On an easy transition from operator dynamics to generating functionals by Clifford algebras

TL;DR

This paper develops a geometric, Clifford algebra-based framework for generating functionals, enabling a more efficient transition from operator dynamics to functionals, demonstrated through spinor field theories.

Contribution

It introduces a Clifford algebraic approach to generating functionals and a new, more efficient transition method from operator dynamics, linking orderings to the functional Hamiltonian.

Findings

Transition method is significantly shorter than standard approaches.

Connection established between orderings and the functional Hamiltonian.

Demonstrated in spinor field theory and QED examples.

Abstract

Clifford geometric algebras of multivectors are treated in detail. These algebras are build 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 also. Based on this 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 corresponding generating functionals, which is based on the algebraic techniques. As a calculational benefit, this transition is considerable 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 non-linear spinor field theory and spinor QED. Antisymmetrized and normal-ordered functional equations are derived in both cases.