Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions

Abstract

The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical (QSM) system, built from abelian class field theory. We introduce a general notion of isomorphism of QSM-systems and prove that it preserves (extremal) KMS equilibrium states. We prove that two number fields with isomorphic quantum statistical mechanical systems are arithmetically equivalent, i.e., have the same zeta function. If one of the fields is normal over Q, this implies that the fields are isomorphic. Thus, in this case, isomorphism of QSM-systems is the same as isomorphism of number fields, and the noncommutative space built from the abelianized Galois group can replace the anabelian absolute Galois group from the theorem of Neukirch and Uchida.