Functional Measure for Lattice Gravity

A procedure is developed for transcription of any measure for the integration over metric fields in the continuum to the Regge-calculus lattice.

application to classical Einstein gravity has existed' for more than twenty years in the form of Regge calculus. Because of its own intrinsic interest and as a result of its connection with the study of random lattices, 2 this subject has had a revival3 4 of interest. Such a discrete formulation of gravity theories permits us to consider numerical studies4 of their quantum counterparts. In addition to a discrete form of the action for such theories we still need a measure for the functional integrals appearing in the Feynman quantization procedure. It is the purpose of this Letter to provide a transcription of a given continuum measure for quantum gravity to the discrete case. The numerical studies referred to previously used ad hoc measures. A prescription for the transferring of an integration measure from the continuum to the discrete case will likewise permit a lattice formulation of the Polyakovs string theory.
In d space-time dimensions quantum theory is obtained by integrating over the [d(d+ I)/2]independent components of the metric tensor g""(x).
For example the vacuum-to-vacuum amplitude is have emphasized that what will be presented in this work is a transcription of a given continuum measure to the discrete form is that there exist various prescriptions for the continuum measure p, (g).

For S[g""] = Jddxig [A -~R ]
the vacuum-to-vacuum amplitude in harmonic coordinates is bein e " is related to the metric tensor g""by As on the lattice we do not have to fix further a coordinate system, it is only the p, (g) of Eq. (6) we wish to transcribe.
In the following part of this work it will prove more convenient to work in the vielbein formalism. The viel with q ts a flat Minkowski metric. In general, lattice calculations are performed in Euclidean space; we shall, however, present our results for the curved space being locally Minkowski. In part this is due to the fact that the transition between a Minkowski and Euclidean formulation of gravity is not as direct as it is for flat metric field theories. The formal transposition plex Swe may take the P~& s'. s emerging from any vertex as the independent set of vielbeins for that simplex. Equation (9) tells us that two vectors P(i s.and P, s, . V '7 associated with the same edge but with different simplices must be related by a Lorentz transformation depending only on the simplices Sand S': p", , = [I (s', s)],i"" (10) of our results to the Euclidean case is straightforward.
The relation between the integration over the metric tensor and integrating over the vielbein variables is dg~"=v g Ide~.
Such a configuration is specified by giving all the edge lengths ii& between neighboring vertices ij T.hese edge lengths are the dynamical variables of this theory. To obtain the functional measure we found it easiest What is the continuum analog of this relation'? The continuum vielbeins satisfy D~e q = (QJ~) pe to work in a vieibein formulation. To this end we will cu") p is the spin connection, an infinitesimal Lorentz develop, within Regge calculus, such formalism. transformation and D" is a vector covariant derivative.
To each edge (;J) of a s,mplex Swe ass, g"a fiat d. Translated to the lattice, this covariant derivative is dimensional vector P, . s, a = 1, 2, . . . , d, satisfying just the difference of the lattice vielbeins between two neighboring simplices. The translation of Eq. (11) to Pij;si (i;sn p= (iii)', the lattice is just Eq. (10). The v'g in the above ensures that the integrations over the spin connections yield a constant independent of the metric.
Noting that Eq. (10) is the lattice version of Eq. (11), the above result may be transcribed to the lattice: The (d -1)-dimensional hypervolume of this subsimplex is denoted by cuss, and the edge dual to it has length i~, . The double prime denotes that this product is over (in view of the first 5 function) any (d -1) independent edges (i,j) common to both S and S'. lss, cuss, is one of the lattice analogs3 of 4g. There is one factor I0) for each edge ij in the fmal product. The integration over the Lorentz transformations and over the 5-function constraints may be performed In this case co(i is the length dual to the (ij) edge. I wish to thank Dr. Herbert Hamber for many useful, detailed discussions. I would also like to thank R. Friedberg for extremely useful comments. This work was supported in part by the National Science Foundation.