Diffeomorphism invariant measure for finite-dimensional geometries

Menotti, Pietro; Peirano, Pier Paolo

doi:10.1016/s0550-3213(97)00017-5

Cited by 28 publications

(24 citation statements)

References 12 publications

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“…There has been considerable debate in the literature on the measure for quantum Regge calculus [75][76][77][78]. One difficulty that makes any analytical calculations hard are the triangle inequalities that make determining the integration range very complicated.…”

Section: Quantum Regge Calculus: the Path Integral Measurementioning

confidence: 99%

3D holography: from discretum to continuum

Bonzom

Dittrich

2016

J. High Energ. Phys.

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Abstract:We study the one-loop partition function of 3D gravity without cosmological constant on the solid torus with arbitrary metric fluctuations on the boundary. To this end we employ the discrete approach of (quantum) Regge calculus. In contrast with similar calculations performed directly in the continuum, we work with a boundary at finite distance from the torus axis. We show that after taking the continuum limit on the boundary -but still keeping finite distance from the torus axis -the one-loop correction is the same as the one recently found in the continuum in Barnich et al. for an asymptotically flat boundary. The discrete approach taken here allows to identify the boundary degrees of freedom which are responsible for the non-trivial structure of the one-loop correction. We therefore calculate also the Hamilton-Jacobi function to quadratic order in the boundary fluctuations both in the discrete set-up and directly in the continuum theory. We identify a dual boundary field theory with a Liouville type coupling to the boundary metric. The discrete set-up allows again to identify the dual field with degrees of freedom associated to radial bulk edges attached to the boundary. Integrating out this dual field reproduces the (boundary diffeomorphism invariant part of the) quadratic order of the Hamilton-Jacobi functional. The considerations here show that bulk boundary dualities might also emerge at finite boundaries and moreover that discrete approaches are helpful in identifying such dualities.

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Section: Quantum Regge Calculus: the Path Integral Measurementioning

confidence: 99%

3D holography: from discretum to continuum

Bonzom

Dittrich

2016

J. High Energ. Phys.

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“…One may hope that in the non-perturbative Regge regime no gauge-fixing is necessary, since the contributions from zero-modes cancel out in the path-integral representation for operator averages [124,122]. Menotti and Peirano [162,164,163,161], following a strategy suggested by Jevicki and Ninomiya [134], have argued vigorously that the functional integral should contain a non-trivial Faddeev-Popov determinant. Their starting point is somewhat different from that adopted in the path-integral simulations (see also [199]).…”

Section: Gauge Invariance In Regge Calculus?mentioning

confidence: 99%

Discrete Approaches to Quantum Gravity in Four Dimensions

Loll

1998

Living Rev. Relativ.

176

175

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The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation; quantum Regge calculus; and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.

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“…where = 2(1 − g) is the Euler characteristic of the manifold M expressed by the number of handles g in M. For a simplicial complex K the Euler characteristic can be computed as 12) where N 0 , N 1 , and N 2 denote the number of sites, links, and triangles, respectively. In higher than two dimensions an R 2 interaction term is sometimes added to guarantee the boundedness of the gravitational action from below.…”

Section: Model and Simulationmentioning

confidence: 99%