Wigner’s symmetry representation theorem

Additive bounded mappings

Chevalier, Proposition 2, Remark 3. An additive bounded mapping \(F\) is a sum of linear and antilinear operators.

Additivity means that \(F(\varphi+\psi)=F(\varphi)+F(\psi)\). We note that for \(n \in \mathbb Z\), \(F(n\varphi)=nF(\varphi)\). Further, \(F(\varphi)=F(n\varphi/n)=n F(\varphi/n)\implies F(\varphi/n)=F(\varphi)/n\). Hence, for \(q \in \mathbb Q\), \(F(q\varphi)=q F(\varphi)\).

The mapping \(F\) is bounded, so there exists a real constant \(M\): \(|F(\varphi)|\lt M |\varphi|,\forall \varphi\). This allows us to derive continuity: \(|F(\varphi+\psi)-F(\varphi)|=|F(\psi)|\lt M |\psi|\). Thus Hilbert space completeness gives: \(|\psi|\rightarrow 0\implies F(\varphi+\psi)\rightarrow F(\varphi)\). From continuity, we get linearity over the reals: \(x \in \mathbb R \implies F(x\varphi)=x F(\varphi)\).

We are ready to define our linear \(L\) and antilinear \(A\) operators: \(L(\varphi)=(F(\varphi)-\mathrm i F(\mathrm i\varphi))/2\) and \(A(\varphi)=(F(\varphi)+\mathrm i F(\mathrm i\varphi))/2\). Clearly, \(F=L+A\). The additivity of \(F\) implies that of \(L,A\).

The final part is to show, for \(z \in \mathbb C\), that \(L(z\varphi)=z L(\varphi),A(z\varphi)=z^* A(\varphi)\). As an intermediate step, we evaluate \(L(\mathrm i \varphi)=(F(\mathrm i \varphi)+\mathrm i F(\varphi))/2=\mathrm i L(\varphi)\). Similarly, \(A(\mathrm i \varphi)=(F(\mathrm i \varphi)-\mathrm i F(\varphi))/2=-\mathrm i A(\varphi)\). Applying the linearity over reals, and these expressions, gives the respective linearity/antilinearity.