Discrete Symmetries

Discrete symmetries are transformations not generated by continuously varying a parameter from the identity. Spatial inversion and time reversal are the central nonrelativistic examples. Charge conjugation and CPT point beyond fixed-particle quantum mechanics toward relativistic field theory, while particle–hole and chiral structures lead toward modern classifications of quantum matter.

The word discrete does not mean simple. Parity is represented by a unitary operator, but time reversal is antiunitary. That distinction changes how complex coefficients transform, how Hamiltonian matrices are constrained, and whether a symmetry can force degeneracy.

A reliable analysis asks four questions:

1. On which Hilbert space or sector does the transformation act?
2. Is the implementing operator unitary or antiunitary?
3. Which observables and external parameters change sign?
4. Does the fixed Hamiltonian remain invariant, or only a parameterized family transform covariantly?

This chapter answers those questions for parity and time reversal, derives the $T^2=\pm I$ distinction, and explains why $T^2=-I$ produces Kramers pairs. It then marks the canonical boundary between nonrelativistic discrete symmetries, field-theoretic $CPT$, and condensed-matter symmetry classes.

Canonical Boundaries

This page owns the chapter map and the comparison among the discrete transformations. Detailed derivations and specialized applications remain at their canonical homes.

Topic	Canonical home	Role here
spatial inversion	Parity	defines $\Pi$, parity eigenstates, and the action on $\mathbf X$, $\mathbf P$, $\mathbf L$, and $\mathbf S$
conceptual time reversal	Time Reversal	separates the Hilbert-space operation from reversing the time parameter
antiunitary derivation	Antiunitary Time Reversal	owns the Schrödinger-equation, commutator, and propagator arguments
scalar-particle representation	Time Reversal for Spinless Particles	develops $\Theta=K$, real Hamiltonians, currents, and $\Theta^2=+I$
spinor representation	Time Reversal for Spin-1/2 Particles	develops $\Theta=-i\sigma_yK$, spin reversal, and $\Theta^2=-I$
protected doublets	Kramers Degeneracy	states and proves the antiunitary degeneracy theorem
particle–antiparticle transformation	Charge Conjugation Preview	explains why $C$ needs charge-conjugate sectors and belongs mainly to relativistic theory
relativistic theorem boundary	CPT Preview	states the theorem’s field-theoretic scope and assumptions
term-by-term model test	Discrete Symmetries in Hamiltonians	tests potentials, spin terms, matrix Hamiltonians, and external fields
tenfold-way vocabulary	Symmetry Classification Preview	introduces time-reversal, particle–hole, and chiral constraints without duplicating quantum matter

The general theory of unitary and antiunitary quantum symmetries belongs to Symmetry Principles. The detailed parity-zero argument belongs to Parity Selection Rules. Relativistic field-theory development belongs beyond this chapter; the transition is mapped in From Discrete Symmetries to CPT.

A Discrete Transformation Is Not Automatically a Symmetry

An operator $S$ may define a meaningful transformation even when a particular Hamiltonian does not preserve it. The fixed-Hamiltonian symmetry test is

$$SHS^{-1}=H.$$

If the Hamiltonian depends on external parameters $\lambda$, a weaker covariance statement may hold:

$$S H(\lambda) S^{-1} = H(S\lambda).$$

These statements answer different questions. For example, time reversal maps a magnetic-field Hamiltonian according to

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

The family is covariant under field reversal. A single Hamiltonian with a fixed nonzero $\mathbf B$ is generally not time-reversal invariant.

The distinction matters whenever an electric field, magnetic field, flux, rotation rate, magnetization, or other background selects a direction. One must say whether the background is transformed as part of the physical system or held fixed as part of the experimental setup.

Unitary and Antiunitary Implementations

A unitary operator $U$ is linear:

$$U\left( a|\psi\rangle+b|\phi\rangle \right) = aU|\psi\rangle+bU|\phi\rangle.$$

An antiunitary operator $\Theta$ is antilinear:

$$\begin{aligned} \Theta\left( a|\psi\rangle+b|\phi\rangle \right) &= a^*\Theta|\psi\rangle \\ &\quad+ b^*\Theta|\phi\rangle. \end{aligned}$$

It preserves transition probabilities through the conjugated inner-product relation

$$\langle\Theta\phi|\Theta\psi\rangle = \langle\phi|\psi\rangle^*.$$

In particular,

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

Parity is unitary and leaves scalar $i$ unchanged. Time reversal is antiunitary and conjugates it. This is why the two operations cannot be treated as merely different sign tables.

In a chosen orthonormal basis, every antiunitary operator can be written

$$\Theta=UK,$$

where $U$ is unitary and $K$ complex-conjugates components in that basis. The decomposition is useful but basis-dependent. The full antiunitary operator $\Theta$, not $K$ by itself, has invariant physical meaning.

Parity: Reverse Space

Parity sends

$$\mathbf x\mapsto-\mathbf x$$

while leaving time unchanged. For a spinless scalar wavefunction,

$$(\Pi\psi)(\mathbf x) = \psi(-\mathbf x).$$

Its defining operator actions are

$$\Pi\mathbf X\Pi^{-1} = -\mathbf X, \qquad \Pi\mathbf P\Pi^{-1} = -\mathbf P.$$

Applying inversion twice gives

$$\Pi^2=I$$

in the ordinary scalar setting, so parity eigenvalues are

$$\pi=\pm1.$$

The corresponding states are even or odd:

$$\Pi|\psi\rangle = \pi|\psi\rangle.$$

In position space,

$$\psi(-\mathbf x) = \pi\psi(\mathbf x).$$

For

$$H = \frac{\mathbf P^2}{2m} +V(\mathbf X),$$

parity is a symmetry when

$$V(-\mathbf x)=V(\mathbf x).$$

If an energy eigenvalue is nondegenerate, its eigenstate can then be chosen with definite parity. In a degenerate eigenspace, one diagonalizes $\Pi$ within that subspace; arbitrary linear combinations need not have definite parity.

Polar and axial vectors

Orbital angular momentum is

$$\mathbf L = \mathbf X\times\mathbf P.$$

Both factors reverse under parity, so

$$\Pi\mathbf L\Pi^{-1} = \mathbf L.$$

Spin is likewise an axial vector in ordinary nonrelativistic quantum mechanics:

$$\Pi\mathbf S\Pi^{-1} = \mathbf S.$$

Thus $\mathbf X$ and $\mathbf P$ are parity-odd polar vectors, while $\mathbf L$ and $\mathbf S$ are parity-even axial vectors. Rotational rank alone does not determine parity.

Parity matrix elements

If

$$\Pi|i\rangle=\pi_i|i\rangle, \qquad \Pi|f\rangle=\pi_f|f\rangle,$$

and

$$\Pi O\Pi^{-1} = \eta_OO,$$

then a nonzero matrix element requires

$$\pi_f\eta_O\pi_i=1.$$

An even operator connects equal parities; an odd operator connects opposite parities. This result is a matrix-element constraint, not a statement that parity by itself forces energy degeneracy.

Time Reversal: Reverse Motion

Time reversal leaves the spatial point unchanged but reverses motion:

$$\Theta\mathbf X\Theta^{-1} = \mathbf X, \qquad \Theta\mathbf P\Theta^{-1} = -\mathbf P.$$

Angular momenta also reverse:

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

This is the first sharp contrast with parity. Orbital angular momentum is parity even but time-reversal odd.

Time reversal is not simply the replacement $t\mapsto-t$. The operator $\Theta$ acts on states and observables. When the Hamiltonian is invariant, the transformed state

$$|\psi_\Theta(t)\rangle = \Theta|\psi(-t)\rangle$$

obeys the same Schrödinger equation as the original state.

Why Time Reversal Must Be Antiunitary

The Schrödinger equation is

$$i\hbar\frac{d}{dt} |\psi(t)\rangle = H|\psi(t)\rangle.$$

Reversing the time parameter changes the sign of the derivative. Preserving the form of the equation requires the symmetry operation to change the sign of $i$ as well:

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

The canonical commutator gives an independent check. Suppose an operator leaves $X$ fixed and reverses $P$. Then

$$[X,-P] = -i\hbar I.$$

A unitary transformation would leave the scalar $i$ unchanged and would instead transform

$$[X,P]=i\hbar I$$

into $+i\hbar I$, a contradiction. Antiunitarity conjugates the right-hand side and restores consistency.

The same sign appears in time evolution:

$$\begin{aligned} \Theta e^{-iHt/\hbar}\Theta^{-1} &= e^{ +i(\Theta H\Theta^{-1})t/\hbar }. \end{aligned}$$

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

$$\Theta U(t)\Theta^{-1} = U(-t).$$

Antiunitarity is therefore not an optional representation choice. It is required by the dynamical and canonical algebras.

Transformation Table

For standard nonrelativistic variables:

Quantity	Parity	Time reversal
$\mathbf X$	$-\mathbf X$	$\mathbf X$
$\mathbf P$	$-\mathbf P$	$-\mathbf P$
$\mathbf L=\mathbf X\times\mathbf P$	$\mathbf L$	$-\mathbf L$
$\mathbf S$	$\mathbf S$	$-\mathbf S$
$i$	$i$	$-i$
external $\mathbf E$	$-\mathbf E$	$\mathbf E$
external $\mathbf B$	$\mathbf B$	$-\mathbf B$

The field rows describe transformation of the physical background. Holding a field fixed while transforming only the quantum degrees of freedom tests a different statement from transforming the whole family of systems.

The table is a starting point, not a substitute for an operator calculation. Products can contain several signs, and antiunitarity also complex-conjugates coefficients and matrix entries.

Spinless Time Reversal

For a scalar particle in the standard position representation,

$$\Theta=K, \qquad (\Theta\psi)(\mathbf x) = \psi(\mathbf x)^*.$$

This representation has

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

Because

$$\mathbf P = -i\hbar\nabla,$$

complex conjugation reverses momentum. In momentum space the same physical operation acts as

$$(\Theta\widetilde\psi)(\mathbf p) = \widetilde\psi(-\mathbf p)^*$$

up to momentum-ket phase conventions. The formula $\Theta=K$ is therefore representation-specific; the reversal of momentum is physical.

For a real scalar potential,

$$H = \frac{\mathbf P^2}{2m} +V(\mathbf X)$$

satisfies

$$KHK^{-1}=H.$$

In a finite-dimensional basis where $\Theta=K$, the same condition is

$$H^*=H.$$

Together with Hermiticity, this makes $H$ real symmetric in that basis.

If a time-reversal-invariant spinless eigenvalue is nondegenerate, its eigenvector can be rephased to be real in the $K$ basis. This does not imply Kramers degeneracy because $\Theta^2=+I$.

Probability current reverses. Without a vector potential,

$$\mathbf j = \frac{\hbar}{m} \operatorname{Im} \left( \psi^*\nabla\psi \right),$$

and complex conjugation gives

$$\mathbf j\mapsto-\mathbf j.$$

Time Reversal for Spin One-Half

Complex conjugation alone does not reverse every spin component. In the usual $S_z$ basis, a convenient convention is

$$\Theta=-i\sigma_yK.$$

It acts on the basis spinors as

$$\Theta|\uparrow\rangle = |\downarrow\rangle, \qquad \Theta|\downarrow\rangle = -|\uparrow\rangle.$$

For

$$|\psi\rangle = \alpha|\uparrow\rangle +\beta|\downarrow\rangle,$$

antilinearity gives

$$\Theta|\psi\rangle = -\beta^*|\uparrow\rangle +\alpha^*|\downarrow\rangle.$$

The operator reverses all Pauli components:

$$\Theta\boldsymbol\sigma\Theta^{-1} = -\boldsymbol\sigma.$$

Most importantly,

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

This sign cannot be changed by multiplying $\Theta$ by an overall phase. If

$$\Theta' = e^{i\chi}\Theta,$$

then antilinearity gives

$$\begin{aligned} (\Theta')^2 &= e^{i\chi}e^{-i\chi} \Theta^2 \\ &= \Theta^2. \end{aligned}$$

For an isolated angular-momentum multiplet,

$$\Theta^2 = (-1)^{2j}I.$$

Integer $j$ therefore has positive square, while half-integer $j$ has negative square. For several degrees of freedom the square is a statement about the complete Hilbert-space sector, not merely about one constituent.

The Square of Time Reversal Is Structural

The two common cases have sharply different consequences:

Sector	Typical representation	Square	Forced degeneracy?
scalar spinless particle	$K$	$+I$	no
single spin-$1/2$	$-i\sigma_yK$	$-I$	yes, if $H$ is time-reversal invariant

An even number of independent spin-$1/2$ factors can have a product time-reversal operator with positive square, while an odd number has the characteristic negative square on the spin sector. In many-electron problems this often becomes an even- versus odd-particle-number distinction.

The sign is also the bridge to modern symmetry classes: time-reversal symmetry with positive square and time-reversal symmetry with negative square impose different matrix structures and support different protected phenomena.

Kramers Degeneracy

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 discrete normalizable eigenspace in that sector has even dimension.

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.

Hamiltonian Constraints in Practice

The practical method is to transform every term, including complex coefficients and external parameters.

Scalar particle

For

$$H = \frac{P^2}{2m}+V(X),$$

a real even $V$ preserves both parity and spinless time reversal. A real asymmetric $V$ preserves spinless time reversal but breaks parity.

Finite spinless matrix

In a two-state basis with $\Theta=K$, write

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

with real coefficients. Since $\sigma_y$ is imaginary,

$$K\sigma_yK^{-1} = -\sigma_y,$$

so time-reversal invariance requires

$$b_y=0.$$

This basis-specific form represents the invariant condition $KHK^{-1}=H$.

Single spin one-half

For one spin-$1/2$,

$$H = aI+\mathbf b\cdot\boldsymbol\sigma.$$

Because all Pauli components are time-reversal odd,

$$\Theta H\Theta^{-1} = aI-\mathbf b\cdot\boldsymbol\sigma.$$

With no time-reversal-odd parameter transformed alongside the spin, invariance requires

$$\mathbf b=\mathbf0.$$

This is the two-dimensional face of Kramers degeneracy: a time-reversal-invariant Hamiltonian on a single spin-$1/2$ space is proportional to the identity.

Magnetic backgrounds

Minimal coupling gives

$$\begin{aligned} H(\mathbf A,\Phi) &= \frac{1}{2m} \left[ \mathbf P -q\mathbf A(\mathbf X) \right]^2 \\ &\quad+ q\Phi(\mathbf X). \end{aligned}$$

Time reversal maps

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

up to gauge convention. Equivalently, $\mathbf B$ reverses and $\mathbf E$ does not. A fixed magnetic flux or field therefore provides a standard time-reversal-breaking background even for a spinless particle.

Parity and Time Reversal Are Independent

The two transformations can be preserved or broken independently:

• a real asymmetric scalar potential can preserve time reversal while breaking parity;
• a symmetric potential with a fixed magnetic field can preserve inversion while breaking time reversal;
• a real even potential can preserve both;
• more general interactions can break both.

Their action on angular momentum makes the distinction vivid:

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

Combining the symbols $P$ and $T$ does not make their assumptions interchangeable. Each operation must be defined and tested separately on the physical Hilbert space.

Charge Conjugation: A Relativistic Boundary

Charge conjugation reverses internal charge:

$$CQC^{-1}=-Q.$$

If

$$Q|q,\alpha\rangle = q|q,\alpha\rangle,$$

then a well-defined $C$ maps the state into a charge-$-q$ sector. An electron-only nonrelativistic Hilbert space is not closed under this operation because it does not contain positron states.

Charge conjugation is usually unitary, unlike time reversal. It is not the same as complex conjugating a wavefunction, and it is not the same as charge conservation:

$$[H,Q]=0$$

states that charge is conserved, whereas

$$CHC^{-1}=H$$

states that the dynamics treat charge-conjugate sectors symmetrically.

The detailed spinor and field transformations belong to relativistic quantum theory and QFT. This chapter supplies only the conceptual signpost.

CPT: State the Theorem’s Assumptions

The CPT theorem says, in its standard physics form, that a local Lorentz-invariant quantum field theory satisfying the usual Hilbert-space, positive-energy, locality, and stability assumptions possesses a combined CPT symmetry.

It does not follow from finite-dimensional quantum mechanics, parity, and time reversal alone. Ordinary fixed-particle Schrödinger theory generally lacks:

• antiparticle sectors;
• local relativistic fields;
• Lorentz covariance;
• microcausality;
• a vacuum supporting particle creation and annihilation.

CPT invariance also does not imply separate $C$, $P$, $T$, or $CP$ invariance. The combined theorem can hold even when several individual transformations are violated.

The right use of this chapter is preparatory: parity and antiunitary time reversal establish the quantum-mechanical language, while CPT Preview and the QFT bridge state where new assumptions enter.

Modern Symmetry Classification Preview

After ordinary commuting unitary symmetries have been used to block-diagonalize a Hamiltonian, three structures often organize a noninteracting fermion or Bogoliubov–de Gennes block.

Time reversal is antiunitary and preserves energy:

$$\mathcal T H(\mathbf k) \mathcal T^{-1} = H(-\mathbf k).$$

Particle–hole structure is antiunitary and reverses the Hamiltonian:

$$\mathcal C H(\mathbf k) \mathcal C^{-1} = -H(-\mathbf k).$$

Chiral or sublattice symmetry is unitary and anticommutes with the Hamiltonian:

$$\mathcal S H(\mathbf k) \mathcal S^{-1} = -H(\mathbf k).$$

The presence or absence of these structures, together with

$$\mathcal T^2=\pm I, \qquad \mathcal C^2=\pm I,$$

gives the ten Altland–Zirnbauer classes.

This classification uses particle–hole in a condensed-matter or Nambu-space sense. It must not be silently identified with relativistic charge conjugation. A symmetry class also does not determine a topological phase by itself; spatial dimension, a spectral gap, locality, interactions, and crystalline symmetries matter.

The full ten-class table and model examples belong to Symmetry Classification Preview. Topological invariants and interacting refinements belong to Quantum Matter and the mathematical topology material.

A Reliable Discrete-Symmetry Workflow

1. Specify the Hilbert space

State the particle-number, charge, spin, momentum, or effective quasiparticle sector. Check that the proposed transformation maps the sector into itself.

2. Write the implementing operator

Give $\Pi$, $\Theta=UK$, or the relevant representation. State the basis in which $K$ means componentwise complex conjugation.

3. Mark unitary versus antiunitary

For an antiunitary symmetry, conjugate scalar coefficients and factors of $i$. Do not apply only a sign table to observables.

4. Transform observables and parameters

Track $\mathbf X$, $\mathbf P$, $\mathbf L$, $\mathbf S$, external fields, fluxes, and complex phases. Distinguish polar from axial vectors.

5. Test the family and the fixed Hamiltonian

First establish

$$S H(\lambda)S^{-1} = H(S\lambda).$$

Then ask whether the chosen parameter value satisfies

$$H(S\lambda)=H(\lambda).$$

6. Compute the square

For antiunitary symmetries, determine $S^2$ on the actual sector. The sign cannot be inferred from the word time reversal alone.

7. Draw only justified consequences

Parity may label states and force matrix-element zeros. Spinless time reversal may permit a real basis. Negative-square time reversal can force Kramers pairs. None of these conclusions should be transferred to a different symmetry algebra without rechecking the assumptions.

Chapter Map

Read the chapter in the following order:

1. Parity establishes the unitary inversion example.
2. Time Reversal introduces the physical sign rules.
3. Antiunitary Time Reversal explains why complex conjugation is unavoidable.
4. Time Reversal for Spinless Particles develops the positive-square case.
5. Time Reversal for Spin-1/2 Particles develops the negative-square case.
6. Kramers Degeneracy turns the negative square into a theorem.
7. Charge Conjugation Preview marks the transition to particle–antiparticle sectors.
8. CPT Preview states the relativistic theorem boundary.
9. Discrete Symmetries in Hamiltonians consolidates term-by-term model tests.
10. Symmetry Classification Preview connects the algebra to the tenfold way.

Reading Paths

First graduate pass

Read parity, the time-reversal overview, antiunitary time reversal, the two concrete time-reversal representations, and Kramers degeneracy. Then use the Hamiltonian page to practice term-by-term tests.

Atomic and molecular physics

Focus on parity, time reversal for spin-$1/2$, Kramers degeneracy, spin–orbit coupling, and external-field tests. Keep charge conjugation and CPT as scoped bridges rather than importing relativistic claims into fixed-particle models.

Quantum matter

Study both signs of $\Theta^2$, Kramers pairs, magnetic-field breaking, and the classification preview. Then continue to Berry curvature, topological invariants, and quantum-matter models.

Relativistic and QFT bridge

Establish parity and antiunitary time reversal carefully, then read the charge-conjugation and CPT previews together with From Discrete Symmetries to CPT.

Distinctions Worth Keeping

Do not conflate	Why
transformation and symmetry	an operator can be defined even when a Hamiltonian is not invariant
fixed Hamiltonian and covariant family	transforming an external field may relate two different members of a family
parity and time reversal	parity reverses position and is unitary; time reversal leaves position fixed and is antiunitary
$K$ and the physical antiunitary operator	$K$ depends on basis; $\Theta=UK$ is the symmetry
$\Theta^2=+I$ and $\Theta^2=-I$	only the negative-square case forces Kramers degeneracy
parity label and degeneracy	parity block-diagonalizes but does not generally pair levels
spin–orbit coupling and magnetic order	the first can preserve time reversal; the second generally breaks it
charge conservation and charge conjugation	$[H,Q]=0$ differs from $CQC^{-1}=-Q$ and $CHC^{-1}=H$
relativistic $C$ and Nambu particle–hole structure	they act in different settings and impose different Hamiltonian relations
CPT theorem and ordinary nonrelativistic symmetry	CPT requires relativistic locality and field-theory assumptions

Common Mistakes

• Treating time reversal as a unitary operator that merely flips momentum.
• Forgetting to complex-conjugate scalar coefficients under an antiunitary transformation.
• Writing $\Theta=K$ without naming the basis or the spinless setting.
• Reversing position under time reversal or failing to reverse angular momentum.
• Assuming $\Theta^2=-I$ can be changed by rephasing $\Theta$.
• Applying Kramers degeneracy in a $\Theta^2=+I$ sector.
• Calling a fixed magnetic-field Hamiltonian time-reversal invariant because the field-reversed family is covariant.
• Treating spin–orbit coupling as time-reversal breaking by itself.
• Assuming parity symmetry forces degenerate even and odd states.
• Applying charge conjugation within a Hilbert space that contains no opposite-charge sector.
• Presenting CPT as a theorem of finite-dimensional or fixed-particle quantum mechanics.
• Identifying Bogoliubov particle–hole redundancy with relativistic charge conjugation.

Cross-Links

• Symmetry Principles
• Antiunitary Symmetries
• Symmetry Constraints on Hamiltonians
• Tensor Operators and Selection Rules
• Parity Selection Rules
• Spin and Spinors
• Spin in Magnetic Fields
• Spin–Orbit Coupling
• Gauge, Phase, and Magnetic Geometry
• Symmetry Breaking and Emergence
• Symmetry-Protected Structure Preview
• From Discrete Symmetries to CPT
• Time-Reversal Operator
• Kramers Degeneracy Reference Card
• Berry Curvature
• Chern Numbers
• Topological Invariants

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.
• M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter, Springer, 2008.
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press, 2000.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.
• A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Physical Review B 55, 1142, 1997.
• 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.

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. Show directly from the canonical commutator that no unitary operator can leave $X$ fixed and send $P$ to $-P$.

Solution

Assume a unitary $U$ satisfies

$$UXU^{-1}=X, \qquad UPU^{-1}=-P.$$

Transforming the commutator by its operators gives

$$U[X,P]U^{-1} = [X,-P] = -i\hbar I.$$

But unitary conjugation does not conjugate scalar $i$, so using $[X,P]=i\hbar I$ gives

$$U[X,P]U^{-1} = i\hbar I.$$

The contradiction shows that the desired transformation cannot be unitary. An antiunitary operator resolves it because

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

3. 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

$$\begin{aligned} KHK^{-1} &= aI +b_x\sigma_x \\ &\quad- b_y\sigma_y +b_z\sigma_z. \end{aligned}$$

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 $K^2=+I$.

4. A single spin-$1/2$ has

$$H(\mathbf B) = -\gamma\mathbf S\cdot\mathbf B.$$

Distinguish the covariance of the Hamiltonian family from symmetry of a fixed nonzero field.

Solution

Time reversal flips spin:

$$\Theta\mathbf S\Theta^{-1} = -\mathbf S.$$

Therefore

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

This is covariance of the family when the physical magnetic field is reversed. If $\mathbf B$ is held fixed and nonzero, then

$$H(-\mathbf B)\ne H(\mathbf B),$$

so the fixed Hamiltonian is not time-reversal invariant. At $\mathbf B=\mathbf0$, the fixed Hamiltonian does satisfy the symmetry.

5. 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.