# parity

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## parity

or## space parity,

in physics, quantity that refers to the relationship between an object or process and the image that it can produce in a mirror. For example, any right-handed object will produce a mirror-image counterpart that is identical to it in every way except that the mirror image is left-handed. A moving particle that spins in a clockwise manner, as would a right-handed screw advancing through space, will possess a mirror-image particle that is identical to it in every way except that it spins counterclockwise, as would a left-handed screw advancing through space. The law of conservation of parity implies that every real object or process has a mirror image that can also exist and that obeys the same physical laws. Although this concept has little significance in classical physics, it is of great importance in atomic and nuclear physics. From this law scientists inferred that all elementary particles**elementary particles,**

the most basic physical constituents of the universe.

**Basic Constituents of Matter**

Molecules are built up from the atom, which is the basic unit of any chemical element. The atom in turn is made from the proton, neutron, and electron.

**.....**Click the link for more information. and their interactions possessed mirror image counterparts that also exist. However, in 1956 T. D. Lee and C. N. Yang published a paper in which they argued that parity was not conserved in weak interactions. Their conjecture was verified the same year by C. S. Wu and coworkers at the U.S. National Bureau of Standards and other institutions in an experiment involving beta decay (see radioactivity

**radioactivity,**

spontaneous disintegration or decay of the nucleus of an atom by emission of particles, usually accompanied by electromagnetic radiation. The energy produced by radioactivity has important military and industrial applications.

**.....**Click the link for more information. ). Parity is still conserved in the strong nuclear interactions and in the electromagnetic interactions. Formally, parity,

*P,*is a quantity that expresses the behavior of the wave function of any system of particles when the spatial coordinates

*x, y, z,*of the wave function are reflected through the origin to −

*x,*−

*y,*−

*z*(see quantum theory

**quantum theory,**

modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics.

**.....**Click the link for more information. ). This mathematical operation is called the parity, or space-inversion, operation. See also symmetry

**symmetry,**

generally speaking, a balance or correspondence between various parts of an object; the term

*symmetry*is used both in the arts and in the sciences. In art and design, it is often used in a somewhat loose sense, to mean a kind of balance in which the

**.....**Click the link for more information. .

## Parity (quantum mechanics)

A physical property of a wave function which specifies its behavior under simultaneous reflection of all spatial coordinates through the origin, that is, when *x* is replaced by -*x*, *y* by -*y*, and *z* by -*z*. If the single-particle wave function &psgr; satisfies Eq. (1), it is said to have even parity. If, on the other hand, Eq. (2) holds, the wave function is said to have odd parity. These two expressions can be combined in Eq. (3),

*P*= ±1 is a quantum number, parity, having only the two values +1 (designated as even parity) and -1 (odd parity). More precisely, parity is defined as the eigenvalue of the operation of space inversion. Parity is a concept that has meaning only for fields or waves and therefore has meaning only in classical field theory or in quantum mechanics.

*See*Quantum mechanics

The conservation of parity follows from the inversion symmetry of space, that is, the invariance of the Schrödinger equation *H*&psgr; = *E*&psgr; (the wave equation satisfied by the wave function &psgr;) to the inversion of space coordinates, **r** → - **r** . The parity (or inversion) operator, which changes **r** to - **r** , has the alternative interpretation that the coordinate values remain unchanged but the coordinate axes are inverted; that is, the positive *x* axis of the new frame points along the old negative *x* axis, and similarly for *y* and *z*. If the original frame was right-handed, then the new frame is left-handed. [A cartesian coordinate system (frame, for short) is called right-handed if it is possible to place the right hand at the origin and point the thumb and first and second fingers along the positive *x*, *y*, and *z* axes, respectively.] Thus, parity would be conserved if the statement of physical laws were independent of the handedness of the coordinate system that was being used. Of course, the fact that most people are right-handed is not a physical law but an accident of evolution; there is nothing in the relevant laws of physics which favors a right-handed over a left-handed human. The same holds for optically active organic compounds, such as the amino acids. However, the statement that the neutrino is left-handed is a physical law. *See* Neutrino

All the strong interactions between hadrons (for example, nuclear forces) and the electromagnetic interactions are symmetrical to inversion, so that parity is conserved by these interactions. As far as is known, only the weak interactions fail to conserve parity. Thus parity is not conserved in the weak decays of elementary particles (including beta decay of nuclei); in all other processes the weak interactions play a small role, and parity is very nearly conserved. Likewise, in energy eigenstates, weak interactions can be neglected to a very good approximation, and parity is very nearly a good quantum number, so that each atomic, nuclear, or hadronic state is characterized by a definite value of parity, and its conservation in reactions is an important principle. *See* Fundamental interactions, Weak nuclear interactions

One of the selection rules which follows from parity conservation is the following: A spin zero boson cannot decay sometimes into two &pgr; mesons and sometimes into three &pgr; mesons, because these final states have different parities, even and odd respectively. But the positive *K* meson is observed to have both these decay modes, originally called the Θ and the &tgr; mesons, respectively, but later shown by the identity of masses and lifetimes to be decay modes of the same particle. This &tgr;-Θ puzzle was the first observation of parity nonconservation. In 1956, T. D. Lee and C. N. Yang made the bold hypothesis that parity also is not conserved in beta decay. They reasoned that the magnitude of the beta-decay coupling is about the same as the coupling which leads to decay of the *K* meson, and so these decay processes may be manifestations of a single kind of coupling. Also, there is a very natural way to introduce parity nonconservation in beta decay, namely, by assuming a restriction on the possible states of the neutrino (two-component theory). They pointed out that no beta-decay experiment had ever looked for the spin-momentum correlations that would indicate parity nonconservation; they urged that these correlations be sought.

In the first experiment to show parity nonconservation in beta decay, the spins of the beta-active nuclei cobalt-60 were polarized with a magnetic field at low temperature; the decay electrons were observed to be emitted preferentially in directions opposite to the direction of the ^{60}Co spin. The magnitude of this correlation shows that the parity-nonconserving and parity-conserving parts of the beta interaction are of equal size, substantiating the two-component neutrino theory.

It was at first somewhat disconcerting to find parity not conserved, for that seemed to imply a handedness of space. But this is not really the situation; the saving thing is that anti-^{60}Co decays in the opposite direction. Thus, after all, there is nothing intrinsically left-handed about the world, just as there is nothing intrinsically positively charged about nuclei. What really exists here is a correlation between handedness and sign of charge.

## Parity

an economic term expressing the ratio between the currencies of various countries.

Until World War I, before which gold or silver circulated as money and paper money could be freely exchanged for gold, the mint parity was used. The ratio between foreign currencies was based on the amount of pure gold or silver contained in the respective monetary units of the different countries. When paper money that was not exchangeable for gold began to circulate after World War I, the mint parity was replaced by currency parity, which reflects the ratio of currencies according to the nominal amount of gold in the national monetary units as officially determined by the respective governments *(see*EXCHANGE RATE). Under the gold standard the parity value of currency and the level of domestic prices changed in conformity with changes in the gold content of the national monetary unit. However, with the abolition of the gold standard in the mid-1930’s, countries were compelled to alter the exchange parity of currencies and the gold content of the national monetary unit according to the movement of domestic prices.

The exchange parity of the currency is adjusted to bring the purchasing power of the national currency closer to the purchasing power of other currencies. The relationship between the currencies of different countries, based on equivalence of purchasing power, expresses the relative purchasing power of different currencies. In foreign-exchange markets currencies are not exchanged at their official rates, or parities, but rather according to an exchange rate that deviates from parity in accordance with fluctuations in the market supply and demand for the particular currency. During monetary crises, these fluctuations become very severe.

In socialist economies, the exchange parity of the currency is established by the state on a planned basis according to the gold content of the currency and a comparison of the purchasing power of the national currency in the domestic and world markets, respectively. The exchange rates of the socialist countries fluctuate only in relation to the capitalist currencies, depending on the market deviations in the exchange rates for the latter.

V. V. SHCHEGOLEV

## Parity

the principle of equal representation of parties. Commissions to review border disputes, for example, are formed on the basis of parity. The principle of parity is most commonly applied in conciliation hearings on labor disputes and conflicts. For example, in the USSR, commissions on labor disputes are formed from an equal number of representatives of the workers and of the administration.

## Parity

a quantum-mechanical characteristic of the state of a physical microparticle, for example, a molecule, an atom, an atomic nucleus, or an elementary particle. Parity represents the symmetry properties of the wave function of such a microparticle with respect to mirror reflection. In processes resulting from strong interactions and from electromagnetic interactions, the law of parity conservation holds; that is, a physical system having mirror symmetry of a definite type in the initial state retains that symmetry at all later moments in time. The conservation of parity leads to a number of selection rules in the electromagnetic radiation of atoms and atomic nuclei, in nuclear reactions, and in reactions involving transformations of elementary particles.

The law of parity conservation may be demonstrated by using the Zeeman effect as an example. When a magnetic field is applied, the intensity of the radiation in individual spectral lines remains symmetric with respect to the plane perpendicular to the field but is no longer identical in all directions. The radiation in the direction of the magnetic field is the same as the radiation in the opposite direction. If we imagine a system—in the form of a current-carrying circular conductor with a specimen placed at the center of the circle—for observing the Zeeman effect, the mirror symmetry of the system becomes obvious, but only on the condition that all the elementary particles composing the system have mirror symmetry. Thus, the law of parity conservation is based on the assumption that all electrons, protons, and other particles are transformed into themselves under mirror reflection.

Instead of mirror symmetry with respect to a plane, it is more convenient to consider an operation that entails the inversion of coordinate axes: r→ –r or *x* →; – *x. y*→ – *y,z*→ – *z* (*see*SPACE INVERSION).

The law of parity conservation governs the transformation properties of physical quantities under the inversion of coordinate axes. Thus, from the assumption that a charged particle— for example, an electron—is transformed into itself under inversion, it follows that electric charge *q* is a scalar, current density **j** and electric field strength **E** are true (or polar) vectors, and magnetic field strength **H** is an axial vector (or a pseudovector): *q* → *q*’ –, **j**′, **E** → – **E** ′ **H** → **H** ′.

The law of parity conservation is violated in weak interactions, which—in particular—are responsible for the beta decay of nuclei. The violation was predicted in 1956 by Tsung Dao Lee and Chen Ning Yang. It was confirmed experimentally in 1957 by Chien Shiung Wu and co-workers in the beta decay of nuclei and by the American physicists L. Lederman and R. Garwin, as well as other scientists, in the decay of muons. Parity is also not conserved in the decays of charged pions, kaons, and hyperons. The Soviet physicists Iu. G. Abov, V. M. Lobashev, and others discovered the nonconservation of parity in weak nucleon-nucleon interactions.

A diagram of Wu’s experiment is shown on the left in Figure 1. A specimen containing the radioactive isotope ^{60}Co was placed in the magnetic field H of a circular current. The field H oriented the relatively large magnetic moments of the ^{60}Co nuclei in the direction parallel to H. The short arrow on the left in Figure 1 indicates the direction of the electron velocities within the conductor. As in the example of the Zeeman effect, Wu’s system was mirror symmetric with respect to the plane in which the circular current flowed. If the law of parity conservation held, the intensity of the electron (*e*^{–})emission during β^{–} decay should have been the same on both sides of the plane in which the current flowed. However, a pronounced asymmetry was observed in the experiment: 40 percent more electrons were emitted on one side of the plane than on the other side.

From Wu’s experiment, it follows that magnetic field strength is a polar vector rather than an axial vector. This is not inconsistent with the equations of electrodynamics if we assume that current density and electric field strength are axial vectors and—at the same time—that electric charge is a pseudoscalar. The identification of charge as a pseudoscalar means that, under mirror reflection, electrons are transformed into positrons (e^{+}) and, in general, all particles are transformed into their respective anti-particles. The possibility of such an interpretation of reflection was pointed out by the American scientists E. Wigner, G. Wick, and A. Wightman as early as 1952.

L. D. Landau referred to mirror reflection accompanied by the replacement of all particles by their antiparticles as combined inversion. The hypothesis that the laws of nature are symmetric with respect to combined inversion is expressed by the law of the conservation of combined parity, which is usually called the law of *CP* conservation. When the law of parity conservation is replaced by the law of *CP* conservation, the scheme of Wu’s experiment is no longer mirror symmetric, since the β^{+} decay of an an-ticobalt nucleus, C͂o, in the magnetic field of a circular positron current is the mirror image of her experiment (Figure 1, on the right); an anticobalt nucleus consists of antiprotons and antineutrons. Since the charge of the positron is positive, the sign of the current changes if the positrons move in the same direction as the electrons. A change in the sign of the current leads to a change in the sign of the magnetic field (**H**′).

Thus, the law of parity conservation is approximate and holds only when weak interactions are ignored. The conventional treatment—where, for example, **H** is an axial vector—of the transformation properties of electromagnetic quantities with respect to the inversion of coordinate axes is valid to the same extent to which the law of parity conservation holds.

In quantum theory, the parity of a state of a system consisting of *n* particles is defined as the eigenvalue of the parity operator *P*, which inverts space. The action of the operator *P* on the state vector ψ(**p**_{1}, . . . , **p**_{n}) consists in a change of the signs of the particle momenta p_{i} and in the multiplication of the parity of the state vector by the product Π_{1} . . . Π_{n} of the intrinsic parities of the particles. Intrinsic parity is an inherent property of a particle and is equal to either + 1 or – 1. Particles for which Π_{k} = 1 are said to have even parity, while particles for which Π_{k} = –1 are said to have odd parity. For example, the intrinsic parity of neutral pions is odd. The intrinsic parities of antiparticles with half-integral spin are the opposite of those of the respective particles. For this reason, in particular, mesons consisting of a quark and an antiquark with zero orbital angular momentum—for example, pions, kaons, rho mesons, and omega mesons—have odd intrinsic parity. The operator *P* does not act on spin projections or on charges. The eigenvalues of the operator *P* are equal to ±1. States with *P* = 1 are called states of even parity; states with *P* = – 1 are known as states of odd parity.

The definition of parity leads to two rules for determining the parities of physical systems consisting of several particles. First, the parity of a system consisting of *n* particles with orbital angular momenta *ℏl*_{1}, . . . , *ℏl _{n}*, where

*ℏ*is Planck’s constant and

*I*, are integers, is equal to

_{i}Π_{1} . . . Π_{2} (– 1)^{l} 1 + . . . + *l _{n}*

Second, the parity Π_{12} of a complex system consisting of two subsystems with respective parities Π_{1} and Π_{2} is equal to Π_{1} Π_{2}(–1)^{L}, where *ℏL* is the orbital angular momentum associated with the relative motion of the subsystems.

Electromagnetic field quanta have neither intrinsic parity nor orbital angular momenta. The parity of a quantum of electromagnetic radiation, or a photon, is determined by the quantum’s multipole order (*see*MULTIPOLE). The parity of an electric 2^{l}-pole is equal to ( –1)*I*, while the parity of a magnetic 2^{l}-pole is equal to (–1)^{l + 1}. Therefore, the parity of a physical system is conserved when an electric multipole quantum with even *l* or a magnetic multipole quantum with odd *l* is emitted or absorbed. It is reversed when an electric multipole quantum with odd *l* or a magnetic multipole quantum with even *l* is emitted or absorbed. The parity selection rules for the electromagnetic radiation of atoms and nuclei result from the fact that, for identical multipole orders with all other conditions being equal, magnetic multipole radiation is substantially weaker than electric multipole radiation. The ratio of the probabilities of magnetic and electric multipole radiation is of the order of (2π*R*/λ)^{2}, where *R* is the linear size of the radiator and λ is the wavelength of the radiated quantum. As a rule, the ratio is much smaller than unity for both nuclei and atoms, so that the parity selection rules are markedly exhibited.

The law of parity conservation, which is also called *P* invariance, is formulated as the conservation of the quantity *P* in strong and electromagnetic interactions.

The concept of the intrinsic parity of a particle and, thus, of the parity of a state contains a certain degree of ambiguity, which is associated with the impossibility of comparing the parities of states that differ in the values of just one of the conserved charges, for example, electric charge or baryon charge. Therefore, in particular, the parity of the vacuum state and the parities of the proton, the neutron, and the electron are arbitrary and may be assigned positive values. However, in this case, the parities of, for example, a pion, the positron, and the antiproton become well defined and must be assigned negative values.

The fundamental question of the symmetry of real space with respect to mirror reflection is closely associated with the concept of parity. The methods of group theory have been used to prove that, if space has mirror symmetry, either the law of parity conservation or the principle of *CP* invariance must be strictly satisfied. The violation of both the law and the principle in weak interactions has been established experimentally. Therefore, grounds exist for postulating that either space does not have symmetry between left and right or such symmetry is broken in certain types of interactions. An example of such interactions is the interactions that result in the decay of the long-lived neutral kaon, K°_{L} → 2π.

### REFERENCES

Lee, T., and C. Wu.*Slabye vzaimodeistviia*. Moscow, 1968. (Translated from English.)

Shirokov, Iu. M., and N. P. Iudin.

*Iadernaia fizika*. Moscow, 1972.

Lee, Tsung Dao, and Chen Ning Yang. In the collection

*Novye svoistva simmetrii elementarnykh chastits*. Moscow, 1957. Page 13. (Translated from English.)

Wu, Chien Shiung [et al.]. In the collection

*Novye svoistva simmetrii elementarnykh chastits*. Moscow, 1957. Page 69. (Translated from English.)

Garwin, R., L. Lederman, and M. Weinrich. In the collection

*Novye svoistva simmetrii elementarnykh chastits*. Moscow, 1957. Page 75. (Translated from English.)

Abov, Yu.G.,etal.

*Physics Letters*, 1968, vol. 27B, no. l,p. 16.

Lobashev, V. M.

*Vestnik AN SSSR*, 1969, no. 2, p. 58.

Wigner, E.

*Uspekhi fizicheskikh nauk*, 1958, vol. 65, issue 2, p. 257.

Wick, G., A. Wightman, and E. Wigner.

*Physical Review*, 1952, vol. 88, p. 101.

Landau, L. D.

*Zhurnal eksperimental’noi i teoreticheskoi fiziki*, 1957, vol. 32, issue 2, p. 405.

Shirokov, Iu. M.

*Zhurnal eksperimental’noi i teoreticheskoi fiziki*, 1958, vol. 34, issue 3, p. 717.

Shirokov, Iu. M.

*Zhurnal eksperimental’noi i teoreticheskoi fiziki*, 1960, vol. 38, issue 1, p. 140.

IU. M. SHIROKOV

## parity

[′par·əd·ē]## parity

**1.**

*Physics*

**a.**a property of a physical system characterized by the behaviour of the sign of its wave function when all spatial coordinates are reversed in direction. The wave function either remains unchanged (

**even parity**) or changes in sign (

**odd parity**)

**b.**a quantum number describing this property, equal to +1 for even parity systems and --1 for odd parity systems.

**2.**

*Maths*a relationship between two integers. If both are odd or both even they have the same parity; if one is odd and one even they have different parity

## parity

(storage, communications)See also longitudinal parity, checksum, cyclic redundancy check.