# Introductory Notes on Non-linear Electrodynamics and its Applications

Dmitri P. Sorokin 111e-mail<EMAIL_ADDRESS>
###### Abstract

In 1933-1934 Born and Infeld constructed the first non-linear generalization
of Maxwell’s electrodynamics that turned out to be a remarkable theory in many
respects. In 1935 Heisenberg and Euler computed a complete effective action
describing non-linear corrections to Maxwell’s theory due to quantum electron-
positron one-loop effects. Since then, these and a variety of other models of
non-linear electrodynamics proposed in the course of decades have been
extensively studied and used in a wide range of areas of theoretical physics
including string theory, gravity, cosmology and condensed matter (CMT). In
these notes I will overview general properties of non-linear electrodynamics
and particular models which are distinguished by their symmetries and physical
properties, such as a recently discovered unique non-linear modification of
Maxwell’s electrodynamics which is conformal and duality invariant. I will
also sketch how non-linear electromagnetic effects may manifest themselves in
physical phenomena (such as vacuum birefringence), in properties of
gravitational objects (e.g. charged black holes) and in the evolution of the
universe, and can be used, via gravity/CMT holography, for the description of
properties of certain conducting and insulating materials.

I.N.F.N. Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy
and
Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi
di Padova

###### Contents

  1. 1 Preface
  2. 2 Maxwell’s electrodynamics
  3. 3 Non-linear electrodynamics
    1. 3.1 Heisenberg-Euler-Kockel theory
    2. 3.2 General structure of NEDs
  4. 4 Born-Infeld theory and its weak and strong field limits
    1. 4.1 Conformal and duality-invariant non-linear electrodynamics of Bialynicki-Birula
  5. 5 ModMax electrodynamics
  6. 6 Other specimens of non-linear electrodynamics and their applications
    1. 6.1 NEDs and exact regular electric and magnetic black holes
    2. 6.2 ModMax effects on black holes
    3. 6.3 Cosmological solutions without primordial singularity smoothed by electromagnetic non-linearities
    4. 6.4 Universe expansion accelerated by NED
    5. 6.5 NEDs and gravity/CMT holography
  7. 7 Birefringence in non-linear electrodynamics
  8. 8 Outlook

##  1 Preface

Models of non-linear electrodynamics (NED for short) have been extensively
studied as possible guides to catch new physics: e.g. for tackling fundamental
cosmological problems (such as physics of black holes, inflation and dark
energy), and for an effective description of properties of certain condensed
matter materials and optical media. Notable examples are the famous Born-
Infeld theory [1] (based on earlier work of Born [2, 3]), which was invented
to ensure that electric field self-energy of charged point particles is finite
but much later turned out to be an important ingredient of String Theory (see
e.g. [4] for a review), and the Heisenberg-Euler effective theory of quantum
electrodynamics [5] (see [6] for a historical review).

In addition to Born-Infeld and Heisenberg-Euler theories, there exists a
variety of models of non-linear electrodynamics containing such instances as
logarithmic, double-logarithmic, exponential, rational, arcsin, power-law,
inverse, quasitopological and other NEDs (see e.g. [7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]). They have been created for
applications in gravity and cosmology, as well as for a gravity/CMT
holographic description of certain strongly-coupled condensed matter systems.
The newest recently found specimen in this “NED zoo” is ModMax, the unique
non-linear modification of Maxwell’s theory that preserves all its symmetries
including electric-magnetic duality and conformal invariance [26]. We came
upon this finding somewhat occasionally when studying conformal limits of
Born-Infeld electrodynamics and its higher-rank gauge field generalizations on
branes in the context of string and M-theory [27].

In these notes I will overview basic properties of generic NEDs and the most
distinguished examples of NEDs in four-dimensional space-time including
Heisenberg-Euler, Born-Infeld, Bialynicki-Birula [28, 29] and ModMax, and will
flash some effects via which non-linearities of these and other models may
manifest themselves in physical phenomena, such as vacuum birefringence, and
their role in the context of gravity and condensed matter theory. I would like
to keep the presentation as simple as possible and refer the reader for more
details to the original literature and more extended reviews listed in a very
incomplete Bibliography.

##  2 Maxwell’s electrodynamics

To set the stage and introduce the notation and main ingredients, let us start
with the very well known example of free Maxwell’s theory in four-dimensional
space-time. Its action can be written in the manifestly Lorentz-invariant and
gauge-invariant form with the use of the field strength
$F_{\mu\nu}=\partial_{\mu}A_{\mu}-\partial_{\mu}A_{\mu}$ of the four-vector
potential $A_{\mu}(x)$, or in the non-manifestly Lorentz invariant form in
terms of the electric and magnetic three-vector fields $\mathbf{E}$ and
$\mathbf{B}$, which are components of $F_{\mu\nu}$ 111In the most of the
expressions we use natural units in which the speed of light $c$, the vacuum
permittivity $\varepsilon_{0}$ and the Plank constant $\hbar$ are set to one.:

$\mathcal{S}=-\frac{1}{4}\int d^{4}x\,F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\int
d^{4}x\,({\mathbf{E}}^{2}-{\mathbf{B}}^{2})$ (2.1)
$E_{i}=\partial_{0}A_{i}-\partial_{i}A_{0},\qquad
B^{i}=\frac{1}{2}\varepsilon^{ijk}F_{jk},\qquad\mu,\nu=(0,i);\qquad
i,j,k=1,2,3\,.$

Maxwell’s equations, including the $F_{\mu\nu}$ Bianchi identities, are

$\displaystyle\partial_{\mu}F^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0,$
(2.2) $\displaystyle\partial_{\mu}\tilde{F}^{\mu\nu}$ $\displaystyle=$
$\displaystyle
0\,,\qquad\qquad\tilde{F}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma},$

or

$\displaystyle\frac{\partial{\mathbf{E}}}{\partial
t}=\nabla\times{\mathbf{B}},\qquad\nabla\cdot{\mathbf{E}=0},$ (2.3)
$\displaystyle\frac{\partial{\mathbf{B}}}{\partial
t}=-\nabla\times{\mathbf{E}},\qquad\nabla\cdot{\mathbf{B}=0}.$

As is well known, the free Maxwell equations are invariant under the
$SO(2)$-duality rotations of $F_{\mu\nu}$ and its Hodge dual
$\tilde{F}_{\mu\nu}$ (or $\mathbf{E}$ and $\mathbf{B}$)

$\begin{pmatrix}F^{\prime\,\mu\nu}\\\
\tilde{F}^{\prime\,\mu\nu}\end{pmatrix}=\begin{pmatrix}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{pmatrix}\,\begin{pmatrix}F^{\mu\nu}\\\
\tilde{F}^{\mu\nu}\end{pmatrix}\,.$

The Maxwell action is not duality-invariant, but there exist formulations in
which the duality invariance gets lifted off-shell, i.e. to the level of the
action. We will not discuss these manifestly duality-symmetric actions in
these notes.

However, as is well known, the Maxwell action is invariant under a larger,
conformal symmetry which includes the Poincaré group, dilatations and special
conformal transformations. For simplicity, let me only present the dilatation
(re-scaling) of the space-time coordinates and the gauge vector potential with
a constant parameter $a$ which leaves the Maxwell action invariant

$x^{\mu}\to a\,x^{\mu},\quad A_{\mu}\to a^{-1}A_{\mu},\quad F_{\mu\nu}\to
a^{-2}F_{\mu\nu}.$ (2.4)

Note that conformal symmetry and duality invariance play an important role in
various areas of theoretical physics, from condensed matter to the theory of
fundamental interactions, and in particular in String Theory. These
fundamental symmetries are among basic principals assumed to be satisfied by a
variety of theoretical models at least in certain limits in which these
symmetries are expected. It is therefore natural to consider implications of
conformal symmetry and duality invariance also in the context of non-linear
generalizations of Maxwell’s electrodynamics.

##  3 Non-linear electrodynamics

The Maxwell action is quadratic in $F_{\mu\nu}$ and Maxwell’s equations are
linear. But Maxwell’s electrodynamics may receive non-linear corrections of
higher orders in $F_{\mu\nu}$ and may thus be extended to a non-linear theory.
The nature and form of these corrections can be different, depending on a
model.

###  3.1 Heisenberg-Euler-Kockel theory

For instance, in the Heisenberg-Euler theory [5] the higher order corrections
appear as a result of the one-loop effects produced by virtual electrons and
positrons circulating in the quantum loops. More precisely the Heisenberg-
Euler action is the full non-perturbative one-loop effective action for
quantum spinor electrodynamics in (all orders of) a uniform background
electromagnetic field and describes QED vacuum polarization effects. Its
Lagrangian density is

$\displaystyle\mathcal{L}_{HE}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbf{E}^{2}-\mathbf{B}^{2})$ $\displaystyle+$
$\displaystyle\frac{2\pi e^{2}}{\hbar
c}\int_{0}^{\infty}\,\frac{d\eta}{\eta^{3}}e^{-\eta}\left\\{{\tt
i}\eta^{2}(\mathbf{E}\cdot\mathbf{B})\,\frac{\cos\left(\frac{\eta}{\mathcal{E}_{c}}\sqrt{\mathbf{E}^{2}-\mathbf{B}^{2}+2{{\rm
i}}(\mathbf{E}\cdot\mathbf{B})}\right)+c.c}{\cos\left(\frac{\eta}{\mathcal{E}_{c}}\sqrt{\mathbf{E}^{2}-\mathbf{B}^{2}+2{{\rm
i}}(\mathbf{E}\cdot\mathbf{B})}\right)-c.c}+\mathcal{E}^{2}_{c}-\frac{\eta^{3}}{3}(\mathbf{E}^{2}-\mathbf{B}^{2})\right\\}\,,$

where the presence of the speed of light $c$ and the Planck constant $\hbar$
has been made explicit, and $\mathcal{E}_{c}=\frac{m_{e}^{2}c^{3}}{e\hbar}\sim
10^{16}V/{\tt cm}$ (with $m_{e}$ being the electron mass) is the critical
electric field strength above which the creation of physical electron-positron
pairs from vacuum occurs.

Up to the fourth order in $F_{\mu\nu}$ the Heisenberg-Euler Lagrangian density
reduces to the effective Lagrangian density computed earlier (in 1935) for low
energy photons $\hbar\omega\ll m_{e}c^{2}$ by Euler and Kockel [30]

$\displaystyle\mathcal{L}_{spinor\,QED}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}+\frac{\hbar^{3}\alpha^{2}}{90\mu_{0}^{2}m_{e}^{4}c^{5}}\left((F_{\mu\nu}F^{\mu\nu})^{2}+\frac{7}{4}(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}\right)+\mathcal{O}\left((FF)^{3}\right)$
(3.2) $\displaystyle=$
$\displaystyle\frac{1}{2}\,({\mathbf{E}}^{2}-{\mathbf{B}}^{2})+\frac{2\hbar^{3}\alpha^{2}}{45\mu_{0}^{2}m_{e}^{4}c^{5}}\left(({\mathbf{E}}^{2}-{\mathbf{B}}^{2})^{2}+7(\mathbf{E}\cdot\mathbf{B})^{2}\right)+\,\cdots,$

where $\alpha=\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{\hbar c}$ is the fine
structure constant and $\mu_{0}$ is the vacuum magnetic impermeability. The
coefficient of the fourth-order quantum corrections in this Lagrangian are
very small

$\frac{2\hbar^{3}\alpha^{2}}{45\mu_{0}m_{e}^{4}c^{5}}=1.32\times
10^{-24}\,({\rm Tesla}^{-2})\,.$

This is why their effects are very hard to measure experimentally.

For comparison with the spinor electrodynamics, let us also look at the form
of the effective Lagrangian density describing one-loop contributions of a
massive charged scalar field to the forth order in $F_{\mu\nu}$ [31, 32]

$\displaystyle\mathcal{L}_{scalar\,QED}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}+\frac{\hbar^{3}\alpha^{2}}{1440\mu_{0}^{2}m_{s}^{4}c^{5}}\left(7(F_{\mu\nu}F^{\mu\nu})^{2}+(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}\right)+\mathcal{O}\left((FF)^{3}\right).$
(3.3)

###  3.2 General structure of NEDs

The most general Lorentz-invariant action for non-linear electrodynamics in
$D=4$, which usually does not include derivatives of $F_{\mu\nu}$ to avoid
ghosts, is a functional of two independent Lorentz invariants, a scalar and a
pseudo-scalar quadratic in $F_{\mu\nu}$ [33, 34]

$S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\,({\mathbf{E}}^{2}-{\mathbf{B}}^{2})\,,\qquad
P=-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}={\mathbf{E}}\cdot{\mathbf{B}}\,.$
(3.4)

So the generic NED action is

$I_{NED}=\int d^{4}x\,\mathcal{L}(S,P)\,.$ (3.5)

Usually, one assumes that $\mathcal{L}(S,P)$ is an analytic function at
$S=P=0$ and that in the limit of weak electromagnetic fields the action
reduces to that of the Maxwell theory. But, as we will see there are
interesting models of non-linear electrodynamics (e.g. the conformal NEDs and
in particular ModMax) which do not obey these assumptions.

Other restrictions on the form of the NED action are imposed by the physical
requirements of causality and unitarity of the theory (see e.g. [35, 36, 37,
38, 39]). These are related to the requirement that the Lagrangian density is
a convex function of the electric field strength $E^{i}$, i.e. that the
Hessian $3\times 3$ matrix

$\mathcal{L}_{ij}=\frac{\partial^{2}\mathcal{L}}{\partial E^{i}\partial
E^{j}}$ (3.6)

has only non-negative eigenvalues for all values of $\mathbf{E}$ and
$\mathbf{B}$. The convexity of the Lagrangian density also insures the
possibility of performing an involutive Legendre transform to the Hamiltonian
formulation of the theory.

In contrast to the Maxwell theory a generic NED is neither conformal, nor
duality-invariant, as is the Heisenberg-Euler theory. However, the effective
Heisenberg-Euler-like Lagrangian for a supersymmetric system of minimally
coupled one spin-1/2 and two spin-0 charged particles of equal mass is duality
invariant (at least) to the fourth order in $F_{\mu\nu}$ [40] and coincides to
this order with the Born-Infled Lagrangian (see eq. (4.1) below) for an
appropriate choice of its coupling constant. The fourth order terms in the
supersymmetric extension of the Heisenberg-Euler Lagrangian are the sum of the
forth order terms in (3.2) and twice the fourth order terms in (3.3).

If one is interested in maintaining the conformal invariance, than the
Lagrangian density of the non-linear electrodynamics must be a homogeneous
function of $S$ and $P$ of degree one under the scale transformations (2.4) of
the coordinates and the gauge vector potential $A_{\mu}$

$\mathcal{L}(S,P)\quad\to\quad\mathcal{L}(a^{-4}S,a^{-4}P)=a^{-4}\mathcal{L}(S,P).$
(3.7)

Then in the action the scaling factor of the Lagrangian density gets
compensated by the opposite re-scaling of the integration measure. Various
conformal non-linear electrodynamics models and their applications (mainly to
black holes but also in the AdS/CFT contexts) have been studied (in different
space-time dimensions) for more than two decades (see e.g. [11, 13, 14, 16,
38, 41, 42, 43, 44]).

As far as the electric-magnetic duality invariance of the field equations of
non-linear electrodynamics is concerned, the condition which this invariance
imposes on the form of the Lagrangian density was first derived by Bialynicki-
Birula in 1983 [28]. It has the following form

$F_{\mu\nu}\tilde{F}^{\mu\nu}-2\varepsilon^{\mu\nu\lambda\rho}\,\frac{\partial\mathcal{L}}{\partial
F^{\mu\nu}}\,\frac{\partial\mathcal{L}}{\partial
F^{\lambda\rho}}=0\qquad\Rightarrow\qquad{\mathcal{L}}_{S}^{2}-\frac{2S}{P}{\mathcal{L}}_{S}{\cal
L}_{P}-{\mathcal{L}}_{P}^{2}=1,$ (3.8)

where $\mathcal{L}_{S}=\frac{\partial\mathcal{L}}{\partial S}$ and
$\mathcal{L}_{P}=\frac{\partial\mathcal{L}}{\partial P}$.

When the condition (3.8) holds, the Euler-Lagrange equations and the Bianchi
identities of the NED

$\displaystyle\partial_{\mu}\left(-2\frac{\partial\mathcal{L}}{\partial
F_{\mu\nu}}\right)=0,$ (3.9) $\displaystyle\partial_{\mu}\tilde{F}^{\mu\nu}=0$

are transformed into each other by the SO(2)-duality rotations

$\begin{pmatrix}-2\frac{{\partial\mathcal{L}(F^{\prime})}}{\partial
F^{\prime}_{\mu\nu}}\\\
\tilde{F}^{\prime\mu\nu}\end{pmatrix}=\begin{pmatrix}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{pmatrix}\,\begin{pmatrix}-2\frac{\partial\mathcal{L}}{\partial
F_{\mu\nu}}\\\ \tilde{F}^{\mu\nu}\end{pmatrix}\,,$ (3.10)

where $\mathcal{L}(F^{\prime})$ has the same form as $\mathcal{L}(F)$. This,
of course, does not mean that the NED Lagrangian is invariant under the
duality rotations (3.10). In fact, it does not but the corresponding energy-
momentum tensor [45] and the Hamiltonian [28] are duality-invariant.

In the paper of 1983 Bialynicki-Birula showed that the Lagrangian density of
the Born-Infeld theory (4.1) satisfies the condition (3.8) and hence that this
theory is duality invariant, although probably the first evidence of the
duality invariance of the Born-Infeld theory appeared in a paper by
Schrödinger of 1935 [46]. The properties of the duality-invariance condition
(3.8) its (general) solutions and an $SL(2,R)$-duality extension to the case
of NEDs coupled to an axion-dilaton were further analyzed in detail in [47,
48] and [49, 50], the latter papers having been the development of earlier
work [45] of Gaillard and Zumino on the general theory of duality-invariant
nonlinear electrodynamics. More explicit examples of duality-invariant NED
Lagrangians were found in [51]. A generalization of this condition to the case
of electrodynamics non-minimally coupled to a higher-curvature gravity was
recently considered in [52, 53].

##  4 Born-Infeld theory and its weak and strong field limits

The Born-Infeld Lagrangian density has the following form

$\displaystyle\mathcal{L}_{BI}$ $\displaystyle=$ $\displaystyle
T-T\sqrt{-\det(\eta_{\mu\nu}+T^{-\frac{1}{2}}F_{\mu\nu})}=T-\sqrt{T^{2}+\frac{T}{2}F_{\mu\nu}F^{\mu\nu}-\frac{1}{16}(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}}\,$
(4.1) $\displaystyle=$
$\displaystyle-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{1}{32\,T}\left((F_{\mu\nu}F^{\mu\nu})^{2}+(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}\right)+\mathcal{O}\left((F^{2})^{3}\right)\,,$

where $T$ is the coupling parameter of the dimension of energy density.

The original motivation of Born and Infeld for constructing this theory was to
ensure finiteness of the electric field self-energy of charged particles,
which is guaranteed by the condition that the expression under the square root
may only have non-negative values. Much later it was discovered that the Born-
Infeld theory is an important ingredient of String Theory. In bosonic open
string theory the Born-Infeld action arises as an effective low-energy action
[54], while in type II superstring theories it describes an effective
worldvolume theory of Dirichlet branes (in ten-dimensional space-time) on
which open strings end [55], and $T$ has the meaning of brane tension. Since
$T$ is dimensional, the Born-Infeld theory is not conformal, but it is
duality-invariant, as we have mentioned above.

###  4.1 Conformal and duality-invariant non-linear electrodynamics of
Bialynicki-Birula

As is well known and not hard to see from (4.1), in the infinite tension limit
$T\to\infty$ (which is equivalent to the weak electromagnetic field limit),
the Born-Infeld Lagrangian density reduces to that of Maxwell’s
electrodynamics (2.1). A much less known limiting theory is the one appearing
in the strong field limit of Born-Infeld theory [28, 29], which is equivalent
to take $T\to 0$. A reason why this limit is less known is that it cannot be
taken in the Lagrangian formulation. Indeed, in this limit the Born-Infeld
Lagrangian density becomes a purely imaginary total derivative

$\mathcal{L}_{BI}|_{T\to 0}=\frac{\tt i}{4}|F_{\mu\nu}\tilde{F}^{\mu\nu}|\,,$

hence the corresponding action vanishes (modulo an un-physical imaginary
boundary Chern-Simons term).

To take the strong field limit it is necessary to pass (via the Legendre
transform) to the Hamiltonian formulation of the Born-Infeld theory as was
noticed by Bialynicki-Birula in 1983 [28, 29]. The Born-Infeld Hamiltonian
density is

$\mathcal{H}_{BI}(\mathbf{D},\mathbf{B})=\mathbf{D}\cdot\mathbf{E}-\mathcal{L}_{BI}(\mathbf{E},\mathbf{B})=\sqrt{T^{2}+T({\bf
D}^{2}+{\bf B}^{2})+|{\bf D}\times{\bf B}|^{2}}-T\,,$ (4.2)

where $\mathbf{D}=\frac{\partial\mathcal{L}}{\partial\mathbf{E}}$ is the
electric displacement vector field playing the role of the canonical momentum
conjugate to the three-vector gauge potential $\mathbf{A}$.

Now the tensionless (strong field) limit $T\to 0$ can be taken and one gets
the Hamiltonian density first obtained by Bialynicki-Birula in 1983

$\mathcal{H}_{BB}=|{\bf D}\times{\bf B}|\,.$ (4.3)

As Bialynicki-Birula discussed in detail in [29], the theory described by this
Hamiltonian is rather remarkable. It is invariant under $SL(2,R)$-duality
(rather than SO(2)), which can be immediately seen from the invariance of the
Hamiltonian density (4.3) under the $SL(2,R)$ transformations of
$\mathbf{B}^{I}=(\mathbf{D},\mathbf{B})$ ($\mathbf{B}^{\prime
I}=L^{I}{}_{J}\,\mathbf{B}^{J},\,\,\det L^{I}{}_{J}=1,\,\,I,J=1,2$). It is
conformal and possesses infinite number of conserved currents with the
associated tensorial conserved charges

$Q^{i_{1}i_{2}\cdots i_{k}}=\int
dx^{3}\,\mathcal{H}_{BB}\,n^{i_{1}}n^{i_{2}}\cdots
n^{i_{k}},\qquad\mathbf{n}=\frac{\mathbf{D}\times\mathbf{B}}{|\mathbf{D}\times\mathbf{B}|},$

which indicates that this theory can be integrable. As is seen from the form
of its Hamiltonian, the Bialynicki-Birula theory does not include Maxwell’s
electrodynamics as a whole. It can be shown that this theory describes all
possible configurations of the so-called null electromagnetic fields whose
scalar and pseudo-scalar Lorentz invariants (3.4) vanish

$S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}(\mathbf{E}^{2}-\mathbf{B}^{2})=0,\qquad
P=-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}=\mathbf{E}\cdot\mathbf{B}=0.$ (4.4)

The Bialyanicki-Birula theory can be associated with an infra-red limit of a
D3-brane [56] and has its counterparts in higher space-time dimensions [27].
For instance, in $D=6$ it is a strong-field limit [57, 27] of the non-linear
Born-Infeld-like theory for a chiral 2-form gauge field [58, 59], while in
$D=10$ the theory whose strong field limit is the BB-like chiral 4-form
electrodynamics is still unknown [27].

A natural question arises whether there are other conformal and duality
invariant models of electrodynamics in addition to the Maxwell and Bialynicki-
Birula theory? The answer is yes, there is but only one more non-linear
electrodynamics with these symmetry properties called ModMax [26].

##  5 ModMax electrodynamics

The Lagrangian density of ModMax has the following form [26, 60].

$\displaystyle{\cal L}_{ModMax}$
$\displaystyle=-\frac{\cosh\gamma}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{\sinh\gamma}{4}\,\sqrt{(F_{\mu\nu}F^{\mu\nu})^{2}+(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}}\,$
$\displaystyle=\frac{\cosh\gamma}{2}\,(\mathbf{E}^{2}-\mathbf{B}^{2})+\frac{\sinh\gamma}{2}\,\sqrt{(\mathbf{E}^{2}-\mathbf{B}^{2})^{2}+4(\mathbf{E}\cdot\mathbf{B})^{2}}\,,$

where $\gamma$ is the dimensionless coupling constant. The conditions of
causality and unitarity require this constant be non-negative $\gamma\geq 0$.
These values of $\gamma$ also ensure that the Lagrangian density is a convex
function of the electric field $E^{i}$ [26] and that its energy-momentum
tensor satisfies the weak, strong and dominant energy conditions [61]. Note
that Maxwell’s electrodynamics is not a weak field limit of ModMax because of
conformal invariance, but is recovered when the ModMax coupling constant tends
to zero $\gamma\to 0$.

The field equations of the theory are

$\cosh\gamma\,\partial_{\mu}F^{\mu\nu}+\sinh\gamma\,\partial_{\mu}\,\Bigl{(}\frac{SF^{\mu\nu}+P\tilde{F}^{\mu\nu}}{\sqrt{S^{2}+P^{2}}}\Bigr{)}=0.$
(5.2)

They are non-linear, but linearize by field configurations for which $P=a\,S$
with $a$ being a constant. So all the solutions of Maxwell’s equations with
$P=a\,S$ are solutions of ModMax theory.

Since ModMax is conformal, its energy-momentum tensor is traceless and
proportional to that of Maxwell’s theory

$\displaystyle T^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(F^{\mu}{}_{\rho}F^{\nu\rho}-\frac{1}{4}\eta^{\mu\nu}(F_{\rho\lambda}F^{\rho\lambda})\right){\mathcal{L}_{S}}\,,$
(5.3) $\displaystyle\mathcal{L}_{S}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
S}=\left(\cosh\gamma-\sinh\gamma\frac{F_{\sigma\nu}F^{\sigma\nu}}{\sqrt{(FF)^{2}+(F\tilde{F})^{2}}}\right)\,,$
$\displaystyle T^{\mu}{}_{\mu}$ $\displaystyle=0\,.$

One can see that the energy-momentum tensor is non-analytic and is not well
defined when the electromagnetic fields are null (4.4). The ambiguity of the
value of the scalar factor $\mathcal{L}_{S}$ of $T^{\mu\nu}$ in the null-field
limit ranges from $e^{-\gamma}$ to $e^{\gamma}$. This might be an issue for
the theory, since the class of solutions for which the electromagnetic fields
are null, such as the plane waves, are not well defined in the ModMax
Lagrangian formulation.222Note, though, that the vacuum (in which
$F_{\mu\nu}=0$) is a well defined solution of the ModMax Lagrangian field
equations (5.2) with zero energy-momentum tensor. However, somewhat
surprisingly, the Hamiltonian formulation of ModMax comes to the rescue [26].
The ModMax Hamiltonian density is obtained from its Lagrangian density by the
Legendre transform and has the following form 333A detailed analysis of the
ModMax Hamiltonian formulation in the phase space associated with the
Plebanski variables [34] has been carried out in [62].

$\mathcal{H}_{MM}=\frac{1}{2}\left(\cosh\gamma\,(\mathbf{D}^{2}+\mathbf{B}^{2})-\sinh\gamma\sqrt{(\mathbf{D}^{2}+\mathbf{B}^{2})^{2}-4(\mathbf{D}\times\mathbf{B})^{2}}\right)+A_{0}\,\boldsymbol{\nabla}\cdot\mathbf{D}\,,$
(5.4)

where in the last term the time component of the $4D$ gauge vector field plays
the role of the Lagrange multiplier that insures the Gauss law constraint
$\boldsymbol{\nabla}\cdot\mathbf{D}=0$.

Note that the Hamiltonian density is the phase-space counterpart of the energy
density

$T^{00}=\frac{1}{2}\left({\mathbf{E}}^{2}+{\mathbf{B}}^{2}\right)\left(\cosh\gamma+\sinh\gamma\frac{{\mathbf{E}}^{2}-{\mathbf{B}}^{2}}{\sqrt{({\mathbf{E}}^{2}-{\mathbf{B}}^{2})^{2}+4({\mathbf{E}}\cdot{\mathbf{B}})^{2}}}\right)\geq
0\,.$ (5.5)

However, in contrast to the Lagrangian energy density, $\mathcal{H}_{MM}$ is
well defined also for the null-field configurations (4.4) for which the
electric displacement field
$\mathbf{D}=\frac{\partial\mathcal{L}}{\partial\mathbf{E}}$ and the magnetic
field $\mathbf{B}$ satisfy the following condition

$\mathbf{D}^{2}+\mathbf{B}^{2}=2\cosh\gamma\,|\mathbf{D}\times\mathbf{B}|\,,$
(5.6)

and the Hamiltonian density reduces to

$\mathcal{H}|_{null-fields}=|\mathbf{D}\times\mathbf{B}|\,.$ (5.7)

As one might expect, for the null fields the ModMax Hamiltonian density
coincides with that of Bialynicki-Birula (4.3).

The condition (5.6) is also the bound on $\mathbf{D}$ and $\mathbf{B}$ above
which the ModMax Hamiltonian density (with $\gamma\geq 0$) is a convex
function of $\mathbf{D}$ [26]. Indeed, it can be shown that the Hessian matrix
$H_{ij}=\frac{\partial{\mathcal{H}_{MM}}}{\partial D^{i}\partial D^{j}}$ has
only non-negative eigenvalues iff

$\mathbf{D}^{2}+\mathbf{B}^{2}\geq
2\cosh\gamma\,|\mathbf{D}\times\mathbf{B}|\,.$ (5.8)

Only if the electric displacement fields $\mathbf{D}$ satisfy eq. (5.8) the
Legendre transform to the Lagrangian density (5) is involutive. This raises an
interesting general question how to handle theories (especially at quantum
level) whose phase space geometry is subject to inequality constraints like
(5.8)?

The null field configurations saturate the convexity bound (5.8). Among these
configurations, the plane waves [26] and topologically non-trivial knotted
electromagnetic fields [63] (generalizing those of Maxwell’s theory [64]) are
exact solutions of the ModMax Hamiltonian equations

$\displaystyle\dot{\bf B}$ $\displaystyle=$
$\displaystyle-\boldsymbol{\nabla}\times{\bf
E}\,,\qquad\boldsymbol{\nabla}\cdot{\bf B}=0\,,$ $\displaystyle\dot{\bf D}$
$\displaystyle=$ $\displaystyle{}\;\;\boldsymbol{\nabla}\times{\bf
H}\,,\qquad\boldsymbol{\nabla}\cdot{\bf D}=0\,,$ (5.9)

where

${\bf E}=\partial{\mathcal{H}_{MM}}/\partial{\bf D}\,,\qquad{\bf
H}=\partial{\mathcal{H}_{MM}}/\partial{\bf B}\,.$ (5.10)

For example, a linearly polarized plane wave has the following form

$\mathbf{E}=e^{-\gamma}\mathbf{D}=\mathbf{E}_{0}\cos(|\mathbf{k}|t-\mathbf{k}\cdot\mathbf{x})=-\frac{\mathbf{k}}{\mathbf{|}\mathbf{k}|}\times\mathbf{B}\,,$
(5.11)

where $\mathbf{k}$ is the wave vector and $\mathbf{E}_{0}$ is the wave
amplitude.444Of course, the linear superposition of two plane waves is not, in
general, a solution of non-linear electrodynamics equations [65, 66].

This demonstrates a non-trivial complementarity of the Hamiltonian and
Lagrangian formulation of ModMax electrodynamics in which the existence of the
null-field solutions is not an obvious fact. In contrast, it is
straightforward to see that all theories of non-linear electrodynamics whose
Lagrangian reduces to that of Maxwell’s theory in the weak field limit
(including Born-Infeld) do have null-field solutions, such as the plane waves
[46, 65, 66, 34]. This is because all the non-linear terms in the field
equations obtained from the Lagrangian (like (3.2) or (4.1)), which can be
expanded in series of positive powers of the Lorentz invariants (3.4), vanish
for the null fields.

In [26, 27] ModMax and Born-Infeld theories were combined into a duality-
symmetric generalized Born-Infeld electrodynamics whose weak field limit is
ModMax and the strong field limit is the Bialynicki-Birula electrodynamics.
The Lagrangian density of this theory has the following form

$\mathcal{L}_{\gamma
BI}=T-\sqrt{T^{2}-2T\mathcal{L}_{MM}-\frac{1}{16}(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}}\,,$
(5.12)

where $\mathcal{L}_{MM}$ is the ModMax Lagrangian density (5).

In [27] these theories were lifted to a six-dimensional space-time where
corresponding new models of a non-linear chiral two-form gauge field were
found, and also generalized to $2n+1$-form gauge field NEDs in $D=4n+4$ with
the use of the manifestly duality-symmetric and space-time covariant PST
construction of their actions [67, 59, 68, 69]. An alternative duality-
symmetric space-time covariant Lagrangian formulation of ModMax was considered
in [70]. A possibility of attributing a brane-like nature to the generalized
Born-Infeld theory (5.12) was discussed in [71]. A four-parameter family of
NEDs which includes ModMax and the generalized Born-Infeld theory was
considered in [72]. It was shown there that for certain values of parameters
the electric field in the center of the charged point particle and its self-
energy are finite. Another deformation of ModMax which breaks duality
invariance was proposed in [73]. $N=1$ supersymmetric extensions of ModMax and
the generalized Born-Infeld electrodynamics were constructed in [74, 39], and
in [75] ModMax has been generalized to four-dimensional duality-invariant
(super)conformal non-linear models of higher-spin gauge fields. A ’t Hooft
anomaly in the ModMax theory has been studied in [76].

##  6 Other specimens of non-linear electrodynamics and their applications

In addition to Born-Infeld-like, Heisenberg-Euler-like and the generic NED, in
the last more than two decades there has been a rather extensive activity in
constructing a variety of other (heuristic) models of non-linear
electrodynamics (see e.g. [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25]) and using them for applications in gravity, cosmology
and CMT. For instance, NEDs coupled to gravity have been used to construct new
black hole solutions and study their properties and stability (see e.g. [77,
78, 79, 80, 8, 81, 82, 10, 83, 9, 84, 35, 85, 12, 86, 87, 88, 89, 90, 91, 92,
93, 94, 95] and also [21, 96, 97, 98, 52, 53, 99, 100, 101, 102, 103, 104,
105, 106] for the latest developments and review), to create cosmological
models (which might avoid initial Big-Bang singularities) [107, 108], to mimic
dark energy and generate an accelerated expansion of the universe [109, 110,
111, 112, 113, 114, 115, 116], as well as in the context of AdS/CFT
correspondence and gravity/CMT holography (see e.g. [117, 118, 119, 14, 120,
121, 122, 123] and [124] for a recent review and more references). For the
study of magnetic brane solutions in modified gravity theories coupled to NEDs
see e.g. [125] and references there in.

A common setup for gravity-related applications of all these models is the
coupling of a NED to General Relativity or its modifications. In the case of
Einstein’s gravity (in general with the cosmological constant $\Lambda$) one
considers Einstein’s equations sourced by the energy-momentum tensor of a
certain NED and some matter

$R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}(R-2\Lambda)=8\pi\,(T^{\mu\nu}_{NED}+T^{\mu\nu}_{matter}),$
(6.1)

and the NED equations of motion sourced, in general, by electric and magnetic
currents

$\displaystyle\nabla_{\mu}\left(-2\frac{\partial\mathcal{L}}{\partial
F_{\mu\nu}}\right)=j^{\mu}_{e},$ (6.2)
$\displaystyle\nabla_{\mu}\tilde{F}^{\mu\nu}=j^{\mu}_{m}\,,$ (6.3)

where $\nabla_{\mu}$ is the covariant derivative containing a metric
connection.

Note that in the presence of magnetic monopoles or dyons, the definition of
$F_{\mu\nu}$ as the curl of the vector potential $A_{\mu}$ should be modified
and include the contribution of the Dirac string (along which $A_{\mu}$ is
singular) for the consistency of the “Bianchi” identity (6.3) (see e.g. [126]
for a review of the quantum field theory of electric and magnetic charges).

One then looks e.g. for static black-hole solutions of these equations
characterized by the metric

$ds^{2}_{BH}=-f(r)\,dt^{2}+\frac{1}{f(r)}\,{dr}^{2}+u(r)\,dM_{2}\,,$ (6.4)

where $f(r)$ and $u(r)$ are functions of the radial coordinate $r$, whose
forms depend on the choice of the electrodynamics model and the ansatz
therein. For instance, for the spherical black holes

$u(r)\,dM_{2}=r^{2}(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\,,$

where $\theta$ and $\varphi$ are the angular coordinates of a two-sphere. In
AdS/CFT applications one often chooses

$u(r)\,dM_{2}=r^{2}(dx^{2}+dy^{2})\,,$

where $x$ and $y$ parametrize the AdS boundary at $r\to\infty$. Of course, the
above choices of the gravitational metric are not the most general ones (for a
more general ansatz see e.g. a recent review [127]). The metric (6.4) appears
in simple examples which we will now consider.

###  6.1 NEDs and exact regular electric and magnetic black holes

As was shown by Penrose [128], the black holes formed as a result of the
gravitational collapse of matter in General Relativity inevitably have
curvature singularities. This indicates that close to the singularity the
validity of the classical Einstein’s gravity breaks down and it should be
replaced by a quantum theory that should resolve the singularity. This issue
raised the question of the existence of regular black holes which may arise as
solutions in some effective (modified) theories of gravity. The first regular
black hole metric (satisfying the weak energy condition) was constructed by
Bardeen in [129] without the notion of the underlying theory whose field
equations it obeys. Only much later first exact electric and magnetic regular
black hole solutions were found and studied in specific models of non-linear
electrodynamics coupled to General Relativity in [8, 81, 82, 10, 83, 9, 12].
See [106] for a review and further refining of constraints on the possibility
of removing black hole singularities with NEDs.

In [10] the Bardeen black hole was interpreted as a magnetic monopole in a
certain gravity-coupled NED which we will consider as an example. The function
$f(r)$ of the Bardeen black hole metric has the following form

$f(r)=1-\frac{Mr^{2}}{(r^{2}+Q_{m}^{2})^{3/2}}\,,$ (6.5)

where $M$ and $Q_{m}$ are integration constants which (upon the result of
[10]) can be regarded, respectively, as the black hole mass and magnetic
charge.

Ayón-Beato and Gracia [10] found that a very specific NED Lagrangian density
which produces the source for the Bardeen black hole in the Einstein equations
has the following form

$\mathcal{L}_{ABG}(S)=\frac{3M}{|Q_{m}|^{3}}\left(\frac{|Q_{m}|\sqrt{-2S}}{1+|Q_{m}|\,\sqrt{-2S}}\right)^{5/2}\,,\qquad
S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$ (6.6)

and the electro-magnetic field strength is that of the magnetic monopole

$\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}=Q_{m}\sin\theta d\theta\wedge
d\varphi.$ (6.7)

Another example of a NED which gives rise to a regular magnetically charged
black hole was considered in [9]. Its Lagrangian density is

$\mathcal{L}=-\frac{S}{\cosh^{2}\left(a|2S|^{\frac{1}{4}}\right)}\,,\qquad
a=const,\qquad S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$ (6.8)

and the function $f(r)$ in the black hole metric sourced by (6.7) has the
following form

$f(r)=1-\frac{|Q_{m}|^{\frac{3}{2}}\left(1-\tanh\left(\frac{\sqrt{|Q_{m}|}}{r}\right)\right)}{ar}\,.$
(6.9)

###  6.2 ModMax effects on black holes

Also effects of the modified Maxwell electrodynamics (5) and its
generalizations [27, 72] on properties and thermodynamics of black holes (e.g.
Taub-NUT and Reissner-Nordström, accelerated ones and others), and
gravitational waves have been explored in a number of papers [130, 131, 132,
133, 98, 134, 102, 135, 136, 137, 138]. Here I will restrict the discussion
only to the simplest example of the Reissner-Nordström black hole.

The Reissner-Nordström black hole sourced by the ModMax energy momentum tensor
has the metric (6.4) with the scalar function $f(r)$ of the following form

$f(r)=1-\frac{2M}{r}+\frac{e^{-\gamma}(Q^{2}_{e}+Q^{2}_{m})}{r^{2}},$ (6.10)

where $M$ is the black hole mass and $Q_{e}$ and $Q_{m}$ are its electric and
magnetic charge. The corresponding electromagnetic potential is

$A_{\mu}dx^{\mu}=-e^{-\gamma}\frac{Q_{e}}{r}\,dt+Q_{m}\,\cos\theta\,d\varphi\,.$
(6.11)

The black hole has two horizons

$f(r)=0\quad\to\quad
r_{\pm}=M\pm\sqrt{M^{2}-(Q_{e}^{2}+Q_{m}^{2})e^{-\gamma}}\quad\to\quad
M^{2}\geq(Q^{2}_{e}+Q^{2}_{m}){e^{-\gamma}}\,,$ (6.12)

the mass squared of the extreme black hole being
$M^{2}=(Q^{2}_{e}+Q^{2}_{m}){e^{-\gamma}}$. One can see that the effect of the
non-linearity of ModMax consists in “screening” the charges by the factor of
$e^{-\gamma}$ (remember that for consistency of the ModMax theory $\gamma\geq
0$). So the mass squared of the extremal ModMax Reissner-Nordström black hole
can be less than the sum of the squares of its charges. This, however depends
on how the physically measurable charges are defined in this theory. As was
shown in [139], the screening factor $e^{-\gamma}$ can be removed from eqs.
(6.10) and (6.12) and similar expressions by an appropriate redefinition of
the physical charges, but it will reappear in other contributions of electric
and magnetic charges e.g. in an accelerated black hole solution considered in
[138].

###  6.3 Cosmological solutions without primordial singularity smoothed by
electromagnetic non-linearities

Many cosmological models, such as Friedmann-Lemaitre-Robertson-Walker (FLRW)
cosmology, face a primordial singularity at the initial time of the
development of the universe which indicates that these models (as in the case
of the conventional black holes) are not applicable in the region close to the
singularity. As was shown in a number of papers (see e.g. [107, 108] and [140]
for a review), non-linear electrodynamics may remove these singularities and
give rise to regular cosmological solutions.

The FLRW metric has the following form

$ds^{2}=-dt^{2}+a(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\right),$
(6.13)

where $k=1$ stands for a closed, $k=0$ for flat and $k=-1$ for an open
universe. The form of the scalar function $a(t)$ depends on the matter which
fills the universe. The FLRW singularity (with infinite density and pressure
of the matter) occurs if $a(t_{0})=0$ at an initial time $t_{0}$. This happens
e.g. in the case of the FLRW universe filled with a perfect fluid or with
Maxwell’s electromagnetic fields. However, as was shown in [108] the
singularity can be lifted by non-linear corrections to the Maxwell Lagrangian.
For instance, in a simple case of the Heisenberg-Euler-Kockel-like model

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\alpha^{2}(F_{\mu\nu}F^{\mu\nu})^{2}+\beta(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}\,,$
(6.14)

where $\alpha$ and $\beta$ are constant parameters, the scalar function $a(t)$
has the following form (for the flat universe)

$a^{2}(t)\propto\sqrt{t^{2}+\alpha^{2}}$ (6.15)

and is therefore never vanishing.

###  6.4 Universe expansion accelerated by NED

A number of papers studied possibilities of accelerating the expansion of the
universe by dominated electromagnetic non-linearities at early time (i.e.
inflation) and later stages of its life (see e.g. [111, 112, 113, 114, 115,
116] and references therein). The expansion of the universe is governed by the
Freedmann equation for the FLRW scalar $a(t)$

$\frac{\ddot{a}}{a}=-\frac{4\pi}{3}(\rho+3p)\,,$ (6.16)

where $\rho$ is the density of a substance that gives a dominant contribution
to Einstein’s equations and $p$ is its pressure. The universe gets accelerated
if $\rho+3p<0$, i.e. when the pressure is negative. As was shown in [111] this
may occur, for instance, if the universe is dominated by non-linear
electromagnetic fields described by the following non-linear Lagrangian
density

$\mathcal{L}=S+\frac{\beta}{S},\qquad S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$
(6.17)

where $\beta$ is a constant parameter.

Another model (called rational NED) which produces a similar accelerating
effect was considered in [113]

$\mathcal{L}=\frac{S}{1+2\beta S}\,.$ (6.18)

In these models the values of $\rho$ and $p$ derived from the structure of the
electromagnetic energy-momentum tensor are such that
$\rho+3p=2\left(\mathcal{L}+|\mathbf{B}|^{2}\frac{\partial\mathcal{L}}{\partial
S}\right)<0$.

###  6.5 NEDs and gravity/CMT holography

Models of non-linear electrodynamics have also been used for the description
of strongly correlated condensed matter systems via gravity/CMT correspondence
(see e.g. [117, 118, 119, 14, 120, 121, 122, 141, 123, 124] and references
therein). Gravity/CMT correspondence (see e.g. [142, 143, 144, 127] for a
review) is a condensed matter oriented ramification of a powerful conceptual
framework called AdS/CFT correspondence, or gauge/gravity duality or
holography which has its origin in String Theory (see e.g. [145] for a
review). This framework provides a toolkit for establishing a dual relation
between a weakly coupled gravity theory in an asymptotically anti-de-Sitter
space-time and a strongly coupled field theory leaving on the AdS boundary. It
prescribes how to translate the physics of the boundary field theory into the
gravitational bulk and vice versa, and thus allows to describe strongly
coupled non-perturbative quantum systems in terms of the quantities of weakly-
coupled gravity, matter and gauge fields.

In applications to condensed matter systems, the holographic set-up in the
gravity bulk includes (non-linear) gauge and matter fields (example, an axion,
dilaton and/or charged scalar fields) and is described by the following action

$I=\int\,d^{4}x\,\sqrt{-g}\left(R-2\Lambda+\mathcal{L}_{NED}(S,P,\phi)+\mathcal{L}(\phi)\right)+I_{bd},$
(6.19)

where $\mathcal{L}_{NED}(S,P,\phi)$ is the Lagrangian density of a non-linear
electrodynamics coupled to scalar fields, $\mathcal{L}(\phi)$ is a scalar
field Lagrangian density and $I_{bd}$ stands for boundary terms which are
required to have a well-defined variational problem. The axion-like fields are
usually introduced to break translational invariance in the bulk by taking
their profiles be linear in spatial coordinates, which in the boundary CMT
corresponds to a momentum dissipation (see e.g. [127] for a review).

As we have already mentioned, the gravitational background assumed to be an
asymptotically AdS black whole whose metric (in many models) has the following
form

$ds^{2}=-f\left(r\right)dt^{2}+\frac{dr^{2}}{f\left(r\right)}+u\left(r\right)\left(dx^{2}+dy^{2}\right)\,,$
(6.20)

where $r$ is the radial holographic coordinate, $x$ and $y$ are associated
with the AdS boundary (at $r\rightarrow\infty$), and $f(r)$ and $u(r)$ are
functions of $r$ whose explicit form depends on the structure of the matter
and gauge sector of the action (6.19). $f(r_{h})=0$ determines the black hole
horizon.

An often used ansatz for the background electromagnetic field is

$A_{\mu}\,dx^{\mu}=A_{t}(r)dt+\frac{h}{2}(xdy-ydx),$ (6.21)

where $h$ is the magnitude of the magnetic field.

The black holes are an important ingredient of the AdS/CMT correspondence
since they are thermal states, and their temperature
$T=\frac{f^{\prime}(r_{h})}{4\pi}$ is related to the temperature in the dual
strongly coupled field theory, while the inclusion of the coupling of a NED to
a charged scalar field generalizes that of [146] for the description of the
transition from the normal to the superconducting phase of the superconductor
living on the AdS boundary, which in the black hole bulk is manifested by the
formation of a scalar field “hair” at and below the critical temperature of
the superconductor.

Gravity/CMT holography with NEDs included in the bulk theory was used to study
properties of holographic superconductors (see e.g. [147, 14, 148] and [149]
for the latest developments and more references) and for understanding in an
analytically controllable way the properties of planar Mott insulators,
materials whose insulating character is due to the electron-electron
interaction (see [120] and e.g. [124] for more references). Mott insulators
form a class of very important not completely understood condensed matter
systems. An important class of Mott insulators is that of the parent compound
of high temperature superconductivity (high-Tc) cuprates with a wide range of
applications.

Within the holographic set-up, non-linear electrodynamics has also been used
for the description of anomalous scaling laws of strange metals (see [121,
122, 123] and [127] for the latest review and references) in which the
dynamics of charged degrees of freedom is described by a non-linear Born-
Infeld action coupled to gravity and axion scalar fields. In this way the
theory reproduces, via non-linear electromagnetic interactions, the anomalous
temperature dependence of the resistivity $\rho\sim T$ and Hall angle
$\cot\theta_{H}\sim T^{2}$ of the cuprate strange metals.555These materials
are called strange because their behavior deviates from the standard
properties of metals. In particular, they are characterized by an anomalous
temperature dependence of resistivity. While the resistivity of the
conventional metals increases with the square of the temperature, the
electrical resistivity of a wide class of strange metals increases linearly
with temperature.

A basic idea behind the holographic description of these materials is that the
electric current operator $j^{\mu}$ describing the charged sector is
holographically dual to the electromagnetic field $A_{\mu}$ in the $AdS$ black
hole bulk. So, current-current interactions in the condensed matter system are
associated with self-interactions of the electromagnetic field via non-linear
terms of $F_{\mu\nu}$ in the NED Lagrangian. The physical characteristics of
the system depend, of course, on the choice of the concrete model of
electrodynamics. For instance, for a wide class of models, in the probe field
limit, the components of the electrical conductivity matrix are given by [150,
151]

$\sigma_{xx}=\sigma_{yy}=\frac{\partial\mathcal{L}\left(S,P,\phi\right)}{\partial
S}|_{r_{h}},\qquad\sigma_{xy}=-\sigma_{yx}=-\frac{\partial\mathcal{L}\left(S,P,\phi\right)}{\partial
P}|_{r_{h}}$

which are evaluated at the black brane horizon $r=r_{h}$.

Using these relations one can compute e.g. the electric resistivity and the
Hall angle of the material

$\rho_{xx}=\frac{\sigma_{xx}}{\sigma^{2}_{xx}+\sigma^{2}_{yy}}\,,\qquad\cot\Theta_{H}=\frac{\sigma_{xx}}{\sigma_{yy}}\,,$
(6.22)

and study their properties in relation to temperature, phase transitions etc.
by applying the holography tools.

##  7 Birefringence in non-linear electrodynamics

Let us now discuss a physical phenomenon which is common to almost all models
of non-linear electrodynamics and which can be tested by an
experiment.666Another phenomenon associated with electromagnetic non-
linearities of the QED is light-by-light scattering which was only quite
recently observed experimentally [152, 153]. This is the birefringence of
light propagating in a uniform strong electromagnetic background.
Birefringence is a phenomenon of double refraction such that a ray of light is
split by polarization, with respect to the optical axis of the material, into
two rays taking different geodesic paths. In the NED, the electromagnetic
background plays the role of an optical material.

To compute the birefringence effect in non-linear electrodynamics one splits
the electromagnetic field into the background field (satisfying the NED
equations of motion) and small fluctuations around this background
$F_{\mu\nu}=F^{ext}_{\mu\nu}+f_{\mu\nu}$, derives the form of the Lagrangian
up-to the second order in the fluctuations and computes the dispersion
relations determining the propagation of plane waves in a stong
electromagnetic background [154, 34, 155]. In general, the dispersion
relations depend on the orientation of the polarization of the propagating
light (its electric field direction) with respect to the axis of the
background field. Let us assume that the background field is purely magnetic
$\mathbf{B}_{ext}$. Then the dispersion relations have the following form

$(\parallel)\quad\omega^{2}=\frac{|{\bf
k}|^{2}}{n^{2}_{\parallel}}\,\,,\qquad(\perp)\quad\omega^{2}=\frac{|{\bf
k}|^{2}}{n_{\perp}^{2}}\,,$ (7.1)

where $n_{\parallel}(\mathbf{k},\mathbf{B}_{ext})$ and
$n_{\perp}(\mathbf{k},\mathbf{B}_{ext})$ are refraction indices. Their
subscripts indicate that the electric field of the propagating linearly
polarized light is parallel or orthogonal to $\mathbf{B}_{ext}$. The values of
the refraction indices depend on properties of the NED and the background. For
instance,

  * •

in Maxwell’s theory $n_{\parallel}=n_{\perp}=1$, the light propagates along a
relativistic geodesic and there is no birefringence.

  * •

The birefringence is also absent in the Born-Infeld theory (4.1) in which
$n_{\parallel}=n_{\perp}$ and (for $\mathbf{k}\perp\mathbf{B}_{ext}$)

$\omega^{2}=|\mathbf{k}|^{2}\left(1+\frac{|\mathbf{B}_{ext}|^{2}}{T}\right)^{-1}\,.$
(7.2)

One can see that the speed of light in the Born-Infeld electromagnetic
background is less than the speed of light in the vacuum. So there is no
problem with acasual propagation. The Born-Infeld theory is the unique
(physically consistent) non-linear electrodynamics which has no birefringence
[154, 34].

  * •

In the Heisenberg-Euler-Kockel effective theory of the spinor QED (for
$\mathbf{k}\perp\mathbf{B}_{ext}$)

$(\parallel)\quad\omega^{2}=|{\bf
k}|^{2}\left(1-14\frac{2\alpha^{2}}{45m_{e}^{4}}|\mathbf{B}_{ext}|^{2}\right)\,,\qquad(\perp)\quad\omega^{2}=|{\bf
k}|^{2}\left(1-8\frac{2\alpha^{2}}{45m_{e}^{4}}|\mathbf{B}_{ext}|^{2}\right)\,,$
(7.3)

and the magnitude of birefringence $(\Delta n=n_{\parallel}-n_{\perp})$

$\Delta n_{QED}=3.96\times 10^{-24}|\mathbf{B}_{ext}|^{2}\,({\rm
Tesla})^{-2}\,,$ (7.4)

depends on the magnitude of the magnetic field.

  * •

In the ModMax theory the dispersion relations are [26, 130]

$(\perp)\quad n_{\perp}=1,\qquad\omega^{2}=|\bf k|^{2}\,,$ (7.5)

which is the standar relativistic light-cone, and another one

$\omega^{2}=|{\bf k}|^{2}(\cos^{2}\varphi+e^{-2\gamma}\sin^{2}\varphi),$ (7.6)

where $\varphi$ is the angle between ${\bf k}$ and $\bf B_{ext}$ . Note that
in ModMax, in contrast to the Born-Infeld and Heisenberg-Euler theory, the
dispersion relations do not depend on the magnitude of the background field
$|\mathbf{B}_{ext}|$ but only on the angle between ${\bf k}$ and $\bf B_{ext}$
and the coupling constant $\gamma$. This is a consequence of conformal
invariance. The propagation of this mode is always sub-luminal if $\gamma>0$.
As in all other cases of non-linear electrodynamics, the effect of the
magnetic background is maximal when the light propagates in a direction
orthogonal to $\bf B_{ext}$ ($\varphi=\frac{\pi}{2}$) while its electric
component is parallel to $\bf B_{ext}$:

$(\parallel)\quad\omega^{2}=|{\bf k}|^{2}e^{-2\gamma}\,,\qquad
n_{\parallel}=e^{\gamma}\,.$ (7.7)

For the rays with the dispersion relations (7.5) and (7.7) the magnitude of
birefringence is

$\Delta n_{ModMax}=e^{\gamma}-1\simeq\gamma+O(\gamma^{2})\,.$ (7.8)

Electromagnetic vacuum birefringence has been searched for in a number of
experiments one of which is PVLAS [156]. The experiment was not able to reach
the predicted value from QED (7.4), but set the current best limit on vacuum
birefringence in an external magnetic field $\mathbf{B}_{ext}$ = 2.5
Tesla777An indirect evidence associated with vacuum birefringence of light
passing through strong magnetic fields generated by a neutron star was
reported in [157]. Another indirect evidence was announced recently by the
STAR collaboration at RHIC [158].

$\Delta n_{exp}\leq(12\pm 17)\times 10^{-23}\,\,\,{\rm
for}\,\,\,|\mathbf{B}_{ext}|=2.5\,\,{\rm Tesla}\,.$ (7.9)

A new PVALS experiment which should come in operation in a near future will
use more powerful magnets which will allow to reach the QED value of
birefringence.

The PVLAS experimental bound (7.9) allows to estimate a bound on the value of
the coupling constant $\gamma$ of the ModMax electrodynamics which (as follows
from (7.8)) is

$\gamma\leq 3\cdot 10^{-22}\,.$

So if ModMax has something to do with fundamental physics this raises the
question why its coupling constant is so small. This of course concerns the
parameters of any NED. On the other hand, in the effective field theory
applications of NEDs to the description of condensed matter systems their
parameters may be large.

One should however note that the birefringence in external electromagnetic
background may be a combined effect of different classical and quantum non-
linear contributions which may either enhance or decrease it (like in the
$N=1$ supersymmetric non-linear electrodynamics in which $\Delta n=0$ for the
non-linearities of the fourth order in $F_{\mu\nu}$). So it is important to
have precise experimental results on the values of $n_{\parallel}$ and
$n_{\perp}$ for being able to falsify one or another NED.

To conclude this section let me note that NED has also been considered in the
context of cosmic birefringence that has long been regarded as a promising
avenue for new physics. A particularly important scenario is Cosmic Microwave
Background (CMB) birefringence induced by the coupling between a
(pseudo)scalar axion-like field and the electromagnetism, which causes a
rotation of the linear polarization plane of CMB photons during propagation
[159]. Cosmological birefringence has nowadays attracted a renewed interest
(see e.g. [160] and references therein), following its claimed extraction from
Planck 2018 data [161]. Cosmic birefringence can also manifest itself due to
Euler-Heisenberg photon-photon forward scattering that produces circular
polarized CMB photons [162, 163] for which new observational constraints have
recently been obtained [164, 165].

For other measurable effects which can be used to test the NEDs see e.g. [166,
167, 168, 169].

##  8 Outlook

For almost 90 years non-linear electrodynamics has been an active area of
research in theoretical physics with applications ranging from the theory of
fundamental interactions and cosmology to condensed matter systems and non-
linear optics. All (or most of) the models of non-linear electrodynamics
should be regarded as effective field theories. Only for few of them, namely
the Born-Infeld and Heisenberg-Euler electrodynamics, we know the
corresponding fundamental theories, such as String Theory and QED. As was
discussed in [35], some NEDs may arise upon a Kaluza-Klein dimensional
reduction of a multi-dimensional higher-curvature gravity, which in turn can
be a low energy limit of String Theory. It would be of interest to look for a
fundamental origin of other NEDs, in particular, of ModMax whose structure is
uniquely determined by two fundamental symmetries, electric-magnetic duality
and conformal invariance. Recently, Nastase [71] (see also [170]) discussed a
possibility of interpreting the generalized Born-Infeld electrodynamics, which
reduce to ModMax in the weak field limit, in terms of a brane-like
construction within a string theory. An origin of ModMax and its Born-Infeld
generalization from a $T\bar{T}$-like deformation of Maxwell’s theory has been
revealed in [171, 170, 172]. One may also wonder whether ModMax arises as a
certain (conformal) limit of a theory in which electromagnetic fields interact
with some matter fields (e.g. axion-dilaton-like) and the latter are
integrated out in the effective action. A related interesting question is
whether ModMax can be consistently quantized.

Work on these and other NED issues and their applications is going ahead.

## Acknowledgements

I am grateful to Kurt Lechner for useful comments on these notes. This work
has been partially supported by the INFN Research Project “String Theory and
Fundamental Interactions” (STEFI), by the Spanish MICINN/FEDER (ERDF EU) grant
PGC2018-095205-B-I00 and by the Basque Government Grant IT-979-16.

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