Continuum Modes of Nonlocal Field Theories

TL;DR

This paper explores nonlocal Lorentzian quantum field theories with a spectrum of continuum massive modes, deriving their path integral formulation and dual local field representation to understand their interactions and potential extensions.

Contribution

It introduces a dual local field picture for nonlocal theories with continuum modes, facilitating analysis of interactions and future extensions beyond scalar fields.

Findings

Derivation of the path integral formulation for nonlocal theories.

Development of a dual local field representation.

Insights into the interaction structure of continuum massive modes.

Abstract

A class of nonlocal Lorentzian quantum field theories is introduced in arXiv:1502.01655 and arXiv:1411.6513, where the d'Alembertian operator $\Box$ is replaced by a non-analytic function of the d'Alembertian, $f(\Box)$. This is inspired by the Causal Set program where such an evolution arises as the continuum limit of a wave equation on causal sets. The spectrum of these theories contains a continuum of massive excitations. This is perhaps the most important feature which leads to distinct/interesting phenomenology. In this paper, we study properties of the continuum massive modes in depth. We derive the path integral formulation of these theories. Meanwhile, this derivation introduces a dual picture in terms of local fields which clearly shows how continuum massive modes of the nonlocal field interact. The dual picture, in principle, provides a path to extension beyond scalar fields and addressing the issue of renormalization.