Thirring–Wess model

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The Thirring–Wess model or Vector Meson model is an exactly solvable quantum field theory describing the interaction of a Dirac field with a vector field in dimension two.


The Lagrangian density is made of three terms:

the free vector field is described by

for and the boson mass must be strictly positive; the free fermion field is described by

where the fermion mass can be positive or zero. And the interaction term is

Although not required to define the massive vector field, there can be also a gauge-fixing term


There is a remarkable difference between the case and the case : the latter requires a field renormalization to absorb divergences of the two point correlation.


This model was introduced by Thirring and Wess as a version of the Schwinger model with a vector mass term in the Lagrangian .

When the fermion is massless (), the model is exactly solvable. One solution was found, for , by Thirring and Wess [1] using a method introduced by Johnson for the Thirring model; and, for , two different solutions were given by Brown[2] and Sommerfield.[3] Subsequently Hagen[4] showed (for , but it turns out to be true for ) that there is a one parameter family of solutions.


  1. ^ Thirring, WE; Wess, JE (1964). "Solution of a field theoretical model in one space one time dimensions". Annals of Physics. 27 (2): 331–337. Bibcode:1964AnPhy..27..331T. doi:10.1016/0003-4916(64)90234-9.
  2. ^ Brown, LS (1963). "Gauge invariance and Mass in a Two-Dimensional Model". Il Nuovo Cimento. 29 (3): 617–643. Bibcode:1963NCim...29..617B. doi:10.1007/BF02827786.
  3. ^ Sommerfield, CM (1964). "On the definition of currents and the action principle in field theories of one spatial dimension". Annals of Physics. 26 (1): 1–43. Bibcode:1964AnPhy..26....1S. doi:10.1016/0003-4916(64)90273-8.
  4. ^ Hagen, CR (1967). "Current definition and mass renormalization in a Model Field Theory". Il Nuovo Cimento A. 51 (4): 1033–1052. Bibcode:1967NCimA..51.1033H. doi:10.1007/BF02721770.

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