Four-dimensional Quantum Gravity Research Articles

QUANTUM GRAVITY IN PLEBANSKI FORMULATION

We present a theory of four-dimensional quantum gravity with massive gravitons which may be essentially renormalizable. In Plebanski formulation of general relativity in which the tetrads, the connection and the curvature are all independent variables (and the usual relations among these quantities are valid only on-shell), we consider the nonperturbative theory of gravity with a nonzero background connection. We predict a tiny value of the graviton mass: mg≈1.5×10-42 GeV and extremely small dimensionless coupling constant of the perturbative gravitational interaction: g~10-60. We put forward the idea by Isimori56 on renormalizability of quantum gravity having multi-gravitons with masses m0, m1, …, mN-1.

Coupling a point-like mass to quantum gravity with causal dynamical triangulations

We present a possibility of coupling a point-like, non-singular, mass distribution to four-dimensional quantum gravity in the nonperturbative setting of causal dynamical triangulations (CDT). In order to provide a point of comparison for the classical limit of the matter-coupled CDT model, we derive the spatial volume profile of the Euclidean Schwarzschild–de Sitter space glued to an interior matter solution. The volume profile is calculated with respect to a specific proper-time foliation matching the global time slicing present in CDT. It deviates in a characteristic manner from that of the pure-gravity model. The appearance of coordinate caustics and the compactness of the mass distribution in lattice units put an upper bound on the total mass for which these calculations are expected to be valid. We also discuss some of the implementation details for numerically measuring the expectation value of the volume profiles in the framework of CDT when coupled appropriately to the matter source.

IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES

IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES

Nonassociative quantum theory on octooctonion algebra

Using octooctonions (i.e., octonions with octonion coefficients O×O), this paper expresses select findings from nonassociative quantum theory in harmonized notation: Nonrelativistic and relativistic spin operators, Pauli and Dirac matrices, Dirac equation with electromagnetic and gravitational field, and dimensional reduction from quaternionic spin. A generalization of the dimensional reduction program is proposed to argue that octooctonion algebra is wide enough to model a speculated quantum theory that contains all symmetries of the Standard Model, together with four-dimensional Euclidean quantum gravity. The most narrow candidate for such a formulation consists of four generalized Dirac matrices and a four-dimensional operator space with associated fields and charges. Algebraic properties of this relation will be discussed, together with a landscape choice between all possible octooctonionic relations of similar kind.

A Immirzi-like parameter for 3D quantum gravity

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.

Effective gauge group of pure loop quantum gravity is SO(3): New estimate of the Immirzi parameter

We argue that the effective gauge group for {\it pure} four-dimensional loop quantum gravity(LQG) is SO(3) (or $SO(3,C)$) instead of SU(2) (or $SL(2,C)$). As a result, links with half-integer spins in spin network states are not realized for {\it pure} LQG, implying a modification of the spectra of area and volume operators. Our observations imply a new value of $\gamma \approx 0.170$ for the Immirzi parameter which is obtained from matching the Bekenstein-Hawking entropy to the number of states from LQG calculations. Moreover, even if the dominant contribution to the entropy is not assumed to come from configurations with the minimum spins, the results of both pure LQG and the supersymmetric extension of LQG can be made compatible when only integer spins are realized for the former, while the latter also contains half-integer spins, together with an Immirzi parameter for the supersymmetric case which is twice the value of the SO(3) theory. We also verify that the $-{1/2}$ coefficient of logarithmic correction to the Bekenstein-Hawking entropy formula is robust, independent of whether only integer, or also half-integer spins, are realized.

Finiteness of quantum gravity coupled with matter in three spacetime dimensions

As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory C . The Newton constant and the marginal parameters of C are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop and energy expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern–Simons U(1) gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity.

Indications for criticality at zero curvature in a 4d Regge model of Euclidean quantum gravity

We re-examine the approach to four-dimensional Euclidean quantum gravity based on the Regge calculus. A cut-off on the link lengths is introduced and consequently the gravitational coupling and the cosmological constant become independent parameters. We determine the zero curvature, $ =0$, line in the coupling constant plane by numerical simulations. When crossing this line we find a strong, probably first order, phase transition line with indications of a second order endpoint. Beyond the endpoint the transition through the $ =0$ line appears to be a crossover. Previous investigations, using the Regge or the Dynamical Triangulation approach, dealt with a limit in which the first order transition prevails.

A “general boundary” formulation for quantum mechanics and quantum gravity

I propose to formalize quantum theories as topological quantum field theories in a generalized sense, associating state spaces with boundaries of arbitrary (and possibly finite) regions of space–time. I further propose to obtain such “general boundary” quantum theories through a generalized path integral quantization. I show how both, non-relativistic quantum mechanics and quantum field theory can be given a “general boundary” formulation. Surprisingly, even in the non-relativistic case, features normally associated with quantum field theory emerge from consistency conditions. This includes states with arbitrary particle number and pair creation. I also note how three-dimensional quantum gravity is an example for a realization of both proposals and suggest to apply them to four-dimensional quantum gravity.

Implementing causality in the spin foam quantum geometry

We analyze the classical and quantum geometry of the Barrett–Crane spin foam model for four-dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transition amplitudes between quantum gravity states, i.e., realizing in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories).

Matter Dependence of the String Susceptibility Exponent in Four-Dimensional Simplicial Quantum Gravity

The dependence of the number of vector and scalar fields on the string susceptibility exponent is evaluated numerically for four-dimensional simplicial quantum gravity using the grand canonical ensemble method. The numerical result indicates that the four-dimensional simplicial quantum gravity reproduces well predictions of the four-dimensional conformal gravity.

Spin foam models of Riemannian quantum gravity

Using numerical calculations, we compare three versions of the Barrett–Crane model of four-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes given by Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model.

Finite spin-foam-based theory of three- and four-dimensional quantum gravity

Starting from Ooguri's construction for $BF$ theory in three (and four) dimensions, we show how to construct a well defined theory with an infinite number of degrees of freedom. The spin network states that are kept invariant by the evolution operators of the theory are exact solutions of the Hamiltonian constraint of quantum gravity proposed by Thiemann. The resulting theory is the first example of a well defined, finite, consistent, spin-foam based theory in a situation with an infinite number of degrees of freedom. Since it solves the quantum constraints of general relativity it is also a candidate for a theory of quantum gravity. It is likely, however, that the solutions constructed correspond to a spurious sector of solutions of the constraints. The richness of the resulting theory makes it an interesting example to be analyzed by forthcoming techniques that construct the semi-classical limit of spin network quantum gravity.

Clear evidence of a continuum theory of 4D Euclidean simplicial quantum gravity

Four-dimensional (4D) simplicial quantum gravity coupled to both scalar fields ( N X ) and gauge fields ( N A ) has been studied using Monte-Carlo simulations. The matter dependence of the string susceptibility exponent γ (4) is estimated. Furthermore, we compare our numerical results with Background-Metric-Indepenent (BMI) formulation conjectured to describe the quantum field theory of gravity in 4D. The numerical results suggest that the 4D simplicial quantum gravity is related to the conformal gravity in 4D. Therefore, we propose a phase structure in detail with adding both scalar and gauge fields and discuss the possibility and the property of a continuum theory of 4D Euclidean simplicial quantum gravity.

Wavefunction of the Universe and Chern–Simons perturbation theory

The Chern–Simons exact solution of four-dimensional quantum gravity with nonvanishing cosmological constant is presented in metric variables as the partition function of Chern–Simons theory with nontrivial source. The perturbative expansion is given, and the wavefunction is computed to the lowest order of approximation for the Cauchy surface which is topologically a 3-sphere. The state is well-defined even at degenerate and vanishing values of the dreibein. Reality conditions for the Ashtekar variables are also taken into account, and remarkable features of the Chern–Simons state and their relevance to cosmology are pointed out.

Universal regular short distance behavior from an interaction with a scale-invariant gravity

We assume that the fourdimensional quantum gravity is scale invariant at short distances. We show through a simple scaling argument that correlation functions of quantum fields interacting with gravity have a universal (more regular) short distance behavior.

Phase Structure of Four-Dimensional Simplicial Quantum Gravity with a U(1) Gauge Field

The phase structure of four-dimensional simplicial quantum gravity coupled to U(1) gauge fields has been studied using Monte-Carlo simulations. The smooth phase is found in the intermediate region between the crumpled phase and the branched polymer phase. This new phase has a negative string susceptibility exponent, even if the number of vector fields (Nv) is 1. The phase transition between the crumpled phase and the smooth phase has been studied by a finite size scaling method. From the numerical results, we expect that this model (coupled to one gauge field) has a higher order phase transition than first order, which means the possibility to take the continuum limit at the critical point. Furthermore, we consider a modification of the balls-in-boxes model for a clear understanding of the relation between the numerical results and the analytical one.

Reply to “Comment on ‘Wilson loops in four-dimensional quantum gravity’ ”

Although the computation by Fredsted is mathematically correct, his expression for the Wilson loop is meaningful only outside the Schwarzschild radius. We observe that this expression is finite and tends to zero when r approaches ${r}_{S}$ from the outside.

Comment on “Wilson loops in four-dimensional quantum gravity”

Comment on “Wilson loops in four-dimensional quantum gravity”

Discrete Lorentzian quantum gravity

Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice models for four-dimensional quantum gravity, I describe a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries. A number of interesting and unexpected results that have been obtained for these dynamically triangulated models in two and three dimensions make discrete Lorentzian gravity a promising candidate for a non-trivial theory of quantum gravity.