Vector Schwinger Model With a Photon Mass Term: Gauge-Invariant Reformulation, Operator Solutions and Hamiltonian and Path Integral Formulations

Abstract

We consider the vector Schwinger model (VSM) describing two-dimensional electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings, with a mass term for the U(1) gauge field and then study its operator solutions and the Hamiltonian and path integral formulations. We emphasize here that although the VSM has been studied in the literature rather widely but only without a photon mass term (which was a consequence of demanding the regularization for the VSM to be gauge-invariant (GI)). The VSM with a photon mass term is seen to be a gauge-noninvariant (GNI) theory. Using the standard Stueckelberg formalism we then construct a GI theory corresponding to the proposed GNI model. From this reformulated GI theory, we further recover the physical contents of the proposed GNI theory under a very special gauge choice. The theory proposed and studied here presents a new class of models in the two-dimensional 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]:…”

“…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:…”

“…In the path integral formulation, the transition to quantum theory is now made by writing the generating functional Z 2 [J i ] for the theory in the presence of the external sources J i as [9][10][11][12][13][14][15][16][17][18][19]:…”

“…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.…”