Nonlocal scalar quantum field theory from causal sets

Stefano Liberati,Dionigi M. T. Benincasa,Alessio Belenchia

doi:10.1007/jhep03(2015)036

Abstract

We study a non-local scalar quantum field theory in flat spacetime derived from the dynamics of a scalar field on a causal set. We show that this non-local QFT contains a continuum of massive modes in any dimension. In 2 dimensions the Hamiltonian is positive definite and therefore the quantum theory is well-defined. In 4-dimensions, we show that the unstable modes of the non-local d'Alembertian are propagated via the so called Wheeler propagator and hence do not appear in the asymptotic states. In the free case studied here the continuum of massive mode are shown to not propagate in the asymptotic states. However the Hamiltonian is not positive definite, therefore potential issues with the quantum theory remain. Finally, we conclude with hints toward what kind of phenomenology one might expect from such non-local QFTs.

Highlights

We study a non-local scalar quantum field theory in flat spacetime derived from the dynamics of a scalar field on a causal set
We have canonically quantised free massless scalar fields, satisfying nonlocal equations of motion, in 2 and 4 dimensions. In both dimensions we find a continuum of massive modes arising from the cut discontinuity along k2 ≤ 0 of the momentum space Green function, f −1(k2), similar to what was found in previously studied free, non-local, massless QFTs [18]
This feature will be present in any dimension, since the existence of the branch cut is a generic feature of the Laplace transform of retarded Lorentz invariant functions [29]

Summary

Nonlocal d’Alembertians

The first construction of a d’Alembertian operator on a causal set appeared in a seminal paper by Sorkin [12]. Note that the complex mass solutions appear in complex conjugate pairs ensuring that the theory is CPT invariant — the pole structure of the propagator in the complex k0-plane has to be symmetric about the real axis. This is expected since our action (see equation (5.1) below) is CPT-invariant: C being trivial for a real scalar field theory, and PT because the non-local d’Alembertian ρ is a function of the spacetime volume only, which is PT-invariant in Minkowski spacetime. Johnston [16] appeared in which he obtains the same corrections as above, and the same power series expressions found in appendix B

Massive extension: nonlocal Klein-Gordon equation

Huygens’ principle and the nonlocal d’Alembertians

Free scalar nonlocal QFT

Renormalisation

Summary and outlook

A Degree of singularity of complex mass poles

B Series form of the non-local operator

C Asymptotic states