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A Constructive Proof of a General Wigner's Theorem.

Daniel D. Spiegel

Published Oct 23, 2018• 0 citations
OpenAlex

Abstract

This paper presents a constructive proof of a non-bijective, non-separable form of Wigner's theorem, that uses only a few basic facts about Hilbert spaces, including the existence of orthonormal bases and the Fourier decomposition of a vector. Our proof is based on a proof by Steven Weinberg from 1995, but improves on that proof with greater simplicity and generality. Furthermore, our proof fills in a few holes where Weinberg is not completely mathematically rigorous.

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This paper presents a constructive proof of a non-bijective, non-separable form of Wigner's theorem, that uses only a few basic facts about Hilbert spaces, including the existence of orthonormal bases and the Fourier decomposition of a vector. Our proof is based on a proof by Steven Weinberg from 1995, but improves on that proof with greater simplicity and generality. Furthermore, our proof fills in a few holes where Weinberg is not completely mathematically rigorous.
Abstract It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H H{\mathcal H}H correspond to direct product decompositions H ≃ A × A ⊥ H≃A×A⊥\mathcal{H}\simeq A\times A^\perpH≃A×A⊥ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.
Orthogonality preserving transformations on the set of nnn-dimensional subspaces of a Hilbert space
We characterize bijective transformations on the set of all nnn-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.
A systematic study of additive isometries on a quaternionic Hilbert space is presented. A number of new results describing the properties of such operators are proved. The work culminates in the first mathematical proof of Wigner’s theorem for quaternionic Hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic Hilbert space is equivalent, in the sense of differing pointwise by a mere phase factor, to a linear isometry. A complete and concise description of the exceptional situation in a two-dimensional quaternionic Hilbert space is given.