On the realization of Symmetries in Quantum Mechanics

TL;DR

This paper provides a simple, geometric proof of Wigner's theorem, showing its connection to projective geometry and emphasizing its importance for the linear structure of quantum theory.

Contribution

It offers a new, accessible geometric proof of Wigner's theorem, highlighting its fundamental role in the linear realization of symmetries in quantum mechanics.

Findings

Wigner's theorem is a corollary of the fundamental theorem of projective geometry.

The geometric proof clarifies the relation between symmetries and projective geometry.

The proof is simple and accessible for elementary quantum mechanics treatments.

Abstract

The aim of this paper is to give a simple, geometric proof of Wigner's theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner's theorem is not fully appreciated in general. It is Wigner's theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner's theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.