Discrete Symmetries

Parity and time reversal impose different restrictions on quantum systems. Parity reverses spatial position and is unitary; time reversal reverses motion and is antiunitary, so it also conjugates complex amplitudes. That distinction determines whether a symmetry can force an energy doublet. This chapter develops the nonrelativistic examples, then introduces charge conjugation, CPT, and the matrix symmetries used in quantum matter.

Helpful background. Symmetry principles distinguishes a transformation from invariance of a Hamiltonian; spinors supplies the Pauli matrices used for spin-$1/2$ time reversal.

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Question	Read next
How does inversion act on states and vectors?	Parity
Why must reversed motion conjugate amplitudes?	Time reversal and antiunitary time reversal
How does the representation depend on spin?	Spinless particles and spin-$1/2$ particles
When is a doublet protected?	Kramers degeneracy
How do the rules restrict a concrete model?	Discrete symmetries in Hamiltonians
What new assumptions enter relativistic theory?	Charge conjugation and CPT
How are time-reversal and particle–hole constraints classified?	Symmetry classification

Discrete operations and fixed external parameters

Here “discrete” refers to the specified physical symmetry group. Inversion is not connected to the identity within the group of ordinary three-dimensional spatial rotations, and antiunitary time reversal cannot be continuously connected to the identity through unitaries. A finite subgroup can also consist of rotations that lie within a larger continuous group. The term does not claim that a unitary parity matrix has no mathematical logarithm; such a logarithm need not generate a continuous physical spatial symmetry.

For a time-independent Hamiltonian, test $SHS^{-1}=H$, including its domain. Distinguish that from a relation between different Hamiltonians. A magnetic-field family may satisfy

$$\Theta H(\mathbf B)\Theta^{-1}=H(-\mathbf B),$$

while the fixed Hamiltonian at nonzero $\mathbf B$ generally breaks time reversal. The apparatus and external field cannot silently be reversed halfway through a fixed-field symmetry test.

Parity labels do not force doublets

For an ordinary scalar wavefunction, $(\Pi\psi)(\mathbf x)=\psi(-\mathbf x)$, and $\Pi^2=I$ gives eigenvalues $\pm1$. An even potential with inversion-invariant domain and boundary conditions preserves parity. Energy states can then be chosen even or odd, but there is no requirement that an even state and an odd state have equal energy.

Position and momentum are polar vectors and reverse sign under inversion. Their cross product $\mathbf L=\mathbf X\times\mathbf P$ is axial and remains unchanged; spin is also parity even. Consequently, $\mathbf L\cdot\mathbf S$ can preserve parity. An operator’s rotational rank alone does not tell its parity. Parity selection rules applies this distinction to matrix elements.

Antiunitarity reverses the complex phase of evolution

An antiunitary operator obeys

$$\Theta(a|\psi\rangle+b|\phi\rangle) =a^*\Theta|\psi\rangle+b^*\Theta|\phi\rangle,$$

and $\langle\Theta\phi|\Theta\psi\rangle=\langle\phi|\psi\rangle^*$. It preserves transition probabilities but sends $i$ to $-i$. This is required when position is unchanged and momentum reverses: $[X,-P]=-i\hbar I$, consistent with antiunitary conjugation of the canonical commutator.

If $\Theta H\Theta^{-1}=H$, then

$$\Theta e^{-iHt/\hbar}\Theta^{-1}=e^{+iHt/\hbar}.$$

Thus $\Theta|\psi(-t)\rangle$ solves the same autonomous Schrödinger equation. Applying only $t\mapsto-t$ to a wavefunction omits the necessary conjugation.

In a chosen basis, write $\Theta=UK$ with $U$ unitary and $K$ componentwise complex conjugation. Then $\Theta^2=UU^*$, not $U^2$. Rephasing an antiunitary does not change its square because the second phase is conjugated.

Spin determines which time-reversal square applies

For a spinless particle without magnetic coupling, the usual position representation has $\Theta=K$ and $\Theta^2=I$. A real potential with a compatible time-reversal-invariant domain admits this symmetry. A nondegenerate eigenfunction can be rephased to be real; no doublet follows.

In the spin-$1/2$ $S_z$ basis, choose

$$\Theta=-i\sigma_yK,\qquad \Theta\begin{pmatrix}\alpha\\\beta\end{pmatrix} =\begin{pmatrix}-\beta^*\\\alpha^*\end{pmatrix}.$$

This reverses every spin component and has $\Theta^2=-I$. In particular, a single-spin Hamiltonian $H=aI+\mathbf b\cdot\boldsymbol\sigma$ with real coefficients is time-reversal invariant at fixed parameters only if $\mathbf b=0$. The two energies must coincide.

The square must be computed on the complete sector in question. Two independent spin-$1/2$ factors have product time reversal with square $+I$. Their rotationally and time-reversal-invariant exchange Hamiltonian $A\mathbf S_1\cdot\mathbf S_2$, with $A>0$, has a nondegenerate singlet ground state. There is no conflict with the single-spin conclusion.

Kramers pairing requires a negative square

Let $H$ be self-adjoint and $\Theta$ antiunitary. Suppose

$$\Theta H\Theta^{-1}=H$$

and, on the sector under consideration,

$$\Theta^2=-I.$$

If

$$H|\psi\rangle=E|\psi\rangle,$$

then

$$H\Theta|\psi\rangle = E\Theta|\psi\rangle.$$

The energy is real because $H$ is self-adjoint. Antiunitarity and the negative square also imply

$$\langle\psi|\Theta\psi\rangle=0.$$

Thus $|\psi\rangle$ and $\Theta|\psi\rangle$ are distinct orthogonal states with the same energy. Every finite-dimensional normalizable energy eigenspace in that sector has even dimension. The theorem pairs normalizable eigenvectors; continuum scattering states require their own spectral treatment.

The theorem needs every hypothesis:

• $\Theta$ is antiunitary;
• $\Theta^2=-I$ on the relevant sector;
• the fixed Hamiltonian is invariant;
• the transformed state remains in the same physical sector.

Spinless time reversal with $\Theta^2=+I$ does not force a doublet. A fixed magnetic field breaks the hypothesis $\Theta H\Theta^{-1}=H$. A random twofold degeneracy need not be a Kramers degeneracy.

Spin–orbit coupling does not automatically break time reversal. Since

$$\Theta\mathbf L\Theta^{-1} = -\mathbf L, \qquad \Theta\mathbf S\Theta^{-1} = -\mathbf S,$$

the scalar product is even:

$$\Theta \left( \mathbf L\cdot\mathbf S \right) \Theta^{-1} = \mathbf L\cdot\mathbf S.$$

Spin–orbit interactions can reorganize levels while preserving Kramers pairs in a negative-square sector.

Charge conjugation, CPT, and particle–hole structure

Charge conjugation maps charge-$q$ sectors to charge-$-q$ sectors. An electron-only nonrelativistic space is not closed under the physical particle–antiparticle operation because it contains no positron sector. Charge conservation, $[H,Q]=0$, is a different statement from invariance under charge conjugation. Complex conjugating a one-particle wavefunction is likewise not, by itself, the full field-theoretic operation.

The CPT theorem belongs to local Lorentz-covariant quantum field theory with its usual spectrum and Hilbert-space assumptions. It does not follow by multiplying three symbols in a finite-dimensional model, and it does not imply separate $C$, $P$, or $T$ invariance. The CPT introduction and QFT bridge identify the required framework; Streater and Wightman provide the axiomatic development.

In noninteracting band and Bogoliubov–de Gennes models, time-reversal, particle–hole, and chiral constraints instead organize the ten Altland–Zirnbauer classes. Particle–hole structure in Nambu space is not simply relativistic charge conjugation. A class alone does not specify a topological phase: the spatial dimension, gap, locality, and allowed perturbations also matter. Continue with symmetry classification for explicit matrices and examples.

Exercises

1. Determine the parity and time-reversal character of the Hermitian dilation operator

$$D = \frac12 \left( \mathbf X\cdot\mathbf P +\mathbf P\cdot\mathbf X \right).$$

Solution

Under parity, both $\mathbf X$ and $\mathbf P$ reverse. Each product therefore remains unchanged:

$$\Pi D\Pi^{-1}=D.$$

The operator is parity even.

Under time reversal, $\mathbf X$ is even and $\mathbf P$ is odd. The coefficient $1/2$ is real, so antiunitarity introduces no additional sign. Both ordered products reverse:

$$\Theta D\Theta^{-1}=-D.$$

Thus $D$ is time-reversal odd. A real term proportional to $D$ can preserve parity while breaking time reversal.

2. Let a spinless two-state Hamiltonian be

$$H = aI +b_x\sigma_x +b_y\sigma_y +b_z\sigma_z,$$

with real coefficients and time reversal represented by $K$. Which coefficients are constrained by time-reversal invariance?

Solution

In the standard Pauli basis, $I$, $\sigma_x$, and $\sigma_z$ are real, while $\sigma_y$ is imaginary. Therefore

$$KHK^{-1} = aI +b_x\sigma_x - b_y\sigma_y +b_z\sigma_z.$$

The condition $KHK^{-1}=H$ requires

$$b_y=0.$$

The real coefficients $a$, $b_x$, and $b_z$ remain unconstrained by this symmetry. No Kramers degeneracy follows because $K^2=+I$.

3. Let $\Theta$ be antiunitary with $\Theta^2=-I$, and suppose

$$\Theta H\Theta^{-1}=H.$$

Show that a normalizable energy eigenstate and its time-reversed partner have the same energy and are orthogonal.

Solution

If

$$H|\psi\rangle=E|\psi\rangle,$$

then

$$\begin{aligned} H\Theta|\psi\rangle &= \Theta H|\psi\rangle \\ &= \Theta\left( E|\psi\rangle \right) \\ &= E^*\Theta|\psi\rangle. \end{aligned}$$

Self-adjointness makes $E$ real, so the partner has the same energy.

Let

$$c=\langle\psi|\Theta\psi\rangle.$$

Antiunitarity gives

$$\langle\Theta\psi|\Theta^2\psi\rangle = c^*.$$

Using $\Theta^2=-I$, the left side is also

$$-\langle\Theta\psi|\psi\rangle = -c^*.$$

Hence $c^*=-c^*$ and $c=0$. The two states are orthogonal and form a Kramers pair.

References

• A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Physical Review B 55, 1142–1161 (1997). DOI.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries,” Reviews of Modern Physics 88, 035005 (2016). DOI.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press, 2000.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, translated by J. J. Griffin, Academic Press, 1959.