Kinematics of arbitrary spin matter fields in loop quantum gravity

Loop quantum gravity envisions a small scale structure of spacetime that is markedly different from that of the classical spacetime continuum. This has ramifications for the excitation of matter fields and for their coupling to gravity. There is a general understanding of how to formulate scalar fields, spin $\frac{1}{2}$ fields, and gauge fields in the framework of loop quantum gravity. The goal of the present work is to investigate kinematical aspects of this coupling. We will study implications of the Gauß and diffeomorphism constraints for the quantum theory: We define and study a less ambiguous variant of the Baez-Krasnov path observables, and we investigate symmetry properties of spin network states imposed by diffeomorphism group averaging. We will do this in a setting which allows for matter excitations of spin $\frac{1}{2}$ and higher. In the case of spin $\frac{1}{2}$, we will also discuss extensions of it by introducing an electromagnetic field and antiparticles. We finally discuss how far the picture with matter excitations of higher spin can be obtained from classical actions for higher spin fields.

Figure 1

Exemplary generalized spin network state. Vertices with a spin $\frac{1}{2}$ particle are depicted by stars and vertices without matter by dots. The spin $\frac{1}{2}$ particles couple to the star intertwiner gauge invariantly. For the sake of clarity, arrows depicting the direction of the edges are omitted.