Clifford geometric parameterization of inequivalent vacua
Oct, 199717 pages
Published in:
- Math.Methods Appl.Sci. 24 (2001) 885-912
e-Print:
- hep-th/9710047 [hep-th]
DOI:
Report number:
- TUE-PREP-97-02
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Abstract:
We propose a geometric method to parameterize inequivalent vacua. Introducing 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. The parameterization of inequivalent vacua takes place as a parameterization of different multivector products which induce different gradings. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of states. The most striking novelty of the geometric approach is the fact, that positivity is not required and 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. The usefulness of this approach to physical situations is also discussed. For the case of a U(2)-symmetry a gap-equation governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements.Note:
- Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.)
- fermion
- quantization
- Hamiltonian formalism
- Fock space
- algebra: Clifford
- geometry
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