Quantization of the Restricted Gauge Theory of QCD 2

Abstract

In this talk, we consider the restricted gauge theory of QCD 2à la Cho et al. and study its quantization using Hamiltonian, path integral and BRST quantization procedures.In this talk, we consider the restricted gauge theory of QCD 2 [1-27]à la Cho et al. and study its quantization using Hamiltonian [28], path integral [29][30][31][32] and BRST [29] quantization procedures, in the usual instant-form (IF) of dynamics (on the hyperplanes: x 0 = t = constant) [33,34]. We first briefly recap the basics of the so-called restricted gauge theory of QCD 2 due to Cho et al. . The theory makes use of the so-called "Cho-decomposition", which is, in fact, the gauge independent decomposition of the non-Abelian potential into the restricted potential and the valence potential and it helps in the clarification of the topological structure of the non-Abelian gauge theory, and it also takes care of the topological characters in the dynamics. The non-Abelian gauge theory has rich topological structures manifested by the non-Abelian monopoles, the multiple vacuua and the instantons and one needs to take into account these topological characters in the nonAbelian dynamics. Since the decomposition of the non-Abelian connection contains these topological degrees explicitly, it can naturally take care of them in the non-Abelian dynamics. An important consequence of the decomposition is that it allows one to view QCD as the restricted gauge theory (made of the restricted potential) which is coupled to a gauge-covariant colored vector field (the valence potential). The restricted potential is defined in such a way that it allows a covariantly constant unit isovectorn everywhere in space-time, which enables one to define the gauge-independent color direction everywhere in space-time and at the same time allows one to define the magnetic potential of the non-Abelian monopoles. Furthermore it has the full SU(2) gauge degrees of freedom, in spite of the fact that it is restricted. Consequently, the restricted QCD made of the restricted potential describes a very interesting dual dynamics of its own, and plays a crucial role in the understanding of QCD. On the other hand, the restricted QCD is a constrained system, due to the presence of the topological fieldn which is constrained to have the unit norm. A natural way to accommodate the topological degrees into the theory is to introduce a topological fieldn of unit norm, and to decompose the connection into the Abelian projection part which leavesn a covariant constant and the remaining part which forms a covariant vector field: A μ = (A μn − 1 gn × ∂ μn + X μ ) = ( μ + X μ ); A μ =n · A μ andn 2 = 1. Here A μ is the "electric" potential and the Abelian projection A μ is precisely the connection which leavesn invariant under the parallel transport and makesn a covariant constant:D μn = (∂ μn + g μ ×n) = 0 Also, under the infinitesimal gauge transformation: δn = − α ×n, δA μ = 1 g D μ α, δ A μ = 1 gn · ∂ μ α, δ μ = 1 gD μ α, δX μ = − α × X μ (1)

Hamiltonian, Path Integral and BRST Formulations of the Restricted Gauge Theory of <i>QCD<sub>2</sub></i>

“…In this work we consider the restricted gauge theory of quantum chromodynamics (QCD) in one-space one-time dimension (QCD 2 ) à la Cho et al [1]- [14], studied rather widely [2]- [23], and study its quantization using Hamiltonian [24], path integral [25] [26] [27] [28] and Becchi-Rouet-Stora and Tyutin (BRST) [29] [30] [31], formulations [24]- [31], in the usual instant-form (IF) of dynamics (on the hyperplanes: 0 constant x t = = ) [32] [33]. We recap the basis of this theory in the next section where we also highlight the motivations for the present study.…”

Light-Front Quantization of the Restricted Gauge Theory of QCD 2

“…Indeed the affine nature of the connection space guarantees that one can describe an arbitrary potential simply by adding a gauge-covariant piece X μ to the restricted potential. Above decomposition is known as Cho-decomposition or the Cho-Faddeev-Niemi decomposition (introduced in an attempt to demonstrate the monopole condensation in QCD), and the importance of this decomposition in clarifying the non-Abelian dynamics has been studied by many authors [1][2][3][4][5][6][7]. The restricted potential A μ actually has a dual structure and the field strength made of the restricted potential is decomposed as:…”

“…In this talk, we study the light-front (LF) quantization (LFQ) of the restricted gauge theory of QCD 2à la Cho et al [1][2][3][4][5][6][7] on the hyperplanes defined by the equal light-cone (LC) time (τ = x + = 1 √ 2 (x 0 + x 1 )) = constant [8][9][10][11], using the Hamiltonian, path integral and BRST [9-13] quantization procedures under the appropriate LC gauge-fixing conditions (GFC's). The theory makes use of the so-called "Cho-decomposition", which is, in fact, the gauge independent decomposition of the non-Abelian potential into the restricted potential and the valence potential and it helps in the clarification of the topological structure of the non-Abelian gauge theory, and it also takes care of the topological characters in the dynamics [2][3][4][5][6].…”

“…The Lagrangian density of the so-called restricted gauge theory made of the Abelian projection without X μ is therefore defined by [2][3][4][5][6][7]:…”

“…We now study the Hamiltonian and path integral quantization of the above restricted gauge theory of QCD 2 (made of the Abelian projection without X μ ) defined by above Lagrangian density re-expressed as [2][3][4][5][6][7]:…”