Unique effective action in two-dimensional quantum gravity with nonlocal action

The gauge dependence of the effective action in two-dimensional induced quantum gravity is investigated. A unique effective action is found, and its dependence on the metric of the configuration space is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 97–100, January, 1992.     

Multi-matrix models from jet coefficients

We present a very natural framework in which to discuss multi-matrix models of two-dimensional quantum gravity. Multi-matrix model actions, string equations, and other quantities can be compactly expressed in terms of the jets of the resolvents of the relevant differential operators. This allows one to write down equations describing minimal matter coupled to two-dimensional quantum gravity directly in terms of known functionals.     

Random surfaces in three-dimensional simplicial gravity

A model of simplicial quantum gravity in three dimensions is investigated numerically based on the technique of the dynamical triangulation (DT). We are concerned with the surfaces appearing on boundaries (i.e., sections) of three-dimensional DT manifold with topology. A new scaling behavior of genus distributions of boundary surfaces is found. Furthermore, these surfaces are compared with the random surfaces generated by the two-dimensional DT method which are well known as a correct discretized method of the two-dimensional quantum gravity.     

The emergence of background geometry from quantum fluctuations

We show how the quantization of two-dimensional gravity leads to an (Euclidean) quantum space–time where the average geometry is that of constant negative curvature and where the Hartle–Hawking boundary condition arises naturally.     

Stability of Semiclassical Gravity Solutions with Respect to Quantum Metric Fluctuations

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.     

Gravitational actions in two dimensions and the Mabuchi functional

The Mabuchi energy is an interesting geometric functional on the space of Kähler metrics that plays a crucial rôle in the study of the geometry of Kähler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.     

Some remarks on the Liouville approach to two-dimensional quantum gravity

We show the equivalence of tow-dimensional quantum gravity interacting with a total number of scalar and Dirac-fermion degrees of freedom less of 26 and quantum Liouvill field theory. An analysis of the two-dimensional effective geometry, as well as of the conditions of non-triviality of the resulting quantum theory, is also included. A general procedure to get the complete one-loop effective action is suggested at the end.     

An interacting geometry model and induced gravity

We propose the theory of quantum gravity with interactions introduced by topological principle. The fundamental property of such a theory is that its energy-momentum tensor is a BRST anticommutator. Physical states are elements of the BRST cohomology group. The model with only topological excitations, introduced recently by Witten, is discussed from the point of view of the induced gravity program. We find that the mass scale is induced dynamically by gravitational instantons. The low-energy effective theory has gravitons, which occur as the collective excitations of geometry, when the metric becomes dynamical. Applications of cobordism theory to quantum gravity are discussed.This essay received the second award from the Gravity Research Foundation for the year 1988.—Ed.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

Operator product expansion in two-dimensional quantum gravity

We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next, by making two of the operators closer, we examine whether there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry.     

Conformal Couplings in Induced Gravity

It is found that the induced gravity with conformal couplings requires the conformal invariance in both classical and quantum levels for consistency. This is also true for the induced gravity with an extended conformal coupling interacting with torsion.     

Gauge-Invariant and Gauge-Fixing Independent Effective Action in One-Loop Quantum Gravity

The one-parameter dependent family of the gauge invariant and gauge fixing independent effective actions is considered in one-loop approximation. The one-loop unique effective action (chosing as the representative of this family) in d = 4 Einstein quantum gravity with scalar field and Brans-Dicke quantum theory in flat space, in d = 4 Einstein gravity on De Sitter background, in higher derivative gravity on d-dimensional torus compactified background is calculated. The configuration-space metric dependence of the unique effective action in these calculations is investigated. The appearing problems (the configuration-space metric dependence of the physical quantities like induced gravitational constant) are discussed.     

Gravity on a multifractal

Despite their diversity, many of the most prominent candidate theories of quantum gravity share the property to be effectively lower-dimensional at small scales. In particular, dimension two plays a fundamental role in the finiteness of these models of Nature. Thus motivated, we entertain the idea that spacetime is a multifractal with integer dimension 4 at large scales, while it is two-dimensional in the ultraviolet. Consequences for particle physics, gravity and cosmology are discussed.     

Induced classical gravity

We review the induced-gravity approach according to which the Einstein gravity is a long-wavelength effect induced by underlying fundamental quantum fields due to the dynamical-scale symmetry breaking. It is shown that no ambiguities arise in the definition of the induced Newton and cosmological constants if one works with the path integral for fundamental fields in the low-scale region. The main accent is on a specification of the path integral which enables us to utilize the unitarity condition and thereby avoid ambiguities. Induced Einstein equations appear from the symmetry condition that the path integral of fundamental fields for a slowly varying metric is invariant under the local GL(4, R)-transformations of a tetrad, which contain the local Euclidean Lorentz, O(4)-rotations as a subgroup. The relationship to induced quantum gravity is briefly outlined.     

Chiral fermions in 2d quantum gravity

Two theories of chiral fermions coupled to different quantum gravities in two dimensions are studied. One employs Jackiw's ansatz for classical gravity by introducing an auxiliary scalar, the other is based on the induced quantum gravity of Polyakov, which has no classical analogue. By investigating a localized theory of the effective action we show that in both cases a limited number of fermions of either chirality may couple consistently. It is stressed that the Weyl variable has to be quantized properly, which is related to recent work done on non-critical strings.     

Quaternion-Loop Quantum Gravity

It is shown that the Riemannian curvature of the 3-dimensional hypersurfaces in space-time, described by the Wilson loop integral, can be represented by a quaternion quantum operator induced by the SU(2) gauge potential, thus providing a justification for quaternion quantum gravity at the Tev energy scale.     

Convenient versus unique effective action formalism in 2d dilaton-Maxwell quantum gravity

The structure of one-loop divergences of two-dimensional dilaton-Maxwell quantum gravity is investigated in two formalisms: one using a convenient effective action and the other a unique effective action. The one-loop divergences (including surface divergences) of the convenient effetive action are calculated in three different covariant gauges: (i) De Witt, (ii) Ω-degenerate De Witt, and (iii) simplest covariant. The on-shell effective action is given by surface divergences only (finiteness of theS-matrix), which yet depend upon the gauge condition choice. Off-shell renormalizability is discussed and classes of renormalizable dilaton and Maxwell potentials are found which coincide in the cases of convenient and unique effective actions. A detailed comparison of both situations, i.e. convenient vs. unique effective action, is given. As an extension of the procedure, the one-loop effective action in two-dimensional dilaton-Yang-Mills gravity is calculated.     

A matrix model for 2D quantum gravity defined by Causal dynamical triangulations

A novel continuum theory of two-dimensional quantum gravity, based on a version of Causal Dynamical Triangulations which incorporates topology change, has recently been formulated as a genuine string field theory in zero-dimensional target space [J. Ambjørn, R. Loll, Y. Watabiki, W. Westra, S. Zohren, arXiv: 0802.0719]. Here we show that the Dyson–Schwinger equations of this string field theory are reproduced by a cubic matrix model. This matrix model also appears in the so-called Dijkgraaf–Vafa correspondence if the superpotential there is required to be renormalizable. In the spirit of this model, as well as the original large-N expansion by 't Hooft, we need no special double-scaling limit involving a fine tuning of coupling constants to obtain the continuum quantum-gravitational theory. Our result also implies a matrix model representation of the original, strictly causal quantum gravity model.     

The Vilkovisky-DeWitt effective action for quantum gravity

The Vilkovisky-DeWitt effective action for gauge theories is reviewed and then discussed in the context of N-dimensional quantum gravity and quantum Kaluza-Klein theory. The formalism gives an effective action which is gauge-independent and gauge and field-parametrization invariant. These features are illustrated for the vacuum energy of N-dimensional gravity. The Bunch-Parker local momentum space approach is used to calculate also the induced Ricci scalar term in the expansion of the effective action in powers of the curvature. The effective field equations are applied to the self-consistent dimensional reduction of five-dimensional Kaluza-Klein theory. A solution exists, but is found to be physically unacceptable.