Constructive Quantum Field Theory and Rigorous Statistical Mechanics via Operator Algebras and Probability Theory -- Guiding Principles and Research Perspectives

TL;DR

This paper discusses an operator-algebraic framework for quantum systems, emphasizing the role of C*-algebras and von Neumann algebras, and explores probabilistic methods for quantum field theory and statistical mechanics.

Contribution

It introduces a hierarchical operator-algebraic approach and highlights the resolvent algebra as a natural object for bosonic systems, guiding future research perspectives.

Findings

Resolvent algebra better captures quantum structures than Weyl algebra.

Center in weak closure reflects phase transition sector structures.

Representation-equivalence links operator algebras to functional integrals.

Abstract

We present a hierarchical viewpoint on the operator-algebraic formulation of quantum systems, in which $C^{*}$-algebras are responsible for the universal and intrinsic description, whereas von Neumann algebras provide the detailed account obtained after fixing a state compatible with the dynamics. From this standpoint, for bosonic many-body systems the resolvent algebra, rather than the Weyl algebra, is the natural object; in particular, its nuclearity, trivial center, and rich ideal structure faithfully reflect purely quantum-mechanical structures. Macroscopic variables or sector structures associated with phase transitions are captured as the center appearing in the weak closure of the GNS representation. Moreover, the equivalence between representations of operator algebras and functional integrals allows powerful probabilistic methods to be employed. Taking these as guiding principles, we outline research perspectives on concrete objects in constructive quantum field theory and rigorous statistical mechanics.