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International Journal of Theoretical Physics

, Volume 44, Issue 12, pp 2303–2316 | Cite as

Symmetry and Topology in Quantum Logic

  • Alexander Wilce
Article

Abstract

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic.

Keywords

quantum logics symmetry orthoalgebras compact groups 

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References

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, School of Natural and Social SciencesSusquehanna UniversitySelinsgrove

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