Transfer Matrix Formalism for Two-Dimensional Quantum Gravity and Fractal Structures of Space-time

We develop a transfer matrix formalism for two-dimensional pure gravity. By taking the continuum limit, we obtain a “Hamiltonian formalism” in which the geodesic distance plays the role of time. Applying this formalism, we obtain a universal function which describes the fractal structures of two dimensional quantum gravity in the continuum limit. Recent developments of two-dimensional gravity have provided us with an unambiguous definition of quantum gravity. This is based on the equivalence of the continuum formulation [1] and the dynamical triangulation.[2] In two dimensions, we thus have a regularized quantum gravity which has a definite continuum limit. The remarkable success of the matrix models [3] further elucidated topological aspects of the theory. However we still lack a general formulation for describing quantum fluctuations of space-time and for evaluating physical observables such as fractal dimensions. In this paper we propose a new formulation which is a kind of Hamiltonian formalism for quantum gravity. We show that a geodesic distance defined on a dynamically triangulated surface can be regarded as the “time” variable for defining the transfer matrix. We then obtain a “Hamiltonian” in the continuum limit, and analyze the fractal structures of the space-time. Let us consider a cylinder with an entrance loop(c) and an exit loop(c). (See Fig. 1.) We introduce the following quantity which is formally defined in the continuum framework: