The Quantum Logic of Direct-Sum Decompositions

35 Pages Posted: 27 Apr 2016

Date Written: April 12, 2016

Abstract

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space -- which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space -- which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over ℤ₂ where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2.

Keywords: quantum logic, direct-sum decompositions, q-analogs

Suggested Citation

Ellerman, David, The Quantum Logic of Direct-Sum Decompositions (April 12, 2016). Available at SSRN: https://ssrn.com/abstract=2770163 or http://dx.doi.org/10.2139/ssrn.2770163

David Ellerman (Contact Author)

University of Ljubljana ( email )

School of Social Science
Ljubljana, CA
Slovenia

HOME PAGE: http://www.ellerman.org

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