THE VOLUME OPERATOR IN DISCRETIZED QUANTUM GRAVITY.pdf

a r X i v : g r - q c / 9 5 0 6 0 1 4 v 1 7 J u n 1 9 9 5 DFF 228/05/95 May 27, 1995 THE VOLUME OPERATOR IN DISCRETIZED QUANTUM GRAVITY R. Loll1 Sezione INFN di Firenze Largo E. Fermi 2 I-50125 Firenze, Italy Abstract We investigate the spectral properties of the volume operator in quantum gravity in the framework of a previously introduced lattice discretization. The presence of a well-defined scalar product in this approach permits us to make definite statements about the hermiticity of quantum operators. We find that the spectrum of the volume operator is discrete, but that the nature of its eigenstates differs from that found in an earlier continuum treatment. 1 Supported by the European Human Capital and Mobility program on “Constrained Dynamical Systems” 1 Introduction One of the most active branches of research into the quantization of 3+1-dimensional gravity of the last few years has been the canonical, operator-based framework of the so- called loop approach. It is non-perturbative in the sense that it is not a priori restricted to the study of geometries close to flat Minkowski space. Its basic variables are (non-local) generalized Wilson loops of the SL(2,C)-valued Ashtekar connection. Also in the quantum theory the state space and operators are labelled by (equivalence classes of) closed curves in three-space, which has led to considerable progress in solving the quantum constraints of the theory. The first, formal solutions to all of the constraints, including the Wheeler-DeWitt equation, were found in this loop formulation [1]. Although since then many of the mathematical ingredients of loop representations have been scrutinized and better understood (see, for example, [2]), one is still lacking a rigor- ous control over the regularization procedure necessary for obtaining a well-defined quantum Hamiltonian. One difficulty is the absence of a natural background metric in the “fully diffeomorphism-invariant phase” of the theory. Secondly, since the