Hamiltonian, Path Integral and BRST Formulations of the Vector Schwinger Model with a Photon Mass Term with Faddeevian Regularization

Abstract

Recently (in a series of papers) we have studied the vector Schwinger model with a photon mass term describing one-space one-time dimensional electrodynamics with mass-less fermions in the so-called standard regularization. In the present work, we study this model in the Faddeevian regularization (FR). This theory in the FR is seen to be gaugenon-invariant (GNI). We study the Hamiltonian and path integral quantization of this GNI theory. We then construct a gauge-invariant (GI) theory corresponding to this GNI theory using the Stueckelberg mechanism and recover the physical content of the original GNI theory from the newly constructed GI theory under some special gauge-choice. Further, we study the Hamiltonian, path integral and Becchi-Rouet-Stora and Tyutin formulations of the newly constructed GI theory under appropriate gauge-fixing conditions.

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.…”

“…The theory is seen to be gauge-invariant (GI) possessing a set of first-class constraints [14]. We quantize this theory under appropriate gauge-fixing conditions (GFC's) using the Hamiltonian and path integral formulations [24] [28].…”

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

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In this talk we study the light-front quantization of the restricted gauge theory of QCD 2à la Cho et alIn 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].

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“…Also, the roman ind-ices a and b here, are the color ind-ices of the gauge theory of QCD 2 . While considering the path integral formulation, the transition to quantum theory is made again by writing the vacuum to vacuum transition amplitude for the theory, called the generating functional Z [J k ] of the theory, following again the Senjanovic procedure for a theory possessing a set of second-class constraints, appropriate for our present theory, considered under the gauge-fixing conditions η i , in the presence of the external sources: J k as follows [10,11]:…”

“…For this, we first enlarge the Hilbert space of our gauge-invariant theory and replace the notion of gauge-transformation, which shifts operators by c-number functions, by a BRST transformation, which mixes operators with Bose and Fermi statistics. We then introduce new anti-commuting variables c andc (Grassman numbers on the classical level and operators in the quantized theory) and a commuting variable b such that [10,11,13]:δ…”