Parity (physics)
In
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
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 of P (in any number of dimensions) has
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.
Simple symmetry relations
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.
The projective representations of any group are isomorphic to the ordinary representations of a
If one adds to this a classification by parity, these can be extended, for example, into notions of
- scalars (P = +1) and pseudoscalars(P = −1) which are rotationally invariant.
- vectors (P = −1) and axial vectors (also called pseudovectors) (P = +1) which both transform as vectors under rotation.
One can define reflections such as
which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing x-, y-, and z-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 the abelian group due to the relation . All Abelian groups have only one-dimensional
Representations of O(3)
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 the group homomorphism which defines the representation. For a matrix
- scalars: , the trivial representation
- pseudoscalars:
- vectors: , the fundamental representation
- pseudovectors:
When the representation is restricted to , scalars and pseudoscalars transform identically, as do vectors and pseudovectors.
Classical mechanics
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 an
In classical
Effect of spatial inversion on some variables of classical physics
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 the number of dimensions of space is either an odd or even number. The categories of odd or even given below for the parity 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.
Odd
Classical variables whose signs flip when inverted in space inversion are predominantly vectors. They include:
- , helicity
- , magnetic flux
- , the positionof a particle in three-space
- , the velocity of a particle
- , the acceleration of the particle
- , the linear momentumof a particle
- , mass flow[a]
- , the forceexerted on a particle
- , electric current density
- , the electric field
- , the electric displacement field
- , electric polarization
- , electromagnetic vector potential
- , the Poynting vector (flow of electromagnetic power).
Even
Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:
- , the time when an event occurs
- , the mass of a particle
- , the energy of a particle
- , power (rate of work done)
- , electric charge density
- , scalar electric potential (voltage)
- , energy density of the electromagnetic field
- , the orbital and spin) (axial vector)
- , the magnetic field (axial vector)
- , the auxiliary magnetic field
- , magnetization
- , the Maxwell stress tensor.
- All masses, charges, coupling constants, and other scalar physical constants, except those associated with the weak force.
Quantum mechanics
Possible eigenvalues
In quantum mechanics, spacetime transformations act on
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 are even 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 for gerade (German: even) and odd states by a subscript u for ungerade (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 is centrosymmetric (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 the beta decay of nuclei) because the weak nuclear interaction violates parity.[4]
The parity of the states of a particle moving in a spherically symmetric external field is determined by the angular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.[3]
Consequences of parity symmetry
When parity generates the Abelian 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 are invariant (symmetric) under a parity transformation if commutes 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:
- If and have the same parity, then where is the position operator.
- For a state of orbital angular momentum with z-axis projection , then .
- If , then atomic dipole transitions only occur between states of opposite parity.[5]
- If , then a non-degenerate eigenstate of is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of is either invariant to or is changed in sign by .
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, the
Many-particle systems: atoms, molecules, nuclei
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.
Atoms
Atomic orbitals have parity (−1)ℓ, where the exponent ℓ is the azimuthal 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 symbol 4So, 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 is 4P (without an o superscript).[6]
Molecules
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 by Longuet-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 homonuclear
Nuclei
In atomic nuclei, the state of each nucleon (proton or neutron) has even or odd parity, and nucleon configurations can be predicted using the nuclear 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, the isotopes of oxygen include 17O(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.[11]
Quantum field theory
If one can show that the
To show that quantum 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 that canonical 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]
A straightforward extension of these arguments to scalar field theories shows that scalars have even parity. That is, , since
With fermions, there is a slight complication because there is more than one spin group.
Parity in the Standard Model
Fixing the global symmetries
Applying the parity operator twice leaves the coordinates unchanged, meaning that P2 must act as one of the internal symmetries of the theory, at most changing the phase of a state.[12] For example, the Standard Model has three global U(1) symmetries with charges equal to the baryon number B, the lepton number L, and the electric charge Q. Therefore, the parity operator satisfies P2 = eiαB+iβL+iγQ for some choice of α, β, and γ. This operator is also not unique in that a new parity operator P' can always be constructed by multiplying it by an internal symmetry such as P' = P eiαB for some α.
To see if the parity operator can always be defined to satisfy P2 = 1, consider the general case when P2 = Q for some internal symmetry Q present in the theory. The desired parity operator would be P' = PQ−1/2. If Q is part of a continuous symmetry group then Q−1/2 exists, but if it is part of a discrete symmetry then this element need not exist and such a redefinition may not be possible.[13]
The Standard Model exhibits a (−1)F symmetry, where F is the fermion number operator counting how many fermions are in a state. Since all particles in the Standard Model satisfy F = B + L, the discrete symmetry is also part of the eiα(B + L) continuous symmetry group. If the parity operator satisfied P2 = (−1)F, then it can be redefined to give a new parity operator satisfying P2 = 1. But if the Standard Model is extended by incorporating Majorana neutrinos, which have F = 1 and B + 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 satisfies P4 = 1 so the Majorana neutrinos would have intrinsic parities of ±i.
Parity of the pion
In 1954, a paper by William Chinowsky and Jack Steinberger demonstrated that the pion has negative parity.[14] They studied the decay of an "atom" made from a
into two neutrons ().Neutrons are fermions and so obey Fermi–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 a
Parity violation
Although parity is conserved in electromagnetism and gravity, it is violated in weak interactions, and perhaps, to some degree, in strong interactions.[15][16] The Standard Model incorporates parity violation by expressing the weak interaction as a chiral 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 a hidden mirror sector exists in which parity is violated in the opposite way.
An obscure 1928 experiment, undertaken by
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 physicists
The discovery of parity violation explained the outstanding τ–θ puzzle in the physics of kaons.
In 2010, it was reported that physicists working with the
Intrinsic parity of hadrons
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 any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as rho meson decay to pions.
See also
- C-symmetry
- CP violation
- Electroweak theory
- Mirror matter
- Molecular symmetry
- T-symmetry
References
Footnotes
Citations
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Bransden, B. H.; Joachain, C. J. (2003). Physics of Atoms and Molecules (2nd ed.). ISBN 978-0-582-35692-4.
- ^ NIST Atomic Spectrum Database To read the nitrogen atom energy levels, type "N I" in the Spectrum box and click on Retrieve data.
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- ISBN 0-7503-0941-5[1]
- ^ Pique, J. P.; et al. (1984). "Hyperfine-Induced Ungerade-Gerade Symmetry Breaking in a Homonuclear Diatomic Molecule near a Dissociation Limit:I at the − Limit". Phys. Rev. Lett. 52 (4): 267–269. .
- ^ Critchley, A. D. J.; et al. (2001). "Direct Measurement of a Pure Rotation Transition in H". Phys. Rev. Lett. 86 (9): 1725–1728. PMID 11290233.
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- ^ Chinowsky, W.; Steinberger, J. (1954). "Absorption of Negative Pions in Deuterium: Parity of the Pion". .
- ^ Gardner, Martin (1969) [1964]. The Ambidextrous Universe; Left, Right, and the Fall of Parity (rev. ed.). New York: New American Library. p. 213.
- ^ a b
Muzzin, S.T. (19 March 2010). "For one tiny instant, physicists may have broken a law of nature". PhysOrg. Retrieved 5 August 2011.
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Roy, A. (2005). "Discovery of parity violation". S2CID 124880732.
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Caijian, Jiang (1 August 1996). Wu jian xiong-wu li ke xue de si yi fu ren 吳健雄: 物理科學的第一夫人 [Wu Jianxiong: The first lady of physical sciences] (in Chinese). 江才健 (author/biographer). 時報文化出版企業股份有限公司 (Times Culture Publishing Enterprise). p. 216. ISBN 957-13-2110-9
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Kharzeev, D.E.; Liao, J. (2 January 2019). "Isobar collisions at RHIC to test local parity violation in strong interactions". Nuclear Physics News. 29 (1): 26–31. S2CID 133308325.
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Zhao, Jie; Wang, Fuqiang (July 2019). "Experimental searches for the chiral magnetic effect in heavy-ion collisions". Progress in Particle and Nuclear Physics. 107: 200–236. S2CID 181517015.
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