NONPERTURBATIVE QUANTUM GEOMETRY AND TORSION

Abstract

The relevance of torsion in nonperturbative quantum geometry is studied here from the viewpoint of the equivalence of the loop space to the space of gauge potentials modulo gauge transformations satisfying Mandelstam constraints. The topological feature associated with the gauge orbit space of a non-Abelian gauge theory when the topological θ term is introduced in the Lagrangian corresponds to a vortex line and the gauge orbit space appears to be multiply connected in nature. This has an implication in the loop space formalism, in the sense that the latter involves nonlocality and there is no way we could arrive at a corresponding continuum limit. In the gravity without the metric formalism of Capovilla, Jacobson and Dell, the θ term in the Lagrangian corresponds to torsion. This suggests that in the construction of a solution of functionals annihilated by the Hamiltonian constraint, any regularization procedure which destroys the gauge invariance of the loop space 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 provides the link between covariant and canonical quantization of gravity and demonstrates explicitly the role of the arrow of time in nonperturbative quantum geometry also when we take the gauge-invariant holonomy variables as the fundamental entity.