Hamilton-Jacobi analysis of the four-dimensional BF model with cosmological term

Abstract

In this work we perform the Hamilton-Jacobi constraint analysis of the four dimensional Background Field (BF ) model with cosmological term. We obtain the complete set of involutive Hamiltonians that guarantee the integrability of the system and identify the reduced phase space. From the fundamental differential we recover the equations of motion and obtain the generators of the gauge and shift transformations.

Covariant momentum map for non-Abelian topological BF field theory

“…Besides, within the multisymplectic approach, we also constructed the generator of the slicing on the multimomenta phase-space of the theory. In particular, we showed at the multiisymplectic level that the superposition of the covariant momentum maps associated to the gauge and topological symmetries of the BF theory projects into the instantaneous phase-space generator of infinitesimal gauge transformations of the system and, and we also realized that the zero level set of the above projected momentum map coincides with the surface on the instantaneous phase-space defined by the first class constraints of the non-Abelian topological BF field theory, which arise within the Dirac-Hamiltoniana analysis of the model as described in [10,14]. Furthermore, we found that since the generator of the slicing on the multimomenta phase-space generates covariant canonical transformations, its associated covariant momentum map projects to the extended Hamiltonian of this topological field theory also obtained by means of the Dirac's algorithm in the instantaneous canonical analysis [14].…”

“…Besides, we also construct the covariant momentum maps associated to the BF model. Finally, following the works developed in [16,18,21,22,26], after performing the space plus time decomposition for the BF theory, we are able to recover not only the extended Hamiltonian, but also the generator of the infinitesimal gauge transformations as well as the first and second class constraints of the non-Abelian topological BF theory, which explicitly correspond to the results obtained by means of the Dirac's algorithm in references [10,14].…”

“…This relation gives rise to the so-called BF gravity models [5,6]. Recently, BF theories have been exhaustively explored in the literature from different perspectives and approaches at both the classical and the quantum levels [1][2][3][7][8][9][10][11][12], mainly due to the features that distinguish these topological field theories which provide an extended framework to explore different aspects of gauge theories. In this context, of particular interest to us is the case of the 4-dimensional non-Abelian topological BF theory, which at the classical level has been studied from different perspectives including the Lagrangian [1,13], the pure Dirac-Hamiltonia [14], the Hamilton-Jacobi [10], and the covariant canonical [11] formalisms.…”

Topological features of the quantum vacuum

“…Let us start with a brief review of the relevant properties of BF theory [22][23][24][25][26][27][28][29], a topological field theory formulated on the principal bundle of a group G over the spacetime manifold M . In this principal bundle we can define a connection A and the corresponding curvature, a 2-form F .…”

The Hamilton–Jacobi characteristic equations for three-dimensional Ashtekar gravity

“…The identification of noninvolutive Hamiltonians allows us to construct the so-called generalized brackets which are a generalization of the Poisson brackets, at the end of the procedure the fundamental differential will be expressed in terms of involutive Hamiltonians and the generalized brackets. The HJ method has been applied for studying several gauge systems, in particular systems with general covariance just like BF theories [23,24], topological invariants [25] and field theories [26,27]. In this respect, it has been showed that the development of the HJ scheme is more economical with respect either Dirac or the FJ approaches.…”