Form factors, spectral and K\"all\'en-Lehmann representation in nonlocal quantum gravity

TL;DR

This paper analyzes the spectral and K"allén-Lehmann representations in nonlocal quantum gravity, demonstrating that these representations retain positive spectral density and local limits, thus supporting unitarity and consistency of the theory.

Contribution

It provides a detailed calculation of spectral and K"allén-Lehmann representations for nonlocal theories with entire form factors, extending known results to interacting theories.

Findings

Spectral density remains positive-definite and matches local theory spectrum.

The K"allén-Lehmann representation generalizes to nonlocal theories with interactions.

Local limit is smoothly recovered as the fundamental length scale approaches zero.

Abstract

We discuss the conical region of convergence of exponential and asymptotically polynomial form factors and their integral representations. Then, we calculate the spectral representation of the propagator of nonlocal theories with entire form factors, in particular, of the above type. The spectral density is positive-definite and exhibits the same spectrum as the local theory. We also find that the piece of the propagator corresponding to the time-ordered two-point correlation function admits a generalization of the K\"all\'en-Lehmann representation with a standard momentum dependence and a spectral density differing from the local one only in the presence of interactions. These results are in agreement with what already known about the free theory after a field redefinition and about perturbative unitarity of the interacting theory. The spectral and K\"all\'en-Lehmann representations have the same standard local limit, which is recovered smoothly when sending the fundamental length scale $\ell_*$ in the form factor to zero.