Thirring model

Not to be confused withThirring–Wess modelorGross–Neveu model.

TheThirring modelis an exactly solvable quantum field theory which describes the self-interactions of aDirac fieldin (1+1) dimensions.

Definition

The Thirring model is given by theLagrangian density

${\mathcal {L}}={\overline {\psi }}(i\partial \!\!\!/-m)\psi -{\frac {g}{2}}\left({\overline {\psi }}\gamma ^{\mu }\psi \right)\left({\overline {\psi }}\gamma _{\mu }\psi \right)\$

where$\psi =(\psi _{+},\psi _{-})$is the field,gis thecoupling constant,mis themass,and$\gamma ^{\mu }$,for$\mu =0,1$,are the two-dimensionalgamma matrices.

This is the unique model of (1+1)-dimensional, Dirac fermions with a local (self-)interaction. Indeed, since there are only 4 independent fields, because of thePauliprinciple, all the quartic, local interactions are equivalent; and all higher power, local interactions vanish. (Interactions containing derivatives, such as$({\bar {\psi }}\partial \!\!\!/\psi )^{2}$,are not considered because they are non-renormalizable.)

The correlation functions of the Thirring model (massive or massless) verify the Osterwalder–Schrader axioms, and hence the theory makes sense as aquantum field theory.

Massless case

The massless Thirring model is exactly solvable in the sense that a formula for the$n$-points field correlation is known.

Exact solution

After it was introduced byWalter Thirring,many authors tried to solve the massless case, with confusing outcomes. The correct formula for the two and four point correlation was finally found by K. Johnson;then C. R. Hagenand B. Klaiberextended the explicit solution to any multipoint correlation function of the fields.

Massive Thirring model, or MTM

Themass spectrumof the model and thescattering matrixwas explicitly evaluated byBethe ansatz.An explicit formula for the correlations isnotknown. J. I. Cirac, P. Maraner and J. K. Pachos applied the massive Thirring model to the description of optical lattices.

Exact solution

In one space dimension and one time dimension the model can be solved by theBethe ansatz.This helps one calculate exactly the mass spectrum andscattering matrix.Calculation of the scattering matrix reproduces the results published earlier byAlexander Zamolodchikov.The paper with the exact solution of Massive Thirring model by Bethe ansatz was first published in Russian.Ultravioletrenormalizationwas done in the frame of the Bethe ansatz. The fractional charge appears in the model during renormalization as a repulsion beyond the cutoff.

Multi-particle production cancels on mass shell.

The exact solution shows once again the equivalence of the Thirring model and the quantumsine-Gordon model.The Thirring model isS-dualto thesine-Gordon model.The fundamental fermions of the Thirring model correspond to thesolitonsof thesine-Gordon model.

Bosonization

S. Colemandiscovered an equivalence between the Thirring and thesine-Gordon models.Despite the fact that the latter is a pure boson model, massless Thirring fermions are equivalent to free bosons; besides massive fermions are equivalent to the sine-Gordon bosons. This phenomenon is more general in two dimensions and is calledbosonization.

See also

• Dirac equation
• Gross–Neveu model
• Nonlinear Dirac equation
• Soler model

External links

• On the equivalence between sine-Gordon Model and Thirring Model in the chirally broken phase