Zilch (electromagnetism)

In physics, zilch (or zilches) is a set of ten conserved quantities of the source-free electromagnetic field, which were discovered by Daniel M. Lipkin in 1964.<ref name="Lipkin"/> The name refers to the fact that the zilches are only conserved in regions free of electric charge, and therefore have limited physical significance. One of the conserved quantities (Lipkin' <math>Z^0</math>) has an intuitive physical interpretation and is also known as optical chirality.

In particular, first, Lipkin observed that if he defined the quantities

<math>

\begin{align} Z^0 & = \mathbf{E}\cdot \nabla \times\mathbf{E} + \mathbf{B} \cdot \nabla \times\mathbf{B} \\ \mathbf{Z} & = \frac{1}{c}\left ( \mathbf{E}\times\frac{d}{dt}\mathbf{E} + \mathbf{B} \times\frac{d}{dt} \mathbf{B} \right ) \end{align} </math>

Optical chirality

The free Maxwell equations imply that

<math>\partial_0 Z^0 + \nabla \cdot \mathbf{Z} = 0</math>

The precedent equation implies that the quantity <math>\int Z^0 \, d^3x</math> is constant.

This time-independent quantity is known as the zilch, but, more precisely, it is one of the ten zilches discovered by Lipkin . Nowadays, the quantity <math>\int Z^0 \, d^3x</math> is widely known as optical chirality (up to a factor of 1/2).<ref name="Tang">Tang, Y.; Cohen, A.E. (2010). "Optical Chirality and Its Interaction with Matter". Physical Review Letters. 104 (16): 163901–1–4. Bibcode:2010PhRvL.104p3901T. doi:10.1103/PhysRevLett.104.163901. PMID 20482049.</ref>

The quantity <math>{Z}^{0}</math> is the spatial density of optical chirality, while <math>\mathbf{Z}</math> is the optical chirality flux.<ref name="Tang" /> Generalizing the aforementioned differential conservation law for <math>Z^0</math>, Lipkin found other nine conservation laws, all unrelated to the stress–energy tensor. He collectively named these ten conserved quantities the zilch (nowadays, they are also called the zilches<ref name="Smith">Smith, G; Strange, P (2018). "Lipkin's conservation law in vacuum electromagnetic fields" (PDF). Journal of Physics A: Mathematical and Theoretical. 51 (43): 435204. Bibcode:2018JPhA...51Q5204S. doi:10.1088/1751-8121/aae15f. S2CID 125795220.</ref>) because of the apparent lack of physical significance.<ref name="Lipkin">Lipkin, D.M. (1964). "Existence of a New Conservation Law in Electromagnetic Theory". Journal of Mathematical Physics. 5 (696): 696–700. Bibcode:1964JMP.....5..696L. doi:10.1063/1.1704165.</ref><ref>Wheeler, N.A. Classical electrodynamics course notes. Reed College. 1980/81. p. 241-245 </ref>

Properties of zilch tensor

The zilch are often described in terms of the zilch tensor, <math>Z^\mu_{\nu\rho}</math>. The latter can be expressed using the dual electromagnetic tensor <math>\hat{F}^{\mu\nu}=(1/2)\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}</math> as <math>Z^\mu_{\nu\rho} = \hat{F}^{\mu\lambda}F_{\lambda\nu,\rho} - F^{\mu\lambda} \hat{F}_{\lambda\nu,\rho}</math>.<ref name="kibble">Kibble, T.W.B. (1965). "Conservation Laws for Free Fields". Journal of Mathematical Physics. 6 (7): 1022–1026. Bibcode:1965JMP.....6.1022K. doi:10.1063/1.1704363.</ref>

The zilch tensor is symmetric under the exchange of its first two indices, <math>\mu</math> and <math>\nu</math>, while it is also traceless with respect to any two indices, as well as divergence-free with respect to any index.<ref name="kibble" />

The conservation law <math>\partial_{\rho}Z^{\mu \nu \rho}=0</math> means that the following ten quantities are time-independent:

<math>

\int d^{3}x Z^{\mu \nu 0}=\int d^{3}x Z^{\nu \mu 0}. </math>

These are the ten zilches (or just the zilch) discovered by Lipkin.<ref name="Lipkin" /> In fact, only nine zilches are independent.<ref name="kibble" />

The time-independent quantity <math>\int d^{3}x Z^{000}</math> is known as the 00-zilch <ref name="Lipkin" /> and is equal to the aforementioned optical chirality

<math>\int Z^0 \, d^3x</math> (<math> Z^{000}=Z^{0}</math>).

In general, the time-independent quantity <math>\int d^{3}x Z^{\mu \nu 0}</math> is known as the <math>\mu \nu</math>-zilch <ref name="Lipkin" /> (the indices <math>\mu, \nu</math> run from 0 to 3) and it is clear that there are ten such quantities (nine independent).

It was later demonstrated that Lipkin's zilch is part of an infinite number of zilch-like conserved quantities, a general property of free fields.<ref name=kibble/>

History

One of the zilches has been rediscovered. This is the zilch known as "optical chirality". This name was given by Tang and Cohen since this zilch determines the degree of chiral asymmetry in the rate of excitation of a small chiral molecule by an incident electromagnetic field.<ref name=Tang/> A further physical insight of optical chirality was offered in 2012; optical chirality is to the curl or time derivative of the electromagnetic field what helicity, spin and related quantities are to the electromagnetic field itself.<ref>Cameron, R. P.; Barnett, Stephen M.; Yao, Alison M (2012). "Optical helicity, optical spin and related quantities in electromagnetic theory". New Journal of Physics. 14 (5): 053050. Bibcode:2012NJPh...14e3050C. doi:10.1088/1367-2630/14/5/053050. S2CID 54593793.</ref> The physical interpretation of all zilches for topologically non-trivial electromagnetic fields was investigated in 2018.<ref name=Smith/>

Since the discovery of the ten zilches in 1964, there is an important open mathematical question concerning their relation with symmetries. (Recently, the full answer to this question seems to have been found <ref name=Letsios/>). The question is:

What are the symmetries of the standard Maxwell action functional :

<math>S[A_{\mu}]= -\frac{1}{4} \int d^{4}x F_{\mu \nu}F^{\mu \nu}</math> (with <math>F_{\mu \nu}=\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}</math>,

where <math>A_{\mu}</math> is the dynamical field variable) that give rise to the conservation of all zilches using Noether's theorem? Until recently, the answer to this question had been given only for the case of optical chirality by Philbin in 2013.<ref>Philbin, T.G. (2013). "Lipkin's conservation law, Noether's theorem, and the relation to optical helicity". Phys. Rev. A. 87 (4): 043843. arXiv:1303.0687. Bibcode:2013PhRvA..87d3843P. doi:10.1103/PhysRevA.87.043843.</ref> This open question was also emphasized by Aghapour, Andersson and Rosquist in 2020,<ref name="Aghapour">Aghapour, Sajad; Andersson, Lars; Rosquist, Kjell (2020). "The zilch electromagnetic conservation law revisited". Journal of Mathematical Physics. 61 (12): 122902. arXiv:1904.08639. Bibcode:2020JMP....61l2902A. doi:10.1063/1.5126487.</ref> while these authors found the symmetries of the duality-symmetric Maxwell action underlying the conservation of all zilches. (Aghapour, Andersson and Rosquist did not find the symmetries of the standard Maxwell action, but they speculated that such symmetries should exist <ref name="Aghapour" />). There are also earlier works studying the conservation of zilch in the context of duality-symmetric electromagnetism,<ref>Cameron, R.P.; Barnett, S.M. (2012). "Electric–magnetic symmetry and Noether's theorem". New Journal of Physics. 14 (12): 123019. Bibcode:2012NJPh...14l3019C. doi:10.1088/1367-2630/14/12/123019.</ref> but the variational character of the corresponding symmetries was not established.

The full answer to the aforementioned question seems to have been given for the first time in 2022,<ref name=Letsios>A. Letsios, V. (2022). "Continuity equations for all Lipkin's zilches from symmetries of the standard electromagnetic action and Noether's theorem". arXiv:2211.06798v1. {{cite journal}}: Cite journal requires |journal= (help)</ref> where the symmetries of the standard Maxwell action underlying the conservation of all zilches were found. According to this work, there is a hidden invariance algebra of free Maxwell equations in potential form that is related to the conservation of all zilches.

See also

• Conservation law
• Noether's theorem