ar X iv : h ep - t h / 93 10 15 2 v 1 2 2 O ct 1 99 3 SCALAR FIELDS COUPLED TO FOUR - DIMENSIONAL LATTICE GRAVITY ∗

I discuss some results we have obtained recently in a lattice model for quantized gravity coupled to scalar matter in four dimensions. We have looked at how the continuous phase transition separating the smooth from the rough phase of gravity is influenced by the presence of the scalar field. We find that close to the critical point, where the average curvature approaches zero, the effects of the scalar field are small and the coupling of matter to gravity seems to be weak. The nature of the phase diagram and the values for the critical exponents would suggest that gravitational interactions increase with distance. Introduction Of course any serious attempt at understanding the ground state properties of quantized gravity has to include at some stage the consideration of the effects of matter fields. While there are many choices for the matter fields and for their interactions, the simplest actions to deal with in the framework of a lattice model for gravity are the ones that represent one (or more) scalar fields [1]. In this talk I will briefly review these results. Regge’s lattice model for gravity is the natural discretization for quantized gravity in four dimensions. At the classical level, it is completely equivalent to general relativity, and the correspondence is particularly transparent in the lattice weak field expansion, with the invariant edge lengths playing the role of infinitesimal geodesics in the continuum. Recent work based on Regge’s simplicial formulation of gravity has shown, in pure gravity without matter, the appearance in four dimensions of a phase transition in the bare Newton’s constant, separating a smooth phase with small negative average curvature from a rough phase with large positive curvature [2]. Here I will discuss a study we have performed to determine the nature and size of the effects that appear when a scalar field is coupled to gravity. As will be discussed below, our results seem to indicate that the ’vacuum polarization’ effects due to one single scalar field of small mass are rather small for the observables we have investigated, when compared to the dominant pure gravitational contribution. Action and Measure For the gravitational field the following lattice action is used [3]