Logic of infinite quantum systems

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 Lukasiewicz, i.e., algebras of propositions in Ulam's "twenty questions" game with lies.