The Faddeev–Jackiw Approach and the Conformal Affine sl(2) Toda Model Coupled to the Matter Field

Abstract

The conformal affine sl(2) Toda model coupled to the matter field is treated as a constrained system in the context of Faddeev Jackiw and the (constrained) symplectic schemes. We recover from this theory either the sine-Gordon or the massive Thirring model, through a process of Hamiltonian reduction, considering the equivalence of the Noether and topological currrents as a constraint and gauge fixing the conformal symmetry. Academic Press

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.…”

Singular Lagrangians affine in velocities

“…Once the symplectic structure has been got and, therefore, Poisson brackets have been defined, we can make use of the canonical quantization procedure. In particular, the FJ-method uses the reduction technique and has been applied in many different fields, running form condensed matter [17][18][19] and astrophysics [20], until fluid dynamics [21] and, of course, high-energy physics [22]. Even the own Schrödinger equation can be derived using this method [23].…”

Generalized sine-Gordon/massive Thirring models and soliton/particle correspondences

“…10 Off-critical submodels, such as the sl(2) ATM, can be obtained at the classical or quantum mechanical level through some convenient reduction processes starting from CATM. 4,11 In the sl(2) case, using bosonization techniques, it has been shown that the classical equivalence between the U(1) vector and topological currents holds true at the quantum level, and then leads to a bag model like mechanism for the confinement of the spinor fields inside the solitons; in addition, it has been shown that the sl(2) ATM theory decouples into a sine-Gordon model ͑SG͒ and a free scalar. 3,12 These facts indicate the existence of a sort of duality in these models involving solitons and particles.…”

“…3,12 These facts indicate the existence of a sort of duality in these models involving solitons and particles. 6 The symplectic structure of the sl(2) ATM model has recently been studied 11 in the context of Faddeev-Jackiw ͑FJ͒ 13 and ͑constrained͒ symplectic methods. 14,15 Imposing the equivalence between the U(1) vector and topological currents as a constraint there have been obtained the SG or the massive Thirring ͑MT͒ model.…”