Fractions and Fakeons in Quantum Field Theory

TL;DR

This paper explores quantum field theories with fractional kinetic terms, examining their unitarity, classical limits, and multiple formulations, including fakeon prescriptions and continuum decompositions.

Contribution

It introduces novel formulations of fractional quantum field theories using fakeon prescriptions and continuum decompositions, analyzing their properties and potential issues.

Findings

Demonstrates unitarity and classical limits at tree level and for bubble diagrams.

Shows multiple inequivalent Minkowskian formulations with the same Euclidean theory.

Provides methods to handle continuous powers in gauge and gravity theories.

Abstract

We investigate formulations of quantum field theories whose kinetic terms involve fractional or continuous powers of the d'Alembert operator. The primary requirements are perturbative unitarity and a well-defined classical limit with a finite number of initial conditions. A direct approach consists of continuing the correlation functions from Euclidean space to Minkowski spacetime using the fakeon prescription for the fractional part of the power. Alternative formulations arise through decomposition, in which the fractional part is represented as a continuum of ordinary fakeons. These options are infinite in number and yield inequivalent Minkowskian theories with the same Euclidean counterpart. We demonstrate these features at tree level and for bubble diagrams. We also point out potential pitfalls in the calculations. Finally, we show how to treat continuous powers of covariant d'Alembertians in fractional gauge and gravity theories. The Ward and Cutkosky identities hold in all formulations.