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An elementary proof for the non-bijective version of Wigner’s theorem. (English) Zbl 1331.46066
Summary: 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.

MSC:
46N50 Applications of functional analysis in quantum physics
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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