Discrete Symmetries: Parity and Time Reversal

Alongside the continuous symmetries whose generators give conservation laws, quantum mechanics has crucial discrete symmetries — most importantly parity (spatial inversion) and time reversal. These cannot be built up from infinitesimal transformations, and time reversal introduces a genuinely new structure: an antiunitary operator. Their consequences range from selection rules to the Kramers degeneracy of half-integer-spin systems.

1. Parity

Parity $\hat{\Pi }$ inverts spatial coordinates, $\hat{\Pi }\hat{\mathbf{r}}{\hat{\Pi }}^{-1}=-\hat{\mathbf{r}}$, $\hat{\Pi }\hat{\mathbf{p}}{\hat{\Pi }}^{-1}=-\hat{\mathbf{p}}$, while leaving angular momentum $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times \hat{\mathbf{p}}$ invariant (it is an axial vector). Since ${\hat{\Pi }}^2=\hat{1}$, its eigenvalues are $\pm 1$ (even/odd parity). When $[\hat{\Pi },\hat{H}]=0$ — true for any central potential — stationary states have definite parity: the spherical harmonics satisfy $\hat{\Pi }{Y}_{\ell }^m=(-1{)}^{\ell }{Y}_{\ell }^m$, so orbitals alternate even/odd with $\ell$.

Parity yields powerful selection rules: a matrix element $⟨f|\hat{O}|i⟩$ vanishes unless the parities multiply to that of $\hat{O}$. Since the electric-dipole operator $\hat{\mathbf{r}}$ is odd, dipole transitions connect only states of opposite parity ($\Delta \ell$ odd) — Laporte's rule (see time-dependent perturbation theory). Parity is conserved by the electromagnetic and strong interactions but violated by the weak interaction (Wu experiment, 1957) — one of the deepest asymmetries in nature.

2. Time Reversal Is Antiunitary

Time reversal $\hat{\Theta }$ reverses motion: $\hat{\Theta }\hat{\mathbf{r}}{\hat{\Theta }}^{-1}=\hat{\mathbf{r}}$ but $\hat{\Theta }\hat{\mathbf{p}}{\hat{\Theta }}^{-1}=-\hat{\mathbf{p}}$ and $\hat{\Theta }\hat{\mathbf{J}}{\hat{\Theta }}^{-1}=-\hat{\mathbf{J}}$. Consistency with the canonical commutator $[\hat{x},\hat{p}]=i\hslash$ forces $\hat{\Theta }$ to also complex-conjugate scalars:

$$\hat{\Theta }(i){\hat{\Theta }}^{-1}=-i.$$

By Wigner's theorem a symmetry is unitary or antiunitary; time reversal is the physical realization of the antiunitary case. For spinless particles $\hat{\Theta }=\hat{K}$ (complex conjugation) in the position basis, so time reversal simply sends $\psi (\mathbf{r},t)\to {\psi }^{\ast }(\mathbf{r},-t)$. Antiunitarity is why there is no conserved "time-reversal charge" — the continuous-symmetry ⇒ conservation-law argument does not apply to $\hat{\Theta }$.

3. Kramers Degeneracy

For a particle with spin, $\hat{\Theta }=e^{-i\pi {\hat{S}}_y/\hslash }\hat{K}$, and a direct computation gives

$${\hat{\Theta }}^2=(-1{)}^{2s}\hat{1}=\{\begin{array}{ll}+\hat{1},& s\text{ integer},\\ -\hat{1},& s\text{ half-integer}.\end{array}$$

The half-integer case has a striking consequence — Kramers' theorem: in any time-reversal-invariant system with an odd number of half-integer-spin particles, every energy level is at least doubly degenerate, and no purely electric (time-reversal-even) perturbation can split it. A magnetic field, which breaks time reversal, is required to lift the degeneracy. This protects the two-fold degeneracy of electronic states in crystals without magnetic order and underlies the stability of Kramers doublets exploited in spin qubits and topological insulators.

See also

• Symmetries & conservation — continuous symmetries and Wigner's theorem.
• Spin — the ${\hat{\Theta }}^2=-1$ property of half-integer spin.
• Time-dependent perturbation theory — parity selection rules.