# Electrodynamics of superconductors:
from Lorentz to Galilei at zero temperature

Luca Salasnich1,2,3 1Dipartimento di Fisica e Astronomia ”Galileo Galilei” and
Padua QTech Center, Universita di Padova, Via Marzolo 8, 35131, Padova, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8,
35131, Padova, Italy
3Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche, Via
Nello Carrara 2, 50127 Sesto Fiorentino, Italy

###### Abstract

We discuss the derivation of the electrodynamics of superconductors coupled to
the electromagnetic field from a Lorentz-invariant bosonic model of Cooper
pairs. Our results are obtained at zero temperature where, according to the
third law of thermodynamics, the entropy of the system is zero. In the
nonrelativistic limit we obtain a Galilei-invariant superconducting system
which differs with respect to the familiar Schrödinger-like one. From this
point of view, there are similarities with the Pauli equation of fermions
which is derived from the Dirac equation in the nonrelativistic limit and has
a spin-magnetic field term in contrast with the Schrödinger equation. One of
the peculiar effects of our model is the decay of a static electric field
inside a superconductor exactly with the London penetration length. In
addition, our theory predicts a modified D’Alembert equation for the massive
electromagnetic field also in the case of nonrelativistic superconducting
matter. We emphasize the role of the Nambu-Goldstone phase field which is
crucial to obtain the collective modes of the superconducting matter field. In
the special case of a nonrelativistic neutral superfluid we find a gapless
Bogoliubov-like spectrum, while for the charged superfluid we obtain a
dispersion relation that is gapped by the plasma frequency.

##  I Introduction

There is a renewed interest in the phenomenological description of the
superconductive electrodynamics taking explicitly in account relativistic
effects stenuit2001 ; hirsch2004 ; tajmar2008 ; hirsch2015 ; grigorishin2021
or the crucial role of the Nambu-Goldstone phase field nambu1960 ;
goldstone1961  writing low-frequency and long-wavelength Lagrangians for
neutral and charged fermionic superfluids popov1972 ; popov-book ; witten1989
; schakel1990 ; schakel1994 ; zhu1995 ; son2006 ; schakel-book ; sala2008 ;
sala2009 ; sala2013 ; tureci2023 . Quite surprisingly, in the relativistic
models of Refs. stenuit2001 ; hirsch2004 ; tajmar2008 ; hirsch2015 ;
grigorishin2021  the nonrelativistic limit of the relativistic matter field
was not considered, somehow forgetting that the electrons, and the Cooper
pairs, move at nonrelativistic velocities in the experimentally measured
superconducting materials on earth mermin-book ; annett-book ; degennes-book ;
ketterson-book .

In this paper we fill this gap by investigating the nonrelativistic limit of a
relativistic phenomenological model of bosonic Cooper pairs minimally coupled
to the electromagnetic field. Quite remarkably, from the initial Lorentz-
invariant setting we obtain a Galilei-invariant theory for the superconducting
matter field which cointains a crucial electromagnetic coupling term that is
absent in the standard minimally-coupled nonrelativistic Schrödinger field.
This is exactly the analog of the coupling between the spin and the magnetic
field one finds in the Pauli equation, which can be derived from the Dirac
equation in the nonrelativistic limit drell-book . By using our improved
nonrelativistic formulation of the charged matter field, and explicitly taking
into account the role of the Nambu-Goldstone phase field, working at zero
temperature, where the entropy of the system is also zero, we predict effects
which should be measurable at very low temperatures, close to the absolute
zero. Some of them have been previously suggested stenuit2001 ; hirsch2004 ;
tajmar2008 ; hirsch2015 ; grigorishin2021 , but only assuming a quite-
unphysical relativistic matter field inside the superconductor. In particular,
we suggest the decay of a static electric field inside a superconductor
exactly with the London penetration depth. In addition, we obtain a modified
D’Alembert equation for the massive electromagnetic waves inside the
nonrelativistic superconducting matter. We finally derive a gapped spectum for
the density oscillations of the charged superfluid made of Cooper pairs. It is
important to stress that many classical well-known experimental and
theoretical results of superconductivity mermin-book ; annett-book ; degennes-
book ; ketterson-book , such as the spontaneous symmetry breaking of gauge
invariance, the London penetration depth of the magnetic field, and the
collective modes of neutral superfluids, are fully recovered by our formalism.

##  II Relativistic Cooper pairs and minimal coupling

We assume that at zero temperature the Cooper pairs in a superconductor are
described by a relativistic Klein-Gordon klein1926 ; gordon1926  complex
scalar field $\varphi({\bf r},t)$ with Lagrangian density

$\mathscr{L}_{\rm 0}={\hbar^{2}\over
2mc^{2}}|\partial_{t}\varphi|^{2}-{\hbar^{2}\over
2m}|{\boldsymbol{\nabla}}\varphi|^{2}-{mc^{2}\over 2}|\varphi|^{2}-{\cal
E}(|\varphi|^{2})\;,$ (1)

where $m=2m_{e}$ is the mass of a Cooper pair with $m_{e}$ the electron mass,
$\hbar$ is the reduced Planck constant, and $c$ is the speed of light in
vacuum. Here ${\cal E}(|\varphi|^{2})$ is the bulk internal energy of the
system. The Lagrangian (1) is invariant with respect to Lorentz
transformations. A similar model was developed by Govaerts, Bertrand, and
Stenuit stenuit2001  and by Grigorishin grigorishin2021 . However, in Refs.
stenuit2001  and grigorishin2021  the system is supposed to be close to the
critical temperature $T_{c}$, with the temperature-dependent internal energy
${\cal E}(|\varphi|^{2})$ given by the familiar quadratic-quartic Mexican hat
potential. Here, instead, we work at zero temperature and, contrary to the
previous papers, we want to emphasize the emerging properties in the
nonrelativistic limit, where we obtain Maxwell-Proca equations stenuit2001 ;
grigorishin2021 ; tajmar2008  for the electromagnetic field coupled to the
nonrelativistic superconducting matter. Quite remarkably, we find that the
coupling between the electromagnetic field and the nonrelativistic matter
contains a term that is absent by applying the minimal coupling to the
electromagnetic field directly into a Schrödinger Lagrangian density. Remind
that in the nonrelativistic limit from the Dirac equation of fermions one gets
the Pauli equation (which has the spin) and not the Schrödinger equation
(which does not have the spin) drell-book . What is found here is the bosonic
analog of that phenomenon.

The Cooper pair has the electric charge $q=-2e$ with $e>0$ the modulus of the
electron charge. The coupling with the electromagnetic field is obtained with
the minimal substitution

$\displaystyle{\partial_{t}}$ $\displaystyle\to$
$\displaystyle{\partial_{t}}+i{q\over\hbar}\Phi$ (2)
$\displaystyle{\boldsymbol{\nabla}}$ $\displaystyle\to$
$\displaystyle{\boldsymbol{\nabla}}-i{q\over\hbar}{\bf A}$ (3)

where $\Phi({\bf r},t)$ is the electromagnetic scalar potential and ${\bf
A}({\bf r},t)$ is the electromagnetic vector potential, such that

$\displaystyle{\bf E}$ $\displaystyle=$
$\displaystyle-{\boldsymbol{\nabla}}\Phi-\partial_{t}{\bf A}$ (4)
$\displaystyle{\bf B}$ $\displaystyle=$
$\displaystyle{\boldsymbol{\nabla}}\wedge{\bf A}$ (5)

with ${\bf E}({\bf r},t)$ the electric field and ${\bf B}({\bf r},t)$ the
magnetic field.

In this way, the total Lagrangian density $\mathscr{L}_{\rm tot}$ of the
system is given by

$\displaystyle\mathscr{L}_{\rm
tot}=\mathscr{L}_{\rm{shift}}+\mathscr{L}_{\rm{em}}\;,$ (6)

where

$\displaystyle\mathscr{L}_{\rm{shift}}$ $\displaystyle=$
$\displaystyle{\hbar^{2}\over
2mc^{2}}|(\partial_{t}+i{q\over\hbar}\Phi)\varphi|^{2}-{\hbar^{2}\over
2m}|({\boldsymbol{\nabla}}-i{q\over\hbar}{\bf A})\varphi|^{2}$ (7)
$\displaystyle-$ $\displaystyle{mc^{2}\over 2}|\varphi|^{2}-{\cal
E}(|\varphi|^{2})$ $\displaystyle=$ $\displaystyle\mathscr{L}_{\rm
0}+{q^{2}\over 2mc^{2}}|\varphi|^{2}\Phi^{2}-{iq\hbar\over
2mc^{2}}(\varphi^{*}\partial_{t}\varphi-\varphi\partial_{t}\varphi^{*})\Phi$
$\displaystyle-$ $\displaystyle{mc^{2}\over 2}|\varphi|^{2}-{q^{2}\over
2m}|\varphi|^{2}{\bf A}^{2}+{iq\hbar\over
2m}(\varphi^{*}{\boldsymbol{\nabla}}\varphi-\varphi{\boldsymbol{\nabla}}\varphi^{*})\cdot{\bf
A}$

is the shifted Lagrangian density of relativistic Cooper pairs and

$\mathscr{L}_{\rm{em}}={\epsilon_{0}\over 2}{\bf E}^{2}-{1\over 2\mu_{0}}{\bf
B}^{2}$ (8)

is the Lagrangian density of the free electromagnetic field, with
$\epsilon_{0}$ the dielectric constant in the vacuum and $\mu_{0}$ the
paramagnetic constant in the vacuum. Remember that
$c=1/\sqrt{\epsilon_{0}\mu_{0}}$. The Lagrangian (6) is invariant with respect
the local $U(1)$ gauge transformation $\varphi({\bf r},t)\to\varphi({\bf
r},t)\ e^{i\alpha({\bf r},t)}$.

As well known, the total Lagrangian density (6) can be used to develop a zero-
temperature quantum field theory by introducing the real-time partition
function

${\cal Z}=\int{\cal D}[\varphi,\Phi,{\bf A}]\ e^{{i\over\hbar}\int
dt\,d^{3}{\bf r}\,\mathscr{L}_{\rm{tot}}}$ (9)

of the system, within a functional integral formalism popov-book ; schakel-
book . The theory can also be extended at finite temperature by performing a
Wick rotation from real to imaginary time. However, in this paper we do not
consider the finite-temperature effects of the entropy-dependent normal
component of the charged superfluid. The quantum expectation value of the
scalar field $\varphi({\bf r},t)$ is defined as

$\langle\varphi\rangle={1\over{\cal Z}}\int{\cal D}[\varphi,\Phi,{\bf A}]\
\varphi\ e^{{i\over\hbar}\int dt\,d^{3}{\bf r}\,\mathscr{L}_{\rm{tot}}}\;.$
(10)

The transition to the superconducting state is the breaking of the $U(1)$
local gauge invariance, namely $\langle{\varphi}({\bf r},t)\rangle\neq 0$
popov-book ; schakel-book . In this paper we work within the saddle-point
approximation, where

${\cal Z}\simeq e^{{i\over\hbar}\int dt\,d^{3}{\bf
r}\,\mathscr{L}_{\rm{tot}}}$ (11)

and the fields are the ones which extermize the action functional

$S_{\rm{tot}}=\int dt\,d^{3}{\bf r}\,\mathscr{L}_{\rm{tot}}\;.$ (12)

In this mean-field framework $\langle{\varphi}({\bf
r},t)\rangle={\varphi}({\bf r},t)$ and quantum fluctuations are not taken into
account.

##  III From Lorentz to Galilei

The standard way to obtain a Galilei-invariant Schrödinger matter field
$\psi({\bf r},t)$ from the Lorentz-invariant Klein-Gordon field $\varphi({\bf
r},t)$ is to set

$\varphi({\bf r},t)=\psi({\bf r},t)\,e^{-imc^{2}t/\hbar}\;.$ (13)

Inserting this ansatz into (1) and (7) we find

$\displaystyle\mathscr{L}_{\rm 0}$ $\displaystyle=$
$\displaystyle{\hbar^{2}\over 2mc^{2}}|\partial_{t}\psi|^{2}+{mc^{2}\over
2}|\psi|^{2}+{i\hbar\over
2}(\psi^{*}\partial_{t}\psi-\psi\partial_{t}\psi^{*})$ (14) $\displaystyle-$
$\displaystyle{\hbar^{2}\over 2m}|{\boldsymbol{\nabla}}\psi|^{2}-{\cal
E}(|\psi|^{2})$

and

$\displaystyle\mathscr{L}_{\rm{shift}}$ $\displaystyle=$
$\displaystyle\mathscr{L}_{\rm 0}+{q^{2}\over
2mc^{2}}|\psi|^{2}\Phi^{2}-q|\psi|^{2}\Phi$ (15) $\displaystyle-$
$\displaystyle{iq\hbar\over
2mc^{2}}(\psi^{*}\partial_{t}\psi-\psi\partial_{t}\psi^{*})\Phi-{q^{2}\over
2m}|\psi|^{2}{\bf A}^{2}$ $\displaystyle+$ $\displaystyle{iq\hbar\over
2m}(\psi^{*}{\boldsymbol{\nabla}}\psi-\psi{\boldsymbol{\nabla}}\psi^{*})\cdot{\bf
A}$

At this point the matter Lagrangian density is still Lorentz invariant.
However, under the assumption

${mc^{2}\over 2}|\psi|^{2}\ll{i\hbar\over
2}(\psi^{*}\partial_{t}\psi-\psi\partial_{t}\psi^{*})$ (16)

we obtain the approximated nonrelativistic Galilei-invariant Lagrangians

$\tilde{\mathscr{L}}_{\rm 0}={i\hbar\over
2}(\psi^{*}\partial_{t}\psi-\psi\partial_{t}\psi^{*})-{\hbar^{2}\over
2m}|{\boldsymbol{\nabla}}\psi|^{2}-{\cal E}(|\psi|^{2})$ (17)

and

$\displaystyle\tilde{\mathscr{L}}_{\rm{shift}}$ $\displaystyle=$
$\displaystyle\tilde{\mathscr{L}}_{\rm 0}+{q^{2}\over
2mc^{2}}|\psi|^{2}\Phi^{2}-q|\psi|^{2}\Phi-{q^{2}\over 2m}|\psi|^{2}{\bf
A}^{2}$ (18) $\displaystyle+$ $\displaystyle{iq\hbar\over
2m}(\psi^{*}{\boldsymbol{\nabla}}\psi-\psi{\boldsymbol{\nabla}}\psi^{*})\cdot{\bf
A}\;.$

Notice that in Eq. (17) we have omitted the term $mc^{2}|\psi|^{2}/2$ which
does not affect the dynamics. As expected, (17) is the Lagrangian density of a
complex Schrödinger field $\psi({\bf r},t)$. Instead, quite remarkably, Eq.
(18) contains the crucial term
$q^{2}\Phi^{2}|\psi|^{2}/(2mc^{2})=\epsilon_{0}\mu_{0}q^{2}\Phi^{2}|\psi|^{2}/(2m)$.
This term is absent by applying the minimal coupling to the electromagnetic
field directly into a Schrödinger Lagrangian density. To better emphasize this
relevant result, let us write the Euler-Lagrange equation of (18) with respect
to $\psi^{*}({\bf r},t)$, which is given by the following nonlinear
Schrödinger equation

$\displaystyle i\hbar(\partial_{t}+i{q\over\hbar}\Phi)\psi$ $\displaystyle=$
$\displaystyle\left[-{\hbar^{2}\over 2m}(\nabla-i{q\over\hbar}{\bf
A})^{2}+{\mu(|\psi^{2}|)}\right]\psi$ (19) $\displaystyle-$
$\displaystyle{q^{2}\over mc^{2}}\Phi^{2}\,\psi\;,$

where

$\mu(|\psi|^{2})={\partial{\cal E}\over\partial|\psi|^{2}}(|\psi|^{2})\;$ (20)

is the chemical potential of the bulk system as a function of the local number
density $|\psi|^{2}$. In Eq. (19) it is the last term which makes the
nonrelativistic limit of the Klein-Gordon equation coupled to the
electromagnetic field not equivalent to the fully nonrelativistic Schrödinger
equation coupled to the electromagnetic field. This phenomenon is the analog
of the Dirac equation coupled to the electromagnetic field: in the non-
relativistic limit from the Dirac equation one gets the Pauli equation (which
has the spin) and not the Schrödinger equation (which does not have the spin)
drell-book . Note that the term
$q^{2}\Phi^{2}/(mc^{2})=\epsilon_{0}\mu_{0}\Phi^{2}/m$ can be discarded only
if $|q\Phi|\ll mc^{2}$ but in nonrelativistic superconductors this is not the
case.

##  IV Density-phase Lagrangian

The Schrödinger field $\psi({\bf r},t)$ of Eqs. (17) and (18) is the order
parameter of nonrelativistic Cooper pairs. We now set

$\psi=\sqrt{n_{s}}\,e^{i\theta}\;,$ (21)

where $n_{s}({\bf r},t)$ is the number density of Cooper pairs of mass $m$ and
electric charge $q$, while $\theta({\bf r},t)$ is the Nambu-Goldstone phase
field nambu1960 ; goldstone1961 .

Several authors popov1972 ; popov-book ; witten1989 ; schakel1990 ;
schakel1994 ; zhu1995 ; son2006 ; schakel-book ; tureci2023  adopted the idea
of writing a low-frequency and long-wavelength Lagrangian density
$\mathscr{L}$ of a nonrelativistic superfluid in terms of $\theta({\bf r},t)$.
In all these approaches the local superfluid velocity field ${\bf v}_{s}({\bf
r},t)$ is related to $\theta({\bf r},t)$ by the fundamental relationship

${\bf v}_{s}={\hbar\over m}{\boldsymbol{\nabla}}\theta\;.$ (22)

This equation ensures that the fluid is irrotational, i.e.
${\boldsymbol{\nabla}}\wedge{\bf v}_{s}={\bf 0}$ apart from a set of zero
measure of quantized vortices. Indeed in the presence of a quantized vortex
with integer quantum number $\kappa$, around it the circulation of the
superfluid velocity is such that landau-book ; leggett-book

$\oint{\bf v}_{s}\cdot d{\bf r}={\hbar\over
m}\oint{\boldsymbol{\nabla}}\theta\cdot d{\bf r}={\hbar\over
m}\int_{0}^{{2\pi}\kappa}d\theta={2\pi\hbar\over m}\kappa\;.$ (23)

Inserting Eq. (21) into Eqs. (17) and (18) we obtain the following total
Lagrangian density

$\mathscr{L}_{\rm tot}=\tilde{\mathscr{L}}_{\rm 0}+\mathscr{L}_{\rm
em}+\tilde{\mathscr{L}}_{\rm I}\;,$ (24)

where

$\tilde{\mathscr{L}}_{\rm
0}=-n_{s}\,\hbar{\partial_{t}\theta}-n_{s}\,{\hbar^{2}\over
2m}({\boldsymbol{\nabla}}\theta)^{2}-{\hbar^{2}\over
8m}{({\boldsymbol{\nabla}}n_{s})^{2}\over n_{s}}-{\cal E}(n_{s})$ (25)

is our nonrelativistic density-phase Lagrangian, $\mathscr{L}_{\rm em}$ is the
Lagrangian of the free electromagnetic field, given by Eq. (8), and

$\tilde{\mathscr{L}}_{\rm I}=n_{s}\,\epsilon_{0}\mu_{0}{q^{2}\Phi^{2}\over
2m}-n_{s}\,q\Phi-n_{s}\,{q^{2}{\bf A}^{2}\over 2m}+n_{s}\,q\,{\hbar\over
m}{\boldsymbol{\nabla}}\theta\cdot{\bf A}\;$ (26)

is the Lagrangian of the interaction between the Cooper pairs and the
electromagnetic field.

We observe that in Eq. (25) it appears the von Weizsäcker-like term von1935

$\tilde{\mathscr{L}}_{{\rm 0,W}}=-{\hbar^{2}\over
8m}{({\boldsymbol{\nabla}}n_{s})^{2}\over n_{s}}$ (27)

which takes into account the energy cost due to variations of the superfluid
density. As we will see, this term modifies the dispersion relation of
collective modes of both neutral and charged superfluids. In a very recent
paper tureci2023  it has been suggested that Eq. (27) is crucial to obtain a
negative electrohydrostatic pressure between superconducting bodies at zero
temperature.

###  IV.1 Including the ion background

In the ground state of superconductor there is a compensation between the
negative electric charge density $\rho_{s}=qn_{s}=-2en_{s}$ of Cooper pairs
and the positive electric charge density $\rho_{\rm bg}=-q{\bar{n}}_{\rm
bg}=2e{\bar{n}}_{\rm bg}$ of the background of ions with average number
density ${\bar{n}}_{\rm ions}$. Taking into account this fact, similarly to
the Jellium model of a metallic conductor mermin-book , the full Lagrangian of
our model is given by

$\mathscr{L}_{\rm full}=\mathscr{L}_{\rm tot}+\mathscr{L}_{\rm bg}\;,$ (28)

where

$\mathscr{L}_{\rm bg}={\bar{n}}_{\rm bg}\,q\Phi\;.$ (29)

Notice that we are assuming that the number density ${\bar{n}}_{\rm bg}$ of
the ion background is space-time independent.

For the sake of clarity, we stress that in this nonrelativistic framework it
is the Lagrangian density (28) that must be used to obtain the path-integral
partition function

${\cal Z}_{\rm full}=\int{\cal D}[n_{s},\theta,\Phi,{\bf A}]\
e^{{i\over\hbar}\int dt\,d^{3}{\bf r}\,\mathscr{L}_{\rm{full}}}\;,$ (30)

where the functional integration is done with respect to the local number
density $n_{s}({\bf r},t)$ of Cooper pairs, the Nambu-Goldstone field
$\theta({\bf r},t)$, and the electromagnetic potentials $\Phi({\bf r},t)$ and
${\bf A}({\bf r},t)$.

###  IV.2 Charge density and current density

It is impossible to calculate analytically Eq. (30). However, the saddle-point
(mean-field) solution is the set of Euler-Lagrange equations which are
obtained by extremizing the action functional

$S_{\rm full}=\int dt\,d^{3}{\bf r}\,\mathscr{L}_{\rm{full}}\;.$ (31)

The Euler-Lagrange equations of the full Lagrangian (28) with respect to the
scalar potential $\Phi({\bf r},t)$ and the vector potential ${\bf A}({\bf
r},t)$ are nothing else than the Maxwell equations

$\displaystyle{\boldsymbol{\nabla}}\cdot{\bf E}$ $\displaystyle=$
$\displaystyle{\rho\over\epsilon_{0}}$ (32)
$\displaystyle{\boldsymbol{\nabla}}\cdot{\bf B}$ $\displaystyle=$
$\displaystyle 0$ (33) $\displaystyle{\boldsymbol{\nabla}}\wedge{\bf E}$
$\displaystyle=$ $\displaystyle-{\partial_{t}{\bf B}}$ (34)
$\displaystyle{\boldsymbol{\nabla}}\wedge{\bf B}$ $\displaystyle=$
$\displaystyle\mu_{0}\ {\bf j}+{\epsilon_{0}\mu_{0}}\ \partial_{t}{\bf E}$
(35)

where the expressions of the local charge density $\rho({\bf r},t)$ and the
local current density ${\bf j}({\bf r},t)$ are given by

$\displaystyle{\rho}$ $\displaystyle=$
$\displaystyle-{\partial\left(\tilde{\mathscr{L}}_{\rm I}{+\mathscr{L}_{\rm
bg}}\right)\over\partial\Phi}$ (36) $\displaystyle{\bf j}$ $\displaystyle=$
$\displaystyle{\partial\tilde{\mathscr{L}}_{\rm I}\over\partial{\bf A}}$ (37)

Thus, one gets

$\displaystyle{\rho}$ $\displaystyle=$ $\displaystyle
q\,n_{s}{-q\,{\bar{n}}_{\rm bg}}-\epsilon_{0}\,{q^{2}n_{s}\mu_{0}\over
m}\,\Phi$ (38) $\displaystyle{\bf j}$ $\displaystyle=$ $\displaystyle
q\,n_{s}{\bf v}_{s}-{1\over\mu_{0}}{q^{2}n_{s}\mu_{0}\over m}\,{\bf A}\;,$
(39)

where the first term

$\rho_{s}=q\,n_{s}$ (40)

in Eq. (38) is the electric charge density of Cooper pairs, the second term is
the electric charge density of the ion background, and the third term

$\rho_{I}=-{\epsilon_{0}\mu_{0}q^{2}n_{s}\over m}\,\Phi$ (41)

in Eq. (38) is the interaction charge density due to the coupling between the
Cooper pairs and the electromagnetic scalar potential $\Phi$. Instead, the
first term

${\bf j}_{s}=q\,n_{s}{\bf v}_{s}$ (42)

in Eq. (39) is the electric current density of Cooper pairs, which contains
the superfluid velocity ${\bf v}_{s}$ defined in Eq. (22). The second term

${\bf j}_{I}=-{q^{2}n_{s}\over m}\,{\bf A}$ (43)

in Eq. (39) is nothing else than the London current london1935  due to the
interaction between Cooper pairs and the electromagnetic vector potential
${\bf A}$. In 1935 Fritz and Heinz London london1935  introduced a
nonrelativistic model where the interaction density $\rho_{I}$ was included in
the total electric charge density $\rho$. However, due to the lack of
experimental evidences london1936 , subsequently Fritz London discarded this
term in his book london-book . In recent years, it has been suggested by
Hirsch within an alternative model hirsch2004 ; hirsch2015  that, at very low
temperatures, the interaction density $\rho_{I}$ could be effective and
measurable.

We underline that, as well know, manipulating Eqs. (32) and (35) one finds the
continuity equation for the electric charge density $\rho$ and the electric
current density ${\bf j}$, namely jackson-book

$\partial_{t}\rho+{\boldsymbol{\nabla}}\cdot{\bf j}=0\;.$ (44)

This result will be used later in combination with a similar, but not equal,
continuity equation for the superconductive charge density $\rho_{s}$ of
Cooper pairs and the electric current density ${\bf j}$.

Eqs. (32) and (33) equipped with Eqs. (38) and (39) are nothing else than the
Maxwell-Proca equations for the electrodynamics of supercoductors previously
discussed in Refs. stenuit2001 ; tajmar2008 . However, in Ref. stenuit2001
the Maxwell-Proca equations are obtained from a finite-temperature
relativistic model while in Ref. tajmar2008  these equations are heuristically
introduced without a derivation. Here we will analyzed the consequences of the
Maxwell-Proca equations for superconductors at zero temperature, where the
normal density is absent. Moreover, we will investigate the collective modes
of charged superfluid.

Deep inside a superconductor both magnetic field ${\bf B}$ and electric field
${\bf E}$ are zero annett-book . As a consequence, from our Eqs. (32) and (38)
it follows that for the ground state, characterized by a uniform and constant
number density ${\bar{n}}_{s}$ of Cooper pairs and a vanishing electromagnetic
potential $\Phi=0$, the total electric charge density $\rho$ is zero, namely

$0=q\left({\bar{n}}_{s}-{\bar{n}}_{\rm bg}\right)\;,$ (45)

and consequently ${\bar{n}}_{s}={\bar{n}}_{\rm bg}$. As previously discussed,
the ion background neutralizes the system.

###  IV.3 London penetration depth for the static magnetic field

As discussed above, Eq. (39) was obtained for the first time by the London
brothers london1935  and it gives rise to the expulsion of a magnetic field
from a superconductor (Meissner-Ochsenfeld effect) meissner1933 .

In a static configuration with a zero superfluid velocity ${\bf v}_{s}$ and in
the absence of the electric field, i.e. ${\bf E}={\bf 0}$, the curl of Eq.
(35) gives

$-\nabla^{2}{\bf B}=\mu_{0}\
{\boldsymbol{\nabla}}\wedge\left(-{q^{2}n_{s}\over m}{\bf A}\right)\;,$ (46)

taking into account that

${\boldsymbol{\nabla}}\wedge({\boldsymbol{\nabla}}\wedge{\bf
B})=-\nabla^{2}{\bf B}+{\boldsymbol{\nabla}}({\boldsymbol{\nabla}}\cdot{\bf
B})=-\nabla^{2}{\bf B}$ (47)

due to the Gauss law, Eq. (33). Assuming that the local density $n_{s}({\bf
r})$ is uniform, i.e. $n_{s}({\bf r})={\bar{n}}_{s}$, by using Eq. (5) we get

$\nabla^{2}{\bf B}={q^{2}{\bar{n}}_{s}\mu_{0}\over m}{\bf B}\;.$ (48)

Choosing the magnetic field as ${\bf B}=B(x)\,{\bf u}$, with ${\bf u}$ a unit
vector, the previous equation can be written as

${\partial^{2}\over\partial x^{2}}B={q^{2}\bar{n}_{s}\mu_{0}\over m}B$ (49)

which has the following physically relevant solution for a superconducting
slab defined in the region $x\geq 0$:

$B(x)=B(0)\ e^{-x/\lambda_{L}}\;,$ (50)

where

$\lambda_{L}=\sqrt{m\over q^{2}\bar{n}_{s}\mu_{0}}$ (51)

is the so-called London penetration depth, which is typically around $100$
nanometers annett-book . The meaning of Eq. (50) is that inside a
superconductor the static magnetic field decays exponentially. This is the
Meissner-Ochsenfeld effect: the expulsion of a magnetic field from a
superconductor, experimentally observed for the first time in 1933
meissner1933 .

###  IV.4 London penetration depth for the static electric field

It is well know that normal metals screen an external electric field ${\bf
E}$, which can penetrate at most few angströms (Thomas-Fermi screening length)
mermin-book . For superconducting materials, our equations (32), (33), (38),
and (39) suggest that the electric field ${\bf E}$ exponentially decays inside
a zero-temperature superconductor with the much larger London penetration
depth $\lambda_{L}$. Let us show how to derive this relevant result within our
theoretical framework.

In a static configuration, in the absence the magnetic field, i.e. B=0, and
assuming a uniform number density, the gradient of Eq. (32), with Eq. (38) and
Eq. (51), gives

$\nabla^{2}{\bf E}=-{1\over\lambda_{L}^{2}}{\boldsymbol{\nabla}}\Phi\;,$ (52)

taking into account that

${\boldsymbol{\nabla}}({\boldsymbol{\nabla}}\cdot{\bf E})=\nabla^{2}{\bf
E}-{\boldsymbol{\nabla}}\wedge({\boldsymbol{\nabla}}\wedge{\bf
E})=\nabla^{2}{\bf E}\;.$ (53)

Notice that to get Eq. (52) it is crucial to assume an uniform background
$n_{\rm bg}$. In addition, due to Eq. (4) we find

$\nabla^{2}{\bf E}={1\over\lambda_{L}^{2}}{\bf E}\;.$ (54)

Choosing ${\bf E}=E(x)\,{\bf u}$, with ${\bf u}$ a unit vector, the previous
equation can be written as

${\partial^{2}\over\partial x^{2}}E={1\over\lambda_{L}^{2}}E$ (55)

which has the following physically relevant solution for a superconducting
slab defined in the region $x\geq 0$:

$E(x)=E(0)\ e^{-x/\lambda_{L}}\;.$ (56)

The meaning of Eq. (56) is that inside a zero-temperature superconductor the
static electric field decays exponentially with a characteristic decay length
that is exactly the London penetration depth $\lambda_{L}$.

###  IV.5 Modified D’Alembert equation for electromagnetic waves

We investigate what happens to an electromagnetic wave when it is suddenly
applied to a superconductor in its ground state. In full generality, from the
Maxwell equations (32), (33), (34), and (35) one obtains the inhomogeneous
wave equations jackson-book

$\displaystyle\left({1\over c^{2}}{\partial^{2}\over\partial
t^{2}}-\nabla^{2}\right){\bf E}$ $\displaystyle=$
$\displaystyle-{1\over\epsilon_{0}}{\boldsymbol{\nabla}}\rho-\mu_{0}\partial_{t}{\bf
j}$ (57) $\displaystyle\left({1\over c^{2}}{\partial^{2}\over\partial
t^{2}}-\nabla^{2}\right){\bf B}$ $\displaystyle=$
$\displaystyle\mu_{0}{\boldsymbol{\nabla}}\wedge{\bf j}\;.$ (58)

Under the assumption that the local number density of Cooper pairs remains
approximately constant and uniform, i.e. $n_{s}({\bf
r},t)\simeq{\bar{n}}_{s}$, and with a zero superfluid velocity, i.e. ${\bf
v}_{s}({\bf r},t)\simeq{\bf 0}$, after remembering Eqs. (4), (5), (36), and
(37), from Eqs. (57) and (58) we obtain

$\displaystyle\left({1\over c^{2}}{\partial^{2}\over\partial
t^{2}}-\nabla^{2}+{1\over\lambda_{L}^{2}}\right){\bf E}={\bf 0}$ (59)
$\displaystyle\left({1\over c^{2}}{\partial^{2}\over\partial
t^{2}}-\nabla^{2}+{1\over\lambda_{L}^{2}}\right){\bf B}={\bf 0}$ (60)

that are the modified D’Alembert equation for the electromagnetic waves inside
the superconductor with $\lambda_{L}$ is the London penetration depth of Eq.
(51).

The Fourier transform of Eqs. (59) and (60) in the frequency-wavevector domain
$(\omega,{\bf k})$ gives the dispersion relation

$\omega=\sqrt{\omega_{p}^{2}+c^{2}k^{2}}\;,$ (61)

where

$\omega_{p}={c\over\lambda_{L}}=\sqrt{q^{2}{\bar{n}}_{s}\over m\epsilon_{0}}$
(62)

is the Plasma frequency mermin-book . Thus, the photon spectrum becomes gapped
or, in other words, the photon acquires a mass. This is nothing else than the
Anderson-Higgs mechanism anderson1962 ; higgs1964 ; englert1964 , which
survives in our model also in the context of nonrelativistic superconducting
matter. Notice that Eq. (61) appears also in Refs. grigorishin2021 ;
hirsch2004 . As discussed in Ref. grigorishin2021 , the dispersion relation
(61) can be also written as

$k={1\over c}\sqrt{\omega^{2}-\omega_{p}^{2}}=\sqrt{{\omega^{2}\over
c^{2}}-{1\over\lambda_{L}^{2}}}\;.$ (63)

Consequently, the electromagnetic plane wave, that is proportional to
$e^{i({\bf k}\cdot{\bf r}-\omega t)}$ propagates without dissipation inside
the superconductor for $\omega>\omega_{p}$. Instead, for $\omega<\omega_{p}$
the electromagnetic wave is damped as $e^{-{\bf u}\cdot{\bf
r}/\lambda_{\omega}}\,e^{-i\omega t}$ in the interior of the superconductor,
where ${\bf k}=i{\bf u}/\lambda_{\omega}$ with ${\bf u}$ a unit vector and

$\lambda_{\omega}={\lambda_{L}\over\sqrt{1-\left({\omega\over\omega_{p}}\right)^{2}}}$
(64)

is the frequency-dependent penetration depth. Clearly,
$\lambda_{\omega}\to\lambda_{L}$ as $\omega\to 0$. Moreover, indicating with
$\Delta(0)$ the energy gap of Cooper pairs at zero temperature, for
$\omega>2\Delta(0)/\hbar$ the charged superfluid becomes a normal charged
fluid due to the breaking of Cooper pairs.

##  V Euler-Lagrange equations of superconductors

The Euler-Lagrange equation of the full Lagrangian (28) with respect to the
Nambu-Goldstone field $\theta({\bf r},t)$ reads

$\partial_{t}n_{s}+{\boldsymbol{\nabla}}\cdot\left(n_{s}{\bf
v}_{s}-{qn_{s}\over m}{\bf A}\right)=0\;.$ (65)

This is nothing else than the continuity equation

$\partial_{t}\rho_{s}+{\boldsymbol{\nabla}}\cdot{\bf j}=0\;,$ (66)

where the local superconducting charged density $\rho_{s}({\bf r},t)$ is given
by Eq. (40) and the local charged current density ${\bf j}({\bf r},t)$ is
given by Eq. (39). Comparing Eq. (66) with Eq. (44) it follows that

$\partial_{t}({\rho}-\rho_{s})=0\;,$ (67)

i.e. the interaction charge density $\rho_{I}=\rho-\rho_{s}$ given by Eq. (41)
must be time independent or, equivalently

$\Phi\,\partial_{t}n_{s}=-n_{s}\,\partial_{t}\Phi\;.$ (68)

Instead, the Euler-Lagrange equation for the local number density $n_{s}({\bf
r},t)$ leads to

$\displaystyle\hbar\,\partial_{t}\theta$ $\displaystyle+$ $\displaystyle
q\Phi-{\epsilon_{0}\mu_{0}q^{2}\over 2m}\Phi^{2}-{q\over
m}{\boldsymbol{\nabla}}\theta\cdot{\bf A}+{q^{2}{\bf A}^{2}\over
2m}+{\hbar^{2}\over 2m}({\boldsymbol{\nabla}}\theta)^{2}$ (69)
$\displaystyle+$ $\displaystyle{\partial{\cal E}\over\partial
n_{s}}-{\hbar^{2}\over 2m\sqrt{n_{s}}}\nabla^{2}\sqrt{n_{s}}=0\;.$

By applying the gradient operator $\boldsymbol{\nabla}$ to Eq. (69) one finds

$\displaystyle m\partial_{t}{\bf v}_{s}$ $\displaystyle+$
$\displaystyle{\boldsymbol{\nabla}}\big{[}{1\over 2}m{\bf
v}_{s}^{2}+\mu(n_{s})-{\hbar^{2}\over
2m\sqrt{n_{s}}}\nabla^{2}\sqrt{n_{s}}+q\Phi$ (70) $\displaystyle-$
$\displaystyle{\epsilon_{0}\mu_{0}q^{2}\over 2m}\Phi^{2}-{q\over\hbar}{\bf
v}_{s}\cdot{\bf A}+{q^{2}{\bf A}^{2}\over 2m}\big{]}={\bf 0}\;,$

where $\mu(n_{s})$, given by Eq. (20), is the chemical potential of the bulk
system as a function of the local number density $n_{s}({\bf r},t)$.

###  V.1 Gapless collective modes of neutral superfluids

In the very special case of a neutral superfluid, i.e. if $q=0$, the previous
equations become much simpler and it is quite easy to determine the collective
modes of the zero-temperature neutral superfluid. We set

$\displaystyle n_{s}({\bf r},t)={\bar{n}}_{s}+{\delta n}_{s}({\bf r},t)$ (71)
$\displaystyle{\bf v}_{s}({\bf r},t)={\bf 0}+{\delta{\bf v}}_{s}({\bf r},t)$
(72)

assuming that ${\delta n}_{s}({\bf r},t)$ and ${\delta{\bf v}}_{s}({\bf r},t)$
are small perturbations with respect to the ground-state configuration with
uniform number density ${\bar{n}}_{s}$ and zero superfluid velocity.

Under the condition $q=0$, the linearized version of Eqs. (65) and (70) are
then given by

$\displaystyle\partial_{t}{\delta
n}_{s}+{\bar{n}}_{s}{\boldsymbol{\nabla}}\cdot{\delta{\bf v}}_{s}$
$\displaystyle=$ $\displaystyle 0\;,$ (73)
$\displaystyle{\bar{n}}_{s}\partial_{t}{\delta{\bf
v}}_{s}+c_{s}^{2}{\boldsymbol{\nabla}}{\delta n}_{s}-{\hbar^{2}\over
4m^{2}}{\boldsymbol{\nabla}}(\nabla^{2}{\delta n}_{s})$ $\displaystyle=$
$\displaystyle{\bf 0}\;,$ (74)

where

$c_{s}=\sqrt{{\bar{n}_{s}\over m}{\partial\mu\over\partial n}({\bar{n}}_{s})}$
(75)

is the speed of sound. By applying the time derivative $\partial_{t}$ to Eq.
(73) and the divergence ${\boldsymbol{\nabla}}\cdot$ to Eq. (74) and
subtracting the two resulting equations we find

$\left(\partial_{t}^{2}-c_{s}^{2}\nabla^{2}+{\hbar^{2}\over
4m^{2}}\nabla^{4}\right){\delta n}_{s}=0\;,$ (76)

The Fourier transform of Eq. (76) in the frequency-wavevector domain
$(\omega,{\bf k})$ gives the dispersion relation

$\omega=\sqrt{c_{s}^{2}k^{2}+{\hbar^{2}k^{4}\over 4m^{2}}}\;.$ (77)

This dispersion relation is a gapless Bogoliubov-like spectrum bogoliubov1947
, which reduces to the phonon spectrum

$\omega=c_{s}k$ (78)

at very low wavenumbers. while for large wavenumbers one finds

$\omega={\hbar k^{2}\over 2m}$ (79)

that is the single-particle spectrum of free massive particles. For the sake
of completeness, we underline that Eq. (77) is fully consistent with our
previous results sala2008 ; sala2009 ; sala2013  for the collective modes of
nonrelativistic neutral fermionic superfluids with the inclusion of the von
Weizsäcker-like term, Eq. (27).

###  V.2 Gapped collective modes of charged superfluids

We now analyze the collective modes of a zero-temperature superconductor. In
this case $q\neq 0$ and, in addition to Eqs. (71) and (72), we must set

$\displaystyle{\Phi}({\bf r},t)={0}+\delta\Phi({\bf r},t)$ (80)
$\displaystyle{\bf A}({\bf r},t)={\bf 0}+\delta{\bf A}({\bf r},t)$ (81)

assuming that ${\delta\Phi}({\bf r},t)$ and ${\delta{\bf A}}({\bf r},t)$ are
small perturbations with respect to the ground-state electromagnetic
configuration of zero scalar potential and zero vector potential.

Under the condition $q\neq 0$, the linearized version of Eqs. (65) and (70)
are then given by

$\displaystyle\partial_{t}({\delta n}_{s})$ $\displaystyle+$
$\displaystyle{\bar{n}}_{s}{\boldsymbol{\nabla}}\cdot{\delta{\bf
v}}_{s}-{q{\bar{n}}_{s}\over m}{\boldsymbol{\nabla}}\cdot\delta{\bf A}=0\;,$
(82) $\displaystyle{\bar{n}}_{s}\partial_{t}({\delta{\bf v}}_{s})$
$\displaystyle+$ $\displaystyle c_{s}^{2}{\boldsymbol{\nabla}}({\delta
n}_{s})-{\hbar^{2}\over 4m^{2}}{\boldsymbol{\nabla}}(\nabla^{2}{\delta
n}_{s})$ (83) $\displaystyle+$ $\displaystyle{q{\bar{n}}_{s}\over
m}{\boldsymbol{\nabla}}(\delta{\Phi})={\bf 0}\;.$

Similarly, the linearized version of the Maxwell equations (32), (33), (34),
and (35) reads

$\displaystyle{\boldsymbol{\nabla}}\cdot\delta{\bf E}$ $\displaystyle=$
$\displaystyle{\delta\rho\over\epsilon_{0}}$ (84)
$\displaystyle{\boldsymbol{\nabla}}\cdot\delta{\bf B}$ $\displaystyle=$
$\displaystyle 0$ (85) $\displaystyle{\boldsymbol{\nabla}}\wedge\delta{\bf E}$
$\displaystyle=$ $\displaystyle\partial_{t}(\delta{\bf B})$ (86)
$\displaystyle{\boldsymbol{\nabla}}\wedge\delta{\bf B}$ $\displaystyle=$
$\displaystyle\mu_{0}\ \delta{\bf j}+{\epsilon_{0}\mu_{0}}\
\partial_{t}(\delta{\bf E})$ (87)

where

$\displaystyle\delta{\bf E}$ $\displaystyle=$
$\displaystyle-{\boldsymbol{\nabla}}(\delta\Phi)-\partial_{t}(\delta{\bf A})$
(88) $\displaystyle\delta{\bf B}$ $\displaystyle=$
$\displaystyle{\boldsymbol{\nabla}}\wedge\delta{\bf A}$ (89)

and

$\displaystyle\delta{\rho}$ $\displaystyle=$ $\displaystyle q\,\delta
n_{s}-\epsilon_{0}{q^{2}{\bar{n}}_{s}\mu_{0}\over m}\,\delta\Phi$ (90)
$\displaystyle\delta{\bf j}$ $\displaystyle=$ $\displaystyle
q\,{\bar{n}}_{s}\delta{\bf v}_{s}-{1\over\mu_{0}}{q^{2}\mu_{0}\over m}{\bf
A}\,\delta n_{s}-{1\over\mu_{0}}{q^{2}{\bar{n}}_{s}\mu_{0}\over m}\,\delta{\bf
A}.$ (91)

Needless to say, finding analytical solutions of the coupled equations from
(82) to (91) seems not easy. However, we are able to obtain some interesting
result. By applying the time derivative $\partial_{t}$ to Eq. (82) and the
divergence ${\boldsymbol{\nabla}}\cdot$ to Eq. (83) and subtracting the two
resulting equations we find

$\left(\partial_{t}^{2}-c_{s}^{2}\nabla^{2}+{\hbar^{2}\over
4m^{2}}\nabla^{4}\right){\delta n}_{s}+{q{\bar{n}}_{s}\over
m}{\boldsymbol{\nabla}}\cdot\delta{\bf E}=0\;,$ (92)

taking into account Eq. (88). Then, from the first Maxwell equation (84) and
Eq. (90) we get

${\boldsymbol{\nabla}}\cdot\delta{\bf
E}={\delta\rho\over\epsilon_{0}}={q\over\epsilon_{0}}\delta
n_{s}+{\delta\rho_{I}\over\epsilon_{0}}\;.$ (93)

Remembering that $\rho_{I}=\rho-\rho_{s}$ is constant in time, as shown by Eq.
(67), by applying the operator $\partial_{t}$ to Eq. (92) we obtain

$\left(\partial_{t}^{3}-c_{s}^{2}\partial_{t}\nabla^{2}+{\hbar^{2}\over
4m^{2}}\partial_{t}\nabla^{4}+\omega_{p}^{2}\partial_{t}\right){\delta
n}_{s}=0\;,$ (94)

which gives the dispersion relation $\omega=0$ but also

$\omega=\sqrt{\omega_{p}^{2}+c_{s}^{2}k^{2}+{\hbar^{2}k^{4}\over 4m^{2}}}$
(95)

that is a gapped generalization of Eq. (77). As expected, the gap is exactly
due to the plasma frequency $\omega_{p}$ of Eq. (62).

##  VI Conclusions

We have analyzed several consequences of a time-dependent relativistic model
of bosonic charged Cooper pairs minimally coupled to the electromagnetic
field. While our model shares similarities with other relativistic treatments
of superconductivity stenuit2001 ; hirsch2004 ; tajmar2008 ; hirsch2015 ;
grigorishin2021 , it differs in at least three important ways. First, our
results have been obtained at zero temperature where, at least for clean
superconductors, the normal component of the superconducting electrons is zero
and a real-time description of the supercondictive bosonic field is fully
justified varlamov-book . Second, we have explicitly discussed the derivation
of the nonrelativistic model for the matter field from the relativistic one,
emphasizing the crucial role of a term which couples the density of Cooper
pairs with the electromagnetic scalar potential. This term can be also
obtained babaev-book  from the nonrelativistic low-frequency and long-
wavelength Popov’s action popov1972 ; popov-book  of a charged superfluid, but
only performing a quantum-mechanical functional integration with respect to
the density field within the saddle-point approximation. Third, we have
obtained the full set of equations for nonrelativistic charged superfluids
coupled to the (relativistic) electromagnetic field in terms of the superfluid
density and the superfluid velocity, that is directly related to the gradient
of the Nambu-Goldstone phase field. In these equations, in addition to the
previously discussed coupling term, there is a von Weizsäcker-like term
von1935  which takes into account the energy cost due to variations of the
superfluid density and modifies the dispersion relation of superfluid
collective modes.

Our model supports the claim hirsch2004 ; hirsch2015 ; grigorishin2021  that,
very close to zero temperature, it should be possible to experimentally
measure the decay of a static electric field inside a superconductor with a
characteristic length that is the London penetration depth instead of the
Thomas-Fermi screening length. A recent experimental attempt to measure this
effect by using atomic force microscopy on a niobium sample was inconclusive
due to limited accuracy peronio2016 . We expect that near-future experiments
could test also other zero-temperature predictions discussed in this paper: a
gapped spectrum of the electromagnetic waves inside the superconductor and the
gapped spectrum of the supercondicting density oscillations. To achieve these
goals it is necessary to work at extremely low temperatures, where the normal
component, containing the entropy and the viscosity of the system, is
negligible. This is the main experimental problem that needs to be overcome.

###### Acknowledgements.

The author thanks Andrea Bardin, Luca Dell’Anna, Koichiro Furutani, Francesco
Lorenzi, Maria Pelizzo, Andrea Perali, and Dima Sorokin for useful discussions
and suggestions. This work is partially supported by the European Union-
NextGenerationEU within the National Center for HPC, Big Data and Quantum
Computing [Project No. CN00000013, CN1 Spoke 10: “Quantum Computing”], by the
BIRD Project ”Ultracold atoms in curved geometries” of the University of
Padova, by “Iniziativa Specifica Quantum” of Istituto Nazionale di Fisica
Nucleare, and by the European Quantum Flagship Project ”PASQuanS 2”. The
projects ”Frontiere Quantistiche” (Dipartimenti di Eccellenza) and PRIN 2022
”Quantum Atomic Mixtures: Droplets, Topological Structures, and Vortices” have
been financed by the Italian Ministry for University and Research and the
European Union-NextGenerationEU. The author declares no conflict of interest.

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