Electric–magnetic symmetry and Noether's theorem

The invariance in form of Maxwell's equations (3.1)–(3.4) under a duality rotation (3.5) is the embodiment of electric–magnetic symmetry and was, perhaps, the first symmetry observed in electromagnetic theory [1, 2, 13]. We begin our investigation here.

From (3.10), the infinitesimal transformation

$$\begin{eqnarray} A^\alpha \rightarrow A^{\alpha'}&=&A^\alpha+\theta C^\alpha, \nonumber \\\\ C^\alpha \rightarrow C^{\alpha'}&=&C^\alpha-\theta A^\alpha, \nonumber \end{eqnarray} \tag{ 6.1 }$$

where θ is an infinitesimal Lorentz pseudoscalar angle, yields, in all gauges:

$$\begin{eqnarray} F^{\alpha\beta}\rightarrow F^{\alpha\beta'}&=&F^{\alpha\beta}+\theta G^{\alpha\beta}, \nonumber \\\\ G^{\alpha\beta}\rightarrow G^{\alpha\beta'}&=&G^{\alpha\beta}-\theta F^{\alpha\beta},\nonumber \end{eqnarray} \tag{ 6.2 }$$

which is the infinitesimal form of a duality rotation (3.5). From (6.1) and our form (4.3) of Noether's theorem, with $\Delta \mathcal {L}=0$, we obtain

$$\begin{eqnarray} &&\theta \partial_\gamma\left[\textstyle\frac{1}{2}\left(A_\alpha G^{\gamma\alpha}-C_\alpha F^{\gamma\alpha}\right)\right]=0. \end{eqnarray} \tag{ 6.3 }$$

As θ ≠ 0, the conservation law

$$\begin{equation} \partial_\gamma h^\gamma =0, \quad h^\gamma= \textstyle\frac{1}{2}\left(A_\alpha G^{\gamma\alpha}-C_\alpha F^{\gamma\alpha}\right) \end{equation} \tag{ 6.4 }$$

follows immediately from (6.3). Although the conservation law (6.4) holds in all gauges, the density, h⁰, of the conserved quantity is itself not gauge invariant. When we integrate h⁰ over all space, however, we obtain the quantity

$$\begin{eqnarray} &&\mathcal{H}=\int\!\!\int\!\!\int h^0 \, {\mathrm{d}}^3\mathbf{r}=\int\!\!\int\!\!\int \frac{1}{2}\left(\mathbf{A}\cdot\mathbf{B}-\mathbf{C}\cdot\mathbf{E}\right) \, {\mathrm{d}} ^3\mathbf{r}, \end{eqnarray} \tag{ 6.5 }$$

which is both conserved (${\mathrm {d}} \mathcal {H}/{\mathrm {d}} t=0$, assuming that the field falls off suitably as |r| → ∞) and gauge invariant. The latter property follows from the fact that the integral over all space of the dot product of a transverse and a longitudinal field vanishes [11]. Thus, it is only the transverse, gauge invariant [11] pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials that contribute to (6.5) such that

$$\begin{eqnarray} &&\mathcal{H}=\int\!\!\int\!\!\int \frac{1}{2}\left(\mathbf{A}^\bot\cdot\mathbf{B}-\mathbf{C}^\bot\cdot\mathbf{E}\right) \, {\mathrm{d}} ^3\mathbf{r}. \end{eqnarray} \tag{ 6.6 }$$

We recognize (6.6) as the optical helicity [22, 23]. It is equivalent to the 'screw-action' introduced by Candlin [24]. The association of duality rotations (3.5) with the optical helicity (6.6) was recognized by Calkin [25] (see also [7–9, 22, 23, 26–33]). The optical helicity (6.6) is a Lorentz pseudoscalar with the dimensions of an angular momentum and is, as its name suggests, the total photon helicity of the field [22, 23]. For a plane wave, a duality rotation (3.5) simply rotates the electric and magnetic field vectors about the direction of propagation [22, 23]. It seems natural that such a symmetry transformation should be associated with the conservation of the optical helicity (6.6).

Interestingly, this is not the end of our story. Consider the infinitesimal transformation

$$\begin{eqnarray} F^{\alpha\beta}\rightarrow F^{\alpha\beta '}&=&F^{\alpha\beta}+\phi F^{\alpha\beta}, \nonumber \\\\ G^{\alpha\beta}\rightarrow G^{\alpha\beta' }&=&G^{\alpha\beta}+\phi G^{\alpha\beta},\nonumber \end{eqnarray} \tag{ 6.7 }$$

where we interpret the infinitesimal Lorentz scalar ϕ as an angle. Just as (6.2) is, for a plane wave, an infinitesimal rotation of the electric and magnetic field vectors about the direction of propagation through an angle θ, (6.7) is an infinitesimal boost of the electric and magnetic field vectors in the direction of propagation with rapidity ϕ, leaving the spacetime distribution of the wave unchanged (this interpretation also holds for the finite form of (6.7)). Thus, (6.2) and (6.7) are 'partners'. Noether's theorem leads us to associate (6.7) with the conservation law

$$\begin{equation} \partial_\gamma d^\gamma =0, \quad d^\gamma= \textstyle\frac{1}{2}\left(A_\alpha F^{\alpha\gamma}+C_\alpha G^{\alpha\gamma}\right), \end{equation} \tag{ 6.8 }$$

which holds in all gauges. The density, d⁰, of the conserved quantity has the dimensions of a boost angular momentum per unit volume, in line with our interpretation of (6.7). Although the density, d⁰, is gauge dependent, we can obtain a gauge invariant quantity by integrating over all space:

$$\begin{eqnarray} \mathcal{D}=\int\!\!\int\!\!\int d^0 \, {\mathrm{d}}^3\mathbf{r}&=&-\int\!\!\int\!\!\int \frac{1}{2}\left(\mathbf{A}^\bot\cdot\mathbf{E}+\mathbf{C}^\bot\cdot\mathbf{B}\right) \, {\mathrm{d}}^3\mathbf{r}. \end{eqnarray} \tag{ 6.9 }$$

This quantity is, however, trivial, as

$$\begin{eqnarray} \mathcal{D}&=&-\int\!\!\int\!\!\int \frac{1}{2}\left(-\mathbf{A}^\bot\cdot\boldsymbol{\nabla}\times\mathbf{C}^\bot+\mathbf{C}^\bot\cdot\boldsymbol{\nabla}\times\mathbf{A}^\bot\right) \, {\mathrm{d}}^3\mathbf{r} \nonumber \\ &=&-\int\!\!\int\!\!\int \frac{1}{2}\left[-\mathbf{A}^\bot\cdot\boldsymbol{\nabla}\times\mathbf{C}^\bot+\mathbf{A}^\bot\cdot\boldsymbol{\nabla}\times\mathbf{C}^\bot+\boldsymbol{\nabla}\cdot\left(\mathbf{A}^\bot\times\mathbf{C}^\bot\right)\right] \, {\mathrm{d}}^3\mathbf{r} \nonumber \\ &=&0. \end{eqnarray} \tag{ 6.10 }$$

In going from the second line to the third line, we have used Gauss's theorem to convert the final term into a surface integral which vanishes, assuming that the transverse pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials fall off suitably as |r| → ∞. Related observations have been made by Fushchich and Nikitin [34, 35], Drummond [7, 8] and Anco and The [9].

The idea that symmetries exist in pairs, only one member of which is associated with a non-trivial conserved quantity, appears to hold with generality, as we shall see in what follows.

Bessel-Hagen [36] was the first to apply Noether's theorem in electromagnetic theory. Equipped with the standard Lagrangian density (3.9), he obtained the conservation laws associated with the 15 parameter group of conformal symmetries. We now consider these symmetries and, as a check on our present approach, confirm that our electric–magnetic Lagrangian density (3.11) leads us to the same conservation laws obtained by Bessel-Hagen.

An infinitesimal conformal transformation of the field is invoked, in all gauges, by transforming the potentials as

$$\begin{eqnarray} A_\alpha \rightarrow A_\alpha'&=&A_\alpha-A_\beta\partial_\alpha X^\beta-X^\beta \partial_\beta A_\alpha, \nonumber \\\\ C_\alpha \rightarrow C_\alpha'&=&A_\alpha-C_\beta\partial_\alpha X^\beta-X^\beta \partial_\beta C_\alpha, \nonumber \end{eqnarray} \tag{ 6.11 }$$

where X^α is [37, 38]

$$\begin{equation} X^\alpha=t^\alpha+w^\alpha_{\ \, \beta} x^\beta+\vartheta x^\alpha +\left(2x^\alpha x^\beta-x_\mu x^\mu\eta^{\alpha\beta}\right) a_\beta, \end{equation} \tag{ 6.12 }$$

with x^α = (t,r) the position four-vector. The infinitesimal components of t^α and w^αβ = −w^βα define a translation in spacetime and a rotation and boost, respectively. Such transformations constitute the Poincaré group [12, 39]. In addition, ϑ and the components of a^α, also infinitesimal, define a scale transformation and a so-called special conformal transformation, respectively. The physical significance of such transformations, due to Bateman [40, 41] and Cunningham [42], is, it seems, not entirely understood (see e.g. [39, 43–45]). Their independence from the transformations of the Poincaré group has been questioned by Fushchich and Nikitin [34]. Note that the transformation (6.11) is an active one, like all transformations in the present paper. In this active form, t^α, w^αβ = −w^βα, ϑ and a^α are tensors, with suitable dimensions, so that X^α is itself a four-vector [46]. Plybon has claimed that the 15 conformal symmetry transformations are the only ones that assume the form of (6.11), referred to by him as being 'geometric' [46].

From (6.11) and our form of Noether's theorem (4.3), we deduce, following some simple manipulations, that

$$\begin{equation} \partial_\gamma \left[\textstyle\frac{1}{2} X^\mu \left(F^{\alpha\gamma}F_{\alpha\mu}+G^{\alpha\gamma}G_{\alpha\mu}\right)\right]=0. \end{equation}\noindent \tag{ 6.13 }$$

As t^α,w^αβ = −w^βα,ϑ and a^α are independent, we obtain from (6.12) and (6.13), a total of 15 conservation laws:

$$\begin{eqnarray} \partial_\gamma T^{\alpha\gamma}&=&0, \quad T^{\alpha\gamma}=\textstyle\frac{1}{2}\left(F_{\ \, \mu}^\alpha F^{\mu\gamma}+G_{\ \, \mu}^\alpha G^{\mu\gamma}\right), \end{eqnarray}\noindent \tag{ 6.14 }$$

$$\begin{eqnarray} \partial_\gamma M^{\alpha\beta\gamma}&=&0, \quad M^{\alpha\beta\gamma}= x^\alpha T^{\beta\gamma}-x^\beta T^{\alpha\gamma}, \end{eqnarray}\noindent \tag{ 6.15 }$$

$$\begin{eqnarray} \partial_\gamma D^\gamma&=&0, \quad D^\gamma=x_\alpha T^{\alpha\gamma}, \end{eqnarray}\noindent \tag{ 6.16 }$$

$$\begin{eqnarray} \partial_\gamma I^{\alpha\gamma}&=&0, \quad I^{\alpha\gamma}=2 x^\alpha x^\mu T^{\gamma}_{\mu}-x_\mu x^\mu T^{\alpha\gamma}, \end{eqnarray}\noindent \tag{ 6.17 }$$

corresponding to spacetime translations, rotations and boosts, scale transformations and special conformal transformations, respectively [47]. Our results (6.14)–(6.17) are essentially the ones advocated by Bessel-Hagen [36], as desired.

As is well-known, (6.14) expresses the conservation of energy and linear momentum, whilst (6.15) expresses the conservation of rotation and boost angular momenta, to which we return in section 7. Of the remaining conservation laws, (6.16) and (6.17), Bessel-Hagen commented that 'the future will show if they have any physical significance' [36, 48]. It appears that their physical significance is still not understood, a point noted by Rohrlich [39] and Fulton et al [43], by Plybon [38] and, more recently, by Ibragimov [48]. The independence of the conservation law (6.17) from the others has been questioned by Plybon [38].

In passing, we remark that the conserved quantity with density D⁰ has the dimensions of a boost angular momentum. For a single plane wave, the conservation law (6.16) can be interpreted as a statement of the familiar dispersion relation, ω = |k|. This relation connects a time (the period of the wave) with a (wave)length which is somewhat appropriate given that the invariance of Maxwell's equations (3.1)–(3.4) under a scale transformation (which, importantly, invokes a dilation or contraction of temporal and spatial properties of the field in equal measure) is itself a reflection of the fact that all periods and wavelengths of light are equally welcome, provided, of course, that they are related such that ω = |k|.

An infinitesimal conformal symmetry transformation, as invoked by (6.11), possesses a (non-geometric) partner that is invoked, in all gauges, by transforming the potentials as

$$\begin{eqnarray} A_\alpha \rightarrow A_\alpha'&=&A_\alpha-C_\beta\partial_\alpha Y^\beta-Y^\beta \partial_\beta C_\alpha, \nonumber \\\\ C_\alpha \rightarrow C_\alpha'&=&C_\alpha+A_\beta\partial_\alpha Y^\beta+Y^\beta \partial_\beta A_\alpha.\nonumber \end{eqnarray}\noindent \tag{ 6.18 }$$

The four-pseudovector Y^α is

$$\begin{equation} Y^\alpha=g^\alpha+q^\alpha_{\beta} x^\beta+\psi x^\alpha +\left(2x^\alpha x^\beta-x_\mu x^\mu\eta^{\alpha\beta}\right) b_\beta, \end{equation}\noindent \tag{ 6.19 }$$

where the components of the pseudotensors g^α, q^αβ = −q^βα, ψ and b^α are infinitesimal and have suitable dimensions. This symmetry has also been recognized by Krivskii and Simulik [49, 50] and Anco and The [9]. Through Noether's theorem, we find that (6.18) is associated with trivial conserved quantities, as noted by Anco and The [9].

In 1964, Lipkin [51] introduced a rank-three pseudotensor which describes a set of nine independent conserved quantities, referred to collectively as the 'zilch' of the field. Considering a monochromatic plane wave, he demonstrated that the components of his pseudotensor are dependent upon the wave's sense of circular polarization, suggesting a connection with the helicity of the photon. Lipkin also observed, however, that his zilches do not have the dimensions of an angular momentum, but rather, that of a force, displaying a highly unusual frequency dependence. The situation was clarified shortly afterwards by Candlin [24], who effectively introduced the optical helicity (6.6) itself and conjectured that Lipkin's zilches are simply members of an infinite hierarchy of higher-order extensions of the optical helicity (6.6). Related generalizations of Lipkin's discovery were also made by Morgan [52], O'Connell and Tompkins [53], and Kibble [54].

Despite these facts, the zilches have recently been reintroduced into the literature by Tang, Cohen and Yang [55, 56]. In particular, they have referred to Lipkin's 00-zilch density as the 'optical chirality', advocating it as a measure of the chirality of the field, an interpretation that has been utilized to predict and describe the results of experiments [55–60]. Considering a strictly monochromatic field, Bliokh and Nori [61] recognized, much as Lipkin himself did, that the 00-zilch and the 0i-zilches are, in the chosen frame of reference, proportional to, but not equal to, the helicity of the field and the components of the spin of the field, respectively. The proportionality factor is the square of the associated angular frequency. Such proportionalities were also observed by Andrews and Coles [62–64].

We have recently demonstrated elsewhere [22, 23] that Lipkin's zilches are indeed members of an infinite hierarchy of extensions of the optical helicity (6.6) and related quantities, as Candlin suggested [24]. For a strictly monochromatic field in a given frame of reference, the apparent similarity that exists between, for example, the optical helicity (6.6) and Lipkin's 00-zilch can be traced to a remarkable self-similarity inherent in Maxwell's equations (3.1)–(3.4) [22]. We return to this observation below.

For a general polychromatic field, there are no proportionalities like the ones seen by Bliokh and Nori [61] and Coles and Andrews [62–64]. Thus, there is no sense in which the optical helicity (6.6) and related quantities are 'equivalent' to Lipkin's zilches [22, 23]. It is the optical helicity (6.6) in particular, which has the dimensions of an angular momentum and is the conserved Lorentz pseudoscalar associated with a duality rotation (6.2), that provides a physically meaningful description of photon helicity. Other quantities such as Lipkin's 00-zilch lack both the dimensions and the Lorentz transformation properties that are required in this context.

We can further highlight the fact that Lipkin's zilches do not describe the angular momentum of the field by identifying their associated symmetry. The infinitesimal transformation

$$\begin{eqnarray} A_\alpha \rightarrow A_{\alpha}'&=&A_\alpha+\zeta^{\mu\nu} \partial_\mu G_{\nu\alpha}, \nonumber \\\\ C_\alpha \rightarrow C_{\alpha}'&=&C_\alpha-\zeta^{\mu\nu} \partial_\mu F_{\nu\alpha},\nonumber \end{eqnarray} \tag{ 6.20 }$$

where the components of the pseudotensor ζ^αβ = ζ^βα are infinitesimal and have the dimensions of a squared length (not an angle!), yields

$$\begin{eqnarray} F_{\alpha\beta}\rightarrow F_{\alpha\beta}'&=&F_{\alpha\beta}+\zeta^{\mu\nu} \partial_\mu\partial_\nu G_{\alpha\beta}, \nonumber \\\\ G_{\alpha\beta}\rightarrow G_{\alpha\beta}'&=&G_{\alpha\beta}-\zeta^{\mu\nu} \partial_\mu \partial_\nu F_{\alpha\beta}\nonumber \end{eqnarray} \tag{ 6.21 }$$

in all gauges. The infinitesimal transformation (6.21) resembles an infinitesimal duality rotation (6.2), but differs crucially through the appearance of second derivatives and is not a rotation of any kind. This obscure symmetry is, in fact, the one associated with Lipkin's zilches. From (6.20) and our form (4.3) of Noether's theorem, acknowledging the independence of the ζ^αβ = ζ^βα, we obtain, as claimed, the conservation law:

$$\begin{equation} \partial_\gamma Z^{\alpha\beta\gamma}=0, \quad Z^{\alpha\beta\gamma}=\textstyle\frac{1}{2}\left(G^{\gamma\mu}\partial^\alpha F_{\mu}^{\ \, \beta}-F^{\gamma\mu} \partial^\alpha G_{\mu}^{\ \, \beta}\right),\ \ \end{equation} \tag{ 6.22 }$$

where Z^αβγ is the form of Lipkin's zilch pseudotensor recognized by Morgan [52] and Kibble [54].

The symmetries associated with the individual zilches have been identified variously by Calkin [25] and Przanowski et al [31]. Frequent incorrect identifications of the symmetries associated with Lipkin's zilches [48–50, 65] can be traced to the use of 'Lagrangians' that do not have the dimensions of an energy.

The infinitesimal zilch symmetry transformation (6.21) possesses a partner which is

$$\begin{eqnarray} F_{\alpha\beta}\rightarrow F_{\alpha\beta}'&=&F_{\alpha\beta}+\xi^{\mu\nu} \partial_\mu \partial_\nu F_{\alpha\beta}, \nonumber \\\\ G_{\alpha\beta}\rightarrow G_{\alpha\beta}'&=&G_{\alpha\beta}+\xi^{\mu\nu} \partial_\mu \partial_\nu G_{\alpha\beta}, \nonumber \end{eqnarray} \tag{ 6.23 }$$

where the components of the tensor ζ^μν = ζ^νμ are infinitesimal and have the dimensions of a squared length. Through Noether's theorem, we find that this symmetry is associated with trivial conserved quantities which have also emerged in the work of Fradkin [66].

We now demonstrate that the field possesses an infinite number of local symmetries and associated tensor and pseudotensor conservation laws.

We have seen that the infinitesimal symmetry transformations:

$$\begin{eqnarray} \Delta F^{\alpha\beta}&=&\theta G^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=-\theta F^{\alpha\beta}, \end{eqnarray}\noindent \tag{ 6.24 }$$

$$\begin{eqnarray} \Delta F^{\alpha\beta}&=&g^\mu\partial_\mu G^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=-g^\mu\partial_\mu F^{\alpha\beta}, \end{eqnarray}\noindent \tag{ 6.25 }$$

$$\begin{eqnarray} \Delta F^{\alpha\beta}&=&\zeta^{\mu\nu}\partial_\mu\partial_\nu G^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=-\zeta^{\mu\nu}\partial_\mu\partial_\nu F^{\alpha\beta} \end{eqnarray} \tag{ 6.26 }$$

are associated with the helicity four-pseudovector (6.4) (of rank-one), a trivial pseudotensor (of rank-two) and the zilch pseudotensor (6.22) (of rank-three), respectively. In addition, we have observed a complimentary structure in that the infinitesimal symmetry transformations:

$$\begin{eqnarray} \Delta F^{\alpha\beta}&=&\phi F^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=\phi G^{\alpha\beta}, \end{eqnarray}\noindent \tag{ 6.27 }$$

$$\begin{eqnarray} \Delta F^{\alpha\beta}&=&t^\mu\partial_\mu F^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=t^\mu\partial_\mu G^{\alpha\beta}, \end{eqnarray}\noindent \tag{ 6.28 }$$

$$\begin{eqnarray} \Delta F^{\alpha\beta}&=&\xi^{\mu\nu}\partial_\mu\partial_\nu F^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=\xi^{\mu\nu}\partial_\mu\partial_\nu G^{\alpha\beta} \end{eqnarray} \tag{ 6.29 }$$

are associated with a trivial four-vector (6.8) (of rank-one), the energy momentum tensor, T^αβ (6.14) (of rank-two) and a trivial tensor (of rank-three), respectively.

These observations are readily generalized. The infinitesimal transformation

$$\begin{equation} \Delta F^{\alpha\beta}=\theta^{\mu\nu\ldots\kappa}\partial_\mu\partial_\nu\ldots \partial_\kappa G^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=-\theta^{\mu\nu\ldots\kappa}\partial_\mu\partial_\nu\ldots \partial_\kappa F^{\alpha\beta} \end{equation} \tag{ 6.30 }$$

constitutes a symmetry for any number of derivatives provided, of course, that the components of the pseudotensor $\theta ^{\mu \nu \ldots \kappa }$, which are infinitesimal, possess suitable dimensions. Without loss of generality, we take $\theta ^{\mu \nu \ldots \kappa }$ to be symmetric in all of its indices. This, (6.30), is the generalization of (6.24)–(6.26). Through Noether's theorem, we find that the conservation laws associated with (6.30), for one or more derivatives, can be expressed as

$$\begin{equation} \partial_\gamma H^{\alpha\beta\ldots\kappa\gamma}=0, \quad H^{\alpha\beta\ldots\kappa\gamma}=\textstyle\frac{1}{2}\left(G^{\gamma\mu}\partial^\beta\ldots\partial^\kappa F_{\mu}^{\ \, \alpha}-F^{\gamma\mu}\partial^\beta\ldots\partial^\kappa G_{\mu}^{\ \, \alpha}\right), \end{equation} \tag{ 6.31 }$$

where we have assumed that the components of the symmetric pseudotensor $\theta ^{\mu \nu \ldots \kappa }$ are otherwise independent. The existence of this infinite hierarchy of pseudotensor conservation laws was observed by Morgan [52] (although the helicity four-pseudovector (6.4), which lies 'lowest' amongst these pseudotensors, being of rank-one, escaped Morgan's attention). We have now tied them to their associated symmetries (6.30).

In a similar vein, the infinitesimal transformation

$$\begin{equation} \Delta F^{\alpha\beta}=\tau^{\mu\nu\ldots\kappa}\partial_\mu\partial_\nu\ldots \partial_\kappa F^{\alpha\beta}, \quad \Delta G^{\alpha\beta}=\tau^{\mu\nu\ldots\kappa}\partial_\mu\partial_\nu\ldots \partial_\kappa G^{\alpha\beta} \end{equation} \tag{ 6.32 }$$

constitutes a symmetry for any number of derivatives provided that the components of the tensor $\tau ^{\mu \nu \ldots \kappa }$, which are infinitesimal, possess suitable dimensions. Without loss of generality, we take $\tau ^{\mu \nu \ldots \kappa }$ to be symmetric in all of its indices. Through Noether's theorem, we find that the conservation laws associated with (6.32), for one or more derivatives, can be expressed as

$$\begin{equation} \partial_\gamma T^{\alpha\beta\ldots\kappa\gamma}=0, \quad T^{\alpha\beta\ldots\kappa\gamma}=\textstyle\frac{1}{2}\left(F^{\gamma\mu}\partial^\beta\ldots\partial^\kappa F_{\mu}^{\ \, \alpha}+G^{\gamma\mu}\partial^\beta\ldots\partial^\kappa G_{\mu}^{\ \, \alpha}\right), \end{equation} \tag{ 6.33 }$$

where we have assumed that the components of the symmetric tensor $\tau ^{\mu \nu \ldots \kappa }$ are otherwise independent. The existence of this infinite hierarchy of tensor conservation laws was also observed by Morgan [52] (although the trivial four-vector (6.8), which lies 'lowest' amongst these tensors, being of rank-one, also escaped Morgan's attention). We have now tied them to their associated symmetries (6.32).

It seems that the pseudotensors (6.31) and tensors (6.33) of even and odd rank, respectively, describe trivial conserved quantities whilst those of odd and even rank, respectively, describe conserved quantities that are dependent upon the difference and sum, respectively, of photon numbers of opposite circular polarization. Thus, we identify a kind of 'alternation' as we ascend rank. This pattern, whose first three 'layers' are

$$\begin{eqnarray} \mathcal{H}&=&\int\!\!\int\!\!\int h^{0}\, {\mathrm{d}}^3\mathbf{r}=\int\!\!\int\!\!\int \hbar \left[n_{\mathrm{L}}(\mathbf{k})-n_{\mathrm{R}}(\mathbf{k})\right] \, {\mathrm{d}}^3\mathbf{k}, \end{eqnarray} \tag{ 6.34 }$$

$$\begin{eqnarray} \mathcal{T}^\alpha&=&\int\!\!\int\!\!\int T^{\alpha 0}\, {\mathrm{d}}^3\mathbf{r}=\int\!\!\int\!\!\int \hbar k^\alpha \left[n_{\mathrm{L}}(\mathbf{k})+n_{\mathrm{R}}(\mathbf{k})\right] \, {\mathrm{d}}^3\mathbf{k}, \end{eqnarray} \tag{ 6.35 }$$

$$\begin{eqnarray} \mathcal{Z}^{\alpha\beta}&=&\int\!\!\int\!\!\int Z^{\alpha\beta 0}\, {\mathrm{d}}^3\mathbf{r}=\int\!\!\int\!\!\int \hbar k^\alpha k^\beta \left[n_{\mathrm{L}}(\mathbf{k})-n_{\mathrm{R}}(\mathbf{k})\right] \, {\mathrm{d}}^3\mathbf{k}, \end{eqnarray} \tag{ 6.36 }$$

appears to extend indefinitely and is, in fact, the pattern the existence of which was conjectured by Candlin [24]. Here, $k^\alpha =\left (\omega ,\mathbf {k}\right)$ is the wave four-vector [13] whilst n_L(k) and n_R(k) are, respectively, the classical limits of the photon numbers of the left- and right-handed circular polarizations associated with the wavevector k [11]. The pieces of this pattern that are dependent upon the difference in photon numbers of opposite circular polarization in particular have recently been examined within the framework of quantum electrodynamics by Coles and Andrews [67].

The simple picture that we have painted is enlivened by the existence of an infinite number of conserved tensors and pseudotensors that depend explicitly upon time and position. Consider, for example, the conserved tensors (6.15)–(6.17) which are constructed from the energy momentum tensor, T^αγ, and the position four-vector, x^α. Conserved pseudotensors can also be constructed from the zilch tensor, Z^αβγ, and the position four-vector x^α, a fact that has been observed by Krivskii and Simulik [49]. In general, such quantities are obscure and we shall not consider them in any detail here.

Amongst the tensor and pseudotensor conservation laws considered above, which are infinite in number, there are but a small handful, of low rank, that describe conserved quantities that possess familiar dimensions. In particular, we can readily appreciate the physical significance of the optical helicity (6.6) as well as the energy momentum tensor (6.14) and the angular momentum tensor (6.15).

We should also comment, however, on those higher-order tensor and pseudotensor conservation laws, amongst them most of (6.31) and (6.33), that describe conserved quantities with unfamiliar dimensions. We suggest that these conserved quantities describe properties of various derivatives of the electric and magnetic fields. To illustrate the idea, suppose that we define, in terms of various first derivatives of the electric and magnetic fields, a pseudovector ${\bf G}={\boldsymbol{\nabla}}\times {\bf E}=-\partial {\bf B}/\partial t$ and a vector ${\bf M}={\boldsymbol{\nabla}}\times {\bf B}=\partial {\bf E}/\partial t$. We then find that these obey the equations:

$$\begin{eqnarray} \boldsymbol{\nabla}\cdot\mathbf{G}&=&0, \nonumber \\ \boldsymbol{\nabla}\cdot\mathbf{M}&=&0, \nonumber \\\\ \boldsymbol{\nabla}\times\mathbf{G}&=&-\frac{\partial \mathbf{M}}{\partial t}, \nonumber \\ \boldsymbol{\nabla}\times\mathbf{M}&=&\frac{\partial \mathbf{G}}{\partial t},\nonumber \end{eqnarray} \tag{ 6.37 }$$

which are identical in form to Maxwell's equations (3.1)–(3.4) themselves. Furthermore, we find that various second derivatives of the electric and magnetic fields also obey a set of Maxwell-like equations. In fact, such patterns recur indefinitely as we consider derivatives of ever-increasing order [7, 8, 22]. Superficially, at least, it seems that it is this self-similarity that is being reflected in the existence of the infinite hierarchies of conserved quantities considered above. For example, we have explained elsewhere [22] that Lipkin's 00-zilch (as obtained from (6.36) with αβ = 00) is to G and M what the optical helicity (6.34) is to the electric and magnetic fields, E and B, themselves. That the Lorentz transformation properties of Lipkin's 00-zilch (6.36) differ from those of the optical helicity (6.34) is, perhaps, a reflection of the fact that the Lorentz transformation properties of G and M differ from those of the electric and magnetic fields, E and B. Indeed, the equations (6.37) can be expressed as

$$\begin{eqnarray} \partial_\beta \partial_0 F^{\alpha\beta}&=& 0, \nonumber \\\\ \partial_\beta \partial_0 G^{\alpha\beta}&=& 0,\nonumber \end{eqnarray} \tag{ 6.38 }$$

and may be seen to follow from the general observation that all contractions of the tensor $\partial _\alpha \partial _\beta \ldots \partial _\gamma F^{\mu \nu }$ and the pseudotensor $\partial _\alpha \partial _\beta \ldots \partial _\gamma G^{\mu \nu }$ vanish, irrespective of the number of derivatives present.

In the present section we devote ourselves to our chosen frame of reference. Some of the quantities that we are about to meet make explicit reference to the transverse, gauge invariant [11] pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials. These pieces can be expressed as non-local functions of the electric and magnetic fields [68]. It seems appropriate, therefore, for us to consider our electric–magnetic Lagrangian, L, rather than our electric–magnetic Lagrangian density, $\mathcal {L}$, in what follows. Integrating our result (4.3) over all space gives us the change, ΔL, induced in our electric–magnetic Lagrangian by the transformation (4.1):

$$\begin{eqnarray} \Delta L &=& \frac{{\mathrm{d}}}{{\mathrm{d}} t}\int\!\!\int\!\!\int\frac{1}{2}\left[-\mathbf{E}\cdot \left(\Delta\mathbf{A}\right)^\bot-\mathbf{B}\cdot \left(\Delta\mathbf{C}\right)^\bot \right]\, {\mathrm{d}}^3\mathbf{r}. \end{eqnarray} \tag{ 7.1 }$$

Note that this result (7.1) has been obtained with generality, making no gauge assumptions. Evidently, however, our electric–magnetic Lagrangian is sensitive only to the transverse pieces, (ΔA)^⊥ and (ΔC)^⊥, of the changes in the vector and pseudovector potentials and is not sensitive to changes in the scalar and pseudoscalar potentials. This is a reflection of both its gauge invariant form and the fact that the time derivatives of A⁰ and C⁰ do not make an appearance in our electric–magnetic Lagrangian density (3.11), so that the scalar and pseudoscalar potentials are themselves not true dynamical variables [11, 18].

As explained in section 5, all meaningful transformations (4.1) of the potentials constitute, automatically, symmetries of both Maxwell's equations and our electric–magnetic Lagrangian. Their associated global conservation laws are obtained from (7.1) with ΔL = 0. This is the form of Noether's theorem that we adopt in the present section. In light of the observations discussed in the preceding paragraph, we focus our attention upon transformations of the transverse, gauge invariant [11] pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials, making no transformations of their longitudinal pieces or the scalar and pseudoscalar potentials.

An infinitesimal rotation of the field about the origin is invoked by taking [69, 70]:

$$\begin{eqnarray} \mathbf{E}&\rightarrow&\mathbf{E}'=\mathbf{E}+\left(\boldsymbol{\theta}\times\mathbf{E}\right)-\boldsymbol{\theta}\cdot \left(\mathbf{r}\times\boldsymbol{\nabla}\right)\mathbf{E}, \nonumber \\\\ \mathbf{B}&\rightarrow&\mathbf{B}'=\mathbf{B}+\left(\boldsymbol{\theta}\times\mathbf{B}\right)-\boldsymbol{\theta}\cdot \left(\mathbf{r}\times\boldsymbol{\nabla}\right)\mathbf{B},\nonumber \end{eqnarray} \tag{ 7.2 }$$

where the orientation and magnitude of the infinitesimal pseudovector θ define an axis and an angle of rotation about that axis, respectively. The first contributions to (7.2) rotate the electric and magnetic field vectors whilst the second contributions rotate the spatial distribution of the field. Of course, (7.2), the form of which may be deduced readily from (6.11), is a local transformation that leaves Maxwell's equations (3.1)–(3.4) invariant in form. The (rotation) angular momentum

$$\begin{equation} {\boldsymbol{\mathcal{J}}}=\int\!\!\int\!\!\int \mathbf{r}\times\left(\mathbf{E}\times\mathbf{B}\right)\, {\mathrm{d}}^3\mathbf{r} \end{equation}\noindent \tag{ 7.3 }$$

of the field is the conserved quantity associated with this symmetry. Indeed, the conservation law

$$\begin{equation} \frac{{\mathrm{d}}{\boldsymbol{\mathcal{J}}}}{{\mathrm{d}} t}=0 \end{equation} \tag{ 7.4 }$$

follows from the space-like components of the tensor equation (6.15).

An infinitesimal boost of the field 'about' the origin is invoked by taking [70]

$$\begin{eqnarray} \mathbf{E}\rightarrow\mathbf{E}'&=&\mathbf{E}-\left(\boldsymbol{\phi}\times\mathbf{B}\right)-\boldsymbol{\phi}\cdot\left(t\boldsymbol{\nabla}+\mathbf{r}\frac{\partial}{\partial t}\right)\mathbf{E}, \nonumber \\\\ \mathbf{B}\rightarrow\mathbf{B}'&=&\mathbf{B}+\left(\boldsymbol{\phi}\times\mathbf{E}\right)-\boldsymbol{\phi}\cdot\left(t\boldsymbol{\nabla}+\mathbf{r}\frac{\partial}{\partial t}\right)\mathbf{B}, \nonumber\end{eqnarray} \tag{ 7.5 }$$

where the orientation and magnitude of the infinitesimal vector ϕ define the direction of the boost and its rapidity, respectively. The first contributions to (7.5) mix the electric and magnetic field vectors whilst the second contributions 'rotate' the spacetime distribution of the field in a hyperbolic manner [12, 14]. The local transformation (7.5), the form of which may be deduced readily from (6.11), also leaves Maxwell's equations (3.1)–(3.4) invariant in form. The boost angular momentum

$$\begin{equation} {\boldsymbol{\mathcal{K}}}=\int\!\!\int\!\!\int\left[t\,\mathbf{E}\times\mathbf{B}-\frac{1}{2}\mathbf{r}\left(E^2+B^2\right)\right]\, {\mathrm{d}}^3\mathbf{r} \end{equation} \tag{ 7.6 }$$

of the field is the conserved quantity associated with this symmetry. Indeed, the conservation law

$$\begin{equation} \frac{{\mathrm{d}}{\boldsymbol{\mathcal{K}}}}{{\mathrm{d}} t}=0 \end{equation} \tag{ 7.7 }$$

follows from the mixed components of the tensor equation (6.15) and is essentially a statement of the uniform motion of the field's centre of energy [13, 14, 39].

Although the boost angular momentum (7.6) of the field is, perhaps, less familiar than the angular momentum (7.3) of the field, it is important to note that relativity places these quantities on equal footing as, taken together, they form an antisymmetric rank-two tensor [70]. This fact is inherent in (6.15). Let us now look more closely at these properties of the field in our chosen frame of reference. We begin with the more familiar angular momentum (7.3) of the field before turning to the boost angular momentum (7.6) of the field.

In optics, the idea is well-established that light can possess both 'spin' and 'orbital' angular momenta [71]. The spin angular momentum is intrinsic and is associated with polarization [72] whilst the orbital angular momentum is extrinsic in general and is associated with the spatial distribution of the field. Helical phase fronts, in particular, give rise to an orbital contribution [73] that has been the subject of much research in recent years [74]. Regarding the description of these properties of light in electromagnetic theory, it is also well-established that the angular momentum (7.3) of the field can be recast, using integration by parts, as the sum of a conserved spin-like contribution, $\boldsymbol{\mathcal{S}}$, and a conserved orbital-like contribution, $\boldsymbol{\mathcal{O}}$ [11, 69, 70, 75]:

$$\begin{equation} {\boldsymbol{\mathcal{J}}}={\boldsymbol{\mathcal{S}}}+{\boldsymbol{\mathcal{O}}}. \end{equation} \tag{ 7.8 }$$

The spin-like contribution in (7.8) is

$$\begin{equation} {\boldsymbol{\mathcal{S}}}=\int\!\!\int\!\!\int\frac{1}{2}\left(\mathbf{E}\times\mathbf{A}^\bot+\mathbf{B}\times\mathbf{C}^\bot\right)\, {\mathrm{d}}^3\mathbf{r}. \end{equation} \tag{ 7.9 }$$

This gauge invariant and intrinsic quantity, which we refer to as the optical spin, displays a critical dependence upon the polarization of light. We have recently suggested elsewhere [22, 23] that the optical spin (7.9) is most meaningfully thought of as a quantity that describes photon helicity, in addition to the optical helicity (6.6) itself of course. Indeed, the integrand of the optical spin (7.9), which may be thought of as the spin density of the field, is also the helicity flux density, as given by the spatial components of the helicity four-pseudovector (6.4) in the Coulomb gauge. Note that the components of the optical spin are not equivalent, in any sense, to Lipkin's 0i-zilches [51] which are to G and M in (6.37) what the components of the optical spin are to the electric and magnetic fields, E and B, themselves [22, 23]. The remaining orbital-like contribution in (7.8) is

$$\begin{equation} {\boldsymbol{\mathcal{O}}}=\int\!\!\int\!\!\int\frac{1}{2}\left[E_i \left(\mathbf{r}\times\boldsymbol{\nabla}\right)A_i^\bot+B_i \left(\mathbf{r}\times\boldsymbol{\nabla}\right) C_i^\bot\right]\, {\mathrm{d}}^3\mathbf{r}, \end{equation}\noindent \tag{ 7.10 }$$

and is both gauge invariant and extrinsic.

At first glance, it is tempting, perhaps, to associate the conservation of the optical spin (7.9) with the rotation of the electric and magnetic field vectors in (7.2) and the conservation of the orbital-like contribution (7.10) with the rotation of the spatial distribution of the field in (7.2). In fact, these pieces of (7.2), when considered separately, do not constitute symmetries of Maxwell's equations (3.1)–(3.4). In particular, they do not respect the transversality of the field [69]. Such (incorrect) identifications have led to suggestions that the optical spin (7.9) and the orbital-like contribution (7.10) are not separately meaningful [11, 39, 76].

This situation was clarified by Van Enk and Nienhuis [77, 78] who established that the optical spin (7.9) and the orbital-like contribution (7.10) are separately meaningful, but noted that neither is a 'true' angular momentum, as the components of their quantized forms do not satisfy the usual angular momentum commutation relations. More recently, it has been demonstrated [69] that the infinitesimal transformation associated with the optical spin (7.9) is, in fact

$$\begin{eqnarray} \mathbf{E}\rightarrow\mathbf{E}'&=&\mathbf{E}+\left(\boldsymbol{\theta}\times\mathbf{E}\right)^\bot, \nonumber \\\\ \mathbf{B}\rightarrow\mathbf{B}'&=&\mathbf{B}+\left(\boldsymbol{\theta}\times\mathbf{B}\right)^\bot,\nonumber \end{eqnarray} \tag{ 7.11 }$$

which does leave Maxwell's equations (3.1)–(3.4) invariant in form [22, 69]. Physically, the spin symmetry transformation (7.11) differs from the first contributions in (7.2) in that it is the closest approximation to an infinitesimal rotation of the electric and magnetic field vectors, in the sense defined by the infinitesimal pseudovector θ, leaving the spatial distribution of the field unchanged, that is consistent with the requirement of transversality [69, 77, 78]. In a similar vein, it has been demonstrated that the infinitesimal transformation associated with the orbital-like contribution (7.10) is

$$\begin{eqnarray} \mathbf{E}\rightarrow\mathbf{E}'&=&\mathbf{E}-\left[\boldsymbol{\theta}\cdot\left(\mathbf{r}\times\boldsymbol{\nabla}\right) \mathbf{E}\right]^\bot, \nonumber \\\\ \mathbf{B}\rightarrow\mathbf{B}'&=&\mathbf{B}-\left[\boldsymbol{\theta}\cdot\left(\mathbf{r}\times\boldsymbol{\nabla}\right) \mathbf{B}\right]^\bot, \nonumber \end{eqnarray}\noindent \tag{ 7.12 }$$

which also leaves Maxwell's equations (3.1)–(3.4) invariant in form. Physically, the orbital symmetry transformation (7.12) differs from the second contributions in (7.2) in that it is the closest approximation to an infinitesimal rotation of the spatial distribution of the field about the origin, in the sense defined by the infinitesimal pseudovector θ, leaving the orientations of the electric and magnetic field vectors unchanged, that is consistent with the requirement of transversality [69]. Note that both (7.11) and (7.12) constitute non-local symmetries in that the transverse pieces indicated are obtainable, in general, through the evaluation of integrals that extend over all space [11, 69]. Let us now approach such ideas using Noether's theorem. As may be confirmed simply by taking its time derivative, the infinitesimal transformation

$$\begin{eqnarray} \mathbf{A}^\bot&\rightarrow& \mathbf{A}^{\bot'}=\mathbf{A}^\bot+\left(\boldsymbol{\theta}\times \mathbf{A}^\bot\right)^\bot, \nonumber \\\\ \mathbf{C}^\bot&\rightarrow& \mathbf{C}^{\bot'}=\mathbf{C}^\bot+\left(\boldsymbol{\theta}\times \mathbf{C}^\bot\right)^\bot, \nonumber \end{eqnarray} \tag{ 7.13 }$$

of the transverse pieces of the vector and pseudovector potentials gives rise to (7.11). Following some simple manipulations, observing the independence of the components of θ, we deduce from (7.13) and our form (7.1) of Noether's theorem, that

$$\begin{equation} \frac{{\mathrm{d}}{\boldsymbol{\mathcal{S}}}}{{\mathrm{d}} t}=0, \end{equation} \tag{ 7.14 }$$

justifying the association of the spin symmetry transformation (7.11) with the conservation of the optical spin (7.9), as expected. An analogous derivation allows us to deduce, from (7.12) that

$$\begin{equation} \frac{{\mathrm{d}}{\boldsymbol{\mathcal{O}}}}{{\mathrm{d}} t}=0, \end{equation} \tag{ 7.15 }$$

justifying the association of the orbital symmetry transformation (7.12) with the conservation of the orbital-like contribution (7.10).

Although the components of their quantized forms do not satisfy the usual angular momentum commutation relations [77, 78], we can suggest that the optical spin (7.9) and the orbital-like contribution (7.10) are angular momenta in the more liberal sense that they are pseudovectors with the dimensions of an angular momentum, the conservation of which is associated with the transversality maintaining rotations (7.11) and (7.12), respectively. A similar mentality is applicable, perhaps, to the optical helicity (6.6) which, despite the fact that it is a Lorentz pseudoscalar and not a pseudovector, is a quantity with the dimensions of an angular momentum, the conservation of which is associated with duality rotations (6.2).

We now turn our attention to the boost angular momentum (7.6) of the field. The question has been posed recently [70]: is it possible to divide the boost angular momentum (7.6) of the field into conserved 'spin' and 'orbital' contributions which, if only by analogy with the angular momentum (7.3) of the field, we might separately associate with the mixing of electric and magnetic field vectors and with the hyperbolic rotation of the spacetime distribution of the field exhibited in the first and second contributions, respectively, to (7.5)? It has been shown [70] that (7.6) can be recast, using integration by parts, as

$$\begin{equation} {\boldsymbol{\mathcal{K}}}={\boldsymbol{\mathcal{V}}}+{\boldsymbol{\mathcal{Y}}}. \end{equation} \tag{ 7.16 }$$

The first contribution to (7.16),

$$\begin{equation} {\boldsymbol{\mathcal{V}}}=\int\!\!\int\!\!\int\frac{1}{2}\left(\mathbf{B}\times\mathbf{A}^\bot-\mathbf{E}\times\mathbf{C}^\bot\right)\, {\mathrm{d}}^3\mathbf{r}, \end{equation} \tag{ 7.17 }$$

is gauge invariant and does not make explicit reference to t or r. The second contribution to (7.16),

$$\begin{equation} {\boldsymbol{\mathcal{Y}}}=\int\!\!\int\!\!\int\frac{1}{2}\left[-A_i^\bot\left(t\boldsymbol{\nabla}+\mathbf{r}\frac{\partial}{\partial t}\right) E_i-C_i^\bot\left(t\boldsymbol{\nabla}+\mathbf{r}\frac{\partial}{\partial t}\right)B_i\right]\, {\mathrm{d}}^3\mathbf{r}, \end{equation} \tag{ 7.18 }$$

is gauge invariant and does make explicit reference to t and r. It seems natural, perhaps, to identify (7.17) and (7.18) as the spin and orbital contributions, respectively, to the boost angular momentum (7.6) of the field.

We can now examine the separation (7.16) using Noether's theorem. As may be confirmed simply by taking its time derivative, the infinitesimal transformation

$$\begin{eqnarray} \mathbf{A}^\bot\rightarrow\mathbf{A}^{\bot '}&=&\mathbf{A}^\bot-\left(\boldsymbol{\phi}\times\mathbf{C}^\bot\right)^\bot, \nonumber \\\\ \mathbf{C}^\bot\rightarrow\mathbf{C}^{\bot '}&=&\mathbf{C}^\bot+\left(\boldsymbol{\phi}\times\mathbf{A}^\bot\right)^\bot \nonumber \end{eqnarray} \tag{ 7.19 }$$

of the transverse pieces of the vector and pseudovector potentials invokes the infinitesimal transformation:

$$\begin{eqnarray} \mathbf{E}\rightarrow\mathbf{E}'&=&\mathbf{E}-\left(\boldsymbol{\phi}\times\mathbf{B}\right)^\bot, \nonumber \\\\ \mathbf{B}\rightarrow\mathbf{B}'&=&\mathbf{B}+\left(\boldsymbol{\phi}\times\mathbf{E}\right)^\bot \nonumber \end{eqnarray} \tag{ 7.20 }$$

of the electric and magnetic fields which leaves Maxwell's equations (3.1)–(3.4) invariant in form. Let us identify (7.20) as our boost spin symmetry transformation. Comparing it with (7.5), we see that (7.20) is the closest approximation to an infinitesimal mixing of the electric and magnetic field vectors, in the sense defined by the infinitesimal vector ϕ, leaving the spacetime distribution of the field unchanged, that is consistent with the requirement of transversality. In a similar vein, we identify the infinitesimal transformation:

$$\begin{eqnarray} \mathbf{E}\rightarrow\mathbf{E}'&=&\mathbf{E}-\left[\boldsymbol{\phi}\cdot\left(t\boldsymbol{\nabla}+\mathbf{r}\frac{\partial}{\partial t}\right)\mathbf{E}\right]^\bot, \nonumber \\\\ \mathbf{B}\rightarrow\mathbf{B}'&=&\mathbf{B}-\left[\boldsymbol{\phi}\cdot\left(t\boldsymbol{\nabla}+\mathbf{r}\frac{\partial}{\partial t}\right)\mathbf{B}\right]^\bot, \nonumber \end{eqnarray} \tag{ 7.21 }$$

which leaves Maxwell's equations (3.1)–(3.4) invariant in form, as our boost orbital symmetry transformation. Comparing it with (7.5), we see that (7.21) is the closest approximation to an infinitesimal hyperbolic rotation of the spacetime distribution of the field about the origin, in the sense defined by the infinitesimal vector ϕ, without mixing the electric and magnetic field vectors, that is consistent with the requirement of transversality. The effects of the symmetry transformations (7.20) and (7.21) on a linearly polarized plane wave are depicted in figures 1 and 2, respectively.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Following some simple manipulations, observing the independence of the components of ϕ, we deduce from (7.19) and our form (7.1) of Noether's theorem, that

$$\begin{equation} \frac{{\mathrm{d}} {\boldsymbol{\mathcal{V}}}}{{\mathrm{d}} t}=0, \end{equation} \tag{ 7.22 }$$

which, pleasingly, expresses the conservation of our candidate (7.17) for the spin contribution to the boost angular momentum (7.6) of the field. As was noted originally [70], however, this candidate (7.17) is a vanishing quantity ($\boldsymbol{\mathcal{V}}=0$). It appears, therefore, that the separation of the boost angular momentum (7.6) of the field into separately conserved spin and orbital contributions ultimately fails in that our candidate (7.18) for the orbital contribution constitutes the entirety of the boost angular momentum (7.6) of the field ($\boldsymbol{\mathcal{K}}=\boldsymbol{\mathcal{Y}}$).

The boost spin symmetry transformation (7.20) that we have identified is the natural partner of the spin symmetry transformation (7.11). The vanishing of our candidate (7.17) for the spin contribution to the boost angular momentum (7.6) of the field thus falls in line with our general observations regarding such symmetry pairs. For completeness, we note that the partner symmetries of (7.12) and (7.21) are seemingly obscure and are associated with trivial conserved quantities.

We have seen that the optical spin (7.9) and the orbital-like contribution (7.10), which have the dimensions of an angular momentum, are associated with non-local symmetry transformations, as necessitated by the requirement of transversality. Other conserved quantities that possess familiar dimensions and are associated with non-local symmetries also exist, for example the ij-infra-zilches which we have considered elsewhere [22].

We recognize, in fact, that there are an infinite number of non-local symmetries and associated conserved quantities. Unlike, for example, the optical spin (7.9), the orbital-like contribution (7.10) and the ij-infra-zilches [22], however, the majority of these conserved quantities have unfamiliar dimensions. To illustrate this claim, we present the infinitesimal transformation:

$$\begin{eqnarray} \mathbf{E}&\rightarrow&\mathbf{E}'= \mathbf{E}+\alpha\mathbf{A}^\bot, \nonumber \\\\ \mathbf{B}&\rightarrow&\mathbf{B}'= \mathbf{B}+\alpha \mathbf{C}^\bot, \nonumber\end{eqnarray} \tag{ 7.23 }$$

where α is an infinitesimal scalar with the dimensions of an inverse length. Note that (7.23) is a non-local transformation in the sense that the transverse pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials are expressible as non-local integral functions of the electric and magnetic fields, E and B [68], as noted earlier. That (7.23) constitutes a symmetry follows from the fact that the transverse pieces of the vector and pseudovector potentials obey the equations:

$$\begin{eqnarray} \boldsymbol{\nabla}\cdot\mathbf{A}^\bot&=& 0, \nonumber \\ \boldsymbol{\nabla}\cdot\mathbf{C}^\bot&=& 0, \nonumber \\\\ \boldsymbol{\nabla}\times\mathbf{A}^\bot&=&-\frac{\partial \mathbf{C}^\bot}{\partial t}, \nonumber \\ \boldsymbol{\nabla}\times\mathbf{C}^\bot&=&\frac{\partial \mathbf{A}^\bot}{\partial t},\nonumber \end{eqnarray}\noindent \tag{ 7.24 }$$

which are identical in form to Maxwell's equations (3.1)–(3.4) themselves. Such self-similarity recurs indefinitely as we delve into the realms of various integrals of the electric and magnetic fields and is quite analogous to the self-similarity seen as we consider various derivatives of the electric and magnetic fields (compare (7.24) with (6.37)) [22]. In light of this structure, one can readily deduce the existence of an infinite hierarchy of non-local symmetries, as claimed. Through Noether's theorem, we find that (7.23), for example, is associated with the conservation law:

$$\begin{equation} \frac{{\mathrm{d}}\mathcal{Q}}{{\mathrm{d}} t}=0, \ \ \ \ \ \ \ \ \mathcal{Q}=\int\!\!\int\!\!\int\frac{1}{2}\left(\mathbf{A}^\bot\cdot\mathbf{A}^\bot+\mathbf{C}^\bot\cdot\mathbf{C}^\bot\right) \, {\mathrm{d}}^3\mathbf{r}. \end{equation}\noindent \tag{ 7.25 }$$

A comparison of (7.24) and (7.25) with (3.1)–(3.4) and (3.13), respectively, leads us to identify the conserved quantity, $\mathcal {Q}$, above, which has unfamiliar dimensions, as the 'energy' of the transverse pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials, in much the same way that Lipkin's 00-zilch (6.36) is the 'helicity' of G and M in (6.37), for example. This particular conserved quantity (7.25) has also been recognized by Drummond [7, 8]. We emphasize that (7.25) is but one of an infinite number of conserved quantities with unfamiliar dimensions that are associated with non-local symmetries and appear to describe properties of various integrals of the electric and magnetic fields.

As with local symmetries, it seems that non-local symmetries exist in pairs, only one member of which is associated with a non-trivial conserved quantity. Through Noether's theorem, we find that the partner symmetry of (7.23), for example, is associated with a trivial conserved quantity. The existence of various non-local symmetries has also been recognized by Fushchich and Nikitin [34, 35].

In proof, an anonymous referee highlighted the fact that the time derivative of the 'energy', $\mathcal {Q}$, (7.25) of the transverse pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials is essentially equal to the trivial quantity, $\mathcal {D}$, that we met earlier (6.9), in that ${\mathrm {d}} \mathcal {Q}/{\mathrm {d}} t=2\mathcal {D}$. Thus, the vanishing (6.10) of the latter quantity ($\mathcal {D}=0$) can be viewed as a statement of the conservation law ${\mathrm {d}}\mathcal {Q}/{\mathrm {d}} t\propto \mathcal {D}=0$ deduced above (7.25). Such observations are readily generalized. In particular, we find, following the discussion above, that the (conserved) 'linear momentum' of the transverse pieces, A^⊥ and C^⊥, of the vector and pseudovector potentials is

$$\begin{equation} {\boldsymbol{\mathcal{R}}}=\int\!\!\int\!\!\int\left(\mathbf{A}^\bot\times\mathbf{C}^\bot\right)\, {\mathrm{d}}^3\mathbf{r}. \end{equation} \tag{ 7.26 }$$

Note that the form of the integrand in (7.26) is analogous to Poynting's vector, E × B [13, 79]. We observe that the time derivative of (7.26) is essentially equal to our candidate, $\boldsymbol{\mathcal{V}}$, for the spin contribution (7.17) to the boost angular momentum (7.6) of the field in that ${\mathrm{d}} \boldsymbol{\mathcal{R}} / {\mathrm{d}} t=2 \boldsymbol{\mathcal{V}}$. It appears, therefore, that the vanishing of our candidate ($\boldsymbol{\mathcal{V}}=0$) can be viewed as a statement of the conservation law ${\mathrm{d}}\boldsymbol{\mathcal{R}}/{\mathrm{d}} t\propto \boldsymbol{\mathcal{V}}=0$.