Path integral measure in Regge calculus from the functional Fourier transform

AbstractThe problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it is “best approximation” to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in treating the measure as functional on the space of the metric functionals. We require coincidence of the measures for the discrete and continuous versions of the theory on some sufficiently large (dense) set of metric functionals which exist and admit exact definitions and calculation in the both versions. This set consists of generalisation of the usual finite-dimensional plane waves to the functional space so that the discrete measure follows by means of the functional Fourier transform. The possibility for such set to exist is due to the Regge manifold being a particular case of general Riemannian one (Regge calculus is a minisuperspace theory). Only a certain continuum measure among the local ones (the scale invariant Misner measure) is found to be reduciable in this way to the well defined Regge discretisation, and we find the two versions for the latter depending on what metric tensor, covariant or contravariant one, is taken as fundamental field variable. The closed expressions for the measure are obtained in the two simple cases of Regge manifold. These turn out to be quite reasonable one of them indicating to possibility of passing in backward direction when appropriately defined continuum limit of the Regge measure would reproduce the original continuum GR measure