Two dimensional quantum gravity coupled to matter (original) (raw)

QUANTUM GEOMETRY OF 2d GRAVITY COUPLED TO UNITARY MATTER

We show that there exists a divergent correlation length in 2d quantum gravity for the matter fields close to the critical point provided one uses the invariant geodesic distance as the measure of distance. The corresponding reparameterization invariant two-point functions satisfy all scaling relations known from the ordinary theory of critical phenomena and the KPZ exponents are determined by the power-like fall off of these two-point functions. The only difference compared to flat space is the appearance of a dynamically generated fractal dimension d_h in the scaling relations. We analyze numerically the fractal properties of space-time for Ising and three-states Potts model coupled to 2d dimensional quantum gravity using finite size scaling as well as small distance scaling of invariant correlation functions. Our data are consistent with d_h=4, but we cannot rule out completely the conjecture d_H = -2\alpha_1/\alpha_{-1}, where \alpha_{-n} is the gravitational dressing exponent o...

A quantum mechanical framework for pure two-dimensional gravity

Physics Letters B, 1991

A natural quantum mechanical framework for the study of pure 2D quantum gravity arises from the observation that a Fokker-Planck potential fixes the critical behavior of the system. This clarifies the Marinari-Parisi suggestion of defining the zero-dimensional bosonic theory as the dimensional reduction of a supersymmetric one-dimensional counterpart, and can be used in all its strength to approach the D=0 problem.

February 1992 QMW/PH / 91/22 Canonical Quantization of 2d Gravity Coupled to

1992

We study 2d gravity coupled to c < 1 matter through canonical quantization of a free scalar field, with background charge, coupled to gravity. Various features of the theory can be more easily understood in the canonical approach, like gauge indipendence of the path-integral results and the absence of the local physical degrees of freedom. By performing a noncanonical transformation of the phase space variables, we show that the theory takes a free-field form, i.e. the constraints become the free-field Virasoro constraints. This implies that the David-Distler-Kawai results can be derived in a gauge indipendent way, and also proves the free-field assumption which was used for obtaining the spectrum of the theory in the conformal gauge. A discussion of the physical spectrum of the theory is presented, with an analysis of the unitarity of the discrete momentum states. 1

A four-dimensional theory for quantum gravity with conformal and non-conformal explicit solutions

Classical and Quantum Gravity, 1995

The most general version of a renormalizable d = 4 theory corresponding to a dimensionless higher-derivative scalar field model in curved spacetime is explored. The classical action of the theory contains 12 independent functions, which are the generalized coupling constants of the theory. We calculate the one-loop beta functions and then consider the conditions for finiteness. The set of exact solutions of power type is proven to consist of precisely three conformal and three nonconformal solutions, given by remarkably simple (albeit nontrivial) functions that we obtain explicitly. The finiteness of the conformal theory indicates the absence of a conformal anomaly in the finite sector. The stability of the finite solutions is investigated and the possibility of renormalization group flows is discussed as well as several physical applications.

Quantum Gravity in 2+ε Dimensions

Progress of Theoretical Physics Supplement, 1993

We formulate a renormalizable quantum gravity in 2 + E dimensions by generalizing the nonlinear sigma model approach to string theory. We find that the theory possesses the ultraviolet stable fixed point if the central charge of the matter sector is in the range 0 < c < 25. This may imply the existence of consistent quantum gravity theory in 3 and 4 dimensions. We compute the scaling dimensions of the relevant operators in the theory at the ultraviolet fixed point. We obtain a scaling relation between the cosmological constant and the gravitational constant, which is crucial for searching for the continuum limit in the constructive approach to quantum gravity.

Geometry of a two-dimensional quantum gravity: Numerical study

Nuclear Physics B, 1991

A two-dimensional quantum gravity is simulated by means of the dynamical triangulation model. The size of the lattice was up to hundred thousand triangles. Massively parallel simulations and recursive sampling were implemented independently and produced similar results. Wherever the analytical predictions existed, our results confirmed them. The cascade process of baby universes formulation a la Coleman-Hawking scenario in a two-dimensional case has been observed. We observed that there is a simple universal inclusive probability for a baby universe to appear. This anomalous branching of surfaces led to a rapid growth of the integral curvature inside a circle. The volume of a disk in the internal metric has been proven to grow faster than any power of radius. The scaling prediction for the mean square extent given by the Liouville theory has been confirmed. However, the naive expectation for the average Liouville lagrangian < f (p~6)2 > is about 1 order of magnitude different from the results. This apparently points out to some flaws in the current definition of a Liouville model.

CANONICAL QUANTIZATION APPROACH TO 2D GRAVITY COUPLED TO c < 1 MATTER

International Journal of Modern Physics A, 1994

We show that all important features of 2D gravity coupled to c < 1 matter can be easily understood from the canonical quantization approach à la Dirac. Furthermore, we construct a canonical transformation which maps the theory into a free field form, i.e. the constraints become free field Virasoro generators with background charges. This implies the gauge independence of the David–Distler–Kawai results, and also proves the free field assumption which was used for obtaining the spectrum of the theory in the conformal gauge. A discussion of the unitarity of the physical spectrum is presented and we point out that the scalar products of the discrete states are not well defined in the standard Fock space framework.

Two-dimensional quantum gravity

This chapter contains a short review the correspondence between large N matrix models and critical phenomena on lattices with fluctuating geometry.

The Constraints of 2+1 Quantum Gravity

The study of the gravitational field in 2+1 spacetime dimensions (2 space, 1 time) has blossomed in the last few years into a substantial industry, after important contributions by Leutwyler [1], Deser, Jackiw and 'tHooft [2] and Witten .

Scaling exponents in quantum gravity near two dimensions

Nuclear Physics B, 1993

We formulate quantum gravity in 2 + ǫ dimensions in such a way that the conformal mode is explicitly separated. The dynamics of the conformal mode is understood in terms of the oversubtraction due to the one loop counter term. The renormalization of the gravitational dressed operators is studied and their anomalous dimensions are computed. The exact scaling exponents of the 2 dimensional quantum gravity are reproduced in the strong coupling regime when we take ǫ → 0 limit. The theory possesses the ultraviolet fixed point as long as the central charge c < 25, which separates weak and strong coupling phases. The weak coupling phase may represent the same universality class with our Universe in the sense that it contains massless gravitons if we extrapolate ǫ up to 2.