Speaker
Description
Loop quantum gravity (LQG) in its current formulation is a the quantisation of the SU(2) gauge theory of gravity in Ashtekar-Barbero variables. It started out as an SL(2,C) gauge theory in Ashtekar's selfdual variables, but the quantisation program was never fully carried out in this formulation. The two main obstacles are the non-compactness of the gauge group SL(2,C) and the necessity to implement complicated reality conditions. The latter ensure reality of the spatial metric and its evolution.
We revisit the original formulation by considering the selfdual part of complexified general relativity in Ashtekar variables. These are a complex flux and an SL(2,C) connection. We show that one is lead to a classical theory that is holomorphic in the canonical variables, in order to have a non-degenerate symplectic structure. This does not allow to implement the reality conditions as additional constraints in the action, they have to be added by hand during the quantisation. We describe first steps to extend the holomorphic character also to the quantum theory, with SL(2,C) holonomies, holomorphic derivatives, and a notion of holomorphic spin networks. Thus, working in a holomorphic setup turns out to be natural, as anticipated by Ashtekar and others in early works on the selfdual theory. We will also comment on the implementation of the reality conditions.
