Wigner's theorem for an infinite set

  • 1 Department of Mathematical Sciences, NM 88003, Las Cruces, USA
John Harding
  • Corresponding author
  • Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

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 HA×A of the Hilbert space, a result that lies at the heart of the superposition principle. In [] 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.

  • [1]

    BIRKHOFF, G.—VON NEUMANN, J.: The logic of quantum mechanics, Ann. of Math. (2) 37 (4) (1936), 823–843.

  • [2]

    BURRIS, S.—SANKAPPANAVAR, H. P.: A Course in Universal Algebra. Graduate Texts in Math. 78, Springer, 1981.

  • [3]

    GREECHIE, R. J.: A particular non-atomistic orthomodular poset, Comm. Math. Phys. 14 (1969), 326–328.

  • [4]

    CHEVALIER, G.: Automorphisms of an orthomodular poset of projections, Internat. J. Theoret. Phys. 44 (7) (2005), 985–998.

  • [5]

    CHEVALIER, G.: The orthomodular poset of projections of a symmetric lattice, Internat. J. Theoret. Phys. 44 (11) (2005), 2073–2089.

  • [6]

    CHEVALIER, G.: Wigner’s theorem and its generalizations. In: The Handbook of Quantum Logic and Quantum Structures, Engesser, Gabbay, and Lehmann (eds.), Elsevier, 2007, pp. 429–475.

  • [7]

    CHEVALIER, G.: Wigner type theorems for projections, Internat. J. Theoret. Phys. 47 (1) (2008), 69–80.

  • [8]

    DIXON, J.—MORTIMER, B.: Permutation Groups. Graduate Texts in Math. 163, Springer, 1996.

  • [9]

    HANNAN, T.—HARDING, J.: Automorphisms of decompositions, Math. Slovaca, to appear.

  • [10]

    HARDING, J.: Decompositions in quantum logic, Trans. Amer. Math. Soc. 348 (5) (1996), 1839–1862.

  • [11]

    HARDING, J.: Regularity in quantum logic, Internat. J. Theoret. Phys. 37 (4) (1998), 1173–1212.

  • [12]

    HARDING, J.: Axioms of an experimental system, Internat. J. Theoret. Phys. 38 (6) (1999), 1643–1675.

  • [13]

    HARDING, J.: A link between quantum logic and categorical quantum mechanics, Internat. J. Theoret. Phys. 48 (3) (2009), 769–802.

  • [14]

    HARDING, J.: Orthomodularity of decompositions in a categorical setting, Internat. J. Theoret. Phys. 45 (6) (2006), 1117–1127.

  • [15]

    HARDING, J.: Dynamics in the decompositions approach to quantum mechanics, Internat. J. Theoret. Phys., to appear.

  • [16]

    HARDING, J.—YANG, T.: Sections in orthomodular structures of decompositions, Houston J. Math., to appear.

  • [17]

    HARDING, J.—YANG, T.: The logic of bundles, Internat. J. Theoret. Phys., to appear.

  • [18]

    KALMBACH, G.: Orthomodular Lattices. London Math. Soc. Monogr. 18, Academic Press, Inc. London, 1983.

  • [19]

    MACKEY, G. W.: The Mathematical Foundations of Quantum Mechanics, A Lecture-Note Volume by W. A. Benjamin, Inc., New York-Amsterdam, 1963.

  • [20]

    McKENZIE, R.—McNULTY, G.—TAYLOR, W.: Algebras, Lattices, Varieties, vol. 1, Wadsworth & Brooks/Cole, 1987.

  • [21]

    MUSHTARI, D. K.: Projection logics in Banach spaces, Soviet Math. (Iz. VUZ) 33 (1989), 59–70.

  • [22]

    OVCHINNIKOV, P.: Automorphisms of the poset of skew projections, J. Funct. Anal. 115 (1993), 184–189.

  • [23]

    PTA´ K, P.—PULMANNOVA´ , S.: Orthomodular Structures as Quantum Logics. Fundam. Theor. Phys. 44, Kluwer Academic Publishers Group, Dordrecht, 1991.

  • [24]

    SINGER, S. F.: Linearity, Symmetry, and Prediction in the Hydrogen Atom, Springer, 2005.

  • [25]

    UHLHORN, U.: Representation of symmetry transformations in quantum mechanics, Ark. Fys. 23 (1963), 307–340.

  • [26]

    VARADARAJAN, V. S.: Geometry of Quantum Theory, second ed., Springer-Verlag, 1985.

  • [27]

    WIGNER, E. P.: Group Theory and its Applications to the Quantum Mechanics of the Atomic Spectra, Academic Press, 1959.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


Journal + Issues

Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

Search