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

Abstract

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.

Figures (11)

• Figure 1: Illustration of a spin-network state. At 3-valent nodes the intertwiner is uniquely specified by the corresponding spins. At 4 or higher valent nodes an intertwiner has to be specified. Choosing an intertwiner corresponds to decompose the $n$-valent node in terms of $3$-valent ones adding new virtual links (dashed lines) and their corresponding spins. This is illustrated explicitly in the figure for the $4$-valent node.
• Figure 2: A set of discrete transitions representing one of the contributing histories implied by our regularization of the generalized projection $P$ in Equation (\ref{['PS']}); from left to right in two rows.
• Figure 3: Spin foam representation of the transition between to loop states. Our regularization of the generalized projection operator $P$ produces (in the path integral representation) a continuous transition between embedded spin networks. Here we illustrate the result at three different slicing.
• Figure 4: A set of discrete transitions representing one of the contributing histories implied by our regularization of the generalized projection $P$ in Equation (\ref{['PS']}); from left to right in two rows.
• Figure 5: The regularization of the generalized projector $P$ produces a continuous sequence of transitions through spin-network states that can be pictured in the form of a continuous 2-complex.