Area expectation values in quantum area Regge calculus

V.M Khatsymovsky

doi:10.1016/s0370-2693(03)00439-8

Abstract

Regge calculus generalised to independent area tensor variables is considered. Continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. Quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as the time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas.

Highlights

The standard canonical quantisation prescription requires continuous coordinate which would play the role of time
The latter are the 4D generalisations of the 3D Ponzano-Regge model of quantum gravity [3] where the partition function of
As for the standard canonical quantisation approaches issuing directly from the action, these are applicable to the theories with discrete space but continuous time and are faced with the complication consisting in the fact that discrete constraints of general relativity (GR) are generally second class

Summary

Introduction

The standard canonical quantisation prescription requires continuous coordinate which would play the role of time. It turns out that the quantum measure for the completely discrete theory does exist which possesses the following property (of equivalence of the different coordinates). Where πσab2 are the tensors of the 2-faces σ2 which in the particular case of the usual link vector formulation (not area modified) should reduce to ǫabcdl1cl2d, l1a, l2a being 4-vectors of some two edges of the triangle σ2 in the local frame of a certain 4-tetrahedron containing σ2.

Results

Conclusion