Riemann-Silberstein representation of the complete Maxwell equations set
M. V. Cheremisin
Complete set of Maxwell equations is represented by single equation using the Riemann-Silberstein vector ${F}={E}+i{B}$. We demonstrate that the Fourier form of invariants $E^2-B^2$ and ${EB}$ is proportional to dissipated power and equal to zero respectively.
hep-th0310036.tex
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\begin{document}
\title{Riemann-Silberstein representation of the complete Maxwell equations set}
\author{M. V. Cheremisin}
\affiliation{A.F.Ioffe Physical-Technical Institute,
St.Petersburg, Russia}
\date{\today}
\begin{abstract}
Complete set of Maxwell equations is represented by single
equation using the Riemann-Silberstein vector ${\bf F}={\bf
E}+i{\bf B}$. We demonstrate that the Fourier form of invariants
$E^2-B^2$ and ${\bf EB}$ is proportional to dissipated power and
equal to zero respectively.
\end{abstract}
\pacs{03.50.-z, 03.50.De} \maketitle The relative orientation of
the magnetic and electric fields in a particular co-ordinate
system is of great importance since both the real and the
imaginary parts of the product ${\bf F}^2=E^2-B^2+2i{\bf EB}$
remain invariants along upon the Lorentz relativistic
transformation. In this paper we represent Maxwell equations set
by a single equation using Riemann-Silberstein \cite{Riemann},
\cite{Silberstein} formalism. Then, Fourier analysis allows us to
eliminate the role playing by both invariants.

We use the set of conventional Maxwell equations

\begin{eqnarray}
\text{rot}{\bf B} =\frac{1}{c}\frac{\partial {\bf E}}{\partial
t}+\frac{4\pi}{c}{\bf
J} \qquad \text{div}{\bf B}=0 \eqnum{1} \label{maxwell_1}\\
\text{rot}{\bf E}=-\frac{1}{c}\frac{\partial {\bf B}}{\partial t}
\qquad \text{div}{\bf E}=4\pi \rho. \nonumber
\end{eqnarray}
Here, the total current ${\bf J}={\bf J}^{\text{free}} + c \cdot
\text{rot}{\bf M} + \frac {\partial \bf P}{\partial t}$ and charge
density $\rho=\rho^{\text{free}}- \text{div}{\bf P}$ are modified
with respect to those for free carriers ${\bf J}^{\text{free}},
\rho^{\text{free}}$ taking into account the polarization $\bf P$
and magnetization $\bf M$ of the system \cite{Tamm}.

With the help of conventional definition ${\bf
B}=\text{rot}{\mathbf A}$, ${\bf E}= - \frac {1} {c} \frac
{\partial {\bf A}} {\partial t}-\nabla {\phi}$ widely used to
represent vector and electrical auxiliary potentials respectively
Eqs.(\ref{maxwell_1}) yield

\begin{equation}
\fbox{} {\bf A} =- \frac{4\pi}{c} {\bf J}+\nabla {\psi} \qquad
\fbox{} \phi =- 4\pi \rho- \frac{1}{c}\frac{\partial
\psi}{\partial t},
\label{vector}\\
\end{equation}
where $\psi=\text{div}{\bf A}+\frac{1}{c}\frac{\partial
\phi}{\partial t}$, $\fbox{}$ is the d'Lambert operator. We argue
that Eqs.(\ref{vector}) provide the gauge transformation ${\bf A}
\rightarrow {\bf A}+\nabla f, \phi \rightarrow {\bf
\phi}-\frac{1}{c}\frac{\partial f}{\partial t}$ is already
satisfied.

It is to be noted that Eqs.(\ref{vector}) differ with respect to
those known in literature. Usually, upon the extra condition
$\psi=0$ one can readily recover the conventional Lorentz gauge
\cite{Lorentz} $\text{div}{\bf A}+\frac{1}{c}\frac{\partial
\phi}{\partial t}=0$, therefore Eqs.(\ref{vector}) become
uncoupled. One can check, however, that the gauge
transformation(see above), if applied, results in additional
equation $\fbox{}f=0$, which is always disregarded
\cite{Stratton}.

We now represent Maxwell equations by means of Riemann-Silberstein
vector ${\bf F}$. Eqs.(\ref{maxwell_1}) yield
\begin{equation}
\text{rot}{\bf F} =\frac{i}{c}({\bf \dot{F}}+ 4\pi {\bf J}),
\qquad \text{div}{\bf F}=4\pi \rho
\label{maxwell_2}\\
\end{equation}
and finally
\begin{equation}
\fbox{} {\bf F} = {\bf \dot{I}}-ic \cdot \text{rot}\bf{I},
\label{maxwell_3}\\
\end{equation}
where we introduce auxiliary vector ${\bf \dot{I}}= 4\pi({\bf
\dot{J}}/c^2+ \nabla \rho)$. One can complete the transformed
Maxwell equations set implied above using the conventional
boundary conditions as follows
\begin{equation}
F_{2n}-F_{1n}=4\pi \sigma, \qquad [{\bf nF_{2}}]-[{\bf
nF_{1}}]=\frac{4\pi i}{c} {\bf j},
\label{boundary}\\
\end{equation}
where $F_{1,2}$ is the complex field above(below) the interface,
$\bf n$ the normal to the interface. Then, we introduce the unit
area related surface charge density
$\sigma=\sigma^{free}-(P_{2n}-P_{1n})$ and surface current density
${\bf j}={\bf j}^{free}+c([{\bf nM_{2}}]-[{\bf nM_{1}}])$.

The advantage of the Eqs.(\ref{maxwell_3}),(\ref{boundary}) is, in
fact, the compact representation for both the current(charge)
sources of the electromagnetic field. In particular case of
electric(magnetic) dipole irradiation Eq.(\ref{maxwell_3}) can
reproduce, for example, the conventional result \cite{Landau} as
follows

\begin{equation}
\fbox{} {\bf F} = -\frac{4\pi i}{c}\text{rot}({\bf \dot{K}}-ic
\cdot \text{rot{\bf K}} )-4\pi \fbox{}{\bf P}.
\label{irradiation}\\
\end{equation}
Here, we neglect free charge(current) densities, and, then
introduce the complex electric(magnetic) dipole moment of the
system ${\bf K }={\bf P}+i{\bf M}$.

With the complex field ${\bf F}$ specified above the energy
conservation law may be represented as follows
\begin{equation}
\frac {1}{8\pi} \frac{\partial} {\partial t} ({\bf
FF^{*}})=-\text{div}{\bf S}-{\bf J}Re{\bf F},
\label{energy}\\
\end{equation}
where ${\bf F}^{*}={\bf E}-i{\bf B}$, ${\bf S}=\frac{ic}{8
\pi}[{\bf FF}^{*}]=\frac{c}{4 \pi}[{\bf EB}]$ is the conventional
Poynting energy flux density.

Within Fourier formalism ${\bf F}=\int \limits_{-\infty}^{+\infty}
{{\bf F}_{k,\omega}}e^{i\omega t-i{\bf kr}} \frac{d^{3}{\bf
k}}{(2\pi)^3} \frac{d\omega}{2\pi}$ Eq.(\ref{maxwell_3}) yields
\begin{equation}
{\bf F}_{k,\omega}=\frac{c}{\kappa^2}[{\bf
kI}_{k,\omega}]-\frac{i\omega}{\kappa^2}{\bf I}_{k,\omega},
\label{Fourier}\\
\end{equation}
where ${\bf I}_{k,\omega}=4\pi(\frac{{\bf
J}_{k,\omega}}{c^2}-\frac{\rho_{k,\omega}{\bf k}}{\omega})$, then
$\kappa^2=k^2-\omega^2/c^2$. Since $({\bf I}_{k,\omega}[{\bf
kI}_{k,\omega}])=0$, one obtains

\begin{equation}
\text{Re}{\bf F}^2_{k,\omega}=\frac{4\pi}{\kappa^2} ({\bf
J}_{k,\omega} {\bf I}_{k,\omega}),\qquad \text{Im}{\bf
F}^2_{k,\omega}=0. \label{invariants}
\end{equation}
The latter result, namely that $({\bf E}_{k,\omega}{\bf
B}_{k,\omega})=0$ can be easily derived using
Eqs.(\ref{maxwell_1}) as well. It's worthwhile to mention that
${\bf E}_{k,\omega}=-\frac{i \omega}{\kappa^2}{\bf I}_{k,\omega}$,
and thus $\text{Re}{\bf F}^2_{k,\omega}$ is proportional to
dissipated energy $({\bf J}_{k,\omega}{\bf E}_{k,\omega})$. For
$\rho=0$ Eqs.(\ref{invariants}) correspond to electromagnetic wave
propagating in vacuum when ${\bf EB}=0, E^2-B^2=0$.

In conclusion, we represent complete set of Maxwell equations by
single equation using Riemann-Silberstein formalism. Fourier form
of invariants $E^2-B^2$ and ${\bf EB}$ is proportional to
dissipated power and equal to zero respectively.

This work was supported by RFBR (grant 03-02-17588) and
LSF(HPRI-CT-2001-00114)

\begin{references}

\bibitem{Riemann}H.Weber, Die partiellen Differential-Gleichungen
der Mathematischen Physik nach Riemann's Vorlesungen (Friedrich
Vieweg und Sohn, Braunschweig), 348, 1901
\bibitem{Silberstein} L.Silberstein, Ann. d Phys.(Leipzig) {\bf 22}, 579 1907; {\bf 24}, 783, 1907
\bibitem{Tamm} I.E.Tamm, Osnovi teorii elektrichestva, Moskow, 1957
\bibitem{Lorentz} H.A.Lorentz, Encyklopedie der mathematischen Wissenschaffen, {\bf 5}, 156, 1903
\bibitem{Stratton} J.A.Stratton, Electromagnetic theory, New York, 1941
\bibitem{Landau} L.D.Landau, E.M.Lifshits, The classical theory of fields, Pergamon, New
York,1966
\end{references}
\end{document}

hep-th0310036.extracted_bib
\bibitem{Riemann}H.Weber, Die partiellen Differential-Gleichungen
der Mathematischen Physik nach Riemann's Vorlesungen (Friedrich
Vieweg und Sohn, Braunschweig), 348, 1901

\bibitem{Silberstein} L.Silberstein, Ann. d Phys.(Leipzig) {\bf 22}, 579 1907; {\bf 24}, 783, 1907

\bibitem{Tamm} I.E.Tamm, Osnovi teorii elektrichestva, Moskow, 1957

\bibitem{Lorentz} H.A.Lorentz, Encyklopedie der mathematischen Wissenschaffen, {\bf 5}, 156, 1903

\bibitem{Stratton} J.A.Stratton, Electromagnetic theory, New York, 1941

\bibitem{Landau} L.D.Landau, E.M.Lifshits, The classical theory of fields, Pergamon, New
York,1966
