Half-Differentials and Fermion Propagators

TL;DR

This paper explores the geometric structure of massless spinors in 3+1 dimensions using half-differentials, deriving covariant constraints and extending fermion propagators consistent with Lorentz symmetry.

Contribution

It introduces a geometric framework based on half-differentials for analyzing fermions and derives covariant constraints and propagator extensions.

Findings

Derived primary correlators for fermion propagators.

Established covariant holomorphy conditions for spinors.

Extended fermion propagators compatible with Lorentz covariance.

Abstract

From a geometric point of view, massless spinors in $3+1$ dimensions are composed of primary fields of weights $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$, where the weights are defined with respect to diffeomorphisms of a sphere in momentum space. The Weyl equation thus appears as a consequence of the transformation behavior of local sections of half--canonical bundles under a change of charts. As a consequence, it is possible to impose covariant constraints on spinors of negative (positive) helicity in terms of (anti--)holomorphy conditions. Furthermore, the identification with half--differentials is employed to determine possible extensions of fermion propagators compatible with Lorentz covariance. This paper includes in particular the full derivation of the primary correlators needed in order to determine the fermion correlators.