\section{Introduction}
Let us consider the Robertson-Walker line-element written using one of
its appropriate coordinates:

\begin{equation}
\label {1.1}
ds^2=-c^2dt^2+F^2(t)N^2(r)\delta_{ij}dx^idx^j, \quad
N(r)=1/(1+kr^2/4), \quad r=\sqrt{\delta_{kl}x^kx^l}
\end{equation}
to which we shall refer as the standard coordinates of the
Robertson-Walker metrics.
We shall assume that the scale factor $F$ is dimensionless, that
$t$ has dimensions of time $T$ and that $r$ has dimensions of space $L$.
The curvature
constant $k$ will have therefore dimensions $L^{-2}$ and the speed of
light $c$ dimensions $LT^{-1}$.

From both the geometrical and physical points of view the
line-element \ref{1.1} is strictly equivalent to the following one:

\begin{equation}
\label {1.2}
d\tau^2=dt^2-\frac{F^2(t)}{c^2}N^2(r)\delta_{ij}dx^idx^j,
\end{equation}
which can be written also in the following suggestive form:

\begin{equation}
\label {1.3}
d\tau^2=dt^2-\frac{1}{c_{eff}^2}N^2(r)\delta_{ij}dx^idx^j,
\end{equation}
where:

\begin{equation}
\label {1.4}
c_{eff}=c/F(t)
\end{equation}

For any local physical process whose evolution is fast compared
with the time evolution of the scale factor, and is also local in
space or $k=0$, the line-element \ref{1.3} is telling us that this process will
evolve as if the Universe were Minkowski space-time
with an effective speed of light given by \ref{1.4}. If the time scale
of the evolution of the system that is considered is not short, like
for instance if the system is the Universe itself, the conclusion
above does not follow from the form of the line-element \ref{1.3}, but
it is an heuristic idea\,\footnote{This idea was already mentioned in
\cite{AB}. It has recently been substantiated in different contexts. See for
example \cite{BM} and references therein} that in our opinion deserves to be
analyzed.

More precisely, let $\rho(t)$ and $p(t)$ be the density and the
pressure of the fluid filling homogeneously and isotropically the
Universe. We are going to assume that in the phase during which the
Universe is dominated by radiation the equation of state is not $p=1/3\rho
c^2$ as it is usually assumed, but:

\begin{equation}
\label {1.5}
p=\frac{1}{3}\rho c^2/F^2=1/3\rho c_{eff}^2
\end{equation}

Section 2 will set up the basic equations that can be derived from
Einstein's equations and the equation of state above for any interval
of time where it can be relevant. In section 3 we shall justify this
equation of state from the point of view of Relativistic kinetic
theory by proposing an appropriate re-definition of the
energy-momentum tensor of a gas of zero mass particles. In section 4
we shall set up the equivalence of the equation of state \ref{1.5}
with a particular inflationary scalar field in the very early phase
of the evolution of the Universe. In section 5 we shall consider the
very simple model that follows from assuming this equation of state
all the way during the life-span of the Universe.

\section{The field equations}

For a Robertson-Walker line element, Einstein's field equations with
a perfect fluid energy-momentum source term are\,\footnote{There are
many good books on cosmology. Two of them are \cite{Weinberg} and
\cite{Islam}}

\begin{equation}
\label {3.1}
S_{\alpha\beta}-\Lambda g_{\alpha\beta}=\frac{8\pi G}{c^2}T_{\alpha\beta}, \quad
T_{\alpha\beta}=\rho u_\alpha u_\beta + p\hat g_{\alpha\beta}, \quad
\hat g_{\alpha\beta}=g_{\alpha\beta}+ u_\alpha u_\beta
\end{equation}
where $u^\alpha$ is the unit time-like vector of the world-lines of
the cosmological fluid. Its components in standard coordinates are:

\begin{equation}
\label {3.2}
u^0=1, \quad u^i=0
\end{equation}
$\rho$ is the mass density with dimensions $ML^{-3}$ and $p$ is the
pressure with dimensions $ML^{-1}T^{-2}$.

Eqs. \ref{3.1} can be reduced to the following two equations, a dot meaning a
derivative with respect to $t$:

\begin{equation}
\label {3.3a}
{\dot F}^2=1/3(8\pi G\rho +c^2\Lambda)F^2-c^2k
\end{equation}
\begin{equation}
\label {3.3b}
2F\ddot F+{\dot F}^2+kc^2=-\frac{8\pi G}{c^2}pF^2+c^2\Lambda F^2
\end{equation}
These two equations imply the conservation equation:

\begin{equation}
\label {3.4}
\dot \rho+3(\rho+p/c^2)\dot F/F=0
\end{equation}
which is also equivalent to:

\begin{equation}
\label {3.5}
\frac{d}{dF}(\rho F^3)=-(3p/c^2)F^2
\end{equation}

We shall use the standard notation for the Hubble and the deceleration
functions:
\begin{equation}
\label {3.6}
H=\dot F/F, \quad q=-F\ddot F/{\dot F}^2
\end{equation}
and for a while we shall use units of length, time and mass such that:

\begin{equation}
\label {3.7}
c=8\pi G=1, \quad H_0 =1
\end{equation}
where $H_0$ is the present value of $H$.

As it was said in the introductory section we are going to discuss a
model of the universe for which the equation of state \ref{1.5}
holds. The equation \ref{3.5} can be written as:

\begin{equation}
\label {3.8}
\frac{d}{dF}\ln \rho=-(F^{-2}+3)F^{-1}
\end{equation}
which integrated yields the $F$ dependence of $\rho$:

\begin{equation}
\label {3.9}
\rho=AF^{-3}\exp[1/(2F^2)]
\end{equation}
where $A$ is a constant of integration. The $t$ dependence of $F$ can
be inferred from the result of integrating Eq. \ref{3.3a}:

\begin{equation}
\label {3.10}
t(F_2)-t(F_1)=\int^{F_2}_{F_1}\frac{dF}{[1/3(\rho+ \Lambda)F^2-k]^{1/2}}
\end{equation}.

Eq. \ref{3.3a} can be re-written:

\begin{equation}
\label {3.11}
k=F^2(H^2-\Lambda/3)(\Omega-1)
\end{equation}
where as usual we have defined $\Omega$ as the quotient of $\rho$ and
the critical density $\rho_c$:

\begin{equation}
\label {3.12}
\Omega=\rho/\rho_c, \quad  \rho_c=3H^2-\Lambda
\end{equation}

Eq. \ref{3.3b} becomes:

\begin{equation}
\label {3.14}
-2F^2qH^2+F^2H^2\Omega-1/3F^2\Lambda\Omega-2/3\Lambda F^2+H^2\Omega
-1/3\Lambda\Omega=0
\end{equation}
Solving the system of Eqs. \ref{3.12} and \ref{3.14} for $\Omega$ and
$\Lambda$ we get:

\begin{equation}
\label {3.15}
\Omega=\frac{2F^2\rho}{6F^2H^2(q+1)-\rho(F^2+1)}
\end{equation}
and:
\begin{equation}
\label {3.13}
\Lambda=-\frac{6F^2qH^2-\rho(F^2+1)}{2F^2}
\end{equation}

The two equations \ref{3.11} and \ref{3.13} would yield directly the
values of the two free parameters of the model derived from the
the density and the Hubble and deceleration functions at some
given time.

\section{Relativistic kinetic theory}

Let $f(x^\alpha, k_\beta)$ be the distribution function of a gas of
particles with mass $m=0$. The energy momentum tensor of such gas has always been defined
as\,\footnote{Relativistic kinetic
theory for a gas of massless particles filling the Universe at some
stage has been thoroughly studied in
\cite{EGS}. A gas of massive particles may behave also as a gas of
massless particles at some stages of the evolution of the Universe;
see the preceding reference and
\cite{Hakim}. See also \cite{Bel} for a discussion of the concept of
equilibrium in a cosmological context}
:

\begin{equation}
\label {2.1}
T_{\alpha\beta}(x^\mu) =\int_{C^+_x} f(x^\mu, k_\nu)k_\alpha k_\beta
\sqrt{-g}\frac{dk^1\wedge dk^2\wedge dk^3}{-k_0}
\end{equation}
where ${C^+_x}$ is the future-poynting light-cone with vertex at the
event $x$.

A kinetic theory model of a domain of space-time filled with radiation
is then governed by a system of equations which are:

\begin{itemize}
\item Einstein's equations \ref{3.1} where $T_{\alpha\beta}$ is defined
as above and
\item A kinetic equation for the distribution function $f(x^\alpha,
k_\beta)$.
\end{itemize}
This equation is the Liouville equation if one assumes
either 1) that the radiation is so diluted that collisions can be
neglected or ii) that there is equilibrium balance. This
equation then reads:

\begin{equation}
\label {2.3}
Df\equiv k^\alpha\partial_\alpha f-
\Gamma^\alpha_{\lambda\mu}u^\lambda u^\mu
\frac{\partial f}{\partial u^\alpha}=0
\end{equation}
where $\Gamma^\alpha_{\lambda\mu}$ are the Christoffel symbols of the
line-element \ref{1.1}
In a cosmological Universe filled with black-body radiation these equations
reduce to two groups of equations which using a system of standard
coordinates are:
\begin{itemize}
\item Einstein's equations \ref{3.3a}, \ref{3.3b} or/and \ref{3.4}

where now $\rho$ and $p$ are given by:

\begin{equation}
\label {2.4a}
\rho=4\pi\int_0^\infty f(t,\nu )\nu^3\, d\nu
\end{equation}
\begin{equation}
\label {2.4b}
p=4\pi\int_0^\infty f(t,\nu )\nu \,d\nu
\end{equation}

where $f(t,\nu)$ is Planck's distribution function:

\begin{equation}
\label {2.5}
f=\frac{2h}{\exp((h\nu/k_BT(t))-1)}
\end{equation}
$k_B$ being Boltzman's constant and $T(t)$ the local temperature
function:

\begin{equation}
\label {2.5bis}
T(t)=T_0F(t)
\end{equation}
$T_0$ being the temperature at some particular time.

\item The Liouville equation that in this particular situation reads:

\begin{equation}
\label {2.4c}
Df=\nu\partial_t f-H\nu^2\frac{\partial f}{\partial\nu}=0
\end{equation}
\end{itemize}

Since $k^\alpha$ is a null vector the trace of the energy momentum
tensor \ref{2.1} is zero, $g^{\alpha\beta}T_{\alpha\beta}=0$, and this
is equivalent to the equation of state $p=1/3\rho$
and not to the equation of state \ref{1.5}.

Let us remind that the world-lines of the cosmological fluid in a
Robertson-Walker universe are the world-lines of a conformal Killing
vector. That is to say that there exists a function $\xi(x^\alpha)$
such that the vector field $\xi^\alpha=\xi u^\alpha$ satisfies the
following equations:

\begin{equation}
\label {2.8}
\nabla_\alpha \xi_\beta+\nabla_\beta\xi_\alpha=2\Psi g_{\alpha\beta}
\quad \Psi=1/4\nabla_\sigma\xi^\sigma
\end{equation}

Consider the new definition of the energy-momentum tensor:

\begin{equation}
\label {2.7}
T_{\alpha\beta}(x^\sigma) =\int_{C^+_x} f(x^\mu, k_\nu)
\bar k_\alpha \bar k_\beta
\sqrt{-g}\frac{dk^1\wedge dk^2\wedge dk^3}{-k_0}
\end{equation}
where $\bar k_\mu$ is

\begin{equation}
\label {2.8bis}
\bar k_\mu=\xi^{-1}(k_\mu+(1-\xi^{-1})(k^\sigma u_\sigma) u_\mu)
\end{equation}
Consider now the following metric:

\begin{equation}
\label {2.9}
\bar g^{\alpha\beta}=g^{\alpha\beta}+(1-\xi^{-2})u^\alpha u^\beta
\end{equation}
We now have:

\begin{equation}
\label {2.10}
\bar g^{\alpha\beta}T_{\alpha\beta}=0
\end{equation}
and taking into account the fact that in standard coordinates we have
$\xi=F$ from Eq. \ref{2.10} it follows that the new expressions for $\rho$:

\begin{equation}
\label {2.11}
\rho(t)=aT^4(t), \quad a=\frac{8\pi^5k_B^4}{15h^3}
\end{equation}
and $p$ satisfy the equation of state \ref{1.5}.

The fact that a small modification of the standard formalism of
Relativistic kinetic theory for a gas of zero mass particles allows
to derive Eq. \ref{1.5} is, we believe, a sufficient justification for
this equation of state. Nevertheless one should be aware of an
important conceptual innovation. Contrary to what happened with
\ref{2.1}, the modified energy-momentum tensor \ref{2.7} will not
satisfy automatically the conservation equation \ref{3.4} as a
consequence of the Liouville equation \ref{2.3}. It will satisfy this
conservation equation only as a consequence of Einstein's equations
\ref{3.3a} and \ref{3.3b} and this means: i) that the Liouville equation
can not be implemented and ii) that the local temperature function
$T(t)$ in Planck's distribution function \ref{2.5} has to be derived by equating the
expression of the density above and that that we derived before in section 2, i.e.
Eq. \ref{3.9}:

\begin{equation}
\label {2.12}
AF^{-3}\exp[1/(2F^2)]=aT^4
\end{equation}
This temperature function corresponds to a distribution function that
satisfies a kinetic equation that can be written in many ways; a
possibility being:

\begin{equation}
\label {2.13}
Df(t,\nu)=1/8H(F^2-1)F^{-2}\nu h^{-1} f(f+2h)\ln(1+2hf^{-1})
\end{equation}

Therefore while we can still speak about a black-body radiation, the
equilibrium described by the distribution function \ref{2.5} is now
local and not global. This is a satisfactory situation for an expanding
Universe.

\section{Inflationary equivalent model}
It is well known
that it is not necessary to impose
that the source of a Robertson-Walker space-time is a perfect fluid
because this follows directly from Einstein's equations and the
assumptions that lead to the line-element \ref{1.1}. This is only
necessary if one wants to specify local conditions, like for example
choosing a particular equation of state. But by the same
token we can always assume that the source of a Robertson-Walker
universe is a scalar field  for which the energy-momentum tensor is:

\begin{equation}
\label {4.1}
T_{\alpha\beta}=\partial_\alpha\Phi\partial_\beta\Phi
-g_{\alpha\beta}[\partial_\mu\partial^\mu\Phi-V(\Phi)]
\end{equation}
where $V(\Phi)$ is a potential function depending on the model being
considered. In standard coordinates
the equivalence between a perfect fluid source and
a scalar field $\Phi(t)$ depending on the time only is set up by
writing:

\begin{equation}
\label {4.2}
\rho=\frac12{\dot\Phi}^2+V(\Phi), \quad p=\frac12{\dot\Phi}^2-V(\Phi)
\end{equation}
equations that could be considered as the parametric representation
of an equation of state.

Let us assume that the model presented in the preceding sections is a
good description of the Universe during an early period of time for
which the scale factor $F$ is very small. It is an elementary
exercise to obtain the potential function $V(\Phi)$ that would make this model
look as a particular inflationary model. From \ref{4.2} and the above
equations we get:

\begin{equation}
\label {4.3}
\ddot\Phi+3H\dot\Phi+ V^\prime=0, \quad
V^\prime=\frac{\partial V}{\partial\Phi}
\end{equation}
Deriving the first Eq. \ref{4.2} with respect to time and using the
equation above we obtain:

\begin{equation}
\label {4.4}
{\dot\Phi}^2=-\frac{\dot\rho}{3H}
\end{equation}
which substituted in the first Eq. \ref{4.2} yields:

\begin{equation}
\label {4.5}
V=\rho+\frac{\dot\rho}{6H}
\end{equation}

Now we consider our particular solution characterized by a density
function \ref{3.9}. Deriving this function with respect to $t$ we obtain:

\begin{equation}
\label {4.6}
\dot\rho=-\frac{\rho}{F^3}(3F^2+1)\dot F
\end{equation}
and substituting into \ref{4.4} and \ref{4.5} we get:

\begin{equation}
\label {4.7b}
{\dot\Phi}^2=\frac{\rho}{3F^2}(3F^2+1)
\end{equation}

\begin{equation}
\label {4.7a}
V=\frac{\rho}{6F^2}(3F^2-1)
\end{equation}
Let us consider the scalar field as a function $\Phi(F)$ of the scale
factor. Deriving this function with respect to $t$ and using Eqs.
\ref{4.7b} and \ref{3.3a} we obtain:

\begin{equation}
\label {4.8}
\frac{d\Phi}{dF}=\frac1F\sqrt{\frac{3F^2+1}{F^2+(\Lambda F^2-3k)/\rho)}}
\end{equation}
If in the interval under consideration $F$ is small enough
then the r-h-t can be approximated to $1/F^2$ and
therefore we obtain:

\begin{equation}
\label {4.10}
\Phi(F_2)-\Phi(F_1)=
\mp(\frac{1}{F_2}-\frac{1}{F_1})
\end{equation}
and therefore, up to a sign factor and an arbitrary additive
constant, we get:

\begin{equation}
\label {4.11}
\Phi=\frac{1}{F}
\end{equation}
in the largest interval for which the assumptions made are valid.

Using the expression above into \ref{4.7a} and keeping only the largest
term we obtain the inflationary
potential\,\footnote{Concerning the general ideas about inflation see
for instance \cite{Guth} or \cite{Islam}} that would be equivalent to the equation of state
considered in \ref{1.5}:

\begin{equation}
\label {4.12}
V(\Phi)=-\frac{A}{6}\Phi^5\exp[(1/2)\Phi^2]
\end{equation}

\section{A pure radiation very simple model}

Either from \ref{1.5} but still better from \ref{3.9} one can see that for
large
values of $F(t)$ a model based on the equation of state
\ref{1.5} will be equivalent to a dust model, i.e. a matter model with
zero pressure. In fact under this condition the density
behaves as:

\begin{equation}
\label {5.1}
\rho(F)=AF^{-3}, \quad
\end{equation}
It is therefore tempting to examine the properties of a Universe that
would satisfy the equation of state \ref{1.5} all the way during its
life-span. Such a model contains only two parameters: the
curvature constant $k$ and the cosmological constant $\Lambda$. We
shall determine them using the available theory presented before and
the present observable values of the Hubble constant $H_0$, the
deceleration parameter $q_0$ and
the present density.

The point of view that we presented in the introductory section
raises the following important question: What is the constant $c$
appearing in the Robertson-Walker metric but also in Einstein's
constant, $8\pi G/c^2$, and in the energy-momentum tensor. It is
called the speed of light in vacuum\,\footnote{It should be called
otherwise because it is actually a property of space-time in the
sense that it belongs to all causal interactions} but we are also
saying here that since measuring the speed of light is a local
process the result of our measure is actually $c_{eff}$, not c. It
turns out then that for fixed units of length and time c can take any
value as long as we choose the present value of $F$ to be:
$F_0=c/c_{eff}$. In particular we mankind, leaving for a very short
period of the history of the Universe, can choose to avoid the use of
two different quantities $c$ and $c_{eff}$ and decide to answer our
question by saying that $c$ is the measured value of the speed of
light in vacuum with the proviso that we take the present value of
$F$ to be:

\begin{equation}
\label {5.3}
F_0=1
\end{equation}
But remember this is not a choice of units condition as that that we made
in \ref{3.7}. This is a
conceptual decision.

From \ref{3.11} we have:

\begin{equation}
\label {5.5}
k=(1-\Lambda/3)(\Omega_0-1)
\end{equation}
and from \ref{3.15} and \ref{3.13} we get:

\begin{equation}
\label {5.6}
\Omega_0=\frac{2\rho_0}{6(q_0+1)-2\rho_0}
\end{equation}

\begin{equation}
\label {5.7}
\Lambda=-3q_0-\rho_0
\end{equation}
where $q_0$ and $\rho_0$ are the present observed values.

We have now to make our mind about the present
density of the Universe.  To keep the model as simple as possible we
offer two possibilities:

i) To imagine that the gas of zero mass particles that we have
considered is in fact a faithful physical equivalent of the present
mixture of radiation and massive particles as if, so to speak, the latter
were still diluted in the form of radiation. From this point of view
it would be safe to consider the present density, as a free parameter.

ii)To assume, as it has been repeatedly proposed, and as observations have
always consistently suggested\,\footnote{See for example \cite{Mandelbrot} and
references therein}, that the mass distribution in the
Universe is highly hierarchical. From this point of view there is no
objection in considering that the density of the mass distribution
could become much smaller than the density of the background
black-body radiation at some scale of distances as if, so to speak,
everything, from elementary particles to clusters of clusters of
clusters of galaxies, were nothing more than the {\it Froth of the
Universe}.

In both cases the present density and the present temperature
would be given by the Stefan-Boltzman law:
\begin{equation}
\label {5.8}
\rho_0=aT_0^4
\end{equation}
but the nicer aspect of the second choice is that for this case $T_0$
is a known quantity: the present temperatute of the background
black-body radiation. We give below a few numerical results
corresponding to this case.

Using the following values: $q_0=.1$, $T_0$=2.7 K, and assuming
that $H_0^{-1}$ is $13$ billion years ($H_0=75\, \mbox{km/s/Mpc}$) we obtain
using normalized units, i.e. those satisfying
the conditions \ref{3.7}:

\begin{equation}
\label {5.10}
\rho_0=1.3\times 10^{-4}, \quad \Omega_0=3.8\times 10^{-5}
\end{equation}
\begin{equation}
\label {5.9}
\Lambda=-.3, \quad k=-1.1
\end{equation}
Reverting to the MKS system of units these values correspond to the
following ones:

\begin{equation}
\label {5.11}
\rho_0=4.5\times 10^{-31}\, \mbox{kg/m}^3
\end{equation}

\begin{equation}
\label {5.12}
\Lambda=-2.\times 10^{-51} \mbox{m}^{-2}, \quad
k=-7.3\times 10^{-51} \mbox{m}^{-2}
\end{equation}
Eq. \ref{3.10} with $F_1=0$ and $F_2=1$ gives the age of the Universe for this model:

\begin{equation}
\label {5.14}
\mbox{Age}=.77(H_0^{-1}), \hbox{      or       }
\mbox{Age}=9.83\times 10^9\, \mbox{yr}
\end{equation}

The maximum value $F_{max}$ of $F$ is given by the
root of the Eq. ${\dot F}^2=0$. Using the data above we have
$F_{max}=3.32$ and therefore Eq. \ref{3.10} with $F_1=0$ and $F_2=F_{max}$ gives half the life-span of the Universe, or:

\begin{equation}
\label{5.16}
\mbox{Life-span}=9.4, \hbox{     or     }
\mbox{Life-span}=12.2\times 10^{10}\, \mbox{yr}
\end{equation}
Notice that although $F_{max}$ is not a large number it is still
justified to claim that the energy density behaves as in \ref{5.1}
in the interval, say, $F=F_{max}\pm 1$.
\section{Concluding remarks}

This paper is based on three ideas. Namely:

\begin{itemize}
\item that the effective speed of light for any local process, in the
cosmological sense, is given by \ref{1.4}
\item that since the equation of state \ref{1.5} leads to an
inflationary behavior during the early stages of the Universe it
might deserve consideration, for the same reasons that one has for
other inflationary models,

\item and that since this same equation of state is formally
equivalent during
the latest stages of the evolution, to
assuming that the Universe is filled with dust matter, it is tempting
to consider a model for which this equation of state would be valid
from the beginning to the end.
\end{itemize}

We think that we have shown that these three ideas are connected. But
nothing forbids to consider them separately. We could for instance
keep the first one and forget the second and third. From this point
of view the single interest of the idea would be to predict a
variation of the effective speed of light given by:

\begin{equation}
\label {6.1}
\dot c_{eff}=-c_{eff}H_0
\end{equation}
but it is doubtful that a local uncorrupted test of this prediction, of
cosmological origin, could be performed.

We could also keep the second idea and consider the Eq. \ref{1.5}
valid only during an early fraction of the history of the Universe to
be connected with other models at some other stages. And so on. Any
cosmological model can be improved making it more complicated. The
model of section 5 is just a model incorporating the three ideas
above in the simplest possible way. It has to be considered as a
template ready to be modified to fit other particular points of view.

\section*{Acknowledgements}
Many of the misprints and Latex errors of the initial manuscript were pointed out to me by A. Molina. Thanks Alfred !


\section*{Acknowledgments}
The authors wish to acknowledge the contribution provided by Ioannis Tsohantjis in the early stages of this work, as well as discussions with A J Bracken.


\section{Introduction}
\label{sec1}

Three-dimensional gauge theories play an important role
for high temperature particle physics, since they
constitute the Matsubara zero mode sector of finite temperature quantum field
theories in the imaginary time formalism. In particular, if the
temperatures of interest are larger than the mass scales of the
theory, fermions and non-zero modes may be integrated
out by dimensional reduction \cite{ap} to leave a three-dimensional
effective theory describing all static properties and the equilibrium
thermodynamics of the theory under consideration. While dimensional
reduction is a perturbative procedure, the resulting three-dimensional
gauge theories require non-perturbative techniques in order to
calculate physical quantities. In particular, in the cases of QCD and
the symmetric phase of the electroweak sector of the Standard Model,
the three-dimensional effective theories are confining and thus
not accessible by perturbative methods. It is then expedient
to split the problem into two parts: perform dimensional
reduction perturbatively, and study the resulting three-dimensional
theory on the lattice. This approach has been particularly successful in
clarifying the order and strength of the electroweak phase transition
\cite{ew}, and is also applied to the more difficult case
of QCD and the quark gluon plasma, e.g. \cite{qcd}.
An important question
connected with non-perturbative physics concerns the screening properties
of non-Abelian plasmas like Debye screening, and the much less
understood question of magnetic screening and a corresponding
``magnetic mass" \cite{linde}.

In this contribution, the concept of screening masses is considered
in the framework of three-dimensional gauge theories.
Starting point is the argument that any static physical quantity
of the finite temperature field theory
must have a corresponding quantity in the three-dimensional theory, with
which it agrees up to a (perturbative) part
due to the non-zero Matsubara modes.
In general, a static screening length in finite temperature field theory
is defined as the exponential decay of some {\it spatial} correlation function.
In the dimensionally reduced theory,
the direction of the correlation may be taken to correspond to (euclidean) time,
and hence the same quantity appears in the spectrum or some other
physical property of the (2+1) dimensional theory.
In order to fully understand screening in non-Abelian plasmas, it is
therefore necessary to understand the non-perturbative physics
of the underlying three-dimensional theories.
Here I shall discuss the physcial properties of (2+1) dimensional
gauge theories like the mass spectrum, the static potential, the string
tension, screening of the static potential
and the various correlation functions that serve to
study them. The connection of these quantities to the
magnetic mass and the Debye mass is discussed.
Special emphasis is put on the non-perturbative, confining
nature of the theories.
Correspondingly, most of the presented results are drawn from lattice
simulations.

\section{The SU(2) Higgs model}
\label{sec2}
As a prominent example corresponding to the dimensionally reduced
electroweak Standard Model \cite{smred},
consider the SU(2) Higgs model in (2+1) dimensions,
\begin{eqnarray}\label{l3d}
S & =  \int d^3x \; \mbox{Tr} & \left({1\over 2}F_{ij}F_{ij} +
(D_{i}\Phi)^{\dag} D_{i}\Phi \right. \nonumber \\
& & \left. + m_3^2 \Phi^{\dag} \Phi
+ 2 \lambda_3 (\Phi^{\dag} \Phi)^2 \right) \, ,
\end{eqnarray}
where all fields are in a $2\times 2$ matrix notation. The gauge
coupling $g_3$ and the scalar coupling $\lambda_3$
have mass dimension 1/2 and 1,
respectively, so the physical properties of the theory are determined
by two dimensionless parameters, $x=\lambda_3/g_3^2$ and $y=m_3^2/g_3^4$.
In the framework of dimensional reduction, these parameters
are of course determined by the four-dimensional couplings $g,\lambda$
and temperature \cite{smred}. In the present context the interest is merely in
three-dimensional dynamics and this dependence will be supressed.

The phase diagram of the model is schematically shown in Fig.~\ref{pd}.
The perturbative Higgs phase and the non-perturbative symmetric or
confinement phase are for small $x$ separated by a line of first order
phase transitions. With increasing $x$ the transition becomes
weaker until it ends in a point with a second order
phase transition \cite{ks}. Beyond this point the
tansition disappears entirely to become a smooth crossover, so the phases
are analytically connected. This phase diagram was only established
non-perturbatively in the last few years \cite{bp,kl} and answered
many relevant questions concerning the thermodynamics
of the electroweak phase transition.
Here we are mainly concerned with the non-perturbative properties
of the confinement phase. Due to the connected nature of the phase diagram,
it is possible to smoothly vary the parameters from some point in
the Higgs region, where the physics is perturbative and well understood,
to the confinement regime and also pure gauge theory, the latter by making
the scalars infinitely heavy so that they decouple.
\begin{figure}\epsfxsize=8cm
\centerline{\epsfbox{phased.ps}}
\vspace{-3cm}
\caption{Schematic phase diagram of the SU(2) Higgs model with a line
of first order phase transitions.}
\label{pd}
\end{figure}

\section{Mass spectrum}

All physical properties of the theory are encoded in
gauge-invariant n-point functions. In particular, the mass spectrum
is computed from the exponential fall-off of two-point correlation functions
\begin{equation}
\lim_{\mid x-y \mid \rightarrow \infty} \langle \phi^{\dag}(x)\phi(y)\rangle
\sim {\rm e}^{-M\mid x-y \mid},
\end{equation}
where $\phi$ generically denotes some gauge-invariant operator
with quantum numbers $J^{PC}$.
For example, operators with quantum number $0^{++}$ are
\begin{eqnarray} \label{op}
R & \sim & \mbox{Tr} \left ( \Phi^{\dag}(x)\Phi(x)\right), \nonumber \\
L & \sim & \mbox{Tr} \left( (D_i\Phi)^{\dag}(x) D_{i} \Phi(x) \right),\nonumber \\
P & \sim & \mbox{Tr} \left( F^a_{ij}(x) F^a_{ij}(x)\right),
\end{eqnarray}
where $R$ and $L$ are typically used to measure e.g. the Higgs mass,
whereas $P$ serves to compute
glueball masses in pure gauge theory.
From each of these operator types analogues with
quantum numbers other than $0^{++}$ can be constructed
by taking suitable linear combinations with the desired
rotational and $PC$-symmetry properties.
A gauge-invariant $1^{--}$ operator, used to compute the mass of
the W-boson, is given by
\begin{equation} \label{wop}
V^a_{i}\sim \mbox{Tr} \left(\sigma^a \Phi^{\dag}(x) D_{i} \Phi(x) \right).
\end{equation}

A method to compute the mass eigenstates including the spectrum of
higher excitations
is to measure the full correlation matrix
between all available operators in a given quantum number channel,
$\langle \phi^{\dag}_i (x) \phi_j(y) \rangle$, where $\phi_i \in \{R,L,P\}$,
and to diagonalise it by a variational calculation to find the
mass eigenstates of the Hamiltonian (for details see \cite{ptw}
and references therein),
\begin{equation} \label{oleq}
\varphi_i=\sum_k a_{ik} \phi_k.
\end{equation}
The coefficients $a_{ik}$ are a measure for the overlap or wavefunction
between a given mass eigenstate $\varphi_i$ and the original operators
$\phi_k$.

The results of a lattice calculation \cite{ptw} of the lowest states
of the spectrum with
quantum numbers $0^{++}$, $2^{++}$ and $1^{--}$ at some points in
the Higgs and confinement region of the phase diagram are displayed
in Fig.\ref{spec}.
\begin{figure}\epsfxsize=6cm
\centerline{\epsfbox{spec1s.ps}}
\vskip4mm
\caption{ \label{spec} \it
Mass spectrum of the SU(2) Higgs model for $x=\lambda_3/g_3^2=0.0239$ and
two values of $y=m_3^2/g_3^4$, located in the Higgs (right)
and confinement (left) regions.}
\end{figure}
In the Higss region we see the familiar and perturbatively
calculable Higgs and W-boson in the
$0^{++}$ and $1^{--}$ channels, respectively. There is a large gap
to the higher excitations which are scattering states.
In the confinement region, in contrast, there is
a dense spectrum of bound states in all channels, very much
resembling the situation in QCD.
Open symbols denote bound states of scalar fields, whereas
full symbols represent the low lying glueballs.
This identification is based on a
detailed mixing analysis employing the overlaps (\ref{ol}),
as shown in Fig.~\ref{ol} for the $0^{++}$ channel.
\begin{figure}\epsfxsize=6cm
\centerline{\epsfbox{op.ps}}
\vskip4mm
\caption[]{ \label{ol} \it
The coefficients $a_{ik}$, eq.~(\ref{oleq}), of the operators
eq.~(\ref{op}) for the lowest three $0^{++}$ states
in the confinement region (ground state at the bottom).
Indices on the operators refer
to different smearing levels \cite{ptw}.}
\end{figure}
There it is easy to see that the ground state is a bound state of
scalar fields, with almost exclusive contributions from $R,L$-type
operators. The first excitation is a mixture of scalar and gluonic
contributions. The second excited state, in contrast,
has almost exclusively gluonic content and hence corresponds to a
glueball.

Furthermore, the numerical values of the glueball
masses agree within less than 5\%  with those obtained
in lattice simulations of pure SU(2) gauge theory \cite{mike1}.
In the latter, the physical spectrum consists of
glueballs only. It is very interesting to note that adding scalar matter
fields to the pure gauge theory leaves the glueball spectrum practically
undisturbed, but simply adds an apparently disjoint sector of mesonic
states to the spectrum. This observation is further corroborated by
the fact that the string tension in the confinement region of the
Higgs model is measured to be about 97\% of the one in pure gauge theory
\cite{ptw,mike}.
It has further been established in \cite{ptw} that the glueball masses
as well as the string tension in the confinement phase are insensitive
to variations of the parameters $x,y$, which only influence the masses
of the bound states of scalars. There are preliminary results indicating
that this behaviour is repeated in the SU(2) Higgs model with
scalar fields in the adjoint representation \cite{hp}.

In summary, the mass spectrum in three-dimensional, non-Abelian
gauge Higgs models consists of a copy of the glueball
spectrum of the corresponding
pure gauge theory and additional bound states of matter fields.

\section{Magnetic mass vs. physical mass spectrum}

A motivation for the occurrence of the magnetic mass in hot non-Abelian
gauge theories has been given in the contribution of Nair \cite{nair}.
In the following sections the question is considered, whether the magnetic mass
can be identified non-perturbatively in a three-dimensional gauge theory,
and how it might be related to physical quantities of the theory.

In the three-dimensional theory, the magnetic mass according to its
standard definition appears as a pole in the gluon propagator,
\begin{equation} \label{ww}
\langle A^a_i(x)A^b_j(y)\rangle \sim {\rm e}^{-m_A \mid x-y \mid}.
\end{equation}
This definition has a caveat. The pole of the gluon propagator
can be shown to be gauge-invariant order by order in (resummed) perturbation
theory \cite{kkr}, but this does not guarantee that the full
non-perturbative propagator has a pole. On the other hand, since
the gluon propagator itself is gauge-variant,
a fully non-perturbative lattice calculation requires gauge fixing.
Since that proceeds numerically, gauge-invariance of the mass
is not explicit.
Furthermore, due to the nature of gauge fixing procedures on the lattice,
it is also not immediately evident whether the mass signal extracted really
corresponds to a pole or some other analytic structure.
However, it is important to note that it is quite possible for
a pole in the full gluon propagator to exist, without contradicting
the fact that no asymptotic gluon states can be observed \cite{hoo}.
A physical asymptotic state is indicated by a pole in the
correlation function of a gauge-invariant operator, and clearly the
gluon propagator does not belong to this class.

The correlator (\ref{ww}) has been evaluated for pure gauge theory
by means of gap equations
in various different resummation schemes
at the one loop level \cite{bp,an,jp}, as discussed in \cite{nair}
and summarised in Table \ref{mmag}.
The difference in the results is rather small given that this
is the leading order result for a fully non-perturbative quantity.
A recent two-loop calculation \cite{eb} employing the resummation of \cite{bp}
finds corrections of about 15\% and thus one may hope for
convergence of the resummed perturbation series.
There are also lattice simulations of the gluon propagator in Landau gauge,
see Table \ref{mmag}. In the simulations in \cite{kar2} different
gauges were employed and no notable gauge dependence was found.
At present, the reliability and
accuracy of the calculations has not yet reached a fully satisfactory
quantitative level.
But there is evidence that the gluon propagator
in three dimensions can be evaluated by non-perturbative methods
in a fixed gauge, and that it produces a mass scale
of the order of $m_A \sim 0.35 - 0.46 g_3^2$.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|cc|cc|}
               & ref. && $m_A/g_3^2$& \\ \hline
1-loop gap eq. & \cite{an} &   & 0.38 & \\
               & \cite{bp,jp}& & 0.28 &\\
               & \cite{cor}  & & 0.25 &\\
2-loop gap eq. & \cite{eb}   & & 0.34 &\\ \hline
lattice        & \cite{kar1} & & 0.35(1)& \\
(Landau gauge) & \cite{kar2} & & 0.46(3)&
\end{tabular}
\end{center}
\caption[]{\label{mmag}
\it Comparison of gluon propagator masses from
gap equations and lattice simulations.}
\end{table}

Given the difficulties with the propagator pole
the question arises whether the magnetic mass can be identified in
another manifestly gauge-invariant and non-perturbative way.
If a pole exists in the full gluon propagator,
does it show up in the spectrum or another physical property of the theory?
Consider again the SU(2) Higgs model discussed above.
In the Higgs phase, there are
free point like Higgs and W-bosons described by the gauge-invariant
operators $R$ and $V$. The masses extracted from the
correlation functions of these operators agree perfectly
with those extracted from the poles of the propagators \cite{kar1},
as is expected in a perturbative Higgs regime.
As one moves around the phase diagram to
the confinement phase, the gauge-invariant correlation functions and
the propagators split up. In particular, $m_A$ extracted from
the gluon propagator remains constant, i.e.~it is independent
of the values of the scalar parameters $x,y$ (as it should if it
represents a property of pure gauge theory), whereas
the mass of the $1^{--}$ state rises with $x$ and $y$ to become
much heavier than $m_A$, c.f.~Fig.~\ref{spec}. This is in accord
with its interpretation as a bound state of scalars.
Thus the operator $V$ is not suitable to describe the magnetic mass in
the symmetric phase of the Higgs model, in pure gauge theory
without scalars it cannot even be defined.
Likewise it is apparent from Fig.~\ref{spec} that
$m_A$ cannot be identified with any glueball mass (those with quantum
numbers other than $0^{++}, 2^{++}$ are even heavier \cite{mike1}).
In other words, if there is a mass scale associated with
the propagator, it does not show up directly in the spectrum.

\section{A constituent picture}

An indirect way of identifying the propagator mass in the spectrum
by means of a constituent picture has been proposed in \cite{bp1}.
In the constituent picture, the gauge-invariant composite operators
describe the asymptotic mass eigenstates, whereas the propagator
masses are identified as ``constituent" masses.
In the Higgs phase of the SU(2) Higgs model
the asymptotic states correspond to point particles, i.e. the
Higgs and W-bosons exist as free particles,
and hence propagators and gauge-invariant operators
give an equivalent description.
Moving into the confinement phase, the coupling becomes strong
and what were free particles now become constituents to form bound states.
The latter are the asymptotic states with poles in the correlation
functions of the composite gauge-invariant operators, whereas
the propagator poles give the constituent masses which do not
correspond to asymptotic states.
This is a unified description where only gauge-invariant operators
correspond to asymptotic states. The fact that asymptotic states
in the Higgs phase can also be described by the gauge-variant
propagators is a consequence of the perturbative Higgs nature
of that regime \cite{bp1}.

To the extent that such a constituent picture holds, the
bound state masses should be the sum of the constituent
masses plus some binding energy.
Counting $\Phi$ as a constituent scalar and $D_{\mu}$ as a
constituent vector particle, the naive mass formulae for the operators
in eqs. (\ref{op}), (\ref{wop}) are,
\begin{eqnarray}
& & m_R \simeq 2 m_{\Phi}\,\quad m_L \simeq 2 m_{\Phi} + 2m_{A}\, \nonumber \\
& & m_P \simeq 4 m_A\, \quad m_V \simeq 2 m_{\Phi} + m_A\, .
\end{eqnarray}
A comparison with the numerical values from lattice simulations
is given in Table \ref{comp}.
For a test of the constituent picture in the SU(2) adjoint Higgs model,
see \cite{kar2}.
\begin{table}
\begin{center}
\begin{tabular}{|c|cccc|c|}
    &&$J^{PC}=0^{++}$ & & &$J^{PC}=1^{--}$    \\
    &$ R $ &$ L $ & &$ P $ &$ V $  \\ \hline\hline
lattice \cite{ptw}
& 0.839(15) & 1.47(4) & & 1.60(4) & 1.27(6) \\ \hline
const. model & -  & 1.76$\;$ & & 1.67$\;$ & 1.29$\;$
\end{tabular}
\end{center}
\caption[]{\label{comp}
\it Comparison of screening masses from lattice simulations
and a constituent model using $m_A/g_3^2=0.46$, $m_R$
is used to fix the constituent scalar mass.}
\end{table}

Clearly, this naive counting rule works only for operators
that have very good overlap with a definite mass eigenstate.
This is the case for $R,P,V$ but not for $L$, c.f.~Fig.~\ref{ol}.
Since all binding effects have been neglected, such a constituent
picture can at best be qualitatively correct. Moreover, it is not suitable
to describe higher excitations.
Its purpose in
the present context is not to give precise predictions for the
mass eigenstates, but to furnish
a possible connection between propagator masses and the spectrum of
gauge-invariant operators.

\section{Static potential and screening}

Another interesting quantity determining the physical properties
of confining theories is the potential energy of static colour sources,
which is calculated from the exponential decay of large Wilson loops,
\begin{equation}
V(r)=-\lim_{t\to\infty}
\ln\left[\frac{W(r,t)}{W(r,t-1)}\right].
\end{equation}
As in four dimensions, a string of colour flux binds the sources
(in fundamental as well as adjoint representation)
causing a linear rise in the potential with growing separation $r$.
For sources in the fundamental representation, this linear rise
continues to infinity in pure gauge theory. If fundamental matter
fields are present as in the Higgs model, the string breaks at
some scale $r_b$, when its energy is large enough to
produce a pair of scalars. The dynamical scalars are then bound
to the static sources forming a pair of static-light ``mesons", and the
Wilson loop obeys a perimeter law $W(r,t) \sim \exp[-2M(r+t)]$ \cite{bock}
resulting in a saturation of the potential at a constant
value corresponding to twice the static-light meson energy,
\begin{equation} \label{asymp}
V(R\rightarrow\infty)=2 M.
\end{equation}
A gauge-invariant two-point function describing the correlation
of such a static-light meson is
\begin{eqnarray}
G_\phi(x,y)& = &
 \langle \mbox{Tr} \phi^\dagger(t) U^{\rm fund}(x,y)\phi(y)\rangle , \nonumber \\
U^{\rm fund}(x,y) & = & {\cal P} \exp({ig\int_x^y\! dx_i A_i^a T^a}).
\label{GPhi}
\end{eqnarray}
At large separations this two-point function falls off with the
same mass parameter as
in the Wilson loop perimeter law \cite{bock},
\begin{equation}
G_\phi(x,y)\sim {\rm e}^{-M\mid x-y \mid}.
\end{equation}

Considering instead the static potential in adjoint representation,
string breaking occurs already in pure gauge theory, because the
adjoint string can couple to gluons that may be pair produced.
The corresponding final state corresponds to a bound state of
the static source and some gluonic constituent and is described by
\begin{eqnarray}
G_{F}(x,y)&=&
\langle F^a_{ij}(x) U^{\rm adj}_{ab}(x,y)F^b_{kl}(y)\rangle , \nonumber \\
U^{\rm adj}_{ab}(x,y) &=&
2 \mbox{Tr} T^a U^{\rm fund}(x,y)T^b [U^{\rm fund}(x,y)]^\dagger ,
\label{GA}
\end{eqnarray}
with now an adjoint Wilson line connecting the field strength tensors.

The correlators eqs.~(\ref{GPhi}) and (\ref{GA}) are manifestly gauge-invariant
because of the string inserted between the fields. If the string is chosen
to be a straight line, then it can be gauged away to be unity in some
axial gauge. In such a gauge and in a perturbative regime
the correlators coincide with the propagators for the scalar and
gauge fields.
Does this connection also hold in a non-perturbative
regime? If so, the masses of the dynamical fields should be related
to the constituents discussed in the last section.
However, there is one subtlety involved, in that the masses extracted
from the correlators $G_{\phi},G_{F}$ contain a logarithmic
divergence due to the self-energy of the Wilson
line representing the static source. Fortunately, due to
superrenormalisability in three dimensions,
the divergence can be computed perturbatively and subtracted \cite{ay,mo},
leaving us with a finite number that is only defined up to a finite
additive renormalisation that has to be fixed by some prescription.
\begin{figure}\epsfxsize=8cm
\centerline{\epsfbox{mkappa.eps}}
\vspace{-3cm}
\caption{The lowest
     physical $1^{--}$ state (the W-boson in the Higgs phase) and
     the mass $M''$ extracted from $G_{F}(x,y)$.
}
\label{giprop}
\end{figure}

Fig.\ref{giprop} shows the mass values extracted from a lattice
simulation of the F-F correlator in comparison with the $1^{--}$
bound state mass \cite{mo}.
In the Higgs phase, the correlator indeed
reproduces the mass of the W-boson as also obtained from the propagator
or the gauge-invariant operator $V$, eq.~(\ref{wop}), after its finite
part has been fixed by matching it to the $V$-correlator at $y=-0.02$.
In the symmetric phase
it stays at a practically constant value of $\sim 0.75 g_3^2$,
which is about half the glueball mass $m_P=1.60(4)g_3^2$ \cite{ptw},
and about twice the propagator mass $m_A=0.46(3) g_3^2$ \cite{kar2}.
Since the field strength tensor contains two covariant
derivatives, $F_{ik}=[D_i,D_k]$, this is again in good agreement with
the naive constituent picture expectation and may be taken as
another piece of evidence for a gluonic mass unit entering gauge-invariant
quantities.

Finally, another manifestly gauge-invariant and physical quantity
of three-dimensional gauge theories is the string breaking
scale $r_b$ itself. Its size depends on the string tension and the
mass of the dynamical particles that have to be produced to break
the string. The breaking scale for the fundamental representation
potential in the SU(2) Higgs model
has been calculated in a recent lattice simulation \cite{pw},
by extracting it from the turnover of the potential
as shown in Fig.~\ref{break}. The continuum extrapolation of
those results gives
\begin{equation} \label{rb}
r_b g_3^2\approx 8.5,\quad  r^{-1}_b\approx 0.12 g_3^2 \,.
\end{equation}
Again for comparison,
the lightest scalar bound state from Fig.~\ref{spec} is
$m_{R}=0.839(15)g_3^2$, and the lightest glueball $m_{P}=1.60(4)g_3^2$.
A calculation of the adjoint representation potential in pure gauge theory
is still in progress, but
a fairly good estimate for its breaking scale may be obtained by
comparing the final state energy extracted from the F-F correlator
with the energy of the linear part in the potential extrapolated
from Wilson loop calculations.
The result \cite{mo} is of the same order of magnitude as that
given in eq.(\ref{rb}).

In summary, pure gauge theory as well as the Higgs model contain
non-perturbative, dynamical mass scales that are much smaller than
any of the masses present in the spectrum. As of yet, it is not understood
what is the origin for those small scales, but intriguing to
speculate whether they may have
something to do with the previously discussed propagator mass scales.
\begin{figure}\epsfxsize=9cm
\vspace{-3cm}
\centerline{\epsfbox{beta7_pot.ps}}
\vskip4mm
\caption{\label{break}
The energies of the ground state and the first excited state
for the static potential in fundamental representation.
The dashed lines indicate
the location of twice the energy of the single meson state, as
extracted from $G_{\phi}$.}
\end{figure}

\section{The Debye mass}

An important concept in the phenomenology of high temperature
QCD is the static electric screening mass, or the Debye mass $m_D$.
Although it has a leading order contribution that is perturbative
\cite{rebhan}, it couples to the three-dimensional
magnetic sector in next-to-leading
order, and hence requires a non-perturbative treatment as well.
The Debye mass can be expanded as
\begin{equation}
m_D = m_D^\rmi{LO}+{Ng_3^2\over4\pi}\ln{m_D^\rmi{LO}\over
g_3^2} +
c_N g_3^2 + {\cal O}(g^3T),
\label{md4d}
\end{equation}
where $m_D^\rmi{LO}=(N/3+N_f/6)^{1/2}gT$
and $N_f$ is the number of flavours.
The logarithmic part of the ${\cal O}(g^2)$ correction
can be extracted perturbatively \cite{rebhan},
but $c_N$ and the higher terms are non-perturbative.
To allow for a lattice determination, a non-perturbative
definition was formulated in~\cite{ay}, employing
the SU(N) adjoint Higgs model as
the dimensionally reduced effective theory.
By integrating out the heavy adoint Higgs
this can be further reduced to the pure SU(N) theory in three dimensions.
The statement of \cite{ay} is that the coefficient $c_N$ entering the Debye
mass can be determined from the exponential fall-off
of an adjoint Wilson line
with appropriately chosen adjoint charge operators at the ends.
The F-F correlator discussed in the last section is precisely
such an operator. From its measurement in a lattice simulation
of 3d pure SU(2) in \cite{mo} one finds the complete non-perturbative
$O(g_3^2)$ corrections to the Debye mass with high precision
to be $c_2=1.06(4)$. Judging from the precision of the result,
the correlator $G_F$ seems to be much more amenable to the analysis
than other operators that have been studied in \cite{lai}.
A calculation for SU(3) is currently in progress.

\section{Conclusions}

Electric and magnetic screening properties of hot non-Abelian
plasmas are non-perturbative phenomena and closely related to
the physics of confinement in three-dimensional gauge theories.
The physical properties of SU(2) pure gauge theory and the SU(2) Higgs model
in three dimensions have been studied by means of lattice simulations,
and the mass spectrum as well as the static potential
are known with rather good precision.
The inverse string breaking scales of the potentials in adjoint and
fundamental representation constitute physical mass scales which in
both cases are much smaller than those of the lightest states in the
spectrum.
It was attempted to establish a connection
between the magnetic mass, as obtained from simulations of the gluon propagator,
and the mass spectrum as well as the static potential
by means of a constituent picture. Although suggestive, the proposed
connection is not conclusive at this level. In order to make progress
in this direction, it would be necessary to find
a gauge-invariant correlation function giving just one
gluon constituent mass and thus reproducing the propagator simulations.
Regarding the Debye mass, the full non-perturbative corrections of order
$\sim g_3^2$ arising from three-dimensional pure gauge dynamics have
been determined following the method of \cite{ay}.

\section{Acknowledgments}

It is a pleasure to thank U. Heinz and
the organisers for a stimulating conference.


\section{Introduction}

\vspace{-2mm}
Over the last two decades a variety of attempts
in field theory have been aiming for a qualitative understanding and modeling
of two basic properties of QCD: quark confinement and
chiral symmetry breaking. The most prominent schemes are the instanton
liquid model \cite{ILM} and the dual superconductor picture of the
QCD vacuum.\cite{DSC}
While the first model explains chiral symmetry breaking and solves the $U_A(1)$
problem, the second
one provides a simple idea for the confinement mechanism.
In this scenario, where the vacuum is viewed as a dual superconductor,
condensation of color magnetic monopoles leads to confinement
of color charges through a dual Meissner effect.
The superconductor picture was substantiated by a large number of
lattice simulations over the last years. So it was shown that
in the confinement phase monopoles percolate through the $4D$
volume \cite{BORN}
and are responsible for the dominant contribution to the string
tension.\cite{SUZU}
At present, more and more groups
characterize their lattice vacuum in accordance to the instanton liquid
picture.\cite{NEGELE98}

Both models rest on the existence of very different
kinds of topological excitations,
instantons and color magnetic monopoles.
For a long time they have  been treated
independently, only recently some deeper connection among those different
objects has been pointed out,
both on the lattice and in the continuum.\cite{MONINST}
Instantons are localized solutions of the  Euclidean equations of motion
in Yang-Mills theory carrying action and integer topological charge.
Even though it is difficult to detect instantons and antiinstantons among
quantum fluctuations, there is no problem to study these well-defined
objects in classical
or semiclassical (heated) configurations on the lattice.
The situation for monopoles is more difficult.
Following 't Hooft, monopoles should be searched for
as pointlike singularities of some gauge transformation dictated by a local,
gauge
covariant composite field. The standard prescription, however, is localizing
monopoles in QCD as Abelian monopoles via an Abelian projection from some
gauge (for example the maximally Abelian gauge). This leads to monopole
trajectories which are dependent on the gauge chosen.
Note however, that the condensation mechanism of monopoles itself seems to be
gauge independent.\cite{DIGI2} \\
In this contribution we relate gauge invariant observables to monopole
trajectories, with the intention to further understand the semiclassical
vacuum structure in terms of monopoles and lumps of topological charge
and their role for the confinement problem.
We will comment on a new way of monopole identification on the lattice
which evades serious problems of previous methods, and which might have
a close formal relationship to 't Hooft-Polyakov monopoles.

\vspace{-1mm}
\section{Smoothing}

\vspace{-1mm}
To resolve semiclassical structures in gauge field configurations
provided by lattice simulations, the cooling method has been used,
which locally minimizes the action.
However, even improved versions of cooling rapidly destroy monopole
percolation and reduce the string tension.
From the instanton point of view cooling is known to destroy small
instantons and instanton-antiinstantons pairs, such that the true
topological structure is accessible at best by a backward
extrapolation to zero cooling steps.
Up to now, most lattice studies are performed with the Wilson action,
for which the lattice definitions of the topological charge $Q$ are
known to violate the bound valid for the continuum action:
$S\geq 8\pi^2|Q|/g^2$,
where $g$ is the coupling constant. The Wilson action is known to
decrease with smaller size $\rho$ of an instanton, such that
isolated instantons are
unstable under cooling. In contrast to this, improved
cooling finds instantons stabilized within a size interval
$\rho > 2~a$ ($a$ is the lattice spacing). Other methods like APE smearing
let instantons grow.

\begin{figure}
\vspace{-0.3cm}
\begin{tabular}{cc}
\hspace{-3.0mm} \epsfxsize=4.5cm\epsffile{top_l51.eps}&
\hspace{-9mm} \epsfxsize=4.5cm\epsffile{top_l51mon.eps}
\end{tabular}
\caption{Regions of low modulus of the auxiliary Higgs field (dots),
	 which {\it should} mark the trajectories of monopoles according to
	 LAG, are found very close to the trajectories of DGT monopoles
         obtained by Abelian projection. For clarity DGT monopoles are
         also shown alone (right).
}
\vspace{-0.1cm}
\label{fig1}
\end{figure}
To avoid these ambiguities we have used a method of `constrained
smoothing' \cite{FEUE} which is based on the concept of perfect
actions.\cite{HASE}
These actions respect the above bound for the topological charge
and lead to a theoretically consistent `inverse blocking'
operation.
Inverse blocking is a method to find a smooth interpolating field on a
fine lattice by constrained minimization of the perfect action, provided
a configuration is given on a coarse lattice.
This makes an unambiguous definition of topological charge possible.
Constrained smoothing is a renormalization group motivated method
which first blocks fields $\{U\}$, sampled on a fine lattice with lattice
spacing $a$, to a coarse lattice $\{V\}$ with lattice spacing $2~a$
by a standard blockspin transformation. Then inverse blocking is used to
find a smoothed field $\{U^{\mathrm{sm}}\}$ replacing $\{U\}$.

An important feature of this method is that it
does not drive configurations into classical fields as unconstrained
minimization of the action would do. It saves the long range structure of the
Monte Carlo configuration in $\{V\}$, such that the smooth background
contains semiclassical objects deformed by classical and quantum interaction.
The upper blocking scale roughly defines the border line between 'long and
short range'\footnote{The iterative application of this method,
`cycling',\cite{BOUL} obscures the idea of a definite blocking scale while
it still preserves rather well features of long range physics as the string
tension.}.
In this work we used a simplified fixed-point action \cite{PRD98} for Monte
Carlo sampling and for constrained smoothing before the configurations
were analyzed.
\begin{figure}[thb]
\epsfxsize=8.5cm\epsffile{asymmetrie.eps}
\vspace{-2mm}
\caption{Total monopole length (top) and space-time asymmetry (bottom)
         as a function of $\beta$ for monopoles obtained in different gauges.
         $\beta_c = 1.545(10)$ is the deconfinement point.  }
\label{fig2}
\vspace{-1mm}
\end{figure}

\section{Gauge Fixing}

The most popular gauge to study monopoles on the lattice
is the maximally Abelian gauge (MAG).\cite{KRON87} This gauge is enforced by
an iterative minimization procedure, which can get stuck in local minima,
so-called {\it technical} Gribov copies. The Laplacian Abelian gauge
(LAG) \cite{SIJS} is not afflicted by this problem.
MAG and LAG can be understood along the same lines.
The gauge functional of the MAG can be written:
\begin{eqnarray}
F(\Omega) & = &\sum_{x,\mu} (1-\frac{1}{2} \, tr \,
          (\sigma_3 U^{(\Omega)}_{x,\mu}
           \sigma_3 U^{(\Omega)\dag}_{x,\mu} ) ) \nonumber \\
= & &  \sum_{x,\mu,a} (X_x^a - \sum_{b} R^{a,b}_{x,\mu}
          X_{x+\hat \mu}^b )^2 \rightarrow \int_V (D_{\mu} X)^2 \, ,
\end{eqnarray}
with the gauge transformation $\Omega_x$ acting on $\{U\}$
\begin{displaymath}
U^{(\Omega)}_{x,\mu}  = \Omega_x U_{x,\mu} \Omega^{\dag}_{x+\hat \mu}
\end{displaymath}
encoded in an {\it auxiliary} adjoint Higgs field
\begin{displaymath}
\Phi_x  =  \Omega^{\dag}_x \sigma_3 \Omega_x = \sum_a X^a_x \sigma_a
\end{displaymath}
subject to local constraints $\sum_a (X^a_x)^2 =  1$
and with adjoint links
\begin{displaymath}
R^{a,b}_{x,\mu}  =  \frac{1}{2} \, tr \, (\sigma_a U_{x,\mu} \sigma_b
                                               U_{x,\mu}^{\dag}) \, .
\end{displaymath}
In LAG the local constraints are relaxed and
replaced by a global normalization:
$ \sum_{x,a} (X^a_x)^2 = V $, such that
Eq. (1) can be further written:
\begin{equation}
\int_V (D_{\mu} X)^2 \rightarrow \sum_{x,a} \sum_{y,b}
X^a_x\{-\Box_{x,y}^{a,b}(R) \} X^b_y \, .
\end{equation}
Then the minimization reduces to a search for  the lowest eigenmode
of the covariant lattice Laplacian.
LAG is unambiguously defined, except for degenerate
lowest eigenmodes, which correspond to {\it true} Gribov copies.
For both MAG and LAG, the gauge transformation is finally performed
by diagonalization of the field $\Phi_x$.
Quite similarly we enforce  the
Polyakov gauge (PG) by diagonalization of Polyakov loops.
\begin{figure}[thb]
\begin{tabular}{c}
\hspace{0.4cm} \epsfxsize=8.4cm\epsffile{actbinmon_allec.eps}\\
\epsfxsize=8.0cm\epsffile{qbinmon_allec.eps}
\end{tabular}
\caption{Average occupation number of monopoles $<m>$ nearest to sites
with action density
$s_{\rm site}$ (top) and topological charge density $q$ (bottom) in
the confinement phase.}
\label{fig3}
\vspace{-0mm}
\end{figure}

After the Abelian gauge of choice has been fixed one extracts the Abelian
degrees of freedom (Abelian projection). The Abelian link angles
can then be used for the identification of monopoles,
like in compact $U(1)$ theory, searching for the ends of Dirac strings.
Monopoles identified in this manner are generally referred to as
DeGrand-Toussaint (DGT) monopoles.
The Higgs field indroduced in the LAG provides an alternative
for monopole identification which is more satisfactory from
a physical point of view.
Lines of $\rho_x=|\Phi_x|=0$ where $\rho_x$ is defined as
\begin{equation}
 X^a_x = \rho_x \hat X^a_x \,\,\ , \,\,\,
\rho_x = \sqrt{\sum_{a=1}^3 (X^a_x)^2 }  \, ,
\end{equation}
directly define lines of gauge fixing
singularities (mo\-no\-po\-les),
more in the original spirit of 't Hooft.\footnote{For the 't Hooft-Polyakov
monopole regions with $\rho=0$ of the {\it physical} Higgs field are identified
with the centers of such monopoles.} Note here that this way of monopole
identification does not require to perform the actual gauge fixing and
Abelian projection!

In Fig. 1 we show that both methods of monopole identification
turn out to be quite related.
Regions of small $\rho$ are highly correlated with trajectories of monopoles
identified by the DGT method.
\begin{figure}[htb]
\begin{tabular}{c}
\epsfxsize=7.5cm\epsffile{S_ex_ahoch0c.eps}\\
\epsfxsize=7.5cm\epsffile{q_ex_ahoch0c.eps}\\
\end{tabular}
\vspace{-1mm}
\caption{Excess action (top) and charge (bottom)  as a  function of
temperature.   }
\vspace{-1mm}
\end{figure}

\section{Physical Properties of Monopoles}

The following results were obtained from simulations
of pure $SU(2)$ theory
on a $12^3\times 4$ lattice. Observables were computed on
50 independent configurations per $\beta$.
Different $\beta$ values were considered
to study the behavior slightly below and above the
deconfinement phase transition. For the particular action used,\cite{PRD98}
$\beta_c=1.545(10)$.
Global properties like
the total loop length and the space-time asymmetry
are shown in Fig. 2.
DGT monopoles extracted from the MAG and the LAG behave qualitatively similar.
Those from the the PG show no change
at the deconfinement phase transition. This reflects the
fact that PG monopoles should be static.

In Fig. 3 we present the average occupation number of monopoles on dual links
nearest to a given site as a function of the local
action $s_{site}(x)$ and charge $q(x)$,
for different gauges.
One observes
that the probability of finding monopoles increases
with the amount of action/charge  density at the same lattice position.
This result is practically independent of the  gauge  used
to define the (DGT) monopoles.

If monopoles are physical objects, one  expects that they can
be characterized by a local excess of the (gauge invariant) action.
We define such an  excess action of monopoles by
\begin{equation}
S_{\rm ex}= \frac{< S_{\rm monopole}-S_{\rm no monopole} > }{<S_{\rm no monopole} >}  \,\, ,
\end{equation}
where $S_{\rm monopole}$ is the action contained in a three-dimensional
cube which corresponds to the dual link occupied by a monopole.
Replacing the action in the above expression by the modulus of the
topological charge density according to the L\"uscher method
we obtain the charge excess $q_{\rm ex}$.
For details of the definition of the local operators see Ref. 11.
Fig. 4 shows that just below $T_c$ the excess action and charge for the
MAG and LAG monopoles are clearly above one,
indicating an excess of action of more than a factor of two compared to the
bulk average (background).
The large error bars above $T_c$ reflect the fact that the
topological activity diminishes  in the deconfinement phase.
These results are somewhat enhanced in comparison
to  a $T=0$ study with Wilson action
without cooling or smoothing.\cite{BAKK98}

\begin{figure}[htb]
\vspace{-0mm}
\epsfxsize=7.5cm\epsffile{nachlapmag.eps}
\caption{Static quark-antiquark potentials obtained from
Pol\-ya\-kov-line correlators after smoothing.
The slope of the Abelian potential in the MAG
after Abelian projection is about 5\% less than that of $SU(2)$ gauge field.
Another 5\% are lost in the LAG. Still the LAG carries 90\%
of the original string tension, indicating Abelian dominance
for the LAG.}
\label{fig4}
\end{figure}

In Fig. 5 we display  static quark-antiquark potentials  obtained from
Polyakov-Antipolyakov correlators, for the $SU(2)$ fields
and, in the case of LAG and MAG,
after Abelian projection.
The Abelian string tension of the MAG is about 5\% less than that
the $SU(2)$ field. The Abelian string tension of LAG is a little smaller
than for the MAG but still exhibits Abelian dominance.
The Abelian string tension of PG is trivially identical with that measured
on the smoothed $SU(2)$ configurations.

Finally we present an intuitive  argument, that mono\-poles also should
carry electric charge, that they are dyons.
Consider the
selfduality equations $F_{\mu \nu} = \pm \tilde F_{\mu \nu}$.
From the trivial relation
\begin{equation}
\int d^4 x {\rm Tr} [(F_{\mu \nu} \pm \tilde F_{\mu \nu})^2] \geq 0
\end{equation}
it immediately follows that
\begin{equation}
S \geq \frac{8 \pi}{g^2} |Q|
\end{equation}
and (anti)selfdual fields saturate the identity.
Fig. 6 depicts the probability
distribution of topological charge density for a given local action and
shows that for
$s_{site}>0.3$ the local action obeys a local version of Eq. (6)
near to saturation,
$s_{site}(x) \sim \frac{8 \pi}{g^2} |q(x)|$.
The plot was obtained after one constrained smoothing step,
and exhibits that the gauge fields are already sufficiently smooth to
expose semi-classical structure.  This is suggested by the relatively clear
ridges indicating approximate local selfduality for large enough action density.
In Fig. 3 we provided evidence that  monopoles are found predominantly
in regions of large action.
We thus conclude that monopoles also carry electric
charge and should be interpreted as dyons.
Note that this way of argumentation is a shortcut, to be more precise,
one would have to test for local selfdualtiy along individual monopole
trajectories.

\begin{figure}[htb]
\vspace{-1mm}
\begin{tabular}{c}
\epsfxsize=9.0cm\epsffile{qact140.eps}\\
\end{tabular}
\vspace{-1mm}
\caption{Probability distribution for finding a topological
charge density $q(x)$ at a lattice site $x$ if the local action density
equals $s_{site}(x)$. The ridges follow the lines
$s_{site}(x) = \frac{8 \pi}{g^2} |q(x)|$ where local
(anti)selfduality is satisfied.}
\vspace{-1mm}
\end{figure}

\section{Conclusions}

We have demonstrated that the renormalization group smoothing technique
with an (approximate) classically perfect action provides a powerful tool
to investigate the semiclassical vacuum structure. Analyzing trajectories
of monopoles identified in various gauges we found that monopoles appear
preferably in regions  which are characterized by enhanced action and
topological charge density. We showed that in exactly those regions local
(anti)selfduality of the gauge fields is prevailing.
This is further evidence that monopoles should be
addressed as dyons.
We demonstrated that almost the complete string tension
can be recovered from the Abelian projected field corresponding to
various Abelian gauges, indicating Abelian dominance also for the LAG.
This is trivially true for the PG, but the corresponding
monopoles do not change at the deconfinement transition.
We have shown that monopole trajectories carry an excess action of
about twice the background action density of smoothed gauge fields.
Similarly, monopoles also carry excess topological charge.
In the confinement phase this observation is rather independent of the
gauge chosen for identifying Abelian monopoles, but the behavior of
PG monopoles is different in the deconfinement.
We therefore conclude that MAG and LAG monopoles
behave similar physically and can be
interpreted  as physical objects which carry action and topological charge.
\\
\\
\noindent
This work was supported in part by FWF,  No. P11456.

\section*{References}


\section*{ABSTRACT}
Via solution of appropriate variational problem it is shown that
light beams with Gaussian spatial profile and sufficiently short
duration provide maximal destruction of global coherence under
nonlinear self-modulation.

{\bf Keywords:} nonlinear phase self-modulation, controllable
coherence degradation.

\section{INTRODUCTION}

Earlier it was demonstrated \cite{KarelinLazaruk},
that self-modulation of light fields
in a nonlinear medium can serve as a convenient
tool for controllable coherence manipulation. Process of local
interaction of coherent incident field
$E_{in}({\bf r},t)={\cal E}({\bf r})e(t)$ with ``optically
thin'' nonlinear layer, being described by Raman-Nath approximation,
causes different nonstationary phase shifts
$$
E_{out}({\bf r},t)=E_{in}({\bf r},t)
\exp \left\{
i\Phi(|E_{in}({\bf r},t)|^2)
\right\},
\eqno(1)
$$
where the phase is determined by media parameters and intensity of
input light in every particular point ${\bf r}$ of a beam
cross-section. The resulting degradation of spatial coherence can be
treated as a decay of initially single-mode (but
nonuniform) radiation into a number of mutually incoherent,
orthogonal modes. Such a process imitates the action of
moving phase diffuser and can in principle be used for speckle-noise
reduction in experiments with short light pulses.

The main characteristic of output field in discussed process is
an overall degree of coherence
$$
\mu={1 \over U^2}\int dt_1 \int dt_2 \big|K(t_1,t_2) \big|^2,
\eqno(2)
$$
where
$$
K(t_1,t_2) = \int d^2r\, E_{out}({\bf r},t_1) E^*_{out}({\bf r},t_2)
\eqno(3)
$$
is a spatially averaged temporal correlation function, and
$$
U=\int d^2r \int dt |E_{in}({\bf r},t)|^2 = \int dt\, K(t,t)
$$
--- pulse energy (nonlinear medium is supposed to be absorbtionless).
The value (2) determines the contrast of all
interference phenomena (including speckles) and it is closely
connected with coherent-mode structure of optical
fields. Modal treatment is an analogue of Karhunen-Lo\'eve
transformation, and the main parameter of such approach
--- effective number of modes --- is $N_{eff}=1/\mu$
(see Ref. 2, 3 for further details).

\section{OPTIMISATION OVER SPATIAL BEAM DISTRIBUTION}

The consideration in paper \cite{KarelinLazaruk} was carried out
for the model of
cubic nonlinearity with exponential relaxation and
speckled input field. Maximal coherence destruction there is
achieved in the ultimate case of inertial interaction with
infinite memory
$$
\Phi(|E_{in}({\bf r},t)|^2)=
\eta \int^t_{-\infty} dt |E_{in}({\bf r},t)|^2 .
\eqno(4)
$$
For this limit the value (2) does not depend on temporal shape of
input pulse, and every initial field is equivalent to
rectangular pulse of duration $T$:
$$
E_{in}({\bf r},t)=\left\{
	{{\cal E}(r)/\sqrt{T} , \; 0<t\le T \atop
              0, \; {\it elsewhere}}
\right.
$$

The interaction with infinite memory provides maximal total phase
shift, so one can expect, that the case (4) will
result in maximal coherence destruction for any input beam. Hence
the main aim of the present analysis is to optimise
the transformation (1), (4) over possible spatial distributions
of initial field ${\cal E}({\bf r})$, what can be done via solving
variational problem on minimum of double integral
$$
\mu U^2=2\int d^2 r_1 \int d^2 r_2 \, I({\bf r}_1) I({\bf r}_2)
\frac{1-\cos(\eta T[I({\bf r}_1)-I({\bf r}_2)])}
     {\eta^2 T^2 [I({\bf r}_1)-I({\bf r}_2)]^2}
\eqno(5)
$$
under additional constrain of constant energy
$$
\int d^2 r I({\bf r})=U,
\eqno(6)
$$
where $I({\bf r})=|{\cal E}({\bf r})|^2$ --- input intensity profile.

So far the above two functionals do depend on incident intensity
only, the task can be simplified by transition to
integration over beam intensity with proper introduction of
quasi-distribution function $P(I)$:
$$
K(t_1,t_2)={1 \over T}\int^{I_0}_0 dI \, P(I) I \exp \{i \eta I(t_1 - t_2)\},
\qquad
(0 \le t_1, t_2 \le T),
\eqno(3a)
$$
$$
\mu U^2=2\int^{I_0}_0 dI_1 \int^{I_0}_0 dI_2 \, P(I_1) P(I_2) I_1 I_2
\frac{1-\cos(\eta T[I_1-I_2])}{\eta^2 T^2 (I_1 - I_2)^2},
\eqno(5a)
$$
$$
\int^{I_0}_0 dI \, P(I) \, I = U.
\eqno(6a)
$$
Function $P(I)$ has a meaning of measure of contribution into these integrals
due to field points of given intensity. In
particular important case of axially symmetric and monotonous
dependence $I(r)$ it is easy to see that
\hbox{$P(I)=2\pi r(I) |\partial r/\partial I|$}. Here additional
condition of finite maximal intensity $I_0$ is implied. It is reasonable
for any real field distribution (but it does not the
case for speckle model).

At sufficiently high values of $\eta$ one can do further simplification
on the base of equality
$$
(1-\cos \eta x)/\eta^2x^2 \approx \delta(x) \pi/\eta,
$$
and then (5a) takes the form
$$
\mu U^2 = \frac{2\pi}{\eta T}\int^{I_0}_0 dI\, P^2(I)\,I^2.
\eqno(5b)
$$
Then it is readily seen that maximal coherence destruction is achieved when
$$
P_{MAX}(I)={U \over I_0}{1 \over I},
$$
what in coordinate representation takes the form of Gaussian beam
$$
I_{MAX}(r)=I_0 \exp(-\pi r^2 I_0/U),
\eqno(7)
$$
and appropriate overall degree of coherence is
$$
\mu = \frac{2\pi}{\eta T I_0}.
\eqno(8)
$$

\section{DISCUSSION}

The derived profile of input beam has some unique features, that makes
it especially attractive for nonlinear
speckle-noise reduction. First, such fields are easy to generate,
to control and to operate.

Second, in bulk a layer of nonlinear media
\cite{KarelinLazaruk} diffractional mixing
diminish the resulting decoherence of a field.
Gaussian beam has absolutely minimal diffractional divergence and
consequently it allows to use thicker layers,
increasing the efficiency of coherence transformation.

At last, it should be noted that according to paper
\cite{LazarukKarelin} zones of
identical intensity (for (7) these are concentric
circles) belong to one mode or, in other words, produce coherent
radiation. In order to generate output light with
desired structure of spatial coherence function one just need
to mix the field after nonlinear media on suitable
stationary diffuser.

\section*{ACKNOWLEDGEMENTS}

This work was supported by the Belarusian Foundation for Basic
Research under grant No. F97-253.