Notes on the (2+1)-Dimensional Wheeler-DeWitt Equation

In contrast to other approaches to (2+1)-dimensional quantum gravity, the Wheeler-DeWitt equation appears to be too complicated to solve explicitly, even for simple spacetime topologies. Nevertheless, it is possible to obtain a good deal of information about solutions and their interpretation. In particular, strong evidence is presented that Wheeler-DeWitt quantization is not equivalent to reduced phase space quantization. ∗email: [email protected] In the continuing search for a realistic quantum theory of gravity, it has often proven useful to explore simpler models that share the basic conceptual features of general relativity. Gravity in 2+1 dimensions is one such model, providing a fully diffeomorphism-invariant theory of spacetime geometry that nevertheless avoids many of the technical difficulties of realistic (3+1)-dimensional gravity. The goal of this paper is to explore the ramifications of one popular approach to quantum gravity, the Wheeler-DeWitt equation, in this simple setting. We shall see that even with the simplifications of 2+1 dimensions, Wheeler-DeWitt quantization is considerably more complicated than one might guess, and that at least in its simplest interpretations, it is not equivalent to other known approaches to quantization. 1. Canonical Gravity in 2+1 Dimensions Before trying to construct a quantum theory, let us briefly review the canonical formalism for (2+1)-dimensional gravity, as described by Moncrief [1] and Hosoya and Nakao [2]. We begin with a spacetime with the topology IR×Σ, where Σ is a closed surface. The standard (2+1)-dimensional ADM variables are then a spatial metric gij and its canonical momentum π . By a classical result of Riemann surface theory, any metric on Σ can be written in the form gij = e f ḡij, (1.1) where f is a spatial diffeomorphism and ḡij is a “standard” metric of constant curvature 1 (if Σ is a sphere), 0 (if Σ is a torus), or −1 (if Σ is a surface of genus g > 1). For a surface of genus g > 1, the standard metrics ḡij comprise a (6g − 6)-dimensional family; for the torus, the family is two-dimensional, and we can choose ds̄ = τ2 |dx+ τdy|, (1.2) where x and y are angular coordinates with period 1 and τ = τ1+iτ2 is a complex parameter, the modulus. Corresponding to the decomposition (1.1), the momentum π can be written as π = e √ ḡ ( p + 1 2 ḡπ/ √ ḡ +∇Y j +∇Y i − ḡ∇kY k ) , (1.3) where ∇i is the covariant derivative for the connection compatible with ḡij, indices are now raised and lowered with ḡij, and p ij is a transverse traceless tensor with respect to ∇i (in the language of Riemann surface theory, a holomorphic quadratic differential). Roughly speaking, p is canonically conjugate to ḡij , π to λ, and Y i to f . More precisely, if we consider a cotangent vector δgij in the space of metrics, δgij = ∇iδξj +∇jδξi + 2eḡijδλ+ eδḡij, (1.4) the symplectic structure can be read off from the expression ∫