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$\mathcal{L}_{EM} = -{1\over 4} F_{\mu \nu}F^{\mu \nu} - J^{\mu}A_{\mu}$
It is a model in the framework of classical field theory.
Fields in physics are something which associate with each point in space and with each instance in time a quantity. In case of electromagnetism this is a quantity describing the electric and magnetic properties at this point. Each of these two properties turn out to have a strength and a direction. Thus the electric and magnetic fields associate with each point in space and time an electric and a magnetic magnitude and a direction. For a magnetic field this is well known from daily experience. Go around with a compass. As you move, the magnetic needle will arrange itself in response to the geomagnetic field. Thus, this demonstrates that there is a direction involved with magnetism. That there is also a strength involved you can see when moving two magnets closer and closer together. How much they pull at each other depends on where they are relative to each other. Thus there is also a magnitude associated with each point. The same actually applies to electric fields, but this is not as directly testable with common elements. Ok, so it is now clear that electric and magnetic fields have a direction and a magnitude. Thus, at each point in space and time six numbers are needed to describe them: two magnitudes and two angles each to determine a direction.
When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism.
In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as Maxwell equations in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with quantum electro dynamics - QED. http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html
Classical electrodynamics describes the interplay of light and charged objects via the Maxwell equations (the static limit is known as Coulomb force law) and additionally the Lorentz Force Law,
Recommended Textbooks:
The standard textbook is Classical Electrodynamics by John David Jackson
Coordinate Free Electrodynamics
It is possible to formulate electrodynamics in a completely coordinate-free way with the help of Differential Forms.
This is described perfectly in “Geometrical methods of mathematical physics” by Schutz starting at page 175. Alternatively a great discussion of this modern approach to classical electrodynamics can be found in Gauge Field Knots and Gravity by John Baez and Javier P Muniain
An important aspect of this modern perspective is how the homogeneous Maxwell equations follow directly as generalized Bianchi identities, if we invoke Noether's second theorem from the invariance of the action under the electromagnetic gauge group.
Geometry of Electrodynamics
The fundamental symmetry at the heart of electrodynamics is the $U(1)$ gauge symmetry. We have a copy of the $U(1)$ symmetry above each spacetime point. Geometrically $U(1)$ is the unit circle. Thus, we can imagine that we have a circle above each point. The set of all these circles is known as a fiber bundle. Depending on the system under investigation the fiber bundle looks different. In this geometric interpretation, the electromagnetic field can be understood as the curvature on the fiber bundle.
Electrodynamics was for a long time the best theory to describe the behavior of light and electrically charged objects.
It describes electric and magnetic fields, and as a consequence also X-rays, and every other form of electromagnetic wave.
In addition, electrodynamics explains how electrodynamics and magnetism are related. It has now been superseded by Quantum Electrodynamics. However, classical electrodynamics is still an important approximation for many engineering applications.
For example, classical electrodynamics is crucial for radio transmissions.
Premetric Electrodynamics
A field of active research is whether Maxwell's theory of electrodynamics is really the most general theory possible and sufficient to describe all possible systems.
One possibility to go beyond Maxwell's theory is to start with the fundamental facts of the theory, like the conservation of electrical charge, and then derive what theories of electrodynamics are, in principle, possible.
This is known as premetric approach and is described in detail in the book "Foundations of Classical Electrodynamics: Charge, Flux, and Metric" by F. W. Hehl and Yu. N. Obukhov.
In addition, the electrodynamics and especially the Maxwell equations are modified if axions exist. This is known as axion electrodynamics and was first noted in F. Wilczek, Phys. Rev. Lett. 58, 1799 (1987). The main idea here is that if the theta parameter can change in space and time $\theta=\theta(x,t)$, the term $\theta F \tilde F$ directly influences the equations of motion. This is discussed explicitly in Section 1.2 "The Theta Term" in Tong's lectures on gauge theory.
The Maxwell equations in the presence of axions read
\begin{align} {\nabla}\cdot {\bf E} &= \rho - g_{a\gamma} {\bf B}\cdot{\nabla} a\,, \label{eq:Maxwell-a}\\ {\nabla}{\times}{\bf B}- \dot{\bf E} &= {\bf J} +g_{a\gamma}({\bf B}\, \dot a -{\bf E}{ \times}{\nabla} a)\,,\label{eq:Maxwell-b}\\ {\nabla}\cdot{\bf B}&= 0\,, \label{eq:Maxwell-cone}\\ {\nabla}{\times}{\bf E}+\dot{\bf B}&=0\,, \label{eq:Maxwell-d}\\ \ddot a-{\nabla}^2 a +m_a^2 a &= g_{a\gamma} {\bf E}\cdot {\bf B}\,.\label{eq:Maxwell-c} \end{align}
Axiomatically stated electrodynamics is richer than Maxwell's theory. Symmetries in Fundamental Physics byKurt Sundermeyer
We formulated our deductions of the Maxwell-Lorentz spacetime relation already in such a way that the possible generalizations of the Maxwell-Lorentz framework are apparent: Have a dilaton, have an axion, have a skewon, either one of them, some of them, or all of them. […]
Let us come back to the possible extensions of the Maxwell-Lorentz electrodynamics by means of introducing dilaton, axion and/or skewon fields. There are two distinct classes. Those that respect the light cone /one could call them the harmless extensions) and those that don't. To the former belong, axion, dilaton, and axion-dilaton electrodynamics, to the latter skewon electrodynamics with possible admixtures of axion and/or dilaton fields.
As soon as one mixes a skewon field into the spacetime relation one cannot recover the light cone any longer and the Lorentz group is dissolved. Several such attempts are discussed in the literature.. If one wants to look for possible new physics violating Lorentz ivnariance, then the assumption of a skewon appears to be the most natural possibility. "Foundations of Classical Electrodynamics: Charge, Flux, and Metric" by F. W. Hehl and Yu. N. Obukhov
For more details and see here and here.
According to its definition, the skewon field is some kind of permeability/permittivity of spacetime — and this in a premetric setting when the metric has not yet ”condensed”. In this sense, the skewon field is an elementary electromagnetic property of spacetime. As such, it influences light propagation.
The skewon field contributes non-trivially to the electromagnetic energy. In particular, it induces an asymmetric electromagnetic energy-momentum tensor, which can cause specific gravitational effects as a source term in the Einstein-Cartan-Maxwell system (with skewon). http://www.jct.ac.il/sites/default/files/Research/GIF4/HehlSkewonJerusalem02.pdf
It was the case for Maxwell’s highly mathematical treatise "A Dynamical Theory of the Electromagnetic Field", which gives the foundation of modern electromagnetism and predicts electromagnetic waves travelling at the speed of light. “For more than twenty years, his theory of electromagnetism was largely ignored,” recounts Dyson. “It was regarded as an obscure speculation without much experimental evidence to support it.” [53]
[53] is F. J. Dyson, Why is Maxwell’s Theory so hard to understand? Proceedings of 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh 2007.