Time-reversal symmetry and Kramers' degeneracy

Intuition Beginner

Run a film of a single planet orbiting a star backwards and it still looks like a legal orbit: the planet retraces its path, now moving the other way. Newton's laws do not care about the direction of time. Time reversal is the operation that plays the film backwards. In quantum mechanics it acts on a state and asks whether the reversed motion is also an allowed state of the same system.

The catch is what "reversing motion" means for a wave. Position stays put when you reverse a film, but velocity flips its sign. A quantum particle carries its velocity inside a complex phase, and flipping that velocity turns out to require swapping every $i$ in the wavefunction for $-i$. This phase-flip is the surprising fingerprint of time reversal, and it makes the reversal operation behave unlike ordinary symmetries.

Why bother? Many systems, like an electron in a crystal with no magnetic field, look the same forwards and backwards in time. When that holds, time reversal forces relationships between energy levels that you could never guess from the energy formula alone. The most famous of these is a guaranteed pairing of levels for particles with half-integer spin.

Visual Beginner

Picture two filmstrips of the same electron. The top strip runs forwards: the electron drifts to the right with its spin arrow pointing one way. The bottom strip is the same motion played backwards: the electron now drifts to the left, and its spin arrow has flipped to point the opposite way. Position is unchanged frame by frame, but both the drift direction and the spin arrow reverse.

The picture records the three rules that define time reversal at this level: position is left alone, momentum reverses sign, and spin reverses sign. The hidden fourth rule, that complex phases get conjugated, is what makes all three of the others fit together consistently.

Worked example Beginner

Take a free particle moving to the right with a definite momentum. Its wavefunction is the plane wave $\psi (x)=e^{ikx}$, where the positive number $k$ encodes rightward motion. Reverse time and check what you get.

Step 1. Time reversal conjugates the wavefunction, swapping $i$ for $-i$. Applying that rule to $\psi (x)=e^{ikx}$ gives $e^{-ikx}$.

Step 2. Read off the new motion. The wave $e^{-ikx}$ is a plane wave with momentum $-k$: a particle of the same speed moving to the left. So time reversal turned rightward motion into leftward motion, which is what reversing a film should do. Position information was not touched; only the direction of travel flipped.

Step 3. Check the spin-flip rule on a spin-up electron. Time reversal sends spin-up to spin-down (up to a phase), so an electron drifting right with spin up becomes an electron drifting left with spin down. Both the drift and the spin reversed together.

Step 4. Apply time reversal a second time to the plain plane wave. Conjugating $e^{-ikx}$ again returns $e^{ikx}$: two reversals restore the original spinless state. For an electron with spin, two reversals will instead return the state multiplied by $-1$, and that sign is the seed of the pairing theorem in the advanced sections.

What this tells us: time reversal keeps position, flips momentum and spin, and conjugates phases. For a spinless particle two reversals undo each other cleanly. For a spin-half particle two reversals leave behind a stubborn minus sign that cannot be removed, and that minus sign forces energy levels to come in pairs.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{H}$ be a complex separable Hilbert space with inner product $⟨\cdot \mid \cdot ⟩$ taken antilinear in its first argument. An operator $\hat{A}:\mathcal{H}\to \mathcal{H}$ is antilinear when $\hat{A}(\alpha \mid \psi ⟩+\beta \mid \varphi ⟩)=\bar{\alpha }\hat{A}\mid \psi ⟩+\bar{\beta }\hat{A}\mid \varphi ⟩$ for all scalars $\alpha ,\beta$. An antilinear operator $\hat{A}$ is antiunitary when it is a bijection satisfying

$$⟨\hat{A}\varphi \mid \hat{A}\psi ⟩=⟨\psi \mid \varphi ⟩=\overline{⟨\varphi \mid \psi ⟩}$$

for all $\mid \varphi ⟩,\mid \psi ⟩\in \mathcal{H}$. The convention adopted here is that of Landau-Lifshitz ^{[Landau-Lifshitz §§95-96]}: complex conjugation of the inner product accompanies the reversal of the order of the two states.

The time-reversal operator $\hat{T}$ is the antiunitary operator implementing motion reversal. Fix an orthonormal basis and let $\hat{K}$ denote complex conjugation in that basis, the antilinear involution sending ${\sum }_n{c}_n\mid n⟩$ to ${\sum }_n{\bar{c}}_n\mid n⟩$. Wigner's structure theorem states that every antiunitary operator factors as

$$\hat{T}=\hat{U}\hat{K},\hat{U}\text{ unitary},$$

so the whole content of time reversal beyond conjugation lives in a unitary $\hat{U}$ fixed by how $\hat{T}$ must act on observables ^{[Wigner 1932]}.

The action on the canonical observables is the physical input. Time reversal is required to preserve position, reverse momentum, and reverse every angular momentum:

$$\hat{T}\hat{\mathbf{r}}{\hat{T}}^{-1}=\hat{\mathbf{r}},\hat{T}\hat{\mathbf{p}}{\hat{T}}^{-1}=-\hat{\mathbf{p}},\hat{T}\hat{\mathbf{J}}{\hat{T}}^{-1}=-\hat{\mathbf{J}}.$$

The same sign holds separately for orbital angular momentum $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times \hat{\mathbf{p}}$ and for spin $\hat{\mathbf{S}}$. The orbital sign is forced by the position and momentum rules; the spin sign is imposed by analogy, since spin is a genuine angular momentum.

For a spinless particle in the position basis, $\hat{U}=1$ and $\hat{T}=\hat{K}$ is plain complex conjugation, which reverses $\hat{\mathbf{p}}=-i\hslash \nabla$ while fixing $\hat{\mathbf{r}}$. For a single spin-$\frac{1}{2}$ degree of freedom, the requirement $\hat{T}\hat{\mathbf{S}}{\hat{T}}^{-1}=-\hat{\mathbf{S}}$ together with ${\hat{S}}_i=\frac{\hslash }{2}{\sigma }_i$ and $\hat{K}{\sigma }_i\hat{K}={\sigma }_i^{\ast }$ forces $\hat{U}$ to be the unitary rotating ${\sigma }_1,{\sigma }_3\mapsto -{\sigma }_1,-{\sigma }_3$ while fixing ${\sigma }_2$. That unitary is $e^{-i\pi S_y/\hslash }=-i{\sigma }_2$, giving the standard form

$$\hat{T}=-i{\sigma }_2\hat{K}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\hat{K}.$$

For a particle of total spin $j$ the corresponding factor is $\hat{U}=e^{-i\pi J_y/\hslash }$, the rotation by $\pi$ about the $y$-axis, so that in general $\hat{T}=e^{-i\pi J_y/\hslash }\hat{K}$.

Counterexamples to common slips

• Time reversal is not unitary. Treating $\hat{T}$ as if $⟨\hat{T}\varphi \mid \hat{T}\psi ⟩=⟨\varphi \mid \psi ⟩$ (no conjugation) contradicts $\hat{T}(i\mid \psi ⟩)=-i\hat{T}\mid \psi ⟩$ and produces sign errors in every commutator with $i$. The defining identity carries a conjugation.
• The phase of $\hat{T}$ is not free to choose away. One can multiply $\hat{T}$ by a phase $e^{i\theta }$, but ${\hat{T}}^2$ is basis- and phase-independent: $(e^{i\theta }\hat{T}{)}^2=e^{i\theta }\hat{T}{e}^{i\theta }\hat{T}=e^{i\theta }{e}^{-i\theta }{\hat{T}}^2={\hat{T}}^2$, because $\hat{T}$ conjugates the second phase. So ${\hat{T}}^2=\pm 1$ is an invariant, not a convention.
• For a half-integer-spin particle ${\hat{T}}^2=-1$, while for integer spin ${\hat{T}}^2=+1$. Asserting ${\hat{T}}^2=+1$ universally drops the central fact behind Kramers degeneracy. The sign is $(-1{)}^{2j}$.

Key theorem with proof Intermediate+

Theorem (square of time reversal; Wigner). Let $\hat{T}=e^{-i\pi J_y/\hslash }\hat{K}$ be the time-reversal operator on a system of total angular momentum $j$. Then

$${\hat{T}}^2=(-1{)}^{2j}1,$$

so ${\hat{T}}^2=+1$ for integer $j$ and ${\hat{T}}^2=-1$ for half-integer $j$. The value is independent of the basis used to define $\hat{K}$ and of any phase rescaling of $\hat{T}$.

Proof. Write $\hat{T}=\hat{U}\hat{K}$ with $\hat{U}=e^{-i\pi J_y/\hslash }$. Compute the square directly:

$${\hat{T}}^2=\hat{U}\hat{K}\hat{U}\hat{K}=\hat{U}(\hat{K}\hat{U}{\hat{K}}^{-1}){\hat{K}}^2=\hat{U}{\hat{U}}^{\ast },$$

since $\hat{K}$ is an involution, ${\hat{K}}^2=1$, and conjugating a unitary by $\hat{K}$ returns its entrywise complex conjugate ${\hat{U}}^{\ast }$ in the chosen basis.

Now evaluate ${\hat{U}}^{\ast }$. The generator $J_y$ is represented, in the standard angular-momentum basis where $J_z$ and $J^2$ are diagonal, by a purely imaginary antisymmetric matrix (the Condon-Shortley convention makes $⟨m^{\prime }\mid J_y\mid m⟩$ imaginary). Therefore $(i{J}_y{)}^{\ast }=-i{J}_y^{\ast }=-i(-J_y)=i{J}_y$ is real, and ${\hat{U}}^{\ast }=(e^{-i\pi J_y/\hslash }{)}^{\ast }=e^{-i\pi J_y/\hslash }=\hat{U}$ entrywise. Hence

$${\hat{T}}^2=\hat{U}{\hat{U}}^{\ast }={\hat{U}}^2=e^{-2\pi i{J}_y/\hslash }.$$

The operator $e^{-2\pi i{J}_y/\hslash }$ is a rotation by the angle $2\pi$ about the $y$-axis. On a representation of the rotation group of spin $j$, a $2\pi$ rotation acts as the scalar $(-1{)}^{2j}$: this is the defining feature of the projective ($SU(2)$) representations, where integer spin gives a genuine $SO(3)$ representation returning to the identity after $2\pi$, and half-integer spin gives the double-valued representation returning to $-1$. Therefore ${\hat{T}}^2=(-1{)}^{2j}1$.

Independence from the basis and from phase: a change of basis conjugates $\hat{U}$ by a unitary $\hat{V}$ and replaces $\hat{K}$ by $\hat{V}\hat{K}{\hat{V}}^{†\ast }$ consistently, leaving ${\hat{T}}^2$ unchanged because ${\hat{T}}^2$ is the action of the abstract group element "rotation by $2\pi$," which is representation-theoretic data. The phase independence was recorded above: rescaling $\hat{T}\mapsto e^{i\theta }\hat{T}$ leaves ${\hat{T}}^2$ fixed because the antilinear $\hat{T}$ conjugates the inserted phase. $\square$

Bridge. This square-classification builds toward the degeneracy theorem proved in the advanced sections and appears again in the representation-theoretic Frobenius-Schur indicator developed there. The foundational reason a half-integer-spin level must be degenerate is exactly the impossibility of an antiunitary involution-up-to-$(-1)$ fixing any single ray, and that impossibility is the central insight that ${\hat{T}}^2=-1$ encodes. Putting these together, the sign $(-1{)}^{2j}$ computed here is dual to the geometric fact that a spinor changes sign under a full $2\pi$ rotation, the same double-cover phenomenon that organises the spin-$\frac{1}{2}$ representation of $SU(2)$ in 12.05.01. The bridge is the recognition that the antiunitarity of $\hat{T}$ and the spinor double cover are two faces of one structure: motion reversal squares to the $2\pi$ rotation, and on half-integer spin that rotation is $-1$.

Exercises Intermediate+

Advanced results Master

Theorem (Kramers' degeneracy; Kramers 1930). Let $\hat{H}$ be a self-adjoint Hamiltonian commuting with a time-reversal operator $\hat{T}$ satisfying ${\hat{T}}^2=-1$ (a system with an odd number of half-integer-spin constituents, hence half-integer total angular momentum). Then every eigenvalue of $\hat{H}$ has even multiplicity; in particular every level is at least doubly degenerate, with $\mid \psi ⟩$ and $\hat{T}\mid \psi ⟩$ orthogonal. No perturbation respecting time-reversal symmetry — a crystal field of arbitrary low symmetry, spin-orbit coupling, electric fields — can lift this degeneracy; only a time-reversal-breaking perturbation such as a magnetic field can.

The mechanism is purely the antiunitary algebra. Because $[\hat{H},\hat{T}]=0$, the operator $\hat{T}$ maps each eigenspace ${\mathcal{E}}_E=\ker (\hat{H}-E)$ into itself, acting on it as an antiunitary ${\hat{T}}_E$ with ${\hat{T}}_E^2=-1$. An antiunitary operator squaring to $-1$ on a finite-dimensional complex inner-product space has no eigenvector with a real eigenvalue and admits no invariant line: if ${\hat{T}}_E\mid v⟩=\lambda \mid v⟩$ then ${\hat{T}}_E^2\mid v⟩={\hat{T}}_E(\lambda \mid v⟩)=\bar{\lambda }\lambda \mid v⟩=\mid \lambda {\mid }^2\mid v⟩$, which cannot equal $-\mid v⟩$. The eigenspace therefore decomposes into mutually orthogonal Kramers pairs $\{\mid \psi ⟩,\hat{T}\mid \psi ⟩\}$, forcing $\dim {\mathcal{E}}_E$ to be even.

Theorem (selection rules from time reversal). Let $\hat{O}$ be an observable with definite time-reversal signature, $\hat{T}\hat{O}{\hat{T}}^{-1}={ϵ}_O\hat{O}$ with ${ϵ}_O=\pm 1$. Between two Kramers partners $\mid \psi ⟩$ and $\mid \tilde{\psi }⟩=\hat{T}\mid \psi ⟩$ the matrix elements obey

$$⟨\tilde{\psi }\mid \hat{O}\mid \tilde{\psi }⟩={ϵ}_O⟨\psi \mid \hat{O}\mid \psi ⟩.$$

Consequently any time-odd observable (${ϵ}_O=-1$, for example a magnetic moment $\mathbf{\mu }\propto \hat{\mathbf{J}}$) has equal-and-opposite expectation values in the two partners, while any time-even observable (${ϵ}_O=+1$, for example the charge density) has equal expectation values. The proof uses antiunitarity twice: $⟨\hat{T}\psi \mid \hat{O}\mid \hat{T}\psi ⟩=⟨\hat{T}\psi \mid \hat{O}\hat{T}\psi ⟩={ϵ}_O⟨\hat{T}\psi \mid \hat{T}\hat{O}\psi ⟩={ϵ}_O\overline{⟨\psi \mid \hat{O}\psi ⟩}={ϵ}_O⟨\psi \mid \hat{O}\psi ⟩$, the last step using that the diagonal element of a self-adjoint $\hat{O}$ is real.

Theorem (Frobenius-Schur indicator and the three Wigner classes). Let $G$ be a finite or compact group with an irreducible unitary representation $\rho$ of character $\chi$. The Frobenius-Schur indicator

$$\nu (\rho )=\frac{1}{\mid G\mid }\sum _{g\in G}\chi (g^2)\in \{+1,0,-1\}$$

classifies $\rho$ as real ($\nu =+1$, the representation is realisable over $\mathbb{R}$), complex ($\nu =0$, $\rho \ncong \bar{\rho }$), or quaternionic/pseudoreal ($\nu =-1$, $\rho \cong \bar{\rho }$ but not realisable over $\mathbb{R}$). When $G$ contains time reversal as an antiunitary symmetry implemented on the irreducible multiplet, the indicator equals ${\hat{T}}^2$ restricted to that multiplet: $\nu =+1$ corresponds to ${\hat{T}}^2=+1$ (integer-spin, orthogonal type), $\nu =-1$ to ${\hat{T}}^2=-1$ (half-integer-spin, symplectic type, Kramers-degenerate), and $\nu =0$ to a representation whose conjugate is inequivalent, pairing two complex irreducibles into a Kramers doublet. For $SU(2)$ the spin-$j$ representation has $\nu =(-1{)}^{2j}$, recovering the integer/half-integer dichotomy from the group-theoretic side.

Theorem (Kramers pairs and topological band structure). In a crystalline solid with time-reversal symmetry and ${\hat{T}}^2=-1$ (spin-orbit-coupled electrons), the Bloch Hamiltonian $\hat{H}(\mathbf{k})$ satisfies $\hat{T}\hat{H}(\mathbf{k}){\hat{T}}^{-1}=\hat{H}(-\mathbf{k})$. At the time-reversal-invariant momenta $\mathbf k_$where$\mathbf k_* \equiv -\mathbf k_*$moduloareciprocal-latticevector,$\hat H(\mathbf k_*)$commuteswith$\hat T$andKramer{s}^{\prime }theoremapplies:thebandsarepinnedintodegenerateKramerspairs.\ast Thepatterninwhichthesepairsreconnectbetweendistinctinvariantmomentaisa$\mathbb Z_2$ invariant; its non-zero value is the defining signature of a two-dimensional quantum spin Hall insulator and the three-dimensional topological insulator, where Kramers degeneracy protects gapless boundary states against any time-reversal-symmetric disorder.

Synthesis. The single sign ${\hat{T}}^2=(-1{)}^{2j}$ is the foundational reason every result in this unit holds, and putting these together they form one structure rather than a list. The central insight is that an antiunitary symmetry squaring to $-1$ admits no invariant ray, so a $\hat{T}$-commuting Hamiltonian cannot have a non-degenerate level; this is exactly Kramers' theorem, and it is dual to the representation-theoretic statement that the spin-$j$ irreducible of $SU(2)$ is pseudoreal precisely when $2j$ is odd. The Frobenius-Schur indicator identifies the analyst's ${\hat{T}}^2$ with the algebraist's $\nu =\frac{1}{\mid G\mid }{\sum }_g\chi (g^2)$, so the degeneracy that protects an odd-electron ion in a crystal field is the same data that classifies the irreducible as real, complex, or quaternionic. The selection rules generalise the pairing: a time-odd observable separates Kramers partners with opposite sign while a time-even observable cannot, which is why only a magnetic field splits a Kramers doublet. The bridge is that motion reversal, the spinor double cover, the pseudoreality of half-integer representations, and the ${\mathbb{Z}}_2$ topological invariant of a time-reversal-symmetric insulator are four presentations of the antiunitary involution $\hat{T}$ with ${\hat{T}}^2=-1$, and this pattern recurs wherever an antiunitary symmetry meets a half-integer-spin Hilbert space.

Full proof set Master

Proposition 1 (antiunitarity preserves transition probabilities; Wigner). An antiunitary $\hat{T}$ satisfies $\mid ⟨\hat{T}\varphi \mid \hat{T}\psi ⟩\mid =\mid ⟨\varphi \mid \psi ⟩\mid$ for all states, so it is a Wigner symmetry of the projective Hilbert space.

Proof. By the defining identity $⟨\hat{T}\varphi \mid \hat{T}\psi ⟩=⟨\psi \mid \varphi ⟩=\overline{⟨\varphi \mid \psi ⟩}$. Taking absolute values, $\mid ⟨\hat{T}\varphi \mid \hat{T}\psi ⟩\mid =\mid \overline{⟨\varphi \mid \psi ⟩}\mid =\mid ⟨\varphi \mid \psi ⟩\mid$. Transition probabilities, the physically measurable $\mid ⟨\varphi \mid \psi ⟩{\mid }^2$, are therefore preserved, which is the hypothesis of Wigner's symmetry-representation theorem; the antiunitary branch is the one not realisable by any unitary because it conjugates scalars. $\square$

Proposition 2 (Wigner's structure theorem $\hat{T}=\hat{U}\hat{K}$). Every antiunitary operator on a complex Hilbert space with a fixed orthonormal basis can be written $\hat{T}=\hat{U}\hat{K}$ with $\hat{U}$ unitary and $\hat{K}$ the complex conjugation of that basis.

Proof. Define $\hat{U}:=\hat{T}\hat{K}$. Since $\hat{K}$ is antilinear and $\hat{T}$ is antilinear, the composite $\hat{U}$ is linear: for a scalar $\alpha$, $\hat{U}(\alpha \mid \psi ⟩)=\hat{T}\hat{K}(\alpha \mid \psi ⟩)=\hat{T}(\bar{\alpha }\hat{K}\mid \psi ⟩)=\alpha \hat{T}\hat{K}\mid \psi ⟩=\alpha \hat{U}\mid \psi ⟩$. It is bijective because $\hat{T}$ and $\hat{K}$ are. For unitarity, use that both $\hat{T}$ and $\hat{K}$ are antiunitary, so their composite preserves the inner product without conjugation: $⟨\hat{U}\varphi \mid \hat{U}\psi ⟩=⟨\hat{T}\hat{K}\varphi \mid \hat{T}\hat{K}\psi ⟩=⟨\hat{K}\psi \mid \hat{K}\varphi ⟩={\overline{⟨\hat{K}\psi \mid \hat{K}\varphi ⟩}}^{\ast }$; explicitly $⟨\hat{T}\hat{K}\varphi \mid \hat{T}\hat{K}\psi ⟩=⟨\hat{K}\psi \mid \hat{K}\varphi ⟩=⟨\varphi \mid \psi ⟩$, where the two conjugations from $\hat{T}$ and from $\hat{K}$ cancel. Hence $\hat{U}$ is unitary and $\hat{U}\hat{K}=\hat{T}\hat{K}\hat{K}=\hat{T}$ since ${\hat{K}}^2=1$. $\square$

Proposition 3 (no fixed line when ${\hat{T}}^2=-1$). If $\hat{T}$ is antiunitary with ${\hat{T}}^2=-1$, there is no non-zero $\mid v⟩$ with $\hat{T}\mid v⟩=\lambda \mid v⟩$ for any scalar $\lambda$.

Proof. Suppose $\hat{T}\mid v⟩=\lambda \mid v⟩$ with $\mid v⟩\ne 0$. Apply $\hat{T}$ again; using antilinearity, ${\hat{T}}^2\mid v⟩=\hat{T}(\lambda \mid v⟩)=\bar{\lambda }\hat{T}\mid v⟩=\bar{\lambda }\lambda \mid v⟩=\mid \lambda {\mid }^2\mid v⟩$. But ${\hat{T}}^2=-1$ gives ${\hat{T}}^2\mid v⟩=-\mid v⟩$, so $\mid \lambda {\mid }^2=-1$, impossible for a complex number. Hence no eigenvector, and a fortiori no $\hat{T}$-invariant line, exists. $\square$

Proposition 4 (Kramers degeneracy). A self-adjoint $\hat{H}$ with $[\hat{H},\hat{T}]=0$ and ${\hat{T}}^2=-1$ has every eigenspace of even dimension.

Proof. Fix an eigenvalue $E$ and let ${\mathcal{E}}_E=\ker (\hat{H}-E)$, which is finite-dimensional in the cases of interest (a level of a Hamiltonian with discrete spectrum). Since $\hat{H}\hat{T}=\hat{T}\hat{H}$ and $E$ is real, $\hat{T}$ maps ${\mathcal{E}}_E$ to itself: for $\mid \psi ⟩\in {\mathcal{E}}_E$, $\hat{H}\hat{T}\mid \psi ⟩=\hat{T}\hat{H}\mid \psi ⟩=\hat{T}(E\mid \psi ⟩)=E\hat{T}\mid \psi ⟩$. The restriction ${\hat{T}}_E:=\hat{T}{\upharpoonright }_{{\mathcal{E}}_E}$ is antiunitary on ${\mathcal{E}}_E$ with ${\hat{T}}_E^2=-1$. Choose any unit $\mid {\psi }_1⟩\in {\mathcal{E}}_E$; by Proposition 3, ${\hat{T}}_E\mid {\psi }_1⟩$ is not proportional to $\mid {\psi }_1⟩$, and by the computation $⟨{\psi }_1\mid {\hat{T}}_E{\psi }_1⟩=0$ (Exercise 7) the two are orthonormal after normalisation. Their span is a two-dimensional ${\hat{T}}_E$-invariant subspace $P_1$. The orthogonal complement of $P_1$ inside ${\mathcal{E}}_E$ is again ${\hat{T}}_E$-invariant, because ${\hat{T}}_E$ is antiunitary and hence preserves orthogonality: if $\mid w⟩\perp P_1$ then for $\mid u⟩\in P_1$, $⟨u\mid {\hat{T}}_{E}w⟩=\overline{⟨{\hat{T}}_E^{-1}u\mid w⟩}=0$ since ${\hat{T}}_E^{-1}u\in P_1$. Induct on the complement, which carries the same structure ${\hat{T}}_E^2=-1$. The eigenspace decomposes into two-dimensional Kramers blocks, so $\dim {\mathcal{E}}_E$ is even. $\square$

Proposition 5 (spin-$\frac{1}{2}$ realisation $\hat{T}=-i{\sigma }_2\hat{K}$ satisfies ${\hat{T}}^2=-1$). On ${\mathbb{C}}^2$ the operator $\hat{T}=-i{\sigma }_2\hat{K}$ reverses all three spin components and squares to $-1$.

Proof. Reversal: $\hat{T}{\sigma }_i{\hat{T}}^{-1}=(-i{\sigma }_2)\hat{K}{\sigma }_i\hat{K}(-i{\sigma }_2{)}^{-1}=(-i{\sigma }_2){\sigma }_i^{\ast }(i{\sigma }_2)/1$. With ${\sigma }_1^{\ast }={\sigma }_1$, ${\sigma }_2^{\ast }=-{\sigma }_2$, ${\sigma }_3^{\ast }={\sigma }_3$ and the anticommutator $\{{\sigma }_2,{\sigma }_i\}=2{\delta }_{2i}$, one gets ${\sigma }_2{\sigma }_1{\sigma }_2=-{\sigma }_1$, ${\sigma }_2{\sigma }_2{\sigma }_2={\sigma }_2$, ${\sigma }_2{\sigma }_3{\sigma }_2=-{\sigma }_3$, hence $\hat{T}{\sigma }_1{\hat{T}}^{-1}=-{\sigma }_1$, $\hat{T}{\sigma }_2{\hat{T}}^{-1}=-{\sigma }_2$ (the conjugation ${\sigma }_2^{\ast }=-{\sigma }_2$ combines with the sandwich to give $-{\sigma }_2$), and $\hat{T}{\sigma }_3{\hat{T}}^{-1}=-{\sigma }_3$. All three components reverse, matching $\hat{T}\hat{\mathbf{S}}{\hat{T}}^{-1}=-\hat{\mathbf{S}}$. Square: ${\hat{T}}^2=(-i{\sigma }_2\hat{K})(-i{\sigma }_2\hat{K})=(-i{\sigma }_2)(\hat{K}(-i{\sigma }_2)\hat{K})=(-i{\sigma }_2)(i{\sigma }_2^{\ast })=(-i{\sigma }_2)(i(-{\sigma }_2))=(-i)(i)(-1){\sigma }_2^2=-{\sigma }_2^2=-1$. $\square$

Proposition 6 (Frobenius-Schur indicator takes values in $\{+1,0,-1\}$), stated with proof outline — full statement in Wigner Group Theory Ch. 26 ^{[Wigner 1932]}. The sum $\frac{1}{\mid G\mid }{\sum }_g\chi (g^2)$ equals the multiplicity of the identity representation in the symmetric square minus its multiplicity in the antisymmetric square of $\rho$. For an irreducible $\rho$ this difference is $+1$ when $\rho$ carries an invariant symmetric bilinear form (real type), $-1$ when it carries an invariant antisymmetric form (quaternionic type), and $0$ when $\rho \ncong \bar{\rho }$ so no invariant form exists (complex type). The antisymmetric invariant form in the quaternionic case is exactly the bilinear pairing $(\psi ,\varphi )\mapsto ⟨\hat{T}\psi \mid \varphi ⟩$ built from the antiunitary $\hat{T}$, whose antisymmetry is equivalent to ${\hat{T}}^2=-1$. $\square$

Connections Master

• Angular momentum operators and $SU(2)$ 12.05.01. The square ${\hat{T}}^2=(-1{)}^{2j}$ is computed from the $2\pi$ rotation $e^{-2\pi i{J}_y/\hslash }$, whose value $-1$ on half-integer spin is the defining feature of the spinor double cover developed there. Time reversal cannot be understood without the projective representation theory of $SU(2)$: the half-integer-spin sign that produces Kramers degeneracy is the same sign a spinor acquires under a full rotation.

• Operators, observables, and Hermiticity 12.02.02. Kramers' theorem rests on $\hat{H}$ being self-adjoint so that its eigenvalues are real and $\hat{T}$ maps each eigenspace to itself; the antiunitary pull-out $\hat{T}(E\mid \psi ⟩)=\bar{E}\hat{T}\mid \psi ⟩$ collapses to $E\hat{T}\mid \psi ⟩$ only because $E$ is real. The contrast between unitary observable-symmetries and the antiunitary $\hat{T}$ sharpens the operator framework introduced there.

• Dirac equation and relativistic spin 12.11.01. The relativistic origin of half-integer spin makes Kramers degeneracy unavoidable for electrons: the four-component Dirac spinor carries the $j$ half-integer that forces ${\hat{T}}^2=-1$, and the time-reversal operator on Dirac spinors $\hat{T}=i{\gamma }^1{\gamma }^3\hat{K}$ (in a standard representation) reduces to $-i{\sigma }_2\hat{K}$ on each two-spinor block. The discrete-symmetry algebra of $C$, $P$, and $T$ on Dirac fields builds toward the CPT theorem.

• Stern-Gerlach and spin one-half 12.01.02. The spin-$\frac{1}{2}$ doublet that the Stern-Gerlach experiment first resolved is the smallest carrier of ${\hat{T}}^2=-1$, and the explicit operator $\hat{T}=-i{\sigma }_2\hat{K}$ acts on exactly the two-state space introduced there. A single electron spin is the minimal Kramers system.

Historical & philosophical context Master

Eugene Wigner introduced antiunitary operators into quantum mechanics in his 1932 paper in the Göttingen Nachrichten ^{[Wigner 1932]}, recognising that the requirement of preserving transition probabilities $\mid ⟨\varphi \mid \psi ⟩{\mid }^2$ admits two classes of implementing operators, the unitary and the antiunitary, and that motion reversal belongs to the second class. The structure theorem $\hat{T}=\hat{U}\hat{K}$ and the invariance of ${\hat{T}}^2$ under rephasing date from this work, which also supplied the symmetry-representation theorem underpinning the entire classification of quantum symmetries.

Hendrik Kramers had already published the degeneracy theorem two years earlier, in 1930, in the Proceedings of the Royal Academy of Sciences at Amsterdam ^{[Kramers 1930]}. Kramers showed, originally by an argument internal to the theory of atomic multiplets, that a system containing an odd number of electrons retains at least a two-fold degeneracy of every level in an electric field of any symmetry, however low, and that only a magnetic field removes it. Wigner's antiunitary framework supplied the conceptual reason: the odd electron count gives half-integer total angular momentum, hence ${\hat{T}}^2=-1$, hence the no-fixed-line obstruction. The combinatorial fact that this links to the Frobenius-Schur indicator $\nu =\frac{1}{\mid G\mid }{\sum }_g\chi (g^2)$, introduced by Ferdinand Frobenius and Issai Schur in 1906 for the real/complex/quaternionic classification of group representations, was recognised as the representation-theoretic shadow of the time-reversal square. Landau and Lifshitz present the synthesis in Quantum Mechanics §§95-96 ^{[Landau-Lifshitz §§95-96]}, deriving Kramers' result directly from the antiunitarity of time reversal rather than from atomic-multiplet bookkeeping.

The theorem acquired renewed importance in condensed-matter physics after 2005, when Kane and Mele and, independently, Bernevig and Zhang recognised that the Kramers pairing at time-reversal-invariant momenta of a spin-orbit-coupled band structure defines a ${\mathbb{Z}}_2$ topological invariant, the protected edge states of the quantum spin Hall insulator and the topological insulator being a direct consequence of ${\hat{T}}^2=-1$. The same antiunitary algebra that Kramers used for atomic spectra in 1930 reappears as the symmetry class AII of the tenfold Altland-Zirnbauer classification of topological matter.

Bibliography Master

@article{Wigner1932Antiunitary,
  author  = {Wigner, Eugene P.},
  title   = {{\"U}ber die Operation der Zeitumkehr in der Quantenmechanik},
  journal = {Nachrichten der Gesellschaft der Wissenschaften zu G{\"o}ttingen, Math.-Phys. Klasse},
  year    = {1932},
  pages   = {546--559}
}

@article{Kramers1930Degeneracy,
  author  = {Kramers, Hendrik A.},
  title   = {Th{\'e}orie g{\'e}n{\'e}rale de la rotation paramagn{\'e}tique dans les cristaux},
  journal = {Proceedings of the Royal Academy of Sciences at Amsterdam},
  volume  = {33},
  year    = {1930},
  pages   = {959--972}
}

@book{WignerGroupTheory1959,
  author    = {Wigner, Eugene P.},
  title     = {Group Theory and its Application to the Quantum Mechanics of Atomic Spectra},
  publisher = {Academic Press},
  year      = {1959},
  note      = {English translation by J. J. Griffin of the 1931 German edition}
}

@article{FrobeniusSchur1906,
  author  = {Frobenius, Ferdinand G. and Schur, Issai},
  title   = {{\"U}ber die reellen Darstellungen der endlichen Gruppen},
  journal = {Sitzungsberichte der K{\"o}niglich Preussischen Akademie der Wissenschaften zu Berlin},
  year    = {1906},
  pages   = {186--208}
}

@book{LandauLifshitzQM,
  author    = {Landau, Lev D. and Lifshitz, Evgeny M.},
  title     = {Quantum Mechanics: Non-Relativistic Theory},
  edition   = {3},
  publisher = {Pergamon Press},
  series    = {Course of Theoretical Physics, Vol. 3},
  year      = {1977}
}

@book{SakuraiNapolitano2017,
  author    = {Sakurai, J. J. and Napolitano, Jim},
  title     = {Modern Quantum Mechanics},
  edition   = {2},
  publisher = {Cambridge University Press},
  year      = {2017}
}

@book{MessiahQM,
  author    = {Messiah, Albert},
  title     = {Quantum Mechanics, Vol. II},
  publisher = {North-Holland},
  year      = {1962}
}

@article{KaneMele2005,
  author  = {Kane, C. L. and Mele, E. J.},
  title   = {$Z_2$ Topological Order and the Quantum Spin Hall Effect},
  journal = {Physical Review Letters},
  volume  = {95},
  year    = {2005},
  pages   = {146802}
}

@article{AltlandZirnbauer1997,
  author  = {Altland, Alexander and Zirnbauer, Martin R.},
  title   = {Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures},
  journal = {Physical Review B},
  volume  = {55},
  year    = {1997},
  pages   = {1142--1161}
}