A Note on the Resolvent Algebra and Functional Integral Approach to the Free Bose Einstein Condensation

TL;DR

This paper systematically analyzes Bose-Einstein condensation in the free Bose gas using operator algebra and functional integrals, clarifying the structure of phase transitions and establishing their equivalence.

Contribution

It provides a rigorous framework connecting operator-algebraic and functional integral approaches to BEC, elucidating phase transition features in the free Bose gas.

Findings

Clarified the representation-theoretic structure of finite-temperature BEC states

Established the equivalence between measure decomposition and operator algebra representations

Provided a foundation for analyzing phase transitions in quantum field theory

Abstract

We present a systematic description of the structure of Bose-Einstein condensation (BEC) in the free Bose gas from the viewpoint of the correspondence between the operator-algebraic formulation based on the resolvent algebra and the functional integral representation. By clarifying the representation-theoretic structure of finite-temperature BEC states and rigorously analyzing the correspondence between their direct integral decomposition and the ergodic decomposition of the associated probability measures, we provide a framework in which general features of phase transitions-such as the emergence of order parameters, the decomposition of states, and clustering properties-are explicitly described using BEC in the free Bose gas as a concrete example. Furthermore, we construct in detail the correspondence between the decomposition of measures in the functional integral approach and that of operator-algebraic representations, thereby establishing the equivalence between the probabilistic and algebraic aspects, and providing a guiding principle for isolating the essential structures by disentangling the additional mathematical complications arising from the treatment of infrared singularities in interacting systems. These results lay a foundation for the rigorous analysis of phase transitions in non-relativistic constructive quantum field theory and quantum statistical mechanics, and serve as a starting point for extensions to interacting models.