The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in two dimension.
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Definition
The Thirring model is given by the Lagrangian density
where ψ = (ψ + ,ψ − ) is the field, g is the coupling constant, m is the mass, and γμ, for μ = 0,1, are the two-dimensional gamma matrices.
History
After it was introduced by Walter Thirring [1], 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 Hagen and Klaiber extended the explicit solution to any multipoint correlation function of the fields.
Very early people realized a nice relation between Thirring and Sine-Gordon models. Despite the fact that the latter is a pure boson model, massless Thirring fermions are equivalent to a free bosons; while massive fermions are equivalent to the Sine-Gordon bosons. This phenomenon is more general in two-dimension and is called bosonization.
Since the spectrum of the Hamiltonian and the scattering matrix of the Sine-Gordon model can be explicitly evaluated by Bethe Ansatz, also the massive Thirring model is said exactly solvable, though an explicit formula for the correlations is not known.
Only recently it has been possible to prove that the correlations functions of the Thirring model (massive or massless, at imaginary time) verify the Osterwalder-Schrader axioms, and hence the theory makes sense as a quantum field theory.
Exact solution
In one space and one time dimension the model can be solved by Bethe ansatz. This helps to calculate exactly mass spectrum and scattering matrix. Calculation of the scattering matrix reproduce the results published earlier by Alexander Zamolodchikov. The paper with exact solution of Massive Thirring model by Bethe Ansatz was first published in Russian language in Theoretical and mathematical Physics vol 41, page 169, 1979 [1] . The paper was translated into English in Theoretical and Mathematical Physics, 1979, 41:2, 953–967 [2]. Ultraviolet renormalization was done in the frame of Bethe ansatz. The fractional charge appears in the model during renormalization as a repulsion beyond cutoff.
Multi particle production cancels on mass shell.
Exact solution shows once again the equivalence of Thirring model and quantum sine-Gordon model. The Thirring model is S-dual to the sine-Gordon model. The fundamental fermions of the Thirring model correspond to the solitons of the sine-Gordon model.
References
- ^ Thirring, Walter (1958). "A Soluble relativistic field theory?" (PDF). Annals Phys. 3: 91–112. http://www.slac.stanford.edu/spires/find/hep/www?j=APNYA,3,91.
External links
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