Faddeev–Jackiw quantization of four dimensional BF theory

Abstract

The symplectic analysis of a four dimensional BF theory in the context of the Faddeev-Jackiw symplectic approach is performed. It is shown that this method is more economical than Dirac's formalism. In particular, the complete set of Faddeev-Jackiw constraints and the generalized FaddeevJackiw brackets are reported. In addition, we show that the generalized Faddeev-Jackiw brackets and the Dirac ones coincide to each other. Finally, the similarities and advantages between FaddeevJackiw method and Dirac's formalism are briefly discussed.PACS numbers: 98.80.-k,98.80.Cq

Perturbative BF theory

“…The gauge structure group of the BF theory naturally suggests possible connections with (2+1) gravity [6,7,8,9,10,11,12,13,14,15,16,17,18,19], and applications of the BF formalism in the context of loop quantum gravity have also been studied [20,21,22,23,24,25,26]. Generalizations of the BF models in higher dimensions have been considered [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. Quite recently, the use of the BF field theory has been envisaged for the description of topological effects in condensed matter [43,44,45,46,47,48,49,50,51].…”

Canonical and symplectic analysis for three dimensional gravity without dynamics

“…For that reason, the study of toy models with a similar canonical structure to that present in gravity becomes to be an interesting topic. In this respect, there are several examples of toy models with a close relation with gravity just as topological theories [8][9][10][11][12][13][14][15][16] and models as those reported by V. Hussain where there is not dynamics [17,18], this means, in the canonical formulation of those models there is not an analog to the Hamiltonian constraint that is present in real gravity theory [19]. In this respect, the study of toy models is a interesting subject because those are good laboratories for testing classical and quantum ideas that could be applied to General…”

Symplectic analysis for the Holst action with Dirac fields

“…The symplectic analysis of gravitation theories with Immirzi parameters has been made in the context of Faddeev-Jackiw formalism, with the focus on obtaining the generalized brackets [13,14], with results generally coinciding with those obtained in analysis via Dirac's formalism [15], although some differences arise in regards to the set of constraints of the theory [14]. There are also symplectic analysis of theories with half-integer spin variables not coupled to gravity [16].…”