Nontopological Methods for Determining Topological Charge for Bosons and Fermions in Flat Spacetime

TL;DR

This paper introduces a relativistic quantum field theory-based method to determine topological charge, revealing that fermions' topological charge cannot be quantized, unlike bosons, thus providing new insights beyond traditional instanton solutions.

Contribution

It presents a novel covariant approach to topological charge calculation that distinguishes between bosons and fermions, unlike previous instanton-based methods.

Findings

Topological charge for bosons is quantized, consistent with instanton predictions.

Topological charge for fermions cannot be quantized using the new method.

The new technique yields results previously unobtainable with traditional methods.

Abstract

An alternative method to the topological instanton solution for deriving an expression for the topological charge is presented. This alternative method involves the use of relativistic quantum field theory and covariant electrodynamics. In the case of bosons, this method is consistent with the instanton solution in predicting that topological charge is quantized. But furthermore, this method led to the new results that topological charge for fermions cannot be quantized, whereas the instanton solution cannot distinguish between bosons (quantized) and fermions (not quantized). Thus the new technique produced results that were previously unobtainable. Mathematics Subject Classifications (1991): // Key words: Topological charge, Dirac quantization condition, Klein-Gordon equation