Obtaining non-Abelian field theories via the Faddeev–Jackiw symplectic formalism

Abstract

In this work we have shown that it is possible to construct non-Abelian field theories employing, in a systematic way, the Faddeev-Jackiw symplectic formalism. This approach follows two steps. In the first step, the original Abelian fields are modified in order to introduce the non-Abelian algebra. After that, the Faddeev-Jackiw method is implemented and the gauge symmetry relative to some non-Abelian symmetry group, is introduced through the zero-mode of the symplectic matrix. We construct the SU (2) and SU (2)⊗U (1) Yang-Mills theories having as starting point the U (1) Maxwell electromagnetic theory.

On the Faddeev–Jackiw symplectic framework for topologically massive gravity

“…Furthermore, this approach not only has been useful to study non-Abelian systems [32], hidden symmetries [33], and self-dual fields [34], but also to quantize massive non-Abelian Yang-Mills fields [35] fields and to study the extended Horava-Lifshitz gravity [36]. For other work on the F-J symplectic approach we refer the interested reader to Refs.…”

Higher-derivative non-Abelian gauge fields via the Faddeev–Jackiw formalism

“…At this point one should introduce convenient gauge (subsidiary) conditions, like a constraint; and the twoform matrix becomes, therefore, invertible. This extension was proposed and developed by Barcelos-Neto and Wotzasek [4,5] and by Montani and Wotzasek [6], and this was studied in several models [7][8][9][10]. It basically is in the spirit of Dirac's work, with proposal works by imposing the stability of the constraints under time evolution.…”

Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

“…Furthermore, the matrix (22) is still singular; however, we have shown that there are no more constraints and that the theory has a gauge symmetry. In order to construct a symplectic tensor, we need to fix the gauge [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], and thus we will fix the temporal gauge, say, 0 = 0 = 0, which means thaṫ= 0 anḋ= 0. In this manner, the fixing gauge will be added to the symplectic Lagrangian via Lagrange multipliers, Θ and Ξ .…”

“…In this manner, with the antecedents mentioned above, in this paper, we will study the P-CS theory from a symplectic point of view. For this aim, we will use the symplectic formalism of Faddeev-Jackiw [FJ] [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], due basically to the fact that the FJ approach is more economical than Dirac's method. In fact, the FJ is a symplectic description where all relevant information of the theory can be obtained through a symplectic tensor, which is constructed from the symplectic variables that are identified from the Lagrangian.…”