Lorentz invariant exact solution of the anomalous chiral Schwinger model

We present an exact solution of the anomalous chiral Schwinger model using Fermionic variables. We implement infrared regularization by considering the model on a spatial circleS ¹. Quantum effects modify the gauge constraints through the appearance of Schwinger terms in the gauge algebra. We perform a careful analysis of the resulting second class gauge constraints by implementing Dirac's method at the quantum level and obtain the spectrum of the theory. We get a consistent unitary Lorentz invariant theory for particular values of the counterterms. We find that when we regulate the fermionic sector of the model without reference to the gauge fields Lorentz invariance requires that we add both Lorentz variant and gauge variant counterterms.     

Batalin-Tyutin quantization of the chiral Schwinger model

We quantize the chiral Schwinger model by using the Batalin-Tyutin formalism. We show that one can systematically construct the first-class constraints and the desired involutive Hamiltonian, which naturally generates all secondary constraints. Fora>1, this Hamiltonian gives the gauge invariant Lagrangian including the well-known Wess-Zumino terms, while fora=1 the corresponding Lagrangian has the additional new type of the Wess-Zumino terms, which are irrelevant to the gauge symmetry.     

Anomalies of a chiral Schwinger model

I consider the connection between anomalies of the ordinary and chiral Schwinger model and identify the regulating operator used to obtain these anomalies in the Fujikawa approach. The role of Lorentz invariance in determining a regularization procedure is discussed.     

Hamiltonian formulation of the anomalous chiral Schwinger model

We construct the hamiltonian formulation of the anomalous chiral Schwinger model, which has recently been shown to yield a consistent unitary theory. The impact of the anomaly on the constraints of the system is exhibited and the system is quantized using an appropriate hamiltonian consistent with the constraints.     

Operator solutions of the bosonized chiral Schwinger model

Starting from the modified Lagrangian of the bosonized chiral Schwinger model, operator solutions are obtained under three types of gauge fixing conditions. We show that the physical spectrum consists of a massive free boson and a massless excitation. We emphasize that the “longitudinal” component of the gauge field must be treated properly.     

Perturbation theory for the anomaly-free chiral Schwinger model

We apply perturbation theory to the gauge invariant version of the chiral Schwinger model. The cancellation of anomalies is shown explicitly in terms of Feynman diagrams. We calculate the exact propagators for the gauge field, for the Wess-Zumino field and for the mixing between these fields. Using these propagators, we demonstrate the existence of a massive state.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

Quantization of the anomalous,chiral, Schwinger model

We implement the recent proposal of Faddeev [6] and present a quantization of the anomalous, chiral, Schwinger model. We carry out a Schrödinger representation, hamiltonian formulation quantization, on a circle. We expose the structure of the fermionic Hilbert bundle as a functional of the background gauge fields. We find that, although a unitary and consistent quantum field theory is obtained, Lorentz invariance is lost.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

On the breaking of chiral symmetry in the Schwinger model

The Schwinger model is reinvestigated in the A₁ = 0 gauge. Based on the explicit operator solution, it is suggested that the breaking of chiral symmetry is dynamical and the vacuum is unique contrary to the arguments by Kogut and Susskind.     

The anomalous chiral Schwinger model and its quantization

We solve the anomalous chiral Schwinger model in the hamiltonian formulation implementing the Faddeev proposal. We diagonalize the hamiltonian by functional techniques and construct the eigenfunctionals of the quantized system. The spectrum is non Lorentz invariant since the Poincarè algebra is not closed. We point out that, relinquishing the Faddeev condition, there would be an infinite degeneration of the hamiltonian. Among the degenerate 1-particle eigenfunctionals it is possible to find a state with relativistic spectrum. We also examine the meaning of a non local modification of the hamiltonian which restores Lorentz invariance.