scholarly journals A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY

2010 ◽  
Vol 25 (23) ◽  
pp. 4475-4492 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Dynamical Variable ◽  
Renormalization Procedure ◽  
Emergent Phenomena ◽  
Tensor Models ◽  
Theories Of Gravity

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

scholarly journals TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

2011 ◽  
Vol 26 (19) ◽  
pp. 3249-3258 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Algebraic Structure ◽  
The Other ◽  
Invariant Form ◽  
Infinitesimal Symmetry ◽  
Tensor Models ◽  
Symmetry Transformations

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

One-loop renormalization in quantum gravity

2021 ◽  
pp. 474-487
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
General Relativity ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Detailed Analysis ◽  
Polynomial Model ◽  
Similar Calculation ◽  
Higher Derivative ◽  
Gauge Fixing ◽  
Fourth Derivative ◽  
Detailed Derivation

This chapter demonstrates the basic methods of one-loop calculations in quantum gravity. Basing its discussion on the general results obtained in chapter 10, it first presents a detailed analysis of the gauge-fixing dependence of one-loop divergences in quantum general relativity and higher-derivative models of quantum gravity. After that, a detailed derivation of divergences in quantum general relativity is given, with the simplest parametrization of the quantum metric and minimal gauge fixing. One-loop divergences in the general (non-conformal) fourth-derivative quantum gravity are then derived in less detail. For a similar calculation in the superrenormalizable polynomial model (superrenormalizable gravity), the chapter just presents and discusses the final result.

scholarly journals GRAVITY AS BF THEORY PLUS POTENTIAL

2009 ◽  
Vol 24 (15) ◽  
pp. 2776-2782 ◽  
Author(s):  
KIRILL KRASNOV
Keyword(s):  
General Relativity ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Alternative Formulation ◽  
Potential Term ◽  
Bf Theory ◽  
Spin Foam Models ◽  
Spin Foam ◽  
To Come

Spin foam models of quantum gravity are based on Plebanski's formulation of general relativity as a constrained BF theory. We give an alternative formulation of gravity as BF theory plus a certain potential term for the B-field. When the potential is taken to be infinitely steep one recovers general relativity. For a generic potential the theory still describes gravity in that it propagates just two graviton polarizations. The arising class of theories is of the type amenable to spin foam quantization methods, and, we argue, may allow one to come to terms with renormalization in the spin foam context.

scholarly journals THE LOWEST MODES AROUND GAUSSIAN SOLUTIONS OF TENSOR MODELS AND THE GENERAL RELATIVITY

2008 ◽  
Vol 23 (24) ◽  
pp. 3863-3890 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Two Dimensions ◽  
Tensor Model ◽  
Momentum Dependence ◽  
Two Dimensional ◽  
Renormalization Procedure ◽  
Metric Tensor ◽  
Tensor Models ◽  
General Coordinate Transformation ◽  
Metric Fluctuations

In the paper arXiv:0706.1618[hep-th], the number distribution of the low-lying spectra around Gaussian solutions representing various dimensional fuzzy tori of a tensor model was numerically shown to be in accordance with the general relativity on tori. In this paper, I perform more detailed numerical analysis of the properties of the modes for two-dimensional fuzzy tori, and obtain conclusive evidences for the agreement. Under a proposed correspondence between the rank-3 tensor in tensor models and the metric tensor in the general relativity, conclusive agreement is obtained between the profiles of the low-lying modes in a tensor model and the metric modes transverse to the general coordinate transformation. Moreover, the low-lying modes are shown to be well on a massless trajectory with quartic momentum dependence in the tensor model. This is in agreement with that the lowest momentum dependence of metric fluctuations in the general relativity will come from the R2-term, since the R-term is topological in two dimensions. These evidences support the idea that the low-lying low-momentum dynamics around the Gaussian solutions of tensor models is described by the general relativity. I also propose a renormalization procedure for tensor models. A classical application of the procedure makes the patterns of the low-lying spectra drastically clearer, and suggests also the existence of massive trajectories.

scholarly journals NEW SPIN FOAM MODELS OF QUANTUM GRAVITY

2005 ◽  
Vol 20 (17n18) ◽  
pp. 1305-1313
Author(s):  
A. MIKOVIĆ
Keyword(s):  
General Relativity ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Path Integral ◽  
Critical Review ◽  
Loop Quantum Gravity ◽  
Spin Networks ◽  
Spin Foam Models ◽  
Spin Foam Model ◽  
Spin Foam

We give a brief and a critical review of the Barret-Crane spin foam models of quantum gravity. Then we describe two new spin foam models which are obtained by direct quantization of General Relativity and do not have some of the drawbacks of the Barret-Crane models. These are the model of spin foam invariants for the embedded spin networks in loop quantum gravity and the spin foam model based on the integration of the tetrads in the path integral for the Palatini action.

scholarly journals CANONICAL TENSOR MODELS WITH LOCAL TIME

2012 ◽  
Vol 27 (05) ◽  
pp. 1250020 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Field Theory ◽  
General Relativity ◽  
Local Time ◽  
String Field Theory ◽  
Dynamical Variable ◽  
Tensor Model ◽  
Canonical Variables ◽  
Hamilton Formalism ◽  
Tensor Models ◽  
Constraint Algebra

It is an intriguing question how local time can be introduced in the emergent picture of space–time. In this paper, this problem is discussed in the context of tensor models. To consistently incorporate local time into tensor models, a rank-three tensor model with first class constraints in Hamilton formalism is presented. In the limit of usual continuous spaces, the algebra of constraints reproduces that of general relativity in Hamilton formalism. While the momentum constraints can be realized rather easily by the symmetry of the tensor models, the form of the Hamiltonian constraints is strongly limited by the condition of the closure of the whole constraint algebra. Thus the Hamiltonian constraints have been determined on the assumption that they are local and at most cubic in canonical variables. The form of the Hamiltonian constraints has similarity with the Hamiltonian in the c < 1 string field theory, but it seems impossible to realize such a constraint algebras in the framework of vector or matrix models. Instead these models are rather useful as matter theories coupled with the tensor model. In this sense, a three-index tensor is the minimum-rank dynamical variable necessary to describe gravity in terms of tensor models.

scholarly journals Renormalization Group Approach to the Continuum Limit of Matrix Models of Quantum Gravity With Preferred Foliation

2021 ◽  
Vol 9 ◽  
Author(s):  
Alicia Castro ◽  
Tim Andreas Koslowski
Keyword(s):  
Renormalization Group ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Fixed Points ◽  
Renormalization Group Equation ◽  
Two Dimensions ◽  
Gravity Models ◽  
Group Equation ◽  
Tensor Models ◽  
The Matrix

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson [1], which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

scholarly journals Breaking the cosmological background degeneracy by two-fluid perturbations in f(R) gravity

2015 ◽  
Vol 24 (07) ◽  
pp. 1550053 ◽  
Author(s):  
Amare Abebe
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Structure Formation ◽  
Background Level ◽  
Cosmological Perturbations ◽  
Gauge Invariant ◽  
First Order ◽  
Cosmological Background ◽  
Theories Of Gravity ◽  
Two Fluid

One of the exact solutions of f(R) theories of gravity in the presence of different forms of matter exactly mimics the ΛCDM solution of general relativity (GR) at the background level. In this work we study the evolution of scalar cosmological perturbations in the covariant and gauge-invariant formalism and show that although the background in such a model is indistinguishable from the standard ΛCDM cosmology, this degeneracy is broken at the level of first-order perturbations. This is done by predicting different rates of structure formation in ΛCDM and the f(R) model both in the complete and quasi-static regimes.

scholarly journals QUANTUM GRAVITY BY THE COMPLEX CANONICAL FORMULATION

1992 ◽  
Vol 01 (03n04) ◽  
pp. 439-523 ◽  
Author(s):  
HIDEO KODAMA
Keyword(s):  
General Relativity ◽  
Quantum Gravity ◽  
Conventional Treatment ◽  
Quantum States ◽  
Hamiltonian Constraint ◽  
Canonical Formulation ◽  
Recent Developments ◽  
Canonical Approach ◽  
New Formulation ◽  
Basic Features

The basic features of the complex canonical formulation of general relativity and the recent developments in the quantum gravity program based on it are reviewed. The exposition is intended to be complementary to the review articles already available and some original arguments are included. In particular the conventional treatment of the Hamiltonian constraint and quantum states in the canonical approach to quantum gravity is criticized and a new formulation is proposed.

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