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advanced_notions:chern-simons [2017/11/19 07:05]
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>​
  
 +
 +<​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
 +
 +<note tip>​Chern-Simons terms describe topological properties of systems. A topological property is something that remains unchanged under small geometric changes. ​
 +
  
-<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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 invariance: For two linked tubes, it is proportional invariance: For two linked tubes, it is proportional
 to the product of the two circulations to the product of the two circulations
-(each conserved by Kelvin’s circulation ​</​blockquote>​+(each conserved by Kelvin’s circulation ​
 theorem), whereas for a single knotted tube, theorem), whereas for a single knotted tube,
 or a deformed unknotted tube, it is proportional or a deformed unknotted tube, it is proportional
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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>:​ Can one write the Chern-Simons term as a total
 +derivative, so that (as befits a topological quantity) the spatial volume integral becomes a
 +surface integral. An argument that this should be possible is the following: The Chern-Simons
 +term is a 3-form on 3-space, hence it is maximal and its exterior derivative vanishes because
 +there are no 4-forms on 3-space. This establishes that on 3-space the Chern-Simons term
 +is closed, so one can expect that it is also exact, at least locally, that is, it can be written
 +as a total derivative. Of course, such a representation for the Chern-Simons term requires
 +expressing the potentials in terms of “pre-potentials”,​ since the formulas (7), (8) show no
 +evidence of derivative structure. [Recall that the total derivative formulas (5), (6) for the axial
 +anomaly also require using potentials to express F.]
 +There is a physical, practical reason for wanting the Abelian Chern-Simons term to be a
 +total derivative. It is known in fluid mechanics that there exists an obstruction to constructing
 +a Lagrangian for Euler’s fluid equations, and this obstruction is just the kinetic helicity $\int d^3r \vec v \cdot \vec \omega$, that is, the volume integral of the Abelian Chern-Simons term, constructed from
 +the velocity 3-vector $\vec v$. This obstruction is removed when the integrand is a total derivative,
 +because then the kinetic helicity volume integral is converted to a surface integral by
 +Gauss’ theorem. When the integral obtains contributions only from a surface, the obstruction
 +disappears from the 3-volume, where the fluid equation acts [4].<​cite>​[[https://​arxiv.org/​pdf/​hep-th/​0011274.pdf|John Bell’s Observations on the Chiral Anomaly and Some Properties of Its Descendants]] by R. Jackiw</​cite></​blockquote>​
 +
 +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 ​
 +
 +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
 +
 +<--
 +
 +-->In Electrodynamics#​
 +
 +<​blockquote>​The magnetic helicity is the flux of the magnetic field through the surface bounding the volume,
 +with θ acting as a modulating factor.<​cite>​https://​arxiv.org/​pdf/​hep-th/​0103017.pdf</​cite></​blockquote>​
 +
 +In electrodynamics the Chern-Simons term is known as helicity and described the linking of magnetic flux lines. This can be seen by interpreting the magnetic field as an incompressible fluid flow, with vector potential $\vec A$: $B= \Delta \times A$. 
 +
 +For more on this, see [[https://​books.google.de/​books?​id=lA8tgLMRu2kC&​lpg=PP1&​hl=de&​pg=PA52#​v=onepage&​q&​f=false|Helicity and Electomagnetic Field Topology]] by Gerald E. Marsh
 +
 +<​blockquote>​It has been suggested that primordial magnetic fields can develop large correlation
 +lengths provided they carry nonvanishing “magnetic helicity”
 +R
 +d
 +3
 +r
 +a
 +·
 +b, a quantity known
 +to particle physicists as the Abelian, Euclidean Chern-Simons term. Here
 +a is an Abelian
 +gauge potential for the magnetic field
 +b
 +=
 +
 +×
 +a. If there exists a period of decaying
 +turbulence in the early universe, which can occur after a first-order phase transition, a
 +magnetic field with nonvanishing helicity could have relaxed to a large-scale configuration,​
 +which enjoys force-free dynamics (source currents for the magnetic fields proportional to
 +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>​
 +<--
 +
 +-->In Non-Abelian Gauge Theories#
 +
 +"​anomalous currents are sourced by gauge field configurations with nonzero Chern-Simons number. The
 +sphaleron reactions are therefore mediated by non-perturbative,​ extended gauge field configurations.
 +In the SU(3)c sector the strong sphaleron mediates a vacuum-to-vacuum transition which induces a
 +chirality-violating process among the colored particles [32]. We say that the transition is “vacuum-to-vacuum”
 +because the SU(3)c gauge field strength tensor vanishes asymptotically. [...] An anomalous reaction associated with the U(1)Y hypercharge sector is not usually included
 +in leptogenesis calculations. This may be related to the fact that the hypercharge sphaleron, unlike
 +the strong and weak sphalerons, is not a vacuum-to-vacuum transition. This is a consequence of the
 +trivial topology of the vacuum manifold of the Abelian gauge theory, and for this case the ChernSimons
 +number is proportional to the field strength tensor [38]. As a result, if the Abelian ChernSimons
 +number changes in a process, then either the initial or the final state (or both) cannot be the
 +vacuum. This exchange of energy between the particle sector and the gauge sector provides the basis
 +of magnetogenesis. As we will see, the energy exchange is minimal, perhaps justifying the neglect
 +of the hypercharge sphaleron in leptogenesis calculations. However, we should emphasize that field
 +configurations with nonzero Chern-Simons number are known to exist, e.g., hypermagnetic knots
 +[20, 21] and linked magnetic flux tubes [11]. We use the term “hypercharge sphaleron” to refer to any
 +U(1)Y field configuration that interpolates between a vacuum configuration and a configuration with
 +non-zero Chern-Simons number, which in MHD corresponds to magnetic field “helicity” (see below). Such a field configuration will source the anomaly equation for each Standard Model fermion field
 +since they all carry hypercharge. Particle production due to the anomaly can also be understood as
 +the emergence of filled states from the Dirac sea into the positive energy sector, just as in the case of
 +the Abelian chiral anomaly in 3+1 dimensions [39]." [[https://​arxiv.org/​pdf/​1309.2315.pdf|Source]]
 +
 +Ref 39 is C. Adam, B. Muratori, and C. Nash, Particle creation via relaxing hypermagnetic knots, Phys.Rev.
 +D62 (2000) 105027, [hep-th/​0006230].
 +
 +Ref 21 is [[http://​xxx.lanl.gov/​abs/​hep-th/​9911072|Creation and evolution of magnetic helicity]] by R. Jackiw, So-Young Pi 
  
 +<--
 <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>​
advanced_notions/chern-simons.1511071536.txt.gz · Last modified: 2017/12/04 08:01 (external edit)