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" parametr… Show more
“…It is straightforward to recover the original results of [14] in d = 2, 4 as the simplest cases of the analysis reported in [15].…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 82%
“…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…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 99%
“…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…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 99%
“…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.…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 99%
“…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.…”
We investigate the spectral dimension obtained from non-local continuum d'Alembertians derived from causal sets. We find a universal dimensional reduction to 2 dimensions, in all dimensions. We conclude by discussing the validity and relevance of our results within the broader context of quantum field theories based on these nonlocal dynamics.
“…It is straightforward to recover the original results of [14] in d = 2, 4 as the simplest cases of the analysis reported in [15].…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 82%
“…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…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 99%
“…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…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 99%
“…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.…”
Section: B Nonlocal D'alembertiansmentioning
confidence: 99%
“…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.…”
We investigate the spectral dimension obtained from non-local continuum d'Alembertians derived from causal sets. We find a universal dimensional reduction to 2 dimensions, in all dimensions. We conclude by discussing the validity and relevance of our results within the broader context of quantum field theories based on these nonlocal dynamics.
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