2.2 Wigner’s theorem
According to E. Wigner, a symmetry transformation \(T\) acting on a quantum system is a bijection \[\begin{equation} \tag{2.6} T: \mathbb{P}\mathcal{H} \to \mathbb{P}\mathcal{H}, \end{equation}\] that preserves the transition probability (2.3). We call a bijection \(U\) on \(\mathcal{H}\) is compatible with \(T\) if \(U\) induces the same action on \(\mathbb{P}\mathcal{H}\) as \(T\).
Then, Wigner’s theorem states:
Theorem 2.1 (Wigner 1931) Given a symmetry transformation \(T\), there exists a bijection \(U_T\) on \(\mathcal{H}\) compatible with \(T\). This \(U_T\) is either linear (over \(\mathbb{C}\)) and unitary, or anti-linear and anti-unitary. If \(\mathrm{dim}\mathcal{H}\ge 2\), \(U_T\) is unique up to a overall phase redefinition \(U_T \mapsto e^{\mathrm{i}\alpha}U_T\). (When \(\mathrm{dim}\mathcal{H} = 1\), \(T\) is unique, and \(U_T\) can be taken either of unitary or anti-unitary one. Once the choice is fixed, it is up to the phase.)
This is why we usually care about unitary operator (like Pauli matrices). The proof can be found somewhere, e.g. [8]. We will see examples soon.
We also have an action of \(T\) on the algebra of operators \(\mathcal{A}\) through this theorem: \[\begin{align} \tag{2.7} T \curvearrowright \mathcal{A} & \to \mathcal{A} \\ \mathcal{O} & \mapsto U_T \mathcal{O} U^\dagger_T. \end{align}\] Note that this action is independent of the phase freedom of \(U_T\) and thus uniquely defined. (2.5) is a special case of this.
Finally, we call a symmetry \(T\) is preserved by the Hamiltonian if \[\begin{equation} \tag{2.8} [U_T,H] = 0. \end{equation}\]