On an easy transition from operator dynamics to generating functionals by Clifford algebras
Year: 1998
Type: article
Abstract: Clifford geometric algebras of multivectors are treated in detail. These algebras are built over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear that opposite Clifford algebras have to be used for obtaining all endomorphisms. Based on these mathematics, we give a fully Clifford algebraic account on generating functionals, which is thereby geometric. The field operators are shown to be Clifford and opposite Clifford maps. This picture, relying on geometry, does not need positivity in principle. Furthermore, we propose a transition from operator dynamics to generating functionals, which is based on the algebraic techniques. As a calculational benefit, this transition is considerably short compared to standard ones. The transition is not injective (unique) and depends additionally on the choice of an ordering. We obtain a direct and constructive connection between orderings and the explicit form of the functional Hamiltonian. These orderings depend on the propagator of the theory and thus on the ground state. This is invisible in path integral formulations. The method is demonstrated within two examples, a nonlinear spinor field theory and spinor QED. Antisymmetrized and normal-ordered functional equations are derived in both cases.
Source: Journal of Mathematical Physics
DOI: 10.1063/1.532500
Author: Bertfried Fauser
Institution: University of Tübingen
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2092957152