Electromagnetic helicity

This definition is related to the difference between left- and right-handed photons because B(3) switches sign between left and right circularly polarized electromagnetic radiation. Therefore, H and B(3) constitute electromagnetic helicities of a knot, and there is also a link between B(3) and the Sachs theory [1], as shown in the review [6] by Evans, linking 0(3) electrodynamics and the Sachs theory. 

B. The Case of Maxwell s Theory in Vacuum Electromagnetic Helicity... 

The electromagnetic helicity has also been studied by Evans [54—57], especially its consequences for his new non-Abelian SO(3) gauge version of QED (quantum electrodynamics). 

Given any Maxwell field in vacuum, we define the current density of electromagnetic helicity Jf x as one-half times the sum of the current densities of electric and magnetic helicities (41) and (42) ... 

By construction, and taking into account Eqs. (43), the density of electromagnetic helicity is a conserved current for any Maxwell field in vacuum (with the above indicated behavior at infinity) ... 

Furthermore, it was shown in Section II.C that the semisum of the two helicities = (/im + he) = na, which we call the electromagnetic helicity, is a constant of the motion for any standard electromagnetic field in empty space ... 

In the previous section, it was shown that the constant a must be equal to [hc in order to obtain the right quantization of the electromagnetic helicity. This implies that the topological model predicts that the fundamental charge, either electric or magnetic, has the value... 

One of these topological constants of the motion is the electromagnetic helicity, defined as the semisum of the magnetic and electric helicities, which is equal to the linking number of the force lines... 

In electromagnetic theory, we replace W 1 by GM the relativistic helicity of the field. Therefore, Eq. (770) forms a fundamental Lie algebra of classical electrodynamics within the Poincare group. From first principles of the Lie algebra of the Poincare group, the field B is nonzero. 

Helicity is defined as a qualitative measurement of how a topological configuration is linked, knotted, or twisted. If A is the vector potential of electromagnetism, the quantity then defines magnetic helicity. In order to obtain a nonnull value for K, the condition A (V A) /- 0 must be verified in the volume... 

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

A very important property is that the magnetic and electric lines of an electromagnetic knot are the level curves of the scalar fields 4>(r, t) and 0(r, f), respectively. Another is that the magnetic and the electric helicities are topological constants of the motion, equal to the common Hopf index of the corresponding pair of dual maps constant with dimensions of action times velocity. 

We will write h(X) or simply h if there is no risk of confusion. The helicity (24) is especially useful in two physical contexts (1) in fluid dynamics, where Y is the flow velocity v(r, t), X is the vorticity w = V x v, and h w, D) is called vortex helicity, and (2) in plasma physics, or in general in electromagnetism, under the form of magnetic helicity, dehned as... 

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