Standard and Z2-Regge theory in two dimensions

Abstract

We qualitatively compare two versions of quantum Regge calculus by means of Monte Carlo simulations. In Standard Regge Calculus the quadratic link lengths of the triangulation vary continuously, whereas in the Z2-Regge Model they are restricted to two possible values. The goal is to determine whether the computationally more easily accessible Z2 model retains the characteristics of standard Regge theory.Standard Regge Calculus (SRC) [1] provides an interesting method to explore quantum gravity in a non-perturbative fashion [2]. Although its code can be efficiently vectorized for large scale computing, it is still a very time demanding enterprise. One therefore seeks for suitable approximations which will simplify the SRC and yet retain most of its universal features. The Z2-Regge Model (Z2RM) [3] could be such a desired simplification. Here the quadratic link lengths of the simplicial complexes are restricted to take on only the two valuesin close analogy to the ancestor of all lattice models, the Ising-Lenz model. To test whether this simpler model is in a reasonable sense still similar to SRC, we study both models in two dimensions and compare a number of observables for one particular lattice size. Starting point for both SRC and Z~.RM is Regge's discrete description of General Relativity [1] in which a given continuum manifold is replaced by a piecewise flat simplicial space. In two dimensions this procedure is easily illustrated by choosing a triangulation of the surface under consideration. Every triangle then represents a part of a piecewise linear manifold. In principle the functional integration should extend over all metrics on all possible topologies, but, as is usually done, we restrict ourselves to one specific topology, the torus, whose Euler characteristic is zero. Consequently the action in the exponent of (2) consists only of a cosmological constant A times the sum over all triangle areas At. Moreover the path-integral approach suffers from a non-uniqueness of the integration measure and it is even claimed that the true measure is of non-local nature [4]. We used as a trial functional integration measure the expression within the square brackets of (2) with m E IR permitting to investigate a 1-parameter family of measures. The function ~" constrains the integration to those Euclidean configurations of link lengths which do not violate the triangle inequalities. In the Z2RM [3] the squared link lengths as well as functions of them are rewritten with respect to (1). Thus the area of a triangle with edges ql, q2, qt is expressed as At = co + Cl(al + ~r2 + az) + c2(~rla2 + 3uO'10"l "~-0"20"/) 71-C30"ldr20r / •The coefficients ci depend on e only and impose the condition e < ~ = e,~,~ in order