Quantisation of O(N) invariant nonlinear sigma model in the Batalin-Tyutin formalism

Abstract

We quantise the O(N ) nonlinear sigma model using the Batalin Tyutin (BT) approach of converting a second class system into first class. It is a nontrivial application of the BT method since the quantisation of this model by the conventional Dirac procedure suffers from operator ordering ambiguities. The first class constraints, the BRST Hamiltonian and the BRST charge are explicitly computed. The partition function is constructed and evaluated in the unitary gauge and a multiplier (ghost) dependent gauge.Over the last few years a method of generalised canonical quantisation of constrained dynamical systems has been developed by Fradkin and collaborators [1,2] as an alternative to the pioneering formulation of Dirac [3]. This method [1,2], which has been reasonably well established for systems with first class constraints only, has been very recently extended to include systems with second class constraints [4,5]. We shall henceforth refer to this later method as the Batalin-Fradkin (BF) [4] and Batalin-Tyutin (BT) [5] schemes. The basic idea of this method is to convert the second class system into first class by extending the phase space and then to use the familiar machinary valid for first class systems [1,2]. While the BT [5] method remains unexplored (as far as quantisation of specific models is concerned), Some applications of the BF [4] formalism have been reported recently [6][7][8][9]. It is noteworthy, however, that these applications are confined to examples like the chiral gauge theories [6,7], the chiral boson theory [8], the massive Maxwell [7] and the massive Yang-Mills [9] theories. In all these models the Dirac brackets among the canonical variables are very simple (i.e. there are no operator ordering problems) and the quantisation can be, and indeed has earlier been [10,11], just carried out by the classical method of Dirac [3]. Such examples are, therefore, pedagogic exercises and do not reveal the complete power or flexibility of either the BF [4] or BT [5] approaches. Furthermore, the quantisation presented in ref. [7] is not a systematic application of the BF or BT method [4,5].The motivation of this paper is to provide a non-trivial application of the BT procedure [5]. We shall consider the quantisation of the O(N) invariant nonlinear sigma model. This model, which is a second class system, is known to have (quadratic) field dependent Dirac brackets among the canonical variables. Consequently quantisation by Dirac's [3] procedure is riddled with operator ordering ambiguities. The conventional approach is to work in the configuration space functional integral formalism [11]. In this paper we show that an ambiguity free operator quantisation of the model can be performed by using the BT method [5] of converting the second class system into first class. The involutive (i.e. first class) Hamiltonian contains an infinite number of terms, although the number of additional (unphysical) fields introduced to extend the phase space is finite. A remarkable series of cancellations allows us to expres...

First class models from linear and nonlinear second class constraints

“…Most papers about these models are focused on the consistent canonical quantization and their quantum spectrum. This family of models were considered in several approaches including: the symplectic embedding [8,9,10,11], the BFT formalism [9,12,13,17,14,15,16], Stuckelberg field shifting [19,18] or mixed approaches based on first principles of the making gauge systems [9,18,20,21,22].…”

Gauging the nonlinearσmodel through a non-Abelian algebra

“…This occurs because we do not know a priori what is the best choice we can make to go from one step to another. Sometimes it is possible to figure out a convenient choice for X ab in order to obtain a first-class (Abelian) constraint algebra in the first stage of the process [8,9]. It is opportune to mention that in the work of reference [4], the use of a non-Abelian algebra was in fact a way of avoiding to consider higher order of the iterative method.…”

BFT Hamiltonian Embedding of Massive Theory with One- and Two-Form Gauge Fields

“…In order for studying the whole constraints efficiently, we further need to modify the constraints (2.7) as 9) which are equivalent to the original ones, Λ, Λ i on the constraint surface. Then, the set of these new constraints makes the constraint algebra (2.8) much simpler and concise as 10) while the others identically vanish 4 . As results, we have obtained the fully second-class constraints (2.6) and (2.9) for the first order Lagrangian of the topologically massive theory.…”

“…With the above choice of ω αβ , we solve the Eq. (3.5) and find a simple solution as 10) where I 3×3 denotes a 3 × 3 identity matrix.…”

“…[2,3,4]. This procedure has been established by Batalin, Fradkin, and Tyutin (BFT) [5,6] and extensively studied in the canonical formalism for various models including massive gauge fields [7], spontaneously broken gauge theories [8], SU(2)/SU(3) Skyrmion models [9], non-linear sigma models [10], noncommutative systems [11], and many others [12]. The BFT embedding formalism has been also widely applied to obtain the WessZumino (WZ) actions [13] showing that original theory can be regarded as a gauge-fixed version of extended gauge system, while verifying dual equivalent descriptions [14] in particular gauges from the phase space partition function corresponding to the BFT embedded involutive Hamiltonian.…”