Proof
By the definition of an antiunitary operator, , where and are vectors in . Replacing and and using that , we get which implies that .
Consequently, if a Hamiltonian is time-reversal symmetric, i.e. it commutes with , then all its energy eigenspaces have even degeneracy, since applying to an arbitrary energy eigenstate gives another energy eigenstate that is orthogonal to the first one. The orthogonality property is crucial, as it means that the two eigenstates and represent different physical states. If, on the contrary, they were the same physical state, then for an angle , which would imply
To complete Kramers degeneracy theorem, we just need to prove that the time-reversal operator acting on a half-odd-integer spin Hilbert space satisfies . This follows from the fact that the spin operator represents a type of angular momentum, and, as such, should reverse direction under :
Concretely, an operator that has this property is usually written as
where is the spin operator in the direction and is the complex conjugation map in the spin basis.[2]
Since has real matrix components in the basis, then
Hence, for half-odd-integer spins , we have . This is the same minus sign that appears when one does a full rotation on systems with half-odd-integer spins, such as fermions.