Lattice Gravity Research Articles

Gauge invariance and functional integration measure in lattice gravity

We compute to one loop a Slavnov-Taylor identity related to the reparametrization transformations in lattice Einstein gravity at zero external momentum. With a proper choice of the local integration measure such an identity is satisfied in a completely gauge invariant way thus supporting the recovery of reparametrization invariance in the continuum limit. This calculation moreover fixes some perturbative features of the functional integration measure but, due to the lack of asymptotic freedom, remnants of the non-rotational invariance prevent us from inferring the general form of the non-perturbative measure.

Functional measure for lattice gravity.

Author(s): Bander, M | Abstract: A procedure is developed for transcription of any measure for the integration over metric fields in the continuum to the Regge-calculus lattice. © 1986 The American Physical Society.

Simplicial quantum gravity with higher derivative terms: Formalism and numerical results in four dimensions

Higher derivative terms for Regge's formulation of lattice gravity are discussed. The analytic weak-field expansion for the regular tessellation α 5 of the four-sphere is presented. Preliminary numerical results for some computations in four dimensions are also discussed.

Lattice gravity and strings

We are concerned with applications of the simplicial discretization method (Regge calculus) to two-dimensional quantum gravity with emphasis on the physically relevant string model. Beginning with the discretization of gravity and matter we exhibit a discrete version of the conformal trace anomaly. Proceeding to the string problem we show how the direct approach of (finite difference) discretization based on Nambu action corresponds to unsatisfactory treatment of gravitational degrees. Based on the Regge approach we then propose a discretization corresponding to the Polyakov string. In this context we are led to a natural geometric version of the associated Liouville model and two-dimensional gravity.

Planar diagrams, two-dimensional lattice gravity and surface models

Abstract Some discrete lattice models for quantum two-dimensional euclidean gravity are shown to be equivalent to zero-dimensional planar field theories. Explicit expressions are given for partition functions. A universal continuum limit exists for open surfaces, but not for closed ones, and is argued to describe a space with negative average curvature. Extensions to those models to higher dimensions and to surface models are briefly discussed.

A correlation decay theorem at high temperature

We derive a theorem of exponential decay of correlation functions at high temperature for a general statistical mechanical system following the approach introduced by L. Gross. The theorem is formulated so as to be useful for locality problems in lattice quantum gravity.

Lattice gravity near the continuum limit

We prove that the lattice gravity always approaches the usual continuum limit when the link length ℓ → 0, provided that certain general boundary conditions are satisfied. This result holds for any lattice, regular or irregular. Furthermore, for a given lattice, the deviation from its continuum limit can be expressed as a power series in ℓ 2. General formulas for such a perturbative calculation are given, together with a number of illustrative examples, including the graviton propagator. The lattice gravity satisfies all the invariance properties of Einstein's theory of general relativity. In addition, it is symmetric under a new class of transformations that are absent in the usual continuum theory. The possibility that the lattice theory (with a nonzero ℓ) may be more fundamental is discussed.

Rotationally invariant field theory on lattices. III. Quantizing gravity by means of lattices

Whether one views the lattice as merely regulatory or physically real, lattice gravity is different from continuum gravity because the symmetry group of coordinate transformations is destroyed by the lattice. Since negative-norm kinetic terms can no longer be removed by specifying a gauge, the functional integrals will not converge. We suggest including a quartic potential in the action which obliges the action to converge regardless of the sign of those terms. This potential must depend on a background metric in order that a number of desirable features may hold. For example, consider the conformal factor: One needs a quartic term to cancel the wrong-sign kinetic energy. But one also wants the theory to have asymptotically flat manifolds and to admit the flat manifold as a solution. This requires a canceling quadratic. This in turn requires a background structure. This structure is supplied in a natural way by the lattice itself. The approach to the continuum is examined.