Fermionic Operators from Bosonic Fields in 3+1 Dimensions

TL;DR

This paper constructs fermionic operators in 3+1 dimensional QED using bosonic electric fields and their conjugates, extending methods from lower dimensions and illustrating their transformation properties.

Contribution

It introduces a novel method to represent fermionic operators in 3+1 dimensions via bosonic fields, generalizing previous 2+1 dimensional constructions.

Findings

Fermionic operators are expressed as products of charge creation and t'Hooft loop operators.

The construction demonstrates how axial U(1) symmetry is realized.

The approach extends lower-dimensional techniques to 3+1 dimensions.

Abstract

We present a construction of fermionic operators in 3+1 dimensions in terms of bosonic fields in the framework of $QED_4$. The basic bosonic variables are the electric fields $E_i$ and their conjugate momenta $A_i$. Our construction generalizes the analogous constuction of fermionic operators in 2+1 dimensions. Loosely speaking, a fermionic operator is represented as a product of an operator that creates a pointlike charge and an operator that creates an infinitesimal t'Hooft loop of half integer strength. We also show how the axial $U(1)$ transformations are realized in this construction.