Examples Of Quantum Field Theory Models

φ4

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 ) for a real field φ 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 ) for a complex field φ. See the analytization trick. See phi to the fourth.

QED

\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} See the analytization trick See QED

Schwinger model

See Schwinger model

The Yukawa model

\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 See Yukawa model

Yang-Mills theory

\mathcal{L}=-\frac{1}{4g^2}TrF_{\mu\nu}F^{\mu\nu} See Yang-Mills, Quantum Yang-Mills theory

The Yang-Mills-Higgs model

See Yang-Mills-Higgs model

Nonlinear sigma models

S\phi=\int d^dx \left\phi^i(x) \partial_\mu \phi^j(x) - V(\phi(x))\right for some positive definite tensor G acting bilinearly upon the tangent space of the target manifold and a potential V bounded from below. See nonlinear sigma model

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

Wess-Zumino-Witten model

 

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