In physics, 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 or point inversion):
P
:
(
x
y
z
)
↦
(
−
x
−
y
−
z
)
.
{\displaystyle \mathbf {P} :{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\-y\\-z\end{pmatrix}}.}
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 transformation, a phenomenon known as P-symmetry. As established by the Wu experiment conducted at the US National Bureau of Standards by Chinese-American scientist Chien-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 in radioactive 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 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.
Contents
Simple symmetry relations
Under rotations, classical geometrical objects can be classified into scalars, vectors, and other tensors of rank 2 or higher. (Technically, scalars, such as electrical potential are tensors of rank 0; vectors are tensors of rank 1; tensors of rank 2 or higher are called "tensors" both in the generic sense, as are scalars and vectors, and in the specific sense of being order higher than vectors. At the low-energy quantum level, gravitational force is a second order tensor.) 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 central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO(3), are ordinary representations of the special unitary group SU(2). Projective representations of the rotation group that are not representations are called spinors 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
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
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
ρ
{\displaystyle \ \rho \ }
which defines the representation. For a matrix
R
∈
O
(
3
)
,
{\displaystyle \ R\in {\text{O}}(3)\ ,}
When the representation is restricted to
S
O
(
3
)
,
{\displaystyle \ {\mathsf {SO}}(3)\ ,}
both scalars and pseudoscalars transform identically, as do vectors and pseudovectors.
Classical mechanics
Newton's equation of motion
F
=
m
a
{\displaystyle \ \mathbf {F} =m\ \mathbf {a} \ }
(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
L
{\displaystyle \ \mathbf {L} \ }
is an axial vector,
L
=
r
×
p
P
^
(
L
)
=
(
−
r
)
×
(
−
p
)
=
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 under space inversion are predominantly vectors. They include:
Even
Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:
Quantum mechanics
Possible eigenvalues
In quantum mechanics, spacetime transformations act on quantum states. The parity transformation,
P
^
{\displaystyle {\hat {\mathcal {P}}}}
, is a unitary operator, in general acting on a state
ψ
{\displaystyle \psi }
as follows:
P
^
ψ
(
r
)
=
e
i
ϕ
/
2
ψ
(
−
r
)
{\displaystyle {\hat {\mathcal {P}}}\,\psi {\left(r\right)}=e^{{i\phi }/{2}}\psi {\left(-r\right)}}
.
One must then have
P
^
2
Consequences of parity symmetry
When parity generates the Abelian group
Z
2
{\displaystyle \mathbb {Z} _{2}}
, 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
P
^
{\displaystyle {\hat {\mathcal {P}}}}
commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any scalar potential, i.e.,
V
=
V
(
r
)
{\displaystyle V=V{\left(r\right)}}
, hence the potential is spherically symmetric. The following facts can be easily proven:
If
|
φ
⟩
{\displaystyle |\varphi \rangle }
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).
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) 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 diatomic molecules as well as certain symmetric molecules such as ethylene, benzene, xenon tetrafluoride and sulphur hexafluoride. For centrosymmetric molecules, the point group contains the operation i which is not to be confused with the parity operation. The operation i involves the inversion of the electronic and vibrational displacement coordinates at the nuclear centre of mass. For centrosymmetric molecules the operation i 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 operation i, or they are changed in sign by i. The former are denoted by the subscript g and are called gerade, while the latter are denoted by the subscript u and are called ungerade. The complete electromagnetic Hamiltonian of a centrosymmetric molecule
does not commute with the point group inversion operation i because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of g and u vibronic states (called ortho-para mixing) and give rise to ortho-para transitions
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 oxygen isotope 17O(5/2+) has spin 5/2 and even parity. 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.
Quantum field theory
If one can show that the vacuum state is invariant under parity,
P
^
|
0
⟩
=
|
0
⟩
{\displaystyle {\hat {\mathcal {P}}}\left|0\right\rangle =\left|0\right\rangle }
, the Hamiltonian is parity invariant
[
H
^
,
P
^
]
{\displaystyle \left[{\hat {H}},{\hat {\mathcal {P}}}\right]}
and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction.
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:
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. 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.
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.
They studied the decay of an "atom" made from a deuteron (21H+) and a negatively charged pion (π− ) in a state with zero orbital angular momentum
L
=
0
{\displaystyle ~\mathbf {L} ={\boldsymbol {0}}~}
into two neutrons (
n
{\displaystyle n}
).
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
L
=
1
.
{\displaystyle ~L=1~.}
The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function
(
−
1
)
L
.
{\displaystyle ~\left(-1\right)^{L}~.}
Parity violation
Although parity is conserved in electromagnetism and gravity, it is violated in weak interactions, and perhaps, to some degree, in strong interactions. 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 R. T. Cox, G. C. McIlwraith, and B. Kurrelmeyer, had in effect reported parity violation in weak decays, but, since the appropriate concepts had not yet been developed, those results had no impact. In 1929, Hermann Weyl explored, without any evidence, the existence of a two-component massless particle of spin one-half. This idea was rejected by Pauli, because it implied parity violation.
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 Tsung-Dao Lee and Chen-Ning Yang went further, showing 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. They were mostly ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards.
Wu, Ambler, Hayward, Hoppes, and Hudson (1957) found a clear violation of parity conservation in the beta decay of cobalt-60. 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. Three of them, R. L. Garwin, L. M. Lederman, and R. M. Weinrich, modified an existing cyclotron experiment, and immediately verified the parity violation. 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.
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.