Quantum gauge theory

See gauge theory for the classical preliminaries.

Quantization

In quantum physics, in order to quantize a gauge theory, like for example Yang–Mills theory, Chern–Simons theory or the BF model, one method is to perform a gauge fixing. This is done in the BRST and Batalin-Vilkovisky formulation.

Another is to factor out the symmetry by dispensing with vector potentials altogether (they're not physically observable anyway) and work directly with Wilson loops, Wilson lines contracted with other charged fields at its endpoints and spin networks.

An alternative approach using lattice approximations is covered in (Wick rotated) lattice gauge theory.

Older approaches to quantization for Abelian models use the Gupta-Bleuler formalism with a "semi-Hilbert space" with an indefinite sesquilinear form. However, it is much more elegant to work with the quotient space of vector field configurations by gauge transformations.

To establish the existence of the Yang-Mills theory and a mass gap is one of the seven Millennium Prize Problems of the Clay Mathematics Institute.

Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Local quantum field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum hadrodynamics Quantum Yang–Mills theory Schwinger model sine-Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model

Related	History of quantum field theory Axiomatic quantum field theory Quantum field theory in curved spacetime

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