Wigner’s symmetry representation theorem

Bargmann 1 — lemma 1

Chevalier, Lemma 1. Let \(H,H^{\prime}\) be complex inner product spaces and consider a mapping \(f\) from a subset \(D\) of \(H\) into \(H^{\prime}\) which preserves the modulus of the inner product. For any pair \(\varphi_{1,2}\) of nonzero mutually orthogonal vectors of \(D\) and any pair of complex numbers \(\alpha,\beta\) such that \(\alpha\varphi_{1}+\beta\varphi_{2}\in D\) there exist \(\alpha^{\prime},\beta^{\prime}\in\mathbb{C}\) such that \[f\left(\alpha\varphi_{1}+\beta\varphi_{2}\right)=\alpha^{\prime}f\varphi_{1}+\beta^{\prime}f\varphi_{2}\]Moreover, \(\left|\alpha\right|=\left|\alpha^{\prime}\right|,\left|\beta\right|=\left|\beta^{\prime}\right|\).

The mapping \(f\) satisfies \(\left|\langle f\psi|f\varphi\rangle\right|=\left|\langle\psi|\varphi\rangle\right|\) for \(\varphi,\psi\in D\).

We have \(\left|\langle f\alpha\varphi|f\varphi\rangle\right|=\left|\langle\alpha\varphi|\varphi\rangle\right|=\left|\alpha\right|\left\Vert \varphi\right\Vert ^{2}=\left\Vert f\alpha\varphi\right\Vert \left\Vert f\varphi\right\Vert\). The extremes can only be equal if the vectors are linearly dependent (Schwarz inequality). Thus \(f\alpha\varphi=\alpha^{\prime}f\varphi\). The inner product also requires \(\left|\alpha\right|=\left|\alpha^{\prime}\right|\).

If \(\langle\varphi_{1}|\varphi_{2}\rangle=0\), i.e. the vectors are ‘orthogonal’, then \(f\varphi_{1,2}\) are also orthogonal. We can Fourier expand: \[\left|f\left(\alpha\varphi_{1}+\beta\varphi_{2}\right)\right\rangle =\frac{\left|f\varphi_{1}\right\rangle \langle f\varphi_{1}|f\left(\alpha\varphi_{1}+\beta\varphi_{2}\right)\rangle}{\left\Vert f\varphi_{1}\right\Vert ^{2}}+\frac{\left|f\varphi_{2}\right\rangle \langle f\varphi_{2}|f\left(\alpha\varphi_{1}+\beta\varphi_{2}\right)\rangle}{\left\Vert f\varphi_{2}\right\Vert ^{2}}\]

The two sides have the same norm, \(\left|\alpha\right|^{2}\left\Vert \varphi_{1}\right\Vert ^{2}+\left|\beta\right|^{2}\left\Vert \varphi_{2}\right\Vert ^{2}\), so there are no missing vector components in the subspace orthogonal to \(f\varphi_{1,2}\). [Chevalier’s presentation assumes that if a vector is in the domain, a corresponding unit vector is too.]