Realizability of the Lorentzian (n, 1)-simplex

Tate, Kyle; Visser, Matt

doi:10.1007/jhep01(2012)028

Cited by 19 publications

(18 citation statements)

References 15 publications

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“…One question that can be asked is what are the generalized Lorentzian signature triangle inequalities for these Lorentzian simplices? One can show that the (4,1) simplex has its 4 time-like edge lengths completely unconstrained, provided its Euclidean tetrahedron is realizable, while the (3,2) simplex has its edge lengths subject to complicated constraints [34]. Thus, it is safe to say that the constrained Lorentzian configuration space is very different from that of the Euclidean configuration space in any dimension.…”

Section: + 1 and Beyondmentioning

confidence: 99%

Fixed-topology Lorentzian triangulations: Quantum Regge Calculus in the Lorentzian domain

Tate

Visser

2011

J. High Energ. Phys.

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A key insight used in developing the theory of Causal Dynamical Triangulations (CDTs) is to use the causal (or light-cone) structure of Lorentzian manifolds to restrict the class of geometries appearing in the Quantum Gravity (QG) path integral. By exploiting this structure the models developed in CDTs differ from the analogous models developed in the Euclidean domain, models of (Euclidean) Dynamical Triangulations (DT), and the corresponding Lorentzian results are in many ways more "physical".In this paper we use this insight to formulate a Lorentzian signature model that is analogous to the Quantum Regge Calculus (QRC) approach to Euclidean Quantum Gravity. We exploit another crucial fact about the structure of Lorentzian manifolds, namely that certain simplices are not constrained by the triangle inequalities present in Euclidean signature. We show that this model is not related to QRC by a naive Wick rotation; this serves as another demonstration that the sum over Lorentzian geometries is not simply related to the sum over Euclidean geometries. By removing the triangle inequality constraints, there is more freedom to perform analytical calculations, and in addition numerical simulations are more computationally efficient.We first formulate the model in 1+1 dimensions, and derive scaling relations for the pure gravity path integral on the torus using two different measures. It appears relatively easy to generate "large" universes, both in spatial and temporal extent. In addition, loop-to-loop amplitudes are discussed, and a transfer matrix is derived. We then also discuss the model in higher dimensions.

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Section: + 1 and Beyondmentioning

confidence: 99%

Fixed-topology Lorentzian triangulations: Quantum Regge Calculus in the Lorentzian domain

Tate

Visser

2011

J. High Energ. Phys.

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“…Apart from a few works [11,13,14,15,16,12], the path integrals of Lorentzian simplicial quantum gravity have not been studied much in the past. 2 Because of the numerical sign problem, naive Monte Carlo simulations do not work efficiently in the Lorentzian as in the Euclidean.…”

Section: Introductionmentioning

confidence: 99%

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Jia

2021

Preprint

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Evaluating gravitational path integrals in the Lorentzian has been a longstanding challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity.In order to allow complex deformations of the integration contour, we provide a manifestly holomorphic formula for Lorentzian simplicial gravity. This leads to a complex version of simplicial gravity that generalizes the Euclidean and Lorentzian cases. Outside the context of numerical computation, complex simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.

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“…The study of gravitational path integrals defined in terms of simplicial spacetime configurations á la Regge [9] has a long history [10,11,12,13,14]. While previous works focused on the Euclidean theory, there has been a growing interest in the Lorentzian theory in recent years [15,16,17,18,19,8,20]. As in causal dynamical triangulation, the light ray paths on a piecewise flat simplicial spacetime configuration is obtainable, in particular using Lorentzian trigonometry which we illustrate in Section 3.1.…”

Section: Introductionmentioning

confidence: 99%

Light ray fluctuations in simplicial quantum gravity

Jia

2022

Preprint

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A non-perturbative study on the quantum fluctuations of light ray propagation through a quantum region of spacetime is long overdue. Within the theory of Lorentzian simplicial quantum gravity, we compute the probabilities for a test light ray to land at different locations after travelling through a symmetry-reduced box region in 2,3 and 4 spacetime dimensions. It is found that for fixed boundary conditions, light ray fluctuations are generically large when all coupling constants are relatively small in absolute value. For fixed coupling constants, as the boundary size is decreased light ray fluctuations first increase and then decrease in a 2D theory with the cosmological constant, Einstein-Hilbert and R-squared terms. While in 3D and 4D theories with the cosmological constant and Einstein-Hilbert terms, as the boundary size is decreased light ray fluctuations just increase. Incidentally, when studying 2D quantum gravity we show that the global time-space duality with the cosmological constant and Einstein-Hilbert terms noted previously also holds when arbitrary even powers of the Ricci scalar are added. We close by discussing how light ray fluctuations can be used in obtaining the continuum limit of non-perturbative Lorentzian quantum gravity.

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Realizability of the Lorentzian (n, 1)-simplex

Cited by 19 publications

References 15 publications

Fixed-topology Lorentzian triangulations: Quantum Regge Calculus in the Lorentzian domain

Fixed-topology Lorentzian triangulations: Quantum Regge Calculus in the Lorentzian domain

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Light ray fluctuations in simplicial quantum gravity

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