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Wigner's theorem for an infinite set

  • John Harding EMAIL logo
Published/Copyright: October 20, 2018
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 68 Issue 5

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 correspond to direct product decompositions H≃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.

MSC 2010: Primary 81P10; Secondary 06C15; 20C35; 51A05; 81R99
Keywords: Wigner’s theorem; orthomodular poset; automorphism; group representation; direct product decomposition

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Received: 2016-07-24
Accepted: 2017-08-31
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0178
Keywords for this article
Wigner’s theorem; orthomodular poset; automorphism; group representation; direct product decomposition

Articles in the same Issue

  1. Prof. RNDr. Beloslav Riečan, DrSc., Dr.h.c. *NOV. 10, 1936 – †AUG. 13, 2018
  2. On the n × n × n Rubik's Cube
  3. Congruences involving alternating harmonic sums modulo pαqβ
  4. On the proximity of large primes
  5. On the factorizations of cubic polynomials with the same discriminant modulo a prime
  6. Some results on abstract convexity of functions
  7. Remarks on b-Metric and metric-preserving functions
  8. Generalization of Ostrowski inequality for convex functions
  9. On microscopic sets and Fubini Property in all directions
  10. New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator
  11. Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations
  12. Approximate sentinels for diffusion phenomena with pollution
  13. Remarks on a semilinear system in ℝn motivated by difference equations
  14. Oscillation tests for difference equations with several non-monotone deviating arguments
  15. On the regularity of one-sided fractional maximal functions
  16. A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space
  17. A note about Volterra operator
  18. Some reverse and numerical radius inequalities
  19. A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds
  20. Sequential decreasing strong size properties
  21. Approximation of Information Divergences for Statistical Learning with Applications
  22. Wigner's theorem for an infinite set
  23. A note on automorphisms of lie ideals in prime rings
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