Proof of Wigner’s Theorem

Pankaj Sharan

Máté Győry

György Pál Gehér

Daniel D. Spiegel

This paper presents a constructive proof of a non-bijective, non-separable form of Wigner's theorem, that uses only a few basic facts about Hilbert spaces, including the existence of orthonormal bases and the Fourier decomposition of a vector. Our proof is based on a proof by Steven Weinberg from 1995, but improves on that proof with greater simplicity and generality. Furthermore, our proof fills in a few holes where Weinberg is not completely mathematically rigorous.

Daniel D. Spiegel

This expository note presents a constructive proof of Wigner's theorem using only a few basic facts about Hilbert spaces, such as the existence of orthonormal bases and the Fourier decomposition of a vector. Our proof is based on a proof by Steven Weinberg found in the first volume of his series of textbooks on quantum field theory, but differs in a few places for the sake of greater simplicity and rigor.