The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.

Definition

The Thirring model is given by the Lagrangian density

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

where ${\displaystyle \psi =(\psi _{+},\psi _{-})}$ is the field, g is the coupling constant, m is the mass, and ${\displaystyle \gamma ^{\mu }}$, for ${\displaystyle \mu =0,1}$, are the two-dimensional gamma 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 the Pauli principle, all the quartic, local interactions are equivalent; and all higher power, local interactions vanish. (Interactions containing derivatives, such as ${\displaystyle ({\bar {\psi }}\partial \!\!\!/\psi )^{2}}$, 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 a quantum field theory.

Massless case

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

Massive Thirring model, or MTM

The mass spectrum of the model and the scattering matrix was explicitly evaluated by Bethe Ansatz. An explicit formula for the correlations is not known. J. I. Cirac, P. Maraner and J. K. Pachos applied the massive Thirring model to the description of optical lattices.^[5]

Bosonization

S. Coleman ^[7] discovered an equivalence between the Thirring and the sine-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 called bosonization.

See also

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

References

External links

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