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The optical chirality of the electromagnetic field is a conserved quantity, analogous to the energy density, linear momentum density, and angular momentum density, which describes how chiral the EM field is at each point, and it is given by $$ C=\frac{\varepsilon_0}{2}\mathbf E\cdot\left(\nabla\times\mathbf E\right)+\frac{1}{2\mu_0}\mathbf B\cdot\left(\nabla\times\mathbf B\right). $$ It is a time-even pseudoscalar, which puts it in a nice little niche of its own. It was originally discovered by Lipkin [J. Math. Phys. 5, 696 (1964)], who called it "zilch" for lack of a better name, and it was rediscovered and put to good use recently by Tang and Cohen [Phys. Rev. Lett. 104, 163901 (2010)], who showed that in the linear regime and the magnetic dipole approximation, all chiral interactions of a molecule with the electromagnetic field produce effects proportional to the optical chirality $C$ times a molecular chirality.

Lipkin himself inscribed $C$ as a component of a bigger tensor, $Z^{\lambda\mu}_{\ \ \ \ \nu}$, and Kibble extended it shortly thereafter [J. Math. Phys. 6, 1022 (1965)] into an infinite series of related conservation laws. That is, the optical chirality $C$ is really the first term in a hierarchy of measures of chirality, somewhat reminiscent of a multipole expansion.

This hierarchy was systematized fairly recently by Anco and Pohjanpelto [Acta Appl. Math. 69, 285 (2001)]; they provide a completeness result so it seems that any relevant conservation law must be contained within their scheme. Anco and Pohjanpelto also point to the previous work by Fushchich and Nikitin [Symmetries of Equations of Quantum Mechanics (Allerton Press, New York, 1994), pp 288-296; eprint] who provide a complete classification, I believe, in relatively more readable terms.


I am studying a field configuration which I believe to be chiral, but it turns out that the optical chirality $C$ vanishes identically for it. This is, by itself, not fatal: the electric field of a static, chiral configuration of charges is also in general chiral, but for it $C$ vanishes as well. As Tang and Cohen point out, in these cases there is (or ought to be) a hierarchy of higher-order time-even pseudoscalar optical chiralities, which can in general pick out chiral components given by higher-order derivatives of the electric and magnetic fields.

I am looking for the first few terms in this hierarchy. That is, I am looking for time-even pseudoscalar conserved quantities, involving higher order derivatives, in the first few rungs of this ladder. Moreover, I am looking for explicit expressions in terms of the electric and magnetic field vectors.

I must confess, however, that I find Anco and Pohjanpelto's work to be pretty heavy going, and particularly I am finding the Killing vector language pretty deadly. That is: I roughly understand what's going on, but I am having trouble implementing the relations they describe to obtain concrete examples. (Which means, of course, that I don't actually understand what's going on all that well.) I find the classification by Fushchich and Nikitin more accessible but I am still struggling to find expressions I am confident about, and I'm pretty lost as to how one would look for the first nontrivial time-even pseudoscalar beyond the terms they explicitly enumerate.

I am therefore hoping someone more fluent with this sort of language will have an easier time decoding these things. Any help will be deeply appreciated.

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  • $\begingroup$ Related question on this hierarchy: Optical chirality and its possible hierarchy of generalizations. $\endgroup$ – Emilio Pisanty Dec 8 '15 at 17:39
  • $\begingroup$ Please, see the following recent review by T.G. Philibin where the conserved quantities generalizing the zilch are given in equations (21-22). These quantities correspond to the Noether symmetries given in equation (19). arxiv.org/abs/1303.0687 $\endgroup$ – David Bar Moshe Dec 15 '15 at 16:13
  • $\begingroup$ @David Yeah, we've seen that one, but I hadn't made it that far in - you're right, it does include the quantities of interest. See also this question and this chat for where we are at the moment. It's a bit curious that Philbin doesn't discuss the rest of the components of the zilch tensor but for this question indeed it's only the 000...0 components that matter. We found this paper by Cameron and Barnett most useful. $\endgroup$ – Emilio Pisanty Dec 15 '15 at 16:23
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Over the days I've had an extensive discussion with Emilio and ACuriousMind in this chatroom. There, we discussed a bunch of issues related to this question and this related question by Emilio. In my recent answer to the latter, I laid out much of the fundamental ideas that I will be using in this answer. Thus, I refer back to it for most of the background. I furthermore note that this answer is not the only way to approach the question: A related yet distinct approach was outlined here, as pointed out in the comments on the question.

The basic setup

The zilch tensor is related to duality rotations (which are just rotations on the vector $(F_{\mu\nu},*F_{\mu\nu})$ in electromagnetism, which are a symmetry of Maxwell's equations:

$$ \partial_\mu F^{\mu\nu}=0 \qquad \qquad \partial_\mu * F^{\mu\nu}=0$$

but regrettably not of the standard action for electrodynamics:

$$ S_\text{Maxwell}=-\frac{1}{4}\int\mathrm d^4x F^{\mu\nu}F_{\mu\nu} $$

One nice way of getting rid of this problem is introduced in the answer I linked earlier, and can be summarized as follows: Treat $F_{\mu\nu}$ and its dual as independent variables in the action (for notational clarity, we introduce $G^{\mu\nu}\equiv *F^{\mu\nu}$, but keep in the back of your mind the constraint, which we can impose on all equations we derive, that they are each other's duals. This leads one to consider a modified action

$$S_{\text{duality}}=-\frac{1}{8}\int \mathrm d^4x F_{\mu\nu}F^{\mu\nu}+G_{\mu\nu}G^{\mu\nu} $$

Along with $G^{\mu\nu}$ (which, again, is really just the dual field strength tensor in disguise), we introduce a dual (electric) four-potential $C^\mu$ such that $G_{\mu\nu}=\partial_{[\mu}C_{\nu]}$.

Symmetries

The new action and associated Lagrangian $\mathcal L$ are not only nice because they are manifestly invariant under duality rotations (since $F$ and $G$ appear completely symmetrically). In fact, it generally reflects the symmetries of Maxwell's equations better than the standard Maxwell Lagrangian. To clarify this remark, we note the following:

Consider a general transformation of the fundamental fields underlying the Lagrangian:

$$ A^\mu\mapsto A'^\mu=A^\mu+\delta A^\mu \qquad \qquad C^\mu\mapsto C'^\mu=C^\mu+\delta C^\mu $$

(where we must always ensure that the variation does not violate the constraint $G_{\mu\nu}\equiv *F_{\mu\nu}$). Inserting this into the Lagrangian and using $F_{\mu\nu}\delta F^{\mu\nu}=-*F_{\mu\nu}\delta *F^{\mu\nu}$, we find:

$$\delta \mathcal L\propto \delta G_{\mu\nu}(G'^{\mu\nu}-*F'^{\mu\nu}) $$

which precisely vanishes whenever we remain on the constraint surface. Hence, a transformation that does not violate the constraint induces a symmetry of the action and hence of Maxwell's equations, and also gives rise to a conserved current through Noether's theorem. As we saw, this includes transformations that do not correspond to symmetries of the Maxwell action. To derive the general expression for a Noether current, we perform the standard derivation (using derivatives of $\mathcal L$) and obtain:

$$\tag{$\star$} \delta \mathcal L = \frac{1}{2}\partial_\nu (F^{\mu\nu}\delta A_\mu+ G^{\mu\nu}\delta C_\mu) $$

Infinite hierarchies

Now, we are in a position to finally determine the infinite hierarchy of symmetries of Maxwell's equations that includes the zilch tensor $Z^\mu_{\ \ \nu\rho}$ as a special case. We start from the basic duality rotation:

$$ \delta A_\mu = -\theta C_\mu \qquad \qquad \delta C_\mu = \theta A_\mu $$

Using our general formula ($\star$), we find:

$$\delta \mathcal L = \frac{\theta}{2}\partial_\nu(-F^{\mu\nu}C_\mu + G^{\mu\nu}A_\mu)\equiv 0 $$

and hence obtain a conserved current

$$ \kappa^\nu=G^{\mu\nu}A_\mu - F^{\mu\nu}C_\mu $$

Now, the crucial insight is that we can easily generalize this procedure to generate new conserved currents with more indices. On the level of field strengths, the duality rotation induces a variation

$$ \delta F_{\mu\nu}=-\theta G_{\mu\nu} \qquad\qquad \delta G_{\mu\nu}=\theta F_{\mu\nu} $$

Now, it is not that far-fetched to look for a related current which induces a similar variation

$$ \delta F_{\mu\nu}=-\zeta^\alpha \partial_\alpha G_{\mu\nu} \qquad \qquad \delta G_{\mu\nu}= \zeta^\alpha\partial_\alpha F_{\mu\nu} $$

instead. It is easy to check that this variation is induced by the following variation of the potentials:

$$\delta A_\mu=-\zeta^\alpha \partial_\alpha C_\mu \qquad \qquad \delta C_\mu = \zeta^\alpha \partial_\alpha A_\mu $$

We note that, once again, the transformed $F'$ and $G'$ satisfy the equations of motion (as is easily checked by using the equations of motion for the old field strengths and the fact that partial derivatives commute). We can see a pattern beginning to emerge here, which easily generalizes to yield the infinite hierarchy of symmetries that was alluded to in the question! The variation of $F_{\mu\nu}$ (and $G_{\mu\nu}$) is of the same form as the variation of the corresponding potential, after replacing $A$ ($C$) by $F$ ($G$). Thus, we can consider a variation of the more general form

$$\delta A_\mu=-\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_n} C_\mu \qquad \qquad \delta C_\mu=\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_n} A_\mu$$

which induces a variation on the level of field strengths of the form:

$$\delta F_{\mu\nu}=-\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_n} G_{\mu\nu} \qquad \qquad \delta G_{\mu\nu}=\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_n} F_{\mu\nu}$$ which will clearly give rise to an infinite hierarchy of conserved currents. We note, as a side remark, that $\xi$ will automatically be symmetric in all of its indices (because of the derivatives that it is contracted with).

However, there is a problem with this idea because the current, given by ($\star$), will not be gauge invariant (one can see that $A$ and $C$ will appear explicitly, and this is typically a bad sign!). Fortunately, this is not the only way to induce the above variation of the field strength tensors. One can alternatively consider variations of the following form:

$$ \delta A_\mu=-\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} G_{\alpha_n \mu} \qquad \qquad \delta C_\mu=\xi^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} F_{\alpha_n \mu}$$

It is clear that the Noether current corresponding to this variation will, in fact, only feature $F$ and $G$ and hence it will be gauge-invariant. A quick check shows that, indeed, the variation of the field strength is identical:

$$\delta F_{\mu\nu}=-\xi^{\alpha_1\cdots \alpha_n} \partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}(\partial_\mu G_{\alpha_n \nu}-\partial_\nu G_{\alpha_n\mu}) = -\xi^{\alpha_1\cdots \alpha_n} \partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}\partial_{\alpha_n}G_{\mu\nu} $$

where we expanded $G_{\mu\nu}$ in terms of $C_\mu$ to perform the last step. Of course, the variation of $G_{\mu\nu}$ is similarly checked. However, that is not all!

As explained in the paper by Cameron & Barnett linked in my afore-mentioned answer (on which this answer is based!), one can obtain another infinite sequence of symmetries, by considering variations

$$\delta A_\mu=\kappa^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} F_{\alpha_n \mu} \qquad \qquad \delta C_\mu=\kappa^{\alpha_1\cdots \alpha_n}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}} G_{\alpha_n \mu} $$

which again generate symmetries of the equations of motion, as can be seen from

$$\delta F_{\mu\nu}=\kappa^{\alpha_1\cdots \alpha_n} \partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}(\partial_\mu F_{\alpha_n \nu}-\partial_\nu F_{\alpha_n\mu}) = \kappa^{\alpha_1\cdots \alpha_n} \partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}\partial_{\alpha_n}F_{\mu\nu} $$

Conserved quantities

Now, we can proceed with the investigation of our gauge-invariant currents. It is easily seen from ($\star$) that our first infinite chain of symmetries gives rise to

$$\delta \mathcal L=\frac{\xi^{\alpha_1\cdots \alpha_n}}{2}\partial_\nu (G^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}F_{\alpha_n \mu} -F^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}G_{\alpha_n \mu}) $$

whence we conclude that we have the following infinite sequence of conserved currents:

$$ Z^\nu_{\ \ \alpha_1\cdots\alpha_n}:= G^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}F_{\alpha_n \mu} -F^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}G_{\alpha_n \mu}$$

Furthermore, our second sequence yields:

$$\delta \mathcal L=\frac{\kappa^{\alpha_1\cdots \alpha_n}}{2}\partial_\nu (F^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}F_{\alpha_n \mu} +G^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}G_{\alpha_n \mu}) $$

which corresponds to the "ladder" of conserved quantities

$$\tau^\nu_{\ \ \alpha_1\cdots \alpha_n}:= F^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}F_{\alpha_n \mu} +G^{\mu\nu}\partial_{\alpha_1}\cdots \partial_{\alpha_{n-1}}G_{\alpha_n \mu} $$

Without derivation, we note that the ladders of $Z$ and $\tau$-tensors are dual to each other in some sense: We define conserved charges through the usual method, i.e.

$$ Q^Z_{\alpha_1\cdots \alpha_n}:= \int \mathrm d^3 x Z^0_{\alpha_1\cdots\alpha_n} $$

and similarly for $\tau$-tensors. For the case $n=1$ it turns out that the corresponding $Z$-tensor defines a conserved charge that vanishes identically, while the $\tau$-charges are nontrivial. However, for the case $n=2$ the situation is reversed: The charges defined by the $\tau$-tensor are trivial, but the "partner" $Z$-tensor is of considerably more interest. It is given by

$$ Z^\nu_{\alpha\beta}=*F^{\mu\nu}\partial_\alpha F_{\beta\mu} - F^{\mu\nu}\partial_\alpha *F_{\beta\mu}$$

which we can immediately identify with the zilch tensor mentioned in the question. The corresponding conserved charges are non-trivial, with the best-known example being:

\begin{align*} Q^Z_{00}&=\int \mathrm d^3 x Z^0_{\ \ 00} =\int \mathrm d^3 x *F^{i0}\partial_0 F_{0i} - F^{i0}\partial_0 *F_{0i}\\ &=\int \mathrm d^3x \vec E\cdot (\vec \nabla\times \vec E) +\vec B \cdot (\vec \nabla\times \vec B)=\int \mathrm d^3x C \end{align*}

where we used Maxwell's equations in the vacuum to convert time derivatives into curls. the integral of the optical chirality $C$ (note that I have set $c=\epsilon_0=1$ throughout). The general pattern is the following: For even (odd) $n$, the $Z$-tensor ($\tau$-tensor) defines a non-trivial charge while its partner defines a trivial one. All the non-trivial charges thus obtained are argued, in the paper by Cameron & Barnett, to admit an interpretation in terms of "the difference [(for $Z$-charges)] and sum [(for $\tau$-charges)] of photon numbers of opposite circular polarization". This is yet another piece of evidence that the conserved $Z$-tensors yield information about the chirality of the system under consideration.

Given all of this, it is not too hard to find the next possible time-even, pseudo-scalar conserved chiral quantities from the ladder of $Z$-charges: We consider the cases $n=4,6$, with corresponding non-trivial charge densities:

\begin{align*} \mathcal Q^Z_{\mu\nu\rho\sigma} &=Z^0_{\mu\nu\rho\sigma} = G^{0i}\partial_\mu \partial_\nu\partial_\rho F_{\sigma i} -F^{0i}\partial_\mu\partial_\nu\partial_\rho G_{\sigma i}\\ \mathcal Q^Z_{\mu\nu\rho\sigma\kappa\lambda} &=Z^0_{\mu\nu\rho\sigma\kappa\lambda} = G^{0i}\partial_\mu \partial_\nu\partial_\rho\partial_\kappa\partial_\lambda F_{\sigma i} -F^{0i}\partial_\mu\partial_\nu\partial_\rho \partial_\kappa\partial_\lambda G_{\sigma i} \end{align*}

Luckily, we do not have to consider many components of these tensors: We are only interested in pseudoscalar quantities, and hence we must impose that all free indices equal $0$ (else we will get a quantity that transforms like a vector or tensor under $O(3)$). We furthemore note that we do not lose anything by only considering $Z$-charges here, since the non-trivial $\tau$-charges (which occur when $n=3,5$) will have an odd number of time-derivatives: They cannot be time-even!

Hence, the first two local quantities of interest (after the optical chirality $C$), arising from the constructed hierarchy of conserved chiral quantities, are:

\begin{align*} C^{(2)}&:= \mathcal Q^Z_{0000} = G^{0i}\partial_0^3 F_{0 i} -F^{0i}\partial_0^3 G_{0 i}\\ C^{(3)}&:=\mathcal Q^Z_{000000} = G^{0i}\partial_0^5 F_{0 i} -F^{0i}\partial_0^5 G_{0i} \end{align*}

which can be written in terms of the electric and magnetic fields as:

\begin{align*} C^{(i+1)}&=\vec B \cdot (\partial_0^2)^i(\vec\nabla \times\vec B)+\vec E \cdot (\partial_0^2)^i(\vec\nabla\times\vec E) \end{align*}

This formula is easily seen to generalize to all integers $i\geq 0$.

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