Gauge-Invariant Reformulation of the Vector Schwinger Model With a Photon Mass Term and Its Hamiltonian, Path Integral and BRST Formulations

Abstract

Using the Stueckelberg formalism, we construct a gauge-invariant version of the vector Schwinger model (VSM) with a photon mass term studied by one of us recently. This model describes two-dimensional massive electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings. This model describing the 2D massive electrodynamics becomes gaugenoninvariant (GNI). This is in contrast to the case of the massless VSM which is a gauge-invariant (GI) theory (as a consequence of demanding the regularization for the theory to be GI). In this work we first construct a GI theory corresponding to this model describing the 2D massive electrodynamics, using the Stueckelberg formalism and then we recover the physical contents of the original GNI theory studied earlier, under some special gauge choice. We then study the Hamiltonian, path integral and BRST formulations of this GI theory under appropriate gauge-fixing. The theory presents a new class of models in the 2D quantum electrodynamics with massless fermions but with a photon mass term.

Hamiltonian and Path Integral Quantization of the Conformally Gauge-Fixed Polyakov D1 Brane Action in the Presence of a Scalar Dilation Field

“…Further, in the path integral formulation, the transition to the quantum theory is made by writing the vacuum to vacuum transition amplitude for the theory called the generating functional Z 1 [J μ] of the theory in the presence of external sources J μ as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]:…”

“…Here the symbol ≈ denotes a weak equality (WE) in the sense of Dirac [9][10][11][12][13][14][15][16][17][18][19], and it implies that these above constraints hold as strong equalities only on the reduced hypersurface of the constraints and not in the rest of the phase space of the classical theory (and similarly one can consider it as a weak operator equality (WOE) for the corresponding quantum theory) [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

“…The nonsingular nature of the matrix R αβ signifies that the set of constraints ρ i is second class [24] and consequently the theory described by the action S 2 is gauge noninvariant (GNI) [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] as is expected because it is gauge-fixed theory (under the conformal gauge given by (2)) and therefore GNI. It is important to recapitulate here that the original Polyakov D1 brane action defined by the actionS described the propagation of the D-string in a curved background and is gaugeinvariant (GI) possessing the well known three local (or gauge) symmetries defined by the 2-dimensional WS reparametrization invariance and the Weyl invariance [1][2][3][4][5][6][7][8][9][10][11][12].…”

“…However, the present theory under our current investigation is GNI because it is gauge-fixed theory being studied under the so-called conformal gauge defined by (2) and is consequently GNI as it should be. Now, following the Dirac quantization procedure in the Hamiltonian formulation [9][10][11][12][13][14][15][16][17][18][19] the nonvanishing equal WS-time Dirac brackets of the theory are formally obtained as:…”

“…hyperplanes defined by the WS-time σ 0 = τ = constant [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] using the standard constraint quantization techniques [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In the next section we briefly recapitulate some basics of the model without the scalar dilation field in the CG.…”