Hilbert space built over connections with a non-compact structure group

Abstract

Quantization of general relativity in terms of -connections (i.e. in terms of the complex Ashtekar variables) is technically difficult because of the non-compactness of . The difficulties concern the construction of a diffeomorphism invariant Hilbert space structure on the space of cylindrical functions of the connections. We present here a ‘toy’ model of such a Hilbert space built over connections whose structure group is the group of real numbers. We show that in the case of any Hilbert space built analogously over connections with any non-compact structure group (this includes some models presented in the literature) there exists an obstacle which does not allow us to define a ∗-representation of cylindrical functions on the Hilbert space by the multiplication map which is the only known way to define a diffeomorphism invariant representation of the functions.