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

2014/06/01 by György Pál Gehér, Gy. P. Gehér · 69 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Analytic proof #Bijection #Discrete mathematics #Elementary proof #Graph theory and applications #Hilbert space #Isometry (Riemannian geometry) #Mathematical proof #Mathematics #Pure mathematics #math-ph #math.MP #msc:46C05 #msc:46C50

paper · pdf · doi:10.1016/j.physleta.2014.05.039

published in Physics Letters A 378(30-31), 2054-2057 (Elsevier BV) · 4 pages

openalex publication_date 2014/06/01 · arxiv created 2014/07/02 · arxiv updated 2014/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• An algebraic approach to Wigner's unitary-antiunitary theorem
• Note on Wigner's Theorem on Symmetry Operations
• Symmetry Groups in Quantum Mechanics and the Theorem of Wigner on the Symmetry Transformations