Lorentz-covariant Hamiltonian analysis of BF gravity with the Immirzi parameter

Abstract

We perform the Lorentz-covariant Hamiltonian analysis of two Lagrangian action principles that describe general relativity as a constrained BF theory and that include the Immirzi parameter. The relation between these two Lagrangian actions has been already studied through a map among the fields involved. The main difference between these is the way the Immirzi parameter is included, since in one of them the Immirzi parameter is included explicitly in the BF terms, whereas in the other (the CMPR action) it is in the constraint on the B fields. In this work we continue the analysis of their relationship but at the Hamiltonian level. Particularly, we are interested in seeing how the above difference appears in the constraint structure of both action principles. We find that they both possess the same number of firstclass and second-class constraints and satisfy a very similar (off-shell) Poisson-bracket algebra on account of the type of canonical variables employed. The two algebras can be transformed into each other by making a suitable change of variables.

Perturbative BF theory

“…16) Expression(8.16) is in agreement with the general structure of the BF knot invariant, in which the whole dependence of W C on the framing C f of the knot C is given by the overall multiplicative factor BF framing factor = e −i ℓk(C,C f )[Λr−(g/2)Λ 2 ] . (8.17) …”

Canonical analysis with no second-class constraints of BF gravity with Immirzi parameter

“…This splitting was already noted in Ref. [8], where the constraint (26b) was treated as a primary constraint of the theory; its time evolution then generated a secondary constraint, and both constraints turned out to be second class.…”

“…II and an alternative form of the action in Sec. III, whose relation has already been established at Lagrangian [20] and Hamiltonian [8] frameworks, although involving secondclass constraints in the latter case. The strategy we followed in both cases consisted in first performing the 3+1 decomposition of the action principles and then appropriately handling the constraint imposed by the Lagrange multiplier ψ IJKL on the B field to make contact with the homologous canonical analysis of the Holst action reported in Ref.…”

“…This is achieved by reexamining the solution of the constraint on the B field in each action principle considered in Ref. [8]. In this way, we establish that the manifestly Lorentz-covariant canonical formulation of general relativity involving only first-class constraints [14,15] can also be derived, without introducing second-class constraints in the process, from the formulation of general relativity as a constrained BF theory with Immirzi parameter and a cosmological constant.…”

“…It is at this point where the canonical analysis performed here deviates from the canonical analysis reported in Ref. [8]. There, the constraint (26b) is considered as a primary constraint of the theory and handled according to Dirac's method.…”