Gravitational Wilson loop in discrete quantum gravity

Hamber, Herbert W.; Williams, Ruth M.

doi:10.1103/physrevd.81.084048

Cited by 24 publications

(22 citation statements)

References 61 publications

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“…The hope is that the present approach will allow the use of a different set of approximation methods, and numerical algorithms, to explore what in some instances are largely known issues, but now from an entirely different perspective. Among the problems one might want to consider, we list: the description of invariant correlation functions [41,42], the behavior of the fundamental gravitational correlation length as a function of the coupling G, the approach to the lattice continuum limit at G c , estimates for the critical exponents in the vicinity of the fixed point, and the large-scale behavior of the gravitational Wilson loop [43]. Note that what is sometimes referred to as the problem of time does not necessarily affect the above issues, which in our opinion can be settled by looking exclusively at certain types of invariant correlations along the spatial directions only.…”

Section: Discussionmentioning

confidence: 99%

Discrete Wheeler-DeWitt equation

Hamber

Williams²

2011

Phys. Rev. D

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We present a discrete form of the Wheeler-DeWitt equation for quantum gravitation, based on the lattice formulation due to Regge. In this setup, the infinite-dimensional manifold of 3-geometries is replaced by a space of three-dimensional piecewise linear spaces, with the solutions to the lattice equations providing a suitable approximation to the continuum wave functional. The equations incorporate a set of constraints on the quantum wave functional, arising from the triangle inequalities and their higher-dimensional analogs. The character of the solutions is discussed in the strong-coupling (large-G) limit, where it is shown that the wave functional only depends on geometric quantities, such as areas and volumes. An explicit form, determined from the discrete wave equation supplemented by suitable regularity conditions, shows peaks corresponding to integer multiples of a fundamental unit of volume. An application of the variational method using correlated product wave functions suggests a relationship between quantum gravity in n þ 1 dimensions, and averages computed in the Euclidean path integral formulation in n dimensions. The proposed discrete equations could provide a useful, and complementary, computational alternative to the Euclidean lattice path integral approach to quantum gravity.

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Section: Discussionmentioning

confidence: 99%

Discrete Wheeler-DeWitt equation

Hamber

Williams²

2011

Phys. Rev. D

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“…Just to name a few: noncommutative geometry [25], causal sets approach [26], gravitational Wilson loops [27], Regge calculus [28], path integral over geometries [29], spin foam model [30], and categories [31]. The discreteness may translate at the semi-classical level into a foamy structure of space.…”

Section: Discussionmentioning

confidence: 99%

“…Let us consider the eigenvalue problem for the operatorŵ in the set of functions of the form (27). We haveŵ…”

Section: Eigenvalue Problemmentioning

confidence: 99%

Turning big bang into big bounce: II. Quantum dynamics

Małkiewicz¹,

Piechocki²

2010

Class. Quantum Grav.

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We analyze the big bounce transition of the quantum FRW model in the setting of the nonstandard loop quantum cosmology (LQC). Elementary observables are used to quantize composite observables. The spectrum of the energy density operator is bounded and continuous. The spectrum of the volume operator is bounded from below and discrete. It has equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semiclassical level which may be detected in astro-cosmo observations. The nonstandard LQC method has a free parameter that should be fixed in some way to specify the big bounce transition.

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“…[26]). The scaling invariant region g c , scaling law (199) and correlation length are important to study our present Universe (see Ref.…”

Section: Some Remarksmentioning

confidence: 93%

Detailed discussions and calculations of quantum Regge calculus of Einstein-Cartan theory

Xue

2010

Phys. Rev. D

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This article presents detailed discussions and calculations of the recent paper ''Quantum Regge calculus of Einstein-Cartan theory'' in [9]. The Euclidean space-time is discretized by a four-dimensional simplicial complex. We adopt basic tetrad and spin-connection fields to describe the simplicial complex. By introducing diffeomorphism and local Lorentz invariant holonomy fields, we construct a regularized Einstein-Cartan theory for studying the quantum dynamics of the simplicial complex and fermion fields. This regularized Einstein-Cartan action is shown to properly approach to its continuum counterpart in the continuum limit. Based on the local Lorentz invariance, we derive the dynamical equations satisfied by invariant holonomy fields. In the mean-field approximation, we show that the averaged size of 4-simplex, the element of the simplicial complex, is larger than the Planck length. This formulation provides a theoretical framework for analytical calculations and numerical simulations to study the quantum Einstein-Cartan theory.

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Gravitational Wilson loop in discrete quantum gravity

Cited by 24 publications

References 61 publications

Discrete Wheeler-DeWitt equation

Discrete Wheeler-DeWitt equation

Turning big bang into big bounce: II. Quantum dynamics

Detailed discussions and calculations of quantum Regge calculus of Einstein-Cartan theory

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