We study the semiclassical expansion of the effective action for a Regge state-sum model and its dependence on the choice of the path-integral measure and the spectrum of the edge lengths. If the positivity of the edge lengths is imposed in the effective action equation, we find that the semiclassical expansion is not possible for the power-law measures, while the exponential measures allow the semiclassical expansion. Furthermore, a slightly generalized exponential measure can generate the cosmological term in the effective action as a quantum correction, with a naturally small value of the cosmological constant. We also find that in the case of a discrete length spectrum, the semiclassical expansion is allowed only if the spectrum gap is much smaller than the Planck length.
The review paper "Discrete Structures in Physics", written in 2000, describes how Regge's discretization of Einstein's theory has been applied in classical relativity and quantum gravity. Here, developments since 2000 are reviewed briefly, with particular emphasis on progress in quantum gravity through spin foam models and group field theories.
The Regge-Teitelboim formulation of gravity, which utilizes dynamical embeddings in a background space, effectively introduces source terms in the standard Einstein equations that are not attributable to the energy-momentum tensor. We show that for a simple class of embeddings of the Robertson-Walker metric, these source terms naturally generate cosmic acceleration.
We argue that Hartle-Hawking states in the Regge quantum gravity model generically contain non-trivial entanglement between gravity and matter fields. Generic impossibility to talk about "matter in a point of space" is in line with the idea of an emergent spacetime, and as such could be taken as a possible candidate for a criterion for a plausible theory of quantum gravity. Finally, this new entanglement could be seen as an additional "effective interaction", which could possibly bring corrections to the weak equivalence principle.
Theory of gravity is considered in the Regge-Teitelboim approach in which the pseudo-Riemannian space is treated as a surface isometrically embedded in an ambient Minkowski space of higher dimension. This approach is formulated in terms of a field theory in which the original pseudo-Rimannian space is defined by the field constant-value surfaces. The symmetry properties of the proposed theory are investigated, and possible structure of the field-theoretical Lagrangian is discussed.
We consider the formulation of the gravity theory first suggested by Regge and Teitelboim where the space-time is a four-dimensional surface in a flat ten-dimensional space. We investigate a canonical formalism for this theory following the approach suggested by Regge and Teitelboim. Under constructing the canonical formalism we impose additional constraints agreed with the equations of motion. We obtain the exact form of the first-class constraint algebra. We show that this algebra contains four constraints which form a subalgebra (the ideal), and if these constraints are fulfilled, the algeb…
This paper is devoted to the approach to gravity as a theory of a surface embedded in a flat ambient space. After the brief review of the properties of original theory by Regge and Teitelboim we concentrate on its field-theoretic reformulation, which we call splitting theory. In this theory embedded surfaces are defined through the constant value surfaces of some set of scalar fields in high-dimensional Minkowski space. We obtain an exact expressions for this scalar fields in the case of Friedmann universe. We also discuss the features of quantisation procedure for this field theory.
We consider spinfoam quantum gravity for general triangulations in the regime $l_P^2\ll a\ll a/γ$, namely in the combined classical limit of large areas $a$ and flipped limit of small Barbero-Immirzi parameter $γ$, where $l_P$ is the Planck length. Under few working hypotheses we find that the flipped limit enforces the constraints that turn the spinfoam theory into an effective Regge-like quantum theory with lengths as variables, while the classical limit selects among the possible geometries the ones satisfying the Einstein equations. Two kinds of quantum corrections appear in terms of power…
The equations governing the perturbations of the Schwarzschild metric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the technique introduced in [2], we prove an integrated local energy decay estimate for both the Regge-Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the $r^p$ hierarchy estimates [13, 32] to prove that both the Regge-Wheeler and Zerilli variables decay as $t^{-\frac{3}{2}}$ in fixed regions of $r$.
We show that it is possible to solve the cosmological constant (CC) problem in a discrete quantum gravity theory based on Regge calculus by using the effective action approach and a special path-integral measure. The effective cosmological constant is given as a sum of 3 terms: the classical CC, the quantum gravity CC and the matter CC. Since the observations can only measure the sum of these 3 terms, we can choose the classical CC to be equal to the negative value of the matter CC. Hence the effective CC is given only by the quantum gravity CC, which is determined by the path-integral measure…
We review the construction of the path integral and the corresponding effective action for the Regge formulation of General Relativity under the assumption that the short-distance structure of the spacetime is not a smooth 4-manifold, but a picewise linear manifold based on a triangulation of a smooth 4-manifold. We point out that the exponentially damped 4-volume path-integral measure does not give a finite path integral, although it can be used for the construction of the perturbative effective action. We modify the 4-volume measure by multiplying it by an inverse power of the product of the…
This is the first of a series of papers dedicated to the study of the partition function of three-dimensional quantum gravity on the twisted solid torus with the aim to deepen our understanding of holographic dualities from a non-perturbative quantum gravity perspective. Our aim is to compare the Ponzano-Regge model for non-perturbative three-dimensional quantum gravity with the previous perturbative calculations of this partition function. We begin by reviewing the results obtained in the past ten years via a wealth of different approaches, and then introduce the Ponzano--Regge model in a sel…
We introduce a physical piecewise linear metric associated to a Regge triangulation of a smooth 4-manifold. We describe the basic properties of the corresponding geometry in the cases of the Euclidean and the Minkowski signature. In the Minkowski case, we describe the Regge action and how to define the corresponding path integral for the casual triangulations. We also discus the Regge path integral for a triangulation associated to the Friedman-Lemaitre-Robertson-Walker cosmological model and briefly study the corresponding wavefunctions, namely the Hartle-Hawking and the Vilenkin wavefunction…
A number of approaches to 4D quantum gravity, such as holography and loop quantum gravity, propose areas instead of lengths as fundamental variables. The Area Regge action, which can be defined for general 4D triangulations, is a natural choice for an action based on areas. It does indeed appear in the semi-classical limit of spin foam models. The Area Regge action does however only lead to a discrete version of the gravitational equations of motion, if one implements constraints, that ensure that the areas are compatible with a consistent length assignment to the edges of the triangulation. T…
