An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time

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 Möbius structures. We describe natural principal bundle structures associated with Möbius 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 ⊂ SU(2, 2) of a principal bundle associated with a given Möbius structure where , while being a subset of SU(2, 2), is also isomorphic to SL(2, C) × (1). The analysis requires the use of an intertwining operator between the action of on R 4 and the adjoint action of on su(2, 2) and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.

A Spectral Calculus for Lorentz Invariant Measures on Minkowski Space

“…Firstly one can obtain complete, unencumbered and "elegant" results. Secondly, many the distributional objects of interest in QFT (such as correlation functions) can, through Wick's theorem, or else the operator product expansion, be represented in terms of Feynman propagators and the propagators are Lorentz invariant measures (or else K invariant matrix-valued measures whose trace is Lorentz invariant) [16].…”

Quantum chromodynamics through the geometry of Möbius structures

“…In a previous paper (Mashford, 2017a) we have presented a theory in which spacetime is modeled as a (causal) locally conformally flat Lorentzian manifold so we are considering a subset of the set of space-times considered in GR. Actually we do not specify a metric, we only specify the causal, or interaction, structure.…”

“…Let, for any x ∈ X, I x denote {i ∈ I : x ∈ U i }. If U and V are open subsets of Minkowski space R 4 and f : U → V is a conformal transformation (diffeomorphism), let C(f ) ∈ C(1, 3) be the unique maximal conformal transformation g ∈ C(1, 3) such that f ⊂ g, where C(1, 3) denotes the group of maximal conformal transformations in Minkowski space (see (Mashford, 2017a)).…”

“…The category of Möbius structures has the same logical status as the category of smooth manifolds (which forms the starting point of GR) except that the pseudogroup of transformations (Kobayashi and Nomizu, 1963, p. 1) is the conformal pseudogroup Γ(1, 3) (Mashford, 2017a) rather than the pseudogroup of general coordinate transformations, i.e. the diffeomomorphisms between open subsets of Minkowski space.…”

“…In (Mashford, 2017a) we described how there is a natural principal bundle with structure group C(1, 3) associated with any Möbius structure which is obtained by taking the overlap conformal diffeomorphisms as (constant) transition functions. A natural principal bundle with structure group which we called K, where K is locally isomorphic to SL(2, C) × U (1), can be obtained by reduction of structure group.…”