Clifford geometric parameterization of inequivalent vacua

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.

Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.)

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Cited by in corpus (9)

• Bilinear Covariants and Spinor Fields Duality in Quantum Clifford Algebras
• Quantum field theory and Hopf algebra cohomology
• A Hopf laboratory for symmetric functions
• On the transposition anti-involution in real Clifford algebras I: The transposition map
• On an easy transition from operator dynamics to generating functionals by Clifford algebras
• Cliffordization, Spin and Fermionic Star Products
• Products, coproducts and singular value decomposition
• On the bundle of Clifford algebras over the space of quadratic forms
• Complexifying the spacetime algebra by means of an extra timelike dimension: Pin, Spin and algebraic spinors