Quantum gauge theory

See gauge theory for the classical preliminaries.

Quantization

Gauge Fixing

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.

Wilson loops

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.

Lattices

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

Older approaches

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.

Yang-Mills Theory

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.