Quantum gauge theory

See gauge theory for the classical preliminaries.

Quantization

Gauge fixing

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

Wilson loops

Another method is to factor out the symmetry by dispensing with vector potentials altogether (since they are not physically observable) and by working 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.

Quantum Yang–Mills theory
See also: 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.

A positive estimate from below of the mass gap in the spectrum of quantum Yang-Mills Hamiltonian has been already established.[1]

References
  1. Dynin, A. (January 2017). "Mathematical quantum Yang-Mills theory revisited". Russian Journal of Mathematical Physics. 24 (1): 26–43. arXiv:1308.6571. Bibcode:2017RJMP...24...19D. doi:10.1134/S1061920817010022.


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