The use of projective limits in classical statistical mechanics and Euclidean quantum field theory

Abstract

The Gibbs states of an infinite classical statistical system correspond to the states of “reservoir at infinity”. It is shown that its configuration space can be thought of as a generalized projective limit of configuration spaces of remote reservoirs. This notion of projective limit is defined and it is noted that it can also be used e.g. for proofs of the existence of Gibbs states in the thermodynamic limit and their decomposition into pure phase. A similar approach to (nonperturbative) Euclidean quantum field theory is suggested and connections with the concept of renormalizability are found.