Clifford geometric parameterization of inequivalent vacua

TL;DR

This paper introduces a geometric method using quantum Clifford algebras to uniquely parameterize inequivalent vacua in quantum field theory, allowing for non-statistical states and analyzing phase transitions.

Contribution

It presents a novel geometric framework for vacuum parameterization that does not require positivity and connects to algebraic and physical concepts like BCS theory.

Findings

Explicit parameterization of vacua from propagator matrices

Application to U(2)-symmetry and phase transitions

Relation established with BCS theory and Bogoliubov transformations

Abstract

We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras --as Clifford algebras-- by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*-algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non-statistical, non-definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U(2)-symmetry producing a gap-equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov-Valatin transformations is given.