An elementary proof for the non-bijective version of Wigner's theorem

TL;DR

This paper provides a new, simple, and short proof of the non-bijective Wigner's theorem, showing that maps preserving transition probabilities on pure states are induced by linear or antilinear isometries, without assuming bijectivity or separability.

Contribution

The paper introduces an elementary and concise proof of the non-bijective Wigner's theorem, removing the need for assumptions like bijectivity or space separability.

Findings

Proof is elementary and very short

Does not assume bijectivity of the map

Applicable to non-separable spaces

Abstract

The non-bijective version of Wigner's theorem states that a map which is defined on the set of self-adjoint, rank-one projections (or pure states) of a complex Hilbert space and which preserves the transition probability between any two elements, is induced by a linear or antilinear isometry. We present a completely new, elementary and very short proof of this famous theorem which is very important in quantum mechanics. We do not assume bijectivity of the mapping or separability of the underlying space like in many other proofs.