NONPERTURBATIVE QUANTUM GEOMETRY AND TORSION

The relevance of torsion in nonperturbative quantum geometry is studie d here from the viewpoint of the equivalence of the loop space to the space of gauge potentials modulo gauge transformations satisfying Mand elstam constraints. The topological feature associated with the gauge orbit space of a non-Abelian gauge theory when the topological a term is introduced in the Lagrangian corresponds to a vortex line and the g auge orbit space appears to be multiply connected in nature. This has an implication in the loop space formalism, in the sense that the latt er involves nonlocality and there is no way we could arrive at a corre sponding continuum limit. In the gravity without the metric formalism of Capovilla, Jacobson and Dell, the theta term in the Lagrangian corr esponds to torsion. This suggests that in the construction of a soluti on of functionals annihilated by the Hamiltonian constraint, any regul arization procedure which destroys the gauge invariance of the loop sp ace variables also destroys the topology of the gauge orbit space and the continuum limit can be achieved only by removing the vortex line. Thus the constraint equations of canonical quantization of gravity can be achieved only in the limit of torsion tending to zero. This provid es the link between covariant and canonical quantization of gravity an d demonstrates explicitly the role of the arrow of time in nonperturba tive quantum geometry also when we take the gauge-invariant holonomy v ariables as the fundamental entity.