advanced_notions:chern-simons
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advanced_notions:chern-simons [2017/11/20 13:20] jakobadmin [Student] |
advanced_notions:chern-simons [2017/11/22 10:46] (current) jakobadmin [Student] |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| + | <blockquote>The applications [of Chern-Simons terms] range from the mathematical | ||
| + | characterization of knots to the physical description of electrons in the quantum Hall | ||
| + | effect [5], vivid evidence for the deep significance of the Chern-Simons structure and of its | ||
| + | antecedent, the chiral anomaly.<cite>[[https://arxiv.org/pdf/hep-th/0103017.pdf|Collaborating with David Gross; Descendants of the Chiral Anomaly]] by R. Jackiw</cite></blockquote> | ||
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| <blockquote>Electroweak baryogenesis proceeds via changes in the non-Abelian Chern-Simons number.<cite>[[http://xxx.lanl.gov/pdf/astro-ph/0101261v3|Estimate of the primordial magnetic field helicity]] by Tanmay Vachaspati </cite></blockquote> | <blockquote>Electroweak baryogenesis proceeds via changes in the non-Abelian Chern-Simons number.<cite>[[http://xxx.lanl.gov/pdf/astro-ph/0101261v3|Estimate of the primordial magnetic field helicity]] by Tanmay Vachaspati </cite></blockquote> | ||
| <tabbox Layman> | <tabbox Layman> | ||
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| <tabbox Student> | <tabbox Student> | ||
| + | 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 | ||
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| + | <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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| - | <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. | + | 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". | ||
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| 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, | ||
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| 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 | ||
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| 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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