A version of quantum measure in Regge calculus in three dimensions

Abstract

We propose a version of quantum measure in 3D Regge calculus based on principles of analyticity, canonical quantization and equivalence of different coordinates. This measure is specified by the following requirement. Suppose a Regge manifold is made continuous in some direction by tending corresponding dimensions of simplices to zero. Then the measure of interest is required to tend to that corresponding to the canonical quantization with continuous coordinate playing the role of time. (If a continuous coordinate has spacelike signature, the same is required to hold for analytical continuation of the measure of interest obtained by multiplying link-lengths along the considered direction by .) By equivalence of different coordinates this property of measure is assumed to hold for any coordinate being made continuous. We construct the above measure in three dimensions using the tetrad-connection form of Regge calculus. It is remarkable that in the 3D case such a measure does really exist and, moreover, is defined by the considered requirement practically uniquely (in fact, a family of similar measures is obtained). Averaging with the help of the constructed measure gives finite expectation values for links.