Large quantum gravity effects and nonlocal variables

An interesting effect found recently, that points out to potential problems and/or surprises that could be found in quantum gravity, is the presence, in the quantization of certain midisuperspaces, of what has been called “large quantum gravity effects” (LE). In the original analysis of Ashtekar [1], the model chosen is rotationally symmetric 2+1 gravity coupled to a massless scalar field (a spacetime that may be obtained by symmetry reducing Einstein-Rosen waves). As shown in [1], in the quantized version of this model, coherent states for the scalar field emerge as natural candidates for a set of coherent states for the metric. It is then noted that only at low frequencies the mean value of the metric in these states is sharply peaked around its classical value. This implies that there are classical solutions that are spurious: they do not arise as classical limits of the quantum theory. And, curiously, even states very close to those that minimize the uncertainties on the metric cause huge fluctuations on the later. On the other hand, Gambini and Pullin have analyzed a different family of states, for which the uncertainties on the metric can be diminished at the expense of increased fluctuations on the matter fields [2]. Beetle [3] and Varadarajan [4] have also analyzed other models, along lines similar to those of Ashtekar, and found results in agreement to those in [1]. Although the usual formulation of General Relativity (GR) is in terms of local variables, e.g., metric, connection, etc., there exists a reformulation, the Null Surface Formulation of GR (NSF), whose basic variable can be taken as a “Z ′′ function, that, a posteriori turns out to describe light cone cuts at future null infinity of a metric that satisfies Einstein equations. The purpose of this paper is to reconsider the model analyzed by Ashtekar, but now focusing our attention, not on the metric, but on Z. As we shall see, coherent states for the scalar field are also coherent states for Z, in exactly the same regime as that for the metric. But, while small deviations from this regime are amplified in the metric, they are damped in Z. A recent review of NSF, together with a complete list of references, can be found in [6], and some results of classical NSF in 2 + 1 in [7]. Let us start by briefly discussing some classical aspects of the model. Coordinates (“Weyl coordinates”) can be chosen, on which the scalar field Ψ decouples from the metric. The equations for Ψ are those of a scalar field on a flat fiducial spacetime. Once Ψ is known, the metric can be obtained by quadratures, and it takes the form