Symmetry and Topology in Quantum Logic |
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Authors: | Email author" target="_blank">Alexander?WilceEmail author |
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Institution: | (1) Department of Mathematical Sciences, School of Natural and Social Sciences, Susquehanna University, Selinsgrove, Pennsylvania, 17870 |
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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.
PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta. |
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Keywords: | quantum logics symmetry orthoalgebras compact groups |
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