Examples of quantum field theory models

Contents

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

See 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.

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.

Nambu-Jona-Lasinio model

See Nambu-Jona-Lasinio

Wess-Zumino-Witten model

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