Physics:Quantum gauge theory: Difference between revisions
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Quantum gauge theory is a Book I topic in the Quantum Collection. It studies quantum field theories whose interactions are organized by local symmetry transformations. Gauge symmetry introduces fields that mediate forces, such as the electromagnetic field in quantum electrodynamics and the gluon field in quantum chromodynamics. Quantizing a gauge theory requires special care because not all field components represent physical degrees of freedom. Gauge fixing, BRST symmetry, path integrals, Wilson loops, and lattice methods are tools for handling this structure. The topic links symmetry, conservation laws, particles, fields, and the mathematical foundations of modern fundamental physics. | |||
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[[File:Quantum_gauge_theory_concept_map.svg|thumb|280px|gauge theory in the Quantum Collection.]] | |||
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==Quantization== | ==Quantization== | ||
===Gauge fixing=== | ===Gauge fixing=== | ||
In quantum physics, in order to | 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=== | ===Wilson loops=== | ||
Another method is to factor out the symmetry by dispensing with | 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=== | ===Lattices=== | ||
An alternative approach using lattice approximations is covered in (Wick rotated) | An alternative approach using lattice approximations is covered in (Wick rotated) lattice gauge theory. | ||
===Older approaches=== | ===Older approaches=== | ||
Older approaches to quantization for | 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{{clarify|date=May 2016}} to work with the quotient space of vector field configurations by gauge transformations. | ||
== Quantum Yang–Mills theory == | == Quantum Yang–Mills theory == | ||
To establish the existence of the Yang-Mills theory and a | 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.<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> | 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> | ||
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==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
[[Category:Quantum field theory]] | [[Category:Quantum field theory]] | ||
{{Sourceattribution|Quantum gauge theory|1}} | {{Sourceattribution|Quantum gauge theory|1}} | ||
Latest revision as of 23:34, 23 May 2026
Quantum gauge theory is a Book I topic in the Quantum Collection. It studies quantum field theories whose interactions are organized by local symmetry transformations. Gauge symmetry introduces fields that mediate forces, such as the electromagnetic field in quantum electrodynamics and the gluon field in quantum chromodynamics. Quantizing a gauge theory requires special care because not all field components represent physical degrees of freedom. Gauge fixing, BRST symmetry, path integrals, Wilson loops, and lattice methods are tools for handling this structure. The topic links symmetry, conservation laws, particles, fields, and the mathematical foundations of modern fundamental physics.
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[clarification needed] to work with the quotient space of vector field configurations by gauge transformations.
Quantum 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
- ↑ Dynin, A. (January 2017). "Mathematical quantum Yang-Mills theory revisited". Russian Journal of Mathematical Physics 24 (1): 26–43. doi:10.1134/S1061920817010022. Bibcode: 2017RJMP...24...19D.
Source attribution: Quantum gauge theory
