The Schwinger Model on the Null-Plane

Abstract

We study the Schwinger Model on the null-plane using the Dirac method for constrained systems. The fermion field is analyzed using the natural null-plane projections coming from the γ-algebra and it is shown that the fermionic sector of the Schwinger Model has only second class constraints. However, the first class constraints are exclusively of the bosonic sector. Finally, we establish the graded Lie algebra between the dynamical variables, via generalized Dirac bracket in the null-plane gauge, which is consistent with every constraint of the theory.

Podolsky’s electromagnetic theory on the null-plane gauge

“…The second-class constraints do not appear in the instant-form dynamics for this theory, thus, they are a common effect of the null-plane dynamics [7].…”

“…Now let us eliminate the arbitrary functions from our theory by using gauge invariance to fix the remaining three degrees of freedom corresponding to the first class constraints (7). Let us choose the null-plane gauge defined by [8] A − ≈ 0.…”

The canonical structure of Podolsky's generalized electrodynamics on the null-plane

“…It will also determine an unique inverse of the second class constraint matrix which allows to obtain the correct Dirac Brackets among the fundamental variables. Thus, in the study of the Podolsky's theory we follow the same tune outlined in [31,32,33].…”

Functional Analysis for Gauge Fields on the Front-Form and the Light-Cone Gauge

“…Now, we can follow the steps of [6][7][8] to analyse the constraints a la Dirac [9] and find that there are two of the first-class type, i.e., which are known as the null-plane gauge conditions [6][7][8]. It is this set of appropriate non-covariant gauge conditions that fixes the first-class constraints of the theory.…”

“…Note that the relation A a − ≈ 0 alone does not define the null-plane gauge; the subsidiary condition π − a + D x − ab A b + ≈ 0, which comes from the consistency of A a − ≈ 0, is necessary in order to fix the first-class constraints of the theory. Appropriate boundary conditions were imposed on the fields [3][4][5][6][7][8]29] and then the transition amplitudes expressed in terms of the physical components. The results found are thoroughly consistent with the ones reported in the literature [33][34][35]38].…”