An approach to classical quantum field theory based on the geometry of locally conformally flat space-time

TL;DR

This paper introduces a geometric framework for classical quantum field theory using locally conformally flat space-time structures, linking principal bundles, fermion and boson fields, and the Dirac and Maxwell equations.

Contribution

It develops a geometric approach to classical quantum field theory based on M"{o}bius structures and principal bundles, providing new insights into the mathematical foundation of field interactions.

Findings

Classical quantum field theory derived from M"{o}bius structures.

Identification of principal bundle structures related to fermion and boson fields.

Demonstration of the intertwining property of the Feynman slash operator.

Abstract

This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e. are manifolds) and hence are M\"{o}bius structures. We describe natural principal bundle structures associated with M\"{o}bius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell's equations) is obtained by considering representations of the structure group $K\subset U(2,2)$ of a principal bundle associated with a given M\"{o}bius structure where $K$, while being a subset of $U(2,2)$, is also isomorphic to $SL(2,{\bf C})\times U(1)$. The analysis requires the use of an intertwining operator between the action of $K$ on ${\bf R}^4$ and the adjoint action action of $K$ on $u(2,2)$ and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.