New Vacuum of Bethe Ansatz Solutions in Thirring Model

Abstract

We find a new vacuum of the Bethe ansatz solutions in the massless Thirring model. This vacuum breaks the chiral symmetry and has the lower energy than the well-known symmetric vacuum energy. Further, we evaluate the energy spectrum of the one particle-one hole ($1p-1h$) states, and find that it has a finite gap. The analytical expressions for the true vacuum as well as for the lowest $1p-1h$ excited state are also found. Further, we examine the bosonization of the massless Thirring model and prove that the well-known procedure of bosonization of the massless Thirring model is incomplete because of the lack of the zero mode in the boson field.Comment: 7 page

Massless Thirring Model in Canonical Quantization Scheme

“…, -for the phase with spontaneously broken chiral symmetry [13,15], further we consider here only the first possibility. The corresponding BP allows to operate with functionals of boson fields instead of fermion operators and forms a powerful tool for obtaining non-perturbative solutions in various two-dimensional models [9,13,16,24].…”

“…(4.7)-(4.9) and (5.14), under the conditions: 15) reproduces the bosonization relations (3.8), (3.9) and (4.1) as following: 20) and finds: aa = πc (Ψ) , a − a = gc (Ψ) , (5.21) in agreement with [3][4][5]. On the other hand, in accordance with [8,24,25], the algebra of the Heisenberg operator of the conserved fermionic charge, by virtue of (5.17) …”

“…Despite a considerable age the two-dimensional Thirring model [1][2][3] is still remained as important touchstone for non-perturbative methods of quantum field theory [4][5][6][7][8] revealing new features both in the well-known [9][10][11][12][13] and in newly obtained solutions [15]. At the same time the methods of integration of such two-dimensional models provide a key for understanding some non-linear theories of higher dimension [13].…”

“…At the same time the methods of integration of such two-dimensional models provide a key for understanding some non-linear theories of higher dimension [13]. In particular the Thirring model turns out to be a two-dimensional analog of the well-known Nambu-Jona-Lasinio model [13,15] and together with the Schwinger model provides an important example of using the well-known bosonization procedure (BP) [7][8][9][10][11][12][13][14][15][16].…”

“…The problem of Schwinger terms in the currents commutator [4], being closely related to BP [9][10][11][12][13][14][15][16], also finds here a natural solution [24] in fact borrowed from QED [25], where it is also sufficient to define this commutator only for the free fields in corresponding "interaction representation".…”