Non Local Theories: New Rules for Old Diagrams

TL;DR

This paper extends Wick's theorem to non-local quantum field theories, introducing a modified propagator and diagrammatic rules that accommodate violations of locality and causality, while preserving the local case as a special limit.

Contribution

It develops a generalized Wick theorem and diagrammatic expansion for non-local QFTs, including twisted products, with a modified propagator and Feynman rules.

Findings

A generalized Wick theorem applicable to non-local theories.

A diagrammatic expansion with modified Feynman rules.

Recovery of local theory as a special case.

Abstract

We show that a general variant of the Wick theorems can be used to reduce the time ordered products in the Gell-Mann & Low formula for a certain class on non local quantum field theories, including the case where the interaction Lagrangian is defined in terms of twisted products. The only necessary modification is the replacement of the Stueckelberg-Feynman propagator by the general propagator (the ``contractor'' of Denk and Schweda) D(y-y';tau-tau')= - i (Delta_+(y-y')theta(tau-tau')+Delta_+(y'-y)theta(tau'-tau)), where the violations of locality and causality are represented by the dependence of tau,tau' on other points, besides those involved in the contraction. This leads naturally to a diagrammatic expansion of the Gell-Mann & Low formula, in terms of the same diagrams as in the local case, the only necessary modification concerning the Feynman rules. The ordinary local theory is easily recovered as a special case, and there is a one-to-one correspondence between the local and non local contributions corresponding to the same diagrams, which is preserved while performing the large scale limit of the theory.