An Analysis on the Solution of Curvature Constraint of 2 + 1 Dimensional BF Model by Constructing a Projector P

Abstract

In this paper we analyze the structure of the physical Hilbert space of a 2+1 dimensional BF model starting from a Hilbert space H0 of wave functionals which are gauge invariant, ie, the Gauss constraint G is considered to be already fulfilled. We see then that in H0 we have only the empty vector as a solution of the curvature constraint and we have to construct a larger space. This will be the dual S ⋆ of a dense subspace S ⊂ H0. It is convenient to define an operator P, sometimes called “projector”, which is a surjective mapping from the space S to the physical Hilbert space Hfis , subspace of the dual S ⋆ . If we can build the “projector” P we have enough to define the inner product in space Hfis . In this work we see that the imposition of the curvature constraint F on the wave functional |Ψi can be explicitly calculated with the aid of the Schwartz’s t heory of distributions. We use the technique known as group averaging in order to construct the projector and the physical Hilbert space, from a generalization to the present situation of the solution of a system of equations for Schwartz’s distributions of one rea l variable.