In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry) while its spatial coordinates are inverted ("mirror" or P symmetry). The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.


It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present universe, and in the study of weak interactions in particle physics.


Until the 1950s, parity conservation was believed to be one of the fundamental geometric conservation laws (along with conservation of energy and conservation of momentum). After the discovery of parity violation in 1956, CP-symmetry was proposed to restore order. However, while the strong interaction and electromagnetic interaction seem to be invariant under the combined CP transformation operation, further experiments showed that this symmetry is slightly violated during certain types of weak decay.


Only a weaker version of the symmetry could be preserved by physical phenomena, which was CPT symmetry. Besides C and P, there is a third operation, time reversal T, which corresponds to reversal of motion. Invariance under time reversal implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one and occurs at the same rate forwards and backwards.


The combination of CPT is thought to constitute an exact symmetry of all types of fundamental interactions. Because of the CPT symmetry, a violation of the CP-symmetry is equivalent to a violation of the T symmetry. CP violation implied nonconservation of T, provided that the long-held CPT theorem was valid. In this theorem, regarded as one of the basic principles of quantum field theory, charge conjugation, parity, and time reversal are applied together. Direct observation of the time reversal symmetry violation without any assumption of CPT theorem was done in 1998 by two groups, CPLEAR and KTeV collaborations, at CERN and Fermilab, respectively. Already in 1970 Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell-Steinberger unitarity relation.


The idea behind parity symmetry was that the equations of particle physics are invariant under mirror inversion. This led to the prediction that the mirror image of a reaction (such as a chemical reaction or radioactive decay) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang revealed that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests.


The first test based on beta decay of cobalt-60 nuclei was carried out in 1956 by a group led by Chien-Shiung Wu, and demonstrated conclusively that weak interactions violate the P symmetry or, as the analogy goes, some reactions did not occur as often as their mirror image. However, parity symmetry still appears to be valid for all reactions involving electromagnetism and strong interactions.


Overall, the symmetry of a quantum mechanical system can be restored if another approximate symmetry "S" can be found such that the combined symmetry "PS" remains unbroken. This rather subtle point about the structure of Hilbert space was realized shortly after the discovery of "P" violation, and it was proposed that charge conjugation, "C", which transforms a particle into its antiparticle, was the suitable symmetry to restore order.


In 1956 Reinhard Oehme in a letter to Yang and shortly after, Ioffe, Okun and Rudik showed that the parity violation meant that charge conjugation invariance must also be violated in weak decays.


Charge violation was confirmed in the Wu experiment and in experiments performed by Valentine Telegdi and Jerome Friedman and Garwin and Lederman who observed parity non-conservation in pion and muon decay and found that C is also violated. Charge violation was more explicitly shown in experiments done by John Riley Holt at the University of Liverpool.


Oehme then wrote up a paper with Lee and Yang in which they discussed the interplay of non-invariance under P, C and T. The same result was also independently obtained by B.L. Ioffe, Okun and A.P. Rudik. Both groups also discussed possible CP violations in neutral kaon decays.


Lev Landau proposed in 1957 "CP-symmetry", often called just "CP" as the true symmetry between matter and antimatter. "CP-symmetry" is the product of two transformations: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the mirror image of the original proces and so the combined CP symmetry would be conserved in the weak interaction.


In 1962, a group of experimentalists at Dubna, on Okun's insistence, unsuccessfully searched for CP-violating kaon decay.


In 1964, James Cronin, Val Fitch and coworkers provided clear evidence from kaon decay that CP-symmetry could be broken. This work won them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity, but also their combination. The discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today. The lack of an exact CP-symmetry, but also the fact that it is so close to a symmetry, introduced a great puzzle.


The kind of CP violation discovered in 1964 was linked to the fact that neutral kaons can transform into their antiparticles (in which each quark is replaced with the other's antiquark) and vice versa, but such transformation does not occur with exactly the same probability in both directions; this is called "indirect" CP violation.


Despite many searches, no other manifestation of CP violation was discovered until the 1990s, when the NA31 experiment at CERN suggested evidence for CP violation in the decay process of the very same neutral kaons ("direct" CP violation). The observation was somewhat controversial, and final proof for it came in 1999 from the KTeV experiment at Fermilab and the NA48 experiment at CERN.


Starting in 2001, a new generation of experiments, including the BaBar experiment at the Stanford Linear Accelerator Center (SLAC) and the Belle Experiment at the High Energy Accelerator Research Organisation (KEK) in Japan, observed direct CP violation in a different system, namely in decays of the B mesons. A large number of CP violation processes in B meson decays have now been discovered. Before these "B-factory" experiments, there was a logical possibility that all CP violation was confined to kaon physics. However, this raised the question of why CP violation did "not" extend to the strong force, and furthermore, why this was not predicted by the unextended Standard Model, despite the model's accuracy for "normal" phenomena.


In 2011, a hint of CP violation in decays of neutral D mesons was reported by the LHCb experiment at CERN using 0.6 fb−1 of Run 1 data. However, the same measurement using the full 3.0 fb−1 Run 1 sample was consistent with CP symmetry.


In 2013 LHCb announced discovery of CP violation in strange B meson decays.


In March 2019, LHCb announced discovery of CP violation in charmed formula_1 decays with a deviation from zero of 5.3 standard deviations.


In 2020, the T2K Collaboration reported some indications of CP violation in leptons for the first time.


In this experiment, beams of muon neutrinos () and muon antineutrinos () were alternately produced by an accelerator. By the time they got to the detector, a significantly higher proportion of electron neutrinos () were detected from the beams, than electron antineutrinos () were from the beams. The results were not yet precise enough to determine the size of the CP violation, relative to that seen in quarks. In addition, another similar experiment, NOvA sees no evidence of CP violation in neutrino oscillations and is in slight tension with T2K.


"Direct" CP violation is allowed in the Standard Model if a complex phase appears in the CKM matrix describing quark mixing, or the PMNS matrix describing neutrino mixing. A necessary condition for the appearance of the complex phase is the presence of at least three generations of quarks. If fewer generations are present, the complex phase parameter can be absorbed into redefinitions of the quark fields. A popular rephasing invariant whose vanishing signals absence of CP violation and occurs in most CP violating amplitudes is the "Jarlskog invariant",


The reason why such a complex phase causes CP violation is not immediately obvious, but can be seen as follows. Consider any given particles (or sets of particles) formula_3 and formula_4, and their antiparticles formula_5 and formula_6. Now consider the processes formula_7 and the corresponding antiparticle process formula_8, and denote their amplitudes formula_9 and formula_10 respectively. Before CP violation, these terms must be the "same" complex number. We can separate the magnitude and phase by writing formula_11. If a phase term is introduced from (e.g.) the CKM matrix, denote it formula_12. Note that formula_10 contains the conjugate matrix to formula_9, so it picks up a phase term formula_15.


Physically measurable reaction rates are proportional to formula_18, thus so far nothing is different. However, consider that there are "two different routes": formula_19 and formula_20 or equivalently, two unrelated intermediate states: formula_21 and formula_22. Now we have:


Thus, we see that a complex phase gives rise to processes that proceed at different rates for particles and antiparticles, and CP is violated.


From the theoretical end, the CKM matrix is defined as = . , where and are unitary transformation matrices which diagonalize the fermion mass matrices and , respectively.


Thus, there are two necessary conditions for getting a complex CKM matrix:


There is no experimentally known violation of the CP-symmetry in quantum chromodynamics. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as the strong CP problem.


QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory in which the gauge fields couple to chiral currents constructed from the fermionic fields, the gluons couple to vector currents. Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create the electric dipole moment of the neutron which would be comparable to 10−18 e·m while the experimental upper bound is roughly one trillionth that size.


This is a problem because at the end, there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry.


For a nonzero choice of the θ angle and the chiral phase of the quark mass θ′ one expects the CP-symmetry to be violated. One usually assumes that the chiral quark mass phase can be converted to a contribution to the total effective formula_27 angle, but it remains to be explained why this angle is extremely small instead of being of order one; the particular value of the θ angle that must be very close to zero (in this case) is an example of a fine-tuning problem in physics, and is typically solved by physics beyond the Standard Model.


There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory, involving new scalar particles called axions. A newer, more radical approach not requiring the axion is a theory involving two time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev.


The universe is made chiefly of matter, rather than consisting of equal parts of matter and antimatter as might be expected. It can be demonstrated that, to create an imbalance in matter and antimatter from an initial condition of balance, the Sakharov conditions must be satisfied, one of which is the existence of CP violation during the extreme conditions of the first seconds after the Big Bang. Explanations which do not involve CP violation are less plausible, since they rely on the assumption that the matter–antimatter imbalance was present at the beginning, or on other admittedly exotic assumptions.


The Big Bang should have produced equal amounts of matter and antimatter if CP-symmetry was preserved; as such, there should have been total cancellation of both—protons should have cancelled with antiprotons, electrons with positrons, neutrons with antineutrons, and so on. This would have resulted in a sea of radiation in the universe with no matter. Since this is not the case, after the Big Bang, physical laws must have acted differently for matter and antimatter, i.e. violating CP-symmetry.


If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new physics beyond the Standard Model would be required to explain additional sources of CP violation. Adding new particles and/or interactions to the Standard Model generally introduces new sources of CP violation since CP is not a symmetry of nature.


Sakharov proposed a way to restore CP-symmetry using T-symmetry, extending spacetime "before" the Big Bang. He described complete "CPT reflections" of events on each side of what he called the "initial singularity". Because of this, phenomena with an opposite arrow of time at "t" < 0 would undergo an opposite CP violation, so the CP-symmetry would be preserved as a whole. The anomalous excess of matter over antimatter after the Big Bang in the orthochronous (or positive) sector, becomes an excess of antimatter before the Big Bang (antichronous or negative sector) as both charge conjugation, parity and arrow of time are reversed due to CPT reflections of all phenomena occurring over the initial singularity:


Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: "positive" and "negative" (commonly carried by protons and electrons respectively). Like charges repel each other and unlike charges attract each other. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.


Electric charges produce electric fields. A moving charge also produces a magnetic field. The interaction of electric charges with an electromagnetic field (combination of electric and magnetic fields) is the source of the electromagnetic (or Lorentz) force, which is one of the four fundamental forces in physics. The study of photon-mediated interactions among charged particles is called quantum electrodynamics.


The SI derived unit of electric charge is the coulomb (C) named after French physicist Charles-Augustin de Coulomb. In electrical engineering it is also common to use the ampere-hour (Ah). In physics and chemistry it is common to use the elementary charge ("e" as a unit). Chemistry also uses the Faraday constant as the charge on a mole of electrons. The lowercase symbol "q" often denotes charge.


Charge is the fundamental property of matter that exhibit electrostatic attraction or repulsion in the presence of other matter with charge. Electric charge is a characteristic property of many subatomic particles. The charges of free-standing particles are integer multiples of the elementary charge "e"; we say that electric charge is "quantized". Michael Faraday, in his electrolysis experiments, was the first to note the discrete nature of electric charge. Robert Millikan's oil drop experiment demonstrated this fact directly, and measured the elementary charge. It has been discovered that one type of particle, quarks, have fractional charges of either − or +, but it is believed they always occur in multiples of integral charge; free-standing quarks have never been observed.


By convention, the charge of an electron is negative, "−e", while that of a proton is positive, "+e". Charged particles whose charges have the same sign repel one another, and particles whose charges have different signs attract. Coulomb's law quantifies the electrostatic force between two particles by asserting that the force is proportional to the product of their charges, and inversely proportional to the square of the distance between them. The charge of an antiparticle equals that of the corresponding particle, but with opposite sign.


The electric charge of a macroscopic object is the sum of the electric charges of the particles that make it up. This charge is often small, because matter is made of atoms, and atoms typically have equal numbers of protons and electrons, in which case their charges cancel out, yielding a net charge of zero, thus making the atom neutral.


An "ion" is an atom (or group of atoms) that has lost one or more electrons, giving it a net positive charge (cation), or that has gained one or more electrons, giving it a net negative charge (anion). "Monatomic ions" are formed from single atoms, while "polyatomic ions" are formed from two or more atoms that have been bonded together, in each case yielding an ion with a positive or negative net charge.


During the formation of macroscopic objects, constituent atoms and ions usually combine to form structures composed of neutral "ionic compounds" electrically bound to neutral atoms. Thus macroscopic objects tend toward being neutral overall, but macroscopic objects are rarely perfectly net neutral.


Even when an object's net charge is zero, the charge can be distributed non-uniformly in the object (e.g., due to an external electromagnetic field, or bound polar molecules). In such cases, the object is said to be polarized. The charge due to polarization is known as bound charge, while the charge on an object produced by electrons gained or lost from outside the object is called "free charge". The motion of electrons in conductive metals in a specific direction is known as electric current.


The SI derived unit of quantity of electric charge is the coulomb (symbol: C). The coulomb is defined as the quantity of charge that passes through the cross section of an electrical conductor carrying one ampere for one second. This unit was proposed in 1946 and ratified in 1948. In modern practice, the phrase "amount of charge" is used instead of "quantity of charge". The lowercase symbol "q" is often used to denote a quantity of electricity or charge. The quantity of electric charge can be directly measured with an electrometer, or indirectly measured with a ballistic galvanometer.


The amount of charge in 1 electron (elementary charge) is defined as a fundamental constant in the SI system of units, (effective from 20 May 2019). The value for elementary charge, when expressed in the SI unit for electric charge (coulomb), is "exactly" .


After finding the quantized character of charge, in 1891 George Stoney proposed the unit 'electron' for this fundamental unit of electrical charge. This was before the discovery of the particle by J. J. Thomson in 1897. The unit is today referred to as , , or simply as . A measure of charge should be a multiple of the elementary charge "e", even if at large scales charge seems to behave as a real quantity. In some contexts it is meaningful to speak of fractions of a charge; for example in the charging of a capacitor, or in the fractional quantum Hall effect.


The unit faraday is sometimes used in electrochemistry. One faraday of charge is the magnitude of the charge of one mole of electrons, i.e. 96485.33289(59) C.


In systems of units other than SI such as cgs, electric charge is expressed as combination of only three fundamental quantities (length, mass, and time), and not four, as in SI, where electric charge is a combination of length, mass, time, and electric current.


In late 1100s, the substance jet, a compacted form of coal, was noted to have an amber effect, and in the middle of the 1500s, Girolamo Fracastoro, discovered that diamond also showed this effect. Some efforts were made by Fracastoro and others, especially Gerolamo Cardano to develop explanations for this phenomenon.


Around 1663 Otto von Guericke invented what was probably the first electrostatic generator, but he did not recognize it primarily as an electrical device and only conducted minimal electrical experiments with it. Other European pioneers were Robert Boyle, who in 1675 published the first book in English that was devoted solely to electrical phenomena. His work was largely a repetition of Gilbert's studies, but he also identified several more "electrics", and noted mutual attraction between two bodies.


Up until about 1745, the main explanation for electrical attraction and repulsion was the idea that electrified bodies gave off an effluvium.


It is now known that the Franklin model was fundamentally correct. There is only one kind of electrical charge, and only one variable is required to keep track of the amount of charge.


Until 1800 it was only possible to study conduction of electric charge by using an electrostatic discharge. In 1800 Alessandro Volta was the first to show that charge could be maintained in continuous motion through a closed path.


In 1833, Michael Faraday sought to remove any doubt that electricity is identical, regardless of the source by which it is produced. He discussed a variety of known forms, which he characterized as common electricity (e.g., static electricity, piezoelectricity, magnetic induction), voltaic electricity (e.g., electric current from a voltaic pile), and animal electricity (e.g., bioelectricity).


In 1838, Faraday raised a question about whether electricity was a fluid or fluids or a property of matter, like gravity. He investigated whether matter could be charged with one kind of charge independently of the other. He came to the conclusion that electric charge was a relation between two or more bodies, because he could not charge one body without having an opposite charge in another body.


In 1838, Faraday also put forth a theoretical explanation of electric force, while expressing neutrality about whether it originates from one, two, or no fluids. He focused on the idea that the normal state of particles is to be nonpolarized, and that when polarized, they seek to return to their natural, nonpolarized state.


In developing a field theory approach to electrodynamics (starting in the mid-1850s), James Clerk Maxwell stops considering electric charge as a special substance that accumulates in objects, and starts to understand electric charge as a consequence of the transformation of energy in the field. This pre-quantum understanding considered magnitude of electric charge to be a continuous quantity, even at the microscopic level.


The role of charge in static electricity.


Static electricity refers to the electric charge of an object and the related electrostatic discharge when two objects are brought together that are not at equilibrium. An electrostatic discharge creates a change in the charge of each of the two objects.


When a piece of glass and a piece of resin—neither of which exhibit any electrical properties—are rubbed together and left with the rubbed surfaces in contact, they still exhibit no electrical properties. When separated, they attract each other.


A second piece of glass rubbed with a second piece of resin, then separated and suspended near the former pieces of glass and resin causes these phenomena:


This attraction and repulsion is an "electrical phenomenon", and the bodies that exhibit them are said to be "electrified", or "electrically charged". Bodies may be electrified in many other ways, as well as by friction. The electrical properties of the two pieces of glass are similar to each other but opposite to those of the two pieces of resin: The glass attracts what the resin repels and repels what the resin attracts.


If a body electrified in any manner whatsoever behaves as the glass does, that is, if it repels the glass and attracts the resin, the body is said to be "vitreously" electrified, and if it attracts the glass and repels the resin it is said to be "resinously" electrified. All electrified bodies are either vitreously or resinously electrified.


An established convention in the scientific community defines vitreous electrification as positive, and resinous electrification as negative. The exactly opposite properties of the two kinds of electrification justify our indicating them by opposite signs, but the application of the positive sign to one rather than to the other kind must be considered as a matter of arbitrary convention—just as it is a matter of convention in mathematical diagram to reckon positive distances towards the right hand.


No force, either of attraction or of repulsion, can be observed between an electrified body and a body not electrified.


The role of charge in electric current.


Electric current is the flow of electric charge through an object, which produces no net loss or gain of electric charge. The most common charge carriers are the positively charged proton and the negatively charged electron. The movement of any of these charged particles constitutes an electric current. In many situations, it suffices to speak of the "conventional current" without regard to whether it is carried by positive charges moving in the direction of the conventional current or by negative charges moving in the opposite direction. This macroscopic viewpoint is an approximation that simplifies electromagnetic concepts and calculations.


At the opposite extreme, if one looks at the microscopic situation, one sees there are many ways of carrying an electric current, including: a flow of electrons; a flow of electron holes that act like positive particles; and both negative and positive particles (ions or other charged particles) flowing in opposite directions in an electrolytic solution or a plasma.


Beware that, in the common and important case of metallic wires, the direction of the conventional current is opposite to the drift velocity of the actual charge carriers; i.e., the electrons. This is a source of confusion for beginners.


The total electric charge of an isolated system remains constant regardless of changes within the system itself. This law is inherent to all processes known to physics and can be derived in a local form from gauge invariance of the wave function. The conservation of charge results in the charge-current continuity equation. More generally, the rate of change in charge density "ρ" within a volume of integration "V" is equal to the area integral over the current density J through the closed surface "S" = ∂"V", which is in turn equal to the net current "I":


Thus, the conservation of electric charge, as expressed by the continuity equation, gives the result:


The charge transferred between times formula_2 and formula_3 is obtained by integrating both sides:


where "I" is the net outward current through a closed surface and "q" is the electric charge contained within the volume defined by the surface.


Aside from the properties described in articles about electromagnetism, charge is a relativistic invariant. This means that any particle that has charge "q" has the same charge regardless of how fast it is travelling. This property has been experimentally verified by showing that the charge of one helium nucleus (two protons and two neutrons bound together in a nucleus and moving around at high speeds) is the same as two deuterium nuclei (one proton and one neutron bound together, but moving much more slowly than they would if they were in a helium nucleus).


In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.


A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume.


From Noether's theorem, each conservation law is associated with a symmetry in the underlying physics.


Conservation laws as fundamental laws of nature.


Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering.


Most conservation laws are exact, or absolute, in the sense that they apply to all possible processes. Some conservation laws are partial, in that they hold for some processes but not for others.


One particularly important result concerning conservation laws is Noether's theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry of nature. For example, the conservation of energy follows from the time-invariance of physical systems, and the conservation of angular momentum arises from the fact that physical systems behave the same regardless of how they are oriented in space.


A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely "have never been proven to be violated:"


There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.


In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge "q" is


where ∇⋅ is the divergence operator, "ρ" is the density of "q" (amount per unit volume), j is the flux of "q" (amount crossing a unit area in unit time), and "t" is time.


If we assume that the motion u of the charge is a continuous function of position and time, then


In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation:


where the dependent variable "y" is called the "density" of a "conserved quantity", and "A"("y") is called the "current Jacobian", and the subscript notation for partial derivatives has been employed. The more general inhomogeneous case:


is not a conservation equation but the general kind of balance equation describing a dissipative system. The dependent variable "y" is called a "nonconserved quantity", and the inhomogeneous term "s"("y","x","t") is the-"source", or dissipation. For example, balance equations of this kind are the momentum and energy Navier-Stokes equations, or the entropy balance for a general isolated system.


In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the "advection" form:


where the dependent variable "y"("x","t") is called the density of the "conserved" (scalar) quantity, and "a"("y") is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density of the conserved quantity "j"("y"):


In this case since the chain rule applies:


the conservation equation can be put into the current density form:


In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:


where the "conserved quantity" is "y"(r,"t"), formula_11 denotes the scalar product, "∇" is the nabla operator, here indicating a gradient, and "a"("y") is a vector of current coefficients, analogously corresponding to the divergence of a vector current density associated to the conserved quantity j("y"):


This is the case for the continuity equation:


Here the conserved quantity is the mass, with density "ρ"(r,"t") and current density "ρ"u, identical to the momentum density, while u(r,"t") is the flow velocity.


In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:


where y is called the "conserved" (vector) quantity, ∇ y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrix J(y):


and the conservation equation can be put into the form:


For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:


It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:


Conservation equations can be also expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions. By integrating in any space-time domain the current density form in 1-D space:


and by using Green's theorem, the integral form is:


In a similar fashion, for the scalar multidimensional space, the integral form is:


where the line integration is performed along the boundary of the domain, in an anticlockwise manner.


Moreover, by defining a test function "φ"(r,"t") continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:


Note that in the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.


In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always "conserved". Charge conservation, considered as a physical conservation law, implies that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a continuity equation between charge density formula_1 and current density formula_2.


This does not mean that individual positive and negative charges cannot be created or destroyed. Electric charge is carried by subatomic particles such as electrons and protons. Charged particles can be created and destroyed in elementary particle reactions. In particle physics, charge conservation means that in reactions that create charged particles, equal numbers of positive and negative particles are always created, keeping the net amount of charge unchanged. Similarly, when particles are destroyed, equal numbers of positive and negative charges are destroyed. This property is supported without exception by all empirical observations so far.


Although conservation of charge requires that the total quantity of charge in the universe is constant, it leaves open the question of what that quantity is. Most evidence indicates that the net charge in the universe is zero; that is, there are equal quantities of positive and negative charge.


Charge conservation was first proposed by British scientist William Watson in 1746 and American statesman and scientist Benjamin Franklin in 1747, although the first convincing proof was given by Michael Faraday in 1843.


Mathematically, we can state the law of charge conservation as a continuity equation:


where formula_4 is the electric charge accumulation rate in a specific volume at time , formula_5 is the amount of charge flowing into the volume and formula_6 is the amount of charge flowing out of the volume; both amounts are regarded as generic functions of time.


The integrated continuity equation between two time values reads:


The general solution is obtained by fixing the initial condition time formula_8, leading to the integral equation:


The condition formula_10 corresponds to the absence of charge quantity change in the control volume: the system has reached a steady state. From the above condition, the following must hold true:


therefore, formula_5 and formula_6 are equal (not necessarily constant) over time, then the overall charge inside the control volume does not change. This deduction could be derived directly from the continuity equation, since at steady state formula_14 holds, and implies formula_15.


In electromagnetic field theory, vector calculus can be used to express the law in terms of charge density (in coulombs per cubic meter) and electric current density (in amperes per square meter). This is called the charge density continuity equation


The term on the left is the rate of change of the charge density at a point. The term on the right is the divergence of the current density at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a current of charge to flow into or out of the point. This statement is equivalent to a conservation of four-current.


The net current into a volume is


where is the boundary of oriented by outward-pointing normals, and is shorthand for , the outward pointing normal of the boundary . Here is the current density (charge per unit area per unit time) at the surface of the volume. The vector points in the direction of the current.


From the Divergence theorem this can be written


Charge conservation requires that the net current into a volume must necessarily equal the net change in charge within the volume.


The total charge "q" in volume "V" is the integral (sum) of the charge density in "V"


Since this is true for every volume, we have in general


Charge conservation can also be understood as a consequence of symmetry through Noether's theorem, a central result in theoretical physics that asserts that each conservation law is associated with a symmetry of the underlying physics. The symmetry that is associated with charge conservation is the global gauge invariance of the electromagnetic field. This is related to the fact that the electric and magnetic fields are not changed by different choices of the value representing the zero point of electrostatic potential formula_24. However the full symmetry is more complicated, and also involves the vector potential formula_25. The full statement of gauge invariance is that the physics of an electromagnetic field are unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field formula_26:


In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle:


so gauge invariance is equivalent to the well known fact that changes in the phase of a wavefunction are unobservable, and only changes in the magnitude of the wavefunction result in changes to the probability function formula_29. This is the ultimate theoretical origin of charge conservation.


Gauge invariance is a very important, well established property of the electromagnetic field and has many testable consequences. The theoretical justification for charge conservation is greatly strengthened by being linked to this symmetry. For example, gauge invariance also requires that the photon be massless, so the good experimental evidence that the photon has zero mass is also strong evidence that charge is conserved.


Even if gauge symmetry is exact, however, there might be apparent electric charge non-conservation if charge could leak from our normal 3-dimensional space into hidden extra dimensions.


Simple arguments rule out some types of charge nonconservation. For example, the magnitude of the elementary charge on positive and negative particles must be extremely close to equal, differing by no more than a factor of 10−21 for the case of protons and electrons. Ordinary matter contains equal numbers of positive and negative particles, protons and electrons, in enormous quantities. If the elementary charge on the electron and proton were even slightly different, all matter would have a large electric charge and would be mutually repulsive.


The best experimental tests of electric charge conservation are searches for particle decays that would be allowed if electric charge is not always conserved. No such decays have ever been seen.


The best experimental test comes from searches for the energetic photon from an electron decaying into a neutrino and a single photon:


but there are theoretical arguments that such single-photon decays will never occur even if charge is not conserved.


Charge disappearance tests are sensitive to decays without energetic photons, other unusual charge violating processes such as an electron spontaneously changing into a positron,


and to electric charge moving into other dimensions.


The best experimental bounds on charge disappearance are:


Conservation of energy can be rigorously proven by Noether's theorem as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time.


A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist, that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings. For systems which do not have time translation symmetry, it may not be possible to define "conservation of energy". Examples include curved spacetimes in general relativity or time crystals in condensed matter physics.


In 1605, Simon Stevinus was able to solve a number of problems in statics based on the principle that perpetual motion was impossible.


In 1639, Galileo published his analysis of several situations—including the celebrated "interrupted pendulum"—which can be described (in modern language) as conservatively converting potential energy to kinetic energy and back again. Essentially, he pointed out that the height a moving body rises is equal to the height from which it falls, and used this observation to infer the idea of inertia. The remarkable aspect of this observation is that the height to which a moving body ascends on a frictionless surface does not depend on the shape of the surface.


The fact that kinetic energy is scalar, unlike linear momentum which is a vector, and hence easier to work with did not escape the attention of Gottfried Wilhelm Leibniz. It was Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with "motion" (kinetic energy). Using Huygens' work on collision, Leibniz noticed that in many mechanical systems (of several masses, "mi" each with velocity "vi"),


was conserved so long as the masses did not interact. He called this quantity the "vis viva" or "living force" of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Many physicists at that time, such as Newton, held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum:


was the conserved "vis viva". It was later shown that both quantities are conserved simultaneously, given the proper conditions such as an elastic collision.


In 1687, Isaac Newton published his "Principia", which was organized around the concept of force and momentum. However, the researchers were quick to recognize that the principles set out in the book, while fine for point masses, were not sufficient to tackle the motions of rigid and fluid bodies. Some other principles were also required.


The law of conservation of vis viva was championed by the father and son duo, Johann and Daniel Bernoulli. The former enunciated the principle of virtual work as used in statics in its full generality in 1715, while the latter based his "Hydrodynamica", published in 1738, on this single conservation principle. Daniel's study of loss of vis viva of flowing water led him to formulate the Bernoulli's principle, which relates the loss to be proportional to the change in hydrodynamic pressure. Daniel also formulated the notion of work and efficiency for hydraulic machines; and he gave a kinetic theory of gases, and linked the kinetic energy of gas molecules with the temperature of the gas.


This focus on the vis viva by the continental physicists eventually led to the discovery of stationarity principles governing mechanics, such as the D'Alembert's principle, Lagrangian, and Hamiltonian formulations of mechanics.


Engineers such as John Smeaton, Peter Ewart, , Gustave-Adolphe Hirn and Marc Seguin recognized that conservation of momentum alone was not adequate for practical calculation and made use of Leibniz's principle. The principle was also championed by some chemists such as William Hyde Wollaston. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics, but in the 18th and 19th centuries, the fate of the lost energy was still unknown.


Gradually it came to be suspected that the heat inevitably generated by motion under friction was another form of "vis viva". In 1783, Antoine Lavoisier and Pierre-Simon Laplace reviewed the two competing theories of "vis viva" and caloric theory. Count Rumford's 1798 observations of heat generation during the boring of cannons added more weight to the view that mechanical motion could be converted into heat and (as important) that the conversion was quantitative and could be predicted (allowing for a universal conversion constant between kinetic energy and heat). "Vis viva" then started to be known as "energy", after the term was first used in that sense by Thomas Young in 1807.


which can be understood as converting kinetic energy to work, was largely the result of Gaspard-Gustave Coriolis and Jean-Victor Poncelet over the period 1819–1839. The former called the quantity "quantité de travail" (quantity of work) and the latter, "travail mécanique" (mechanical work), and both championed its use in engineering calculation.


In a paper "Über die Natur der Wärme"(German "On the Nature of Heat/Warmth"), published in the "Zeitschrift für Physik" in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called "Kraft" [energy or work]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."


A key stage in the development of the modern conservation principle was the demonstration of the "mechanical equivalent of heat". The caloric theory maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable.


In the middle of the eighteenth century, Mikhail Lomonosov, a Russian scientist, postulated his corpusculo-kinetic theory of heat, which rejected the idea of a caloric. Through the results of empirical studies, Lomonosov came to the conclusion that heat was not transferred through the particles of the caloric fluid.


In 1798, Count Rumford (Benjamin Thompson) performed measurements of the frictional heat generated in boring cannons, and developed the idea that heat is a form of kinetic energy; his measurements refuted caloric theory, but were imprecise enough to leave room for doubt.


The mechanical equivalence principle was first stated in its modern form by the German surgeon Julius Robert von Mayer in 1842. Mayer reached his conclusion on a voyage to the Dutch East Indies, where he found that his patients' blood was a deeper red because they were consuming less oxygen, and therefore less energy, to maintain their body temperature in the hotter climate. He discovered that heat and mechanical work were both forms of energy and in 1845, after improving his knowledge of physics, he published a monograph that stated a quantitative relationship between them.


Meanwhile, in 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. In the most famous, now called the "Joule apparatus", a descending weight attached to a string caused a paddle immersed in water to rotate. He showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle.


Over the period 1840–1843, similar work was carried out by engineer Ludwig A. Colding, although it was little known outside his native Denmark.


Both Joule's and Mayer's work suffered from resistance and neglect but it was Joule's that eventually drew the wider recognition.


In 1844, William Robert Grove postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single "force" ("energy" in modern terms). In 1846, Grove published his theories in his book "The Correlation of Physical Forces". In 1847, drawing on the earlier work of Joule, Sadi Carnot and Émile Clapeyron, Hermann von Helmholtz arrived at conclusions similar to Grove's and published his theories in his book "Über die Erhaltung der Kraft" ("On the Conservation of Force", 1847). The general modern acceptance of the principle stems from this publication.


In 1850, William Rankine first used the phrase "the law of the conservation of energy" for the principle.


In 1877, Peter Guthrie Tait claimed that the principle originated with Sir Isaac Newton, based on a creative reading of propositions 40 and 41 of the "Philosophiae Naturalis Principia Mathematica". This is now regarded as an example of Whig history.


Matter is composed of atoms and what makes up atoms. Matter has "intrinsic" or "rest" mass. In the limited range of recognized experience of the nineteenth century it was found that such rest mass is conserved. Einstein's 1905 theory of special relativity showed that rest mass corresponds to an equivalent amount of "rest energy". This means that "rest mass" can be converted to or from equivalent amounts of (non-material) forms of energy, for example kinetic energy, potential energy, and electromagnetic radiant energy. When this happens, as recognized in twentieth century experience, rest mass is not conserved, unlike the "total" mass or "total" energy. All forms of energy contribute to the total mass and total energy.


For example, an electron and a positron each have rest mass. They can perish together, converting their combined rest energy into photons having electromagnetic radiant energy, but no rest mass. If this occurs within an isolated system that does not release the photons or their energy into the external surroundings, then neither the total "mass" nor the total "energy" of the system will change. The produced electromagnetic radiant energy contributes just as much to the inertia (and to any weight) of the system as did the rest mass of the electron and positron before their demise. Likewise, non-material forms of energy can perish into matter, which has rest mass.


Thus, conservation of energy ("total", including material or "rest" energy), and conservation of mass ("total", not just "rest"), each still holds as an (equivalent) law. In the 18th century these had appeared as two seemingly-distinct laws.


The discovery in 1911 that electrons emitted in beta decay have a continuous rather than a discrete spectrum appeared to contradict conservation of energy, under the then-current assumption that beta decay is the simple emission of an electron from a nucleus. This problem was eventually resolved in 1933 by Enrico Fermi who proposed the correct description of beta-decay as the emission of both an electron and an antineutrino, which carries away the apparently missing energy.


For a closed thermodynamic system, the first law of thermodynamics may be stated as:


where formula_10 is the quantity of energy added to the system by a heating process, formula_11 is the quantity of energy lost by the system due to work done by the system on its surroundings and formula_12 is the change in the internal energy of the system.


Entropy is a function of the state of a system which tells of limitations of the possibility of conversion of heat into work.


For a simple compressible system, the work performed by the system may be written:


where formula_18 is the pressure and formula_19 is a small change in the volume of the system, each of which are system variables. In the fictive case in which the process is idealized and infinitely slow, so as to be called "quasi-static", and regarded as reversible, the heat being transferred from a source with temperature infinitesimally above the system temperature, then the heat energy may be written


where formula_21 is the temperature and formula_22 is a small change in the entropy of the system. Temperature and entropy are variables of state of a system.


If an open system (in which mass may be exchanged with the environment) has several walls such that the mass transfer is through rigid walls separate from the heat and work transfers, then the first law may be written:


where formula_24 is the added mass and formula_25 is the internal energy per unit mass of the added mass, measured in the surroundings before the process.


With the discovery of special relativity by Henri Poincaré and Albert Einstein, the energy was proposed to be one component of an energy-momentum 4-vector. Each of the four components (one of energy and three of momentum) of this vector is separately conserved across time, in any closed system, as seen from any given inertial reference frame. Also conserved is the vector length (Minkowski norm), which is the rest mass for single particles, and the invariant mass for systems of particles (where momenta and energy are separately summed before the length is calculated).


The relativistic energy of a single massive particle contains a term related to its rest mass in addition to its kinetic energy of motion. In the limit of zero kinetic energy (or equivalently in the rest frame) of a massive particle, or else in the center of momentum frame for objects or systems which retain kinetic energy, the total energy of particle or object (including internal kinetic energy in systems) is related to its rest mass or its invariant mass via the famous equation formula_26.


Thus, the rule of "conservation of energy" over time in special relativity continues to hold, so long as the reference frame of the observer is unchanged. This applies to the total energy of systems, although different observers disagree as to the energy value. Also conserved, and invariant to all observers, is the invariant mass, which is the minimal system mass and energy that can be seen by any observer, and which is defined by the energy–momentum relation.


In general relativity, energy–momentum conservation is not well-defined except in certain special cases. Energy-momentum is typically expressed with the aid of a stress–energy–momentum pseudotensor. However, since pseudotensors are not tensors, they do not transform cleanly between reference frames. If the metric under consideration is static (that is, does not change with time) or asymptotically flat (that is, at an infinite distance away spacetime looks empty), then energy conservation holds without major pitfalls. In practice, some metrics such as the Friedmann–Lemaître–Robertson–Walker metric do not satisfy these constraints and energy conservation is not well defined. The theory of general relativity leaves open the question of whether there is a conservation of energy for the entire universe.


In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and is its velocity (also a vector quantity), then the object's momentum is:<br>


In SI units, momentum is measured in kilogram meters per second (kg⋅m/s).


Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a "conserved" quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in special relativity (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. It is an expression of one of the fundamental symmetries of space and time: translational symmetry.


Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is generalized momentum, and in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function. The momentum and position operators are related by the Heisenberg uncertainty principle.


In continuous systems such as electromagnetic fields, fluid dynamics and deformable bodies, a momentum density can be defined, and a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.


Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see multiple dimensions).


The momentum of a particle is conventionally represented by the letter . It is the product of two quantities, the particle's mass (represented by the letter ) and its velocity ():


The unit of momentum is the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s). In cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second (g⋅cm/s).


Being a vector, momentum has magnitude and direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground.


The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses and , and velocities and , the total momentum is


The momenta of more than two particles can be added more generally with the following:


A system of particles has a center of mass, a point determined by the weighted sum of their positions:


If one or more of the particles is moving, the center of mass of the system will generally be moving as well (unless the system is in pure rotation around it). If the total mass of the particles is formula_6, and the center of mass is moving at velocity , the momentum of the system is:


This is known as Euler's first law.


If the net force applied to a particle is constant, and is applied for a time interval , the momentum of the particle changes by an amount


In differential form, this is Newton's second law; the rate of change of the momentum of a particle is equal to the instantaneous force acting on it,


If the net force experienced by a particle changes as a function of time, , the change in momentum (or impulse ) between times and is


Impulse is measured in the derived units of the newton second (1 N⋅s = 1 kg⋅m/s) or dyne second (1 dyne⋅s = 1 g⋅cm/s)


Under the assumption of constant mass , it is equivalent to write


hence the net force is equal to the mass of the particle times its acceleration.


"Example": A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3 newtons due north. The change in momentum is 6 kg⋅m/s due north. The rate of change of momentum is 3 (kg⋅m/s)/s due north which is numerically equivalent to 3 newtons.


In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant. This fact, known as the "law of conservation of momentum", is implied by Newton's laws of motion. Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If the particles are numbered 1 and 2, the second law states that and . Therefore,


with the negative sign indicating that the forces oppose. Equivalently,


If the velocities of the particles are and before the interaction, and afterwards they are and , then


This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.


Momentum is a measurable quantity, and the measurement depends on the motion of the observer. For example: if an apple is sitting in a glass elevator that is descending, an outside observer, looking into the elevator, sees the apple moving, so, to that observer, the apple has a non-zero momentum. To someone inside the elevator, the apple does not move, so, it has zero momentum. The two observers each have a frame of reference, in which, they observe motions, and, if the elevator is descending steadily, they will see behavior that is consistent with those same physical laws.


Suppose a particle has position in a stationary frame of reference. From the point of view of another frame of reference, moving at a uniform speed , the position (represented by a primed coordinate) changes with time as


This is called a Galilean transformation. If the particle is moving at speed in the first frame of reference, in the second, it is moving at speed


Since does not change, the accelerations are the same:


Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or Galilean invariance.


A change of reference frame, can, often, simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame, is the center of mass frame – one that is moving with the center of mass. In this frame,


By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision. Another property of the motion, kinetic energy, must be known. This is not necessarily conserved. If it is conserved, the collision is called an "elastic collision"; if not, it is an "inelastic collision".


An elastic collision is one in which no kinetic energy is absorbed in the collision. Perfectly elastic "collisions" can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps them apart. A slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls is a good example of an "almost" totally elastic collision, due to their high rigidity, but when bodies come in contact there is always some dissipation.


A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are and before the collision and and after, the equations expressing conservation of momentum and kinetic energy are:


In general, when the initial velocities are known, the final velocities are given by


If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.


In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as heat or sound). Examples include traffic collisions, in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the Franck–Hertz experiment); and particle accelerators in which the kinetic energy is converted into mass in the form of new particles.


In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are and before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity after the collision. The equation expressing conservation of momentum is:


If one body is motionless to begin with (e.g. formula_23), the equation for conservation of momentum is


In a different situation, if the frame of reference is moving at the final velocity such that formula_26, the objects would be brought to rest by a perfectly inelastic collision and 100% of the kinetic energy is converted to other forms of energy. In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless.


One measure of the inelasticity of the collision is the coefficient of restitution , defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula:


The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. Rockets also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.


Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with axes, velocity has components in the -direction, in the -direction, in the -direction. The vector is represented by a boldface symbol:


Similarly, the momentum is a vector quantity and is represented by a boldface symbol:


The equations in the previous sections, work in vector form if the scalars and are replaced by vectors and . Each vector equation represents three scalar equations. For example,


The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the magnitude of the vector, for example,


Each vector equation represents three scalar equations. Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.


A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision (as in the figure).


The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas. In analyzing such an object, one treats the object's mass as a function that varies with time: . The momentum of the object at time is therefore . One might then try to invoke Newton's second law of motion by saying that the external force on the object is related to its momentum by , but this is incorrect, as is the related expression found by applying the product rule to :


This equation does not correctly describe the motion of variable-mass objects. The correct equation is


where is the velocity of the ejected/accreted mass "as seen in the object's rest frame". This is distinct from , which is the velocity of the object itself as seen in an inertial frame.


This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass ("dm"). When considered together, the object and the mass ("dm") constitute a closed system in which total momentum is conserved.


Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance. It also results in a prediction that the speed of light can vary from one reference frame to another. This is contrary to observation. In the special theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light is invariant. As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.


Consider, for example, one reference frame moving relative to another at velocity in the direction. The Galilean transformation gives the coordinates of the moving frame as


Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the "inertial mass" of an object a function of velocity:


Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, is approximately equal to , the Newtonian expression for momentum.


In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example for position. The expression for the "four-momentum" depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of proper time, , defined by


is invariant under Lorentz transformations (in this expression and in what follows the metric signature has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by ; or by keeping time a real quantity and embedding the vectors in a Minkowski space. In a Minkowski space, the scalar product of two four-vectors and is defined as


In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by


where is the invariant mass. If (in Minkowski space), then


Using Einstein's mass-energy equivalence, , this can be rewritten as


Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.


The magnitude of the momentum four-vector is equal to :


and is invariant across all reference frames.


The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting it follows that


In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.


The four-momentum of a planar wave can be related to a wave four-vector


For a particle, the relationship between temporal components, , is the Planck–Einstein relation, and the relation between spatial components, , describes a de Broglie matter wave.


In Lagrangian mechanics, a Lagrangian is defined as the difference between the kinetic energy and the potential energy :


If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or Euler–Lagrange equations) are a set of equations:


If a coordinate is not a Cartesian coordinate, the associated generalized momentum component does not necessarily have the dimensions of linear momentum. Even if is a Cartesian coordinate, will not be the same as the mechanical momentum if the potential depends on velocity. Some sources represent the kinematic momentum by the symbol .


In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as


Each component is said to be the "conjugate momentum" for the coordinate .


Now if a given coordinate does not appear in the Lagrangian (although its time derivative might appear), then


This is the generalization of the conservation of momentum.


Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism.


In Hamiltonian mechanics, the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as


where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are


As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.


Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem. For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include curved spacetimes in general relativity or time crystals in condensed matter physics.


In Maxwell's equations, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force ("Lorentz force") on a particle with charge due to a combination of electric field and magnetic field is


It has an electric potential and magnetic vector potential .


In the non-relativistic regime, its generalized momentum is


The quantity formula_60 is sometimes called the "potential momentum". It is the momentum due to the interaction of the particle with the electromagnetic fields. The name is an analogy with the potential energy formula_61, which is the energy due to the interaction of the particle with the electromagnetic fields. These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called hidden-momentum of the electromagnetic fields


In Newtonian mechanics, the law of conservation of momentum can be derived from the law of action and reaction, which states that every force has a reciprocating equal and opposite force. Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions. Nevertheless, the combined momentum of the particles and the electromagnetic field is conserved.


The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.


In a vacuum, the momentum per unit volume is


where is the vacuum permeability and is the speed of light. The momentum density is proportional to the Poynting vector which gives the directional rate of energy transfer per unit area:


If momentum is to be conserved over the volume over a region , changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If is the momentum of all the particles in , and the particles are treated as a continuum, then Newton's second law gives


and the equation for conservation of each component of the momentum is


The term on the right is an integral over the surface area of the surface representing momentum flow into and out of the volume, and is a component of the surface normal of . The quantity is called the Maxwell stress tensor, defined as


The above results are for the "microscopic" Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density is modified to


where the H-field is related to the B-field and the magnetization by


The electromagnetic stress tensor depends on the properties of the media.


In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.


For a single particle described in the position basis the momentum operator can be written as


where is the gradient operator, is the reduced Planck constant, and is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in momentum space the momentum operator is represented as


where the operator acting on a wave function yields that wave function multiplied by the value , in an analogous fashion to the way that the position operator acting on a wave function yields that wave function multiplied by the value "x".


For both massive and massless objects, relativistic momentum is related to the phase constant formula_72 by


Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy).


In fields such as fluid dynamics and solid mechanics, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a continuum in which there is a particle or fluid parcel at each point that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density and velocity that depend on time and position . The momentum per unit volume is .


Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is , where is the gravitational acceleration. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure . The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is


If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative because the fluid in a given volume changes with time. Instead, the material derivative is needed:


Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to advection as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to . This is equal to the net force on the droplet.


Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a shear stress , exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or strain rate. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the direction varies with , the tangential force in direction per unit area normal to the direction is


where is the viscosity. This is also a flux, or flow per unit area, of x-momentum through the surface.


Including the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are


These are known as the Navier–Stokes equations.


The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction and force in direction , there is a stress component . The nine components make up the Cauchy stress tensor , which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation:


The Cauchy momentum equation is broadly applicable to deformations of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see Types of viscosity).


A disturbance in a medium gives rise to oscillations, or waves, that propagate away from their source. In a fluid, small changes in pressure can often be described by the acoustic wave equation:


where is the speed of sound. In a solid, similar equations can be obtained for propagation of pressure (P-waves) and shear (S-waves).


The flux, or transport per unit area, of a momentum component by a velocity is equal to . In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero. However, nonlinear effects can give rise to a nonzero average. It is possible for momentum flux to occur even though the wave itself does not have a mean momentum.


The work of Philoponus, and possibly that of Ibn Sīnā, was read and refined by the European philosophers Peter Olivi and Jean Buridan. Buridan, who in about 1350 was made rector of the University of Paris, referred to impetus being proportional to the weight times the speed. Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.


René Descartes believed that the total "quantity of motion" () in the universe is conserved, where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more important, he believed that it is speed rather than velocity that is conserved. So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion. Galileo, in his "Two New Sciences", used the Italian word "impeto" to similarly describe Descartes' quantity of motion.


Leibniz, in his "Discourse on Metaphysics", gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances. He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.


Christiaan Huygens concluded quite early that Descartes's laws for the elastic collision of two bodies must be wrong, and he formulated the correct laws. An important step was his recognition of the Galilean invariance of the problems. His views then took many years to be circulated. He passed them on in person to William Brouncker and Christopher Wren in London, in 1661. What Spinoza wrote to Henry Oldenburg about them, in 1666 which was during the Second Anglo-Dutch War, was guarded. Huygens had actually worked them out in a manuscript "De motu corporum ex percussione" in the period 1652–6. The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He published them in the "Journal des sçavans" in 1669.


In physics, zilch is a conserved quantity of the electromagnetic field.


Daniel M. Lipkin observed that if he defined the quantities


which implies that the total "zilch" formula_3 is constant (formula_4 is the "zilch current"). Generalising the result, Lipkin found nine related conservation laws, all unrelated to the stress–energy tensor. He named the quantity zilch because of the apparent lack of physical significance.


Zilch can also be expressed using the dual electromagnetic tensor formula_5 as


It was later demonstrated that zilch is part of an infinite number of zilch-like conserved quantities, a general property of free fields.


Zilch has occasionally been rediscovered. It has been called "optical chirality", since it determines the degree of chiral asymmetry in the rate of excitation of a small chiral molecule by an incident electromagnetic field. A physical interpretation of zilch was offered in 2012; zilch is to the curl or time derivative of the electromagnetic field what helicity, spin and related quantities are to the electromagnetic field itself. The conservation of zilch is not associated with duality transformations, but instead with a more subtle symmetry transformation, which has no special name.


Magnetic braking is a theory explaining the loss of stellar angular momentum due to material getting captured by the stellar magnetic field and thrown out at great distance from the surface of the star. It plays an important role in the evolution of binary star systems.


The currently accepted theory of the solar system's evolution states that the Solar System originates from a contracting gas cloud. As the cloud contracts, the angular momentum formula_1 must be conserved. Any small net rotation of the cloud will cause the spin to increase as the cloud collapses, forcing the material into a rotating disk. At the dense center of this disk a protostar forms, which gains heat from the gravitational energy of the collapse. As the collapse continues, the rotation rate can increase to the point where the accreting protostar can break up due to centrifugal force at the equator.


Thus the rotation rate must be braked during the first 100,000 years of the star's life to avoid this scenario. One possible explanation for the braking is the interaction of the protostar's magnetic field with the stellar wind. In the case of our own Sun, when the planets' angular momenta are compared to the Sun's own, the Sun has less than 1% of its supposed angular momentum. In other words, the Sun has slowed down its spin while the planets have not.


As ionized material follows the Sun's magnetic field lines, due to the effect of the field lines being frozen in the plasma, the charged particles feel a force formula_2 of the magnitude:


where formula_4 is the charge, formula_5 is the velocity and formula_6 is the magnetic field vector. This bending action forces the particles to "corkscrew" around the magnetic field lines while held in place by a "magnetic pressure"formula_7, or "energy density", while rotating together with the Sun as a solid body:


Since magnetic field strength decreases with the cube of the distance there will be a place where the kinetic gas pressure formula_9 of the ionized gas is great enough to break away from the field lines:


where n is the number of particles, m is the mass of the individual particle and v is the radial velocity away from the Sun, or the speed of the solar wind.


Due to the high conductivity of the stellar wind, the magnetic field outside the sun declines with radius like the mass density of the wind, i.e. decline as an inverse square law. The magnetic field is therefore given by


where formula_12 is the magnetic field on the surface of the sun and formula_13 is its radius. The critical distance where the material will break away from the field lines can then be calculated as the distance where the kinetic pressure and the magnetic pressure are equal, i.e.


If the solar mass loss is omni-directional then the mass loss formula_17; plugging this into the above equation and isolating the critical radius it follows that


This leads to a critical radius formula_23. This means that the ionized plasma will rotate together with the Sun as a solid body until it reaches a distance of nearly 15 times the radius of the Sun; from there the material will break off and stop affecting the Sun.


The amount of solar mass needed to be thrown out along the field lines to make the Sun completely stop rotating can then be calculated using the specific angular momentum:


It has been suggested that the sun lost a comparable amount of material over the course of its lifetime.


In 2016 scientists at Carnegie Observatories published a research suggesting that stars at a similar stage of life as the Sun were spinning faster than magnetic braking theories predicted. To calculate this they pinpointed the dark spots on the surface of stars and tracked them as they moved with the stars' spin. While this method has been successful for measuring the spin of younger stars, the "weakened" magnetic braking in older stars proved harder to confirm, as the latter notoriously have fewer star spots. In a study published in Nature Astronomy in 2021, researchers at the University of Birmingham used a different approach, namely asteroseismology, to confirm that older stars do appear to rotate faster than expected.


In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument.


is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle "θ"


the function, after some cancellation of terms, takes exactly the same form


The rotation of coordinates can be expressed using matrix form using the rotation matrix,


or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is


In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. For a real-valued function of three or more real variables, this expression extends easily using appropriate rotation matrices.


The concept also extends to a vector-valued function f of one or more variables;


In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself.


which maps elements from a subset "X" of the real line ℝ to itself, rotational invariance may also mean that the function commutes with rotations of elements in "X". This also applies for an operator that acts on such functions. An example is the two-dimensional Laplace operator


which acts on a function "f" to obtain another function ∇2"f". This operator is invariant under rotations.


If "g" is the function "g"("p") = "f"("R"("p")), where "R" is any rotation, then (∇2"g")("p") = (∇2"f" )("R"("p")); that is, rotating a function merely rotates its Laplacian.


In physics, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.


In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger's equation. That is


for any rotation "R". Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have ["R", "H"] = 0.


For infinitesimal rotations (in the "xy"-plane for this example; it may be done likewise for any plane) by an angle "dθ" the (infinitesimal) rotation operator is


in other words angular momentum is conserved.


In astrodynamics, the "vis-viva" equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight.


"Vis viva" (Latin for "living force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the total work of the accelerating forces of a system and that of the retarding forces is equal to one half the "vis viva" accumulated or lost in the system while the work is being done.


For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the "vis-viva" equation is as follows:


The product of GM can also be expressed as the standard gravitational parameter using the Greek letter μ.


Derivation for elliptic orbits (0 ≤ eccentricity < 1).


In the vis-viva equation the mass "m" of the orbiting body (e.g., a spacecraft) is taken to be negligible in comparison to the mass "M" of the central body (e.g., the Earth). The central body and orbiting body are also often referred to as the primary and a particle respectively. In the specific cases of an elliptical or circular orbit, the vis-viva equation may be readily derived from conservation of energy and momentum.


Specific total energy is constant throughout the orbit. Thus, using the subscripts "a" and "p" to denote apoapsis (apogee) and periapsis (perigee), respectively,


Recalling that for an elliptical orbit (and hence also a circular orbit) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires specific angular momentum formula_4, thus formula_5:


Isolating the kinetic energy at apoapsis and simplifying,


From the geometry of an ellipse, formula_11 where "a" is the length of the semimajor axis. Thus,


Substituting this into our original expression for specific orbital energy,


Thus, formula_14 and the vis-viva equation may be written


Therefore, the conserved angular momentum L = mh can be derived using formula_17 and formula_18,


where a is semi-major axis and b is semi-minor axis of the elliptical orbit, as follows -


Given the total mass and the scalars "r" and "v" at a single point of the orbit, one can compute "r" and "v" at any other point in the orbit.


Given the total mass and the scalars "r" and "v" at a single point of the orbit, one can compute the specific orbital energy formula_23, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "suborbital" (a ballistic missile, for example), having enough energy to be "orbital", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and/or go to infinity (as a meteor, for example).


The formula for escape velocity can be obtained from the Vis-viva equation by taking the limit as formula_24 approaches formula_25:


In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of "one" spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection):


It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions.


A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is "not" a parity transformation; it is the same as a 180°-rotation.


In quantum mechanics, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are odd functions.


Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group.


Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. The word "projective" refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.