Time reversal and Kramers degeneracy

Take a time-independent Hamiltonian and write the reversed state as $\Theta|\psi(-t)\rangle$. The time derivative contributes a minus sign. Antilinearity supplies a second sign by conjugating $i$, allowing the reversed state to obey the same equation when $\Theta H\Theta^{-1}=H$. A purely unitary map with unchanged Hamiltonian would generally fail this test. For a spinless particle in a real scalar potential, component conjugation in the position basis is a convenient representation of time reversal.

For spin one half choose $\Theta=i\sigma_yK$. An overall phase of this operator is conventional; because it is antiunitary, changing that phase does not change its square. Applied to a column $(a,b)^T$, the chosen operator gives $(b^*,-a^*)^T$. It reverses every spin expectation, and applying it twice returns minus the original column. Complex conjugation alone would reverse the y component but leave the x and z components unchanged, so it is insufficient for a physical spin-one-half reversal.

Suppose a normalizable energy eigenstate belongs to a time-reversal-invariant Hamiltonian and the relevant state space has $\Theta^2=-I$. Since energy is real, applying the antiunitary operator to the eigenvalue equation gives another state with the same energy. To prove it is different, use the antiunitary inner-product identity on the pair consisting of the state and its reversal. The identity turns the overlap into its negative once the operator square is inserted, forcing the overlap to vanish.

This proof does not require rotational symmetry, a conserved spin component, or a spherically symmetric potential. Spin-orbit interactions can mix individual spin directions while preserving time reversal and the paired spectrum. In ordinary nonrelativistic systems, an odd number of spin-one-half fermions gives the required square, while an even number need not. The decisive theorem hypothesis is the operator square on the relevant sector, rather than a claim that every object described informally as magnetic must be doubly degenerate.

A fixed external magnetic field is odd under time reversal. Reversing the spin while holding that field fixed generally changes the Zeeman term, so the same Hamiltonian loses the symmetry and Kramers protection no longer applies. Reversing both the state and the field relates two different experimental settings; it does not prove degeneracy at one setting. A perturbation breaking time reversal permits a splitting, but need not produce a nonzero splitting for every specially chosen level or parameter value.

In a translation-invariant crystal, reversal also changes crystal momentum from $\mathbf k$ to $-\mathbf k$. Equal energies at these two labels do not automatically mean two distinct states at the same generic momentum. A same-momentum Kramers pair is guaranteed at time-reversal-invariant momenta, where the labels agree modulo a reciprocal lattice vector, under the relevant symmetry assumptions. Additional spatial symmetries can strengthen degeneracy elsewhere. For $\Theta^2=+I$, a time-invariant Hamiltonian can have an entirely nondegenerate spectrum.