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<p><b>New page</b></p><div>{{hatnote|See [[Physics:Gauge theory|gauge theory]] for the classical preliminaries.}}<br />
==Quantization==<br />
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===Gauge fixing===<br />
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In quantum physics, in order to [[Physics:Quantization|quantize]] a [[Physics:Gauge theory|gauge theory]], for example the [[Yang–Mills theory]], [[Physics:Chern–Simons theory|Chern–Simons theory]] or the [[Physics:BF model|BF model]], one method is to perform [[Physics:Gauge fixing|gauge fixing]]. This is done in the BRST and Batalin-Vilkovisky formulation.<br />
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===Wilson loops===<br />
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Another method is to factor out the symmetry by dispensing with [[Physics:Vector potential|vector potential]]s altogether (since they are not physically [[Physics:Observable|observable]]) and by working directly with [[Physics:Wilson loop|Wilson loop]]s, Wilson lines contracted with other charged fields at its endpoints and [[Physics:Spin network|spin network]]s.<br />
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===Lattices===<br />
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An alternative approach using lattice approximations is covered in (Wick rotated) [[Physics:Lattice gauge theory|lattice gauge theory]].<br />
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===Older approaches===<br />
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Older approaches to quantization for [[Abelian group|Abelian]] models use the Gupta-Bleuler formalism with a "[[Semi-Hilbert space|semi-Hilbert space]]" with an indefinite [[Sesquilinear form|sesquilinear form]]. However, it is much more elegant{{clarify|date=May 2016}} to work with the [[Quotient space (topology)|quotient space]] of vector field configurations by gauge transformations.<br />
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== Quantum Yang–Mills theory ==<br />
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To establish the existence of the Yang-Mills theory and a [[Physics:Mass gap|mass gap]] is one of the seven Millennium Prize Problems of the Clay Mathematics Institute.<br />
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A positive estimate from below of the mass gap in the spectrum of quantum Yang-Mills Hamiltonian has been already established.<ref>{{cite journal |last=Dynin |first=A. |date=January 2017 |title=Mathematical quantum Yang-Mills theory revisited |journal=Russian Journal of Mathematical Physics |volume=24 |issue=1 |pages=26–43|bibcode=2017RJMP...24...19D |arxiv=1308.6571 |doi=10.1134/S1061920817010022 }}</ref><br />
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==References==<br />
{{reflist}}<br />
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{{Quantum field theories}}<br />
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[[Category:Quantum field theory]]<br />
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{{Sourceattribution|Quantum gauge theory|1}}</div>imported>WikiHarold