Revisiting loop quantum gravity with selfdual variables: Hilbert space and first reality condition

We consider the quantization of gravity as an <?CDATA SL(2,C)?> S L ( 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. It is then natural to push for a quantization in terms of holomorphic wave functions. Thus we consider holomorphic cylindrical wave functions over <?CDATA SL(2,C)?> S L ( 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 in the space of holomorphic cylindrical functions. We describe a class of cylindrically consistent measures that implements RCI. We show that spin networks with <?CDATA SU(2)?> S U ( 2 ) intertwiners form a basis for gauge invariant states. They are still mutually orthogonal, but the normalisation is different than for the Ashtekar–Lewandowski measure for <?CDATA SU(2)?> S U ( 2 ) . We do not consider RCII in the present article. Work on RCII is ongoing and will be presented elsewhere.