Flux formulation of loop quantum gravity: classical framework

Abstract

We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…This fact makes the construction of states describing large scale geometries extremely complicated. This motivated the construction of a new quantum geometry realization [15][16][17] based on the BF topological field theory [14], which involves a vacuum peaked on vanishing curvature instead. This vacuum indeed solves three-dimensional gravity without a cosmological constant.…”

“…The spin network basis diagonalizes Wilson loops along the boundary of the triangles of the triangulation, which represent exponentiated flux operators [15][16][17][40][41][42][43][44]. The curvature basis diagonalizes Wilson loops around the edges of the triangulation and represent holonomy operators, measuring curvature.…”

“…The entire construction rather assumed a quantum deformed BF vacuum and we can therefore expect that the operators we consider here are cylindrically consistent with respect to this vacuum. Thus it should be straightforward to construct a continuum Hilbert space based on the techniques in [15][16][17].…”

“…In the loop quantum gravity interpretation [51,52] this gives the square of the area of the triangle enclosed by the Wilson loop. The limit (5.4) for the eigenvalues of the normalized Wilson loop operators suggest a geometric interpretation for these operators: in SU (2) one can approximate the Casimir operator of the group via a sum over the cosine function applied to the Lie algebra generators [15][16][17]. Due to the exponentiation this operator does however violate gauge invariance.…”

“…Due to the exponentiation this operator does however violate gauge invariance. One can however project the operator back to a gauge invariant one, and the resulting spectrum approximates the Casimir spectrum j(j + 1) for sufficiently small representation labels [15][16][17]. For larger j the bound imposed by taking the cosine of the generators sets in.…”