Zilch (electromagnetism)

In physics, zilch is a conserved quantity of the electromagnetic field.

Daniel M. Lipkin observed that if he defined the quantities

${\displaystyle {\begin{aligned}Z^{0}&=\mathbf {E} \cdot \nabla \times \mathbf {E} +\mathbf {B} \cdot \nabla \times \mathbf {B} \\[4pt]\mathbf {Z} &={\frac {1}{c}}\left(\mathbf {E} \times {\frac {d}{dt}}\mathbf {E} +\mathbf {B} \times {\frac {d}{dt}}\mathbf {B} \right)\end{aligned}}}$

then the Maxwell equations imply that

${\displaystyle \partial _{0}Z^{0}+\nabla \cdot \mathbf {Z} =0}$

which implies that the total "zilch" ${\displaystyle \int Z^{0}\,d^{3}x}$ is constant (${\displaystyle \mathbf {Z} }$ is the "zilch current"). Generalising the result, Lipkin found nine related conservation laws, all unrelated to the stress-energy tensor. He named the quantity zilch because of the apparent lack of physical significance.^[1][2]

Zilch can also be expressed using the dual electromagnetic tensor ${\displaystyle {\hat {F}}^{\mu \nu }=(1/2)\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}$ as ${\displaystyle Z_{\nu \rho }^{\mu }={\hat {F}}^{\mu \lambda }F_{\lambda \nu ,\rho }-F^{\mu \lambda }{\hat {F}}_{\lambda \nu ,\rho }}$.^[3]

It was later demonstrated that zilch is part of an infinite number of zilch-like conserved quantities, a general property of free fields.^[3]

Zilch has occasionally been rediscovered. It has been called "optical chirality", since it determines the degree of chiral asymmetry in the rate of excitation of a small chiral molecule by an incident electromagnetic field.^[4] A physical interpretation of zilch was offered in 2012; zilch is to the curl or time derivative of the electromagnetic field what helicity, spin and related quantities are to the electromagnetic field itself.^[5] The conservation of zilch is not associated with duality transformations, but instead with a more subtle symmetry transformation, which has no special name. 

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.