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International Journal of Theoretical Physics

Boolean transfer from coherent quantum logics to quantum logics with continuous superselection rules

Hirokazu Nishimura

Published May 1, 1992• 7 citations
Publisher
DOI
10.1007/bf00678550

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From the logic approach to quantum and classical mechanics, the W*−algebraic approach is deduced in dependence of a suitable ’’prestate.’’ An algebraic representation of the logic description is in fact constructed in a framework in which continuous superselection rules can be present. Logic propositions, observables, and states are represented by decomposable projections, decomposable self−adjoint operators, and normal states in a direct integral of Hilbert spaces. In this representation each algebraic term becomes the representative of a homologous logic one and the expectation values as well as the superselection rules are conserved. When a principle of ’’undistinguishability’’ is taken into account, the representation is faithful. In the classical case, the representation results in Koopman’s formalism.
Superselection rules (SSRs) limit the mechanical and quantum processing resources represented by quantum states. However, SSRs can be violated using reference systems to break the underlying symmetry. We show that there is a duality between the ability of a system to do mechanical work and to act as a reference system. Further, for a bipartite system in a globally symmetric pure state, we find a triality between the system's ability to do local mechanical work, its ability to do ``logical work'' due to its accessible entanglement, and its ability to act as a shared reference system.