Wigner's theorem revisited

TL;DR

This paper provides a comprehensive solution to a functional equation involving norms in inner product spaces, extending Wigner's theorem to a broader class of equations with potential applications in quantum mechanics.

Contribution

It offers the general solution to a specific norm-based functional equation in inner product spaces, utilizing a real version of Wigner's theorem, and explores related equations.

Findings

Solved the functional equation in inner product spaces

Extended Wigner's theorem to new functional contexts

Provided insights into related norm equations

Abstract

In this paper, we give the general solution of the functional equation $$\big\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\big\}=\big\{\|x+y\|,\|x-y\|\big\}\qquad(x,y\in X)$$ where $f:X\to Y$ and $X,Y$ are inner product spaces. Related equations are also considered. Our main tool is a real version of Wigner's unitary--antiunitary theorem.