our own galaxy will have been extinguished and they, along with all of the planets, will have been devoured by black holes. Inhabitants of a planet may see their fate coming. Despite the name, black holes in the universe today aren’t invisible but are surrounded by a disc of material swirling around their equator rather like a spiral galaxy, before falling in. With luck, this will enable any spacefaring beings to escape before the planet crosses the black hole’s event horizon – an imaginary sphere surrounding the hole, and from within which the gravity is too strong for even light to escape. Once the planet crosses this threshold, there is no return. As the planet nears the singularity at the black hole’s core, the gravitational forces will increase extremely rapidly. The difference between the forces pulling on one side of the planet and the other will become enormous. And this will stretch out the planet, and everything on it, into a long spaghetti-like strand that will be consumed in an instant and then gone. And that’s why, from 10,000 billion billion billion billion years, the universe is officially in the black-hole era. Though it won’t be like this for ever. As Hawking famously proved, black holes ain’t so black. In what was probably the defining achievement of his scientific career, Hawking proved mathematically that black holes aren’t purely the devourers of all that comes their way, but that they actually emit a tiny stream of particles and radiation in the opposite direction. We know that, because of quantum physics, virtual particles are popping in and out of existence in empty space. Outside the event horizon of a black hole, the exact same process is taking place; but whereas the virtual particles normally annihilate one another after a short time, near a black hole one particle will occasionally get sucked in by the hole’s gravity before the two have had chance to recombine. When this happens, one particle escapes from the black hole and carries away positive energy, while the other falls in, carrying in an equal but opposite energy, and thus lowering the black hole’s mass ever so slightly. The process is known as black-hole evaporation. [image: f0239] Pairs of virtual particles are continually creating and annihilating outside the event horizon of a black hole. Sometimes both particles in the pair will recombine (case 1) and sometimes both will fall into the black hole before this can happen (case 2). Occasionally though, one particle falls in and the other escapes, creating a net flow of matter and energy out of the black hole. This is Hawking’s black-hole evaporation. In our universe today, the quantity of matter carried away from black holes by this process is tiny compared with the amount falling in through natural gravitational attraction. But during the universe’s black-hole era, when black holes exist alone in utter isolation, with no other matter for them to feed on, then black-hole evaporation is a constant erosion that steadily wears the mass of every black hole in the universe away to zero. Admittedly, the process is extremely slow, but by approximately 10 billion billion billion billion billion billion billion billion billion billion billion years after the Big Bang – a staggeringly long number of years – every black hole in the universe has evaporated, and the particles and radiation that they’ve given off have been stretched out by the expansion of the universe and spread as a wafer-thin smear across space. At this point, the universe has entered its final phase, the cosmic dark era. It’s now in the most disordered state possible, with matter and radiation distributed across space with perfect randomness – all of the neat partitioning of material into stars, galaxies, even atoms and molecules has been erased. It’s rather like my messy desk taken to the ultimate extreme, where every book, every coffee cup and every article of detritus has been ground down into atoms, mixed up thoroughly and spread across the desk with absolute uniformity. The second law of thermodynamics says that entropy, the degree of disorder of the universe, must always increase (see here). In the cosmic dark era, the universe has reached its maximum entropy state, and here physics must cease. This is the real meaning of the term Heat Death. It refers to a final state in which every corner of the universe is exactly the same temperature as every other. In this state, there can be no flow of heat, or any other form of energy, from one place to another. It derives from the work of the nineteenth-century Scottish-Irish physicist William Thomson (also known as Lord Kelvin) who laid much of the groundwork for the field of thermodynamics. During the 1850s, he developed the idea that mechanical energy must ultimately dissipate into heat. Applied on the large scale, it meant there would come a point where the mechanical energy of the whole universe – the energy required for any sort of process in physics to take place – would fall away to zero. In an article published in 1862, entitled ‘On the Age of the Sun’s Heat’, Thomson described the universe as ‘running down like a clock, and stopping forever’. One side-effect of the Heat Death scenario is that it’s not just about the death of the universe but also the death of cosmology as a testable science. As the expansion of space carries the other galaxies in the universe over our cosmological horizon and out of view, so we lose access to all of the tests and measures against which we control our theories of how the universe works. That means that if the knowledge we’ve accumulated about the universe is for any reason lost, there will be no way in the cosmic far future for it to be rediscovered. For example, the only reason we know that the expansion of the universe is accelerating is because we can observe distant galaxies and use their recession rates, as inferred from their redshifts, to chart how the expansion rate of the universe has evolved over cosmic time (see Chapter 7). By the time the universe is 2,000 billion years old, ours is the only galaxy we can see. There are no cosmic mileposts against which to gauge the expansion of space, and how that expansion has changed since the Big Bang. It’s not just the expansion rate of the universe that becomes unmeasurable during a Heat Death. That other great window on the nature of the Big Bang, the cosmic microwave background, will finally close on us too. The 13.8 billion years of expansion between the birth of the universe and the present day has redshifted the temperature of the universe from an infinitely large value down to just 2.73 degrees above absolute zero. As the universe continues to expand, the CMB is redshifted further. Calculations made in 2007 by the US cosmologists Lawrence Krauss and Robert Scherrer showed that by the time the universe is 700 billion years old, the CMB will have become so dilute that we’ll no longer be able to detect it above the background hum of the gas that hangs between the stars in our galaxy. Even perhaps the strongest evidential pillar for the Big Bang theory, the abundance of the light elements produced during primordial nucleosynthesis – the creation of hydrogen, helium and lithium by nuclear fusion during the Big Bang – will have been largely washed away after 1,000 billion years of cosmic expansion. The key piece of information here is the relative abundance of hydrogen to helium, which is both predicted and observed today to be around 3:1. But the Big Bang isn’t the only thing capable of converting hydrogen into helium – stars are also notoriously good at it. Nuclear fusion in a star welds hydrogen nuclei together into helium and when the star reaches the end of its life – either blowing itself to bits in a supernova, or puffing itself up to become a red giant – this material gets flung off into space, where it enriches the interstellar clouds that feed subsequent generations of stars. This will mean that, by the time the universe has reached 1,000 billion years old, the actual hydrogen to helium ratio will have evolved considerably from 3:1, taking a value more like 1:3. And with no way of measuring the universe’s expansion history, it’ll be difficult to work out how many years of stellar pollution need to be accounted for to wind that ratio back in time and get a number representative of the universe as it was when it emerged from the Big Bang. Krauss and Scherrer speculate that any would-be cosmologists in new civilizations appearing 1,000 billion years from now will probably be forced to conclude that we live in a static and unchanging universe, much as Einstein believed before Hubble and Humason’s discovery that the universe is expanding. Indeed, it would be fascinating to see how the course of science, physics in particular, might have progressed (or not) without all those cues and prompts from observational cosmology. We saw earlier how dark energy could potentially flip a Heat Death into a Big Crunch – if, for example, the gravitational force created by dark energy could evolve to become less negative, or even positive like ordinary matter. But what if the opposite could happen, and dark energy was allowed to become even more repulsive? This is the basic premise of an idea first put forward in 2003 by a group of American cosmologists led by Robert Caldwell of Dartmouth College, New Hampshire. The scenario is called the Big Rip and it supposes that the expansion of space becomes so rapid that material structures are literally torn apart. Back in Chapter 8, you may remember we met something called the energy conditions – hypothetical constraints that physicists have placed on the nature of the matter filling space and time. As we saw, cosmic inflation violates something called the strong energy condition, which amounts to saying that negative-pressure material is allowed to exist. If dark energy is caused by vacuum fluctuations, the energy in empty space, then it too must violate the strong energy condition. But what Caldwell and his colleagues were imagining was a form of matter even more extreme than this. Known as phantom energy, its pressure is much more negative than that of vacuum fluctuations; and, accordingly, it violates all of the energy conditions in the physicist’s book (there are four of them in total). Phantom energy causes the very rate at which cosmic expansion is accelerating to accelerate and gather pace. A typical example of a Big Rip scenario has the cosmos bowing out around 22 billion years from now, considerably sooner than the 10-billion-billion-billion-billion-billion-billion-billion-billion-billion-billion-billion-year lifespan it has in the case of a classic Heat Death – but probably still no cause for immediate panic. Around a billion years before the end, the expansion rate of the universe would begin to pick up, becoming fast enough to pull clusters of galaxies apart, overcoming the gravitational forces binding each galaxy to the next. If our astronomers were staying vigilant and checking their records as the aeons passed, they would see the redshifts of the galaxies in our Local Group begin to dramatically increase as the phantom energy tightened its grip on them. As time passed, galaxies would become isolated entities in deep space, unbound to any other structures in the cosmos. At 60 million years before the end, the expansion would be capable of wreaking havoc within individual galaxies such as the Milky Way. Stars are stretched apart from one another and galaxies are rapidly smeared out and erased. The night sky on a clear evening would quickly become dark as the spectacular band of the Milky Way dispersed and stars drifted apart into the blackness. At three months to go, the same thing happens to solar systems, as planets are ripped away from their parent stars and dragged off into space. Any life forms reliant on heat from their star rapidly perish. For others, there is only brief respite. With just thirty minutes to go, stars and planets themselves succumb, torn apart into gas and rubble by the inexorable cosmic expansion. At around 10-billion-billionths of a second before the Big Rip the force exerted by the expansion of space is ferocious enough to be felt even on quantum scales, so that atoms and molecules are reduced to their composite protons, neutrons and electrons. And just the merest shred of a moment later, these too are pulverized down into the most fundamental quantum entities before being scattered to the corners of the cosmos. For the briefest instant, the universe has attained the sublime emptiness of the Heat Death scenario. But then it’s all gone. The expansion becomes too much for even space to bear and the infinite gravitational forces tear it apart. The universe has become everywhere a singularity, a region of infinite spacetime curvature, much like the Big Bang singularity whence it came, and where the infinite force of gravity causes physics itself to break down. Welcome to the end of the universe. Most of Our Universe is Missing ‘You know, dark matter matters.’ Looking up at the sky on a clear, dark night reveals a blaze of light from nearby stars and far-away galaxies. And yet astrophysicists believe this astonishing display represents just 5 per cent of the actual matter present in our universe. The remaining 95 per cent is dark stuff lurking tantalizingly out of sight – non-luminous or otherwise undetectable material pervading space alongside the bright material that we can see. Ninety-five per cent of the universe is a big deal – easily enough to decide its ultimate fate. That all hinges on the total amount of matter out there in space – whether there’s sufficient to ultimately halt cosmic expansion or whether the universe is gravitationally unbound and will continue to expand for ever. It also turns out to be an important consideration at the other end of the cosmic timeline, strongly influencing the formation of galaxies and clusters of galaxies shortly after the Big Bang. Nobody’s quite sure what this dark stuff is actually made of – despite many science-budget dollars and cosmologist hours being spent in the quest to find out. The one thing most researchers are fairly sure about is that it has to be there. Gravity makes things move – the more gravity there is, the faster they tend to go. And numerous instances are now known where stars and galaxies seem to move much faster than they should, given the amount of gravity inferred from just the bright material – stars and glowing nebulae – alone. Bizarrely, scientists came up with a name for the missing material before there was any palpable evidence for it. In 1906, French mathematician Henri Poincaré used the term dark matter to describe non-luminous material in our Milky Way galaxy. One of the first to claim to have caught a glimpse of it was the Dutch astronomer Jan Oort, in 1932. He measured the speeds of stars moving up and down through the disc of the Milky Way. The gravity of the matter present in the disc makes the stars bob up and down like carousel horses as they circle around the galaxy’s centre. But Oort saw that the stars weren’t moving as far above or below the plane of the disc as they should be – it was as if the gravitational force holding them back was stronger than could be produced by the amount of material visible in the galactic disc. The following year, Swiss astronomer Fritz Zwicky provided further evidence. Zwicky was a brilliant astronomer who coined the term ‘supernova’ for the explosions marking the deaths of massive stars, predicted the existence of neutron stars (the superdense remnant left behind after a supernova) and later pioneered the use of supernovae for gauging distances across the universe. He also made contributions to the classification and cataloguing of galaxies and correctly predicted that Einstein’s work on the bending of light in general relativity meant that massive objects could curve and focus light like a lens, magnifying any objects behind them – this gravitational lensing effect, as it became known, was subsequently observed in 1979. Zwicky was also a famously cantankerous personality, frequently falling out with colleagues and acquaintances. His favourite insult was to refer to those he disliked as ‘spherical bastards’ – because, ‘No matter how you look at them, they are just bastards.’ Zwicky was studying a group of more than 1,000 galaxies known as the Coma Cluster, 323 million light-years away from Earth in the constellation of Coma Berenices. Zwicky measured how fast the galaxies in the cluster were moving. The simple fact that the galaxies weren’t all flying off into space meant that there must be enough gravity in the cluster to hold them all together as a bound system. It’s rather like firing a cannon into the air on Earth. If I see the cannon ball eventually stop and fall back to the ground then I know that the Earth’s gravity, and hence its mass, must be sufficient to halt the ball’s ascent. If I could fire a number of cannon balls, then in principle I could measure the speed of the slowest ball that’s able to escape the Earth’s pull and from this use Newton’s law of gravity to deduce the planet’s mass. But when Zwicky did this for the galaxies in the Coma Cluster, he found that the mass of the cluster, calculated from its gravity, didn’t square at all with the figure he got by simply adding up the estimated masses of all its galaxies. Estimating the mass of a distant galaxy isn’t straightforward. If the galaxy is nearby, you can measure the motions of its stars and, again like the cannon ball analogy, get a handle on how much gravitating material must be present in order to prevent the stars whizzing off into space. However, at hundreds of millions of light-years distant, the individual stars making up the galaxies in the Coma Cluster could not be resolved. Instead, Zwicky used what astronomers refer to as a mass-to-light ratio. By studying a number of typical nearby galaxies – measuring their brightnesses and estimating their masses from the speeds of their constituent stars – an average ratio of the amount of mass present per unit of brightness can be deduced. Zwicky could then apply these ratios to the observed brightness of each of the galaxies in the Coma Cluster to derive an estimate of its total mass. When he did this and found that the value he got for the cluster’s mass was hundreds of times smaller than that inferred from the galaxy speeds, Zwicky pronounced that the only resolution was for there to be a large quantity of material in the cluster that couldn’t be seen. Zwicky’s numbers were in fact a little wide of the mark, mainly because 1930s estimates of the Hubble constant (see Chapter 3), which governs the expansion rate of the universe, were way too big. This led Zwicky to overestimate the distance to the Coma Cluster (remember, the distance is given by the measured recession speed multiplied by the Hubble constant). And this in turn threw out his estimate of the quantity of dark matter required to hold it together, though his broad conclusions – that most of the cluster’s mass is invisible – were later borne out. During the 1970s, American astronomer Vera Rubin, working with colleague Kent Ford, both at the Carnegie Institution of Washington, uncovered evidence for dark matter existing on smaller scales. They were studying spiral galaxies – that is, swirling collections of stars, much like our own Milky Way – and in particular their rotation curves, graphs showing how the speed at which stars are circling around the galaxy varies with the distance from its centre. Newton’s law of gravity (see Chapter 2) dictated that outside the galaxies’ bright central nucleus the rotation speeds should decrease with the distance from the centre. Close to the centre, Rubin’s observations agreed with theoretical predictions. But in the outer region of a galaxy – known as the halo – theory and observation went their separate ways. Rubin and Ford found the rotation speed in the haloes to be roughly constant, not decreasing at all. (In fact, there are very few stars in the halo of a galaxy and so Rubin measured the rotation rate here by tracking the motion of glowing clouds of hydrogen gas.) The only way this could happen is if the halo was chock-full of hidden dark matter, making up at least 50 per cent of each galaxy’s total mass. [image: f0099] In the 1970s, Vera Rubin and Kent Ford discovered that the rotation speeds of spiral galaxies deviate considerably from theoretical predictions. The implication was that galaxy haloes contain huge amounts of invisible ‘dark’ matter. In the late 1990s, international collaborations of astronomers using images and data gathered by the Earth-orbiting Hubble Space Telescope (HST) began to study the gravitational lensing by clusters of far-away quasars. These are very distant galaxies, also known as quasi-stellar objects – so named because they’re so far away that they appear as pinpricks of light when observed through a telescope, resembling a star rather than the usual fuzzy form of a galaxy. They are ferociously bright, each powered by a giant black hole at its core weighing anything up to several billion times the mass of the sun. The black hole is devouring the stars and gas in the host galaxy, which spews out radiation as it gets compressed and heated by the intense gravity. At the time of writing, in spring 2018, the most distant quasar known lies an astonishing 13 billion light-years away, and the light that we see from it was released when the universe was just a few hundred million years old. Although point-like when seen in a normal telescope, viewing a quasar through the gigantic natural telescope formed by a gravitational lens deforms the image into one or more banana-shaped arcs. Einstein showed that an exact alignment between the observer, the quasar and the cluster, and with all of the cluster’s mass concentrated in a point at the centre, would stretch the image into a perfect ring. In practice, these simplifications are rarely valid. But for astronomers studying dark matter that’s a good thing. By modelling the gravitational influence of the lensing cluster on a computer, they’re able to invert the observed structure of the arcs in the image, to reconstruct how the mass of the cluster is distributed within it. Doing this reveals sharp peaks of mass corresponding to the individual galaxies, superimposed on to a smooth background of matter between them. This smooth background is invisible dark matter, which is estimated to account for upwards of 85 per cent of the cluster’s mass. So what exactly might this strange stuff be made of? The initial temptation may be to think that dark matter is just ordinary matter that’s not giving off any light. After all, you and I are both made of ordinary matter and we’re not luminous. Asteroids plying the depths of the solar system are non-luminous, unless we happen to catch a glint of sunlight reflected from them. And we know for a fact that there are dark dust clouds out in space – think, for instance, of the iconic image of the Horsehead Nebula. This is a cloud of hydrogen gas in the constellation of Orion that’s being illuminated and caused to glow bright pink by a nearby bright star. Part of the glowing gas cloud is obscured by a blotch of intervening dust to form a striking silhouette in the shape of a horse’s head. The trouble is that so much dark matter is required that if it was made from ordinary matter it would reveal itself to us. If the dark matter was dust we’d see it obscuring the light from many more stars and bright nebulae – in exactly the same way that we can see the Horsehead. Even if it was in the form of small dark bodies, we would still see them. For example, not so long ago it was thought that some of the dark matter in galaxies could exist in the form of what are called massive compact astrophysical halo objects, or MACHOs, swirling around in the galaxies’ outer halos. MACHOs are a broad class of objects. They include brown dwarfs – essentially failed stars, each a sphere of hydrogen gas (exactly like an embryonic star) but which has failed to grow massive enough to spark up nuclear fusion in its core. Also included are stellar remnants – that is, superdense objects (including black holes and cooled neutron stars and white dwarfs, which we’ll come to shortly) left behind after a star has ended its life. It was proposed that we might be able to detect MACHOs by gravitational microlensing, which is like the gravitational lensing of a distant quasar by an intervening cluster of galaxies but on an altogether smaller scale. The idea was that when a MACHO passed in front of a background star, it would cause a momentary brightening of the star, which in principle could be detected. However, telescopic searches have failed to turn up enough microlensing events for MACHOs to make anywhere near a significant contribution to the dark matter in galaxies. More enlightening and compelling evidence followed from detailed measurements of the cosmic microwave background (CMB). As we discovered in the previous chapter, the CMB is the relic echo from the fireball of the Big Bang, or rather from the last scattering surface, 380,000 years after the Big Bang. This surface represents the furthest that we can see back into the cosmic past – prior to this, matter and radiation were interacting, making the universe opaque. But after last scattering, they decoupled,
28a07a19431a20da	My watch list Orbital hybridisation In chemistry, hybridisation or hybridization (see also spelling differences) is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the qualitative description of atomic bonding properties. Hybridised orbitals are very useful in the explanation of the shape of molecular orbitals for molecules. It is an integral part of valence bond theory. Although sometimes taught together with the valence shell electron-pair repulsion (VSEPR) theory, valence bond and hybridization are in fact not related to the VSEPR model.[1] Additional recommended knowledge Historical development The hybridisation theory was promoted by chemist Linus Pauling[2] in order to explain the structure of molecules such as methane (CH4). Historically, this concept was developed for such simple chemical systems but the approach was later applied more widely, and today it is considered an effective heuristic for rationalizing the structures of organic compounds. Hybridisation theory is not as practical for quantitative calculations as Molecular Orbital Theory. Problems with hybridisation are especially notable when the d orbitals are involved in bonding, as in coordination chemistry and organometallic chemistry. Although hybridisation schemes in transition metal chemistry can be used, they are not generally as accurate. It is important to note that orbitals are a model representation of the behavior of electrons within molecules. In the case of simple hybridisation, this approximation is based on the atomic orbitals of hydrogen. Hybridised orbitals are assumed to be mixtures of these atomic orbitals, superimposed on each other in various proportions. Hydrogen orbitals are used as a basis for simple schemes of hybridisation because it is one of the few examples of orbitals for which an exact analytic solution to its Schrödinger equation is known. These orbitals are then assumed to be slightly, but not significantly, distorted in heavier atoms, like carbon, nitrogen, and oxygen. Under these assumptions is the theory of hybridisation most applicable. It must be noted that one does not need hybridisation to describe molecules, but for molecules made up from carbon, nitrogen and oxygen (and to a lesser extent, sulfur and phosphorus) the hybridisation theory/model makes the description much easier. The hybridisation theory finds its use mainly in organic chemistry, and mostly concerns C, N and O (and to a lesser extent P and S). Its explanation starts with the way bonding is organized in methane. sp3 hybrids Hybridisation describes the bonding atoms from an atom's point of view. That is, for a tetrahedrally coordinated carbon (e.g. methane, CH4), the carbon should have 4 orbitals with the correct symmetry to bond to the 4 hydrogen atoms. The problem with the existence of methane is now this: Carbon's ground-state configuration is 1s2 2s2 2px1 2py1 or perhaps more easily read: C\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\downarrow}{2s}\; \frac{\uparrow\,}{2p_x}\; \frac{\uparrow\,}{2p_y}\; \frac{\,\,}{2p_z} (Note: The 1s orbital is lower in energy than the 2s orbital, and the 2s orbital is lower in energy than the 2p orbitals) The valence bond theory would predict, based on the existence of two half-filled p-type orbitals (the designations px py or pz are meaningless at this point, as they do not fill in any particular order), that C forms two covalent bonds. CH2. However, methylene is a very reactive molecule (see also: carbene) and cannot exist outside of a molecular system. Therefore, this theory alone cannot explain the existence of CH4. Furthermore, ground state orbitals cannot be used for bonding in CH4. While exciting a 2s electron into a 2p orbital would theoretically allow for four bonds according to the valence bond theory, (which has been proved experimentally correct for systems like O2) this would imply that the various bonds of CH4 would have differing energies due to differing levels of orbital overlap. Once again, this has been experimentally disproved: any hydrogen can be removed from a carbon with equal ease. To summarise, to explain the existence of CH4 (and many other molecules) a method by which as many as 12 bonds (for transition metals) of equal strength (and therefore equal length) can be created was required. The first step in hybridisation is the excitation of one (or more) electrons (we consider the carbon atom in methane, for simplicity of the discussion): C^{}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{2s}\; \frac{\uparrow\,}{2p_x} \frac{\uparrow\,}{2p_y} \frac{\uparrow\,}{2p_z} The proton that forms the nucleus of a hydrogen atom attracts one of the valence electrons on carbon. This causes an excitation, moving a 2s electron into a 2p orbital. This, however, increases the influence of the carbon nucleus on the valence electrons by increasing the effective core potential (the amount of charge the nucleus exerts on a given electron = Charge of Core − Charge of all electrons closer to the nucleus). The combination of these forces creates new mathematical functions known as hybridised orbitals. In the case of carbon attempting to bond with four hydrogens, four orbitals are required. Therefore, the 2s orbital (core orbitals are almost never involved in bonding) mixes with the three 2p orbitals to form four sp3 hybrids (read as s-p-three). See graphical summary below. becomes C^{}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp^3}\; \frac{\uparrow\,}{sp^3} \frac{\uparrow\,}{sp^3} \frac{\uparrow\,}{sp^3} In CH4, four sp3 hybridised orbitals are overlapped by hydrogen's 1s orbital, yielding four σ (sigma) bonds. The four bonds are of the same length and strength. This theory fits our requirements. translates into An alternative view is: View the carbon as the C4− anion. In this case all the orbitals on the carbon are filled: C^{4-}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\downarrow}{2s}\; \frac{\uparrow\downarrow}{2p_x} \frac{\uparrow\downarrow}{2p_y} \frac{\uparrow\downarrow}{2p_z} If we now recombine these orbitals with the empty s-orbitals of 4 hydrogens (4 protons, H+) and allow maximum separation between the 4 hydrogens (i.e. tetrahedral surrounding of the carbon), we see that at any orientation of the p-orbitals, a single hydrogen has an overlap of 25% with the s-orbital of the C, and a total of 75% of overlap with the 3 p-orbitals (see that the relative percentages are the same as the character of the respective orbital in an sp3-hybridisation model, 25% s- and 75% p-character). According to the orbital hybridisation theory the valence electrons in methane should be equal in energy but its photoelectron spectrum [3] shows two bands, one at 12.7 eV (one electron pair) and one at 23 eV (three electron pairs). This apparent inconsistency can be explained when one considers additional orbital mixing taking place when the sp3 orbitals mix with the 4 hydrogen orbitals. sp2 hybrids Other carbon based compounds and other molecules may be explained in a similar way as methane. Take, for example, ethene (C2H4). Ethene has a double bond between the carbons. The Lewis structure looks like this: Carbon will sp2 hybridise, because hybrid orbitals will form only σ bonds and one π (pi) bond is required for the double bond between the carbons. The hydrogen-carbon bonds are all of equal strength and length, which agrees with experimental data. C^{}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp^2}\; \frac{\uparrow\,}{sp^2} \frac{\uparrow\,}{sp^2} \frac{\uparrow\,}{p} forming a total of 3 sp2 orbitals with one p-orbital remaining. In ethylene the two carbon atoms form a σ bond by overlapping two sp2 orbitals and each carbon atom forms two covalent bonds with hydrogen by ssp2 overlap all with 120° angles. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. The amount of p-character is not restricted to integer values, i.e. hybridisations like sp2.5 are also readily described. In this case the geometries are somewhat distorted from the ideally hybridised picture. For example, as stated in Bent's rule, a bond tends to have higher p-character when directed toward a more electronegative substituent. sp hybrids The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization. C^{}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp}\; \frac{\uparrow\,}{sp} \frac{\uparrow\,}{p} \frac{\uparrow\,}{p} In this model the 2s orbital mixes with only one of the three p-orbitals resulting in two sp orbitals and two remaining unchanged p orbitals. The chemical bonding in acetylene (C2H2) consists of spsp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by pp overlap. Each carbon also bonds to hydrogen in a sigma ssp overlap at 180° angles. Hybridisation and molecule shape Hybridisation, along with the VSEPR theory, helps to explain molecule shape: • AX1 (e.g., LiH): no hybridisation; trivially linear shape • AX2 (e.g., BeCl2): sp hybridisation; linear or diagonal shape; bond angles are cos−1(−1) = 180° • AX3 (e.g., BCl3): sp2 hybridisation; trigonal planar shape; bond angles are cos−1(−1/2) = 120° • AX4 (e.g., CCl4): sp3 hybridisation; tetrahedral shape; bond angles are cos−1(−1/3) ≈ 109.5° • AX5 (e.g., PCl5): sp3d hybridisation; trigonal bipyramidal shape • AX6 (e.g., SF6): sp3d2 hybridisation; octahedral (or square bipyramidal) shape This holds if there are no lone electron pairs on the central atom. If there are, they should be counted in the Xi number, but bond angles become smaller due to increased repulsion. For example, in water (H2O), the oxygen atom has two bonds with H and two lone electron pairs (as can be seen with the valence bond theory as well from the electronic configuration of oxygen), which means there are four such 'elements' on O. The model molecule is, then, AX4: sp3 hybridisation is utilized, and the electron arrangement of H2O is tetrahedral. This agrees with the experimentally-determined shape for water, a non-linear, bent structure, with a bond angle of 104.5 degrees (the two lone-pairs are not visible). In general, for an atom with s and p orbitals forming hybrids hi and hj with included angle θ, the following holds: 1 + λiλj cos(θ) = 0. The p-to-s ratio for hybrid i is λi2, and for hybrid j it is λj2. In the special case of equivalent hybrids on the same atom, again with included angle θ, the equation reduces to just 1 + λ2 cos(θ) = 0. For example, BH3 has a trigonal planar geometry, three 120o bond angles, three equivalent hybrids about the boron atom, and thus 1 + λ2 cos(θ) = 0 becomes 1 + λ2 cos(120o) = 0, giving λ2 = 2 for the p-to-s ratio. In other words, sp2 hybrids, just as expected from the list above. Controversy regarding d-orbital participation Hybridisation theory has failed in a few aspects, notably in explaining the energy considerations for the involvement of d-orbitals in chemical bonding (See above for sp3d and sp3d2 hybridisation). This can be well-explained by means of an example. Consider, for instance, how the theory in question accounts for the bonding in phosphorus pentachloride (PCl5). d-orbitals are large, comparatively distant from the nucleus and high in energy. Radial distances of orbitals from the nucleus seem to reveal that d-orbitals are far too high in energy to 'mix' with s- and p-orbitals. 3s - 0.47 , 3p - 0.55, 3d - 2.4 (in angstroms). Thus, at first sight, it seems improbable for sp3d hybridisation to occur. However, a deeper look into the factors that affect orbital size(and energy) reveals more. Formal charge on the central atom is one such factor, and it's obvious that the P atom in PCl5 has quite a large partial positive charge on itself. Thus the 3d orbital contracts in size to enough of an extent so that hybridisation may occur with s and p orbitals. Further, note the cases in which d-orbital participation was proposed in hybridisation: SF6(sulfur hexafluoride), IF7, XeF6; in all these molecules, the central atom is surrounded by the highly electronegative fluorine atom, thus making hybridisation probable among s, p and d orbitals. A further study reveals that orbital size also depends on the number of electrons occupying it. And even further, coupling of d orbital electrons also results in contraction, albeit to a smaller extent. The molecular orbital theory, however, offers a clearer insight into the bonding in these molecules. Hybridisation theory vs. MO theory Hybridisation theory is an integral part of organic chemistry and in general discussed together with molecular orbital theory in advanced organic chemistry textbooks although for different reasons. One textbook notes that for drawing reaction mechanisms sometimes a classical bonding picture is needed with 2 atoms sharing two electrons [4]. It also comments that predicting bond angles in methane with MO theory is not straightforward. Another textbook treats hybridisation theory when explaining bonding in alkenes [5] and a third [6] uses MO theory to explain bonding in hydrogen but hybridisation theory for methane. Although the language and pictures arising from Hybridisation Theory, more widely known as Valence Bond Theory, remain widespread in synthetic organic chemistry, this qualitative analysis of bonding has been largely superseded by molecular orbital theory in other branches of chemistry. For example, inorganic chemistry texts have all but abandoned instruction of hybridisation, except as a historical footnote.[7][8] One specific problem with hybridisation is that it incorrectly predicts the photoelectron spectra of many molecules, including such fundamental species such as methane and water. From a pedagogical perspective, hybridisation approach tends to over-emphasize localisation of bonding electrons and does not effectively embrace molecular symmetry as does MO Theory. 1. ^ "It is important to recognize that the VSEPR model provides an approach to bonding and geometry based on the Pauli principle that is completely independent of the valence bond (VB) theory or of any orbital description of bonding." Gillespie, R. J. J. Chem. Educ. 2004, 81, 298-304. 2. ^ L. Pauling, J. Am. Chem. Soc. 53 (1931), 1367 3. ^ photo electron spectrum of methane 1 photo electron spectrum of methane 2 4. ^ Organic Chemistry. Jonathan Clayden, Nick Greeves, Stuart Warren, and Peter Wothers 2001 ISBN 0-19-850346-6 6. ^ Organic Chemistry 3rd Ed. 2001 Paula Yurkanis Bruice ISBN 0-13-017858-6 8. ^ Shriver, D. F.; Atkins, P. W.; Overton, T. L.; Rourke, J. P.; Weller, M. T.; Armstrong, F. A. “Inorganic Chemistry” W. H. Freeman, New York, 2006. ISBN 0-7167-4878-9. See also This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Orbital_hybridisation". A list of authors is available in Wikipedia.
cb85788956a50430	The Schrödinger equation The fathers of matrix quantum mechanics believed that the quantum particles are unvisualizable and pop into existence only when measured. Challenging this view, in 1926 Erwin Schrödinger developed his wave equation that describes the quantum particles as packets of quantum probability amplitudes evolving in space and time. Thus, Schrödinger visualized the unvisualizable and lifted the veil that has been obscuring the wonders of the quantum world. The Schrödinger equation governs the time evolution of closed quantum physical systems \begin{equation}\imath\hbar\frac{\partial}{\partial t}\,\|\Psi(x,y,z,t)\rangle=\hat{H}\,\|\Psi(x,y,z,t)\rangle\label{eq:1}\end{equation} where $\imath$ is the imaginary unit, $\hbar$ is the reduced Planck constant, $\frac{\partial}{\partial t}$ indicates a partial derivative with respect to time, $\Psi$ is the wave function of the quantum system, $x$, $y$, $z$, $t$ are the three position coordinates and time respectively, and $\hat{H}$ is the Hamiltonian operator corresponding to the total energy of the system. The solution of the Schrödinger equation is the quantum wave function $\Psi$ of the system. At each point $(x,y,z)$ in space at a time $t$, the value of the quantum wave function is a complex number $\Psi(x,y,z,t)$, referred to as a quantum probability amplitude. Since the quantum wave function provides a complete description of the quantum state of a physical system, it follows that the fabric of the quantum world is made of quantum probability amplitudes $\Psi(x,y,z,t)$. The squared modulus $\left\|\Psi(x,y,z,t)\right\|^{2}$ of each quantum probability amplitude gives a corresponding quantum probability for a physical event to occur at the given point in space and time. The quantum probabilities do not arise due to our ignorance of what the state of the quantum system is, but rather represent inherent propensities of the quantum systems to produce certain outcomes under experimental measurement. Despite that the Schrödinger equation can be easily written in its general mathematical form, solving it for an arbitrary Hamiltonian $\hat{H}$ is extremely difficult and usually done only approximately with a finite numerical precision. Understanding the general properties of the Schrödinger equation and its solutions is essential for modern developments in theoretical physics, quantum information theory, quantum chemistry, biology, neuroscience, cognitive science and artificial intelligence.
3895c523c2b65e57	CRC 235 Emergence of Life Breadcrumb Navigation Principal Investigators alimProf. Dr. Karen Alim Department of Physics (TUM) Phone: +49 89 289 12192 Flow networks are a fundamental building block of life, whether as blood circulation, bronchioles in the lungs, nephron in the kidneys, intestines, lymphatic system or as the hydrothermal vents providing the conditions for the emergence of life. We want to decipher the physical principles that govern flow network architecture through the interaction between network architecture and the flows flowing through it. To this end we investigate flows, transport and reactions within and remodeling flow networks. boekhovenProf. Dr. Job Boekhoven Department of Chemistry (TUM) Phone: +49 89 289 10592 We look for life-like behavior in fully synthetic molecular assemblies. Inspired by biology, we couple the self-assembly of our building blocks to networks of irreversible chemical reactions. In the resulting far-from-equilibrium assemblies we search for robustness, adaptivity and self-replication, in an attempt to find life-like behavior using prebiotic chemistry. braun_2Prof. Dr. Dieter Braun Systems Biophysics (LMU) Phone: +49 89 2180 1484 Can we recreate molecular living systems in the lab? Which nonequilibrium settings can generate and trigger dead molecules to perform autonomous, Darwinian evolution? Temperature differences across open pores of rock are thermally cycling genetic molecules for replication while thermophoresis enhances polymerization and selects long over short genes. Despite studying the physics of thermophoresis, the movement of molecules in a thermal gradient, its application for biomolecule affinity detection has found widespread application through the Startup NanoTemper. caselli_2Prof. Dr. Paola Caselli Center for Astrochemical Studies Director - Max Planck Institute for Extraterrestrial Physics Phone: +49 89 30000 3400 Life has emerged on Earth and maybe in other planets/moons in our Solar System and beyond. We find organic molecules in comets and pre-biotic materials such as carbonaceous chondrites or interplanetary dust particles. When stars reach the end of their lives, heavy elements will be incorporated into future generations of stars and planets. Understanding this cycle is fundamental to understand how stars and planets form, how organic material evolves and how life could originate. dietzProf. Dr. Hendrik Dietz Department of Physics (TUM) Phone: +49 89 289 11615 dingwellProf. Dr. Don Dingwell Department of Earth and Environmental Sciences Section of Mineralogy, Petrology and Geochemistry (LMU) Phone: +49 89 2180 4136 Experimental Volcanology is seeking an adequate understanding of the nature and extent of physico-chemical processes involved in explosive volcanism. In recent years much effort has focused on high silica (rhyolitic) melts under conditions relevant to explosive volcanism. Operating four experimental volcanoes, we study the origin, transport and extraction of magmas under different conditions of pressure, temperature and oxygen fugacity. Our experimental methods include the study of mineral synthesis and physicochemical characterisation of geomaterials under high pressure and high temperature conditions. Immediate applications include volcano monitoring, eruption forecasting, reconstructing the history of volcanic events through geologic time and understanding the role of explosive volcanism and volcanic ash in the earth system and its biosphere. eisenreichapl. Prof. Dr. Wolfgang Eisenreich Biochemistry (TUM) Phone: +49 89 289 13336 We focus on the detailed quantitative analysis of metabolism networks and use incorporation experiments with stable isotope labeled precursors followed by GC-MS or NMR isotope analysis of key metabolites (carbohydrates, amino acids). Instead of analyzing metabolism using a linear and unidirectional view, the complexities of biological networks and their crosstalk with host cells, we assess the isotope distribution of multiple metabolites. The identification of metabolic key reactions can serve as novel targets for antibiotics. ercolanoProf. Dr. Barbara Ercolano University Observatory (LMU) Phone: +49 89 2180 6974 We want to understand how planets are formed and how they evolve in their natal environment, which is the circumstellar disc surrounding newly-born stars. The star and planet formation process are intertwined and we aim at obtaining a holistic picture of this complex phenomenon. The atmospheric chemistry of a planet is dramatically influenced by the formation process within the disc and by the irradiation from the central star. We aim at developing and applying comprehensive numerical models to state-of-art-ground-based and space-borne observations to understand the star-disc-planet interaction that leads to the formation of habitable worlds. frey_2Prof. Dr. Erwin Frey Statistical and Biological Physics (LMU) Phone: +49 89 2180 4538 We focus on collective effects in the dynamics of complex systems. We are interested in how the interplay between stochastic fluctuations, geometries and interactions shape system-level properties and their functional characteristics. The aim is to develop a mechanistic understanding of system features. We use and develop analytical and numerical methods from non-equilibrium statistical physics and nonlinear dynamics. gerland_2Prof. Dr. Ulrich Gerland Head of Theory of Complex Biosystems (TUM) Phone: +49 89 289 12380 We want to understand how the laws of physics constrain the implementation of biological function. In physics, interactions between particles follow laws. In biology, interactions between biomolecules serve a function. For instance, spatial arrangement and coordination of enzymes determine the efficiency of a multi-step reaction. Functional trade-offs emerge, which must be characterized to understand the optimization of such systems. heckl_2Prof. Dr. Wolfgang Heckl Oskar-von-Miller Chair of Scientific Communication(TUM) Director General of the Deutsches Museum Phone: +49 89 2179 313/314 The Deutsches Museum with its branch museums is an outstanding place for communicating scientific and technical knowledge and for a constructive dialogue between science and society. Together with the members of the CRC 235 we try to make the Emergence of Life scientific work more accessible to general public. heuerDr. Amelie Heuer-Jungemann Emmy-Noether-Research Group "DNA Hybridnanomaterials" Max Planck Institute of Biochemistry Phone: +49 89 8578 2434 huberDr. Claudia Huber Lehrstuhl für Biochemie (TUM) Phone: +49 89 289 13044 Analytical techniques (GC/MS, LC/MS, NMR) are used for various research topics. Focus is set on the origin of biomolecules on the early earth at volcanic hydrothermal settings. Reactions of various gases (eg. CO, COS, HCN, H2C2, H2S, H2, …) in aqueous solutions, on the surface of transition metal catalysts (eg. FeS, NiS, ….) under primordeal conditions (anaerobic, high temperature and pressure) are investigated. The resulting biomolecules (eg. hydroxy-, amino- and fatty acids) are considered to be basic constituents of a “metabolism first” origin of life on earth. Stable isotopes are used for product verification and mechanistical insights. Prof. Dr. Andres Jäschkejäschke Bioorganic Chemistry (Uni Heidelberg) Phone: +49 6221 - 54 4851 Our laboratory explores unknown roles of RNA modifications, in particular RNA-linked coenzymes, in biology. Furthermore, we develop methods for imaging and microscopy of RNA in living cells, with a focus on super-resolution techniques. In another research area we develop, characterize and apply photoswitchable biomolecules. We also have a long-standing interest in the origin of life. Our work combines organic synthesis with molecular and cellular biology, biochemistry, bioinformatics and modern bioanalytical methods. liedlProf. Dr. Tim Liedl Soft Condensed Matter (LMU) Phone: +49 89 2180 3725 We focus on molecular self-assembly processes in general and particularly in the engineering of functional DNA devices using the DNA origami. The integration of self-assembled DNA into hybrid materials and its characterization will help us to understand the mechanical and chemical interactions across multiple levels of organization between the different molecular components in living cells and tissues. mast_2Dr. Christof Mast Systems Biophysics (LMU) Phone: +49 89 2180 1484 We are interested in the role of non-equilibrium physical systems in processes relevant to the origin of life. Specifically, we are investigating how heat fluxes can separate and concentrate prebiotically relevant chemicals as well as the influence of UV radiation on sequence evolution in prebiotic polymer pools and combine these systems with realistic geological boundary conditions. mutschlerProf. Dr. Hannes Mutschler Biomimetic Systems (TU Dortmund) Phone: +49 231 755 8696 We are interested in the bottom-up reconstitution of minimal genetic systems capable of recursive replication and Darwinian Evolution. Moreover, we are exploring different prebiotic environments for their ability to assist the emergence primitive RNA-enzymes during the Origin of Life. We employ a variety of methods from protein and nucleic acid biochemistry, genome assembly, directed evolution, and deep sequencing. ochsenfeld_2Prof. Dr. Christian Ochsenfeld Department of Chemistry (LMU) Chair of Theoretical Chemistry Email: christian.ochsenfeld(at) Phone: +49 89 2180 77920 Prof. Ochsenfeld's expertise is molecular dynamics, especially large scale quantum mechanical calculations solving the Schrödinger equations with efficient scaling on graphics card clusters. He has worked already on various biological systems, including DNA repair and is interested to learn about opportunities to apply his methods. His approach fundamentally differs both from force field and DFT calculations and he will be happy to describe and probe its advantages in future collaborations. orsiProf. Dr. William Orsi Department of Earth and Environmental Sciences (LMU) Palaeontology & Geobiology Phone: +49 89 2180 6598 We are focused on the biological and biochemical mechanisms underlying microbial loops: the biogeochemical cycling of carbon and its processing through microbial food webs. At deep sea low temperature hydrothermal vents chemolithoautotrophic microbes are able to fix CO2 in the absence of sunlight, fueled by geological energy sources. These settings exhibit water-rock interactions and abiotic chemistry called serpentinization, whereby the abiotic synthesis of organic molecules occurs in a naturally exiting proton gradient. This is hypothesized to be analogous to the metabolism of the first cell and the chemolithoautotrophic microbes living at these settings today obtain energy through a similar carbon fixation and bioenergetic pathway. We are interested in the genomic evolution of this ancient bioenergetic pathway throughout the geological record, its role in the origins of life, and the mechanisms by which it supports life and ecosystems. richertProf. Dr. Clemens Richert Institute of Organic Chemistry (Uni Stuttgart) Phone: +49 711 685 64311 The Richert Research Group studies potentially prebiotic processes from the point of view of organic chemistry. This includes reactions involving nucleotides, amino acids and cofactors in aqueous solution containing a condensing agent and buffer salts. Product analysis is performed using spectroscopic, spectrometric and chromatographic techniques to shed light on the reactivity of fundamental biomolecules that are found in all kingdoms of life. bettyProf. Dr. Bettina Scheu Department for Earth and Environmental Sciences Section for Mineralogy, Petrology und Geochemistry Phone: +49 89 3187 3246 Volcanoes are the most spectacular expression of geologic dynamism on Earth as well as on other planets. Geological evidences show that volcanoes on Earth have been active before the appearance of life above and below the oceans, thus contributing to the conditions for which life has originated and subsequently proliferated. We experimentally investigate the conditions of generation and dispersion of solid aerosols, their electrification and their reactivity with volcanic and other gases as well as the high temperature and pressure conditions of hydrothermal vents that might have contributed to the origin of life on our planet. schmitt-kopplinProf. Dr. Philippe Schmitt-Kopplin HelmholtzZentrum München Research Unit Analytical BioGeoChemistry Phone: +49 89 2180 4259 We describe the small molecule chemistry of Life (Metabolomics), Life-decay (organic geochemistry) and “pre-Life” (prebiotic and meteoritic chemistry) using an analytical combination of high resolving power. We profile the compositional space of these organic environments with ultrahigh resolution mass spectrometry (ICR-FT-MS) and the functional space with high field nuclear magnetic resonance spectroscopy (NMR). Our output is a process understanding on a molecular level and/or the description of molecular markers of biological, geochemical or astrochemical relevance. schwille-2Prof. Dr. Petra Schwille Cellular and Molecular Biophysics Director - Max Planck Institute of Biochemistry Phone: +49 89 8578 2900 Our ambition is to quantitatively understand living systems on the scale of individually active and interactive molecules such as proteins, lipids and nucleic acids. To unravel the underlying basic principles of biological phenomena, we follow a bottom-up approach of minimal systems in the framework of synthetic biology. The very far goal of such approaches could be the in vitro reconstitution of a self-replicating biomimetic system. simmelProf. Dr. Friedrich Simmel Synthetic Biological Systems (TUM) Phone: +49 89 289 - 11611 The Simmel Group explores the pysicochemical properties of natural and artificial biomolecular systems and their applications in bionanotechnology and synthetic biology. The goal is the realization of self-organizing molecular and cellular systems that are able to respond to their environment, compute, move, take action. On the long term, we envision autonomous systems that are reconfigurable, that can evolve and develop. trapp_2Prof. Dr. Oliver Trapp Department of Chemistry (LMU) Phone: +49 89 2180 77461 Are there universal principles which have led to the origin of life on Earth and how can we identify them? We are addressing fundamental questions about the formation of chemical networks that can form living systems. Major focus is the understanding of mechanisms that led to the homochirality of organic compounds, e.g. amino acids and sugars. weidendorferProf. Dr. Daniel Weidendorfer Department for Earth and Environmental Sciences Section for Mineralogy, Petrology und Geochemistry Phone: +49 89 2180 4274 zeymerProf. Dr. Cathleen Zeymer Department of Chemistry (TUM) Phone: +49 89 289 13343 The Zeymer group focuses on the design and laboratory evolution of artificial metalloenzymes and photoenzymes. We engineer novel biocatalysts and want to understand their structure and molecular mechanism. In CRC 235, we aim to explore the potential and plausibility of simple metallopeptides as prebiotic catalysts. We will study these minimalist versions of modern enzymes in the context of reaction cycles that drive the formation of molecular assemblies.
dabc43a0260e3c3b	Helium atom Systematic IUPAC name 3D model (JSmol) EC Number • 231-168-5 MeSH Helium RTECS number • MH6520000 UN number 1046 • InChI=1S/He checkY • [He] Molar mass 4.002602 g·mol−1 Appearance Colourless gas Boiling point −269 °C (−452.20 °F; 4.15 K) 126.151-126.155 J K−1 mol−1 V03AN03 (WHO) ☒N verify (what is checkY☒N ?) A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Schematic termscheme for Para- and Orthohelium with one electron in ground state 1s and one excited electron. The quantum mechanical description of the helium atom is of special interest, because it is the simplest multi-electron system and can be used to understand the concept of quantum entanglement. The Hamiltonian of helium, considered as a three-body system of two electrons and a nucleus and after separating out the centre-of-mass motion, can be written as where is the reduced mass of an electron with respect to the nucleus, and are the electron-nucleus distance vectors and . The nuclear charge, is 2 for helium. In the approximation of an infinitely heavy nucleus, we have and the mass polarization term disappears. In atomic units the Hamiltonian simplifies to It is important to note, that it operates not in normal space, but in a 6-dimensional configuration space . In this approximation (Pauli approximation) the wave function is a second order spinor with 4 components , where the indices describe the spin projection of both electrons (z-direction up or down) in some coordinate system.[2][better source needed] It has to obey the usual normalization condition . This general spinor can be written as 2×2 matrix and consequently also as linear combination of any given basis of four orthogonal (in the vector-space of 2×2 matrices) constant matrices with scalar function coefficients as . A convenient basis consists of one anti-symmetric matrix (with total spin , corresponding to a singlet state) and three symmetric matrices (with total spin , corresponding to a triplet state) It is easy to show, that the singlet state is invariant under all rotations (a scalar entity), while the triplet can be mapped to an ordinary space vector , with the three components Since all spin interaction terms between the four components of in the above (scalar) Hamiltonian are neglected (e.g. an external magnetic field, or relativistic effects, like angular momentum coupling), the four Schrödinger equations can be solved independently.[3][better source needed] The spin here only comes into play through the Pauli exclusion principle, which for fermions (like electrons) requires antisymmetry under simultaneous exchange of spin and coordinates Parahelium is then the singlet state with a symmetric function and orthohelium is the triplet state with an antisymmetric function . If the electron-electron interaction term is ignored, both spatial functions can be written as linear combination of two arbitrary (orthogonal and normalized) one-electron eigenfunctions : or for the special cases of (both electrons have identical quantum numbers, parahelium only): . The total energy (as eigenvalue of ) is then for all cases (independent of the symmetry). This explains the absence of the state (with ) for orthohelium, where consequently (with ) is the metastable ground state. (A state with the quantum numbers: principal quantum number , total spin , angular quantum number and total angular momentum is denoted by .) If the electron-electron interaction term is included, the Schrödinger equation is non separable. However, even if it is neglected, all states described above (even with two identical quantum numbers, like with ) cannot be written as a product of one-electron wave functions: — the wave function is entangled. One cannot say, particle 1 is in state 1 and the other in state 2, and measurements cannot be made on one particle without affecting the other. Nevertheless, quite good theoretical descriptions of helium can be obtained within the Hartree–Fock and Thomas–Fermi approximations (see below). The Hartree–Fock method is used for a variety of atomic systems. However it is just an approximation, and there are more accurate and efficient methods used today to solve atomic systems. The "many-body problem" for helium and other few electron systems can be solved quite accurately. For example, the ground state of helium is known to fifteen digits. In Hartree–Fock theory, the electrons are assumed to move in a potential created by the nucleus and the other electrons. Perturbation method The Hamiltonian for helium with two electrons can be written as a sum of the Hamiltonians for each electron: where the zero-order unperturbed Hamiltonian is while the perturbation term: is the electron-electron interaction. H0 is just the sum of the two hydrogenic Hamiltonians: Eni, the energy eigenvalues and , the corresponding eigenfunctions of the hydrogenic Hamiltonian will denote the normalized energy eigenvalues and the normalized eigenfunctions. So: Neglecting the electron-electron repulsion term, the Schrödinger equation for the spatial part of the two-electron wave function will reduce to the 'zero-order' equation This equation is separable and the eigenfunctions can be written in the form of single products of hydrogenic wave functions: The corresponding energies are (in atomic units, hereafter a.u.): Note that the wave function An exchange of electron labels corresponds to the same energy . This particular case of degeneracy with respect to exchange of electron labels is called exchange degeneracy. The exact spatial wave functions of two-electron atoms must either be symmetric or antisymmetric with respect to the interchange of the coordinates and of the two electrons. The proper wave function then must be composed of the symmetric (+) and antisymmetric(−) linear combinations: This comes from Slater determinants. The factor normalizes . In order to get this wave function into a single product of one-particle wave functions, we use the fact that this is in the ground state. So . So the will vanish, in agreement with the original formulation of the Pauli exclusion principle, in which two electrons cannot be in the same state. Therefore, the wave function for helium can be written as Where and use the wave functions for the hydrogen Hamiltonian.[a] For helium, Z = 2 from where E(0) = −4 a.u. which is approximately −108.8 eV, which corresponds to an ionization potential V(0) = 2 a.u. (≅54.4 eV). The experimental values are E0 = −2.90 a.u. (≅ −79.0 eV) and Vp = 0.90 a.u. (≅ 24.6 eV). The energy that we obtained is too low because the repulsion term between the electrons was ignored, whose effect is to raise the energy levels. As Z gets bigger, our approach should yield better results, since the electron-electron repulsion term will get smaller. So far a very crude independent-particle approximation has been used, in which the electron-electron repulsion term is completely omitted. Splitting the Hamiltonian showed below will improve the results: V(r) is a central potential which is chosen so that the effect of the perturbation is small. The net effect of each electron on the motion of the other one is to screen somewhat the charge of the nucleus, so a simple guess for V(r) is where S is a screening constant and the quantity Ze is the effective charge. The potential is a Coulomb interaction, so the corresponding individual electron energies are given (in a.u.) by and the corresponding wave function is given by If Ze was 1.70, that would make the expression above for the ground state energy agree with the experimental value E0 = −2.903 a.u. of the ground state energy of helium. Since Z = 2 in this case, the screening constant is S = 0.30. For the ground state of helium, for the average shielding approximation, the screening effect of each electron on the other one is equivalent to about of the electric charge.[5] The variational method To obtain a more accurate energy the variational principle can be applied to the electron-electron potential Vee using the wave function After integrating this, the result is: This is closer to the experimental value, but if a better trial wave function is used, an even more accurate answer could be obtained. An ideal wave function would be one that doesn't ignore the influence of the other electron. In other words, each electron represents a cloud of negative charge which somewhat shields the nucleus so that the other electron actually sees an effective nuclear charge Z that is less than 2. A wave function of this type is given by: Treating Z as a variational parameter to minimize H. The Hamiltonian using the wave function above is given by: After calculating the expectation value of and Vee the expectation value of the Hamiltonian becomes: The minimum value of Z needs to be calculated, so taking a derivative with respect to Z and setting the equation to 0 will give the minimum value of Z: This shows that the other electron somewhat shields the nucleus reducing the effective charge from 2 to 1.69. So we obtain the most accurate result yet: Where again, E1 represents the ionization energy of hydrogen. By using more complicated/accurate wave functions, the ground state energy of helium has been calculated closer and closer to the experimental value −78.95 eV.[6] The variational approach has been refined to very high accuracy for a comprehensive regime of quantum states by G.W.F. Drake and co-workers[7][8][9] as well as J.D. Morgan III, Jonathan Baker and Robert Hill[10][11][12] using Hylleraas or Frankowski-Pekeris basis functions. One needs to include relativistic and quantum electrodynamic corrections to get full agreement with experiment to spectroscopic accuracy.[13][14] Experimental value of ionization energy Helium's first ionization energy is −24.587387936(25) eV.[15] This value was derived by experiment.[16] The theoretic value of Helium atom's second ionization energy is −54.41776311(2) eV.[15] The total ground state energy of the helium atom is −79.005151042(40) eV,[15] or −2.90338583(13) Atomic units a.u., which equals −5.80677166 (26) Ry. See also 1. ^ For n = 1, ℓ = 0 and m = 0, the wavefunction in a spherically symmetric potential for a hydrogen electron is .[4] In atomic units, the Bohr radius equals 1, and the wavefunction becomes . 1. ^ "Helium - PubChem Public Chemical Database". The PubChem Project. USA: National Center for Biotechnology Information. 2. ^ Rennert, P.; Schmiedel, H.; Weißmantel, C. (1988). Kleine Enzyklopädie Physik (in German). VEB Bibliographisches Institut Leipzig. pp. 192–194. ISBN 3-323-00011-0. 3. ^ Landau, L. D.; Lifschitz, E. M. (1971). Lehrbuch der Theoretischen Physik (in German). Vol. Bd. III (Quantenmechanik). Berlin: Akademie-Verlag. Kap. IX, pp. 218. OCLC 25750516. 4. ^ "Hydrogen Wavefunctions". Hyperphysics. Archived from the original on 1 February 2014. 5. ^ Bransden, B. H.; Joachain, C. J. Physics of Atoms and Molecules (2nd ed.). Pearson Education. 6. ^ Griffiths, David I. (2005). Introduction to Quantum Mechanics (Second ed.). Pearson Education. 7. ^ Drake, G.W.F.; Van, Zong-Chao (1994). "Variational eigenvalues for the S states of helium". Chemical Physics Letters. Elsevier BV. 229 (4–5): 486–490. Bibcode:1994CPL...229..486D. doi:10.1016/0009-2614(94)01085-4. ISSN 0009-2614. 8. ^ Yan, Zong-Chao; Drake, G. W. F. (1995-06-12). "High Precision Calculation of Fine Structure Splittings in Helium and He-Like Ions". Physical Review Letters. American Physical Society (APS). 74 (24): 4791–4794. Bibcode:1995PhRvL..74.4791Y. doi:10.1103/physrevlett.74.4791. ISSN 0031-9007. PMID 10058600. 9. ^ Drake, G. W. F. (1999). "High Precision Theory of Atomic Helium". Physica Scripta. IOP Publishing. T83 (1): 83–92. Bibcode:1999PhST...83...83D. doi:10.1238/physica.topical.083a00083. ISSN 0031-8949. 10. ^ J.D. Baker, R.N. Hill, and J.D. Morgan III (1989), "High Precision Calculation of Helium Atom Energy Levels", in AIP ConferenceProceedings 189, Relativistic, Quantum Electrodynamic, and Weak Interaction Effects in Atoms (AIP, New York),123 11. ^ Baker, Jonathan D.; Freund, David E.; Hill, Robert Nyden; Morgan, John D. (1990-02-01). "Radius of convergence and analytic behavior of the 1/Z expansion". Physical Review A. American Physical Society (APS). 41 (3): 1247–1273. Bibcode:1990PhRvA..41.1247B. doi:10.1103/physreva.41.1247. ISSN 1050-2947. PMID 9903218. 12. ^ Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions" (PDF). Phys. Rev. A. 75 (6): 060101. Bibcode:2007PhRvA..75f0101S. doi:10.1103/PhysRevA.75.060101. hdl:11383/1679348. 13. ^ Drake, G. W. F.; Yan, Zong-Chao (1992-09-01). "Energies and relativistic corrections for the Rydberg states of helium: Variational results and asymptotic analysis". Physical Review A. American Physical Society (APS). 46 (5): 2378–2409. Bibcode:1992PhRvA..46.2378D. doi:10.1103/physreva.46.2378. ISSN 1050-2947. PMID 9908396. S2CID 36134307. 14. ^ G.W.F. Drake (2006). "Springer Handbook of Atomic, molecular, and Optical Physics", Edited by G.W.F. Drake (Springer, New York), 199-219. [1] 15. ^ a b c Kramida, A.; Ralchenko, Yu.; Reader, J., and NIST ASD Team. "NIST Atomic Spectra Database Ionization Energies Data". Gaithersburg, MD: NIST.((cite web)): CS1 maint: multiple names: authors list (link) 16. ^ D. Z. Kandula, C. Gohle, T. J. Pinkert, W. Ubachs, and K. S. E. Eikema (2010). "Extreme Ultraviolet Frequency Comb Metrology". Phys. Rev. Lett. 105 (6): 063001. arXiv:1004.5110. Bibcode:2010PhRvL.105f3001K. doi:10.1103/PhysRevLett.105.063001. PMID 20867977. S2CID 2499460.((cite journal)): CS1 maint: uses authors parameter (link)
977e7d47730e5ba0	Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be satisfied so that this symmetry can be gauged? To give an example, the free Schrödinger field theory is given by the Lagrangian $$ \mathscr L=\psi^\dagger\biggl(i\partial_t+\frac{\nabla^2}{2m}\biggr)\psi. $$ Apart from the usual $U(1)$ phase transformation, the action is also invariant under independent shifts, $\psi\to\psi+\theta_1+i\theta_2$, of the real and imaginary parts of $\psi$. It seems that such shift symmetry cannot be gauged, although I cannot prove this claim (correct me if I am wrong). This seems to be related to the fact that the Lie algebra of the symmetry necessarily contains a central charge. So my questions are: (i) how is the (im)possibility to gauge a global symmetry related to central charges in its Lie algebra?; (ii) can one formulate the criterion for "gaugability" directly in terms of the Lagrangian, without referring to the canonical structure such as Poisson brackets of the generators? (I have in mind Lagrangians with higher field derivatives.) N.B.: By gauging the symmetry I mean adding a background, not dynamical, gauge field which makes the action gauge invariant. • 4 $\begingroup$ Would it be accurate to say that one could rephrase your question as follows: "Under what conditions can one take a Lagrangian leading to an action which possesses a global Lie-algebraic symmetry and find a new Lagrangian depending on a gauge field that leads to an action with local Lie-algebraic symmetry such that the new Lagrangian agrees with the old Lagrangian when the gauge field is set to zero and the symmetry parameter is constant on spacetime?" $\endgroup$ Apr 16, 2013 at 9:02 • 2 $\begingroup$ Yes, this is exactly what I have in mind! I just wanted to avoid being too formal. But sometimes it is better if one wants to be precise :) $\endgroup$ Apr 16, 2013 at 9:07 • $\begingroup$ Ok well in that case, I'm not sure Lubos' answer is very satisfying. I have to think a bit more about this, but I think the answer, if phrased in the manner of my first comment above, is that given any action of a Lie algebra on the fields that is either (i) an internal symmetry or (ii) a spacetime symmetry or both, making the symmetry parameter local, introducing a gauge field, introducing the corresponding covariant derivative, and replacing partial derivatives with covariant derivatives will do the trick. $\endgroup$ Apr 16, 2013 at 9:19 • $\begingroup$ Replacing ordinary derivatives with covariant ones is indeed what one usually does. But that doesn't work in this trivial example. The covariant derivative for the shift symmetry will be something like $D_\mu\psi=\partial_\mu\psi-A_\mu$, and the term with the time derivative will not be gauge invariant. (There are theories whose action is gauge invariant yet the gauge field does not appear solely in covariant derivatives. An example is the low-energy effective theory for a ferromagnet. Generally this happens when the Lagrangian changes under the gauge transformation by a surface term.) $\endgroup$ Apr 16, 2013 at 9:38 • $\begingroup$ I think that one can gauge this symmetry, which is the responsible of the renormalizability of Proca theory physics.stackexchange.com/q/16931 Have you read the opposite? Ref or link? $\endgroup$ Apr 17, 2013 at 6:19 4 Answers 4 The guiding principle is: "Anomalous symmetries cannot be gauged". The phenomenon of anomalies is not confined to quantum field theories. Anomalies exist also in classical field theories (I tried to emphasize this point in my answer on this question). (As already mentioned in the question), in the classical level, a symmetry is anomalous when the Lie algebra of its realization in terms of the fields and their conjugate momenta (i.e., in term of the Poisson algebra of the Lagrangian field theory) develops an extension with respect to its action on the fields. This is exactly the case of the complex field shift on the Schrödinger Lagrangian. In Galilean (classical) field theories, the existence of anomalies is accompanied by the generation of a total derivative increment to the Lagrangian, which is again manifested in the case of the field shift symmetry of the Schrödinger Lagrangian, but this is not a general requirement. (please see again my answer above referring to the generation of mass as a central extension in Galilean mechanics). In a more modern terminology, the impossibility of gauging is termed as an obstruction to equivariant extensions of the given Lagrangians. A nontrivial family of classical Lagrangians, exhibiting nontrivial obstructions, are Lagrangians containing Wess-Zumino-Witten terms. Given these terms only anomaly free subgroups of the symmetry groups can be gauged (classically). These subgroups consist exactly of the anomaly free ones. The gauging and the obstruction to it can be obtained using the theory of equivariant cohomology, please see the following article by Compean and Paniagua and its reference list. • $\begingroup$ I see, so my guess that the obstruction to gauging is related to the central charge (in the commutator of the generators of real and imaginary shifts of $\psi$) was in the right direction. Thanks for a clear and general statement! I will study your previous answer carefully. Apparently, there is a lot they didn't tell me at school :) $\endgroup$ Apr 17, 2013 at 13:10 • $\begingroup$ I read your previous answer and got the point with the classical anomaly, thanks! Before I get into the mathematical details, is there an intuitive way to understand why an obstruction like a central charge in the Poisson brackets of the symmetry generators prevents gauging the symmetry on the classical level? Also, is there an elementary way to decide whether the symmetry can be gauged which does not use the canonical structure? (I think of low-energy effective Lagrangians which can contain an arbitrary number of derivatives.) $\endgroup$ Apr 17, 2013 at 15:03 • $\begingroup$ @Tomáš Brauner: Yes, the obstruction to gauging can be understood intuitively by means of the Dirac's constraint theory. If the theory can be gauged, the currents couple to the gauge field. Since the time component of the gauge field is non-dynamical, the charge densities of the symmetry currents will become constraints, i.e., must become zero on the gauge surface which can be thought as a reformulation of the theory with gauge invariant fields. But, then how can the bracket of two vanishing quantities give a nonzero constant. $\endgroup$ Apr 17, 2013 at 15:56 • $\begingroup$ I think I have understood your point, thanks once again! $\endgroup$ Apr 18, 2013 at 7:46 I) The topic of gauging global symmetries is a quite large subject, which is difficult to fit in a Phys.SE answer. Let us for simplicity only consider a single (and thus necessarily Abelian) continuous infinitesimal transformation$^1$ $$ \tag{1} \delta \phi^{\alpha}(x)~=~\varepsilon(x) Y^{\alpha}(\phi(x),x), $$ where $\varepsilon$ is an infinitesimal real parameter, and $Y^{\alpha}(\phi(x),x)$ is a generator, such that the transformation (1) is a quasi-symmetry$^2$ of the Lagrangian density $$ \tag{2} \delta {\cal L} ~=~ \varepsilon d_{\mu} f^{\mu} + j^{\mu} d_{\mu}\varepsilon $$ whenever $\varepsilon$ an $x$-independent global parameter, such that the last term on the rhs. of eq. (2) vanishes. Here $j^{\mu}$ and $$ \tag{3} J^{\mu}~:=~j^{\mu}-f^{\mu}$$ are the bare and the full Noether currents, respectively. The corresponding on-shell conservation law reads$^3$ $$ \tag{4} d_{\mu}J^{\mu}~\approx~ 0, $$ cf. Noether's first Theorem. Here $f^{\mu}$ are so-called improvement terms, which are not uniquely defined from eq. (2). Under mild assumptions, it is possible to partially fix this ambiguity by assuming the following technical condition $$ \tag{5}\sum_{\alpha}\frac{\partial f^{\mu}}{\partial(\partial_{\nu}\phi^{\alpha})}Y^{\alpha}~=~(\mu \leftrightarrow \nu), $$ which will be important for the Theorem 1 below. We may assume without loss of generality that the original Lagrangian density $$ \tag{6} {\cal L}~=~{\cal L}(\phi(x), \partial \phi(x); A(x),F(x);x). $$ depends already (possibly trivially) on the $U(1)$ gauge field $A_{\mu}$ and its Abelian field strength $$\tag{7} F_{\mu\nu}~:=~\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$$ The infinitesimal Abelian gauge transformation is defined to be $$ \tag{8} \delta A_{\mu}~=~d_{\mu}\varepsilon. $$ Let us introduce the covariant derivative $$ \tag{9} D_{\mu}\phi^{\alpha}~=~\partial_{\mu}\phi^{\alpha} - A_{\mu}Y^{\alpha}, $$ which transforms covariantly $$ \tag{10} \delta (D_{\mu}\phi)^{\alpha}~=~\varepsilon (D_{\mu}Y)^{\alpha}$$ under gauge transformations (1) and (8). One may then prove under mild assumptions the following Theorem 1. Theorem 1. The gauge transformations (1) and (8) are a quasi-symmetry for the following so-called gauged Lagrangian density $$ \tag{11} \widetilde{\cal L}~:=~ \left.{\cal L}\right\|_{\partial\phi\to D\phi}+\left. A_{\mu} f^{\mu}\right\|_{\partial\phi\to D\phi}. $$ II) Example: Free Schrödinger field theory. The wavefunction $\phi$ is a complex (Grassmann-even) field. The Lagrangian density reads (putting $\hbar=1$): $$ \tag{12} {\cal L} ~=~ \frac{i}{2}(\phi^\partial_0\phi - \phi \partial_0\phi^) - \frac{1}{2m} \sum_{k=1}^3(\partial_k\phi)^\partial^k\phi. $$ The corresponding Euler-Lagrange equation is the free Schrödinger equation $$ 0~\approx~\frac{\delta S}{\delta\phi^} ~=~ i\partial_0\phi~+\frac{1}{2m}\partial_k\partial^k\phi $$ $$ \tag{13} \qquad \Leftrightarrow \qquad 0~\approx~\frac{\delta S}{\delta\phi} ~=~ -i\partial_0\phi^~+\frac{1}{2m}\partial_k\partial^k\phi^. $$ The infinitesimal transformation is $$ \tag{14} \delta \phi~=~Y\varepsilon \qquad \Leftrightarrow \qquad \delta \phi^~=~Y^\varepsilon^, $$ where $Y\in\mathbb{C}\backslash\{0\}$ is a fixed non-zero complex number. Bear in mind that the above Theorem 1 is only applicable to a single real transformation (1). Here we are trying to apply Theorem 1 to a complex transformation, so we may not succeed, but let's see how far we get. The complexified Noether currents are $$ \tag{15} j^0~=~ \frac{i}{2}Y\phi^, \qquad j^k~=~-\frac{1}{2m}Y\partial^k\phi^, \qquad k~\in~\{1,2,3\},$$ $$ \tag{16} f^0~=~ -\frac{i}{2}Y\phi^, \qquad f^k~=~0, $$ $$ \tag{17} J^0~=~ iY\phi^, \qquad J^k~=~-\frac{1}{2m}Y\partial^k\phi^, $$ and the corresponding complex conjugate relations of eqs. (15)-(17). The infinitesimal complex gauge transformation is defined to be $$ \tag{18} \delta A_{\mu}~=~d_{\mu}\varepsilon \qquad \Leftrightarrow \qquad \delta A_{\mu}^~=~d_{\mu}\varepsilon^. $$ The Lagrangian density (11) reads $$ \widetilde{\cal L} ~=~\frac{i}{2}(\phi^D_0\phi - \phi D_0\phi^) - \frac{1}{2m} \sum_{k=1}^3(D_k\phi)^D^k\phi +\frac{i}{2}(\phi Y^ A_0^* - \phi^Y A_0) $$ $$ \tag{19} ~=~\frac{i}{2}\left(\phi^(\partial_0\phi-2Y A_0) - \phi (\partial_0\phi-2YA_0)^\right) - \frac{1}{2m} \sum_{k=1}^3(D_k\phi)^D^k\phi . $$ We emphasize that the Lagrangian density $\widetilde{\cal L}$ is not just the minimally coupled original Lagrangian density $\left.{\cal L}\right\|_{\partial\phi\to D\phi}$. The last term on the rhs. of eq. (11) is important too. An infinitesimal gauge transformation of the Lagrangian density is $$ \tag{20} \delta\widetilde{\cal L}~=~\frac{i}{2}d_0(\varepsilon^Y^\phi -\varepsilon Y\phi^) + i\|Y\|^2(\varepsilon A_0^ - \varepsilon^* A_0) $$ for arbitrary infinitesimal $x$-dependent local gauge parameter $\varepsilon=\varepsilon(x)$. Note that the local complex transformations (14) and (18) is not a (quasi) gauge symmetry of the Lagrangian density (19). The obstruction is the second term on the rhs. of eq. (20). Only the first term on the rhs. of eq. (20) is a total time derivative. However, let us restrict the gauge parameter $\varepsilon$ and the gauge field $A_{\mu}$ to belong to a fixed complex direction in the complex plane, $$ \tag{21}\varepsilon,A_{\mu}~\in ~ e^{i\theta}\mathbb{R}.$$ Here $e^{i\theta}$ is some fixed phase factor, i.e. we leave only a single real gauge d.o.f. Then the second term on the rhs. of eq. (20) vanishes, so the gauged Lagrangian density (19) has a real (quasi) gauge symmetry in accordance with Theorem 1. Note that the field $\phi$ is still a fully complex variable even with the restriction (21). Also note that the Lagrangian density (19) can handle both the real and the imaginary local shift transformations (14) as (quasi) gauge symmetries via the restriction construction (21), although not simultaneously. III) An incomplete list for further studies: 1. Peter West, Introduction to Supersymmetry and Supergravity, 1990, Chap. 7. 2. Henning Samtleben, Lectures on Gauged Supergravity and Flux Compactifications, Class. Quant. Grav. 25 (2008) 214002, arXiv:0808.4076. $^1$ The transformation (1) is for simplicity assumed to be a so-called vertical transformation. In general, one could also allow horizontal contributions from variation of $x$. $^2$ For the notion of quasi-symmetry, see e.g. this Phys.SE answer. $^3$ Here the $\approx$ symbol means equality modulo equation of motion (e.o.m). The words on-shell and off-shell refer to whether e.o.m. is satisfied or not. • $\begingroup$ Thanks a lot, this is very helpful! I still have to check how (and whether) this works in the non-Abelian case, but I have already learned quite something from you :) $\endgroup$ Apr 17, 2013 at 9:04 – Qmechanic Apr 17, 2013 at 16:22 • $\begingroup$ This is again very instructive, thanks! So to summarize this example, the free Schrödinger field theory has a global ISO(2) symmetry (phase rotations plus two translations in the plane of complex $\psi$). Any of the three one-parametric Abelian subgroups can be gauged, but this breaks the remaining global symmetry. Two whole ISO(2) group, or even the normal subgroup of translations, cannot be gauged. It's good to see this so explicitly. $\endgroup$ Apr 18, 2013 at 8:13 First of all, one can't gauge a symmetry without modifying (enriching) the field contents. Gauging a symmetry means to add a gauge field and the appropriate interactions (e.g. by covariantize all terms with derivatives, in the case of both Yang-Mills and diffeomorphism symmetries). Global and gauge symmetries are different entities when it comes to their physical interpretations; but they're also different entities when it comes to the degree of symmetry they actually carry. Concerning the latter difference, a symmetry is a gauge symmetry if the parameters of the transformations $\lambda$ may depend on the spacetime coordinates, $\lambda = \lambda(\vec x,t)$. If they can, they can and the theory has a gauge symmetry; if they can't, they can't and the theory has at most a global symmetry. There can't be any ambiguity here; you can't "gauge a symmetry by changing nothing at all". Concerning the former difference, the gauge symmetries must be treated as redundancies: the physical configurations (classically) or quantum states (quantum mechanically) must be considered physically identical if they only differ by a gauge transformation. For Lie gauge symmetries, this is equivalent to saying that physical states must be annihilated by the generators of the gauge symmetries. For any local symmetry as described in the previous paragraph, one typically generates unphysical states (of negative norm etc.) and they have to be decoupled – by being classified as unphysical ones. In the Yang-Mills case, a global symmetry may be gauged but the final spectrum should be anomaly-free because gauge anomalies are physical inconsistencies exactly because gauge symmetries are just redundancies and one isn't allowed to "break" them spontaneously because they really reduce the physical spectrum down to a consistent one. In this respect, they differ from the global symmetries that can be broken. Of course, even an anomalous global symmetry may be gauged by adding the gauge fields and other fields that are capable of canceling the gauge anomaly. Finally, the shift invariance of the massless Dirac $\psi$ in your example physically corresponds to the possibility to add a zero-momentum, zero-energy fermion into the system. It's just a way to find a "new solution" of this theory which is possible because $\psi$ is only coupled via derivative terms. The symmetry wouldn't be a symmetry if there were a mass term. You may easily gauge this symmetry by replacing $\psi$ with $\psi+\theta$ everywhere in the action and promoting $\theta$ to a new field – which plays a similar role as the new gauge field $A_\mu$ if you're gauging a Yang-Mills-like global symmetry. By doing so, you will have twice as many off-shell fermionic degrees of freedom but the action won't depend on one of them, $\psi+\theta$ (the opposite sign), at all. So this field will create ghostly particles that don't interact with anything else – in fact, they don't even have kinetic terms. Clearly, these dynamically undetermined quanta shouldn't be counted in a physical theory (although, in some sense, they "just" increase the degeneracy of each state of the physical fields by an infinite extra factor) so the right way to treat them, as always in gauge theories, is to require that physical states can't contain any such quanta. This requirement effectively returns you to the original theory, just with $\psi$ renamed as $\psi+\theta$. You won't get a new interesting theory in this way and there's no reason why gauging a symmetry should always produce such an interesting new theory. The case of Yang-Mills theories or generally covariant theories is different because they are interesting: with a Lorentz-covariant field content, one may create theories with no ghosts (negative-norm states) despite the fact that they predict the existence of spin-one or spin-two particles (from the gauge field – which is the metric tensor in the spin-two case). But this is only possible because these theories are special and the action of the symmetry transformations is less trivial than in your case. "Shift" symmetries may only be gauged in a way that renames or erases whole fields so they just can't lead to interesting new possibilities. • $\begingroup$ Hi Luboš, thanks for the comprehensive answer! Concerning the first six paragraphs: I do know these things :) As to the rest: I'm afraid I had something else in mind. I want to add an auxiliary gauge field, i.e. no new physical degrees of freedom. The point is that I want to construct a generating functional for the conserved currents of the theory. This is just a mathematical trick; the physical symmetry of the theory remains global. E.g. for a massless Lorentz scalar, the shift symmetry can be gauged as $\mathscr L=\frac12(\partial_\mu\phi-A_\mu)^2$, but this doesn't work in my case. $\endgroup$ Apr 16, 2013 at 9:20 • $\begingroup$ Thanks, Tomáš! Unfortunately, I can't help you with this because I don't understand your theory that is both gauge-symmetric as well as physically globally symmetric only. It's like the Princess Koloběžka (Scooter?) the First, right? ;-) youtube.com/watch?v=mBC9vr3nuiI What does it mean for a field to be called a "gauge field" if the normally associated with it gauge symmetry doesn't exist at all? $\endgroup$ Apr 16, 2013 at 17:28 • $\begingroup$ Suppose you have an action $S[\phi]$ with a global symmetry. I'm looking for an action $S'[\phi,A_\mu]$ such that $S'[\phi,0]=S[\phi]$. If I manage to make $S'[\phi,A_\mu]$ gauge-invariant, then $e^{-W[A_\mu]}=\int[d\phi]e^{-S'[\phi,A_\mu]}$ will be a generating functional of Green's functions of conserved currents of the original theory. That's why I say that $A_\mu$ is not a dynamical field and the physical symmetry, obtained by setting it to zero, is still global. You see the scooter there? :) $\endgroup$ Apr 17, 2013 at 7:40 • $\begingroup$ Dear Tomáši, first, it's easy to add fields so that the new theory will have a gauge symmetry but in a typical case, one doesn't get an interesting theory because the new gauge symmetry just removes some degrees of freedom and it's more useful to erase them immediately with the symmetry, anyway. Second, I don't understand why you would consider extra fields to find out whether the original theory has conserved currents. If $S$ admits conserved currents, it does, otherwise it doesn't. $\endgroup$ Apr 18, 2013 at 9:00 • $\begingroup$ Third, you may always add $J\cdot A$ fields to an action so that you obtain a generating functional for correlation functions of $J$ in the original theory: the theory with the extra term doesn't have to have a gauge symmetry. The field $A$ is auxiliary. Fourth, treating it as auxiliary is less constraning because if it is dynamical, you have to impose the eqn of motion from varying $A$. Fifth, the equation of motion is $J=0$ unless you manage to write new "kinetic" terms for $A$ as well which is what makes the gauge symmetry physical and interesting but it's not guaranteed to exist. $\endgroup$ Apr 18, 2013 at 9:02 Assuming that the following manipulations are correct the translational symmetry of your Lagrangian can be gauged by including a scalar gauge field $\phi$ and a one form gauge field $A_{\mu}$. First of all, assuming that the boundary terms do not contribute we can write the Lagrangian density as $$ \mathscr L=\psi^\dagger i\partial_t\psi-\frac{1}{2m}(\partial^{\mu}\psi)^{\dagger}\partial_{\mu}\psi. $$ Now writing $\psi$ as $\left[\begin{array}{cc}\psi(x)\\1\end{array}\right]$ the translation of $\psi$ can be written as $\left[ \begin{array}{cc} 1 & \theta_1+i\theta_2\\ 0 & 1\\ \end{array}\right]\left[\begin{array}{cc}\psi(x)\\1\end{array}\right]$ Lie algebra of the group of matrices of the form $\left[ \begin{array}{cc} 1 & \theta_1+i\theta_2\\ 0 & 1\\ \end{array}\right]$ is the set of matrices $\left[ \begin{array}{cc} 0 & a+ib\\ 0 & 0\\ \end{array}\right]$ Now to gauge this symmetry introduce a Lie algebra valued one form $A=A_{\mu}dx^{\mu}$ which under a gauge transformation $\left[\begin{array}{cc}\psi(x)\\1\end{array}\right]\rightarrow \left[ \begin{array}{cc} 1 & \theta_1(x)+i\theta_2(x)\\ 0 & 1\\ \end{array}\right]\left[\begin{array}{cc}\psi(x)\\1\end{array}\right]$ transform as $A_{\mu}\rightarrow g(x)A_{\mu}g(x)^{-1}+(\partial_{\mu}g(x))g(x)^{-1} $ However we note that the Lagrangian $$ \mathscr L=\psi^\dagger i(\partial_t-A_t)\psi-\frac{1}{2m}((\partial^{\mu}-A^{\mu})\psi)^{\dagger}(\partial_{\mu}-A_{\mu})\psi. $$ is not gauge invariant and neither real. Obstruction to gauge invariance is the fact that $\psi^{\dagger}$ doesn't transform by right multiplication by $g(x)^{-1}$ but rather by right multiplication by $g(x)^{\dagger}$ To repair gauge invariance one may introduce a matrix valued scalar gauge field $\phi$ whose exponential under change of gauge changes as $exp(\phi(x))\rightarrow (g(x)^{\dagger})^{-1}exp(\phi(x))g(x)^{-1}$ (how does $\phi$ change? I am not sure) Then we see that the Lagrangian $$ \mathscr L=\psi^\dagger iexp(\phi(x))(\partial_t-A_t)\psi-\frac{1}{2m}((\partial^{\mu}-A^{\mu})\psi)^{\dagger}exp(\phi(x))(\partial_{\mu}-A_{\mu})\psi. $$ is gauge invariant. However still the Lagrangian is not real. To repair that we can include complex conjugate of each term in it. • $\begingroup$ Nice construction! In effect, it boils down to introducing a extra field $\chi$ which transforms as $\chi\to\chi-\theta_1-i\theta_2$, and replacing $\psi^\dagger i\partial_t\psi$ with $\frac i2[(\psi^\dagger+\chi^\dagger)\partial_t\psi-\partial_t\psi^\dagger(\psi+\chi)]$ and subsequently ordinary derivatives with covariant ones. I'm not sure if this is what I want though, since even after setting $A=0$, this theory has a different global symmetry than the theory I started with. Thanks anyway, you made me formulate more precisely what I'm looking for! $\endgroup$ Apr 17, 2013 at 13:02 Your Answer
e23cfe4ba1552d01	Does Information Carry Mass? If information carries mass, could it be the dark matter physicists are craving? The existence of dark energy and dark matter was inferred in order to correctly predict the expansion of the universe and the rotational velocity of galaxies. In this view, dark energy could be the source of the centrifugal force expanding the universe (it is what accounts for the Hubble constant in the leading theories), while dark matter could be the centripetal force (an additional gravity source) necessary to stabilize galaxies and clusters of galaxies, since there isn’t enough ordinary mass to keep them together. Among other hypotheses, dark energy and dark matter are believed to be related to the vacuum fluctuations, and huge efforts have been devoted to detecting it. The fact that no evidence has yet been found calls for a change of perspective that could be due to information theory. How could we measure the mass of information? Dr. Melvin Vopson, of the University of Portsmouth, has a hypothesis he calls the mass-energy-information equivalence. It extends the already existing information-energy equivalence by proposing information has mass. Initial works on Shannon’s classical information theory, its applications to quantum mechanics by Dr. Wheeler, and Landauer’s principle predicting that erasing one bit of information would release a tiny amount of heat, connect information to energy. Therefore, through Einstein’s equivalence between mass and energy, information – once created – has mass. The figure below depicts the extended equivalence principle. In order to find the mass of digital information, one would start with an empty data storage device, measuring its total mass with a highly sensitive device. Once the information is recorded in the device, its mass is measured again. The next step is to erase one file and measure again. The limiting step is the fact that such an ultra-sensitive device doesn’t exist yet. In his paper published in the journal AIP Advances, Vopson proposes that this device could be in the form of an interferometer similar to LIGO, or a weighing machine like a Kibble balance. In the same paper, Vopson describes the mathematical basis for the mechanism and physics by which information acquires mass, and formulates this powerful principle, proposing a possible experiment to test it. In regard to dark matter, Vopson says that his estimate of the ‘information bit content’ of the universe is very close to the number of bits of information that the visible universe would contain to make up all the missing dark matter, as estimated by M.P. Gough and published in 2008,. This idea is synchronistic with the recent discovery that sound carries mass (, i.e., phonons are massive. Vopson is applying for a grant in order to design and build the measurement device and perform the experiments. We are so looking forward to his results! RSF in perspective Both dark matter and dark energy have been inferred as a consequence of neglecting spin in the structure of space-time. In the frame of the Generalized Holographic approach, spin is the natural source of centrifugal and centripetal force that emerges from the gradient density across scales, just as a hurricane emerges due to pressure and temperature gradients. The vacuum energy of empty space – the classical or cosmological vacuum – has been estimated to be 10−9 joules per cubic meter. However, vacuum energy density at quantum scale is 10113joules per cubic meter. Such a discrepancy of 122 orders of magnitude difference in vacuum densities between micro and cosmological scales is known as the vacuum catastrophe. This extremely large density gradient in the Planck field originates spin at all scales. Additionally, the holographic model explains mass as an emergent property of an information transfer potential between the information-energy stored in a confined volume and the information-energy in the surface or boundary of that volume, with respect to the size or volume of a bit of information. Each bit of information-energy voxelating the surface and volume is spinning at an extremely fast speed. Space is composed of these voxels, named Planck Spherical Units (PSU), which are a quanta of action. The expressed or unfolded portion of the whole information is what we call mass. For more details on how the holographic approach explains dark mass and dark energy, please see our RSF article on the Vacuum Catastrophe ( Link original: Quantum entanglement realized between distant large objects A team of researchers from the University of Copenhagen’s Niels Bohr Institute has successfully entangled two very distinct quantum particles. The findings, which were reported in Nature Physics, have various possible applications in ultra-precise sensing and quantum communication. Quantum communication and quantum sensing are both based on entanglement. It’s a quantum link between two items that allows them to act as if they’re one quantum object. Researchers were able to create entanglement between a mechanical oscillator—a vibrating dielectric membrane—and a cloud of atoms, each serving as a small magnet, or «spin,» according to physicists. By joining these disparate entities with photons, or light particles, they were able to entangle. The membrane—or mechanical quantum systems in general—can be used to process quantum information, and the membrane—or mechanical quantum systems in general—can be used to store quantum information. Professor Eugene Polzik, the project’s leader, says: «We’re on our way to pushing the boundaries of entanglement’s capabilities with this new technique. The larger the objects, the further away they are, and the more different they are, the more intriguing entanglement becomes from both a basic and an applied standpoint. Entanglement between highly diverse things is now conceivable thanks to the new result.» Imagine the position of the vibrating membrane and the tilt of the total spin of all atoms, similar to a spinning top, to explain entanglement using the example of spins entangled with a mechanical membrane. A correlation occurs when both items move randomly yet are observed travelling right or left at the same moment. The so-called zero-point motion—the residual, uncorrelated motion of all matter that occurs even at absolute zero temperature—is generally the limit of such correlated motion. This limits our understanding of any of the systems. Eugene Polzik’s team entangled the systems in their experiment, which means they moved in a correlated way with more precision than zero-point motion. «Quantum mechanics is a double-edged sword—it gives us amazing new technology, but it also restricts the precision of measurements that would appear simple from a classical standpoint,» explains Micha Parniak, a team member. Even if they are separated by a large distance, entangled systems can maintain perfect correlation, a fact that has perplexed academics since quantum physics’ inception more than a century ago. Christoffer stfeldt, a Ph.D. student, elaborates: «Consider the many methods for manifesting quantum states as a zoo of diverse realities or circumstances, each with its own set of features and potentials. If, for example, we want to construct a gadget that can take advantage of the many attributes they all have and perform different functions and accomplish different tasks, we’ll need to invent a language that they can all understand. For us to fully utilise the device’s capabilities, the quantum states must be able to communicate. This entanglement of two zoo elements has demonstrated what we are presently capable of.» Quantum sensing is an example of distinct perspectives on entangling different quantum things. Different objects have different levels of sensitivity to external pressures. Mechanical oscillators, for example, are employed in accelerometers and force sensors, while atomic spins are used in magnetometers. Entanglement permits only one of the two entangled objects to be measured with a sensitivity not restricted by the object’s zero-point fluctuations when only one of the two is subject to external perturbation. The approach has the potential to be used in sensing for both small and large oscillators in the near future. The first detection of gravity waves, performed by the Laser Interferometer Gravitational-wave Observatory, was one of the most significant scientific breakthroughs in recent years (LIGO). LIGO detects and monitors extremely faint waves produced by deep-space astronomical events such as black hole mergers and neutron star mergers. The waves can be seen because they shake the interferometer’s mirrors. However, quantum physics limits LIGO’s sensitivity since the laser interferometer’s mirrors are likewise disturbed by zero-point fluctuations. These variations produce noise, which makes it impossible to see the tiny movements of the mirrors induced by gravitational waves. It is theoretically possible to entangle the LIGO mirrors with an atomic cloud and so cancel the reflectors’ zero-point noise in the same manner that the membrane noise is cancelled in the current experiment. Due to their entanglement, the mirrors and atomic spins have a perfect correlation that can be used in such sensors to almost eliminate uncertainty. It’s as simple as taking data from one system and applying what you’ve learned to the other. In this method, one may simultaneously learn about the position and momentum of LIGO’s mirrors, entering a so-called quantum-mechanics-free subspace and moving closer to unlimited precision in motion measurements. A model experiment demonstrating this principle is on the way at Eugene Polzik’s laboratory. Link original: Can humans travel through wormholes in space? Two new studies examine ways we could engineer human wormhole travel. Imagine if we could cut paths through the vastness of space to make a network of tunnels linking distant stars somewhat like subway stations here on Earth? The tunnels are what physicists call wormholes, strange funnel-like folds in the very fabric of spacetime that would be—if they exist—major shortcuts for interstellar travel. You can visualize it in two dimensions like this: Take a piece of paper and bend it in the middle so that it makes a U shape. If an imaginary flat little bug wants to go from one side to the other, it needs to slide along the paper. Or, if there were a bridge between the two sides of the paper the bug could go straight between them, a much shorter path. Since we live in three dimensions, the entrances to the wormholes would be more like spheres than holes, connected by a four-dimensional “tube.” It’s much easier to write the equations than to visualize this! Amazingly, because the theory of general relativity links space and time into a four-dimensional spacetime, wormholes could, in principle, connect distant points in space, or in time, or both. The idea of wormholes is not new. Its origins reach back to 1935 (and even earlier), when Albert Einstein and Nathan Rosen published a paper constructing what became known as an Einstein-Rosen bridge. (The name ‘wormhole’ came up later, in a 1957 paper by Charles Misner and John Wheeler, Wheeler also being the one who coined the term ‘black hole.’) Basically, an Einstein-Rosen bridge is a connection between two distant points of the universe or possibly even different universes through a tunnel that goes into a black hole. Exciting as the possibility is, the throats of such bridges are notoriously unstable and any object with mass that ventures through it would cause it to collapse upon itself almost immediately, closing the connection. To force the wormholes to stay open, one would need to add a kind of exotic matter that has both negative energy density and pressure—not something that is known in the universe. (Interestingly, negative pressure is not as crazy as it seems; dark energy, the fuel that is currently accelerating the cosmic expansion, does it exactly because it has negative pressure. But negative energy density is a whole other story.) If wormholes exist, if they have wide mouths, and if they can be kept open (three big but not impossible ifs) then it’s conceivable that we could travel through them to faraway spots in the universe. Arthur C. Clarke used them in “2001: A Space Odyssey”, where the alien intelligences had constructed a network of intersecting tunnels they used as we use the subway. Carl Sagan used them in “Contact” so that humans could confirm the existence of intelligent ETs. “Interstellar” uses them so that we can try to find another home for our species. Two recent papers try to get around some of these issues. Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu use normal matter with electric charge to stabilize the wormhole, but the resulting throat is still of submicroscopic width, so not useful for human travel. It is also hard to justify net electric charges in black hole solutions as they tend to get neutralized by surrounding matter, similar to how we get shocked with static electricity in dry weather. Juan Maldacena and Alexey Milekhin’s paper is titled ‘Humanly Traversable Wormholes’, thus raising the stakes right off the bat. However, they are open to admitting that “in this paper, we revisit the question [of humanly traversable wormholes] and we engage in some ‘science fiction.’” The first ingredient is the existence of some kind of matter (the “dark sector”) that only interacts with normal matter (stars, us, frogs) through gravity. Another point is that to support the passage of human-size travelers, the model needs to exist in five dimensions, thus one extra space dimension. When all is set up, the wormhole connects two black holes with a magnetic field running through it. And the whole thing needs to spin to keep it stable, and completely isolated from particles that may fall into it compromising its design. Oh yes, and extremely low temperature as well, even better at absolute zero, an unattainable limit in practice. Maldacena and Milekhins’ paper is an amazing tour through the power of speculative theoretical physics. They are the first to admit that the object they construct is very implausible and have no idea how it could be formed in nature. In their defense, pushing the limits (or beyond the limits) of understanding is what we need to expand the frontiers of knowledge. For those who dream of humanly traversable wormholes, let’s hope that more realistic solutions would become viable in the future, even if not the near future. Or maybe aliens that have built them will tell us how. Link Original: Hidden philosophy of the Pythagorean theorem A cosmos of geometry Everything is a right triangle Pythagorean theorem goes beyond squares Today, the Pythagorean theorem is taught using squares. Pythagoras and the philosophy of cosmology Ask Ethan: What should everyone know about quantum mechanics? Quantum physics isn’t quite magic, but it requires an entirely novel set of rules to make sense of the quantum universe. The most powerful idea in all of science is this: The universe, for all its complexity, can be reduced to its simplest, most fundamental components. If you can determine the underlying rules, laws, and theories that govern your reality, then as long as you can specify what your system is like at any moment in time, you can use your understanding of those laws to predict what things will be like both in the far future as well as the distant past. The quest to unlock the secrets of the universe is fundamentally about rising to this challenge: figuring out what makes up the universe, determining how those entities interact and evolve, and then writing down and solving the equations that allow you to predict outcomes that you have not yet measured for yourself. In this regard, the universe makes a tremendous amount of sense, at least in concept. But when we start talking about what, precisely, it is that composes the universe, and how the laws of nature actually work in practice, a lot of people bristle when faced with this counterintuitive picture of reality: quantum mechanics. That’s the subject of this week’s Ask Ethan, where Rajasekaran Rajagopalan writes in to inquire: “Can you please provide a very detailed article on quantum mechanics, which even a… student can understand?” Let’s assume you’ve heard about quantum physics before, but don’t quite know what it is just yet. Here’s a way that everyone can — at least, to the limits that anyone can — make sense of our quantum reality. Before there was quantum mechanics, we had a series of assumptions about the way the universe worked. We assumed that everything that exists was made out of matter, and that at some point, you’d reach a fundamental building block of matter that could be divided no further. In fact, the very word “atom” comes from the Greek ἄτομος, which literally means “uncuttable,” or as we commonly think about it, indivisible. These uncuttable, fundamental constituents of matter all exerted forces on one another, like the gravitational or electromagnetic force, and the confluence of these indivisible particles pushing and pulling on one another is what was at the core of our physical reality. The laws of gravitation and electromagnetism, however, are completely deterministic. If you describe a system of masses and/or electric charges, and specify their positions and motions at any moment in time, those laws will allow you to calculate — to arbitrary precision — what the positions, motions, and distributions of each and every particle was and will be at any other moment in time. From planetary motion to bouncing balls to the settling of dust grains, the same rules, laws, and fundamental constituents of the universe accurately described it all. Until, that is, we discovered that there was more to the universe than these classical laws. 1.) You can’t know everything, exactly, all at once. If there’s one defining characteristic that separates the rules of quantum physics from their classical counterparts, it’s this: you cannot measure certain quantities to arbitrary precisions, and the better you measure them, the more inherently uncertain other, corresponding properties become. • Measure a particle’s position to a very high precision, and its momentum becomes less well-known. • Measure the angular momentum (or spin) of a particle in one direction, and you destroy information about its angular momentum (or spin) in the other two directions. • Measure the lifetime of an unstable particle, and the less time it lives for, the more inherently uncertain the particle’s rest mass will be. These are just a few examples of the weirdness of quantum physics, but they’re sufficient to illustrate the impossibility of knowing everything you can imagine knowing about a system all at once. Nature fundamentally limits what’s simultaneously knowable about any physical system, and the more precisely you try and pin down any one of a large set of properties, the more inherently uncertain a set of related properties becomes. 2.) Only a probability distribution of outcomes can be calculated: not an explicit, unambiguous, single prediction. Not only is it impossible to know all of the properties, simultaneously, that define a physical system, but the laws of quantum mechanics themselves are fundamentally indeterminate. In the classical universe, if you throw a pebble through a narrow slit in a wall, you can predict where and when it will hit the ground on the other side. But in the quantum universe, if you do the same experiment but use a quantum particle instead — whether a photon, and electron, or something even more complicated — you can only describe the possible set of outcomes that will occur. Quantum physics allows you to predict what the relative probabilities of each of those outcomes will be, and it allows you do to it for as complicated of a quantum system as your computational power can handle. Still, the notion that you can set up your system at one point in time, know everything that’s possible to know about it, and then predict precisely how that system will have evolved at some arbitrary point in the future is no longer true in quantum mechanics. You can describe what the likelihood of all the possible outcomes will be, but for any single particle in particular, there’s only one way to determine its properties at a specific moment in time: by measuring them. 3.) Many things, in quantum mechanics, will be discrete, rather than continuous. This gets to what many consider the heart of quantum mechanics: the “quantum” part of things. If you ask the question “how much” in quantum physics, you’ll find that there are only certain quantities that are allowed. • Particles can only come in certain electric charges: in increments of one-third the charge of an electron. • Particles that bind together form bound states — like atoms — and atoms can only have explicit sets of energy levels. • Light is made up of individual particles, photons, and each photon only has a specific, finite amount of energy inherent to it. In all of these cases, there’s some fundamental value associated with the lowest (non-zero) state, and then all other states can only exist as some sort of integer (or fractional integer) multiple of that lowest-valued state. From the excited states of atomic nuclei to the energies released when electrons fall into their “hole” in LED devices to the transitions that govern atomic clocks, some aspects of reality are truly granular, and cannot be described by continuous changes from one state to another. 4.) Quantum systems exhibit both wave-like and particle-like behaviors. And which one you get — get this — depends on if or how you measure the system. The most famous example of this is the double slit experiment: passing a single quantum particle, one-at-a-time, through a set of two closely-spaced slits. Now, here’s where things get weird. • If you don’t measure which particle goes through which slit, the pattern you’ll observe on the screen behind the slit will show interference, where each particle appears to be interfering with itself along the journey. The pattern revealed by many such particles shows interference, a purely quantum phenomenon. • If you do measure which slit each particle goes through — particle 1 goes through slit 2, particle 2 goes through slit 2, particle 3 goes through slit 1, etc. — there is no interference pattern anymore. In fact, you simply get two “lumps” of particles, one each corresponding to the particles that went through each of the slits. It’s almost as if everything exhibits wave-like behavior, with its probability spreading out over space and through time, unless an interaction forces it to be particle-like. But depending on which experiment you perform and how you perform it, quantum systems exhibit properties that are both wave-like and particle-like. 5.) The act of measuring a quantum system fundamentally changes the outcome of that system. According to the rules of quantum mechanics, a quantum object is allowed to exist in multiple states all at once. If you have an electron passing through a double slit, part of that electron must be passing through both slits, simultaneously, in order to produce the interference pattern. If you have an electron in a conduction band in a solid, its energy levels are quantized, but its possible positions are continuous. Same story, believe it or not, for an electron in an atom: we can know its energy level, but asking “where is the electron” is something can only answer probabilistically. So you get an idea. You say, “okay, I’m going to cause a quantum interaction somehow, either by colliding it with another quantum or passing it through a magnetic field or something like that,” and now you have a measurement. You know where the electron is at the moment of that collision, but here’s the kicker: by making that measurement, you have now changed the outcome of your system. You’ve pinned down the object’s position, you’ve added energy to it, and that causes a change in momentum. Measurements don’t just “determine” a quantum state, but create an irreversible change in the quantum state of the system itself. 6.) Entanglement can be measured, but superpositions cannot. Here’s a puzzling feature of the quantum universe: you can have a system that’s simultaneously in more than one state at once. Schrodinger’s cat can be alive and dead at once; two water waves colliding at your location can cause you to either rise or fall; a quantum bit of information isn’t just a 0 or a 1, but rather can be some percentage “0” and some percentage “1” at the same time. However, there’s no way to measure a superposition; when you make a measurement, you only get one state out per measurement. Open the box: the cat is dead. Observe the object in the water: it will rise or fall. Measure your quantum bit: get a 0 or a 1, never both. But whereas superposition is different effects or particles or quantum states all superimposed atop one another, entanglement is different: it’s a correlation between two or more different parts of the same system. Entanglement can extend to regions both within and outside of one another’s light-cones, and basically states that properties are correlated between two distinct particles. If I have two entangled photons, and I wanted to guess the “spin” of each one, I’d have 50/50 odds. But if I measured the spin of one, I would know the other’s spin to more like 75/25 odds: much better than 50/50. There isn’t any information getting exchanged faster than light, but beating 50/50 odds in a set of measurements is a surefire way to show that quantum entanglement is real, and affect the information content of the universe. 7.) There are many ways to “interpret” quantum physics, but our interpretations are not reality. This is, at least in my opinion, the trickiest part of the whole endeavor. It’s one thing to be able to write down equations that describe the universe and agree with experiments. It’s quite another thing to accurately describe just exactly what’s happening in a measurement-independent way. Can you? I would argue that this is a fool’s errand. Physics is, at its core, about what you can predict, observe, and measure in this universe. Yet when you make a measurement, what is it that’s occurring? And what does that means about reality? Is reality: • a series of quantum wavefunctions that instantaneously “collapse” upon making a measurement? • an infinite ensemble of quantum waves, were measurement “selects” one of those ensemble members? • a superposition of forwards-moving and backwards-moving potentials that meet up, now, in some sort of “quantum handshake?” • an infinite number of possible worlds, where each world corresponds to one outcome, and yet our universe will only ever walk down one of those paths? If you believe this line of thought is useful, you’ll answer, “who knows; let’s try to find out.” But if you’re like me, you’ll think this line of thought offers no knowledge and is a dead end. Unless you can find an experimental benefit of one interpretation over another — unless you can test them against each other in some sort of laboratory setting — all you’re doing in choosing an interpretation is presenting your own human biases. If it isn’t the evidence doing the deciding, it’s very hard to argue that there’s any scientific merit to your endeavor t all. If you were to only teach someone the classical laws of physics that we thought governed the universe as recently as the 19th century, they would be utterly astounded by the implications of quantum mechanics. There is no such thing as a “true reality” that’s independent of the observer; in fact, the very act of making a measurement alters your system irrevocably. Additionally, nature itself is inherently uncertain, with quantum fluctuations being responsible for everything from the radioactive decay of atoms to the initial seeds of structure that allow the universe to grow up and form stars, galaxies, and eventually, human beings. The quantum nature of the universe is written on the face of every object that now exists within it. And yet, it teaches us a humbling point of view: that unless we make a measurement that reveals or determines a specific quantum property of our reality, that property will remain indeterminate until such a time arises. If you take a course on quantum mechanics at the college level, you’ll likely learn how to calculate probability distributions of possible outcomes, but it’s only by making a measurement that you determine which specific outcome occurs in your reality. As unintuitive as quantum mechanics is, experiment after experiment continues to prove it correct. While many still dream of a completely predictable universe, quantum mechanics, not our ideological preferences, most accurately describes the reality we all inhabit. China’s New Quantum Computer Has 1 Million Times the Power of Google’s It appears a quantum computer rivalry is growing between the U.S. and China. Physicists in China claim they’ve constructed two quantum computers with performance speeds that outrival competitors in the U.S., debuting a superconducting machine, in addition to an even speedier one that uses light photons to obtain unprecedented results, according to a recent study published in the peer-reviewed journals Physical Review Letters and Science Bulletin. China has exaggerated the capabilities of its technology before, but such soft spins are usually tagged to defense tech, which means this new feat could be the real deal. China’s quantum computers still make a lot of errors The supercomputer, called Jiuzhang 2, can calculate in a single millisecond a task that the fastest conventional computer in the world would take a mind-numbing 30 trillion years to do. The breakthrough was revealed during an interview with the research team, which was broadcast on China’s state-owned CCTV on Tuesday, which could make the news suspect. But with two peer-reviewed papers, it’s important to take this seriously. Pan Jianwei, lead researcher of the studies, said that Zuchongzhi 2, which is a 66-qubit programmable superconducting quantum computer is an incredible 10 million times faster than Google’s 55-qubit Sycamore, making China’s new machine the fastest in the world, and the first to beat Google’s in two years. The Zuchongzhi 2 is an improved version of a previous machine, completed three months ago. The Jiuzhang 2, a different quantum computer that runs on light, has fewer applications but can run at blinding speeds of 100 sextillion times faster than the biggest conventional computers of today. In case you missed it, that’s a one with 23 zeroes behind it. But while the features of these new machines hint at a computing revolution, they won’t hit the marketplace anytime soon. As things stand, the two machines can only operate in pristine environments, and only for hyper-specific tasks. And even with special care, they still make lots of errors. «In the next step we hope to achieve quantum error correction with four to five years of hard work,» said Professor Pan of the University of Science and Technology of China, in Hefei, which is in the southeastern province of Anhui. China’s quantum computers could power the next-gen advances of the coming decades «Based on the technology of quantum error correction, we can explore the use of some dedicated quantum computers or quantum simulators to solve some of the most important scientific questions with practical value,» added Pan. The circuits of the Zuchongzhi have to be cooled to very low temperatures to enable optimal performance for a complex task called random walk, which is a model that corresponds to the tactical movements of pieces on a chessboard. The applications for this task include calculating gene mutations, predicting stock prices, air flows in hypersonic flight, and the formation of novel materials. Considering the rapidly increasing relevance of these processes as the fourth industrial revolution picks up speed, it’s no exaggeration to say that quantum computers will be central in key societal functions, from defense research to scientific advances to the next generation of economics. Link Original: Los científicos a menudo se refieren al neutrino como la «partícula fantasma. «Los neutrinos fueron una de las partículas más abundantes en el origen del universo y lo siguen siendo hoy en día. Las reacciones de fusión en el sol producen vastos ejércitos de ellos, que vierten sobre la Tierra todos los días. Trillones pasan a través de nuestros cuerpos cada segundo, luego vuelan a través de la Tierra como si no estuviera allí.Aunque fue postulado por primera vez hace casi un siglo y detectado por primera vez hace 65 años, los neutrinos permanecen envueltos en el misterio debido a su reticencia a interactuar con la materia», dijo Alessandro Lovato, un físico nuclear del Departamento de Energía de los Estados Unidos (DO E) Laboratorio Nacional Argonne.Lovato es miembro de un equipo de investigación de cuatro laboratorios nacionales que ha construido un modelo para abordar uno de los muchos misterios acerca de los neutrinos – cómo interactúan con los núcleos atómicos, sistemas complicados hechos de protones y neutrones («núcleo ns») unidos por la fuerza fuerte. Este conocimiento es esencial para desentrañar un misterio aún más grande — por qué durante su viaje a través del espacio o la materia los neutrinos se transforman mágicamente de uno en otro de tres posibles tipos o «sabores. «Para estudiar estas oscilaciones, se han llevado a cabo dos series de experimentos en el Laboratorio Nacional de Accelerator Fermi (MiniBooNE y NOvA). En estos experimentos, los científicos generan una intensa corriente de neutrinos en un acelerador de partículas, luego los envían a detectores de partículas durante un largo período de tiempo (MiniBooNE) o a quinientas millas de la fuente (NOvA).Conociendo la distribución original de los sabores de neutrinos, los experimentalistas recogen datos relacionados con las interacciones de los neutrinos con los núcleos atómicos en los detectores. A partir de esa información, pueden calcular cualquier cambio en los sabores de neutrinos a lo largo del tiempo o la distancia. En el caso de los detectores MiniBooNE y NOvA, los núcleos son del isótopo carbono-12, que tiene seis protones y seis neutrones.»Nuestro equipo entró en escena porque estos experimentos requieren un modelo muy preciso de las interacciones de los neutrinos con los núcleos detectores en un gran rango de energía», dijo Noemi Rocco, una posdoctora de la división de Física de Argonne y Fermilab. Dada la esquividad de los neutrinos, lograr una descripción completa de estas reacciones es un desafío formidable.El modelo de física nuclear del equipo de interacciones de neutrinos con un solo nucleón y un par de ellos es el más preciso hasta ahora. «El nuestro es el primer enfoque para modelar estas interacciones a un nivel tan microscópico», dijo Rocco. «Los enfoques anteriores no eran tan finos. «Uno de los hallazgos importantes del equipo, basado en los cálculos llevados a cabo en la supercomputadora Mira ahora retirada en la Argonne Leadership Computing Facility (ALCF), fue que la interacción del par de nucleones es crucial para modelar las interacciones de neutrinos con nu Clei con exactitud. El ALCF es una instalación de usuario de la Oficina de Ciencia DOE.»Cuanto más grandes son los núcleos en el detector, mayor es la probabilidad de que los neutrinos interactúen con ellos», dijo Lovato. «En el futuro, planeamos extender nuestro modelo a datos de núcleos más grandes, a saber, los de oxígeno y argón, en apoyo de experimentos planeados en Japón y los EE. UU.».Rocco añadió que «Para esos cálculos, nos basaremos en computadoras ALCF aún más potentes, el sistema Theta existente y la próxima máquina exascale, Aurora. «Los científicos esperan que, eventualmente, surja una imagen completa de oscilaciones de sabor tanto para neutrinos como para sus antipartículas, llamados «antineutrinos». «Ese conocimiento puede arrojar luz sobre por qué el universo se construye a partir de materia en lugar de antimateria — una de las preguntas fundamentales sobre el universo. Link Original:Quantum Physics News This Simple Experiment Could Challenge Standard Quantum Theory A deceptively simple experiment that involves making precise measurements of the time it takes for a particle to go from point A to point B could spark a breakthrough in quantum physics. The findings could focus attention on an alternative to standard quantum theory called Bohmian mechanics, which posits an underworld of unseen waves that guide particles from place to place. A new study, by a team at the Ludwig Maximilian University of Munich (LMU) in Germany, makes precise predictions for such an experiment using Bohmian mechanics, a theory formulated by theoretical physicist David Bohm in the 1950s and augmented by modern-day theorists. Standard quantum theory fails in this regard, and physicists have to resort to assumptions and approximations to calculate particle transit times. “If people knew that a theory that they love so much—standard quantum mechanics—cannot make [precise] predictions in such a simple case, that should at least make them wonder,” says theorist and LMU team member Serj Aristarhov. It is no secret that the quantum world is weird. Consider a setup that fires electrons at a screen. You cannot predict exactly where any given electron will land to form, say, a fluorescent dot. But you can predict with precision the spatial distribution, or pattern, of dots that takes shape over time as the electrons land one by one. Some locations will have more electrons; others will have fewer. But this weirdness hides something even stranger. All else being equal, each electron will reach the detector at a slightly different time, its so-called arrival time. Just like the positions, the arrival times will have a distribution: some arrival times will be more common, and others will be less so. But textbook quantum physics has no mechanism for precisely predicting this temporal distribution. “Normal quantum theory is only concerned with ‘where’; they ignore the ‘when,’” says team member and theorist Siddhant Das. “That’s one way to diagnose that there’s something fishy.” There is a deep reason for this curious shortcoming. In standard quantum theory, a physical property that can be measured is called an “observable.” The position of a particle, for example, is an observable. Each and every observable is associated with a corresponding mathematical entity called an “operator.” But the standard theory has no such operator for observing time. In 1933 Austrian theoretical physicist Wolfgang Pauli showed that quantum theory could not accommodate a time operator, at least not in the standard way of thinking about it. “We conclude therefore that the introduction of a time operator … must be abandoned fundamentally,” he wrote. But measuring particle arrival times and or their “time of flight” is an important aspect of experimental physics. For example, such measurements are made with detectors at the Large Hadron Collider or instruments called mass spectrometers that use such information to calculate the masses and momenta of particles, ions and molecules. Even though such calculations concern quantum systems, physicists cannot use unadulterated quantum mechanics all the way through. “You would have no way to come up with [an unambiguous] prediction,” Das says. Instead they resort to assumptions to arrive at answers. For example, in one method, experimenters assume that once the particle leaves its source, it behaves classically, meaning it follows Newton’s equations of motion. This results in a hybrid approach—one that is part quantum, part classical. It starts with the quantum perspective, where each particle is represented by a mathematical abstraction called a wave function. Identically prepared particles will have identical wave functions when they are released from their source. But measuring the momentum of each particle (or, for that matter, its position) at the instant of release will yield different values each time. Taken together, these values follow a distribution that is precisely predicted by the initial wave function. Starting from this ensemble of values for identically prepared particles, and assuming that a particle follows a classical trajectory once it is emitted, the result is a distribution of arrival times at the detector that depends on the initial momentum distribution. Standard theory is also often used for another quantum mechanical method for calculating arrival times. As a particle flies toward a detector, its wave function evolves according to the Schrödinger equation, which describes a particle’s changing state over time. Consider the one-dimensional case of a detector that is a certain horizontal distance from an emission source. The Schrödinger equation determines the wave function of the particle and hence the probability of detecting that particle at that location, assuming that the particle crosses the location only once (there is, of course, no clear way to substantiate this assumption in standard quantum mechanics). Using such assumptions, physicists can calculate the probability that the particle will arrive at the detector at a given time (t) or earlier. “From the perspective of standard quantum mechanics, it sounds perfectly fine,” Aristarhov says. “And you expect to have a nice answer from that.” There is a hitch, however. To go from the probability that the arrival time is less than or equal to t to the probability that it is exactly equal to tinvolves calculating a quantity that physicists call the quantum flux, or quantum probability current—a measure of how the probability of finding the particle at the detector location changes with time. This works well, except that, at times, the quantum flux can be negative even though it is hard to find wave functions for which the quantity becomes appreciably negative. But nothing “prohibits this quantity from being negative,” Aristarhov says. “And this is a disaster.” A negative quantum flux leads to negative probabilities, and probabilities can never be less than zero. Using the Schrödinger evolution to calculate the distribution of arrival times only works when the quantum flux is positive—a case that, in the real world, only definitively exists when the detector is in the “far field,” or at a considerable distance from the source, and the particle is moving freely in the absence of potentials. When experimentalists measure such far-field arrival times, both the hybrid and quantum flux approaches make similar predictions that tally well with experimental findings. But they do not make clear predictions for “near field” cases, where the detector is very close to the source. Dissatisfied with this flawed status quo, in 2018 Das and Aristarhov, along with their then Ph.D. adviser Detlef Dürr, an expert on Bohmian mechanics at LMU who died earlier this year, and their colleagues, began working on Bohmian-based predictions of arrival times. Bohm’s theory holds that each particle is guided by its wave function. Unlike standard quantum mechanics, in which a particle is considered to have no precise position or momentum prior to a measurement—and hence no trajectory—particles in Bohmian mechanics are real and have squiggly trajectories described by precise equations of motion (albeit ones that differ from Newton’s equations of motion). Among the researchers’ first findings was that far-field measurements would fail to distinguish between the predictions of Bohmian mechanics and those of the hybrid or quantum flux approaches. This is because, over large distances, Bohmian trajectories become straight lines, so the hybrid semi-classical approximation holds. Also, for straight far-field trajectories, the quantum flux is always positive, and its value is predicted exactly by Bohmian mechanics. “If you put a detector far enough [away], and you do Bohmian analysis, you see that it coincides with the hybrid approach and the quantum flux approach,” Aristarhov says. The key, then, is to do near-field measurements, but those have been considered impossible. “The near-field regime is very volatile. It’s very sensitive to the initial wave function shape you have created,” Das says. Also, “if you come very close to the region of initial preparation, the particle will just be detected instantaneously. You cannot resolve [the arrival times] and see the differences between this prediction and that prediction.” To avoid this problem, Das and Dürr proposed an experimental setup that would allow particles to be detected far away from the source while still generating unique results that could distinguish the predictions of Bohmian mechanics from those of the more standard methods. Conceptually, the team’s proposed setup is rather simple. Imagine a waveguide—a cylindrical pathway that confines the motion of a particle (an optical fiber is such a waveguide for photons of light, for example). On one end of the waveguide, prepare a particle—ideally an electron or some particle of matter—in its lowest energy, or ground, state and trap it in a bowl-shaped electric potential well. This well is actually the composite of two adjacent potential barriers that collectively create the parabolic shape. If one of the barriers is switched off, the particle will still be blocked by the other that remains in place, but it is free to escape from the well into the waveguide. Das pursued the painstaking task of fleshing out the experiment’s parameters, performing calculations and simulations to determine the theoretical distribution of arrival times at a detector placed far away from a source along a waveguide’s axis. After a few years of work, he had obtained clear results for two different types of initial wave functions associated with particles such as electrons. Each wave function can be characterized by something called its spin vector. Imagine an arrow associated with the wave function that can be pointing in any direction. The team looked at two cases: one in which the arrow points along the axis of the waveguide and another in which it is perpendicular to that axis. The team showed that, when the wave function’s spin vector is aligned along the waveguide’s axis, the distribution of arrival times predicted by the quantum flux method and by Bohmian mechanics are identical. But they differ significantly from the hybrid approach. When the spin vector is perpendicular, however, the distinctions become starker. With help from their LMU colleague Markus Nöth, the researchers showed that all the Bohmian trajectories will strike the detector at or before this cutoff time. “This was very unexpected,” Das says. Again, the Bohmian prediction differs significantly from the predictions of the semi-classical hybrid theory, which do not exhibit such a sharp arrival-time cutoff. And crucially, in this scenario, the quantum flux is negative, meaning that calculating arrival times using Schrödinger evolution becomes impossible. The standard quantum theorists “put their hands up when [the quantum flux] becomes negative,” Das says. Quantum theorist Charis Anastopoulos of the University of Patras in Greece, an expert on arrival times, who was not involved with this work, is both impressed and circumspect. “The setup they are proposing seems plausible,” he says. And because each approach to calculating the distribution of arrival times involves a different way of thinking about quantum reality, a clear experimental finding could jolt the foundations of quantum mechanics. “It will vindicate particular ways of thinking. So in this way, it will have some impact,” Anastopoulos says. “If it [agrees with] Bohmian mechanics, which is a very distinctive prediction, this would be a great impact, of course.” At least one experimentalist is gearing up to make the team’s proposal a reality. Before Dürr’s death, Ferdinand Schmidt-Kaler of the Johannes Gutenberg University Mainz in Germany had been in discussions with him about testing arrival times. Schmidt-Kaler is an expert on a type of ion trap in which electric fields are used to confine a single calcium ion. An array of lasers is used to cool the ion to its quantum ground state, where the momentum and position uncertainties of the ion are at their minimum. The trap is a three-dimensional bowl-shaped region created by the combination of two electric potentials; the ion sits at the bottom of this “harmonic” potential. Switching off one of the potentials creates conditions similar to what is required by the theoretical proposal: a barrier on one side and a sloping electric potential on the other side. The ion moves down that slope, accelerates and gains velocity. “You can have a detector outside the trap and measure the arrival time,” Schmidt-Kaler says. “That is what made it so attractive.” For now, his group has done experiments in which the researchers eject the ion out of its trap and detect it outside. They showed that the time of flight is dependent on a particle’s initial wave function. The results were published in New Journal of Physics this year. Schmidt-Kaler and his colleagues have also performed not yet published tests of the ion exiting the trap only to be reflected back in by an “electric mirror” and recaptured—a process the setup achieves with 98 percent efficiency, he says. “We are underway,” Schmidt-Kaler says. “Of course, it is not tuned to optimize this measurement of the time of flight distribution, but it could be.” That is easier said than done. The detector outside the ion trap will likely be a sheet of laser light, and the team will have to measure the ion’s interaction with the light sheet to nanosecond precision. The experimentalists will also need to switch off one half of the harmonic potential with similar temporal precision—another serious challenge. These and other pitfalls abound on the tortuous path that must be traversed between theoretical prediction and experimental realization. Still, Schmidt-Kaler is excited about the prospects of using time-of-flight measurements to test the foundations of quantum mechanics. “This has the attraction of being completely different from other [kinds of] tests. It really is something new,” he says. “This will go through many iterations. We will see the first results, I hope, in the next year. That’s my clear expectation.” Meanwhile Aristarhov and Das are reaching out to others, too. “We really hope that the experimentalists around the world notice our work,” Aristarhov says. “We will join forces to do the experiments.” And a conclusion written by Dürr in a yet to be published paper features final words that could almost be an epitaph: “It should be clear by now that the chapter on time measurements in quantum physics can only be written if genuine quantum mechanical time-of-flight data become available,” he wrote. Which theory will the experimental data pick out as correct—if any? “It’s a very exciting question,” Dürr added. Link Original: A unique brain signal may be the key to human intelligence Though progress is being made, our brains remain organs of many mysteries. Among these are the exact workings of neurons, with some 86 billion of them in the human brain. Neurons are interconnected in complicated, labyrinthine networks across which they exchange information in the form of electrical signals. We know that signals exit an individual neuron through a fiber called an axon, and also that signals are received by each neuron through input fibers called dendrites. Understanding the electrical capabilities of dendrites in particular — which, after all, may be receiving signals from countless other neurons at any given moment — is fundamental to deciphering neurons’ communication. It may surprise you to learn, though, that much of everything we assume about human neurons is based on observations made of rodent dendrites — there’s just not a lot of fresh, still-functional human brain tissue available for thorough examination. For a new study published January 3 in the journal Science, however, scientists got a rare chance to explore some neurons from the outer layer of human brains, and they discovered startling dendrite behaviors that may be unique to humans, and may even help explain how our billions of neurons process the massive amount of information they exchange. A puzzle, solved? Electrical signals weaken with distance, and that poses a riddle to those seeking to understand the human brain: Human dendrites are known to be about twice as long as rodent dendrites, which means that a signal traversing a human dendrite could be much weaker arriving at its destination than one traveling a rodent’s much shorter dendrite. Says paper co-author biologist Matthew Larkum of Humboldt University in Berlin speaking to LiveScience, “If there was no change in the electrical properties between rodents and people, then that would mean that, in the humans, the same synaptic inputs would be quite a bit less powerful.” Chalk up another strike against the value of animal-based human research. The only way this would not be true is if the signals being exchanged in our brains are not the same as those in a rodent. This is exactly what the study’s authors found. The researchers worked with brain tissue sliced for therapeutic reasons from the brains of tumor and epilepsy patients. Neurons were resected from the disproportionately thick layers 2 and 3 of the cerebral cortex, a feature special to humans. In these layers reside incredibly dense neuronal networks. Without blood-borne oxygen, though, such cells only last only for about two days, so Larkum’s lab had no choice but to work around the clock during that period to get the most information from the samples. “You get the tissue very infrequently, so you’ve just got to work with what’s in front of you,” says Larkum. The team made holes in dendrites into which they could insert glass pipettes. Through these, they sent ions to stimulate the dendrites, allowing the scientists to observe their electrical behavior. In rodents, two type of electrical spikes have been observed in dendrites: a short, one-millisecond spike with the introduction of sodium, and spikes that last 50- to 100-times longer in response to calcium. In the human dendrites, one type of behavior was observed: super-short spikes occurring in rapid succession, one after the other. This suggests to the researchers that human neurons are “distinctly more excitable ” than rodent neurons, allowing them to successfully traverse our longer dendrites. In addition, the human neuronal spikes — though they behaved somewhat like rodent spikes prompted by the introduction of sodium — were found to be generated by calcium, essentially the opposite of rodents. An even bigger surprise The study also reports a second major finding. Looking to better understand how the brain utilizes these spikes, the team programmed computer models based on their findings. (The brains slices they’d examined could not, of course, be put back together and switched on somehow.) The scientists constructed virtual neuronal networks, each of whose neurons could could be stimulated at thousands of points along its dendrites, to see how each handled so many input signals. Previous, non-human, research has suggested that neurons add these inputs together, holding onto them until the number of excitatory input signals exceeds the number of inhibitory signals, at which point the neuron fires the sum of them from its axon out into the network. However, this isn’t what Larkum’s team observed in their model. Neurons’ output was inverse to their inputs: The more excitatory signals they received, the less likely they were to fire off. Each had a seeming “sweet spot” when it came to input strength. What the researchers believe is going on is that dendrites and neurons may be smarter than previously suspected, processing input information as it arrives. Mayank Mehta of UC Los Angeles, who’s not involved in the research, tells LiveScience, “It doesn’t look that the cell is just adding things up — it’s also throwing things away.” This could mean each neuron is assessing the value of each signal to the network and discarding “noise.” It may also be that different neurons are optimized for different signals and thus tasks. Much in the way that octopuses distribute decision-making across a decentralized nervous system, the implication of the new research is that, at least in humans, it’s not just the neuronal network that’s smart, it’s all of the individual neurons it contains. This would constitute exactly the kind of computational super-charging one would hope to find somewhere in the amazing human brain. Link Original: Liquid’ light shows social behaviour Could photons, light particles, really condense? And how will this «liquid light» behave? Condensed light is an example of a Bose-Einstein condensate: The theory has been there for 100 years, but University of Twente researchers have now demonstrated the effect even at room temperature. For this, they created a micro-size mirror with channels in which photons actually flow like a liquid. In these channels, the photons try to stay together as group by choosing the path that leads to the lowest losses, and thus, in a way, demonstrate «social behavior.» The results are published in Nature Communications. A Bose-Einstein condensate (BEC) is typically a sort of wave in which the separate particles can not be seen anymore: There is a wave of matter, a superfluid that typically is formed at temperatures close to absolute zero. Helium, for example, becomes a superfluid at those temperatures, with remarkable properties. The phenomenon was predicted by Albert Einstein almost 100 years ago, based on the work of Satyendra Nath Bose; this state of matter was named for the researchers. One type of elementary particle that can form a Bose-Einstein condensate is the photon, the light particle. UT researcher Jan Klärs and his team developed a mirror structure with channels. Light traveling through the channels behaves like a superfluid and also moves in a preferred direction. Extremely low temperatures are not required in this case, and it works at room temperature. The structure is the well-known Mach-Zehnder interferometer, in which a channel splits into two channels, and then rejoins again. In such interferometers, the wave nature of photons manifests, in which a photon can be in both channels at the same time. At the reunification point, there are now two options: The light can either take a channel with a closed end, or a channel with an open end. Jan Klärs and his team found that the liquid decides for itself which path to take by adjusting its frequency of oscillation. In this case, the photons try to stay together by choosing the path that leads to the lowest losses—the channel with the closed end. You could call it «social behavior,» according to researcher Klärs. Other types of bosons, like fermions, prefer staying separate. The mirror structure somewhat resembles that of a laser, in which light is reflected back and forth between two mirrors. The major difference is in the extremely high reflection of the mirrors: 99.9985 percent. This value is so high that photons don’t get the chance to escape; they will be absorbed again. It is in this stadium that the photon gas starts taking the same temperature as room temperature via thermalization. Technically speaking, it then resembles the radiation of a black body: Radiation is in equilibrium with matter. This thermalization is the crucial difference between a normal laser and a Bose-Einstein condensate of photons. In superconductive devices at which the electrical resistance becomes zero, Bose-Einstein condensates play a major role. The photonic microstructures now presented could be used as basic units in a system that solves mathematical problems like the Traveling Salesman problem. But primarily, the paper shows insight into yet another remarkable property of light. Link Original:
0ebec3f1b2677f72	Structure of Atom Class 11 Notes 8 minute read Structure of Atom Class 11 Notes All matter in the universe is made up of atoms – it is the building block of everything. For students of Science stream, knowledge of atoms, its structure and constituent particles is of utmost importance. The chapter of class 11 on Structure of an Atom, which is part of the class 11 chemistry syllabus, thus aims to familiarise students with this topic. Struggling with this chapter? Here are some easy to understand Structure of Atom class 11 notes that will help you in grasping all the concepts in this chapter! What are Subatomic Particles? The word atom is derived from the Greek word “a-tomio”, meaning “non-divisible”. Atoms are the most crucial thing in building blocks of matter, and therefore it is required to understand the structure of the atom and its essential parts i.e., protons, neutrons, and electrons. The chapter of class 11 on the structure of an atom covers all the essential topics in detail. In the year 1850, Michael Faraday experimented with studying the electrical discharge in cathode ray discharge tubes. Under this experiment, the pressure of different gases was adjusted with glass tubes’ help, and a high voltage was applied across the electrodes. As a result, the current started flowing from the negative electrode to the positive electrode, and it was called cathode ray particles. The experiment showed that cathode rays’ behaviour was like that of negatively charged particles and concluded that cathode rays consist of negatively charged particles called electrons. Thus, it was derived that electrons are the primary constituent of all the atoms. You can refer to the chapter of class 11 on the structure of an atom for further clarification. Respiration in Plants Class – 11 Notes Protons and Neutrons An electrical discharge in the cathode ray tube led to the discovery of canal rays that were positively charged. These positively charged particles had multiple fundamental electrical charge units and were behaving opposite to that of cathode rays. Thus, the lightest and smallest positive ion obtained from hydrogen was called a proton, and those particles that had mass higher than that of protons was called neutrons. You can also refer to the chapter of class 11 on structure of an atom for getting further clarification on this topic. Also Read: Class 11 Hydrogen Notes Atomic Models Different atomic models were proposed by different scientists to explain charged particles’ movement and behaviour in an atom. While some models explained the stability of atoms, some failed to do so. It is necessary to understand the behaviour of elements in terms of physical and chemical properties and how electromagnetic radiations are emitted and absorbed by the atoms. Let us have a look at some popular Atomic Models covered in Structure of Atom class 11 notes. Thomson Model of Atom In 1898, scientist J.J. Thomson proposed that an atom has a spherical shape where the positive charge is distributed uniformly. Because of this, this model was also called plum pudding, raisin pudding, or watermelon model. This model visualized the watermelon of a positive charge that had seeds like electrons embedded into it. J.J. Thomson’s atom model explained the overall neutrality of the atom while detailing how an atom is uniformly distributed over the other atom. Go through the chapter of class 11 on the structure of an atom for more details. What is Charge to Mass Ratio of Electron? By using the cathode ray tube, British physicist J.J. Thomson applied the electrical and magnetic field perpendicular to each other to find the ratio of electrical charge to the mass of the electron. During the experiment, he finds out that it is possible to bring back the electron to the predefined path by balancing the electrical and magnetic field strength. He also concluded that the mass of the particle affects the electrical and magnetic field. Class 11 States of Matter Study Notes and Important Questions Rutherford’s Nuclear Model of Atom Scientist Rutherford and his students conducted the alpha particle scattering experiment by directing the high energy alpha particles from a radioactive source to a thin gold metal foil. When the alpha particles were struck on the fluorescent zinc sulphide screen around the gold foil, it produced a tiny flash of light. The experiment showed that most of the alpha particles left undeflected, and only a small fraction of particles were deflected. Rutherford concluded that most of the atom’s space is not utilized and left empty, and the electrostatic force of attraction holds the nucleus and electrons. You can also refer to the chapter of class 11 on structure of an atom for more details. What is Atomic Number and Mass Number? A nucleus has a positive charge because of protons, and the total number of protons in an atom is known as the atomic number. It is essential because it is unique for every atom in an element. To calculate an atomic number, calculate the number of protons in the nucleus of an atom or the number of electrons present in a neutral atom. A nucleon is a combination of protons and neutrons present in the nucleus. The total number of nucleons in an atom is called a mass number of the atom. The mass number equals the number of protons and neutrons. What are Isobars and Isotopes? Isobars are the atoms with similar mass numbers but have a different atomic number, while Isotopes have the same proton numbers, but different neutron numbers. The difference between the isotopes happens because of the presence of different neutron numbers present in the nucleus. The chapter of class 11 on the structure of an atom provides comprehensive knowledge on these topics. Developments leading to Bohr’s Model of Atom Wave Nature of Electromagnetic Radiation Thermal radiations consist of electromagnetic waves that vary in frequency and wavelength. In 1870, James Maxwell gave a comprehensive explanation that other electrical and magnetic fields are produced when electrically charged particles are moved with acceleration. These magnetic fields, when transmitted in the form of a wave, are called electromagnetic radiation or electromagnetic waves. Unlike the other waves that require a medium to move, electromagnetic waves do not require any medium and can move in a vacuum. When these electromagnetic radiations differ in terms of wavelengths, they form the electromagnetic spectrum. The chapter of class 11 on the structure of an atom covers all the topics about electromagnetic waves and radiations. Also Read: Class 11 Redox Reactions Planck’s Quantum Theory When an ideal body uniformly emits and absorbs the radiation of all the frequencies, it is called a black body and the radiation from such a body is called as black body radiation. German Physicist Max Planck in 1900 given the first concrete explanation on how black body radiation occurs and its effect. Based on this, he suggested that atoms and molecules can emit and absorb the energy in discrete quantities and not necessarily in a continuous manner. He named the smallest quantity of the energy as a quantum that can emit and absorb the energy in the form of electromagnetic radiation and named the theory as Planck’s Quantum theory. What is the Photoelectric Effect? The photoelectric effect is an experiment performed by scientist H. Hertz in 1887. In the experiment, the electrons were ejected when metals like potassium, rubidium, caesium, etc. were exposed to a light beam. According to this experiment, there was no time lag between the striking of the light beam and the ejection of the electrons. The number of electrons ejected was proportional to the brightness of the light. Thus, a photoelectric effect shows the emission of electrons when it gets hit by electromagnetic radiation. Also Read: Class 11 Chapter 10 S Block Elements Bohr’s Model The first person to explain the general features of the structure of an atom of hydrogen quantitatively was Neils Bohr. He developed this model in 1913, using the concept of quantisation of energy developed by Planck. The model is based on a structure that is similar to the solar system – the electrons move around a dense nucleus in circular paths or orbits. it also mentions that the energy of an electron does not undergo any changes although it may move from a lower to a higher stationary state or vice versa. Developments towards Quantum Mechanical Model of Atom The shortcomings of the bohr’s model necessitated the development towards a more acceptable atomic model. Some important developments in this regard were as follows. Dual Behavior Of Matter In 1924, de Broglie, a French physicist proposed that just like radiation, matter should also display dual behavior – both particle and wave-like properties. it meant that electrons should also have both momentum and wavelength, just like photons. This led to him developing a formula of the relations between wavelength and momentum of any particle of matter. Heisenberg Uncertainty Principle As a consequence of radiation and the dual behavior of matter, in 1927 a German physicist Werner Heisenberg developed his uncertainty principle. This principle states that it is not possible to simultaneously determine the exact momentum and exact position of an electron. Quantum Mechanical Model of Atom Quantum mechanics is the branch of science that takes into account the dual behavior of matter. It deals with studying the motions of microscopic objects that possess both particle-like and wave-like properties. The quantum mechanical model of the atom is developed through the application of the Schrödinger equation to the atoms. Some of the main features of this model include: • The energy of electrons within an atom is limited to certain specific values • The quantized energy level is a result of the wavelike properties of the electrons • The exact position and exact momentum of an electron cannot be determined simultaneously Structure of Atom- Practice Questions The chapter of class 11 on the structure of an atom is relatively easy if you work on practice questions as part of your preparation. Below are some vital question sets for practice: 1. What is Bohr’s model for hydrogen atoms? 2. What does the negative electronic energy for hydrogen atoms mean? 3. What is the difference between electromagnetic radiations and waves? Structure of Atom Class 11 Notes – Solved Questions Question 1: How many protons and neutrons are there in the nuclei 13C6? Carbon mass number = 13 => atomic number of carbons = no. of proton in one atom of carbon = 6 => total number of neutrons in 1 atom of carbon = mass number – atomic number => 13-6 = 7 Question 2: Write the possible values of n, l, and ml, if the electron is in one of the 3d orbitals. Value of n = 3, Value of l = 2, and Value of ml = -2, -1, 0, 1, 2 You can go through the chapter of class 11 on the structure of an atom for more practice questions. Were these comprehensive structure of atom class 11 notes helpful? Let us know in the comment section below. If you are confused about which course to pursue after your 12th boards, reach out to our experts at Leverage Edu who can help you identify the best courses in the Science stream. Sign up for a free 30-minute counselling session today! Leave a Reply Required fields are marked * Talk to an expert
86d607af2c29092c	7+ Things You Might Not Know about Mathematician Terence Tao 7+ Things You Might Not Know about Mathematician Terence Tao We are searching data for your request: Forums and discussions: Manuals and reference books: Data from registers: Wait the end of the search in all databases. Terence Tao might just be one of the greatest mathematicians of our time. His work has garnered him many awards and recognition amongst his peers and math maniacs around the world. We briefly explore the man's life and highlight some of his main contributions to math throughout his career. Who is Terence Tao? Terence Tao is an Australian-American mathematician best known for his enormous contributions to the field of mathematics. He is both the Fellow of the Australian Academy of Science and the Fellow of the Royal Society. He has been interested in math from a very young age. His most notable contributions to math include, but are not limited to: • 'Green-Tao theorem.' • 'Tao’s inequality.' • 'Kakeya Conjecture' • 'Horn Conjecture' For his work, Tao has collected various awards throughout his career and is also a published author. He currently focuses on several branches of mathematics, including geometric combinatorics, harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinators, compressed sensing, and analytic number theory. He currently works at the Department of Mathematics at UCLA and is also a member of the so-called "Analysis Group" at UCLA. He also edits various mathematical journals. What is Terence Tao famous for? Terence Tao is a very accomplished mathematician whose work is characterized by his high-degree of originality. He also regularly crosses many research boundaries with an innate ability to collaborate with specialists in other fields. Terence's main work is on the theory of partial differential equations. These are the principle equations that tend to be used in mathematical physics, among other fields. By way of example, the non-linear Schrödinger equation is vital for modeling light transmission in fiber optic cables. Interestingly, despite their widespread real-world application, it is often very hard to prove that such equations have provable solutions or that they have the required properties. Tao's work, along with his peers' on concepts such as nonlinear equations, established crucial existence theorems. He has also conducted important work on waves that have been applied to gravitational waves, as predicted in Einstein's Theory of General Relativity. Terence Tao has also been able to show, along with British mathematician Ben Green, that a set of prime numbers appear to contain arithmetic progressions of any length. Terence is a prolific mathematician and authored or co-authored over 275 research papers. One of his most notable works is his results on 3-D Navier-Stokes existence and smoothness. For his work, he has received many important awards, including the Salem Prize and an American Mathematical Society Bocher Memorial Prize. He is the second-ever ethnic Chinese mathematician to win the International Mathematical Union's Fields medal after Shing-Tung Yau. He is also the first Australian mathematician to win the medal. Where does Terence Tao teach, and what is he working on? Terence Tao currently works as a Professor at the Department of Mathematics at UCLA. "[He works] in a number of mathematical areas, but primarily in harmonic analysis, PDE, geometric combinatorics, arithmetic combinatorics, analytic number theory, compressed sensing, and algebraic combinatorics. [He is] part of the Analysis Group at UCLA, and also an editor or associate editor at several mathematical journals." - Terence Tao's biography, UCLA. Where is Terence Tao from? Terence Tao was born in Adelaide, Australia on the 17th of July 1975. His father, Dr. Billy Tao, worked as a pediatrician and was born in Shanghai, China, and earned his medical degree at the University of Hong Kong in 1969. His mother, Grace, was born in Hong Kong and received a first-class honors degree in physics and math from the University of Hong Kong. His parents met at the same university. Prior to leaving China, his mother worked as a secondary school teacher in math and physics in Hong Kong. In 1972, Terence's parents emigrated to Australia. After completing his undergraduate studies in his mid-teens, Terence emigrated to the United States in 1992 to undertake his Ph.D. at Princeton University. Tao's wife, Laura Tao, has worked as an electrical engineer at NASA's Jet Propulsion Laboratory. They have one son, William, and a daughter Madeleine, and the family lives in Los Angeles, California. Some interesting takeaway facts about Terence Tao Here are some interesting facts about mathematician Terence Tao. 1. Terence was born on the 17th of July 1975 in Adelaide, Australia. Terence's father, Dr. Billy Tao, and his mother, Grace, lived in Hong Kong before they as a family emigrated to Australia in 1972. Terence is the oldest of their three children, including his two brothers, Trevor and Nigel. 2. When Terence was only two years old, his parents noticed he was very different from other children of his age. By the age of 5, he was able to teach other children how to spell and complete basic addition problems. When asked how he knew the things he was teaching, he replied that he learned them from Sesame Street on TV. 3. When Terence was eight years old, he attended Blackwood High School in Adelaide. It was not uncommon to find him, even then, poring over hardback books and mathematics, on subjects like Calculus. 4. By the age of 11, Terence was attending classes at Flinders University in Adelaide, along with his normal classes at Blackwood High School. Terence was taught by Professor Garth Gaudry at Flinders. 5. Terence was able to complete his very first research paper by the age of 15. He would later complete his Bachelor's degree with Honors in 1991 and a Master's degree in 1992 at Flinders University. His Master's thesis was on the "Convolution operators generated by right-monogenic and harmonic kernels." For his extraordinary skills, he was awarded the University Medal at Flinders University and a Fulbright Postgraduate Scholarship. Both of these helped him win a place to undertake research in the U.S. 6. Terence completed his Ph.D. at the tender age of 21. He soon joined the University of California faculty and by the age of 24, had become a full-time professor at UCLA. This made him the youngest person to ever do so. 7. In 2007, Tao was nominated for the Australian of the Year. He also published his 'Tao's inequality' principle the same year. Watch the video: 97. Why is Terrence Tao The Worlds Best Mathematician? (July 2022). 1. Bartalan 2. Branson Let's check it out ... 3. Boukra Exactly the messages 4. Jakeem Your idea is simply excellent 5. Cenon what touching phrase :) 6. Reagan Bookmarked it. Write a message
61a9049f55e06ab6	Field of Science Spooky factions at a distance For me, a highlight of an otherwise ill-spent youth was reading mathematician John Casti’s fantastic book “Paradigms Lost“. The book came out in the late 1980s and was gifted to my father who was a professor of economics by an adoring student. Its sheer range and humor had me gripped from the first page. Its format is very unique – Casti presents six “big questions” of science in the form of a courtroom trial, advocating arguments for the prosecution and the defense. He then steps in as jury to come down on one side or another. The big questions Casti examines are multidisciplinary and range from the origin of life to the nature/nurture controversy to extraterrestrial intelligence to, finally, the meaning of reality as seen through the lens of the foundations of quantum theory. Surprisingly, Casti himself comes down on the side of the so-called many worlds interpretation (MWI) of quantum theory, and ever since I read “Paradigms Lost” I have been fascinated by this analysis. So it was with pleasure and interest that I came across Sean Carroll’s book that also comes down on the side of the many worlds interpretation. The MWI goes back to the very invention of quantum theory by pioneering physicists like Niels Bohr, Werner Heisenberg and Erwin Schrödinger. As exemplified by Heisenberg’s famous uncertainty principle, quantum theory signaled a striking break with reality by demonstrating that one can only talk about the world only probabilistically. Contrary to common belief, this does not mean that there is no precision in the predictions of quantum mechanics – it’s in fact the most accurate scientific framework known to science, with theory and experiment agreeing to several decimal places – but rather that there is a natural limit and fuzziness in how accurately we can describe reality. As Bohr put it, “physics does not describe reality; it describes reality as subjected to our measuring instruments and observations.” This is actually a reasonable view – what we see through a microscope and telescope obviously depends on the features of that particular microscope or telescope – but quantum theory went further, showing that the uncertainty in the behavior of the subatomic world is an inherent feature of the natural world, one that doesn’t simply come about because of uncertainty in experimental observations or instrument error. At the heart of the probabilistic framework of quantum theory is the wave function. The wave function is a mathematical function that describes the state of the system, and its square gives a measure of the probability of what state the system is in. The controversy starts right away with this most fundamental entity. Some people think that the wave function is “epistemic”, in the sense that it’s not a real object and is simply related to our knowledge – or our ignorance – of the system. Others including Carroll think it’s “ontological”, in the sense of being a real entity that describes features of the system. The fly in the ointment concerns the act of actually measuring this wave function and therefore the state of a quantum system, and this so-called “measurement problem” is as old as the theory itself and kept even the pioneers of quantum theory awake. The problem is that once a quantum system interacts with an “observer”, say a scintillation screen or a particle accelerator, its wave function “collapses” because the system is no longer described probabilistically and we know for certain what it’s like. But this raises two problems: Firstly, how do you exactly describe the interaction of a microscopic system with a macroscopic object like a particle accelerator? When exactly does the wave function “collapse”, by what mechanism and in what time interval? And who can collapse the wave function? Does it need to be human observers for instance, or can an ant or a computer do it? What can we in fact say about the consciousness of the entity that brings about its collapse? The second problem is that contrary to popular belief, quantum theory is not just a theory of the microscopic world – it’s a theory of everything except gravity (for now). This led Erwin Schrödinger to postulate his famous cat paradox which demonstrated the problems inherent in the interpretation of the theory. Before measurement, Schrödinger said, a system is deemed to exist in a superposition of states while after measurement it exists only in one; does this mean that macroscopic objects like cats also exist in a superposition of entangled states, in case of his experiment in a mixture of half dead-half alive states? The possibility bothered Schrödinger and his friend Einstein to no end. Einstein in particular refused to believe that quantum theory was the final word, and there must be “hidden variables” that would allow us to get rid of the probabilities if only we knew what they were; he called the seemingly instantaneous entanglement of quantum states “spooky action at a distance”. Physicist John Bell put that particular objection to rest in the 1960s, proving that at least local quantum theories could not be based on hidden variables. Niels Bohr and his group of followers from Copenhagen were more successful in their publicity campaign. They simply declared the question of what is “real” before measurement irrelevant and essentially pushed the details of the measurement problem under the rug by saying that the act of observation makes something real. The cracks were evident even then – the physicist Robert Serber once pointedly pointed out problems with putting the observer on a pedestal by asking if we might regard the Big Bang unreal because there were no observers back then. But Bohr and his colleagues were widespread and rather zealous, and most attempts by physicists like Einstein and David Bohm met with either derision or indifference. Enter Hugh Everett who was a student of John Wheeler at Princeton. Everett essentially applied Occam’s Razor to the problem of collapse and asked a provocative question: What are the implications if we simply assume that the wave function does not collapse? While this avoids asking about the aforementioned complications with measurement, it creates problems of its own since we know for a fact that we can observe only one reality (dead vs alive cat, an electron track here rather than there) while the wave function previously described a mixture of realities. This is where Everett made a bold and revolutionary proposal, one that was as courageous as Einstein’s proposal of the constancy of the speed of light: he surmised that when there is a measurement, the other realities encoded in the wavefunction split off from our own. They simply don’t collapse and are every bit as real as our own. Just like Einstein showed in his theory of relativity that there are no privileged observers, Everett conjectured that there are no privileged observer-created realities. This is the so-called many-worlds interpretation of quantum mechanics. Everett proposed this audacious claim in his PhD thesis in 1957 and showed it to Wheeler. Wheeler was an enormously influential physicist, and while he was famous for outlandish ideas that influenced generations of physicists like Richard Feynman and Kip Thorne, he was also a devotee of Bohr’s Copenhagen school – he and Bohr had published a seminal paper explaining nuclear fission way back in 1939, and Wheeler regarded Bohr’s Delphic pronouncements akin to those of Confucius – that posited observer-generated reality. He was sympathetic to Everett but could not support him in the face of Bohr’s objections. Everett soon left theoretical physics and spent the rest of his career doing nuclear weapons research, a chain-smoking, secretive, absentee father who dropped dead of an unhealthy lifestyle in 1982. After a brief resurrection by Everett himself at a conference organized by Wheeler, many-worlds didn’t see much popular dissemination until writers like Casti and the physicist David Deutsch wrote about it. As Carroll indicates, the MWI has a lot of things going for it. It avoids the prickly, convoluted details of what exactly constitutes a measurement and the exact mechanism behind it; it does away with especially thorny details of what kind of consciousness can collapse a wavefunction. It’s elegant and satisfies Occam’s Razor because it simply postulates two entities – a wave function and a Schrödinger equation through which the wave function evolves through time, and nothing else. One can calculate the likelihood of each of the “many worlds” by postulating a simple rule proposed by Max Born that assigns a weight to every probability. And it also avoids an inconvenient split between the quantum and the classical world, treating both systems quantum mechanically. According to the MWI, when an observer interacts with an electron, for instance, the observer’s wave function becomes entangled with the electron’s and continues to evolve. The reason why we still see only one Schrödinger’s cat (dead or alive) is because each one is triggered by distinct random events like the passage of photons, leading to separate outcomes. Carroll thus sees many-worlds as basically a logical extension of the standard machinery of quantum theory. In fact he doesn’t even see the many worlds as “emerging” (although he does see them as emergent); he sees them as always present and intrinsically encoded in the wave function’s evolution through the Schrödinger equation. A scientific theory is of course only as good as its experimental predictions and verification – as a quote ascribed to Ludwig Boltzmann puts it, matters of elegance should be left to the tailor and the cobbler. Does MWI postulate elements of reality that are different from those postulated by other interpretations? The framework is on shakier ground here since there are no clear observable predictions except those predicted by standard quantum theory that would truly privilege it over others. Currently it seems that the best we can say is that many worlds is consistent with many standard features of quantum mechanics. But so are many other interpretations. To be accepted as a preferred interpretation, a theory should not just be consistent with experiment, but uniquely so. For instance, consider one of the very foundations of quantum theory – wave-particle duality. Wave-particle duality is as counterintuitive and otherworldly as any other concept, but it’s only by postulating this idea that we can ever make sense of disparate experiments verifying quantum mechanics, experiments like the double-slit experiment and the photoelectric effect. If we get rid of wave-particle duality from our lexicon of quantum concepts, there is no way we can ever interpret the results of thousands of experiments from the subatomic world such as particle collisions in accelerators. There is thus a necessary, one-to-one correspondence between wave-particle duality and reality. If we get rid of many-worlds, however, it does not make any difference to any of the results of quantum theory, only to what we believe about them. Thus, at least as of now, many-worlds remains a philosophically pleasing framework than a preferred scientific one. Many-worlds also raises some thorny questions about the multiple worlds that it postulates. Is it really reasonable to believe that there are literally an infinite copies of everything – not just an electron but the measuring instrument that observes it and the human being who records the result – splitting off every moment? Are there copies of me both writing this post and not writing it splitting off as I type these words? Is the universe really full of these multiple worlds, or does it make more sense to think of infinite universes? One reasonable answer to this question is to say that quantum theory is a textbook example of how language clashes with mathematics. This was well-recognized by the early pioneers like Bohr: Bohr was fond of an example where a child goes into a store and asks for some mixed sweets. The shopkeeper gives him two sweets and asks him to mix them himself. We might say that an electron is in “two places at the same time”, but any attempt to actually visualize this dooms us, because the only notion of objects existing in two places is one that is familiar to us from the classical world, and the analogy breaks down when we try to replace chairs or people with electrons. Visualizing an electron spinning on its axis the way the earth spins on its is also flawed. Similarly, visualizing multiple copies of yourself actually splitting off every nanosecond sounds outlandish, but it’s only because that’s the only way for us to make sense of wave functions entangling and then splitting. Ultimately there’s only the math, and any attempts to cast it in the form of everyday language is a fundamentally misguided venture. Perhaps when it comes to talking about these things, we will have to resort to Wittgenstein’s famous quote – whereof we cannot speak, thereof we must be silent (or thereof we must simply speak in the form of pictures, as Wittgenstein did in his famous ‘Tractatus’). The other thing one can say about many-worlds is that while it does apply Occam’s Razor to elegantly postulating only the wave function and the Schrödinger equation, it raises questions about the splitting off process and the details of the multiple worlds that are similar to those about the details of measurement raised by the measurement problem. In that sense it only kicks the can of complex worms down the road, and in that case believing what particular can to open is a matter of taste. As an old saying goes, nature does not always shave with Occam’s Razor. In the last part of the book, Carroll talks about some fascinating developments in quantum gravity, mainly the notion that gravity can emerge through microscopic degrees of freedom that are locally entangled with each other. One reason why this discussion is fascinating is because it connects many disparate ideas from physics into a potentially unifying picture – quantum entanglement, gravity, black holes and their thermodynamics. These developments don’t have much to do with many-worlds per se, but Carroll thinks they may limit the number of “worlds” that many worlds can postulate. But it’s frankly difficult to see how one can find definitive experimental evidence for any interpretation of quantum theory anytime soon, and in that sense Richard Feynman’s famous words, “I think it is safe to say that nobody understands quantum mechanics” may perpetually ring true. Very reasonably, many-worlds is Carroll’s preferred take on quantum theory, but he’s not a zealot about it. He fully recognizes its limitations and discuss competing interpretations. But while Carroll deftly dissects many-worlds, I think that the real value of this book is to exhort physicists to take what are called the foundations of quantum mechanics more seriously. It is an attempt to make peace between different quantum factions and bring philosophers into the fold. There’s a huge number of “interpretations” of quantum theory, some more valid than others, being separated by each other as much by philosophical differences as by physical ones. There was a time when the spectacular results of quantum theory combined with the thorny philosophical problems it raised led to a tendency among physicists to “shut up and calculate” and not worry about philosophical matters. But philosophy and physics have been entwined since the ancient Greeks, and in one sense, one ends where the other begins. Carroll’s book is a hearty reminder for physicists and philosophers to eat at the same table, otherwise they may well remain spooky factions at a distance when it comes to interpreting quantum theory. 1. "whereof we cannot speak of, therefore we must be silent" Good God, what a butchering. 2. Great post!! However, I think the back half of paragraph 10 is incorrect. Wave-particle duality (i.e. deBroglie's pilot wave) is not an absolutely necessary part of the explanation in light of MWI. For example, Deutsch argues that MWI is the more parsimonious explanation of the double-slit experiment: particles in other universes collide with the observable particles in ours resulting in the interference pattern. Or, in other words, there is no duality: the particles in our universe and their counterparts causing the interference constitute a wave, but it is a wave made up of particles in the multiverse. Markup Key: - <b>bold</b> = bold - <i>italic</i> = italic
fa8f584442216d3d	Theoretical Chemistry Prof. Dr. Jürgen Gauß In quantum chemistry, the electronic structure of molecules is investigated based on a quantum mechanical description. In addition, important molecular properties such as energy, structure, etc. are calculated. For chemically relevant molecular systems consisting of a large number of electrons (up to several hundreds of atoms and several thousands of electrons) the relevant equations, i.e., the electronic Schrödinger equation, cannot be solved in closed form. Therefore, one key aspect of quantum chemistry is the develpoment of numerical approximations for the solution of these equations. Important topics that are under study at present include â¢ development of methods to describe electron-correlation effects â¢ design of methods using analytic energy derivatives for the computation of pro-perties â¢ implementation of techniques of response theory for the study of electronic exci-tations â¢ development of methods to treat relativistic effects on molecular properties Apart from these methodological studies the application of quantum-chemical methods to the investigation of chemically relevant problems constitutes the important field of computational chemistry. Examples for such applications are shown below: Calculated structure of a complex aluminum-chloride cluster consisting of 198 atoms and 1106 electrons. Calculated atomic distances in molecules exhibiting hydrogen bonds. The indicated distances vary from 0.1 to 0.3 nanometers.
61989f9ccf07b31d	Dismiss Notice Join Physics Forums Today! Can all differential equations be derived from a variational principle? 1. Jul 3, 2006 #1 In classical mechanics, for conservative systems, it well knows that the differential laws of motion can be derived from a variational principle called "least action principle". I know also that some non-conservative systems can be derived from a variational principle: the damped harmonic oscillator has a time-dependent Lagragian and an associated least action principle. Physically, I wanted to know if: the least action is something special happening in special conditions, and which conditions or if it is a general rule that applies to (nearly) all differential systems of equations Can all differential equations be derived from a variational principle? I would greatly enjoy your ideas, comments suggestion or any track. Examples could be very useful too. Last edited: Jul 3, 2006 2. jcsd 3. Jul 5, 2006 #2 That is a very interesting question. My initial opinion would be that "the least action is something special happening in special conditions, and which conditions," since I think (and this is opinion only, of course) that the variational principle [itex]\delta S = 0[/itex] has some physical content. But someone may just come and prove me wrong. Maybe it is the second one, and that the real trick is to find which quantity we want to minimize in order to get some diff. equations out. It just so happens that for physical situations, it's generally a simple quantity that we want to minimize. 4. Jul 6, 2006 #3 That's indeed the reason why I asked the question here after a short discussion in the CP section. The limit from CM to QM is a well known physical interpretation. The damped oscillator admits a variational principle too, but I don't see the physical interpretation. The Schrödinger equation can also be derived from a variational principle, but what is the physical meaning then? Actually I don't know about some differential equation that could not be derived from a variational principle, at least in physics. (I will ask that in the CP section) If only a small class of DE could be derived from a variational principle, then it would mean that physical laws are really special and this would deserve a further study. If variational principles are (nearly) the rule, they would be no less interresting, but their status would be more comparable to differential equations. Last edited: Jul 6, 2006 5. Jul 6, 2006 #4 Mathematically, I believe that any second order differential equation could be represented as the minimization (or maximization, as is sometimes the case) of an appropriate integral, but I haven't seen a proof of that so I'm not 100%. Whether this is physically significant.....I'm not sure. All differential equations can be represented as integral equations (I'm 100% on that), but humans don't tend to set them up that way. Any given problem can usually be represented a number of different ways, but certain methods tend to take precendence. I've always thought this said more about people than the physical universe. 6. Jul 6, 2006 #5 inverse problem of the calculus of variations I browsed on the net to find some track. I guess the problem I submitted is known and called inverse problem of the calculus of variations Quickly reading here and there suggested me that it is solved only for second-order differential equations (one variable) and maybe some even-order systems (one variable). I don't understand that quite well, since I am thinking essentially first-order and any number of variables. Here is one of the links I found: http://8icdga.math.slu.cz/PDF/425-434.pdf" [Broken] . Any suggestion, comment, reading, ... ? Last edited by a moderator: May 2, 2017 7. Jul 6, 2006 #6 User Avatar Staff Emeritus Science Advisor Gold Member You need to separate the mathematics from the physics. When you say "all differential equations" you are asking a mathematics question. As a pure mathematical concept DEQs have nothing to do with physics or any principle of physics. It is a very happy coincidence that physical properties can be expressed in terms of a well understood mathematical language. It is observed, that in the physical world, systems involving changes in energy evolve in such a way that energy is minimized. This may be a major factor in why the concept of energy is so useful. When asking this type of question you must remember that ALL DQs can be, and are, studied completely independent of Physics. You do not need ANY physical principles to discuss the mathematics of Differential Equations. 8. Jul 7, 2006 #7 I don't agree with your philosophy, at least as I understand it from your post. (I would better assume that mathematics is physics !) Assume it could be proved that all systems of differential equations can be derived from a variational principle, and eventually how to construct a Lagrangian for it. Then the consequence for the interpretation of physics is very important. This would mean the equivalence of these two questions: Why a least action principle, why a variational principle for CM or QM ... ? Why are differential equations at the root of physics ? It is well known that the transition from quantum mechanics to classical mechanics gives a physical interpretation for the least action principle: "the waves do really have the ability to sense all possible trajectories and concentrate on stationary phase path" -so to speak physically- . But is this interpretation really useful if it could be proved that all DEqs could be derived from a variational principle? Actually, would that not indicate somehow the reverse route from CM to QM ! Now consider the Schrödinger equation. It can also be derived from a variational principle. Same for the Maxwell's equations. Many other examples can be considered. Actually, I don't know one physical theory that cannot be derived from a variational principle. Do you? Therefore my question. Maybe all differential equations can be derived from a variational principle! And there is more: the Lagrangian may even not be unique. (see this paper). Again, then a mathematical problem questions the physics. If the relation between two possible Lagrangian is simply related to a symmetry, this already points to some (known) physics. But could the possibilities be broader? Sorry for the philosophy. This was simply to explain my motivation. The question remains: can all DE be derived from a VP ? Last edited: Jul 7, 2006 9. Jul 8, 2006 #8 As a physics-trained engineer, specialising in CFD research, perhaps I could add in a perspective: 1. Physicists use Conservation Laws to derive relationships between physical quantities. 2. A simple method is to use the Continuum Principle, which lends itself to differential equations applying at each point in the continuum. This can be converted to an integral form if so desired. 3. The usual rules of mathematics are then used to analyse these equations. 4. In many cases, numeric methods need to be employed eg. for Navier-Stokes. So, if I were to address the OP's original question, could this be re-couched in terms of an apparent relationship between the Conservation Laws & the Variational Principle? In other words, "Do the Conservation Laws always express a minimisation of 'some (conserved) quantity'?" 10. Jul 9, 2006 #9 User Avatar Staff Emeritus Science Advisor Gold Member Pretty big assumption isn't it? How could you possibly make this sort of assumption then carry on to a conclusion that means anything. Rather then assuming this, please show us how to prove it. 11. Jul 9, 2006 #10 jim mcnamara User Avatar Staff: Mentor IMO - arguing assumptions for math motivation like this one isn't very productive. There has been a dichotomy in Mathematics for a long time. In general, maths evolved to answer physical questions (fluxions, for example), but as early as Euclid, "pure" math also became defined, probably as an exercise in logic. And this is the point of contention here - call it pure vs applied. 12. Jul 9, 2006 #11 Pure vs. applied... That's an excellent way to put things. As a physicist-type, I tend to see Mathematics as a useful toolbox of tricks I can use to go where I need to go along my research trail. When matters of Mathematical purity emerge, then I conveniently find time to move onto the next tool. :shy: 13. Jul 9, 2006 #12 User Avatar Science Advisor Homework Helper we are getting off the topic, towards personal biases. it was interesting until then. there is to me no reason that avariational principle could not have a purely mathematical formulation, and thus make sense in either context. look for example at the proof of existence of solutions of odes by eulers or picards methods. these rewrite it as an integral equation, or integral operator, and take limits. picards point of view is that of an integral operator transforming one functions into another, and the solution is always a function which is "fixed" or transformed into itself. this is surely some kind of minimization principle. think of the gradient principle in calculus, of loking at the gradient of the function and always moving along the gradient. a minimum is a point where this leaves the point fixed, since the gradient there is zero and the point is "stable" for the transformation. think of a marble at the bottom of a bowl. just brainstorming on this very intersting question. i of course am totally ignorant even if what most of the words mean in this topic, but that does not stop me from babbling analogies. thats how i work. 14. Jul 9, 2006 #13 This is also one of the reason I asked for opinions here. Apparently non conservative systems can also derive from a variational principle. For example, the damped oscillator can be derived from a variational principle. But its lagrangian is time dependent (contains an exponential decay). 15. Jun 15, 2007 #14 Why isn´t it as simple as taking for the general PDE F(x,y,z,f_x,f_y,f_z,f_xy,...) = 0 the functional G(F) := Integral(F^2(f)) This is =0 for the solution f of the PDE, >=0 for all other functions (not solving the PDE) and if there were some notion of continuity this could be interpreted as a minimum problem? Of course it would be a trivial solution and not very useful... 16. Jun 15, 2007 #15 lalbatros: I believe you are using logic in a wrong way. As most DE's in physics are derived on the assumption of energy conservation, they admit a variational formulation, but as others pointed out before, the set of DE's you can write down in a piece of paper is far more extense than the set of DE's that have physicall meaning. I am almost sure that not all DE's admit an energy functional, and I am pretty sure that variational methods do not work on all DE's. Let me do a little research with my variational friends and I'll get back to you. As Integral said, if you have a proof of your statemen, by all means show it, as it should be an award winning paper. This post reminded me of a problem that a brilliant professor of mine once talked about: All ODE's can be written as difference equations by means of Fourier methods. Now, do all difference equations come from a ODE? Last edited: Jun 15, 2007 17. Jun 15, 2007 #16 The question posed by lalbatros is actually extremely important. It basically boils down to wether the diff. equations or the action principle are primordial. The question is very important because the fancy theoretical physicists trying to quantize a system usually start by writing an action, implicitly assuming the system has one. What if it doesn't ? I've heard not all differential equations can be derived by varying some action. Suppose you are given an action S. You derive equations from that action by setting the first functional derivatives to zero. The question is: given the equations, is it always possible to find an action which has those equations as functional derivatives. I make the following loose analogy with ordinary calculus. An analogous question in calculus would be: given some functions, is it always possible to find a function with such partial derivatives. The answer in calculus is NO - the given functions must obey certain consistency conditions (the cross derivatives must be equal) to be partials of the same function. By analogy, the answer is NO in the action case. I wonder if anyone has derived the conditions a set of equations has to satisfy to be derivable by an action (maybe the cross functional derivatives must be equal) analogously to the conditions a set of functions have to satisfy to be partials of a single function. I'm reading a book in classical mechanics and there they say that a system with non-holonomic type of constraints can't be described by action cause the constraints can't be integrated to a function and incorporated in the action by lagrange multiplier. It's another story MOST of the systems in physics we KNOW DO have actions. That doesn't mean the systems we don't know will continue to be derivable by actions. Last edited: Jun 15, 2007 18. Jun 15, 2007 #17 I found an article, <http://www.phy.bme.hu/~van/Publ/VanNyi99a.pdf>, [Broken] which states: "...A strict mathematical theorem tells us the condition of the existence of a variational principle for a given differential (or almost any kind of) equation (see for example in [?, ?])..." Unfortunately the authors don´t give the source of this assertion. Probably it is an unfinished pre-print, but I didn´t find a final version with a bibliography. Maybe one of you has? Last edited by a moderator: May 2, 2017
24786facf3145633	Thursday, May 24 2018 \| Updated at 12:08 AM EDT Stay Connected With Us F T R Feb 28, 2017 09:03 AM EST Princeton University Researchers Discover How To Make An Atom Mimic Another Atom's Light Researchers from Princeton University were able to find a way to have an atom mimic the light emissions of another atom. This can help several scientists in a variety of fields. Light from atoms can be measured when their electrons move between energy levels to identify the presence of an element, reported. An example would be astronomers who can detect the presence of argon in a star by recognizing the unique signature of light that it emits. Princeton University researchers discovered that, in specific instances, atoms can be manipulated to impersonate other atoms. They were able to do this by controlling the light that was fired at a given atom to cause it to emit the signature of another atom. One instance would be when they caused an argon atom to emit the same light wavelength as a hydrogen atom by manipulating the pulse of laser light that was fired at the former atom. This technique can also be used to modify the quantum state of the target atom. While the technique of using light to make atoms react to certain things is not new, the use of light to control an atom's state can be utilized for new applications. One way to use this is to have molecules emit different colors to make it easier to identify them in biological processes. The Princeton University researchers published their study in the journal "Physical Review Letters." It is entitled "How to Make Distinct Dynamical Systems Appear Spectrally Identical" by Andre G. Campos, Denys I. Bondar, Renan Cabrera, and Herschel A. Rabitz. The scientists began their study by creating a model that demonstrated how a single pulse of light can cause any one of several atomic states and changing it in a specific way to change the wavelength of light that it produced. The researchers used the Schrödinger equation to calculate which pulse shape would result to the desired wavelength. Join the Conversation Get Our FREE Newsletters Stay Connected With Us F T R Real Time Analytics
448594a76a397f09	Is quantum mechanics deterministic or random? Quantum mechanics is intrinsically random. If we look at a radioactive element, it will decay with a specific half-life time, but there is no way to predict exactly when it is going to decay. It is a random process. On the other hand, the Schrödinger equation that describes the time evolution of a quantum system is fully deterministic. Given well defined initial conditions for the wave-function at some point in time, we can determine the wave-function in any other time, in the future or in the past \Psi(t) = e^{-\frac{i}{\hbar} \int_0^t H(t) dt} \Psi(0) . How can these two statements coexist in a single consistent theory? The answer is that it is just a question of perspective. If you look at the entire wave-function, you would realize that the theory is deterministic. But if you write down the wave-function as a combination of eigen-states, then each state has a probability associated with it depending on its amplitude, according to the Born rule P(x) = \left\|\Psi(x)\right\| , and therefore from the perspective of the eigen-state the system is random. For example, take a spin which is in a superposition of up state and down state \frac{1}{\sqrt{2}} \left ( \left\|\uparrow\right> + \left\|\downarrow\right> \right) . There is a 50% chance that it is an up spin and 50% chance that it is a down spin. This randomness/probability is intrinsic in quantum mechanic. The randomness is there even without measurement, but it is easier to describe when we talk about a detector that made a measurement. Before the measurement the spin and the detector are uncorrelated. \Psi_{\text{initial}} = \frac{1}{2} \left ( \left\|\uparrow\right> + \left\|\downarrow\right> \right) \left ( \left\|\text{detect}\uparrow\right> + \left\|\text{detect}\downarrow\right> \right) . When our detector measures the spin, it also gets into a superposition of states. There is one state in which the spin is up and the detector measured spin up. There is a second state in which the spin is down and the detector measured spin down \Psi_{\text{final}} = \frac{1}{\sqrt{2}} \left ( \left\|\uparrow\right> \left\|\text{detect}\uparrow\right> + \left\|\downarrow\right> \left\|\text{detect}\downarrow\right> \right) . This measurement process is unitary. The wave-function of the entire system after the measurement can be calculated deterministically. Yet from the point of view of the detector, a purely random event occurred. It is impossible to predict what spin will be detected, because in the final state both spins are detected. But from the point of view of the detector, only one spin is detected, and therefore from its perspective it is a purely random event. Moreover, random processes are irreversible. If a ball bounces of a wall at a random angle, there is no way to trace back in time the motion of the ball before it hit the wall. Indeed, from the point of view of the detector, the measurement is irreversible. If you start from a state of a detector that measured spin up, there is no way to trace back time to the original state of the spin before measurement. On the other hand, if you start with the full final state that includes the superposition of the two states, you can reverse time and recover the original spin state. This picture is also consistent in the context of thermodynamics. In thermodynamics, entropy is not an intrinsic property of the physical system. Entropy is a measure of how much we know about the system, it depends on our perspective. For example, take a classical ideal gas in a box. What happens if you double the volume of the box? If you know the exact position and momentum of every particle in the gas at a certain time, you can predict their future position. Therefore the entropy of the system does not change when the volume increases. But, if you are not aware of the microscopical state, your conclusion would be that the entropy increased with the change in volume. Ideal gas, about to double in volume. Back to our quantum system, from the perspective of the entire wave-function, the entropy is constant. As long as you look at the microscopic details of a closed system, the entropy does not increase over time. This is consistent with the fact that the process is reversible. From the perspective of the detector, there was one bit of random information in the measurement, and therefore the entropy increased exactly by log(2). This is again consistent with the fact that from the point of view of the detector the process is irreversible. We described the exact same physical system from the point of view of two different actors with different knowledge about the system. One description is deterministic and one is random. Both are correct at the same time, it is just a question of what is your perspective. Understanding quantum mechanics For almost a century now, there is an on going debate on the interpretations of quantum mechanics. Two such interpretations are the Copenhagen Interpretation (CI) and the Many Worlds Interpretation (MWI). Beyond that, there is a meta-debate going on for almost as long about whether there is any value in debating these interpretations. Physicists that think this is a pointless endeavor consist of the Shut Up And Calculate (SUAC) camp. In a recent article in the New York Times, Sean Carroll writes that Even Physicists Don’t Understand Quantum Mechanics. The New York Times is not a scientific journal, so normally I would not cite it as an authoritative scientific source. But it is a good source to demonstrate sentiments. This is an example of a physicist studying the foundation of quantum mechanics who feels that too many physicist do not care that they do not understand quantum mechanics, as long as they know how to make calculations. Much of what is written here was inspired by Sabine Hossenfelder’s blog. Specifically, by her post on quantum measurement and the long comment thread that followed. This is also where I was referred to a paper by Lev Vaidman on the Ontology of the wave function and the many-worlds interpretation. My views are closely aligned with the views he presents in this paper. Vaidman’s entire career was researching the foundations of quantum mechanics. It would be hard to blame him that he does not care about the subject. Let me try to convince you that at least some of us in the SUAC camp do care about understanding quantum mechanics. The point is that we actually understand quantum mechanics fairly well. While we might not understand everything, our understanding is deep enough that we are convinced that we do not need to worry about quantum interpretations. We also understand the measurement process. The “measurement problem” is a solved problem. As a baseline, all parties in this debate actually agree about almost everything. We all agree that the theoretical formulation of quantum mechanics is based on Schrodinger’s equation (and all its extensions) together with the Born rule that interprets the wave-function as a probability amplitude. We also agree about all the experimental results that confirm the validity of quantum mechanics. The only disagreement is about the process of measurement, which you could say, is somewhat crucial for connecting theory with experiment. So lets delve deeper into the Measurement Postulate. A measurement involves an observable operator which is an Hermitian operator. Examples of Hermitian operators are position, momentum, energy, angular momentum and spin. An Hermitian operator has real eigen-values that correspond to the real values that we measure. Each eigen-value has a corresponding eigen-state. This is where the problem begins. The Measurement Postulate states that a measurement would result in the system being in the eigen-state corresponding to the eigen-value that was measured. This is a non-unitary, non-local, non-deterministic, irreversible process. Therefore, it can not be described using the standard Hamiltonian time evolution of quantum mechanics. The Copenhagen Interpretation (CI) claims that the wave-function collapses during the measurement. Therefore, the measurement process cannot be described using the standard Hamiltonian time evolution of quantum mechanics. The problem with this interpretation is that no one ever came up with a mechanism for this collapse that is consistent with existing experiments and that makes any new unique predictions. Many Worlds Interpretation (MWI) claims that all branches of the measurement keep on existing, there is no collapse. One could say that the world “splits” during measurement to several worlds, but what MWI really says is that everything is encoded in the wave-function, so there is no special event during measurement. This solves the unitarity problem. We now have a reversible, deterministic, local model. One criticism of MWI is that without collapse the measurement has no effect on our system. Choosing an observable Hermitian operator is equivalent to choosing a basis, it does not change the physics. This is a legitimate criticism, but the problem is not in quantum mechanics. The problem is that in the formulation of the Measurement Postulate, people forget to mention that a measurement requires an interaction between the observable operator and our system. For example, if we want to measure the spin of a particle, we use the Stern–Gerlach experiment as a detector. Such a detector is essentially equivalent to adding an interaction term in the Hamiltonian that describes the coupling between the particle’s spin and a non-uniform magnetic field. A measurement is not just a change of basis. We can model how our detector deflects different spins in different directions. The Stern-Gerlach experiment The Stern-Gerlach experiment To illustrate the effect of the measurement, think about an experiment that starts with a spin pointing up (\left\|\uparrow\right>). The spin is first measured in the X axis and then it is measured in the Y axis. A Stern–Gerlach experiment where the initial state is spin up. First the spin is measured in the X direction and then in the Y direction. If, like the critics claim, the measurement in the X axis had no effect on the spin, it would stay in the up state. Then the measurement in the Y direction would just be an up state (\left\|\uparrow\right>). If the measurement “collapses” the wave-function, there would be a mix of two pure spin states, one pointing left (\left\|\leftarrow\right>) and one pointing right (\left\|\rightarrow\right>). Then, the measurement in the Y axis would measure each such pure state as a combination of up (\left\|\uparrow\right>) and down (\left\|\downarrow\right>) spins, resulting in a mixed state of 4 pure spin states. Four different outcome of our experiment where we vary the strength of the measurement in the X direction. (a) there is no X measurement, we only measure spin up. (b) weak X measurement, there is small separation in X and therefore a small chance of measuring spin down. (c) stronger X measurement, better separation in X and therefore a higher change of measuring spin down. (d) Full separation in X and therefore all four possible outcomes have equal probability. Conceptually, one could calculate the results of this experiment using the Schrödinger equation, modelling explicitly the elusive wave-function collapse. There is no need for any extra measurement postulate. I must admit that I did not perform these calculations. Solving the Schrödinger equation for this case can be done analytically. But matching the boundary conditions and setting initial condition for the wave-packet is a bit more complicated. If I’ll get around to it, I will post a detailed explanation of the calculation. Meanwhile, the plot above illustrates the expected results. Another criticism of MWI is that we cannot have any evidence about other worlds because these worlds have no effect on our world. But we do have evidence. In the double slit experiment the particle goes through one slit in one world and through the other slit in the other world. The interference pattern in the double slit experiment is the manifestation of the effect that different worlds have on each other. Some worry that scaling up the measurement to macroscopic scales would require giving up reductionism. Yet the Stern-Gerlach experiment is an explicit example of how a microscopic spin quantity transforms into a macroscopic spatial separation. Others worry that there is something mysterious going on during decoherence. Again, the Stern-Gerlach experiment shows us exactly how it works. The calculation is time-reversible. Still, the incoming wave-packet is a combination of energy states that come out of the detector with very specific phases for each state. Reversing the experiment by sending in two spins and trying to get out a pure spin state is clearly not practical. The last criticism I can think of is that in MWI probabilities are meaningless if we claim that all outcomes exist. I do not see how this different from CI where all outcomes are possible. Moreover, it is not different from classical probabilities. To summarize, since the introduction of the Schrödinger equation in 1925, and the Born rule in 1926, we have made some progress in our understanding of quantum mechanics. The EPR experiment, Bell’s inequalities and Everett’s many worlds interpretation contributed towards this understanding. This progress convinces us that nature behaves exactly according to the rules quantum mechanics. There is just no shred of evidence for a missing ingredient in our understanding. As physicists we always hope that nature would challenge us with fascinating riddles. Unfortunately, I am fairly convinced that interpretations of quantum mechanics and the Measurement Postulate are not such riddles.
4650cca3c5732a01	Quantum mechanics, field theory are false. How physics becomes wrong ? How did we pick wrong physics ? [ Quantum mechanics is unrealistic and useless. ] (Fig.1) ↓ How did we pick useless quantum mechanics ? Quantum mechanics is unreal, they claim a single electron can be everywhere in fantasy parallel worlds. Why did we have to pick this unreal quantum mechanics ? In 1920s when quantum mechanics was born, there were No computers. So physicists could not compute three-body problem = helium classically. Schrödinger equation in quantum mechanics could not solve Helium, either. They had to use illogical approximate method where they just choose some fake wavefunction with artificially adjustable parameters without solving it, instead, just integrate it. This is the start of unreal quantum mechanics. "Choosing approximate (= fake ) solution" means this method is useless, forever. The word "approximate" is wrong, because we prove it is impossible that Schrödinger equation has exact solutions in multi-electron atoms. Quantum "spin" is fantasy and useless. [ Electron's spin cannot stop ! It doesn't return by 360o rotation ? ] (Fig.2) ↓ Classical orbit and electron spin have the same magnet !? In 1920s when there were no computers, physicists could not compute multi-electron atoms classically. So they had no choice but to use nonphysical Schrödinger equation, which claims electron's orbital angular momentum is zero, so it crashes into nucleus !? Quantum hydrogen has zero angular momentum, so they needed to fabricate electron spin, which accidentally has the same Bohr magneton as classical orbit. The electron spin is uncanny, it cannot stop or slow down even after many reactions ! It needs 720o rotation to return ! An electron is very tiny, so it has to be spinning faster-than-light to generate magnetic field, so spin is Not real "spinning" ? The media misleads people ? The point is, we cannot see this unreal spin itself. All we can detect is its magnetic field, which happens to be the same as classical orbit. The Stern-Gerlach experiment measured silver atom's magnetic field (= by electron's orbital motion ), it cannot measure a single electron's spin. Spin cannot explain Pauli principle, maximum electron number, they keep using fictitious spin in solid physics, which shows "electron spin" is unreal and useless. Solid physics in quantum mechanics is unreal. [ Fake electron, quasiparticle with fake mass in metal ? ] (Fig.3) ↓ Quantum mechanics lacks reality. Schrödinger equation is useless, then, how does quantum mechanics handle solid-state physics with muli-electrons ? Surprisingly, they treat the whole solid as if it were a single fictitious quasi-electron with fake (= effective ) mass, which can be negative or changed in different directions ! In this "fictitious" quantum mechanics, all concepts such as mass, pseudo-spin, momentum are illusion ( this p.12 ). Quantum mechanics has to rely on many unreal quasiparticles with fake mass to explain various phenomena, so wrong. Quantum mechanics uses unreal quasiparticle. [ Quasiparticle is unreal with fake mass and charge. ] (Fig.4) ↓ Phonon is unreal quasiparticle. Quantum mechanis has given up clarifying true microscopic mechanism. Instead, they focus on fabricating fictitious unreal quasiparticle with adjustable fake mass and charge to explain physical macroscopic phenomena. They made up various unreal quasiparticles to describe different phenomena such as virtual fractional-charge anyon, Majorana, skyrmion, phonon, monopole. All these fictitious quantum particles are expressed only as nonphysical math symbols with No real figure, so useless. Quantum mechanics is useless. [ Science stops at "nonphysical" quantum mechanics. ] (Fig.5) Quantum mechanics is useless, unrealistic ↓ Quantum mechanics was invented in old 1920s when there were No computers, so quantum mechanics is Not a theory using computer to describe atomic behavior. Quantum mechanics uses "fictitious things" to describe atomic behavior instead of computers. They can only express each electron as nonphysical "math letter (= c )" with No detailed picture. This electron has fake (= effective ) mass, which can be negative. ← unreal These fake electrons are not enough to explain phenomena, so they invented unreal quasiparticles. For example, they created fictitious "phonon quasiparticle" to express atomic vibration, which is also abstract math letter with no physical picture, so it is useless. They invented fictitious "exciton quasiparticle" combining an electron and a hole to explain solar cell, which is a key technoology for climate change, but useless. This "exciton quasiparticle" is also nonphysical "math letter" with No concrete shape, and they try to increase this fictitious particles in vain. This "fictitious" physics is useless, so physicists introduced crazy idea = parallel universes for "imaginary" quantum computer, which is scam. Science is full of fake news. [ How do the media and academia deceive people and governments ? ] (Fig.6) ↓ Useless quantum mechanics → unreal quantum computer. Quantum mechanics relying on unreal things is useless, forever. Then, how are the media and academia deceiving people and governments now ? They made up fictitious target = unreal quantum computer to make the current useless physics "look" useful. Journals and the media try to link all useless quantum experiments to imaginary quantum computer in the "future (← ? )". They claim quantum computer will use fantasy parallel universes where a single cat can be dead and alive at the same time, but it can never be observed ! There is No proof of this unreal parallel-world computing so far. Quantum computer is still useless and in its infancy even after long-time researches. To get subsidy, academia and corporations need to make up "fake purpose" = quantum computer, which words are always seen in any meaningless papers and science news as "imaginary future target". The truth is the present physics based on quantum fake particles is making No progress. Taxpayers are deceived by academia, the media and corporations. Quantum field theory is unreal. [ Virtual photon, Higgs, quark are all nonphysical with No real figures. ] (Fig.7) ↓ Quantum field theory discards real atomic picture. Quantum mechanics and quantum field theory are too old, so they only describe each particle as nonphysical math letters. ← useless. They can only express Pauli exclusion principle as nonphysical abstract math relation which tells us nothing about its real mechanism ( this p.3 ) "Photon (= particle light ? )" is also just nonphysical math letter. ← useless. Each electron can interact only with unreal virtual photon, which is also nonphysical math letter ( this p.14, this p.11 ). All calculation results by quantum electrodynamics (= QED ) unrealistically diverge to infinity, so they artificially remove its infinity to get tiny finite values. ← nonsense. This renormalization technique ( finite = ∞ - ∞ ) is nonsense, just wrong math trick, of course, useless. Actually nobody uses this QED in daily life. To remove infinity, the bare charge and mass of an electron must be unrealistically infinite ( this p.2 ) ! ← crazy. All other particles such as quarks and Higgs are nonphysical math letters with No real figure ( this p.6, this p.4, this p.11 ). So these quantum field theory can tell us nothing about real physical detailed mechanism, and the current science stops progressing at these old useless theories. The final form of this quantum field theory is unreal 10 ( or 11 ) dimensional quantum gravity. ← nonsense. All quantum particles are unreal. [ Higgs, quark are "imaginary" particles inside collider. ] (Fig.8) They cannot separate Higgs, quark from accelerator = unreal. Almost all particles (← ? ) in standard model are unreal and unnecessary, because we cannot isolate those imaginary particles from particle colliders. Actually Large hadron collider (= LHC ) and Higgs research are still useless and end up wasting taxpayers' money. They just repeat the same kind of meaningless experiments of "Higgs decays into something ?" Very short-lived Higgs or quark cannot be observed directly. They just pick up some electrons and light in a pile of irrelevant garbage. It is said that trillions of protons' collisions are needed to produce only one Higgs. ← insane ! All these reactions must include unreal virtual particles, which contradict the current theory. ← the current theory is wrong ! We cannot isolate or confirm these " imaginary" Higgs or quarks, so we can say these particles are unreal, useless. Like imaginary quantum computer, they made up fictitious target = BigBang to mislead taxpayers into wasting money in useless gigantic colliders. Why science stops advancing ? [ Technology is great, but theory is wrong. ] (Fig.9) The current technology can manipulate a single atom. We already have technology of manipulating a single atom. So we should be able to make and manipulate tiny molecular machine to cure fatal disease. But we cannot Because the current basic physics is useless and discarded real atomic picture. We need only two basic physical conditions to predict all atomic behaviors with the help of modern computers. If there is only Coulomb-force condition, atomic (elecron orbital) radius is not determined. de Broglie wavelength determines atomic size, and it can explain the different sizes between carbon and silicon. These two conditions can explain the maximum electrons' number (= Pauli principle ) correctly, without unreal spin. This simple atomic model (= only Coulomb + de Broglie ) can be easily applied to much bigger, complicated molecules with the help of computer to cure fatal disease. So the first thing we should do now is discard the current old useless theories, and adopt more realistic ones to control each single atom for future nanomachines. 2018/9/24 updated. Feel free to link to this site.
87915f9275bfb4db	Modern Mathematical Physics: what it should beModern Mathematical Physics: what it should be L. D. Faddeev Steklov Mathematical Institute, St Petersburg 191011, Russia When somebody asks me, what I do in science, I call myself a specialist in mathematical physics. As I have been there for more than 40 years, I have some definite interpretation of this combination of words: «mathematical physics.» Cynics or purists can insist that this is neither mathematics nor physics, adding comments with a different degree of malice. Naturally, this calls for an answer, and in this short essay I want to explain briefly my understanding of the subject. It can be considered as my contribution to the discussion about the origin and role of mathematical physics and thus to be relevant for this volume. The matter is complicated by the fact that the term «mathematical physics» (often abbreviated by MP in what follows) is used in different senses and can have rather different content. This content changes with time, place and person. I did not study properly the history of science; however, it is my impression that, in the beginning of the twentieth century, the term MP was practically equivalent to the concept of theoretical physics. Not only Henri Poincaré, but also Albert Einstein, were called mathematical physicists. Newly established theoretical chairs were called chairs of mathematical physics. It follows from the documents in the archives of the Nobel Committee that MP had a right to appear both in the nominations and discussion of the candidates for the Nobel Prize in physics [1]. Roughly speaking, the concept of MP covered theoretical papers where mathematical formulae were used. However, during an unprecedented bloom of theoretical physics in the 20’s and 30’s, an essential separation of the terms «theoretical» and mathematical«occurred. For many people, MP was reduced to the important but auxiliary course «Methods of Mathematical Physics» including a set of useful mathematical tools. The monograph of P. Morse and H. Feshbach [2] is a classical example of such a course, addressed to a wide circle of physicists and engineers. On the other hand, MP in the mathematical interpretation appeared as a theory of partial differential equations and variational calculus. The monographs of R. Courant and D. Hilbert [3] and S. Sobolev [4] are outstanding illustrations of this development. The theorems of existence and uniqueness based on the variational principles, a priori estimates, and imbedding theorems for functional spaces comprise the main content of this direction. As a student of O. Ladyzhenskaya, I was immersed in this subject since the 3rd year of my undergraduate studies at the Physics Department of Leningrad University. My fellow student N. Uraltseva now holds the chair of MP exactly in this sense. MP in this context has as its source mainly geometry and such parts of classical mechanics as hydrodynamics and elasticity theory. Since the 60’s a new impetus to MP in this sense was supplied by Quantum Theory. Here the main apparatus is functional analysis, including the spectral theory of operators in Hilbert space, the mathematical theory of scattering and the theory of Lie groups and their representations. The main subject is the Schrödinger operator. Though the methods and concrete content of this part of MP are essentially different from those of its classical counterpart, the methodological attitude is the same. One sees the quest for the rigorous mathematical theorems about results which are understood by physicists in their own way. I was born as a scientist exactly in this environment. I graduated from the unique chair of Mathematical Physics, established by V. I. Smirnov at the Physics Department ofLeningrad State University already in the 30’s. In his venture V .I. Smirnov got support from V. Fock, the world famous theoretical physicist with very wide mathematical interests. Originally this chair played the auxiliary role of being responsible for the mathematical courses for physics students. However in 1955 it got permission to supervise its own diploma projects, and I belonged to the very first group of students using this opportunity. As I already mentioned, O. A. Ladyzhenskaya was our main professor. Although her own interests were mostly in nonlinear PDE and hydrodynamics, she decided to direct me to quantum theory. During last two years of undergraduate studies I was to read the monograph of K. O. Friedrichs, «Mathematical Aspects of Quantum Field Theory,» and relate it to our group of 5 students and our professor on a special seminar. At the same time my student friends from the chair of Theoretical Physics were absorbed in reading the first monograph on Quantum Electrodynamics by A. Ahieser and V. Berestevsky. The difference in attitudes and language was striking and I was to become accustomed to both. After my graduation O. A. Ladyzhenskaya remained my tutor but she left me free to choose research topics and literature to read. I read both mathematical papers (i.e. on direct and inverse scattering problems by I.M. Gelfand and B. M. Levitan, V. A. Marchenko, M. G. Krein, A. Ya. Povzner) and «Physical Review» (i.e. on formal scattering theory by M. Gell-Mann, M. Goldberger, J. Schwinger and H. Ekstein) as well. Papers by I. Segal, L. Van-Hoveand R. Haag added to my first impressions on Quantum Field Theory taken from K. Friederichs. In the process of this self-education my own understanding of the nature and goals of MP gradually deviated from the prevailing views of the members of the V. Smirnov chair. I decided that it is more challenging to do something which is not known to my colleagues from theoretical physics rather than supply theorems of substantiality. My first work on the inverse scattering problem especially for the many-dimensional Schrödinger operator and that on the three body scattering problem confirm that I really tried to follow this line of thought. This attitude became even firmer when I began to work on Quantum Field Theory in the middle of the 60’s. As a result, my understanding of the goal of MP drastically modified. I consider as the main goal of MP the use of mathematical intuition for the derivation of really new results in the fundamental physics. In this sense, MP and Theoretical Physics are competitors. Their goals in unraveling the laws of the structure of matter coincide. However, the methods and even the estimates of the importance of the results of work may differ quite significally. Here it is time to say in what sense I use the term «fundamental physics.» The adjective «fundamental» has many possible interpretations when applied to the classification of science. In a wider sense it is used to characterize the research directed to unraveling new properties of physical systems. In the narrow sense it is kept only for the search for the basic laws that govern and explain these properties. Thus, all chemical properties can be derived from the Schrödinger equation for a system of electrons and nuclei. Alternatively, we can say that the fundamental laws of chemistry in a narrow sense are already known. This, of course, does not deprive chemistry of the right to be called a fundamental science in a wide sense. The same can be said about classical mechanics and the quantum physics of condensed matter. Whereas the largest part of physical research lies now in the latter, it is clear that all its successes including the theory of superconductivity and superfluidity, Bose-Einstein condensation and quantum Hall effect have a fundamental explanation in the nonrelativistic quantum theory of many body systems. An unfinished physical fundamental problem in a narrow sense is physics of elementary particles. This puts this part of physics into a special position. And it is here where modern MP has the most probable chances for a breakthrough. Indeed, until recent time, all physics developed along the traditional circle: experiment v- theoretical interpretation v- new experiment. So the theory traditionally followed the experiment. This imposes a severe censorship on the theoretical work. Any idea, bright as it is, which is not supplied by the experimental knowledge at the time when it appeared is to be considered wrong and as such must be abandoned. Characteristically the role of censors might be played by theoreticians themselves and the great L. Landau and W. Pauli were, as far as I can judge, the most severe ones. And, of course, they had very good reason. On the other hand, the development of mathematics, which is also to a great extent influenced by applications, has nevertheless its internal logic. Ideas are judged not by their relevance but more by esthetic criteria. The totalitarianism of theoretical physics gives way to a kind of democracy in mathematics and its inherent intuition. And exactly this freedom could be found useful for particle physics. This part of physics traditionally is based on the progress of accelerator techniques. The very high cost and restricted possibilities of the latter soon will become an uncircumventable obstacle to further development. And it is here that mathematical intuition could give an adequate alternative. This was already stressed by famous theoreticians with mathematical inclinations. Indeed, let me cite a paper [5] by P. Dirac from the early 30’s: The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced. This is only natural and to be expected. What, however, was not expected by the scientific workers of the last century was the particular form that the line of advancement of the mathematics would take, namely, it was expected that the mathematics would get more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract. Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalization of the axioms at the base of mathematics rather than with logical development of any one mathematical scheme on a fixed foundation. There are at present fundamental problems in theoretical phy-sics awaiting solution, e.g., the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will therefore have to proceed in a more inderect way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities. Similar views were expressed by C.N. Yang. I did not find a compact citation, but all spirit of his commentaries to his own collection of papers [6]shows this attitude. Also he used to tell this to me in private discussions. I believe that the dramatic history of setting the gauge fields as a basic tool in the description of interactions in Quantum Field Theory gives a good illustration of the influence of mathematical intuition on the development of the fundamental physics. Gauge fields, or YangvMills fields, were introduced to the wide audience of physicists in 1954 in a short paper by C.N. Yang and R. Mills [7], dedicated to the generalization of the electromagnetic fields and the corresponding principle of gauge invariance. The geometric sense of this principle for the electromagnetic field was made clear as early as in the late 20’s due to the papers of V. Fock [8] and H. Weyl [9]. They underlined the analogy of the gauge (or gradient in the terminology of V. Fock) invariance of the electrodynamics and the equivalence principle of the Einstein theory of gravitation. The gauge group in electrodynamics is commutative and corresponds to the multiplication of the complex field (or wave function) of the electrically charged particle by a phase factor depending on the spacevtime coordinates. Einstein’s theory of gravity provides an example of a much more sophisticated gauge group, namely the group of general coordinate transformation. Both H. Weyl and V. Fock were to use the language of the moving frame with spin connection, associated with local Lorentz rotations. Thus the Lorentz group became the first nonabelian gauge group and one can see in [8] essentially all formulas characteristics of nonabelian gauge fields. However, in contradistinction to the electromagnetic field, the spin connection enters the description of the space-time and not the internal space of electric charge. In the middle of the 30’s, after the discovery of the isotopic spin in nuclear physics, and forming the Yukawa idea of the intermediate boson, O. Klein tried to geometrise these objects. His proposal was based on his 5-dimensionalpicture. Proton and neutron (as well as electron and neutrino, there were no clear distinction between strong and weak interactions) were put together in an isovector and electromagnetic field and charged vector meson comprised a 2×2 matrix. However the noncommutative SU(2) gauge group was not mentioned. Klein’s proposal was not received favorably and N. Borh did not recommend him to publish a paper. So the idea remained only in the form of contribution to proceedings of Warsaw Conference «New Theories in Physics» [10]. The noncommutative group, acting in the internal space of charges, appeared for the first time in the paper [8] of C.N. Yang and R. Mills in 1954. There is no wonder that Yang received a cool reaction when he presented his work at Princeton in 1954. The dramatic account of this event can be found in his commentaries [7]. Pauli was in the audience and immediately raised the question about mass. Indeed the gauge invariance forbids the introduction of mass to the vector charged fields and masslessness leads to the long range interaction, which contradicts the experiment. The only known massless particles (and accompaning long range interactions) are photon and graviton. It is evident from Yang’s text, that Pauli was well acquainted with the differential geometry of nonabelian vector fields but his own censorship did not allow him to speak about them. As we know now, the boldness of Yang and his esthetic feeling finally were vindicated. And it can be rightly said, that C. N. Yang proceeded according to mathematical intuition. In 1954 the paper of Yang and Mills did not move to the forefront of high energy theoretical physics. However, the idea of the charged space with noncommutative symmetry group acquired more and more popularity due to the increasing number of elementary particles and the search for the universal scheme of their classification. And at that time the decisive role in the promotion of the YangvMills fields was also played by mathematical intuition. At the beginning of the 60’s, R. Feynman worked on the extension of his own scheme of quantization of the electromagnetic field to the gravitation theory of Einstein. A purely technical difficulty v- the abundance of the tensor indices v- made his work rather slow. Following the advice of M. Gell-Mann, he exercised first on the simpler case of the YangvMills fields. To his surprise, he found that a naive generalization of his diagrammatic rules designed for electrodynamics did not work for the Yang-Mills field. The unitarity of the S-matrix was broken. Feynman restored the unitarity in one loop by reconstructing the full scattering amplitude from its imaginary part and found that the result can be interpreted as a subtraction of the contribution of some fictitious particle. However his technique became quite cumbersome beyond one loop. His approach was gradually developed by B. De-Witt[11]. It must be stressed that % the physical senselessness of the YangvMills field did not preclude Feynman from using it for mathematical construction. The work of Feynman [12] became one of the starting points for my work in Quantum Field Theory, which I began in the middle of the 60’s together with Victor Popov. Another point as important was the mathematical monograph by A. Lichnerowitz [13], dedicated to the theory of connections in vector bundles. From Lichnerowitz’s book it followed clearly that the YangvMills field has a definite geometric interpretation: it defines a connection in the vector bundle, the base being the space-time and the fiber the linear space of the representation of the compact group of charges. Thus, the YangvMills field finds its natural place among the fields of geometrical origin between the electromagnetic field (which is its particular example for theone-dimensional charge) and Einstein’s gravitation field, which deals with the tangent bundle of the Riemannian space-time manifold. It became clear to me that such a possibility cannot be missed and, notwithstanding the unsolved problem of zero mass, one must actively tackle the problem of the correct quantization of the YangvMills field. The geometric origin of the YangvMills field gave a natural way to resolve the difficulties with the diagrammatic rules. The formulation of the quantum theory in terms of Feynman’s functional integral happened to be most appropriate from the technical point of view. Indeed, to take into account the gauge equivalence principle one has to integrate over the classes of gauge equivalent fields rather than over every individual configuration. As soon as this idea is understood, the technical realization is rather straightforward. As a result V. Popov and I came out at the end of 1966 with a set of rules valid for all orders of perturbation theory. The fictitious particles appeared as auxiliary variables giving the integral representation for the nontrivial determinant entering the measure over the set of gauge orbits. Correct diagrammatic rules of quantization of the Yang-Mills field, obtained by V. Popov and me in 1966v1967 [14], [15] did not attract immediate the attention of physicists. Moreover, the time when our work was done was not favorable for it. Quantum Field Theory was virtually forbidden, especially in the Soviet Union, due to the influence of Landau. «The Hamiltonian is dead» v- this phrase from his paper [16], dedicated to the anniversary of W. Pauli v- shows the extreme of Landau’s attitude. The reason was quite solid, it was based not on experiment, but on the investigation of the effects of renormalization, which led Landau and his coworkers to believe that the renormalized physical coupling constant is inevitably zero for all possible local interactions. So there was no way for Victor Popov and me to publish an extended article in a major Soviet journal. We opted for the short communication in «Physics Letters» and were happy to be able to publish the full version in the preprint series of newly opened Kiev Institute of Theoretical Physics. This preprint was finally translated into English by B. Lee as a Fermilab preprint in 1972, and from the preface to the translation it follows that it was known in the West already in 1968. A decisive role in the successful promotion of our diagrammatic rules into physics was played by the works of G. ’t Hooft [17], dedicated to the YangvMills field interacting with the Higgs field (and which ultimately led to a Nobel Prize for him in 1999) and the discovery of dimensional transmutation (the term of S. Coleman [18]). The problem of mass was solved in the first case via the spontaneous symmetry breaking. The second development was based on asymptotic freedom. There exists a vast literature dedicated to the history of this dramatic development. I refer to the recent papers of G. ’t Hooft [19] and D. Gross [20], where the participants in this story share their impressions of this progress. As a result, the Standard Model of unified interactions got its main technical tool. From the middle of the 70’s until our time it remains the fundamental base of high energy physics. For our discourse it is important to stress once again that the paper [14] based on mathematical intuition preceded the works made in the traditions of theoretical physics. The Standard Model did not complete the development of fundamental physics in spite of its unexpected and astonishing experimental success. The gravitational interactions, whose geometrical interpretation is slightly different from that of the YangvMills theory, is not included in the Standard Model. The unification of quantum principles, LorentzvEinstein relativity and Einstein gravity has not yet been accomplished. We have every reason to conjecture that the modern MP and its mode of working will play the decisive role in the quest for such a unification. Indeed, the new generation of theoreticians in high energy physics have received an incomparably higher mathematical education. They are not subject to the pressure of old authorities maintaining the purity of physical thinking and/or terminology. Futhermore, many professional mathematicians, tempted by the beauty of the methods used by physicists, moved to the position of the modern mathematical physics. Let use cite from the manifesto, written by P. MacPherson during the organization of the Quantum Field Theory year at the School of Mathematics of the Institute for Advanced Study at Princeton: The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics, such as quantum field theory. The emphasis will be on developing the intuition stemming from functional integrals. One way to define the goals of the program is by negation, excluding certain important subjects commonly pursued by mathematicians whose work is motivated by physics. In this spirit, it is not planned to treat except peripherally the magnificient new applications of field theory, such as Seiberg-Witten equations to Donaldson theory. Nor is the plan to consider fundamental new constructions within mathimatics that were inspired by physics, such as quantum groups or vertex operator algebras. Nor is the aim to discuss how to provide mathematical rigor for physical theories. Rather, the goal is to develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra. I propose to call the intuition to which MacPherson refers that of mathematical physics. I also recommend the reader to look at the instructive drawing by P. Dijkgraaf on the dust cover of the volumes of lectures given at the School [21]. The union of these two groups constitutes an enormous intellectual force. In the next century we will learn if this force is capable of substituting for the traditional experimental base of the development of fundamental physics and pertinent physical intuition. 1. B. Nagel, The Discussion Concerning the Nobel Prize of Max Planck, Science Technology and Society in the Time of Alfred Nobel (New York: Pergamon, 1982). 2. P. Morse and H. Feshbach, Methods of Theoretical Physics, (New York: McGraw-Hill, 1953). 3. R. Courant and D. Hilbert, Methoden der mathematischen Physik, (Berlin: Springer, 1931). 4. S.L. Sobolev, Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike (Some Applications of Functional Analysis in Mathematical Physics), (Leningrad: Lenigrad. Gos. Univ., 1950). 5. P. Dirac, Quantized Singularities in the Electromagnetic Field, Proc. Roy. Soc. London A 133, 60v72 (1931). 6. C.N. Yang, Selected Papers 1945v1980 with Commentary, (San Francisco: Freeman, 1983). 7. C.N. Yang and R. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191v195i (1954). 8. V. Fock, L’equation d’onde de Dirac et la geometrie de Riemann, J. Phys. et Rad. 70 392v405 (1929). 9. H. Weyl, Electron and Gravitation, Zeit. Phys., 56, 330v352 (1929). 10. O. Klein, On the Theory of Charged Fields: Submitted to the Conference New Theories in Physics, Warsaw, 1938, Surv. High Energy Phys., 1986, 5 269 (1986). 11. B. De-Witt, Quantum Theory of Gravity II: The manifest covariant theory, Phys. Rev., 1967, 162, 1195v1239 (1967). 12. R.P. Feynman, Quantum Theory of Gravitation, Acta Phys. Polon. 24, 697v722 (1963). 13. A. Lichnerowicz, Théorie globale des connexions et des groupes d’holonomie, (Roma: Ed. Cremonese, 1955). 14. L. Faddeev and V. Popov, Feynman Diagrams for the Yang-Mills Field, Phys. Lett. B, 25, 29v30 (1967). 15. V. Popov and L. Faddeev, Perturbation Theory for Gauge-Invariant Fields, Preprint, National Accelerator Laboratory,NAL-THY-57 (1972). 16. L. Landau, in Theoretical Physics in the twentieth century, a memorial volume to Wolfgang Pauli, ed. M. Fierz and V. Weisskopf, (Cambridge, USA, 1956). 17. G. ’t Hooft, Renormalizable Lagrangians for Massive Yang-Mills Fields, Nucl. Phys. B, 35, 167v188 (1971). 18. S. Coleman, Secret Symmetries: An Introduction to Spontaneous Symmetry Breakdown and Gauge Fields: Lecture given at 1973 International Summer School in Physics Ettore Majorana, Erice (Sicily), 1973, Erice Subnucl. Phys., 1973. 19. G. ’t Hooft, When was Asumptotic Freedom Discovered? Rehabilitation of Quantum Field Theory, Preprint, hep-th/9808154 (1998). 20. D. Gross, Twenty Years of Asymptotic Freedom, Preprint, hep-th/9809060 (1998). 21. V. Dijkgraaf, Quantum Fields and Strings: A course for mathematicians, vols. I, II (AMS, IAS, 1999). • Просмотров: 18228
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f0e13972626b74c1	Oregon State UniversitySpecial Collections & Archives Research Center February 28, 2001 Video: Keynote Address: “Timing in the Invisible,” Part 1 Ahmed Zewail Get the Flash Player to see this video. 41:54 - Abstract \| Biography \| More Videos from Session 1 John Westall: It gives me great pleasure this morning to introduce the lead speaker for today's symposium, Professor Ahmed Zewail, Linus Pauling Professor of Chemical Physics and Professor of Physics at the California Institute of Technology, and Nobel Laureate in Chemistry in 1999. No one can be more appropriate to lead off a symposium on Linus Pauling than Ahmed Zewail. While Pauling pioneered the understanding of the structure of the chemical bond, and won a Nobel Prize for that, half a century later Zewail has pioneered the understanding of the dynamics of the chemical bond, and won a Nobel Prize for that. At the risk of oversimplification, let me try to convey to you in just a few words the excitement Zewail's work has caused in the last decade of the 20th century, just as I imagine Pauling's work did in the earlier part of the 20th century. Early work on bonding and structure can be seen in the early part of the 19th century when Berzelius provided a fairly satisfactory explanation of what has become known today as ionic bonding. By the early part of the 20th century, G.N. Lewis had provided the early framework for covalent bonding. When Pauling got involved in the late ‘20s and early ‘30s, there were still large questions unanswered about molecular geometry and the relation of chemical bonding to the emerging atomic theory. Not until Pauling's groundbreaking papers on the nature of the chemical bond in 1931 did the world see how the nature of the structure of molecules could be understood from the ground up with geometry consistent with the fundamental atomic theory. Pauling's success was based on his skill as an experimentalist in the field of x-ray crystallography, his mastery of theory, and his ability to make the appropriate approximation at the appropriate time. A similar evolution and groundbreaking event can be seen in our understanding of our dynamics of the chemical bond. Dynamics encompasses vibrational motions that make or break the chemical bond. Dynamics is at the heart of chemical reactivity, and at the heart of chemistry itself. By the end of the 19th century, Arrhenius had related chemical reaction rates to temperature. Whereas Arrhenius had considered ensembles of molecules, in the 1930s it was Eyring and Polanyi who devised a transition state theory - that is, a state midway between reactants and products, and on a time scale related to the vibrations of the molecules. This time scale is on the order of 10 to 100 femtoseconds, and that is a very, very short time. A femtosecond is to a second as a second is to about 30 million years. [3:56] For the rest of the 20th century, the world worked with this theory of Eyring, but no one ever dreamed of actually trying to observe one of these transition states at such a very short time scale. But that is exactly what Zewail set out to do, to observe the transition state and, in an extraordinary set of experiments in the late 1980s, he succeeded. By using an ingenious combination of experimental technology, theory, and appropriate approximations, he developed what amounts to a high speed camera to image molecules at the femtoscale time scale of chemical reactions, and was actually able to view chemical bonds as they were breaking. The impact of this work has been tremendous. Rudolph Marcus, a Nobel Prize-winner from the early 1990s, has said of Zewail: "Zewail's work has fundamentally changed the way scientists view chemical dynamics. The study of chemical, physical, and biological events that occur on the femtosecond time scale is the ultimate achievement of over a century of effort by mankind, since Arrhenius, to view dynamics of the chemical bond as it actually unfolds in real time." Ahmed Zewail is currently Linus Pauling Professor of Chemistry and Professor of Physics at the California Institute of Technology, and Director of the National Science Foundation Laboratory for molecular science. He received his B.S. and M.S. degrees from Alexandria University in Egypt, and his Ph.D. from the University of Pennsylvania. He was appointed to the faculty of Caltech in 1976, and in 1990 was appointed Linus Pauling Chair at the Institute. Professor Zewail has numerous honorary degrees from all over the world - he is a member of several national academies of science, he has won top awards from national chemistry and physics organizations, only one of which I'll mention, the 1997 Linus Pauling Award from the Northwest sections of the American Chemical Society, which was awarded in Portland. [6:39] Pauling's work culminated in the Nobel Prize for chemistry in 1954 with the citation for his research into the nature of the chemical bond and its application for the elucidation of the structure of complex substances. Zewail won the Nobel Prize in chemistry in 1999 with the citation for his studies on the transition states of chemical reactions using femtosecond spectroscopy. It is now my great pleasure to introduce for today's keynote lecture Professor Ahmed Zewail. [7:20] Ahmed Zewail: It's a great pleasure to be here and to be part of this wonderful centennial. In fact, I think with what you've heard from John I should really sit down now, as you have the whole story of why I'm going to talk today. It's really very special to me, because I do have a very special relationship to Linus Pauling, not only that I sit in his chair at Caltech, but for a variety of reasons. As a matter of fact, I fly tomorrow morning from here because at Caltech we also are celebrating the centennial of Linus Pauling, and I do chair the session in the morning. Everybody knew about Pauling, and everybody knew about Pauling's achievements, but I think, as Steve said, there's something about Pauling in terms of his personality and his way of doing things that people really don't know about. It comes out in the biographies much better, but from my daughters to people I have seen at Caltech who are in the late stage of their lives, who just get so excited when they meet Linus Pauling. He was a fair man, a very decent human being, and the word I use around our house, he was a very civilized human being. When you sit with Linus, you can talk about the world, and you can enjoy talking to him about the world, but I always find in him civility and decency. His scientific contributions are very well known, but I would like to make a link between some of the work that Pauling has achieved in chemistry and biology. It is appropriate to tell you that I didn't know of Pauling when I arrived at Caltech, all that I knew was his name; I arrived at Caltech in 1974. But this is one point that I want to touch on, especially since we have two historians with us today who wrote about the biography of Linus; that's his relation to Caltech. When I arrived in 1974, there were rumors around the campus that Linus was not too happy about his departure from Caltech, and of the incident that took place. Things get messed up in the public press a lot and people say things that are not really true, because I dug into some of the historical minutes of the faculty and what Lee DuBridge said. But as a young assistant professor I didn't know any better so I thought it just didn't make any sense, that Pauling, who really was and still is a giant in chemistry, was not visiting Caltech. I was fortunate to be tenured very shortly at Caltech, when I became part of the establishment, so I thought we should do something about it. And so, I had the great honor to chair and to organize…I like this photo of Linus, actually, giving a lecture at Caltech; this was February 28, 1986, and that was his 85th birthday. I thought that this was a great occasion to bring Linus to campus. He came back and was just so excited; Linus Jr. was with us, and could see his father speaking from the heart. I gave him the stage in the evening to say whatever he wanted to say about Caltech, and he did give it to us. It was just a very special occasion, and I also organized the 90th birthday for Linus at Caltech, and I think if Linus did not come back to Caltech to share his great moments, it would have been a mistake in the history of Caltech and science. I even crowned him the Pharaoh of Chemistry, and I believe that he loved this picture. I think it's here, isn't it? It cost me about $500 to do this, because I had to go to Hollywood and try to fit his face into one of Ramses II. If you want to learn about all of this, I also edited a book on Linus called The Chemical Bond: Structure and Dynamics, and that is really the focus of my lecture today. I want to take you from the wonderful and important era, which Jack Dunitz will be talking to you about, of structures that are static, meaning there is no way of looking at the movement of the atoms in these structures, into a case where we try to understand the dynamics of how these atoms move as a bond breaks and forms. By the way, the highest honor I received was from Linus, when he wrote to me saying he considered me his friend. Linus Jr. must recognize his father's handwriting. In 1991, he sent me a book, and wrote to me "To my friend Ahmed." I treasure this. [14:39] In his lecture at Caltech, Linus showed the structure of sodium chloride that was believed to be by Mr. [William] Barlow in 1898. This is the structure of table salt. I am showing this because there is an analogy between the beginning of Linus' work on structural chemistry, and what we were trying to do in dynamic chemistry. Looking at the structure of sodium chloride with two atoms probably seems trivial, especially during the era of the Braggs and the work that was done with x-ray crystallography. When we started it was also with table salt, sodium chloride, sodium iodide, to try and understand the dynamics of this chemical bond, but also we received a lot of criticism in the early days, that this was trivial; these are two atoms, and not much is going to be found from this. Of course, you know the work on structure by Linus has led, for example, to the structure of hemoglobin. The picture is not as pretty as you normally see, but this is the protein structure of deoxyhemoblogin. And in fact you will see at the end of my talk that we can also look at the dynamics of proteins, DNA and the like. [16:24] It is remarkable to me that I read very recently - I didn't know that Linus wrote this - but he received the Nobel Prize in 1954 and shortly after the award he was asked to reflect on what chemistry will be in the coming 50 years. Remember, he spent all this time studying the structure, the architecture, of molecules, but he had the vision, and I think that this was the first time that anybody has seen this Scientific American article, that "the half century we are just completing has seen the evolution of chemistry from a vast but largely formless body of empirical knowledge into a coordinated science. The new ideas about electrons and atomic nuclei were speedily introduced into chemistry, leading to the formulation of a powerful structural theory which has welded most of the great mass of chemical fact into a unified system. What will the next 50 years bring?" This is Pauling speaking 50 years ago, mid-century. "We may hope that the chemists of the year 2000 will have obtained such penetrating knowledge of the forces between atoms and molecules that he will be able to predict the rate of any chemical reaction." So even though Pauling was immersed in static structures and their study, his eye was on the next century, and maybe the next century Nobel Prize. That is precisely what I want to talk to you about today, is how we look at the forces that control the motions of atoms and molecules, whether it's in chemical or biological systems, and can we distill some concepts out of this, just like Pauling was trying to understand hybridizations and so on, that teach us something about the dynamics of chemical bonding, and hence the forces and how to predict rates and the dynamics. [19:05] The work I will talk about relates to the Nobel Prize that we received in 1999, and incidentally, there is an incident in history here that I think I told some of the speakers at dinner last night. When we received the Nobel Prize, Caltech had a big party, as they did for Linus, and one of my colleagues, Vince McKoy, noted that Linus was born February 28, 1901, and I was born February 26, 1946, so two days away from Linus. He received the prize in 1954, we received the prize in 1999, and both in October, so we were both almost 50 years of age when we received the Nobel Prize. This is remarkable where Caltech is concerned, because both Pauling and myself started at Caltech as Assistant Professors, they did not "buy us from the outside," as they say. I thought you would be intrigued by this: when the prize was announced, it was everywhere because our work also touches on physics, and so many of the physics magazines wrote detailed articles, and here is a distinguished one from England; it surprised me, as you will see, that England would write something like this. "Ahmed Zewail receives the 1999 Nobel Prize for Chemistry…" and it goes on to say "laser-based techniques that allow the motion of atoms inside molecules to be followed during chemical reactions." It goes on, very complimentary. And then it says "Zewail was born in Egypt in 1496." I told the Nobel committee in Sweden that it is really remarkable that it took them 500 years to give me the Nobel Prize. Here is the journey in time. I think for the people in the audience who are not used to seeing this number, you certainly hear and you know by now… Here could be 12 or 15 billion years of the Big Bang, and then you come down to our lifespan, which is about 100 years or so - your heart beats in one second. But to go from here [present day] to there [Big Bang] is about 1015, and I am going to take you from the heart into a molecule inside the heart, or eye specifically, and you have to decrease by 15 orders of magnitude to see the beats of this molecule, as you see the beats of your heart. The timescale is fast, and I wrote a review which is probably useless, but I thought it's interesting that if you go from this age of the universe, and you count back from the age of the Earth to the human lifespan to your heart (1 second), and then you go to the microscopic world (sub-second), into how molecules rotate, vibrate, and how the electrons move. In this whole microscopic world here, we reach 10-15 or so seconds, where on the opposite end you reach 1015, and the remarkable thing is the heart is the geometric average of the two. As humans, we are in a very unique position. [23:50] It's very difficult to study history; I thought it was very easy, but it turns out it is very difficult, and it takes some time to get the facts. I showed this in the Nobel lecture, because it gives five milestones, snapshots, of an evolution over about six millennia. The first time we know of the concept of measuring time is the calendar, which was developed in 4200 BC. In fact some historians say that it is 4240 BC, so it is almost 6000 years ago, which we knew what a year is, what a day is, what a month is. What an hour is was measured using sundials in about 1500 BC. You can use a shadow from an obelisk or a sundial and know what the time of day is. The mechanical clock was developed in Europe around 1500 AD, and all of this was still in seconds or longer, so up until then we could only measure as accurately as a second. One of the most famous experiments in the history of measurements was done by Edward Muybridge in 1887. Mr. [Leland] Stanford, the Governor of California, was very interested in horses and hired Muybridge to find out the actual nature of the motion of the horse as it gallops. Mr. Muybridge designed this camera, which had a shutter speed of about 1/1000th of a second. This was done in Palo Alto, where there is a plaque commemorating it. In so doing, as the horse galloped the shutter was opened, and Muybridge was able to take a snapshot of the horse proving the contention of Mr. Stanford, that all four legs are off the ground at the same time. I believe this was the very beginning of fast photography and motion pictures. In the 1800s, this was a very important development. To go into the word of molecules, and to be able to see transition states, just as Muybridge saw the transition state of the horse moving, you need an increase of about 12 orders of magnitude in time resolution, and we only can do this with these lasers, as you will see in a few minutes. This was done in 1980, and it may be called femtoscopic. [27:16] So, why is it that this is important to chemistry and biology - the dynamics of the chemical bond? It turns out, the dynamics of any chemical bond, whether it is in a chemical system or a protein or DNA, is totally controlled by the fundamental vibrational time scale, as you heard from John. This vibrational time scale, and there is some philosophy that one can dwell on here, determined by Planck's constant and so on, but fundamentally, two atoms connected by a spring will move in a spring-motion on a time scale of 10-13 seconds, and the shortest possible time for any molecule in this universe will be 10-14 seconds. All the way from hydrogen molecules to protein molecules, the time scale will be from 10-14 seconds to 10-12 seconds. Molecules can rotate in the laboratory without you seeing it in a picosecond 10-12 to 10-9, and everything longer than this we consider in the Stone Ages; it's not interesting to us. So on that time scale you will see many interesting phenomena in chemistry and biology that happen. This is the end of time resolution for chemistry and biology, because if you look here, even molecules that are linking undergo collisions on a time scale of 10-14 seconds. A molecule can break a bond and make a bond on this time scale as well. The eye has a molecule called rhodopsin which divides and allows you to see, and that happens in 200 femtoseconds. The way we get photosynthesis to work, and the electron to transfer inside the green plant, is on the order of femtoseconds. So this is the fundamental time scale, and if we were to understand the dynamics of the chemical bond we must understand this time scale. Luckily, that is the reason behind all of this, and to be totally truthful, this is the way we were thinking about it. We were making real approximations - what I like to call the "back of the envelope" type of calculation - and we didn't go to big computers and do quantum calculations and all of that stuff, because I always feel that there is beauty in the simplicity of science, and if we are not able to distill it into the essence of science, it seems to me that we are fuzzy about it. What I just told you is general to any molecular system; if you think of the chemical binding energies that we have, that Pauling would have calculated with his slide ruler, and then if I activate a chemical bond, I can calculate the energy in this bond. And if I have the energy I can calculate the speed at which the atoms should move in principal. For all chemical and biological systems, you'll find that this speed is about one kilometer per second. Even in the spring that I mentioned, the two atoms would collide with each other at about one kilometer per second, or 105 centimeters per second. If we're going to look on the molecular scale to understand the dynamics of the chemical bond, we have to understand that the distance scale is about an Angstrom, or 10-8. You can see, without any fancy quantum mechanics at this point, potentials and forces and all of that, that if you put these two numbers together (105 and 10-8) you come up with a time resolution of 10-13 seconds. Therefore, our idea is simple: if we can reach a time resolution just like x-ray crystallography and electron diffraction, if we can get the de Broglie wavelengths or the special resolution less than the bond distance, you would be able to see bonds and atoms. Here, we can also say that if our time resolution is 10-14 seconds, then we can freeze this motion, we can do what Muybridge did, because now this time scale will be shorter by a factor of ten than the time it takes the atoms to move in the molecule. More scientifically, we say that the time resolution here is shorter than the vibrational period of the molecule, or the rotational period of the molecule. And that, in its totality, defines the field of femtochemistry, and now there is something called femtobiology, and so on. [33:07] So this is a simple picture of the problem, but for the physicists in the audience the problem is much more intriguing and much more fundamental. It is remarkable that Pauling recognized, right after Schrödinger wrote his famous wave equation in 1926 in Zurich - which we shall celebrate in April - very quickly recognized that if you think of matter as particles, you can also think of them as waves. De Broglie, with brilliant intuition, used the quantization equation and Einstein equation and came up with this very simple thesis that lambda is equal to h over p, which Einstein reviewed. So for each momentum of a particle, there is an associated wave, lambda. When people study matter as light, and we made this mistake too, they are always thinking of Schrödinger's Eigen states, the wave character of the system, wave mechanics, Schrödinger wave equation. But that is not what I am interested in, because what I am interested in is to be able to see the atoms in motion, and to understand the fundamentals of these dynamics. I don't want to go to the delocalized wave picture and calculate Schrödinger equations and come at the end without a simple picture describing the particle behavior of these atoms in motion, moving from one configuration into the other. So how can we make matter behave in a fundamental way as a particle and not as a wave? If we look at what is known as the uncertainty principle, which is seen here, that you cannot do precise measurements of the position of the particle and momentum simultaneously. Similarly, you cannot make a precise measurement of time and energy simultaneously. We understand this from Mr. Heisenberg, there is no problem. But the chemist had difficulty understanding how we can shorten the time, Δt, because this would be at the expense of the energy, ΔE, and therefore we ruined the quantum state of the molecule, the nature of this molecule. What was not recognized, was that this is in fact a beautiful way of locating atoms in space, because in a crude way if you think of how ΔtΔE=hbar and ΔxΔp=hbar, don't think of this alone, just think of the two together, and say that the momentum and energy are related (which they are). If you combine the two, you'll find that if you shorten the time, Δt, you can make Δx very small. That is the only way in this universe that we know of to localize atoms in space. We are not in violation of the uncertainty principle here, but if you do it cleverly, combining time and energy, you can show clearly that you will localize this nuclei to better than a tenth of an Angstrom. The chemical bond distance is about an Angstrom, so we have a factor of ten in space and should be able to see the motion of these atoms as the bond breaks and forms. It has a long history in physics, and in fact it goes back to the 1800s - there is an experiment in 1801 by Mr. [Thomas] Young on light, not on matter. If you take two slits, with light behind them, then you will see these fringes, and it is very similar to what we can now do with matter and molecules. We can take a femtosecond pulse, which is acting like the light, and we can take all these wave functions of the molecules - this, by the way, is two atoms and a spring moving - and we can take these waves and try to do this interference experiment on this matter, and all of a sudden you see what's called a "wave packet," meaning that we localize all of these vibration states in one place, which becomes a classical picture. We can speak of a spring in this point in space, and that it is coming back, and we can see this motion. Early in the field, it was thought that this energy is too broad and ugly, and there are occasionally objections over the uncertainty principle. Actually, without the uncertainty principle, it would be impossible to see the nuclei moving on this time scale. This, by the way, was also of concern to the big minds in 1926. This letter was from [Ernest] Lawrence to Schrödinger, pointing out that this idea of connecting the classical mechanics of Newtons, and understanding the particle instead of the wave behavior, "you will be unable to construct wave packets." This was 1926, and in 1980 not only can you do it on chemical systems and create this wave packet, but the field has expanded so much that you can also do it on liquids, solids, gas phase, and clusters - this has been observed by many people around the globe. [40:45] The way we do this experimentally is tricky, because there are no cameras like Muybridge's that will allow you to open and close mechanically in a femtosecond; it's just impossible. The way we do this is to use an idea of creating pulses of light in femtoseconds and then delay, as a series of pulses - so this red here is delayed from the yellow - and the first one, the yellow, will set off the clock and give us a time zero for the change. With a whole series of pulses we can "photograph" what is happening here as a function of time. We do it this way, using the speed of light, because it turns out that when we delay this pulse in relation to this one by one micron in the laboratory, it's equivalent to 3.3 femtoseconds because of the speed of light: 100,000 km per second. [41:54] Watch Other Videos Session 1 Session 2 Return to Main Page
85a1f62a94e7be9d	Anonymous yet personal, this Blog chronicles the daily events and musings of Jim. and serves as a online repository for his random “Deconstructing Jim” is simply here to entertain you, but not intended for college credit. A little about me My photo Chapel Hill, NC, United States Blog Archive Wednesday, March 24, 2010 How ET Ruined Harmony (and why you shouldn't worry about it) Yesterday I caught an interesting lecture at the Longy School of Music by musicologist Ross Duffin. He is the Fynette H. Kulas Professor of Music at Case Western Reserve University in Cleveland, Ohio. His lecture was about his new book, How Equal Temperament Ruined Harmony (and why you should care), which has been called by his critics as "the most subversive book on a musical subject I've ever read." He refers to Equal-Temperament by the acronym "ET." I didn't exactly know what to expect from Professor Duffin. Based on the provocative title of his book, I feared that he would rail against all music written after Bach. But in the end it was scholarly, well-balanced, and rather informative. As a musician who grew up playing keyboard instruments and fretted-string instruments, I never really had to deal too much with subtle tuning issues. In practice, singers and string players had to adjust to me - since I was the one who was "out of tune." Duffin's research aptly summarizes the dysfunction that has existed regarding tuning systems, theories and performance practice since the Renaissance. Is is clear that virtually all of the solutions that have been proposed over the centuries are messy, ad hoc, and less than elegant. If you are the sort of person who likes certainty, uniform standards, and mathematical precision, you should avoid Duffin's book like the plague. In this regard the world of musical temperament is similar to law-making in Washington DC: you really don't want to know how they make the sausage. Here were a few interesting tidbits and takeaways from the lecture... ET is recent invention in music history - a kind of worst-case totalitarian system that arises when everyone is made to suffer for the common good of uniformity and standardization. In the 18th century, Mozart, Haydn, and probably Beethoven thought of the octave as having more than 12 notes. For them, D-sharp was a very different note than E-flat. Sharps were LOWER in pitch than flats. For example E-flat was a higher note than D-sharp. Leopold Mozart, Wolfgang's father had published the definitive treatise on violin playing with charts indicating these very distinctions. And it's not only string players who abided by this system. The flutist, composer, and music theorist Johann Joachim Quantz published a fingering chart indicating different fingerings for enharmonic notes. For him, sharped and flatted notes were quite different. In the 19th century, some musicians - often virtuoso string soloists who played unaccompanied - reversed the paradigm. As performers they tended to focus on the linear aspects of music. For them, D-sharp was played as a leading-tone and would sound HIGHER in pitch than E-flat because of the voice leading. This was the opposite practice of what was done just a century earlier, and this is the current belief, concept, and standard today. It is the convention practiced by the majority of mainstream classical musicians in the 2oth and 21st centuries - although this norm apparently has little acoustical, historical, or theoretical ground to stand on. The current practice of tuning pianos with ET seems to be a bit of a fraud too. When one analyses what piano tuners actually do in terms of temperament, the result is somewhat sketchy and amorphous. Find me two different pianos, and I'll show you two different tuning standards. It's the invisible elephant in the room. Good piano tuners and excellent chefs don't share their secrets. Our contemporary bias in favor of scientific and mathematical clarity with tuning systems too often seems to go against our better musical instincts and natural hearing. While I found Duffin's talk very enlightening, I'm not a music historian. Overall I consider myself pretty liberal when it comes to performance practice. For me, music is not the acoustical properties of the sound, but the ideas behind its presumed imperfect acoustical representation. In my mind, the tuning and temperament purity argument is a little like saying it's better to read a book printed at 1200 dpi than 300 dpi. The higher resolution allows for a better and more accurate representation of the typeface. Isn't that kind of missing the point of what music is all about? Duffin played a few musical examples to illustrate his points. One example utilized an electronically produced and scientifically accurate realization of a piano work in two contrasting temperaments. While I have to say there was a subtle but discernible difference between them - and that the non-ET version sounded less strained, warmer, and had less beating of upper harmonics - I was not overly impressed with the improved version. It wasn't at all like seeing a movie in 3D for the first time after having only known the standard format. Given all of the factors that go into experiencing a work of music, the tuning aspect pales in comparison. To my ears, the version of temperament that is used is fairly trivial. I don't go to concerts to listen to intonation, and "imprecision" in performance normally doesn't bother me (unless it is really, really bad). Another thing I realized from Duffin's presentation is that the social aspects of music making override the theoretical rules that theorists claim exist. It could be that every accomplished musician has their own unique tuning system. This is what makes one great violinist different from another. They just hear notes and intervals differently - as if it were part of their musical DNA or cultural context. In practice the range of expression possible in the production of a major-third, or a perfect-fifth can vary enormously. There are more gradations than even an enharmonic sharp or flat. Ask any microtonalist. ET is no more than an approximation and a guidepost. It has never been more than a musical version of lane-lines painted on the highway. No musician in their right mind would expect all music to conform to such a limited and restrictive tuning system. It's a framework, not a Draconian pitch-grid where your teacher will swat your fingers with a ruler if you go outside of the lines. On the other hand, alternative tuning systems to ET that have been (or likely will be) proposed are also a compromise. I hate to break the news, but no tuning-system Utopia exists - at least with the 12-note to the octave standard. In the end, ANY tuning system will only function as a rough and imperfect road map for the fabulous musical excursions that practicing musicians will inevitably take us on. I don't buy the argument that equal-temperament has ruined harmony, and I don't think we have to worry about it either. There are much bigger bones to pick. You can sleep soundly at night knowing that music will still be there for you the next morning, equal-temperament or not. Saturday, March 20, 2010 Dutch Invasion For March Madness, a Sports Factoid... Towering at 6 feet - 10 inches, Dutchman Kenneth van Kempen plays basketball for the Ohio Bobcats. He is a Senior at Ohio University, but his hometown is Weert in the Netherlands. Sunday, March 14, 2010 Reading Karl Kraus It's a rainy weekend, perfect for catching up on some reading. I'm enjoying Harry Zohn's biography and critical analysis of the Viennese writer and satirist Karl Kraus (1874-1936). Zohn published his book in 1971, and I was a student of his at Brandeis University where I failed to learn even basic German rather quite miserably. But Zohn, who chaired the Germanic and Slavic Language Department, was a real expert in turn of the century Vienna, and in particular the artistic, literary, and musical movements of that fascinating time. Harry Zohn (1923-2001) was an excellent translator - bringing to English such works as Freud's "Delusion and Dream," the complete diaries of Theodor Herzel, and some 40 other volumes. He was a violist in the Brandeis Symphony Orchestra, and made all of his students (including yours truly) sing Viennese wine garden songs in class (I still have the sheet music). We also sampled a wide-variety of German Beer and pub food during a research-oriented field trip to Boston's Jacob Wirth House. Kraus was a creative force of nature who embodied the Zeitgeist of his generation. Today we would probably call him a Performance Artist. For example, he held some 700 recitals in his traveling show billed as "Theatre of Poetry." It included readings of poetry and prose, satire, opera, and lieder. His circle of intellectuals included composers such as Schoenberg and Mahler, painters such as Klimt and Schiele, and scientists and philosophers such as Wittgenstein and Freud. Kraus was not a musician. According to Zohn... Kraus's inability to read music was not compensated for by any great vocal resources. His singing voice was really Sprechgesang in the manner of Schönberg or Berg, but it was considerably enhanced by this great intuition and empathy, his rhythmic acuteness, his talent as an imitator, and his pervasive moral fanaticism. The pianists who accompanied him were among the best, including composers Ernst Křenek and Josef Matthias Hauer. Zohn observes, "In January, 1932, he gave a program of poems and scenes by Bert Brecht, accompanied on the piano by Kurt Weill." Kraus's 60th birthday was celebrated with a musical-literary matinee and a film about him. Composer Alban Berg was in attendance. Musicians Eduard Steuermann and Rudolf Kolisch were also amongst Kraus's close friends. In Zohn's concluding statement ends with the following observation"...Karl Kraus may have been a failure. But surely he was one of the grandest failures in world literature." Zohn's scholarly but assessable book on Kraus ends with "An Aphoristic Sampler." Here are just a few of the choice translations Zohn made of Kraus's work: I can say with pride that I have spent days and nights not reading anything, and that with unflagging energy I use every free moment gradually to acquire an encyclopedic lack of education. I dreamt that I had died for my country. And right way a coffin-lid opener was there, holding out his hand for a tip. Am I to blame if hallucinations and visions are alive and have names and permanent residences? In one ear and out the other: this would make the head a transit station. What I hear has to go out the same ear. I ask no one for a light. I don't want to be beholden to anyone - in life, love, or literature. And yet I smoke. I hear noises which others don't hear and which interfere with the music of the spheres that others don't hear either. I already remember many things that I am experiencing. Kokoschka has made a portrait of me. It could be that those who know me will not recognize me; but surely those who don't know me will recognize me. "He masters the German language" -that is true of a salesman. An artist is a servant of the word. Today's literature: prescriptions written by patients. The superman is a premature ideal, one that presupposes man. A journalist is stimulated by a deadline. He writes worse when he has time. Diplomacy is a game of chess in which the nations are checkmated. A Gourmet once told me that he preferred the scum of the earth to the cream of society. Democracy means the permission to be everyone's slave. Medicine: "Your money and your life!" I do not trust the printing press when I deliver my written words to it. How a dramatist can rely on the mouth of an actor! If the earth had any idea of how afraid the comet is of contact with it! More satirical quotes of Kraus can be found on the web here... Karl Kraus Quotes Friday, March 12, 2010 Yesterday at Harvard University, composer Rob Zuidam delivered the second of three lectures on contemporary Dutch music. The ongoing series is presented by the Harvard Music Department in conjunction with the Erasumus Lectures on History and Civilization of the Netherlands and Flanders. Zuidam focused on what's known as Hoketus - or "ensemble culture" in the Netherlands and how it evolved. It really all began in 1966 when a group of five Dutch composers organized to protest against the artistic direction taken by the Concertgebouw Orchestra. The group called themselves The Five and led a larger group known as Notenkrakers ("Nutcrackers" - which has multiple meanings, including "note" and "nut"). The Notenkrakers wanted the orchestra to hire Bruno Maderna as a second conductor to work along side with their current Music Director, Bernard Haitink. Maderna was perhaps the leading conductor of contemporary music at that time. Over the course of the next few years little progress was made by Concertgebouw to purge the conservatism from their programs and inject music composed by the younger generation of Dutch composers (such as members of The Five). The Notenkrakers made little if any progress with the Concertgebouw managements on this matter. It all came to a head in November of 1969 when The Notenkrakers stormed in and disturbed a concert about to begin in the Concertgebouw. This "notenkrakersactie" (nutcracker action) was a historic event that some say changed level of acceptance of new music in Holland. Just before Haitink was able to complete his initial downbeat, the protesters had skillfully disrupted the concert with their noise makers and megaphones. The group of students passed out leaflets and confronted the orchestra and audience. Peter Schat (1935-2003), a member of The Five, used his megaphone to demand that Bernard Haitink come down off the podium and address The Notenkrakers and the audience in an open public discussion. The confrontation instilled a a minor riot, and the police were soon called to eject the protesters from the concert hall. Besides Peter Schat, members of The Five included Misha Mengelberg (b. 1935), Louis Andriessen (b. 1939), and Reinbert de Leeuw (b. 1938) - all of which had studied with the composer Kees van Baaren. To the disappointment of The Notenkrakers , their protest was denounced by their philosophical mentor: Matthijs Vermeulen. The Five had held Vermeulen (who was the subject of Zuidam's first lecture) in high regard for his harsh reviews of the Concertgebouw and their lack of interest in performing contemporary music. But to their surprise, Vermeulen released a public statement that The Five was off-base. Vermeulen wrote that hiring Bruno Maderna would be impractical and that Concertgebouw actually supported modern music rather well compared to other international orchestras. Undeterred, throughout the 1970s and into the early 80s, members of The Five independently formed and led ad hoc new music ensembles throughout Holland. Effectively, younger Dutch composers had largely abandoned the idea of the symphony orchestra as an instrument in favor of more responsive new music ensembles that were fluid and dynamic. Louis Andriessen's group Hoketus is a prime example of the resulting "ensemble culture" which continues on today. (Link: ) In the end, the incident of the November 1969 notenkrakersactie when The Notenkrakers stormed in at the Concertgebouw resulted in a positive change for musical performance in the Netherlands. The Five has been credited with "shaking Dutch musical life out of its suffocating provincialism." Thursday, March 11, 2010 Elliott Carter Rocks! A short little guitar piece by Elliott Carter (b. 1908)... Shard (1997) I like the fast music at the end, starting at 1'40" It reminds me of Jimi Hendrix. Wednesday, March 10, 2010 Ralph Towner Jazz guitarist Ralph Towner (b. 1940) is coming to the Regattabar in Cambridge on March 23rd. He's performing on baritone and 12-string guitars along with Paolo Fresu. The set starts at 7:30 PM, and tickets are $22. I vaguely remember Towner from his presence as a sideman on Weather Report's jazz fusion album from 1972: I Sing the Body Electric. But his music today is mostly acoustic. When not on tour, Towner lives the good life in Rome. Here is a YouTube clip of Towner performing the jazz standard I Fall in Love Too Easily by Sammy Cahn. He mentions in his introduction that Sammy passed away just recently, and that the well-known jazz guitarist Steve Kahn is his son. I had studied guitar with Steve Kahn when I was still in high school. (Link: ) Spread Spectrum Communications I teach a course in Data Communications (aka Network Standards and Protocols). I enjoy providing a little history when the course gets to the unit covering Spread Spectrum (SS) technology, which has evolved today into Wi-Fi (IEEE 802.11). Spread Spectrum communications uses wide band, noise-like signals. It's been a favorite technology for the military, since SS signals are hard to detect, hard to intercept, hard to jam, and hard to demodulate. Students love their Wi-Fi, and look a little puzzled when I explain how that technology was invented by a famous movie actress and an avant-garde composer. Yeap, that's right. The story is amazing. If you made it into a movie, nobody would believe it. Austrian-born actress Hedy Lamarr (born Hedwig Eva Maria Kiesler) and composer George Antheil are universally credited for discovering and patenting this communications technology. Hedy Lamarr (1914 - 2000), is better known for the many movies she made at MGM Studios in Hollywood. Lamarr was born to Jewish parents in Vienna, and studied ballet and piano at an early age. She later worked with Max Reinhardt in Berlin, who called her the "most beautiful woman in Europe." Later Lamarr would end up marrying a controlling and wealthy Vienna arms manufacturer who was 13 years her senior. He'd lock her up in the residence, known has"Castle Schwarzenau." Lamarr objected to her husbands' support of Hitler's war machine, but both Hitler and Mussolini were frequent guests at their lavish parties in the castle. It is said that Lamaar dressed as one of her maids and fled to Paris. In 1933 Lamarr created a scandal when she appeared nude for an extended period in the movie "Ecstasy." George Antheil (1900 - 1959) is know primarily as a daring modernist composer who's music shocked audiences on both sides of the Atlantic. His most famous piece Ballet Mécanique was originally conceived in the 1920s for 16 synchronized player pianos, two grand pianos, electronic bells, xylophones, bass drums, a siren and three airplane propellers. It's often performed today in a scaled down version for percussion, four pianos, and a recording of an airplane motor. Well-known in both America and in Europe as being an "ultra-modern pianist/composer," Antheil influenced many avant-garde composers, including Edgard Varèse and John Cage. He held wide interests such as publishing articles and books on female endocrinology, and in his spare time wrote a mystery novel (blogs didn't exist yet). It was Antheil's interest in female endocrinology that brought him and Lamarr together. Antheil had a theory about how men could tell the availability of women based on the glandular effects of their appearance. He consulted with Lamarr on this, and the result of this research was published in his book, "The Glandbook for the Questing Male." Their research broadened, and after much joint discussion they devised a secret communication system that is now regarded as the predecessor of the today's Spread Specturm "frequency hopping" technology used in communications systems all around the world. Antheil had already applied this technique to control the 88 keys of player pianos in his strange musical compositions. Lamarr and Antheil submitted their idea to the US Patent office in June of 1941 and were granted a Patent (US Patent # 2,292,387). One of the figures for the patent is show below. It was clear that they believed their technique could be applied to radio-guided torpedoes and aid the war effort. The US Military took notice, but in 1942 it did not possess the technology to implement Spread Spectrum-based control systems to guide their torpedoes. However, in 1962 during the US blockade of Cuba during the infamous "Cuban Missile Crisis." Lamarr and Antheil's technology was deployed and proven to work (although their Patent had long expired). Here is a very short YouTube clip about Hedy Lamarr. (Feel free to explore the others posted on YouTube).... And here is the beginning of George Antheil's Ballet Mécanique in a recent performance by the New Jersey Percussion Ensemble (Peter Jarvis conducting)... Tuesday, March 9, 2010 CD Review: Sheila Mac Donald Regular followers of this blog might be surprised to see this CD review, since I don't regularly write about Folk music. But the fact is that I listen to all kinds of music, and at earlier stages in my life as a guitarist I accompanied and performed widely in Rock, Jazz, C&W, and Folk venues. This review is of a new Folk CD released in January of this year by Boston-based independent recording artist Sheila Mac Donald. The album is titled This Way. She is a songwriter, vocalist, and guitarist. Mac Donald's new CD is comprised of interesting and mysterious songs with lyrics that hint at harsh realities and matters of personal loss. Yet, at least on the surface, the songs transcend the ordinary to provide a rare insight into the human experience. (photo by Karen Holland) There is a high degree of imagination, fantasy, and wisdom in Mac Donald's lyrics. She writes and sings from the heart, even though at times her songs seem deeply philosophical, metaphorical, abstract, or even pensive and dark. Musically, the melodies and accompaniment are spirited, joyful, and very pleasant to listen to. Mac Donald's songs have a simplicity and honest quality that's clearly hard-earned and rings true. It's hard not to tune into and receive the bittersweet message that this artist portrays in her album of songs. Her song "The Blue" is a song about workplace imprisonment. It reflects on the psychological mindset of a creative person eking out a living while working in retail. It is a song that I can fully understand and sympathize with. Mac Donald was born and raised in Quincy, MA, and hails from a combination of Irish and Scottish heritage (her paternal grandparents were from Nova Scotia). This Celtic ancestry can be heard in some of her songs - particularly "Bare Branches" and "Burning Slow" - which have an explicit Irish sound. But the songs are also quite contemporary and individual. When I inquired about the genesis of her music, Mac Donald provided the following bit of information, "Most of the 14 songs are recent but I pulled some out of the notebooks that were older. I wrote Sarah and Sandra in 1993, Grazna, and Burning Slow in 1998." Mac Donald has composed many songs, and recorded several of her works for Fast Folk Musical Magazine (organized by Jack Hardy in the early 90s and available on Smithsonian Folkways). Her song "Night Bird's Song" was covered at a Fast Folk Live at the Bottom Line show in NYC and recorded on CD. Mac Donald's "My Wallet" was included on the Chill Out East Coast Edition, Vol. 10 compilation CD (2008). This Way is Mac Donald's first full length CD. It was recorded, engineered, and mixed at blue fish sound productions in Marblehead, MA. Mac Donald collaborated with some of Boston's finest musicians on this album, which was skillfully produced and arranged by Raymond Gonzalez. Gonzalez plays guitar, bass, mandolin, and keyboards on many of the tracks. Violinist Pam Kuras joins them for two tracks with her Celtic sounding violin playing. The CD is professionally mastered by Toby Mountain at Northeastern Digital, in Southborough, MA. Mac Donald's training includes formal academic studies in music and composition at Brandeis University in the 1980s. At Brandeis, she was awarded two of the music department's most prestigious awards: the Remis and Reiner. She studied composition, counterpoint, harmony, and music theory. I once had her as a student myself. Her avant-garde musical compositions from that period include a wonderful but edgy atonal solo violin work that was performed in concert. But it is in Folk music that she has found her true voice. Mac Donald is a member of ASCAP. "This Way" can be purchased directly on CD Baby at: or from iTunes The Artist's MySpace page provides more information about the singer/songwriter as well as ways to purchase the CD by mail: Information about Mac Donald's Folkways recording: Monday, March 8, 2010 The Bohlen-Pierce scale Boston is hosting an interesting symposium and concert series this week (March 7-9, 2010). It began yesterday and continues through tomorrow evening. The event is drawing a diverse crowd of composers, musicians, mathematicians, music theorists, computer scientists, researchers, neurologists, and musical instrument builders from all over the world - all because of an intriguing and unifying idea. The idea that has caught their imagination is a new scale. The scale was conceived independently by two microwave engineers and a computer scientist in the 1970s and 80s. It is now referred to as the Bohlen-Pierce or "BP" scale. Heinz Bohlen, Kees van Prooijen, and John R. Pierce all had a hand in it's discovery. The BP scale is rather unique. There are several variants, but the primary idea is that BP utilizes the 3:1 ratio instead of the 2:1 ratio that defines the traditional even-tempered scale used in most Western music. With traditional Western music, the octave is a basic and primary interval. It's derived from the 2:1 ratio, and from that we divide the octave into 12 equal steps. BP replaces the octave with something they call a "tritave." Arriving at the tritave in the notes of a rising BP scale does provide a melodic sense of closure or completion. With the BP scale, the 3:1 ratio defines the lower and upper degrees of the scale (which happen to be the span of what we usually think of as an octave and a fifth). That range is then divided into 13 steps. The Bohlen-Pierce Symposium and Concerts are sponsored by the Boston Microtonal Society and Georg Hajdu (Hamburg Hochschule für Musik und Theater) in partnership with the Goethe Institute of Boston, the Berklee College of Music, Northeastern University, and New England Conservatory. The conference website is I was intrigued by the idea of this new scale, and attended the first concert at the Fenway Center at Northeastern University last evening to hear what composers are doing with it. The concert featured nine pieces - although the BP conference in total will showcase 24 premieres of pieces written by composers from around the globe who are utilizing the BP scale. My first impressions of this new musical structure are mixed. While there seems to be a solid theoretical basis for BP, how composers and musicians apply the raw material of sound is the ultimate proof in the pudding. There seems to be a number of aesthetic and logistical issues regarding the execution. First, it's pretty hard to abandon the interval of an octave. It's so universal and ingrained in virtually all the music we have known up to this point, that skipping over the octave seems strange. In the BP sound world, the octave is an an invisible elephant sitting on the stage. Electronic realizations of the BP scale sound so much more convincing than realizations produced by singers or instruments. I'm not convinced that musicians have yet attained the needed hearing and performance skills to accurately render the BP scale. Nor have our ears grown familiar enough with it. For instance, one of the more successful works on the concert was Five Moods by Anthony De Ritis. De Ritis recorded BP clarinetist Amy Advocat and produced a tape piece based on those sounds and pitches. From his piece, I could hear the totality of the scale, and how it has some very consonant properties. All music is cultural. The basic premise that the BP system seems to ride on is a notion that the current 12-note equal-tempered scale is somehow inferior. The BP scale, while still imperfect, strives (at least theoretically) to make a better map onto the frequencies implied by Nature's Grand Dame - the overtone series. My beef with that objective is one of personal bias. Who says that musical systems should follow Nature's lead? Why the heck should human-kind not divide musical intervals as they please. I like to hear my music served up on a plate with "in-harmonic" intervals. I like the sound of notes and their overtones beating "out of tune." I like dissonance. I like tone clusters, scalar symmetry, and the certainty of an equally loaded 12-gage keyboard. The traditional semi-tone is one of my favorite intervals, and I don't desire anything smaller - particularly when it is dispersed using octave-equivalence over several spanning registers. Octave equivalence is an amazing property. As with the Early Music folks, the BP advocates seem to want just intonation and tonal consonance. They believe that the modern day piano is corrupt, evil, and ugly. Frankly, I just don't share that point of view. I'm a composer who works mostly in an atonal universe. For that sound scape, a 12-note even-tempered scalar system works rather nicely. Musicians are trained to hear it, read it, and perform it. It doesn't require relearning a new system. A very good notational system exists for the 12-tone system which has evolved through the collective efforts of musicians over centuries of practical use and applicaton. BP music notation (or the prevailing version of it) uses the standard five-line music staff and traditional clefs that musicians are familiar with, but the seven notes C-D-E-F-G-A-B are augmented with H and J. The thing that throws me off is that while the musicians may be reading the note G - what actually sounds (and the interval it creates with the preceding note) is something entirely different. BP notation and keyboards need to go through a lot more development. Will BP catch on and become mainstream? Probably not. It's hard to create a tradition when only six BP tuned clarinets exist in the entire world (four of which are in Boston for the concert series). The BP clarinet should be the hallmark for this scale, since the clarinet is acoustically ideal for it. It over blows in the correct ratio, and it's "square wave" timbre nicely reinforces the gestalt of the scale by providing energy on the odd-numbered overtones. The bottom line is that composers will end up writing what they want to hear. What they want to hear is, in the end, approximated by whatever system or scale they happen to be using. Musical ideas fall onto a pitch-frequency grid, and while the grid can vary, the musical ideas themselves transcend the surface characteristics of the aural medium. There is no magic bullet, and no substitute for genuine and substantive musical ideas regardless of the tuning system it is constructed on top of. Tuning systems are secondary - almost arbitrary. There were some pieces on last evening's program that caught my attention. For example Liebesleid (2010) , a short work by James Bergin, seemed classical in conception. Bergin took a rather conservative approach to the BP scale, and worked within the constraints of a simple melodic materials. His piece was straight-forward and elegant. His intervals in the BP language sounded large compared to other pieces that I've heard from him, although the same unique composers' voice still comes through regardless of the underline pitch system. Julia Werntz's piece Imperfections (2010) was also for solo BP clarinet. It too was short and simple, and to me sounded like a transcription of her 72-note microtonal music. In fact, she converted the BP scale and notation into a subset of the language she usually composes in (a 72-note system devised by Ezra Sims), and back again after the piece was conceived. Having heard her music before, Imperfections sounded like a subset of her normal sound world. The ending of the piece intrigued me. The wide leaps did point to a coherence in the BP scale. It did make me wonder if BP is actually a universal chord, rather than a scalar set of distinct stand-alone pitches related to one-another. When heard as a chord, all the notes seem to be cut from the same cloth, and seem intuitively related - like members of a family. I want to keep abreast of developments in the BP field. But at present, I don't see it as a new panacea or musical Shangri-La. I'm quite content to keep composing in the system that has done me well over all of these years. I'm find no shortage of relationships to exploit in the 12-pitch even-tempered system. It's not lacking in any way. In fact, 12-pitches per octave is about all that I can handle. Sunday, March 7, 2010 A Concert of Spectral Music On Friday evening March 5th, 2010 the sounds of Spectral Music could be heard at Boston University's Marsh Chapel. That's where the Xanthos Ensemble presented a program featuring music of four composers whom - each in their own way - represent a unique approach to this still emerging movement of contemporary musical composition. Spectral music is a hard to define. The term was coined by the French composer/philosopher Hugues Dufourt. Dufourt used his term Spectral music in an article published in the early 1980s, and the label caught on. The Spectral music movement has evolved into a bona fide musical trend. It has functioned as an aesthetic school of thought since the late 1980s. Composers working in this methodology typically embrace the framework of a generalized musical approach rather than a specific style or "ism." Spectral music does not espouse any particular technique, but rather is passionately indicative of working methods that are rooted deeply in current scientific research, psychoacoustic theory, as well as digital sound analysis and synthesis. Today, Spectral music - at least in the United States, is still a sub-genre of the overall mainstream new music scene. Practitioners of this art form appear to be united, organized, and well-placed. They seem to have plenty to talk about, and easily quote from scientific studies or cite from memory passages of Hermann von Helmholtz's classic treatise On the Sensations of Tone. It seems at times as if they view musical instruments as sound generating devices that can be scientifically manipulated to realize interesting and novel spectral patterns. In their world, cellos and pianos function as hardware that can be called up to execute the coded instructions of their creative software. The Spectral music concept has some merit. Many important discoveries about the nature of sound have emerged in recent decades. Scientific resources and acoustical data are more available to composers today than ever before in history. With the help of ubiquitous software programs for spectral analysis, composers can study sounds, and then re-generate those very specific overtones in a musical work using a combination of traditional instruments and extended performance techniques (such as wind instrument multiphonics and microtonality). Composers working in this discipline could in theory make a cello sound like a screaming chicken, or replicate the hum and buzz of a modern factory - using only traditional musical instruments as their sound generation equipment. But rock solid technique does not automatically translate into a viable artistic movement. Does Spectral music have feet to walk on? It seems to me that the relative success of the Spectral music movement has more to do with filling a void. The public's desire to move on from the staid and entrenched (albeit broadly misunderstood) period of musical exploration that dominated the 1950s and 60s could have been fulfilled by any "ism." According to Wikipedia... Spectral music represented an alternative to the prestige of the serialists and post-serialists as the vanguard of serious musical composition and compositional technique. In the mid-1980s, Spectral music was new, cool, and very European. It engendered an aesthetic predominance practiced by a slew of followers led and inspired by the French composer-conductor Pierre Boulez. They were a new generation of composers, theorists, scientists and technicians who worked at the famed research institute in Paris known as IRCAM. Perhaps too Spectral music replaced Serialism as the latest nexus between musical and scientific thought. It became a space for composers to exercise their proclivities of rational thinking and scientific method in service of their more primitive musical instincts. It allows composers to "bang on a can" while at the same time rationalizing the sound using a host of convincing algorithms, formulas, and/or spectral diagrams. Spectral music seems to play by a slightly different set of rules than traditional modernist music. It avoids traditional notions of form. Pitch and melody tend to function secondarily to the parameter of timbre. Spectral music is obsessed with the micro-events of acoustical phenomenon over the role of traditional musical narrative and development. I did not follow the progress of Spectral music too closely when it hit the music scene with a vengeance in the 80s and 90s. So, as you can imagine, I was interested to hear a concert dedicated to this musical genre. The Xanthos concert was an opportunity to hear four mature pieces by skilled composers working in this specialized field for some time. The composers Xanthos Ensemble program were Tristan Murail (b. 1947), Ronald Bruce Smith, Joshua Fineberg (b. 1969), and Gérard Grisey (1946-1998). Murail and Grisey are two names closely associated with this movement. The first work, "Seven Lakes Drive" (2006) by Tristan Murail was for flute, clarinet, horn, piano, violin, and violoncello. According to the composers' program notes, "The material of the piece is built on the natural resonances of the French horn and the piano." I noticed that the cello often played in the stratosphere, in a register well above the violin. The French horn explored the overtone series, and much of the music resonated in the piano's open shell. The second work on the program was a Boston premiere by the Canadian composer Ronald Bruce Smith. He is currently on the faculty at Northeastern University. His piece Remembrances of a Garden is inspired by paintings and concepts found in painting - namely Paul Klee and Claude Monet. The work is scored for flute/piccolo, clarinet/bass clarinet, percussion, piano, violin, and violoncello. Remembrances of a Garden is a formative work, full of rich detail and interesting sonic occurrences. It presented well in the acoustical space of Marsh Chapel. Joshua Fineberg 's Veils for solo piano was a real discovery. The shimmering sounds of the piano filled the air with vivid colors, vibration, and a blanket of interesting musical objects flowing through time. The composer wrote in his program notes, "It is not the notes (or not only the notes) which draw me to the piano; rather, for me, the real magic of the piano is its resonance." For me, Veils makes a convincing argument in favor of timbre over pitch. Eunyoung Kim was impressive and commanding at the piano, attacking the keyboard at times with Zen-like force and an intuitive conviction about the notes. She can make the piano sing, like a chorus of 10 thousand voices. The concert ended with a Spectral music classic, Talea by Gérard Grisey. Grisey died in 1998 at the age of 52, so we will never know where the trajectory of his music development would have eventually led. But his work Talea (1986) seemed rather traditional to me in many ways. It contains a primary motif in the form of a well defined (but not-pitch specific) gesture. That central idea is developed and transformed in ways that remind us of music of the past. That's not so revolutionary! The musicians of the Xanthos Ensemble met and exceeded my already high expectations of this Boston-based new music sensation. They played with grace and panache. Joanna Goldstein (flute), Alexis Lanz (clarinet), Brenda van der Merwe (violin and viola), Leo Eguchi (cello), Joseph Walker (French horn), Eunyoung Kim (piano), and George Nickson (percussion) were directed by conductor Jeffrey Means. A nice reception followed the concert. Most of the Boston University music department composition faculty were there in force. I look forward to hearing what the Xanthos Ensemble dreams up for concerts in the future. They are the real deal. Xanthos Ensemble Presented by the Boston University College of Fine Arts - School of Music Marsh Chapel, Boston University March 5th, 2010 Friday, March 5, 2010 Musical Palindromes I just finished a new work for flute, Bb clarinet (doubling on bass Cl.), piano, violin, cello, and percussion. It's a pretty big piece - in three movements and lasting over 15 minutes in duration. The first movement exploits a musical technique that I've always wanted to experiment with, but never got around to. It plays with palindromes. I've titled this movement Immagine Speculare. While the musical flow of Immagine Speculare tosses around large phrases of its' thematic material within its' multi-dimensional mirror-image layered texture, the music on the surface progresses in a continuous and transparent way that should appear to the listener as logically constructed and naturally derived. It's also a fast and furious work that demands a lot of physical and mental energy, not to mention calorie expenditure on the part of hard-working performers. But I'm certainly not creating anything original by using this technique, since musical palindrome has been around for a long time. It goes back to at least the "crab canon" where one voice is reversed in time and pitch from the second voice (Crabs walk backward). Haydn's Symphony #47 has a minuet and trio in musical palindrome. Mozart's Scherzo-Duetto plays this game too. One of my favorite musical palindromes is from Alban Berg's opera Lulu. He was influenced by the technical possibilities that were emerging in the avant-garde silent film industry at the time. His opera used palindrome on the level of drama and within the structure of the music. In my view, the interlude contains some of the best music in the opera. Béla Bartók too was very fond of "arch form" which allows the composer to frame the structure of a movement (or the larger work) according to a palindromic pathway of musical association. For example: A B C D C' B' A'. Anton Webern is another shinning example of someone who liked this structure. Webern is one of my musical heroes, but his music at present seems to virtually ignored by the reigning musical establishment. Yet he never met a palindrome that he didn't like. You could say that Webern was symmetry obsessed - choosing symmetry to organize every structure imaginable in the time and pitch domains of his musical works (both vertically and horizontally). His beautiful Symphonie Op. 21 (second movement) has a cool palindrome right near the beginning. Igor Stravinsky's rendition of Edward Leer's famous nonsense poem, The Owl and the Pussy Cat is another good example of palindrome. The composer scored it for boy soprano and piano in 1966 when he was 84 years old, and it turned out to be his final original composition. The music uses the 12-tone technique, yet it is tuneful enough for a child to sing (with some practice). And of course The Owl and the Pussy Cat music utilizes palindromic structure. And then there is the famous example of Der Mondfleck, the 18th movement of Arnold Schönberg's history changing Pierrot Lunaire op. 21. After the piano introduction, the music is palindromic. Here is a spirited performance of Der Mondfleck conducted by Pieter van der Wulp with Ensemble 88 in Holland. The soprano is Bauwien van der Meer. Enjoy... Thursday, March 4, 2010 A few palindromes Yo, banana boy! No lemon, no melon. No cab, no tuna nut on bacon. Dammit, I'm mad. Dennis and Edna sinned. As I pee, sir, I see Pisa. Anne, I vote more cars race Rome to Vienna. A Toyota. Race fast, safe car. A Toyota. Devil lived Boston did not sob. Not so, Boston! If I had hi-fi. Paris, I rap. Nina Ricci ran in. Can I attain a "C"? "La" - not atonal. Satan, oscillate my metallic sonatas! Air an aria. In Italian... odo (to hear) ossesso (obsessed) ottetto (octet) animina (little soul) onorarono (to honor) ingegni (ingeniousness) otto (eight) ala (wing) In Dutch... Legovogel (Lego bird) Today's Google Doodle Today's Google Doodle celebrates the birthday of a great Venetian baroque composer. Thank you Google for bringing a composer to our attention. Happy Birthday Antonio! Q: How old are you now? A: 332 That's old! Going Rogue Waves are everywhere. Most of the time they are very predictable. The mathematics of probability theory show that the size of waves fall within a very clearly defined window of measurable and predictable amplitude. But is size everything? Earlier this week a 26-foot rogue wave came out of nowhere to hit a cruise ship in the French Mediterranean. Two passengers were killed and six people injured. The study and science of "rogue" or "freak" waves is rather new. It's really in the past decade that researchers have had the tools to study the phenomenon in depth. Many didn't even believe that the phenomenon existed, since there was no clear scientific evidence on record. The entire world-wide shipping industry has evolved on the assumption that waves will never attain a height beyond 50-feet. Ships are designed to withstand that "worst case" scenario, but anything taller is likely to inflict damage or disaster. But in the last two decades over 200 supertankers and container ships exceeding 200 metres in length have been damaged or were sunk by this random and freakish act of nature. Then on New Year's day in 1995, the Draupner platform in the North Sea was pounded by one of the mysterious rogue waves. It was massive, and oddly enough, it was the first time a rogue wave had ever been recorded. A rogue wave is not the same as Tsunami. They come out of nowhere, and can occur anywhere in the sea. They also defy our assumptions about how waves are created. The new paradigm regarding rogue waves utilizes a different mathematical model, one that is associated with quantum mechanics. Quantum mechanics does not neatly conform our old rules of classical physics. Rogue waves are not possible in Newton's world In a nutshell, rogue waves are thought to be nonlinear. If you are the sort of person that needs an equation to explain the ocean surf, then you will be comforted to know that the nonlinear Schrödinger equation (NLS) does the job. For those of us who are math-challenged, all you really need to know is that rogue waves start off as a "normal" wave, but then suck up energy from its' adjacent neighbors. The waves just before and after the emerging rogue wave are demoted to mere ripples. Their energy is transferred to the rogue wave, and it becomes a solid wall of moving water. The upper limit is not known, but past encounters have shown that rogue waves can attain a height of well over 100 feet. Think about that the next time you book a vacation cruise. As someone who in inclined to think defensively, I began to wonder if the nonlinear Schrödinger equation might apply to other areas of the physical universe. For example, is there such thing as a rogue sound wave? Is it possible that a non suspecting audience at Symphony Hall attending a concert by the Boston Symphony might be subjected suddenly to a "death wave" of sound emerging randomly from a performance of Beethoven's 5th symphony? I've heard a lot of rogue pieces of music in my life, but so far the random killer piece of music has resulted from acts that are purely human in origin. But it does beg the question since this nonlinear phenomenon is proven to be fact (albeit relatively rare): Is there even a remote possibility that a freakish synergy created by some strange combination of audio frequencies could behave in this deadly fashion? To be honest, I wouldn't rule it out. Wednesday, March 3, 2010 Thoughts about Novell There's a little M&A activity to report in the news, and I'm not talking about Music and Art. Yesterday after the stock market ended its session, a NY-based hedge fund firm called Elliott Associates offered to buy out the software and networking company Novell for an even $2 billion dollars. It was an unsolicited bid (aka hostile takeover). The hedge fund's modus operandi is to take "an activist approach to investing, frequently amassing significant but minority stakes in distressed or under performing companies and attempting to foment change." Elliott Associates (and its' parent Elliott International) manage more than $16 billion of capital for large institutional investors and wealthy individuals. They clearly see an opportunity and bargain in Novell. In after-hours trading Elliott offered to pay $5.75 a share in cash for Novell stock. That's 21 percent higher than Novell’s price at close yesterday. Predictably, there was a upward tick in Novell’s share price: it rose $1.23, or nearly 26 percent, to $5.98 in after-hours trading. Elliott Associates - an investment firm that specializes in "distressed companies" -currently holds more than eight percent of Novell’s stock. It is offering an additional 1.8 billion to take over the company from stock holders. Novell’s Board of Directors received a letter from Elliott Associates explaining the rationale for the firm’s bid: Over the past several years, the company has attempted to diversify away from its legacy division with a series of acquisitions and changes in strategic focus that have largely been unsuccessful. As a result, we believe the company’s stock has meaningfully underperformed all relevant indices and peers. With over 33 years of experience in investing in public and private companies and an extensive track record of successfully structuring and executing acquisitions in the technology space, we believe that Elliott is uniquely situated to deliver maximum value to the company’s stockholders on an expedited basis. While I do not own any Novell stock personally, I have worked with their products over a period that spans decades, and am quite familiar with the company history. Their corporate HQ is actually nearby in Waltham, MA. I've been to the tech giant's "Brain Share" conference in Salt Lake City numerous times, and know or have worked with scores of folks associated with the company. I'd be sorry to see Novell bought and cannibalized for it's remaining cash, trademarks, patents, and software inventory (such as SUSE Linux). In this ailing economy, the vultures are having a field day. There is some truth that Novell has made famous and grandiose mistakes in strategy and marketing over the years, but their technology has for the most part been innovative, ahead of the curve, and rock solid. I was very hopeful when ex-Sun Microsystems CTO Eric Schmidt took over as Novell's CEO in 1997. Schmidt had a clear understanding of the importance of the Internet and how cloud computing would eventually dominate the industry. At Sun he led its Java development efforts. I recell being invited to a top-level executive briefing with Eric Schmidt at Novell's office in Wellseley, MA. Schmidt wanted to hear first-hand from Novell's customer base. The international company I worked for at the time (Zurich Scudder Kemper Investments) was a significant corporate client. The meeting was informative and pleasant. Schmidt had just flown into town on Novell's corporate jet. I recall that he parked his plane at Hansom Field in Bedf0rd, preferring it over Logan International. Schmidt spoke to the small group about Novell's initiatives, listened to our ideas and concerns, and met with us individually. Customers were provided with lunch, and I received a nice leather notebook case and cool ball-point pen with the Novell logo inscribed on it. The future looked rosy. In 2001 things changed. A local IT consulting company that was heavily invested in Microsoft technology - and who's core business was outsourcing software development - was "acquired" by Novell. It was called Cambridge Technology Partners (CTP), and their offices were in a renovated brick factory building along Memorial Drive in Cambridge across the Charles River from BU. The street talk at the time was that the merger was a good idea, since Novell would benefit from the additional talent of CTP's programming staff. In reality, CTP assumed the upper edge, took over Novell's management team, robbed it of cash, and led the company down a confused and ultimately dire path. Schmidt left Novell soon after the acquisition of CTP. The rumor was that he found managing the new Novell quite difficult. The remaining senior management of the company was still entrenched in their old business models, and a myriad of complications had been introduced by the merger with CTP. Lingering complications from Novell's prior mergers (such as WordPerfect) contributed to the dysfunction. The magnitude of the organizational turmoil set Novell firmly into a state of classic management gridlock - and for an IT company that is a bad state to be in. Around this time Google founders Larry Page and Sergy Brin met secretly with Schmidt. They realized that they had a lot in common (such as flying airplanes). The trio hit if off in a way that only geeky software engineers can do, and Schmidt agreed to assist the dynamic duo in transforming Google into a real company. The rest is history. Novell's loss was Google's gain. I still have my spiffy black leather Novell executive case and pen. I retain my obscure Novell certifications of CNE and Master CNE. But although Novell's customers still exist, they are much harder to find. I know that from looking at the employment pages. All of this technology (and the companies that created them) appear to be dissipating into the proverbial "cloud." Tuesday, March 2, 2010 Japan consumes 10 percent of the world's fish. 40 percent of that is imported.
f473c3c5ab8623c2	The Coulomb's force is given by $$ F = {k q^2 \over r^2} $$ When $ r \rightarrow 0 $, $ F \rightarrow \infty $ Does this mean two electrons never touch each other? • 1 $\begingroup$ Define touch. Electrons are standing waves and are known to show interference patterns. I don't know what you mean by them 'touching'. $\endgroup$ – Gerard Aug 4 '13 at 12:05 • $\begingroup$ @user1305192 can there not be free electron in space at some given postion? $\endgroup$ – hasExams Aug 5 '13 at 15:57 • 1 $\begingroup$ Pauli Exclusion . . . $\endgroup$ – Abhimanyu Pallavi Sudhir Aug 9 '13 at 11:55 • 2 $\begingroup$ Pauli Exclusion won't save them from touching if they have different spins $\endgroup$ – Ruslan May 22 '14 at 6:17 • 1 $\begingroup$ @Ruslan so 50% of the time it works every time $\endgroup$ – Jim May 22 '14 at 14:28 Yes, two electrons or two charged particles are not allowed to overlap – the interaction energy goes like $1/r$ so it would diverge in the strict limit and the world doesn't have the infinite energy to do that. Two electrons are forbidden to overlap for another reason, the Pauli exclusion principle. On the other hand, two particles of opposite signs attract by a force that goes to infinity in the $r\to 0$ limit, too. Nevertheless, even in this case, they won't end up overlapping due to the uncertainty principle of quantum mechanics. For example, take the hydrogen atom (or any atom). The electron can't sit exactly at the nucleus even though it would save an infinite amount of energy. The reason is that a very low $r$ implies a very low $\Delta x$ of the same order and, therefore, a high $\Delta p \geq \hbar / \Delta x$. A high $\Delta p$ means a high average $\Delta p^2$, and therefore high kinetic energy, and this one actually wins if the proton-electron distance is too short. If the electrostatic force increased faster than $1/r^2$ at short distances, one could actually beat the kinetic energy and oppositely charged particles would choose to sit on top of each other. • $\begingroup$ @ Lubos Motl what dou you mean by electrostatic force increased faster than 1/r^2 .please clarify on that. $\endgroup$ – Ufomammut Mar 31 '13 at 12:42 • $\begingroup$ The answer is wrong. Correct one is the opposite - see my answer. $\endgroup$ – Ruslan May 22 '14 at 7:40 • $\begingroup$ @AaKASH - I mean if the potential energy were $K/r^n$ for $n\gt 2$ or as any function that is greater than that for all $r\lt \varepsilon$. For example $K/r^3$. This is so huge near $r=0$ that even the huge kinetic energy you get from the inevitable $\Delta p$ is smaller and it's energetically favored for the electron to sit strictly at $r=0$. Of course, it's just a mathematical limit - you won't find those things exactly in Nature. $\endgroup$ – Luboš Motl May 30 '14 at 4:52 • 1 $\begingroup$ Lubos is nothing but a dick. $\endgroup$ – Les Adieux Aug 26 '16 at 17:23 • $\begingroup$ Thanks, you're far from the first person who considers me a reincarnation of dick feynman. $\endgroup$ – Luboš Motl Aug 27 '16 at 15:10 Although this is quite an old question, I have to disagree with answer by Luboš. First, Pauli exclusion principle says that no two fermions can share the same state. But, if the electrons have different spins (i.e. are in so called spin-singlet state), then they can be in the same positional state. Next, indeed, in classical case, two charged particles with equal signs cannot touch, because their potential energy $\sim1/r$ will go to infinity at the collision point. But, as electrons are quantum particles, they obey uncertainty principle, which allows them, in particular, to tunnel to classically restricted locations. To more precisely describe their motion, one has to use Schrödinger equation. Suppose two electrons with different spins are in a parabolic 3D potential well, while interacting by Coulomb force between each other. Schrödinger equation for them can be simplified by separation of variables. Then the part for inter-electron wave function would look like (neglecting dimensional constants): The boundary conditions for $\phi(r)$ are boundedness at $r=0$ and $r\to\infty$. Solving* this equation with $l=0$ (otherwise electrons won't touch because of centrifugal force, which grows faster than Coulomb force), we get the wavefunction for ground state: enter image description here As you can see, the wavefunction doesn't go to zero at $r=0$. Neither will it vanish for excited $S$-states (i.e. excited states with $l=0$). Instead, there's a cusp with a local minimum of probability density at the point of collision. Were the potential to go to infinity at a higher rate, like $r^{-2}$, the wavefunction would indeed vanish at that point. This is what happens for $P, D$ and other states with $l>0$. Note that the reason for this is very similar to reason why electron doesn't have infinite binding energy in atom in $S$ states: the wavefunction there has a local maximum cusp, but it still is bounded. For potentials $\sim -r^{-2}$ the electron would fall onto the nucleus and have infinite binding energy. * See the (paywalled) article for details on separation of variables. ** I solved it via Frobenius method, limiting to 1000 series terms. I'm not sure if there's a closed-form solution for this BVP. • $\begingroup$ An interesting answer, @Ruslan, but I don't know why it implies that they will touch. The probability of strictly $r=0$ is still zero, right? Would you say that the electron touches the nucleus at the 1s state? $\endgroup$ – Luboš Motl May 30 '14 at 4:54 • $\begingroup$ @LubošMotl Of course, probability at $(r,\theta,\phi)=(0,?,?)$ is zero, but so it is for any other $(r,\theta,\phi)$ (because of integration over zero volume). The difference with centrifugal potential is that in the Coulomb case relative probability $P(0,?,?)/P(r,\theta,\phi)$ is non-zero for all $r$, $\theta$ and $\phi$. So, yes, I'd say the electron does touch the nucleus in 1s state. Using point-likeness of particles as a reason for them to not touch would be somewhat vacuous (and then your answer taking potential into account would be redundant). $\endgroup$ – Ruslan May 30 '14 at 8:34 • 1 $\begingroup$ It is not redundant! For attractive potentials stronger than $1/r^2$, like $-1/r^3$, the electron would touch the nucleus even in my, strong sense, as the wave function would be proportional to the delta-function! So the probability for the electron to be strictly at the origin would be positive, finite. I would agree with you that the probability density for the $1s$ state is parametrically higher than for $l\neq 0$ states but it isn't enough to call these things "touching" (in the approximation of a pointlike nucleus; for a finite realistic nucleus, the touching occurs for any $l$). $\endgroup$ – Luboš Motl Jun 3 '14 at 11:26 • $\begingroup$ Indeed. Although, I would't call it a touch, instead it'd be a (permanent and irreversible) fusion :) Also it seems wavefunction won't be proportional to delta function per se, although it would be something qualitatively very similar to it. $\endgroup$ – Ruslan Jun 3 '14 at 11:41 • $\begingroup$ @user104 collision of quantum particles is a fuzzy concept. Usually it just means scattering (elastic or inelastic), and doesn't imply actual touch. $\endgroup$ – Ruslan Feb 14 '16 at 11:59 Your Answer
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Klemm Simon Kochen Hans Kornhuber Stephen Kosslyn Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé Pierre-Simon Laplace David Layzer Benjamin Libet Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch Ulrich Mohrhoff Jacques Monod Emmy Noether Abraham Pais Howard Pattee Wolfgang Pauli Massimo Pauri Roger Penrose Steven Pinker Colin Pittendrigh Max Planck Susan Pockett Henri Poincaré Daniel Pollen Ilya Prigogine Hans Primas Adolphe Quételet Jürgen Renn/a> Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Claude Shannon David Shiang Herbert Simon Dean Keith Simonton B. F. Skinner Lee Smolin Ray Solomonoff Roger Sperry John Stachel Henry Stapp Tom Stonier Antoine Suarez Leo Szilard Max Tegmark William Thomson (Kelvin) Giulio Tononi Peter Tse Vlatko Vedral Heinz von Foerster John von Neumann John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson Stephen Wolfram H. Dieter Zeh Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Free Will Mental Causation James Symposium Copenhagen Interpretation of Quantum Mechanics The idea that there was a Copenhagen way of thinking was christened as the "Kopenhagener Geist der Quantentheorie" by Werner Heisenberg in the introduction to his 1930 textbook The Physical Principles of Quantum Theory, based on his 1929 lectures in Chicago (given at the invitation of Arthur Holly Compton). It is a sad fact that Einstein, who had found more than any other scientist on the quantum interaction of electrons and photons, was largely ignored or misunderstood at this Solvay, when he again clearly described nonlocality At the 1927 Solvay conference on physics entitled "Electrons and Photons," Niels Bohr and Heisenberg consolidated their Copenhagen view as a "complete" picture of quantum physics, despite the fact that they could not, or would not, visualize or otherwise explain exactly what is going on in the microscopic world of "quantum reality." From the earliest presentations of the ideas of the supposed "founders" of quantum mechanics, Albert Einstein had deep misgivings of the work going on in Copenhagen, although he never doubted the calculating power of their new mathematical methods. He described their work as incomplete because it is based on the statistical results of many experiments so only makes probabilistic predictions about individual experiments. Einstein hoped to visualize what is going on in an underlying "objective reality." Bohr seemed to deny the existence of a fundamental "reality," but he clearly knew and said that the physical world is largely independent of human observations. In classical physics, the physical world is assumed to be completely independent of the act of observing the world. In quantum physics, Heisenberg said that the result of an experiment depends on the free choice of the experimenter as to what to measure. The quantum world of photons and electrons might look like waves or look like particles depending on what we look for, rather than what they "are" as "things in themselves." The information interpretation of quantum mechanics says there is only one world, the quantum world. Averaging over large numbers of quantum events explains why large objects appear to be classical Bohr thus put severe epistemological limits on knowing the Kantian "things in themselves," just as Immanuel Kant had put limits on reason. The British empiricist philosophers John Locke and David Hume had put the "primary" objects beyond the reach of our "secondary" sensory perceptions. In this respect, Bohr shared the positivist views of many other empirical scientists, Ernst Mach for example. Twentieth-century analytic language philosophers thought that philosophy (and even physics) could not solve some basic problems, but only "dis-solve" them by showing them to be conceptual errors. Neither Bohr nor Heisenberg thought that macroscopic objects actually are classical. They both saw them as composed of microscopic quantum objects. The exact location of that transition from the quantum to the classically describable world was arbitrary, said Heisenberg. He called it a "cut" (Schnitt). Heisenberg's and especially John von Neumann's and Eugene Wigner's insistence on a critical role for a "conscious observer" has led to a great deal of nonsense being associated with the Copenhagen Interpretation and in the philosophy of quantum physics. Heisenberg may only have been trying to explain how knowledge reaches the observer's mind. For von Neumann and Wigner, the mind was considered a causal factor in the behavior of the quantum system. Today, a large number of panpsychists, some philosophers, and a small number of scientists, still believe that the mind of a conscious observer is needed to cause the so-called "collapse" of the wave function. A relatively large number of scientists opposing the Copenhagen Interpretation believe that there are never any "collapses" in a universal wave function. In the mid 1950's, Heisenberg reacted to David Bohm's 1952 "pilot-wave" interpretation of quantum mechanics by calling his own work the "Copenhagen Interpretation" and the only correct interpretation of quantum mechanics. A significant fraction of working quantum physicists say they agree with Heisenberg, though few have ever looked carefully into the fundamental assumptions of the Copenhagen Interpretation. This is because they pick out from the Copenhagen Interpretation just the parts they need to make quantum mechanical calculations. Most textbooks start the story of quantum mechanics with the picture provided by the work of Heisenberg, Bohr, Max Born, Pascual Jordan, Paul Dirac, and of course Erwin Schrödinger. What Exactly Is in the Copenhagen Interpretation? There are several major components to the Copenhagen Interpretation, which most historians and philosophers of science agree on: • The quantum postulates. Bohr postulated that quantum systems (beginning with his "Bohr atom" in 1913) have "stationary states" which make discontinuous "quantum jumps" between the states with the emission or absorption of radiation. Until at least 1925 Bohr insisted the radiation itself is continuous. Einstein said radiation is a discrete "light quantum" (later called a photon) as early as 1905. Ironically, largely ignorant of the history of quantum mechanics (dominated by Bohr's account), many of today's textbooks teach the "Bohr atom" as emitting or absorbing photons - Einstein light quanta! Also, although Bohr made a passing reference, virtually no one today knows that discrete energy states or quantized energy levels in matter were first discovered by Einstein in his 1907 work on specific heat. • Wave-particle duality. The complementarity of waves and particles, including a synthesis of the particle-matrix mechanics theory of Heisenberg, Max Born, and Pascual Jordan, with the wave mechanical theory of Louis deBroglie and Erwin Schrödinger. Again ironically, wave-particle duality was first described by Einstein in 1909. Heisenberg had to have his arm twisted by Bohr to accept the wave picture. • Indeterminacy principle. Heisenberg sometimes called it his "uncertainty" principle, which could imply human ignorance, implying an epistemological (knowledge) problem rather than an ontology (reality) problem. Bohr considered indeterminacy as another example of his complementarity, between the non-commuting conjugate variables momentum and position, for example, Δp Δx ≥ h (also between energy and time and between action and angle variables). • Correspondence principle. Bohr maintained that in the limit of large quantum numbers, the atomic structure of quantum systems approaches the behavior of classical systems. Bohr and Heisenberg both described this case as when Planck's quantum of action h can be neglected. They mistakenly described this as h -> 0. But h is a fundamental constant. The quantum-to-classical transition is when the action of a macroscopic object is large compared to h . As the number of quantum particles increases (as mass increases), large macroscopic objects behave like classical objects. Position and velocity become arbitrarily accurate as h / m -> 0. Δv Δx ≥ h / m. There is only one world. It is a quantum world. Ontologically it is indeterministic, but epistemically, common sense and everyday experience inclines us to see it as deterministic. Bohr and Heisenberg insisted we must use classical (deterministic?) concepts and language to communicate our knowledge about quantum processes! • Completeness. Schrödinger's wave function ψ provides a "complete" description of a quantum system, despite the fact that conjugate variables like position and momentum cannot both be known with arbitrary accuracy, as they can in classical systems. There is less information in the world than classical physics implies. The wave function ψ evolves according to the unitary deterministic Schrödinger equation of motion, conserving that information. When one possibility becomes actual (discontinuously), new information may be irreversibly created and recorded by a measurement apparatus, or simply show up as a new information structure in the world. By comparison, Einstein maintained that quantum mechanics is incomplete, because it provides only statistical information about ensembles of quantum systems. He also was deeply concerned about nonlocality and nonseparability, things not addressed at all by the Copenhagen interpretation. • Irreversible recording of information in the measuring apparatus. Without this record (a pointer reading, blackened photographic plate, Geiger counter firing, etc.), there would be nothing for observers to see and to know. Information must come into the universe long before any scientist can "observe" it. In today's high-energy physics experiments and space research, the data-analysis time between the initial measurements and the scientists seeing the results can be measured in months or years. All the founders of quantum mechanics mention the need for irreversibility. The need for positive entropy transfer away from the experiment to stabilize new information (negative entropy) so it can be observed was first shown by Leo Szilard in 1929, and later by Leon Brillouin and Rolf Landauer. • Classical apparatus?. Bohr required that the macroscopic measurement apparatus be described in ordinary "classical" language. This is a third "complementarity," now between the quantum system and the "classical apparatus" But Born and Heisenberg never said the measuring apparatus is "classical." They knew that everything is fundamentally a quantum system. Lev Landau and Evgeny Lifshitz saw a circularity in this view, "quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case [correspondence principle], yet at the same time it requires this limiting case for its own formulation. • Statistical interpretation (acausality). Born interpreted the square modulus of Schrödinger's complex wave function as the probability of finding a particle. Einstein's "ghost field" or "guiding field," deBroglie's pilot or guide wave, and Schrödinger's wave function as the distribution of the electric charge density were similar views in much earlier years. Born sometimes pointed out that his direct inspiration was Einstein. All the predicted properties of physical systems and the "laws of nature" are only probabilistic (acausal, indeterministic ). All results of physical experiments are statistical. Briefly, theories give us probabilities, experiments give us statistics. Large numbers of identical experiments provide the statistical evidence for the theoretical probabilities predicted by quantum mechanics. Bohr's emphasis on epistemological questions suggests he thought that the statistical uncertainty may only be in our knowledge. They may not describe nature itself. Or at least Bohr thought that we can not describe a "reality" for quantum objects, certainly not with classical concepts and language. However, the new concept of an immaterial possibilities function (pure information) moving through space may make quantum phenomena "visualizable." Ontological acausality, chance, and a probabilistic or statistical nature were first seen by Einstein in 1916, as Born later acknowledged. But Einstein disliked this chance. He and most scientists appear to have what William James called an "antipathy to chance." • No Visualizability?. Bohr and Heisenberg both thought we could never produce models of what is going on at the quantum level. Bohr thought that since the wave function cannot be observed we can't say anything about it. Heisenberg said probability is real and the basis for the statistical nature of quantum mechanics. Whenever we draw a diagram of the waves impinging on the two-slits, we are in fact visualizing the wave function as possible locations for a particle, with calculable probabilities for each possible location. Today we can visualize with animations many puzzles in physics, including the two-slit experiment, entanglement, and microscopic irreversibility. • No Path?. Bohr, Heisenberg, Dirac and others said we cannot describe a particle as having a path. The path comes into existence when we observe it, Heisenberg maintained. (Die “Bahn” entsteht erst dadurch, dass wir sie beobachten) Einstein's "objective reality" hoped for a deeper level of physics in which particles do have paths and, in particular, they obey conservation principles, though intermediate measurements needed to observe this fact would interfere with the experiments. • Paul Dirac formalized quantum mechanics with these three fundamental concepts, all very familiar and accepted by Bohr, Heisenberg, and the other Copenhageners: • Axiom of measurement. Bohr's stationary quantum states have eigenvalues with corresponding eigenfunctions (the eigenvalue-eigenstate link). • Superposition principle. According to Dirac's transformation theory, ψ can be represented as a linear combination of vectors that are a proper basis for the combined target quantum system and the measurement apparatus. • Projection postulate. The collapse of the wave function ψ, which is irreversible, upon interacting with the measurement apparatus and creating new information. • Two-slit experiment. A "gedanken" experiment in the 1920's, but a real experiment today, exhibits the combination of wave and particle properties. Note that what two-slit experiment really shows is There are many more elements that play lesser roles, some making the Copenhagen Interpretation very unpopular among philosophers of science and spawning new interpretations or even "formulations" of quantum mechanics. Some of these are misreadings or later accretions. They include: • The "conscious observer." The claim that quantum systems cannot change their states without an observation being made by a conscious observer. Does the collapse only occur when an observer "looks at" the system? How exactly does the mind of the observer have causal power over the physical world? (the mind-body problem). Einstein objected to the idea that his bed had diffused throughout the room and only gathered itself back together when he opened the bedroom door and looked in. John von Neumann and Eugene Wigner seemed to believe that the mind of the observer was essential, but it is not found in the original work of Bohr and Heisenberg, so should perhaps not be a part of the Copenhagen Interpretation? It has no place in standard quantum physics today • The measurement problem, including the insistence that the measuring apparatus must be described classically when it is made of quantum particles. There are actually at least three definitions of the measurement problem. 1. The claim that the two dynamical laws, unitary deterministic time evolution according to the Schrödinger equation and indeterministic collapse according to Dirac's projection postulate are logically inconsistent. They cannot both be true, it's claimed. The proper interpretation is simply that the two laws laws apply at different times in the evolution of a quantum object, one for possibilities, the other for actuality (as Heisenberg knew): • first, the unitary deterministic evolution moves through space exploring all the possibilities for interaction, • second, the indeterministic collapse randomly (acausally) selects one of those possibilities to become actual. 2. The original concern that the "collapse dynamics" (von Neumann Process 1) is not a part of the formalism (von Neumann Process 2) but is an ad hoc element, with no rules for when to apply it. If there was a deterministic law that predicted a collapse, or the decay of a radioactive nucleus, it would not be quantum mechanics! 3. Decoherence theorists say that the measurement problem is the failure to observe macroscopic superpositions, such as Schrödinger's Cat. • The many unreasonable philosophical claims for "complementarity:" e.g., that it solves the mind-body problem?, • The basic "subjectivity" of the Copenhagen interpretation. It deals with epistemological knowledge of things, rather than the "things themselves." Opposition to the Copenhagen Interpretation Albert Einstein, Louis deBroglie, and especially Erwin Schrödinger insisted on a more "complete" picture, not merely what can be said, but what we can "see," a visualization (Anschaulichkeit) of the microscopic world. But de Broglie and Schrödinger's emphasis on the wave picture made it difficult to understand material particles and their "quantum jumps." Indeed, Schrödinger and more recent physicists like John Bell and the decoherence theorists H. D. Zeh and Wojciech Zurek deny the existence of particles and the collapse of the wave function, which is central to the Copenhagen Interpretation. Perhaps the main claim of those today denying the Copenhagen Interpretation (and standard quantum mechanics) began with Schrödinger's (nd later Bell's) claim that "there are no quantum jumps." Decoherence theorists and others favoring Everett's Many-Worlds Interpretation reject Dirac's projection postulate, a cornerstone of quantum theory. Heisenberg had initially insisted on his own "matrix mechanics" of particles and their discrete, discontinuous, indeterministic behavior, the "quantum postulate" of unpredictable events that undermine the classical physics of causality. But Bohr told Heisenberg that his matrix mechanics was too narrow a view of the problem. This disappointed Heisenberg and almost ruptured their relationship. But Heisenberg came to accept the criticism and he eventually endorsed all of Bohr's deep philosophical view of quantum reality as unvisualizable. In his September Como Lecture, a month before the 1927 Solvay conference, Bohr introduced his theory of "complementarity" as a "complete" theory. It combines the contradictory notions of wave and particle. Since both are required, they complement (and "complete") one another. Although Bohr is often credited with integrating the dualism of waves and particles, it was Einstein who predicted this would be necessary as early as 1909. But in doing so, Bohr obfuscated further what was already a mysterious picture. How could something possibly be both a discrete particle and a continuous wave? Did Bohr endorse the continuous deterministic wave-mechanical views of Schrödinger? Not exactly, but Bohr's accepting Schrödinger's wave mechanics as equal to and complementing his matrix mechanics was most upsetting to Heisenberg. Bohr's Como Lecture astonished Heisenberg by actually deriving (instead of Heisenberg's heuristic microscope argument) the uncertainty principle from the space-time wave picture alone, with no reference to the acausal dynamics of Heisenberg's picture! After this, Heisenberg did the same derivation in his 1930 text and subsequently completely accepted complementarity. Heisenberg spent the next several years widely promoting Bohr's views to scientists and philosophers around the world, though he frequently lectured on his mistaken, but easily understood, argument that looking at particles disturbs them. His microscope is even today included in many elementary physics textbooks. Bohr said these contradictory wave and particle pictures are "complementary" and that both are needed for a "complete" picture. He co-opted Einstein's claim to a more "complete" picture of an objective" reality, one that might restore simultaneous knowledge of position and momentum, for example. Classical physics has twice the number of independent variables (and twice the information) as quantum physics. In this sense, it does seem more "complete." Many critics of Copenhagen thought that Bohr deliberately and provocatively embraced logically contradictory notions - of continuous deterministic waves and discrete indeterministic particles - perhaps as evidence of Kantian limits on reason and human knowledge. Kant called such contradictory truths "antinomies." The contradictions only strengthened Bohr's epistemological resolve and his insistence that physics required a subjective view unable to reach the objective nature of the "things in themselves." As Heisenberg described it in his explanation of the Copenhagen Interpretation, This again emphasizes a subjective element in the description of atomic events, since the measuring device has been constructed by the observer, and we have to remember that what we observe is not nature in itself but nature exposed to our method of questioning. Our scientific work in physics consists in asking questions about nature in the language that we possess and trying to get an answer from experiment by the means that are at our disposal. Copenhagen Interpretation on Wikipedia Copenhagen Interpretation on Stanford Encyclopedia of Philosophy "Copenhagen Interpretation of Quantum Theory", in Physics and Philosophy, Werner Heisenberg, 1958, pp.44-58 "The Copenhagen Interpretation", American Journal of Physics, 40, p.1098, Henry Stapp, 1972 "The History of Quantum Theory", in Physics and Philosophy, Werner Heisenberg, 1958, pp.30-43 For Teachers For Scholars • Born's statistical interpretation - brings in Schrödinger waves, which upset Heisenberg • uncertainty principle, March 1927 • complementarity - waves and particles, wave mechanics and matrix mechanics, again upsets Heisenberg • the two-slit experiment • measurements, observers, "disturb" a quantum system, - Microscope echo • loss of causality (Einstein knew), unsharp space-time description (wave-packet) • classical apparatus, quantum system • our goal not to understand reality, but to acquire knowledge Rosenfeld quote • Experimenter must choose either particle-like or wave-like experiment - need examples • Heisenberg uncertainty was discontinuity, intrusion of instruments, for Bohr it was "the general complementary character of description" - wave or particle • Complementarity a general framework, Heisenberg particle uncertainty a particular example • Einstein/Schrödinger want a field theory and continuous/waves only? Bohr wants sometimes waves, sometimes particles. Bohr wants always both waves and particles. • Combines Heisenberg's "free choice" of experimenter as to what to measure, with Dirac's "free choice" of Nature with deterministic evolution of possibilities followed by discontinuous and random appearance of one actual from all the possibles. Chapter 1.1 - Creation Chapter 1.3 - Information Home Part Two - Knowledge Normal \| Teacher \| Scholar
21b3feccc58cc631	I'm doing a computational project on neutron scattering and I've found that if you simulate (n,n) collisions on Uranium 238, the program predicts resonances not found in experimental data. The optical model includes imaginary surface or volume potential terms which supposedly damp out these resonances. How should an imaginary potential be interpreted in this context? The time-dependent Schrödinger equation (one dimension will do for this discussion) is \begin{align} V(x) \psi - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi = i\hbar \frac{\partial}{\partial t} \psi \end{align} Usually you can separate the solution $\psi$ into a spatial part $e^{ikx}$ and a time-varying part $e^{-i\omega t}$, so the right-hand side of the Schrödinger equation gives you $\hbar\omega\psi = E\psi$. If your problem includes absorption, you want to include that in your wavefunction as well. A term in the wavefunction $\psi \propto e^{-\Gamma t/2\hbar}$ accounts for the fact that the probability of detecting your particle anywhere, \begin{align} P_\text{anywhere} = \int_\infty\!\! dx\ \|\psi\|^2, \end{align} decreases exponentially, with its time constant $\tau$ inversely proportional to the energy width $\Gamma$. Following the usual recipe for converting from the time-dependent to time-independent Schödinger equations gives \begin{align} V(x) \psi - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi &= \left(\hbar \omega - \frac{i\Gamma}2 \right)\psi \\ \left( V(x) + \frac{i\Gamma}2 \right) \psi - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi &= \hbar \omega \psi \\ \end{align} So the effect of a complex potential (at least, as long as the imaginary part is strictly positive) is to describe solutions where absorption plays an important part in the dynamics. This is by far the most common use of complex potentials in the literature, including in neutron optics. It's not immediately obvious to me how this approach would "damp out resonances," but I can imagine a handwaving argument involving changing narrow, long-lived energy eigenstates from a potential with real $V(x)$ to states with wide intrinsic energy; the probability densities for the short-lived states get smeared over an energy of width $\Gamma$ and therefore don't compete with the short-lived resonances any more. Your Answer
a7c94e6691642dea	Pauli Exclusion Principle Get Pauli Exclusion Principle essential facts below. View Videos or join the Pauli Exclusion Principle discussion. Add Pauli Exclusion Principle to your PopFlock.com topic list for future reference or share this resource on social media. Pauli Exclusion Principle Wolfgang Pauli formulated the law stating that no two electrons can have the same set of quantum numbers. In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: n, the principal quantum number, , the Azimuthal quantum number, m, the magnetic quantum number, and ms, the spin quantum number. For example, if two electrons reside in the same orbital, then their n, , and m values are the same, therefore their ms must be different, and thus the electrons must have opposite half-integer spin projections of 1/2 and -1/2. Particles with an integer spin, or bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser or atoms in a Bose-Einstein condensate. A more rigorous statement is that concerning the exchange of two identical particles the total wave function is antisymmetric for fermions, and symmetric for bosons. This means that if the space and spin coordinates of two identical particles are interchanged, then the wave function changes its sign for fermions and does not change for bosons. The Pauli exclusion principle describes the behavior of all fermions (particles with "half-integer spin"), while bosons (particles with "integer spin") are subject to other principles. Fermions include elementary particles such as quarks, electrons and neutrinos. Additionally, baryons such as protons and neutrons (subatomic particles composed from three quarks) and some atoms (such as helium-3) are fermions, and are therefore described by the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons -- for example helium-3 has spin 1/2 and is therefore a fermion, in contrast to helium-4 which has spin 0 and is a boson.[1]:123-125 As such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the chemical behavior of atoms. "Half-integer spin" means that the intrinsic angular momentum value of fermions is (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics fermions are described by antisymmetric states. In contrast, particles with integer spin (called bosons) have symmetric wave functions; unlike fermions they may share the same quantum states. Bosons include the photon, the Cooper pairs which are responsible for superconductivity, and the W and Z bosons. (Fermions take their name from the Fermi-Dirac statistical distribution that they obey, and bosons from their Bose-Einstein distribution.) In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" by Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an even number of electrons in any given shell, and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube (see: cubical atom).[2] In 1919 chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells around the nucleus.[3] In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".[4]:203 Pauli looked for an explanation for these numbers, which were at first only empirical. At the same time he was trying to explain experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism. He found an essential clue in a 1924 paper by Edmund C. Stoner, which pointed out that, for a given value of the principal quantum number (n), the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.[5][6] Connection to quantum state symmetry The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state and the other in state , and is given by: [clarification needed] and antisymmetry under exchange means that . This implies when , which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric. Conversely, if the diagonal quantities are zero in every basis, then the wavefunction component [clarification needed] is necessarily antisymmetric. To prove it, consider the matrix element This is zero, because the two particles have zero probability to both be in the superposition state . But this is equal to The first and last terms are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey: Advanced quantum theory According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta-function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, the exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum nonlinear Schrödinger equation. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions,[7] as well as for interacting spins and Hubbard model in one dimension, and for other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere. The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate electron shell structure of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An electrically neutral atom contains bound electrons equal in number to the protons in the nucleus. Electrons, being fermions, cannot occupy the same quantum state as other electrons, so electrons have to "stack" within an atom, i.e. have different spins while at the same electron orbital as described below. An example is the neutral helium atom, which has two bound electrons, both of which can occupy the lowest-energy (1s) states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle. However, the spin can take only two different values (eigenvalues). In a lithium atom, with three bound electrons, the third electron cannot reside in a 1s state and must occupy one of the higher-energy 2s states instead. Similarly, successively larger elements must have shells of successively higher energy. The chemical properties of an element largely depend on the number of electrons in the outermost shell; atoms with different numbers of occupied electron shells but the same number of electrons in the outermost shell have similar properties, which gives rise to the periodic table of the elements.[8]:214-218 Solid state properties In conductors and semiconductors, there are very large numbers of molecular orbitals which effectively form a continuous band structure of energy levels. In strong conductors (metals) electrons are so degenerate that they cannot even contribute much to the thermal capacity of a metal.[9]:133-147 Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion. Stability of matter The stability of each electron state in an atom is described by the quantum theory of the atom, which shows that close approach of an electron to the nucleus necessarily increases the electron's kinetic energy, an application of the uncertainty principle of Heisenberg.[10] However, stability of large systems with many electrons and many nucleons is a different matter, and requires the Pauli exclusion principle.[11] It has been shown that the Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. This suggestion was first made in 1931 by Paul Ehrenfest, who pointed out that the electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms, therefore, occupy a volume and cannot be squeezed too closely together.[12] A more rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard, who considered the balance of attractive (electron-nuclear) and repulsive (electron-electron and nuclear-nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.[13][14] The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or Coulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time. Freeman Dyson and Andrew Lenard did not consider the extreme magnetic or gravitational forces that occur in some astronomical objects. In 1995 Elliott Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in neutron stars, although at a much higher density than in ordinary matter.[15] It is a consequence of general relativity that, in sufficiently intense gravitational fields, matter collapses to form a black hole. Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of white dwarf and neutron stars. In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held in hydrostatic equilibrium by degeneracy pressure, also known as Fermi pressure. This exotic form of matter is known as degenerate matter. The immense gravitational force of a star's mass is normally held in equilibrium by thermal pressure caused by heat produced in thermonuclear fusion in the star's core. In white dwarfs, which do not undergo nuclear fusion, an opposing force to gravity is provided by electron degeneracy pressure. In neutron stars, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure, neutron degeneracy pressure, albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higher density than a white dwarf. Neutron stars are the most "rigid" objects known; their Young modulus (or more accurately, bulk modulus) is 20 orders of magnitude larger than that of diamond. However, even this enormous rigidity can be overcome by the gravitational field of a massive star or by the pressure of a supernova, leading to the formation of a black hole.[16]:286-287 See also 1. ^ Kenneth S. Krane (5 November 1987). Introductory Nuclear Physics. Wiley. ISBN 978-0-471-80553-3. 2. ^ "Linus Pauling and The Nature of the Chemical Bond: A Documentary History". Special Collections & Archives Research Center - Oregon State University – via scarc.library.oregonstate.edu. 3. ^ Langmuir, Irving (1919). "The Arrangement of Electrons in Atoms and Molecules" (PDF). Journal of the American Chemical Society. 41 (6): 868-934. doi:10.1021/ja02227a002. Archived from the original (PDF) on 2012-03-30. Retrieved . 4. ^ Shaviv, Glora (2010). The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure. Springer. ISBN 978-3642020872. 5. ^ Straumann, Norbert (2004). "The Role of the Exclusion Principle for Atoms to Stars: A Historical Account". Invited talk at the 12th Workshop on Nuclear Astrophysics. arXiv:quant-ph/0403199. Bibcode:2004quant.ph..3199S. CiteSeerX 6. ^ Pauli, W. (1925). "Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren". Zeitschrift für Physik. 31: 765-783. Bibcode:1925ZPhy...31..765P. doi:10.1007/BF02980631. 7. ^ A. G. Izergin; V. E. Korepin (July 1982). "Pauli principle for one-dimensional bosons and the algebraic bethe ansatz" (PDF). Letters in Mathematical Physics. 6 (4): 283-288. doi:10.1007/BF00400323. 8. ^ Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7 9. ^ Kittel, Charles (2005), Introduction to Solid State Physics (8th ed.), USA: John Wiley & Sons, Inc., ISBN 978-0-471-41526-8 10. ^ Elliot J. Lieb The Stability of Matter and Quantum Electrodynamics 11. ^ This realization is attributed by Lieb and by G. L. Sewell (2002). Quantum Mechanics and Its Emergent Macrophysics. Princeton University Press. ISBN 0-691-05832-6. to F. J. Dyson and A. Lenard: Stability of Matter, Parts I and II (J. Math. Phys., 8, 423-434 (1967); J. Math. Phys., 9, 698-711 (1968) ). 12. ^ As described by F. J. Dyson (J.Math.Phys. 8, 1538-1545 (1967)), Ehrenfest made this suggestion in his address on the occasion of the award of the Lorentz Medal to Pauli. 14. ^ Dyson, Freeman (1967). "Ground-State Energy of a Finite System of Charged Particles". J. Math. Phys. 8 (8): 1538-1545. Bibcode:1967JMP.....8.1538D. doi:10.1063/1.1705389. 15. ^ Lieb, E. H.; Loss, M.; Solovej, J. P. (1995). "Stability of Matter in Magnetic Fields". Physical Review Letters. 75 (6): 985-9. arXiv:cond-mat/9506047. Bibcode:1995PhRvL..75..985L. doi:10.1103/PhysRevLett.75.985. 16. ^ Martin Bojowald (5 November 2012). The Universe: A View from Classical and Quantum Gravity. John Wiley & Sons. ISBN 978-3-527-66769-7. • Dill, Dan (2006). "Chapter 3.5, Many-electron atoms: Fermi holes and Fermi heaps". Notes on General Chemistry (2nd ed.). W. H. Freeman. ISBN 1-4292-0068-5. • Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. • Massimi, Michela (2005). Pauli's Exclusion Principle. Cambridge University Press. ISBN 0-521-83911-4. • Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0. • Scerri, Eric (2007). The periodic table: Its story and its significance. New York: Oxford University Press. ISBN 9780195305739. External links Music Scenes
4727cbee3d48a0d2	Atomistic Models Concepts in Computational Chemistry Ratings : ( 0 ) 126 pages Basic and some selected advanced concepts in computational chemistry and molecular modeling presented on the M.Sc./Ph.D. level for both specialists and non-specialists in theoretical chemistry. This is a free eBook for students Sign up for free access Free 30-day trial Latest eBooks About the author Per-Olof Åstrand was born in Sätila in western Sweden in 1965 and grew up in Tygelsjö just south of Malmö in the very southern part of Sweden. After a compulsory military service, he moved to Lund in 1985 to study at Lund University for a degree in chemical engineering which was completed in 1990.... Chemical modeling on the atomistic scale has become a general tool in chemistry research, but it is not always easy to understand the limitations of the applied methods or to interpret the results. Here, the concepts of computational chemistry is introduced on the M.Sc./Ph.D. level (basic knowledge in physical chemistry is assumed). The focus is on the user's perspective, the concepts needed to use computational chemistry in state-of-the-art research, and not on the method developer's perspective as efficient algorithms or software implementation. This first edition includes chapters on computational quantum chemistry and force-field methods, whereas chapters on statistical thermodynamics and molecular simulations will be included in the next edition. 1. Introduction 1. What is molecular modeling? 2. Briefsummary 2. Molecular quantum mechanics 1. The Schrödinger equation 2. The molecular Hamiltonian 3. Some basic properties of the wavefunction 4. The Born-Oppenheimer approximation 5. Atomic orbitals 6. Molecular orbitals 7. The variational principle 8. Perturbation theory 9. First and second-order electric properties 10. The Hartree-Fock approximation 11. Basis set expansion 12. Electron correlation 13. Density functional theory 3. Force fields 1. Introduction to force fields 2. Force-field terms for covalent bonding 3. Intermolecular interactions 4. Intermolecular forces from quantum mechanics
6c1af4d6bbd2f104	Acta Physica Sinica Citation Search Quick Search ISSN 1000-3290 CN 11-1958/O4 » About APS » Editorial Board » SCI IF » Staff » Contact Browse APS » Accepts » In Press » Current Issue » Past Issues » View by Fields » Top Downloaded » Sci Top Cited » Submit an Article » Manuscript Tracking » Call for Papers » Scope » Instruction for Authors » Copyright Agreement » Templates » Author FAQs » PACS » Review Policy » Referee Login » Referee FAQs » Editor in Chief Login » Office Login HighLights More» Design and fabrication of broadband chirped mirror pair Wang Yan-Zhi, Shao Jian-Da, Yi Kui, Qi Hong-Ji, Wang Ding, Leng Yu-Xin Acta Physica Sinica, 2013, 62 (20): 204207 Hydrodynamic theory of grains, water and air Jiang Yi-Min, Liu Mario Acta Physica Sinica, 2013, 62 (20): 204501 A new approach to depict anisotropy diffusion of water molecule in vivo Zhang Shou-Yu, Bao Shang-Lian, Kang Xiao-Jian, Gao Song Acta Physica Sinica, 2013, 62 (20): 208703 Acta Physica Sinica Acta Physica Sinica--2013, 62 (20) Published: 20 October 2013 Select \| Export to EndNote Improved study on parabolic equation model for radio wave propagation in forest Zhang Qing-Hong, Liao Cheng, Sheng Nan, Chen Ling-Lu Acta Physica Sinica. 2013, 62 (20): 204101 doi: 10.7498/aps.62.204101 Full Text: [PDF 613 KB] Download:(523) Show Abstract Regarding the forest as a mixture of air and plants, the refractive model is used to obtain the effective dielectric constant of forest, and the correctness is verified by comparing with the experimental result. The parabolic equation model of forest can be improved by introducing the method of effective dielectric constant for forest. Compared with the traditional model, the improved model takes into account the effect of various elements of forest on radio wave propagation and is more suitable for the radio propagation problems with different plant species in different regions. Otherwise the non-uniform mesh technique is introduced and improves the computational efficiency effectively. Finally, in this paper we analyze the effects of various elements of forest, such as the volume content of plant, the gravimetric moisture content, etc. on the radio propagation. Fast analyses of electromagnetic scattering characteristics from conducting targets using improved and the adaptive cross approximation algorithm Wang Zhong-Gen, Sun Yu-Fa, Wang Guo-Hua Acta Physica Sinica. 2013, 62 (20): 204102 doi: 10.7498/aps.62.204102 Full Text: [PDF 743 KB] Download:(675) Show Abstract Constructing characteristic basis functions (CBFs) is a key step of characteristic basis function method (CBFM). But it is required to set adequate plane wave excitations in each sub-block, which leads to the increased number of characteristic basis functions and the longer time consumed in singular value decomposition of traditional method. In order to accelerate the construction of CBFs, an improved CBFM is presented, which fully considers the mutual coupling effects among sub-blocks and then the secondary level characteristic basis function (SCBF) is obtained, therefore the number of plane wave excitations is reduced greatly, and so is the number of characteristic basis functions. The adaptive cross approximation algorithm is also used to accelerate the matrix-vector multiplication procedure of generating SCBF and constructing the reduced matrix. Numerical results demonstrate that the proposed method is accurate and efficient. Design and implementation of sub-reflector for Ka band antenna based on quartz wafer coating technology Xia Bu-Gang, Zhang De-Hai, Zhao Xin, Yi Min, Huang Jian Acta Physica Sinica. 2013, 62 (20): 204103 doi: 10.7498/aps.62.204103 Full Text: [PDF 4224 KB] Download:(639) Show Abstract The design, fabrication and performance of a frequency selective surface (FSS) which is required to operate as broadband sub-reflector of Ka band antenna for satellite communication application is proposed and validated experimentally. In order to obtain this spatial filter which exhibits low insertion losses and insensitivities to the variation of oblique incident angle for TE and TM polarized wave, the mode-matching method is applied to the analysis of the geometrical structure and electrical parameters of FSS unit cell, and the fabrication process of this Ka FSS sub-reflector utilizing sophisticated quartz wafer coating technology is described. Electromagnetic field simulations and measurements results demonstrate that this FSS filter has virtually identical spectral responses in the two polarization planes. A simulation method by air and space integrated fusion based on hyper-/multispectral imagery Chen Shan-Jing, Hu Yi-Hua, Sun Du-Juan, Xu Shi-Long Acta Physica Sinica. 2013, 62 (20): 204201 doi: 10.7498/aps.62.204201 Full Text: [PDF 5938 KB] Download:(807) Show Abstract A simulation method by air and space integrated fusion based on hyper-/multispectral imagery is proposed. The transformations of spectra, scale-space, radiative intensity, mixed pixel, noise are adopted in the simulation method on the basis of aviatic multispectral imagery according to parameters of space multispectral imagery, and certain ground objects in the aviatic hyperspectral imagery are mapped into the space multispectral imagery to obtain simulative space multispectral imagery for these ground objects through the simulation method. Experimental results demonstrate that the simulation method is effective and easy to implement; the heavy workload on model establishment of three-dimensional scene and sensor response is reduced in this simulation method; it is successful for the method to simulate space multispectral imagery of certain ground objects. A new domain on simulation method of remote sensing imagery is developed. This simulation method is valuable in research and application. Image sparsity evaluation based on principle component analysis Ma Yuan, Lü Qun-Bo, Liu Yang-Yang, Qian Lu-Lu, Pei Lin-Lin Acta Physica Sinica. 2013, 62 (20): 204202 doi: 10.7498/aps.62.204202 Full Text: [PDF 3931 KB] Download:(1636) Show Abstract In compressive sensing, signal sparsity is an important parameter which influences the number of data sampling in reconstruction process and the quantity of the reconstructed result. But in practice, undersampled and oversampled phenomenon will occur because of the unknown sparsity, which may lose the advantages of compressive sensing. So how to determine the image sparsity quickly and accuratly is significant in the compressive sensing process. In this paper, we calculate the image sparsity based on the data acquired during compressive sensing recontruction projection which sparses the origin image in wavelets domain, but we find that its procession is complex, and the final results are seriously influenced by wavelet basis function and the transform scales. We then introduce the principle component analysis (PCA) theory combined with compressive sensing, and establish a linear relationship between image sparsity and coefficient founction variance based on the assumption that PCA is of approximately normal distribution. Multiple sets of experiment data verify the correctness of the linear relationship mentioned above. Through previous analysis and simulation, the sparsity estimation based on PCA has an important practical value for compressive sensing study. Influence of recording parameters on particle field measurement by digital holographic microscopy：a numerical investigation Zhou Bin-Wu, Wu Xue-Cheng, Wu Ying-Chun, Yang Jing, Gérard Gréhan, Cen Ke-Fa Acta Physica Sinica. 2013, 62 (20): 204203 doi: 10.7498/aps.62.204203 Full Text: [PDF 1126 KB] Download:(516) Show Abstract Digital holographic microscopy plays a key role in micro-fluid measurement,and appears to be a strong contender as the next-generation technology for diagnostics of three-dimensional (3D) particle field. However, various recording parameters, such as the recording distance, the particle size, the wavelength, the size of the CCD chip, the pixel size and the particle concentration, will affect the results of the reconstruction, and may even determine the success or failure of a measurement. In this paper, we numerically investigate the effects of particle concentration and the volume depth on reconstruction efficiency, to evaluate the capability of digital holographic microscopy. Standard particle holograms with all known recording parameters are numerically generated by using a common procedure based on Lorenz-Mie scattering theory. Reconstruction of those holograms are then performed by a wavelet-transform based method. Results show that on the premise that the value of volume depth is 24 μm, the reconstruction efficiency Ep decreases quickly until particle concentration reaches 6.89×105 mm-3, and decreases slowly with the increase of particle concentration from 6.89×105 mm-3 to 55.08×105 mm-3. And on the premise that the value of particle concentration is 13.77×105 mm-3, the reconstruction efficiency Ep decreases linearly with the increase of the volume depth. When shadow density is constant, the variance of the construction efficiency presents a certain regularity. When the volume depth is small, the effect of particle concentration on the reconstruction efficiency becomes larger than one of volume depth, while it comes to a completely opposite result with a larger volume depth. Self-sustained oscillation in controllable quadratic coupling opto-mechanical systems Song Zhang-Dai, Zhang Lin Acta Physica Sinica. 2013, 62 (20): 204204 doi: 10.7498/aps.62.204204 Full Text: [PDF 1059 KB] Download:(493) Show Abstract The traditional opto-mechanical coupling in an opto-mechanical system is a linear coupling which is proportional to the field intensity I and oscillator displacement x. The nonlinear spatial coupling effect becomes obvious and important in a strong cavity field with large oscillating amplitude, and then the nonlinear effect with quadratic coupling in opt-mechanical device is also significant. In this article, we find that a general opto-mechanical system with quadratic coupling will produce a stable self-sustained oscillation when the energy injected by external driving equals that of dissipations in certain parametric regions. We numerically solve the semi-classical equation of motion of the system and find high-dimensional limit circles in its phase space under the control of driving and damping. We verify the high-dimensional limit circles by the closed orbits in all the projective three-dimensional phase space and show a highly controllable topological structure of the phase orbit which is very similar to Lissajous figures formed in a two-dimensional case. The self-sustained oscillations of the driving resonator with controllable amplitudes and frequencies demonstrate a reliable physical application of opto-mechanical system under quadratic coupling. Thermally induced frequency difference characteristics of dual-frequency microchip laser used optical generation millimeter-wave Hu Miao, Zhang Hui, Zhang Fei, Liu Chen-Xi, Xu Guo-Rui, Deng Jing, Huang Qian-Feng Acta Physica Sinica. 2013, 62 (20): 204205 doi: 10.7498/aps.62.204205 Full Text: [PDF 4392 KB] Download:(438) Show Abstract The thermal effect of laser diode end-pumped double longitudinal mode dual-frequency microchip laser on dual-frequency laser spectrum is investigated in detail. Through solving the heat conduction equation of isotropic material, a general expression of temperature field within Nd:YVO4 microchip crystal is obtained, then the thermally induced refractive index change of microchip laser is analyzed, and thus the thermally induced frequency difference change of dual-frequency microchip laser is calculated. According to the theoretical results, an experiment is designed. The experimental results show that with a small pumping power, astable doublelongitudinal mode dual-frequency is obtained; increasing the pumping power, the thermal effect of crystal makes the frequencydifference decrease gradually, and the width of each mode spectrum broaden. The experimental results are in good agreement with the theoretical analyses. Supercontinuum generation in all-normal dispersion photonic crystal fiber Li Shu-Guang, Zhu Xing-Ping, Xue Jian-Rong Acta Physica Sinica. 2013, 62 (20): 204206 doi: 10.7498/aps.62.204206 Full Text: [PDF 506 KB] Download:(775) Show Abstract A kind of lead silicate photonic crystal fiber (PCF) is proposed in this paper. The dispersion characteristics of this SF57 PCF are studied by the finite element method. The reasults demonstrate that the PCF presents all-normal dispersion profiles in the visible and infrared spectral regions. We use an adaptive split-step Fourier method to study the propagations of the 1550 nm and 150 fs laser pulse in the PCF. The resulting spectral profiles are extremely flat from 1300 nm to 1900 nm and have excellent stabibities and coherence properties. Design and fabrication of broadband chirped mirror pair Hot! Acta Physica Sinica. 2013, 62 (20): 204207 doi: 10.7498/aps.62.204207 Full Text: [PDF 523 KB] Download:(766) Show Abstract The chirped mirror (CM) pair is designed to provide group delay dispersion (GDD) of around-60 fs2 with bandwidth 500 nm at a central wavelength of 800 nm. The GDD oscillation decreases from ±100 fs2 to ±20 fs2. CM pair is fabricated using ion beam sputtering. The GDD is determined by using a white light interferometer. The measurement results show that the manufactured CM can meet our requirements. By balancing the extra-cavity dispersion with the fabricated chirped mirrors, the pulse 24-27 fs is compressed to 12 fs. Suppression of the stray light of 2-dimensional gratings combined with an array of periodic slit Yu Miao, Gao Jin-Song, Zhang Jian, Xu Nian-Xi Acta Physica Sinica. 2013, 62 (20): 204208 doi: 10.7498/aps.62.204208 Full Text: [PDF 5131 KB] Download:(1115) Show Abstract An array of periodic slits, combined with classic grid two-dimensional gratings, can acquire a dual band-pass characteristic in radar and optical wave band. The concentrated distribution of stray light energy will severely restrict the applications of classic composite film structure in high precision detection and imaging. To solve this problem, a novel composite film structure is developed in this paper, which is composed of slit elements and annular gratings. A scalar diffractive model is built based on Fraunhofer diffractive theory through contrasting the diffractive properties of two kinds of composite film structures. The theoretical simulations and experimental results both prove that the annular composite film structure can suppress the stray light effectively, for its higher transmittance, lower ratio and more uniform distribution of stray light, and enhance the reliability of composite film structure in the practical optical applications. Shock wave negative pressure characteristics of underwater plasma sound source Liu Xiao-Long, Huang Jian-Guo, Lei Kai-Zhuo Acta Physica Sinica. 2013, 62 (20): 204301 doi: 10.7498/aps.62.204301 Full Text: [PDF 1286 KB] Download:(1015) Show Abstract The propagation process of intense acoustic shock wave, generated by the discharge of underwater plasma sound source, is analyzed based on a modified Rayleigh model. The bunching sound field model of underwater plasma sound source is established by using the Euler equation as the control equations. The formation mechanism of the shock wave negative pressure is analyzed theoretically and intuitively through the sound field charts obtained by simulation. The results demonstrate that the water around the bunching wave will be stretched and form a zone of negative pressure with the combination of the rarefaction wave and the inertia of water. It will make the water form a discontinuous phenomenon if the stretching force is greater than the ultimate tensile strength of the water, the phenomenon of cavitation bubble will appear at this time. Besides that, negative pressure will be aggravated by the diffracted wave generated at the edge of the energy-gathered reflector, and the shock wave negative pressure will reach a maximum value by the superimposition of the edge diffraction wave and the stretch wave. The reasons for the formation of the shock wave negative pressure is testified and revealed further by comparing the waveforms of simulation and experiment. The study results provide a theoretical guide for understanding the propagation law of underwater shock wave and further improving the design of the underwater plasma sound source. Acoustic band fracture variation of low frequency transmission zone in one-dimensional solid-fluid periodic structures under omnidirectional incidence Liu Cong, Xu Xiao-Dong, Liu Xiao-Jun Acta Physica Sinica. 2013, 62 (20): 204302 doi: 10.7498/aps.62.204302 Full Text: [PDF 1013 KB] Download:(603) Show Abstract In order to study the characteristics of acoustic/elastic wave propagation in one-dimensional (1D) solid-fluid periodic structure incited by omnidirectional incidence, a theoretical model of wave propagation in 1D solid-fluid periodic structure is established using a transfer matrix method. Based on this model, the band structures of the infinite case and the transmission properties of the finite one are further calculated and analyzed. The results show that an acoustic band fracture appears in the low frequency transmission zone for a certain incident angle. The corresponding incident angle of the fracture is non-relevant to the mass densities or the thicknesses of the solid and fluid layers constituting the periodic structure, and determined only by the wave velocity of the constituent material. Weak photoacoustic signal detection in photoacoustic cell Xu Xue-Mei, Dai Peng, Yang Bing-Chu, Yin Lin-Zi, Cao Jian, Ding Yi-Peng, Cao Can Acta Physica Sinica. 2013, 62 (20): 204303 doi: 10.7498/aps.62.204303 Full Text: [PDF 1659 KB] Download:(467) Show Abstract The content of pollution gas in atmosphere is generally small, so the photoacoustic signal detected by photoacoustic spectroscopy to monitor is very weak. In this paper, on the basis of Holmes Duffing equation, we propose a new method i.e. variable scale difference method which is quite suitable for photoacoustic signal detection. This method can be used to detect weak signal by transforming the size of signal and then substracting the resulting signal. Theoretical analysis and experimental measurements show that it can well restrain the common-mode noise of system phase space, and it can also highlight the chaos state of critical value. This method has a relative error less than 5%, indicating that it can be effectively used to detect the amplitude of weak photoacoustic signal that has a higher frequency and unknown phase and frequency. Hydrodynamic theory of grains, water and air Hot! Jiang Yi-Min, Liu Mario Acta Physica Sinica. 2013, 62 (20): 204501 doi: 10.7498/aps.62.204501 Full Text: [PDF 274 KB] Download:(653) Show Abstract Starting from the thermodynamic framework of a mixture, granular solid hydrodynamics (GSH), which has been developed in recent years, is generalized to the cases in which water and/or gas are present in the interstitials of a granular solid. A preliminary model for the free energy of the mixture is proposed. The three-phase system of grains, water and air is a material relevant to soil mechanics and rock engineering, especially geological catastrophies, for which the macroscopic physics has not been clarified as yet. The engineering theory used currently for analyzing this mixture contains the Darcy’s law of intersticial flow, the effective stress by Terzaghi, including its equation of motion (i.e., the constitutive relation). Comparing it with the theory of GSH, we clarify that Darcy’s equation represents mass diffusion, and the effective stress can be explained by the specific model of free energy that is volumetric filling.The usual engineering approach and GSH, a theory based on physics, are consistent, but we do find some discrepancies, especially on how to parameterize the model: the engineering appraoch employs varying constitutive relation, but the physical approach considers the free energy and the transport coefficients. Clarifying this, we believe, is important for eventually obtaining a unified continuous mechanical theory of soils,especially nonsaturated ones, which is complete and satisfying from physics’s point of view. Cavitation shape of the three-dimensional slender at a small attack angle in a steady flow Zhang Zhong-Yu, Yao Xiong-Liang, Zhang A-Man Acta Physica Sinica. 2013, 62 (20): 204701 doi: 10.7498/aps.62.204701 Full Text: [PDF 3115 KB] Download:(427) Show Abstract In this paper, based on the boundary element method, the cavitation shape of the three-dimensional slender at a small attack angle in a steady flow is simulated through the iterative method, while Dirichlet boundary conditions and Neumann boundary conditions are satisfied in cavitation and slender respectively. The linear triangular elements are adopted and the control points are arranged in grid nodes. The velocity potential for cavity surface is determined through an iterative method in a local orthogonal coordinate system, and then the distribution of cavitation thickness can be determined by the boundary integral equation. To prevent the remeshing operation in the iterative process, the Lagrange interpolation method is used to determine the thickness at the end of cavity. The numerical results are in good agreement with the experimental data. The influence of those on cavitation shape of the three-dimensional slender are investigated, such as cavitation number, attack angle and cone angle. Numerical results show that the cavitation shape of the three-dimensional slender is asymmetric at an attack angle and is analogous to the cavitation stacking in the lee side. While with the decrease in the cavity number or the increase in cone angle, the asymmetry for the cavity shape will be more serious. The drop oscillation model and the comparison with the numerical simulations Chen Shi, Wang Hui, Shen Sheng-Qiang, Liang Gang-Tao Acta Physica Sinica. 2013, 62 (20): 204702 doi: 10.7498/aps.62.204702 Full Text: [PDF 1746 KB] Download:(2092) Show Abstract Due to the complexity inside the drop after impact on solid surfaces and the interaction among gas, liquid and solid phases, it is difficult to investigate the shape variation of the drop through the theoretical analysis, and most studies have focused on experiments and numerical simulations. In this paper, expressions for empirical coefficients of inertia, viscosity and surface tension are acquired by analyzing the force state. The drop oscillation model after impact is built further. The expression of the drop spreading radius, and the effects of surface tension and viscosity on the spreading process are obtained. Finally, the correction factor in the drop oscillation model is determined and the feasibility of the model is verified by comparing the computational results with the numerical results. Investigation on mobility of single-layer MoS2 at low temperature Dong Hai-Ming Acta Physica Sinica. 2013, 62 (20): 206101 doi: 10.7498/aps.62.206101 Full Text: [PDF 201 KB] Download:(1004) Show Abstract The two-dimensional, single-layer MoS2 with a direct band-gap of 1.8 eV, which makes it very suitable for nanoelectronic applications, such as field-effect transistors, has aroused great interest because of its distinctive electronic, optical, and catalytic properties. In this paper, we present a detailed theoretical study of the electronic transport property of single-layer MoS2 on the basis of the usual momentum-balance equation. We obtain the analytical electric mobility at low temperature. It shows that the electric mobility of MoS2 is linear with respect to substrate dielectric constant squared and the rate between the electron density and charged impurity density at low temperature. It is found that by using relatively high dielectric constant materials as substrates, reducing impurity densities and increasing carrier densities high mobilities in MoS2-substrate wafer systems can be achieved. The study of oxygen and sulfur adsorption on the InSb (110) surface, using first-principle energy calculations Zhang Yang, Huang Yan, Chen Xiao-Shuang, Lu Wei Acta Physica Sinica. 2013, 62 (20): 206102 doi: 10.7498/aps.62.206102 Full Text: [PDF 1538 KB] Download:(522) Show Abstract In this paper, we analyze the difference between oxygen and sulfur adsorption on the InSb (110) surface by using the first-principles energy calculations. The merits of sulfur adsorption are demonstrated theoretically. Thus the reason why the sulfur passivation that is necessary in technology can be well explained. Wetting state transition on hydrophilic microscale rough surface Liu Si-Si, Zhang Chao-Hui, He Jian-Guo, Zhou Jie, Yin Heng-Yang Acta Physica Sinica. 2013, 62 (20): 206201 doi: 10.7498/aps.62.206201 Full Text: [PDF 998 KB] Download:(568) Show Abstract The wetting process and the wetting state transition stages are studied on hydrophilic rough surfaces covered with microscale pillar arrays of different geometrical morphologies and distributions. The effects of geometrical morphology, distribution, parameters, hydrophilicity, and contact angle hysteresis of pillar arrays on wetting state transition are analyzed by an energy method. The results indicate that on the hydrophilic rough surface covered with hexagonal arrays of square pillars, the water droplet tends to stay in a stable Cassie state, or the wetting state transits only from a Cassie state to an intermediate state. Moreover, smaller pillar interval, larger square pillar width or diameter of cylinder, higher pillar height, strong hydrophilicity are beneficial to the stability of Cassie state, therefore, the wetting state could be prevented from transforming to pseudo-Wenzel state or Wenzel state. However, smaller area fraction of solid-liquid interface under the water drop and weaker hydrophilicity is beneficial to increasing the apparent contact angle. Therefore, the stability of wetting state and the large apparent contact angle should be considered in hydrophilic surface design. The contact angle hysteresis gives rise to an opposite effect on wetting state stability and the hydrophobicity or superhydrophobicity of rough solid surface. The results provide a theoretical foundation for designing the substrates covered with hydrophilic rough structures, on which the water droplet will obtain a stable Cassie state. Influence of different annealing temperature and atmosphere on the Ni/Au Ohmic contact to p-GaN Li Xiao-Jing, Zhao De-Gang, He Xiao-Guang, Wu Liang-Liang, Li Liang, Yang Jing, Le Ling-Cong, Chen Ping, Liu Zong-Shun, Jiang De-Sheng Acta Physica Sinica. 2013, 62 (20): 206801 doi: 10.7498/aps.62.206801 Full Text: [PDF 554 KB] Download:(1486) Show Abstract In this paper, we investigate the effect of annealing conditions on the characteristic of Ni/Au Ohmic contact to p-GaN. The specific contact resistivities under different annealing temperature and different annealing atmosphere are tested using the circular transmission line model. It is found that the best annealing temperature is about 500 ℃. The annealing atmosphere of nitrogen-oxygen gas mixture can lead to lower specific contact resistivity than that of pure nitrogen, and the specific contact resistivity has no relationship with the content of oxygen. Finally, we obtain the lowest specific contact resistivity to be 7.65×10-4 Ω·cm2 at the best annealing temperature and atmosphere. Electrical and magnetic properties of graphene nanoribbons with BN-chain doping Wang Ding, Zhang Zhen-Hua, Deng Xiao-Qing, Fan Zhi-Qiang Acta Physica Sinica. 2013, 62 (20): 207101 doi: 10.7498/aps.62.207101 Full Text: [PDF 7415 KB] Download:(972) Show Abstract By using the first-principles method based on the density-functional theory, electrical and magnetic properties of graphene nanoribbons (GNRs) with the BN-chain doping are systematically studied. For the zigzag-edge graphene nanoribbon (ZGNR), its multispin-state properties: spin-unpolarized non-magnetism (NM) state, spin-polarized ferromagnetic (FM), and anti-ferromagnetic (AFM) states, are considered. The emphasis on our investigations is the effect of doping position for a single BN-chain. It is found that the BN-chain doping armchair-edge graphene nanoribbon (AGNR) has an increase in bandgap and becomes semiconductors with various different bandgaps upon the doping positions. When the ZGNR at the NM state is doped by the BN-chain, its metallic property is weakened, and the quasi-metallic property can also occur. The BN-chain doping ZGNR at the AFM state makes it change from a semiconductor to a metal or half-metal, depending on doping positions. And the BN-chain doping ZGNR at the FM state always keeps its metallic property unchanged regardless of the doping positions. These results indicate that the BN-chain doping can effectively modulate the electronic structure to form abundant electrical and magnetic properties for GNRs. It is of important significance for developing various kinds of nanodevices based on GNRs. Non-classical energy state and quantum tunnling coherence dissipation for the two-state system with the spin coupled to the lattice phonon Luo Zhi-Hua Acta Physica Sinica. 2013, 62 (20): 207201 doi: 10.7498/aps.62.207201 Full Text: [PDF 241 KB] Download:(540) Show Abstract Including the spin-two-phonon interaction, for the two-state tunneling system with the spin coupled to the lattice phonon (i.e., spin-lattice phonon coupling model) at a finite temperature, the non-classical energy state and the quantum coherence dissipation are studied by the expansion approach of the correlated squeezed-coherent state of phonon. To restrain the quantum coherence loss caused by the Debye-Waller’s coherent scattering of the particle spin by the coherent phonons, the non-classical correlation effects are used in our research with the special consideration of the spin-two-phonon interaction, i.e., 1) the particle spin-displaced phonon state correlation; 2) the process coherence between the one-phonon coherent state and the phonon squeezed state which originates from the squeezed-coherent state of phonon; 3) the renormalization of the phonon displacement. We find the new phenomena that the phonon squeezed state is enhanced significantly due to the particle spin-two phonon interaction, in particular, at the same time the effects of the squeezed coherent state and the representation correlation will be essentially increased. Therefore, the striking decline in the quantum tunneling (Δ0σx) and the serious quantum coherence loss by the Debye-Waller coherent scattering are restricted more noticeably, as a result, the energy of the non-classical state for the two-level system with spin coupled to the lattice phonon is much lower. Quick analysis of miniaturized-element frequency selective surface that loaded with lumped elements by using an equivalent circuit model Wang Xiu-Zhi, Gao Jin-Song, Xu Nian-Xi Acta Physica Sinica. 2013, 62 (20): 207301 doi: 10.7498/aps.62.207301 Full Text: [PDF 377 KB] Download:(572) Show Abstract In order to quicken the pace of the frequency selective surface (FSS) design and optimization, an equivalent circuit method is used to analyze the miniaturized-element FSS structure loaded with lumped elements. According to the physical structure of the FSS, an equivalent circuit model is established. These parameters values of the equivalent circuit model are obtained by a curve-fitting process using ADS to obtain the best fitting between the circuit response and the full-wave analysis response. The fitting accuracy is improved by increasing the curve frequency and the extrema. By using the circuit model, the frequency responses of the FSS at different LC values of lumped components are obtained. The transmissions at center frequencies calculated by the circuit model are slight higher than the exact results from the full-wave analysis, and the relative errors between the center frequencies and the –3 dB bandwidths are smaller than 10%. This paper proves that it is feasible to analyze the complex FSS structure by the equivalent circuit model based on curve-fitting process. It will give some references to the quick design and optimization of the FSS. Comparison bwtween intrinsic and interfacial electrical pulse induced resistance effects in La0.5Ca0.5MnO3 ceramics Wu Mei-Ling, Shi Da-Wei, Kan Zhi-Lan, Wang Rui-Long, Ding Yi-Min, Xiao Hai-Bo, Yang Chang-Ping Acta Physica Sinica. 2013, 62 (20): 207302 doi: 10.7498/aps.62.207302 Full Text: [PDF 925 KB] Download:(375) Show Abstract In general, the electrical pulse induced resistance (EPIR) effect of perovskite manganite originates from the interfacial Schottky barrier between the metal electrode and the surface of sample. In this work, La0.5Ca0.5MnO3 (LCMO) ceramic samples are synthesized by solid state reaction and the transport properties, especially the EPIR effect are investigated using 4-wire measurement mode. Although the I-V curve of LCMO shows ohmic linearity under the 4-wire measurement mode at room temperature, a stable and remarkable EPIR effect can still be observed when the pulse voltage is more than the critical value. Through the comparison between the intrinsic EPIR under 4-wire mode and the interface one under 2-wire mode, we find that the intrinsic EPIR of LCMO has a smaller critical pulse voltage to induce the effect, but it has a better anti-fatigue property. The intrinsic EPIR effect is a novel one which is observed in rare earth doped manganites. Properties of δ doped GaAs/AlxGa1-xAs 2DEG with embedded InAs quantum dots Wang Hong-Pei, Wang Guang-Long, Yu Ying, Xu Ying-Qiang, Ni Hai-Qiao, Niu Zhi-Chuan, Gao Feng-Qi Acta Physica Sinica. 2013, 62 (20): 207303 doi: 10.7498/aps.62.207303 Full Text: [PDF 806 KB] Download:(411) Show Abstract The δ-doped GaAs/AlxGa1-xAs 2DEG samples are grown with molecular beam epitaxy. In this process, the doping concentration (Nd), spatial isolation layer thickness (Wd) and Al component of AlxGa1-xAs (xAl) are changed separately. Then Hall measurements on the samples are made in the two temperature conditions (300 and 78 K). According to the test results, the relationships of Nd, Wd and xAl to the carrier density and mobility of GaAs/AlxGa1-xAs 2DEG are discussed respectively. The δ-doped GaAs/AlxGa1-xAs 2DEG with embedded InAs quantum dot samples are grown, and InAs quantum dots with different densities are grown with gradient growth method. The Hall measurement results show that the mobility of GaAs/AlxGa1-xAs 2DEG greatly decreases with density of InAs quantum dots steadily increasing. In experiments, the lowest density of 16×108/cm2 InAs quantum dot sample is obtained. The experimental results can provide a reference for the study and application of δ-doped GaAs/AlxGa1-xAs 2DEG with embedded InAs quantum dots. Optimization of the parameters for growth high-qulity GaN film by hydride vapor phase epitaxy Zhang Li-Li, Liu Zhan-Hui, Xiu Xiang-Qian, Zhang Rong, Xie Zi-Li Acta Physica Sinica. 2013, 62 (20): 208101 doi: 10.7498/aps.62.208101 Full Text: [PDF 8466 KB] Download:(847) Show Abstract In this paper, the processing parameters of growing GaN epilayer by hydride vapor phase epitaxy are optimized. The influences of the low-temperature (LT) nucleation layer growth time, V/Ⅲ precursor ratio and the growth temperature on GaN layer are investigated by the high-resolution X-ray diffraction (HRXRD) signature for the asymmetric and symmetric reflections. The investigation finds that the LT-nucleation layer not only supplies the nucleation centers having good crystal quality, but also promotes the lateral growth of the sequent high temperature (HT) growth. The optimal LT nucleation layer growth time, V/Ⅲ precursor ratio and the growth temperature can effectively enhance lateral growth to reduce the crystal defects and are favorable to converting the growth mechanism from three-dimension to two-dimension in HT growth. The structural and optoelectronic properties of the as-grown GaN layer with a thickness of 15 μat the optimal parameters are studied by scanning electron microcopy, atomic force microscopy (AFM), HRXRD, Raman spectra, and photoluminescence (PL) measurements. X-ray rocking curves show that the full widths at half maximum of (002) and (102) are 317 and 343 arcsec, respectively. The surface roughness (rms: root mean square) is 0.334 nm detected using AFM. These characteristics show that the sample has good lattice quality and smooth surface morphology. In PL spectrum, the near band edge emission is dominated by emission from excitons bound to neutral donors (D0X) near 3.478 eV with 11 meV blue-shift and the yellow band emission is very weak. The results indicate that the GaN layer has good crystal quality and excellent optoelectronic properties, but a little biaxial in-plane compressive strain also exists in it due to the lattice and thermal mismatch. Preparation of W-doped VO2/FTO composite thin films by DC magnetron sputtering and characterization analyses of the films Tong Guo-Xiang, Li Yi, Wang Feng, Huang Yi-Ze, Fang Bao-Ying, Wang Xiao-Hua, Zhu Hui-Qun, Liang Qian, Yan Meng, Qin Yuan, Ding Jie, Chen Shao-Juan, Chen Jian-Kun, Zheng Hong-Zhu, Yuan Wen-Rui Acta Physica Sinica. 2013, 62 (20): 208102 doi: 10.7498/aps.62.208102 Full Text: [PDF 4924 KB] Download:(1055) Show Abstract In order to obtain low phase transition temperature and superior thermochromic optical material, W-doped VO2/FTO composite thin films are prepared by depositing metallic vanadium on FTO (F:SnO2) conductive glass substrate in argon atmosphere at room temperature and then annealed in air ambient. XPS, XRD and SEM are used for analyzing the structures and surface morphologies of the films. The results indicate that no mixed oxides of V, W and F are produced during high-temperature thermal oxidation. W is doped by replacing V atoms. Compared with the pure VO2/FTO composite thin film prepared using the same process, the crystal orientation of W-doped VO2 thin film is not changed and still retains preferred crystal orientation in the (110) direction. The phase transition temperature drops down to about 35 ℃, and the thermal hysteresis loop narrows to 4 ℃. The variation of IR transmittance between the high temperature and the low temperature reaches 28%. SEM results show that the crystallinity of the thin film is improved significantly, showing smooth, compact and uniform surface morphology. This brings about many new opportunities for optoelectronic devices and industrial production. Simulation and experiment demonstration of an ultra-thin wide-angle planar metamaterial absorber Lu Lei, Qu Shao-Bo, Su Xi, Shang Yao-Bo, Zhang Jie-Qiu, Bai Peng Acta Physica Sinica. 2013, 62 (20): 208103 doi: 10.7498/aps.62.208103 Full Text: [PDF 5653 KB] Download:(1334) Show Abstract In this paper, we present the simulation and experimental validation of an ultra-thin planar metamaterial absorber, which is composed of Jerusalem crosses loaded by interdigital capacitors. By increasing the coupling capacitance between adjacent unit cells, we are able to significantly lower the operating frequency of the absorber. The measured results indicate that the metamaterial absorber achieves a peak absorption of 88.48% at 1.58 GHz. The total thickness and the unit cell size of the absorber are 2 mm and 11 mm, which are approximately 1/95 and 1/17 of the working wavelength, respectively. Additionally, the Jerusalem crosses and the metallic ground plane are connected by vias, which makes the metamaterial absorber achieve wide-angle absorption for both transverse electric and transverse magnetic polarizations electromagnetic wave. The absorptivity is still large even at the incident angle of 60°, and the frequency of the absorption peak almost has no deviation, which makes the absorber more practical. P-type conductivity and NO2 sensing properties for V-doped W18O49 nanowires at room temperature Qin Yu-Xiang, Liu Kai-Xuan, Liu Chang-Yu, Sun Xue-Bin Acta Physica Sinica. 2013, 62 (20): 208104 doi: 10.7498/aps.62.208104 Full Text: [PDF 1634 KB] Download:(493) Show Abstract Tungsten oxide nanowire has a great potential application to gas sensor with high sensitivity and low power consumption. Its gas-sensing properties can be greatly improved after doping the nanowires. In this paper, vanadium (V)-doped W18O49 nanowires are synthesized by solvothermal method, with WCl6 serving as precursor and NH4VO3 as dopant. The microstructures of the pure and the doped nanowires are characterized by using SEM, TEM, XRD, and XPS techniques, and the NO2-sensing properties are evaluated in a static gas-sensing measurement system. The obtained results indicate that the introduction of V dopant suppresses the growth of one-dimensional nanowires along their axis direction and causes the secondary assembly of nanowires bundles. At room temperature, the V-doped W18O49 nanowires show an abnormal p-type response characteristic upon being exposed to NO2 gas, and a conductivity transition from p-to n-type occurs when operating temperature is raised to about 110 ℃. The doped nanowires-based sensor exhibits obvious sensitivity and good response stability to dilute NO2 gas of 80 ppb at room temperature. The origin for the p-n conductivity transition and the high sensitivity at room temperature for the V-doped W18O49 nanowires are analyzed, and they can be attributed to the strong surface adsorption of oxygen and NO2 molecules due to the large density of unstable surface states. Synthesis of near infrared PbS quantum dots by pyrolysis of organometallic sulfur complex Peng Yong, Luo Xi-Xian, Fu Yao, Xing Ming-Ming Acta Physica Sinica. 2013, 62 (20): 208105 doi: 10.7498/aps.62.208105 Full Text: [PDF 1101 KB] Download:(576) Show Abstract Sulfur metal-organic complex Pb(S2CNEt2)2 is synthesized with Pb(NO3)2 and Na(S2CNEt2)·3H2O as reactants in deionized water. Under the protection of argon, PbS quantum dots are synthesized by pyrolysis of the precursor Pb(S2CNEt2)2 in oleic and octadecene mixed solution. Four samples a, b, c, and d of PbS quantum dots are synthesized on the condition that the reaction times are 30, 60, 90, and120 min, respectively. Infrared spectrum of the precursor Pb(S2CNEt2)2 shows that two sulfur atoms of the ligand Na(S2CNEt2)·3H2O have successfully coordinated with Pb2+. X-ray powder diffraction and transmission electron microscopy results show that the PbS nano crystals are of pure cubic phase structure, and are well-dispersed spherical particles. UV-visible absorption spectrum and photoluminescence spectra of PbS quantum dots show that absorption spectrum and photoluminescence spectra both have red-shift along with reaction time extending. This indicates that absorption spectrum and photoluminescence spectra can be modulated by optimizing the thermal decomposition reaction time. The emission peak of sample is located at 1080 nm, which is matched to the silicon solar cell. It can be used as the fluorescent material of silicon luminescent solar concentrator. Influence of high magnetic fields on phase transition and solidification microstructure in Mn-Sb peritectic alloy Yuan Yi, Li Ying-Long, Wang Qiang, Liu Tie, Gao Peng-Fei, He Ji-Cheng Acta Physica Sinica. 2013, 62 (20): 208106 doi: 10.7498/aps.62.208106 Full Text: [PDF 4012 KB] Download:(728) Show Abstract In recent years, the application of high magnetic field in material processing has received much attention from many researchers. However, most studies focus on single-phase solidification or eutectic solidification. The effect of high magnetic field on peritectic alloy is rarely reported. In this study, the solidification experiments on a Mn-56.5 wt%Sb peritectic alloy are carried out under high magnetic fields up to 11.5 T. According to the temperature curve recorded during solidification, it is revealed that high magnetic field increases the liquidus temperature and this rise increases with magnetic flux density increasing. The liquidus temperature rises by about 3 ℃ when the magnetic flux density is 11.5 T. On the contrary, no obvious change in peritectic temperature is found. In addition, the solidified microstructure is analyzed by quantitative metallographic analysis and the result shows that the amount of MnSb phase decreases markedly by the application of high magnetic field. This result consists with the change of phase transition temperature. By the X-ray diffraction, it is found that the c axis of MnSb crystal and (311) plane of Mn2Sb are perpendicular and parallel to the direction of high magnetic field,respectively. Furthermore, the solidification experiments with different cooling rates are also carried out. The quantitative metallographic analysis reveals that the effect of high magnetic field on solidified microstructure is affected by cooling rate. With the increase in cooling rate, the effect of high magnetic field on the fraction of MnSb phase fraction is weakened. A 3D profile evolution method of ion etching simulation based on compression representation Yang Hong-Jun, Song Yi-Xu, Zheng Shu-Lin, Jia Pei-Fa Acta Physica Sinica. 2013, 62 (20): 208201 doi: 10.7498/aps.62.208201 Full Text: [PDF 1389 KB] Download:(548) Show Abstract In order to study the mechanism of the profile evolution process, a three-dimensional (3D) profile evolution method based on compression representation is proposed to simulate the plasma etching process and consider emphatically ion etching. To solve the problem of large memory requirements of 3D cellular model, the presented method adopts a new data structure, which combines two-dimensional array with dynamic storage, to represent cellular information. The structure realizes the lossless compression of cellular information and keeps the spatial correlation between 3D cells. The experimental results show that the method not only significantly reduces the memory, but also has a higher searching efficiency of surface cell which ion first passes through in high-resolution simulation. The method is applied to 3D profile evolution simulation of silicon etching process. A comparison between the simulation results and the experimental results also verifies the effectiveness of the proposed method. Pyrolysis of CL20-TNT cocrystal from ReaxFF/lg reactive molecular dynamics simulations Liu Hai, Li Qi-Kai, He Yuan-Hang Acta Physica Sinica. 2013, 62 (20): 208202 doi: 10.7498/aps.62.208202 Full Text: [PDF 5442 KB] Download:(2002) Show Abstract ReaxFF/lg reactive force field is the extention of ReaxFF by adding a van der Waals attraction term. It can be used to well describe density and structure of crystal, moreover, the macroscopic property of detonation is significantly influenced by the density of energetic material. We report on the initial thermal decomposition of condensed phase CL20-TNT cocrystal under high temperature here. The time evolution curve of the potential energy can be described reasonably well by a single exponential function from which we obtain the initial equilibration and induction time, overall characteristic time of pyrolysis. Afterward, we also obtain the activation energy Ea (185.052 kJ/mol) from these simulations. All the CL20 molecules are completed before TNT decomposition in our simulations. And as the temperature rises, the TNT decomposition rate is significantly accelerated. The higher the temperature at which complete decomposition occurs, the closer to each other the times needed for CL20 and TNT to be completely decomposed will be. Product identification analysis with the limited time steps shows that the main products are NO2, NO, CO2, N2, H2O, HON, HNO3. C–NO2 and N–NO2 bond homolysis jointly contribute to the results of the NO2. The NO2 yield rapid increases to the peak and then decreases subsequently. This process is accompanied with NO2 participating in other reactions so that the N atom of NO2 enters into the other N-containing molecule. Secondary products are mainly CO, N2O, N2O5, CHO. N2O has a strong oxidation ability, so that the distribution has a dramatic fluctuation characteristics. A Colpitts chaotic oscillator based on metal oxide semiconductor transistors and its synchronizaiton research Wang Chun-Hua, Xu Hao, Wan Zhao, Hu Yan Acta Physica Sinica. 2013, 62 (20): 208401 doi: 10.7498/aps.62.208401 Full Text: [PDF 2275 KB] Download:(911) Show Abstract A Colpitts chaotic oscillator is proposed by replacing the bipolar transistor with metal oxide semiconductor (MOS) transistor. Through a series of variable transformations, a state model of proposed circuit similar to one of traditional Colpitts oscillators with bipolar transistors is established. The system parameters are easily obtained by matching the two similar models. Indexes of the balance point show that chaos mechanisms of the two structures are not the same. After the process of parameter inversion and scaling transformation, the detailed circuit parameters are determined. Chaotic attractor and chaotic signal frequency diagrams are generated on PSpice platform. The simulations show that the proposed structure can work under low voltage and can generate chaotic signal with high frequency. Finally, using the linear error feedback method, a synchronization of two identical chaotic circuits is achieved. Mode-matching analytic method of a coaxial Bragg structure corrugated with rectangular ripples and its experimental verification Lai Ying-Xin, Yang Lei, Zhang Shi-Chang Acta Physica Sinica. 2013, 62 (20): 208402 doi: 10.7498/aps.62.208402 Full Text: [PDF 1368 KB] Download:(532) Show Abstract Based on the mode-matching method, an analytical model with full-wave coupling is presented for the coaxial Bragg structures corrugated with rectangular ripples, where the expressions of the reflectivity and transmission rate for each involved mode are derived. The validity of the analytical model is examined in terms of a reported experiment, and good agreement between the theoretical results and the experimental measurements is demonstrated. Comparative study is carried out between the present model and the published theoretical results. It is found that the approximate treatment adopted by the previous model leads to notable deviation of the transmission response curve due to the neglect of the evanescent modes excited by rectangular ripples. The analytical method presented in this paper can be expected to provide a useful approach to the characteristic investigation and engineering practice of the coaxial Bragg structures with rectangular ripples. Numerical simulation of single-event-transient effects on ultra-thin-body fully-depleted silicon-on-insulator transistor based on 22 nm process node Bi Jin-Shun, Liu Gang, Luo Jia-Jun, Han Zheng-Sheng Acta Physica Sinica. 2013, 62 (20): 208501 doi: 10.7498/aps.62.208501 Full Text: [PDF 14090 KB] Download:(578) Show Abstract Single-event-transient response of 22-nm technology ultra-thin-body fully-depleted silicon-on-insulator transistor is examined by technology computer-aided design numerical simulation. The influences of ground plane doping, heavy ion injection location, gate work function and substrate bias on single-event-transient characteristic are systematically studied and analyzed. Simulation results show that the influences of ground plane doping and quantum effects on single-event-transient (SET) are relatively small. The SET characteristics and collected charge are strike-location sensitive. The most SET-sensitive region in ultra-thin-body fully-depleted silicon-on-insulator transistor is located near the drain region. When gate work function varies from 4.3 eV to 4.65 eV, the transient current peak is reduced from 564 μA to 509 μA and the collected charge decreases from 4.57 fC to 3.97 fC. The transient current peak is strongly affected by substrate bias. In contrast, the total collected charge depends only weakly on substrate bias. Relations between traversal time in ferromagnetic/semiconductor(insulator)/ferromagnetic heterojunction and the relative magnetic moment angle in two ferromagnetic layers Lü Hou-Xiang, Shi De-Zheng, Xie Zheng-Wei Acta Physica Sinica. 2013, 62 (20): 208502 doi: 10.7498/aps.62.208502 Full Text: [PDF 477 KB] Download:(387) Show Abstract Based on the concept of group velocity, the relations between traversal time of spin-polarized electrons in ferromagnetic/semiconductor(insulator)/ferromagnetic heterojunction and relative magnetic moment angle in two ferromagnetic layers are studied. The results show that when the middle layer is semiconductor layer, influenced by the Rashba spin-orbit coupling, the minimum transverse times difference between the spin-up and down electrons can appear if the relative angle values in two ferromagnetic layers are nearly the π/2 and 3π/2, respectively. When the middle layer is insulator, the transverse time difference between the different spin orientations can be varied with the potential barrier heights and flip if the height exceeds a critical value. Kidney segmentation in computed tomography sequences based on energy minimization Zhang Pin, Liang Yan-Mei, Chang Sheng-Jiang, Fan Hai-Lun Acta Physica Sinica. 2013, 62 (20): 208701 doi: 10.7498/aps.62.208701 Full Text: [PDF 5044 KB] Download:(645) Show Abstract With the continuous development of medical imaging technology, medical image processing has played an increasingly prominent role in computer-aided diagnosis and disease management. Kidney segmentation in abdominal computed tomography (CT) sequences is a key step. In this paper, combining with the contextual property of renal tissues, a new energy minimization model based on active contour and graph cuts is proposed for kidney extraction in CT sequence. According to the relationship between the shape difference in adjacent slices and corresponding layer thickness, the optimal search range of the contour evolution is calculated for graph cut optimization. The energy function, combining the geodesic active contour with Chan-Vese model, takes into account the boundary and regional information. Then, graph cut methods are used to optimize the discrete energy function and drive the active contour towards object boundaries. Thirty abdominal CT sequences are used to evaluate the accuracy and effectiveness of the proposed algorithm. The experimental results reveal that this approach can extract renal tissues in CT sequences effectively and the average Dice coefficient reaches about 93.7%. Infrared microscopic observation of trapped absorbing particles Zhang Zhi-Gang, Liu Feng-Rui, Zhang Qing-Chuan, Cheng Teng, Gao Jie, Wu Xiao-Ping Acta Physica Sinica. 2013, 62 (20): 208702 doi: 10.7498/aps.62.208702 Full Text: [PDF 4289 KB] Download:(501) Show Abstract Optical tweezers technology is widely used to trap and manipulate particles, and the trapping mechanism generally accepted by researchers is due to the action of photophoretic force. In this paper, infrared microscopic observation of absorbing particles trapped in gaseous medium is first achieved. When the laser power is about 1.0 W, the temperature of trapped toner particles (diameter is about 7 μm) and toluidine blue particles (diameter is from 1 μm to 20 μm) rises by about 14 K, which provides strong evidence for the photophoretic force mechanism. In addition, the periodic oscillation of trapped particles is first observed by both optical microscope and infrared microscope, and the oscillation principle is analyzed. A new approach to depict anisotropy diffusion of water molecule in vivo Hot! Acta Physica Sinica. 2013, 62 (20): 208703 doi: 10.7498/aps.62.208703 Full Text: [PDF 1341 KB] Download:(400) Show Abstract Diffusion anisotropy indices (DAIs) are parameters derived from diffusion tensor imaging (DTI) data which describe the morphological characteristics of diffusion tensor within a specific range. DAIs are the measurements used to quantitatively describe the diffusion direction and strength of the hydrone in vivo, so that DAIs enable one to indirectly probe the internal structure of an imaging subject. The reliability of DAIs is of great importance for the analysis and interpretation of DTI data. Based on the geometric characteristic of the diffusion tensor ellipsoid, we propose a new DAI, the “ellipsoidal geometric ratio” (EGR), to describe the hydrone diffusion anisotropy property. The analysis results of Monte Carlo simulation and human brain DTI data show that the EGR has better contrast and robustness than fractional anisotropy, the most commonly used DAI, and ellipsoidal area ratio at different noise levels. Furthermore, since EGR makes full use of the ellipsoidal volume information, it is more robust than any other DAIs in the fiber crossing case. EGR may be a superior measure of diffusion anisotropy both in quantifying deep white matter with relatively high anisotropy and pericortical white matter with relatively low anisotropy. Modification to the performance of hydrogenated amorphous silicon germanium thin film solar cell Liu Bo-Fei, Bai Li-Sha, Wei Chang-Chun, Sun Jian, Hou Guo-Fu, Zhao Ying, Zhang Xiao-Dan Acta Physica Sinica. 2013, 62 (20): 208801 doi: 10.7498/aps.62.208801 Full Text: [PDF 412 KB] Download:(675) Show Abstract In this paper, we study hydrogenated amorphous silicon germanium thin film solar cells prepared by the radio frequency plasma-enhanced chemical vapor deposition. In the light of the inherent characteristics of hydrogenated amorphous silicon germanium material, the modulation of the germanium/silicon ratio in silicon germanium alloys can separately control open circuit voltage (Voc) and short circuit current density (Jsc) of a-SiGe:H thin film solar cells. By the structural design of band gap profiling in the amorphous silicon germanium intrinsic layer, hydrogenated amorphous silicon germanium thin film solar cells, which can be used efficiently as the component cell of multi-junction solar cells, are obtained. Analysis on degree characteristics of mobile call network Yu Xiao-Ping, Pei Tao Acta Physica Sinica. 2013, 62 (20): 208901 doi: 10.7498/aps.62.208901 Full Text: [PDF 724 KB] Download:(1375) Show Abstract Mobile phone data record the people’s communication behaviors in detail, and become an important resource for studying people’s social relationships and behavior patterns. The call numbers, call volumes and call durations are the basic properties of the mobile phone data network. Based on the theory of complex networks and the statistic method, studied in this paper are the degree distribution and average values of the call numbers, call volumes, call durations, by using the mobile phone call data of different holidays and workdays and different scales of day and period, which are produced by the 3.3 million subscribers registered in a western city in China. Research shows that at all scales the number degrees, volume degrees and duration degrees are all of power-law distribution, and the power exponents are different depending on the scale, date and index, and fluctuate from 1.3 to 4. In general, the number degree exponent is greater than that of the volume degree and duration degree, and the exponents of three indexes of in-degree are greater than those exponents of out-degree; the exponents in holidays are greater than those of the workdays, and the exponents in working periods are greater than those of non-working periods. Compared with in workdays, in holidays the average number degree and volume degree are small, and the average duration degree is large. It reveals that most of the subscribers call only one number in a day or a period, and the call numbers, call volumes and call durations all decrease on holidays or working periods, however the average call duration increases meanwhile. The problem of periodic solution of nonlinear model in zero-dimensional climate system Chen Li-Juan, Lu Shi-Ping Acta Physica Sinica. 2013, 62 (20): 200201 doi: 10.7498/aps.62.200201 Full Text: [PDF 125 KB] Download:(687) Show Abstract Using Mawhin’s continuation theorem, the existence of periodic solution for a class of nonlinear problem is discussed, and then by using it, the problem of periodic solution of nonlinear model in zero-dimensional climate system is investigated. A result about the existence of periodic solution to the model is obtained. Extended expansion method for (G’/G) and new exact solutions of Zakharov equations Yin Jun-Yi Acta Physica Sinica. 2013, 62 (20): 200202 doi: 10.7498/aps.62.200202 Full Text: [PDF 126 KB] Download:(737) Show Abstract We generalize the (G’/G)-expansion method, introduce new auxiliary equation and add negative power exponent. We obtain some new exact solutions of Zakharov equations using the extended (G’/G)-expansion method. This method can also be applied to other nonlinear evolution equations. Complex information system security risk propagation research based on cellular automata Li Zhao, Xu Guo-Ai, Ban Xiao-Fang, Zhang Yi, Hu Zheng-Ming Acta Physica Sinica. 2013, 62 (20): 200203 doi: 10.7498/aps.62.200203 Full Text: [PDF 1504 KB] Download:(1188) Show Abstract There models of complex information system security risk propagation are proposed in this paper based on cellular automata, and the probabilistic behaviors of security risk propagation in complex information systems are investigated by running the proposed models on nearest-neighbor coupled network, Erdos-Renyi random graph network, Watts-Strogatz small world network and Barabasi-Albert power law network respectively. Analysis and simulations show that the proposed models describe the behaviors of security risk propagation in the above four kinds of networks perfectly. By researching on the propagation threshold of security risks in four kinds of network topology and comparing with the existing research result, the correctness of the models is verified. The relationship between the heterogeneity of degree distribution and the value of the propagation threshold is analyzed and verified in this paper. Through the research on the evolutionary trends of security risk propagation, the relationship between the heterogeneity of degree distribution and the influence sphere and speed of security risk propagation is analyzed and verified as well. Meanwhile, the relationship between the heterogeneity of degree distribution and the effect of the immune mechanism on controlling security risk propagation is pointed out. Furthermore, the result of simulations describes the negative exponent relationship between security risk extinction rate and the propagation rate. The key factors affecting the security risk propagation are analyzed in this paper, providing the guidance for the control of security risk propagation in complex information systems. Approximate analytical solutions of bound states for the double ring-shaped Hulthén potential Lu Fa-Lin, Chen Chang-Yuan, You Yuan Acta Physica Sinica. 2013, 62 (20): 200301 doi: 10.7498/aps.62.200301 Full Text: [PDF 140 KB] Download:(443) Show Abstract The double ring-shaped Hulthén potential is proposed in this paper. By using the analytical method of function, the approximate analytical solutions of bound state solutions of Schrödinger equation for the double ring-shaped Hulthén potential are presented within the framework of an exponential approximation of the centrifugal potential for arbitrary l-states. The normalized angular and radial wave function expressed in terms of hypergeometric polynomials are presented. The energy spectrum equations are obtained. The wave function and the energy spectrum equations of the system are related to three quantum numbers and parameters of double ring-shaped Hulthén potential. The polar angular wave functions of the central potential and ring-shaped potential and the energy spectrum equations of Hulthén potential turn out to be the special cases of the double ring-shaped Hulthén potential. Performance analysis of decoy-state quantum key distribution with a heralded pair coherent state photon source Zhou Yuan-Yuan, Zhang He-Qing, Zhou Xue-Jun, Tian Pei-Gen Acta Physica Sinica. 2013, 62 (20): 200302 doi: 10.7498/aps.62.200302 Full Text: [PDF 456 KB] Download:(475) Show Abstract A comprehensive analysis is made on the performance of decoy-state quantum key distribution with a heralded pair coherent state photon source from the effectiveness, stability and feasibility. The key generation rate, quantum bit error rate, and optimal signal intensity each as a function of secure transmission distance are simulated and analyzed by the three-intensity decoy-state method based on a heralded pair coherent state photon source with four groups of experimental data. Considering the intensity fluctuation, the stability of this method is simulated and discussed. Furthermore, the feasibility of the simple and easy method that is proposed with a heralded pair coherent state photon source is analyzed. The simulation results show that the key generation rate and secure transmission distance obtained from the decoy-state method with a heralded pair coherent state photon source are better than those obtained from the methods with a weak coherent state source and heralded single photon source. With the same intensity fluctuation, the heralded pair coherent state photon source is less stable than the heralded single photon source, but more robust than the weak coherent state source. However, the advantage in the effectiveness of the heralded single photon source can give rise to the shortage of the stability. Moreover, the two same modes of the heralded single photon source provide the feasibility to design a simple and easy passive decoy-state method. Simulation analysis of one-way error reconciliation protocol for quantum key distribution Zhao Feng Acta Physica Sinica. 2013, 62 (20): 200303 doi: 10.7498/aps.62.200303 Full Text: [PDF 206 KB] Download:(409) Show Abstract Higher efficiency of error reconciliation technique is for data post-processing for quantum key distribution. Based on the one way error reconciliation scheme of Hamming syndrome concatenation, the error correction performances of there kinds of Hamming codes are demonstrated by data simulation analysis. The simulation results indicate that when the initial error rates are (-∞, 1.5%], (1.5%, 3%], and (3%, 11%], if using Hamming (31, 26), (15, 11), and (7, 4) codes to correct the errors, respectively, the key generation rate is maximized. Base on these outcomes, we propose a kind of modified error reconciliation scheme which is based on the mixed pattern of Hamming syndrome concatenation. The ability to correct the errors and the key generation rate are verified through data simulation. Meanwhile, using the parameters of the posterior distribution based on the tested data, a simple method of estimating bit error rates (BER) with a confidence interval is estimated. The simulation results show that when the initial bit error rate is 9.50%, after correcting error 8 times error, the error bits are eliminated completely and the key generation rate is 9.94%. The BER expectation is 5.21×10-12, when the confidence is 90% the corresponding upper limit of BER is 2.85×10-11. The key generation rate increased by about 3-fold compared with that of original error reconciliation scheme. Quantum wireless wide-area networks and routing strategy Liu Xiao-Hui, Nie Min, Pei Chang-Xing Acta Physica Sinica. 2013, 62 (20): 200304 doi: 10.7498/aps.62.200304 Full Text: [PDF 757 KB] Download:(677) Show Abstract According to multistage quantum teleportation, we here propose a quantum routing scheme which teleports a quantum state from one quantum device to another wirelessly even though the two devices which do not share entangled photon pairs mutually. By the quantum routing scheme, the model of quantum wireless wide-area networks is presented. In terms of time complexity, the proposed scheme transports a quantum bit in the time almost the same as the quantum teleportation regardless of the number of hops between the source and destination node. From this point of view, the quantum routing scheme is close to optimal scheme in data transmission time. The correlative routing process is proposed under different conditions. The quantum state is transferred by multistage quantum teleportation, therefore, the information transfer between any two nodes in quantum wireless wide-area networks is realized. A generation algorithm of unitary transformation applied in quantum data compression Liang Yan-Xia, Nie Min Acta Physica Sinica. 2013, 62 (20): 200305 doi: 10.7498/aps.62.200305 Full Text: [PDF 165 KB] Download:(487) Show Abstract An algorithm to generate unitary transformation (UT) of two orthogonal base kets is proposed in this paper. Certain requirements that UT must meet are as follows: four typical base kets in the first category can be transformed into states with the last qubit \|0>, and the other four atypical base kets can be transformed into states with the last qubit \|1>. This UT is applied to quantum data compression, with a result that the fidelity of the compression is 0.942. This method provides an important basis for realizing quantum compression and decompression. And it can be an important reference of other UT generation method which must fulfill some requirements. Vortex pattern in spin-orbit coupled spin-1 Bose-Einstein condensate of 23Na Liu Chao-Fei, Wan Wen-Juan, Zhang Gan-Yuan Acta Physica Sinica. 2013, 62 (20): 200306 doi: 10.7498/aps.62.200306 Full Text: [PDF 6656 KB] Download:(524) Show Abstract Using the damped projecting Gross-Pitaevkii align, we study the vortex pattern in the two-dimensional spin-orbit coupled spin-1 Bose-Einstein condensate of 23Na. We concentrate on the influence of spin-orbit coupling on the vortex pattern and find that the periodic vortex lattice which would occur without any spin-orbit coupling, can be completely destroyed although the strength of spin-orbit coupling is not very strong. With a strong spin-orbit coupling, vortexes in the condensate of each spin state tend to form some vortex groups and then they create a flower-like lattice around the center. Global analysis of crises in a Duffing vibro-impact oscillator with non-viscously damping Liu Li, Xu Wei, Yue Xiao-Le, Han Qun Acta Physica Sinica. 2013, 62 (20): 200501 doi: 10.7498/aps.62.200501 Full Text: [PDF 12972 KB] Download:(671) Show Abstract In this paper studied is the crises in the Duffing vibro-impact oscillator with non-viscously damping by the composite cell coordinate system method. It is assumed that the non-viscously damping depends on the past history of the velocities other than the instantaneous generalized velocities. The energy dissipation behaviors of real structural materials can be preferably represented in the non-viscously damping models. Numerical simulations show that as the damping coefficient or the relaxation parameter or the recovery coefficient is varied, there appear two kinds of crises: one is the interior crisis, which results from the collision between a chaotic attractor and a chaotic saddle on the basin boundary, and the other is the regular/chaotic boundary crisis, which is due to the collision of a chaotic attractor with a periodic/chaotic saddle on its basin boundary. All the crises result in a sudden change in size and shape of the attractor. Chaotic dynamics in the forced nonlinear vibration of an axially accelerating viscoelastic beam Ding Hu, Yan Qiao-Yun, Chen Li-Qun Acta Physica Sinica. 2013, 62 (20): 200502 doi: 10.7498/aps.62.200502 Full Text: [PDF 603 KB] Download:(747) Show Abstract In this paper, the chaotic behaviour in the transverse vibration of an axially moving viscoelastic tensioned beam under the external harmonic excitation is studied. The parametric excitation comes from harmonic fluctuations of the moving speed. A nonlinear integro-partial-differential governing equation is established to include the material derivative in the viscoelastic constitution relation and the finite axial support rigidity. Moreover, the longitudinally varying tension due to the axial acceleration is also considered. The nonlinear dynamics of axially moving beam is investigated under incommensurable relationships between the forcing frequency and the parametric frequency. Based on the Galerkin truncation and the Runge-Kutta time discretization, the numerical solutions of the nonlinear governing equation are obtained. The time history of the center of the axially moving viscoelastic beam is chosen to represent the motion of the beam. Based on the time history of the axially moving beam, the Poincaré map is constructed by sampling the displacement and the velocity of the center. The bifurcation diagram of the axially moving beam is used to show the influence of the external excitation. Furthermore, quasi-periodic motions are identified using different methods including the Poincaré map, the phase-plane portrait, and the fast Fourier transforms. Bifurcation and chaos in sliding mode controlled first-order h-bridge inverter based on pulse width modulation Hao Xiang, Xie Rui-Liang, Yang Xu, Liu Tao, Huang Lang Acta Physica Sinica. 2013, 62 (20): 200503 doi: 10.7498/aps.62.200503 Full Text: [PDF 2529 KB] Download:(936) Show Abstract Sliding mode control (SMC) is recognized as a robust controller with fast dynamic response and high stability in a wide range of operating conditions, and therefore it is widely used in the control of inverter. The sliding mode controlled inverter is in nature a nonlinear controlled time-varying nonlinear system, and it has complicated nonlinear behaviors in practice. In this paper the sliding mode controlled first-order H-bridge inverter based on pulse width modulation is taken for example. First, through observation of the waveforms under different SMC parameters, a new type of bifurcation is discovered for the first time, in which diverse multi-period bifurcations exist at the same time. Second, the discrete time iterative model is established for the system and the folded diagram is employed to observe the waveforms. These analyses of current waveforms prove that the new type of bifurcation proposed in this paper is not a route to chaos. Moreover, the stability of the system is much concerned in engineering applications. However, because of the nonlinear characteristics of sliding mode controller, the method using eigenvalues of Jacobian matrix and other analytical methods are unsuitable for the system, and the graphic methods are not accurate enough. Finally, a criterion of fast-scale stability which can accurately distinguish the stability of the system is proposed, and it can be used to provide reliable reference for the parameter design of sliding mode controller. Wronskian solution of general nonlinear evolution equations and Young diagram prove Cheng Jian-Jun, Zhang Hong-Qing Acta Physica Sinica. 2013, 62 (20): 200504 doi: 10.7498/aps.62.200504 Full Text: [PDF 3593 KB] Download:(784) Show Abstract In this paper, we give indirect methods constructed Wronskian solution of a general nonlinear evolution equations. Under the properties of the computing of Young diagram we have proved the proposition of this paper and discuss the relationship between the permutation group character and Young diagram expressions coefficient. A regularized super resolution algorithm based on the double threshold Huber norm estimation Zhou Shu-Bo, Yuan Yan, Su Li-Juan Acta Physica Sinica. 2013, 62 (20): 200701 doi: 10.7498/aps.62.200701 Full Text: [PDF 5790 KB] Download:(950) Show Abstract Multi-frame super-resolution reconstruction is a technology which obtains a high-resolution image from several low-resolution images of the same scene. Among various super resolution methods, the regularized method is widely used since it has advantages for solving the ill-posed problems. However, the super-resolution reconstruction results based on this method strongly depend on the estimation accuracy of the optimum estimator. In this paper, a double-threshold Huber norm based maximum likehood estimator is proposed, which improves the threshold tolerance of the estimator and increases the estimation accuracy. Then a regularized algorithm based on this estimator is presented. The super-resolution reconstruction results of synthetic low resolution images confirm that the proposed algorithm has better performance over the existing algorithms. The proposed algorithm is also used to deal with the low-resolution images obtained from a plenoptic camera. The results confirm the effectiveness of the proposed algorithm. Electrorotation characteristics of gold-coated SU-8 microrods at low frequency Hou Li-Kai, Ren Yu-Kun, Jiang Hong-Yuan Acta Physica Sinica. 2013, 62 (20): 200702 doi: 10.7498/aps.62.200702 Full Text: [PDF 5725 KB] Download:(556) Show Abstract According to the theory of traditional Maxwell-Wagner interface polarization, the metal micro/nano particles have no obvious electrorotation behavior under the alternating current electric field. However, we find the opposite experimental results. In this paper, electrorotation experiments are carried out, and the basic mechanism of gold-coated SU-8 microrods is presented. Therefore the electrorotation characteristics of gold-coated microrod at low frequency are analysed by considering the surface electric double layer at the microrod-electrolyte interface. Specifically, first we establish an approximate ellipsoid model in the electric field, analyze the polarization mechanism of metal particles under the action of solid-liquid interface electric double layer, and then calculate the electrorotation torque and present an electrorotation angular speed formula of the gold-coated microrod. Secondly, electrorotation experiments of gold-coated SU-8 microrods suspended in electrolytes with different conductivities are presented in a frequency range of 100 Hz to 30 MHz. Finally, the experimental results are discussed, and compared with the theoretical analysis, showing the experimental results are in good agreement with theoretical analyses by considering the friction between the microrods and substrate. Investigation of enhancing higher harmonics by changing the shape of atomic force microscope cantilever Yang Hai-Yan, Wang Zhen-Yu, Li Ying-Zi, Zhang Wei-Ran, Qian Jian-Qiang Acta Physica Sinica. 2013, 62 (20): 200703 doi: 10.7498/aps.62.200703 Full Text: [PDF 3595 KB] Download:(1021) Show Abstract Higher harmonics of tapping-mode atomic force microscope carries information about the mechanical properties of the sample on a nanometer scale. Unfortunately, the vibration amplitudes of traditional atomic force microscope (AFM) cantilever at higher harmonics are too small for practical AFM imaging. Ritz method demonstrates that specific cutout on the cantilever can realize internal resonance to enhance higher harmonics. In this paper, by COMSOL finite element simulation, the laws for fundamental frequency, second resonance frequency and their ratio each as a function of the size of the cutout and the position of the cutout on the cantilever are achieved. Using focused ion beam to hole the cantilever makes the second resonance frequency close to 6 times that of the fundamental frequency and also the 6th harmonic enhanced. Moreover, we obtain the image of the 6th harmonic on our home-made higher harmonic system. Gas measurement system of NO and NO2 based on photoacoustic spectroscopy Xu Xue-Mei, Li Ben-Rong, Yang Bing-Chu, Jiang Li, Yin Lin-Zi, Ding Yi-Peng, Cao Can Acta Physica Sinica. 2013, 62 (20): 200704 doi: 10.7498/aps.62.200704 Full Text: [PDF 1641 KB] Download:(1377) Show Abstract NO and NO2 are the common gases and have serious harmfulness to the atmosphere pollution of environment. To detect the concentration of the two gases in pollution, we construct a low cost photoacoustic spectrum gas measurement system based on the thermal radiation light source. Absorption lines of the NO and NO2 between 2500 and 6667 nm are calculated. By modeling the photoacoustic transmission line the relations between quality factor, acoustic pressure and cavity length, cavity radius, modulation frequency are obtained, and the design of geometric construction of photoacoustic cell is also guided. The experiment show that there exists a good linearity between photoacoutic signal detected in the system and gas concentration, and the system ultimate detection sensitivities to the NO and NO2 are 4.01 and 1.07 μL/L respectively. When the emission wavelength of laser is regulated suitably and the edmund optics is chosen reasonably, this system is also suitable for the concentration-detection of other trace gas. Wang Yang, Li Ang, Xie Pin-Hua, Chen Hao, Mou Fu-Sheng, Xu Jin, Wu Feng-Cheng, Zeng Yi, Liu Jian-Guo, Liu Wen-Qing Acta Physica Sinica. 2013, 62 (20): 200705 doi: 10.7498/aps.62.200705 Full Text: [PDF 1188 KB] Download:(808) Show Abstract The inversion method of the vertical profile and vertical column density (VCD) of tropospheric NO2 using multi-axis differential optical absorption spectroscopy (MAX-DOAS) is investigated in this paper. An inversion method of two-step procedure is operated. In this method firstly the aerosol vertical profile is retrieved. Then the vertical distribution of trace gases is retrieved based on the corresponding aerosol status. Nonlinear optimal estimation algorithm is extended to acquire NO2 profile to reduce the dependence of the inversion on priori information. It is more advantageous to automatically obtain trace gases profile. At first we investigate how to calculate some parameters (weighting function, the covariance matrices of measurement, and a priori information) of the algorithm and design nonlinear iteration strategy suited to the region where NO2 vertical distribution usually shows rapid variation. Then this inversion algorithm is verified by computer simulation in the cases of box profile and elevated profile of NO2. It is indicated that the distribution of NO2 below 2 km could be well rebuilt and the retrieval accuracy of surface-near NO2 volume mixing ratio is 0.6%. The study of how accurately this algorithm can rebuild the same true profile in three aerosol status of low aerosol, high aerosol and elevated aerosol indicates that similar retrieval results could be acquired. In addition, the effect of wrong aerosol status on the retrieving of NO2 profile and the error sources of this algorithm are analyzed. After that a continuous observation is reported in the city of Hefei. NO2 VCDs derived from MAX-DOAS are compared with those from satellite observations, and the correlation coefficient is 0.85. The surface-near NO2 concentrations measured by MAX-DOAS are compared with those from LP-DOAS, and the correlation coefficient is 0.76. In addition, the simplified MAX-DOAS inversion method of obtaining the trace gas profile usually uses invariable typical aerosol status as input. The comparison with the tropospheric NO2 VCD from simplified method indicates that the using of invariable typical aerosol status would cause large deviation of NO2 VCD, and its maximum relative deviation is about 112%. So exactly acquiring aerosol status, aerosol optical density especially, is necessary to exactly retrieve tropospheric NO2 vertical column density. Density functional study on the electronic and magnetic properties of two-dimensional hexagonal boron nitride containing vacancy Wei Zhe, Yuan Jian-Mei, Li Shun-Hui, Liao Jian, Mao Yu-Liang Acta Physica Sinica. 2013, 62 (20): 203101 doi: 10.7498/aps.62.203101 Full Text: [PDF 2172 KB] Download:(613) Show Abstract According to density functional first-principles calculations, we study the electronic and magnetic properties of two-dimensional h-BN containing lattice vacancies: B atom vacancy (VB), N atom vacancy (VN) and BN pair vacancies (VBN). In microstructures, the neighbored N atoms around vacancy in VB system are constructed into an isosceles triangle; the neighbored B atoms around vacancy in VN system are constructed into an equilateral triangle; and the neighbored atoms around BN pair vacancies are constructed into a keystone. The calculations on band structures and density of states show that the significant spin polarization exists in two-dimensional h-BN containing single vacancy defects, while the BN pair vacancy system exhibits non-spin polarization. Three kinds of vacancy defects make the band gap change from direct band gap to indirect band gap. For the VB system, the total magnetic moment is 1.00 μB, due to the contribution from the neighbored N atoms around vacancy. The directions of magnetic moment in three neighbored N atoms around vacancy are different. Among them, ferromagnetic and anti-ferromagnetic coupling are coexisting. For the VN system, the total magnetic moment in the entire calculated supercell is also 1.00 μB and presents a certain distribution in the area around the vacancy. Adsorption behavior of inhibitor on copper surface Liu Na-Na, Sun Jian-Lin, Xia Lei, Zeng Ying-Feng Acta Physica Sinica. 2013, 62 (20): 203102 doi: 10.7498/aps.62.203102 Full Text: [PDF 5339 KB] Download:(1321) Show Abstract The global and local activity of benzotrialole (BTA) and 2,5-Dimercapto-1,3,4-thiadiazole (DMTD) are calculated by density function theory. Thermodynamic properties of BTA and DMTD are simulated by molecular dynamics. The corrosion inhibition effect of mixed system inhibitor is studied through corrosion test. The results show that the inhibition efficiency of DMTD is larger than that of BTA. There are several active sites which focus on N and S atoms. So the inhibitors are absorbed on the surface of Cu in the parallel direction. The specific heat capacities of Cu absorbing inhibitor are the same as those without Cu adsorbing inhibitor at room temperature. The specific heat capacities increase with inhibitor increasing. They provide the reference for the selection of the inhibitor. The effect of mixed system inhibitor is better and the best mixed ratio is BTA: DMTD=1:1. Synergistic effects in Fe/N codoped anatase TiO2 (101) surface：a theoretical study based on density functional theory calculation Li Zong-Bao, Wang Xia, Jia Li-Chao Acta Physica Sinica. 2013, 62 (20): 203103 doi: 10.7498/aps.62.203103 Full Text: [PDF 3995 KB] Download:(830) Show Abstract The interaction between implanted nitrogen atom and transition metal iron at the anatase TiO2(101) surface is investigated by the periodic density functional theory calculations. Substitutional and interstitial configurations and formation energies for Fe-doping, and several N and Fe atom codopings at different sites of the (101) surface are considered. Our formation energy calculations suggest that when Fe atom transfers from surface to body, it is subjected to a larger energy barrier while asynergetic effect takes place between the nitrogen and the codoped Fe in the surface. The analyses of the electronic structure and densities of states show that the property of half-metallic appears, with N and Fe codoped. Potential energy curve and spectroscopic properties of PS (X2Π) radical Liu Hui, Xing Wei, Shi De-Heng, Sun Jin-Feng, Zhu Zun Lüe Acta Physica Sinica. 2013, 62 (20): 203104 doi: 10.7498/aps.62.203104 Full Text: [PDF 189 KB] Download:(732) Show Abstract The potential energy curve (PEC) of ground X2Π state of PS radical is studied using highly accurate internally contracted multireference configuration interaction approach with the Davidson modification. The Dunning’s correlation-consistent basis sets are used for the present study.To improve the quality of PECs, scalar relativistic and core-valence correlation corrections are considered. Scalar relativistic correction calculations are performed using the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. Core-valence correlation corrections are calculated with an aug-cc-pCV5Z basis set. All the PECs are extrapolated to the complete basis set limit. Using the PEC, the spectroscopic parameters (Re, ωe, ωexe, ωeye, Be, αe and De) of the X2Π state of PS are determined and compared with those reported in the literature. With the Breit-Pauli operator, the PECs of two Ω states of the ground Λ-S state are calculated. Based on these PECs, the spectroscopic parameters (Te, Re, ωe, ωexe, ωeye, Be and αe) of two Ω states of PS are obtained. Compared with those reported in the literature, the present results are accurate. The vibration manifolds are evaluated for each Ω and Λ-S state of non-rotation PS radical by numerically solving the radical Schrödinger equation of nuclear motion. For each vibrational state, the vibrational level and inertial rotation constants are obtained, which are in excellent accordance with the experimental findings. Up-conversion luminescence in Tm3+/Yb3+ Co-doped NaY(MoO4)2 phosphors by optimal design of experiments Zhai Zi-Hui, Sun Jia-Shi, Zhang Jin-Su, Li Xiang-Ping, Cheng Li-Hong, Zhong Hai-Yang, Li Jing-Jing, Chen Bao-Jiu Acta Physica Sinica. 2013, 62 (20): 203301 doi: 10.7498/aps.62.203301 Full Text: [PDF 369 KB] Download:(482) Show Abstract The method of optimally designing experiments is introduced to study the properties of up-conversion luminescence in Tm3+/Yb3+ co-doped NaY(MoO4)2 phosphors. The design of experiment, data analysis and objective are all optimized. Uniform design is used to seek for the doping concentration range, and quadratic general rotary unitized design is included to build the equation between the luminescent intensity and the Tm3+/Yb3+doping concentration. In order to obtain the best luminescent intensity, genetic algorithm is introduced into our calculation. The optimal Tm3+/Yb3+ co-doped NaY(MoO4)2 phosphor is synthesized by high temperature solid state method. The up-conversion emission spectrum under the 980 nm laser excitation is detected and the luminescence mechanism is proposed. We observe intense blue light (476 nm) and week red light (649 nm) at room temperature, which are emitted from the Tm3+ transition of 1G43H6 and 1G43F4, respectively. In this up-conversion uminescence system, the up-conversion emission of 1G4 is due to the cooperation energy transfer of two photons. We also discuss the temperature effect of the sample, and as the temperature increases its luminescent intensity decreases. Dynamical properties of Rydberg hydrogen atom interacting with a metal surface and an electric field Li Hong-Yun, Yue Da-Guang, Liang Zhi-Qiang, Yi Chang-Hong, Chen Jian-Zhong Acta Physica Sinica. 2013, 62 (20): 203401 doi: 10.7498/aps.62.203401 Full Text: [PDF 3723 KB] Download:(618) Show Abstract The dynamical properties of Rydberg hydrogen atom in an electric field near a metal surface are presented by analyzing the phase space. The dynamical behavior of the excited hydrogen atom depends sensitively on the atom-surface distance and the electric field strength. The evolutions of the Poincaré surface of section and the electric orbit are analyzed for a certain atom-surface distance and different electric field strengths. The results indicate that the electric field accelerates the adsorption of electron. With the increase of the electric field strength, the controlling factor of dynamical behavior changes from the atom-surface distance to the electric field strength. As the electric field strength becomes very large, the system is integrable and all the electric orbits become vibrational type of orbits. Theoretical study on the energy loss induced by electron collisions in weakly ionized air plasma Zhou Qian-Hong, Dong Zhi-Wei Acta Physica Sinica. 2013, 62 (20): 205201 doi: 10.7498/aps.62.205201 Full Text: [PDF 278 KB] Download:(762) Show Abstract The energy loss induced by electron collisions in weakly ionized air plasma is calculated based on the electron energy distribution function that we obtained. Since there are a lot of low-energy-threshold molecular rotation and vibration excitations and the electron-molecule energy transfer is inefficient in elastic collision, the fraction of energy loss for electron elastic collision (less than 6%) is negligible. Among different collision processes the electron energy loss is dominant in different energy regions. As the effective electron temperature (or the reduced electric field) increases, the dominant energy loss process becomes sequentially rotational excitation, vibrational excitation, electronic excitation, collisional ionization, and accelerating ionized electrons. When E/N=1350 Td (or Te=14 eV), the average energy loss per ion-electron pair reaches a minimum value of 57 eV. By controlling the electric field according to the requirement in applications, we can control the electric field to achieve a higher energy efficiency. Theoretical study on the air breakdown by perpendicularly intersecting high-power microwave Zhou Qian-Hong, Dong Zhi-Wei Acta Physica Sinica. 2013, 62 (20): 205202 doi: 10.7498/aps.62.205202 Full Text: [PDF 2516 KB] Download:(537) Show Abstract Air breakdown by perpendicularly intersecting high-power microwave (HPM) is investigated by numerical solution of fluid-based plasma equations coupled with the Maxwell equations. For two coherently intersecting HPM beams, collisional cascade breakdown takes place only when the initial free electrons appear in or arrive at a region of strong electric field, where the electron can be accelerated. At the initial stage of discharge, the filamentary plasma moves along the strong field and forms plasma-filament band. When the plasma-filament band grows long enough, in the vicinity of which the two HPM beams are separated due to its scattering and absorption by plasma. The new plasma-filament bands continue to appear as time increases. It is also found that under the same condition, the plasma region produced by incoherent beams is smaller than by coherent beams. Effects of pulse temporal profile on electron bow-wave injection of electrons in laser-driven bubble acceleration Zhang Guo-Bo, Zou De-Bin, Ma Yan-Yun, Zhuo Hong-Bin, Shao Fu-Qiu, Yang Xiao-Hu, Ge Zhe-Yi, Yin Yan, Yu Tong-Pu, Tian Cheng-Lin, Gan Long-Fei, Ouyang Jian-Ming, Zhao Na Acta Physica Sinica. 2013, 62 (20): 205203 doi: 10.7498/aps.62.205203 Full Text: [PDF 6158 KB] Download:(571) Show Abstract Effects of pulse temporal profile on electron injection and trapping in the electron bow-wave injection regime are investigated by two-dimensional particle-in-cell simulations. It is found that a positive skew pulse can enhance the wake-field amplitude, extends the accelerating aera, and improves the initial velocity of electrons injected into the bubble. Thus more energetic electrons are driven into the bubble accelerating phase. The total injection number for a positive skew pulse is higher than those of negative skew and usual Gaussian pulses in the same conditions, and the quality of electron beam is also improved. The obtained result is very important and beneficial for the future experimental investigation of the laser wake-field acceleration to obtain an energetic electron beam with a large charge quantity. X-ray source produced by laser solid target interaction at kHz repetition rate Huang Kai, Yan Wen-Chao, Li Ming-Hua, Tao Meng-Ze, Chen Yan-Ping, Chen Jie, Yuan Xiao-Hui, Zhao Jia-Rui, Ma Yong, Li Da-Zhang, Gao Jie, Chen Li-Ming, Zhang Jie Acta Physica Sinica. 2013, 62 (20): 205204 doi: 10.7498/aps.62.205204 Full Text: [PDF 2460 KB] Download:(1092) Show Abstract X-ray radiation with an average flux of 1.3×107 photons·sr-1·s-1 is generated by the interaction between ultra-short laser and solid target working at 1 kHz repetition rate. A knife edge is introduced to measure the source size. The X-ray emission shows obvious dependences on the laser contrast and intensity. It is discovered that at lower intensity the Kα yield increases with lower laser contrast. However, by using high contrast and high intensity laser pulses, high flux and high signal-to-noise ratio Kα X-ray can be generated. The generated source is used in proof-of-principle radiograghic imaging of simple specimens. Effect of electron temperature on the characteristics of plasma sheath in Hall thruster Duan Ping, Cao An-Ning, Shen Hong-Juan, Zhou Xin-Wei, Qin Hai-Juan, Liu Jin-Yuan, Qing Shao-Wei Acta Physica Sinica. 2013, 62 (20): 205205 doi: 10.7498/aps.62.205205 Full Text: [PDF 407 KB] Download:(453) Show Abstract In this paper, the effect of electron temperature on the characteristics of plasma sheath in the channel of Hall thruster is studied by using two-dimensional (2D) particle-in-cell simulation method. The change laws of electron number density, sheath potential, electric field and secondary electron emission coefficient at different electron temperatures are discussed. The results show that when the electron temperature is low, electron number density decreases exponentially in the radial direction and reaches a minimum at the wall, the sheath potential drops and variation of electric field in the radial direction is larger, and the wall potential stays at a stable value, the stability of sheath is better. However, when the electron temperature is high, the electron number density inside the sheath region approximates to that at the sheath boundary, but in a narrow area near the wall it increases rapidly and reaches a maximum at the wall, sheath potential changes slowly, the sheath potential drops and variation of electric field in the radial direction is smaller, and the wall potential tends to maintain a persistent oscillation and the stability of sheath is reduced. The influence of electron temperature on electric field in the axial direction is small. With the increase of the electron temperature, wall secondary electron emission coefficient increases in the early stage, and reduces later. Wavefront criterion of small focal spot in optical zooming Huang Xiao-Xia, Gao Fu-Hua, Yuan Qiang, Hu Dong-Xia, Zhang Kun, Zhou Wei, Dai Wan-Jun, Deng Xue-Wei Acta Physica Sinica. 2013, 62 (20): 205206 doi: 10.7498/aps.62.205206 Full Text: [PDF 1189 KB] Download:(408) Show Abstract In the direct drive, since the target is becoming smaller in compression process, the initial focal spot will no longer match the compressed target, leading to large losses of energy from target edge. Optical zooming means making the focal spot smaller when the target compresses. In this way, the coupling efficiency between the beam and target can be improved. It has great significance. The primary problem of solving the optical zooming is the qualification of small focal spot. According to the full-band optical transmission rules, in this paper constructed are low and medium-high frequency wavefront distortion sources under different smoothing conditions, and then given is a scaling relation between wavefront and focal spot size. The scaling relation can be regarded as a wavefront criterion of small focal spot. Using this wavefront criterion, the correction range of low and medium-high frequency wavefront distortion can be known to achieve a small focal spot. The full three-dimensional electromagnetic PIC/MCC numerical algorithm research of Penning ion source discharge Yang Chao, Long Ji-Dong, Wang Ping, Liao Fang-Yan, Xia Meng-Zhong, Liu La-Qun Acta Physica Sinica. 2013, 62 (20): 205207 doi: 10.7498/aps.62.205207 Full Text: [PDF 5372 KB] Download:(854) Show Abstract In this article, we study the physical mechanism of the Penning discharge, develop a full three-dimensional particle simulation software of high-quality algorithm (PIC), design and add the corresponding physical scenario of Monte Carlo collisions (MCC) module, and track electron, hydrogen molecular ion (H2+), hydrogen positive ion (H+), and tri-n-hydrogen ion (H3+) at the same time, and successfully develop a full three-dimensional electromagnetic PIC/MCC numerical algorithm. Combined with the Penning discharge model extensively studied in china, the algorithm is verified through simulation. The simulation results show that the use of effective filtering algorithm can suppress the electromagnetic numerical noise. Electron energy is of Maxwell distribution. Due to the radial drift and accelerate of electrons, the H2+ yield is larger at the top of the ion source. Phase resolved optical emission spectroscopy of dual frequency capacitively coupled plasma Du Yong-Quan, Liu Wen-Yao, Zhu Ai-Min, Li Xiao-Song, Zhao Tian-Liang, Liu Yong-Xin, Gao Fei, Xu Yong, Wang You-Nian Acta Physica Sinica. 2013, 62 (20): 205208 doi: 10.7498/aps.62.205208 Full Text: [PDF 2011 KB] Download:(872) Show Abstract In this article we use phase resolved optical emission spectroscopy to study emission pattern in plasma sheath of dual frequency capacitively coupled plasma in Ar and Ar-O2 discharge. Two emission patterns are found in sheath region of radio frequency coupled powered electrode. The first pattern is related to electron impact excitation because of the sheath expansion. The second pattern is caused by electron impact excitation of secondary electrons. Two emission patterns are also highly modulated with the low frequency cycle. Under the condition of argon discharge, the emission intensities of the two excitation processes are very similar. The emission structure by secondary electrons becomes weak with the increase of O2 content in the gas mixture. In addition, we also use phase resolved optical emission spectroscopy to study low frequency cycle averaged axial emission profile of excited atomic argon at 751 nm in Ar-O2 mixture gas. Distance from the powered electrode (about 3.8 mm) is defined as the boundary sheath of dual frequency capacitively coupled plasma. Preliminary study on similarity of glow discharges in scale-down gaps Fu Yang-Yang, Luo Hai-Yun, Zou Xiao-Bing, Liu Kai, Wang Xin-Xin Acta Physica Sinica. 2013, 62 (20): 205209 doi: 10.7498/aps.62.205209 Full Text: [PDF 446 KB] Download:(409) Show Abstract In order to investigate the validity of the scale-down experiment on gas discharge, the discharge in argon at low pressure is numerically simulated with scale-down discharge gap based on the conjecture of discharge similarity that if the product of gas pressure p and gap length d is kept constant, p1d1=p2d2, and the spatial distributions of the reduced field E/p along these two gaps are the same, the gas discharges in these two discharge gaps would be similar. In the simulation, three scale-down discharge gaps are used. Gap A is 30 mm long and works at a pressure of 1 Torr (1 Torr=133.322 Pa). Gap B is 15 mm long at 2 Torr and gap C 10 mm long at 3 Torr. The results show that the discharges in these three gaps are glow discharges with a cathode fall layer. The values of thickness of the cathode fall layer, dC, for gaps A, B and C are 2.71 mm, 1.35 mm and 0.87 mm, respectively, which corresponds to more or less the same value of pdC≈2.70 Torr·mm that is close to the lowest point of Paschen curve of argon where pd≈2.86 Torr·mm. The proportionalities of the parameters (working voltage, electric field, current density, electron density and ion density) between the discharges in the scale-down gaps are found to be in good agreement with those determined by the discharge similarity. It is concluded that the conjecture of discharge similarity is correct for the glow discharge in argon in the scale-down gap. Effects of clouds and precipitation disturbance on the surface radiation budget and energy balance over loess plateau semi-arid grassland in China Yue Ping, Zhang Qiang, Zhao Wen, Wang Jin-Song, Wang Run-Yuan, Yao Yu-Bi, Wang Sheng, Hao Xiao-Cui, Yang Fu-Lin, Wang Ruo-An Acta Physica Sinica. 2013, 62 (20): 209201 doi: 10.7498/aps.62.209201 Full Text: [PDF 1392 KB] Download:(586) Show Abstract The feedback effect of land surface radiation budget and energy distribution on land-atmosphere system is one of the most important physical processes in climate models. Studying the effects of disturbance of cloud and precipitation on radiation budget and energy distribution is a crucial link to increase the parameterization effect of evaluating land surface radiation budget and energy balance in numerical models. Based on the data observed at the Semi-Arid Climate and Environment Observatory of Lanzhou University in 2008, the weakening influences of disturbance of cloud and precipitation on components in radiation budget and on land surface energy balance are studied. The annual average results showed that cloudy situation could be thought of as climatic background of annual average; weakening influences of cloud and precipitation on short-wave radiation are the strongest, and atmospheric long-wave radiation increases while surface long-wave radiation decreases with cloud amount increasing; the influences of cloud and precipitation on the ratio of net radiation to global radiation is small. The seasonal average results show that daily integral value of short-wave radiation decreases with cloud amount increasing both in growing season and in non-growing season, and weakening influences of cloud and precipitation on short-wave radiation in growing season are obviously stronger than in non-growing season. In growing season, there is no substantial difference among surface long-wave radiations on clear days, partly cloudy days and cloudy days, and atmospheric and surface long-wave radiation decrease apparently on overcast days. In non-growing season, the influences of cloud and precipitation on surface long-wave radiation are smaller, and its daily integral value changes a little, and atmospheric long-wave radiation increases with cloud amount increasing. The surface albedo has obvious diurnal and seasonal variation characteristics and it is higher in winter than in autumn; the diurnal variation curves of albedo present unsymmetrical “V” shapes. In growing season, sensible heat flux and soil heat flux decreased along with cloud amount increasing; latent heat flux increases with cloud amount increasing on clear, partly cloudy and cloudy days; due to the precipitation on obscure days, latent heat flux decreases greatly because net radiation is seriously weakened. In non-growing season, daily integral value of net radiation is greatest on partly cloudy days, and net radiation on clear days is very close to it on cloudy and overcast days; sensible and latent heat flux decreases with cloud amount increasing, average daily integral value of soil heat flux is negative in non-growing season. In growing season, energy closure degree on cloudy days is best, and imbalance energy accounts for 3.9% of net energy; energy closure degree on overcast days is worst, and imbalance energy accounts for 16.8% of net energy; imbalance energy accounts for 7% of net energy under both clear and partly cloud situations. Due to snow effect in non-growing season, energy closure degree in non-growing season is greater obviously than it in growing season. Ionospheric disturbances produced by chemical releases at different release altitudes Hu Yao-Gai, Zhao Zheng-Yu, Zhang Yuan-Nong Acta Physica Sinica. 2013, 62 (20): 209401 doi: 10.7498/aps.62.209401 Full Text: [PDF 16471 KB] Download:(408) Show Abstract As one of important and effective means for modifying the ionosphere, releases of neutral gases including H2O, CO2, H2, SF6, etc. can produce artificial ionospheric holes. In this paper, the ionospheric disturbances produced under various release conditions, including use of different release species (H2O and SF6), different release altitudes, and different amounts of released substance, are investigated based on an improved three-dimensional chemical release dynamics model which includes neutral gas diffusion, chemical reaction and the ambipolar diffusion of the plasma. The effects of release altitudes and release amount on ionospheric disturbances, morphology and the dynamics of ionospheric hole are studied. Furthermore, the results are also briefly discussed. Empirical mode decomposition pulsar signal denoising method based on predicting of noise mode cell Wang Wen-Bo, Wang Xiang-Li Acta Physica Sinica. 2013, 62 (20): 209701 doi: 10.7498/aps.62.209701 Full Text: [PDF 796 KB] Download:(563) Show Abstract In order to improve the de-noising effect of the pulsar signal, an empirical mode decomposition (EMD) denoising algorithm based on the prediction of noise mode cell is put forward. The core steps of the proposed method is as follows: firstly, the noisy pulsar signal is decomposed into a group intrinsic mode function (IMF) by EMD, and the noise mode cell is predicted according to the IMF coefficients statistics and local minimum mean square error criteria. The selected noise mode cells are set to be zero. Then the IMF which has been processed according to noise mode cell prediction is denoised by optimal mode cell proportion shrinking, for removing the noise and retaining the signal details. The experimental results show that compared with the Sure Shrink wavelet threshold algorithm, Bayes Shrink wavelet threshold algorithm and the EMD mode cell proportion shrinking algorithm, the proposed method performs well in removing the pulsar signal noise and retaining the signal details information. The proposed method can achieve a higher signal-to-noise, the lower root mean square error, error of the peak position, relative error of the peak value and phase error. Copyright © Acta Physica Sinica Tel: 010-82649294，82649829，82649863 E-mail:
db11db52b17549a8	Open main menu Quantum tunnelling or tunneling (see spelling differences) is the quantum mechanical phenomenon where a subatomic particle passes through a potential barrier. Quantum tunneling is not predicted by the laws of classical mechanics where surmounting a potential barrier requires enough potential energy. Quantum tunnelling plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun.[1] It has important applications in the tunnel diode,[2] quantum computing, and in the scanning tunnelling microscope. The effect was predicted in the early 20th century, and its acceptance as a general physical phenomenon came mid-century.[3] Fundamental quantum mechanical concepts are central to this phenomenon, which makes quantum tunnelling one of the novel implications of quantum mechanics. Quantum tunneling is projected to create physical limits to the size of the transistors used in microprocessors, due to electrons being able to tunnel past them if the transistors are too small.[4][5] Tunnelling is often explained in terms of the Heisenberg uncertainty principle that the quantum object can be known as a wave or as a particle in general. Quantum tunnelling was developed from the study of radioactivity,[3] which was discovered in 1896 by Henri Becquerel.[6] Radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903.[6] Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlrausch. The idea of the half-life and the possibility of predicting decay was created from their work.[3] Quantum tunneling was first noticed in 1927 by Friedrich Hund when he was calculating the ground state of the double-well potential[6] and independently in the same year by Leonid Mandelstam and Mikhail Leontovich in their analysis of the implications of the then new Schrödinger wave equation for the motion of a particle in a confining potential of a limited spatial extent.[8] Its first application was a mathematical explanation for alpha decay, which was done in 1928 by George Gamow (who was aware of the findings of Mandelstam and Leontovich[9]) and independently by Ronald Gurney and Edward Condon.[10][11][12][13] The two researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunnelling. Introduction to the conceptEdit Animation showing the tunnel effect and its application to an STM A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier. The tunnelling problemEdit The wave function of a particle summarises everything that can be known about a physical system.[16] Therefore, problems in quantum mechanics center on the analysis of the wave function for a system. Using mathematical formulations of quantum mechanics, such as the Schrödinger equation, the wave function can be solved. This is directly related to the probability density of the particle's position, which describes the probability that the particle is at any given place. In the limit of large barriers, the probability of tunnelling decreases for taller and wider barriers. Related phenomenaEdit Tunnelling occurs with barriers of thickness around 1-3 nm and smaller,[18] but is the cause of some important macroscopic physical phenomena. For instance, tunnelling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague high-speed and mobile technology; it is considered the lower limit on how small computer chips can be made.[19] Tunnelling is a fundamental technique used to program the floating gates of flash memory, which is one of the most significant inventions that have shaped consumer electronics in the last two decades. Nuclear fusion in starsEdit Quantum tunneling is essential for nuclear fusion in stars. The temperature in stars' cores is generally insufficient to allow atomic nuclei to overcome the Coulomb barrier and achieve Thermonuclear fusion. Quantum tunneling increases the probability of penetrating this barrier. Though this probability is still low, the extremely large number of nuclei in the core of a star is sufficient to sustain a steady fusion reaction for millions, billions, or even trillions of years - a precondition for the evolution of life in insolation habitable zones.[20] Radioactive decayEdit Astrochemistry in interstellar cloudsEdit By including quantum tunnelling, the astrochemical syntheses of various molecules in interstellar clouds can be explained such as the synthesis of molecular hydrogen, water (ice) and the prebiotic important formaldehyde.[20] Quantum biologyEdit Quantum tunnelling is among the central non trivial quantum effects in quantum biology. Here it is important both as electron tunnelling and proton tunnelling. Electron tunnelling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis while proton tunnelling is a key factor in spontaneous mutation of DNA.[20] Cold emissionEdit Tunnel junctionEdit A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires understanding of quantum tunnelling.[25] Josephson junctions take advantage of quantum tunnelling and the superconductivity of some semiconductors to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields,[24] as well as the multijunction solar cell. Quantum-dot cellular automataEdit QCA is a molecular binary logic synthesis technology that operates by the inter-island electron tunneling system. This is a very low power and fast device that can operate at a maximum frequency of 15 PHz.[26] Tunnel diodeEdit Tunnel field-effect transistorsEdit A European research project has demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunnelling rather than by thermal injection, reducing gate voltage from ≈1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they will significantly improve the performance per power of integrated circuits.[29] Quantum conductivityEdit Scanning tunnelling microscopeEdit The scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, may allow imaging of individual atoms on the surface of a material.[24] It operates by taking advantage of the relationship between quantum tunnelling with distance. When the tip of the STM's needle is brought very close to a conduction surface that has a voltage bias, by measuring the current of electrons that are tunnelling between the needle and the surface, the distance between the needle and the surface can be measured. By using piezoelectric rods that change in size when voltage is applied over them the height of the tip can be adjusted to keep the tunnelling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor.[24] STMs are accurate to 0.001 nm, or about 1% of atomic diameter.[28] Kinetic isotope effectEdit In chemical kinetics, the substitution of a light isotope of an element with a heavier one typically results in a slower reaction rate. This is generally attribute to differences in the zero-point vibrational energies for chemical bonds containing the lighter and heavier isotopes and is generally modeled using transition state theory. However, in certain cases, very large isotope effects are observed that cannot be accounted for by a semi-classical treatment of kinetic isotope effects, and quantum tunneling through the energy barrier is invoked. R. P. Bell has developed a modified treatment of Arrhenius kinetics that is commonly used to model this phenomenon.[30] Faster than lightEdit Some physicists have claimed that it is possible for spin-zero particles to travel faster than the speed of light when tunnelling.[3] This apparently violates the principle of causality, since there will be a frame of reference in which it arrives before it has left. In 1998, Francis E. Low reviewed briefly the phenomenon of zero-time tunnelling.[31] More recently experimental tunnelling time data of phonons, photons, and electrons have been published by Günter Nimtz.[32] Other physicists, such as Herbert Winful,[33] have disputed these claims. Winful argues that the wavepacket of a tunnelling particle propagates locally, so a particle can't tunnel through the barrier non-locally. Winful also argues that the experiments that are purported to show non-local propagation have been misinterpreted. In particular, the group velocity of a wavepacket does not measure its speed, but is related to the amount of time the wavepacket is stored in the barrier. Mathematical discussions of quantum tunnellingEdit The following subsections discuss the mathematical formulations of quantum tunnelling. The Schrödinger equationEdit The solutions of this equation represent travelling waves, with phase-constant +k or -k. Alternatively, if M(x) is constant and positive, then the Schrödinger equation can be written in the form The WKB approximationEdit The wave function is expressed as the exponential of a function: , where is then separated into real and imaginary parts: with the following constraints on the lowest order terms, At this point two extreme cases can be considered. Case 2 If the phase varies slowly as compared to the amplitude, and Keeping only the first order term ensures linearity: Using this approximation, the equation near becomes a differential equation: This can be solved using Airy functions as solutions. Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the two coefficients on one side of a classical turning point, the two coefficients on the other side of a classical turning point can be determined by using this local solution to connect them. where are the two classical turning points for the potential barrier. For a rectangular barrier, this expression is simplified to: See alsoEdit 2. ^ Taylor, J. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. p. 234. ISBN 978-0-13-805715-2. 4. ^ "Quantum Computers Explained – Limits of Human Technology". Kurzgesagt. 8 December 2017. Retrieved 30 December 2017. 5. ^ "Quantum Effects At 7/5nm And Beyond". Semiconductor Engineering. Retrieved 15 July 2018. 8. ^ Mandelstam, L.; Leontowitsch, M. (1928). "Zur Theorie der Schrödingerschen Gleichung". Zeitschrift für Physik. 47 (1–2): 131–136. Bibcode:1928ZPhy...47..131M. doi:10.1007/BF01391061. 9. ^ Feinberg, E. L. (2002). "The forefather (about Leonid Isaakovich Mandelstam)". Physics-Uspekhi. 45 (1): 81–100. Bibcode:2002PhyU...45...81F. doi:10.1070/PU2002v045n01ABEH001126. 12. ^ Bethe, Hans (27 October 1966). "Hans Bethe - Session I". Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA (Interview). Interviewed by Charles Weiner; Jagdish Mehra. Cornell University. Retrieved 1 May 2016. 14. ^ Kolesnikov, Alexander I.; Reiter, George F.; Choudhury, Narayani; Prisk, Timothy R.; Mamontov, Eugene; Podlesnyak, Andrey; Ehlers, George; Seel, Andrew G.; Wesolowski, David J. (2016). "Quantum Tunneling of Water in Beryl: A New State of the Water Molecule". Physical Review Letters. 116 (16): 167802. doi:10.1103/PhysRevLett.116.167802. 15. ^ Davies, P. C. W. (2005). "Quantum tunneling time" (PDF). American Journal of Physics. 73 (1): 23–27. arXiv:quant-ph/0403010. Bibcode:2005AmJPh..73...23D. doi:10.1119/1.1810153. 17. ^ Eddi, A.; Fort, E.; Moisy, F.; Couder, Y. (16 June 2009). "Unpredictable Tunneling of a Classical Wave-Particle Association" (PDF). Physical Review Letters. 102 (24): 240401. Bibcode:2009PhRvL.102x0401E. doi:10.1103/PhysRevLett.102.240401. PMID 19658983. Retrieved 1 May 2016. 18. ^ Lerner; Trigg (1991). Encyclopedia of Physics (2nd ed.). New York: VCH. p. 1308. ISBN 978-0-89573-752-6. 19. ^ "Applications of tunneling" Archived 23 July 2011 at the Wayback Machine. Simon Connell 2006. 24. ^ a b c d e f Taylor, J. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. p. 479. ISBN 978-0-13-805715-2. 25. ^ Lerner; Trigg (1991). Encyclopedia of Physics (2nd ed.). New York: VCH. pp. 1308–1309. ISBN 978-0-89573-752-6. 26. ^ Sinha Roy, Soudip (25 December 2017). Generalized Quantum Tunneling Effect and Ultimate Equations for Switching Time and Cell to Cell Power Dissipation Approximation in QCA Devices. doi:10.13140/rg.2.2.23039.71849. 27. ^ a b Krane, Kenneth (1983). Modern Physics. New York: John Wiley and Sons. p. 423. ISBN 978-0-471-07963-7. 28. ^ a b Knight, R. D. (2004). Physics for Scientists and Engineers: With Modern Physics. Pearson Education. p. 1311. ISBN 978-0-321-22369-2. 30. ^ Percy), Bell, R. P. (Ronald (1980). The tunnel effect in chemistry. London: Chapman and Hall. ISBN 0412213400. OCLC 6854792. 32. ^ Nimtz, G. (2011). "Tunneling Confronts Special Relativity". Found. Phys. 41 (7): 1193–1199. arXiv:1003.3944. Bibcode:2011FoPh...41.1193N. doi:10.1007/s10701-011-9539-2. 33. ^ Winful, H. G. (2006). "Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox". Phys. Rep. 436 (1–2): 1–69. Bibcode:2006PhR...436....1W. doi:10.1016/j.physrep.2006.09.002. Further readingEdit External linksEdit
20ef956e46158ec7	Using Swedenborg to Understand the Quantum World III: Thoughts and Forms Swedenborg Foundation By Ian Thompson, PhD, Nuclear Physicist at the Lawrence Livermore National Laboratory In this series of posts, Swedenborg’s theory of correspondences has been shown to have interesting applications for helping us to better understand the quantum world. In part I, we learned that our mental processes occur at variable finite intervals and that they consist of desire, or love, acting by means of thoughts and intentions to produce physical effects. We in turn came to see the correspondential relationship between these mental events and such physical events that occur on a quantum level: in both cases, there will be time gaps between the events leading up to the physical outcome. So since we find that physical events occur in finite steps rather than continuously, we are led to expect a quantum world rather than a world described by classical physics. In part II, we saw that the main similarity between desire (mental) and energy (physical) is that they both persist between events, which means that they are substances and therefore have the capability, or disposition, for action or interaction within the time gaps between those events. Now we come to the question of how it is that these substances persist during the intervals between events. The events are the actual selection of what happens, so after the causing event and before the resultant effect, what occurs is the exploring of “possibilities for what might happen.” With regard to our mental processes, this exploration of possibilities is what we recognize as thinking. Swedenborg explains in detail how this very process of thinking is the way love gets ready to do things (rather than love being a byproduct of the thinking process, as Descartes would require): Everyone sees that discernment is the vessel of wisdom, but not many see that volition is the vessel of love. This is because our volition does nothing by itself, but acts through our discernment. It first branches off into a desire and vanishes in doing so, and a desire is noticeable only through a kind of unconscious pleasure in thinking, talking, and acting. We can still see that love is the source because we all intend what we love and do not intend what we do not love. (Divine Love and Wisdom §364) When we realize we want something, the next step is to work out how to do it. We first think of the specific objective and then of all the intermediate steps to be taken in order to achieve it. We may also think about alternative steps and the pros and cons of following those different routes. In short, thinking is the exploration of “possibilities for action.” As all of this thinking speaks very clearly to the specific objective at hand, it can be seen as supporting our motivating love, which is one of the primary functions of thought. A focused thinking process such as this can be seen, simplified, in many kinds of animal activities. With humans, however, thinking goes beyond that tight role of supporting love and develops a scope of its own. Not only do our thoughts explore possibilities for action, but they also explore the more abstract “possibilities for those possibilities.” Not only do we think about how to get a drink, but we also, for example, think about the size of the container, how much liquid it contains, and how far it is from where we are at that moment! When we get into such details as volume and distance, we discover that mathematics is the exploration of “possibilities of all kinds,” whether they are possibilities for action or not. So taken as a whole, thought is the exploration of all the many possibilities in the world, whether or not they are for action and even whether or not they are for actual things. For physical things (material objects), this exploration of possibilities is spreading over the possible places and times for interactions or selections. Here, quantum physics has done a whole lot of work already. Physicists have discovered that the possibilities for physical interactions are best described by the wave function of quantum mechanics. The wave function describes all the events that are possible, as well as all the propensities and probabilities for those events to happen. According to German physicist Max Born, the probability of an event in a particular region can be determined by an integral property of the wave function over that region. Energy is the substance that persists between physical events, and all physical processes are driven by energy. In quantum mechanics, this energy is what is responsible for making the wave function change through time, as formulated by the Schrödinger equation, which is the fundamental equation of quantum physics.[1] Returning now to Swedenborg’s theory of correspondences, we recognize that the something physical like thoughts in the mind are the shapes of wave functions in quantum physics. In Swedenborg’s own words: When I have been thinking, the material ideas in my thought have presented themselves so to speak in the middle of a wave-like motion. I have noticed that the wave was made up of nothing other than such ideas as had become attached to the particular matter in my memory that I was thinking about, and that a person’s entire thought is seen by spirits in this way. But nothing else enters that person’s awareness then apart from what is in the middle which has presented itself as a material idea. I have likened that wave round about to spiritual wings which serve to raise the particular matter the person is thinking about up out of his memory. And in this way the person becomes aware of that matter. The surrounding material in which the wave-like motion takes place contained countless things that harmonized with the matter I was thinking about. (Arcana Coelestia §6200)[2] Many people who have tried to understand the significance of quantum physics have noted that the wave function could be described as behaving like a non-spatial realm of consciousness. Some of these people have even wanted to say that the quantum wave function is a realm of consciousness, that physics has revealed the role of consciousness in the world, or that physics has discovered quantum consciousness.[3] However, using Swedenborg’s ideas to guide us, we can see that the wave function in physics corresponds to the thoughts in our consciousness. They have similar roles in the making of events: both thoughts and wave functions explore the “possibilities, propensities, and probabilities for action.” They are not the same, but they instead follow similar patterns and have similar functions within their respective realms. Thoughts are the way that desire explores the possibilities for the making of intentions and their related physical outcomes, and wave functions are the way that energy explores the possibilities for the making of physical events on a quantum level. The philosophers of physics have been puzzled for a long time about the substance of physical things,[4] especially that of things in the quantum realm. From our discussion here, we see that energy (or propensity) is also the substance of physical things in the quantum realm and that the wave function, then, is the form that such a quantum substance takes. The wave function describes the shape of energy (or propensity) in space and time. We can recognize, as Aristotle first did, that a substantial change has occurred when a substance comes into existence by virtue of the matter of that substance acquiring some form.[5] That still applies to quantum mechanics, we now find, even though many philosophers have been desperately constructing more extreme ideas to try to understand quantum objects, such as relationalism[6] or the many-worlds interpretation.[7] So what, then, is this matter of energy (desire, or love)? Is it from the Divine? Swedenborg would say as much: It is because the very essence of the Divine is love and wisdom that we have two abilities of life. From the one we get our discernment, and from the other volition. Our discernment is supplied entirely by an inflow of wisdom from God, while our volition is supplied entirely by an inflow of love from God. Our failures to be appropriately wise and appropriately loving do not take these abilities away from us. They only close them off; and as long as they do, while we may call our discernment “discernment” and our volition “volition,” essentially they are not. So if these abilities really were taken away from us, everything human about us would be destroyed—our thinking and the speech that results from thought, and our purposing and the actions that result from purpose. We can see from this that the divine nature within us dwells in these two abilities, in our ability to be wise and our ability to love. (Divine Love and Wisdom §30) When seeing things as made from substance—from the energy (or desire) that endures between events and thereby creates further events—we note that people will tend to speculate about “pure love” or “pure energy”: a love or energy without form that has no particular objective but can be used for anything. But this cannot be. In physics, there never exists any such pure energy but only energy in specific forms, such as the quantum particles described by a wave function. Any existing physical energy must be the propensity for specific kinds of interactions, since it must exist in some form. Similarly, there never exists a thing called “pure love.” The expression “pure love” makes sense only with respect to the idea of innocent, or undefiled, love, not to love without an object. Remember that “our volition [which is the vessel of love] does nothing by itself, but acts through our discernment.” Ian Thompson is also the author of Starting Science from Godas well as Nuclear Reactions in Astrophysics (Univ. of Cambridge Press) and more than two hundred refereed professional articles in nuclear physics. [1] Wikipedia,ödinger_equation. [2] Secrets of Heaven is the New Century Edition translation of Swedenborg’s Arcana Coelestia. [3] See, for example, [4] Howard Robinson, “Substance,” Stanford Encyclopedia of Philosophy, [5] Thomas Ainsworth, “Form vs. Matter,” Stanford Encyclopedia of Philosophy, [6] Michael Epperson, “Quantum Mechanics and Relational Realism: Logical Causality and Wave Function Collapse,” Process Studies 38.2 (2009): 339–366. [7] J. A. Barrett, “Quantum Worlds,” Principia 20.1 (2016): 45–60. Visit our Swedenborg Studies bookstore page to explore our series of scholarly titles > Read more posts from the Scholars on Swedenborg series > Using Swedenborg to Understand the Quantum World II: Desire and Energy Swedenborg Foundation In the previous post of this series, we saw how Swedenborg’s theory of correspondences could help us to better understand the physical world from a quantum perspective. If our mental processes consist of desire acting by means of thoughts and intentions to produce physical effects, then these physical actions should manifest themselves according to a corresponding pattern. More specifically, if the components of our mental processes occur at variable finite intervals, so too should the expected physical events. According to many thinkers throughout history, mental and physical are not identical but instead are two different kinds of substances that relate with each other. Swedenborg describes the mental (spiritual) and physical (natural) as distinct but says that they interact by discrete degrees: A knowledge of degrees is like a key to lay open the causes of things, and to give entrance into them. . . . For things exterior advance to things interior and through these to things inmost, by means of degrees; not by continuous degrees but by discrete degrees. “Continuous degrees” is a term applied to the gradual lessenings or decreasings from grosser to finer . . . or . . . to growths and increasings from finer to grosser . . . precisely like the gradations of light to shade, or of heat to cold. But discrete degrees are entirely different: they are like things prior, subsequent and final; or like end, cause, and effect. These degrees are called discrete, because the prior is by itself; the subsequent by itself; and the final by itself; and yet taken together they make one. (Divine Love and Wisdom §184) The mental can never be continuously transformed into something physical, nor can the physical be continuously transformed into something mental. They are connected, however, by virtue of their causal relationship: all physical processes are produced, or generated, by something mental. As described in my previous post, this relationship is what gives rise to our correspondences in the first place. Most of us can realize that the mental and the physical are distinct, even though this may be denied by materialists (for whom the mental is merely an emerging product of the physical) and also by monistic idealists (for whom the physical universe is merely a representation in the mind). The latter view is common in many New Age circles today, and it is even thought to be implied by quantum physics. In this series of posts, by contrast, I want to show how Swedenborg’s ideas give us a new understanding of how mental and physical things can both exist in fully-fledged ways and with serious connections between them that are not deflating or reductionist. Mental and physical things can both be substances but, they have very different characteristics: • Mental things are conscious, whereas physical things are unconscious. • Mental beings can think and make deductions using reason, whereas physical beings can only make logical deductions if they are designed that way. • Mental beings can use symbols and language to refer to objects and ideas outside themselves, whereas physical beings have no intrinsic ability to refer to anything. • Mental processes are motivated by purposes and intentions, whereas physical processes are determined by physical causes that supposedly exclude purposes and intentions. • Mental processes tend to produce results according to some conception of what is good, whereas physical processes have no need for any such concept. As already discussed in the previous post, desire is a component of all mental processes, and we recognize “something physical like desire” as energy or propensity. Swedenborg sees desire, or affection, as a specific kind of love: That love and wisdom from the Lord is life can be seen also from this, that man grows torpid as love recedes from him, and stupid as wisdom recedes from him, and that were they to recede altogether he would become extinct. There are many things pertaining to love which have received other names because they are derivatives, such as affections, desires, appetites, and their pleasures and enjoyments. (Divine Love and Wisdom §363) For desire and energy to correspond to each other in the sense that Swedenborg describes, the function of desire as a cause must be similar to the function of energy as a cause. That is, the way in which desire causes mental processes must be similar to the way in which energy causes physical processes. This is not to say that desire is the same as energy but only that desire’s pattern of operation is similar to that of energy. The common pattern is that desire (energy) persists between events, then explores multiple possibilities for those events by means of thoughts (fields of energy), and finally becomes manifest in the physical events produced. Up until now, the idea of substance has been rather obscure in both physics and philosophy, and it has not been developed significantly. From an ontological perspective, substance is that which endures between events. It is what individuates and bears the intrinsic properties of those events. We are not necessarily talking about a substance that endures forever or about a substance that exists independently of everything else. Based on the common pattern described above, we can arrive at the idea of a created substance that persists, or endures, as a thing at least for some finite time between events. And such a substance would be the capability, or disposition, for action or interaction in that time interval. This relates to the idea of “dispositional essentialism” that has been put forth by philosophers in recent years.[1] Dispositional essentialism is the notion that some kind of power or disposition (such as a cause or energy) must be an essential part of something. Some philosophers take this idea even further, saying that disposition must be the individual essence of something. In much the same way, I am saying that disposition is what constitutes the substance of something.[2] So if the main similarity between desire and energy is that they both persist between events, then both desire and energy are substances. By using ideas from Swedenborg to understand the world, we have a new way of grasping the mental and physical and perhaps of understanding quantum physics. Either one of these results would be very useful; to have both is to be extremely fortunate. In the next post of this series, I will discuss how and in what form both desire and energy persist between events. [1] B. Ellis and C. Lierse, “Dispositional Essentialism,” Australasian Journal of Philosophy (72, 1994): 27–45. [2] See Ian J. Thompson, “Power and Substance,” Read more posts from the Scholars on Swedenborg series > Nurturing the Soul Course By Helen Brown published by © July 2009 pp 75 £9.95 Enquiries: contact Helen also leads groups taking this course. The aim is primarily to encourage reflection, experience and exploration of what our ‘soul’ means for each of us. The scope of the course includes music, art, prayer, meditation and energy medicine. This has been a course that was both inspirational and challenging. Throughout the eight sessions in our exploration of the ‘soul’, Helen gently guided us along an inner path in keeping with her book on the subject. We were a small group who throughout our time together formed a very supportive and close connection. At the beginning of each session we sat round a table where there was always a simple but beautiful delicate arrangement of flowers and leaves encircling a candle. As we lit the candle and concentrated on the light we left behind the outside world, forming our own peaceful and tranquil world through meditation and prayer. This made a perfect start to our spiritual journey, discovering and connecting with our own soul. During the course we were encouraged to follow our own path freely, expressing and sharing our thoughts and feelings. It brought in music, art, prayer, meditation and specially chosen readings to help in this search and nourishment of our own soul. It was suggested by Helen that we should keep a journal where we can reflect on the course as it unfolds, giving time for more thought. I chose an artist’s journal as I find both illustrating and writing stimulating. In the quest for the soul I found I was becoming more attuned with my inner self, gaining a deeper understanding of what this long journey involved. There were parts which were challenging and painful; complex issues had to be confronted before moving on. Throughout the sessions this was balanced with the uplifting realization that the soul is a recipient of life from God and that the Divine flows in with Love and Wisdom. The soul is a sacred place where God dwells. This knowledge is inspirational, but before we can find and nurture the soul we have first to reach down into the recesses of ourselves, seek and find God and with love bring Him into our daily lives, letting His light shine in our hearts. I have found this course exhilarating and plan to continue with this spiritual journey. As we sat around the table for our last meditation, it came to me that our group represented a lotus flower. I visualized each one of us as petals of the flower joined together as one with the candle in the centre radiating light. As we lent forward to blow out the flame we sent love and peace to the world. “OM. In the centre of the castle of Brahman, our own body, there is a small shrine in the form of a lotus-flower and within can be found a small space, we should find who dwells there, and we should want to know him” (Chandogya Upanishad). Rosella Williams, Course member Here are some additional comments from others in the group: ‘its been a spiritual journey, a journey of discovery’, ‘ I found it opened doors and brought up things that I hadn’t thought about before’, ‘enlightening’, ‘gave me peaceful thoughts – we shared the journey’, ‘helpful and reassuring’.
e7bdec5c08834e54	Senin, 20 Mei 2013 A "Quantum Micrsocope" To Look At The Hydrogen Wavefunction The hydrogen wavefuction is one of the few systems that we can solve analytically. That is why we teach them in undergraduate QM classes. Yet, the ability to actually view such wavefunction isn't trivial and is part of the fundamental aspect of QM. This latest work looks at the nodal structure of a hydrogen atomic orbitals using photoionization. In the process, the authors have provided a significant step in developing a "quantum microscope". Writing in Physical Review Letters, Aneta Stodolna, of the FOM Institute for Atomic and Molecular Physics (AMOLF) in the Netherlands, and her colleagues demonstrate how photoionization microscopy directly maps out the nodal structure of an electronic orbital of a hydrogen atom placed in a dc electric field. This experiment—initially proposed more than 30 years ago—provides a unique look at one of the few atomic systems that has an analytical solution to the Schrödinger equation. To visualize the orbital structure directly, the researchers utilized an electrostatic lens that magnifies the outgoing electron wave without disrupting its quantum coherence. The authors show that the measured interference pattern matches the nodal features of the hydrogen wave function, which can be calculated analytically. The demonstration establishes the microscopy technique as a quantum probe and provides a benchmark for more complex systems. The link above provides free access to the paper. Tidak ada komentar: Posting Komentar
b62c101d68a37ae5	What is Real? There’s a new popular book out this week about the interpretation of quantum mechanics, Adam Becker’s What is Real?: The Unfinished Quest for the Meaning of Quantum Physics. Ever since my high school days, the topic of quantum mechanics and what it really means has been a source of deep fascination to me, and usually I’m a sucker for any book such as this one. It’s well-written and contains some stories I had never encountered before in the wealth of other things I’ve read over the years. Unfortunately though, the author has decided to take a point of view on this topic that I think is quite problematic. To get an idea of the problem, here’s some of the promotional text for the book (yes, I know that this kind of text sometimes is exaggerated for effect): A mishmash of solipsism and poor reasoning, [the] Copenhagen [interpretation] claims that questions about the fundamental nature of reality are meaningless. Albert Einstein and others were skeptical of Copenhagen when it was first developed. But buoyed by political expediency, personal attacks, and the research priorities of the military industrial complex, the Copenhagen interpretation has enjoyed undue acceptance for nearly a century. The text then goes to describe Bohm, Everett and Bell as the “quantum rebels” trying to fight the good cause against Copenhagen. Part of the problem with this good vs. evil story is that, as the book itself explains, it’s not at all clear what the “Copenhagen interpretation” actually is, other than a generic name for the point of view the generation of theorists such as Bohr, Heisenberg, Pauli, Wigner and von Neumann developed as they struggled to reconcile quantum and classical mechanics. They weren’t solipsists with poor reasoning skills, but trying to come to terms with the extremely non-trivial and difficult problem of how the classical physics formalism we use to describe observations emerges out of the more fundamental quantum mechanical formalism. They found a workable set of rules to describe what the theory implied for results of measurements (collapse of the state vector with probabilities given by the Born rule), and these rules are in every textbook. That there is a “measurement problem” is something that most everyone was aware of, with Schrodinger’s cat example making it clear. Typically, for the good reason that it’s complicated and they have other topics they need to cover, textbooks don’t go into this in any depth (other than often telling about the cat). As usual these days, the alternative to Copenhagen being proposed is a simplistic version of Everett’s “Many Worlds”: the answer to the measurement problem is that the multiverse did it. The idea that one would also like the measurement apparatus to be described by quantum mechanics is taken to be a radical and daring insight. The Copenhagen papering over of the measurement problem by “collapse occurs, but we don’t know how” is replaced by “the wavefunction of the universe splits, but we don’t know how”. Becker pretty much ignores the problems with this “explanation”, other than mentioning that one needs to explain the resulting probability measure. String theory, inflation and the cosmological multiverse are then brought in as supporting Many Worlds (e.g. that probability measure problem is just like the measure problem of multiverse cosmology). There’s the usual straw man argument that those unhappy with the multiverse explanation are just ignorant Popperazzi, unaware of the subtleties of the falsifiability criterion: Ultimately, arguments against a multiverse purportedly based on falsifiability are really arguments based on ignorance and taste: some physicists are unaware of the history and philosophy of their own field and find multiverse theories unpalatable. But that does not mean that multiverse theories are unscientific. For a much better version of the same story and much more serious popular treatment of the measurement problem, I recommend a relatively short book that is now over 20 years old, David Lindley’s Where does the Weirdness Go?. Lindley’s explanation of Copenhagen vs. Many Worlds is short and to the point: The problem with Copenhagen is that it leaves measurement unexplained; how does a measurement select one outcome from many? Everett’s proposal keeps all outcomes alive, but this simply substitutes one problem for another: how does a measurement split apart parallel outcomes that were previously in intimate contact? In neither case is the physical mechanism of measurement accounted for; both employ sleight of hand at the crucial moment. Lindley ends with a discussion of the importance of the notion of decoherence (pioneered by Dieter Zeh) for understanding how classical behavior emerges from quantum mechanics. For a more recent serious take on the issues involved, I’d recommend reading something by Wojciech Zurek, for instance this article, a version of which was published in Physics Today. Trying to figure out what “interpretation” Zurek subscribes to, I notice that he refers to an “existential interpretation” in some of his papers. I don’t really know what that means. Unlike most discussions of “interpretations”, Zurek seems to be getting at the real physical issues involved, so I think I’ll adopt his (whatever it means) as my chosen “interpretation”. Update: For another take on much the same subject, out in the UK now is Philip Ball’s Beyond Weird. The US version will be out in the fall, and I think I’ll wait until then to take a look. In the meantime, Natalie Wolchover has a review at Nature. Update: There’s a new review of What is Real? at Nature. Update: Jim Holt points out that David Albert has a review of the Becker book in the latest New York Review of Books. I just read a print copy last night, presumably it should appear online soon here [Review now available here]. Update: Some comments from Adam Becker, the author of the book. I won’t try to rebut everything Peter has said about my book—there are some things we simply disagree about—but I would like to clear up two statements he makes about the book that are possibly misleading: Peter says that I claim the answer to the measurement problem is that “the multiverse did it.” But I don’t advocate for the many-worlds interpretation in my book. I merely lay it out as one of the reasonable available options for interpreting quantum mechanics (and I discuss some of its flaws as well). I do spend a fair bit of time talking about it, but that’s largely because my book takes a historical approach to the subject, and many-worlds has played a particularly important role in the history of quantum foundations. But there are other interpretations that have played similarly important roles, such as pilot-wave theory, and I spend a lot of time talking about those interpretations too. I am not a partisan of any particular interpretation of quantum mechanics. Second, I don’t think it’s quite fair to say that I paint Bohr as a villain. I mention several times in my book that Bohr was rather unclear in his writing, and that sussing out his true views is dicey. But what matters more that Bohr’s actual views is what later generations of physicists generally took his views to be, and the way Bohr’s work was uncritically invoked as a response to reasonable questions about the foundations of quantum mechanics. It’s true that this subtlety is lost in the jacket flap copy, but that’s publishing for you. Also, for what it’s worth, I do like talking about reality as it relates to quantum mechanics. But I suppose that’s hardly surprising, given that I just wrote a book on quantum foundations titled “What Is Real?”. I’d be happy to discuss all of this further over email if anyone is interested (though I’m pretty busy at the moment and it might take me some time to respond). This entry was posted in Book Reviews, Quantum Mechanics. Bookmark the permalink. 74 Responses to What is Real? 1. Carl Zetie says: My 13 year old son has been reading some of the introductory texts on QM and his reaction on discovering Everett was to ask me, “So Everett’s Many Worlds has the same problem as the Copenhagen interpretation, plus an enormous number of additional universes that are not detectable. How is that better?”. Good kid, but I’m going to need to find him a math tutor soon because he is overtaking my undergrad education. 2. Jim Baggott says: Give me a little more time. I hope to be able to set the record straight in a book to be published probably in 2020. You can read the preamble here: http://www.jimbaggott.com/articles/a-game-of-theories/ 3. Dear Prof. Woit, I don’t understand what you mean by “the wavefunction of the universe splits, but we don’t know how”. We know exactly how; this is the whole point of Many-Worlds: it is just normal Hamiltonian evolution. What do you think is lacking? 4. Ben Jones says: It’s fine to say that ‘in either interpretation, we don’t know what happens at the moment of measurement’. But I disagree that this leaves MW and Copenhagen on an equal footing. Doesn’t Copenhagen posit an additional _mechanism_, that of ‘collapse’, whereas Everett’s view wouldn’t? And are there not additional problems with Copenhagen, such as the ‘selection’ of a single branch is the only non-deterministic element of the whole system? @Carl the non-detectability of the ‘other universes’ follows naturally from how we understand QM to work, it doesn’t count as a piece of evidence against it if we wouldn’t _expect_ to be able to detect them. An unreasonable swish of Occam’s Razor. 5. David Brown says: “Popperazzi” (play on Italian “paparazzi”) is the preferred spelling. https://www.philosophersmag.com/index.php/footnotes-to-plato/77-string-theory-vs-the-popperazzi “String Theory vs the Popperazzi” by Massimo Pigliucci, 2015 6. Peter Woit says: David Brown, Thanks. Fixed. Mateus Araujo, Just saying “the Schrodinger equation does it” doesn’t solve the measurement problem (for instance, the preferred basis problem). All you’re doing is saying that, you don’t know how, but the Schrodinger equation is going to somehow give precisely the same implications for physics as the collapse postulate. Ben Jones, Copenhagen says collapse happens in a measurement, it’s silent on what the theory of collapse is (adding specific new physics to explain collapse is not Copenhagen but something else). 7. hdz says: Peter, I do not intend to enter the details of your discussion, since I have got tired about it. (If somebody should be interested, he/she may look at my website http://www.zeh-hd.de – especially the first two papers under “Quantum Theory”.) However, as a historical remark let me point out that in my understanding, von Neumann and Wigner were never part of the Copenhagen interpretation – rather they objected to it more to less openly (starting in Como). When Wigner used the term “orthodox interpretation”, he exclusively meant von Neumann’s book (including the collapse as a physical process – not just a “normal” increase of information), and I am told that he complained that, as a consequence, he was never invited to Copenhagen. Bohr always disagreed with any attempt to analyze the measurement problem in physical terms. Essentially, I agree with Mateus Aroujo and and Ben Jones Regards, Dieter 8. Jim Baggott says: This kind of discussion can get hopelessly confused very quickly. To my knowledge, the ‘collapse of the wavefunction’ was never part of the Copenhagen interpretation, which is based on some kind of unexplained ‘separation’ between the quantum and classical realms, or what John Bell would refer to as the ‘shifty split’. I believe the notion of a ‘collapse’ was introduced as a ‘projection postulate’ by von Neumann in his book Mathematical Foundations of Quantum Mechanics, first published in German in 1932 (in my English translation the projection postulate – a statistical, discontinuous process in contrast to the continuous, unitary evolution of the wavefunction – appears on p. 357). Influenced, I believe, by Leo Szilard, von Neumann speculates that the collapse might be triggered by the intervention of a (human) consciousness – what he refers to on p. 421 as the observer’s ‘abstract ego’. All this really shouldn’t detract from the main point. The formalism is the formalism and we know it works (and we know furthermore that it doesn’t accommodate local or crypto non-local hidden variables). The formalism is, for now, empirically unassailable. All interpretations of the formalism are then exercises in metaphysics, based on different preconceptions of how we think reality could or should be, such as deterministic (‘God does not play dice’). Of course, the aim of such speculations is to open up the possibility that we might learn something new, and I believe extensions which seek to make the ‘collapse’ physical, through spacetime curvature and/or decoherence, are well motivated. But until such time as one interpretation or extension can be demonstrated to be better than the other through empirical evidence, the debate (in my opinion) is a philosophical one. I’m just disappointed (and rather frustrated) by the apparent rise of a new breed of Many Worlds Taliban who claim – quite without any scientific justification – that the MWI is the only way and the one true faith. 9. Peter Woit, There are dozens of papers explaining how you can model the measurement process in unitary quantum mechanics via entanglement and decoherence; Zeh’s and Żurek’s, among them. It’s not as if anybody is saying “the Schrödinger equation did it” without explaining how. And the measurement problem is usually defined as the incompatibility between collapse and unitary evolution. So even just saying “the Schrödinger equation did it” does solve the problem, on this level. Of course, one must also explain emergence of classicality, permanence of records, probability, etc., but this is not the measurement problem. 10. Peter Woit says: Thanks Dieter. I should point out that Dieter Zeh is one of the major subjects of the book. He’s perhaps the best example of the story Becker wants to fit everything into: a “quantum rebel” who made significant advances in our understanding of foundations and the measurement problem, advances which were not initially recognized due to an entrenched “Copenhagen” ideology that denigrated any such work. Partly due to his work, I think for many decades now mindless Copenhagen ideology has not been such a problem (but we’re well on our way to mindless Many Worlds ideology being a major problem). As he notes and the book describes, there’s controversy over who was on the Copenhagen bus, partly because it was never clear exactly what the Copenhagen Interpretation was, and partly because many people took somewhat different points of view at different times. 11. Peter Woit says: Mateus Araújo, To me the (non-trivial) measurement problem is exactly the “emergence of classicality, permanence of records, probability, etc.,” problems you list, with entanglement and decoherence some of the insights needed to solve them. If you define the measurement problem as you do, then the solution you give by having quantum mechanics apply to the macroscopic world is a trivial one which most everyone expected anyway. 12. Peter Woit, I’m claiming that this is not “my” definition of the measurement problem, but rather the standard definition. See for example Maudlin’s “Three measurement problems”, a canonical reference on the subject. Of course, the solution is trivial; you merely need to give up collapse, or introduce hidden variables, or introduce physical collapse. The problem is that most people don’t want to do any of these, hence they are stuck with the measurement problem. Look, I agree wholeheartedly with you that the interesting problems are those I listed; I’m just insisting on narrow definitions and standard terminology, otherwise it becomes impossible to talk to each other. 13. Peter Woit says: Mateus Araújo, This whole subject is full of endless and heated arguments over problems that are meaningless and/or lacking in substance. Your definition of “the measurement problem” seems to me to insist on sticking to the substance-free aspect of a substantive problem, thus encouraging substance-free-discussion. What’s the correct term to refer to the substantive problem? By the way, on the meaningless side, Becker’s book has a lot about “What is Real?” and claims that the problem with Copenhagen is that it denies the reality of the microscopic world, a discussion I thought it best to just ignore. 14. Peter Woit, Indeed it is! And I think a great many heated and meaningless arguments happen because the parties are talking about different things =) But I don’t know which problem you refer to as “the substantive problem”. Do you have in mind a collective name for the three I listed? I don’t think there exists a standard name for this set, except maybe as “problems of the Many-Worlds interpretation”. They are different problems, that are usually studied separately, and referred to by their individual names. 15. Peter Woit says: Mateus Araújo, I guess I always thought “what happens to the cat?” was the substantive measurement problem, so I’m looking for a name for that. 16. Blake Stacey says: Asher Peres once wrote that there are at least as many Copenhagen Interpretations as people who use the term, probably more. Among other problems, saying “the Copenhagen Interpretation” glosses over substantial differences between Heisenberg and Bohr. Despite this issue being discussed by Pauli, von Neumann, …. 17. Peter Woit, You want unitarity to hold, so that atoms are described by quantum mechanics, and you want collapse to happen, so that the cat is definitely dead, but you can’t have both: the measurement problem! And the solution is again trivial, threefold. Give up on collapse: there are worlds with live cats and worlds with dead cats. Introduce hidden variables: the cat was always dead, you just didn’t know it. Introduce physical collapse: the superposition collapsed on the level of the geiger counter, so that the cat was always dead. Maybe the measurement problem is like the problem of having your cake and eating it too. You know it’s impossible, but you really really want to! 18. Jim Baggott says: I think we can agree that this is all a lot of fun. I’ve been studying these problems for more than 25 years, and I can discern relatively little progress in this time. But I’ve come to believe that it is helpful to distinguish between ‘reality’ (however you want to think about it) and the reality or otherwise of the representation we use to describe it. We can likely all agree that the moon is there when nobody looks, and even that invisible entities like electrons really do exist independently of observation (‘if you can spray them then they are real’, as philosopher Ian Hacking once declared). But this doesn’t necessarily mean that the concepts we use in our representation of this reality should be taken literally as real, physical things. If we choose to agree that the wavefunction isn’t real (as Carlo Rovelli argues in his relational interpretation), then the QM formalism is simply an algorithm for coding what we know about a physical system so that can make successful predictions. All the mystery then goes away – there is no problem of non-locality, no collapse of the wavefunction, and no ‘spooky’ action at a distance. I don’t particularly like this interpretation, as it is obviously instrumentalist and can’t answer our burning questions about how nature actually does that. But it does help to convince me that this endless debate over interpretation is really a philosophical debate, driven by everybody’s very different views on what ‘reality’ ought to be like. And, as such, we’re unlikely to see a resolution anytime soon… 19. Peter Woit says: I’d rather do almost anything with my time than try and moderate a discussion of what is “real” and what isn’t. Any further discussion of ontology will be ruthlessly suppressed. 20. Per Östborn says: Trying to stick to physical issues, I have always wanted to see proponents of the many worlds interpretation derive the spectrum of the hydrogen atom from some clearly defined postulates (or solve some other basic physical problem). Until then it is not clear to me what the theory actually is saying. If anyone can refer to such a calculation I would be glad. Clearly it is not enough to postulate the unitary evolution of the Schrödinger equation, but I have never been able to pinpoint the other postulates of the many worlds interpretation. 21. George Herold says: Thanks for the nice post and discussion. I’m an experimentalist, who is more comfortable with opamps than operators. I have perhaps a naive question; Do any of these books discuss the DeBroglie-Bohm (Pilot wave) theory? Since all these interpretations are just a matter of taste, I find this hidden variable type theory easiest to swallow. In a double slit, the particle ‘knows’ about both slits, but still goes through (only) one of them. Is there any reason not to be happy with this picture? 22. Peter Woit says: Per Östborn, The calculation in Many Worlds is exactly the same textbook calculation as in Copenhagen. It’s the same Schrodinger equation and you solve for its energy eigenvalues the same way. That is the problem: there’s no difference from the standard QM textbook. The Many Worlds people might claim as an advantage over Copenhagen that you can imagine doing much harder calculations about how a Hydrogen atom interacts with its environment during a measurement process, giving insight into the question of why we see energy eigenstates and the Born rule. My point of view would be that Copenhagen never said you shouldn’t do exactly those calculations if you wanted to better understand what was happening during a measurement, it was just a rule for when you couldn’t do such calculations. 23. Peter Woit says: George Herold, Becker’s book has a long and detailed section about the story of Bohm and Bohmian mechanics which you might find of interest. Personally I don’t find Bohmian mechanics compelling, both for reasons Becker mentions and for others having to with the much greater mathematical simplicity of the conventional formalism when applied to our best fundamental theories. Sorry, but I really don’t want to carry on more discussion here of Bohmian mechanics. I realize a lot of people are interested in it, but I’m not at all interested, so such a discussion should be conducted elsewhere. 24. Lee Smolin says: Dear Peter and all, Can I suggest a basic distinction between different approaches to quantum foundations? The first class hypothesizes that the standard quantum mechanics is incomplete as a physical theory, because it fails to give a complete description of individual phenomena. If so, the theory requires a completion, that is a theory which incorporates additional degrees of freedom, and/or additional dynamics. Pilot wave theory and physical collapse models are examples of these. The second class of approaches take it as given that the theory is complete, so the foundational puzzles are to be addressed by modifying how we interpret the equations of the theory. People engaged in the first class of approaches are trying to solve a very different kind of research problem than those in the second class. I would submit that both are worth pursuing, but that progress in physics will eventually depend on the success of the first kind of approach. 25. Peter Woit says: Thanks Lee, That’s a very useful distinction. The Becker book contains a lot of material about such attempts to complete quantum mechanics with new physics that would entail a different resolution of the measurement problem. I didn’t discuss these here, largely because I’m much less optimistic than you that the search for this kind of completion will be fruitful. 26. Per Östborn says: Dear Peter, It is not clear to me, but my impression is that MWI proponents claim that some of the standard postulates of QM can be removed (or replaced by other ones). Then their calculations from first principles must be different from the textbook ones, since they are forbidden to use the removed postulates, of course. As you write, they seem to want to gain insight into why we see energy eigenvalues and why we should use the Born rule. This suggests that they argue that these features of QM can be derived from a smaller or different set of postulates than the standard one. 27. Peter Woit says: Per Östborn, I’m no expert and don’t want to get into the details of exactly what “Many Worlds” is, I think there are many versions. One aspect of it though is yes, the idea that you can derive the relation to classical observables and the Born rule, not postulate them. There’s active research and debate on the extent to which you really can do this. I just want to point out that you naturally ask exactly the same questions in Copenhagen, whenever you decide you want to analyze in more detail what’s happening during a measurement. It’s exactly the same equation and physics. 28. Edward says: I think the most interesting Copenhagen/anti-Copenhagen split was between Heisenberg, Bohr, and others on the one hand and Einstein, Schrodinger, and to some extent de Broglie on the other. There was a generational split and Heisenberg’s views may have prevailed because the older physicists died off. 29. Narad says: Any further discussion of ontology will be ruthlessly suppressed. I think I’m going to have to have that printed on a T-shirt. 30. Chris W. says: I think it would be fair to say that the acceptance of the Copenhagen interpretation, or what various people took it to be, was substantially a function of the high regard in which Bohr and Heisenberg were held as pioneers in the development of quantum mechanics, combined with a strong desire to just “get on with it”—find a way to state problems and do calculations that most people felt they understood well enough to do research and make sensible progress. Later, when a great deal of research had been done and generally accepted progress had been made, it’s understandable that the “yes, but…” questions about quantum theory would be resurgent, especially with the conundrums of quantum gravity looming in the background, along with the growing exploration of quantum phenomena at mesoscopic scales. To my mind the current situation is reminiscent in some ways of Mach’s objections to the conventional understanding of Newtonian dynamics at a time (the mid- to late 19th century) when such concerns had little apparent significance for most working scientists. All this appeared in a different light with the advent of special and general relativity. 31. Peter Woit, I think this is pretty much correct; one can also simply postulate the Born rule in Many-Worlds, as done in single-world theories, but it feels a bit wrong, as one should also explain what probabilities are. Hence the interest in deriving the Born rule in Many-Worlds, as this should shed some light on the issue. One can, though, adapt these derivations also to single-world quantum mechanics, as done by Saunders here. I don’t think, however, that the derivations we have are completely satisfactory, and I’m personally trying to improve on them. 32. Peter Shor says: David Mermin, who didn’t particularly like the Copenhagen interpretation, found himself driven to use it when he was teaching quantum mechanics to computer scientists who didn’t know much about physics. (See his article Copenhagen Computation.) Bohr, at Copenhagen, similarly was teaching quantum mechanics to physicists who didn’t know much about quantum mechanics. So maybe this explained why he gravitated towards the Copenhagen interpretation. And then maybe Bohr came up with all this weird extraneous philosophy to try to convince himself that what he was teaching them actually made some sense. 33. Paddy says: Jim Baggott (or anyone else for that matter): You attribute “collapse” or reduction of the wavepacket to von Neumann (1932). A clear statement (albiet much shorter) is in Dirac’s Principles, Sect 10. Unfortunately our University Library foolishly left the first edition (1930) on the open shelves, and it has long since vanished, so I’ve only traced this statement back to the 2nd ed… My question to anyone who can lay hands on a copy is, is this bit in the original edition, or did Dirac add it after reading von Neumann? As I probably learned from one of Jim’s books, the “reduction” of the wavepacket is Heisenberg’s terminology, from the uncertainty principle paper, so von Neumann or Dirac were in any case just sharpening up Heisenberg’s insight. 34. Tim Maudlin says: It is a bit hard to know how to comment on a discussion of a book called “What is Real?” when it has been asserted that Any further discussion of ontology will be ruthlessly suppressed.” The question “What is real?” just is the question “What exists?” which is in turn just the question “What is the true physical ontology?” which is identical to the question “Which physical theory is true?”. Peter Woit begins by writing “Ever since my high school days, the topic of quantum mechanics and what it really means has been a source of deep fascination to me…”. But that just is the question: What might the empirical success of the quantum formalism imply about what is real? or What exists? or What is the ontology of the world? To say you are interested in understanding the implications of quantum mechanics for physical reality but then ruthlessly suppress discussions of ontology is either to be flatly self-contradictory or to misunderstand the meaning of “ontology” or of “real”. That is also reflected in the quite explicit rejection of any discussion of two of the three possible solutions to the Measurement Problem: pilot wave theories and objective collapse theories. Has Foundations made real progress since Copenhagen? Absolutely! We have two key theorems: Bell’s theorem and the PBR theorem. The first tells us that non-locality is here to stay, so Einstein’s main complaint about the “standard” account—namely its spooky action-at-a-distance—cannot be avoided. Anyone who thinks that they have a way around it is mistaken. That lesson has not yet been learned, even though Bell’s result is half a century old. PBR proves that the wavefunction assigned to a system reflects some real physical aspect of the individual system: two systems assigned different wavefunctions are physically different. So if Carlo Rovelli or the QBists or Rob Spekkens thinks that the wavefunction isn’t real, in the sense that it does not reflect some real physical feature of an individual system, then PBR has proven them wrong. Bell and PBR lay waste to many approaches to understanding quantum theory, including Copenhagen and QBism. Jim Baggot suggests that we can “choose to agree” that the wavefunction isn’t real and somehow thereby eliminate non-locality and spooky action at a distance. No you can’t, for reasons given by both Bell and PBR. A theorem is a theorem, and we are not free to ignore it. The choices are: additional (non-hidden!) variables (e.g. Bohm); objective collapses (e.g. GRW); Many Worlds (e.g. Everett). That’s it. Anything else has been ruled out by the empirical success of the quantum predictions. And this is no more a “philosophical” debate than any other dispute in physics is. It is a debate about what the correct physical understanding of the world is. Only once these real advances in our understanding have been generally acknowledged will we be in a position to make further progress. 35. Peter Woit says: Tim Maudlin, Your comment has some similarities to Lee Smolin’s, wanting to focus on the “real” question of whether QM is all there is (his “second class”, your “Everett” choice) or new physics is needed (his “first class”, your Bohm or GRW). While there’s a lot in Becker’s book about the history of “new physics” proposals, and you and Smolin are quite right that this is a true fundamental question, more significant than empty discussions of “interpretations” corresponding to identical physics, the problem here is that I’m just not interested. As with any proposals for “new physics”, people make their own evaluations based on different experiences, and it’s a good thing that others who see things differently think more about such proposals. This is a case where no such new physics proposals come with any experimental evidence in their favor, so personal criteria of what is worth spending time on are all one has. My own criteria for paying attention to speculative proposals weight heavily the mathematical structures involved. The proposals I’ve seen for supplementing QM invoke mathematical structures that to me seem quite a bit more complicated and ugly than the ones of QM, thus my lack of interest. Maybe someday someone will come up with a different proposal that’s more appealing. Til then, I’ll keep deciding not to spend more time thinking about such things. 36. hdz says: I do not always agree with Tim Maudlin, but in this case I do almost completely. Clearly, the philosophical debate about reality is worthless for physicists, but physicists usually (and tacitly) understand it in the sense of a conceptually consistent description of “Nature” (of what we observe with our senses). So it is certainly not compatible with complementarity. Of course, everybody is free to propose new concepts and theories, but I have decided to wait until such author presents some empirical success (what else can I do?). Mathematical structures have to be formally consistent, Peter, but that is no sufficient argument for their application to the empirical world. (Just consider Tegmark’s unreasonable Level IV of multiverses.) This is not only an argument against Sting Theory! (I have written a comment against Strings (or M-theory) in German long before “Not even wrong” appeared; it can be found on my website under “Deutsche Texte”.) So my conclusion from Tim’s three choices are that the first two ones are well possible (though not more), but we have to wait for empirical support. Then only Everett remains (until being falsified) as yet as a global unitary theory. Let me add that the concept of decoherence, which has clearly been experimentally confirmed, was derived from an extension of unitarity to the environment, while Everett is its further extension to the observer. So why is it “unmotivated” or incomplete? In my opinion there are no arguments but only emotions against Everett! So please take some time to study my website! 37. a1 says: Thanks Peter for pointing that the MWI is basically an inversion of the standard one, an evidence rarely mentioned. So the ‘real’ problem remains untouched: what are probabilities? Actually, surprisingly few physicists seem to be aware of the existence of Bertand’s pardox, something which has disturbing implications. The stance ‘it doesn’t matter what probabilities are as long as we know how to calculate them’ or saying ‘there is an axiomatic for them’ are just ways to avoid the question. Does something physical collapse or is it our ignorance that ends in a measurement? It is rather obvious that a meaningful distinction between a description and its referent (to avoid saying ‘reality’) can become badly twisted when it is not clear what is not known. 38. Dave Miller says: Peter Shor, You wrote: Does anyone ever teach Intro QM without implicitly using the Copenhagen interpretation? Or, to put it the other way around, does anyone teach Intro QM using MWI or the Bohm model? To connect with your own work, it does seem to me that MWI is the “natural” way to think about quantum computing. Not that I think MWI is true: I think it has two fatal flaws — the “preferred basis” problem that Peter Woit has mentioned and the probability-measure problem. Do you yourself see any affinity between MWI and quantum computing? Dave Miller 39. @Paddy Here is the first edition of Dirac for you to check: https://archive.org/details/in.ernet.dli.2015.177580 40. Peter Woit says: Commenter atreat wants to argue with Tim Maudlin, but also points to a 319 comment thread at Scott Aaronson’s blog which includes (besides a lot of other interesting comments) extensive discussion with Maudlin about these topics. I encourage those interested in this argument to visit and not to try and restart it here. 41. GoletaBeach says: To me, the more interesting question is not about whether theory is complete, but… what experiment will displace or extend QFT as the underpinning of all microscopic theory? David Mermin sometimes pointed out that the fact we still use QFT with no new constants of nature was a failing of experimental high energy physics. I tend to agree with him, but still… increasing energy is still an enormously attractive route to finding real chinks in QFT. Also, delayed choice types of measurements are attractive. Unattractive to me… characterizing experiment as a servant of whatever trend is popular in the theory community… strings, multiverses, etc. 42. Jim Baggott says: It’s certainly true that the idea that a quantum system ‘jumps’ into some eigenstate as the result of an observation or measurement is implicit in the early history of quantum mechanics, arguably as far back as Bohr’s 1913 atomic theory. Despite statements by Heisenberg and Dirac implying such ‘jumps’ as part of the measurement process, the reason we know this as ‘von Neumann’s theory of measurement’ is because he was the first to put it forward as an axiom of the theory, the collapse or projection being a statistical ‘process of the first kind’ vs. the unitary evolution of the wavefunction as a causal ‘process of the second kind’. I tend to trust Max Jammer on this – see his ‘Philosophy of Quantum Mechanics’, p. 474. Incidentally, von Neumann treated the apparatus as a quantum system, so departed from the Copenhagen interpretation’s ‘shifty split’ between quantum/classical domains. I sense the discussion of your review of Adam Becker’s book is now pretty well exhausted, but if you will allow I’d like to make a couple of final observations. I fully understand why you like to keep the discussion focused on what you consider to be meaningful questions – as posed here by Lee Smolin and to an extent by Tim Maudlin. I don’t want to make this seem too black-and-white but, generally, if you don’t think the wavefunction is ‘real’ then you’re likely to be satisfied that quantum mechanics is complete (Copenhagen, Rovelli, consistent histories, QBism). But if you prefer to think that the wavefunction is physically real (Einstein, Schrodinger, Bell, Leggett) then obviously something is missing – quantum mechanics is incomplete and the task is to find ways to reinterpret or extend the theory to explain away the mystery, and hopefully deepen our understanding of nature at the same time. Although the MWI in principle adds nothing to the QM formalism, I’d argue that it ‘completes’ QM by adding an infinite or near-infinite number of parallel universes. But all this comes back to a judgement about the ontological status of the wavefunction and hence a choice of philosophical position. I don’t think any kind of ‘new mathematics’ will help to resolve these questions. In fact, my studies of the historical development of QM suggest instead a triumph of physical intuition over mathematical rigour and consistency, which is why von Neumann felt the need to step in and fix things in 1932. My money is therefore on new physics, as and when this might become available, perhaps as part of the search for evidence to support a quantum theory of gravity. Can I leave you with a last thought? In the famous experiments of Alain Aspect and his colleagues in the early 80s, they prepared an entangled two-photon state using cascade emission from excited Ca atoms. We write this as a linear superposition of left and right circularly polarised photons, based on our experience with this kind of system. The apparatus then measures correlations between horizontally and vertically polarised photon pairs, detected within a short time window. So we re-write the wavefunction in terms of the measurement eigenstates, and we use the modulus-squares of the projection amplitudes to tell us what to expect. Now I can’t look at this procedure without getting a very bad feeling that all we’re really doing here is mathematically coding our knowledge in a way that gives us correct predictions (which we then confirm through experience). I really don’t like it, but I confess I’m very worried. 43. Peter Woit says: My semi-joking threat to suppress ontology is due to the fact that I seriously don’t know what you or others mean when they say something is/is not “real”, or “physically real”. I have the same problem with Lee Smolin when he writes about time being “real/not real”. Hanging one’s treatment of a complex issue on this highly ambiguous four-letter word seems to me to just obscure issues (I started to write “real issues”…). About MWI, I’m on board with the “everything evolves according to Schrodinger’s eq.” part, not so much with the “add an infinite number of parallel universes” part. While I think new deep mathematical ideas may inspire progress on unification, in the case of the measurement problem, the argument from mathematical depth and beauty goes the other way. The basic QM formalism already is based on very deep and powerful mathematics. Here my problem is with things like Bohmian mechanics and dynamical collapse models which deface this beauty by adding ugly complexity not forced by experiment. The measurement problem to me is essentially the problem of understanding how the effective classical theory emerges from the fundamental quantum formalism. Mathematics may or may not be helpful here, I don’t know. The confusions I see in most discussions of QM interpretations aren’t ones that mathematics will resolve, they need to be resolved by careful examination of the physics one is discussing. Some people seem to expect that the new physics needed to get quantum gravity will resolve the measurement problem, but I don’t see it. The measurement problem is there when you do a very low energy experiment, whereas the quantum gravity problem is about not understanding Planck-scale physics. I just don’t see how the quantization of gravitational degrees of freedom has anything at all to do with the measurement problem. About the photon experiments. I don’t think the mysterious thing is the “re-write the wavefunction in terms of the measurement eigenstates” part, that is very simple and completely understood. The mystery is the “The apparatus then measures correlations between horizontally and vertically polarised photon pairs” part, where a macroscopic apparatus described in classical terms is coming into play. 44. Lee Smolin says: Dear Peter, I agree the statement “Time is real” is misleading and I regret using it. The more precise claim is that “time is fundamental” in the sense that there is no deeper formulation of the laws of physics that does not involve the evolution of the present state or configuration, by the continual creation of novel events out of present events. ie at no level is time emergent from a timeless formulation of fundamental physics. Were the Wheeler-deWitt equation correct this would not be the case. I view my papers on the real ensemble formulation and relational hidden variables to be modest attempts to develop and explore new hypotheses to complete quantum mechanics. To answer Peter, these are inspired by developments in quantum gravity, particularly the hypotheses that if time is not emergent, space may be emergent from a network of dynamically evolving relationships, as it is in several approaches to quantum gravity such as spin foam models, causal dynamical triangulations and causal set models. But if space is emergent, so is locality and hence so must be non-locality. Hence we may aim to show the origin of Bell-non-locality from disorderings of locality which follows from the emergent nature of locality. This may be a way to realize the idea, which is currently popular, but was first stated by Penrose in his papers on spin networks, that the geometry of space and quantum entanglement may have a common origin. 45. Jim Baggott says: I honestly think you’ve answered your own question. You say “I don’t think the mysterious thing is the re-write the wavefunction in terms of the measurement eigenstates part – that is very simple and completely understood”. So what is it that exists? A quantum state made of circularly polarised states? A quantum state made of linearly polarised states? Or both, but involving a process that gets from one to the other that doesn’t appear anywhere in the formalism except as a postulate, based on a mechanism we can’t fathom? Or are we just using these ‘states’ as a convenient way of connecting one physical situation with another? 46. Blake Stacey says: I noticed an item in the latest quant-ph update that may be of interest to the folks participating here: a critical evaluation of Wallace’s attempt to get meaningful probabilities out of (neo-)Everettian QM. 47. Peter Woit says: The question “what exists?” is just as ill-defined as “what is real?”. The fundamental issue here is that I think I understand completely what a “quantum state” is (any one of the mathematically isomorphic descriptions), but I don’t understand completely what a “physical situation” is (it may involve some system under study, some macroscopic apparatus, my consciousness, and how they are interacting ). Everett tells me that a “physical situation” is also just a quantum state, and I’m willing to believe him since I don’t have evidence against this, but that doesn’t change my lack of full understanding of the “physical situation” I’m somehow part of. 48. Doug McDonld says: I fully agree with you in this. The problem is, explaining “how” this happens. I don’t mean in a philosophical sense, I mean describing it using, somehow, just plain quantum mechanics of increasingly complicated sets of interactions that end up in a macroscopic apparatus that everybody agrees is classical. This is doable. Its where the payoff is eventually going to come. The hard part is explaining it. By increasingly complicated, I mean in terms of say measuring the energy of a high energy photon slamming into an LHC calorimeter in terms of a cascade of events. Each even is small in quantum energy terms, collectively they add up to the answer, including both quantum uncertainty and statistical uncertainty. Its not called a calorimeter for nothing … that says its classical. Normally on blogs like this one my viewpoint usually gets moderated out. 49. Laurence Lurio says: When you are stuck in a philosophical morass, the solution, if one exists, is likely to come from experiment. There is a growing experimental community working on building quantum computers who are worrying about wave function collapse as a very practical matter. It will be interesting to see which interpretation that community gravitates to. Comments are closed.
b8ac13a97ac58e9d	Topological insulators From Scholarpedia Shoucheng Zhang (2015), Scholarpedia, 10(7):30275. doi:10.4249/scholarpedia.30275 revision #151969 [link to/cite this article] Jump to: navigation, search Metals and insulators The electronic band structure determines the conductivity of metals and insulators. The band theory of solid describes the electronic structure of such states, which exploits the 'discrete' translational symmetry of the crystal to classify electronic states in terms of their crystal momentum $\mathbf{k}$, defined in a periodic Brillouin zone. As stated by the Bloch theorem, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves \begin{equation} H\|\psi_{n}(\mathbf{k})\rangle=E_{n}(\mathbf{k})\|\psi_{n}(\mathbf{k})\rangle \end{equation} with \begin{equation} \|\psi_{n}(\mathbf{k})\rangle = e^{i\mathbf{k}\cdot\mathbf{r}}\|u_n(\mathbf{k})\rangle, \end{equation} where $\|u_n({\mathbf{k})}\rangle$ is defined in a single unit cell of the crystal, $\mathbf{r}$ is the position operator. They are eigenstates of the Bloch Hamiltonian $H(\mathbf{k})$ with $H(\mathbf{k})\|u_n({\mathbf{k})}\rangle=E_n(\mathbf{k})\|u_n({\mathbf{k})}\rangle$, with $H=\sum_{\mathbf{k}}\psi^{†}(\mathbf{k})H(\mathbf{k})\psi(\mathbf{k})$, where $\psi(\mathbf{k})$ is a basis. In a crystal with two sublattices, $\psi(\mathbf{k})$ would be a two component vector of annihilation operator $(c_{A\mathbf{k}}, c_{B\mathbf{k}})^T$. The eigenvalues $E_n(\mathbf{k})$ define energy bands that collectively form the band structure, where $n$ is the band index. Fig. 1 illustrates the simplest case of two energy bands (valence and conduction band) and the band gap. Energy bands and the gaps between them determine the conductivity and other properties of solids. • Insulators have a fully occupied valence band and unoccupied conduction band with an energy gap (sometimes a few eV). There are no gapless electronic states available for electrical conduction. • Metals have a partially filled band. Gapless electronic states in the partially filled band are responsible for electrical conduction, even at $T=0$ K.s electronic states in the partially filled band are responsible for electrical conduction, even at $T=0$ K. Figure 1: The energy bands of metals and insulators. For the insulators, the filled valence band is separated from the conduction band by an energy gap. On the other hand, the lower energy band in metals is partially filled with electrons. Conventional surface states of insulators Surface states are electronic states found at the surface of materials. The termination of a crystal obviously causes deviation from perfect periodicity, in this case, $\mathbf{k}$ becomes complex, leading to the formation of a surface state. The wave function of a surface state decays monotonically in the direction of vacuum and damps in an oscillatory fashion in the direction of the crystal, and it is localized close to the crystal surface. There are two kinds of surface states, conventionally called Shockley states and Tamm states. However, there is no real physical distinction between the two terms, only the mathematical approach in describing surface states is different. Shockley states are states that arise due to the change in the electron potential associated solely with the crystal termination, this approach is suited to describe normal metals and some narrow gap semiconductors. While surface states that are calculated in the framework of a tight-binding model are often called Tamm states, they are suitable to describe also transition metals and insulators. These surface states are called topologically trivial in the modern sense, since they originate from one bulk band and return to the same bulk band, say the conduction band, as illustrated in Figure 2a. Potential disorder and oxidation can easily remove these trivial surface states. There also exist surface states on metal surfaces, such as Cu (100) and Ag (110) surface. Figure 2: Schematic representation of the surface energy levels of a crystal in either 2D or 3D, as a function of surface crystal momentum. a, conventional insulator; b, topological insulator. The shaded region shows the bulk continuum states, and the lines show discrete surface (or edge) bands localized near one of the surfaces. b occurs in topological insulators and guarantees that the surface bands cross any Fermi level inside the bulk gap. Topological insulators A topological insulator is a material with time reversal symmetry and topologically protected surface states. These surface states continuously connect bulk conduction and valence bands, as illustrated in Figure 2b. Topological insulator has an energy gap in the bulk interior, just as in an ordinary insulator, but it contains conducting states localized on its surface. Although ordinary band insulators can also support conductive surface states, the surface states of topological insulators are special since they are topologically protected by the time reversal symmetry, and no non-magnetic disorder can destroy them. In the bulk of a non-interacting topological insulator, the electronic band structure resembles an ordinary band insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a topological insulator there are special states that fall within the bulk energy gap and allow metallic conduction on the surface. An odd number of surface states connects conduction band with valence bands, and crossings at time-reversal-invariant points in Brillouin zone (as shown in Figure 2b), which leads to topologically protected metallic boundary states (Zhang et al.(2011),Hasan et al.(2010)). Time-reversal symmetry is represented by an antiunitary operator $\mathcal{T}=\exp(i\pi S_y/\hbar)K$, where $S_y$ is the spin operator and $K$ is complex conjugation. For spin $1/2$ electrons, $\mathcal{T}$ has the property $\mathcal{T}^2=-1$. This leads to an important constraint, known as Kramers' theorem, which states that all eigenenergies of a time-reversal invariant Hamiltonian are at least twofold degenerate. This follows because if a nondegenerate state $\|u\rangle$ existed then $\mathcal{T}\|u\rangle=c\|u\rangle$ for some constant $c$. This would mean $\mathcal{T}^2\|u\rangle=\|c\|^2\|u\rangle$, which is not allowed because $\|c^2\|\neq-1$. If we apply the Kramers' theorem to the Bloch wave in an insulator, we find that for any Bloch state $\psi_{n}(\mathbf{k},\sigma)$, there is another state $\mathcal{T}\psi_{n}(\mathbf{k},\sigma)$ which has the same energy, \begin{equation} \mathcal{T}\psi_{n}(\mathbf{k},\sigma)=\psi'_n(-\mathbf{k},-\sigma). \end{equation} where $\sigma$ is the spin index. In general, the Kramers doublet are located at different momentum point $\mathbf{k}$ and $-\mathbf{k}$. However, if $\mathbf{k}=-\mathbf{k}+\mathbf{G}$, where $\mathbf{G}$ is reciprocal lattice vector, i.e. at the time-reversal-invariant points, every single energy level is double degenerate. This guarantees the surface states must cross at time-reversal-invariant points in Brillouin zone. Two-dimensional topological insulator In 2005, Kane and Mele identified a new type of topological property characterizing two dimensional insulators, different from the property of the integer quantum Hall effect (Kane et al.(2005)). Two groups independently introduced the models for two-dimensional topological insulators (also synonymously called the quantum spin Hall insulator). Adapting an earlier model for graphene by F. Duncan M. Haldane (Haldane(1988)) which exhibits an integer quantum Hall effect, Kane and Mele proposed a quantum spin Hall model in graphene (Kane et al.(2005)), where the spin up electron exhibits a chiral integer quantum Hall effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect. Independently, a quantum spin Hall model was proposed by Bernevig and Zhang (Bernevig et al.(2005)) in strained GaAs, with an intricate strain architecture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons. The main ingredient of both proposals is the existence of spin-orbit coupling, which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron. This theoretical prediction was soon confirmed experimentally, leading to the first discovery of the topological insulator in nature. Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state under realistic conductions. In 2006, Bernevig, Hughes and Zhang (Bernevig et al.(2006)) predicted that mercury telluride quantum wells are topological insulators beyond a critical thickness $d_c=6.5$~nm. The general mechanism for topological insulators is band inversion, in which the usual ordering of the conduction band and valence band is inverted by spin-orbit coupling. In most common semiconductors, the conduction band is formed from electrons in $s$ orbitals and the valence band is formed from electrons in $p$ orbitals. In HgTe, however, the relativistic effect including spin-orbit coupling is so large that the bands are inverted-that is, the $p$-orbital dominated valence band is pushed above the $s$-orbital dominated conduction band. HgTe quantum wells are grown by sandwiching the material between CdTe, which has a similar lattice constant but much weaker spin-orbit coupling with normal band ordering. Therefore, increasing the thickness $d$ of the HgTe layer increases the strength of the spin-orbit coupling for the entire quantum well. For a thin quantum well, as shown in the left column of Figure 3a, the CdTe has the dominant effect and the bands have a normal ordering: The $s$-like conduction subband E1 is located above the $p$-like valence subband H1. In a thick quantum well, as shown in the right column, the opposite ordering occurs due to increased thickness $d$ of the HgTe layer. Figure 3: HgTe quantum wells are two-dimensional topological insulators. (a) The behavior of a HgTe/CdTe quantum well depends on the thickness $d$ of the HgTe layer. Here the blue curve shows the potential-energy well experienced by electrons in the conduction band; the red curve is the barrier for holes in the valence band. Electrons and holes are trapped laterally by those potentials but are free in the other two dimensions. For quantum wells thinner than a critical thickness $d_c\simeq 6.5 {\rm nm}$, the energy of the lowest energy conduction subband, labeled E1, is higher than that of the highest-energy valence band, labeled H1. But for $d>d_c$, those electron and hole bands are inverted. (b) The energy spectra of the quantum wells. The thin quantum well has an insulating energy gap, but inside the gap in the thick quantum well are edge states, shown by red and blue lines. (c) Experimentally measured resistance of thin and thick quantum wells, plotted against the voltage applied to a gate electrode to change the chemical potential. The thin quantum well has a nearly infinite resistance within the gap, whereas the thick quantum well has a quantized resistance plateau at $R = h/2e^2$, due to the perfectly conducting edge states. Moreover, the resistance plateau is the same for samples with different widths, from 0.5 $\mu$m (red) to 1.0 $\mu$m (blue), proof that only the edges are conducting. The QSH state in HgTe can be described by a simple model for the E1 and H1 subbands (Bernevig et al.(2006)) (in appendix). Explicit solution of that model gives one pair of edge states for $d>d_c$ in the inverted regime and no edge states in the $d<d_c$, as shown in Figure 3b. The pair of edge states (called helical states) carry opposite spins and disperse all the way from valence band to conduction band, and backscattering by nonmagnetic impurities is forbidden. A simple semiclassical picture illustrates why single-particle backscattering is forbidden for degrees of freedom with half-odd-integer angular momentum (Figure 4). The mechanism is analogous to the way anti-reflective coatings on eyeglasses and camera lenses work. In such a system, reflected light from the top and bottom surfaces of the antireflective coating interfere destructively, suppressing the overall amount of reflected light (Figure 4a). This effect is, however, not robust, as it requires precise matching of the coating thickness to the wavelength of the light. Just as photons can be reflected from an interface between two dielectrics, so can electrons be backscattered by an impurity, and different backscattering paths will interfere with each other (Figure 4b). On a quantum spin Hall edge, the two paths correspond to the electron going around the impurity in either a clockwise or counterclockwise fashion, with the spin rotating by an angle of $\pi$ or $-\pi$, respectively. Consequently, the phase difference between the two paths is a full $\pi-(-\pi)=2\pi$ rotation of the electron spin. However, the wave function of a spin-$1/2$ particle picks up a minus sign under a full $2\pi$ rotation. Therefore, the two backscattering paths related by time-reversal always interfere destructively, leading to perfect transmission. Figure 4: (a) On a lens with antireflection coating, light waves reflected by the top (blue line) and the bottom (red line) surfaces interfere destructively, which leads to suppressed reflection. (b) A quantum spin Hall edge state can be scattered in two directions by a nonmagnetic impurity. Going clockwise along the blue curve, the spin rotates by $\pi$; counterclockwise along the red curve, by $-\pi$. A quantum mechanical phase factor of $-1$ associated with that difference of $2\pi$ leads to destructive interference of the two paths---the backscattering of electrons is suppressed in a way similar to that of photons off the antireflection coating. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is $G_{xx}=2e^2/h$ in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to the width of the sample. In 2007, a team at the University of Würzburg led by Laurens Molenkamp observed the quantum spin Hall effect in HgTe quantum wells grown by molecular-beam epitaxy (Köonig et al.(2007)). The transport experiment shows finite conductance of $2e^2/h$ for quantum spin Hall insulator indicating the contribution from helical edge states. In contrast, a trivial insulator phase is indeed insulating, with vanishing conductance [Figure 3c]. In 2008, Zhang's group predicted that the type II quantum well GaSb/InAs is a two-dimensional topological insulator (Liu et al.(2008)). The conduction and valence band are spatially separated in two materials with inverted order and weakly coupled in this broken gap quantum well. Soon after the theoretical prediction, Rui-Rui Du's group in Rice University observed the quantum spin Hall effect in GaSb/InAs quantum well (Knez et al.(2011)). Three dimensional topological insulator The two-dimensional topological insulator with a one-dimensional helical edge state can be simply generalized to a three-dimensional topological insulator (Fu et al. (2007),Moore et al. (2007),Roy (2009)), for which the surface state consists of a single two-dimensional massless Dirac fermion and the dispersion forms a so-called Dirac cone, as illustrated in Figure 5. Three dimensional topological insulator was first realized in the alloy ${\rm Bi_{1-x}Sb_x}$ with a special range of $x$ (Liang et al.(2007),Hsieh et al. (2008)). However, the surface states and the underlying mechanism turn out to be extremely complex. Soon afterwards, bulk crystals of ${\rm Bi_2Te_3}$, ${\rm Bi_2Se_3}$, and ${\rm Sb_2Te_3}$ are predicted to be three dimensional topological insulators with bulk energy gap as large as $0.3$~eV, with the topological surface states consisting of a single Dirac cone (Zhang et al. (2009),Xia et al.(2009)). The single Dirac-cone surface state of ${\rm Bi_2Se_3}$ has been observed in the Angle-Resolved PhotoEmission Spectroscopy (ARPES) experiments by a Princeton based group (Xia et al.(2009)). Furthermore, the group's spin-resolved ARPES measurements showed that the electron spin lies in the plane of the surface and is always perpendicular to the momentum. A pure topological insulator phase without bulk carriers was first observed in ${\rm Bi_2Te_3}$ by a Stanford based group in ARPES experiments (Chen et al.(2009)). As shown in Figure 5c, the observed surface states indeed disperse linearly, crossing at the point with zero momentum. By mapping all of momentum space, the ARPES experiments show convincingly that the surface states of ${\rm Bi_2Te_3}$ and ${\rm Bi_2Se_3}$ consist of a single Dirac cone. Similar to HgTe, the nontrivial topology of the ${\rm Bi_2Te_3}$ family is due to band inversion between two orbitals with opposite parity, driven by the strong spin-orbit coupling of Bi and Te. Due to such similarity, that family of 3D topological insulators can be described by a 3D version of the HgTe model (see the appendix). First-principle calculations show that the materials have a single Dirac cone on the surface. The spin of the surface state lies in the surface plane and is always perpendicular to the momentum, as shown in Figure 2b. Further generalization of topological insulators can be made to higher dimensions with different symmetries, and the complete classifications are first given by Schnyder et al. (2008) and Kitaev (2009). Figure 5: (a) The crystal structure of the 3D topological insulator ${\rm Bi_2Te_3}$ consists of stacked quasi-2D layers of Te-Bi-Te-Bi-Te. The arrows indicate the lattice basis vectors. The surface state is predicted to consist of a single Dirac cone (Zhang et al.(2009)). (b) This ARPES plot of energy versus wavenumber in ${\rm Bi_2Te_3}$ shows the linearly dispersing surface-state band (SSB) above the bulk valence band (BVB). The top two dashed green lines denote the bulk insulating gap; the bottom line marks the point of the Dirac cone (Chen et al.(2009)). (c) Angle-resolved photoemission spectroscopy maps the energy states in momentum space. Spin dependent ARPES of the related compound ${\rm Bi_2Se_3}$ reveals that the spins (red) of the surface states lie in the surface plane and are perpendicular to the momentum (Xia et al.(2009)) Topology studies the shapes of manifolds or spaces. The topology of a space is preserved under continuous deformations (e.g. stretching and bending, but not tearing or gluing), and is usually characterized by topological numbers. For example, closed oriented two-dimensional surfaces (spaces) are characterized by genus $g$, the number of holes of the closed surface. A sphere has $g=0$, and a torus has $g=1$ as shown in Figure 6. The genus can be calculated by integration of the two-dimensional curvature $R$ over the space, given by: \begin{equation} 2-2g=\frac{1}{2\pi}\int\limits_{M}RdS \end{equation} Figure 6: Sphere, $g=0$; and torus, $g=1$ Instead of the Gaussian curvature defined above, one can define the Berry curvature in the Brillouin zone, which can be viewed as a torus for a two dimensional insulator. The Berry curvature $F$ can be defined as \begin{equation} F = (\nabla_{\mathbf{k}}\times\mathbf{A})_z, \end{equation} where $\mathbf{A}$ is the Berry connection of Bloch states defined as \begin{equation} \mathbf{A} = i\langle u(\mathbf{k})\|\nabla_{\mathbf{k}}\|u(\mathbf{k})\rangle. \end{equation} This enables us to define a (topological) Chern number $N$ for the insulator \begin{equation} N=\frac{1}{2\pi}\int F dk_xk_y=\frac{1}{2\pi}\oint_C\mathbf{A}\cdot d\mathbf{k}, \end{equation} where the integration is over the Brillouin Zone. In the absence of magnetic fields, an insulator with $N\neq0$ is called a quantum anomalous Hall insulator. As a consequence, a quantum anomalous Hall insulator has $\|N\|$ gapless chiral edge states on the edge. A quantum anomalous Hall insulator breaks the time reversal symmetry. To make the physical picture of Chern number clearer, the simplest case of a two-band model can be studied as an example. The Hamiltonian of a two-band model can be generally written as $h(\mathbf{k})=\sum\limits_{a=1}^3d_a(\mathbf{k})\sigma^a+\epsilon(\mathbf{k})1$. The energy spectrum is easily obtained: $E_{\pm}(\mathbf{k})=\epsilon(\mathbf{k})\pm\sqrt{\sum_ad_a^2(\mathbf{k})}$. When $\sum_ad_a^2(\mathbf{k})>0$ for all $\mathbf{k}$ in the Brillouin zone, the two bands never touch each other. In the single particle Hamiltonian $h(\mathbf{k})$, the vector $d(\mathbf{k})$ acts as a Zeeman field applied to a pseudospin $\sigma_i$ of a two level system. The occupied band satisfies $[\mathbf{d}(\mathbf{k})\cdot\boldsymbol{\sigma}]\|-,\mathbf{k}\rangle=-\|\mathbf{d}(\mathbf{k})\|\|-,\mathbf{k}\rangle$ which thus corresponds to the spinor with spin polarization in the $-\mathbf{d}(\mathbf{k})$ direction. Thus Berry's phase gained by $\|-,\mathbf{k}\rangle$ during an adiabatic evolution along some path $C$ in $\mathbf{k}$ space is equal to Berry's phase a spin-1/2 particle gains during the adiabatic rotation of the magnetic field along the path $\mathbf{d}(C)$. This is known to be half of the solid angle subtended by $\mathbf{d}(C)$. Consequently, the first Chern number $N$ is determined by the winding number of $\mathbf{d}(\mathbf{k})$ around the origin. For time reversal invariant two-dimensional insulators, the Chern number $N$ is always 0. However, with the time reversal invariant points in the Brillouin Zone identified, a $Z_2$ topological number $(-1)^{N_2}$ can be defined Kane et al. (2005). An insulator is trivial if $N_2=0$, and is called a quantum spin Hall if $N_2=1$. Consequentially, a trivial insulator has gapped edge, while a quantum spin Hall has a pair of gapless helical edge states carrying opposite spins. Topological field theory To describe the novel topological properties of a single Dirac cone on the surface of a topological insulator. One can use the topological field theory in terms of elementary concepts in electromagnetism (Qi et al.(2008)). Inside an insulator, the electric field ${\bf E}$ and the magnetic field ${\bf B}$ are both well defined. In a Lagrangian-based field theory, the insulator's electromagnetic response can be described by the effective action $S_0=1/8\pi\int d^3xdt(\epsilon E^2-B^2/\mu)$, with $\epsilon$ the electric permittivity and $\mu$ the magnetic permeability, from which Maxwell's equations can be derived. The integrand depends on geometry, though, so it is not topological. To see that dependence, one can write the action in terms of the field strength tensor $F_{\mu\nu}$, the relativistic electromagnetic field tensor: $S_0=1/16\pi \int d^3xdtF_{\mu\nu}F^{\mu\nu}$. The implied summation over the repeated indices $\mu$ and $\nu$ depends on the metric tensor---that is, on geometry. (Indeed, it is that dependence that leads to the gravitational lensing of light.) There is, however, another possible term called axion term Wilczek (1987) in the action of the electromagnetic field: \begin{eqnarray} S_\theta&=&\frac{\theta\alpha}{4\pi^2}\int d^3xdt{\bf E\cdot B}\equiv\frac{\theta\alpha}{32\pi^2} \int d^3xdt \epsilon_{\mu\nu\rho\tau}F^{\mu\nu}F^{\rho\tau}\nonumber\\ &=&\frac{\theta}{2\pi}\frac{\alpha}{4\pi}\int d^3xdt \partial^\mu (\epsilon_{\mu\nu\rho\tau} A^\nu\partial^\rho A^\tau) \tag{1} \end{eqnarray} where $\alpha=e^2/\hbar c\simeq 1/137$ is the fine-structure constant, $\theta$ is an angular variable which is defined modulo $2\pi$, and $\epsilon_{\mu\nu\rho\tau}$ is the fully antisymmetric 4D Levi-Civita tensor. Unlike the Maxwell action, $S_\theta$ is a topological term---it depends only on the topology of the underlying space, not on the geometry. The axion term in general breaks time-reversal symmetry and inversion symmetry. However, for an insulator with time reversal symmetry, the parameter $\theta$ can only take values of $0$ and $\pi$. $\theta=0$ corresponds to a conventional insulator, and $\theta=\pi$ corresponds to a topological insulator. The definition is generally valid for interacting topological insulators as well. Solving Maxwell's equations with the topological term included leads to predictions of novel physical properties. It should be mentioned that such topological term is an integral of a total derivative, so it only has an effect on the boundary where $\theta$ jumps from $\theta=0$ to $\theta=\pi$. Otherwise, it does not enter into the Maxwell equations. A point charge above the surface of a 3D topological insulator is predicted to induce not only an image electric charge but also an image magnetic monopole below the surface (Qi et al.(2009)). If we uniformly cover the surface with a thin ferromagnetic film, an insulating gap also opens up at the Dirac point giving rise to $\frac{1}{2}e^2/h$ Hall quantum Hall conductance (Qi et al.(2008)). Possible applications The topological protection of the surface state or edge state of topological insulators made them promising candidate for potential applications. One possible application is to use the edge channels of the quantum anomalous Hall insulator as interconnects for integrated circuits Zhang et al. (2012). For these edge channels propagate along the insulator edges without any dissipation, they could greatly reduce heat dissipation in integrated circuits. Especially, the quantum anomalous Hall insulator with multi-channels lowers the contact resistance, significantly improving the performance of the interconnect devices (Wang et al. (2013)). Topological insulators are interesting for thermoelectrics due to their unique electronic structure. In fact, many topological insulators were known as excellent thermoelectric materials, for topological insulator and thermoelectric compounds usually favor the same material features, such as heavy elements and the smaller energy gap. Bi$_2$Te$_3$ and Bi$_2$Se$_3$ comprise some of the best performing room temperature thermoelectrics with a temperature-independent thermoelectric effect, $ZT$, between $0.8$ and $1.0$, where the carrier density is between a semiconductor and a metal. Zhang's group predicted in 2014 that $ZT$ is strongly size dependent in topological insulators ( Xu et al. 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Seeing the magnetic monopole through the mirror of topological surface states. scn 323: 1184. • Xiao Zhang and Shou-Cheng Zhang (2012). Chiral interconnects based on topological insulators. MICRO- AND NANOTECHNOLOGY SENSORS 8373: 837309. • Wang Jing, Lian Biao, Zhang Haijun, Xu Yong and Zhang Shou-Cheng (Sep 2013). Quantum Anomalous Hall Effect with Higher Plateaus. Phys. Rev. Lett. (American Physical Society) 111 (13): 136801. doi:10.1103/PhysRevLett.111.136801. • Xu (Jun 2014). Enhanced Thermoelectric Performance and Anomalous Seebeck Effects in Topological Insulators. Phys. Rev. Lett. (American Physical Society) 112 (22): 226801. doi:10.1103/PhysRevLett.112.226801. Models of topological insulators The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. The two-dimensional topological insulator HgTe can be described by an effective Hamiltonian that is essentially a Taylor expansion in the wave vector ${\bf k}$ of the interactions between the lowest conduction band and the highest valence band (Bernevig et al.(2006)): \begin{eqnarray}\tag{2} H(k)&=&\epsilon({\bf k})\mathbb{I}+\left(\begin{array}{cccc}M({\bf k})&A(k_x+ik_y)&0&0\\A(k_x-ik_y)&-M({\bf k})&0&0\\ 0&0&M({\bf k})&-A(k_x-ik_y)\\0&0&-A(k_x+ik_y)&-M({\bf k})\end{array}\right)\nonumber\\ \epsilon({\bf k})&=&C+D{\bf k}^2,~M({\bf k})=M-B{\bf k}^2 \end{eqnarray} where the upper $2\times 2$ block describes spin-up electrons in the $s$-like E1 conduction and the $p$-like H1 valence bands, and the lower block describes the spin-down electrons in those bands. The term $\epsilon({\bf k})\mathbb{I}$ is an unimportant bending of all the bands ($\mathbb{I}$ is the identity matrix). The energy gap between the bands is $2M$, and $B$, typically negative, describes the curvature of the bands; $A$ incorporates interband coupling to lowest order. For $M/B < 0$, the eigenstates of the model describe a trivial insulator. But for thick quantum wells, the bands are inverted, $M$ becomes negative, and the solution yields the edge states of a quantum spin Hall insulator. The three-dimensional topological insulator in the ${\rm Bi_2Te_3}$ family can be described by a similar model (Zhang et al. (2009)): \begin{eqnarray}\tag{3} H(k)&=&\epsilon({\bf k})\mathbb{I}+\left(\begin{array}{cccc}M({\bf k})&A_2(k_x+ik_y)&0&A_1 k_z\\A_2(k_x-ik_y)&-M({\bf k})&A_1 k_z&0\\ 0&A_1 k_z&M({\bf k})&-A_2(k_x-ik_y)\\A_1 k_z&0&-A_2(k_x+ik_y)&-M({\bf k})\end{array}\right)\nonumber\\ \epsilon({\bf k})&=&C+D_1k_z^2+D_2k_\perp^2,~M({\bf k})=M-B_1k_z^2-B_2k_\perp^2 \end{eqnarray} in the basis of the Bi and Te bonding and antibonding $p_z$ orbitals with both spins. The curvature parameters $B_1$ and $B_2$ have the same sign. As in the two-dimensional model, the solution for $M/B_1 < 0$ describes a trivial insulator, but for $M/B_1 > 0$, the bands are inverted and the system is a topological insulator. Personal tools Focal areas
aef7bf34bcfb36f6	February 28, 2006 by James N. Gardner Originally published in the International Journal of Astrobiology May 2005. Reprinted on February 28, 2006. Goal 7 of the NASA Astrobiology Roadmap states: “Determine how to recognize signatures of life on other worlds and on early Earth. Identify biosignatures that can reveal and characterize past or present life in ancient samples from Earth, extraterrestrial samples measured in situ, samples returned to Earth, remotely measured planetary atmospheres and surfaces, and other cosmic phenomena.” The cryptic reference to “other cosmic phenomena” would appear to be broad enough to include the possible identification of biosignatures embedded in the dimensionless constants of physics. The existence of such a set of biosignatures—a life-friendly suite of physical constants—is a retrodiction of the Selfish Biocosm (SB) hypothesis. This hypothesis offers an alternative to the weak anthropic explanation of our indisputably life-friendly cosmos favored by (1) an emerging alliance of M-theory-inspired cosmologists and advocates of eternal inflation like Linde and Weinberg, and (2) supporters of the quantum theory-inspired sum-over-histories cosmological model offered by Hartle and Hawking. According to the SB hypothesis, the laws and constants of physics function as the cosmic equivalent of DNA, guiding a cosmologically extended evolutionary process and providing a blueprint for the replication of new life-friendly progeny universes. The notion that we inhabit a universe whose laws and physical constants are fine-tuned in such a way as to make it hospitable to carbon-based life is an old idea (Gardner, 2003). The so-called “anthropic” principle comes in at least four principal versions (Barrow and Tipler, 1988) that represent fundamentally different ontological perspectives. For instance, the “weak anthropic principle” is merely a tautological statement that since we happen to inhabit this particular cosmos it must perforce by life-friendly or else we would not be here to observe it. As Vilenkin put it recently (Vilenkin, 2004), “the ‘anthropic’ principle, as stated above, hardly deserves to be called a principle: it is trivially true.” By contrast, the “participatory anthropic principle” articulated by Wheeler and dubbed “it from bit” (Wheeler, 1996) is a radical extrapolation from the Copenhagen interpretation of quantum physics and a profoundly counterintuitive assertion that the very act of observing the universe summons it into existence. All anthropic cosmological interpretations share a common theme: a recognition that key constants of physics (as well as other physical aspects of our cosmos such as its dimensionality) appear to exhibit a mysterious fine-tuning that optimizes their collective bio-friendliness. Rees noted (Rees, 2000) that virtually every aspect of the evolution of the universe—from the birth of galaxies to the origin of life on Earth—is sensitively dependent on the precise values of seemingly arbitrary constants of nature like the strength of gravity, the number of extended spatial dimensions in our universe (three of the ten posited by M-theory), and the initial expansion speed of the cosmos following the Big Bang. If any of these physical constants had been even slightly different, life as we know it would have been impossible: The [cosmological] picture that emerges—a map in time as well as in space—is not what most of us expected. It offers a new perspective on a how a single “genesis event” created billions of galaxies, black holes, stars and planets, and how atoms have been assembled—here on Earth, and perhaps on other worlds—into living beings intricate enough to ponder their origins. There are deep connections between stars and atoms, between the cosmos and the microworld…. Our emergence and survival depend on very special “tuning” of the cosmos—a cosmos that may be even vaster than the universe that we can actually see. As stated recently by Smolin (Smolin, 2004), the challenge is to provide a genuinely scientific explanation for what he terms the “anthropic observation”: The anthropic observation: Our universe is much more complex than most universes with the same laws but different values of the parameters of those laws. In particular, it has a complex astrophysics, including galaxies and long lived stars, and a complex chemistry, including carbon chemistry. These necessary conditions for life are present in our universe as a consequence of the complexity which is made possible by the special values of the parameters. There is good evidence that the anthropic observation is true. Why it is true is a puzzle that science must solve. It is a daunting puzzle indeed. The strangely (and apparently arbitrarily) biophilic quality of the physical laws and constants poses, in Greene’s view, the deepest question in all of science (Greene, 2004). In the words of Davies (Gardner, 2003), it represents “the biggest of the Big Questions: why is the universe bio-friendly?” Modern History of Anthropic Reasoning Modern statements of the cosmological anthropic principle date from the publication of a landmark book by Henderson in 1913 entitled The Fitness of the Environment (Henderson, 1913). Henderson’s book was an extended reflection on the curious fact that there are particular substances present in the environment—preeminently water—whose peculiar qualities rendered the environment almost preternaturally suitable for the origin, maintenance, and evolution of organic life. Indeed, the strangely life-friendly qualities of these materials led Henderson to the view that “we were obliged to regard this collocation of properties in some intelligible sense a preparation for the process of planetary evolution…. Therefore the properties of the elements must for the present be regarded as possessing a teleological character.” Thoroughly modern in outlook, Henderson dismissed this apparent evidence that inanimate nature exhibited a teleological character as indicative of divine design or purpose. Indeed, he rejected the notion that nature’s seemingly teleological quality was in any way inconsistent with Darwin’s theory of evolution through natural selection. On the contrary, he viewed the bio-friendly character of the inanimate natural environment as essential to the optimal operation of the evolutionary forces in the biosphere. Absent the substrate of a superbly “fit” inanimate environment, Henderson contended, Darwinian evolution could never have achieved what it has in terms of species multiplication and diversification. The mystery of why the physical qualities of the inanimate universe happened to be so oddly conducive to life and biological evolution remained just that for Henderson—an impenetrable mystery. The best he could do to solve the puzzle was to speculate that the laws of chemistry were somehow fine-tuned in advance by some unknown cosmic evolutionary mechanism to meet the future needs of a living biosphere: Henderson’s iconoclastic vision was far ahead of its time. His potentially revolutionary book was largely ignored by his contemporaries or dismissed as a mere tautology. Of course there should be a close match-up between the physical requirements of life and the physical world that life inhabits, contemporary skeptics pointed out, since life evolved to survive the very challenges presented by that pre-organic world and to take advantage of the biochemical opportunities it offered. While lacking broad influence at the time, Henderson’s pioneering vision proved to be the precursor to modern formulations of the cosmological anthropic principle. One of the first such formulations was offered by British astronomer Fred Hoyle. A storied chapter in the history of the principle is the oft-told tale of Hoyle’s prediction of the details of the triple-alpha process (Mitton 2005). This prediction, which seems to qualify as the first falsifiable implication to flow from an anthropic hypothesis, involves the details of the process by which the element carbon (widely viewed as the essential element of abiotic precursor polymers capable of autocatalyzing the emergence of living entities) emerges through stellar nucleosynthesis. As noted by Livio (Livio, 2003): Carbon features in most anthropic arguments. In particular, it is often argued that the existence of an excited state of the carbon nucleus is a manifestation of fine-tuning of the constants of nature that allowed for the appearance of carbon-based life. Carbon is formed through the triple-alpha process in two steps. In the first, two alpha particles form an unstable (lifetime ~10-16s)8Be. In the second, a third alpha particle is captured, via 8Be(α,γ)12C. Hoyle argued than in order for the 3α reaction to proceed at a rate sufficient to produce the observed cosmic carbon, a resonant level must exist in 12C, a few hundred keV about the 8Be+4He threshold. Such a level was indeed found experimentally. Other chapters in the modern history of the anthropic principle are treated comprehensively by Barrow and Tipler (Barrow and Tipler, 1988) and will not be revisited here. The New Urgency of Anthropic Investigation Two recent developments have imparted a renewed sense of urgency to investigations of the anthropic qualities of our cosmos. The first is the discovery that the value of dark energy density is exceedingly small but not quite zero—an apparent happenstance, unpredictable from first principles, with profound implications for the bio-friendly quality of our universe. As noted recently by Goldsmith (Goldsmith, 2004): A relatively straightforward calculation [based on established principles of theoretical physics] does yield a theoretical value for the cosmological constant, but that value is greater than the measured one by a factor of about 10120—probably the largest discrepancy between theory and observation science has ever had to bear. If the cosmological constant had a smaller value than that suggested by recent observations, it would cause no trouble (just as one would expect, remembering the happy days when the constant was thought to be zero). But if the constant were a few times larger than it is now, the universe would have expanded so rapidly that galaxies could not have endured for the billions of years necessary to bring forth complex forms of life. The second development is the realization that M-theory—arguably the most promising contemporary candidate for a theory capable of yielding a deep synthesis of relativity and quantum physics—permits, in Bjorken’s phrase (Bjorken, 2004), “a variety of string vacuua, with different standard-model properties.” M-theorists had initially hoped that their new paradigm would be “brittle” in the sense of yielding a single mathematically unavoidable solution that uniquely explained the seemingly arbitrary parameters of the Standard Model. As Susskind has put it (Susskind, 2003): The world-view shared by most physicists is that the laws of nature are uniquely described by some special action principle that completely determines the vacuum, the spectrum of elementary particles, the forces and the symmetries. Experience with quantum electrodynamics and quantum chromodynamics suggests a world with a small number of parameters and a unique ground state. For the most part, string theorists bought into this paradigm. At first it was hoped that string theory would be unique and explain the various parameters that quantum field theory left unexplained. This hope has been dashed by the recent discovery that the number of different solutions permitted by M-theory (which correspond to different values of Standard Model parameters) is, in Susskind’s words, “astronomical, measured not in millions or billions but in googles or googleplexes.” This development seems to deprive our most promising new theory of fundamental physics of the power to uniquely predict the emergence of anything remotely resembling our universe. As Susskind puts it, the picture of the universe that is emerging from the deep mathematical recesses of M-theory is not an “elegant universe” but rather a Rube Goldberg device, cobbled together by some unknown process in a supremely improbable manner that just happens to render the whole ensemble fit for life. In the words of University of California theoretical physicist Steve Giddings, “No longer can we follow the dream of discovering the unique equations that predict everything we see, and writing them on a single page. Predicting the constants of nature becomes a messy environmental problem. It has the complications of biology.”[1] Two Contemporary Restatements of the Weak Anthropic Principle: Eternal Inflation Plus M-Theory and Many-Worlds Quantum Cosmology There have been two principal approaches to the task of enlisting the weak anthropic principle to explain the mysteriously small (and thus bio-friendly) value of the density of dark energy and the apparent happenstance by which our bio-friendly universe was selected from the enormously large “landscape” of possible solutions permitted by M-theory, only a tiny fraction of which correspond to anything resembling the Standard Model prevalent in our cosmos. Eternal Inflation Meets M-Theory The first approach, favored by Susskind (Susskind, 2003). Linde (Linde, 2002), Weinberg (Weinberg, 1999), and Vilenkin (Vilenkin, 2004) among others, overlays the model of eternal inflation with the key assumption that M-theory-permitted solutions (corresponding to different values of Standard Model parameters) and dark energy density values will vary randomly from bubble universe to bubble universe within an eternally expanding ensemble variously termed a multiverse or a meta-univers. Generating a life-friendly cosmos is simply a matter of randomly reshuffling the set of permissible parameters and values a sufficient number of times until a particular Big Bang yields, against odds of perhaps a googleplex-to-one, a permutation that just happens to possess the right mix of Standard Model parameters to be bio-friendly. Sum-Over-Histories Quantum Cosmological Model The second approach invokes a quantum theory-derived sum-over-histories cosmological model inspired by Everett’s “many worlds” interpretation of quantum physics. This approach, which has been prominently embraced by Hawking (Hawking and Hertog, 2002), was summarized as follows by Hogan (Hogan, 2004): In the original formulation of quantum mechanics, it was said that an observation collapsed a wavefunction to one of the eignestates of the observed quantity. The modern view is that the cosmic wavefunction never collapses, but only appears to collapse from the point of view of observers who are part of the wavefunction. When Schrödinger’s cat lives or dies, the branch of the wavefunction with the dead cat also contains observers who are dealing with a dead cat, and the branch with the live cat also contains observers who are petting a live one. Although this is sometimes called the “Many Worlds” interpretation of quantum mechanics, it is really about having just one world, one wavefunction, obeying the Schrödinger equation: the wavefunction evolves linearly from one time to the next based on its previous state. Anthropic selection in this sense is built into physics at the most basic level of quantum mechanics. Selection of a wavefunction branch is what drives us into circumstances in which we thrive. Viewed from a disinterested perspective outside the universe, it looks like living beings swim like salmon up their favorite branches of the wavefunction, chasing their favorite places. Hawking and Hertog (Hawking and Hertog, 2002) have explicitly characterized this “top down” cosmological model as a restatement of the weak anthropic principle: We have argued that because our universe has a quantum origin, one must adopt a top down approach to the problem of initial conditions in cosmology, in which histories that contribute to the path integral, depend on the observable being measured. There is an amplitude for empty flat space, but it is not of much significance. Similarly, the other bubbles in an eternally inflating spacetime are irrelevant. They are to the future of our past light cone, so they don’t contribute to the action for observables and should be excised by Ockham’s razor. Therefore, the top down approach is a mathematical formulation of the weak anthropic principle. Instead of starting with a universe and asking what a typical observer would see, one specifies the amplitude of interest. Critique of Contemporary Restatements of the Weak Anthropic Principle Apart from the objections on the part of those who oppose in principle any use of the anthropic principle in cosmology, there are at least three reasons why both the Hawking/Hogan and the Susskind/Linde/Weinberg restatements of the weak anthropic principle are objectionable. First, both approaches appear to be resistant (at the very least) to experimental testing. Universes spawned by Big Bangs other than our own are inaccessible from our own universe, at least with the experimental techniques currently available to science. So too are quantum wavefunction branches that we cannot, in principle, observe. Accordingly, both approaches appear to be untestable—perhaps untestable in principle. For this reason, Smolin recently argued (Smolin, 2004) “not only is the Anthropic Principle not science, its role may be negative. To the extent that the Anthropic Principle is espoused to justify continued interest in unfalsifiable theories, it may play a destructive role in the progress of science.” Second, both approaches violate the mediocrity principle. The mediocrity principle, a mainstay of scientific theorizing since Copernicus, is a statistically based rule of thumb that, absent contrary evidence, a particular sample (Earth, for instance, or our particular universe) should be assumed to be a typical example of the ensemble of which it is a part. The Susskind/Linde/Weinberg approach, in particular, flouts this principle. Their approach simply takes refuge in a brute, unfathomable mystery—the conjectured lucky roll of the dice in a crap game of eternal inflation—and declines to probe seriously into the possibility of a naturalistic cosmic evolutionary process that has the capacity to yield a life-friendly set of physical laws and constants on a nonrandom basis. Third, both approaches extravagantly inflate the probabilistic resources required to explain the phenomenon of a life-friendly cosmos. (Think of a googleplex of monkeys typing away randomly until one of them, by pure chance, accidentally composes a set of equations that correspond to the Standard Model.) This should be a hint that something fundamental is being overlooked and that there may exist an unknown natural process, perhaps functionally akin in some manner to terrestrial evolution, capable of effecting the emergence and prolongation of physical states of nature that are, in the abstract, vanishingly improbable. The Darwinian Precedent Hogan (Hogan, 2004) has analogized the quantum theory-inspired sum-over-histories version of the weak anthropic principle to Darwinian theory: This blending of empirical cosmology and fundamental physics is reminiscent of our Darwinian understanding of the tree of life. The double helix, the four-base codon alphabet and the triplet genetic code for amino acids, any particular gene for a protein in a particular organism—all are frozen accidents of evolutionary history. It is futile to try to understand or explain these aspects of life, or indeed any relationships in biology, without referring to the way the history of life unfolded. In the same way that (in Dobzhansky’s phrase), “nothing in biology makes sense except in the light of evolution,” physics in these models only makes sense in the light of cosmology. Ironically, Hogan misses the key point that neither the branching wavefunction nor the eternal inflation-plus-M-theory versions of the weak anthropic principle hypothesize the existence of anything corresponding to the main action principle of Darwin’s theory: natural selection. Both restatements of the weak anthropic principle are analogous, not to Darwin’s approach, but rather to a mythical alternative history in which Darwin, contemplating the storied tangled bank (the arresting visual image with which he concludes The Origin of Species), had confessed not a magnificent obsession with gaining an understanding of the mysterious natural processes that had yielded “endless forms most beautiful and most wonderful,” but rather a smug satisfaction that of course the earthly biosphere must have somehow evolved in a just-so manner mysteriously friendly to humans and other currently living species, or else Darwin and other humans would not be around to contemplate it. Indeed, the situation that confronts cosmologists today is reminiscent of that which faced biologists before Darwin propounded his revolutionary theory of evolution through natural selection. Darwin confronted the seemingly miraculous phenomenon of a fine-tuned natural order in which every creature and plant appeared to occupy a unique and well-designed niche. Refusing to surrender to the brute mystery posed by the appearance of nature’s design, Darwin masterfully deployed the art of metaphor[2] to elucidate a radical hypothesis—the origin of species through natural selection—that explained the apparent miracle as a natural phenomenon. A significant lesson drawn from Darwin’s experience is important to note at this point. Answering the question of why the most eminent geologists and naturalists had, until shortly before publication of The Origin of Species, disbelieved in the mutability of species, Darwin responded that this false conclusion was “almost inevitable as long as the history of the world was thought to be of short duration.” It was geologist Charles Lyell’s speculations on the immense age of Earth that provided the essential conceptual framework for Darwin’s new theory. Lyell’s vastly expanded stretch of geological time provided an ample temporal arena in which the forces of natural selection could sculpt and reshape the species of Earth and achieve nearly limitless variation. The central point for purposes of this paper is that collateral advances in sciences seemingly far removed from cosmology (complexity theory and evolutionary theory among them) can help dissipate the intellectual limitations imposed by common sense and naïve human intuition. And, in an uncanny reprise of the Lyell/Darwin intellectual synergy, it is a realization of the vastness of time and history that gives rise to the novel theoretical possibility to be discussed subsequently. Only in this instance, it is the vastness of future time and future history that is of crucial importance. In particular, sharp attention must be paid to the key conclusion of Wheeler: most of the time available for life and intelligence to achieve their ultimate capabilities lie in the distant cosmic future, not in the cosmic past. As Tipler (Tipler, 1994) has stated, “Almost all of space and time lies in the future. By focusing attention only on the past and present, science has ignored almost all of reality. Since the domain of scientific study is the whole of reality, it is about time science decided to study the future evolution of the universe.” The next section of this paper describes an attempt to heed these admonitions. The Selfish Biocosm Hypothesis In a paper published in Complexity (Gardner, 2000), I first advanced the hypothesis that the anthropic qualities which our universe exhibits might be explained as incidental consequences of a cosmic replication cycle in which the emergence of a cosmologically extended biosphere could conceivably supply two of the logically essential elements of self-replication identified by von Neumann (von Neumann, 1948): a controller and a duplicating device. The hypothesis proposed in that paper was an attempt to extend and refine Smolin’s conjecture (Smolin, 1997) that the majority of the anthropic qualities of the universe can be explained as incidental consequences of a process of cosmological replication and natural selection (CNS) whose utility function is black hole maximization. Smolin’s conjecture differs crucially from the concept of eternal inflation advanced by Linde (Linde, 1998) in that it proposes a cosmological evolutionary process with a specific and discernible utility function—black hole maximization. It is this aspect of Smolin’s conjecture rather than the specific utility function he advocates that renders his theoretical approach genuinely novel. As demonstrated previously (Rees, 1997; Baez, 1998), Smolin’s conjecture suffers from two evident defects: (1) the fundamental physical laws and constants do not, in fact, appear to be fine-tuned to favor black hole maximization and (2) no mechanism is proposed corresponding to two logically required elements of any von Neumann self-replicating automaton: a controller and a duplicator.[3] The latter are essential elements of any replicator system capable of Darwinian evolution, as noted by Dawkins (Gardner, 2000) in a critique of Smolin’s conjecture: Note that any Darwinian theory depends on the prior existence of the strong phenomenon of heredity. There have to be self-replicating entities (in a population of such entities) that spawn daughter entities more like themselves than the general population. Theories of cosmological eschatology previously articulated (Kurzweil, 1999; Wheeler, 1996; Dyson, 1988) predict that the ongoing process of biological and technological evolution is sufficiently robust and unbounded that, in the far distant future, a cosmologically extended biosphere could conceivably exert a global influence on the physical state of the cosmos. A related set of insights from complexity theory (Gardner, 2000) indicates that the process of emergence resulting from such evolution is essentially unbounded. A synthesis of these two sets of insights yielded the two key elements of the Selfish Biocosm (SB) hypothesis. The essence of that synthesis is that the ongoing process of biological and technological evolution and emergence could conceivably function as a von Neumann controller and that a cosmologically extended biosphere could, in the very distant future, function as a von Neumann duplicator in a hypothesized process of cosmological replication. In a paper published in Acta Astronautica (Gardner, 2001) I suggested that a falsifiable implication of the SB hypothesis is that the process of the progression of the cosmos through critical epigenetic thresholds in its life cycle, while perhaps not strictly inevitable, is relatively robust. One such critical threshold is the emergence of human-level and higher intelligence, which is essential to the eventual scaling up of biological and technological processes to the stage at which those processes could conceivably exert a global influence on the state of the cosmos. Four specific tests of the robustness of the emergence of human-level and higher intelligence were proposed. In a subsequent paper published in the Journal of the British Interplanetary Society (Gardner, 2002) I proposed that an additional falsifiable implication of the SB hypothesis is that there exists a plausible final state of the cosmos that exhibits maximal computational potential. This predicted final state appeared to be consistent with both the modified ekpyrotic cyclic universe scenario (Khoury, Ovrut, Seiberg, Steinhardt, and Turok, 2001; Steinhardt and Turok, 2001) and with Lloyd’s description (Lloyd, 2000) of the physical attributes of the ultimate computational device: a computer as powerful as the laws of physics will allow. Key Retrodiction of the SB Hypothesis: A Life-Friendly Cosmos The central assertions of the SB hypothesis are: (1) that highly evolved life and intelligence play an essential role in a hypothesized process of cosmic replication and (2) that the peculiarly life-friendly laws and physical constants that prevail in our universe—an extraordinarily improbable ensemble that Pagels dubbed the cosmic code (Pagels, 1983)—play a cosmological role functionally equivalent to that of DNA in an earthly organism: they provide a recipe for cosmic ontogeny and a blueprint for cosmic reproduction. Thus, a key retrodiction of the SB hypothesis is that the suite of physical laws and constants that prevail in our cosmos will, in fact, be life-friendly. Moreover—and alone among the various cosmological scenarios offered to explain the phenomenon of a bio-friendly universe—the SB hypothesis implies that this suite of laws and constants comprise a robust program that will reliably generate life and advanced intelligence just as the DNA of a particular species constitutes a robust program that will reliably generate individual organisms that are members of that particular species. Indeed, because the hypothesis asserts that sufficiently evolved intelligent life serves as a von Neumann duplicator in a putative process of cosmological replication, the biophilic quality of the suite emerges as a retrodicted biosignature of the putative duplicator and duplication process within the meaning of Goal 7 of the NASA Astrobiology Roadmap, which provides in pertinent part: Does this retrodiction qualify as a valid scientific test of the validity of the SB hypothesis? I propose that it may, provided two additional qualifying criteria are satisfied: • The underlying hypothesis must enjoy consilience[4] with mainstream scientific paradigms and conjectural frameworks (in particular, complexity theory, evolutionary theory, M-theory, and theoretically acceptable conjectures by mainstream cosmologists concerning the feasibility, at least in principle, of “baby universe” fabrication); and • The retrodiction must be augmented by falsifiable predictions of phenomena implied by the SB hypothesis but not yet observed. Retrodiction as a Tool for Testing Scientific Hypotheses There is a lively literature debating the propriety of employing retrodiction as a tool for testing scientific hypotheses (Cleland, 2002; Cleland, 2001; Gee, 1999; Oldershaw, 1988). Oldershaw (Oldershaw, 1988) has discussed the use of falsifiable retrodiction (as opposed to falsifiable prediction) as a tool of scientific investigation: A second type of prediction is actually not a prediction at all, but rather a “retrodiction.” For example, the anomalous advance of the perihelion of Mercury had been a tiny thorn in the side of Newtonian gravitation long before general relativity came upon the scene. Einstein found that his theory correctly “predicted,” actually retrodicted, the numerical value of the perihelion advance. The explanation of the unexpected result of the Michelson-Morley experiment (constancy of the velocity of light) in terms of special relativity is another example. As he went on to note, “Retrodictions usually represent falsification tests; the theory is probably wrong if it fails the test, but should not necessarily be considered right if it passes the test since it does not involve a definitive prediction.” Despite their legitimacy as falsification tests of hypotheses, falsifiable retrodictions are qualitatively inferior to falsifiable predictions, in Oldershaw’s view: But, in the final analysis, only true definitive predictions can justify the promotion of a theory from being viewed as one of many plausible hypotheses to being recognized as the best available approximation of how nature actually works. A theory that cannot generate definitive predictions, or whose definitive predictions are impossible to test, can be regarded as inherently untestable.” A less sympathetic view concerning the validity of retrodiction as a scientific tool was offered by Gee (Gee, 1999), who dismissed the legitimacy of all historical hypotheses on the ground that “they can never be tested by experiment, and so they are unscientific…. No science can ever be historical.” This viewpoint, in turn, has been challenged by Cleland (Cleland, 2001) who contends that “when it comes to testing hypotheses, historical science is not inferior to classical experimental science” but simply exploits the available evidence in a different way: There [are] fundamental differences in the methodology used by historical and experimental scientists. Experimental scientists focus on a single (sometimes complex) hypothesis, and the main research activity consists in repeatedly bringing about the test conditions specified by the hypothesis, and controlling for extraneous factors that might produce false positives and false negatives. Historical scientists, in contrast, usually concentrate on formulating multiple competing hypotheses about particular past events. Their main research efforts are directed at searching for a smoking gun, a trace that sets apart one hypothesis as providing a better causal explanation (for the observed traces) than do the others. These differences in methodology do not, however, support the claim that historical science is methodologically inferior, because they reflect an objective difference in the evidential relations at the disposal of historical and experimental researchers for evaluating their hypotheses. Cleland’s approach has the merit of preserving as “scientific” some of the most important hypotheses advanced in such historical fields of inquiry as geology, evolutionary biology, cosmology, paleontology, and archaeology. As Cleland has noted (Cleland, 2002): Experimental research is commonly held up at the paradigm of successful (a.k.a.good) science. The role classically attributed to experiment is that of testing hypotheses in controlled laboratory settings. Not all scientific hypotheses can be tested in this manner, however. Historical hypotheses about the remote past provide good examples. Although fields such as paleontology and archaeology provide the familiar examples, historical hypotheses are also common in geology, biology, planetary science, astronomy, and astrophysics. The focus of historical research is on explaining existing natural phenomena in terms of long past causes. Two salient examples are the asteroid-impact hypothesis for the extinction of the dinosaurs, which explains the fossil record of the dinosaurs in terms of the impact of a large asteroid, and the “big-bang” theory of the origin of the universe, which explains the puzzling isotropic three-degree background radiation in terms of a primordial explosion. Such work is significantly different from making a prediction and then artificially creating a phenomenon in a laboratory. In a paper presented to the 2004 Astrobiology Science Conference (Cleland, 2004), Cleland extended this analytic framework to the consideration of putative biosignatures as evidence of the past or present existence of extraterrestrial life. Acknowledging that “because biosignatures represent indirect traces (effects) of life, much of the research will be historical (vs. experimental) in character even in cases where the traces represent recent effects of putative extant organisms,” Cleland concluded that it was appropriate to employ the methodology that characterizes successful historical research: Successful historical research is characterized by (1) the proliferation of alternative competing hypotheses in the face of puzzling evidence and (2) the search for more evidence (a “smoking gun”) to discriminate among them. From the perspective of the evidentiary standards applicable to historical science in general and astrobiology in particular, the key retrodiction of the SB hypothesis—that the fundamental constants of nature that comprise the Standard Model as well as other physical features of our cosmos (included the number of extended physical dimensions and the extremely low value of dark energy) will be collectively bio-friendly—appears to constitute a legitimate scientific test of the hypothesis. Moreover, within the framework of Goal 7 of the NASA Astrobiology Roadmap, the retrodicted biophilic quality of our universe appears, under the SB hypothesis, to constitute a possible biosignature. Caution Regarding the Use of Retrodiction to Test the SB Hypothesis Because the SB hypothesis is radically novel and because the use of falsifiable retrodiction as a tool to test such an hypothesis creates at least the appearance of a “confirmatory argument resemble[ing] just-so stories (Rudyard Kipling’s fanciful stories, e.g., how leopards got their spots)” (Cleland, 2001), it is important (as noted previously) that two additional criteria be satisfied before this retrodiction can be considered a legitimate test of the hypothesis: • The SB hypothesis must generate falsifiable predictions as well as falsifiable retrodictions; and • The SB hypothesis must be consilient with key theoretical constructs in such “adjoining” area of scientific investigation as M-theory, cosmogenesis, complexity theory, and evolutionary theory. As argued at length elsewhere (Gardner, 2003), the SB hypothesis is both consilient with central concepts in these “adjoining” fields and fully capable of generating falsifiable predictions. Concluding Remarks In his book The Fifth Miracle (Davies, 1999) Davies offered this interpretation of NASA’s view that the presence of liquid water on an alien world was a reliable marker of a life-friendly environment: In claiming that water means life, NASA scientists are… making—tacitly—a huge and profound assumption about the nature of nature. They are saying, in effect, that the laws of the universe are cunningly contrived to coax life into being against the raw odds; that the mathematical principles of physics, in their elegant simplicity, somehow know in advance about life and its vast complexity. If life follows from [primordial] soup with causal dependability, the laws of nature encode a hidden subtext, a cosmic imperative, which tells them: “Make life!” And, through life, its by-products: mind, knowledge, understanding. It means that the laws of the universe have engineered their own comprehension. This is a breathtaking vision of nature, magnificent and uplifting in its majestic sweep. I hope it is correct. It would be wonderful if it were correct. But if it is, it represents a shift in the scientific world-view as profound as that initiated by Copernicus and Darwin put together. An emerging consensus among mainstream physicists and cosmologists is that the particular universe we inhabit appears to confirm what Smolin calls the “anthropic observation”: the laws and constants of nature seem to be fine-tuned, with extraordinary precision and against enormous odds, to favor the emergence of life and its byproduct, intelligence. As Dyson put it eloquently more than two decades ago (Dyson, 1979): The more I examine the universe and study the details of its architecture, the more evidence I find that the universe in some sense must have known that we were coming. There are some striking examples in the laws of nuclear physics of numerical accidents that seem to conspire to make the universe habitable. Why this should be so remains a profound mystery. Indeed, the mystery has deepened considerably with the recent discovery of the inexplicably tiny value of dark energy density and the realization that M-theory encompasses an unfathomably vast landscape of possible solutions, only a minute fraction of which correspond to anything resembling the universe that we inhabit. Confronted with such a deep mystery, the scientific community ought to be willing to entertain plausible explanatory hypotheses that may appear to be unconventional or even radical. However, such hypotheses, to be taken seriously, must: • be consilient with the key paradigms of “adjoining” scientific fields, • generate falsifiable predictions, and • generate falsifiable retrodictions. The SB hypothesis satisfies these criteria. In particular, it generates a falsifiable retrodiction that the physical laws and constants that prevail in our cosmos will be biophilic—which they are. Baez, J. 1998 on-line commentary on The Life of the Cosmos (available at Barrow, J. and Tipler, F. 1988 The Anthropic Cosmological Principle, Oxford University Press. Bjorken, J. 2004 “The Classification of Universes,” astro-ph/0404233. Cleland, C. 2001 “Historical science, experimental science, and the scientific method,” Geology, 29, pp. 978-990. Cleland, C. 2002 “Methodological and Epistemic Differences Between Historical Science and Experimental Science,” Philosophy of Science, 69, pp. 474-496. Cleland, C. 2004 “Historical Science and the Use of Biosignatures,” unpublished summary of presentation abstracted in International Journal of Astrobiology, Supplement 2004, p. 119. Davies, P. 1999 The Fifth Miracle, Simon & Schuster. Dyson, F. 1979 Disturbing the Universe, Harper & Row. Dyson, F. 1988 Infinite in All Directions, Harper & Row. Gardner, J. 2000 “The Selfish Biocosm: Complexity as Cosmology,” Complexity, 5, no. 3, pp. 34-45.. Gardner, J. 2001 “Assessing the Robustness of the Emergence of Intelligence: Testing the Selfish Biocosm Hypothesis,” Acta Astronautica, 48, no. 5-12, pp. 951-955. Gardner, J. 2002 “Assessing the Computational Potential of the Eschaton: Testing the Selfish Biocosm Hypothesis,” Journal of the British Interplanetary Society 55, no. 7/8, pp. 285-288. Gardner, J. 2003 Biocosm, Inner Ocean Publishing. Gee, H. 1999 In Search of Deep Time, The Free Press. Goldsmith, D. 2004 “The Best of All Possible Worlds,” Natural History, 5, no. 6, pp. 44-49. Greene, B. 2004 The Fabric of the Cosmos, Knopf. Hawking, S. and Hertog, T. 2002 “Why Does Inflation Start at the Top of the Hill?” hep-th/0204212. Henderson, L. 1913 The Fitness of the Environment, Harvard University Press. Hogan, C. 2004 “Quarks, Electrons, and Atoms in Closely Related Universes,” astro-ph/0407086. Khoury, J., Ovrut, B. A., Seiberg, N., Steinhardt, P., and Turok, N. 2001 “From Big Crunch to Big Bang,” hep-th/0108187. Kurzweil, R. 1999 The Age of Spiritual Machines, Viking. Linde, A. 2002 “Inflation, Quantum Cosmology and the Anthropic Principle,” hep-th/0211048. Linde, A.1998 “The Self-Reproducing Inflationary Universe,” Scientific American, 9(20), pp. 98-104. Livio, M. 2003 “Cosmology and Life,” astro-ph/0301615. Lloyd, S. 2000 “Ultimate Physical Limits to Computation,” Nature, 406, pp. 1047-1054. Mitton, S. 2005 Conflict in the Cosmos: Fred Hoyle’s Life in Science, Joseph Henry Press. Oldershaw, R. 1988 “The new physics: physical or mathematical science?” American Journal of Physics, 56(12). Pagels, H. 1983 The Cosmic Code, Bantam. Rees, M. 1997 Before the Beginning, Addison Wesley. Rees, M. 2000 Just Six Numbers, Basic Books. Smolin, L. 1997 The Life of the Cosmos, Oxford University Press. Smolin, L. 2004 “Scientific Alternatives to the Anthropic Principle,” hep-th/0407213. Steinhardt, P. and Turok, N. 2001 “Cosmic Evolution in a Cyclic Universe,” hep-th/0111098. Susskind, L. 2003 “The Anthropic Landscape of String Theory,” hep-th/0302219. Tipler, F. 1994 The Physics of Immortality, Doubleday. Vilenkin, A. 2004 “Anthropic predictions: The Case of the Cosmological Constant,” astro-ph/0407586. von Neumann, J. 1948 “On the General and Logical Theory of Automata.” Weinberg, S. 21 October 1999 “A Designer Universe?” New York Review of Books. Wheeler, J. 1996 At Home in the Universe, AIP Press. Wilson, E. O. 1998 “Scientists, Scholars, Knaves and Fools,” American Scientist, 86, pp. 6-7. [2] The metaphor furnished by the familiar process of artificial selection was Darwin’s crucial stepping stone. Indeed, the practice of artificial selection through plant and animal breeding was the primary intellectual model that guided Darwin in his quest to solve the mystery of the origin of species and to demonstrate in principle the plausibility of his theory that variation and natural selection were the prime movers responsible for the phenomenon of speciation. [3] Both defects were emphasized by Susskind in a recent on-line exchange with Smolin which appears at Smolin has argued that his CNS hypothesis has not been falsified on the first ground (Smolin, 2004) but conceded that his conjecture lacks any hypothesized mechanism that would endow the putative process of proliferation of black-hole-prone universes with a heredity function: The hypothesis that the parameters p change, on average, by small random amounts, should be ultimately grounded in fundamental physics. We note that this is compatible with string theory, in the sense that there are a great many string vacua, which likely populate the space of low energy parameters well. It is plausible that when a region of the universe is squeezed to Planck densities and heated to Planck temperatures, phase transitions may occur leading to a transition from one string vacua to another. But there have so far been no detailed studies of these processes which would check the hypothesis that the change in each generation is small. As Smolin noted in the same paper, it is crucial that such a mechanism exist in order to avoid the conclusion that each new universe’s set of physical laws and constants would constitute a merely random sample of the vast parameter space permitted by the extraordinarily large “landscape” of M-theory-allowed solutions: It is important to emphasize that the process of natural selection is very different from a random sprinkling of universes on the parameter space P. This would produce only a uniform distribution prandom(p). To achieve a distribution peaked around the local maxima of a fitness function requires the two conditions specified. The change in each generation must be small so that the distribution can “climb the hills” in F(p) rather than jump around randomly, and so it can stay in the small volume of P where F(p) is large, and not diffuse away. This requires many steps to reach local maxima from random starts, which implies that long chains of descendants are needed. [4] Wilson has identified consilience as one of the “diagnostic features of science that distinguishes it from pseudoscience” (Wilson, 1998): The explanations of different phenomena most likely to survive are those that can be connected and proved consistent with one another. © 2005 James N. Gardner. Reprinted with permission.
30b20dd23c89d8ae	%0 Journal Article %J Phys. Rev. Lett. %D 2020 %T Efficiently computable bounds for magic state distillation %A Xin Wang %A Mark M. Wilde %A Yuan Su %X Magic state manipulation is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to magic state manipulation is the resource theory of magic states, for which one of the goals is to characterize and quantify quantum "magic." In this paper, we introduce the family of thauma measures to quantify the amount of magic in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of magic states. As a first application, we use the min-thauma to bound the regularized relative entropy of magic. As a consequence of this bound, we find that two classes of states with maximal mana, a previously established magic measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of magic states and reveals a fundamental difference between the resource theory of magic states and other resource theories such as entanglement and coherence. As a second application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable magic, which in turn leads to a variety of bounds on the rate at which magic can be distilled, as well as on the overhead of magic state distillation. Finally, we prove that the max-thauma can outperform mana in benchmarking the efficiency of magic state distillation. %B Phys. Rev. Lett. %V 124 %8 3/6/2020 %G eng %U https://arxiv.org/abs/1812.10145 %N 090505 %R https://doi.org/10.1103/PhysRevLett.124.090505 %0 Journal Article %J Quantum %D 2020 %T Time-dependent Hamiltonian simulation with L1-norm scaling %A Dominic W. Berry %A Andrew M. Childs %A Yuan Su %A Xin Wang %A Nathan Wiebe %X The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For the case of sparse Hamiltonian simulation, the gate complexity scales with the L1 norm ∫t0dτ∥H(τ)∥max, whereas the best previous results scale with tmaxτ∈[0,t]∥H(τ)∥max. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. By leveraging the L1-norm information, we obtain polynomial speedups for semi-classical simulations of scattering processes in quantum chemistry. %B Quantum %V 4 %8 4/20/2020 %G eng %U https://arxiv.org/abs/1906.07115 %N 254 %R https://doi.org/10.22331/q-2020-04-20-254 %0 Journal Article %J New Journal of Physics %D 2019 %T Quantifying the magic of quantum channels %A Xin Wang %A Mark M. Wilde %A Yuan Su %X %B New Journal of Physics %V 21 %8 10/8/2019 %G eng %U https://arxiv.org/abs/1903.04483 %N 103002 %R https://doi.org/10.1088/1367-2630/ab451d %0 Journal Article %D 2019 %T Resource theory of asymmetric distinguishability for quantum channels %A Xin Wang %A Mark M. Wilde %X This paper develops the resource theory of asymmetric distinguishability for quantum channels, generalizing the related resource theory for states [arXiv:1006.0302, arXiv:1905.11629]. The key constituents of the channel resource theory are quantum channel boxes, consisting of a pair of quantum channels, which can be manipulated for free by means of an arbitrary quantum superchannel (the most general physical transformation of a quantum channel). One main question of the resource theory is the approximate channel box transformation problem, in which the goal is to transform an initial channel box (or boxes) to a final channel box (or boxes), while allowing for an asymmetric error in the transformation. The channel resource theory is richer than its counterpart for states because there is a wider variety of ways in which this question can be framed, either in the one-shot or n-shot regimes, with the latter having parallel and sequential variants. As in [arXiv:1905.11629], we consider two special cases of the general channel box transformation problem, known as distinguishability distillation and dilution. For the one-shot case, we find that the optimal values of the various tasks are equal to the non-smooth or smooth channel min- or max-relative entropies, thus endowing all of these quantities with operational interpretations. In the asymptotic sequential setting, we prove that the exact distinguishability cost is equal to channel max-relative entropy and the distillable distinguishability is equal to the amortized channel relative entropy of [arXiv:1808.01498]. This latter result can also be understood as a solution to Stein's lemma for quantum channels in the sequential setting. Finally, the theory simplifies significantly for environment-seizable and classical--quantum channel boxes. %8 07/15/2019 %G eng %U https://arxiv.org/abs/1907.06306 %0 Journal Article %D 2019 %T Resource theory of entanglement for bipartite quantum channels %A Stefan Bäuml %A Siddhartha Das %A Xin Wang %A Mark M. Wilde %X The traditional perspective in quantum resource theories concerns how to use free operations to convert one resourceful quantum state to another one. For example, a fundamental and well known question in entanglement theory is to determine the distillable entanglement of a bipartite state, which is equal to the maximum rate at which fresh Bell states can be distilled from many copies of a given bipartite state by employing local operations and classical communication for free. It is the aim of this paper to take this kind of question to the next level, with the main question being: What is the best way of using free channels to convert one resourceful quantum channel to another? Here we focus on the the resource theory of entanglement for bipartite channels and establish several fundamental tasks and results regarding it. In particular, we establish bounds on several pertinent information processing tasks in channel entanglement theory, and we define several entanglement measures for bipartite channels, including the logarithmic negativity and the κ-entanglement. We also show that the max-Rains information of [Bäuml et al., Physical Review Letters, 121, 250504 (2018)] has a divergence interpretation, which is helpful for simplifying the results of this earlier work. %8 07/08/2019 %G eng %U https://arxiv.org/abs/1907.04181 %0 Journal Article %D 2019 %T α-Logarithmic negativity %A Xin Wang %A Mark M. Wilde %X The logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory, due to the fact that it is easy to compute and serves as an upper bound on distillable entanglement. More recently, the κ-entanglement of a bipartite state was shown to be the first entanglement measure that is both easily computable and operationally meaningful, being equal to the exact entanglement cost of a bipartite quantum state when the free operations are those that completely preserve the positivity of the partial transpose. In this paper, we provide a non-trivial link between these two entanglement measures, by showing that they are the extremes of an ordered family of α-logarithmic negativity entanglement measures, each of which is identified by a parameter α∈[1,∞]. In this family, the original logarithmic negativity is recovered as the smallest with α=1, and the κ-entanglement is recovered as the largest with α=∞. We prove that the α-logarithmic negativity satisfies the following properties: entanglement monotone, normalization, faithfulness, and subadditivity. We also prove that it is neither convex nor monogamous. Finally, we define the α-logarithmic negativity of a quantum channel as a generalization of the notion for quantum states, and we show how to generalize many of the concepts to arbitrary resource theories. %8 04/23/2019 %G eng %U https://arxiv.org/abs/1904.10437 %0 Journal Article %D 2018 %T Exact entanglement cost of quantum states and channels under PPT-preserving operations %A Xin Wang %A Mark M. Wilde %X This paper establishes single-letter formulas for the exact entanglement cost of generating bipartite quantum states and simulating quantum channels under free quantum operations that completely preserve positivity of the partial transpose (PPT). First, we establish that the exact entanglement cost of any bipartite quantum state under PPT-preserving operations is given by a single-letter formula, here called the κ-entanglement of a quantum state. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum states. Notably, this is the first time that an entanglement measure for general bipartite states has been proven not only to possess a direct operational meaning but also to be efficiently computable, thus solving a question that has remained open since the inception of entanglement theory over two decades ago. Next, we introduce and solve the exact entanglement cost for simulating quantum channels in both the parallel and sequential settings, along with the assistance of free PPT-preserving operations. The entanglement cost in both cases is given by the same single-letter formula and is equal to the largest κ-entanglement that can be shared by the sender and receiver of the channel. It is also efficiently computable by a semidefinite program. %G eng %U https://arxiv.org/abs/1809.09592 %0 Journal Article %D 2018 %T Quantum Channel Simulation and the Channel's Smooth Max-Information %A Kun Fang %A Xin Wang %A Marco Tomamichel %A Mario Berta %X We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the mutual information of a quantum channel. We provide an exact operational interpretation of the channel's smooth max-information as the one-shot quantum simulation cost under no-signalling assisted codes. Third, we derive the asymptotic equipartition property of the channel's smooth max-information, i.e., it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. This implies the quantum reverse Shannon theorem in the presence of no-signalling correlations. Finally, we explore the simulation cost of various quantum channels. %G eng %U https://arxiv.org/abs/1807.05354
96a35cc6269243ee	And the History of Physics in Two The history of creation can be expressed in three words (not even words but mere conjunctions): “and,” “or,” and “and/or,” whereas the history of physics can be expressed in two of them: “or” and “and/or.” Although it will take slightly more than three words to make the connections—my apologies—allow me to explain. The History of Creation The history of creation began with . . . well, the Creator, of course. Most people, including Jewish mystics, believe that the most characteristic and unique property of G‑d is His infinity. Thus, the Kabbalists call G‑d Ein Sof—literally, “without end,” or “infinite.” This property is unique to the Creator insofar as we do not find infinity in nature—nothing physical can be infinite. However, in mathematics, infinities are ubiquitous, and mathematicians are very comfortable dealing with them. We can make some nontrivial observations about infinities and even compare them. There are infinitely many natural numbers (positive integers like 1, 2, 3,…) and infinitely many real numbers (points on a line, say, between 0 and 1). However, there are infinitely more real numbers than natural numbers, because between any two natural numbers we can fit infinitely many real numbers. The mathematician Georg Cantor[1] introduced new transfinite numbers (that are greater than any finite number but not necessarily infinite), which are a generalization of the number of elements in the set (cardinality of the set) on infinite sets. These numbers are traditionally denoted by the letter aleph (א) with a subscript. Some Christian theologians attacked Cantor’s mathematics for challenging G‑d’s unique infinity. But the use of infinity in mathematics does not pose a challenge to Judaism. The most characteristic and truly unique property of G‑d is not infinity but His self-referential and self-contradictory nature. Or, looking at this from another angle, the self-contradictory nature of G-d stems from His absolute perfection (shelemut), because He lacks nothing. The First “And The self-contradictory nature of G‑d can be expressed by the coordinating conjunction “and.” For any thesis A and antithesis not-A, G‑d can be said to possess both A and not-A, because He lacks neither A nor not-A. Thus He has the potential for both infinitude (ko’ach bli gvul) and finitude (ko’ach hagvul), because He cannot be limited by anything, including by His infinitude. As a perfect being, G-d lacks neither infinitude nor finitude. The most mind-bending statement about G‑d is that He exists and “does not exist,” as it were, at the same time,[2] because He is not limited by His existence. G‑d can be in any state and its opposite state, because He lacks neither state. And He can be in a superposition of all states because nothing is impossible for G‑d. The sages termed this self-contradictory aspect of G‑d as nimna hanimna’ot(literally “restricting [all] restrictions”), that is, the “paradox of paradoxes.”[3] God is the ultimate paradox.[4] This paradoxical ability of G‑d to be in any state and its opposite state is summed up by the coordinating conjunction “and.” The “Or As it arose in the mind of G‑d to create this world, He desired to manifest Himself and His attributes in the lower worlds (tachtonim).[5] The tachtonim (lower created worlds) are characterized by the gevulim (“finiteness,” or “finite measures”), as well as by their false sense of independence. There is nothing infinite in the created worlds. Therefore, there is no room for paradoxes in the lower worlds. Any thesis excludes its antithesis and vice versa: it is either the thesis or the antithesis. This physical world cannot endure any paradoxes. One way to avoid paradoxes is to create time so that two contradictory things happen at different times without canceling each other out. That’s how the doorbell works—its electrical circuit is self-referential and self-contradictory, but it works because the contradictory elements occur one after another, not simultaneously.[6] At any given moment in time, only one of two contradictory statements can be true, but never both. So, at each moment, it is either one or the other. Therefore, this physical world is characterized by the correlative conjunction “or.” The Psalmists said: How manifold are Thy works, O Lord. Psalms 104:24 The fundamental question regarding creation is, How did such an incredible variety of forms of life and matter come from one G‑d? By way of analogy, just as a pure white light after traveling through a prism splits into seven rainbow colors, so too, when Infinite G‑d, who is unlimited and is so-to-say in a superposition of all possible (and impossible) states, wishes to manifest Himself in a finite (lower) world, each of these states has to have its own embodiment, because a limited world cannot contain contradictions. Thus, contradictory states must be realized either at different times, in different places, or be embodied by different objects. This world reflects its divine source. However, unlike its divine source, it cannot combine contradictory states—it’s either one or the other. For this reason, we label this stage in the creation by the correlative conjunction “or.”[7] The “And/Or This world is not intended to remain in the same state forever. Any purposeful project must have an end goal, which represents the fulfillment of the purpose. A purposeful creation also has an end goal—what is called the messianic era (a.k.a. the world-to-come or the hereafter). According to the Lurianic Kabbalah, in the present pre-messianic state of the creation, the Light of the Infinite (Or Ein Sof) shins through the limited vessels (kelim), which greatly screen, diminish, and conceal the godly light. Thus, we receive only a dim glimmer of that infinite light. This is necessary, because the limited vessels of the physical world could not withstand infinite light and would burn up if not for the intense contraction and screening of the light. In the messianic era, the Light of the Infinite will be able to manifest itself in the full measure of its unmitigated infinitude despite the vessels that remain finite. How is this possible? As we discussed above, G-d possesses the potential of infinitude and finitude. The main characteristic of the messianic era is the revelation of G-d’s essence, which contains the potential of infinitude and finitude. It follows that this paradoxical aspect will be revealed and both infinitude and finitude will be manifest in this physical world. That means that the physical world will remain physical and limited as it is now, but after being purified and rectified, its vessels will be able to withhold—and, indeed, benefit from—the infinite light. This is the ultimate contradiction and a miracle of the messianic time, reflecting the contradictory nature of the Creator, who possesses the power of infinitude as well as finitude. The unnatural (and self-contradictory) combination of the Light of the Infinite (Or Ein Sof), symbolized by the coordinating conjunction “and,” on the one hand, and limited vessels (kelim), symbolized by the correlative conjunction “or” on the other hand, creates an unusual combination of “and” and “or,” which we are going to represent as “and/or.”[8] The History of Physics The “Or The history of physics begins with Isaac Newton, who formulated the first rigorous scientific theories of physics—mechanics, optics, and the universal law of gravity, which are still used today in science and engineering. In Newtonian physics, any physical system (a collection of physical objects) is characterized by a state. And by “state,” we do not mean classical states of matter—solid, liquid, and gas (and plasma); by “state,” we mean a set of values of all measurable properties of the physical system. For example, a single particle has two primary properties—its coordinates and its momentum. The values of the coordinates and the momentum of the particle at a particular time define the state of that particle. A system of several particles will have a state defined by the values of the coordinates and the momenta of all particles in the system. Thermodynamical systems have other primary characteristics, such as volume, pressure, and temperature. The values of these characteristics define the state of the thermodynamical system at a particular time. A spinning top has angular momentum as its primary characteristic. Its angular momentum shows which direction the top is spinning (clockwise or counterclockwise) and the angular velocity of rotation. And so on. The fundamental principle of classical physics is that no system can be in two different states simultaneously. Thus, a particle cannot be here and there—in two different locations in space—at the same time. A gas cannot have two different values of its temperature at the same time. And a top cannot spin clockwise and counterclockwise at the same time—it can spin either clockwise or counterclockwise. For this reason, we shall represent this stage of physics development by the correlative conjunction “or.” The “And/Or As it turned out, Newtonian physics is only accurate for ordinary objects we encounter in our daily lives. When it comes to the microworld of atoms and subatomic particles, Newtonian physics gives way to quantum mechanics—the most fundamental and accurate theory of science known to date. Quantum physics is where our intuition goes to die. This theory is famous for its weirdness and mind-bending concepts, but it never fails to predict the results of the experiment. It is the most tested theory in all of science, and we can rely on it to correctly describe subatomic physics as well as molecular chemistry, which is based on it. The weirdness of quantum mechanics comes from the fact that a physical system can be in more than one state at a time. In fact, it can be in a superposition of many states (some of which can be mutually exclusive). In quantum mechanics, a top can spin clockwise and counterclockwise simultaneously, and a cat can be simultaneously dead and alive. This fundamental property of quantum mechanics led one science writer to aptly say, “The pleasure and pain of quantum theory began when an “or” became an “and.”[9] In quantum mechanics, a physical system is described by a wave function that obeys the Schrödinger equation. This equation is linear. As we know from high school algebra, a linear combination of solutions to such an equation is also a solution (in fact, this property defines a linear equation). Therefore, any linear combination of states is also an allowed state in quantum mechanics. This mathematical fact leads to a possibility of a quantum particle having its spin up and down at the same time (a quantum equivalent to a classical top spinning clockwise and counterclockwise simultaneously). This mind-bending property was enigmatically captured by Ervin Schrödinger in his famous thought experiment with the eponymous cat locked in a box with radioactive material that can randomly decay, killing the cat. In that experiment, the cat in a box (until looked at) is in the state of superposition of being dead and being alive. This strange phenomenon can be classified using the coordinating conjunction “and.” However, any act of measurement collapses the wave function of the system, which is always found either in one state or the other—a phenomenon described by the correlative conjunction “or.” Indeed, when we open the box and look inside, we find the poor feline either dead or alive, but never both. This contradiction is called “the measurement problem.” We are going to denote this state by the “and/or” moniker.[10] The Parallel If you haven’t noticed the parallel yet, here it is, staring straight at us: Before CreationFirst StageUltimate Stage The history of Creation:andorand/or The history of physics: orand/or The History of Creation and History of Physics on Parallel Tracks The history of physics tracks the history of creation precisely, with one exception—the initial “and” is missing. But this is fully expected because the initial “and” in the history of creation characterizes the Creator. Physics doesn’t study the Creator (the subject of theological inquiry); it studies the world as it was created. Far from coincidental, this parallel is highly significant. We would expect to see progress in science tracking the development of the creation, and indeed it does. This enigmatic state of nimna hanimna’ot—of being in contradictory states A and not-A—is reminiscent of what is called in quantum mechanics the “superposition of states.” Indeed, God, as an Infinite Being, can be and is in a superposition of all states.[11] The enigmatic state of nimna hanimna’ot would manifest itself only in miracles that violate the laws of nature. A well-documented example of such a miracle is found in the description of the measurements of the Bet HaMikdash (the Holy Temple in Jerusalem) recorded in the Talmud. Aron HaBrit—the “Ark of the Covenant”—was the golden box containing the tablets (Luchot) with the Ten Commandments, placed in the Holy of Holies—the innermost sanctum of the Bet HaMikdash (the “Holy Temple” in Jerusalem). According to the Talmud, the Ark did not occupy any space. The Holy of Holies measured twenty cubits by twenty cubits. The Ark itself measured two-and-a-half cubits. When measured from the south wall to the nearest side of the Ark, the distance was ten cubits. When measured from the side of the Ark to the northern wall, the distance was also ten cubits. It appears that the Ark occupied no space, although its width was 2.5 cubits.[12] This miraculous state of affairs will once again be found in the messianic era. While we can expect and accept miracles in the hereafter, we cannot accept miracles in physics here and now. The fact that we still do not understand the nature of the collapse of the wave function—the measurement problem—is indeed a problem indicative of the fact that the messianic age has not fully arrived yet, albeit very close. The measurement problem is what separates us from the hereafter. When this problem is resolved, this will herald redemption, at least on the intellectual level. In the meantime, on a social level, the bunker mentality “me against them” or “us against them” tribalism expressed by the correlative “or” (it is either them or us) is distinctly un-messianic. While physicists are hard at work hastening the redemption by trying to understand the nature of the measurement problem, the rest of us have to do our part by shifting the mode of our personal relationships from the attitude expressed by the correlative conjunction “or” to the attitude symbolized by the coordinating conjunction “and”—from “I or you” to “I and you.” Only together—you and I—can we bring about redemption. [1] Georg Ferdinand Ludwig Philipp Cantor (1845–1918), a German mathematician, the founder of the branch of mathematics called set theory. [2] The “non-existence” of G-d is only relative to our existence. It simply means that His existence, which according to Maimonides is a necessary existence, is of an entirely different nature and cannot be compared to our existence in any way. When using such expressions as “at the same time” with respect to G‑d, we must remember that G‑d transcends time, which He created, and such expressions are meaningless and are used only as a figure of speech. Moreover, our existence is necessarily temporal. A divine being that transcends time cannot be said to exist because the word “exist” necessarily means to exist in time. [3] Responsa of the Rashba (Shu”t HaRashb”a), vol. I, sec. 418; see also Sefer HaChakirah by the Tzemach Tzedek, p. 34b ff. [4] See more on this in my essay, “Physics of Tzimtzum II — Collapse of the Wave Function” (2020), available at [5] This concept called dira b’tachtonim (a “dwelling place” [for G‑d]), based on the Midrash Tanhuma, Naso, is the cornerstone of the theology of Chasidism. [6] See my essay “On the Nature of Time and the Age of the Universe” (2005), available at [7] In classical logic, this is called an exclusive disjunction. [8] And/or is not used here as a conjunction that indicates that two words or expressions are to be taken together or individually, as it is normally used in the English language. We are using “and/or” here merely as an emblem of an unusual combination of two metaphysical concepts symbolized respectively by “and” and “or.” [9] Richard Webb, “What Makes Quantum Theory So Strange?” in “Quantum Frontiers,” New Scientist, August 28, 2021. [10] See footnote 7 above. As there, our use of “and/or” here is not intended as a conjunction, but as an emblem of a paradoxical clash of two incompatible situations—a superposition of states before measurement denoted as “and” vs. the collapsed state after measurement denoted by “or.” [11] A quantum mechanical system can be in a superposition of all possible states. G-d, on the other hand, can be “in superposition” of all states, possible and “impossible,” because nothing is impossible for God. Moreover, we must not forget that this is just a crude allegory—G-d is not in any state that we can think of, as it says, “I am G-d and I do not change” (Malachi 3:6). Only our idea of G-d can be in different states—He can be angry and forgiving at the same time, for example. [12] See Talmud, Yoma 21a; Megillah 10b; Bava Batra 99a. As the sages explained, this is an example of G‑d’s presence occupying and not occupying physical space—nimna hanimna’ot. See Rabbi Menachem M. Schneerson, Maamar Gadol Yiheyeh Kavod HaBayis HaZeh. See also Sefer HaMa’amarim 5643, p. 100; and loc. cit. 5665 p. 185. Printer Friendly
fa89c95e2d23c48f	During the past few decades, enormous resources have been invested in research efforts to discover new energetic materials with improved performance, thermodynamic stability, and safety. A key goal of these efforts has been to find replacements for a handful of energetics which have been used almost exclusively in the world’s arsenals since World War II - HMX, RDX, TNT, PETN, and TATB1. While hundreds of new energetic materials have been synthesized as a result of this research, many of which have remarkable properties, very few compounds have made it to industrial production. One exception is CL-202,3, the synthesis of which came about as a result of development effort that lasted about 15 years1. After its initial synthesis, the transition of CL-20 to industrial production took another 15 years1. This time scale (20–40 years) from the initial start of a materials research effort until the successful application of a novel material is typical of what has been found in materials research more broadly. Currently, the development of new materials requires expensive and time consuming synthesis and characterization loops, with many synthesis experiments leading to dead ends and/or yielding little useful information. Therefore, computational screening and lead generation is critical to speeding up the pace of materials development. Traditionally screening has been done using either ad-hoc rules of thumb, which are usually limited in their domain of applicability, or by running large numbers of expensive quantum chemistry calculations which require significant supercomputing time. Machine learning (ML) from data holds the promise of allowing for rapid screening of materials at much lower computational cost. A properly trained ML model can make useful predictions about the properties of a candidate material in milliseconds rather than hours or days4. Recently, machine learning has been shown to accelerate the discovery of new materials for dielectric polymers5, OLED displays6, and polymeric dispersants7. In the realm of molecules, ML has been applied successfully to the prediction of atomization energies8, bond energies9, dielectric breakdown strength in polymers10, critical point properties of molecular liquids11, and exciton dynamics in photosynthetic complexes12. In the materials science realm, ML has recently yielded predictions for dielectric polymers5,10, superconducting materials13, nickel-based superalloys14, elpasolite crystals15, perovskites16, nanostructures17, Heusler alloys18, and the thermodynamic stabilities of half-Heusler compounds19. In the pharmaceutical realm the use of ML has a longer history than in other fields of materials development, having first been used under the moniker of quantitative structure-property relationships (QSPR). It has been applied recently to predict properties of potential drug molecules such as rates of absorption, distribution, metabolism, and excretion (ADME)20, toxicity21, carcinogenicity22, solubility, and binding affinity23. As a general means of developing models for relationships that have no known analytic forms, machine learning holds great promise for broad application. However, the evidence to date suggest machine learning models require data of sufficient quality, quantity, and diversity which ostensibly limits the application to fields in which datasets are large, organized, and/or plentiful. Published studies to date use relatively large datasets, with N = 10,000–100,000 being typical for molecular studies and larger datasets appearing in the realm of materials. When sufficiently large datasets are available, very impressive results can be obtained. For example, recently it has been shown that with enough data (N = 117,00024 or N = 435,00025) machine learning can reproduce properties calculated from DFT with smaller deviations from DFT values than DFT’s deviation from experiment24,25. Compared to other areas of materials research, the application of ML methods to energetic materials has received relatively little attention likely due to the scarcity of quality energetics data. Thus energetics may be a suitable platform to consider the factors that effect machine learning performance when limited to small data. While machine learning has been applied to impact sensitivity26,27,28,29, there is little or no previously published work applying ML to predict energetic properties such as explosive energy, detonation velocity, and detonation pressure. While there is previous work applying ML to heat of formation30,31 and detonation velocity & pressure32,33,34,35, the studies are restricted to energetic materials in narrow domains, such as series of molecules with a common backbone. Furthermore, the aforementioned studies have been limited to the use of hand picked descriptors combined with linear modeling. Therefore, in this work we wish to challenge the assumption that large data sets are necessary for ML to be useful by doing the first comprehensive comparison of ML methods to energetics data. We do this using a dataset of 109 energetic compounds computed by Huang & Massa36. While we later introduce additional data from Mathieu37 for most of our work we restrict our study to the Huang & Massa data to demonstrate for the first time how well different ML models & featurizations work with small data. The Huang & Massa data contains molecules from ten distinct compound classes and models trained on it should be relatively general in their applicability. The diverse nature of the Huang & Massa data is important as ultimately we wish our models to be applicable to wide range of candidate energetic molecules. To obtain the energetic properties in their dataset, Huang & Massa calculated gas phase heats of formation using density functional theory calculations at the B3LYP/6-31 G(d,p) level, and calculated the heat of sublimation using an empirical packing energy formula38. These two properties were used to calculate the heat of formation of the solid ${\rm{\Delta }}{H}_{f}^{{\rm{solid}}}$. They obtained densities primarily from experimental crystallographic databases. Huang & Massa then used their calculated heats of formation and densities as inputs to a thermochemistry code which calculates energetic properties under the Chapman-Jouguet theory of detonation. They validated the accuracy of their method with experimental data38. The result is a dataset with nine properties - density, heat of formation of the solid, explosive energy, shock velocity, particle velocity, sound velocity, detonation pressure, detonation temperature, and TNT equivalent per cubic centimeter. Several of these properties have significant correlations between them (a correlation matrix plot is given in Supplementary Fig. S1). Featurization Methods In the case of small data featurization is more critical than selecting a model, as with larger data models can learn to extract complex and/or latent features from a general purpose (materials agnostic) featurization. Feature vectors must be of reasonable dimensionality $d\ll N$, where d is the number of dimensions of the feature vector and N is the number of molecules in the training set, to avoid the curse of dimensionality and the so-called “peaking phenomena”39. Some featurizations make chemically useful information more transparent than others. More abstract general purpose featurizations, such as SMILES strings & Coulomb matrices (discussed below) only excel with large data and deep learning models, which can learn to extract the useful information. With small data, great gains in accuracy can sometimes be gained by hand selecting features using chemical intuition and domain expertise. For example, the number of azide groups in a molecule is known to increase energetic performance while also making the energetic material more sensitive to shock. While the number of azide groups is implicitly contained in the SMILES string and Coulomb matrix for the molecule, ML models such as neural networks typically need a lot of data to learn how to extract that information. To ensure that azide groups are being used by the model to make predictions with small data, an explicit feature corresponding to the number of such groups can be put in the feature vector. Oxygen balance It is well known that the energy released during an explosion is largely due to the reaction of oxygen with the fuel atoms carbon and hydrogen. Based on this fact, Martin & Yallop (1957) found a linear relationship between detonation velocity and a descriptor called oxygen balance40. It can be defined either of two ways: $$\,{{\rm{OB}}}_{1600}\equiv \frac{1600}{{m}_{{\rm{mol}}}}({n}_{{\rm{O}}}-2{n}_{{\rm{C}}}-{n}_{{\rm{H}}}/2)\,{\rm{OB}}{}_{100}\equiv \frac{100}{{n}_{{\rm{atoms}}}}({n}_{{\rm{O}}}-2{n}_{{\rm{C}}}-{n}_{{\rm{H}}}/2)$$ here nC, nH, and nO are the number of carbons, hydrogens, and oxygens respectively, mmol is the molecular weight, and natoms is the number of atoms. An oxygen balance close to zero is sometimes used as requirement in determining if material may be useful as a novel energetic41. While it is easy to calculate and provides a useful rule of thumb, oxygen balance has limitations, which will become clear when compared to more sophisticated featurizations. One limitation of oxygen balance is that it neglects the large variation in bond strengths found in different molecules. It also neglects additional sources of energy released in an explosion, such as from nitrogen recombination (formation of N2), halogen reactions, and the release of strain energy. Furthermore, oxygen balance is built on the assumption that oxygen reacts completely to form CO2 and H2O. More sophisticated calculations take into account the formation of CO (which commonly occurs at high temperatures or in low-oxygen compounds) as well as H2 and trace amounts of unreacted O2 and solid carbon which may appear. Predicting the proportion of such products requires complex thermochemical codes. Custom descriptor set By a “descriptor” we mean any function that maps a molecule to a scalar value. There are many types of descriptors, ranging from simple atom counts to complex quantum mechanical descriptors that describe electron charge distributions and other subtle properties. There are many different algorithms for generating a descriptor set of a specified size from a large pool. Since a brute force combinatorial search for the best set is often prohibitively expensive computationally, usually approximate methods are employed such as statistical tests, greedy forward elimination, greedy backwards elimination, genetic algorithms, etc. For instance, Fayet et al. used Student’s t-test to rank and filter descriptors from a pool of 300 possible ones30. A pitfall common to the t-test and other statistical tests is that two descriptors that have low ranks on their own may be very useful when combined (eg. through multiplication, addition, subtraction, division). Given many of the difficulties in descriptor set selection, Guyon and Elisseeff recommend incorporating physical intuition and domain knowledge whenever possible42. We chose to design our descriptor set based on physical intuition and computational efficiency (we ignore descriptors which require physics computations). The first descriptor we chose was oxygen balance. Next we included the nitrogen/carbon ratio (n N /n C ), a well known predictor of energetic performance43. Substituting nitrogens for carbon generally increases performance, since N=N bonds yield a larger heat of formation/enthalpy change during detonation compared to C-N and C=N bonds43. In addition to raw nitrogen content, the way the nitrogen is incorporated into the molecule is important. For instance, the substitution of N in place of C-H has the extra effect of increasing crystal density. Nitrogen catenation, both in N=N and N-N≡N (azide) is known to greatly increase performance but also increase sensitivity. On the other hand, nitrogen in the form of amino groups (NH2) is known to decrease performance but also decrease sensitivity43. The way oxygen is incorporated into a molecule is similarly important - oxygen that is contained in nitro groups (NO2) release much more energy during detonation than oxygen that is already bonded to a fuel atom. Martin & Yallop (1958) distinguish four oxygen types that are present in energetic materials40. Based on all of this, we distinguished different types of nitrogen, oxygen, and flourine based on their bond configurations: $$\begin{array}{cc}{N-N-O}_{2}({\rm{nitrogen}}\,{\rm{nitro}}\,{\rm{group}}) & {\rm{C}}=N-O({\rm{fulminate}}\,{\rm{group}})\\ {C-N-O}_{2}({\rm{carbon}}\,{\rm{nitro}}\,{\rm{group}}) & C-N={\rm{N}}({\rm{azo}}\,{\rm{group}})\\ {O-N-O}_{2}({\rm{oxygen}}\,{\rm{nitro}}\,{\rm{group}}) & {C-N-H}_{2}({\rm{amine}}\,{\rm{group}})\\ O-N={\rm{O}}({\rm{nitrite}}\,{\rm{group}}) & C-N(-O)-C(N-\text{oxide}\,{\rm{nitrogen}})\\ {\rm{C}}=N-F(\text{nitrogen}-\text{flourine}\,{\rm{group}}) & C-F(\text{carbon}-\text{flourine}\,{\rm{group}})\\ C-O-H({\rm{hydroxyl}}\,{\rm{oxygen}}) & {\rm{N}}={\rm{O}}({\rm{nitrate}}\,{\rm{or}}\,{\rm{nitrite}}\,{\rm{oxygen}}),\\ N-O-C(N-\text{oxide}\,{\rm{oxygen}}) & {\rm{C}}={\rm{O}}(\text{keton}/\text{carboxyl}\,{\rm{oxygen}})\end{array}$$ We also included raw counts of carbon and nitrogen, hydrogen, and fluorine. We tested using ratios instead of raw counts (n x /ntotal) but found this did not improve the performance of the featurization. All together, our custom descriptor set feature vector is: $$\begin{array}{ll}{{\boldsymbol{x}}}_{{\rm{CDS}}}= & [\,{{\rm{OB}}}_{100},{n}_{{\rm{N}}}/{n}_{{\rm{C}}},{n}_{{\rm{NNO}}2},{n}_{{\rm{CNO}}2},{n}_{{\rm{ONO}}2},{n}_{{\rm{ONO}}},{n}_{{\rm{CNO}}},{n}_{{\rm{CNN}}},{n}_{{\rm{NNN}}},{n}_{{\rm{CNH}}2},\\ & {n}_{{\rm{CN}}({\rm{O}}){\rm{C}}},{n}_{{\rm{CNF}}},{n}_{{\rm{CF}}},{n}_{{\rm{C}}},{n}_{{\rm{N}}},{n}_{{\rm{NO}}},{n}_{{\rm{COH}}},{n}_{{\rm{NOC}}},{n}_{{\rm{CO}}},{n}_{{\rm{H}}},{n}_{{\rm{F}}},\,]\end{array}$$ Sum over bonds Based on the intuition that almost all of the latent heat energy is stored in chemical bonds, we introduce a bond counts feature vector. The bond counts vectors are generated by first enumerating all of the bond types in the dataset and then counting how many of each bond are present in each molecule. There are 20 bond types in the Huang & Massa dataset: N-O, N:O, N-N, N=O, N=N, N:N, N#N, C-N, C-C, C-H, C:N, C:C, C-F, C-O, C=O, C=N, C=C, H-O, H-N, F-N. We use the SMARTS nomenclature for bond primitives (‘-’ for single bond, ‘=’ for double bond, ‘#’ for triple bond, and ‘:’ for aromatic bond). Following Hansen, et al.44 we call the resulting feature vector “sum over bonds”. Coulomb matrices An alternative way of representing a molecule is by the Coulomb matrix featurization introduced by Rupp et al.8,45. The Coulomb matrix finds its inspiration in the fact that (in principle) molecular properties can be calculated from the Schrödinger equation, which takes the Hamiltonian operator as its input. While extensions of the Coulomb matrix for crystals have been developed46, it is designed for treating gas phase molecules. The Hamiltonian operator for an isolated molecule can be uniquely specified by the nuclear coordinates R i and nuclear charges Z i . Likewise, the Coulomb matrix is completely specified by {R i , Z i }. The entries of the Coulomb matrix M for a given molecule are computed as: $${{\bf{M}}}_{ij}=\{\begin{array}{cc}0.5{Z}_{i}^{2.4}, & i=j\\ \frac{{Z}_{i}{Z}_{j}}{\|{{\bf{R}}}_{i}-{{\bf{R}}}_{j}\|} & i\ne j\end{array},$$ The diagonal elements of M correspond to a polynomial fit of the potential energies of isolated atoms, while the off-diagonals elements correspond to the energy of Coulombic repulsion between different pairs of nuclei in the molecule. It is clear by construction that M is invariant under translations and rotations of the molecule. However, Coulomb matrices are not invariant under random permutations of the atom’s indices. To avoid this issue, one can use the eigenvalue spectrum of the Coulomb matrix since the eigenvalues of a matrix are invariant under permutation of columns or rows. In this approach, the Coulomb matrix is replaced by a feature vector of the eigenvalues, $\langle {\lambda }_{1},\cdots ,{\lambda }_{d}\rangle $, sorted in a descending order, as illustrated in Fig. 1. Figure 1 figure 1 Transformation of (x, y, z) coordinates and nuclear charges to the Coulomb matrix eigenvalue spectra representation of a molecule. To obtain fixed length feature vectors, the size is of the vectors is set at the number of atoms in the largest molecule in the dataset d = dmax and the feature vectors for molecules with number of atoms less than dmax are padded with zeros. Using the eigenvalues implies a loss of information from the full matrix. Given this fact, we also compare with the “raw” Coulomb matrix. Since it is symmetric, we take only the elements from the diagonal and upper triangular part of the matrix and then put them into a feature vector (which we call “Coulomb matrices as vec”). The “raw” Coulomb matrix is quite a bit different than other feature vectors as the physical meaning of each specific element in the feature vector differs between molecules. This appears to be problematic, especially for kernel ridge regression (KRR) and the other variants on linear regression, which treat each element separately and for which it is usually assumed that each element has the same meaning in every sample. Bag of bonds The bag of bonds featurization was introduced by Hansen et al. in 201544. It is inspired by the “bag of words” featurization used in natural language processing. In bag of words, a body of text is featurized into a histogram vector where each element, called a “bag”, counts the number of times a particular word appears. Bag of bonds follows a similar approach by having “bags” that correspond to different types of bonds (such as C-O, C-H, etc). Bonds are distinguished by the atoms involved and the order of the bond (single, double, triple). However bag of bonds differs from bag of words in several crucial ways. First, each “bag” is actually a vector where each element is computed as ${Z}_{i}{Z}_{j}/\|{{\bf{R}}}_{{\bf{i}}}-{{\boldsymbol{R}}}_{{\bf{j}}}\|$. The bag vectors between molecules are enforced to have a fixed length by padding them with zeros. The entries in each bag vector are sorted by magnitude from highest to lowest to ensure a unique representation. Finally, all of the bag vectors are concatenated into a final feature vector. Molecular fingerprints were originally created to solve the problem of identifying isomers47, and later found to be useful for rapid substructure searching and the calculation of molecular similarity in large molecular databases. In the past two decades, fingerprints have been used as an alternative to descriptors for QSPR studies. Fingerprinting algorithms transform the molecular graph into a vector populated with bits or integers. In this work we compare several fingerprints found in RDKit, a popular cheminformatics package–Atom-Pair48, Topological Torsion49, Extended Connectivity Fingerprints (ECFPs)50, E-state fingerprints51, Avalon fingerprints52, RDKit graph fingerprints53, ErG fingerprints54, and physiochemical property fingerprints55. A few general features of fingerprints can be noted. First of all, all the fingerprinting algorithms start with atom level descriptors, each of which encode atom-specific information into an integer or real number. The simplest atom descriptor is the atomic number, but in most fingerprints this is augmented with additional information that describes the local environment of the atom. A second general point is that some fingerprinting algorithms can generate either integer vectors or bit vectors. In extended connectivity fingerprints (ECFPs) for instance, count information on the number of times a feature/fragment appears is lost when bit vectors are used. In RDKit, fingerprints return bit vectors by default. The length of bit vector based fingerprint representations can be tuned (to avoid the curse of dimensionality) through a process called folding. Another point is that the fingerprints we study contain only 2D graph information, and do not contain information about 3D structure/conformation. Comparisons between 2D fingerprints and analogous 3D fingerprints have found that 3D fingerprints generally do not yield superior performance for similarity searching56, or binding target prediction57, although they are useful for certain pharmaceutical applications58. Other featurizations Two other featurizations that have been developed for molecules are smooth overlap of atomic positions (SOAP)59,60, and Fourier series of atomic radial distribution functions61. We choose not to investigate these featurizations due to their poorer performance in past comparisons. Another featurization strategy, the random walk graph kernel, may be promising for future exploration but is computationally expensive to properly implement62. Very recently, several custom deep learning architectures with built-in featurization have been developed - custom graph convolutional fingerprints63,64, deep tensor networks65, message passing neural networks4, and hierarchically interacting particle neural nets66. We attempted to train a graph convolutional fingerprint using the “neural fingerprint” code of Duvenaud et al.63 but were not able achieve accuracy that was competitive with any of the other featurizations (for example the best we achieved for shock velocity was a MAE of ≈0.35 km/s as opposed to 0.30 km/s for sum over bonds + KRR). Additional data and hyperparameter optimization would likely improve the competitiveness of custom convolutional fingerprinting. Comparison of fingerprints Figure 2 shows fingerprint mean absolute error of explosive energy in out-of-sample test data using kernel ridge regression vs. the fingerprint length in bits. The E-state fingerprint, which has a fixed length, performs the best, with the Avalon fingerprint nearly matching it. The Estate fingerprint is based on electrotopological state (E-state) indices67, which encode information about associated functional group, graph topology, and the Kier-Hall electronegativity of each atom. E-state indices were previously found to be useful for predicting the sensitivity (h50 values) of energetic materials with a neural network68. The E-state fingerprint differs from all of the other fingerprints we tested as it is fixed length, containing a vector with counts of 79 E-state atom types. Thus, technically it is more akin to a descriptor set than a fingerprint. We found that only 32 of the atom types were present in the energetic materials we studied (types of C,N,H,O,F), so we truncated it to a length of 32 (ie. threw out the elements that were always zero). The E-state fingerprint can also be calculated as a real-valued vector which sums the E-state indices for each atom, however we found the predictive performance was exactly the same as with the count vector version. It is not immediately clear why E-state performs better than the other fingerprints, but we hypothesize that it is due to the fact that the E-state atom types are differentiated by valence state and bonding pattern, including specifically whether a bond to a hydrogen is present. Thus a NH2 nitrogen is considered a specific atom type. In the atom-pair, topological torsion, Morgan, and RDKit topological fingerprint the atom types are differentiated soley by atomic number, number of heavy atom neighbors, and number of pi electrons. The Avalon fingerprint, which does almost as well as E-state, likely does so because it contains a large amount of information including atom pairs, atom triplets, graph topology, and bonding configurations. Figure 2 figure 2 Mean errors in explosive energy obtained with different fingerprinting methods at different fingerprint lengths, using Bayesian ridge regression and leave-5-out cross validation. The E-state fingerprint has a fixed length and therefore appears as a flat line. Comparison of featurizations Table 1 shows a comparison of the featurization schemes discussed in the previous sections, using kernel ridge regression with hyperparameter optimization performed separately via grid search with cross validation for each featurization. Hyperparameter optimization was done for all hyperparameters - the regularization parameter α, kernel width parameter and kernel type (Laplacian/L1 vs Gaussian/L2). Fairly dramatic differences are observed in the usefulness of different featurizations. The sum over bonds featurization always performs the best, but additional gains can be made by concatenating featurizations. The gains from concatenation are especially reflected in the correlation coefficient r, which increases from 0.65 to 0.78 after E-state and the custom descriptor set is concatenated with sum over bonds. As expected, the two different oxygen balance formulae perform nearly the same, and Coulomb matrix eigenvalues perform better than the raw Coulomb matrix. Summed bag of bonds (summing each bag vector) performs nearly as well as traditional bag of bonds, but with a much more efficient representation $({\boldsymbol{x}}\in {{\mathbb{R}}}^{20}{\rm{vs}}\,{\boldsymbol{x}}\in {{\mathbb{R}}}^{2527})$. Table 1 Detailed comparison of 13 different featurization schemes for prediction of explosive energy with kernel ridge regression, ranked by MAEtest. These variance in MAEs between folds was less than 0.01 in all cases. Hyperparameter optimization was used throughout with nested 5-fold cross validation. The metrics are averaged over 20 train-test sets using shuffle split with 80/20 splitting. Comparison of machine learning models Table 2 presents a comparison of five different ML models & seven featurization methods for each target property in the Huang & Massa dataset (a complete comparison with 5 additional models and additional evaluation metrics can be found in Supplementary Tables S1, S2, S3, and S4 and in Supplementary Fig. S2). The mean average error was averaged over 20 random train-test splits (with replacement) with a train/test ratio of 4:1. We also calculated 95% confidence intervals (shown in Supplementary Table S) which show the values are well converged and meaningful comparisons can be made. Hyperparameter optimization was performed on all models using grid search. We found that careful hyperparameter optimization, especially of the regularization parameter, is critical to properly comparing and evaluating models. We found that LASSO regression and Bayesian ridge regression performed nearly as well as ridge regression, and gradient boosted trees performed nearly as well as random forest, so we omitted them from the table. Two key observations can be made. The first is that the sum over bonds featurization is the best for all target properties with the exception of the speed of sound, where bag of bonds does slightly better. The second is that kernel ridge and ridge regression performed best. Other models might become competitive with the addition of more data. The gap between the train and test MAEs indicates overfitting is present. Table 2 Average mean absolute errors (MAEs) in the test sets for different combinations of target property, model and featurization. Hyperparameter optimization was used throughout with nested 5-fold cross validation. The test MAEs are averaged over 20 test sets using shuffle split with 80/20 splitting. The properties are density, heat of formation of the solid, explosive energy, shock velocity, particle velocity, sound velocity, detonation pressure, detonation temperature, and TNT equivalent per cubic centimeter. The models are kernel ridge regression (KRR), ridge regression (Ridge), support vector regression (SVR), random forest (RF), k-nearest neighbors (kNN), and a take-the-mean dummy predictor. The last row gives the average value for each property in the dataset. Incorporating dimensionality reduction Dimensionality reduction can often improve model performance, especially when nfeatures ≈ nexamples39. Our first test was done with Coulomb matrices, where we found the error converged with D = 15 or more principle components. A similar convergence at D = 15 was observed for E-state and for the combined featurization Estate + SoB + CDS. We also experimented with some other dimensionality reduction techniques such as t-SNE, PCA followed by fast independent component analysis (ICA), and spectral embedding (Laplacian eigenmaps). In all cases, however, the error was not improved by dimensionality reduction (see supplementary information for more detail). This indicates that while our feature vectors could be compressed without loosing accuracy, the regularization in our models is capable of handling the full dimensionality of our feature vectors without loss in performance. Analysis and Discussion Machine learning performance vs. training data quantity and diversity To compare our diverse-data case with how machine learning works in a narrow, non-diverse context we fit the dataset of Ravi et al. which contains 25 pyrazole-based molecules (Table 3). Ravi et al. performed DFT calculations to obtain the heat of formation with respect to products and then used the Kamlet-Jacobs equations to predict detonation velocity and detonation pressure. Table 3 shows how very high accuracy in predicting energetic properties is achieved. Not surprisingly, the custom descriptor set slightly outperforms other featurizations here, since the only thing that is changing within the dataset is the type and number of functional groups attached to the pyrazole backbone. While accurate, the trained models will not generalize beyond the class of molecules they were trained since they cannot capture significant differences between classes35. Similarly, insights gained from the interpretation of such models may not generalize and may be contaminated by spurious correlations that often occur in small datasets. Table 3 Mean absolute errors and Pearson correlation coefficients for ML on the dataset of Ravi et al.34, which contains 25 nitropyrazole molecules. 5-fold cross validation was used, so Ntrain = 20 and Ntest = 5. The part of chemical space where a model yields accurate predictions is known as the applicability domain69. The concept of applicability domain is inherently subjective as it depends on a choice of accuracy one considers acceptable. Mapping the applicability domain of a model by generating additional test data is typically expensive given the high dimensionality of chemical space. There are several heuristics for estimating an outer limit to the applicability domain of a model, such as using the convex hull of the training data, a principle components based bounding box, or a cutoff based on statistical leverage69. While useful, applicability domains determined from such methods all suffer from the possibility that there may be holes inside where the model performs badly. To spot holes, distance based methods can be used, which average the distance to the k nearest neighbors in the training data under a given metric and apply a cutoff. There is no universal way of determining a good cutoff, however, so determining a practical cutoff requires trial and error. Residual analysis Useful insights into the applicability of our models can be gained by looking at a model’s residuals (ypred – ytrue) in the test set. We chose to look at residuals in leave-one-out cross validation using the sum over bonds featurization and kernel ridge regression (Fig. 3). We suspected that our model may have difficulty predicting the explosive energy for cubane-derived molecules, since they release a large amount of strain energy in explosion - something which is not significant in other molecules except for the CL20 group, and which is not explicitly contained in any of our featurizations. In fact, the three worst performing molecules were heptanitrocubane, the linear molecule QQQBRD02, and cubane. The model overestimates the energy of cubane and underestimates the explosive energy of nitrocubane. The mean absolute errors and average residuals of different groups in leave-one-out CV are shown in Table 4. In order to visualize the distribution of data in high dimensional space one can utilize various embedding techniques that can embed high dimensional data into two dimensions. Often one finds that high dimensional data lies close to a lower dimensional manifold. The embeddings shown in the supplementary information (t-SNE, PCA, & spectral) show that the cubane molecules are quite separated from the rest of the data (Supplementary Information Fig. S3). Based on this observation, we tried removing the cubanes. We found that the accuracy of prediction of explosive energy with kernel ridge regression and sum over bonds remained constant (MAE = 0.36 kJ/cc) while the r value actually decreased significantly from 0.76 to 0.68. Figure 3 figure 3 Residuals in leave-one-out cross validation with kernel ridge regression and the sum over bonds featurization (left), and some of the worst performing molecules (right). Table 4 The mean absolute error, Pearson correlation and average residual in different groups, for prediction of explosive energy (kJ/cc) with leave-one-out CV on the entire dataset using sum over bonds and kernel ridge regression. The groups are sorted by r value rather than MAE since the average explosive energy differs significantly between groups. Learning curves Insights into the data-dependence of ML modeling can be obtained from plots of cross-validated test error vs number of training examples, which are known as learning curves. While we were able to obtain good learning curves from just the Huang & Massa dataset, to ensure their accuracy we supplemented the Huang & Massa data with 309 additional molecules from the dataset given in the supplementary information of Mathieu et al., which includes detonation velocity and detonation pressure values calculated from the Kamlet-Jacobs equations37. A few of the molecules are found in both datasets, but most are unique, yielding a total of ≈400 unique molecules. We assume detonation velocity is equivalent to the shock velocity found in the Huang & Massa data. The method of calculation of properties differs between the two datasets, possibly introducing differing bias between the sets, so we shuffle the data beforehand. Figure 4 shows the learning curves for detonation velocity and detonation pressure. The gap between training and test score curves indicates error from overfitting (variance) while the height of the curves indicates the degree of error from the choice of model (bias). As the quantity of data increases, the gap between the training and test curves should decrease. The flattening out of the training accuracy curve indicates a floor in accuracy for the given choice of model & featurization. Even with much more data, it is very likely that creating predictions more accurate than such a floor would require better choices of model and featurization. Empirically, learning curves are known to have a $A{N}_{{\rm{train}}}^{-\beta }$ dependence70. In the training of neural networks typically $1 < \beta < 2$71, while we found values between 0.15 − 0.30 for the properties studied. Since different types of models can have different learning curves, we also looked at random forest, where we found similar plots with β ≈ 0.20. Figure 4 figure 4 The learning curves for predicting detonation velocity (left) and detonation pressure (right) for the combined (N = 418) dataset plotted on a log-log plot. Shaded regions show the standard deviation of the error in 5-fold cross validation. Conclusion and Future Directions We have presented evidence that machine learning can be used to predict energetic properties out of sample after being trained on a small yet diverse set of training examples (Ntrain = 87, Ntest = 22). For all the properties tested, either kernel ridge or ridge regression were found to have the highest accuracy, and out of the base featurizations we compared the sum over bonds featurization performed best. Small improvements could be gleaned by concatenating featurizations. The best r values achieved were 0.94 for heat of formation, 0.74 for density, 0.79 for explosive energy, and 0.78 for shock velocity. With kernel ridge regression and sum over bonds we obtained mean percentage errors of 11% for explosive energy, 4% for density, 4% for detonation velocity and 11% for detonation pressure. By including ≈300 additional molecules in our training we showed how the mean absolute errors can be pushed lower, although the convergence with number of molecules is slow. There are many possible future avenues of research in the application of machine learning methods to the discovery of new energetic materials. One possible way to overcome limitations of small data is with transfer learning23,72. In transfer learning, a model is first trained for a task where large amounts of data is available, and then the model’s internal representation serves as a starting point for a prediction task with small data. Another way our modeling might be improved without the need for more data is by including in the training set CNOHF type molecules that are non-energetic, which may require overcoming sampling bias by reweighting the energetic molecules. Additional work we have undertaken investigates how our models can be interpreted to illuminate structure-property relationships which may be useful for guiding the design of new energetic molecules73. Finally, a promising future direction of research involves coupling the property predicting models developed here with generative models such as variational autoencoders or generative adversarial networks to allow for molecular generation and optimization. We used in-house Python code for featurization and the scikit-learn package ( for machine learning. Parts of our code has been used to establish a library we call the Molecular Machine Learning (MML) toolkit. The MML toolkit is open source and available online at Model evaluation There are many different ways to evaluate model performance. The simplest scoring function is the mean absolute error (MAE): $$\,{\rm{MAE}}=\frac{1}{N}\sum _{i}^{N}\|{y}_{i}^{{\rm{true}}}-{y}_{i}^{{\rm{pred}}}\|$$ A related scoring function is the root mean squared error (RMSE), also called the standard error of prediction (SEP), which is more sensitive to outliers than MAE: $$\,{\rm{RMSE}}=\sqrt{\frac{1}{N}\sum _{i}{({y}_{i}^{{\rm{true}}}-{y}_{i}^{{\rm{pred}}})}^{2}}$$ We also report the mean average percent error (MAPE), although this score is often misleadingly inflated when ${y}_{i}^{{\rm{true}}}$ is small: $$\,{\rm{MAPE}}=\frac{100}{N}\sum _{i}^{N}\|\frac{{y}_{i}^{{\rm{true}}}-{y}_{i}^{{\rm{pred}}}}{{y}_{i}^{{\rm{true}}}}\|$$ It is also important to look at the Pearson correlation coefficient r: $$r=\frac{\sum _{i}({y}_{i}^{{\rm{true}}}-{\bar{y}}_{i}^{{\rm{true}}})({y}_{i}^{{\rm{pred}}}-{\bar{y}}_{i}^{{\rm{pred}}})}{\sqrt{\sum _{i}{({y}_{i}^{{\rm{true}}}-{\bar{y}}_{i}^{{\rm{true}}})}^{2}\sum _{i}{({y}_{i}^{{\rm{pred}}}-{\bar{y}}_{i}^{{\rm{pred}}})}^{2}}}$$ Additionally, we report the coefficient of determination: $${R}^{2}=1-\frac{\sum _{i}{({y}_{i}^{{\rm{true}}}-{y}_{i}^{{\rm{pred}}})}^{2}}{\sum _{i}{({y}_{i}^{{\rm{true}}}-{\bar{y}}_{i}^{{\rm{true}}})}^{2}}$$ Unlike the Pearson correlation coefficient r, R2 can assume any value between −∞ and 1. When reported for test or validation data, this scoring function is often called Q2. While a bad Q2 indicates a bad model, a good Q2 does not necessarily indicate a good model, and models should not be evaluated using Q2 in isolation74. Data gathering With a the exception of the Coulomb-matrix and bag-of-bonds featurizations, the only input required for our machine learning featurizations is the molecular connectivity graph. For easy encoding of the molecular graph we used Simplified Molecular-Input Line-Entry System (SMILES) strings75. SMILES strings are a non-unique representation which encode the molecular graph into a string of ASCII characters. SMILES strings for 56 of the Huang & Massa molecules were obtained from the Cambridge Structure Database Python API76, and the rest were generated by hand using a combination of the Optical Structure Recognition Application77, the molecule builder, and the Open Babel package78, which can convert.mol files to SMILES. Since Coulomb matrices require atomic coordinates, 3D coordinates for the molecules were generated using 2D → 3D structure generation routines in the RDKit53 and Open Babel78 python packages. After importing the SMILES into RDKit and adding hydrogens to them, an embedding into three dimensions was done using distance geometry, followed by a conjugate gradient energy optimization. Next, a weighted rotor search with short conjugate gradient minimizations was performed using Open Babel to find the lowest energy conformer. Finally, a longer conjugate gradient optimization was done on the lowest energy conformer. For one molecule (the cubane variant ‘EAT07’) we used the obgen utility program of Open Babel to do the coordinate embedding (in retrospect, obgen could have been used to yield good enough results for all molecules). All energy minimizations were done with the MMFF94 forcefield79. The accuracy of the generated structures was verified by visually comparing the generated structures of 56 of the compounds to x-ray crystallographic coordinates obtained from the Cambridge Structure Database. Data Availability The compiled property data from Huang & Massa (2010) and Mathieu (2017) and the generated SMILES strings are included in the Supplementary Information. The full Mathieu (2017) dataset is available free of charge on the ACS Publications website at : DOI:10.1021/acs.iecr.7b02021 Python Jupyter notebooks for reproducing all of our results have been open sourced and are available at The Python notebooks use the open source Molecular Machine Learning toolkit developed in conjunction with this work, which can be found at
6d9128fee16a2784	Toward quantum-like modeling of financial processes Olga Choustova International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of Växjö, S-35195, Sweden We apply methods of quantum mechanics for mathematical modeling of price dynamics at the financial market. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. Trajectories of prices are determined by two financial potentials: classical-like (”hard” market conditions, e.g., natural resources) and quantum-like (behavioral market conditions). On the one hand, our Bohmian model is a quantum-like model for the financial market, cf. with works of W. Segal, I. E. Segal, E. Haven, E. W. Piotrowski, J. Sladkowski. On the other hand, (since Bohmian mechanics provides the possibility to describe individual price trajectories) it belongs to the domain of extended research on deterministic dynamics for financial assets (C.W. J. Granger, W.A. Barnett, A. J. Benhabib, W.A. Brock, C. Sayers, J. Y. Campbell, A. W. Lo, A. C. MacKinlay, A. Serletis, S. Kuchta, M. Frank, R. Gencay, T. Stengos, M. J. Hinich, D. Patterson, D. A. Hsieh, D. T. Caplan, J.A. Scheinkman, B. LeBaron and many others). 1 Deterministic and Stochastic Models for Financial Market 1.1 Stochastic models and the efficient market hypothesis In economics and financial theory, analysts use random walk and more general martingale techniques to model behavior of asset prices, in particular share prices on stock markets, currency exchange rates and commodity prices. This practice has its basis in the presumption that investors act rationally and without bias, and that at any moment they estimate the value of an asset based on future expectations. Under these conditions, all existing information affects the price, which changes only when new information comes out. By definition, new information appears randomly and influences the asset price randomly. Corresponding continuous time models are based on stochastic processes (this approach was initiated in the thesis of L. Bachelier [1] in 1890), see, e.g., the books of R. N. Mantegna and H. E. Stanley [45] and A. Shiryaev [3] for historical and mathematical details. This practice was formalized through the efficient market hypothesis which was formulated in the sixties, see P. A. Samuelson [4] and E. F. Fama [5] for details: A market is said to be efficient in the determination of the most rational price if all the available information is instantly processed when it reaches the market and it is immediately reflected in a new value of prices of the assets traded. Mathematically the efficient market hypothesis was supported by investigations of Samuelson [4]. Using the hypothesis of rational behavior and market efficiency he was able to demonstrate how the expected value of price of a given asset at time is related to the previous values of prices through the relation Typically there is introduced the -algebra generated by random variables The condition (1) is written in the form: Stochastic processes of such a type are called martingales [3]. Alternatively, the martingale model for the financial market implies that the is a “fair game” (a game which is neither in your favor nor your opponent’s): On the basis of information, which is available at the moment one cannot expect neither 1.2 Deterministic models for dynamics of prices First we remark that empirical studies have demonstrated that prices do not completely follow random walk. Low serial correlations (around 0.05) exist in the short term; and slightly stronger correlations over the longer term. Their sign and the strength depend on a variety of factors, but transaction costs and bid-ask spreads generally make it impossible to earn excess returns. Interestingly, researchers have found that some of the biggest prices deviations from random walk result from seasonal and temporal patterns, see the book [45]. There are also a variety of arguments, both theoretical and obtained on the basis of statistical analysis of data, which question the general martingale model (and hence the efficient market hypothesis), see, e.g., [6][13]. It is important to note that efficient markets imply there are no exploitable profit opportunities. If this is true then trading on the stock market is a game of chance and not of any skill, but traders buy assets they think are unevaluated at the hope of selling them at their true price for a profit. If market prices already reflect all information available, then where does the trader draw this privileged information from? Since there are thousands of very well informed, well educated asset traders, backed by many data researchers, buying and selling securities quickly, logically assets markets should be very efficient and profit opportunities should be minimal. On the other hand, we see that there are many traders whom successfully use their opportunities and perform continuously very successful financial operations, see the book of G. Soros [14] for discussions.111It seems that G.Soros is sure he does not work at efficient markets. There were also performed intensive investigations on testing that the real financial data can be really described by the martingale model, see [6][13]. Roughly speaking people try to understand on the basis of available financial data: Do financial asset returns behave randomly (and hence they are unpredictable) or deterministically (and in the latter case one may hope to predict them and even to construct a deterministic dynamical system which would at least mimic dynamics of the financial market)? Predictability of financial asset returns is a broad and very active research topic and a complete survey of the vast literature is beyond the scope of this work. We shall note, however, that there is a rather general opinion that financial asset returns are predictable, see [6][13]. 1.3 Behavioral finance and behavioral economics We point out that there is no general consensus on the validity of the efficient market hypothesis. As it was pointed out in [11]: “… econometric advances and empirical evidence seem to suggest that financial asset returns are predictable to some degree. Thirty years ago this would have been tantamount to an outright rejection of market efficiency. However, modern financial economics teaches us that others, perfectly rational factors may account for such predictability. The fine structure of securities markets and frictions in trading process can generate predictability. Time-varying expected returns due to changing business conditions can generate predictability. A certain degree of predictability may be necessary to reward investors for bearing certain dynamic risks.” Therefore it would be natural to develop approaches which are not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. In particular, there are two well established (and closely related ) fields of research behavioral finance and behavioral economics which apply scientific research on human and social cognitive and emotional biases222Cognitive bias is any of a wide range of observer effects identified in cognitive science, including very basic statistical and memory errors that are common to all human beings and drastically skew the reliability of anecdotal and legal evidence. They also significantly affect the scientific method which is deliberately designed to minimize such bias from any one observer. They were first identified by Amos Tversky and Daniel Kahneman as a foundation of behavioral economics, see, e.g., [15]. Bias arises from various life, loyalty and local risk and attention concerns that are difficult to separate or codify. Tversky and Kahneman claim that they are at least partially the result of problem-solving using heuristics, including the availability heuristic and the representativeness. to better understand economic decisions and how they affect market prices, returns and the allocation of resources. The fields are primarily concerned with the rationality, or lack thereof, of economic agents. Behavioral models typically integrate insights from psychology with neo-classical economic theory. Behavioral analysis are mostly concerned with the effects of market decisions, but also those of public choice, another source of economic decisions with some similar biases. Since the 1970s, the intensive exchange of information in the world of finances has become one of the main sources determining dynamics of prices. Electronic trading (that became the most important part of the environment of the major stock exchanges) induces huge information flows between traders (including foreign exchange market). Financial contracts are performed at a new time scale that differs essentially from the old ”hard” time scale that was determined by the development of the economic basis of the financial market. Prices at which traders are willing to buy (bid quotes) or sell (ask quotes) a financial asset are not only determined by the continuous development of industry, trade, services, situation at the market of natural resources and so on. Information (mental, market-psychological) factors play a very important (and in some situations crucial) role in price dynamics. Traders performing financial operations work as a huge collective cognitive system. Roughly speaking classical-like dynamics of prices (determined) by ”hard” economic factors are permanently perturbed by additional financial forces, mental (or market-psychological) forces, see the book of J. Soros [14]. 1.4 Quantum-like model for behavioral finance In this thesis we develop a new approach that is not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. Our approach can be considered as a special econophysical [45] model in the domain of behavioral finance. In our approach information about the financial market (including expectations of agents of the financial market) is described by an information field financial wave. This field evolves deterministically perturbing the dynamics of prices of stocks and options. Dynamics is given by Schrödinger’s equation on the space of prices of shares. Since the psychology of agents of the financial market gives an important contribution into the financial wave our model can be considered as a special psycho-financial model. This thesis can be also considered as a contribution into applications of quantum mechanics outside microworld, see [16], [17], [18]. This thesis is fundamentally based on investigations of D. Bohm, B. Hiley, and P. Pylkkänen [19], [20] on the active information interpretation of Bohmian mechanics [21], [22] and its applications to cognitive sciences, see also Khrennikov [18]. In this thesis we use methods of Bohmian mechanics to simulate dynamics of prices at the financial market. We start with the development of the classical Hamiltonian formalism on the price/price-change phase space to describe the classical-like evolution of prices. This classical dynamics of prices is determined by ”hard” financial conditions (natural resources, industrial production, services and so on). These conditions as well as ”hard” relations between traders at the financial market are mathematically described by the classical financial potential. As we have already remarked, at the real financial market ”hard” conditions are not the only source of price changes. The information and market psychology play important (and sometimes determining) role in price dynamics. We propose to describe those”soft” financial factors by using the pilot wave (Bohmian) model of quantum mechanics. The theory of financial mental (or psychological) waves is used to take into account market psychology. The real trajectories of prices are determined (by the financial analogue of the second Newton law) by two financial potentials: classical-like (”hard” market conditions) and quantum-like (”soft” market conditions). Our quantum-like model of financial processes was strongly motivated by consideration by J. Soros [14] of the financial market as a complex cognitive system. Such an approach he called the theory of reflexivity. In this theory there is a large difference between market that is ”ruled” by only ”hard” economical factors and a market where mental factors play the crucial role (even changing the evolution of the ”hard” basis, see [14]). J. Soros rightly remarked that the ”non mental” market evolves due to classical random fluctuations. However, such fluctuations do not provide an adequate description of mental market. He proposed to use an analogy with quantum theory. However, it was noticed that directly quantum formalism could not be applied to the financial market [14]. Traders differ essentially from elementary particles. Elementary particles behave stochastically due to perturbation effects provided by measurement devices, cf. [24], [25]. According to J. Soros, traders at the financial market behave stochastically due to free will of individuals. Combinations of a huge number of free wills of traders produce additional stochasticity at the financial market that could not be reduced to classical random fluctuations (determined by non mental factors). Here J. Soros followed to the conventional (Heisenberg, Bohr, Dirac, see, e.g., [24], [25]) viewpoint to the origin of quantum stochasticity. However, in the Bohmian approach (that is nonconventional one) quantum statistics is induced by the action of an additional potential, quantum potential, that changes classical trajectories of elementary particles. Such an approach gives the possibility to apply quantum formalism to the financial market. We remark that applications of the pilot-wave theory to financial option pricing were considered by E. Haven in [31], see also [32]. 1.5 Review on investigations in quantum econophysics There were performed numerous investigations on applying quantum methods to financial market, see, e.g., E. Haven [26][29], that were not directly coupled to behavioral modeling, but based on the general concept that randomness of the financial market can be better described by the quantum mechanics, see, e.g., W. Segal and I. E. Segal [30]: ”A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects.” Non-Bohmian quantum models for the financial market (in particular, based on quantum games) were developed by E. W. Piotrowski, J. Sladkowski, M. Schroeder, A. Zambrzycka, see [33][42]. Some of those models can be also considered as behavioral quantum-like models. An interesting contribution to behavioral quantum-like modeling is theory of non-classical measurements in behavioral sciences (with applications to economics) which was developed by V. I. Danilov and A. Lambert-Mogiliansky [43], [44]. 2 A Brief Introduction to Bohmian Mechanics In this section we present the basic notions of Bohmian mechanics. This is a special model of quantum mechanics in that, in the opposition to the conventional Copenhagen interpretation, quantum particles (e.g., electrons) have well defined trajectories in physical space. By the conventional Copenhagen interpretation (that was created by N. Bohr and W. Heisenberg) quantum particles do not have trajectories in physical space. Bohr and Heisenberg motivated such a viewpoint to quantum physical reality by the Heisenberg uncertainty relation: where is the Planck constant, and are the position and momentum, respectively, and and are uncertainties in determination of and Nevertheless, David Bohm demonstrated [21], see also [22], that, in spite of Heisenberg’s uncertainty relation (4), it is possible to construct a quantum model in that trajectories of quantum particles are well defined. Since this thesis is devoted to mathematical models in economy and not to physics, we would not go deeper into details. We just mention that the root of the problem lies in different interpretations of Heisenberg’s uncertainty relation (4). If one interpret and as uncertainties for the position and momentum of an individual quantum particle (e.g., one concrete electron) then (4), of course implies that it is impossible to create a model in that the trajectory is well defined. On the other hand, if one interpret and as statistical deviations then there is no direct contradiction between Heisenberg’s uncertainty relation (4) and the possibility to consider trajectories. There is a place to such models as Bohmian mechanics. Finally, we remark (but without comments) that in real experiments with quantum systems, one always uses the statistical interpretation (5) of and . We emphasize that the conventional quantum formalism cannot say anything about the individual quantum particle. This formalism provides only statistical predictions on huge ensembles of particles. Thus Bohmian mechanics provides a better description of quantum reality, since there is the possibility to describe trajectories of individual particles. However, this great advantage of Bohmian mechanics was not explored so much in physics. Up to now there have not been done experiments that would distinguish predictions of Bohmian mechanics and conventional quantum mechanics. In this thesis we shall show that the mentioned advantages of Bohmian mechanics can be explored in applications to the financial market. In the latter case it is really possible to observe the trajectory of the price or price-change dynamics. Such a trajectory is described by equations of the mathematical formalism of Bohmian mechanics. We now present the detailed derivation of the equations of motion of a quantum particle in the Bohmian model of quantum mechanics. Typically in physical books it is presented very briefly. But, since this thesis is oriented to economists and mathematicians, who are not so much aware about quantum physics, we shall present all calculations. The dynamics of the wave function is described by Schrödinger’s equation Here is a complex valued function. At the moment we prefer not to discuss the conventional probabilistic interpretation of We consider as just a field.333We recall that by the probability interpretation of (which was proposed by Max Born) the quantity gives the probability to find a quantum particle at the point at the moment We consider the one-dimensional case, but the generalization to the multidimensional case, is straightforward. Let us write the wave function in the following form: where and is the argument of the complex number We put (7) into Schrödinger’s equation (6). We have and hence: We obtain the differential equations: By multiplying the right and left-hand sides of the equation (8) by and using the trivial equalities: we derive the equation for : We remark that if one uses the Born’s probabilistic interpretation of the wave function, then gives the probability. Thus the equation (10) is the equation describing the dynamics of the probability distribution (in physics it is called the continuity equation). The second equation can be written in the form: Suppose that and that the contribution of the term can be neglected. Then we obtain the equation: From the classical mechanics, we know that this is the classical Hamilton-Jacobi equation which corresponds to the dynamics of particles: where particles moves normal to the surface David Bohm proposed to interpret the equation (11) in the same way. But we see that in this equation the classical potential is perturbed by an additional ”quantum potential” Thus in the Bohmian mechanics the motion of a particle is described by the usual Newton equation, but with the force corresponding to the combination of the classical potential and the quantum one The crucial point is that the potential is by itself driven by a field equation - Schrödinger’s equation (6). Thus the equation (14) can not be considered as just the Newton classical dynamics (because the potential depends on as a field parameter). We shall call (14) the Bohm-Newton equation. We remark that typically in books on Bohmian mechanics [21], [22] it is emphasized that the equation (14) is nothing else than the ordinary Newton equation. This make impression that the Bohmian approach give the possibility to reduce quantum mechanics to ordinary classical mechanics. However, this is not the case. The equation (14) does not provide the complete description of dynamics of a systems. Since, as was pointed out, the quantum potential is determined through the wave function and the latter evolves according to the Schrödinger equation, the dynamics given by Bohm-Newton equation can not be considered independent of the Schrödinger’s dynamics. 3 Classical Econophysical Model for Financial Market 3.1 Financial phase-space Let us consider a mathematical model in that a huge number of agents of the financial market interact with one another and take into account external economic (as well as political, social and even meteorological) conditions in order to determine the price to buy or sell financial assets. We consider the trade with shares of some corporations (e.g., VOLVO, SAAB, IKEA,...).444Similar models can be developed for trade with options, see E. Haven [32] for the Bohmian financial wave model for portfolio. We consider a price system of coordinates. We enumerate corporations which did emissions of shares at the financial market under consideration: (e.g., VOLVO: SAAB: IKEA:,…). There can be introduced the -dimensional configuration space of prices, where is the price of a share of the th corporation. Here is the real line. Dynamics of prices is described by the trajectory in the configuration price space Another variable under the consideration is the price change variable: see, for example, the book [45] on the role of the price change description. In real models we consider the discrete time scale Here we should use a discrete price change variable We denote the space of price changes (price velocities) by the symbol with coordinates As in classical physics, it is useful to introduce the phase space namely the price phase space. A pair = (price, price change) is called the state of the financial market. Later we shall consider quantum-like states of the financial market. A state which we consider at the moment is a classical state. We now introduce an analogue of mass as the number of items (in our case shares) that a trader emitted to the market.555 ‘Number’ is a natural number – the price of share, e.g., in the US-dollars. However, in a mathematical model it can be convenient to consider real This can be useful for transitions from one currency to another. We call the financial mass. Thus each trader (e.g., VOLVO) has its own financial mass (the size of the emission of its shares). The total price of the emission performed by the th trader is equal to (this is nothing else than market capitalization). Of course, it depends on time: To simplify considerations, we consider a market at that any emission of shares is of the fixed size, so does not depend on time. In principle, our model can be generalized to describe a market with time-dependent financial masses, We also introduce financial energy of the market as a function If we use the analogue with classical mechanics. (Why not? In principle, there is not so much difference between motions in ”physical space” and ”price space”.), then we could consider (at least for mathematical modeling) the financial energy of the form: is the kinetic financial energy and is the potential financial energy, is the financial mass of th trader. The kinetic financial energy represents efforts of agents of financial market to change prices: higher price changes induce higher kinetic financial energies. If the corporation has higher financial mass than the corporation so then the same change of price, i.e., the same financial velocity is characterized by higher kinetic financial energy: We also remark that high kinetic financial energy characterizes rapid changes of the financial situation at market. However, the kinetic financial energy does not give the attitude of these changes. It could be rapid economic growth as well as recession. The potential financial energy describes the interactions between traders (e.g., competition between NOKIA and ERICSSON) as well as external economic conditions (e.g., the price of oil and gas) and even meteorological conditions (e.g., the weather conditions in Louisiana and Florida). For example, we can consider the simplest interaction potential: The difference between prices is the most important condition for arbitrage. We could never take into account all economic and other conditions that have influences to the market. Therefore by using some concrete potential we consider the very idealized model of financial processes. However, such an approach is standard for physical modeling where we also consider idealized mathematical models of real physical processes. 3.2 Classical dynamics We apply the Hamiltonian dynamics on the price phase space. As in classical mechanics for material objects, we introduce a new variable the price momentum variable. Instead of the price change vector we consider the price momentum vector The space of price momentums is denoted by the symbol The space will be also called the price phase space. Hamiltonian equations of motion on the price phase space have the form: If the financial energy has form (15) then the Hamiltonian equations have the form The latter equation can be written in the form: The quantity is natural to call the price acceleration (change of price change). The quantity is called the (potential) financial force. We get the financial variant of the second Newton law: ”The product of the financial mass and the price acceleration is equal to the financial force.” In fact, the Hamiltonian evolution is determined by the following fundamental property of the financial energy: The financial energy is not changed in the process of Hamiltonian evolution: We need not restrict our considerations to financial energies of form (15). First of all external (e.g. economic) conditions as well as the character of interactions between traders at the market depend strongly on time. This must be taken into account by considering time dependent potentials: Moreover, the assumption that the financial potential depends only on prices, is not so natural for the modern financial market. Financial agents have the complete information on price changes. This information is taken into account by traders for acts of arbitrage, see [45] for the details. Therefore, it can be useful to consider potentials that depend not only on prices, but also on price changes: or in the Hamiltonian framework: In such a case the financial force is not potential. Therefore, it is also useful to consider the financial second Newton law for general financial forces: Remark 1. (On the form of the kinetic financial energy) We copied the form of kinetic energy from classical mechanics for material objects. It may be that such a form of kinetic financial energy is not justified by real financial market. It might be better to consider our choice of the kinetic financial energy as just the basis for mathematical modeling (and looking for other possibilities). Remark 2. (Domain of price-dynamics) It is natural to consider a model in that all prices are nonnegative, Therefore financial Hamiltonian dynamics should be considered in the phase space where is the set of nonnegative real numbers. We shall not study this problem in details, because our aim is the study of the corresponding quantum dynamics. But in the quantum case this problem is solved easily. One should just consider the corresponding Hamiltonian in the space of square integrable functions Another possibility in the classical case is to consider centered dynamics of prices: The centered price evolves in the configuration space 3.3 Critique of classical econophysical model The model of Hamiltonian price dynamics on the price phase space can be useful to describe a market that depends only on “hard” economic conditions: natural resources, volumes of production, human resources and so on. However, the classical price dynamics could not be applied (at least directly) to modern financial markets. It is clear that the stock market is not based only on these “hard” factors. There are other factors, soft ones (behavioral), that play the important and (sometimes even determining) role in forming of prices at the financial market. Market’s psychology should be taken into account. Negligibly small amounts of information (due to the rapid exchange of information) imply large changes of prices at the financial market. We can consider a model in that financial (psychological) waves are permanently present at the market. Sometimes these waves produce uncontrollable changes of prices disturbing the whole market (financial crashes). Of course, financial waves also depend on “hard economic factors.” However, these factors do not play the crucial role in forming of financial waves. Financial waves are merely waves of information. We could compare behavior of financial market with behavior of a gigantic ship that is ruled by a radio signal. A radio signal with negligibly small physical energy can essentially change (due to information contained in this signal) the motion of the gigantic ship. If we do not pay attention on (do not know about the presence of) the radio signal, then we will be continuously disappointed by ship’s behavior. It can change the direction of motion without any ”hard” reason (weather, destination, technical state of ship’s equipment). However, if we know about the existence of radio monitoring, then we could find information that is sent by radio. This would give us the powerful tool to predict ship’s trajectory. We now inform the reader that this example on ship’s monitoring was taken from the book of D. Bohm and B. Hiley [19] on so called pilot wave quantum theory (or Bohmian quantum mechanics). 4 Quantum-like Econophysical Model for Financial Market 4.1 Financial pilot waves If we interpret the pilot wave as a field, then we should pay attention that this is a rather strange field. It differs crucially from “ordinary physical fields,” i.e., the electromagnetic field. We mention some of the pathological features of the pilot wave field, see [21], [19], [22] for the detailed analysis. In particular, the force induced by this pilot wave field does not depend on the amplitude of wave. Thus small waves and large waves equally disturb the trajectory of an elementary particle. Such features of the pilot wave give the possibility to speculate, see [19], [20], that this is just a wave of information (active information). Hence, the pilot wave field describes the propagation of information. The pilot wave is more similar to a radio signal that guides a ship. Of course, this is just an analogy (because a radio signal is related to an ordinary physical field, namely, the electromagnetic field). The more precise analogy is to compare the pilot wave with information contained in the radio signal. We remark that the pilot wave (Bohmian) interpretation of quantum mechanics is not the conventional one. As we have already noted, there are a few critical arguments against Bohmian quantum formalism: 1. Bohmian theory gives the possibility to provide the mathematical description of the trajectory of an elementary particle. However, such a trajectory does not exist according to the conventional quantum formalism. 2. Bohmian theory is not local, namely, via the pilot wave field one particle ”feels” another on large distances. We say that these disadvantages of theory will become advantages in our applications of Bohmian theory to financial market. We also recall that already Bohm and Hiley [19] and Hiley and Pilkkänen[20] discussed the possibility to interpret the pilot wave field as a kind of information field. This information interpretation was essentially developed in works of Khrennikov [18] that were devoted to pilot wave cognitive models. Our fundamental assumption is that agents at the modern financial market are not just “classical-like agents.” Their actions are ruled not only by classical-like financial potentials but also (in the same way as in the pilot wave theory for quantum systems) by an additional information (or psychological) potential induced by a financial pilot wave. Therefore we could not use the classical financial dynamics (Hamiltonian formalism) on the financial phase space to describe the real price trajectories. Information (psychological) perturbation of Hamiltonian equations for price and price change must be taken into account. To describe such a model mathematically, it is convenient to use such an object as a financial pilot wave that rules the financial market. In some sense describes the psychological influence of the price configuration to behavior of agents of the financial market. In particular, the contains expectations of agents.666 The reader may be surprised that there appeared complex numbers However, the use of these numbers is just a mathematical trick that provides the simple mathematical description of dynamics of the financial pilot wave. We underline two important features of the financial pilot wave model: 1. All shares are coupled on the information level. The general formalism [21], [19], [22] of the pilot wave theory says that if the function is not factorized, i.e., then any changing the price will automatically change behavior of all agents of the financial market (even those who have no direct coupling with -shares). This will imply changing of prices of -shares for At the same time the ”hard” economic potential need not contain any interaction term. For example, let us consider at the moment the potential The Hamiltonian equations for this potential – in the absence of the financial pilot wave – have the form: Thus the classical price trajectory does not depend on dynamics of prices of shares for other traders (for example, the price of shares of ERIKSSON does not depend on the price of shares of NOKIA and vice versa).777Such a dynamics would be natural if these corporations operate on independent markets, e.g., ERIKSSON in Sweden and NOKIA in Finland. Prices of their shares would depend only on local market conditions, e.g., on capacities of markets or consuming activity. However, if, for example, the wave function has the form: where is some normalization constant, then financial behavior of agents at the financial market is nonlocal (see further considerations). 2. Reactions of the market do not depend on the amplitude of the financial pilot wave: waves will produce the same reaction. Such a behavior at the market is quite natural (if the financial pilot wave is interpreted as an information wave, the wave of financial information). The amplitude of an information signal does not play so large role in the information exchange. The most important is the context of such a signal. The context is given by the shape of the signal, the form of the financial pilot wave function. 4.2 The dynamics of prices guided by the financial pilot wave In fact, we need not develop a new mathematical formalism. We will just apply the standard pilot wave formalism to the financial market. The fundamental postulate of the pilot wave theory is that the pilot wave (field) induces a new (quantum) potential which perturbs the classical equations of motion. A modified Newton equation has the form: We call the additional financial force a financial mental force. This force determines a kind of collective consciousness of the financial market. Of course, the depends on economic and other ‘hard’ conditions given by the financial potential . However, this is not a direct dependence. In principle, a nonzero financial mental force can be induced by the financial pilot wave in the case of zero financial potential, So does not imply that Market psychology is not totally determined by economic factors. Financial (psychological) waves of information need not be generated by some changes in a real economic situation. They are mixtures of mental and economic waves. Even in the absence of economic waves, mental financial waves can have a large influence to the market. By using the standard pilot wave formalism we obtain the following rule for computing the financial mental force. We represent the financial pilot wave in the form: where is the amplitude of (the absolute value of the complex number ) and is the phase of (the argument of the complex number ). Then the financial mental potential is computed as and the financial mental force as These formulas imply that strong financial effects are produced by financial waves having essential variations of amplitudes. Example 1. (Financial waves with small variation have no effect). Let us start with the simplest example: Then the financial (behavioral) force As it is impossible to change expectations of the whole financial market by varying the price of one fixed type of shares, The constant information field does not induce psychological financial effects at all. As we have already remarked the absolute value of this constant does not play any role. Waves of constant amplitude as well as produce no effect. Let now consider the case: This is a linear function; variation is not so large. As the result here also. No financial behavioral effects. Example 2. (Speculation) Let (it does not depend on the amplitude !) and The quadratic function varies essentially more strongly than the linear function, and, as a result, such a financial pilot wave induces a nontrivial financial force. We analyze financial drives induced by such a force. We consider the situation: (the starting price) and The financial force stimulates the market (which works as a huge cognitive system) to decrease the price. For small prices, If the financial market increases the price for shares of this type, then the negative reaction of the financial force becomes stronger and stronger. The market is pressed (by the financial force) to stop increasing of the price However, for large prices, If the market can approach this range of prices (despite the negative pressure of the financial force for relatively small then the market will feel decreasing of the negative pressure (we recall that we consider the financial market as a huge cognitive system). This model explains well the speculative behavior of the financial market. Example 3. Let now Here the behavior of the market is more complicated. Set If the price is changing from to then the market is motivated (by the financial force to increase the price. The price is critical for his financial activity. By psychological reasons (of course, indirectly based on the whole information available at the market) the market ”understands” that it would be dangerous to continue to increase the price. After approaching the price the market has the psychological stimuli to decrease the price. Financial pilot waves with that are polynomials of higher order can induce very complex behavior. The interval is split into a collection of subintervals such that at each price level the trader changes his attitude to increase or to decrease the price. In fact, we have considered just a one-dimensional model. In the real case we have to consider multidimensional models of huge dimension. A financial pilot wave on such a price space induces splitting of into a large number of domains The only problem which we have still to solve is the description of the time-dynamics of the financial pilot wave, We follow the standard pilot wave theory. Here is found as the solution of Schrödinger’s equation. The Schrödinger equation for the energy has the form: with the initial condition Thus if we know then by using Schrödinger’s equation we can find the pilot wave at any instant of time Then we compute the corresponding mental potential and mental force and solve Newton’s equation. We shall use the same equation to find the evolution of the financial pilot wave. We have only to make one remark, namely, on the role of the constant in Schrödinger’s equation, cf. E. Haven [27], [29], [32]. In quantum mechanics (which deals with microscopic objects) is the Planck constant. This constant is assumed to play the fundamental role in all quantum considerations. However, originally appeared as just a scaling numerical parameter for processes of energy exchange. Therefore in our financial model we can consider as a price scaling parameter, namely, the unit in which we would like to measure price change. We do not present any special value for There are numerous investigations into price scaling. It may be that there can be recommended some special value for related to the modern financial market, a fundamental financial constant. However, it seems that evolves depending on economic development. We suppose that the financial pilot wave evolves via the financial Schrödinger equation (an analogue of Schrödinger’s equation) on the price space. In the general case this equation has the form: where is self-adjoint operator corresponding to the financial energy given by a function on the financial phase space. Here we proceed in the same way as in ordinary quantum theory for elementary particles. 4.3 On the choice of a measure of classical fluctuations As the mathematical basis of the model we use the space of square integrable functions where is the configuration price space, or some domain (for example, Here is the Lebesque measure, a uniform probability distribution, on the configuration price space. Of course, the uniform distribution is not the unique choice of the normalization measure on the configuration price space. By choosing we assume that in the absence of the pilot wave influence, all prices “have equal rights.” In general, this is not true. If there is no financial (psychological) waves the financial market still strongly depends on “hard” economic conditions. In general, the choice of the normalization measure must be justified by a real relation between prices. So in general the financial pilot wave belongs to the space of square integrable functions with respect to some measure on the configuration price space: In particular, can be a Gaussian measure: where is the covariance matrix and is the average vector. The measure describes classical random fluctuations in the financial market that are not related to ‘quantum’ (behavioral) effects. The latter effects are described in our model by the financial pilot wave. If the influence of this wave is very small we can use classical probabilistic models; in particular, based on the Gaussian distribution. 5 Comparing with conventional models for the financial market Our model of the stocks market differs crucially from the main conventional models. Therefore we should perform an extended comparative analysis of our model and known models. This is not a simple tass and it takes a lot of efforts. 5.1 The stochastic model Since the pioneer paper of L. Bachelier [1], there was actively developed various models of the financial market based on stochastic processes. We recall that Bachelier determined the probability of price changes by writing down what is now called the Chapman-Kolmogorov equation. If we introduce the density of this probability distribution: so then it satisfies to the Cauchy problem of the partial differential equation of the second order. This equation is known in physics as Chapman’s equation and in probability theory as the direct Kolmogorov equation. In the simplest case the underlying diffusion process is the Wiener process (Brownian motion), this equation has the form (the heat conduction equation): We recall again that in the Bachelier paper [1], was the price change variable. For a general diffusion process we have the direct Kolmogorov equation: This equation is based on the diffusion process where is the Wiener process. This equation should be interpreted as a slightly colloquial way of expressing the corresponding integral equation We pay attention that Bachelier original proposal of Gaussian distributed price changes was soon replaced by a model of in which prices of stocks are log-normal distributed, i.e., stocks prices are performing a geometric Brownian motion. In a geometric Brownian motion, the difference of the logarithms of prices are Gaussian distributed. We recall that a stochastic process is said to follow a geometric Brownian motion if it satisfies the following stochastic differential equation: where is a Wiener process (=Brownian motion) and (“the percentage drift”) and (“the percentage volatility”) are constants. The equation has an analytic solution: The depends on a random parameter this parameter will be typically omitted. The crucial property of the stochastic process is that the random variable is normally distributed. In the opposition to such stochastic models our Bohmian model of the stocks market is not based on the theory stochastic differential equations. In our model the randomness of the stocks market cannot be represented in the form of some transformation of the Wiener process. We recall that the stochastic process model was intensively criticized by many reasons, see, e.g., [45]. First of all there is a number of difficult problems that could be interpreted as technical problems. The most important among them is the problem of the choice of an adequate stochastic process describing price or price change. Nowadays it is widely accepted that the GBM-model provides only a first approximation of what is observed in real data. One should try to find new classes of stochastic processes. In particular, they would provide the explanation to the empirical evidence that the tails of measured distributions are fatter than expected for a geometric Brownian motion. To solve this problem, Mandelbrot proposed to consider the price changes that follow a Levy distribution [45]. However, the Levy distribution has a rather pathological property: its variance is infinite. Therefore, as was emphasized in the book of R. N. Mantegna and H. E. Stanley [45], the problem of finding a stochastic process providing the adequate description of the stocks market is still unsolved. However, our critique of the conventional stochastic processes approach to the stocks market has no direct relation to this discussion on the choice of an underlying stochastic process. We are more close to scientific groups which criticize this conventional model by questioning the possibility of describing of price dynamics by stochastic processes at all. 5.2 The deterministic dynamical model In particular, there was done a lot in applying of deterministic nonlinear dynamical systems to simulate financial time series, see [45] for details. This approach is typically criticized through the following general argument: “the time evolution of an asset price depends on all information affecting the investigated asset and it seems unlikely to us that all this information can be essentially described by a small number of nonlinear equations,” [45]. We support such a viewpoint. We shall use only critical arguments against the hypothesis of the stochastic stocks market which were provided by adherents of the hypothesis of deterministic (but essentially nonlinear) stocks market. Only at the first sight is the Bohmian financial model is a kind of deterministic model. Of course, dynamics of prices (as well as price changes) are deterministic. It is described by the Newton second law, see the ordinary differential equation (17). It seems that randomness can be incorporated into such a model only through the initial conditions: where are random variables (initial distribution of prices and momenta) and here is a chance parameter. However, the situation is not so simple. Bohmian randomness is not reduced to randomness of initial conditions or chaotic behavior of the equation (17) for some nonlinear classical and quantum forces. These are classical impacts to randomness. But a really new impact is given by the essentially quantum randomness which is encoded in the -function (=pilot wave=wave function). As we know, the evolution of the -function is described by an additional equation – Schrödinger’s equation – and hence the -randomness could be extracted neither from the initial conditions for (25) nor from possible chaotic behavior. In our model the -function gives the dynamics of expectations at the financial market. These expectations are a huge source of randomness at the market – mental (psychological) randomness. However, this randomness is not classical (so it is a non-Kolmogorov probability model). Finally, we pay attention that in quantum mechanics the wave function is not a measurable quantity. It seems that a similar situation we have for the financial market. We are not able to measure the financial -field (which is an infinite dimensional object, since the Hilbert space has the infinite dimension). This field contains thoughts and expectations of millions agents and of course it could not be “recorded” (in the opposition to prices or price changes). 5.3 The stochastic model and expectations of the agents of the financial market Let us consider again the model of the stocks market based on the geometric Brownian motion: We pay attention that in this equation there is no term describing the behavior of agents of the market. Coefficients and do not have any direct relation to expectations and the market psychology. Moreover, if we even introduce some additional stochastic processes describing behavior of agents and additional coefficients (in stochastic differential equations for such processes) we would be not able to simulate the real market. A finite dimensional vector cannot describe the “mental state of the market” which is of the infinite complexity. One can consider the Bohmian model as the introduction of the infinite-dimensional chance parameter And this chance parameter cannot be described by the classical probability theory. 5.4 The efficient market hypothesis and the Bohmian approach to financial market The efficient market hypothesis was formulated in sixties, see [4] and [5] for details: The efficient market hypothesis is closely related to the stochastic market hypothesis. Mathematically the efficient market hypothesis was supported by investigations of Samuelson [4]. Using the hypothesis of rational behavior and market efficiency he was able to demonstrate how the expected value of price of a given asset at time is related to the previous values of prices through the relation Thus the efficient market hypothesis implies that the financial market is described by a special class of stochastic processes - martingales, see A. Shiryaev [3]. Since the Bohmian quantum model for the financial market is not based on the the stochastic market hypothesis, the efficient market hypothesis can be neither used as the basis of the Bohmian quantum model. The relation between the efficient market model and the Bohmian quantum market model is very delicate. There is no direct contradiction between these models. Since classical randomness is also incorporated into the Bohmian quantum market model (through randomness of initial conditions), we should agree that “the available information is instantly processed when it reaches the market and it is immediately reflected in a new value of prices of the assets traded.” However, besides the available information there is information encoded through the -function describing the market psychology. As was already mentioned, -function is not measurable, so the complete information encoded in this function is not available. Nevertheless, some parts of this information can be extracted from the -function by some agents of the financial market (e.g., by those who “feel better the market psychology”). Therefore classical forming of prices based on the available information is permanently disturbed by quantum contributions to prices of assets. Finally, we should conclude that the real financial market (and not it idealization based on the mentioned hypothesizes) is not efficient. In particular, it determine not the most rational price. It may even induce completely irrational prices through quantum effects. 6 On Views of G. Soros: Alchemy of Finances or Quantum Mechanics of Finances? G. Soros is unquestionably the most powerful and profitable investor in the world today. He has made a billion dollars going against the British pound. Soros is not merely a man of finance, but a thinker and philosopher as well. Surprisingly he was able to apply his general philosophic ideas to financial market. In particular, the project Quantum Fund inside Soros Fund Management gained hundreds millions dollars and has 6 billion dollars in net assets. The book “Alchemy of Finance” [14] is a kind of economic-philosophic investigation. In my thesis I would like to analyze philosophic aspects of this investigation. The book consists of five chapters. In fact, only the first chapter - “Theory of Reflexivity” - is devoted to pure theoretical considerations. J. Soros studied economics in college, but found that economic theory was highly unsatisfactory. He says that economics seeks to be a science, but science is supposed to be objective. And it is difficult to be scientific when the subject matter, the participant in the economic process, lacks objectivity. The author also was greatly influenced by Karl Popper’s ideas on scientific method, but he did not agree with Popper’s “unity method.” By this Karl Popper meant that methods and criteria which can be applied to the study of natural phenomena also can be applied to the study of social events. George Soros underlined a fundamental difference between natural and social sciences: The events studied by social sciences have thinking participants and natural phenomena do not. The participants’ thinking creates problems that have no counterpart in natural science. There is a close analogy with QUANTUM PHYSICS, where the effects of scientific observations give rise to Heisenberg uncertainty relations and Bohr’s complementarity principle. But in social events the participants’ thinking is responsible for the element of uncertainty, and not an external observer. In natural science investigation of events goes from fact to fact. In social events the chain of causation does not lead directly from fact to fact, but from fact to participants’ perceptions and from perceptions to fact. This would not create any serious difficulties if there were some kind of correspondence or equivalence between facts and perceptions. Unfortunately, that is impossible, because the participants perceptions do not relate to facts, but to a situation that is contingent on their own perceptions and therefore cannot be treated as a fact. In order to appreciate the problem posed by thinking participants, Soros takes a closer look at the way scientific method operates. He takes Popper’s scheme of scientific method, described in technical terms as “deductive-nomological” or “D-N” model. The model is built on three kinds of statements: specific initial conditions, specific final conditions, and generalizations of universal validity. Combining a set of generalizations with known initial conditions yields predictions, combining them with known final conditions provides explanations; and matching known initial with known final conditions serves as testing for generalizations involved. Scientific theories can only be falsified, never verified. The asymmetry between verification and falsification and the symmetry between prediction and explanation are two crucial features of Popper’s scheme. The model works only if certain conditions are fulfilled. It is the requirement of universality. That is, if a given set of conditions recurred, it would have to be followed or predicted by the same set of conditions as before. The initial and final conditions must consist of observable facts governed by universal laws. It is this requirement that is so difficult to meet when a situation has thinking participants. Clearly, a single observation by a single scientist is not admissible. Exactly because the correspondence between facts and statements is so difficult to establish, science is a collective enterprize where the work of each scientist has to be open to control and criticism by others. Individual scientists often find the conventions quite onerous and try various shortcuts in order to obtain a desired result. The most outstanding example of the observer trying to impose his will on his subject matter is the attempt to convert base metal into gold. Alchemists struggled long and hard until they were finally persuaded to abandon their enterprize by their lack of success. The failure was inevitable because the behavior of base metals is governed by laws of universal validity which cannot be modified by any statements, incantations, or rituals. And now, we can at least understand why J. Soros called his book Alchemy of Finance, see [14]. Soros considers the behavior of human beings. Do they obey universally valid laws that can be formulated in accordance with “D-N” model? Undoubtedly, there are many aspects of human behavior, from birth to death and in between, which are amenable to the same treatment as other natural phenomena. But there is one aspect of human behavior which seems to exhibit characteristics which are different from those of the phenomena from the subject matter of natural science: the decision making process. An imperfect understanding of the situation destroys the universal validity of scientific generalizations: given a set of conditions is not necessary preceded or succeeded by the same set every time, because the sequence of events is influenced by participants’ thinking. The “D-N” model breaks down. But social scientists try to maintain the unity of method but with little success. In a sense, the attempt to impose the methods of natural science on social phenomena is comparable to efforts of alchemists who sought to apply the methods of magic to the field of natural science. And here J. Soros presents (in my opinion) a very interesting idea about the “alchemy method in social science.” He says, that while the failure of the alchemists was total, social scientists have managed to make a considerable impact on their subject matter. Situations which have thinking participants may be impervious to the methods of natural science, but they are susceptible to the methods of alchemy. The thinking of participants, exactly because it is not governed by reality, is easily influenced by theories. In the field of natural phenomena, scientific method is effective only when its theories are valid, but in social, political, and economic matters, theories can be effective without being valid. Whereas alchemy has failed in natural sciences, social science can succeed as alchemy. The relationship between the scientist and his subject matter is quite different in natural science as opposed to social science. In natural science the scientist’s thinking is, in fact, distinct from its subject matter. The scientists can influence the subject matter only by actions, not by thoughts, and the scientists actions are guided by the same laws as all other natural phenomena. Specifically, a scientist can do nothing do when she wants to turn base metals into gold. Social phenomena are different. The imperfect understanding of the participant interferes with the proper functioning of the “D-N” model. There is much to be gained by pretending to abide by conventions of scientific method without actually doing so. Natural science is held in great esteem: the theory that claims to be scientific can influence the gullible public much better than one which frankly admits its political or ideological bias. Soros mentions here Marxism, psychoanalysis and laissez-faire capitalism with its reliance on the theory of perfect competition as typical examples. Soros underlined that Marx and Freud were vocal in protesting their scientific status and based many of their conclusions on authority they derived from being “scientific.” And Soros says, that once this point sinks in, the very expression “social science” became suspect. He compares the expression “social science” with a magic word employed by social alchemists in their effort to impose their will on their subject matter by incantation. And it seems to Soros there is only one way for the “true” practitioners of scientific method to protect themselves against such malpractice - to deprive social science of the status it enjoys on account of natural science. Social science ought to be recognized as a false metaphor. I cannot agree with this Soros statement. First of all there are a lot of boundary sciences. For example, psychoanalysis can be considered as a part of medicine. But medicine accompanied with biology and chemistry are of course the natural sciences. And S. Freud was famous and like many doctors succeeding him they helped many patients. And J. Soros explains by himself his unusual statement. He says that it does not mean that we must give up the pursuit of truth in exploring social phenomena. It means only that the pursuit of truth requires to recognize that the “D-N” model can not be applied to situations with thinking participants. He asks us to abandon the doctrine of the unity of method and to cease the slavish imitation of natural sciences. He says that there are some new scientific methods in all kinds of science as quantum physics has shown. Scientific method is not necessarily confined to the “D-N” model: statistical, probabilistic generalizations may be more fruitful. Nor should we ignore the possibility of developing novel approaches which have no counterpart in natural science. Given the differences in subject matter, there ought to be differences in the method of study. Soros shows us the main distinction between the “D-N” model and his own approach. He says that the world of imperfect understanding does not land itself to generalizations which can be used to explain and to predict specific events. The symmetry between explanation and prediction prevails only in the absence of thinking participants. On the other hand, past events are just as final as in the “D-N” model; thus explanations turn out to be an easier task than prediction. In another part of his book , see [14], Soros abandons the constraint that predictions and explanations are logically reversible. He builds his own theoretical framework. He says that his theory can not be tested in the same way as these theories which fit into Popper’s logical structure, but that is not to say that testing must be abandoned. And he does tests in a real-time experiment, chapter 3, for his model. He uses the theory of perfect competition for the investigation of financial market, but he takes into account thinking of participants of this market. G. Soros proved his theory by becoming one of the most powerful and profitable investors in the world today. In my own work I use methods of classical and quantum mechanics for mathematical modeling of price dynamics at financial market and I use Soros’ statement about cognitive phenomena at the financial market. 7 Existence Theorems for Non-smooth Financial Forces 7.1 The problem of smoothness of price trajectories In the Bohmian model for price dynamics the price trajectory can be found as the solution of the equation with the initial condition Here we consider a ”classical” (time dependent) force and ”quantum-like” force where is the quantum potential, induced by the Schrödinger dynamics. In Bohmian mechanics for physical systems the equation (27) is considered as an ordinary differential equation and as the unique solution (corresponding to the initial conditions of the class is assumed to be twice differentiable with continuous One of possible objections to apply the Bohmian quantum model to describe dynamics of prices (of e.g. shares) at the financial market is smoothness of trajectories. In financial mathematics it is commonly assumed that the price-trajectory is not differentiable, see, e.g., [45], [3]. 7.2 Mathematical model and reality Of course, one could simply reply that there are no smooth trajectories in nature. Smooth trajectories belong neither to physical nor financial reality. They appear in mathematical models which can be used to describe reality. It is clear that the possibility to apply a mathematical model with smooth trajectories depends on a chosen time scale. Trajectories that can be considered as smooth (or continuous) at one time scale might be nonsmooth (or discontinuous) at a finer time scale. We illustrate this general philosophic thesis by the history of development of financial models. We recall that at the first stage of development of financial mathematics, in the Bachelier model and the Black and Scholes model, there were considered processes with continuous trajectories: the Wiener process and more general diffusion processes. However, recently it was claimed that such stochastic models (with continuous processes) are not completely adequate to real financial data, see, e.g., [45], [3] for the detailed analysis. It was observed that at finer time scales some Levy-processes with jump-trajectories are more adequate to data from the financial market. Therefore one could say that the Bohmian model provides a rough description of price dynamics and describes not the real price trajectories by their smoothed versions. However, it would be interesting to keep the interpretation of Bohmian trajectories as the real price trajectories. In such an approach one should obtain nonsmooth Bohmian trajectories. The following section is devoted to theorems providing existing of nonsmooth solutions. 7.3 Picard’s theorem and its generalization We recall the standard uniqueness and existence theorem for ordinary differential equations, Picard’s theorem, that gives the guarantee of smoothness of trajectories, see, e.g., [46]. Theorem 1. Let be a continuous function and let satisfy the Lipschitz condition with respect to the variable : Then, for any point ( there exists the unique -solution of the Cauchy problem: on the segment where depends and We recall the standard proof of this theorem, because the scheme of this proof will be used later. Let us consider the space of continuous functions where is a number which will be determined. Denote this space by the symbol The Cauchy problem (29) for the ordinary differential equation can be written as the integral equation: The crucial point for our further considerations is that continuity of the function with respect to the pair of variables implies continuity of for any continuous But the integral is differentiable for any continuous and is also continuous. The basic point of the standard proof is that, for a sufficiently small the operator maps the functional space into and it is a contraction in this space: for any two trajectories such that and Here, to obtain the interval should be chosen sufficiently small, see further considerations. Here and The contraction condition, implies that the iterations converge to a solution of the integral equation (30). Finally, we remark that the contraction condition (32) implies that the solution is unique in the space Roughly speaking in Theorem 1 the Lipschits condition is ”responsible” for uniqueness of solution and continuity of for existence. We also recall the well known Peano theorem, [46]: Theorem 2. Let be a continuous function. Then, for any point ( there exists locally a -solution of the Cauchy problem (29). We remark that Peano’s theorem does not imply uniqueness of solution. It is clear that discontinuous financial forces can induce price trajectories which are not smooth: more over, price trajectories can even be discontinuous! From this point of view the main problem is not smoothness of price trajectories (and in particular the zero covariation for such trajectories), but the absence of an existence and uniqueness theorem for discontinuous financial forces. We shall formulate and prove such a theorem. Of course, outside the class of smooth solutions one could not study the original Cauchy problem for an ordinary differential equation (29). Instead of this one should consider the integral equation (30). We shall generalize Theorem 1 to discontinuous Let us consider the space consisting of bounded measurable functions Thus: b) for any Borel subset its preimage is again a Borel subset in Lemma 1. The space of trajectories is a Banach space Proof. Let be a sequence of trajectories that is a Cauchy sequence in the space Thus, for any Hence, for any the sequence of real numbers is a Cauchy sequence in But the space is complete. Thus, for any there exists which we denote by In this way we constructed a new function We now write the condition (33) through the -language: We now fix and take the limit in the inequality (34). We obtain: This is nothing else than the condition: Therefore in the space We remark that the trajectory is bounded, because: for any fixed and since we finally get We also remark that is a measurable function as the unifrom limit of measurable functions, see [3]. Thus the space is a complete normed space - a Banach space. Theorem 3. Let be a measurable bounded function and let satisfy the Lipschitz condition with respect to the -variable, see (28). Then, for any point there exists the unique solution of the integral equation (30) of the class where depends on and Proof. We shall determine later. Let be any function of the class Then the function is measurable (since and are measurable) and it is bounded (since is bounded). Thus Any bounded and measurable function is integrable with respect to the Lebesque measure on see [3]. Therefore
e92668b3745050f8	Matter and Energy: A False Dichotomy Matt Strassler [April 12, 2012] It is common that, when reading about the universe or about particle physics, one will come across a phrase that somehow refers to “matter and energy”, as though they are opposites, or partners, or two sides of a coin, or the two classes out of which everything is made. This comes up in many contexts. Sometimes one sees poetic language describing the Big Bang as the creation of all the “matter and energy” in the universe. One reads of “matter and anti-matter annihilating into `pure’ energy.” And of course two of the great mysteries of astronomy are “dark matter” and “dark energy”. As a scientist and science writer, this phraseology makes me cringe a bit, not because it is deeply wrong, but because such loose talk is misleading to non-scientists. It doesn’t matter much for physicists; these poetic phrases are just referring to something sharply defined in the math or in experiments, and the ambiguous wording is shorthand for longer, unambiguous phrases. But it’s dreadfully confusing for the non-expert, because in each of these contexts a different definition for `matter’ is being used, and a different meaning — in some cases an archaic or even incorrect meaning of `energy’ — is employed. And each of these ways of speaking implies that either things are matter or they are energy — which is false. In reality, matter and energy don’t even belong to the same categories; it is like referring to apples and orangutans, or to heaven and earthworms, or to birds and beach balls. On this website I try to be more precise, in order to help the reader avoid the confusions that arise from this way of speaking. Admittedly I’m only partly successful, as I’ll mention below. Summing Up This article is long, but I hope it is illuminating and informative for those of you who want details. Let me give you a summary of the lessons it contains: • Matter and Energy really aren’t in the same class and shouldn’t be paired in one’s mind. • Matter, in fact, is an ambiguous term; there are several different definitions used in both scientific literature and in public discourse. Each definition selects a certain subset of the particles of nature, for different reasons. Consumer beware! Matter is always some kind of stuff, but which stuff depends on context. • Energy is not ambiguous (not within physics, anyway). But energy is not itself stuff; it is something that all stuff has. • The term Dark Energy confuses the issue, since it isn’t (just) energy after all. It also really isn’t stuff; certain kinds of stuff can be responsible for its presence, though we don’t know the details. • Photons should not be called `energy’, or `pure energy’, or anything similar. All particles are ripples in fields and have energy; photons are not special in this regard. Photons are stuff; energy is not. • The stuff of the universe is all made from fields (the basic ingredients of the universe) and their particles. At least this is the post-1973 viewpoint. What’s the Matter (and the Energy)? First, let’s define (or fail to define) our terms. The word Matter. “Matter” as a term is terribly ambiguous; there isn’t a universal definition that is context-independent. There are at least three possible definitions that are used in various places: • “Matter” can refer to atoms, the basic building blocks of what we think of as “material”: tables, air, rocks, skin, orange juice — and by extension, to the particles out of which atoms are made, including electrons and the protons and neutrons that make up the nucleus of an atom. • OR it can refer to what are sometimes called the elementary “matter particles” of nature: electrons, muons, taus, the three types of neutrinos, the six types of quarks — all of the types of particles which are not the force particles (the photon, gluons, graviton and the W and Z particles.) Read here about the known apparently-elementary particles of nature. [The Higgs particle, by the way, doesn’t neatly fit into the classification of particles as matter particles and force particles, which was somewhat artificial to start with; I have a whole section about this classification below.] • OR it can refer to classes of particles that are found out there, in the wider universe, and that on average move much more slowly than the speed of light. With any of these definitions, electrons are matter (although with the third definition they were not matter very early in the universe’s history, when it was much hotter than it is today.) With the second definition, muons are matter too, and so are neutrinos, even though they aren’t constituents of ordinary material. With the third definition, some neutrinos may or may not be matter, and dark matter is definitely matter, even if it turns out to be made from a new type of force particle. I’m really sorry this is so confusing, but you’ve no choice but to be aware of these different usages if you want to know what “matter” means in different people’s books and articles. Now, what about the word Energy. Fortunately, energy (as physicists use it) is a well-defined concept that everyone in physics agrees on. Unfortunately, the word in English has so many meanings that it is very easy to become confused about what physicists mean by it. I’ve briefly describe the various forms of energy that arise in physics in more detail in an article on mass and energy. But for the moment, suffice it to say that energy is not itself an object. An atom is an object; energy is not. Energy is something which objects can have, and groups of objects can have — a property of objects that characterizes their behavior and their relationships to one another. [Though it should be noted that different observers will assign different amounts of energy to a given object — a tricky point that is illustrated carefully in the above-mentioned article on mass and energy.] And for this article, all we really need to know is that particles moving on their own through space can have two types of energy: mass-energy (i.e., E= mc2 type of energy, which does not depend on whether and how a particle moves) and motion-energy (energy that is zero if a particle is stationary and becomes larger as a particle moves faster). Annihilation of Particles and Antiparticles Isn’t Matter Turning Into Energy Let’s first examine the notion that “matter and anti-matter annihilate to pure energy.” This, simply put, isn’t true, for several reasons. In the green paragraphs above, I gave you three different common definitions of “matter.” In the context of annihilation of particles and anti-particles, speakers may either be referring to the first definition or the second. Here I want to discuss the annihilation of electrons and anti-electrons (or “positrons”), or the annihilation of muons and anti-muons. I’ve described this in detail in an article on Particle/Anti-Particle Annihilation. You’ll need it to understand what I say next, so I’m going to assume that you have read it. Once you’ve done that, you’re ready to try to understand where the (false) notion that matter and antimatter annihilate into pure energy comes from. What is meant by “pure energy”? This is almost always used in reference to photons, commonly in the context of an electron and a positron (or some other massive particle and anti-particle) annihilating to make two photons (recall the antiparticle of a photon is also a photon.) But it’s a terrible thing to do. Energy is something that photons have; it is not what photons are. [I have height and weight; that does not mean I am height and weight.] The term “pure energy” is a mix of poetry, shorthand and garbage. Since photons have no mass, they have no mass-energy, and that means their energy is “purely motion-energy”. But that does not mean the same thing, either in physics or intuitively to the non-expert, as saying photons are “pure energy”. Photons are particles just as electrons are particles; they both are ripples in a corresponding field, and they both have energy. The electron and positron that annihilated had energy too — the same amount of energy as the photons to which they annihilate, in fact, since energy is conserved (i.e. the total amount does not change during the annihilation process.) (See Figure 3 of the particle/anti-particle annihilation article. Moreover (see Figures 1 and 2 of the particle/anti-particle annihilation article), the process muon + anti-muon → two photons is on exactly the same footing and occurs with almost exactly the same probability as the process muon + anti-muon → electron + positron — which is matter and anti-matter annihilating into another type of matter and anti-matter. So no matter how you want to express this, it is certainly not true that matter and anti-matter always annihilate into anything you might even loosely call `energy’; there are other possibilities. For these reasons I don’t use the “matter and energy” language on this website when speaking about annihilation. I just call this type of process what it is: • particle 1 + anti-particle 1 → particle 2 + anti-particle 2 With this plain-spoken terminology it is clear why a muon and anti-muon annihilating to two photons, or to an electron and a positron, or to a neutrino and an anti-neutrino, are all on the same footing. They are all the same class of process. And we need not make distinctions that don’t really exist and that obscure the universality of particle/anti-particle annihilation. Not Everything is Matter or Energy, By a Long Shot Why do people sometimes talk about “matter and energy” as though everything is either matter or energy? I don’t know the context in which this expression was invented. Maybe one of my readers knows? Language reflects history, and often reacts slowly to new information. Part of the problem is that enormous changes in physicists’ conception of the world and its ingredients occurred between 1900 and 1980. This has mostly stopped for now; it’s been remarkably stable throughout my career. [String theorists might argue with what I’ve just said, pointing out that their great breakthroughs occurred during the 1980s and 1990s. That’s true, but since string theory hasn’t yet established itself as reality through experimental verification, one cannot say that it has yet been incorporated into our conception of the world.] Our current conception of the physical world is shaped by a wide variety of experiments and discoveries that occurred during the 1950s, 1960s and 1970s. But previous ways of thinking and talking about particle physics partially stuck around even as late as the 1980s and 1990s, while I was being trained as a young scientist. This isn’t surprising; it takes a while for people who grew up with an older vision to come around to a new prevailing point of view, and some never do. And it also takes a while for a newer version to come into sharp focus, and for little niggling problems with it to be resolved. Today, if one wants to talk about the world in the context of our modern viewpoint, one can speak first and foremost of the “fields and their particles.” It is the fields that are the basic ingredients of the world, in today’s widely dominant paradigm. We view fields as more fundamental than particles because you can’t have an elementary particle without a field, but you can have a field without any particles. [I still owe you a proper article about fields and particles; it’s high on the list of needed contributions to this website.] However, it happens that every known field has a known particle, except possibly the Higgs field (whose particle is not yet certain to exist, though [as of the time of writing, spring 2012] there are significant experimental hints.) What do “fields and particles” have to do with “matter and energy”? Not much. Some fields and particles are what you would call “matter”, but which ones are matter, and which ones aren’t, depends on which definition of “matter” you are using. Meanwhile, all fields and particles can have energy; but none of them are energy. Matter Particles and Force Particles — Well… On this website, I’ve divided the known particles up into “matter particles” and “force particles”. I wasn’t entirely happy doing this, because it’s a bit arbitrary. This division works for now; the force particles and their anti-particles are associated with the four forces of nature that we know so far, and the matter particles and their anti-particles are all of the others. And there are many situations in which this division is convenient. But at the Large Hadron Collider [LHC] we could easily discover particles that don’t fit into this categorization; even the Higgs particle poses a bit of a problem, because it arguably is in neither class. There’s an alternate (but very different) division that makes sense: what I called matter particles all happen to be fermions, and what I called force particles all happen to be bosons. But this could change too with new discoveries. What this really comes down to is that all the particles of nature are simply particles, some of which are each other’s anti-particles, and there isn’t a unique way to divide them up into classes . The reason I used “matter” and “force” is that this is a little less abstract-sounding than “fermions” and “bosons” — but I may come to regret my choice, because we might discover particles at the LHC, or elsewhere, that break this distinction down. Matter and Energy in the Universe Another place we encounter words of this type is in the history and properties of the cosmos as a whole. We read about matter, radiation, dark matter, and dark energy. The use of the words by cosmologists is quite different from what you might expect — and it actually involves two or three different meanings, and depends strongly on context. Matter vs. Anti-Matter: when you hear people talk this way, they’re talking about the first definition within the green paragraphs above. They are typically referring to the imbalance of matter over anti-matter in our universe — the fact that the particles that make up ordinary material (electrons, protons and neutrons in particular) are much more abundant than any of their anti-particles. Matter vs. Radiation: if you hear this distinction, you’re dealing with the third definition of `matter’. The universe has a temperature; it was very hot early on and has been gradually cooling, now at 2.7 Celsius-degrees above absolute zero. If you have a gas (or plasma) of particles at a given temperature T, and you measure the energies of these particles, you will find that the average motion-energy per particle is given by k T, where k is Boltzmann‘s famous constant. Now matter, in this context, is any particle whose mass-energy mc2 is large compared to this average motion energy kT; such particles will have velocity much slower than the speed of light. And radiation is any particle whose mass-energy is small compared to kT, and is consequently moving close to the speed of light. Notice what this means. In this context, what is matter, and what is not, is temperature-dependent and therefore time-dependent! Early in the universe, when the temperature was trillions of degrees and even hotter, the electron was what cosmologists consider radiation. Today, with the universe much cooler, the electron is in the category of matter. In the present universe at least two of the three types of neutrinos are matter, and maybe all three, by this definition; but all the neutrinos were radiation early in the universe. Photons have always been and will always be radiation, since they are massless. What is Dark Matter? We can tell from studying the motions of stars and other techniques that most of the mass of a galaxy comes from something that doesn’t shine, and lots of hard work has been done to prove that known particles behaving in ordinary ways cannot be responsible. To explain this effect, various speculations have been proposed, and many have been shown (through observation of how galaxies look and behave, typically) to be wrong. Of the survivors, one of the leading contenders is that dark matter is made from heavy particles of an unknown type. But we don’t know much more than that as yet. Experiments may soon bring us new insights, though this is not guaranteed. [Note also there may be not be any meaning to dark anti-matter; the particles of dark matter, like photons and Z particles, may well be their own anti-particles.] And Dark Energy? It was recently discovered that the universe is expanding faster and faster, not slower and slower as was the case when it was younger. What is presumably responsible is called “dark energy”, but unfortunately, it’s actually not energy. As my colleague Sean Carroll is fond of saying, it is tension, not energy — a combination of pressure and energy density. So why do people call it “energy”? Part of it is public relations. Dark energy sounds cool; dark tension sounds weird, as does any other word you can think of that is vaguely appropriate. At some level this is harmless. Scientists know exactly what is being referred to, so this terminology causes no problem on the technical side; most of the public doesn’t care exactly what is being referred to, so arguably there’s no big problem on the non-technical side. But if you really want to know what’s going on, it’s important to know that dark-energy isn’t a dark form of energy, but something more subtle. Moreover, like energy, dark-energy isn’t an object or set of objects, but a property that fields, or combinations of fields, or space-time itself can have. We don’t yet know what is responsible for the dark-energy whose presence we infer from the accelerating universe. And it may be quite a while before we do. By the way, do you know what an astronomer means by “metals”? It’s not what you think… You might conclude from this article that modern physicists and their relatives have not been very inventive, creative, or careful with language. Apparently it’s not our collective strong suit. Big Bang? Black Hole? The world’s poets will never forgive us for choosing such dull names for such fantastic things…. 294 thoughts on “Matter and Energy: A False Dichotomy” 1. Thanks again for an excellent review. Even though most of it is known as bits and pieces by us the lay readers, this clearly, and coherently explains the intricacies of the commonly misused terms. You write that all fields have corresponding particles and only Higgs field and it’s particle is in doubt but LHC has tantalizing clues of its presence. Wouldn’t gravitational field is something for which no known particle exists even though it is hypothesized? 2. “Energy is something which objects can have”. I can’t say I’m happy with this. The energy of an object, in classical or SR models, at least depends on the frame of reference. I don’t see the Hamiltonian as the generator of time-like translations anywhere here, which at least deserves a mention if we move to QM models. Energy in GR is different again. If we move to signal processing, time-series analysis, we can define an energy of a signal mathematically, but that may not be reducible to phase space concepts (this is arguably engineering, but Physicists do use such concepts in some measurement contexts). I don’t see energy to be as settled in Physics as you suggest. • I think I understand your first point. Energy is a property of an object (or system) but the amount depends on the observer. I emphasized this in my mass and energy article, but not here; based on your comment I’ve added a remark to that effect in the text. Your second point about the Hamiltonian: this technical point is not appropriate for the main readers of this article. The fact that energy conservation is related to the time-independence of physical law has been covered on this site elsewhere; this particular article wasn’t the place for it. The notions of Lagrangian and Hamiltonian are above the assumed knowledge of my readership. Energy in general relativity is an advanced subject, again beyond the scope of this article. Finally: engineering notions of energy in the context of signal processing are beyond the scope of this website. It never arises in any of my research or that of any of my immediate colleagues. I am always referring to that conserved quantity (even in general relativity) that follow from the (local, in GR) time-independence of physical laws. • Just read your article. I think you are right about one thing: We still use language left over from another time. I think you are guilty of this yourself when you use the word “particle.” Sorry, there is no such thing as a particle. Everything is energy. Energy exists as a wave. A wave has momentum. Either that momentum is dedicated to travelling through space at some fraction of the speed of light, or it is turned in on itself as a standing wave to stay in one place as “mass.” Even as mass, it is still a wave. That wave is still energy, it’s just energy that stays in one place rather than travelling through space. When I put my hand on the tabletop, I’m not actually touching it. The electron waves of all the atoms on the tables surface repel the electron waves on the surface of my hand. There is no physical mass or particles touching, merely waves of negatively charged energy repulsing other negatively charged waves of energy. There is no such thing as a “particle.” That description is outdated. • interesting way of explaining the outdated term “particle “…. but isn’t “mass” made of “particles” ? how different is a particle from mass ? Isn’t a standing wave everywhere ? • No. The wave function is just a mathematical concept used to describe the momentum of a particle. It predicts the possibility of where one particle may exist in a single frame of time. It is not the same thing to say that the particle IS the momentum OR the mass. It doesn’t make any sense. This is exactly what the writer is referring to. The use of the words such as mass and energy which are fairly consistent in the macrocosmic world of ours fail to describe anything in the subatomic regions. None of which the writer says is outdated, but a lot of people who don’t really understand the underlying mechanism try to hijack the bandwagon with vague mystical sayings such as ‘Everything is Energy’ and the forth. Energy describes how the particle functions, it is the property of a particle. It is not the same thing as the particle itself. • Strassler made it pretty clear that fields, and not particles, are the fundamental ingredients of the world. He called particles “ripples” in these fields. You saying “everything is energy” leads me to believe you actually wanted to say “all things are fields”. • Hello. I am not a physicist at all. I came upon this article in a search because I could not accept the concept that energy and matter are separate entities and it’s what I keep reading over and over again. That matter has mass but energy doesn’t. Then, how can E=mc^2??? I realize it has been several years since this article has been published online, but I’m still not satisfied with the explanation. How can matter and energy not “be” the same thing? One can’t exist without the other. Where is the example of matter that has no “energy” associated with it? To me its like saying that lung tissue is not “human” even though it contains all of the same DNA as every other cell in the body. The lung tissue is just expressing or taking on the function that it needs to do. Otherwise, how could stem cells be used for various items. I feel the same way about matter and energy. How can you accept E=mc^2 but not accept that matter and energy are just expressions of the same thing? Or like when you say that you “have” height and weight, but you are “not” height and weight. Then what exactly are you? You don’t carry height and weight around as a separate entity in your hand that you can walk away from. It is you, just as the energy that animates our human body “is” each one of us. I am by NO way a scientist, but these concepts have been driving me insane for years. I guess what I’m getting at, if you still access this site: Where is a book, an essay, a study I can turn to that provides some kind of proof? I would be interested to study these concepts further. Thank you. • Even through Quantum Physics and the ToE we find that an ‘unexplained’ conscious ‘force’ is necessary for existence to occur; or something to that effect. The answers will only be found in the quantum realm and common sense. Every atom is built with frequency, energy and vibration which is ‘energy’ regardless of how the textbooks warp the mind. • Sarah, it has been a long time since you asked the questions here. I can answer one of them. You ask “How can matter and energy not “be” the same thing? One can’t exist without the other. Where is the example of matter that has no “energy” associated with it?” This last question should be reversed. There is no matter that has no energy, but there is energy that is not associated with matter. Photons, for instance, have energy, but they are not matter by anyone’s definition. The same is true for gravitational waves; they have energy, but I’ve never heard anyone suggest they should be called matter. Do not be confused by E=mc^2. First, “m” is mass, not mattter, and not all matter has mass, either. Second, “E” does not refer to all possible forms of energy. E=mc^2 means only that: the rest mass m of a particle that has some arises from E, energy, stored inside it. • Hi Matt, I am also bothered by the same question as Sarah. I understand your distinction of matter from energy and, as I’m sure you’re familiar with, the commonly surrendered definition of energy is “the ability to do work”, so its treatment as a property is entirely sensible. The part that spins me for a loop – and I suspect this is why people commonly insist that everything IS energy – is that I don’t know of anything that can exist without energy as a basic property. Even quantum fields have zero-point energy at their lowest-energy state. Then there’s the idea that energy is that thing which is the particle-producing ripple in the field. So if: a) everything Has energy as an irremovable and universal property, and b) things are defined (categorized and distinguished) by their properties, does it not follow that the essence of all things is energy? Even those conceptions which are themselves the absence of things (ie. shadows, absolute zero) are defined by their quality of energy. Maybe I am failing to consider something. Is there anything that can exist without energy? Or is the problem really about the practical issues with using the concept of energy so broadly? Or am I misunderstanding a subtle idea very seriously? Really appreciate your feedback, and the article! • Every human has age. We cannot imagine a human without it. Does that mean every human IS age? Every human has volume. All ordinary objects have volume, in fact. Does that mean the essence of objects is volume? We must not confuse properties that objects have — sometimes even necessary properties — with what those objects are. The fact that energy is a property of all objects and even non-objects reflects the fact that time is a fundamental property of the universe. It tells us nothing about the nature of those objects. Does this help? Nothing is made from energy, but everything has energy. 3. Can we say that the building stuff of the cosmos are mere 2 types of vibrations — organized ripples and psuedo ripples — which we call real or virtual particles ?in something we call fields ? • No — you really don’t want to approach it this way. The basic stuff is fields. Now you want to ask what fields can do and how that contributes to the stuff we see around us. You can’t reduce that the two types. • Is field stuff though? Field is more of an information matrix. Strictly speaking, you have a field of mass or a field of properties, but that doesn’t mean the field itself is either mass or energy. I don’t think we should give definition to ‘the basic stuff’ just as yet. I think we need to learn how to distinguish fields from mass-energy, and mass-energy from its carrier particles for this discussion to have any merit. And why are we so intent on reducing everything to one thing anyway? What good does it even do? Even the scientific world sometimes… • In my experience, a field is a mathematical construct, nothing more. Problems arise when one attempts to interpret a field as some sort of physical entity. This is not to say physical phenomena do not exist. Rather, fields are a mathematical device (the best we presently have) to describe physical reality at the micro level. It’s simply a theoretical framework based on mathematics. 4. There are a contradiction here : 1- E is something mass have 2- Mass is E/c^2 3- E is something E/c^2 have !!?? • You’re making some classic mistakes: First, #2 is false. Mass is only E/c^2 for a particle at rest and sitting on its own. [If you define m to be E/c^2, then you’re using the archiac notion of “relativistic mass”, which particle physicists avoid for several reason (to be explained soon.) And I’m not using “m” in this way anywhere on this website.] For a particle on its own, but moving, E = mc^2 + motion energy . For a more general system of particles and fields, you also have to account for other contributions to the total energy that cannot be assigned to any one particle. Second, #1 is false. Energy is something that stuff can have. Mass is also something that stuff can have. Stuff is not mass; it’s something that can have mass. But not necessarily. Some stuff has no mass — photons, for instance. And it is an accident that electrons are massive; remember that the Higgs field being non-zero on average is responsible for this. Electrons would still be stuff even if the Higgs field were zero on average and electrons were massless. [Experts: I know there’s a tiny subtlety with this statement; let it pass. To make the above statement precise I should also turn off some other interactions when I turn off the Higgs field’s value. But the substance of the remark is true.] Part of the point of this article (and my earlier particle-antiparticle annihilation and mass-and-energy articles) is that photons and electrons are both particles. They are both stuff. It happens that photons are massless and electrons are massive, so they behave quite differently. But the equations that govern them are very similar, and one should not think of electrons as stuff and photons as something else. Mass is something they may or may not have; energy is something they have too. • E=mc^2… 1 the energy content isnt determined by either. As you point out energy is intrinsic to the stuff aka it allready poses order to reject the fact photons have 0 rest mass you assign it velocity to give it relative mass . The problem is if the universe is a closed system there is an objective standard & all frames of reference consolidate into 1 objective frame of reference. Relative speed then becomes an objective speed & the phase space oscilates in a uniform fashion (higgs field). If it were not as such higgs field tensor values would varry & we can only imagine the chaos that would ensue as the phenomena of matter would dissolve along with our physical existence. 5. You say that “matter is always some kind of stuff …” and “photons are stuff”; this would appear to mean that photons are matter (unless photons are stuff that is not matter, of course). Yet photons fit none of your green-paragraph definitions of “matter”. I’m mindful that the whole thrust of the article is that matter is an ill-defined concept (and, as usual, I learned some things from it — thanks!); I am perhaps reinforcing that point by noting that your green-paragraph definitions are yet not enough. • Photons are “stuff”, but they are not matter — for almost every definition of “matter”. Matter is a subset of stuff, though which subset depends on context. I do know of one or two contexts where photons would be called “matter” too — but these settings are ones that you won’t come across often, and usually different terminology is used anyway. What I mean by “stuff”, in general, needs a little more working out. A silly but useful working definition is that something is stuff if it can be used to damage other stuff. I can’t damage your cells with mass or energy — I can’t make an “energy beam” or something like that. I have to make a beam of photons or a beam of electrons or muons or protons or neutrinos. That “stuff” carries energy, sure, but the energy has to be carried by a physical object — a particle — stuff — for it to be able to do anything. (The particles of dark matter — whatever they turn out to be — are stuff too, though I’d need one heck of a beam of dark matter particles to damage anything!) Fields are stuff too: they can be used to pull other stuff apart. Maybe you can point out a flaw in that definition? A challenge to the reader… • Space-time can be curved in ways to ripple ‘big’ stuff apart. That means spacetime itself is a “stuff”? If thats the case, can stuff ‘damage’ spacetime? • Sure. That’s Einstein’s point. You can make two black holes, made entirely from space-time curvature, and arrange for them to orbit each other. The two orbiting black holes form a system — a sort of “atom” — which can be stable for a fairly long time. A sufficiently powerful beam of photons, or electrons, could break that system of two black holes apart. It’s not very different from disrupting an ordinary atom using a extremely powerful gravitational wave, which is also possible in principle. In the first example you would be using something which is obviously stuff to damage an object made entirely from curvature of space and time. In the second you would be using space-time to damage an object made from other stuff. Not that doing either of these is practical — but in principle you could do either one. 6. Haha, that is true! I’m not sure Lemaitre’s “primeval atom” was much better than “Big Bang.” All the really dramatic names I can think of smack of religion. Besides, who wouldn’t want to write a poem about the charmed quark? It’s so… charming. (Yes, I’ll be here all night, folks.) I’ve also heard Dark Energy described as a sort of cosmic “pressure”. I’m not sure how valid that comparison turns out to be. • Yet, his ‘atom primitif’ suggestion sounded better than his other suggestion to name it the ‘cosmic egg’ which was probably inspired by one of many eastern mythologies. That would really have been the worst possible name. 7. I am an IT pro and we have trouble with overloaded terminology just within our field. As with particle physics, our objects keep splitting into pieces, too. The only thing to do is have fun with it! What are your thoughts about the popular vision of an object made of matter meeting an object made of antimatter? Now that I’ve read your article, I realize that “an object made of antimatter” is not a well-defined concept, and it might fall apart if we look at what the statement really means, compared with how matter is really composed. Being that “antimatter” is at the center of the popular imagination, what are your comments on hypothetical, macroscopic anti-objects? • If you could collect enough anti-matter [definition #1] and shield it from all the matter [definition #1] that’s flying around (stray electrons and the like), then there would be no problem in principle to construct anti-salt and anti-steel and anti-cells and anti-cars. The laws of nature are sufficiently symmetric (not exactly, but very close) that anything you can do with matter you could do with anti-matter. And indeed if you brought any significant amount of matter in contact with any significant amount of anti-matter you’d make an explosion. But no one has any practical reason to try to construct large amounts of anti-matter. The closest we’ve gotten is making powerful beams of anti-protons and anti-electrons (positrons) for use in particle physics experiments. But the amounts of energy that you could obtain by slamming those beams into a wall is small. In fact that’s exactly what happens to those beams when we’re done with them; we slam them into a wall, underground somewhere. The wall does heat up, but mostly because the anti-electrons or anti-protons are traveling really fast and have lots of motion-energy — not because of the energy released when they find an electron or proton and annihilate to something else (photons or pions, for instance.) And as far as we can tell, the part of the universe that we can see does not naturally have large amounts of anti-matter anywhere. [If there were regions of the universe with large amounts of anti-matter, at the border between regions of matter and regions of anti-matter you’d expect to see large quantities of photons with very particular energies emitted. We don’t see signs of such borders.] • Thanks, Professor. Two things about our hypothetical antimatter #1, that I hope might surprise me. One, is that I don’t understand what happens when two macro objects “touch,” but I’ve been told it has mostly to do with the electromagnetic force. Are there subtle effects, like the exclusion principle, that changes the way antielectron shells would behave near electron shells? Two, is that I’ve read some interesting history about the early atomic energy laboratories, and the way that fission materials went “prompt critical” almost always interrupted the nuclear reaction, milliseconds after it began. It was as if the nature of the reaction was to defuse itself, contrary to popular imagination. So when the first few leptons meet their anti-partners, and throw off some lightweight particles, what happens next? Their atoms become ions and their molecules would break apart, for one, if they have time. But how do we expect the nucleii to come in contact? Would we get more of a chemical than a nuclear reaction? And if there was an explosive force, wouldn’t it force the macroscopic objects apart? That’s assuming they were solid objects, as in the popular imagination; if gas met anti-gas, or liquid met anti-liquid, the macroscopic dynamics would be quite different. That’s why I wonder if, perhaps, objects and anti-objects might behave differently together than the popular imagination dictates. • Ah, I see what you’re asking. There is no question if you took two cubes, one of matter and one of antimatter, and brought them safely together, the actual contact would be instantly different from the contact between matter and matter. At the surfaces of contact, the electrons on the outskirts of the atoms and the positrons (anti-electrons) on the outskirts of the anti-atoms would start finding each other on very short microscopic time scales, and immediately begin turning into pairs of photons of 511,000 electron-volts of energy each. These “gamma rays” would then create an electromagnetic shower of particles: lower-energy electrons, positrons and photons. Since it only takes a few electron-volts of energy to rip the electrons off an atom (or positrons off an anti-atom), all the atoms and the anti-atoms near the surface would be quickly disrupted, vaporizing the material in that region. The force from the released energetic particles smashing into the remaining atoms of the cubes would most definitely push the cubes apart, just as in the fission experiments you mentioned. So there’d be an explosion, but only of the material nearest the surface of contact, and unless the two cubes were slammed together with enormous energy, as in a fission bomb, you’d only get annihilation near the contact surface. • I am sure that someday soon people will make many millions of anti-atoms. One just has to remember that a glass of water has something like a million million million million atoms of hydrogen. I might be off by a factor of a thousand or so (I didn’t check the number carefully), but you get the point. 8. Now you reduced everything to a word ; stuff , which can have mass , energy , can do work ……but what is stuff , you said what it can do but if we want to go deeper is it the end of our search ? it is all some kind of circular definition ….some kind of dna-protein closed cycle !! where we are lost. 9. I think is quite misleading to say such a thing as “you can’t have an elementary particle without a field, but you can have a field without any particles” because the concept of particle depends on the observer, what is vacuum in one frame of reference has lots of particles in another (accelerated) frame. If the Higgs is the caveat of the phrase then, in my mind, not founding the corresponding particle indicates that such a field does not exist. But I’m by no means an expert in elementary particles and could be (very) wrong about this statement. As a complementary note on the anti-matter in the large scale, there are experiments on producing some dozen anti-hydrogen atoms also. Of course nothing macroscopical. Sorry for being somewhat pedantic. • I did not explain what I meant here, because it is technical, but since you ask… A field that strongly interacts with itself, or with other fields, may have no particle states at all. This is a well known fact about conformal field theory. There are many concrete examples in the context of solid state physics, and many hypothetical examples that arise in high-energy physics. Said another way: a non-interacting field always has well-behaved ripples (which in quantum mechanics are made from quanta), a weakly-interacting field has largely well-behaved ripples (though these may have a finite lifetime), but a strong-interacting field may have nothing resembling a ripple at all. So no, what I said is not misleading — it is the particle picture of fields, which assumes that fields are weakly interacting, that is misleading. And you used that weakly-interacting-field intuition in your comment. • Hum, that’s quite interesting and a sign that I should be studying a lot more. My comment was really made with non-interacting fields in mind, so I see what I got wrong. Just to make clear what I meant was that for non-interacting fields the concept of particle is observer-dependent because of things like Unruh Effect. But I couldn’t agree more with you that fields are the essential concept and not particles (the point I was trying to make). Thank you for the explanation. 10. According to all of your articles the end most fundamental stuff is the ripples in fields , i mean if fields are extended stuff , ripples are ” real concrete stuff relative to our senses ” which can have mass , energy , etc. , but if fields are stuff then we reach an absurdity : what is stuff ? stuff is stuff !!?? • Fields are not traditional ‘stuff’ as we know it. They’re almost like things that can have stuff in them. (The ripples.) I think that a nice definition of ‘stuff’ would be ‘particles’ (Ripples.) since they’re the things that ‘have’ all the other things, such as speed, mass, energy and so on which we usually associate with ‘stuff’. But in the end ‘stuff’ is just a label we attach to things. Matter, particles, objects, squiggles, these are all just names that we use. Nature does not care about our neat little ordering systems and so sometimes things can get a bit confusing. 11. Great stuff, Matt. Thanks. A parenthetical question: This is a whole new way of thinking for me, having spent my career in applied physics before 1980. In my struggle to adapt to your post-classical, if not post-modern, way of thinking, I am trying to understand where and how the Higgs mechanism appears in the mathematical constructs that underlie the standard model (what are those mathematical constructs?) and how we know what decay products to expect from Higgs particle decay. If I wanted to find an answer to those two question on Google, what should I google? • Do you understand superconductivity? The photon obtains a mass inside a superconductor when Cooper pairs (represented by a charge-2e scalar field) condense. The W and Z particles obtain a mass within the universe when the Higgs field condenses. The mathematics is almost the same — relativistic instead of non-relativistic, non-Abelian instead of Abelian, but it is the same idea. The most important difference is that we know that for the superconductor, the charge-2e scalar is a kind of bound state of two electrons, but for the universe, we don’t know what the Higgs field is yet — whether it is a composite of something else or not, whether it is just one field or several — and that’s what the LHC is aiming to find out. As for how the Higgs decays; that’s a little more complicated. Did you read what I wrote on Standard Model Higgs decays? situation; we have developed some nice methods and we are looking to swap solutions with others, be sure to shoot me an e-mail if interested. • Hello! This post couldn’t be written any better! Reading I will forward this write-up to him. Fairly certain he will have a good read. Thank you for sharing! 12. Is mass constant? Or should I be asking is the Higgs field constant? In the beginning of time (and space) there was finite temperature within the absolute maximum of Fermi spheres, the singularity that created the Big Bang. The expansion of this sphere created time and space and hence a field (presumably the gravitational field). With further expansion and lowering of temperature (energy densities) resonances where created and hence more fields, like EMF and Higgs. Now, if energy is a conserved quantity then the sum of all fields must also be constant. So with the continued expansion of time and space (spacetime) the magnitudes of the various ripples as created by the various fields should be reducing. (I say should because the densities continue to lower both absolutely (larger expanse of the entire universe) and locally around the smaller gravity wells. • 1) We do not understand the Big Bang at the very earliest times with the precision that you suggest. 2) Energy is not in any simple sense a conserved quantity in a rapidly expanding universe. Even if it were, fields are not energies and the sum of all fields isn’t even well defined, so it certainly need not be constant. 3) The mass of particles like electrons is not believed to have been constant over time, because the Higgs field was not constant over time. At very high temperatures the Higgs field would have been zero on average. As the universe cooled, an “electro-weak phase transition”, which must have happened a microscopic time after the Big Bang began, must have taken place — its details are not yet known, because we don’t know enough about the Higgs field yet. But the Higgs field is believed to have changed very rapidly and then settled down to its present non-zero value during that transition. 4) As far as we know, the Higgs field’s value, and the electron mass, have not changed a bit since then. In principle they might have varied over space and time, but there is neither experimental evidence nor good theoretical reason to think they actually did, at least within the part of the universe that we can see and over the time since the electroweak phase transition. (For instance, the success of Big Bang Nucleosynthesis in predicting the original helium to hydrogen ratio of the universe could easily have been messed up had the electron or proton or neutron masses been significantly different from what they are today.) Scientists continue to try experimentally and observationally to test whether there is any sign of variation. 13. Hi I was woundering if energy is conserved in general relativity. what i have read on the internet has left me confused. Also woundering where the energy of the cmb has gone. • Ummm… I don’t think I can give a very good answer here that I’m ready to stand behind. Let me say two things. 1) LOCALLY (that is, in any small region of space, for a short time) energy and momentum are conserved the way we are used to. 2) GLOBALLY (this is, across regions or across eons where gravitational fields are very important, such as the universe as a whole) you have to be much more careful about how you define the total energy of a system. It’s quite subtle, and can’t always be done. And I’m not expert enough to answer off the cuff and get all the details right as to when you can and when you can’t, and how you do it in the cases when you can. [Been too long since I reviewed this subject, which is mostly outside my research…] If what you mean, in asking about the energy of the cosmic microwave background radiation (cmb), is how did the photons lose energy as they cooled off during the universe’s expansion, one way to answer that is to say that the expansion of space itself took the energy from the photons, and from all the other particles too. But I’m not sure that’s super-intuitive. I know an intuitive answer that isn’t really correct (namely, to consider how photons in a box lose energy if the box expands). Maybe a general relativity expert here knows a better intuitive and also correct response. H = U + pV H is the enthalpy of the system U is the internal energy of the system p is the pressure at the boundary of the system and its environment V is the volume of the system. As lup mentioned this equation is somewhat confusing. It can be interpreted as the energy is spent to apply pressure to the boundary and expand the volume. But that interpretation implies there is an external environment outside our universe. Are we living in a bubble within a bigger bubble, in a asymmetric phase within a symmetric environment. Are there other asymmetric bubbles nearby? The thing that amazes me Professor is regardless of which theory you believe non explain the real nature of energy. Why was the temperature so high at the Big Bang? Is the symmetric phase an “infinitely” stretched spring that when broken releases all its energy at a very short time (space) and hence the large energy density, temperature? • So energy is conserved when cmb photons have thier wavelength stretched by the expanding universe? is this energy helping to expand the universe further or is it more like a stored potential energy? • The thing about energy in general relativity is that for it to be defined globally the spacetime must be stationary, which is a form of saying that “the space is the same at every time”. The problem is that exapanding universes are not the same at every time, so that energy is not conserved. With that in mind the cmb energy has not gone anywhere but just disappeared. We can’t really ask where has it gone because that implies it is conserved. Since it’s not, the energy has just gone away. I don’t think this has any intuitive answer. The closest to intuitive I have heard is the “photon in a box” thing, which I kind of like even if is not precise. The only thing one must remember is that in general relativity there is no “potential energy” related to gravitational fields, so the energy of photons is not stored in gravity, it has really gone away. To sum everything: the energy of cmb is not conserved when the wavelenght of photons is increased by universe expansion, and the energy is not “stored” in any potential form; it has really gone away. • I’m sure you have heard of the seemingly obsolete theory of “tired “light – could the “hot”photon’s energy not drive the expansion of the universe : in other words red-shifting because the wave packets are themselves expanding (such as water waves dissapate their energy over distance) – rather than being stretched to red because an unknown dark energy is pulling them apart? • Einstein’s equations for the expansion rate of the universe work very well; they are used in deriving the prediction for how much helium there is relative to hydrogen, and in predicting the cosmic microwave background spectrum, both of which agree very well with data. [A nice (if a little out of date) discussion of the latter appears here: ] This is all done without appealing to anything like “tired light”. Moreover, the dominant energy density in the universe hasn’t been from light (and other very light-weight particles such as neutrinos) since the universe was 100,000 years old or so. So it’s hard to suggest that light could be essential to the expansion during most of the last 13.7 billion years. I’m not the world’s expert, but I suspect that with the current precision available in cosmology, there isn’t a lot of room left for exotic theories of why and how the universe is expanding. Of course we don’t know for sure what got the Big Bang started, though there are plenty of theories of that (under the name of “reheating following inflation”). • yes-but as far as I understand Einstein introduced the cosmological constant to make his equations fit the observable reality (non-collapsing universe) rather than this being a mathematical necessity per se and as you point out he seemed have done so particularly well. I also did not want to imply that dark energy be the dominant energy of the universe – it only has to be stronger than the universal gravitational force and supposedly wins the upper hand the further the mass particles are apart. But on the same note, if the universe continues to expand eventually the photons of the background radiation will be stretched beyond the temperature of absolute zero – do they then cease to exist?, do they blend into and become (non-particular) elements of the electromagnetic field? – or could that result in the end of the expansion phase and gravity, however weak, but now unopposed lead to the beginning of the hypothesised big crunch? • Thanks heaps…thats certainly answered my question. The only remaining puzzle for me is that i thought that energy conservation was a consequence of a the laws of physics dont change with time? • So the subtlety here is: if time is just something that sits there and the laws of physics operate within it, then yes, conservation of total energy follows from Noether’s theorem. But once time starts participating in the physics — as it does in Einstein’s theory, where space and time are actually part of the physical phenomena — then to ask whether the laws of nature change with time becomes subtle. For example, you can’t even necessary define a global notion of time in a sufficiently curved space. What survives of our usual notion is that within sufficiently small and weakly-curved regions of space and time, the laws of nature do behave in a time- and space-independent way, to a sufficiently good approximation that energy and momentum are conserved there. This is the notion of LOCAL conservation of energy and momentum. There is still a Noether theorem and still a conservation law, but it applies locally, not globally across all of space and all of time (except in special circumstances, such as a time-independent space-time.) (Technically: there are energy- and momentum-currents that are locally conserved.) • Thanks Matt, thats cleared up a lot of things. Cesar mentioned that the cmb photon energy is really just going away (ceasing to exist) as the universe expands. Can we have the opposite – can we have energy appear that wasnt there before? • Well lup, yes we can have the opposite and it’s pretty much standard. The idea behind energy “going away” is that since the universe is expanding it “stretches” the photon wavelenght and consequentially it’s energy decreases. In order to have the opposite effect all we need is a contracting universe, like those cyclic universe models. To be very clear, in those cases there is so much mass in the whole universe that at some point is stops expanding and starts collapsing. During the contracting phase the photon wavelenght is “compressed” by the contraction of space and the energy increases. It’s really like watching the expanding universe going backwards in time. But remmember that this is just a model, the current astrophysical data supports the accelerated expanding universe which kind of eliminates the recollapsing universes in such a naive way. But from the theoretical point of view there are no problems with photons “heating up”. 14. Hi Matt ! An experiment consisting of a holometer is currently in the design stages at the Fermilab centre. Headed by Dr Craig Hogan, the aim is to determine if spacetime has holographic properties by attempting to measure a quantifiable planck unit. I know that a holographic field is a constructive interference pattern with information encoded in the boundary of the pattern.I also know that I am voyaging into the realms of speculation by suggesting this, but could the fermions be nodal points of a standing wave? Certainly, the fact that they are non-locally connected seems consistent with this. If they are, then could this also mean that mass is a measurement of the negative interferometric-visibility density of a Higgs scalar field instead of a particle ? Could it also imply that the laws of physics are an hierarchical layer of anti-nodal displacement cycles, generated by an underlying constructive interference pattern state, with the gauge bosons being the anti-nodes themselves ? I apologise if these questions seem rather wacky, but ever since I heard of the Fermilab experiment I’ve been racking my brains trying to figure out how the holographic principle would work in practice, and how it could relate to the search for a non-standard model Higgs effect ! • Much as I like and respect Craig Hogan, I’m pretty skeptical about this experiment, I must say. I suspect that the effects he’s looking for are vastly too small to be detected. As far as I can see, there’s absolutely no need for there to be any connection between the Holographic Principle and the Higgs mechanism. One operates on the weak nuclear force at the energy scale of 250 GeV or so; the other relates to space-time viewed at the energy scale of 10,000,000,000,000,000 GeV or so. I’m afraid that the answers to all of your questions are “no” — or better, “be a lot more precise”. Remember particles aren’t little marbles; they’re ripples in fields, and you have to explain the fields first. We have very successful quantum field theory for fermion fields and gauge boson fields, working in some cases to one part in a trillion; so you have to tell me how you could reformulate all of what we know about quantum fields to make the fermions come out as nodes in some kind of standing wave and interacting properly with gauge bosons that come out as anti-nodes in some kind of standing wave. Sounds like an extremely tall mathematical order and I have no idea whatsoever how you would start to make that work. You can’t do theoretical physics with words, because you end up just speaking ambiguities. You have to do math — that’s both the essential part and the hard part, not just because math is technical but because most math you try to write down will be self-inconsistent or inconsistent with existing experiments. Einstein is widely quoted in our culture. But if you read his papers, you’ll find that all those nice-sounding words are backed up with solid calculations — and that’s why it isn’t ambiguous what he means when he starts talking about the subtleties of special and general relativity. And he always checks that what he’s proposed is consistent with existing experiments. 15. In all your posts you mentioned fields so many times , you never explained what is it , the max. you said is ; fields are stuff that can have E , m , charges…. but is this all of what we can define a field ? what are fields , i know that fundamentals cannot be defined as there are nothing more basic to refer to it . Are we to stop at a word ; field? Are fields an ontological reality of a mathematical representation of our observations? Forgive my insistence but i cannot stop at mid road. • In the current way of thinking about the world, fields are about as far as you can go. In specific attempts to go beyond this way of thinking, fields can be manifestations of other things. For example, in theories with extra dimensions, some (but not all) fields can be manifestations of the shapes and sizes of dimensions that are too small for us to see. In other attempts, some of the known fields can be themselves made out of other fields which are more fundamental. But all of these attempts are speculative; we don’t know which ones are correct. So I would say that the fields are currently viewed as the fundamental ingredients; that is where things currently stop. Even space and time are to be understood in terms of gravitational fields. However, knowledge accumulates over time, and what we think is fundamental may change. There can also be multiple equivalent interpretations of the same information — two ways of looking at a problem that have exactly the same mathematics and the same physical predictions. Philosophers are frustrated by this ambiguity, but theoretical physicists have learned that we have to remain light on our feet. • Is it valid to say fields are any change (quantitatively and/or qualitatively) of one or more quantum numbers over the space and/or time domain? In other words, they are defined by the topography of spacetime. I speculate this way because as fundamentals they must all be derived back to the initial field, whether it is gravitation or something more fundamental like vortices created by the rotations of the three dimensional space. This is why in my early post I speculated that the sum of all fields must be constant, because spacetime has an orderly progression. So my logic is there must be a constant constraint that drives order otherwise we would have global chaos. • It’s very important to distinguish what is known (by combining experiments with a clear theoretical framework) from what is speculation and may not end up surviving experimental tests. You say: “as fundamentals they must all be derived back to the initial field”. We don’t know that. You say: “spacetime has an orderly progression. So my logic is there must be a constant constraint that drives order otherwise we would have global chaos.” Again, we don’t know that. You can’t really talk about fields as a change in quantum numbers, no. Quantum numbers are very specific things: they are labels of certain quantum states; in particular, they are eigenvalues of overall quantum operators that are meaningful in specific quantum states. Quantum fields are a more general concept. For example, the electric charge of an electron is a quantum number; but there is no field for that. And conversely, most states involving the electromagnetic field do not have a definite value for the field, and so there’s no sense in which there is a suitable quantum number associated to those states. The electromagnetic field from freshman year physics is the best place to start really understanding classical fields. Quantum fields are like classical fields in that they can support waves; they are unlike them in that the waves cannot come with arbitrary amplitude, but instead must have an amplitude equal to a minimal value (one quantum) times any integer. I’m afraid they are unlike them in a lot of other subtle ways too. • “.. but instead must have an amplitude equal to a minimal value (one quantum) times any integer …” Interesting, is that because we over simplified the math by introducing renormalization because of our inability to visualize the physics, (reached our limit of our conscious awareness to map nature’s secrets). I know that Dirac and Feynman were concern with renormalization because it would contaminate the math in to most of fundamental ways and get us trapped in a our own math. Is renormalization prevent us to involve naturally into more advance physics? Dirac’s criticism was the most persistent. As late as 1975, he was saying: “Most physicists are very satisfied with the situation. They say: ‘Quantum electrodynamics is a good theory and we do not have to worry about it any more.’ I must say that I am very dissatisfied with the situation, because this so-called ‘good theory’ does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!” Another important critic was Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in 1985: “The shell game that we play … is technically called ‘renormalization’. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.” I am inclined to agree with Dirac and Feynman in that we need a better handling of the infinities and constraints to make real progress. I am afraid our ability to experiment is getting very quickly to a stagnation point because of the limits of our machines. I also want to take this opportunity to stress how important it is to convince NASA and their handlers that more resources should be spent in sensors, telescopes, and space experiments than the very expensive human spaceflight projects. • If you think renormalization is about infinities (or that it has to do with an inability to visualize the physics), you have not understood it. Unfortunately most textbooks still talk about it in terms of removing infinities. This is deeply unfortunate and misleading, because in fact, even in theories with no infinities, there is renormalization. Even in quantum mechanics, in the anharmonic oscillator, there is a perfectly finite renormalization. Once you have understood that, then you can understand that renormalization (both perturbative and non-perturbative) has nothing to do with the infinities themselves, but with something more physical and deeper; and you can also see which types of infinities are acceptable in quantum field theory and what types are not. Dirac clearly never grasped this point. As for Feynman, I am sorry I never got to ask him exactly what he meant by his comment. But let’s just say that our comprehension of quantum field theory has come a long way since 1985. Unfortunately this is an extremely subtle and technical subject (which is why almost nobody does it in a sensible way) and I doubt I will be able to explain it on this website. At some point I may write a short technical monograph about it. But suffice it to say that I think you’re not even close to being on the right track. • There are interesting discussions going on below this very nice article 🙂 In particular, I feel really affected by the second paragraph in Prof. Strassler s above comment ending with “… fields can be manifestations of the shapes and sizes of dimensions that are too small for us to see.” 😛 😉 🙂 • Hi Matt, Since you mentioned that “Even space and time are to be understood in terms of gravitational fields”, I want to remark that I have found this concept very tricky to explain to non-experts. We’re so used to thinking about fields as things that “live in” static space and time that understanding space and time themselves in terms of fields is hard to wrap our minds around. If someday you have the time and inclination to tackle an article about this issue, I think many of your readers would appreciate it, and I would be interested to see what approach you take. 16. I read in an article about quantum gravity in stanford encyclopedia of philosophy that fields are specific properties of space-time itself , is this correct ? • No. They can be. But often they are not, even in string theory. My advice: don’t get your physics from philosophy encyclopedias (and I wouldn’t get your philosophy from physics encyclopedias either.) 17. But my dear friend , physics is an ontological science , it is interconnected to philosophy, any thing you say about fundamental level of existence IS philosophy. • (THANKS). This is an interesting point, and I do think I disagree. Physics is interconnected to philosophy, yes. But physics is a quantitative, predictive enterprise. Theoretical physicists will often accept levels of understanding or ambiguity that are unacceptable to philosophers. (And to mathematicians!) Collectively, we are typically much more practically minded than our colleagues in either of these subjects. This allows us to make rapid, but often very ragged, progress. Often when we learn how to calculate something, it is some years, even decades, before we understand what we’ve actually learned well enough for either mathematicians on the one hand, and philosophers on the other, to engage with it. For instance: what people said about particles and fields in the 1950s, before quantum field theory was understood at the much deeper level that is available to us today, was in many cases deeply misleading philosophically. The story I tell you today is based on insights that emerged in the 60s, 70s, 80s and 90s. Indeed, a good chunk of what I learned in grad school was misleading philosophically. Meanwhile mathematicians still haven’t figured out what we’re doing in quantum field theory… and I wish they had, because there are many puzzles about it that we can’t solve. • “physics is a quantitative, predictive enterprise” up to the point where we decide what DoFs and what formalism to use when we construct models for experiments. If we choose the wrong number of DoFs or formalism, we find the accuracy is OK, but not great, but the inaccuracy gives us precious little clue as to what different DoFs or formalism to use. We can’t do much better than guess again. Hence the enormous disparity of models that are published in journals. We also can’t measure the Lagrangian in a comparable way to measuring the field strength, say, supposing we have managed to guess the right DoFs and formalism. The process of guessing again is done slightly better by some Physicists than by others, but that x-factor is not as quantitative and predictive as we’d like it to be. We can call this guessing Foundations of Physics (or we could call it Philosophy done by experienced Physicists, but that’s just names, silly to argue over), in contrast to Philosophy of Physics (Physics done by Philosophers), but there’s a lot of crossover between these two academic communities, with good ideas on both sides being taken seriously by people on both sides. Of course it’s much easier to generate bad ideas, when the subject matter is beyond computationally complex because we don’t know what questions are possible. In the end, however, we can waste 50 years in quantitative predictions if we are using the wrong DoFs and formalism, so it’s worth some people working concurrently on what we think we’re doing, even if many of them waste their time. 18. I think no time is wasted once they are on the grand march to understand existence , no one will put his hand on the ultimate truth , but science and philosophy are 2 faces of same coin……the sacred search for what IS and our place in it , it is the most precious effort humans can perform, even a simple layperson question can lead to some great answer it is our duty and our destiny as humans to think ,to reflect , and to wonder ……nothing is greater than our feeling of awe in front of beauty , design , perfection in a realm which is not perfect itself ….this is the meaning of being human…to enjoy life , to love , to care , to share , to embrace the whole of creation. 19. once upon a time in space,i tried to write a thesis.I went back in time and forward in space…today became yesterday and zero was squared…BANG!!! the singularity of zero was infinite(-0.0000….)and eternal (+0.0000….)…and thus the first law was laid,from which the universe arose…the expanding singularity of infinite totality,,,a total of 1,and a value of 0…as a totality of one singularity,one could not be measured therefore totality was 0-,or 0+ depending on direction as it relates to the opposing direction…giving rise to every thing that followed,,what i speak of is a singularity of time,,,but could have easily mistaken it for the higgs and thee higgs field…or possibly or on a larger scale dark energy,and dark matter…its very hard to make strong case to include all the that this theory covers,without producing a thesis…so just a little snippet for now,as i reduce it to simplicity,,,its a binary universes…where zero has two values,,0,and minus 0,from minus0,0 has the value of enxpanding 1,or 0.9999……and from 0,minus 0 negative 1infinte,or -0.9999 the simplest way to put it is ‘for every action is an equal and opposite reaction’…sir issac newton’s theory of pure simplicity standing the test of time,,,governing totality,and every singularity…eternity is a very long time in space,any inbalance beteween space time no matter how insignificantly minute has catastrophic cosequences…if for example; the mass value of the universe increased by as little as one quark for every googlemilllion lightyears,eventually the entire universe would become one block of mass…(please note) if you think im being a little flippant to exagerate,im absolutely not!!mass has an eternity to do it in…as does energy.the first most fundamental law governs from totality to infinite singularity…the higgs bozon,and the higgs field,interact with mass in the same manner as time in space,… Time is the equal and opposite reaction,preceeding the action of forming space,,,just as all mass is basically congealed energy,all of space,is congealed time,both the positive,and negative…quantum at the particular level,yet in general its relative… thank you for reading,and i value any input 20. A good direction to take the discussion: Why do fermions feel more like “stuff” than bosons? This brings in a little quantum mechanics, which you might not be ready to do yet. But the “rule” that two fermions cannot be in the same quantum state, while bosons can, seems central to a working definition of “matter.” Intuitively, we expect matter to “take up space,” which fermions do, but bosons not so much. Of course fermions don’t take up space by themselves, since any collection of fermions is always accompanied by bosons transmitting forces between fermions, and so on. Thanks for all the great articles! • Well, if you keep running with this too far you start to stumble… For example: in a world with only gluons and no quarks, the gluons would bind together to make hadrons that are called “glueballs”. Those take up some space. Granted you couldn’t make anything macroscopic out of them. If the electron were a boson, you’d still have hydrogen. That takes up space. And hydrogen’s a fermion; you could make things out of it. In our world, hydrogen is a boson, deuterium (heavy hydrogen, whose nucleus contains a neutron as well as a proton) is a fermion. Is one of them matter and the other not? In certain hypothetical universes, you could make a proton-like fermion out of combining a fermion and a boson. And you can make bosons out of combining fermions. Even in our world, protons contain quarks (fermions, which you’d like to call matter) and gluons (which you’d like to say aren’t part of matter so much as representing the force that holds the fermions together.) But you can imagine a world in which some quarks were fermions and others were bosons, and the number of types of gluons was different — and then you could get fermionic proton-like objects which were made from quark fermions and quark bosons as well as gluons. What would you call the quark bosons? Matter or not? The theory of supersymmetry causes a problem, because it combines fermions and bosons in pairs. It doesn’t make sense to say these pairs are part matter and part force; when you write the equations down, the boson-fermion pair appear in a single mathematical object. And in string theory all of these particles can be made from the same type of string. Let’s not forget what we call “dark matter” may be made from bosons. So there’s all sorts of ways in which this line of thinking can break down. At different layers of structure, what you’d want to call matter might change in a profound way. I don’t view the current distinction as likely to survive into the future too much further. • Which is valid, matter is standing spherical waves oscillating at the Compton wavelength or is matter a Fermi sphere with such a radius so as to give a Fermi energy equivalent to the mass-energy of that particular particle? I can understand Pauli exclusion principle via the Fermi sphere definition but not with the standing wave theory. Are we missing some math? • Well, calling fermions the essential ingredient to “matter that takes up space” still seems consistent to me, as long as we add a caveat about how closely you look. An atom or molecule can have the properties of a boson when viewed from the appropriate distance, but when you get “too close,” the properties of the fermions that it is made of become important. How about, for now, say “matter that takes up space” must contain quarks, leptons, or both. Whether or not it has the properties of a boson when viewed from a sufficient distance does not keep it from being “matter that takes up space.” • In a theory with no quarks, the gluons would bind up into hadrons called “glueballs”. These hadrons would take up some space too, individually. Granted I couldn’t make a lattice out of glueballs. But I suspect there are bosonic systems where I could do make a lattice, if I could arrange for some short-distance repulsion from a short-range force. And it is possible to build fermions as solitons in a theory with only fundamental bosons. So this still raises questions about your dichotomy… I think you’re still relying on very specific properties of our universe that would not necessarily be true in other ones. • How do theoretical physicists verify the validity of using the gamma function and the rather very simple energy – momentum (dispersion relationship, E = ap^s) for deriving the thermal de Broglie wavelength. Does this derivation fall in the argument of whether the renormalization is appropriate for quantisation of the “quantum nature of gas? This approximation seems so critical in defining the true natural of the vacuum, how has it been verified and conversely how do you rule out dimensions above the usual 3 that we perceive to live in? 21. Now if the mass of the proton is all of its quarks E divided by c^2 , how can we express the mass of electron or any primary ” particle ” ? if an electron is at “rest” –whatever that means — what is the meaning of its mass ? even what is its E then? are we running in a circle ? • The mass of the proton is not merely all of its quarks’ energies E divided by c^2 (and to the extent a more precise statement is true, it is true only of a stationary proton.) It’s more complicated; that article is coming. An electron can have definite momentum (including zero) as long it is in a state where all position information is lost. So it can be at rest, yes. And I can figure out its mass-energy E_rest in many ways experimentally. One way is to bring an anti-electron close by, watch the two annihilate into two photons, and measure the energies of the photons (which are pure motion-energy, since the photon has no mass). Since energy is conserved, and the initial energy was (to a very good approximation) the mass-energy of the electron plus the mass-energy of the positron, which is 2 E_rest, the energy of each of the two photons is equal to E_rest for the electron. Divide by c^2 and you have the mass of the electron. See for example 22. Hello Matt, Incredible what a interesting subject physics really is, how much there is to say about it and what thorough knowledge you have of every detail of physics! It reminds me of Feynman. For me having studied theoretical physics but not actually working as a physicist, the first two volumes of Weinberg’s “The Quantum Theory of Fields” were a gift from heaven. Thanks to these book I understand physics at a much more fundamental level than when I was a student, when they talked about gauge invariance, fields, particles, representations, CPT and renormalizability and I had no idea how they were all linked together. A remark: To call matter particles fermions is a good thing because of the stability they give to matter due to the Fermi exclusion principle. Usually we see the bosons as force particles and usually this is not too bad a picture. But what about light-light scattering. Here in the box diagram the fermions are the force particles. • Notice Mark Wallace’s comment here, and my reply. You’ll see that I am a little cautious about calling fermions “matter particles” because it runs you into trouble. In fact, your remark suggests another problem. Indeed, pairs of virtual fermions (more precisely, appropriate quantum disturbances in a fermion field) can cause forces. (Though we should recall that virtual particles aren’t really particles: ) In fact, the force which holds a nucleus together is, from some points of view, due to pions — which are bosons, but are made from fermions (quarks and anti-quarks.) So now we have a force particle made from matter particles. Which means our naming scheme is a mess. There is also a (subtle) way to make fermions from bosons (the word Skyrmion appears here.) You see that this distinction just causes problems. At some point I think you have to take the physical phenomena for what they are and not spend too much time worrying about finding the perfect naming scheme for them. 23. Now i really wonder , what is the source of the intrinsic movements / momentum in all sub-atomic entities , what “pushed ” the quarks or electrons to always be in motion , you said – nothing is at rest – well , what physical mechanism is the CAUSE of all that movements , i am beyond some equations equilibrium , i am at the CAUSE , THE URGE , THE DESIRE !!!!! TO BE IN PERPETUAL MOVEMENT!!!! conservation of energy ?but if movement was generated , what caused its generation ? 24. Let me be bold and see if I, an “outside observer”, can “cause” a “disturbance” and create some “fire” in this “medium”. I say, for the lack of a better theory, God caused the universe to ignite and hence the Big Bang. The boundary of our math only goes to the time-energy uncertainty principle was given in 1945 by L. I. Mandelshtam and I. E. Tamm, as follows. For a quantum system in a non-stationary state ψ and an observable B represented by a self-adjoint operator , the following formula holds: σE x (σB / \| dB / dt \| ) >= Planck’s reduced constant / 2 where σE is the standard deviation of the energy operator in the state ψ, σB stands for the standard deviation of B. Although, the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters Schrödinger equation. It is a lifetime of the state ψ with respect to the observable B. In other words, this is the time after which the expectation value changes appreciably. This principle says the quantum state ψ cannot stay the same forever since that means infinite energy, I hope I am right, 🙂 So, once an external disturbance ignites the charged initial state change will continue until all the useful work is spent. 25. One thing has always puzzled me about dark energy. It is supposed to contribute 74% of the mass-energy density of the universe, but at least naively it doesn’t even seem like it should be commensurate with energy. Obviously that’s wrong. I’m dimly aware that there’s such a thing as the stress-energy tensor. But I would have thought “density” would be a component of the tensor, but it seems like the 74-22-4 decomposition must be based on some kind of norm? I guess my real question is, what does that 74-22-4 decomposition mean? 26. One of the great properties i like much in what Matt. write is his total freedom of any prejudice ane pre-conceptions , he gives us the state of the matter with true honest description , so let me state what — being non-physicist — i understand as the core of all what have been said : 1- We only know that there are something /stuff which is the most – maybe- fundamental kind of physical existence which we call fields of which reality –the thing as it is- we know nothing. 2- That stuff have properties we can observe we call m , E , p , interacting according to pre set codes we call theories , these theories are our POINT OF VIEW AS PER TODAY never to be confused with ultimate reality. Now the main essential difference is : Matt. never claimed that what he says is the final ultimate truth in contrast to 99% of articles and books where they claim that what they present is such……………thanks Matt. , this is the true , honest way of doing science. • I think you are more or less stating my point of view, yes. I don’t know what ultimate reality is and I have no idea how I would come into contact with it. I just know that we have found ways of classifying the objects in the world such that we can predict in great detail how they behave. I can’t tell you that classification is unique or complete. Along these lines I think it is also important to keep in mind that we, as minds, never come in contact with anything physical at all. See the table across the room? What do you know about this table? The only thing you know is the image created somehow in your brain, formed on the basis of electrical impulses down your optic nerve from your eye, which in turn are based on photons impinging on your eye, some of which came down from the sun or from the light bulb in the room, and bounced off the table in just the right direction to enter your retina. There are many steps from your image of the table to the table itself. Even when you touch the table, what you feel is in your brain, created from nerve impulses sent down from your fingers, from nerves firing in response to the deformation of your skin by the inter-atomic forces between your skin and the table. Your brain is not in contact with the table. What you feel is in your brain, not in your fingers. Our senses are no different from the measuring equipment used by scientists, allowing us the ability to detect aspects of the world around us. What we know of the world, through our natural sense organs and through the artificial sense organs of scientific experiments, is always indirect. 27. Objection : NO images or feelings are IN the brain , for the first time in all your presentations you decreed/decided on something where no shred of evidence exists as to the ultimate reality of consciousness / feelings / concepts…..etc. David chalmers wrote an article titled ( consciousness and its place in NATURE ) , that was a prejudice , i sent to him asking : how can you decree that its place IS in nature while your article is searching for its place with no conclusion reached ? that is what i call pre-conception/prejudice where a scientist decide / decree a point of view as fact , i really hope that some day science can free itself from decrees based ONLY on relative time dependent observations far from final absolute knowledge. • You are right, I did not phrase this well. Your statement is, I think, a little too strong — we do know that there are electrical phenomena occurring in the brain that are related in some way to the things we see and think, and we know that damage to areas of the brain (strokes, direct injury, disease) result in correlated damage to conscious experience. But how they are related, we do not know. So I would say that it’s not that there’s “no shred” of evidence — just that there’s no clear understanding of the meaning of the evidence. In any case, all I really wanted to say is that conscious experience does not in any sense involve a direct encounter with the physical objects of which we are conscious. What conscious experience itself arises from, I certainly don’t know. • Neurons —> Bosons Synapse chemicals —> Fermions Eddy currents across the neuron membranes —> Field(s) (EMF) Transfer of chemicals across synapses —> Fields (Colors) Synapse firings —> Atoms (type) Sequence of synaptic firings —> Ensemble of atoms (hadrons) Thoughts —> Relativistic structures Hence, one can speculate that our consciousness is part of the universal consciousness and our body, including our wiring in our brain is just one more fiber and/or groups of fibers of the overall cosmic quilt. • Um — You can speculate all you want, but there’s no math on the left-hand side of your correspondance, while there’s a huge amount of detailed, predictive mathematics on the right side. The reason we take the right-hand side seriously is that it goes along with mathematical equations which predict, correctly, the results of millions of experiments. If you can’t make the link from the right-hand side’s math to some corresponding math for the left-hand side, then we have no reason to believe any correspondance of the form you suggests exists. For example, fermions satisfy a Pauli Exclusion Principle. Are you suggesting synapse chemicals satisfy a similar principle? Do Neurons form condensates the way bosons do? Atoms can be bosons or fermions; are you suggesting that synapse firings can be neurons or synapse chemicals? I insist on precise statements. Because that’s what’s needed for science to get done. • I am glad you brought up Pauli’s Exclusion Principle. I can understand Pauli Exclusion Principle via the Fermi sphere definition but not with the standing wave theory. Are we missing some math? • I think you are confusing some things. Atoms involve a nucleus surrounded by electrons which are standing spherical waves (or more complicated standing waves.) The Pauli Exclusion Principle simply says that no two electrons can be in the same standing wave if their spins have the same orientation. Metals involve matter in which the electrons in a given volume form a Fermi sphere. NOTE: this is not a sphere in physical space. It is a sphere in an abstract space (“momentum-space.”) [The standing waves that electrons in atoms occupy are spheres in physical space.] The vacuum does not have an associated Fermi sphere. The mass-energies of particles in empty space are not associated with a Fermi energy. • Ok, thank you for clarifying. It’s been awhile since I done some of this math so I am quickly trying to catch up before diving into B-E and F-D statistics. So to help me visualize, Pauli Exclusion Principle says if the standing waves are in-phase they cannot interfere due to possible ‘beating” and hence tend to infinite resonance (infinite energy which is not allowed by conservation?) Conversely, a standing wave of 180 degree phase shift will cancel the electron’s wave and bring it down to the ground state, (Dirac’s antiparticle? What does he mean when he describes it, antiparticle, as a hole? Is not opposite phase the same thing?) The different fermions are basically standing waves of different amplitudes and frequencies? Why are the half lives different? So, the Fermi sphere is the “mechanism” of transferring momentum from one standing wave to another? How does that work, superposition? Final question, does the standing wave’s ripples propagate to infinity or do they reduce down to ground state at some radius? Is there any association between this radius and quantum entanglement? • I’m afraid that’s not what the Pauli Exclusion Principle says. You have to first work out, for a particular physical system, what its one-electron states are; then the exclusion principle says that nature cannot put two electrons in the same state. This is not something that I know how to visualize, because it involves a quantum mechanical effect for which there is no visualizable analogue. Not all facts about quantum theory can be represented by a picture in the mind; this is part of what makes it hard. • PS; I apologies for my hieroglyphics but my brain works better with images. i can sit a study a math page and get very quickly but the next day I would forget the details and only the images that were created remain, for a long long time, 🙂 Can one interpret this as saying the second “electron” wave cannot superimpose over the first “electron” wave because the “electric charge” imbalance will “push” the second wave to a different phase, higher state? So it is the combined nucleus-electron interaction that prevents the second “electron” wave from coming too close to the nucleus? What about two loose electrons, can they superimpose or is it that the probability of two electron waves coming close to each other is nil? Does the exclusion principle have any thing to do about the Z and W having mass or is it solely because of the variants of the spin states of these bosons? If I can speculate again, and I have seen the derivation of the equations for the Hiiggs field and I somewhat understand the definition of symmetry “breaking” to give the mass term. But physically specking, is it correct to say that it requires the interaction of two bosons to create a higher enough resonance in the composite wave to give it “mass”? But, again, why do different fermions have different half lives? In other other words why is the electron so stable of the other fermions decay so quickly? • One physical object we do experience more directly is our own brain – or at least parts of it. i think that because certain brain activity appears to have SUBJECTIVE sensations associated with it, there is a certain aspect of the reality around us that is not accessable to our scientific instruments. what do u think? • lup, you say .. I wonder. There are some very interesting ways of configuring “scientific instruments” to do a lot our own brains can achieve. Take a look at this very cool stuff: 28. Back on the subject of photons losing energy in cosmic expansion. If we’re to view that as the result of the absence of any time-like Killing vector and hence no energy conservation, how are we to view gravitational redshift in a Schwarzschild geometry, where is time-translation innvariance? 29. Let us have a mental experiment , a static closed universe where only one electron at rest reside : What is the physical meaning then of its m ? E/ c^2 ? well , what is E ? mc^2 ? see what the problem is ? in all your posts E and m were never defind independent of each other , so , what is E ALONE ? and so for m ……. here we have one electron really at rest , what is the physical reality of its E or of its m are both interconnected in a closed circle where they are in principle beyond our understanding? does E and m really have no independent physical essence that we can ever grasp ? then what is m that belongs to that single static electron ?is it a scale by which we measure the ” amount ” / ” quantity ” of ripples ? 30. Most people are making a major category mistake w.r.t. N.N.W. , neuron networks are designed to achieve specific goals within the chemoelectrical medium of brain and body , N.N.W. are systems of equations designed so that with a system of input -output and pre-set goal it converge to accomodate similar goals. What is missed by most people is the fundamental category chasm between chemoelectrical output and sensations or feelings ……. you can never in principle get feeling as output while your input was electro-chemical impulses , only monists say that nonsense , for a very simple reason : this nonsense is the ONLY one within which materialism is possible. I was obliged to write this to clarify some mistaken comments written above. • If I may concur with this in a simple way: one must distinguish between systems that appear to be conscious and systems that actually are conscious. I believe the rest of you experience consciousness because you are similar to me, and I know I experience it. But I don’t have any way to actually check experimentally that you are experiencing what, from the outside, you appear to be. That fundamental problem — the lack of an experimental test of the experience itself — will make it difficult to determine whether a machine which behaves as though conscious actually is conscious. And a question for which there is no experimental test, even in principle, lies outside the reach of science… so until someone invents a convincing test that establishes whether a particular creature, natural or artificial, experiences its apparent consciousness, I cannot easily be convinced that the issue is ever going to be understood scientifically. • The thing that we are most familiar with is also the most difficult and puzzling of all known phenomena. but nature creates it effortlessly every day. 31. Neurons and neuron networks can be reduced to the realm of forces and particles while sensations and feelings can never be reduced to any thing within material universe , as such all talks about machine consciousness are complete void ………beware of A.I. deception with completely material category claimed to represent completely non material category , but if science retreats , here comes logic and rationality …..a law ….a necessity : any non-material category can never in principle be reduced to material category ….sensations and feelings can never obtained from fields , particles and forces . This is a settled fact once all concerned be free of pre-conceptions , prejudice and materialistic naturalistic philosophy , for any individual , he is free in his stand , but once he addresses the public no personal world view is allowed to pollute the innocents . 32. TO LUP : Again you are trapped in the fallacy of deciding/decreeing a stand where no real scientific proof exists , and all logical / rational understanding converge…… you decreed that NATURE create consciousness , this is total void statement : What is nature ? laws ?fields? both have no power to implement any thing in reality. Nature is the material universe while consciousness is beyond m and E . What is your proof for your decree ? See what i mean ? this is exactly the stand 99% of scientists adopt …to claim the unproved as the ultimate fact, we need to be humble in front of the majestic creation of GOD seeing that human consciousness is the ultimate awe inspiring reality as it is the object of feeling and the feeling itself. You need to read lots of sources just to start to feel the awesome tremendous meaning of the reality of what made you understand and decide………..good luck my friend. • Did you really have to bring god into this! Nature is what I would regard as the knowable reality what ever affects or had observable affects in our reality. Laws describe what we observe in reality and fields are what we call certain entities with certain apparent characteristics. The things we are describing to affect our reality but we can only name the properties they possess which is equivalent to saying this is how they affect our reality. 33. To I was just trying to make the point that we have no idea how to build a robot that is conscious but a fertilised human egg will divide many times over the course of 9 months according to the laws of chemistry/nature etc and produce a baby that is conscious. And its done it countless times since the dawn of the human race. I agree that conscious experience appears to be something completely different from what we call the materlistic world (for a start is a private and subjective not objective ‘thing’) but where ever it appears (at least in this universe/reality) there is also this biological organ we call a brain present. Surley u see nature as GODs design – a nature that is able to support the emergence of consciouness experience. Perhaps that is a better way to say it. • Sorry but consciousness is a manifestations derived from the billions of permutations of electrical / biologic signals. When we are awake we can control our thinking via negative biofeedback processing of billions of “digitized” thoughts. And when we sleep, because we have no conscious control, our dreams become more chaotic and random in nature, but we can remember those which are channeled in conscious thoughts, make sense to us because we created them during our awaken phase. If there is a universal consciousness, (and I believe there could be due to the striking similarity between the universe and our own brain structures. see a wonderful video below) it would not belong to God since the universal consciousness would be like our own, manifestations derived from the billions of permutations of particle interactions. 34. If Matt has trouble defining the rules of the game, perhaps we are just goldfish in a bowl. No. I refuse to give up on science. 35. I was reading through this and several comments, and all seem to agree that Z and W bosons of the weak force act as massive particles because of the Higgs field, but prior I was reading Fear of Physics by Lawrence M. Krauss, and came to the conclusion that these bosons act as massive particles because of virtual particles. As virtual particles pop into existence, it takes energy to do so, so pop back out to conserve energy and momentum, but if two virtual particles are attracted to each other, and instead of one pair, fill the entire system being observed, if these particles were to be adjusted in a way to have less energy than in a system with no such particles, due to the fact that two objects bound together have less energy than two separate objects, the system could fill with these particles, which could in turn affect the weak force bosons, causing them to act massive, for the same reason that photons act as massive particles in superconductors. Either I am entirely wrong (very likely) or is it possible that either these virtual particles are themselves the Higgs field, or that these virtual particles are waves in the Higgs field, rather than the specific weak force boson field? P.S. this is a great sight for the novice high school student of 16! Thank you for such a great resource professor! That conclusion is not correct. One thing you should know: `virtual particles’ are not particles (Krauss is not alone in not explaining this well); you should read at some point. Also, have you read the Higgs FAQ yet? If not you should. Now — there are two separate questions: 1) What does the Higgs field do? 2) What is the Higgs field made from (if anything)? Answer to 1) The Higgs field gives masses to the W and Z particles; and a Higgs-like field (often called the Landau-Ginsburg field) gives mass to the photon inside a superconductor. Answer to 2) It could be a single fundamental scalar field, or several such fields, or it could be made as a bound state (simple or complicated) of other particles. The Landau-Ginsburg field turns out to be a field made from electron-electron bound states (Cooper pairs) bound together by phonons. That need not have been the case; it could have been a fundamental field of spin two, or a much more complex object that Krauss would have had even more trouble describing, and it would still have given mass to photons in a superconductor and given us all the phenomena of superconductivity. The Higgs field CANNOT be of a precisely similar form as the Landau-Ginsburg field; Cooper-pair-like objects would break Einstein’s relativity equations. There is something analogous (called `technicolor’) but then the binding between the virtual particles that make up the Higgs field must in such a case be much, much stronger, relatively speaking, than in a typical superconductor. (Incidentally, if there really is a Higgs particle with a mass around 125 GeV/c-squared, technicolor is significantly disfavored.) So you see that there is no contradiction between the two statements; they are simply operating a different levels. Question (1) is about what the Higgs field does, which we know, but knowing the answer does not answer question (2); question (2) is about the very nature of the Higgs field, which we do NOT know, and that’s why we have a Large Hadron Collider, to help us answer this question. Similarly, for superconductors, the answer to question (1) was known long before the answer to question (2). 36. I am a bit confused: I find your distinctions very insightful and helpful, but my understanding is not satisfied: 1. What is an energy what is matter? I think you should mention positive definitions so one can refute them otherwise they are too vague. 2. Nothing wrong with vagueness though, words as said are problematic, but when words have an intrinsic failure, I tend to attribute it to the failure of ‘way to observe the world as human’. I.e. when our senses cannot grab the world. EX: Time in SR as 4th dimension (or 11 dimensions in the strings theory, or particle-vague duality) – the 4th dimension is nothing but a poetic metaphor of the universe that help me to understand the meaning of it, knowing that I am condemn to observe my surrounding in 3 dimensions only and cannot really grab this dimension. Although in physics these concepts makes perfect sense, they are limited to mathematical language and cannot be translated into human one. YOU are doing a precious job in helping me and other to have a better understanding but if words fail it is because it just does not match our human perception. BTW, no problem for me, as after Copernicus I lost hope… 3. A characteristic of an object is also an object UNLESS this characteristic in a a-physical attribute. Ex: A wooden table vs. nice table. As I can assume you refereed only to physical attributes, I cannot see how they are not part of the object itself. 4. “…I have height and weight; that does not mean I am height and weight” – this is at best a problematic example. What is the I is a long philosophical question and unless you are a pure materialist the I to which you refer is not a physical object and therefore cannot be defined by these adjectives (if you are a materialist you should define what that ‘I’ is). • you’re in reality a excellent webmaster. The web site loading speed Also, The contents are masterwork. you have done a magnificent task in this subject! 37. you say that a Photon’s energy is equal to it’s motion, every calculation I have seen, says it’s energy is dependent on it’s frequency, not it’s motion. You also say that A Photon is not pure energy, but that it is a particle made up out of stuff, since a resting photon has never been seen, are you sure that a photon is not a wave/particle potential of pure energy. It seems to me that the reason a Photon can travel at C is that it is mass less, hence it experiences no “drag”, per se, as it travels. If you are saying that a Photon is not energy but has energy, then it can not be mass less as a particle it seems that in a Photons wave/particle duality. As a particle it has mass and interacts, as a wave it is pure energy and has only momentum.. A photon as pure energy explains how it can be mass less and yet still carry information as a wave. 38. Okay, I checked out the links you provided, very good information, any knowledge into this subject I welcome into my deposit of knowledge. I understand your phrase of “motion energy” in the context it was used now. As far as mass experiencing drag per se, I was using the word in context to a Photon moving in space at “C”, having no mass it experiences no loss of motion, maybe the word “drag” is not the best to use, maybe friction, or another one is more appropriate. As far as particle/wave duality, my understanding on a Photon is that it can either be a wave or a particle depending upon observation and/or.interaction as illustrated in the double slit experiment. As far as “mass” I see this the same as the emergence of consciousness, for example one neuron can not create consciousness, just as one particle can not create significant mass, but it is the number of neurons and the complexity of the neural networks that create consciousness, I see object mass in the same way, once a grouping of particles come together into a certain “level” of complexity, then “mass’ follows in the same way that consciousness follows, that is how I see the “Higgs” field, this field is the result of this complexity, the more complex the structure and the more mass it has, the more it is affected by Gravity, Hence the heavier elements are more dense, thereby they have more mass, gold or lead being a good example, but there are of course many exceptions. And as far as a Photon traveling at “c” as a wave of energy I use this observation as my basis, as one example. This link shows a cluster of stars 1 Billion light years away, which means that each Photon from this star had to travel for 1 Billion years in order to reach our solar system. (More amazing to me, is the fact that some of those stars probably do not even exist anymore, yet we still are able to see them as they did exist 1 Billion years ago). In order for the Photons to travel over space and time for 1 Billion years, it seems to me the probability that a particle could travel these distance and time frames at ‘c” for 1 Billion years and not decay is remote. I see a Photon as being a ‘timeless” carrier of information, and I do not see how a particle could travel at “c” for 1 Billion years, and not decay. It seems more probable that in order to do that, it would have to be a wave of energy to accomplish this amazing feat. Of course since no human being has ever “observed” a Photon traveling at “c” or observed one at “rest”, Ie. snapped a Photo of one in either state I think it is too early to come to any conclusions as to exactly how a Photon for example can travel through space and time for 1 Billion years all the time, maintaining “C”, and not decay. I appreciate you quick response to my inquiry, and realize that so many of these answers are not going to be available until the technology to settle some of these big issues comes into play. But I am always striving to improve my outlook and knowledge on these matters, and value any and all inputs I can gather.. Thank You 39. as a side note : The total energy contained in an object is identified with its mass, and energy cannot be created or destroyed. When matter (ordinary material particles) is changed into energy (such as energy of motion, or into radiation), the mass of the system does not change through the transformation process. this does allow for a Photon to change from a wave of energy then into a particle, and so on. Maybe this is how it travels at “c” and never decay’s as a wave of energy it is basically eternal and there fore could travel at “c” for infinity, or for 1 Billion years for example, and then as interaction or observation dictates it changes into a particle, and interacts as dictated by the contact. your thoughts on this . 40. Another side note ( should of put this all on one message, I apologize ): .the initial energy for a Photon comes from its source ( for example a star ) Energy may be stored in systems without being present as matter, or as kinetic or electromagnetic energy. Stored energy is created whenever a particle has been moved through a field it interacts with (requiring a force to do so), but the energy to accomplish this is stored as a new position of the particles in the field—a configuration that must be “held” or fixed by a different type of force (otherwise, the new configuration would resolve itself by the field pushing or pulling the particle back toward its previous position). This type of energy “stored” by force-fields and particles that have been forced into a new physical configuration in the field by doing work on them by another system, is referred to as potential energy Any form of energy may be transformed into another form. For example, all types of potential energy are converted into kinetic energy when the objects are given freedom to move to different position This mathematical entanglement of energy and time also results in the uncertainty principle – it is impossible to define the exact amount of energy during any definite time interval. The uncertainty principle should not be confused with energy conservation – rather it provides mathematical limits to which energy can in principle be defined and measured. It seems to me that as a wave of (electromagnetic) energy a photon travels at ‘C” then changes to a particle upon observation or interaction. Everything I read leads to this conclusion. 41. sorry, I just seen this quote from one the the links you gave me. your quote’s : light waves (and the waves of any relativistic field satisfying the relativistic Class 0 equation) move at the speed c. the energy stored in the wave is E = (n+1/2) h ν where h is Planck’s constant, which always appears when quantum mechanics is important. In other words, the energy associated with each quantum of oscillation depends only on the frequency of oscillation of the wave, and equals E = h ν (for each additional quantum of oscillation) This relation was first suggested, for light waves specifically, by Einstein, in 1905, in his proposed explanation of the photo-electric effect. Please explain how this does not agree with my statements. 42. Date: 05 November 2012 Time: 07:29 AM ET Now, for the first time, a new type of experiment has shown light behaving like both a particle and a wave simultaneously, providing a new dimension to the quandary that could help reveal the true nature of light, Depending on which type of experiment is used, light, or any other type of particle, will behave like a particle or like a wave. So far, both aspects of light’s nature haven’t been observed at the same time. But still, scientists have wondered, does light switch from being a particle to being a wave depending on the circumstance? Or is light always both a particle and a wave simultaneously? 43. You said in this article “Early in the universe, when the temperature was trillions of degrees and even hotter, the electron was what cosmologists consider radiation. Today, with the universe much cooler, the electron is in the category of matter.” Why does “high temperature” make the Higgs (and gluon?) field being “zero” on average early in the universe? Why/how does “temperature” affect these fields? Why does “high temperature” prevent these forces (gluon, Higgs, etc.) acting on particles? And why do these different forces start acting on particles at “different temperatures”? 44. Thank you for your excellent article, and website. I would like to ask how are ‘mass energy’ and ‘motion energy’ related? Can motion energy ever be converted into mass energy and/or vice versa? You seem to be saying in your replies to other posts that ‘E’ in Einstein’s equation E=MC2 doesn’t include motion energy. If not then why does science use the same word ‘energy’ for both concepts when to do so leads to confusion? • you’re truly a excellent webmaster. The website loading velocity is incredible. It seems that you’re doing any distinctive trick. In addition, The contents are masterwork. you’ve done a fantastic job in this topic! 45. Everything you said about matter, stuff, energy, I agree with. A learner has to create separation between the two concepts in order to learn these. Its just like learning the language. A child will break up a long word into sylabils and memorize its pronunciation and meaning that way. And since its almost impossible to define either one of the terms (matter or, energy), why worry about it. Or, a layperson might reach a wrong conclusion that physicist in general are afraid of the word ‘spirit’ (pure energy). I meant spirits, like God for example. Physicists may cringe but the truth is this ‘stuff’ falls into category all on it own and being ethereal it can not be detected, tested or, experimented on in LHC. There is also a big likelihood that this ‘stuff’ disapproves of humans splitting the atom and destroying something he created for the benefit of all humans_ the blue planet, third rock from the Sun. 46. Most people do not have a clue as to the things they speak, nor can they prove them. But they speak none the less, it is all theory, opinion, best guesses, until it is proven to be true. And anyone that claims to understand Quantum Mechanics and the Sub-Atomic plane, is the same person that claims to be \/\/ise, anyone can claim to be anything, but it is one thing to “understand” the “truth”, and one thing to guess about it. But it is fun to read other peoples opinions, but it be better to read the truths, but those are yet hidden. Even though much is understood, he that speaks as if understanding is a given, speaks in circles, because they do not understand the truth. It is ok professor, no one understands it completely, or has the truth. but your opinions do have merit. • Are you trying to reassure me? 🙂 No reassurance needed; my knowledge is used to make cell phones, rocket ships and lasers. What are your alternatives used for? • cell phones, rocket ships, and lasers, are great technologies for the advancement of understanding. But they are each only different manipulations of the same energy that is present in all. A photon of light travels at the speed of light throughout the universe carrying the information of its source for eternity, unless interaction upon another force disrupts or absorbs its energy, such as in vision, if not for this information your eye could not interpret the origins of said photon to turn that information into a “picture” for your neurons to process. In order to travel at the speed of light a photon needs to be a particle, a wave, or both at the same time, and is basically mass-less, but yet still has the ability to transfer information, as Darpa has shown transferring information upon photons. As soon as a Photon “interacts: upon the human eye for example, it has to assume the form of a particle in order for the information to be “read” and transported to neurons for interpretation, this biological process does not occur in non-living matter, also in order to travel at the speed of light, a photon needs to maintain the mass less form of energy in order to reach the speed of light threshold, as no particle of mass may reach this thresh hold, as I am sure you understand completely during experiments preformed using accelerated particles, none of them have achieved 100% equal to the speed of a photon, that reason is established. It has been observed in many recent experiments that light ( a photon ) can be any or all forms at once, or that a photon can change forms as needed in its travels. Decay being the key, for Photons have been collected from telescopes that collected them after they traveled billions of light years across space and time, there are no particles that can achieve this velocity unless they change into pure energy to achieve this speed. Experiments at Cern have even held this constant to be true.. so for you to say that light is not pure energy, is disputed by many recent experiments, and by many old experiments… • Wow! ya why listen at all? Especially when you have everything figured out in your own head and can just vomit streams of metaphysics all over the comment section. No wonder the good professor stopped responding to you 47. I know you use “mass” to mean rest mass, but aren’t there some cases where we really need to talk about the mass corresponding to the total energy of a system? Like the fact that a system as a whole is literally heavier (has greater mass, as one can measure by the inertia or gravitation of the system as a whole) when DeltaE of any type is added to it (without allowing any energy to escape of course): let in some light, add heat, set a top spinning that was at rest before, compress a spring, pull positive and negatively charged objects further away from one another within the system, etc. I’m assuming that we’re examining this system in a single inertial frame in which the system as a whole is at rest, with no external forces or fields, so we don’t have to think about the kinetic or potential energy of the system as a body itself within its environment. The extra mass we’ll measure will be given of course by DeltaE/c^2, but here the E and m both refer to the total energy of the system. 48. Thank you for a solid blog on an important subject. I’m 65 and have a question based on the above mentioned “pre-1973” science era I grew up in. In today’s Physics, is the fundamental essence of the Universe still considered to be Space, Time, and Matter, or, as I gather from the above, something more akin to one of the following: A. Space, Time, Energy B. Space-Time, Matter-Energy C. Energy (when stretched out=Space, when condensed=Matter, and otherwise simply Energy) D. None of the above 49. Excellent article, thank you Matt! I think the abuse of terminology is common to layman description of all technical fields, and is not a prerogative of physics. Part of it comes from public relations (try explaining your discovery to a reporter, and, what is way more difficult, making sure it is published without inaccuracies!), and part of a linguistic inertia. I think as the boldest example of the latest we should recall the fact that in numerous languages (less in English, though) the word “ether” is used to indicate the broadcast of a radio or tv program. “In ether” is used in these languages as a synonym to “on air”. This archaic heritage of the long-obsolete luminiferous ether theory still remains in our terminology, and even modern network technology that has nothing to do with ether bears the name of “Ethernet”. 50. Hi Matt, Below I quote two things that you have said about dark energy and ask for a bit more explanation:- “Dark energy is a property that fields, or combinations of fields, can have.” If there is one thing you’ve got into my head is the idea of “fields and ripples”. A field can have a ripple, an object (stuff) like an electron or a photon. “Dark energy isn’t an object or a set of objects.” Do you say this because the effect in the field is more subtle, like the distrubance in space-time that gives us the gravitational field (stuff, but not defined enough to be called an object)? And if gravity can be visualised as a heavy ball on a trampoline, could dark matter be visualised as something bending its field(s) to make a hill as opposed to gravity’s valley? 51. Matt, You say “it happens that every known field has a known particle, except possibly the Higgs field (whose particle is not yet certain to exist, …” I had always thought that evidence for the existence of the Higgs field and that of the Higgs particle were more or less on equal footing, in fact, that acceptance of the Higgs field’s existence was predicated on detection of the Higgs particle. But your statement above suggests to me that even if a Higgs particle were not to be found (or somehow proved to not exist), then the prevailing point of view would be that the Higgs field exists nonetheless, only having no minimum eigenstate. Is this slight (and very respectful) criticism anywhere close to accurate? I feel blessed to have stumbled upon your site, and will be an ardent reader of your posts/articles. You have an extraordinary talent for expressing complex ideas in a clear, yet scientifically responsible, manner. This is very, very rare. – Doc 52. A Photon is an eternal carrier of information that can transfer and receive energy through contact and interaction upon various fields of mass. This ability can best be represented as you gaze upon a full moon at night, for you do see the moon light shining, and you do see the moon, but in reality you are seeing photons that came from the Sun, interacted upon the mass field of the moon, then imprint the information of this mass field upon the photon, then as it enters your optic nerves the information your neurons receive is that of the physical image of the moon shining from photons originating inside of the Sun. The eternal energy and motion of these photons is lost as each transfers its energy into the organic neural/electrical network becomes a particle transfers its energy ( information ), and ceases to exist in its previous form. Photons have been captured from telescopes that have traveled billions of lights years across space and time, yet they retain there speed only slightly influenced by gravitational fields of fields of mass, and they retain the information of there source of origin, and can attain constant updates to this information as it interacts upon various fields of mass, sometimes exchanging its energy, and sometimes transferring its energy to another medium. this vast universal communication system is recently being discovered as quantum mechanics and physics are relating it to the vary method used by the brain to communicate along vast and complicated neural networks, so, as above, then so below. Energy is eternal it can not be created nor destroyed, only transferred between fields of mass…. 53. Also it is my opinion that dark energy is a result of entropy caused by the transfer of energy not being 100 % efficient during the energy transfer process. And as the universe has expanded and aged, the entropy has increased and that has caused an increase in dark energy that is in a chaotic state. This “dark energy’ then behaves as regular energy, but instead creates dark matter as a result of its chaotic state. energy in is greater then energy lost in most efficient energy systems, and this loss/entropy is the reason for the related increase and amount of dark energy in the universe, and the amount shall ever increase in relation to entropy due to inefficiency. just my opinion though… 54. Well Sir, After reading all this “stuff” , off course not that stuff which you have been mentioning about, I’ve got a feeling that some of my doubts can be cleared by this well learned friend of mine namely Mr. Matt. The following are my doubts Sir, Please address to them: 1. Is the light a form of energy as every one is thinking as on now? 2. If it happens to be right, Is the energy incident on earth’s surface from sun is in form of sunlight? 3. If it is only partially right, what are the other forms in which the energy is being conveyed to earth from the sun? Based on the answers for this queeries there may be further observations you have to answer Sir, Please reply. 55. sorry but I haven’t read the full website … can you succintly put everything in a simple short phrase ?? would it be fair to say that everything is energy and it manifests only in two ways: matter and force fields ?? 56. Hello Professor Strassler. In your post you say that energy is something that objects have, that energy is not an object itself. But I have heard string theorists like Brian Greene say on science tv shows that if string theory is true then everything is made of strings. And what are these strings? He says they are strings of energy. He says these strings are made of energy. But if strings can be made of energy, wouldn’t that mean energy is stuff? And wouldn’t that mean matter is energy at the fundamental level? To hear Brian Greene say this watch these short videos: 57. If energy is neither created nor destroyed than it is neither present nor absent in a particular space and time.Energy and matter are one and same.There is nothing called dark matter and dark energy.All divisions are like dreams.For mind time is understanding through thought/memory.Mind cannot differentiate between dream and reality because it functions by continuous reciprocal causation.One cannot prove in unreal{dream like} time and space. 58. Not sure if this will make sense to someone with knowledge, but when the big bang happened you had pure energy and massive temperatures, as the universe expanded it cooled and turned into (matter) could dark energy be the remaining energy and temperature in the universe wanting to turn into (matter)? Also when the universe started it was extremely hot its cooled so much why did it stop cooling? Why did it stop just above absolute zero in deep space? These are probably stupid questions just thought I’d try to better understand something that interests me. 59. thank you that was very informative and has given me a lot to think about , i just don’t understand how light speed or fster than light speed was achieved in the first few seconds of the start , in whatever state of energy or near mass everything was :_) 60. Is it possible for energy to act within and upon itself? Is it at least possible that consciousness is a form of energy? Or, are there reasons i tcannot be? 61. def(matter)by Google “physical substance in general, (in physics) that which occupies space and possesses rest mass, especially as distinct from energy.” It all depends on your perspective. I meant matter in general, not A specific bit. I also meant that when matter takes a form, that form IS a result of energy. So, for example, the form of a balloon, and the material components of the balloon, are shaped by the forces acting on it and them. “…especially as distinct from energy.” Matter IS distinct from energy, but is not separate from energy, since no matter can exist without energy. Just to emphasize the ambiguity of language, your phrase, “matter is A form that energy CAN take”, implies that energy can take other forms. Of course, you can say that energy takes different forms, such as EM and gravitational, taking “form” in a different sense. When you experience matter, EM energy, and gravity, are you experiencing energy? What about mass? When you perceive mass are you perceiving energy? Aren’t mater and mass simply phantasms composed of energy? In fact, it may be that energy is all there is. : ) 62. not only “may be that energy is all there is” but energy is all there is, and its different forms (force fields and “matter”) 63. And now I can say, consciousness is aware-ized energy, and, all energy is aware-ized. Of course, only subjective experience can tell. I think Matt is missing out on that part. Science has bound up the minds of even its own even most original thinkers, like Matt, for they dare not stray from certain scientific principles. When his private experience of himself does not correlate with what he is told by science, then he may become familiarized with the roots of ego consciousness. This will be done under the direction of an enlightened and expanding egotistical awareness. Then, he could use his talents to organize the hereto neglected knowledge. Not a judgement, just an observation. 64. Incredible story and string – and I always thought it was only we economists who were all screwed up with ‘on the one hand, this’ but ‘on the other hand, that’, ‘but then again maybe it’s…….’ and so on and on. How refreshing. 65. Is consciousness matter or energy, or some combination? Well, actually, you don’t know. So, I will tell you. Consciousness is aware-ized energy, and all energy is aware-ized, whether you believe me or not. : ) That statement is scientific heresy. Subjectivity cannot be demonstrated within the context of current science. How do I know? I know by experiencing my own consciousness in many ways. Learn by doing. 66. Hello there, You’ve done a great job. I’ll certainly digg it and personally suggest to my friends. I am confident they’ll be benefited from this website. 67. Do fields extend endlessly across or through the known universe? Can any distinction be made between fields and their particles on the one hand and the phenomenal world we actually experience by way of our sense modalities? When you suggest that “objects” have (energy), are you arguing for “entities” that have an independent, autonomous existence and an intrinsic identify? I haven’t encountered such as yet, at least not in the phenomenal sense, wherein a thing exists ONLY in relation to other things. – Cal 68. recalling my highscholl physics where the teacher would tell us that the electrons quantum jumps between orbits ? Where does it go ? 69. Two words regarding the matter\energy equivalence debate, “conserved quantities”… energy and matter are interchangeable. There is no dichotomy. Matter is an observed point particle at a specific place and time within a collapsed wave function, just an observed probability. We know that where there is energy we can observe matter popping in and out of existence (for want of a better word)ad nauseam. In my mind the day will come where we realise that scale-symmetry will clarify unequivocally a fundamental view of “what stuff is”. we will drop the idea of super-symmetry and accept that matter having physical dimensions e.g the current zoo of point particles we observe is just a misleading by-product of fundamental symmetry breaking. last month at the International Conference of High-Energy Physics in Valencia, Spain, researchers analysing the largest data set yet from the LHC said and I quote “we have found no evidence of super symmetric particles”. Yes you can always say that at higher energies we can expect to see a super massive primordial particle beyond the combined 14GeV the LHC can pump out, now we have 5 sigma results on the Higgs what’s next? I will retract this when I see a 5 sigma result on a Electron\Selectron pair stopping the Higgs boson mass inflating exponentially being observed … To quote the matrix… “there is no spoon”, but again in my mind there is (at the scales we observe from the observable universe down to the Planck length) a bloody good 4D observable representation of that spoon gaining mass from our old friend Mr H Boson…or if your a M Theorist an 11/6D one bent Uri Geller like through an unobservable Calibi Yau Manifold ;o) Or to put it more accurately the energy that constitutes it appears on our narrow scale of observation to be a spoon, because the interaction it has with the Higgs field ascribes mass to it on our scale … please understand my tongue is placed firmly in my cheek here… ;o) thanks for the article I enjoyed it and the comments below it very much… has a lot of exclusive content I’ve either written myself or outsourced but it seems a lot of it is popping it up all over the internet without my authorization. Do you know any ways to help reduce content from being stolen? I’d genuinely appreciate it. 71. I knew someone who had exactly this problem, in fact I myself discovered the problem while searching the net for a particular literary piece he had written. I put several keywords into a search engine, and lo and behold, I was directed to another site. Worse still, the owner of this other site was passing off his intellectual property as her own. He had a lawyer draw up a cease and desist order (really just a letter), and the plagiarized material was gone within the next 48 hours. I’m not sure an actual lawyer is necessary, since you could probably draw up a convincing letter yourself and self-promote to “Esquire” 😉 72. Hey There. I discovered your blog the usage of msn. That more of your useful information. Thanks for the post. I will certainly return. 73. Polarity is required for thinking. So, for example, if you don’t know what up is, you don’t know what down is. If there is no up, there is no down. Energy has been a difficult concept to define because no one knows its polar opposite. Feynman has said that science has no idea about what energy is, and that makes sense, because there is no polar opposite to energy. I suggest science and philosophy will have the same problem with the objectivity and subjectivity. Like energy, science, by its own requirements, cannot define objectivity, because, there is no polar opposite. Subjectivity, and hence consciousness, can only be defined by science in objective terms. According to current science, as I understand it, subjectivity is only a kind of objectivity we do not understand yet. Hence, since there is no real polar opposite to objectivity, objectivity cannot be defined. You know, and like “consciousness,”like “energy”. 74. Sorry everyone but you are over complicating ! Consider particles are constricted energy, energy seen is variation in dark matter energy field but what is energy? 75. hello, I agree with the cringing feeling you mentioned: I wish there was a cleanly defined ontology for physics too, as there are many for other scientific fields (“”). I also agree that the dichotomy matter/energy is inappropriate (a correct one would rather be mass/energy, as they can be tranformed into each other, right?), as matter is “stuff” and mass or energy are properties of that stuff. But then another dichotomy would be stuff/void or matter/spacetime, right? • ok, no replies… Maybe ontologies are out of scope here! but that’s strange, since both “here” (the web) and ontologies are by Tim Barners Lee, a physician… Anyway, to strictly speak your language and tell about things you know/care… The below is what I wondered about, when I studied (as an hobby) Susskind’s “Special Relativity and Electrodynamics”: So, just take the SR’s invariant (a metric!): dTau^2 = dt^2 – dx^2 – dy^2 – dz^2, then divide it by dTau^2, as Susskind does in one of those lectures. You get an equation on the components of the four-velocity, but what does it say? It says that the square of time component of the 4-velocity, minus the sum of the squares of the spacial components of the 4-velocity is always equal to one. Now, what does this mean/imply? Let’s take “me”, or any “material thing”: I am always “at rest” in my own frame of reference, right? So the spatial components of my 4-velocity shall all be 0, right? That then means the time component shall be 1, right? I, in my own frame or reference, have “some kind of motion” in time, indeed I see it “passing by on my clock”, it’s “moving” with respect to me and vice-versa: the time component of my 4-velocity is 1! To me, that is basically equivalent to say “I ‘keep existing’ as time goes by”. But there’s not just things like “me”, not just (massive) “particles”, out there, right? There also are things for which “time does not pass by” and “all space is compressed in the direction of motion”, so that they cannot have a frame of reference of their own… pretty weird, right? If they could have a reference frame of their own, they would say about themselves that their time component of the 4-velocity is 0 and their space components squared sum up to -1, so if you take the motion along one single space component, their speed is i=sqrt(-1)… too weird, right? BTW: to me, this is pretty much the same thing that happens in EMF with the components of the 4-current: – take the time component: we call it charge density – take the space components: we call them currents why do we do so? To me, it’s us introducing this dichotomy, nature hasn’t (this one). 76. Hello Dr. Strassler, I love your site and the scintific facts and knowledge you share. I would like you to please explain a phenomenon I’m interested in for many years: what cause the matter to became a matter? In psychological language (Jungian and others), it is the anima (the feminine aspect), that sedused the masculin aspect to became a matter. Okay, I can understand this. But what is the equivalent scientific explanation, for lay people like me. very comlicated indeed. Would very much appreciate your reply. 77. I believe we are missing an an equation to satisfy a problem. Matter to energy must have something quantifiable to keep a constant. Since energy can not be created nor destroyed we are left with doing the math to convert both in a dance of harmony. Dark matter is the gorilla in the room. Unless we can move past the speed of light, we can never know. 78. Dr. Strassler: As a novice and a much older high school dropout, any of the sciences leaves me in shambles. However I do ask many questions which usually get me in trouble. So here goes. Might there be an antithesis of energy at the very core of this universe growing exponentially as our material universe dies at an ever increasing speedy death of attrition? While modern science tells me our universe has no center, I have my doubts. I am hoping this may turn into a good and constructive discussion? Thanks, LeftyR 79. Hi, to relieve your mind of the contradiction and ambiguity 🙂 I can tell you this: First of all, this dichotomy clearly originates from before the early twentieth century, when the dual nature of light, and with that of all matter, was not yet apparent. I mean, from before the de-Broglie equation. This dichotomy of matter and energy, actually originates more from chemists than from physicists. In chemistry we use it primarily to define the traditional study of chemistry, as opposed to the study of physics. And we frequently say, chemistry is the study of matter – properties, changes… etc. and physics is the study of everything else (energy). when we say matter, we mean atomic matter. This is not accurate, and modern chemistry and physics overlap quite a lot, but what we mean when we say this is: properties of a substance -> properties of matter -> chemistry A reaction between two substances -> changes of matter -> chemistry energy changes during a reaction -> iffy, could be physics or chemistry, but chemists study it differently intermolecular and intramolecular forces -> same as above other forces -> physics the electromagnetic spectrum -> physics subatomic particles -> physics 80. Very helpful. I landed here with a question . Assuming the properties of only a gravitational interaction, why not just call it like unbound gravity or something? I mean we won’t ever see anything else there. 81. Thank you. But I had a question. Accordingto your article: what is energy in a field? Its amplitude or its wave velocity (I guess this would be C)? 82. Let me begin by saying that every atom in the entire universe has one intrinsic characteristic and that is, Magnetism. No matter how thin we slice or dice it, the atom, even down to what we believe is its lowest common denominator, quarks, has one thing in common, spin, angular momentum or, magnetism. Is it possible that energy itself is derived from something below this level, or in the way atoms get bounced around? 83. As a novice, in trying to make sense of matter vs. energy I have read several papers on the subject and concluded that the word “energy” is nothing more than a way of describing an action and reaction on mass/matter. An example is 2 iron balls, of 5# and 10# respectively, hung at equal heights in a vacuum. Dropped simultaneously, both will hit the ground at the same time with one exception, the heavier ball will dig a deeper hole. And why? Because of “ENERGY” generated by the weight of the mass. 84. Hi, I’m in my late 50’s and have taken a fascination with science, and am thoroughly enjoying learning about it. I found your article certain helpful, although much was over my head. I have saved it and will re read when will understand more. Thank You, 85. Matt, Stumbled onto the post searching for publications on this topic. Find much of your presentation w/r to matter and energy is agreeable to me. We have recently published a short article on this topic, and perhaps should have used the term “radiation”, as you did, rather than “radiant energy” as we wrote. Don’t hold your breathe expecting solid evidence for “dark energy” or “dark matter”. 86. Hello ALL, I bumped into this article and Matter-Energy Blog just on this Thanksgiving Day. Happy Thanksgiving to all. I am searching for particles /energy that explain physical and spiritual dimensions. Is Higgs’ Filed/Bosons the fundamental physical particle representing the Cosmos and we are still after the Holy-Grail, Spritual/Atman Particle? I am convinced that Higgs’ Field explains the material construction. 87. My Question is for Mr Matt Strassler – You say there are field without mass but no mass without field. What is the basic thing for which field obtain mass and also field obtain energy. Ultimately we can not ignore E=mc^2, which means mass and energy are convertible. Also could not understand proper relation between mc^2 and kT. How they are related if say the velocity of the particle (or field?) is v? Is there any mathematical formulae for the same. I also have a question on the concept of “Point of Singularity”. All the fundamental particles which form into “matter” have some basic dimension, however small it may be, which is why they have mass (leaving the particles with zero mass, as matter can not be formed with particles of zero mass alone). Therefore, even if we reduce particles to such a level a physical space would be available for the particle. Therefore, I conclude that there may not be any point of singularity at all – it would have some minimum space consideration which is what we may assume as “singularity” although it will not be same as mathematical singularity. So it may happen that our very assumption that big bang happened from a singularity may not be a valid assumption. However, supposing such a point of singularity to even exist, how does one visualize existence of energy in a concept of singularity. 88. as a developer, I’d say that: – mass is an “excess” of position information (stuff moving in a stationary way) – energy is an “excess” of speed/momentum information (stuff moving “freely”) 89. you might find this interesting to think about ” the universe being made by energy” instead of “being made of energy” this allows for the creation of non physical properties such as space time. Try to start by thinking of energy as a wave and a particle as the work being done, where the work being done is turning nothing into something. 90. Try thinking of energy in an evolution way maybe something like. Energy starts with zero complexity then increases in complexities until it finds symmetry with entropy thus returning energy to zero complexity. Then you can map complexity on an evolutionary tree. It soon becomes clear that gravity is the least complex form of all energy and maybe the source of all energy, from a zero point singularity formed by the single characteristic of nothing. “You cannot separate objects without space time.” A zero point singularity is inevitable in a zero dimension. 91. start with least complexity space time. how energy might create it? it should answer why the speed of light is consonant whither traveling toward or away from the source in very simple terms. then understand the function of spin and you have it. so simple 92. One last question. Is it time or distance that dilates at velocity? All quantum weirdness should vanish if you get all the above right. The end result could be, not a big bang but the continuous bang of a “white hole” and not a big crunch but a continuous crunch of entering a zero dimension a “black hole”. 93. How does one explain to children the difference between solids, liquids, kinetic energy, potential energy…but oh by the way…everything we know about matter vs energy is bogus? How should one approach educating children about the distinction between matter and energy or the types of energy and matter if this is an incorrect dichotomy? For example, how would I describe the basic state of a car vs the sun or electricity or sound or motion…etc. now that doing so will only lead ultimately to misleading information – and that our basic knowledge regarding energy and matter is a false paradigm? Thanks 94. energy is nonmatter, as matter possess energy and energy do not possess matter they are 2 different sides of a single coin , and do not get confused in terms of photons particle , as photon particles as photon particles are stuff of no mass and infinite energy…their mass converts into energy…..Thanx ….by Abhinav…a highschool student…..India ….u.p..alld….ghoorpur 96. Can one describe energy as a measure of the matter which acts on a system to put it back to it’s balanced state…and if so, that without matter energy would be undetectable? 97. “All particles are ripples in fields and have energy…” As it stands this statement is blather. A field of what? A ripple caused by what? One can substitute ‘strings’ for ‘fields’ in this statement and many a Ph.D. physicist might agree. Yet both versions are without any proof; see Lee Smolin ‘The Trouble with Physics’ 98. Energy as in photons always moves at light speed. Matter cannot move at light speed. How can they be two different sides of the same coin? Sure matter has energy inherent in it, as in e=mc2, but I think if you release all that energy, the matter is still there. Dead matter, so to speak. Atomic dust if you will, with no motion and no charge so probably impossible to detect. Photons if anything are a wave. This always moving at light speed suggest they are not a sphere since no part is going to move faster or slower than light speed, whatever happens to it. It is not going to rotate. The photon has two speed components. It moves at light speed in the direction of travel, and moves from side to side, so giving us wavelength and frequency, which seems to work out at one third of light speed (one metre wavelength gives a frequency of 10^8 Hz, so one hundred million metres per second). The more energy a photon has, the smaller it’s wavelength, so the side to side movement in turn becomes faster. The constant here gives us that effect; that what it adds to the speed, the frequency, it must take from the width, the wavelength. I know what I mean but I am not good at explaining this. • Isnt frequency simply the duration of the wave? And don’t photons have different frequency? As in visable light verses microwave? • Frequency is the number of waves that pass through a fixed point over a given unit of time (so not a time measurement per se). Any yes, photons have different frequencies. 99. Seventh. It is knowable by itself being an integrated being. And, it reiterates, it generates the knowledge of itself, by itself. Eighth. Its properties are immanent to it, that is, they constitute it, and were not created. Ninth. Life is the property of dynamic matter. This property manifests as organic life when a dynamic entity reaches a certain level of evolutionary integration, that of algae, for example, in which the phenomenon of individuation appears for the first time in the history of the process of evolutionary integration. And much later, in due time, consciousness appears in the beings who reached the last level of evolutionary integration, achieved to date. 1/ The definition of atom that is used is: the atom as basic and indivisible block that composes the matter of the universe, that was postulated by the atomistic school in Ancient Greece, Century V A.C., being Democritus one of its exponents. Vázquez-Reyna, Mario (2008). Teoría general de la Materia. Borrador. Ciudad de México. Matter has the following fundamental properties: First. It is indestructible, and exists in infinite quantity. For which the nothing has not existed, does not exist, and will never exist, and the laws of conservation are generated. The universal order comes from the immutable character of the properties of the primordial components of matter. Second. Matter is impenetrable at very short distances. It is and has its own space. At great distances the spaces of atoms are penetrable, being fields and overlapping, generating ordinary space, which is, as it is said, property of matter, and can not be empty, since it is itself matter. Third. Matter is composed of atoms. The atomic components of Matter have energy or inertia, are heterogeneous with each other, and the character of heterogeneity is energetic or inertial, so they generate movement and opposition to it, the tendency to repose, universal, and Universal diversity. Fourth. By virtue of their energy or inertia, the atoms of matter with energy and the atoms of matter with inertia are contradictory among them; they apply forces out of themselves and therefore become ponderable. These atoms are components of the so-called chemical elements. Fifth. The atoms are associated with each other by their energy, or by their inertia, that is, they integrate with each other those of the same essence, and by their association, of those who move, those of the dynamic essence, generate universal movement, Life, the universal process of evolutionary integration and universal diversity, and those of the static essence, which oppose life, generate gravity and inertia and try to stop progress by tending to rest, without respite. Sixth. The existence and form of being of matter do not depend on whether any Being is observing it, that is, it exists by itself and is, and therefore has an objective character. 101. I came across this discussion because I was looking up the word ‘thing’. To me a thing is something that I can either see, or hear, or touch/feel, or taste, or smell. So, aren’t matter and energy both things?
28342aa42c1b8994	It seems that quantum computing is often taken to mean the quantum circuit method of computation, where a register of qubits is acted on by a circuit of quantum gates and measured at the output (and possibly at some intermediate steps). Quantum annealing at least seems to be an altogether different method to computing with quantum resources1, as it does not involve quantum gates. What different models of quantum computation are there? What makes them different? To clarify, I am not asking what different physical implementations qubits have, I mean the description of different ideas of how to compute outputs from inputs2 using quantum resources. 1. Anything that is inherently non-classical, like entanglement and coherence. 2. A process which transforms the inputs (such as qubits) to outputs (results of the computation). The adiabatic model This model of quantum computation is motivated by ideas in quantum many-body theory, and differs substantially both from the circuit model (in that it is a continuous-time model) and from continuous-time quantum walks (in that it has a time-dependent evolution). Adiabatic computation usually takes the following form. 1. Start with some set of qubits, all in some simple state such as $\lvert + \rangle$. Call the initial global state $\lvert \psi_0 \rangle$. 2. Subject these qubits to an interaction Hamiltonian $H_0$ for which $\lvert \psi_0 \rangle$ is the unique ground state (the state with the lowest energy). For instance, given $\lvert \psi_0 \rangle = \lvert + \rangle^{\otimes n}$, we may choose $H_0 = - \sum_{k} \sigma^{(x)}_k$. 3. Choose a final Hamiltonian $H_1$, which has a unique ground state which encodes the answer to a problem you are interested in. For instance, if you want to solve a constraint satisfaction problem, you could define a Hamiltonian $H_1 = \sum_{c} h_c$, where the sum is taken over the constraints $c$ of the classical problem, and where each $h_c$ is an operator which imposes an energy penalty (a positive energy contribution) to any standard basis state representing a classical assignment which does not satisfy the constraint $c$. 4. Define a time interval $T \geqslant 0$ and a time-varying Hamiltonian $H(t)$ such that $H(0) = H_0$ and $H(T) = H_1$. A common but not necessary choice is to simply take a linear interpolation $H(t) = \tfrac{t}{T} H_1 + (1 - \tfrac{t}{T})H_0$. 5. For times $t = 0$ up to $t = T$, allow the system to evolve under the continuously varying Hamiltonian $H(t)$, and measure the qubits at the output to obtain an outcome $y \in \{0,1\}^n$. The basis of the adiabatic model is the adiabatic theorem, of which there are several versions. The version by Ambainis and Regev [ arXiv:quant-ph/0411152 ] (a more rigorous example) implies that if there is always an "energy gap" of at least $\lambda > 0$ between the ground state of $H(t)$ and its first excited state for all $0 \leqslant t \leqslant T$, and the operator-norms of the first and second derivatives of $H$ are small enough (that is, $H(t)$ does not vary too quickly or abruptly), then you can make the probability of getting the output you want as large as you like just by running the computation slowly enough. Furthermore, you can reduce the probability of error by any constant factor just by slowing down the whole computation by a polynomially-related factor. Despite being very different in presentation from the unitary circuit model, it has been shown that this model is polynomial-time equivalent to the unitary circuit model [ arXiv:quant-ph/0405098 ]. The advantage of the adiabatic algorithm is that it provides a different approach to constructing quantum algorithms which is more amenable to optimisation problems. One disadvantage is that it is not clear how to protect it against noise, or to tell how its performance degrades under imperfect control. Another problem is that, even without any imperfections in the system, determining how slowly to run the algorithm to get a reliable answer is a difficult problem — it depends on the energy gap, and it isn't easy in general to tell what the energy gap is for a static Hamiltonian $H$, let alone a time-varying one $H(t)$. Still, this is a model of both theoretical and practical interest, and has the distinction of being the most different from the unitary circuit model of essentially any that exists. Measurement-based quantum computation (MBQC) This is a way to perform quantum computation, using intermediary measurements as a way of driving the computation rather than just extracting the answers. It is a special case of "quantum circuits with intermediary measurements", and so is no more powerful. However, when it was introduced, it up-ended many people's intuitions of the role of unitary transformations in quantum computation. In this model one has constraints such as the following: 1. One prepares, or is given, a very large entangled state — one which can be described (or prepared) by having some set of qubits all initially prepared in the state $\lvert + \rangle$, and then some sequence of controlled-Z operations $\mathrm{CZ} = \mathrm{diag}(+1,+1,+1,-1)$, performed on pairs of qubits according to the edge-relations of a graph (commonly, a rectangular grid or hexagonal lattice). 2. Perform a sequence of measurements on these qubits — some perhaps in the standard basis, but the majority not in the standard basis, but instead measuring observables such as $M_{\mathrm{XY}}(\theta) = \cos(\theta) X - \sin(\theta) Y$ for various angles $\theta$. Each measurement yields an outcome $+1$ or $-1$ (often labelled '0' or '1' respectively), and the choice of angle is allowed to depend in a simple way on the outcomes of previous measurements (in a way computed by a classical control system). 3. The answer to the computation may be computed from the classical outcomes $\pm 1$ of the measurements. As with the unitary circuit model, there are variations one can consider for this model. However, the core concept is adaptive single-qubit measurements performed on a large entangled state, or a state which has been subjected to a sequence of commuting and possibly entangling operations which are either performed all at once or in stages. This model of computation is usually considered as being useful primarily as a way to simulate unitary circuits. Because it is often seen as a means to simulate a better-liked and simpler model of computation, it is not considered theoretically very interesting anymore to most people. However: • It is important among other things as a motivating concept behind the class IQP, which is one means of demonstrating that a quantum computer is difficult to simulate, and Blind Quantum Computing, which is one way to try to solve problems in secure computation using quantum resources. • There is no reason why measurement-based computations should be essentially limited to simulating unitary quantum circuits: it seems to me (and a handful of other theorists in the minority) that MQBC could provide a way of describing interesting computational primitives. While MBQC is just a special case of circuits with intermediary measurements, and can therefore be simulated by unitary circuits with only polynomial overhead, this is not to say that unitary circuits would necessarily be a very fruitful way of describing anything that one could do in principle in a measurement-based computation (just as there exists imperative and functional programming languages in classical computation which sit a little ill-at-ease with one another). The question remains whether MBQC will suggest any way of thinking about building algorithms which is not as easily presented in terms of unitary circuits — but there can be no question of a computational advantage or disadvantage over unitary circuits, except one of specific resources and suitability for some architecture. • 1 $\begingroup$ MBQC can be seen as the underlying idea behind some error correcting codes, such as the surface code. Mainly in the sense that the surface code corresponds to a 3d lattice of qubits with a particular set of CZs between them that you then measure (with the actual implementation evaluating the cube layer by layer). But perhaps also in the sense that the actual surface code implementation is driven by measuring particular stabilizers. $\endgroup$ – Craig Gidney Sep 17 '18 at 15:51 • 1 $\begingroup$ However, the way in which the measurement outcomes are used differ substantially between QECCs and MBQC. In the idealised case of no or low rate of uncorrelated errors, any QECC is computing the identity transformation at all times, the measurements are periodic in time, and the outcomes are heavily biased towards the +1 outcome. For standard constructions of MBQC protocols, however, the measurements give uniformly random measurement outcomes every time, and those measurements are heavily time-dependent and driving non-trivial evolution. $\endgroup$ – Niel de Beaudrap Sep 17 '18 at 15:57 • 1 $\begingroup$ Is that a qualitative difference or just a quantitative one? The surface code also has those driving operations (e.g. braiding defects and injecting T states), it just separates them by the code distance. If you set the code distance to 1, a much higher proportion of the operations matter when there are no errors. $\endgroup$ – Craig Gidney Sep 17 '18 at 16:07 • 1 $\begingroup$ I would say that the difference occurs at a qualitative level as well, from my experience actually considering the effects of MBQC procedures. Also, it seems to me that in the case of braiding defects and T-state injection that it is not the error correcting code itself, but deformations of them, which are doing the computation. These are certainly relevant things one may do with an error corrected memory, but to say that the code is doing it is about the same level as saying that it is qubits which do quantum computations, as opposed to operations which one performs on those qubits. $\endgroup$ – Niel de Beaudrap Sep 17 '18 at 16:21 The Unitary Circuit Model This is the best well-known model of quantum computation. In this model one has constraints such as the following: 1. a set of qubits initialised to a pure state, which we denote $\lvert 0 \rangle$; 2. a sequence of unitary transformations which one performs on them, which may depend on a classical bit-string $x\in \{0,1\}^n$; 3. one or more measurements in the standard basis performed at the very end of the computation, yielding a classical output string $y \in \{0,1\}^k$. (We do not require $k = n$: for instance, for YES / NO problems, one often takes $k = 1$ no matter the size of $n$.) Minor details may change (for instance, the set of unitaries one may perform; whether one allows preparation in other pure states such as $\lvert 1 \rangle$, $\lvert +\rangle$, $\lvert -\rangle$; whether measurements must be in the standard basis or can also be in some other basis), but these do not make any essential difference. Discrete-time quantum walk A "discrete-time quantum walk" is a quantum variation on a random walk, in which there is a 'walker' (or multiple 'walkers') which takes small steps in a graph (e.g. a chain of nodes, or a rectangular grid). The difference is that where a random walker takes a step in a randomly determined direction, a quantum walker takes a step in a direction determined by a quantum "coin" register, which at each step is "flipped" by a unitary transformation rather than changed by re-sampling a random variable. See [ arXiv:quant-ph/0012090 ] for an early reference. For the sake of simplicity, I will describe a quantum walk on a cycle of size $2^n$; though one must change some of the details to consider quantum walks on more general graphs. In this model of computation, one typically does the following. 1. Prepare a "position" register on $n$ qubits in some state such as $\lvert 00\cdots 0\rangle$, and a "coin" register (with standard basis states which we denote by $\lvert +1 \rangle$ and $\lvert -1 \rangle$) in some initial state which may be a superposition of the two standard basis states. 2. Perform a coherent controlled-unitary transformation, which adds 1 to the value of the position register (modulo $2^n$) if the coin is in the state $\lvert +1 \rangle$, and subtracts 1 to the value of the position register (modulo $2^n$) if the coin is in the state $\lvert -1 \rangle$. 3. Perform a fixed unitary transformation $C$ to the coin register. This plays the role of a "coin flip" to determine the direction of the next step. We then return to step 2. The main difference between this and a random walk is that the different possible "trajectories" of the walker are being performed coherently in superposition, so that they can destructively interfere. This leads to a walker behaviour which is more like ballistic motion than diffusion. Indeed, an early presentation of a model such as this was made by Feynmann, as a way to simulate the Dirac equation. This model also often is described in terms of looking for or locating 'marked' elements in the graph, in which case one performs another step (to compute whether the node the walker is at is marked, and then to measure the outcome of that computation) before returning to Step 2. Other variations of this sort are reasonable. To perform a quantum walk on a more general graph, one must replace the "position" register with one which can express all of the nodes of the graph, and the "coin" register with one which can express the edges incident to a vertex. The "coin operator" then must also be replaced with one which allows the walker to perform an interesting superposition of different trajectories. (What counts as 'interesting' depends on what your motivation is: physicists often consider ways in which changing the coin operator changes the evolution of the probability density, not for computational purposes but as a way of probing at basic physics using quantum walks as a reasonable toy model of particle movement.) A good framework for generalising quantum walks to more general graphs is the Szegedy formulation [ arXiv:quant-ph/0401053 ] of discrete-time quantum walks. This model of computation is strictly speaking a special case of the unitary circuit model, but is motivated with very specific physical intuitions, which has led to some algorithmic insights (see e.g. [ arXiv:1302.3143 ]) for polynomial-time speedups in bounded-error quantum algorithms. This model is also a close relative of the continuous-time quantum walk as a model of computation. • 1 $\begingroup$ if you want to talk about DTQWs in the context of QC you should probably include references to the work of Childs and collaborators (e.g. arXiv:0806.1972. Also, you are describing how DTQWs work, but not really how you can use them to do computation. $\endgroup$ – glS Mar 26 '18 at 17:23 • 2 $\begingroup$ @gIS: indeed, I will add more details at some point: when I first wrote these it was to quickly enumerate some models and remark on them, rather than give comprehensive reviews. But as for how to compute, does the last paragraph not represent an example? $\endgroup$ – Niel de Beaudrap Mar 26 '18 at 20:05 • 1 $\begingroup$ @gIS: Isn't that work by Childs et al. actually about continuous-time quantum walks, anyhow? $\endgroup$ – Niel de Beaudrap Oct 29 '18 at 20:22 Quantum circuits with intermediary measurements This is a slight variation on "unitary circuits", in which one allows measurements in the middle of the algorithm as well as the end, and where one also allows future operations to depend on the outcomes of those measurements. It represents a realistic picture of a quantum processor which interacts with a classical control device, which among other things is the interface between the quantum processor and a human user. Intermediary measurement is practically necessary to perform error correction, and so this is in principle a more realistic picture of quantum computation than the unitary circuit model. but it is not uncommon for theorists of a certain type to strongly prefer measurements to be left until the end (using the principle of deferred measurement to simulate any 'intermediary' measurements). So, this may be a significant distinction to make when talking about quantum algorithms — but this does not lead to a theoretical increase in the computational power of a quantum algorithm. • 2 $\begingroup$ I think this should go with the "unitary circuit model" post, they are both really just variations of the circuit model, and one does not usually really distinguish them as different models $\endgroup$ – glS Mar 26 '18 at 17:20 • 1 $\begingroup$ @gIS: it is not uncommon to do so in the CS theory community. In fact, the bias is very much towards unitary circuits in particular. $\endgroup$ – Niel de Beaudrap Mar 26 '18 at 20:00 Quantum annealing Quantum annealing is a model of quantum computation which, roughly speaking, generalises the adiabatic model of computation. It has attracted popular — and commercial — attention as a result of D-WAVE's work on the subject. Precisely what quantum annealing consists of is not as well-defined as other models of computation, essentially because it is of more interest to quantum technologists than computer scientists. Broadly speaking, we can say that it is usually considered by people with the motivations of engineers, rather than the motivations of mathematicians, so that the subject appears to have many intuitions and rules of thumb but few 'formal' results. In fact, in an answer to my question about quantum annealing, Andrew O goes so far as to say that "quantum annealing can't be defined without considerations of algorithms and hardware". Nevertheless, "quantum annealing" seems is well-defined enough to be described as a way of approaching how to solve problems with quantum technologies with specific techniques — and so despite Andrew O's assessment, I think that it embodies some implicitly defined model of computation. I will attempt to describe that model here. Intuition behind the model Quantum annealing gets its name from a loose analogy to (classical) simulated annealing. They are both presented as means of minimising the energy of a system, expressed in the form of a Hamiltonian: $$ \begin{aligned} H_{\rm{classical}} &= \sum_{i,j} J_{ij} s_i s_j \\ H_{\rm{quantum}} &= A(t) \sum_{i,j} J_{ij} \sigma_i^z \sigma_j^z - B(t) \sum_i \sigma_i^x \end{aligned} $$ With simulated annealing, one essentially performs a random walk on the possible assignments to the 'local' variables $s_i \in \{0,1\}$, but where the probability of actually making a transition depends on • The difference in 'energy' $\Delta E = E_1 - E_0$ between two 'configurations' (the initial and the final global assignment to the variables $\{s_i\}_{i=1}^n$) before and after each step of the walk; • A 'temperature' parameter which governs the probability with which the walk is allowed to perform a step in the random walk which has $\Delta E > 0$. One starts with the system at 'infinite temperature', which is ultimately a fancy way of saying that you allow for all possible transitions, regardless of increases or decreases in energy. You then lower the temperature according to some schedule, so that time goes on, changes in state which increase the energy become less and less likely (though still possible). The limit is zero temperature, in which any transition which decreases energy is allowed, but any transition which increases energy is simply forbidden. For any temperature $T > 0$, there will be a stable distribution (a 'thermal state') of assignments, which is the uniform distribution at 'infinite' temperature, and which is which is more and more weighted on the global minimum energy states as the temperature decreases. If you take long enough to decrease the temperature from infinite to near zero, you should in principle be guaranteed to find a global optimum to the problem of minimising the energy. Thus simulated annealing is an approach to solving optimisation problems. Quantum annealing is motivated by generalising the work by Farhi et al. on adiabatic quantum computation [arXiv:quant-ph/0001106], with the idea of considering what evolution occurs when one does not necessarily evolve the Hamiltonian in the adiabatic regime. Similarly to classical annealing, one starts in a configuration in which "classical assignments" to some problem are in a uniform distribution, though this time in coherent superposition instead of a probability distribution: this is achieved for time $t = 0$, for instance, by setting $$ A(t=0) = 0, \qquad B(t=0) = 1 $$ in which case the uniform superposition $\def\ket#1{\lvert#1\rangle}\ket{\psi_0} \propto \ket{00\cdots00} + \ket{00\cdots01} + \cdots + \ket{11\cdots11}$ is a minimum-energy state of the quantum Hamiltonian. One steers this 'distribution' (i.e. the state of the quantum system) to one which is heavily weighted on a low-energy configuration by slowly evolving the system — by slowly changing the field strengths $A(t)$ and $B(t)$ to some final value $$ A(t_f) = 1, \qquad B(t_f) = 0. $$ Again, if you do this slowly enough, you will succeed with high probability in obtaining such a global minimum. The adiabatic regime describes conditions which are sufficient for this to occur, by virtue of remaining in (a state which is very close to) the ground state of the Hamiltonian at all intermediate times. However, it is considered possible that one can evolve the system faster than this and still achieve a high probability of success. Similarly to adiabatic quantum computing, the way that $A(t)$ and $B(t)$ are defined are often presented as a linear interpolations from $0$ to $1$ (increasing for $A(t)$, and decreasing for $B(t)$). However, also in common with adiabatic computation, $A(t)$ and $B(t)$ don't necessarily have to be linear or even monotonic. For instance, D-Wave has considered the advantages of pausing the annealing schedule and 'backwards anneals'. 'Proper' quantum annealing (so to speak) presupposes that evolution is probably not being done in the adiabatic regime, and allows for the possibility of diabatic transitions, but only asks for a high chance of achieving an optimum — or even more pragmatically still, of achieving a result which would be difficult to find using classical techniques. There are no formal results about how quickly you can change your Hamiltonian to achieve this: the subject appears mostly to consist of experimenting with a heuristic to see what works in practise. The comparison with classical simulated annealing Despite the terminology, it is not immediately clear that there is much which quantum annealing has in common with classical annealing. The main differences between quantum annealing and classical simulated annealing appear to be that: • In quantum annealing, the state is in some sense ideally a pure state, rather than a mixed state (corresponding to the probability distribution in classical annealing); • In quantum annealing, the evolution is driven by an explicit change in the Hamiltonian rather than an external parameter. It is possible that a change in presentation could make the analogy between quantum annealing and classical annealing tighter. For instance, one could incorporate the temperature parameter into the spin Hamiltonian for classical annealing, by writing $$\tilde H_{\rm{classical}} = A(t) \sum_{i,j} J_{ij} s_i s_j - B(t) \sum_{i,j} \textit{const.} $$ where we might choose something like $A(t) = t\big/(t_F - t)$ and $B(t) = t_F - t$ for $t_F > 0$ the length of the anneal schedule. (This is chosen deliberately so that $A(0) = 0$ and $A(t) \to +\infty$ for $t \to t_F$.) Then, just as an annealing algorithm is governed in principle by the Schrödinger equation for all times, we may consider an annealing process which is governed by a diffusion process which is in principle uniform with tim by small changes in configurations, where the probability of executing a randomly selected change of configuration is governed by $$ p(x \to y) = \max\Bigl\{ 1,\; \exp\bigl(-\gamma \Delta E_{x\to y}\bigr) \Bigr\} $$ for some constant $\gamma$, where $E_{x \to y}$ is the energy difference between the initial and final configurations. The stable distribution of this diffusion for the Hamiltonian at $t=0$ is the uniform distribution, and the stable distribution for the Hamiltonian as $t \to t_F$ is any local minimum; and as $t$ increases, the probability with which a transition occurs which increases the energy becomes smaller, until as $t \to t_F$ the probability of any increases in energy vanish (because any of the possible increase is a costly one). There are still disanalogies to quantum annealing in this — for instance, we achieve the strong suppression of increases in energy as $t \to t_F$ essentially by making the potential wells infinitely deep (which is not a very physical thing to do) — but this does illustrate something of a commonality between the two models, with the main distinction being not so much the evolution of the Hamiltonian as it is the difference between diffusion and Schrödinger dynamics. This suggests that there may be a sharper way to compare the two models theoretically: by describing the difference between classical and quantum annealing, as being analogous to the difference between random walks and quantum walks. A common idiom in describing quantum annealing is to speak of 'tunnelling' through energy barriers — this is certainly pertinent to how people consider quantum walks: consider for instance the work by Farhi et al. on continuous-time quantum speed-ups for evaluating NAND circuits, and more directly foundational work by Wong on quantum walks on the line tunnelling through potential barriers. Some work has been done by Chancellor [arXiv:1606.06800] on considering quantum annealing in terms of quantum walks, though it appears that there is room for a more formal and complete account. On a purely operational level, it appears that quantum annealing gives a performance advantage over classical annealing (see for example these slides on the difference in performance between quantum vs. classical annealing, from Troyer's group at ETH, ca. 2014). Quantum annealing as a phenomenon, as opposed to a computational model Because quantum annealing is more studied by technologists, they focus on the concept of realising quantum annealing as an effect rather than defining the model in terms of general principles. (A rough analogy would be studying the unitary circuit model only inasmuch as it represents a means of achieving the 'effects' of eigenvalue estimation or amplitude amplification.) Therefore, whether something counts as "quantum annealing" is described by at least some people as being hardware-dependent, and even input-dependent: for instance, on the layout of the qubits, the noise levels of the machine. It seems that even trying to approach the adiabatic regime will prevent you from achieving quantum annealing, because the idea of what quantum annealing even consists of includes the idea that noise (such as decoherence) will prevent annealing from being realised: as a computational effect, as opposed to a computational model, quantum annealing essentially requires that the annealing schedule is shorter than the decoherence time of the quantum system. Some people occasionally describe noise as being somehow essential to the process of quantum annealing. For instance, Boixo et al. [arXiv:1304.4595] write Unlike adiabatic quantum computing[, quantum annealing] is a positive temperature method involving an open quantum system coupled to a thermal bath. It might perhaps be accurate to describe it as being an inevitable feature of systems in which one will perform annealing (just because noise is inevitable feature of a system in which you will do quantum information processing of any kind): as Andrew O writes "in reality no baths really help quantum annealing". It is possible that a dissipative process can help quantum annealing by helping the system build population on lower-energy states (as suggested by work by Amin et al., [arXiv:cond-mat/0609332]), but this seems essentially to be a classical effect, and would inherently require a quiet low-temperature environment rather than 'the presence of noise'. The bottom line It might be said — in particular by those who study it — that quantum annealing is an effect, rather than a model of computation. A "quantum annealer" would then be best understood as "a machine which realises the effect of quantum annealing", rather than a machine which attempts to embody a model of computation which is known as 'quantum annealing'. However, the same might be said of adiabatic quantum computation, which is — in my opinion correctly — described as a model of computation in its own right. Perhaps it would be fair to describe quantum annealing as an approach to realising a very general heuristic, and that there is an implicit model of computation which could be characterised as the conditions under which we could expect this heuristic to be successful. If we consider quantum annealing this way, it would be a model which includes the adiabatic regime (with zero-noise) as a special case, but it may in principle be more general. Your Answer
dfaf9fc2d8675e88	World Library Flag as Inappropriate Email this Article Time in physics Article Id: WHEBN0019595664 Reproduction Date: Title: Time in physics Author: World Heritage Encyclopedia Language: English Subject: Time, Spacetime, Acceleration, Century leap year, Sidereal year Collection: Philosophy of Physics, Time, Time in Physics, Timekeeping Publisher: World Heritage Encyclopedia Time in physics Foucault's pendulum in the Panthéon of Paris can measure time as well as demonstrate the rotation of Earth. Time in physics is defined by its measurement: time is what a clock reads.[1] In classical, non-relativistic physics it is a scalar quantity and, like length, mass, and charge, is usually described as a fundamental quantity. Time can be combined mathematically with other physical quantities to derive other concepts such as motion, kinetic energy and time-dependent fields. Timekeeping is a complex of technological and scientific issues, and part of the foundation of recordkeeping. • Markers of time 1 • The unit of measurement of time: the second 2 • The state of the art in timekeeping 2.1 • Conceptions of time 3 • Regularities in nature 3.1 • Mechanical clocks 3.1.1 • Galileo: the flow of time 3.2 • Newton's physics: linear time 3.3 • Thermodynamics and the paradox of irreversibility 3.4 • Electromagnetism and the speed of light 3.5 • Einstein's physics: spacetime 3.6 • Time in quantum mechanics 3.7 • Dynamical systems 4 • Signalling 5 • Technology for timekeeping standards 6 • Time in cosmology 7 • Reprise 8 • See also 9 • References 10 • Further reading 11 Markers of time Before there were clocks, time was measured by those physical processes[2] which were understandable to each epoch of civilization:[3] • the first appearance (see: heliacal rising) of Sirius to mark the flooding of the Nile each year[3] • the periodic succession of night and day, one after the other, in seemingly eternal succession[4] • the position on the horizon of the first appearance of the sun at dawn[5] • the position of the sun in the sky[6] • the marking of the moment of noontime during the day[7] • the length of the shadow cast by a gnomon[8] Eventually,[9][10] it became possible to characterize the passage of time with instrumentation, using operational definitions. Simultaneously, our conception of time has evolved, as shown below.[11] The unit of measurement of time: the second In the International System of Units (SI), the unit of time is the second (symbol: \mathrm{s}). It is a SI base unit, and it is currently defined as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." [12] This definition is based on the operation of a caesium atomic clock. The state of the art in timekeeping The UTC timestamp in use worldwide is an atomic time standard. The relative accuracy of such a time standard is currently on the order of 10−15[13] (corresponding to 1 second in approximately 30 million years). The smallest time step considered observable is called the Planck time, which is approximately 5.391×10−44 seconds - many orders of magnitude below the resolution of current time standards. Conceptions of time Andromeda galaxy (M31) is two million light-years away. Thus we are viewing M31's light from two million years ago,[14] a time before humans existed on Earth. Both Galileo and Newton and most people up until the 20th century thought that time was the same for everyone everywhere. This is the basis for timelines, where time is a parameter. Our modern conception of time is based on Einstein's theory of relativity, in which rates of time run differently depending on relative motion, and space and time are merged into spacetime, where we live on a world line rather than a timeline. Thus time is part of a coordinate, in this view. Physicists believe the entire Universe and therefore time itself[15] began about 13.8 billion years ago in the big bang. (See Time in Cosmology below) Whether it will ever come to an end is an open question. (See philosophy of physics.) Regularities in nature In order to measure time, one can record the number of occurrences (events) of some periodic phenomenon. The regular recurrences of the seasons, the motions of the sun, moon and stars were noted and tabulated for millennia, before the laws of physics were formulated. The sun was the arbiter of the flow of time, but time was known only to the hour for millennia, hence, the use of the gnomon was known across most of the world, especially Eurasia, and at least as far southward as the jungles of Southeast Asia.[16] In particular, the astronomical observatories maintained for religious purposes became accurate enough to ascertain the regular motions of the stars, and even some of the planets. At first, timekeeping was done by hand by priests, and then for commerce, with watchmen to note time as part of their duties. The tabulation of the equinoxes, the sandglass, and the water clock became more and more accurate, and finally reliable. For ships at sea, boys were used to turn the sandglasses and to call the hours. Mechanical clocks Richard of Wallingford (1292–1336), abbot of St. Alban's abbey, famously built a mechanical clock as an astronomical orrery about 1330.[17][18] By the time of Richard of Wallingford, the use of ratchets and gears allowed the towns of Europe to create mechanisms to display the time on their respective town clocks; by the time of the scientific revolution, the clocks became miniaturized enough for families to share a personal clock, or perhaps a pocket watch. At first, only kings could afford them. Pendulum clocks were widely used in the 18th and 19th century. They have largely been replaced in general use by quartz and digital clocks. Atomic clocks can theoretically keep accurate time for millions of years. They are appropriate for standards and scientific use. Galileo: the flow of time In 1583, Galileo Galilei (1564–1642) discovered that a pendulum's harmonic motion has a constant period, which he learned by timing the motion of a swaying lamp in harmonic motion at mass at the cathedral of Pisa, with his pulse.[19] In his Two New Sciences (1638), Galileo used a water clock to measure the time taken for a bronze ball to roll a known distance down an inclined plane; this clock was Galileo's experimental setup to measure the literal flow of time, in order to describe the motion of a ball, preceded Isaac Newton's statement in his Principia: I do not define time, space, place and motion, as being well known to all.[21] The Galilean transformations assume that time is the same for all reference frames. Newton's physics: linear time In or around 1665, when Isaac Newton (1643–1727) derived the motion of objects falling under gravity, the first clear formulation for mathematical physics of a treatment of time began: linear time, conceived as a universal clock. The water clock mechanism described by Galileo was engineered to provide laminar flow of the water during the experiments, thus providing a constant flow of water for the durations of the experiments, and embodying what Newton called duration. In this section, the relationships listed below treat time as a parameter which serves as an index to the behavior of the physical system under consideration. Because Newton's fluents treat a linear flow of time (what he called mathematical time), time could be considered to be a linearly varying parameter, an abstraction of the march of the hours on the face of a clock. Calendars and ship's logs could then be mapped to the march of the hours, days, months, years and centuries. Lagrange (1736–1813) would aid in the formulation of a simpler version[23] of Newton's equations. He started with an energy term, L, named the Lagrangian in his honor, and formulated Lagrange's equations: \frac{d}{dt} \frac{\partial L}{\partial \dot{\theta}} - \frac{\partial L}{\partial \theta} = 0. The dotted quantities, {\dot{\theta}} denote a function which corresponds to a Newtonian fluxion, whereas denote a function which corresponds to a Newtonian fluent. But linear time is the parameter for the relationship between the {\dot{\theta}} and the of the physical system under consideration. Some decades later, it was found that the second order equation of Lagrange or Newton can be more easily solved or visualized by suitable transformation to sets of first order differential equations. Lagrange's equations can be transformed, under a Legendre transformation, to Hamilton's equations; the Hamiltonian formulation for the equations of motion of some conjugate variables p,q (for example, momentum p and position q) is: \dot p = -\frac{\partial H}{\partial q} = \{p,H\} = -\{H,p\} \dot q =~~\frac{\partial H}{\partial p} = \{q,H\} = -\{H,q\} in the Poisson bracket notation and clearly shows the dependence of the time variation of conjugate variables p,q on an energy expression. This relationship, it was to be found, also has corresponding forms in quantum mechanics as well as in the classical mechanics shown above. These relationships bespeak a conception of time which is reversible. Thermodynamics and the paradox of irreversibility By 1798, Benjamin Thompson (1753–1814) had discovered that work could be transformed to heat without limit - a precursor of the conservation of energy or In 1824 Sadi Carnot (1796–1832) scientifically analyzed the steam engines with his Carnot cycle, an abstract engine. Rudolf Clausius (1822–1888) noted a measure of disorder, or entropy, which affects the continually decreasing amount of free energy which is available to a Carnot engine in the: Thus the continual march of a thermodynamic system, from lesser to greater entropy, at any given temperature, defines an arrow of time. In particular, Stephen Hawking identifies three arrows of time:[24] • Psychological arrow of time - our perception of an inexorable flow. • Thermodynamic arrow of time - distinguished by the growth of entropy. • Cosmological arrow of time - distinguished by the expansion of the universe. Entropy is maximum in an isolated thermodynamic system, and increases. In contrast, Erwin Schrödinger (1887–1961) pointed out that life depends on a "negative entropy flow".[25] Ilya Prigogine (1917–2003) stated that other thermodynamic systems which, like life, are also far from equilibrium, can also exhibit stable spatio-temporal structures. Soon afterward, the Belousov-Zhabotinsky reactions[26] were reported, which demonstrate oscillating colors in a chemical solution.[27] These nonequilibrium thermodynamic branches reach a bifurcation point, which is unstable, and another thermodynamic branch becomes stable in its stead.[28] Electromagnetism and the speed of light In 1864, James Clerk Maxwell (1831–1879) presented a combined theory of electricity and magnetism. He combined all the laws then known relating to those two phenomenon into four equations. These vector calculus equations which use the del operator (\nabla) are known as Maxwell's equations for electromagnetism. In free space (that is, space not containing electric charges), the equations take the form (using SI units):[29] \nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \nabla \cdot \mathbf{E} = 0 \nabla \cdot \mathbf{B} = 0 ε0 and μ0 are the electric permittivity and the magnetic permeability of free space; c = 1/\sqrt{\epsilon_0 \mu_0} is the speed of light in free space, 299 792 458 m/s; E is the electric field; B is the magnetic field. These equations allow for solutions in the form of electromagnetic waves. The wave is formed by an electric field and a magnetic field oscillating together, perpendicular to each other and to the direction of propagation. These waves always propagate at the speed of light c, regardless of the velocity of the electric charge that generated them. The fact that light is predicted to always travel at speed c would be incompatible with Galilean relativity if Maxwell's equations were assumed to hold in any inertial frame (reference frame with constant velocity), because the Galilean transformations predict the speed to decrease (or increase) in the reference frame of an observer traveling parallel (or antiparallel) to the light. It was expected that there was one absolute reference frame, that of the luminiferous aether, in which Maxwell's equations held unmodified in the known form. The Michelson-Morley experiment failed to detect any difference in the relative speed of light due to the motion of the Earth relative to the luminiferous aether, suggesting that Maxwell's equations did, in fact, hold in all frames. In 1875, Hendrik Lorentz (1853–1928) discovered Lorentz transformations, which left Maxwell's equations unchanged, allowing Michelson and Morley's negative result to be explained. Henri Poincaré (1854–1912) noted the importance of Lorentz' transformation and popularized it. In particular, the railroad car description can be found in Science and Hypothesis,[30] which was published before Einstein's articles of 1905. The Lorentz transformation predicted space contraction and time dilation; until 1905, the former was interpreted as a physical contraction of objects moving with respect to the aether, due to the modification of the intermolecular forces (of electric nature), while the latter was thought to be just a mathematical stipulation. Einstein's physics: spacetime Main articles: special relativity (1905), general relativity (1915). Albert Einstein's 1905 special relativity challenged the notion of absolute time, and could only formulate a definition of synchronization for clocks that mark a linear flow of time: If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an "A time" and a "B time." We have not defined a common "time" for A and B, for the latter cannot be defined at all unless we establish by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A. Let a ray of light start at the "A time" tA from A towards B, let it at the "B time" tB be reflected at B in the direction of A, and arrive again at A at the “A time” tA. In accordance with definition the two clocks synchronize if t_\text{B} - t_\text{A} = t'_\text{A} - t_\text{B}\text{.}\,\! —Albert Einstein, "On the Electrodynamics of Moving Bodies" [31] Einstein showed that if the speed of light is not changing between reference frames, space and time must be so that the moving observer will measure the same speed of light as the stationary one because velocity is defined by space and time: \mathbf{v}={d\mathbf{r}\over dt} \text{,} where r is position and t is time. Indeed, the Lorentz transformation (for two reference frames in relative motion, whose x axis is directed in the direction of the relative velocity) \begin{cases} t' &= \gamma(t - vx/c^2) \text{ where } \gamma = 1/\sqrt{1-v^2/c^2} \\ x' &= \gamma(x - vt)\\ y' &= y \\ z' &= z \end{cases} can be said to "mix" space and time in a way similar to the way a Euclidean rotation around the z axis mixes x and y coordinates. Consequences of this include relativity of simultaneity. More specifically, the Lorentz transformation is a hyperbolic rotation \begin{pmatrix} ct' \\ x' \end{pmatrix} = \begin{pmatrix} \cosh \phi & - \sinh \phi \\ - \sinh \phi & \cosh \phi \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix} \text{ where } \phi = \operatorname{artanh}\,\frac{v}{c} \text{,} which is a change of coordinates in the four-dimensional Minkowski space, a dimension of which is ct. (In Euclidean space an ordinary rotation \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} is the corresponding change of coordinates.) The speed of light c can be seen as just a conversion factor needed because we measure the dimensions of spacetime in different units; since the metre is currently defined in terms of the second, it has the exact value of 299 792 458 m/s. We would need a similar factor in Euclidean space if, for example, we measured width in nautical miles and depth in feet. In physics, sometimes units of measurement in which c = 1 are used to simplify equations. Time in a "moving" reference frame is shown to run more slowly than in a "stationary" one by the following relation (which can be derived by the Lorentz transformation by putting ∆x′ = 0, ∆τ = ∆t′): \Delta t= • τ is the time between two events as measured in the moving reference frame in which they occur at the same place (e.g. two ticks on a moving clock); it is called the proper time between the two events; • t is the time between these same two events, but as measured in the stationary reference frame; • v is the speed of the moving reference frame relative to the stationary one; • c is the speed of light. Moving objects therefore are said to show a slower passage of time. This is known as time dilation. These transformations are only valid for two frames at constant relative velocity. Naively applying them to other situations gives rise to such paradoxes as the twin paradox. That paradox can be resolved using for instance Einstein's General theory of relativity, which uses Riemannian geometry, geometry in accelerated, noninertial reference frames. Employing the metric tensor which describes Minkowski space: Einstein developed a geometric solution to Lorentz's transformation that preserves Maxwell's equations. His field equations give an exact relationship between the measurements of space and time in a given region of spacetime and the energy density of that region. Einstein's equations predict that time should be altered by the presence of gravitational fields (see the Schwarzschild metric): T=\frac{dt}{\sqrt{\left( 1 - \frac{2GM}{rc^2} \right ) dt^2 - \frac{1}{c^2}\left ( 1 - \frac{2GM}{rc^2} \right )^{-1} dr^2 - \frac{r^2}{c^2} d\theta^2 - \frac{r^2}{c^2} \sin^2 \theta \; d\phi^2}} T is the gravitational time dilation of an object at a distance of r. dt is the change in coordinate time, or the interval of coordinate time. G is the gravitational constant M is the mass generating the field Or one could use the following simpler approximation: \frac{dt}{d\tau} = \frac{1}{ \sqrt{1 - \left( \frac{2GM}{rc^2} \right)}}. Time runs slower the stronger the gravitational field, and hence acceleration, is. The predictions of time dilation are confirmed by particle acceleration experiments and cosmic ray evidence, where moving particles decay more slowly than their less energetic counterparts. Gravitational time dilation gives rise to the phenomenon of gravitational redshift and delays in signal travel time near massive objects such as the sun. The Global Positioning System must also adjust signals to account for this effect. According to Einstein's general theory of relativity, a freely moving particle traces a history in spacetime that maximises its proper time. This phenomenon is also referred to as the principle of maximal aging, and was described by Taylor and Wheeler as:[32] "Principle of Extremal Aging: The path a free object takes between two events in spacetime is the path for which the time lapse between these events, recorded on the object's wristwatch, is an extremum." Einstein's theory was motivated by the assumption that every point in the universe can be treated as a 'center', and that correspondingly, physics must act the same in all reference frames. His simple and elegant theory shows that time is relative to an inertial frame. In an inertial frame, Newton's first law holds; it has its own local geometry, and therefore its own measurements of space and time; there is no 'universal clock'. An act of synchronization must be performed between two systems, at the least. Time in quantum mechanics There is a time parameter in the equations of quantum mechanics. The Schrödinger equation[33] is One solution can be \| \psi_e(t) \rangle = e^{-iHt / \hbar} \| \psi_e(0) \rangle . where e^{-iHt / \hbar} is called the time evolution operator, and H is the Hamiltonian. But the Schrödinger picture shown above is equivalent to the Heisenberg picture, which enjoys a similarity to the Poisson brackets of classical mechanics. The Poisson brackets are superseded by a nonzero commutator, say [H,A] for observable A, and Hamiltonian H: \frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_\mathrm{classical}. This equation denotes an uncertainty relation in quantum physics. For example, with time (the observable A), the energy E (from the Hamiltonian H) gives: \Delta E \Delta T \ge \frac{\hbar}{2} \Delta E is the uncertainty in energy \Delta T is the uncertainty in time \hbar is Planck's constant The more precisely one measures the duration of a sequence of events the less precisely one can measure the energy associated with that sequence and vice versa. This equation is different from the standard uncertainty principle because time is not an operator in quantum mechanics. Corresponding commutator relations also hold for momentum p and position q, which are conjugate variables of each other, along with a corresponding uncertainty principle in momentum and position, similar to the energy and time relation above. Quantum mechanics explains the properties of the periodic table of the elements. Starting with Otto Stern's and Walter Gerlach's experiment with molecular beams in a magnetic field, Isidor Rabi (1898–1988), was able to modulate the magnetic resonance of the beam. In 1945 Rabi then suggested that this technique be the basis of a clock[34] using the resonant frequency of an atomic beam. Dynamical systems See dynamical systems and chaos theory, dissipative structures One could say that time is a parameterization of a dynamical system that allows the geometry of the system to be manifested and operated on. It has been asserted that time is an implicit consequence of chaos (i.e. nonlinearity/irreversibility): the characteristic time, or rate of information entropy production, of a system. Mandelbrot introduces intrinsic time in his book Multifractals and 1/f noise. Signalling is one application of the electromagnetic waves described above. In general, a signal is part of communication between parties and places. One example might be a yellow ribbon tied to a tree, or the ringing of a church bell. A signal can be part of a conversation, which involves a protocol. Another signal might be the position of the hour hand on a town clock or a railway station. An interested party might wish to view that clock, to learn the time. See: Time ball, an early form of Time signal. Evolution of a world line of an accelerated massive particle. This worldline is restricted to the timelike top and bottom sections of this spacetime figure and can not cross the top (future) nor the bottom (past) light cone. The left and right sections, outside the light cones are spacelike. We as observers can still signal different parties and places as long as we live within their past light cone. But we cannot receive signals from those parties and places outside our past light cone. Along with the formulation of the equations for the electromagnetic wave, the field of telecommunication could be founded. In 19th century telegraphy, electrical circuits, some spanning continents and oceans, could transmit codes - simple dots, dashes and spaces. From this, a series of technical issues have emerged; see Category:Synchronization. But it is safe to say that our signalling systems can be only approximately synchronized, a plesiochronous condition, from which jitter need be eliminated. That said, systems can be synchronized (at an engineering approximation), using technologies like GPS. The GPS satellites must account for the effects of gravitation and other relativistic factors in their circuitry. See: Self-clocking signal. Technology for timekeeping standards The primary time standard in the U.S. is currently NIST-F1, a laser-cooled Cs fountain,[35] the latest in a series of time and frequency standards, from the ammonia-based atomic clock (1949) to the caesium-based NBS-1 (1952) to NIST-7 (1993). The respective clock uncertainty declined from 10,000 nanoseconds per day to 0.5 nanoseconds per day in 5 decades.[36] In 2001 the clock uncertainty for NIST-F1 was 0.1 nanoseconds/day. Development of increasingly accurate frequency standards is underway. In this time and frequency standard, a population of caesium atoms is laser-cooled to temperatures of one microkelvin. The atoms collect in a ball shaped by six lasers, two for each spatial dimension, vertical (up/down), horizontal (left/right), and back/forth. The vertical lasers push the caesium ball through a microwave cavity. As the ball is cooled, the caesium population cools to its ground state and emits light at its natural frequency, stated in the definition of second above. Eleven physical effects are accounted for in the emissions from the caesium population, which are then controlled for in the NIST-F1 clock. These results are reported to BIPM. Additionally, a reference hydrogen maser is also reported to BIPM as a frequency standard for TAI (international atomic time). The measurement of time is overseen by BIPM (Bureau International des Poids et Mesures), located in Sèvres, France, which ensures uniformity of measurements and their traceability to the International System of Units (SI) worldwide. BIPM operates under authority of the Metre Convention, a diplomatic treaty between fifty-one nations, the Member States of the Convention, through a series of Consultative Committees, whose members are the respective national metrology laboratories. Time in cosmology The equations of general relativity predict a non-static universe. However, Einstein accepted only a static universe, and modified the Einstein field equation to reflect this by adding the general relativity, that the universe originated in a primordial explosion. At the fifth Solvay conference, that year, Einstein brushed him off with "Vos calculs sont corrects, mais votre physique est abominable."[37] (“Your math is correct, but your physics is abominable”). In 1929, Edwin Hubble (1889–1953) announced his discovery of the expanding universe. The current generally accepted cosmological model, the Lambda-CDM model, has a positive cosmological constant and thus not only an expanding universe but an accelerating expanding universe. If the universe were expanding, then it must have been much smaller and therefore hotter and denser in the past. Fred Hoyle (1915–2001), who invented the term 'Big Bang' to disparage it. Fermi and others noted that this process would have stopped after only the light elements were created, and thus did not account for the abundance of heavier elements. WMAP fluctuations of the cosmic microwave background radiation.[38] Gamow's prediction was a 5–10 kelvin black body radiation temperature for the universe, after it cooled during the expansion. This was corroborated by Penzias and Wilson in 1965. Subsequent experiments arrived at a 2.7 kelvin temperature, corresponding to an age of the universe of 13.8 billion years after the Big Bang. This dramatic result has raised issues: what happened between the singularity of the Big Bang and the Planck time, which, after all, is the smallest observable time. When might have time separated out from the spacetime foam;[39] there are only hints based on broken symmetries (see Spontaneous symmetry breaking, Timeline of the Big Bang, and the articles in Category:Physical cosmology). General relativity gave us our modern notion of the expanding universe that started in the Big Bang. Using relativity and quantum theory we have been able to roughly reconstruct the history of the universe. In our epoch, during which electromagnetic waves can propagate without being disturbed by conductors or charges, we can see the stars, at great distances from us, in the night sky. (Before this epoch, there was a time, 300,000 years after the big bang, during which starlight would not have been visible.) Ilya Prigogine's reprise is "Time precedes existence". He contrasts the views of Newton, Einstein and quantum physics which offer a symmetric view of time (as discussed above) with his own views, which point out that statistical and thermodynamic physics can explain irreversible phenomena[40] as well as the arrow of time and the Big Bang. See also 1. ^ Considine, Douglas M.; Considine, Glenn D. (1985). Process instruments and controls handbook (3 ed.). McGraw-Hill. pp. 18–61. 2. ^ For example, Galileo measured the period of a simple harmonic oscillator with his pulse. 3. ^ a b Otto Neugebauer The Exact Sciences in Antiquity. Princeton: Princeton University Press, 1952; 2nd edition, Brown University Press, 1957; reprint, New York: Dover publications, 1969. Page 82. 4. ^ See, for example William Shakespeare Hamlet: " ... to thine own self be true, And it must follow, as the night the day, Thou canst not then be false to any man." 5. ^ "Heliacal/Dawn Risings". Retrieved 2012-08-17. 6. ^ Farmers have used the sun to mark time for thousands of years, as the most ancient method of telling time. 7. ^ Eratosthenes used this criterion in his measurement of the circumference of Earth 8. ^ Fred Hoyle (1962), Astronomy: A history of man's investigation of the universe, Crescent Books, Inc., London LC 62-14108, p.31 9. ^ The Mesopotamian (modern-day Iraq) astronomers recorded astronomical observations with the naked eye, more than 3500 years ago. P. W. Bridgman defined his operational definition in the twentieth c. 10. ^ Naked eye astronomy became obsolete in 1609 with Galileo's observations with a telescope. Galileo Galilei Linceo, Sidereus Nuncius (Starry Messenger) 1610. 11. ^ Today, automated astronomical observations from satellites and spacecraft require relativistic corrections of the reported positions. 12. ^ "Unit of time (second)". SI brochure. 13. ^ S. R. Jefferts et al., "Accuracy evaluation of NIST-F1". 14. ^ Fred Adams and Greg Laughlin (1999), Five Ages of the Universe ISBN 0-684-86576-9 p.35. 15. ^ See 16. ^ Charles Hose and William McDougall (1912) The Pagan Tribes of Borneo, Plate 60. Kenyahs measuring the Length of the Shadow at Noon to determine the Time for sowing PADI p. 108. This photograph is reproduced as plate B in Fred Hoyle (1962), Astronomy: A history of man's investigation of the universe, Crescent Books, Inc., London LC 62-14108, p.31. The measurement process is explained by: Gene Ammarell (1997), "Astronomy in the Indo-Malay Archipelago", p.119, Encyclopaedia of the history of science, technology, and medicine in non-western cultures, Helaine Selin, ed., which describes Kenyah Tribesmen of Borneo measuring the shadow cast by a gnomon, or tukar do with a measuring scale, or aso do. 19. ^ Jo Ellen Barnett, Time's Pendulum ISBN 0-306-45787-3 p.99. 20. ^ Galileo 1638 Discorsi e dimostrazioni matematiche, intorno á due nuoue scienze 213, Leida, Appresso gli Elsevirii (Louis Elsevier), or Mathematical discourses and demonstrations, relating to Two New Sciences, English translation by Henry Crew and Alfonso de Salvio 1914. Section 213 is reprinted on pages 534-535 of On the Shoulders of Giants:The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4 21. ^ Newton 1687 Philosophiae Naturalis Principia Mathematica, Londini, Jussu Societatis Regiae ac Typis J. Streater, or The Mathematical Principles of Natural Philosophy, London, English translation by Andrew Motte 1700s. From part of the Scholium, reprinted on page 737 of On the Shoulders of Giants:The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4 22. ^ Newton 1687 page 738. 23. ^ "Dynamics is a four-dimensional geometry." --Lagrange (1796), Thèorie des fonctions analytiques, as quoted by Ilya Prigogine (1996), The End of Certainty ISBN 0-684-83705-6 p.58 24. ^ pp. 182-195. Stephen Hawking 1996. The Illustrated Brief History of Time: updated and expanded edition ISBN 0-553-10374-1 25. ^ Erwin Schrödinger (1945) What is Life? 26. ^ G. Nicolis and I. Prigogine (1989), Exploring Complexity 27. ^ R. Kapral and K. Showalter, eds. (1995), Chemical Waves and Patterns 28. ^ Ilya Prigogine (1996) The End of Certainty pp. 63-71 29. ^ Clemmow, P. C. (1973). An introduction to electromagnetic theory. CUP Archive. pp. 56–57. , Extract of pages 56, 57 30. ^ Henri Poincaré, (1902). Science and Hypothesis Eprint 31. ^ Einstein 1905, Zur Elektrodynamik bewegter Körper [On the electrodynamics of moving bodies] reprinted 1922 in Das Relativitätsprinzip, B.G. Teubner, Leipzig. The Principles of Relativity: A Collection of Original Papers on the Special Theory of Relativity, by H.A. Lorentz, A. Einstein, H. Minkowski, and W. H. Weyl, is part of Fortschritte der mathematischen Wissenschaften in Monographien, Heft 2. The English translation is by W. Perrett and G.B. Jeffrey, reprinted on page 1169 of On the Shoulders of Giants:The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4 32. ^ 33. ^ E. Schrödinger, Phys. Rev. 28 1049 (1926) 34. ^ A Brief History of Atomic Clocks at NIST 35. ^ D. M. Meekhof, S. R. Jefferts, M. Stepanovíc, and T. E. Parker (2001) "Accuracy Evaluation of a Cesium Fountain Primary Frequency Standard at NIST", IEEE Transactions on Instrumentation and Measurement. 50, no. 2, (April 2001) pp. 507-509 36. ^ James Jespersen and Jane Fitz-Randolph (1999). From sundials to atomic clocks : understanding time and frequency. Washington, D.C. : U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology. 308 p. : ill. ; 28 cm. ISBN 0-16-050010-9 37. ^ John C. Mather and John Boslough (1996), The Very First Light ISBN 0-465-01575-1 p.41. 38. ^ cosmic microwave background radiation 39. ^ Martin Rees (1997), Before the Beginning ISBN 0-201-15142-1 p.210 40. ^ Prigogine, Ilya (1996), The End of Certainty: Time, Chaos and the New Laws of Nature. ISBN 0-684-83705-6 On pages 163 and 182. Further reading • Boorstein, Daniel J., The Discoverers. Vintage. February 12, 1985. ISBN 0-394-72625-1 • Dieter Zeh, H., The physical basis of the direction of time. Springer. ISBN 978-3-540-42081-1 • Kuhn, Thomas S., The Structure of Scientific Revolutions. ISBN 0-226-45808-3 • Mandelbrot, Benoît, Multifractals and 1/f noise. Springer Verlag. February 1999. ISBN 0-387-98539-5 • Prigogine, Ilya (1984), Order out of Chaos. ISBN 0-394-54204-5 • Serres, Michel, et al., "Conversations on Science, Culture, and Time (Studies in Literature and Science)". March, 1995. ISBN 0-472-06548-3 • Stengers, Isabelle, and Ilya Prigogine, Theory Out of Bounds. University of Minnesota Press. November 1997. ISBN 0-8166-2517-4
8a744b083792c769	Skip to main content Characterization and modelling Characterization and modelling A broader range of experiments 20 Feb 2020 Margaret Harris Anatole von Lilienfeld is a professor of physical chemistry at the University of Basel, Switzerland, and a project leader in the Swiss National Center for Computational Design and Discovery of Novel Materials (MARVEL). He is also the editor-in-chief of a new journal called Machine Learning: Science and Technology, which (like Physics World) is published by IOP Publishing. He spoke to Margaret Harris about the role that machine learning plays in science, the purpose of the journal and how he thinks the field will develop. A photo of Anatole von Lilienfeld holding circuit boards The term “machine learning” means different things to different people. What’s your definition? It’s a term used by many communities, but in the context of physics I would stick to a rather technical definition. Machine learning can be roughly divided into two different domains, depending on the problems one wants to attack. One domain is called unsupervised learning, which is basically about categorizing data. This task can be nontrivial when you’re dealing with high-dimensional, heterogeneous data of varying fidelity. What unsupervised learning algorithms do is try to determine whether these data can be grouped into different clusters in a systematic way, without any bias or heuristic, and without introducing spurious artefacts. Problems of this type are ubiquitous: all quantitative sciences encounter them in one way or another. But one example involves proteins, which fold in certain shapes that depend on their amino acid sequences. When you measure the X-ray spectra of protein crystals, you find something interesting: the number of possible folds is large, but finite. So if somebody gave you some sequences and their corresponding folds, a good unsupervised learning algorithm might be able to cluster new sequences to help you determine which of them are associated with which folds. The second branch of machine learning is called supervised learning. In this case, rather than merely categorizing the data, the algorithms also try to predict values outside the dataset. An example from materials science would be that if you have a bunch of materials for which a property has been measured – the formation energy of some inorganic crystals, say – you can then ask, “I wonder what the formation energy would be of a new crystal?” Supervised learning can give you the statistically most likely estimate, based on the known properties of the existing materials in the dataset. These are the two main branches of machine learning, and the thing they have in common is a need for data. There’s no machine learning without data. It’s a statistical approach, and this is sort of implied when you’re talking about machine learning: these techniques are mathematically rigorous ways to arrive at statistical statements in a quantitative manner. You’ve given a couple of examples of machine-learning applications within materials science. I know this is the subject closest to your heart, but the new journal you’re working on covers the whole of science. What are some applications in other fields? Of course, I’m biased towards materials science, but other domains face similar problems. Here’s an example. One of the most important equations in materials science is the electronic Schrödinger equation. This differential equation is difficult to solve even with computers, but machine learning enables us to circumvent the need to solve it for new materials. Similarly, many scientific domains require solutions to the Navier-Stokes equations in various approximations. These equations can describe turbulent flow, which matters for combustion, for climate modelling, for engineering aerodynamics or for ship construction (among other areas). These equations are also hard to solve numerically, so this is a place where machine learning can be applied. We have a unique opportunity to give people from all these different domains a place to discuss developments of machine learning in their fields Anatole von Lilienfeld Another area of interest is medical imaging. The scanning techniques used to detect tumours and malignant tissues are good applications of unsupervised learning – you want to cluster healthy tissue versus unhealthy tissue. But if you think about it, there is hardly any quantitative domain within the physical sciences where machine learning cannot be applied. With this journal, we have a unique opportunity to give people from all these different domains a place to discuss developments of machine learning in their fields. So if there’s a major advancement in image recognition of, say, lung tumours, maybe materials scientists will learn something from it that will help them interpret X-ray spectra, or vice versa. Traditionally, people would publish such work within their own disciplines, so it would be hidden from everyone else. You talked about machine learning as an alternative to computation for finding solutions to equations. In your editorial for the first issue of Machine Learning: Science and Technology, you say that machine learning is emerging as a fourth pillar of science, alongside experimentation, theory and computation. How do you see these approaches fitting together? Humans began doing experimentation very early. You could view the first tools as being the result of experiments. Theory developed later. Some would say the Greeks started it, but other cultures also developed theories; the Maya, for example, had theories of stellar movement and calendars. All this work culminated in the modern theories of physics, to which many brilliant scientists contributed. But that wasn’t the end, because many of these brilliant theories had equations that could not be solved using pen and paper. There’s a famous quote from the physicist Paul Dirac where he says that all the equations predicting the behaviour of electrons and nuclei are known. The troubled was that no human could solve those equations. However, with some reasonable approximations, computers could. Because of this, simulation has gained tremendous traction over the last decades, and of course it helps that Moore’s Law has meant that you can buy an exponentially increasing amount of computing power for a constant number of dollars. I think the next step is to use machine learning to build on theory, experiment and computation, and thus to make even better predictions about the systems we study. When you use computation to find numerical solutions to equations, you need a big computer. However, the outcome of that big computation can then feed into a dataset and be used for machine learning, and you can feed in experimental data alongside it. Over the next few years, I think we’ll start to see datasets that combine experimental results with simulation results obtained at different levels of accuracy. Some of these datasets may be incredibly heterogeneous, with a lot of “holes” for unknown quantities and different uncertainties. Machine learning offers a way to integrate that knowledge, and to build a unifying model that enables us to identify areas where the holes are the largest or the uncertainties are the greatest. These areas could then be studied in more detail by experiments or by additional simulations. What other developments should we expect to see in machine learning? I think we’ll see a feedback loop develop, similar to the one we have now between experiment and theory. As experiments progress, they create an incentive for proposing hypotheses, and then you use that theory to make a prediction that you can verify experimentally. Historically, some experiments were excluded from that because the equations were too difficult to solve. But then computation arrived, and suddenly the scope of experimental design widened tremendously. I think the same thing is going to happen with machine learning. We’re already seeing it in materials science, where – with the help of supervised learning – we’ve made predictions within milliseconds about how a new material will behave, whereas previously it would have taken hours to simulate on a supercomputer. I believe that will soon be true for all the physical sciences. I’m not saying we will be able to perform all possible experiments, but we’ll be able to design a much broader range of experiments than we could previously. Related events Copyright © 2020 by IOP Publishing Ltd and individual contributors
7f8558ea0e3d52be	Many-worlds interpretation From Wikiquote Jump to navigation Jump to search The Many-worlds interpretation (also referred to as MWI, the relative state formulation, the Everett interpretation, the theory of the universal wavefunction, many-universes interpretation, or just many-worlds) is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternate histories and futures are real, each representing an actual "world" (or "universe"). In layman's terms, the hypothesis states there is a very large—perhaps infinite—number of universes, and everything that could possibly have happened in our past, but did not, has occurred in the past of some other universe or universes. • The “many worlds interpretation” seems to me an extravagant, and above all an extravagantly vague, hypothesis. I could almost dismiss it as silly. And yet…It may have something distinctive to say in connection with the “Einstein Podolsky Rosen puzzle,” and it would be worthwhile, I think, to formulate some precise version of it to see if this is really so. And the existence of all possible worlds may make us more comfortable about the existence of our own world…which seems to be in some ways a highly improbable one. • John S. Bell, "Six possible worlds of quantum mechanics", Proceedings of the Nobel Symposium 65: Possible Worlds in Arts and Sciences. (1986) • Yes, I have strong feelings against it, but I have to qualify that by saying that in this particular Einstein-Podolsky-Rosen situation there is some merit in the many-universes interpretation, in tackling the problem of how something can apparently happen far away sooner than it could without faster-than-light signalling. If, in a sense, everything happens, all choices are realized (somewhere among all the parallel universes), and no selection is made between the possible results of the experiment until later (which is what one version of the many-universes hypothesis implies), then we get over this difficulty. • It's extremely bizarre, and for me that would already be enough reason to dislike it. The idea that there are all those other universes which we can't see is hard to swallow. But there are also technical problems with it which people usually gloss over or don't even realize when they study it. The actual point at which a branching occurs is supposed to be the point at which a measurement is made. But the point at which the measurement is made is totally obscure. The experiments at CERN for example take months and months, and at which particular second on which particular day the measurement is made and the branching occurs is perfectly obscure. So I believe that the many-universes interpretation is a kind of heuristic, simplified theory, which people have done on the backs of envelopes but haven't really thought through. When you do try to think it through it is not coherent. • In fact the physicists have no good point of view. Somebody mumbled something about a many-world picture, and that many-world picture says that the wave function ψ is what's real, and damn the torpedos if there are so many variables, NR. All these different worlds and every arrangement of configurations are all there just like our arrangement of configurations, we just happen to be sitting in this one. It's possible, but I'm not very happy with it. • There is, I think, no sense at all to be made of the splitting of worlds-plus-agents in many worlds. Of course, one can repeat the words over and over until one becomes deaf to the nonsense, but it remains nonsense nevertheless. Curiously, those who favor this interpretation concentrate their defense on dealing with some obvious technical issues: preferred basis, getting the right probabilities via “measures of existence” (or the like), questions of identity and individuation across worlds, and so on. But the fundamental question is just to explain what it means to talk of splitting worlds, and why we should not just write it off, à la Wittgenstein, as language on holiday. (Einstein once described the writings of Hegel as “word-music.” Perhaps that would be a gentler way of dismissing many worlds.) • Arthur Fine, in M. Schlosshauer (ed.), Elegance and Enigma, The Quantum Interviews (2011) • The greatest danger I see in the many-worlds/one-Hilbert-space point of view (beside the ridiculous silliness of it all) is the degree to which it is a dead end. The degree to which it is morally bankrupt. Charlie, by thinking that he has taken some of the anthropocentrism out of the picture, has actually emptied the world of all content. Beyond that though, I think, many-worlds empties the world of content in a way that’s even worse than classical determinism. Let me explain. In my mind, both completely deterministic ontologies and completely indeterministic ones are equally unpalatable. This is because, in both, all our consciousnesses, all our great works of literature, everything that we know, even the coffee maker in my kitchen, are but dangling appendages, illusions. In the first case, the only truth is the Great Initial Condition. In the second, it is the great “I Am That I Am.” But many-worlds compounds that trouble in a far worse fashion by stripping away even those small corners of mystery. It is a world in which anything goes, and everything does. What could be more empty than that? • Christopher A. Fuchs, Letters to Herb Bernstein, “Epiphenomena Chez Dyer”, 02 August 1999, published in Coming of Age with Quantum Information: Notes on a Paulian Idea (2011) • The many-worlds theory is incoherent for reasons which have been often pointed out: since there are no frequencies in the theory there is nothing for the numerical predictions of quantum theory to mean. This fact is often disguised by the choice of fortuitous examples. A typical Schrödinger-cat apparatus is designed to yield a 50 percent probability for each of two results, so the “splitting” of the universe in two seems to correspond to the probabilities. But the device could equally be designed to yield a 99 percent probability of one result and 1 percent probability of the other. Again the world “splits” in two; wherein lies the difference between this case and the last? Defenders of the theory sometimes try to alleviate this difficulty by demonstrating that in the long run (in the limit as one repeats experiments an infinite number of times) the quantum probability assigned to branches in which the observed frequencies match the quantum predictions approaches unity. But this is a manifest petitio principii. If the connection between frequency and quantum “probability” has not already been made, the fact that the assigned “probability” approaches unity cannot be interpreted as approach to certainty of an outcome. All of the branches in which the observed frequency diverges from the quantum predictions still exist, indeed they are certain to exist. It is not highly likely that I will experience one of the frequencies rather than another, it is rather certain that for each possible frequency some descendants of me (descendants through world-splitting) will see it. And in no sense will “more” of my descendants see the right frequency rather than the wrong one: just the opposite is true. So approach of some number to unity cannot help unless the number already has the right interpretation. It is also hard to see how such limiting cases help us: we never get to one since we always live in the short run. If the short-run case can be solved, the theorems about limits are unnecessary; if they can’t be then the theorems are irrelevant. • Tim Maudlin, Quantum Non-Locality and Relativity (3rd ed., 2011), Introduction • I regard this last issue as a problem in the interpretation of quantum mechanics, even though I do not believe that consciousness (as a physical phenomenon) collapses (as a physical process) the wave packet (as an objective physical entity). But because I do believe that physics is a tool to help us find powerful and concise expressions of correlations among features of our experience, it makes no sense to apply quantum mechanics (or any other form of physics) to our very awareness of that experience. Adherents of the many-worlds interpretation make this mistake. So do those who believe that conscious awareness can ultimately be reduced to physics, unless they believe that the reduction will be to a novel form of physics that transcends our current understanding, in which case, as Rudolf Peierls remarked, whether such an explanation should count as “physical” is just a matter of terminology. • N. David Mermin, in M. Schlosshauer (ed.), Elegance and Enigma, The Quantum Interviews (2011) • The philosophical moral behind my question is this: once you give up the distinction between actuality and possibility—as the Many Worlds interpretation in effect does, by postulating that all the quantum mechanical possibilities are actualized, each in its own physical universe—once you say that all possible outcomes are, ontologically speaking, equally actual— the notion of ‘probability’ loses all meaning. ‘No collapse and no hidden variables’ is incoherent. • Some very good theorists seem to be happy with an interpretation of quantum mechanics in which the wavefunction only serves to allow us to calculate the results of measurements. But the measuring apparatus and the physicist are presumably also governed by quantum mechanics, so ultimately we need interpretive postulates that do not distinguish apparatus or physicists from the rest of the world, and from which the usual postulates like the Born rule can be deduced. This effort seems to lead to something like a "many worlds" interpretation, which I find repellent. Alternatively, one can try to modify quantum mechanics so that the wavefunction does describe reality, and collapses stochastically and nonlinearly, but this seems to open up the possibility of instantaneous communication. I work on the interpretation of quantum mechanics from time to time, but have gotten nowhere. • Another aspect of the realist approach, which some physicists find implausible, is that it seems to lead inevitably to the “many-worlds interpretation” of quantum mechanics, presented originally in the 1957 Princeton Ph.D. thesis of Hugh Everett (1930–1982). In this approach, the state vector does not collapse; it continues to be governed by the deterministic time-dependent Schrödinger equation, but different components of the state vector of the system studied become associated with different components of the state vector of the measuring apparatus and observer, so that the history of the world effectively splits into different paths, each characterized by different results of the measurement. • Steven Weinberg, Lectures on Quantum Mechanics (2nd ed. 2015), Chap. 3 : General Principles of Quantum Mechanics • In addition to its other problems, the realist approach faces the challenge of deriving the Born rule. If measurement is really described by quantum mechanics, then we ought to be able to derive such formulas by applying the time-dependent Schrödinger equation to the case of repeated measurement. This is not just a matter of intellectual tidiness, of wanting to reduce the postulates of physical theory to the minimum number needed. If the Born rule cannot be derived from the time-dependent Schrödinger equation, then something else is needed, something outside the scope of quantum mechanics, and in this respect the many-worlds interpretation would share the inadequacies of the instrumentalist and Copenhagen interpretations. • When a physicist measures the spin of an electron, say in the north direction, the wave function of the electron and the measuring apparatus and the physicist are supposed, in the realist approach, to evolve deterministically, as dictated by the Schrödinger equation; but in consequence of their interaction during the measurement, the wave function becomes a superposition of two terms, in one of which the electron spin is positive and everyone in the world who looks into it thinks it is positive, and in the other the spin is negative and everyone thinks it is negative. Since in each term of the wave function everyone shares a belief that the spin has one definite sign, the existence of the superposition is undetectable. In effect the history of the world has split into two streams, uncorrelated with each other. … This is strange enough, but the fission of history would not only occur when someone measures a spin. In the realist approach the history of the world is endlessly splitting; it does so every time a macroscopic body becomes tied in with a choice of quantum states. • There is another thing that is unsatisfactory about the realist approach, beyond our parochial preferences. In this approach the wave function of the multiverse evolves deterministically. We can still talk of probabilities as the fractions of the time that various possible results are found when measurements are performed many times in any one history; but the rules that govern what probabilities are observed would have to follow from the deterministic evolution of the whole multiverse. If this were not the case, to predict probabilities we would need to make some additional assumption about what happens when humans make measurements, and we would be back with the shortcomings of the instrumentalist approach. Several attempts following the realist approach have come close to deducing rules like the Born rule that we know work well experimentally, but I think without final success. See also[edit] External links[edit] Wikipedia has an article about:
787541fe8003d5b3	Nanostructural problem-solvers December 2009 Filed under: Lawrence Berkeley, The preeminent physicist-futurist Richard Feynman famously declared in a 1959 address to the American Physical Society that “there’s plenty of room at the bottom.” He then invited them to enter the strange new world of nanoscale materials, none of which had actually been invented, except in Feynman’s fantastical imagination. It took another generation of scientists before nanotechnology emerged, but Feynman’s assertion still rings true. There’s plenty of room at the nanoscale and scientists at Lawrence Berkeley National Laboratory (LBNL) in California are at the forefront in constructing new materials there. Paul Alivisatos, director of LBNL’s Materials Science Division, is a world leader in nanostructures and inventor of many technologies using quantum dots — special kinds of semiconductor nanocrystals. Quantum dots, which are one ten-millionth of an inch in diameter, fluoresce brightly, are exceedingly stable and don’t interfere with biological processes because they are made of inert minerals. Alivisatos and his colleagues have constructed dozens of variations in which the fluorescent color changes with the dot’s size. Today life-science researchers use quantum dots as markers, allowing them to visualize with extreme accuracy individual genes, proteins and other small molecules inside living cells and fulfilling a prediction Feynman made in his famous lecture. LBNL physicist Lin-Wang Wang likes to say that some day we will view the 21st century as the “nanostructure age.” Wang and LBNL colleague Andrew Canning, a computational physicist who helped pioneer the application of parallel computing to material science, want to use computational methods to understand the emergent behaviors of novel materials built from exceedingly small blocks. “There are a lot of challenges and there are still many mysteries to be solved,” Wang says. “For example, we still don’t quite understand the dynamics of the electron inside a quantum dot or a quantum rod. There is a lot of surface area in a quantum structure, much more than the same material in bulk. So how the surface is coupled with the interior states and how this affects the nanostructure properties is not well understood. The research team is not starting from scratch, of course. There are established equations that predict the behavior of the electron wave function in these materials. The devil lies in the size of the problem. “In terms of computation the nanostructure is challenging. For example, if you have a bulk material the crystal structure is a very small unit cell, just a few atoms, that repeats itself many, many times,” Wang says. “So computationally, you can treat bulk structures by calculating one unit cell — you only deal with a few atoms. With only a few atoms, you can represent the whole, much larger structure of the material. However, for a quantum dot or a quantum wire you have to treat the whole system together. These systems usually contain a few thousand to tens of thousands of atoms, and that makes the computation challenging.” To solve a problem containing thousands of atoms requires new algorithms that handle the physics differently without compromising accuracy and parallel computing on a massive scale. Says Canning, “We know we need to solve the Schrödinger equation for these problems, but to do so fully is exceedingly computationally expensive. What we did was make advances to approximate, solve the problem, and still get the physics right.” Canning collaborated with Steven Louie’s group at the University of California-Berkeley, to improve the Parallel Total Energy Code (Paratec), an ab initio, quantum- mechanical, total energy program. The program runs on Franklin, the Cray XT4 at LBNL’s National Energy Research Scientific Computing Center (NERSC). The massively parallel system has 9,660 compute nodes, but is due to receive an upgrade, increasing its processing capability to a theoretical peak of about 360 teraflops. “Paratec enables us to calculate thousand-atom nanosystems,” Canning says. “The calculation is fast and scales to the cube of the system, rather than exponentially, as a true solution of the many-body Schrödinger equation.” Besides massive parallelization of the codes, the researchers also developed many new algorithms for nanostructure calculations. For example, Wang devised a linear scaling method, called the folded spectrum method, for use on large-scale electronic structure calculations. The conventional methods in Paratec must calculate thousands of electron wave functions, but the Escan code uses the folded spectrum method to calculate only a few states near the nanostructure energy band gap. That means the computation scales linearly to the size of the problem — a critical requirement for efficient nanoscience computation. Wang and Canning recently worked with Osni Marques at LBNL and Jack Dongarra’s group at the University of Tennessee, Knoxville, to reinvestigate and significantly improve the Escan code by adding more advanced algorithms.colleagues also have recently invented a linear scaling three-dimensional fragment (LS3DF) method, which can be hundreds of times faster than a conventional method in calculating the total energy of a given nanostructure. The code has run at 107 teraflops on 137,072 processors of Intrepid, Argonne National Laboratory’s IBM Blue Gene/P. The researchers have in essence designed a new algorithm to solve an existing physical problem with petascale computation. Wang says the LS3DF program is designed for materials science applications such as studying material defects, metal alloys and large organic molecules. Within a nanostructure, the physicists are interested mainly in the location and energy level of electrons in the system because that determines the properties of a nanomaterial. For example, Wang says, electrons within a quantum rod or dot can occupy a series of quantum energy states or levels as they orbit the atomic nucleus and interact with each other. The color emitted by the material typically depends on these energy states. Specifically, the scientists focus on two quantum energy levels: the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), which the Escan code can calculate. The energy difference between these two levels determines the material’s color. The color also changes with the quantum dot’s size, providing one way to engineer its properties. In principle, knowing the electronic properties of a given material lets the researchers predict how a new nanostructure will behave before actually spending the time and money to make it. It’s a potentially less expensive way to experiment with new nanomaterials, Wang says. This article originally appeared in Deixis: The CSGF Annual, 2008-09. (Visited 690 times, 1 visits today) About the Author Karyn Hede is news editor of the Nature Publishing Group journal Genetics in Medicine and a correspondent for the Journal of the National Cancer Institute. Her freelance writing has appeared in Science, New Scientist, Technology Review and elsewhere. She teaches scientific writing at the University of North Carolina, Chapel Hill, where she earned advanced degrees in journalism and biology. Leave a Comment
d59a9c4d7e24bf7f	Quantum Gravity and String Theory 0703 Submissions [2] viXra:0703.0018 [pdf] submitted on 18 Mar 2007 The Theory of Cold Quantum (Counter Gravitation Theory) Authors: Cao junfeng, Cao Rui Comments: recovered from sciprint.org Category: Quantum Gravity and String Theory [1] viXra:0703.0017 [pdf] submitted on 10 Mar 2007 Gravitational Schrödinger Equation from Ginzburg-Landau Equation, and Its Noncommutative Spacetime Coordinate Representation Authors: V. Christianto Comments: recovered from sciprint.org Despite known analogy between condensed matter physics and various cosmological phenomena, a neat linkage between low-energy superfluid and celestial quantization is not yet widely accepted in literature. In the present article we argue that gravitational Schrödinger equation could be derived from time-dependent Ginzburg-Landau (or Gross-Pitaevskii) that is commonly used to describe superfluid dynamics. The solution for celestial quantization takes the same form with Nottale equation. Provided this proposed solution corresponds to the facts, and then it could be used as alternative solution to predict celestial orbits from quantized superfluid vortice dynamics. Furthermore, we also discuss a representation of the wavefunction solution using noncommutative spacetime coordinate. Some implications of this solution were discussed particularly in the context of offering a plausible explanation of the physical origin of quantization of motion of celestial objects. Category: Quantum Gravity and String Theory

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