BFFT Nonlinear Constraints Abelianization of a Prototypical Second-Class System

Abstract

We apply the BFFT formalism to a prototypical second-class system, aiming to convert its constraints from second-to first-class. The proposed system admits a consistent initial set of second-class constraints and an open potential function providing room for applications to mechanical models as well as field theory such as the non-linear sigma model. The constraints can be arbitralily non-linear, broadly generalizing previously known cases. We obtain a sufficient condition for which a simple closed expression for the Abelian converted constraints and modified involutive Hamiltonian can be achieved. As explicit examples, we discuss a particle on a torus and a spontaneous Lorentz symmetry breaking vector model, obtaining in both situations the full first-class abelianized constraints in closed form and the corresponding involutive Hamiltonian.

General Solution and Canonical Quantization of the Conic Path Constrained Second-Class System

“…One can then benefit from the general solution obtained here in order to better understand those BRST/BFV aspects and to strength the comparative analysis with specific field theory models as in references [13,16,17]. The conic path system has recently inspired the construction of a gauge-invariant prototypical mechanical system [21,22] with direct application to quantum field theory models such as the Lorentz symmetry breaking bumblebee model [21] and the nonlinear sigma model [22]. The existence of the general solution presented here should also bring important clues for a deeper analysis of that recently proposed prototypical system.…”

“…where correspond to the functions defined in equation (21). Furthermore, note that the remaining momenta-independent part of {χ 4 , λ} is given by…”

“…Starting from two consistent Lagrangian and Hamiltonian mechanical models, the BRST quantization of a conic constrained particle is fully implemented in [5] whereas a symplectic approach via FJBW 4 can be seen in [6]. Despite being constructed initially for comparative purposes, the model introduced in references [5,6,13] grew up and acquired interest in itself being subsequently used as a starting point for further discussions regarding (anti-)BRST symmetry [16], superfield formalism [17] and BFFT 5 Abelian conversion [21], giving rise to interesting applications in field theory such as the Lorentz breaking bumblebee model [21] and the non-linear sigma model [22].…”

BFV quantization and BRST symmetries of the gauge invariant fourth-order Pais-Uhlenbeck oscillator

“…To convert the constraints from second to first class, we shall use the standard Batalin-Fradkin-Fradkina-Tyutin (BFFT) conversional approach [35][36][37][38][39][40]. As a first step, we introduce a pair of auxiliary fields ϕ a , a = 1, 2, satisfying the extended Poisson bracket relations…”

“…where ǫ ab denotes the totally antisymmetric Levi-Civita symbol with the convention ǫ 12 = 1. Following references [39,40], in order to calculate the first order correction to the constraints (17), we look for a solution of X ab in…”