Modular group, operator ordering, and time in (2+1)-dimensional gravity.

A choice of time-slicing in classical general relativity permits the construction of time-dependent wave functions in the “frozen time” Chern-Simons formulation of (2 + 1)-dimensional quantum gravity. Because of operator ordering ambiguities, however, these wave functions are not unique. It is shown that when space has the topology of a torus, suitable operator orderings give rise to wave functions that transform under the modular group as automorphic functions of arbitrary weights, with dynamics determined by the corresponding Maass Laplacians on moduli space. email: [email protected] Despite decades of research, physicists have not yet managed to construct a workable quantum theory of gravity. The considerable effort that has gone into this venture has not been wasted, however: we have gained a much better insight into the questions that must be addressed, and we now know many of the ingredients that are likely to be important in the final theory. The purpose of this Letter is to bring together three such fragments — the “problem of time,” operator orderings, and the mapping class group — in the simplified context of (2 + 1)-dimensional gravity. The “problem of time” in quantum gravity appears in many guises, but it takes its sharpest form in “frozen time” formulations such as Chern-Simons quantization in 2 + 1 dimensions. Translations in (coordinate) time are diffeomorphisms, which are exact symmetries of the action of general relativity. Operators that commute with the constraints — for instance, the holonomies of the Chern-Simons formulation — are consequently timeindependent. In this context, the basic problem can be posed quite simply: how does one describe dynamics when all observables are constants of motion? Attempts to address this problem, even in very simple models, have been plagued by ambiguities in operator ordering [1, 2]. As Kuchař has stressed [3], such ambiguities can hide a multitude of sins, and no theory should be considered complete unless it offers a clear ordering prescription. So far, the only hint of such a prescription has come from the behavior of the mapping class group, the group of “large” diffeomorphisms. In (2 + 1)-dimensional gravity with a sufficiently simple topology, it is known that the requirement of good behavior under the action of this group places strong restrictions on possible operator orderings and quantizations [4]. In this aspect, (2 + 1)-dimensional quantum gravity resembles twodimensional rational conformal field theory, where the representation theory of the mapping class group plays an important role in limiting the range of possible models [5]. In this Letter, we shall explore these restrictions in more detail. As we shall see, even the simplest nontrivial topology, [0, 1]× T , is rich enough to illustrate both the importance of the mapping class group in determining operator orderings and its limitations. 1. Chern-Simons and ADM Quantization A systematic exploration of the relationship between Chern-Simons and Arnowitt-DeserMisner (ADM) quantization of (2+1)-dimensional gravity was begun in references [4] and [6]. In this section we briefly summarize the results; for more details, the reader is referred to the original papers. In the ADM formulation of canonical (2 + 1)-dimensional gravity [7, 8], one begins by specifying a time-slicing. A convenient choice is York’s “extrinsic time,” in which spacetime is foliated by surfaces of constant mean extrinsic curvature TrK = T . For a spacetime with the topology [0, 1] × T , a slice of constant T is a torus with an intrinsic geometry that can be characterized by a complex modulus τ = τ1 + iτ2 and a conformal factor. Moncrief has shown that the conformal factor is uniquely determined by the constraints [7], so the The notation in this Letter has been changed slightly to conform to the mathematical literature. The modulus of a torus T , previously denoted by m, is now τ , while the trace of the extrinsic curvature, previously τ , is now T .