On the Laplace transforms of retarded, Lorentz-invariant functions

Domínguez, A.; Trione, Susana Elena

doi:10.1016/0001-8708(79)90019-7

Cited by 21 publications

(15 citation statements)

References 6 publications

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“…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%

Spectral dimension from nonlocal dynamics on causal sets

et al. 2016

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Section: B Nonlocal D'alembertiansmentioning

confidence: 99%

Spectral dimension from nonlocal dynamics on causal sets

et al. 2016

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“…Evaluating χ(p, ρ) amounts to computing the Laplace transform of a retarded, Lorentzinvariant function, which has been done in [7]. It follows from their result that…”

Section: Spectrummentioning

confidence: 99%

“…For a fixed sign of p 0 , g (D) ρ (p) is only a function of p · p, making (3.25) the Laplace transform of a Lorentz-invariant function. Similarly to how we derived (3.7), we use the result of [7] to compute G(x − y):…”

Section: Uv Behaviour and The Retarded Green's Functionmentioning

confidence: 99%

“…We were not able to find any, but there are an infinite number of such operators and a definitive search could only be conducted by analytical means. 7 Finally, it bears repeating that there might be more reliable indicators of instability than simply the existence of an exponentially growing plane-wave solution, which a priori tells us nothing about the behavior of solutions of limited spatial extent. For that reason, it would be worthwhile to analyze directly the late-time behavior of the Green function G R (x − y) which is inverse to (D) ρ .…”

Section: Jhep06(2014)024mentioning

confidence: 99%

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Generalized causal set d’Alembertians

Aslanbeigi

Saravani

Sorkin

2014

J. High Energ. Phys.

105

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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.

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“…A wealth of effective works on the diamond kernel of Marcel Riesz were presented by Kananthai [5,6,7,8,9] and Sritanratana and Kananthai [10]. In 1978, Dominguez and Trione [11] introduced the distributional functions H α (P ± i0, n), which are causal (anti-causal) analogues of the elliptic kernel of Marcel Riesz [12]. Later, Cerutti and Trione [13] defined the causal (anti-causal) generalized Marcel Riesz potentials of order α, α ∈ C, by…”

Section: Introductionmentioning

confidence: 99%