On the equivalence between sine-Gordon model and Thirring model in the chirally broken phase of the Thirring model

Abstract

We investigate the equivalence between Thirring model and sine-Gordon model in the chirally broken phase of the Thirring model. This is unlike all other available approaches where the fermion fields of the Thirring model were quantized in the chiral symmetric phase. In the path integral approach we show that the bosonized version of the massless Thirring model is described by a quantum field theory of a massless scalar field and exactly solvable, and the massive Thirring model bosonizes to the sine-Gordon model with a new relation between coupling constants. We show that the non-perturbative vacuum of the chirally broken phase in the massless Thirring model can be described in complete analogy with the BCS ground state of superconductivity. The Mermin-Wagner theorem and Coleman's statement concerning the absence of Goldstone bosons in the 1+1-dimensional quantum field theories are discussed. We investigate the current algebra in the massless Thirring model and give a new value of the Schwinger term. We show that the topological current in the sine-Gordon model coincides with the Noether current responsible for the conservation of the fermion number in the Thirring model. This allows to identify the topological charge in the sine-Gordon model with the fermion number. *

Is the energy density of the ground state of the sine-Gordon model unbounded from below for 2> 8 ?

“…The first two terms describe the contribution of the kinetic energy which should be always taken in the normal-ordered form 1 . In quantum field theory the potential energy, given by the last two terms in (3.1), should be normal ordered as well as the kinetic one.…”

“…As has been discussed in [1], the relation β 2 > 8π leads to a 1+1-dimensional world populated mainly by soliton and antisoliton states [1], which are classical solutions of the equations of motion of the sine-Gordon model (1.2) ✷ϑ(x) + α 0 β sin βϑ(x) = 0 (1.5)…”

“…The Lagrangian of the massless Thirring model is given by [1]- [3] L Th (x) =ψ(x)iγ µ ∂ µ ψ(x) − 1 2 gψ(x)γ µ ψ(x)ψ(x)γ µ ψ(x) + σ(x)ψ(x)ψ(x), (1.1) where σ(x) is an external source of the scalar densityψ(x)ψ(x) of the Thirring fermion fields and g is the coupling constant, which we treat in the attractive case. For σ(x) = −m [2], where m can be interpreted as a mass of Thirring fermion fields, the Thirring model (1.1) bosonizes to the sine-Gordon model with the Lagrangian [1,2] L SG (x) = 1 2 ∂ µ ϑ(x)∂ µ ϑ(x) + α 0 β 2 (cos βϑ(x) − 1), (1.2) where α 0 and β are positive parameters [1,2,4].…”

“…For σ(x) = −m [2], where m can be interpreted as a mass of Thirring fermion fields, the Thirring model (1.1) bosonizes to the sine-Gordon model with the Lagrangian [1,2] L SG (x) = 1 2 ∂ µ ϑ(x)∂ µ ϑ(x) + α 0 β 2 (cos βϑ(x) − 1), (1.2) where α 0 and β are positive parameters [1,2,4]. The parameter α 0 has the meaning of a squared mass of the quantum of the sine-Gordon field and β is a coupling constant.…”

“…The parameter α 0 has the meaning of a squared mass of the quantum of the sine-Gordon field and β is a coupling constant. For the Thirring fermion fields quantized in the chirally broken phase the coupling constants g and β are related by [1] 8π…”