Parity (physics)

Parity is the transformation property of a physical quantity or state under spatial inversion: the replacement of every spatial coordinate by its negative. In quantum mechanics, it also denotes a quantum number distinguishing states that remain unchanged from those that change sign under inversion. Parity conservation means that a system’s dynamics are invariant under this transformation. Electromagnetic and strong interactions conserve parity within experimental limits, whereas the weak interaction violates it. (damtp.cam.ac.uk)

Spatial inversion and physical quantities

In three spatial dimensions, parity acts as

while leaving time unchanged. It reverses the handedness of a coordinate system and cannot be achieved by a spatial rotation alone. Reflection in a single plane differs from full inversion by a rotation; consequently, for rotationally invariant laws, mirror symmetry and parity symmetry are equivalent tests. The transformation must apply to the complete physical configuration, including relevant external fields. (farside.ph.utexas.edu)

Physical quantities behave differently under inversion. Polar vectors, including position and momentum, reverse direction. Axial vectors, or pseudovectors, do not: orbital angular momentum, $\mathbf L=\mathbf r\times\mathbf p$, is unchanged because both factors reverse. Spin angular momentum likewise remains unchanged. In electromagnetism, the electric field is a polar vector, whereas the magnetic field is axial. A scalar is unchanged; a pseudoscalar changes sign. Thus $\mathbf S\cdot\mathbf p$ is parity-odd, providing a useful quantity for detecting parity violation. (farside.ph.utexas.edu)

The quantum parity operator

For a spinless particle with no additional internal transformation, the parity operator acts on its wave function as

Applying inversion twice restores the original function, so $\hat P^2=1$. This operator is unitary and Hermitian, with eigenvalues $+1$ and $-1$. A parity eigenstate therefore satisfies

The two cases are called even and odd parity. A general wave function need not possess definite parity, although it can be decomposed into even and odd parts. (damtp.cam.ac.uk)

If parity commutes with the Hamiltonian operator,

the evolution preserves parity. Energy eigenstates can then be chosen to have definite parity, including within degenerate energy eigenspaces. For a particle in a scalar potential, the condition $V(\mathbf r)=V(-\mathbf r)$ supplies the required inversion symmetry. Parity is a discrete symmetry: its conservation should not be confused with the continuous-symmetry conservation laws ordinarily associated with Noether’s theorem. (damtp.cam.ac.uk)

Orbital and intrinsic parity

For a central potential, the angular dependence of a state is described by spherical harmonics, which obey

Its orbital parity is therefore $(-1)^\ell$: even orbital angular momentum gives even parity, and odd orbital angular momentum gives odd parity. This result underlies the classification of atomic orbitals and many nuclear states. (damtp.cam.ac.uk)

Particles may also carry intrinsic parity, associated with their internal transformation rather than their orbital motion. For two particles of intrinsic parities $\eta_1,\eta_2$ and relative orbital angular momentum $\ell$, the total parity is

Parity is thus a multiplicative quantum number. Composite-particle assignments combine constituent intrinsic parities with orbital factors; parity conservation constrains possible production and decay channels. (damtp.cam.ac.uk)

Selection rules

Parity provides selection rules for quantum transitions. If an operator $\hat O$ has parity $\epsilon_O=\pm1$, its matrix element between states of definite parity can be nonzero only when

This condition follows by applying inversion to the matrix element. It is necessary, not sufficient: angular momentum and other symmetries may impose additional restrictions. (damtp.cam.ac.uk)

The electric dipole operator is parity-odd, so electric-dipole transitions connect states of opposite parity. This rule is central to spectroscopy. A transition forbidden in the electric-dipole approximation is not necessarily impossible; weaker channels or symmetry-breaking perturbations may permit it. Such transitions do not, by themselves, demonstrate fundamental parity violation. (damtp.cam.ac.uk)

Discovery of parity violation

In 1956, Tsung-Dao Lee and Chen Ning Yang examined the evidence for parity conservation and identified the absence of decisive tests in weak processes. They proposed experiments rather than assuming that the symmetry established for other interactions applied universally. (nobelprize.org)

Chien-Shiung Wu and collaborators at the US National Bureau of Standards tested beta decay using cobalt-60 nuclei aligned at very low temperature. The emitted electrons showed an unequal distribution along and against the nuclear spin. Because inversion reverses electron momentum but not spin, this asymmetry contradicted parity conservation. Results emerged in late December 1956, and the experimental paper appeared on February 15, 1957. Lee and Yang received the 1957 Nobel Prize in Physics for their investigation of parity laws. (nist.gov)

Parity in fundamental interactions

In the Standard Model, charged weak currents couple to left-chiral fermion fields, rather than treating left- and right-chiral fields symmetrically. Weak neutral currents can also produce parity-violating effects. These include atomic parity violation and differences between scattering cross sections for opposite electron helicities, measured through asymmetries such as

Such measurements test electroweak couplings. (pdg.lbl.gov)

Parity violation is distinct from CP violation, which concerns the combined operation of charge conjugation and inversion. Neither implies failure of the CPT theorem: the combined transformation of charge conjugation, parity, and time reversal remains an exact symmetry under the standard assumptions of local, Lorentz-invariant quantum field theory with a suitable Hermitian Hamiltonian. (pdgweb.lbl.gov)