Generalized causal set d’Alembertians

Abstract

We introduce a family of generalized d'Alembertian operators in D-dimensionalMinkowski spacetimes M D which are manifestly Lorentz-invariant, retarded, and non-local, the extent of the nonlocality being governed by a single parameter ρ. The prototypes of these operators arose in earlier work as averages of matrix operators meant to describe the propagation of a scalar field in a causal set. We generalize the original definitions to produce an infinite family of "Generalized Causet Box (GCB) operators" parametrized by certain coefficients {a, b n }, and we derive the conditions on the latter needed for the usual d'Alembertian to be recovered in the infrared limit. The continuum average of a GCB operator is an integral operator in M D , and it is these continuum operators that we mainly study. To that end, we compute their action on plane waves, or equivalently their Fourier transforms g(p) [p being the momentum-vector]. For timelike p, g(p) has an imaginary part whose sign depends on whether p is past or future-directed. For small p, g(p) is necessarily proportional to p · p, but for large p it becomes constant, raising the possibility of a genuinely Lorentzian perturbative regulator for quantum field theory in M D . We also address the question of whether or not the evolution defined by the GCB operators is stable, finding evidence that the original 4D causal set d'Alembertian is unstable, while its 2D counterpart is stable.

Spectral dimension from nonlocal dynamics on causal sets

“…It is straightforward to recover the original results of [14] in d = 2, 4 as the simplest cases of the analysis reported in [15].…”

“…al ( [15]) generalised the original constructions of nonlocal d'Alembertians in [13,14,18], to include an infinite family of nonlocal d'Alembertians for any dimension d. The nonlocal d'Alembertians…”

“…This divergence can be regularised by subtracting the constant (aρ 2/d ) −1 from the momentum space Green function [15]. Inverting back gives a regularised momentum space d'Alembertian…”

“…The operators we started with (1) are retarded Lorentz invariant operators, and their Laplace transforms are therefore defined in the limit (k 0 ) → 0 + [15,20,21], i.e. in the upper half complex k 0 -plane.…”

“…In Section 2 we briefly introduce causal sets, the notion of sprinklings and how the non-locality referred to above arises within this context. We then introduce the family of d'Alembertians defined in [15] in both position and Fourier space, and discuss some of their properties. Sec-tion 3 is devoted to computing the spectral dimension of Minkowski spacetime.…”