Three dimensional loop quantum gravity: coupling to point particles

Abstract

We consider the coupling between three dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovelli's generalized projection operator which can be represented as a sum over spin foam amplitudes. In addition we provide an explicit expression of the (physical) distance operator between two particles which is defined as a Dirac observable.

Figures (12)

• Figure 1: Example of a generalized spin-network observable.
• Figure 2: Example of a boundary vertex. The boundary $\ell$ is associated to a representation $j_\ell$ virtually represented by an extra link oriented toward the boundary; the magnetic number associated to the departure point is fixed to the spin $s$ of the particle. The boundary is also colored with an intertwiner $\iota_\ell$ from in-coming representations (containing the virtual one) to out-going representations.
• Figure 3: The two particles spin-network state. Note that the intertwiners associated to the two particles $v_{\ell_1}$ and $v_{\ell_2}$ are different.
• Figure 4: Distance observables and generalized spin-network states. The distance operator between $\wp_1$ and $\wp_2$ is represented by a thick line; holonomies defining generalized spin-network states are represented by thin lines.
• Figure 5: Action of the distance operator on a spin-network state. The action on the left is weakly equal to the action on the right. By weakly, we mean equal on the constraints surface, i.e. one can set any holonomy (with free ends) of the connection to be trivial on any contractible open set of $\Sigma$.