Fermions, loop quantum gravity and geometry

摘要

We discuss here the geometry associated with the loop quantum gravity when it is considered to be generated from fermionic degrees of freedom. It is pointed out that a closed loop having the holonomy associated with the SU(2) gauge group is realized from the rotation of the direction vector associated with the quantization of a fermion depicting the spin degrees of freedom. During the formation of a loop a noncyclic path with open ends can be mapped onto a closed loop when the holonomy involves q-deformed gauge group SUq(2). In this case, the spinorial variable attached to a node of a link is a quasispinor equipped with quasispin associated with the SUq(2) group. The quasispinors essentially correspond to the fermions attached to the end points of an open path in loop space. We can consider adiabatic iteration such that the quasispin associated with the SUq(2) group gradually evolves as the time dependent deformation parameter q changes and we have the holonomy associated with the SU(2) group in the limit q = 1. In this way we can have a continuous geometry developed through a sequence of q-deformed holonomy-flux phase space variables which leads to a continuous gravitational field. Also it is pointed out that for a truncated general relativity given by loop quantum gravity on a fixed graph we can achieve twisted geometry and Regge geometry.