# Definition The time-reversal operator $\Theta$ satisfies the following postulates. Since probability must be conserved under time-reversal, we have: $ \Theta^\dagger \Theta = 1 $ Moreover, it satisfies the conjugation rules (based on classical mechanics): $ \Theta \pmb{x} \Theta^\dagger = \pmb{x} \qquad\Theta \pmb{p} \Theta^\dagger = -\pmb{p} $ This implies $ \Theta \pmb{L} \Theta^\dagger = -\pmb{L} $ We also postulate $ \Theta \pmb{S} \Theta^\dagger = -\pmb{S} $ Thus, for any angular momentum $\pmb{J}$, we have: $ \Theta \pmb{J} \Theta^\dagger = -\pmb{J} $ These postulates lead to the conclusion that $\Theta$ cannot be unitary, i.e., it is an antiunitary operator (an antilinear operator that satisfies $\Theta^\dagger \Theta = 1$). # Properties From the definition of antilinearity: $ \Theta c = c^* \Theta $ Moreover, we have: $ (\bra{\phi} \Theta) \ket{\psi} = [\bra{\phi} (\Theta \ket{\psi})]^* $ i.e., with antilinear operators it matters whether it acts on the right or the left in the matrix element. If we change directions, we must conjugate the matrix element. We can also conclude $ \bra{\phi} (\Theta^\dagger \ket{\psi}) = [(\bra{\psi} \Theta) \ket{\phi}]^* $ # Spin ## Spinless Systems In spinless systems, we simply take the time-reversal operator as $ \Theta = K $ where $K$ is complex conjugation. Note that $\Theta^2 = 1$. ## Spin-1/2 systems Here we have $ \Theta = K(-i \sigma_y) $ Note that $\Theta^2 = -1$. This leads to [[Kramers Theorem|Kramers degeneracy]]. *Resources:* [Robert Littlejohn's Notes](http://bohr.physics.berkeley.edu/classes/221/notes/timerev.pdf)