Signature of the simplicial supermetric

Hartle, James B.; Miller, Warner A.; Williams, Ruth M.

doi:10.1088/0264-9381/14/8/013

Cited by 10 publications

(13 citation statements)

References 19 publications

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“…Its signature is crucial for determining spacelike surfaces in superspace, which are important in Dirac quantisation and in quantum cosmology. In the continuum, there are limited results on the signature and this led to the possibility of investigating it in the discrete case [34], where the analogue is the Lund-Regge supermetric [35]. This supermetric was constructed for some simple manifolds (S 3 and T 3 ) and its signature calculated.…”

Section: Some Quantum Applicationsmentioning

confidence: 99%

Discrete structures in gravity

Regge

Williams²

2000

Journal of Mathematical Physics

164

293

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Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of 3-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalising these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalisations are explored for the original formulation of discrete gravity using edge-length variables. I Introduction to discrete gravity A Basic formalismThe original motivation for the development of a discrete formalism for gravity [1] arose from a number of problems with the continuum formulation of general relativity. These included the difficulty of solving Einstein's equations for general systems without a large degree of symmetry, the problems of representing complicated topologies and the need for considerable geometric insight and capacity for visualisation. It turned out, as we shall see, that the discretisation scheme to be described not only helped with these problems but also found a vital rôle in numerical relativity and in attempts at a formulation of quantum gravity.The related branches of mathematics which found their application to physics in this formulation of gravity are those of piecewise-linear spaces and topology and the geometric notion of intrinsic curvature on polyhedra. The immediate aim was to develop an approach to general relativity which avoided the 1

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Section: Some Quantum Applicationsmentioning

confidence: 99%

Discrete structures in gravity

Regge

Williams²

2000

Journal of Mathematical Physics

164

293

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“…Then by varying the squared volume of a given simplex σ in d dimensions to quadratic order in the metric (in the continuum), or in the squared edge lengths belonging to that simplex (on the lattice), one is led to the identification [24,25] …”

Section: Continuum and Discrete Wheeler-dewitt Equationmentioning

confidence: 99%

Wheeler-DeWitt equation in3+1dimensions

2013

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Physical properties of the quantum gravitational vacuum state are explored by solving a lattice version of the Wheeler-DeWitt equation. The constraint of diffeomorphism invariance is strong enough to uniquely determine part of the structure of the vacuum wave functional in the limit of infinitely fine triangulations of the three-sphere. In the large fluctuation regime the nature of the wave function solution is such that a physically acceptable ground state emerges, with a finite non-perturbative correlation length naturally cutting off any infrared divergences. The location of the critical point in Newton's constant G c , separating the weak from the strong coupling phase, is obtained, and it is inferred from the general structure of the wave functional that fluctuations in the 1 e-mail address : Herbert.Hamber@aei.mpg.de 2 e-mail address : RToriumi@uci.edu 3 e-mail address : R.M.Williams@damtp.cam.ac.uk 1 curvatures become unbounded at this point. Investigations of the vacuum wave functional further suggest that for weak enough coupling, G < G c , a pathological ground state with no continuum limit appears, where configurations with small curvature have vanishingly small probability. One would then be lead to the conclusion that the weak coupling, perturbative ground state of quantum gravity is non-perturbatively unstable, and that gravitational screening cannot be physically realized in the lattice theory. The results we find tend to be in general agreement with the Euclidean lattice gravity results, and would suggest that the Lorentzian and Euclidean lattice formulations of gravity ultimately describe the same underlying non-perturbative physics.

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“…It leads to an alternative way of deriving the lattice measure in Eq. (1.10), by considering the discretized distance between induced metrics g ij (s) [18], 19) with the inverse of the lattice DeWitt supermetric now given by the expression 20) and with again λ = −2/d. This procedure defines a metric on the tangent space of positive real symmetric matrices g ij (s).…”

Section: Standard Measurementioning

confidence: 99%

On the measure in simplicial gravity

Hamber

Williams

1999

Phys. Rev. D

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Functional measures for lattice quantum gravity should agree with their continuum counterparts in the weak field, low momentum limit. After showing that the standard simplicial measure satisfies the above requirement, we prove that a class of recently proposed non-local measures for lattice gravity do not satisfy such a criterion, already to lowest order in the weak field expansion. We argue therefore that the latter cannot represent acceptable discrete functional measures for simplicial geometries. ͓S0556-2821͑99͒00206-4͔

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Signature of the simplicial supermetric

Cited by 10 publications

References 19 publications

Discrete structures in gravity

Discrete structures in gravity

Wheeler-DeWitt equation in3+1dimensions

On the measure in simplicial gravity

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