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Logic of infinite quantum systems. (English) Zbl 0799.03019
Summary: Limits of sequences of finite-dimensional (AF) \(C^*\)-algebras, such as the CAR algebra for the ideal Fermi gas, are a standard mathematical tool to describe quantum statistical systems arising as thermodynamic limits of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the thermodynamic potentials, and handle the proliferation of physically inequivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AF \(C^*\)- algebras correspond to countable Boolean algebras, i.e., algebras of propositions in the classical two-valued calculus. We investigate the noncommutative logic properties of general AF \(C^*\)-algebras, and their corresponding systems. We stress the interplay between Gödel incompleteness and quotient structures – in the light of the “nature does not have ideals” program, stating that there are no quotient structures in physics. We interpret AF \(C^*\)-algebras as algebras of the infinite-valued calculus of Łukasiewicz, i.e., algebras of propositions in Ulam’s “twenty questions” game with lies.

MSC:
03B50 Many-valued logic
03G25 Other algebras related to logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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