Wigner’s symmetry representation theorem

Uhlhorn approach

In the Uhlhon approach, it is only assumed that \(T\) maps orthogonal rays to orthogonal rays: \(\perp\)-symmetry. Chevalier begins with a lemma:

Lemma 7

1. In an inner product space \(H\), \(n+1\) vectors \(\varphi_1, \dots, \varphi_{n+1}\) are linearly independent exactly if the \(n\) first vectors are linearly independent and there exists a vector \(\psi\) which is orthogonal to the first \(n\) vectors and non-orthogonal to \(\varphi_{n+1}\).

Applying Gram-Schmidt orthogonalization to the linearly independent sequence, results in a sequence of orthogonal vectors \(\psi_1(=\varphi_1), \dots, \psi_{n+1}\). The last vector \(\psi_{n+1}\) is orthogonal to the first \(n\) vectors of the original sequence, but not \(\varphi_{n+1}\).

Conversely, we have \(\varphi_1, \dots, \varphi_n\) linearly independent, and orthogonal \(\psi\) that is not orthogonal to \(\varphi_{n+1}\). If \(\lambda_1\varphi_1+ \dots+\lambda_{n+1} \varphi_{n+1}=0\), we have \(\langle \psi | \lambda_1\varphi_1+ \dots+\lambda_{n+1} \varphi_{n+1}\rangle=\lambda_{n+1}\langle \psi|\varphi_{n+1}\rangle=0\implies \lambda_{n+1}=0\), since \(\langle \psi|\varphi_{n+1}\rangle\ne0\). Therefore \(\lambda_1\varphi_1+ \dots+\lambda_n \varphi_n=0\implies \lambda_k=0, 1\le k\le n\), since the first \(n\) vectors are linearly independent. Thus:

2. Every bijective mapping \(F:H\rightarrow H'\) preserving orthogonality of vectors in both directions also preserves linear independence of vectors in both directions and its extension to the set of all rays of \(H,H'\) preserves independence of rays in both directions.