Three dimensional loop quantum gravity: physical scalar product and spin foam models

Details Three dimensional loop quantum gravity: physical scalar product and spin foam models Three dimensional loop quantum gravity: physical scalar product and spin foam models

2004-02-25

arxiv.org/abs/gr-qc/0402110v3

Class.Quant.Grav. 22 (2005) 1739-1762

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Typos corrected, version appearing in Class. Quant. Grav

Scientific paper

10.1088/0264-9381/22/9/017

In this paper, we address the problem of the dynamics in three dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space--corresponding to the quantization of the infinite dimensional kinematical configuration space of the theory--to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard `lattice' methods), and second that no ill-defined sum over spins (`bubble' divergences) are present in the spin foam representation.

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