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In order to Quantize a gauge theory, like for example Yang-Mills theory, Chern-Simons or 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 Potential s altogether (they're not physically Observable anyway) and work directly with Wilson Loop s, Wilson Line s contracted with other charged fields at its endpoints and Spin Network s. 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 just work with the Quotient Space of vector field configurations by gauge transformations. An alternative approach using lattice approximations is covered in ( Wick Rotated ) Lattice Gauge 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 . |
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