Time reversal is an antiunitary operator T , that acts on α α c i , (where α α , is an arbitrary complex coefficient, and ∗ ∗ , denotes complex conjugation) as T α α c i T − − 1 = α α ∗ ∗ ( U T ) i j c j . It can be written as T = U T K where K is the complex conjugation operator and U T is a unitary matrix. Either T 2 = 1 or T 2 = − − 1 . A Hamiltonian with time reversal symmetry satisfies T H ^ ^ T − − 1 = H ^ ^ , or on the level of first-quantised matrices, U T H ∗ ∗ U T − − 1 = H , for some choice of U T .