Comment on " Quantization of Diffeomorphism-invariant Theories with Fermions "

In the comment to the article by J. Baez and K. Krasnov (hep– th/9703112) are discussed some topics related with application of certain constructions to non-trivial principal bundles. — In chapter 2 of the article [1] is used following construction: First, define a 'transporter' from the point p to the point q to be a map from P p to P q that commutes with the right action of G on the bundle P. If we trivialize the bundle over p and q, we can think of such a transporter simply as an element of G. A 'generalized connection' A is a map assigning to each oriented analytic path e in Σ a parallel transporter A e : P p → P q , where p is the initial point of the path e and q is the final point. We require that A satisfy certain obvious consistency conditions: A should assign the same transporter to two paths that differ only by an orientation-preserving reparametrization, it should assign to the inverse of any path the inverse transporter, and it should assign to the composite of two paths the composite transporter. An ordinary smooth connection A gives a generalized connection where the parallel transporter A e along any path e is simply the holonomy of A along this path The trivialization here is identification of fibers P p and P q with G (structure group) in the both points p and q, so the 'transporter' can be represented as left action of G on G.