Inphysics, aparity transformation (also calledparity inversion) is the flip in the sign ofonespatialcoordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (apoint reflection):
It can also be thought of as a test forchirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image.
All fundamental interactions ofelementary particles, with the exception of theweak interaction, are symmetric under parity transformation. As established by theWu experiment conducted at the USNational Bureau of Standards by Chinese-American scientistChien-Shiung Wu, the weak interaction is chiral and thus provides a means for probing chirality in physics. In her experiment, Wu took advantage of the controlling role of weak interactions inradioactive decay of atomic isotopes to establish the chirality of the weak force.
By contrast, 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 ofP (in any number of dimensions) hasdeterminant equal to −1, and hence is distinct from arotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign isnot a parity transformation; it is the same as a180° rotation.
In quantum mechanics, wave functions that are unchanged by a parity transformation are described aseven functions, while those that change sign under a parity transformation are odd functions.
Underrotations, classical geometrical objects can be classified intoscalars,vectors, andtensors of higher rank. Inclassical physics, physical configurations need to transform underrepresentations of every symmetry group.
Quantum theory predicts that states in aHilbert space do not need to transform under representations of thegroup of rotations, but only underprojective representations. The wordprojective 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.
The projective representations of any group are isomorphic to the ordinary representations of acentral extension of the group. For example, projective representations of the 3-dimensional rotation group, which is thespecial orthogonal group SO(3), are ordinary representations of thespecial unitary group SU(2). Projective representations of the rotation group that are not representations are calledspinors and so quantum states may transform not only as tensors but also as spinors.
If one adds to this a classification by parity, these can be extended, for example, into notions of
One can definereflections such as
which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performingx-,y-, andz-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used.
Parity forms theabelian group due to the relation. All Abelian groups have only one-dimensionalirreducible representations. For, there are two irreducible representations: one is even under parity,, the other is odd,. These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by anyphase.
An alternative way to write the above classification of scalars, pseudoscalars, vectors and pseudovectors is in terms of the representation space that each object transforms in. This can be given in terms of thegroup homomorphism which defines the representation. For a matrix
When the representation is restricted to, scalars and pseudoscalars transform identically, as do vectors and pseudovectors.
Newton's equation of motion (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.
However, angular momentum is anaxial vector,
In classicalelectrodynamics, the charge density is a scalar, the electric field,, and current are vectors, but the magnetic field, is an axial vector. However,Maxwell's equations are invariant under parity because thecurl of an axial vector is a vector.
The two major divisions of classical physical variables have either even or odd parity. The way into which particular variables and vectors sort out into either category depends on whether thenumber of dimensions of space is either an odd or even number. The categories ofodd oreven given below for theparity transformation is a different, but intimately related issue.
The answers given below are correct for 3 spatial dimensions. In a 2 dimensional space, for example, when constrained to remain on the surface of a planet, some of the variables switch sides.
Classical variables whose signs flip under space inversion are predominantly vectors. They include:
Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:
In quantum mechanics, spacetime transformations act onquantum states. The parity transformation,, is aunitary operator, in general acting on a state as follows:.
One must then have, since an overall phase is unobservable. The operator, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases. If is an element of a continuous U(1) symmetry group of phase rotations, then is part of this U(1) and so is also a symmetry. In particular, we can define, which is also a symmetry, and so we can choose to call our parity operator, instead of. Note that and so has eigenvalues. Wave functions with eigenvalue under a parity transformation areeven functions, while eigenvalue corresponds to odd functions.[1] However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than.
For electronic wavefunctions, even states are usually indicated by a subscript g forgerade (German: even) and odd states by a subscript u forungerade (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled and the next-closest (higher) energy level is labelled.[2]
The wave functions of a particle moving into an external potential, which iscentrosymmetric (potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions.[3]
The law of conservation of parity of particles states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. However this is not true for thebeta decay of nuclei, because theweak nuclear interaction violates parity.[4]
The parity of the states of a particle moving in a spherically symmetric external field is determined by theangular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.[3]
When parity generates theAbelian group, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number.
In quantum mechanics,Hamiltonians areinvariant (symmetric) under a parity transformation ifcommutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any scalar potential, i.e.,, hence the potential is spherically symmetric. The following facts can be easily proven:
Some of the non-degenerate eigenfunctions of are unaffected (invariant) by parity and the others are merely reversed in sign when the Hamiltonian operator and the parity operator commute:
where is a constant, theeigenvalue of,
The overall parity of a many-particle system is the product of the parities of the one-particle states. It is −1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules.
Atomic orbitals have parity (−1)ℓ, where the exponent ℓ is theazimuthal quantum number. The parity is odd for orbitals p, f, ... with ℓ = 1, 3, ..., and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol4So, where the superscript o denotes odd parity. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is4P (without an o superscript).[6]
The complete (rotational-vibrational-electronic-nuclear spin) electromagnetic Hamiltonian of any molecule commutes with (or is invariant to) the parity operation P (or E*, in the notation introduced byLonguet-Higgins[7]) and its eigenvalues can be given the parity symmetry label+ or− as they are even or odd, respectively. The parity operation involves the inversion of electronic and nuclear spatial coordinates at the molecular center of mass.
Centrosymmetric molecules at equilibrium have a centre of symmetry at their midpoint (the nuclear center of mass). This includes all homonucleardiatomic molecules as well as certain symmetric molecules such asethylene,benzene,xenon tetrafluoride andsulphur hexafluoride. For centrosymmetric molecules, the point group contains the operationi which is not to be confused with the parity operation. The operationi involves the inversion of the electronic and vibrational displacement coordinates at the nuclear centre of mass. For centrosymmetric molecules the operationi commutes with the rovibronic (rotation-vibration-electronic) Hamiltonian and can be used to label such states. Electronic and vibrational states of centrosymmetric molecules are either unchanged by the operationi, or they are changed in sign byi. The former are denoted by the subscriptg and are calledgerade,while the latter are denoted by the subscriptu and are calledungerade. The complete electromagnetic Hamiltonian of a centrosymmetric moleculedoes not commute with the point group inversion operationi because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels ofg andu vibronic states (calledortho-para mixing) and give rise toortho-para transitions[8][9]
In atomic nuclei, the state of each nucleon (proton or neutron) has even or odd parity, and nucleon configurations can be predicted using thenuclear shell model. As for electrons in atoms, the nucleon state has odd overall parity if and only if the number of nucleons in odd-parity states is odd. The parity is usually written as a + (even) or − (odd) following the nuclear spin value. For example, theisotopes of oxygen include17O(5/2+), meaning that the spin is 5/2 and the parity is even. The shell model explains this because the first 16 nucleons are paired so that each pair has spin zero and even parity, and the last nucleon is in the 1d5/2 shell, which has even parity since ℓ = 2 for a d orbital.[10]
If one can show that thevacuum state is invariant under parity,, the Hamiltonian is parity invariant and the quantization conditions remain unchanged under parity, then it follows that every state hasgood parity, and this parity is conserved in any reaction.
To show thatquantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume thatcanonical quantization is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator:[citation needed]where denotes the momentum of a photon and refers to its polarization state. This is equivalent to the statement that the photon has oddintrinsic parity. Similarly allvector bosons can be shown to have odd intrinsic parity, and allaxial-vectors to have even intrinsic parity.
A straightforward extension of these arguments to scalar field theories shows that scalars have even parity. That is,, sinceThis is true even for a complex scalar field. (Details ofspinors are dealt with in the article on theDirac equation, where it is shown thatfermions and antifermions have opposite intrinsic parity.)
Withfermions, there is a slight complication because there is more than onespin group.
Applying the parity operator twice leaves the coordinates unchanged, meaning thatP2 must act as one of the internal symmetries of the theory, at most changing the phase of a state.[11] For example, theStandard Model has three globalU(1) symmetries with charges equal to thebaryon numberB, thelepton numberL, and theelectric chargeQ. Therefore, the parity operator satisfiesP2 =eiαB+iβL+iγQ for some choice ofα,β, andγ. This operator is also not unique in that a new parity operatorP' can always be constructed by multiplying it by an internal symmetry such asP' =PeiαB for someα.
To see if the parity operator can always be defined to satisfyP2 = 1, consider the general case whenP2 =Q for some internal symmetry Q present in the theory. The desired parity operator would beP' =PQ−1/2. IfQ is part of a continuous symmetry group thenQ−1/2 exists, but if it is part of adiscrete symmetry then this element need not exist and such a redefinition may not be possible.[12]
The Standard Model exhibits a(−1)F symmetry, whereF is thefermionnumber operator counting how many fermions are in a state. Since all particles in the Standard Model satisfyF =B +L, the discrete symmetry is also part of theeiα(B +L) continuous symmetry group. If the parity operator satisfiedP2 = (−1)F, then it can be redefined to give a new parity operator satisfyingP2 = 1. But if the Standard Model is extended by incorporatingMajorananeutrinos, which haveF = 1 andB +L = 0, then the discrete symmetry(−1)F is no longer part of the continuous symmetry group and the desired redefinition of the parity operator cannot be performed. Instead it satisfiesP4 = 1 so the Majorana neutrinos would have intrinsic parities of±i.
In 1954, a paper byWilliam Chinowsky andJack Steinberger demonstrated that thepion has negative parity.[13]They studied the decay of an "atom" made from adeuteron (2
1H+
) and a negatively charged pion (π−
) in a state with zero orbitalangular momentum into twoneutrons ().
Neutrons arefermions and so obeyFermi–Dirac statistics, which implies that the final state is antisymmetric. Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function Since the orbital momentum changes from zero to one in this process, if the process is to conserve the total parity then the products of the intrinsic parities of the initial and final particles must have opposite sign. A deuteron nucleus is made from a proton and a neutron, and so using the aforementioned convention that protons and neutrons have intrinsic parities equal to they argued that the parity of the pion is equal to minus the product of the parities of the two neutrons divided by that of the proton and neutron in the deuteron, explicitly from which they concluded that the pion is apseudoscalar particle.
Although parity is conserved inelectromagnetism andgravity, it is violated in weak interactions, and perhaps, to some degree, instrong interactions.[14][15] The Standard Model incorporatesparity violation by expressing the weak interaction as achiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in charged weak interactions in the Standard Model. This implies that parity is not a symmetry of our universe, unless ahidden mirror sector exists in which parity is violated in the opposite way.
An obscure 1928 experiment, undertaken byR. T. Cox, G. C. McIlwraith, and B. Kurrelmeyer, had in effect reported parity violation inweak decays, but, since the appropriate concepts had not yet been developed, those results had no impact.[16] In 1929,Hermann Weyl explored, without any evidence, the existence of a two-component massless particle of spin one-half. This idea was rejected byPauli, because it implied parity violation.[17]
By the mid-20th century, it had been suggested by several scientists that parity might not be conserved (in different contexts), but without solid evidence these suggestions were not considered important. Then, in 1956, a careful review and analysis by theoretical physicistsTsung-Dao Lee andChen-Ning Yang[18] went further, showing that while parity conservation had been verified in decays by the strong orelectromagnetic interactions, it was untested in theweak interaction. They proposed several possible direct experimental tests. They were mostly ignored,[citation needed] but Lee was able to convince his Columbia colleagueChien-Shiung Wu to try it.[citation needed] She needed specialcryogenic facilities and expertise, so the experiment was done at theNational Bureau of Standards.
Wu,Ambler, Hayward, Hoppes, and Hudson (1957) found a clear violation of parity conservation in the beta decay ofcobalt-60.[19] As the experiment was winding down, with double-checking in progress, Wu informed Lee and Yang of their positive results, and saying the results need further examination, she asked them not to publicize the results first. However, Lee revealed the results to his Columbia colleagues on 4 January 1957 at a "Friday lunch" gathering of the Physics Department of Columbia.[20] Three of them,R. L. Garwin,L. M. Lederman, and R. M. Weinrich, modified an existing cyclotron experiment, and immediately verified the parity violation.[21] They delayed publication of their results until after Wu's group was ready, and the two papers appeared back-to-back in the same physics journal.
The discovery of parity violation explained the outstandingτ–θ puzzle in the physics ofkaons.
In 2010, it was reported that physicists working with theRelativistic Heavy Ion Collider had created a short-lived parity symmetry-breaking bubble inquark–gluon plasmas. An experiment conducted by several physicists in theSTAR collaboration, suggested that parity may also be violated in the strong interaction.[15] It is predicted that this local parity violation manifests itself bychiral magnetic effect.[22][23]
To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to anyhadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such asrho meson decay topions.
Footnotes
Citations
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