Speaker
Description
We consider the quantization of gravity as an SL(2,C) gauge theory in terms of Ashtekar's selfdual variables and reality conditions for the spatial metric (RCI) and its evolution (RCII).
We start from a holomorphic phase space formulation and consider holomorphic cylindrical wave functions over SL(2,C) connections. We use an overall phase ambiguity of the complex selfdual action to obtain Poisson brackets that mirror those of the real theory. We then show that there is a representation of the corresponding canonical commutation relations the space of holomorphic cylindrical functions.
We describe a class of cylindrically consistent measures that implements RCI. We also consider a regularization of RCII and show that there are no solutions in the class of measures that we are considering.
We end with a comparison to the literature and some general observations on the consistency of reality conditions, commutation relations and use of holonomies as basic variables.