Non-unitarity of Minkowskian non-local quantum field theories

Fabio Briscese,Leonardo Modesto

doi:10.1140/epjc/s10052-021-09525-7

Abstract

We show that Minkowskian non-local quantum field theories are not unitary. We consider a simple one loop diagram for a scalar non-local field and show that the imaginary part of the corresponding complex amplitude is not given by Cutkosky rules, indeed this diagram violates the unitarity condition. We compare this result with the case of an Euclidean non-local scalar field, that has been shown to satisfy the Cutkosky rules, and we clearly identify the reason of the breaking of unitarity of the Minkowskian theory.

Highlights

We show that Minkowskian non-local quantum field theories are not unitary
Analysing a simple one loop diagram, we will show that Cutkosky rules are violated when the theory is defined in Minkowski signature
In order to prove the loss of unitarity in the non-local Minkowskian theory, we consider diagram in Fig. 1 and show that the Cutkosky rules do not give the correct prescription for the imaginary part of the corresponding complex amplitude

Summary

Introduction

The study of non-local quantum field theory has began a long time ago in the contexts of the Standard Model of particles [1,2,3,4,5,6,7,8,9,10,11,12,13] and stochastic quantization [14,15]; and it has been revived recently, as it has been realized that non-locality plays an important role at the interplay of quantum field theory and gravitation. Higher-derivative models were abandoned until it become clear that the occurrence of ghosts could be avoided introducing derivatives of infinite order in a proper manner (or, equivalently, considering non-local interactions) when the quantum theory is defined in Euclidean signature This has led to the formulation of the so called non-local quantum gravity (NLQG) [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71].

Unitarity condition and Cutkosky rules

One-loop diagram in the Minkowskian non-local theory

One-loop diagram in the Euclidean non-local theory

Conclusions