A numerical study of the correspondence between paths in a causal set and geodesics in the continuum

Ilie, R.; B., Thompson, Gregory; Reid, David D.

doi:10.1088/0264-9381/23/10/002

Cited by 15 publications

(21 citation statements)

References 10 publications

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“…This merely strengthens the conclusion that we are seeing the failure of naive spatial distance in our simulations. 15 IV. n-LINKS, SPHERE DISTANCE AND PROXIMITY So far we have explored earlier attempts at defining a spatial distance measure on a causal set.…”

Section: Demonstration Of Failure Of Naive Spatial Distancementioning

confidence: 99%

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Spacelike distance from discrete causal order

Rideout¹,

Wallden²

2009

Class. Quantum Grav.

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Any discrete approach to quantum gravity must provide some prescription as to how to deduce continuum properties from the discrete substructure. In the causal set approach it is straightforward to deduce timelike distances, but surprisingly difficult to extract spacelike distances, because of the unique combination of discreteness with local Lorentz invariance in that approach. We propose a number of methods to overcome this difficulty, one of which reproduces the spatial distance between two points in a finite region of Minkowski space. We provide numerical evidence that this definition can be used to define a 'spatial nearest neighbor' relation on a causal set, and conjecture that this can be exploited to define the length of 'continuous curves' in causal sets which are approximated by curved spacetime. This provides some evidence in support of the "Hauptvermutung" of causal sets.Contents 2 Note that this 'small scale' may be after some coarse graining of the causal set, to get to the macroscopic, continuum-like scales. It is of course quite open what the smallest scales of the causal set will look like.

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Section: Demonstration Of Failure Of Naive Spatial Distancementioning

confidence: 99%

“…Several of the recent developments that have occurred in this approach at the kinematical level are [12,13,14,15], at dynamical level [16,17], and at the phenomenological level [18,19]. For reviews see e.g.…”

Section: A Causal Setsmentioning

confidence: 99%

Spacelike distance from discrete causal order

Rideout¹,

Wallden²

2009

Class. Quantum Grav.

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“…(z) z→∞ −−−→ γ 2 e −2iπa z ν−1 , I 2 (z) z→∞ −−−→ |γ| 2 z ν−1 . (C.24)Given that both quantities have the same scaling with z in this limit, their ratio must converge to a constant when z T → ∞: |N L | 2 e 2iΘ e −2iπa |N L | 2 = cos[π(ν − a)], (C 25). where Θ = π 2 [a + Re(ν)].…”

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confidence: 99%

A preferred ground state for the scalar field in de Sitter space

Aslanbeigi

Buck

2013

J. High Energ. Phys.

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We investigate a recent proposal for a distinguished vacuum state of a free scalar quantum field in an arbitrarily curved spacetime, known as the Sorkin-Johnston (SJ) vacuum, by applying it to de Sitter space. We derive the associated two-point functions on both the global and Poincaré (cosmological) patches in general d + 1 dimensions. In all cases where it is defined, the SJ vacuum belongs to the family of de Sitter invariant α-vacua. We obtain different states depending on the spacetime dimension, mass of the scalar field, and whether the state is evaluated on the global or Poincaré patch. We find that the SJ vacuum agrees with the Euclidean/Bunch-Davies state for heavy ("principal series") fields on the global patch in even spacetime dimensions. We also compute the SJ vacuum on a causal set corresponding to a causal diamond in 1 + 1 dimensional de Sitter space. Our simulations show that the mean of the SJ two-point function on the causal set agrees well with its expected continuum counterpart.

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“…Meanwhile, in the physics literature on discrete gravity it seems to retain the status of a conjecture, [49,29]. A numerical study can be found in [29].…”

Section: The Asymptotic Behavior Of Discrete Spacetimementioning

confidence: 99%

Discrete spacetime and its applications

Bachmat¹

2008

Contemporary Mathematics

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A discrete spacetime is a random partially ordered set. It is obtained by sampling n points from a compact domain in a spacetime manifold w.r.t. the volume form. The partial order is given by restricting the causal structure on the domain to the sampled points. These objects were introduced by physicists in an attempt to reconcile gravity with quantum mechanics, [40,45,19]. They received very little direct attention in the mathematics literature.Recently, discrete spacetimes have appeared in very diverse applications which are very far removed from their original motivation. In this self-contained survey we will present some results and numerical calculations regarding the asymptotic behavior of these objects. Some of the more basic results are stated as theorems since they are particularly useful in the applications. Numerical calculations are presented with the hope that they will incite some theoretical work, aimed at proving some related assertions. We then present a few of the applications by shamelessly following the pattern of the recent survey paper [21], which describes some related material. We will present a short list of seemingly unrelated problems and show how they fit into the discrete spacetime formalism. While this is a survey paper, it does contain some new material, namely, the statement of theorem 1.3, the (simple) solution to the dimension reconstruction problem, the numerical calculations of figure 2, the connect-the-dots application and the relation between discrete spacetime and the literature on maximal layers. A more comprehensive and extensive treatment of the subject along with full proofs of the basic results will be presented in a forthcoming publication.Our own research relating to discrete spacetime owes much to the constant encouragement which was provided by Percy Deift and it is our sincere pleasure to dedicate this paper to him on the occasion of his 60th birthday.

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A numerical study of the correspondence between paths in a causal set and geodesics in the continuum

Cited by 15 publications

References 10 publications

Spacelike distance from discrete causal order

Spacelike distance from discrete causal order

A preferred ground state for the scalar field in de Sitter space

Discrete spacetime and its applications

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