Thirring–Wess model

Solvable 1+1 dimensional quantum field theory

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.

Definition

The Lagrangian density is made of three terms:

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

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

for $F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }$ and the boson mass $\mu$ must be strictly positive; the free fermion field $\psi$ 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 $\alpha \geq 0$

There is a remarkable difference between the case $\alpha >0$ and the case $\alpha =0$ : the latter requires a field renormalization to absorb divergences of the two point correlation.

History

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 $\alpha =1$ , by Thirring and Wess ^[1] using a method introduced by Johnson for the Thirring model; and, for $\alpha =0$ , two different solutions were given by Brown^[2] and Sommerfield.^[3] Subsequently Hagen^[4] showed (for $\alpha =0$ , but it turns out to be true for $\alpha \geq 0$ ) that there is a one parameter family of solutions.

References

^ 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.
^ 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. S2CID 122285105.
^ 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.
^ 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. S2CID 58940957.

External links

Quantum field theories

Theories

Algebraic QFT
Axiomatic QFT
Conformal field theory
Lattice field theory
Noncommutative QFT
Gauge theory
QFT in curved spacetime
String theory
Supergravity
Thermal QFT
Topological QFT
Two-dimensional conformal field theory

Models

Regular	Born–Infeld Euler–Heisenberg Ginzburg–Landau Non-linear sigma Proca Quantum electrodynamics Quantum chromodynamics Quartic interaction Scalar electrodynamics Scalar chromodynamics Soler Yang–Mills Yang–Mills–Higgs Yukawa
Low dimensional	2D Yang–Mills Bullough–Dodd Gross–Neveu Schwinger Sine-Gordon Thirring Thirring–Wess Toda
Conformal	2D free massless scalar Liouville Minimal Polyakov Wess–Zumino–Witten
Supersymmetric	4D N = 1 N = 1 super Yang–Mills Seiberg–Witten Super QCD Wess–Zumino
Superconformal	6D (2,0) ABJM N = 4 super Yang–Mills
Supergravity	4D N = 1 Higher dimensional N = 8 Pure 4D N = 1
Topological	BF Chern–Simons
Particle theory	Chiral Fermi MSSM Nambu–Jona-Lasinio NMSSM Standard Model Stueckelberg