Unitarity of theories containing fractional powers of the d'Alembertian operator

TL;DR

This paper investigates the unitarity of generalized Maxwell gauge theories with fractional d'Alembertian operators in 2+1 dimensions, identifying specific cases (QED3 and PQED) that preserve unitarity and are relevant for condensed matter systems.

Contribution

It demonstrates that only QED3 and PQED with specific fractional powers of the d'Alembertian operator are unitary, clarifying the conditions for unitarity in these generalized gauge theories.

Findings

QED3 (α=0) is unitary.

Pseudo QED (α=1/2) is unitary.

Other fractional powers do not satisfy unitarity.

Abstract

We examine the unitarity of a class of generalized Maxwell U(1) gauge theories in (2+1) D containing the pseudodifferential operator $\Box^{1-\alpha}$, for $\alpha \in [0,1)$. We show that only Quantum Electrodynamics (QED$_3$) and its generalization known as Pseudo Quantum Electrodynamics (PQED), for which $\alpha =0$ and $\alpha = 1/2$, respectively, satisfy unitarity. The latter plays an important role in the description of the electromagnetic interactions of charged particles confined to a plane, such as in graphene or in hetero-junctions displaying the quantum Hall effect.