New excitations in the Thirring model

Abstract

The quantization of the massless Thirring model in the light-cone using functional methods is considered. The need to compactify the coordinate x − in the light-cone spacetime implies that the quantum effective action for left-handed fermions contains excitations similar to abelian instantons produced by composite of left-handed fermions. Righthanded fermions don't have a similar effective action. Thus, quantum mechanically, chiral symmetry must be broken as a result of the topological excitations. The conserved charge associated to the topological states is quantized. Different cases with only fermionic excitations or bosonic excitations or both can occur depending on the boundary conditions and the value of the coupling.In recent years quantization in the light cone frame has been extensively studied in connection with the discovery of new non-perturbative effects that would be unobservable in the standard spacetime quantization [1][2][3].There are several reasons that make the light cone quantization a method radically different compared to the standard quantization. The dispersion relationin the light cone. This implies that the particles and antiparticles occupy disconnected sectors of momentum space. Furthermore, if the momentum k + is discretized by compactifying x − , the energy k − is also quantized and nonzero. Thus, the light cone momentum is never singular if the spacetime topology is S 1 × ℜ [4]. The above comments show that the quantization in the light cone is naturally defined over a manifold with non-trivial topology. As a consequence, one could expect new physical effects originated in the implicit difference in topology from standard spacetime quantization.From this point of view, the massless Thirring model is an example where one could investigate the new effects that emerge in the light-cone frame. We will show below that, contrary to the standard quantization, the non-trivial topology of the light-cone spacetime induces abelian instanton-like excitations, i.e. a purely quantum * E-mail: cortes@leo.unizar.es † E-mail: jgamboa@lauca.usach.cl ‡ E-mail: ischmidt@fis.utfsm.cl § E-mail: jz@cecs.cl mechanical effect that appears in the calculation of the fermionic determinant. Let us consider the massless Thirring model in the light cone framewhere ∂ ± = ∂ ∂x ± and x + and x − play the role of time and space respectively.One should note that (1) is invariant under the charge conjugation symmetry ψ L,R ↔ ψ † L,R , that is expected to be conserved at the quantum level. However in order to quantize the system following the path integral methods it is more convenient to write (1) as followswhereThus, the generating functional,after integrating over the right handed fields is

Light-cone quantization of two dimensional field theory in the path integral approach

“…In [5] we showed that for the Thirring model in the light cone, the composite L −L dx − ψ † L ψ L is quantized and we argued also that this quantization is due to S 1 × ℜ topology induced by the light cone frame. In this section we will show that this quantization rule is quite natural and it is always present in the quantization of physical systems when periodic or antiperiodic boundary conditions are assumed.…”

“…On the other hand a quantization condition similar to the flux quantization condition found in [5] for the Thirring model can be derived directly from the equation of motion for the fermion variable ψ…”

“…In section 3 we study the Thirring model; following the standar representation of the fermion self-coupling by an auxiliary vector field we find a quantization condition on the integral of the vector field along a constant x − closed line. When the relation between the auxiliary field and the fermion current is taken into account and one considers a coupling g 2 such that g 2 /π is a rational number then the quantization condition translates into a restriction on the possible values of the number of left-handed fermions, a result obtained previously [5] by other means. In section 4 similar arguments are applied to the Schwinger model and the same quantization condition of the Thirring model is obtained with the auxiliary field replaced by the gauge field.…”

“…In [5] a first discussion of the possible implications of a compactified light-like coordinate in the quantization of the Thirring model was presented. The study was based on the identification of a relation of the integration over the auxiliary variable (in the ligth-cone quantization) ψ R with a quantum mechanical determinant.…”

Instanton-like excitations in 2D fermionic field theory

“…In this article we review some recent results previously reported in [1,2], where a new type of nonperturbative excitations was identified in the Thirring and Schwinger models.…”