Thirring–Wess model

The Lagrangian density is made of three terms:

the free vector field $A^\backslash mu$ is described by

:$$

{(F^{\mu\nu})^2 \over 4}

+{\mu^2\over 2} (A^\mu)^2

for $F^\{\backslash mu\backslash nu\}=\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial^\backslash nu\; A^\backslash mu$ and the boson mass $\backslash mu$ must be

strictly positive;

the free fermion field $\backslash psi$

is described by

:$$

\overline{\psi}(i\partial\!\!\!/-m)\psi

where the fermion mass $m$ can be positive or zero.

And the interaction term is

:$$

qA^\mu(\bar\psi\gamma^\mu\psi)

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

:$$

{\alpha\over 2} (\partial^\mu A^\mu)^2

for $\backslash alpha\; \backslash ge\; 0$

There is a remarkable difference between the case $\backslash alpha\; >\; 0$ and the case $\backslash alpha\; =\; 0$: 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 ($m\; =\; 0$), the model is exactly solvable. One solution was found, for $\backslash alpha\; =\; 1$, by Thirring and Wess

{{Cite journal

|last1=Thirring | first1=WE

|last2=Wess |first2=JE

|year=1964

|title=Solution of a field theoretical model in one space one time dimensions

|journal=Annals of Physics

|volume=27 |issue=2 |pages=331–337

|bibcode=1964AnPhy..27..331T

|doi=10.1016/0003-4916(64)90234-9

}}

using a method introduced by Johnson for the Thirring model; and, for $\backslash alpha\; =\; 0$, two different solutions were given by Brown

{{Cite journal

|last=Brown |first=LS

|year=1963

|title=Gauge invariance and Mass in a Two-Dimensional Model

|journal=Il Nuovo Cimento

|volume=29 |issue=3 |pages=617–643

|bibcode=1963NCim...29..617B

|doi=10.1007/BF02827786

|s2cid=122285105

}} and Sommerfield.

{{Cite journal

|last=Sommerfield |first=CM

|year=1964

|title=On the definition of currents and the action principle in field theories of one spatial dimension

|journal=Annals of Physics

|volume=26 |issue=1 |pages=1–43

|bibcode=1964AnPhy..26....1S

|doi=10.1016/0003-4916(64)90273-8

}} Subsequently Hagen

{{Cite journal

|last=Hagen |first=CR

|year=1967

|title=Current definition and mass renormalization in a Model Field Theory

|journal=Il Nuovo Cimento A

|volume=51 |issue=4 |pages=1033–1052

|bibcode=1967NCimA..51.1033H

|doi=10.1007/BF02721770

|s2cid=58940957

}} showed (for $\backslash alpha\; =\; 0$, but it turns out to be true for $\backslash alpha\; \backslash ge\; 0$) that there is a one parameter family of solutions.

{{Quantum field theories}}

{{DEFAULTSORT:Thirring-Wess Model}}

Category:Quantum field theory

Category:Exactly solvable models