A Simple Constructive Proof of Wigner's Theorem

TL;DR

This paper provides a straightforward, constructive proof of Wigner's theorem, simplifying the original proof by Weinberg and emphasizing basic Hilbert space properties for clarity and rigor.

Contribution

It introduces a new, simpler constructive proof of Wigner's theorem using fundamental Hilbert space facts, improving accessibility and rigor.

Findings

Proof is constructive and elementary

Relies only on basic Hilbert space facts

Enhances understanding of Wigner's theorem

Abstract

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.