Some General Problems in Quantum Gravity II: The Three-Dimensional Case (original) (raw)

An alternative path integral for quantum gravity

Journal of High Energy Physics, 2016

We define a (semi-classical) path integral for gravity with Neumann boundary conditions in D dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This "Neumann ensemble" perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.

Mathematical Problems of Quantum Gravity

Arxiv preprint gr-qc/9701042, 1997

This pre-print contains the abstracts of seminars (including key references) presented at the ESI workshop on mathematical problems in quantum gravity held during July and August of 1996. Contributors include A

The Measure in Euclidean Quantum Gravity

2011

In this article a description is given of the measure in Euclidean path-integral in quantum gravity, and recent results using the Faddeev-Popov method of gauge fixing. The results suggest that the effective action is finite and positive.

Lorentzian and Euclidean Quantum Gravity — Analytical and Numerical Results

M-Theory and Quantum Geometry

We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of "dynamical triangulations" is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian-and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a 1 The method of dynamical triangulations was introduced in the context of string theory and 2d quantum gravity in [3, 4, 5], and subsequently extended to higher-dimensional Euclidean quantum gravity [6, 7]. An extensive review covering the developments up to 1996 can be found in the book [8]. A more recent summary is contained in [9], while the review [10] deals with a variety of lattice approaches to four-dimensional quantum gravity, including dynamical triangulations. The use of dynamical-triangulations methods in Lorentzian gravity was pioneered in [11, 12, 13].

A Immirzi-like parameter for 3D quantum gravity

Classical and Quantum Gravity, 2008

We study an Immirzi-like ambiguity in three-dimensional quantum gravity. It shares some features with the Immirzi parameter of four-dimensional loop quantum gravity: it does not affect the equations of motion, but modifies the Poisson brackets and the constraint algebra at the canonical level. We focus on the length operator and show how to define it through non-commuting fluxes. We compute its spectrum and show the effect of this Immirzi-like ambiguity. Finally, we extend these considerations to 4d gravity and show how the different topological modifications of the action affect the canonical structure of loop quantum gravity.

New aspects of two-dimensional quantum gravity

2009

Causal dynamical triangulations (CDT) can be used as a regularization of quantum gravity. In two dimensions the theory can be solved anlytically, even before the cutoff is removed and one can study in detail how to take the continuum limit. We show how the CDT theory is related to Euclidean 2d quantum gravity (Liouville quantum gravity), how it can be generalized and how this generalized CDT model has a string field theory representation as well as a matrix model representation of a new kind, and finally how it examplifies the possibility that time in quantum gravity might be the stochastic time related to the branching of space into baby universes.

Quantum gravity anti path integrals

The path-integral method seems to be the most suitable for the quantization of gravity. One would expect the dominant contribution to the path integral to come from metrics which are near background metrics that are solutions of classical Einstein equations. The action of these background metrics gives rise to a new phenomenon in field theory, intrinsic quantum entropy. This is shown to be related to the scaling behavior of the gravitational action and to the topology of the gravitational field. The quadratic terms in the Taylor series of the action about the background metrics give the one-loop corrections. In a supersymmetric theory the quartic and quadratic but not the so-called logarithmic divergences cancel to give a one-loop term that is finite without regularization. From the one-loop term one can obtain the effective energy-momentum tensor on the background metric. In the case of an evaporating black hole, the energy-momentum tensor will be regular on the future horizon. The usual perturbation expansion breaks down for quantum gravity because the higher (interaction) terms in the Taylor series are not bounded by the quadratic (free) ones. To overcome this I suggest that one might replace the path integrals over the terms in the Taylor series by a discrete sum of the exponentials of the actions of all complex solutions of the Einstein equations, each solution being weighted by its one-loop term. This approach seems to give a picture of the gravitational vacuum as a sea of virtual Planck-mass black holes.

Some Quantum Aspects of D= 3 Space-Time Massive Gravity

Arxiv preprint gr-qc/9611049, 1996

We discuss some features of Einstein-Proca gravity in D = 3 and 4 space-times. Our study includes a discussion on the tree-level unitarity and on the issue of light deflection in 3D gravity in the presence of a mass term. ... 1 fcpnunes@cce.ufes.br 2 gentil@if.ufrj.br

Discrete spacetime volume for three-dimensional BF theory and quantum gravity

The Turaev-Viro state sum invariant is known to give the transition amplitude for the three dimensional BF theory with cosmological term, and its deformation parameterh is related with the cosmological constant viah = √ Λ. This suggests a way to find the expectation value of the spacetime volume by differentiating the Turaev-Viro amplitude with respect to the cosmological constant. Using this idea, we find an explicit expression for the spacetime volume in BF theory. According to our results, each labelled triangulation carries a volume that depends on the labelling spins. This volume is explicitly discrete. We also show how the Turaev-Viro model can be used to obtain the spacetime volume for (2+1) dimensional quantum gravity.