Narrow Results

One of the most important mathematical tools necessary for Quantum Field Theory calculations is the field propagator. Applications are always done in terms of plane waves and although this has furnished many magnificent results, one may still be allowed to wonder what is the form of the most general propagator that can be written. In this paper, by exploiting what is called polar form, we find the most general propagator in the case of spinors, whether regular or singular, and we give a general discussion in the case of vectors.

In Quantum Field Theory (QFT), scattering amplitudes are computed from propagators which, for internal lines, are built upon spin/polarization-sum relationships. In turn, these are normally constructed upon plane-wave solutions of the free field equations. A question that may now arise is whether such spin/polarization-sums can be generalized. In the past, there has been a first attempt at generalizing spin sums for fermionic fields in terms of the Michel–Wightman identities. In this paper, we aim to find the most general spin sums for fermionic fields within the range of QFT.

In the most general geometric background, we study the Dirac spinor fields with particular emphasis given to the explicit form of their gauge momentum and the way in which this can be inverted so as to give the expression of the corresponding velocity; we study how Zitterbewegung affects the motion of particles, focusing on the internal dynamics involving the chiral parts; we discuss the connections to field quantization, sketching in what way anomalous terms may be gotten eventually.