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

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.

45 pages, no figures, LaTeX2e, the discussion of results extended

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• Fundamental Structure of Loop Quantum Gravity
• Timelike surfaces in Lorentz covariant loop gravity and spin foam models
• ADM-like Hamiltonian formulation of gravity in the teleparallel geometry
• Quantization of diffeomorphism invariant theories of connections with a non-compact structure group - an example
• Projective Limits of State Spaces II. Quantum Formalism
• Topspin Networks in Loop Quantum Gravity
• Constrained projective quantum states for the degenerate Plebanski gravity