# Quantization of nonlocal fractional field theories via the extension problem

## TL;DR

This paper demonstrates how to quantize nonlocal fractional field theories using an extension problem, translating the nonlocal behavior into a local higher-dimensional framework, and reproduces known particle content and correlators.

## Contribution

It introduces a method to quantize nonlocal fractional fields via a local extension in higher dimensions, connecting to spectral representations and potential applications in gravity.

## Key findings

- Reproduces particle content of fractional propagators
- Derives two-point functions and vacuum energy from the extension
- Establishes a link between nonlocal and local field descriptions

## Abstract

We use the extension problem proposed by Caffarelli and Silvestre to study the quantization of a scalar nonlocal quantum field theory built out of the fractional Laplacian. We show that the quantum behavior of such a nonlocal field theory in $d$-dimensions can be described in terms of a local action in $d+1$ dimensions which can be quantized using the canonical operator formalism though giving up local commutativity. In particular, we discuss how to obtain the two-point correlation functions and the vacuum energy density of the nonlocal fractional theory as a brane limit of the bulk correlators. We show explicitly how the quantized extension problem reproduces exactly the same particle content of other approaches based on the spectral representation of the fractional propagator. We also briefly discuss the inverse fractional Laplacian and possible applications of this approach in general relativity and cosmology.