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

Abstract

We examine the unitarity of a class of generalized Maxwell U(1) gauge theories in ($2+1$) dimensions containing the pseudodifferential operator ${\ensuremath{\square}}^{1\ensuremath{-}\ensuremath{\alpha}}$, for $\ensuremath{\alpha}\ensuremath{\in}[0,1)$. We show that only ${\mathrm{QED}}_{3}$ and its generalization known as pseudo-QED, for which $\ensuremath{\alpha}=0$ and $\ensuremath{\alpha}=1/2$, 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 heterojunctions displaying the quantum Hall effect. Possible implications of our results on the role of unitarity in the framework of the AdS/CFT correspondence are briefly pointed out at the end.