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 Godel 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.