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Chern-Simons Term

Why is it interesting?

Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

Student

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.

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".

This is motivated by the observation that the equations of vacuum electrostatics

$$ \Delta \cdot \vec E =0, \quad \Delta \times \vec E =0,$$ are exactly the same as the equations for the flow of an incompressible frictionless fluid with no visosity and no "vorticity", or curl:

$$ \Delta \cdot \vec v =0, \quad \Delta \times \vec v =0,$$

where $\vec v$ is the velocity field. In addition, even in situations where the vorticity $\Delta \times \vec v$ is not zero, Helmholtz showed in 1858 that the vortex lines, which means the lines of $\Delta \times \vec v$, move in the direction of $\vec v$ as if they had an existence of their own. (Take note that these observation was also what led Kelvin to propose his theory of vortex atoms). (Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 292)

This idea is made precise by the notion "kinetic helicity".

In fluid dynamics, the kinetic helicity is a measure of the degree of knottedness and/or linkage of the vortex lines of the flow.

Fluid helicity is one of the most important conserved quantities in ideal fluid flows, being an invariant of the Euler equations, and a robust quantity of the dissipative Navier-Stokes equations [13]. In ideal conditions its topological interpretation in terms of Gauss linking number was provided by Moffat [23] and extended by Moffat & Ricca [24]. In the context of vortex dynamics (kinetic) helicity is defined by

$$ H \equiv \int_\Omega u \cdot \omega d^3x, $$ where $u$ is the velocity field, $\omega = \Delta \times u$, is the vorticity, defined on $\Omega$, and $x$ is the position vector.

page 96 in How Nature Works: Complexity in Interdisciplinary Research and Applications by Ivan Zelinka

[H]elicity, is an integral over the fluid domain that expresses the correlation between velocity and vorticity, and an invariant of the classical Euler equations of ideal (inviscid) fluid flow. […] It is precisely because vortex lines are frozen in the fluid, thus conserving their topology, that helicity is conserved too. As Kelvin recognized, if two vortex tubes are linked, then that linkage survives in an ideal fluid for all time; if a vortex tube is knotted, then that knot survives in the same way for all time. Helicity is the integral manifestation of this invariance: For two linked tubes, it is proportional to the product of the two circulations (each conserved by Kelvin’s circulation

theorem), whereas for a single knotted tube, or a deformed unknotted tube, it is proportional to its “writhe plus twist,” as encountered in differential geometry—a property of knotted ribbons that is invariant under continuous deformation (4, 5). Thus, for example, if an untwisted ribbon that goes twice round a circle before closing on itself is unfolded and untwisted back to circular form, then its writhe will decrease continuously from 1 to 0, and its twist will increase continuously from 0 to 1, by way of compensation. Such conversion of writhe to twist is familiar to anyone seeking to straighten out a coiled garden hose.

In Electrodynamics

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 Helicity and Electomagnetic Field Topology by Gerald E. Marsh

Researcher

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. Shiing-shen Cern

  • For a nice discussion, see Chapter 4 in Gauge Fields, Knots and Gravity by Baez and Muniain

Examples

Example1
Example2:

FAQ

History

advanced_notions/chern-simons.1511072108.txt.gz · Last modified: 2017/12/04 08:01 (external edit)