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

TL;DR

This paper explores the challenges of constructing a diffeomorphism invariant Hilbert space for non-compact structure groups in connection-based quantum gravity, using a simplified model with the real numbers.

Contribution

It demonstrates an obstacle in defining a *-representation of cylindrical functions on Hilbert spaces built over non-compact groups, highlighting a key technical difficulty.

Findings

Obstruction to *-representation for non-compact groups

Toy model with real numbers illustrating the problem

Implications for quantization of complex Ashtekar variables

Abstract

Quantization of general relativity in terms of SL(2,C)-connections (i.e. in terms of the complex Ashtekar variables) is technically difficult because of the non-compactness of SL(2,C). 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 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.