Current algebra, statistical mechanics and quantum models

TL;DR

This paper revisits free boson systems to clarify current algebra functionals' physical meaning, explores their role in quantum statistical mechanics via diffeomorphism group representations, and applies the approach to 2D fermion pairing.

Contribution

It compares two mathematical methods for constructing current algebra representations and advocates for the Hilbert space approach as more physically intuitive.

Findings

Clarified the physical meaning of current algebra reducible functionals.

Linked current algebra to phase transitions and density fluctuations.

Applied the methodology to pairing in 2D fermion systems.

Abstract

Results obtained in the past for free boson systems at zero and nonzero temperature are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions. To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.