Integrable coupled massive Thirring model with field values in a Grassmann algebra

TL;DR

This paper introduces an integrable coupled massive Thirring model with fields in a Grassmann algebra, constructs its Lax pair, and explores its symmetries, conserved quantities, and nonlocal reductions, expanding the understanding of integrable quantum field theories.

Contribution

The work presents a new integrable model with Grassmann algebra-valued fields, constructs its Lax pair, and develops nonlocal reductions, revealing new classes of integrable systems.

Findings

The model is shown to possess infinite hierarchies of conserved quantities.

The Lax pair and equations of motion are explicitly constructed.

Nonlocal reductions preserve certain symmetries despite breaking Lorentz invariance.

Abstract

A coupled massive Thirring model of two interacting Dirac spinors in $1+1$ dimensions with fields taking values in a Grassmann algebra is introduced, which is closely related to a SU(1,1) version of the Grassmannian Thirring model also introduced in this work. The Lax pair for the system is constructed, and its equations of motion are obtained from a zero curvature condition. It is shown that the system possesses several infinite hierarchies of conserved quantities, which strongly confirms its integrability. The model admits a canonical formulation and is invariant under space-time translations, Lorentz boosts and global U(1) gauge transformations, as well as discrete symmetries like parity and time reversal. The conserved quantities associated to the continuous symmetries are derived using Noether's theorem, and their relation to the lower-order integrals of motion is spelled out. New nonlocal integrable models are constructed through consistent nonlocal reductions between the field components of the general model. The Lagrangian, the Hamiltonian, the Lax pair and several infinite hierarchies of conserved quantities for each of these nonlocal models are obtained substituting its reduction in the expressions of the analogous quantities for the general model. It is shown that, although the Lorentz symmetry of the general model breaks down for its nonlocal reductions, these reductions remain invariant under parity, time reversal, global U(1) gauge transformations and space-time translations.