A class of QFTs with higher derivative field equations leading to standard dispersion relation for the particle excitations

TL;DR

This paper demonstrates how to construct scalar field theories with higher derivative actions that still produce standard particle dispersion relations, including examples incorporating spacetime zero-point-length effects.

Contribution

It introduces a method to create higher derivative scalar field theories with standard dispersion relations, even with infinite-order derivatives, and compares path integral and operator approaches.

Findings

Constructed scalar field theories with infinite derivatives matching standard dispersion relations.

Provided an example incorporating zero-point-length effects in spacetime.

Highlighted advantages of the path integral approach over operator methods.

Abstract

Given any (Feynman) propagator which is Lorentz and translation invariant, it is possible to construct an action functional for a scalar field such that the quantum field theory, obtained by path integral quantization, leads to this propagator. In general, such a theory will involve derivatives of the field higher than two and can even involve derivatives of infinite order. The poles of the given propagator determine the dispersion relation for the excitations of this field. I show that it is possible to construct field theories in which the dispersion relation is the same as that of standard Klein-Gordan field, even though the Lagrangian contains derivatives of infinite order. I provide a concrete example of this situation starting from a propagator which incorporates the effects of the zero-point-length of the spacetime. I compare the path integral approach with an alternative, operator-based approach, and highlight the advantages of using the former.