hep-th0208170.tex
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{\Large\bf On classical electrodynamics in odd-dimensional worlds}
\\[5mm]
{B.\ P.\ Kosyakov}\\[2mm]
{{\sl Russian Federal Nuclear Center, Sarov, 607190 Russia}}\\[2mm]
{E-mail address: \verb|kosyakov@vniief.ru|}\\[5mm]
{\bf Abstract}\\[2mm]
\begin{minipage}[t]{130mm}
\noindent
{\sl Given the conventional Maxwell--Lorentz formulation of classical 
electrodynamics in a 
flat spacetime of arbitrary odd dimension, the retarded vector potential 
$A^\mu$ generated by a point-like charge is found to be pure gauge, 
$A^\mu=\partial^\mu\chi$.
By the Gauss law, the charge vanishes.
Therefore, classical electromagnetism is missing from odd-dimensional worlds.
By contrast, a particle interacting with a scalar field generates nonzero
field strength.
If the electromagnetic action is augmented by the addition of the Chern--Simons 
term, the interaction picture in the three-dimensional world becomes nontrivial.}
\end{minipage}
\end{center}
PACS numbers: 03.50.De, 11.10.Kk.
\vskip5mm
In this note we come to a surprising result that classical 
electromagnetic interactions, realized in the Maxwell--Lorentz sense,
are missing from odd-dimensional worlds. 
It is well known that the four-dimensional 
electrodynamics can be directly extended to any even dimensions to result in
a consistent theory \cite{k99}.
However, one can hardly conceive that similar extension to 
odd-dimensional spacetimes leads to a picture where 
charged particles generate vanishing field strengths, and, by the Gauss
law, the charges are zero.

The classical odd-dimensional physics is of concern to us not only for 
methodological reasons but also in relation to the holographic principle
\cite{k0, k01}.
By this principle, a physical system can be deemed both
classical and quantum, depending on the spacetime which is attributed to it: 
the system behaves classically in the `bulk' (some $D$-dimensional 
spacetime), whereas the picture becomes quantum being holographically 
projected onto a `screen' (part of the $D-1$-dimensional boundary of 
this spacetime).
For example, the hydrogen atom is a typical quantum-mechanical system in
a four-dimensional screen, but its behavior is classical when 
it is 
referred to a pertinent five-dimensional bulk. 
(This classical  five-dimensional description may be regarded as a kind of
the `hidden-variable' theory.)
A characteristic feature of the Schr\"odinger description, the lack
of radiation in stationary states, should be recovered in the dual
classical  description.
It transpires that not only radiation but also the Coulomb-like part of
electromagnetic field is absent in five-dimensional worlds
(the hidden-variable theory
is not so plain which is naively expected).

We begin with the conventional electrodynamic action in Gaussian units
\begin{equation}
{S}=
S_{\rm p}-\int d^{D}\!x\,\left(j^\mu A_\mu+ {\frac{1}{4\,\Omega_{D-2}}}\,
F_{\mu\nu} F^{\mu\nu}\right)  
\label
{S-em}
\end{equation}
where $S_{\rm p}$ is the mechanical part of the action responsible for the
particle behavior, $\Omega_{D-2}$ is the area of the $D-2$-dimensional 
unite sphere, 
$
\Omega_{D-2}=2\,{\pi^{(D-1)/{2}}}/{\Gamma\left[({D-1})/{2}\right]},
$
and the field strength is expressed through the vector potential, 
$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$.
As is well known, the action (\ref{S-em})  for $D=3$ should be 
augmented by the addition of the Chern--Simons term.
However, we wish to compare even-dimensional and odd-dimensional cases 
within a unified framework, so that we 
ignore for a while any augmentations and employ the action (\ref{S-em}),
which is well suited to the even-dimensional electrodynamics \cite{k99}. 

Variation of (\ref{S-em}) with respect to $A_{\mu }$ gives 
the $D$-dimensional Maxwell equations 
\begin{equation}
\partial _{\mu }F^{\mu \nu }=
\Omega _{D-2}j^{\nu }.
\label
{maxw}
\end{equation}
Due to the linearity of Eq.\ (\ref{maxw}), we may restrict our consideration to 
the single-particle case.
A particle with a $\delta$-shaped distribution of the electric charge $e$,
moving along an arbitrary world line $z^\mu(s)$, gives rise to the 
current  
\begin{equation}
j^{\mu}(x)=e\int_{-\infty}^\infty\!ds\,v^{\mu}(s)\,
\delta ^{D}\biggl (x-z(s)\biggr)
\label
{j-em}
\end{equation}
where $v^\mu :=dz^\mu/ds :={\dot z}^\mu$ is the $D$-velocity of the particle.

We impose the Lorentz condition on the vector potential to fix the gauge, 
$\partial_\mu A^\mu=0$; then Eq.\ (\ref{maxw}) reduces to 
\begin{equation}
\Box A^\mu=\Omega_{D-2} j^\mu.
\label
{wave-equation-D}
\end{equation}                                          

A solution to Eq.\ (\ref{wave-equation-D}) is 
\begin{equation}
A^\mu(x)=\int d^{D}\!x\, G_{\rm ret}(x-x')\,j^\mu(x')
\label
{sol}
\end{equation}
where $G_{\rm ret}(x)$ is the retarded Green's function of the wave equation. 
For a flat spacetime of dimension $D=2n+3,\,n=0,1,...$, we have
(see, e.\ g., \cite{boglogtod}) 
\begin
{equation}
G_{\rm ret}(x)=N_D^{-1}\,\theta(x_0)\left(\frac{d}{dx^2}\right)^n
\frac{\theta(x^2)}{\sqrt{x^2}},
\quad
N_D
=\frac{\Gamma\left(n+\frac12\right)}{\sqrt{\pi}}.
\label
{Green-D}
\end{equation}
We insert (\ref{j-em}) and (\ref{Green-D}) in (\ref{sol}).
Denoting
$
R_\mu= x_\mu-z_\mu(s)
$
where $x_\mu$ is the observation point and $z_\mu(s)$ the emission 
point on the world line, we arrive at
\begin{equation}
A^\mu(x)=\frac{e}{N_D}\int^{\infty}_{-\infty} ds\, v^\mu(s)\,\theta(R_0)
\left(\frac{d}{dR^2}\right)^n\frac{\theta(R^2)}{\sqrt{R^2}}.
\label
{sol2}
\end{equation}
We choose $\lambda=R^2$ as the integration variable and observe that 
$d\lambda/ds=-2R\cdot v$ to  obtain
\begin{equation}
A^\mu(x)=
\frac{e}{2N_D}\int^{\infty}_{-\lambda_m} d\lambda\,\frac{v^\mu}{R\cdot v}
\left(\frac{d}{d\lambda}\right)^n\frac{\theta(\lambda)}{\sqrt{\lambda}}
\label
{sol3}
\end{equation}
where $\lambda_m$ is an uttermost distance squared between the observation 
point and the world line.
For $n\ge 1$, the integral (\ref{sol3}) diverges at $\lambda=0$.
This is due to the fact that the integrand of (\ref{sol}) is the product
of two singular distributions.
Since Green's functions are singular in themselves, they are normally 
integrated with smooth sources $j^\mu(x)$. 
To assign mathematical sense to the ill-defined expression (\ref{sol3}), 
we regularize it by the convention that the derivatives should act on the 
left and the surface term is ignored.
Following to this regularization prescription, we find
\begin{equation}
A^\mu(x)=
\frac{e(-1)^n}{2N_D}\int^{\infty}_0 \frac{d\lambda}{\sqrt{\lambda}}\, 
\left(\frac{d}{d\lambda}\right)^n\frac{v^\mu}{R\cdot v}
=
\frac{e(-1)^n}{2N_D}\int^{\infty}_0 \frac{d\lambda}{\sqrt{\lambda}}\, 
\frac{\partial}{\partial x_\mu} \left(\frac{d}{d\lambda}\right)^n
\log ({R\cdot v}).
\label
{sol4}
\end{equation}
With estimation $R^\mu =O(\lambda^{1/2}),\,v^\mu=O(1)$ as 
$\lambda\to\infty$, valid for asymptotically straight timelike world lines, 
the integral (\ref{sol4}) converges uniformly 
for $n\ge 1$ (that is, $D\ge 5$), 
so that the integration and the differentiation with respect to $x^\mu$
are interchangeable.
As to $n=0$  (that is, $D=3$), one may think of Eq.\ (\ref{sol4}) as 
\begin{equation}
A_\mu(x)=\frac{e}{2}\,\lim_{\epsilon\to 0} \,
\frac{\partial}{\partial x^\mu}
\int^{\infty}_0 \frac{d\lambda}{\sqrt{\lambda}}\, 
e^{-\epsilon\lambda}\,\log ({R\cdot v}).
\label
{sol5}
\end{equation}
Be it as it may, whenever  the integration and  ${\partial}/{\partial x_\mu}$ 
are commutative\footnote{In fact, the possibility to 
interchange these operations is a subtle issue.
The interchanging is certainly impermissible (for lack of the uniform 
convergence) when the measure contains a 
singular component like $\delta$-function or its derivatives. 
To illustrate, for $D=4$,  in lieu of (\ref{sol3}),  we have 
\begin{equation}
A^\mu(x)=
e\int^{\infty}_{-\lambda_m} d\lambda\,\delta(\lambda)\,\frac{v^\mu}{R\cdot v}=
e\int^{\infty}_{-\lambda_m} d\lambda\,\delta(\lambda)\,
\frac{\partial}{\partial x^\mu} \log ({R\cdot v}).
\label
{sol-4D}
\end{equation}
The differentiation in (\ref{sol-4D}) must be done prior to the 
integration, and, therefore, we obtain
$e v^\mu/({R\cdot v})|_{\rm ret}$, the Li{\'e}nard--Wiechert
vector potential, not 
$e \partial^\mu [\log({R\cdot v})]_{\rm ret}$ which differs from the 
Li{\'e}nard--Wiechert result in the term
$eR^\mu[(R\cdot{\dot v})-1]/(R\cdot v)|_{\rm ret}$.},
$A^\mu=\partial^\mu\chi$. 


We integrate (\ref{maxw}) over a domain $V$ containing the charged
particle in the hyperplane perpendicular to the particle world line and take
into account the relation $F_{\mu\nu}=0$ to get
\[
e\,{\Omega_{D-2}}=\oint_{V} d^{D-1}\!x\,\,\partial_\mu F^{\mu\nu} v_\nu=
\oint_{\partial{V}} d^{D-2}\!x\,\,n_\mu F^{\mu\nu} v_\nu=0.
\]
Since the coupling $e$ is zero, the action (\ref{S-em})
relates to a trivial picture without interaction.

This result is insensitive to the form of the regularization 
prescription.
[By contrast, in the even-dimensional case,  no regularization renders 
the corresponding integral uniformly convergent; in fact, any
regularization is superfluous here, 
the integrals are well-defined.]

For comparison, we address a particle interacting with 
scalar field $\phi$. 
The action is  \cite{Barut}
\begin{equation}
{S}=S_{\rm p}+
\int d^{D}\!x\,
\left(j \phi+
\frac{1}{2\Omega_{D-2}}\,\partial^\mu\phi\,\partial_\mu\phi\right)  
\label
{W-scalar}
\end{equation}
where
\[
j(x)=g\int^{\infty}_{-\infty} ds\,\sqrt{v\cdot v}\,
\delta ^{D}\biggl (x-z(s)\biggr).
\]

This action leads to the field equation
$
\Box \phi=\Omega_{D-2}j
$.
Following basically the above procedure, we find the retarded solution:
\begin{equation}
\phi(x)=
\frac{g(-1)^n}{2N_D}\int^{\infty}_0 \frac{d\lambda}{\sqrt{\lambda}}\, 
\left(\frac{d}{d\lambda}\right)^n \frac{\sqrt{v\cdot v}}
{R\cdot v}.
\label
{sol-phi}
\end{equation}
There is nothing uncommon with this solution.
In any case, gradients of $\phi$ are nonzero.
Those play the role of the field strength, which is manifested by the
example of the particle Lagrangian $L_{\rm p}=-m\,(v\cdot v)^{1/2}$ leading 
to the equation of
motion for the particle \cite{Barut}
\begin{equation}
\frac{d}{ds}\,[(m+g\phi)\,v^\mu]=g\partial^\mu\phi.
\label
{particle-equation-motion}
\end{equation}                                          
It is likely that this $L_{\rm p}$ must be augmented by the addition 
of terms with higher derivatives for the description to
be self-consistent in the cases $D=5, 7,...$ \cite{k99}, but this fact is 
apparently unrelated to the form of the force appearing in the right-hand 
side of (\ref{particle-equation-motion}).

We finally consider the effect of incorporation of the Chern--Simons term.
We use the $D=3$ electrodynamic action \cite{DeserJackiwTempleton82}  
\begin{equation}
{S}=
S_{\rm p}-\int d^{3}\!x\,\left(j^\mu A_\mu+ {\frac{1}{8\pi}}\,
(F_{\mu\nu} F^{\mu\nu}-\mu\,\epsilon^{\alpha\beta\gamma}A_\alpha 
F_{\beta\gamma})\right).  
\label
{S-CS}
\end{equation}

The field equation which follows from the actions (\ref{S-CS}) is
\[
\partial_\alpha F^{\alpha\beta}+\mu\,{}^\ast\! F^\beta=2\pi j^\alpha
\]
where ${}^\ast\! F^\alpha:={\scriptstyle\frac12}\,\epsilon^{\alpha\beta\gamma}
F_{\beta\gamma}$. 
We rewrite this equation as 
\[
\Lambda^{\alpha\beta}(\partial)\,{}^\ast\! F_\beta :=(\mu\,\eta^{\alpha\beta}+
\epsilon^{\alpha\beta\gamma}\partial_\gamma)\,{}^\ast\! F_\beta=2\pi j^\alpha,
\]
iterate it with $\Lambda(\partial)$, and use the Bianchi identity 
$\partial_\beta{}^\ast\! F^\beta=0$, to obtain
\begin{equation}
(\Box+\mu)\,{}^\ast\! F_\alpha=2\pi \Lambda_{\alpha\beta}(\partial) j^\beta.
\label
{field-eq-CS}
\end{equation}

It is clear from Eq.\ (\ref{field-eq-CS}), even without explicit writing 
its solution, that the field strength  generated by a 
point-like charged particle is nonzero.
Therefore, the scheme as a whole is nontrivial.
Note that Eq.\ (\ref{field-eq-CS}) allows a direct expression of the field 
variables $F_{\alpha\beta}$ trough the source variables $j^\alpha$, unlike 
Eq.\ (\ref{maxw}) which, with this aim in view, must be converted to the 
the vector potential equation (\ref{wave-equation-D}), culminating in
the pure gauge solution.


To summarize, in an effort to obtain a classical description of
odd-dimensional worlds, one should 
not use the action (\ref{S-em}) resulting in zero electromagnetic coupling $e$.
A credible speculation is that some nonelectromagnetic field, e.\ g.,
scalar, is of paramount importance here.
An alternative way out is a modification of the electromagnetic action 
(\ref{S-em}) in the Chern--Simons spirit, which, though, may introduce some 
off-beat properties (massive gauge fields, nonlinearity in higher dimensions, 
etc.).   


\vskip5mm \noindent {\large {\bf Acknowledgments}} \vskip2mm \noindent 
This work is supported by the
International Science and Technology Center, Project 840.


\begin{thebibliography}{99}


\bibitem
{k99}  
B.\ P.\ Kosyakov, 
Teor.\ Mat.\ Fiz. {\bf 119}, 119 (1999)
[Theor.\ Math.\ Phys. {\bf 119}, 493 (1999)]; 
hep-th/0207217.


\bibitem
{k0}  
B. P. Kosyakov, 
Phys.\ Lett. {\bf B 492}, 349, 2000; hep-th/0009071.

\bibitem
{k01}  
B.\ P.\ Kosyakov, in {\it Proceedings of the Sixth Workshop on non-Perturbative
QCD}, Paris, France, 5-9 June 2001, ed.\ H.\ M.\ Fried, Y.\ Gabellini and B.\ 
M\"uller
(World Scientific, Singapore, 2002), pp.\  283-290; hep-th/0109056.

\bibitem
{boglogtod}
{N.\ N.\ Bogoliubov}, {A.\ A.\ Logunov},  and {I.\ T.\ Todorov}, 
{\it Introduction to Axiomatic Quantum Field Theory} 
(Benjamin, 
New York, 1975).


\bibitem
{Barut}
{A.\ O.\ Barut}, {\it Electrodynamics and Classical Theory
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\bibitem
{DeserJackiwTempleton82}  
S.\ Deser, R.\ Jackiw, and S.\ Templeton, 
Phys.\ Rev.\ Lett.
{\bf 48}, 975 (1982).



\end{thebibliography}

\end{document}



hep-th0208170.extracted_bib
\begin{thebibliography}{99}


\bibitem
{k99}  
B.\ P.\ Kosyakov, 
Teor.\ Mat.\ Fiz. {\bf 119}, 119 (1999)
[Theor.\ Math.\ Phys. {\bf 119}, 493 (1999)]; 
hep-th/0207217.


\bibitem
{k0}  
B. P. Kosyakov, 
Phys.\ Lett. {\bf B 492}, 349, 2000; hep-th/0009071.

\bibitem
{k01}  
B.\ P.\ Kosyakov, in {\it Proceedings of the Sixth Workshop on non-Perturbative
QCD}, Paris, France, 5-9 June 2001, ed.\ H.\ M.\ Fried, Y.\ Gabellini and B.\ 
M\"uller
(World Scientific, Singapore, 2002), pp.\  283-290; hep-th/0109056.

\bibitem
{boglogtod}
{N.\ N.\ Bogoliubov}, {A.\ A.\ Logunov},  and {I.\ T.\ Todorov}, 
{\it Introduction to Axiomatic Quantum Field Theory} 
(Benjamin, 
New York, 1975).


\bibitem
{Barut}
{A.\ O.\ Barut}, {\it Electrodynamics and Classical Theory
of Fields and Particles} (Colier-Macmillan, New York, 1964).

\bibitem
{DeserJackiwTempleton82}  
S.\ Deser, R.\ Jackiw, and S.\ Templeton, 
Phys.\ Rev.\ Lett.
{\bf 48}, 975 (1982).



\end{thebibliography}