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

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.

4 pages

References in corpus (1)

• Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

Cited by in corpus (5)

• Wigner symmetries and Gleason's theorem
• Supercomputer simulations of transmon quantum computers
• Maps on classes of Hilbert space operators preserving measure of commutativity
• Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators
• Phase-isometries on real normed spaces