Examples of quantum field theory models
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φ4
- <math>S[\phi]=\int d^dx \left (\frac{1}{2} \partial^\mu \phi(x) \partial_\mu \phi(x) -\frac{1}{2}m^2\phi(x)^2 -\frac{\lambda}{4!}\phi(x)^4\right )<math>
for a real field φ
- <math>S[\phi^*,\phi]=\int d^dx \left (\partial^\mu \phi^*(x) \partial_\mu \phi(x) -m^2\phi^*(x)\phi(x) -\frac{\lambda}{4}(\phi^*(x)\phi(x))^2\right )<math>
for a complex field φ.
See analytization trick.
See phi to the fourth.
QED
- <math>\mathcal{L}(\bar{\psi},\psi,\bold{A})=\bar\psi(i\gamma_\mu D^\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}<math>
See analytization trick.
See QED.
Schwinger model
See Schwinger model.
The Yukawa model
- <math>\mathcal{L}=\frac{1}{2} \partial^\mu \phi(x) \partial_\mu \phi(x) -\frac{1}{2}m_\phi^2\phi(x)^2 +\bar\psi(i\gamma_\mu D^\mu-m_\psi-\lambda\phi)\psi<math>
See Yukawa model.
Yang-Mills theory
- <math>\mathcal{L}=-\frac{1}{4g^2}Tr[F_{\mu\nu}F^{\mu\nu}]<math>
See Yang-Mills, Quantum Yang-Mills theory.
The Yang-Mills-Higgs model
Nonlinear sigma models
- <math>S[\phi]=\int d^dx \left[\frac{1}{2}G_{ij}(\phi(x))\partial^\mu \phi^i(x) \partial_\mu \phi^j(x) - V(\phi(x))\right]<math>
for some positive definite tensor G acting bilinearly upon the tangent space of the target manifold and a potential V bounded from below.
Chiral model
See chiral model.
The Thirring model
See Thirring model.
The Sine-Gordon model
See Sine-Gordon.
The Chern-Simons model
See Chern-Simons model, topological quantum field theory.
The Gross-Neveu model
See Gross-Neveu.