Stability and causality of multi-local theories

TL;DR

This paper investigates the stability and causality of multi-local classical field theories, revealing that non-locality affects conservation laws and stability, but causality can be maintained through specific constructions.

Contribution

It provides a detailed analysis of how non-locality impacts conservation, causality, and stability in multi-local field theories, highlighting conditions for causality preservation.

Findings

Conservation laws are less significant in non-local theories.

Energy conservation does not guarantee stability due to unbounded energy.

Appropriate point splitting can ensure causality in these theories.

Abstract

The regularized theories are non-local at the scale of the cutoff, leading so to the usual difficulties of non-local theories. In this work the conservation laws and causality are investigated for classical field theories with multi-cluster action. The conservation laws are found to play a less significant role than in local theories because due to the non-locality the conserved quantities are not integrals of the motion, and they can exist even without underlying symmetries. Moreover, the conservation of the energy can not prevent instability brought about by the unbounded nature of the energy from below. Hence a sufficient condition of stability is lost. Theories, obtained by appropriate point splitting of local interactions are shown to be causal thereby a necessary condition of stability can be retained.