Kramers' theorem

Kramers' theorem states that a time-reversal-invariant quantum system with half-integer total angular momentum has energy levels that are at least doubly degenerate.

Time-reversal symmetry: A symmetry that reverses time while leaving the physical laws or system dynamics unchanged. The theorem requires the system's Hamiltonian to be invariant under this symmetry.
Spin (physics): An intrinsic form of angular momentum carried by quantum particles. Half-integer spin gives states the sign change that underlies Kramers degeneracy.
Kramers degeneracy: The at-least-twofold degeneracy enforced by time-reversal symmetry for states with half-integer angular momentum. It is the spectral consequence used directly in quantum systems.
Hendrik Kramers: A Dutch physicist whose work in quantum theory included the degeneracy theorem bearing his name. Kramers established the result while analyzing quantum states in the 1930s.
Time-reversal operator: The antiunitary quantum operator that reverses momenta and angular momenta while complex-conjugating amplitudes. Its antiunitary action produces the partner state required by the theorem.
Half-integer: A number of the form n plus one-half, where n is an integer. The theorem applies when total angular momentum takes half-integer values.
Kramers-Weyl fermion: A chiral quasiparticle occurring at a time-reversal-invariant momentum in certain noncentrosymmetric crystals. Kramers degeneracy helps constrain states at time-reversal-invariant momenta.
Quantum mechanics: The physical theory describing matter and radiation through quantum states, observables, and probabilistic predictions. The theorem is a consequence of quantum-state symmetry and angular momentum.
Kramers pair: Two orthogonal quantum states related by time reversal and sharing the same energy in a Kramers-degenerate system. These are the degenerate partners whose existence the theorem guarantees.
Quantum degeneracy: The equality of energy eigenvalues for two or more distinct quantum states. Kramers' theorem guarantees a minimum twofold energy degeneracy.

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Symmetry in quantum mechanicsBroader topic: It is a direct spectral consequence of time-reversal symmetry.