text stringlengths 13 991 |
|---|
This is the Hamiltonian for an infinite number of uncoupled harmonic oscillators. Thus different modes of the field are independent and satisfy the commutation relations: |
This state describes the zero-point energy of the vacuum. It appears that this sum is divergent – in fact highly divergent, as putting in the density factor |
shows. The summation becomes approximately the integral: |
for high values of . It diverges proportional to for large . |
Necessity of the vacuum field in QED. |
The vacuum state of the "free" electromagnetic field (that with no sources) is defined as the ground state in which for all modes . The vacuum state, like all stationary states of the field, is an eigenstate of the Hamiltonian but not the electric and magnetic field operators. In the vacuum state, therefore, the electr... |
In a process in which a photon is annihilated (absorbed), we can think of the photon as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. An atom, for instance, can be considered t... |
or in other words the spectral energy density of the vacuum field: |
The zero-point energy density in the frequency range from to is therefore: |
This can be large even in relatively narrow "low frequency" regions of the spectrum. In the optical region from 400 to 700 nm, for instance, the above equation yields around 220 erg/cm3. |
We showed in the above section that the zero-point energy can be eliminated from the Hamiltonian by the normal ordering prescription. However, this elimination does not mean that the vacuum field has been rendered unimportant or without physical consequences. To illustrate this point we consider a linear dipole oscilla... |
This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate and the canonical momentum of the oscillator are: |
since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative |
For nonrelativistic motion we may neglect the magnetic force and replace the expression for by: |
Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for is found similarly from the Hamiltonian to be: |
In deriving these equations for , , and we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say, ) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture ... |
Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field equation is: |
and therefore the equation for may be written: |
It can be shown that in the radiation reaction field, if the mass is regarded as the "observed" mass then we can take: |
The total field acting on the dipole has two parts, and . is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e., the solution, at the position of the dipole, of the wave equation |
satisfied by the field in the (source free) vacuum. For this reason is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at . is the source field, the field generated by the dipole and acting on the dipole. |
Using the above equation for we obtain an equation for the Heisenberg-picture operator formula_48 that is formally the same as the classical equation for a linear dipole oscillator: |
where . in this instance we have considered a dipole in the vacuum, without any "external" field acting on it. the role of the external field in the above equation is played by the vacuum electric field acting on the dipole. |
Classically, a dipole in the vacuum is not acted upon by any "external" field: if there are no sources other than the dipole itself, then the only field acting on the dipole is its own radiation reaction field. In quantum theory however there is always an "external" field, namely the source-free or vacuum field . |
According to our earlier equation for the free field is the only field in existence at as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at is therefore of the form |
where is the vacuum state of the field and is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero: |
since . however, the energy density associated with the free field is infinite: |
The important point of this is that the zero-point field energy does not affect the Heisenberg equation for since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with . We can therefore drop the zero-point field energy from the Hamiltonian, as is usually done. But the zero-po... |
The free field is in fact necessary for the formal consistency of the theory. In particular, it is necessary for the preservation of the commutation relations, which is required by the unitary of time evolution in quantum theory: |
We can calculate from the formal solution of the operator equation of motion |
and that equal-time particle and field operators commute, we obtain: |
For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e., . Then the integrand above is sharply peaked at and: |
the necessity of the vacuum field can also be appreciated by making the small damping approximation in |
Without the free field in this equation the operator would be exponentially dampened, and commutators like would approach zero for . With the vacuum field included, however, the commutator is at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free ... |
What we have here is an example of a "fluctuation-dissipation elation". Generally speaking if a system is coupled to a bath that can take energy from the system in an effectively irreversible way, then the bath must also cause fluctuations. The fluctuations and the dissipation go hand in hand we cannot have one without... |
The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of , the spectral energy density of the vacuum field must be proportional to the third power of in order for to hold. In the case of a dissi... |
The fact that the canonical commutation relation for a harmonic oscillator coupled to the vacuum field is preserved implies that the zero-point energy of the oscillator is preserved. it is easy to show that after a few damping times the zero-point motion of the oscillator is in fact sustained by the driving zero-point ... |
The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a "non-perturbative" vacuum state, characterized by a non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the ... |
The Higgs mechanism is a type of superconductivity which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged and thus the field has a nonzero vacuum expectation value. Interaction with the vacuum energy filling the space prevents certain forces from propagating over lon... |
A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by Dutch physicist Hendrik Casimir, who considered the quantized electromagnetic field between a pair of grounded, neutral metal plates. The vacuum energy contains contributions f... |
Early experimental tests from the 1950s onwards gave positive results showing the force was real, but other external factors could not be ruled out as the primary cause, with the range of experimental error sometimes being nearly 100%. That changed in 1997 with Lamoreaux conclusively showing that the Casimir force was ... |
In 2009, Munday et al. published experimental proof that (as predicted in 1961) the Casimir force could also be repulsive as well as being attractive. Repulsive Casimir forces could allow quantum levitation of objects in a fluid and lead to a new class of switchable nanoscale devices with ultra-low static friction. |
An interesting hypothetical side effect of the Casimir effect is the Scharnhorst effect, a hypothetical phenomenon in which light signals travel slightly faster than between two closely spaced conducting plates. |
Taking (Planck's constant divided by ), (the speed of light), and (the electromagnetic coupling constant i.e. a measure of the strength of the electromagnetic force (where is the absolute value of the electronic charge and formula_61 is the vacuum permittivity)) we can form a dimensionless quantity called the fine-stru... |
The fine-structure constant is the coupling constant of quantum electrodynamics (QED) determining the strength of the interaction between electrons and photons. It turns out that the fine-structure constant is not really a constant at all owing to the zero-point energy fluctuations of the electron-positron field. The q... |
The Heisenberg inequality where , and , are the standard deviations of position and momentum states that: |
It means that a short distance implies large momentum and therefore high energy i.e. particles of high energy must be used to explore short distances. QED concludes that the fine-structure constant is an increasing function of energy. It has been shown that at energies of the order of the Z0 boson rest energy, 90 GeV, ... |
rather than the low-energy . The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining . All in all, depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The ... |
In the late 1990s it was discovered that very distant supernova were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down. This revived discussion that Einstein's cosmological constant, long disregarded by physicists as being equal to zero, was in fact some small posit... |
There is no natural candidate for what might cause what has been called dark energy but the current best guess is that it is the zero-point energy of the vacuum. One difficulty with this assumption is that the zero-point energy of the vacuum is absurdly large compared to the observed cosmological constant. In general r... |
Cosmic inflation is a faster-than-light expansion of space just after the Big Bang. It explains the origin of the large-scale structure of the cosmos. It is believed quantum vacuum fluctuations caused by zero-point energy arising in the microscopic inflationary period, later became magnified to a cosmic size, becoming ... |
The mechanism for inflation is unclear, it is similar in effect to dark energy but is a far more energetic and short lived process. As with dark energy the best explanation is some form of vacuum energy arising from quantum fluctuations. It may be that inflation caused baryogenesis, the hypothetical physical processes ... |
Physicists overwhelmingly reject any possibility that the zero-point energy field can be exploited to obtain useful energy (work) or uncompensated momentum; such efforts are seen as tantamount to perpetual motion machines. |
Nevertheless, the allure of free energy has motivated such research, usually falling in the category of fringe science. As long ago as 1889 (before quantum theory or discovery of the zero point energy) Nikola Tesla proposed that useful energy could be obtained from free space, or what was assumed at that time to be an ... |
A common assumption is that the Casimir force is of little practical use; the argument is made that the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only tiny force in nature. In 1984 Rober... |
In 1995 and 1998 Maclay et al. published the first models of a microelectromechanical system (MEMS) with Casimir forces. While not exploiting the Casimir force for useful work, the papers drew attention from the MEMS community due to the revelation that Casimir effect needs to be considered as a vital factor in the fut... |
In 1999, Pinto, a former scientist at NASA's Jet Propulsion Laboratory at Caltech in Pasadena, published in "Physical Review" his thought experiment (Gedankenexperiment) for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the Casimir effect was possible, even stating in the abs... |
In 2001, Capasso et al. showed how the force can be used to control the mechanical motion of a MEMS device, The researchers suspended a polysilicon plate from a torsional rod – a twisting horizontal bar just a few microns in diameter. When they brought a metallized sphere close up to the plate, the attractive Casimir f... |
Despite this and several similar peer reviewed papers, there is not a consensus as to whether such devices can produce a continuous output of work. Garret Moddel at University of Colorado has highlighted that he believes such devices hinge on the assumption that the Casimir force is a nonconservative force, he argues t... |
In 2008, DARPA solicited research proposals in the area of Casimir Effect Enhancement (CEE). The goal of the program is to develop new methods to control and manipulate attractive and repulsive forces at surfaces based on engineering of the Casimir force. |
There have been a growing number of papers showing that in some instances the classical laws of thermodynamics, such as limits on the Carnot efficiency, can be violated by exploiting negative entropy of quantum fluctuations. |
Despite efforts to reconcile quantum mechanics and thermodynamics over the years, their compatibility is still an open fundamental problem. The full extent that quantum properties can alter classical thermodynamic bounds is unknown |
In 1986 the U.S. Air Force's then Rocket Propulsion Laboratory (RPL) at Edwards Air Force Base solicited "Non Conventional Propulsion Concepts" under a small business research and innovation program. One of the six areas of interest was "Esoteric energy sources for propulsion, including the quantum dynamic energy of va... |
In 2002 Phantom Works, Boeing's advanced research and development facility in Seattle, approached Evgeny Podkletnov directly. Phantom Works was blocked by Russian technology transfer controls. At this time Lieutenant General George Muellner, the outgoing head of the Boeing Phantom Works, confirmed that attempts by Boei... |
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite sense of rotation or revolution is (in Commonwealth English) anticlockwise (... |
Before clocks were commonplace, the terms "sunwise" and "deasil", "deiseil" and even "deocil" from the Scottish Gaelic language and from the same root as the Latin "dexter" ("right") were used for clockwise. "Widdershins" or "withershins" (from Middle Low German "weddersinnes", "opposite course") was used for countercl... |
The terms clockwise and counterclockwise can only be applied to a rotational motion once a side of the rotational plane is specified, from which the rotation is observed. For example, the daily rotation of the Earth is clockwise when viewed from above the South Pole, and counterclockwise when viewed from above the Nort... |
Occasionally, clocks whose hands revolve counterclockwise are nowadays sold as a novelty. Historically, some Jewish clocks were built that way, for example in some synagogue towers in Europe such as the Jewish Town Hall in Prague, to accord with right-to-left reading in the Hebrew language. In 2014 under Bolivian presi... |
Typical nuts, screws, bolts, bottle caps, and jar lids are tightened (moved away from the observer) clockwise and loosened (moved towards the observer) counterclockwise in accordance with the right-hand rule. |
To apply the right-hand rule, place one's loosely clenched right hand above the object with the thumb pointing in the direction one wants the screw, nut, bolt, or cap ultimately to move, and the curl of the fingers, from the palm to the tips, will indicate in which way one needs to turn the screw, nut, bolt or cap to a... |
The reason for the clockwise standard for most screws and bolts is that supination of the arm, which is used by a right-handed person to tighten a screw clockwise, is generally stronger than pronation used to loosen. |
In trigonometry and mathematics in general, plane angles are conventionally measured counterclockwise, starting with 0° or 0 radians pointing directly to the right (or east), and 90° pointing straight up (or north). However, in navigation, compass headings increase clockwise around the compass face, starting with 0° at... |
A circle defined parametrically in a positive Cartesian plane by the equations and is traced counterclockwise as the angle "t" increases in value, from the right-most point at . An alternative formulation with sin and cos swapped gives a clockwise trace from the upper-most point. |
In general, most card games, board games, parlor games, and multiple team sports play in a clockwise turn rotation in Western Countries and Latin America with a notable resistance to playing in the opposite direction (counterclockwise). Traditionally (and still continued for the most part) turns pass counterclockwise i... |
Notably, the game of baseball is played counterclockwise. |
Most left-handed people prefer to draw circles and circulate in buildings clockwise, while most right-handed people prefer to draw circles and circulate in buildings counterclockwise. While this was theorized to result from dominant brain hemispheres, research shows little correlation and instead attributes it to muscl... |
In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properti... |
While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a ten... |
The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondege... |
Carl Friedrich Gauss in his 1827 "Disquisitiones generales circa superficies curvas" ("General investigations of curved surfaces") considered a surface parametrically, with the Cartesian coordinates , , and of points on the surface depending on two auxiliary variables and . Thus a parametric surface is (in today's term... |
depending on an ordered pair of real variables , and defined in an open set in the -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending t... |
One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the pr... |
If the variables and are taken to depend on a third variable, , taking values in an interval , then will trace out a parametric curve in parametric surface . The arc length of that curve is given by the integral |
where formula_3 represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives: |
The integrand is the restriction to the curve of the square root of the (quadratic) differential |
The quantity in () is called the line element, while is called the first fundamental form of . Intuitively, it represents the principal part of the square of the displacement undergone by when is increased by units, and is increased by units. |
Using matrix notation, the first fundamental form becomes |
Suppose now that a different parameterization is selected, by allowing and to depend on another pair of variables and . Then the analog of () for the new variables is |
The chain rule relates , , and to , , and via the matrix equation |
where the superscript T denotes the matrix transpose. The matrix with the coefficients , , and arranged in this way therefore transforms by the Jacobian matrix of the coordinate change |
A matrix which transforms in this way is one kind of what is called a tensor. The matrix |
with the transformation law () is known as the metric tensor of the surface. |
first observed the significance of a system of coefficients , , and , that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form () is "invariant" under changes in the coordinate system, and that this follows exclusively from the transformation prope... |
Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the dot product of tangent vectors in a mann... |
for suitable real numbers and . If two tangent vectors are given: |
then using the bilinearity of the dot product, |
This is plainly a function of the four variables , , , and . It is more profitably viewed, however, as a function that takes a pair of arguments and which are vectors in the -plane. That is, put |
This is a symmetric function in and , meaning that |
It is also bilinear, meaning that it is linear in each variable and separately. That is, |
for any vectors , , , and in the plane, and any real numbers and . |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.