Nonlocal and quasi-local field theories

TL;DR

This paper explores nonlocal and quasi-local field theories, analyzing their mathematical properties, causality issues, and implications for quantum amplitudes, with a focus on scalar models and potential extensions.

Contribution

It provides a rigorous analysis of classical and quantum properties of nonlocal field theories, including existence of solutions and causality conditions, especially for quasi-local kernels.

Findings

Existence of solutions with loss of uniqueness due to delays

Acausality confined within compact support regions for quasi-local kernels

Generalized causality condition for nonlocal quantum amplitudes

Abstract

We investigate nonlocal field theories, a subject that has attracted some renewed interest in connection with nonlocal gravity models. We study, in particular, scalar theories of interacting delocalized fields, the delocalization being specified by nonlocal integral kernels. We distinguish between strictly nonlocal and quasi-local (compact support) kernels and impose conditions on them to insure UV finiteness and unitarity of amplitudes. We study the classical initial value problem for the partial integro-differential equations of motion in detail. We give rigorous proofs of the existence but accompanying loss of uniqueness of solutions due to the presence of future, as well as past, "delays," a manifestation of acausality. In the quantum theory we derive a generalization of the Bogoliubov causality condition equation for amplitudes, which explicitly exhibits the corrections due to nonlocality. One finds that, remarkably, for quasi-local kernels all acausal effects are confined within the compact support regions. We briefly discuss the extension to other types of fields and prospects of such theories.