<EMAIL_ADDRESS>
# On the origin of the Kerker phenomena

Jon Lasa-Alonso  [  Donostia International Physics Center, Paseo Manuel de
Lardizabal 4, 20018 Donostia-San Sebastián, Spain  Centro de Física de
Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-San Sebastián, Spain
Chiara Devescovi  Donostia International Physics Center, Paseo Manuel de
Lardizabal 4, 20018 Donostia-San Sebastián, Spain  Carlos Maciel-Escudero
Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018
Donostia-San Sebastián, Spain  Centro de Física de Materiales, Paseo Manuel de
Lardizabal 5, 20018 Donostia-San Sebastián, Spain  Aitzol García-Etxarri
Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018
Donostia-San Sebastián, Spain  IKERBASQUE, Basque Foundation for Science,
María Díaz de Haro 3, 48013 Bilbao, Spain  Gabriel Molina-Terriza  Donostia
International Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San
Sebastián, Spain  Centro de Física de Materiales, Paseo Manuel de Lardizabal
5, 20018 Donostia-San Sebastián, Spain  IKERBASQUE, Basque Foundation for
Science, María Díaz de Haro 3, 48013 Bilbao, Spain

###### Abstract

We provide an insight into the origin of the phenomena reported 40 years ago
by Kerker, Wang and Giles (Journal of the Optical Society of America, 73, 6,
pp. 765-767, (1983)). We show that the impedance and refractive index matching
conditions, discussed in Sections II and IV of the seminal paper, are
intimately related with space-time symmetries. We derive our results starting
from the theory of representations of the Poincaré group, as it is the theory
on which one of the most elemental descriptions of electromagnetic waves is
based. We show that fundamental features of electromagnetic waves in material
environments can be derived from group theoretical arguments. In particular,
we identify the Casimir invariants of $P_{\scriptscriptstyle{{3,1}}}$ subgroup
as the magnitudes which describe the nature of monochromatic electromagnetic
waves propagating in matter. Finally, we show that the emergence of the Kerker
phenomena is associated with the conservation of such Casimir invariants in
piecewise homogeneous media.

##  I Introduction

The study of the so-called _Kerker conditions_ Kerker  _et al._ (1983); Giles
and Wild (1982) has gathered an important amount of scientific efforts in the
last decade Nieto-Vesperinas  _et al._ (2011a); García-Etxarri and Dionne
(2013); Pfeiffer and Grbic (2013); Person  _et al._ (2013); Fu  _et al._
(2013); Alaee  _et al._ (2015); Babicheva and Evlyukhin (2017); Bag  _et al._
(2018); Barhom  _et al._ (2019); Lasa-Alonso  _et al._ (2020a); Xu  _et al._
(2020); Lasa-Alonso  _et al._ (2020b); Olmos-Trigo  _et al._ (2020a); Fan  _et
al._ (2021); Gerasimov  _et al._ (2021); Qin  _et al._ (2022); Xiong  _et al._
(2022). The phrase _first Kerker condition_ refers to scatterers that behave
as those described in Section II of Kerker’s seminal paper Kerker  _et al._
(1983). On the other hand, the phrase _second Kerker condition_ has been
usually employed to make reference to one type of scatterers discussed in
Section III. The key characteristics of the scatterers fulfilling either of
these two conditions are related to the preservation of helicity and the
directionality of the emission. In particular, samples fulfilling Kerker’s
first condition are characterized for producing a scattered field with the
same helicity as the illuminating excitation. As preservation of helicity is
related to the restoration of duality symmetry Calkin (1965), scatterers
fulfilling the first Kerker condition are commonly called dual Fernandez-
Corbaton  _et al._ (2013). Scatterers fulfilling Kerker’s second condition,
however, are characterize for producing a field of the opposite helicity.
Consequently, they have usually been denoted as antidual. In addition, in the
specific case of rotationally symmetric scatterers, the directionality of the
emitted field is directly correlated with its helicity. For example,
cylindrical dual scatterers do not emit in the backward direction, whereas
cylindrical antidual scatterers do not emit in the forward direction Zambrana-
Puyalto  _et al._ (2013); Fernandez-Corbaton (2013).

The features described above have frequently placed the discussion about the
two Kerker conditions at the same level. However, we should bear in mind that
scatterers fulfilling the first or the second Kerker conditions have
contradictory behaviours in many important aspects. For instance, dual
scatterers can generally be defined in terms of a condition over the material
constants, i.e. they are achieved whenever the impedance of the sample and the
surrounding medium is matched. Antidual scatterers, however, have only been
identified in terms of multipolar scattering coefficients Zambrana-Puyalto
_et al._ (2013); Lasa-Alonso  _et al._ (2022). Moreover, whereas helicity
preserving scatterers have been experimentally reported several times in the
literature, antidual scatterers have fundamental problems regarding
conservation of energy. These difficulties impulsed the search for alternative
definitions of the second Kerker condition that respected the optical theorem.
The efforts resulted in the notion of the _generalized second Kerker
condition_ , that minimizes the forward scattered emission for a fixed
scattering cross section Nieto-Vesperinas  _et al._ (2011b); Olmos-Trigo  _et
al._ (2020b). Even if the generalized second Kerker condition was shown to be
compatible with the conservation of electromagnetic energy, it is still a
definition based on the Mie scattering coefficients of a sphere. Recently, it
has been shown that antidual scatterers of any size and form have, by
construction, a null extinction cross section Lasa-Alonso  _et al._ (2023).

The aforementioned differences in the description of dual and antidual
scatterers led us to the conclusion that the current description of the two
Kerker conditions is not satisfactory. Even if they have been usually
discussed on the same footing, it is clear that the nature of these two
effects is dissimilar in many important aspects. Here, we propose a solution
to this historical conflict with the introduction of the resonant helicity
mixing condition. This phenomenon is partially studied in Section IV of
Kerker’s original paper, but it was not until recently that its crucial role
has been put forward Lasa-Alonso  _et al._ (2023). As it was shown therein,
the resonant helicity mixing condition is also defined in terms of material
constants, i.e. it is fulfilled in samples with matched refractive indices.
Moreover, it permits the construction of spherical scatterers which flip
helicity of light very efficiently, while still respecting the energy
conservation law. Finally, also in line with impedance-matched materials,
index-matched materials have an associated conserved quantity: the square of
linear momentum. As we show next, the origin of this conserved magnitude lies
in the symmetries of space-time and, thus, its comprehension requires a
systematic approach to the description of electromagnetic waves in terms of
group theory.

In this work, we first revisit the theory of unitary irreducible
representations of the Poincaré group and its link to the description of
electromagnetic waves in vacuum. We place at the forefront of our analysis,
the important group theoretical concept of Casimir operators, applied to some
problems in electromagnetism. In group theory, the Casimir operators of a
particular group are operators that commute with all the generators of the
group and, thus, label the unitary irreducible representations. In this line,
we show that, exactly as Bialynicki-Birula’s photon wave function 111In this
context, we use the term _photon wave function_ as a legacy from Bialynicki-
Birula’s work. We will always deal in this work with classical electromagnetic
fields, and the photon wave functions are electromagnetic wave solutions. is
associated with the unitary irreducible representations of the Poincaré group
Bialynicki-Birula and Bialynicka-Birula (2017), monochromatic electromagnetic
waves propagating in infinitely homogeneous media are associated with the
unitary irreducible representations of $P_{\scriptscriptstyle{{3,1}}}$
subgroup. As a result, such wave solutions are constructed as eigenfunctions
of the three Casimir operators of $P_{\scriptscriptstyle{{3,1}}}$, i.e. the
generator of time-translations, $\hat{P}_{0}$, the helicity operator,
$\hat{\Lambda}$, and the square of the linear momentum operator,
$\hat{\mathbf{P}}^{2}$. Finally, we study the propagation of electromagnetic
waves in piecewise media and we particularly focus on Kerker’s problem, i.e.
the scattering of electromagnetic waves by magnetic spheres. We show that the
two matching conditions reported in the seminal paper are related with the
breaking of the homogeneity of space, while still preserving the Casimir
invariants of $P_{\scriptscriptstyle{{3,1}}}$.

##  II Electromagnetic waves in vacuum

One of the most elemental descriptions of the propagation of electromagnetic
waves in vacuum is due to Eugene Wigner Wigner (1939); Bargmann and Wigner
(1948). His findings not only provided an axiomatic derivation of Maxwell’s
equations in vacuum, but unified the description of all relativistic particles
in terms of the so-called Wigner’s classification. His major achievement is
probably the connection between unitary irreducible representations (UIRs) of
the Poincaré group and wave functions of isolated physical systems in vacuum.
Indeed, he found that the invariant vector spaces of the different UIRs of the
Poincaré group are associated with the wave functions of relativistic
particles Tung (1985). More specifically, Wigner and Bargmann found that the
UIRs of the Poincaré group (and, in fact, of many other continuous groups) can
be labelled by the eigenvalues of the two principal Casimir operators:
$\hat{C}_{1}=\hat{P}_{\nu}\hat{P}^{\nu}$ and
$\hat{C}_{2}=\hat{W}_{\nu}\hat{W}^{\nu}$, which are the modulus of the
$4$-momentum, $\hat{P}_{\nu}$, and the modulus of the Pauli-Lubanski
pseudovector, $\hat{W}_{\nu}$, respectively Bargmann and Wigner (1948);
Bargmann (1947). In practice, the first Casimir operator represents the mass,
whereas the second Casimir operator is associated with the internal degrees of
freedom of the particles. In this Section, we will mostly focus on
electromagnetic waves propagating in vacuum and, thus, we will center our
analysis on the massless and discrete-spin UIRs of the Poincaré group. For
this type of particles, a third Casimir operator is identified Lomont and
Moses (1962): helicity, $\hat{\Lambda}$.

The description of the photon, as a fundamental relativistic particle, is
considered within the $0_{s}$ class of UIRs of the Poincaré group Bargmann and
Wigner (1948). This class describes particles of null mass,
$\text{\emph{M}}=0$, and helicity eigenvalue $\lambda=\pm\text{\emph{S}}$,
where _S_ is an integer or an odd-half-integer Tung (1985). The
electromagnetic case is recovered when fixing $\text{\emph{S}}=1$. The
invariant vector spaces associated with the description of the photon fulfill
the following relations: $\hat{C}_{1}=0$ and
$\hat{W}_{\nu}=\lambda\hat{P}_{\nu}$ Bargmann and Wigner (1948). Even if the
form might be cumbersome, it can be shown that the first equation represents
the wave equation in vacuum. Moreover, the time component of the second
equation represents Faraday-Ampère’s laws in vacuum and, finally, Gauss’ laws
can be derived from the spatial components of the second equation Bacry
(1976); Gersten (1999); Dirac (1936). In our view, the deepest explicit
connection between the UIRs of the Poincaré group and Maxwell’s equations has
been carried out by Iwo Bialynicki-Birula and Zofia Bialynicka-Birula. In
their works, they argue that the electromagnetic object which more closely
follows the UIRs of the Poincaré group is the Riemann-Silberstein (RS) vector,
from which a proper photon wave function can be constructed. Explicitly, they
have proposed the following form of the photon wave function in vacuum
Bialynicki-Birula and Bialynicka-Birula (2017):

$\mathbf{\Psi}^{\lambda}(\mathbf{r},t)=\frac{1}{(2\pi)^{3/2}}\int
d\mathbf{k}~{}f^{\lambda}(\mathbf{k})\mathbf{e}^{\lambda}(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega_{\mathbf{k}}t)},$
(1)

where $\mathbf{r}$ is the position vector and $t$ the time. On the other hand,
$\lambda=\pm 1$ is the helicity label, $\mathbf{k}$ is the wavevector of the
radiation field, $f^{\lambda}(\mathbf{k})$ is an arbitrary complex amplitude
associated with a fixed wavevector and helicity,
$\mathbf{e}^{\lambda}(\mathbf{k})$ is a unitary polarization vector and
$\omega_{\mathbf{k}}=|\mathbf{k}|$ is the angular frequency (we choose natural
units, $\hbar=c=1$). Finally, the integral in Eq. (1) is considered over the
entire reciprocal space.

The expression in Eq. (1) of the photon wave function has been derived as a
particular solution of Maxwell’s equations in terms of the RS vector. However,
let us now show that Eq. (1) can also be derived by constructing the invariant
vector spaces of the massless UIRs of the Poincaré group. This connection
highlights that important features of electromagnetic wave solutions in vacuum
can be derived from pure group theoretical arguments. In this regard, we
closely follow the method indicated by Wu-Ki Tung’s book, _Group Theory in
Physics_ Tung (1985), which indicates that the construction of invariant
spaces can be carried out by operating over a “standard” vector. Indeed, the
basis vectors of the massless and discrete-spin UIRs of the Poincaré group are
constructed as (see Ref. Tung (1985), Chapter 10, Section 4):

$\bm{\Psi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)\equiv\left[\hat{R}_{z}(\phi)\hat{R}_{y}(\theta)\hat{L}_{z}(\xi)\right]\bm{\Psi}^{\lambda}_{\mathbf{k}_{l}}(\mathbf{r},t).$
(2)

where
$\bm{\Psi}^{\lambda}_{\mathbf{k}_{l}}(\mathbf{r},t)=\mathbf{u}^{\lambda}e^{ik_{l}(z-t)}$
represents a monochromatic plane wave propagating in the positive $OZ$
direction with linear momentum $\mathbf{k}_{l}$ and
$\mathbf{u}^{\lambda}=(1,\lambda i,0)/\sqrt{2}$ is a polarization vector with
well-defined helicity $\lambda$. $\hat{R}_{z}(\phi)$ represents a rotation
along $OZ$ axis by an angle $\phi$ and $\hat{R}_{y}(\theta)$ a rotation along
$OY$ axis by an angle $\theta$. $\hat{L}_{z}(\xi)$ represents a Lorentz
transformation along the $OZ$ direction to an inertial frame moving with
velocity $v=\tanh(\xi)$. It can be checked that the application of the Lorentz
transformation modifies the frequency of the standard vector as follows:
$|\mathbf{k}|=e^{-\xi}|\mathbf{k}_{l}|$. Thus, by choosing the boost parameter
$\xi\in(-\infty,\infty)$ one spans all possible values of the wavevector
modulus (see Appendix A). On the other hand, by choosing $\phi\in(0,2\pi)$ and
$\theta\in(0,\pi)$ the rotation transformation makes the monochromatic plane
wave propagate in an arbitrary direction.

If we now consider the integral in Eq. (1) in spherical coordinates of
reciprocal space, we see that the three parameters $\\{\xi,\theta,\phi\\}$
span all the integration domain. In other words, the photon wave function in
vacuum given in Eq. (1) can be written in the following form:

$\mathbf{\Psi}^{\lambda}(\mathbf{r},t)=\frac{1}{(2\pi)^{3/2}}\int
d\mathbf{k}~{}g^{\lambda}(\mathbf{k})\bm{\Psi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t),$
(3)

where the introduction of a different complex function,
$g^{\lambda}(\mathbf{k})=e^{\xi}f^{\lambda}(\mathbf{k})$, is just because the
Lorentz transformation not only changes the frequency, but also modifies the
amplitude of the standard vector by a factor $e^{-\xi}$ (see Appendix A). Also
note that, even if the construction in Eq. (2) starts from a monochromatic
wave, the final state in Eq. (3) is no longer monochromatic. From this result
we conclude that one can retrieve the expression of the photon wave function
in vacuum, in the form proposed by Bialynicki-Birula, just by taking linear
superpositions of the basis vectors of the massless and discrete-spin UIRs of
the Poincaré group. Now, the question arises: following similar group
theoretical arguments, can we study the propagation of electromagnetic waves
in a different medium other than vacuum?

##  III Electromagnetic waves in infinitely homogeneous media

Let us first consider the next simplest case, i.e. a medium where both the
electric permittivity, $\varepsilon$, and magnetic permeability, $\mu$, are
different from the values of vacuum but are constant functions of space
coordinates. If such a medium extends all over the space, we say that it is
infinitely homogeneous. Moreover, we will also consider that the permittivity
and permeability are scalar functions (isotropic) and that they do not change
on time (static).

The key modification of the problem when studying the propagation of
electromagnetic waves in a infinitely homogeneous medium compared to the case
of vacuum is the set of underlying symmetries. An infinitely homogeneous
medium is not invariant under the Poincaré group. This is due to the fact that
an isotropic medium becomes, in general, bianisotropic when performing a
Lorentz transformation (see Fig. 1) Landau and Lifshitz (1984); Kong (1972).
As a result, the presence of a material reduces the symmetry group from the
Poincaré group to a subgroup which does not contain Lorentz transformations.
Such a particular way of addressing physical problems is commonly denoted as
the symmetry breaking principle and it can be compactly stated in the
following terms Winternitz (1977); Patera  _et al._ (1975a, b, 1977); Gagnon
and Winternitz (1988); Beckers  _et al._ (1977); Patera  _et al._ (1976a):
_consider a physical system which is described by a given group $G$ and an
external influence reduces the symmetry from the original $G$ to a subgroup
$G_{i}\subset G$. Then, the subgroup $G_{i}$ can be used to study the
properties of the new modified system. In particular, the generators and
Casimir operators of the subgroup will provide conserved quantities and the
UIRs will determine the new wave functions or, at least, some of their
properties._

Figure 1: Effect of a Lorentz transformation over electromagnetic waves
propagating in vacuum and in a homogeneous medium. Black arrows represent the
propagation directions, $(\theta,\phi)$, red/blue spinning arrows the
helicity, $\lambda$, and different colours indicate the frequency of the
waves, $\omega$. In the left upper panel, a superposition of plane-waves of
different frequencies propagating in vacuum is shown. When switching to the
moving reference frame (left bottom panel), the environment is left unaltered
and the waves change their frequency. In the right upper panel, a
superposition of monochromatic plane-waves is shown. When switching to the
moving reference frame (right bottom panel), the medium is modified and
becomes bianisotropic.

So, let us apply the symmetry breaking principle to study the propagation of
electromagnetic waves in an infinitely homogeneous medium. As stated above
such a medium is not invariant under the full Poincaré group, but only under
the subgroup $P_{\scriptscriptstyle{{3,1}}}$ which includes all the generators
of the Poincaré group except for Lorentz boosts. In other words,
$P_{\scriptscriptstyle{{3,1}}}$ comprises seven generators: the generator of
time translations ($\hat{P}_{0}$), the three generators of space translations
(linear momentum components, $\hat{P}_{i}$) and the three generators of
spatial rotations (total angular momentum components, $\hat{J}_{i}$). In
addition to this, the subgroup has three Casimir operators, i.e. the generator
of time translations, $\hat{P}_{0}$, a magnitude proportional to the helicity,
$\hat{\mathbf{J}}\cdot\hat{\mathbf{P}}$, and the square of linear momentum,
$\hat{\mathbf{P}}^{2}$ Patera  _et al._ (1976a). Helicity and square of linear
momentum are also conserved magnitudes for electromagnetic waves propagating
in infinitely homogeneous media due to their condition of Casimir invariants.
These invariants will be of utmost importance in Sections IV and V, when
discussing the conserved quantities associated with the Kerker phenomena.

Casimir operators play a central role in the determination of the UIRs. This
is because the basis vectors of the UIRs are necessarily eigenvectors of all
such operators Bargmann and Wigner (1948); Bargmann (1947); Tung (1985). Thus,
for the basis vectors associated with the UIRs of
$P_{\scriptscriptstyle{{3,1}}}$ subgroup,
$\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$, we have that:

$\displaystyle\hat{P}_{0}~{}\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$
$\displaystyle=\omega\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$ (4)
$\displaystyle
k^{\scriptscriptstyle{-1}}\hat{\mathbf{J}}\cdot\hat{\mathbf{P}}~{}\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$
$\displaystyle=\lambda\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$ (5)
$\displaystyle\hat{\mathbf{P}}^{2}~{}\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$
$\displaystyle=k^{2}\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t).$ (6)

Setting $\lambda=\pm 1$ and $k=\omega n$, with $n=\sqrt{\varepsilon\mu}$ the
refractive index of the medium, it can be noted that Eqs. (4)-(6) actually
represent the dynamic equations of monochromatic electromagnetic waves
propagating in an infinitely homogeneous medium. Indeed, Eq. (4) determines
the monochromaticity of the fields; on the other hand, Eq. (5) represents all
four time-independent Maxwell’s equations; and, finally, Eq. (6) represents
the wave equation for monochromatic waves, i.e. Helmholtz’s equation. The
connection appears more clearly when employing the explicit form of the
operators Bliokh and Nori (2015); Fernandez-Corbaton  _et al._ (2012); Sarkar
and Halas (1997); Nienhuis (2016): $\hat{P}_{0}=i\partial_{t}$,
$\hat{\Lambda}=k^{\scriptscriptstyle{-1}}\hat{\mathbf{J}}\cdot\hat{\mathbf{P}}=k^{\scriptscriptstyle{-1}}\nabla\times$
and $\hat{\mathbf{P}}^{2}=-\nabla^{2}$. This is in agreement with the results
previously obtained for vacuum, where Casimir operators were also related with
the dynamic equations of electromagnetic waves.

At this stage, following the same procedure as in the case of vacuum, we
should be able to obtain some information of the photon wave function in an
infinitely homogeneous medium by analyzing the UIRs of
$P_{\scriptscriptstyle{{3,1}}}$. Particularly, $P_{\scriptscriptstyle{{3,1}}}$
can be split as the direct product of two of its subgroups which are the
Euclidean group in three dimensions, $E(3)$, and the one parameter subgroup of
time translations, $T$. This is due to the fact that time translations commute
with all the elements of the Euclidean group in three dimensions. As a result,
the UIRs of $P_{\scriptscriptstyle{{3,1}}}$ are constructed as the product of
the UIRs of $E(3)$ and the UIRs of $T$ Mackey (1978); Rossmann (2002). An
intuitive way of understanding this is by noting that the basis vectors
$\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)$ are necessarily built as the
product of a spatial function and a temporal function. Equations (5)-(6)
determine the spatial dependence of the vectors, whereas Eq. (4) determines
their temporal dependence. Thus, the basis vectors of the UIRs of
$P_{\scriptscriptstyle{{3,1}}}$ associated with electromagnetic waves can be
constructed in the following way (see Ref. Tung (1985), Chapter 9, Section 7):

$\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t)\equiv\left[\hat{R}_{z}(\phi)\hat{R}_{y}(\theta)\right]\bm{\Phi}^{\lambda}_{\mathbf{k}_{0}}(\mathbf{r})e^{-i\omega
t}.$ (7)

where
$\bm{\Phi}^{\lambda}_{\mathbf{k}_{0}}(\mathbf{r})=\mathbf{u}^{\lambda}e^{ikz}$
is now the standard vector and $\mathbf{u}^{\lambda}=(1,\lambda i,0)/\sqrt{2}$
is a circular polarization vector.

Note that the basis given in Eq. (7) bears a close resemblance to the basis
previously constructed for the UIRs of the Poincaré group. Indeed, they are
constructed exactly in the same way except for the Lorentz boost applied in
Eq. (2), which permits the modulation of the frequency through parameter
$\xi$. The basis vectors given by Eq. (7), on the other hand, have a fixed
frequency $\omega$ (see Table 1). As a result, we can conclude that the photon
wave function in an infinitely homogeneous medium can be written as:

$\mathbf{\Phi}^{\lambda}(\mathbf{r},t)=\int
d\Omega~{}\varphi^{\lambda}(\Omega)\bm{\Phi}^{\lambda}_{\mathbf{k}}(\mathbf{r},t),$
(8)

where $d\Omega=\sin\theta d\theta d\phi$, with integration limits
$\phi\in(0,2\pi)$ and $\theta\in(0,\pi)$, and $\varphi^{\lambda}(\Omega)$ is
an arbitrary complex function of the azimuthal and polar angles. Note that
such a field is nothing but an abstract representation of the monochromatic RS
vector. As a matter of fact, such an expression of the photon wave function in
an infinitely homogeneous medium has already been employed to construct
solutions of electromagnetic waves with well-defined helicity, in particular,
vector spherical harmonics and Bessel beams Tung (1985); Fernandez-Corbaton
_et al._ (2012); Vavilin and Fernandez-Corbaton (2023). The way in which we
have derived it here indicates that the validity of the expression lies in the
underlying symmetry group and that it goes far beyond the specific examples
previously reported. Our findings indicate that the role of the monochromatic
RS vector is central in the description of optical phenomena in material
environments.

|  $\hat{C}_{0}$ |  $\hat{C}_{1}$ |  $\hat{C}_{2}$ |  $\hat{\Lambda}$ |  $\hat{P}_{0}$ |  $\hat{\mathbf{P}}^{2}$
---|---|---|---|---|---|---
$\bm{\Psi}^{\lambda}(\mathbf{r},t)$ |  |  |  |  |  |
$\bm{\Phi}^{\lambda}(\mathbf{r},t)$ |  |  |  |  |  |
Table 1: Behaviour of the photon wave functions in vacuum,
$\bm{\Psi}^{\lambda}(\mathbf{r},t)$, and in a homogeneous medium,
$\bm{\Phi}^{\lambda}(\mathbf{r},t)$, under different Casimir operators. The
check mark indicates that the wave function is an eigenvector of the indicated
Casimir operator. $\hat{C}_{0}=\hat{P}_{0}/|\hat{P}_{0}|$ is the sign of
frequency, $\hat{C}_{1}=\hat{\mathbf{P}}^{2}-\hat{P}_{0}^{2}$ is the modulus
of the 4-momentum and
$\hat{C}_{2}=\hat{\mathbf{W}}^{2}-(\hat{\mathbf{J}}\cdot\hat{\mathbf{P}})^{2}$
the modulus of the Pauli-Lubanski pseudo-vector.
$\hat{\mathbf{W}}=\hat{P}_{0}\hat{\mathbf{J}}+\hat{\mathbf{K}}\times\hat{\mathbf{P}}$
is the spatial component of the Pauli-Lubanski pseudovector, with
$\hat{\mathbf{K}}$ the boost operator, i.e. the generator of Lorentz
transformations. Finally, note that $\hat{P}_{0}$ and $\hat{\mathbf{P}}^{2}$
operators label different UIRs of the $P_{\scriptscriptstyle 3,1}$ subgroup,
whereas Poincaré UIRs are superpositions of monochromatic waves with different
frequencies.

All the analysis above indicates that the dynamic equations that are employed
to study the propagation of electromagnetic waves in infinitely homogeneous
media can be derived from pure group theoretical arguments. And, in
particular, it highlights the fundamental role that the Casimir operators
$\hat{P}_{0}$, $\hat{\Lambda}$ and $\hat{\mathbf{P}}^{2}$ play in the
description of electromagnetic waves propagating in material environments.
Also, note that the symmetry breaking principle had previously been applied
for non-relativistic massive particles Winternitz (1977); Beckers  _et al._
(1977). Indeed, the time-independent Schrödinger’s equation in an infinitely
homogeneous medium can also be associated with Eqs. (4)-(6) simply by fixing
$\lambda=0$ and $k=\sqrt{2m(E-V)}$, where $E$ is the eigenvalue of
$\hat{P}_{0}$ operator and $V$ represents a constant potential Winternitz
(1977); Beckers  _et al._ (1977); Kalnins  _et al._ (1975); Mackey (1978).
This mathematical analogy between the dynamic equations and wave functions
describing massless and massive particles will aid us later in the discussion
of the Kerker phenomena.

##  IV Electromagnetic waves in piecewise media: impedance and refractive
index matching

In static piecewise homogeneous media there is only one symmetry left, i.e.
the one-parameter subgroup of time translations, $T$. It is so because this
type of environments are not invariant under spatial translations and
rotations in three dimensions (see Fig. 2). As a result, the only symmetry of
the system is due to the static nature of the medium.

Only a few things can be generally stated about the dynamics in this type of
media, i.e. that frequency, $\omega$, is conserved and that the
electromagnetic wave solutions are eigenstates of $\hat{P}_{0}$. Indeed, this
result is extensively employed to solve problems of very different nature in
physics such as time-independent electromagnetic or quantum scattering
problems. The UIRs of the group of time translations can be labelled as (see
Ref. Tung (1985), Chapter 6, Section 6):

$\hat{P}_{0}\bm{\psi}(\mathbf{r},t)=\omega\bm{\psi}(\mathbf{r},t).$ (9)

This condition is equivalent to Eq. (4). As before, it implies that the time
dependence of the solutions in piecewise homogeneous media is fixed and it is
given by a complex exponential. On the contrary, the spatial functional form
of the solutions will strongly depend on the specific symmetries of the
particular system one may consider. The generators of the subgroups of $E(3)$
are not the same and, thus, solutions in piecewise media are, in general,
eigenfunctions of different sets of operators Beckers  _et al._ (1977); Patera
_et al._ (1976a). As a result, conserved quantities are expected to be
strongly dependent on the particular geometries.

In addition, there exist two situations under which electromagnetic fields
have an unusual behaviour in piecewise homogeneous media. Such conditions can
be identified from the form that Maxwell’s equations acquire in terms of the
monochromatic RS vector, $\bm{\Phi}^{\lambda}(\mathbf{r},t)$. Indeed, in a
generic inhomogeneous medium Maxwell’s equations can be expressed as
Bialynicki-Birula (1996); Lasa-Alonso  _et al._ (2023):

$\displaystyle i\partial_{t}\bm{\Phi}^{+}$
$\displaystyle=\frac{1}{\sqrt{n}}\nabla\times\left(\frac{\bm{\Phi}^{+}}{\sqrt{n}}\right)+\frac{1}{n}\nabla\ln\sqrt{Z}\times\bm{\Phi}^{-}$
(10) $\displaystyle i\partial_{t}\bm{\Phi}^{-}$
$\displaystyle=-\frac{1}{\sqrt{n}}\nabla\times\left(\frac{\bm{\Phi}^{-}}{\sqrt{n}}\right)-\frac{1}{n}\nabla\ln\sqrt{Z}\times\bm{\Phi}^{+},$
(11)

where, for convenience, we have omitted the dependence on the position vector
and time. Also, we have defined the local impedance
$Z(\mathbf{r})=\sqrt{\mu(\mathbf{r})/\varepsilon(\mathbf{r})}$ and local
refractive index
$n(\mathbf{r})=\sqrt{\varepsilon(\mathbf{r})\mu(\mathbf{r})}$. Given the form
of Maxwell’s equations in Eqs. (10)-(11), we can identify two specific types
of environments in which the electromagnetic field may have a particular
behaviour: media for which the impedance is constant ($\nabla Z=0$) or
environments for which the refractive index is constant ($\nabla n=0$). We
usually denote these two situations as the impedance matching and the
refractive index matching conditions, respectively, and they are discussed in
Sections II and IV of Kerker’s paper Kerker  _et al._ (1983). Note that, in
piecewise media, the impedance matching condition is fulfilled when
$Z_{j}=\sqrt{\mu_{j}/\varepsilon_{j}}$ is constant in all domains $V_{j}$. On
the other hand, the refractive index matching condition is fulfilled whenever
$n_{j}=\sqrt{\varepsilon_{j}\mu_{j}}$ is constant in all domains $V_{j}$.

In this line, Fernandez-Corbaton and coworkers indicated that the impedance
matching condition leads to the conservation of electromagnetic helicity
Fernandez-Corbaton  _et al._ (2013). It was shown that, whenever
$\mu_{j}/\varepsilon_{j}$ is constant, Maxwell’s equations in the whole
piecewise medium remain invariant under electromagnetic duality
transformations. As helicity had previously been identified as the generator
of the duality transformation Calkin (1965), it was then concluded that
helicity had to be preserved in an impedance-matched piecewise homogeneous
medium. However, there is also a dynamical way of understanding the
conservation of helicity in impedance-matched media. Indeed, given the form of
Maxwell’s equations specified in Eqs. (10)-(11), it can be checked that the
impedance matching condition makes the two helicity components of the
electromagnetic field, $\bm{\Phi}^{+}(\mathbf{r},t)$ and
$\bm{\Phi}^{-}(\mathbf{r},t)$ to be decoupled. This implies that, if we fix
the initial conditions at time $t=0$ to contain a single helicity component,
the solutions in an impedance-matched piecewise medium will have a single
helicity component at times $t>0$. In the case of piecewise media, this
implies that the solutions under the impedance matching conditions fulfill:
$\hat{\Lambda}\bm{\psi}_{\lambda}(\mathbf{r},t)=\lambda\bm{\psi}_{\lambda}(\mathbf{r},t)$,
with $\lambda=\pm 1$. This means that we can also infer the conservation of
helicity in piecewise media from the fact that eigenstates of $\hat{\Lambda}$,
acting as the Casimir operator in Eq. (5), remain eigenstates of
$\hat{\Lambda}$ in dual media.

In this sense, the helicity operator has a double role. On the one hand is the
generator of dual transformations, but as seen in Eq. (5), it also acts as a
Casimir operator. This could bring the question of whether Casimir operators
are fundamentally interesting to study electromagnetic wave dynamics in
piecewise homogeneous media. We will show now that in the case of the
refractive index matching condition, the conserved quantity is not associated
with the generator of a transformation, but with another Casimir operator:
$\hat{\mathbf{P}}^{2}$. In this case, the conservation of a physical magnitude
can be inferred from how electromagnetic wave solutions are built in piecewise
media. Indeed, as it is shown in Fig. 2, solutions of electromagnetic waves
propagating in this type of environments are constructed by solving
Helmholtz’s equation in each domain $V_{j}$ and, then, applying boundary
conditions. This implies that, in a generic piecewise homogeneous medium,
electromagnetic wave solutions fulfill,
$-\nabla^{2}\bm{\psi}(\mathbf{r},t)=k_{j}^{2}\bm{\psi}(\mathbf{r},t)$ for
$\mathbf{r}\in V_{j}$, where $k_{j}=\omega n_{j}$. Note that, for a generic
piecewise homogeneous medium, $k_{j}$ changes depending on the region of space
we may consider. Thus, the eigenvalue of $\hat{\mathbf{P}}^{2}=-\nabla^{2}$
operator varies from one region $V_{j}$ to another $V_{j^{\prime}}$, as long
as $n_{j}\neq n_{j^{\prime}}$. However, whenever the refractive index matching
condition is fulfilled ($n_{j}=n_{j^{\prime}},~{}\forall j,j^{\prime}$), the
wavevector modulus is constant all over the medium and the square of linear
momentum operator fulfills:
$\hat{\mathbf{P}}^{2}\bm{\psi}_{k}(\mathbf{r},t)=k^{2}\bm{\psi}_{k}(\mathbf{r},t)$,
where now $k$ is fixed. This is exactly the way in which the conservation of
square of linear momentum is represented in Eq. (6). In this line, the
refractive index matching condition recovers that same relation, but in
piecewise environments.

Figure 2: Sketch of a piecewise homogeneous medium. Electromagnetic wave
solutions are obtained by solving Helmholtz’s equation and applying boundary
conditions at the interfaces.

Following the previous discussion on helicity conservation, we may try to link
the conservation of $\hat{\mathbf{P}}^{2}$ with the restoration of a symmetry.
This, however, is not possible because the square of linear momentum is not a
generator of any continuous symmetry transformation. Therefore, we are led to
comprehend the conservation of $\hat{\mathbf{P}}^{2}$ from its condition of
Casimir invariant. Indeed, Casimir operators of a group _G_ may remain as
conserved quantities in environments whose symmetry group is a subgroup _G
${}_{i}\subset$ G_ Patera  _et al._ (1976b). This phenomenon is not a
particularity of electromagnetic waves, it is known to occur also in the
dynamics of massive particles. Let us put forward a particular example which
very closely mimics the refractive index matching condition for
electromagnetic waves. As we have shown in Section III, the conservation of
$\hat{\mathbf{P}}^{2}$ for non-relativistic massive particles can also be
related to the symmetries of Euclidean space Mackey (1978). Thus, in
principle, we may expect the square of linear momentum to be exclusively
preserved when massive particles propagate in free space. This, however, is
not true. There is a particular interaction potential, i.e. the hard sphere
potential, for which $\hat{\mathbf{P}}^{2}$ is a conserved quantity (see
Appendix B). The same also holds for classical mechanics, where the kinetic
energy, proportional to the square of the linear momentum, is known to be
preserved provided that the interactions are elastic. To the best of our
knowledge, the conservation of kinetic energy in elastic collisions has not
been linked before with the restoration of a space-time symmetry.

There are more cases in which a Casimir invariant of a group _G_ is preserved
in an environment whose symmetry group is a subgroup _G i_. A well-known
example is the conservation of mass _M_ and spin _S_. As we have shown in
Section II, the conservation of these magnitudes can be concluded from their
condition of Casimir invariants of the Poincaré group. This, in principle,
would imply that _M_ and _S_ are exclusively preserved for relativistic
massive particles propagating in vacuum. We know, however, that these
magnitudes are preserved in a great deal of problems for which the symmetry
group is a subgroup of the Poincaré group. For instance, there are many
situations in which the mass and spin of the particles are preserved in the
framework of non-relativistic quantum scattering theory. Indeed, mass is quite
generally assumed to be conserved in this context, whereas spin may not be
conserved only in particular cases such as spin-orbit or nucleon-nucleon
interactions Taylor (1972). All these examples point in the direction that the
conservation of certain physical magnitudes cannot be related with the
restoration of a symmetry, but to their condition of Casimir invariants.

##  V Origin of the Kerker phenomena

Finally, let us apply the previous analysis on conserved quantities to a case
of special significance, i.e. the emergence of the Kerker phenomena in
magnetic spheres. To be clear in the wording, instead of the usual phrase
_Kerker conditions_ , we employ _Kerker phenomena_ to denote both the
impedance and refractive index matching conditions in magnetic spheres.

Figure 3: Emergence of the Kerker phenomena in a magnetic sphere embedded in a
homogeneous surrounding. The helicity expectation value,
$\langle\Lambda\rangle$, quantifies the helicity of the scattered
electromagnetic field. $s$ is the impedance contrast and $m$ the refractive
index contrast. Frequency, $\omega$, is preserved at every point due to the
static nature of the piecewise system. Helicity, is preserved whenever the
impedance is constant (horizontal dashed line). Square of linear momentum, is
preserved whenever the refractive index is constant (vertical dashed line).
The singular point $[m,s]=[1,1]$ represents an infinitely homogeneous medium.

In Fig. 3, we show the analysis of conserved quantities associated with the
helicity map of a magnetic sphere with size parameter $x=3$. Let us briefly
remind that $\langle\Lambda\rangle$ represents the helicity expectation value,
which is an observable that determines whether electromagnetic helicity of the
incident wave is conserved, $\langle\Lambda\rangle=1$, or completely flipped,
$\langle\Lambda\rangle=-1$, upon scattering Lasa-Alonso  _et al._ (2022).
Along the vertical axis we tune the impedance contrast, $s$, which is the
ratio between the impedances of the sphere and the surrounding medium. In the
horizontal axis, we fix the refractive index contrast, $m$, which is the ratio
between the refractive indices. Due to the static nature of the media
involved, all the solutions represented in the figure are constructed as
eigenstates of the generator of time-translations, $\hat{P}_{0}$. Then, under
the impedance matching condition, along the line $s=1$, helicity is conserved
and, thus, solutions are eigenstates of $\hat{\Lambda}$. In addition to this,
under the index matching condition, whenever $m=1$, square of linear momentum
is conserved and, as a result, solutions are eigenstates of
$\hat{\mathbf{P}}^{2}$ operator. Note that, whenever $m=1$, the helicity of
the scattered field is almost completely flipped (see Fig. 3) and, thus, this
condition has been denoted as the resonant helicity mixing condition Lasa-
Alonso  _et al._ (2023). On the other hand, the line $s=1/m$ represents the
response of nonmagnetic materials, i.e. those for which $\mu=1$. Finally, in
the singular point $[m,s]=[1,1]$ of the colormap solutions are eigenstates of
all three operators $\hat{P}_{0}$, $\hat{\Lambda}$ and $\hat{\mathbf{P}}^{2}$.
This is because the particular case $m=s=1$, represents an infinitely
homogeneous medium and, by means of Eqs. (4)-(6), electromagnetic waves are
constructed as eigenstates of the three Casimir operators of
$P_{\scriptscriptstyle{{3,1}}}$.

The Kerker phenomena can also be interpreted making use of Maxwell’s equations
as expressed in Eq. (10)-(11). Indeed, under both matching conditions, i.e.
$\nabla Z=0$ and $\nabla n=0$, the environment ceases to be inhomogeneous and
it becomes infinitely homogeneous. In this situation, Maxwell’s equations
converge to the analytical form dictated by Eqs. (4)-(6). In other words,
electromagnetic wave solutions are constructed as eigenstates of the operators
$\hat{P}_{0}$, $\hat{\Lambda}$ and $\hat{\mathbf{P}}^{2}$. This situation is
equivalent to the one represented by $[m,s]=[1,1]$ point in Fig. 3. And, then,
it is clear that Eqs. (10)-(11) indicate two preferential directions in which
the homogeneity of space maybe broken. These two preferential directions are
associated with the two matching conditions. On the one hand, homogeneity of
space may be broken while still keeping the impedance constant. In piecewise
homogeneous media, this leads to the conservation of both $\hat{P}_{0}$ and
$\hat{\Lambda}$ and the situation is equivalent to the $s=1$ line in Fig. 3.
On the other hand, homogeneity of space may also be broken while keeping the
refractive index constant. In piecewise homogeneous media, this leads to the
conservation of $\hat{P}_{0}$ and $\hat{\mathbf{P}}^{2}$ and the situation is
equivalent to the $m=1$ line in Fig. 3.

In our view, this analysis provides a fundamental interpretation of Maxwell’s
equations in inhomogeneous media as expressed in Eqs. (10)-(11). Please note
that such a particular form of expressing the equations is obtained through a
simple change of basis. Indeed, instead of the usual electric,
$\mathbf{E}(\mathbf{r},t)$, and magnetic, $\mathbf{H}(\mathbf{r},t)$, fields
we rewrite Maxwell’s equations in terms of the monochromatic RS vector,
$\bm{\Phi}^{\lambda}(\mathbf{r},t)$, which is linked with the UIRs of the
$P_{\scriptscriptstyle{{3,1}}}$ subgroup of the Poincaré group (see Section
III). As a result, we obtain a completely equivalent form of the equations in
which instead of the usual material parameters, derivatives of the local
impedance, $Z(\mathbf{r})$, and refractive index, $n(\mathbf{r})$, appear.
Finally, we find that these material parameters are closely related with the
Casimir invariants of $P_{\scriptscriptstyle{{3,1}}}$. It really seems like
the connection of Maxwell’s equations with space-time symmetries is clarified
when expressing them in terms of the RS vector. In this line, Kerker phenomena
represent an expression of such a fundamental connection in a particular
electromagnetic scattering problem.

##  VI Conclusion

In conclusion, we have revisited the link of Maxwell’s equations with the
theory of representations of continuous groups. Starting from the Poincaré
group, we have shown that its subgroups may be employed to analyze conserved
quantities and wave solutions in different material environments. It is the
first time that this procedure, which had previously been employed in the
framework of quantum mechanics, is systematically applied to the study of
electromagnetic waves.

In this context, we have obtained a set of fundamental results. First, we have
shown that Bialynicki-Birula’s photon wave function in vacuum,
$\bm{\Psi}^{\lambda}(\mathbf{r},t)$, can be constructed from pure group
theoretical arguments. Second, we have shown that Maxwell’s equations in
infinitely homogeneous media can be derived from the Casimir invariants of
$P_{\scriptscriptstyle{{3,1}}}$ subgroup of the Poincaré group. As a result,
we have identified the fundamental role that the monochromatic RS vector,
$\bm{\Phi}^{\lambda}(\mathbf{r},t)$, plays in the description of
electromagnetic waves propagating in matter. Third, we have seen that the
matching conditions in piecewise media are associated with the conservation of
two Casimir invariants of $P_{\scriptscriptstyle{{3,1}}}$, i.e. helicity and
square of linear momentum. Finally, we have shown that Kerker phenomena
constitute a particular example of how the matching conditions emerge in
piecewise homogeneous media.

Our contribution not only gives a sound mathematical foundation to the origin
of the Kerker phenomena, but also invites to the search of new effects based
on similar group theoretical arguments. In addition, we have identified the
monochromatic RS vector in Eq. (8) as the fundamental object associated with
the description of electromagnetic waves propagating in matter. This may be
useful to obtain more insightful descriptions of optical phenomena in
piecewise homogeneous systems.

## Acknowledgments

J.L.A. acknowledges discussions with Jorge Olmos-Trigo. J.L.A., C.D. and A.G.E
acknowledge support from Project No. PID2019-109905GA-C22 of the Spanish
Ministerio de Ciencia, Innovación y Universidades (MICIU), IKUR Strategy under
the collaboration agreement between Ikerbasque Foundation and DIPC on behalf
of the Department of Education of the Basque Government, Gipuzkoa Quantum:
QUAN-000021-01 project from Diputacion Foral de Gipuzkoa, Programa de ayudas
de apoyo a los agentes de la Red Vasca de Ciencia, Tecnología e Innovación
acreditados en la categoría de Centros de Investigación Básica y de Excelencia
(Programa BERC) from the Departamento de Universidades e Investigación del
Gobierno Vasco and Centros Severo Ochoa AEI/CEX2018-000867-S from the Spanish
Ministerio de Ciencia e Innovación. G.M.T received funding from the IKUR
Strategy under the collaboration agreement between Ikerbasque Foundation and
DIPC/MPC on behalf of the Department of Education of the Basque Government.

##  Appendix A Boosts acting over a monochromatic plane wave with well-defined
helicity

Let us analyze the action of the Lorentz boost in the $OZ$ direction,
$\hat{L}_{z}(\xi)$, over the standard vector. In general, if a boost is
carried out in the $\mathbf{b}/|\mathbf{b}|$ direction, a generic
electromagnetic field with well-defined helicity, $\mathbf{G}_{\pm}$, is
transformed as Jackson (1999); Fernández-Corbaton (2014):

$\mathbf{G}^{\prime}_{\pm}=\gamma\left(\mathbf{G}_{\pm}\mp
i\mathbf{b}\times\mathbf{G}_{\pm}\right)-\frac{\gamma^{2}}{\gamma+1}\mathbf{b}\left(\mathbf{b}\cdot\mathbf{G}_{\pm}\right),$
(12)

where $|\mathbf{b}|=v$ and $\gamma=(1-|\mathbf{b}|^{2})^{-1/2}$. Here, we are
interested the transformation properties of the field
$\mathbf{G}_{\pm}=\mathbf{u}_{\pm}e^{i|\mathbf{k}_{l}|(z-t)}$, with
$\mathbf{u}_{\pm}=(1,\pm i,0)$. Moreover, as the boost is along the $OZ$ axis,
we need to fix $\mathbf{b}/|\mathbf{b}|=\hat{z}$ and, as a result, we have
that $i\mathbf{b}\times\mathbf{u}_{\pm}=\pm|\mathbf{b}|\mathbf{u}_{\pm}$. It
can also be checked that $\mathbf{b}$ and $\mathbf{G}_{\pm}$ are orthogonal,
i.e. $\mathbf{b}\cdot\mathbf{G}_{\pm}=0$, which makes the second term in Eq.
(12) vanish. Finally, applying the inverse Lorentz transformation to the
coordinates, i.e. from $(\mathbf{r},t)$ to $(\mathbf{r}^{\prime},t^{\prime})$,
the standard vector finally results in:

$\mathbf{G}^{\prime}_{\pm}=\gamma(1-|\mathbf{b}|)\mathbf{u}_{\pm}e^{i\gamma(1-|\mathbf{b}|)|\mathbf{k}_{l}|(z^{\prime}-t^{\prime})}.$
(13)

The expression in the equation above shows that apart from changing its
amplitude, the standard vector also changes the modulus of its linear momentum
vector as:

$|\mathbf{k}^{\prime}|=\gamma(1-|\mathbf{b}|)|\mathbf{k}_{l}|=(\cosh\xi-\sinh\xi)|\mathbf{k}_{l}|=e^{-\xi}|\mathbf{k}_{l}|.$
(14)

Please note that different sign conventions are considered in Wu-Ki Tung and
Jackson for the Lorentz boosts (see, for instance, Eq. 10.1-9 in Wu-Ki Tung
and Eq. 11.21 in Jackson). The result in Eq. (14) implies that the boost
parameter $\xi$ controls the modulus of the linear momentum,
$\mathbf{k}^{\prime}$, of the monochromatic plane wave for a fixed
$|\mathbf{k}_{l}|$. As the frequency and the modulus of linear momentum are
proportional, one finally gets that $\omega$ is also modulated by the
parameter $\xi$. Importantly, note that the boost $\hat{L}_{z}(\xi)$ does not
modify either the helicity or the direction of propagation of the standard
vector.

##  Appendix B Conservation of $\hat{\mathbf{P}}^{2}$ for quantum non-
relativistic massive particles

In this appendix, we show that the conservation of $\hat{\mathbf{P}}^{2}$ for
quantum non-relativistic massive particles does not respond to the restoration
of a symmetry, but just to the properties of particular scattering potentials.

In our previous discussion we have been focused on the description of
electromagnetic waves in different media. However, the mathematical tool that
has been employed, i.e. the symmetry breaking principle applied to the
Poincaré group, does not mind about the nature of the particle it is
describing. In other words, the mathematical principle operates regardless of
whether the physical theory represents a massless or massive particle Patera
_et al._ (1975a). In this line, we have shown that the identities given by
Eqs. (4)-(6) can also represents the dynamic equations of a non-relativistic
massive particle, i.e. the time-independent Schrödinger’s equation with a
constant potential. This implies that the UIRs of
$P_{\scriptscriptstyle{{3,1}}}$ and its subgroups also play a fundamental role
in the determination of the wave functions of massive particles Beckers  _et
al._ (1977). In the upcoming, we focus on the role of the square of linear
momentum operator: we seek a physical situation for massive particles in which
$\hat{\mathbf{P}}^{2}$ is a conserved quantity, regardless of the geometry of
the problem.

Such a situation would be reached by the scattering of a massive particle in
an inhomogeneous potential, $\hat{V}(\mathbf{r})$, for which the solutions
still remain being eigenstates of the $\hat{\mathbf{P}}^{2}$ operator,
regardless of the symmetries of the problem. This would similarly imply that a
Casimir operator of $P_{\scriptscriptstyle{3,1}}$ group can also be preserved
in inhomogeneous environments. Interestingly, there is a potential that
fulfills the requirements above: the generalized quantum hard sphere
potential, which can be defined as:

$\hat{V}_{hs}(\mathbf{r})=\left\\{\begin{array}[]{lcc}\infty&\text{if}&\mathbf{r}\in
U_{i}\\\ \\\ V_{0}&\text{else},\end{array}\right.$ (15)

where $U_{i}$ with $i=1,2,...,N$, are $N$ arbitrary regions of the three
dimensional space and $V_{0}$ represents a finite and constant potential
energy. Crucially, due to the arbitrariness of the $U_{i}$ regions, such a
potential can be constructed with any possible geometry. The solutions of the
time-independent Schrödinger’s equations in such a potential,
$\psi_{hs}(\mathbf{r})$, are most generally determined by the following
constrains

$\left\\{\begin{array}[]{lcc}\psi_{hs}(\mathbf{r},t)=0&\text{if}&\mathbf{r}\in
U_{i}\\\ \\\
\hat{\mathbf{P}}^{2}\psi_{hs}(\mathbf{r},t)=2m(E-V_{0})\psi_{hs}(\mathbf{r},t)&\text{else}.\end{array}\right.$
(16)

As a result, it can be checked that the wave functions which are solutions to
the generalized quantum hard sphere potential fulfill:
$\hat{\mathbf{P}}^{2}\psi_{hs}(\mathbf{r},t)=p^{2}\psi_{hs}(\mathbf{r},t)$,
$\forall~{}\mathbf{r},t$ with $p=\sqrt{2m(E-V_{0})}$.

From the properties above, it directly follows that the square of linear
momentum is a conserved magnitude in the scattering with a hard sphere
potential. Indeed, one can show that the expected value of
$\hat{\mathbf{P}}^{2}$ remains constant for such a potential, i.e.

$\frac{d}{dt}\left(\int
d\mathbf{r}~{}\psi^{*}_{hs}(\mathbf{r},t)\left[\hat{\mathbf{P}}^{2}\psi_{hs}(\mathbf{r},t)\right]\right)=0.$
(17)

This implies that if at an initial instant solutions are eigenstates of
$\hat{\mathbf{P}}^{2}$, this property continues to hold in the curse of time.
In other words, one usually says that $p$ is a good quantum number. Note that
exactly the same holds for the kinetic energy,
$\hat{T}=\hat{\mathbf{P}}^{2}/2m$, as the mass is a fixed parameter in non-
relativistic quantum mechanics. Both in quantum and classical scattering
theory the conservation of kinetic energy is just invoked as a property of
certain type of interactions. As a result, we conclude that the preservation
of the square of linear momentum is not, in general, associated with the
restoration of a space-time symmetry.

## References

  * Kerker  _et al._ (1983) M. Kerker, D.-S. Wang, and C. L. Giles, J. Opt. Soc. Am. 73, 765 (1983).
  * Giles and Wild (1982) C. L. Giles and W. J. Wild, Applied Physics Letters 40, 210 (1982).
  * Nieto-Vesperinas  _et al._ (2011a) M. Nieto-Vesperinas, R. Gómez-Medina, and J. J. Sáenz, J. Opt. Soc. Am. A 28, 54 (2011a).
  * García-Etxarri and Dionne (2013) A. García-Etxarri and J. A. Dionne, Phys. Rev. B 87, 235409 (2013).
  * Pfeiffer and Grbic (2013) C. Pfeiffer and A. Grbic, Phys. Rev. Lett. 110, 197401 (2013).
  * Person  _et al._ (2013) S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, Nano Letters 13, 1806 (2013).
  * Fu  _et al._ (2013) Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, Nature Communications 4 (2013).
  * Alaee  _et al._ (2015) R. Alaee, R. Filter, D. Lehr, F. Lederer, and C. Rockstuhl, Opt. Lett. 40, 2645 (2015).
  * Babicheva and Evlyukhin (2017) V. E. Babicheva and A. B. Evlyukhin, Laser & Photonics Reviews 11, 1700132 (2017).
  * Bag  _et al._ (2018) A. Bag, M. Neugebauer, P. Woźniak, G. Leuchs, and P. Banzer, Phys. Rev. Lett. 121, 193902 (2018).
  * Barhom  _et al._ (2019) H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, Nano Letters 19, 7062 (2019).
  * Lasa-Alonso  _et al._ (2020a) J. Lasa-Alonso, D. R. Abujetas, A. Nodar, J. A. Dionne, J. J. Sáenz, G. Molina-Terriza, J. Aizpurua, and A. García-Etxarri, ACS Photonics 7, 2978–2986 (2020a).
  * Xu  _et al._ (2020) X. Xu, M. Nieto-Vesperinas, C.-W. Qiu, X. Liu, D. Gao, Y. Zhang, and B. Li, Laser & Photonics Reviews 14, 1900265 (2020).
  * Lasa-Alonso  _et al._ (2020b) J. Lasa-Alonso, M. Molezuelas-Ferreras, J. J. M. Varga, A. García-Etxarri, G. Giedke, and G. Molina-Terriza, New Journal of Physics 22, 123010 (2020b).
  * Olmos-Trigo  _et al._ (2020a) J. Olmos-Trigo, C. Sanz-Fernández, D. R. Abujetas, J. Lasa-Alonso, N. de Sousa, A. García-Etxarri, J. A. Sánchez-Gil, G. Molina-Terriza, and J. J. Sáenz, Physical Review Letters 125, 073205 (2020a).
  * Fan  _et al._ (2021) K. Fan, I. V. Shadrivov, A. E. Miroshnichenko, and W. J. Padilla, Opt. Express 29, 10518 (2021).
  * Gerasimov  _et al._ (2021) V. S. Gerasimov, A. E. Ershov, R. G. Bikbaev, I. L. Rasskazov, I. L. Isaev, P. N. Semina, A. S. Kostyukov, V. I. Zakomirnyi, S. P. Polyutov, and S. V. Karpov, Phys. Rev. B 103, 035402 (2021).
  * Qin  _et al._ (2022) F. Qin, Z. Zhang, K. Zheng, Y. Xu, S. Fu, Y. Wang, and Y. Qin, Phys. Rev. Lett. 128, 193901 (2022).
  * Xiong  _et al._ (2022) L. Xiong, H. Ding, Y. Lu, and G. Li, Journal of Physics D: Applied Physics 55, 185106 (2022).
  * Calkin (1965) M. G. Calkin, American Journal of Physics 33, 958 (1965).
  * Fernandez-Corbaton  _et al._ (2013) I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, Physical Review Letters 111, 060401 (2013).
  * Zambrana-Puyalto  _et al._ (2013) X. Zambrana-Puyalto, I. Fernandez-Corbaton, M. L. Juan, X. Vidal, and G. Molina-Terriza, Opt. Lett. 38, 1857 (2013).
  * Fernandez-Corbaton (2013) I. Fernandez-Corbaton, Optics Express 21, 29885 (2013).
  * Lasa-Alonso  _et al._ (2022) J. Lasa-Alonso, J. Olmos-Trigo, A. García-Etxarri, and G. Molina-Terriza, Mater. Adv. 3, 4179 (2022).
  * Nieto-Vesperinas  _et al._ (2011b) M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, J. Opt. Soc. Am. A 28, 54 (2011b).
  * Olmos-Trigo  _et al._ (2020b) J. Olmos-Trigo, D. R. Abujetas, C. Sanz-Fernández, J. A. Sánchez-Gil, and J. J. Sáenz, Physical Review Research 2, 013225 (2020b).
  * Lasa-Alonso  _et al._ (2023) J. Lasa-Alonso, J. Olmos-Trigo, C. Devescovi, P. Hernández, A. García-Etxarri, and G. Molina-Terriza, Phys. Rev. Res. 5, 023116 (2023).
  * Note (1) In this context, we use the term _photon wave function_ as a legacy from Bialynicki-Birula’s work. We will always deal in this work with classical electromagnetic fields, and the photon wave functions are electromagnetic wave solutions.
  * Bialynicki-Birula and Bialynicka-Birula (2017) I. Bialynicki-Birula and Z. Bialynicka-Birula, Journal of Optics 19, 125201 (2017).
  * Wigner (1939) E. Wigner, Annals of Mathematics 40, 149 (1939).
  * Bargmann and Wigner (1948) V. Bargmann and E. P. Wigner, Proceedings of the National Academy of Sciences 34, 211 (1948).
  * Tung (1985) W.-K. Tung,  _Group Theory in Physics_ (World Scientific, 1985).
  * Bargmann (1947) V. Bargmann, Annals of Mathematics 48, 568 (1947).
  * Lomont and Moses (1962) J. S. Lomont and H. E. Moses, Journal of Mathematical Physics 3, 405 (1962).
  * Bacry (1976) H. Bacry, Il Nuovo Cimento 32 A, 448 (1976).
  * Gersten (1999) A. Gersten, Foundations of Physics Letters 12, 291 (1999).
  * Dirac (1936) P. A. M. Dirac, Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences 155, 447 (1936).
  * Landau and Lifshitz (1984) L. D. Landau and E. M. Lifshitz,  _Electrodynamics of continuous media_ (Pergamon Press, 1984).
  * Kong (1972) J. A. Kong, Proceedings of the IEEE 60, 1036 (1972).
  * Winternitz (1977) P. Winternitz, in  _Group Theoretical Methods in Physics_ , edited by R. T. Sharp and B. Kolman (Academic Press, 1977) pp. 549–572.
  * Patera  _et al._ (1975a) J. Patera, P. Winternitz, and H. Zassenhaus, Journal of Mathematical Physics 16, 1597 (1975a).
  * Patera  _et al._ (1975b) J. Patera, P. Winternitz, and H. Zassenhaus, Journal of Mathematical Physics 16, 1615 (1975b).
  * Patera  _et al._ (1977) J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Journal of Mathematical Physics 18, 2259 (1977).
  * Gagnon and Winternitz (1988) L. Gagnon and P. Winternitz, Journal of Physics A: Mathematical and General 21, 1493 (1988).
  * Beckers  _et al._ (1977) J. Beckers, J. Patera, M. Perroud, and P. Winternitz, Journal of Mathematical Physics 18, 72 (1977).
  * Patera  _et al._ (1976a) J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Journal of Mathematical Physics 17, 977 (1976a).
  * Bliokh and Nori (2015) K. Y. Bliokh and F. Nori, Physics Reports 592, 1 (2015), transverse and longitudinal angular momenta of light.
  * Fernandez-Corbaton  _et al._ (2012) I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, Physical Review A 86, 042103 (2012).
  * Sarkar and Halas (1997) D. Sarkar and N. J. Halas, Phys. Rev. E 56, 1102 (1997).
  * Nienhuis (2016) G. Nienhuis, Phys. Rev. A 93, 023840 (2016).
  * Mackey (1978) G. W. Mackey,  _Unitary Group Representations in Physics, Probability and Number Theory_ (The Benjamin/Cummings Publishing Company, Inc., 1978).
  * Rossmann (2002) W. Rossmann,  _Lie groups: An introduction through linear groups_ (Oxford University Press, 2002).
  * Vavilin and Fernandez-Corbaton (2023) M. Vavilin and I. Fernandez-Corbaton, “The polychromatic t-matrix,” (2023), arXiv:2306.07776 .
  * Kalnins  _et al._ (1975) E. G. Kalnins, J. Patera, R. T. Sharp, and P. Winternitz, in  _Group Theory and its Applications_ , edited by E. M. Loebl (Academic Press, 1975) pp. 369–464.
  * Bialynicki-Birula (1996) I. Bialynicki-Birula, in  _Progress in Optics_ , Vol. 36, edited by E. Wolf (Elsevier, 1996) Chap. 5, pp. 245–294.
  * Patera  _et al._ (1976b) J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Journal of Mathematical Physics 17, 977 (1976b).
  * Taylor (1972) J. R. Taylor,  _Scattering theory: the quantum theory of nonrelativistic collisions_ (John Wiley and Sons, Inc., 1972).
  * Jackson (1999) J. D. Jackson,  _Classical Electrodynamics_ (John Wiley and Sons, Inc., 1999).
  * Fernández-Corbaton (2014) I. Fernández-Corbaton, “Helicity and duality symmetry in light matter interactions: theory and applications,” (2014).