advanced_notions:chern-simons
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advanced_notions:chern-simons [2017/11/20 13:48] jakobadmin [Student] |
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| - | For a great summary see section 2 in [[https://arxiv.org/pdf/hep-th/0103017.pdf|Collaborating with David Gross; Descendants of the Chiral Anomaly]] by R. Jackiw | + | For a great summary see section 2 in [[https://arxiv.org/pdf/hep-th/0103017.pdf|Collaborating with David Gross; Descendants of the Chiral Anomaly]] by R. Jackiw and the [[http://home.mathematik.uni-freiburg.de/cookies17/files/Moffatt_Freiburg%20Lecture%20Notes.pdf|Freiburg Lecture Notes]] by H.K.Moffatt |
| - | <note tip>Chern-Simons terms are known under different names in different branches of physics. In fluid mechanics it is usually called "fluid helicity", in plasma physics and magnetohydrodynamics "magnetic helicity". In the context of field theories it is usually called Chern-Simons term. | + | <note tip>Chern-Simons terms describe topological properties of systems. A topological property is something that remains unchanged under small geometric changes. |
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| + | Chern-Simons terms are known under different names in different branches of physics. In fluid mechanics it is usually called "fluid helicity", in plasma physics and magnetohydrodynamics "magnetic helicity". In the context of field theories it is usually called Chern-Simons term. | ||
| </note> | </note> | ||
| - | **In Fluid Mechanics:** | + | -->In Fluid Mechanics# |
| In the beginning people tried to make a mechanical model of electrodynamics. For example, Maxwell though of Faraday's electric and magnetic field lines as "fine tubes of variable section carrying an incompressible fluid". | In the beginning people tried to make a mechanical model of electrodynamics. For example, Maxwell though of Faraday's electric and magnetic field lines as "fine tubes of variable section carrying an incompressible fluid". | ||
| Line 81: | Line 84: | ||
| to twist is familiar to anyone seeking to | to twist is familiar to anyone seeking to | ||
| straighten out a coiled garden hose. | straighten out a coiled garden hose. | ||
| - | * [[http://www.pnas.org/content/111/10/3663.full.pdf|Helicity and singular structures in fluid dynamics]] by H. Keith Moffatt | + | |
| + | [[http://www.pnas.org/content/111/10/3663.full.pdf|Helicity and singular structures in fluid dynamics]] by H. Keith Moffatt | ||
| </blockquote> | </blockquote> | ||
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| For more information on this obstruction to construct the Lagrangian for Euler's fluid equations, see page 9ff in https://arxiv.org/pdf/hep-th/0004084.pdf and the great review: | For more information on this obstruction to construct the Lagrangian for Euler's fluid equations, see page 9ff in https://arxiv.org/pdf/hep-th/0004084.pdf and the great review: | ||
| - | * [[http://www.annualreviews.org/doi/pdf/10.1146/annurev.fl.20.010188.001301|HAMILTONIAN FLUID MECHANICS]] by Rick Salmon | + | [[http://www.annualreviews.org/doi/pdf/10.1146/annurev.fl.20.010188.001301|HAMILTONIAN FLUID MECHANICS]] by Rick Salmon |
| - | **In Electrodynamics** | + | For an experimental proof that knotted vortices exist indeed in nature, see [[https://www.nature.com/articles/nphys2560|Creation and dynamics of knotted vortices]] by Dustin Kleckner & William T. M. Irvine |
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| + | <-- | ||
| + | |||
| + | -->In Electrodynamics# | ||
| <blockquote>The magnetic helicity is the flux of the magnetic field through the surface bounding the volume, | <blockquote>The magnetic helicity is the flux of the magnetic field through the surface bounding the volume, | ||
| Line 137: | Line 145: | ||
| the fields themselves) thereby avoiding dissipation [1]. | the fields themselves) thereby avoiding dissipation [1]. | ||
| <cite>[[https://arxiv.org/pdf/hep-th/9911072.pdf|Creation and evolution of magnetic helicity]] by R. Jackiw et. al.</cite></blockquote> | <cite>[[https://arxiv.org/pdf/hep-th/9911072.pdf|Creation and evolution of magnetic helicity]] by R. Jackiw et. al.</cite></blockquote> | ||
| + | <-- | ||
| - | **In Non-Abelian Gauge Theories** | + | -->In Non-Abelian Gauge Theories# |
| "anomalous currents are sourced by gauge field configurations with nonzero Chern-Simons number. The | "anomalous currents are sourced by gauge field configurations with nonzero Chern-Simons number. The | ||
| Line 165: | Line 174: | ||
| Ref 21 is [[http://xxx.lanl.gov/abs/hep-th/9911072|Creation and evolution of magnetic helicity]] by R. Jackiw, So-Young Pi | Ref 21 is [[http://xxx.lanl.gov/abs/hep-th/9911072|Creation and evolution of magnetic helicity]] by R. Jackiw, So-Young Pi | ||
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| + | <-- | ||
| <tabbox Researcher> | <tabbox Researcher> | ||
| <blockquote>On a manifold it is necessary to use covariant differentiation; curvature measures its noncommutativitiy. Its combination as a characteristic form measures the nontriviality of the underlying bundle. This train of ideas is so simple and natural that its importance can hardly be exaggerated. <cite>Shiing-shen Cern</cite></blockquote> | <blockquote>On a manifold it is necessary to use covariant differentiation; curvature measures its noncommutativitiy. Its combination as a characteristic form measures the nontriviality of the underlying bundle. This train of ideas is so simple and natural that its importance can hardly be exaggerated. <cite>Shiing-shen Cern</cite></blockquote> | ||
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