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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TITLE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\Large {\bf Motion of a scalar field coupled to a Yang-Mills
field reformulated locally with some gauge invariant variables }}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{0.5cm}
E.~Chopin\\
{\small E-mail:~eric.chopin@wanadoo.fr} \\
{\small PACS number: 11.15-q}
\end{center}
\vspace*{\fill}
\centerline{ {\bf Abstract} }
\baselineskip=14pt
\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small This paper exposes a reformulation of some gauge
theories in terms of explicitly gauge-invariant variables.
We show in the case of Scalar QED that the classical theory
can be reformulated locally with some gauge invariant variables.
We discuss the form of some
realistic asymptotic solutions to these equations. The
equations of motion are then also reformulated in the non-abelian
case.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{\fill}
%\rightline{hep-ph/0001125}
%\rightline{Jan. 2000}
\rightline{Report No:THP/00/001}
\rightline{Journal ref:JHEP 03 (2000) 022}
\end{titlepage}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% INTRODUCTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Introduction}
The gauge symmetry is known to render the calculation
of the elements of the S matrix very intricate.
In some future colliders like LHC or NLC,
some very complicated scattering processes will be studied.
Phenomenologists will
have to consider processes with 3, 4 or more
particles in the final state. The scattering amplitudes for
these processes are in general very complicated because
of the very large number of Feynman graphs, and the numerical
evaluation of these amplitudes in Monte-Carlo programs
suffer from numerical instabilities due for a large
part to some huge compensations between the different
graphs, which arise from the gauge symmetry.
To avoid these numerical instabilities, there are two common methods.
The first one consists in using a specific gauge which
simplifies the different vertices and
propagators~\cite{georges,fujikawa2,hhh,bfm.other,bckgdenner}. The
second one consists in using some algorithms acting on each
Feynman graph, based on Ward identities, in order to simplify the
expression of the graphs~\cite{pinch}. Both methods lead to the
elimination of most of these huge compensations. \\
In this paper, we consider this problem
from another point of view, at the core of
Quantum Field Theory. Basically, we raise the question
of whether the calculations of the elements of the S~matrix can be done
directly using some gauge invariant variables. This question can
be studied in the context of both methods cited above, using as a
fundamental tool the Ward identities. These identities depend on
the gauge fixing procedure used in the calculations. We rather
look here for a method in which
there is no need to break temporarily the gauge symmetry.
As a consequence, we must start our
formulation from the very beginning of gauge theories, that is to
say from the equations of motion. We therefore show in this paper
that one can reformulate these equations in terms of
local gauge invariant variables for the case where matter fields
are scalar.
\\
This new approach may have some interesting applications
regarding the quantization of fields. In standard
field theory, the quantization procedure is done first on free fields,
and therefore matter fields and gauge fields are considered
separately, though they are coupled in the equations of motion.
A significant consequence is that it is irrelevant to consider
the evolution of a free field from a time $t$ to an
interacting field at time $t'>t$ through a unitary transformation,
because Haag's theorem says that the field considered at time $t'$
must be also free (for a good review, see~\cite{bogoliubov}).
Quantum Field Theory is therefore doomed to describe
only the transition between asymptotic fields through the
LSZ formalism. In experiments where the time variable
plays a fundamental role (CP violation experiments in $K^0_S/K^0_L$,
neutrino oscillations,...) one must use a mixed theory,
based in part on classical quantum mechanics (Rabi precession,...)
and in part on quantum field theory for the computation of the decay
width of the particles. A single theory which would
describe completely such experiments is still missing. Since
Haag's theorem does not apply to the case of two constantly
interacting fields, the approach presented in this paper opens
the prospect of finding an evolution operator between two
finite times for an
interacting system. That is to say,
an asymptotic electron would be
described both by its matter field and its surrounding
electromagnetic field, in some sense. So we must also find some
``realistic'' asymptotic solutions to the coupled equations of
motion in replacement of the plane waves that are used in standard
quantum field theory. The word ``realistic'' means here that we
look for solutions that have a finite conserved momentum. We show
that solitons are not possible in this context (for the $U(1)$
case), but we conjecture that some periodic-in-time solutions may
probably exist. \\
What is the basic idea of our approach? We know that for a given
field-strength tensor, one can compute a corresponding gauge
field using the basic cohomological formulas that are reviewed in
the appendix. Some authors have already tried to reformulate the
Yang-Mills Theory using only the Field-Strength tensor as a basic
variable in place of the gauge
field~\cite{halpern.fst,seo-okawa.fst,itabashi.fst,mendel.fst}.
The results of these studies are generally not covariant and
non-local, due to the fact that the cohomological formulas are
essentially of a non-local nature. In this paper, we rather
consider the gauge-currents as fundamental variables, and we keep
both locality and covariance of the equations. \\
The paper is therefore organized as follows: \\
The first section is devoted to the reformulation
of $U(1)$ scalar QED in terms of gauge invariant variables. \\
The second section
contains a discussion on asymptotic solutions of the $U(1)$
scalar QED. We first show that periodic solutions of Klein Gordon
do not have a finite energy, contrary to what is claimed in a
recent paper, and therefore we need to consider the coupled
equations. We show the impossibility of
soliton solutions in this context, and discuss the possibility of
periodic (in time) solutions. \\
The third section presents the non-abelian case, where
the gauge group is in a certain class of subgroups of $U(N)$.
It turns out that the results presented in this paper
are in fact a simpler version
of some results given by Lunev in 1994~\cite{lunev-gauge},
with in addition
the coupling to a scalar multiplet (he only considered
a pure Yang-Mills theory). \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% SECTION 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A possible reformulation of classical SQED}
\label{sqed}
In this section, we reformulate the classical theory of
a scalar field coupled to a $U(1)$ gauge field (SQED)
in terms of gauge invariant variables. We will then
demonstrate
the difficulties appearing when one wants to find
some ``realistic'' asymptotic solutions, which would generate a
Fock-like space. Using such a space, one could then construct a
new formalism for computing cross sections.
Let us start with the classical scalar QED lagrangian:
\be
{\cal L} = (D_\mu\phi)^* D^\mu \phi -m^2 \phi^*\phi -\frac{1}{2}
\partial_\mu A_\nu (\partial^\mu A^\nu - \partial^\nu A^\mu)
\label{sqed-lagr}
\ee
with $D_\mu = \partial_\mu + ieA_\mu$. The electrical current is
given by $J_\mu = ie(\phi^*(D_\mu\phi) - (D_\mu\phi)^*\phi)$
and the probability density $\rho = \phi^*\phi$. Both
$J_\mu$ and $\rho$ are gauge invariant variables and we will show
how to rewrite the previous lagrangian as a function of these
variables (this treatment will have to be modified in the
non-abelian case in which the corresponding expression for these
variables are not gauge invariant but gauge ``covariant''). First,
we shall review the standard equations of motion when $\phi$ and
$A^\mu$ are taken as field variables:
\bea
0 &=& (D_\mu D^\mu+m^2)\phi \label{scalar0} \\
0 &=& (\Box+m^2)\phi
+ 2ieA_\mu\partial^\mu \phi
+ie(\partial\cdot A)\phi -e^2(A\cdot A)\phi \label{scalar} \\
\partial^\alpha F_{\alpha\beta} &=& ie(\phi^*(D_\beta\phi) -
(D_\beta\phi)^*\phi ) = J_\beta \label{gf}
\ena
We shall first note that if one computes $\phi^*(\ref{scalar}) -
(\ref{scalar})^*\phi$, one obtains $\partial_\mu J^\mu =0$, which
we would have already obtained by taking the divergence of
eq.~\ref{gf}. The redundancy between the last two equations can
therefore be removed by making use of $\phi^*(\ref{scalar}) +
(\ref{scalar})^*\phi$ instead of Eq.~\ref{scalar}. After some
algebra, it is not a hard task to make $J_\mu$ and $\rho$ appear
in the equations as we will see later, but for the derivation of
the new equations, we rather choose to start from the lagrangian.
For this purpose, we will use the following relations:
\bea
-\frac{J^2}{e^2} &=& (\phi^*(D_\mu\phi) + (D_\mu\phi)^*\phi)^2
-4 (D_\mu\phi)^*\phi\phi^*(D_\mu\phi) \label{jsquarre} \\
\Rightarrow (D_\mu\phi)^* D^\mu \phi &=& \frac{1}{4\rho}
\left( (\partial_\mu\rho)^2 + \frac{J^2}{e^2} \right) \label{jsquarre2} \\
&=& (\partial_\mu\sqrt{\rho})^2 + \frac{J^2}{4e^2\rho} \label{jsq2}
\ena
Throughout the paper, we will conveniently
define $v^\mu$ such that $J^\mu = 2e^2 \rho v^\mu$
and set $z(x) = \sqrt{\rho(x)}$.
From the definition of the current, one can also extract
the expression of the field strength tensor:
\bea
2e^2F_{\mu\nu} &=& \partial_\mu\left(\frac{J_\nu}{\rho}\right)
- \partial_\nu\left(\frac{J_\mu}{\rho}\right) \label{ym-from-c} \\
F_{\mu\nu} &=& \partial_\mu(v_\nu) - \partial_\nu(v_\mu)
\label{def-fmunu}
\ena
Equations~\ref{jsq2} and~\ref{def-fmunu} are the fundamental tools
of our formalism. With these, we can write the lagrangian as a
function of $z$ and $v^\nu$ in the following way:
\be
{\cal L} = (\partial_\mu z)^2 -m^2 z^2 + e^2 z^2 v^2
-\frac{1}{4}(\partial_\mu(v_\nu) - \partial_\nu(v_\mu))^2
\label{new-lagr}
\ee
We have now re-expressed the lagrangian in terms
of gauge invariant quantities, and
as a by-product the ``effective'' coupling constant is $e^2 =
4\pi\alpha$ instead of $e$. This means that the sign of $e$ is not
relevant. Although this does not mean that in a perturbative
expansion of some solutions, the relevant expansion parameter is
necessarily $e^2$, it may also be $\sqrt{4\pi\alpha}$ or $|e|$.
From this new lagrangian we can derive the following equations of
motion thanks to the Euler-Lagrange equations:
\bea
(\Box+m^2)z &=& 4\pi\alpha zv^2 \label{sqed-gauge-invar1} \\
\Box(v_\nu) -\partial_\nu (\partial\cdot v) &=&
- 8\pi\alpha z^2 v_\nu \label{sqed-gauge-invar2}
\ena
\subsection{The Energy-Momentum Tensor}
We will further look for asymptotic solutions to the
coupled equations with a finite conserved momentum.
The symmetrized energy momentum tensor (or Belinfante tensor)
can be rewritten this way:
\bea
T_{\mu\nu} &\stackrel{def}{=}& (D_\mu\phi)^\dagger D_\nu\phi +
(D_\nu\phi)^\dagger D_\mu\phi -F_{\mu\lambda}F_\nu^{~\lambda}
-g_{\mu\nu}\left((D_\mu\phi)^\dagger D^\mu\phi
-m^2\phi^\dagger\phi-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}
\right)\label{em-tensor} \\
T_{\mu\nu}&=& 2e^2z^2v_\mu v_\nu +2\partial_\mu z
\partial_\nu z -\left(\partial_\mu (v_\lambda)
-\partial_\lambda (v_\mu)\right)
\left(\partial_\nu (v^\lambda)
-\partial^\lambda (v_\nu)\right) \nonumber \\
&& -g_{\mu\nu}\left((\partial_\mu z)^2 -m^2 z^2 + e^2 z^2 v^2
-\frac{1}{4}(\partial_\mu(v_\nu) - \partial_\nu(v_\mu))^2
\right)\label{em-tensor2} \\
P_\mu &=& \int_\Sigma d\sigma^\nu T_{\nu\mu} \label{cons-momentum}
\ena
In Eq.~\ref{cons-momentum}, $\Sigma$ represents any space-like
hyper-surface in the Minkowsky space time, and
for the sake of simplicity, we will generally take the $t=0$
hypersurface for the computation of $P_\mu$. \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% SECTION 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Solitons solutions are not normalizable}
\label{solitons}
In general, the spatial extent of the wave
function of a free particle (obeying the Klein Gordon equation)
increases in time. Here, we will rather look
for the possibility to find
``soliton-like'' solution to the coupled field equations of scalar
QED (i.e. Eq.~\ref{sqed-gauge-invar1} and
Eq.~\ref{sqed-gauge-invar2}). First, we will show that for the
Klein Gordon equation, we can find some simple ``soliton-like''
solutions but these solutions are not normalizable (similarly to
plane waves). We obtain the same result when the interaction is
taken into account, but the arguments used to reject this case are
different from the free case.
For this reason, the free case is also presented, even if it can
be seen as a particular case of the interacting one.
\subsection{Generalities about solitons}
We will say that a function $f(x)$ defined on space-time
is a soliton if we can find a time-like momentum
$p^\mu$ such that:
\be
p^\mu\partial_\mu f = 0 \label{soliton-def}
\ee
This time-like momentum represents the global momentum of the wave
which moves without deformation. To see this trivial fact,
Eq.~\ref{soliton-def} simply means that if we are placed in a
frame where $p^\mu = (m_0,\vec 0)$, then the shape of the wave
function does not depend on time ($\partial_0 f=0$). Suppose now
that at time $t=0$ we look at the shape of the wave function. It
is reasonable to say that for an asymptotic solution (supposed to
describe a free scalar particle) the probability density is
spherically symmetric. We can deduce from that that the function
$f$ is a function of only one variable. To be more specific, let
us consider the two variables \fbox{$u=(p\cdot x)^2-p^2x^2$} and
\fbox{$\tau=p\cdot x$}. In the ``rest frame'', where $p^\mu =
(m_0,\vec 0)$ \footnote{We shall remark that $p^2=m_0^2$, where
the mass $m_0$ is {\it a priori} different from the mass $m$
appearing in the Klein Gordon equation.}, then:
\bea
u &=& (p\cdot x)^2-p^2x^2 \label{udef} \\
&=& (m_0t)^2-m_0^2(t^2-\vec x^2) = m_0^2\vec x^2 \label{ucalc} \\
\tau &=& p\cdot x = m_0 t \label{vcalc}
\ena
A ``spherically symmetric'' scalar function $f$ is therefore
a function of $u$ and $\tau$ only. We have seen that
$u\geq 0$ for any $x$ and we will often write
\fbox{$y=\sqrt{u}$}. We have by construction
$u(x^\mu+\lambda p^\mu)
= u(x^\mu)$, which means that a function of the variable $u$ is
invariant under any translation in the $p^\mu$ direction. \\
For convenience, we will also use the following notations:
\bea
\lambda^\alpha &=& p^\alpha(p\cdot x)-p^2x^\alpha
= \frac{1}{2}\partial^\alpha u(x) \label{lambda} \\
\lambda^2 &=& -p^2 u(x) = -m_0^2 u(x) \quad \quad\quad
p\cdot\lambda = 0 \label{sl01} \\
\partial_\alpha \lambda_\beta &=& p_\alpha p_\beta-m_0^2
g_{\alpha\beta} = \tau_{\alpha\beta}\label{sl03a} \\
\partial^\alpha \lambda_\alpha &=& \tau^\alpha_\alpha =
-3m_0^2 \label{sl03b} \\
p^\alpha \tau_{\alpha\beta} &=& 0 \quad \quad\quad
\lambda^\alpha \tau_{\alpha\beta} = -m_0^2 \lambda_\beta
\label{sl03c}\\
\partial_\mu \sqrt{u} &=& \frac{\lambda_\mu}{\sqrt{u}}~~~~~~~~~
\partial^\mu \left(\frac{\lambda_\mu}{\sqrt{u}}\right)=
-\frac{2m_0^2}{\sqrt{u}} \label{sl03d}\\
\ena
And in the rest frame, \fbox{$\lambda^\mu = -m_0^2(0,\vec x)$}.
Then, if the scalar function $f$ is a ``spherically symmetric''
soliton, we have:
\bea
f(x) &=& g(\tau, u=(p\cdot x)^2-p^2x^2) \label{s02} \\
0=p^\mu\partial_\mu f &=& p^\mu(p_\mu\partial_0 g
+2\lambda_\mu \partial_1 g) \label{s03} = p^2\partial_0 g
\ena
Therefore $g$ does not depend on $\tau$, but only on $u$.
We therefore obtain a covariant formulation of the notion of a
``spherically symmetric'' soliton.
\subsection{Periodic solutions to the Klein Gordon equation}
Before we look for some asymptotic solutions to the
coupled equations, we must explain why we cannot
have some realistic asymptotic states in the free case.
Of course finite energy solutions to the
Klein-Gordon equation exist, consisting in wave-packets
with square integrable momentum densities. But one
of the criteria we set in order to define a ``realistic''
asymptotic field is that the wave-packet must be ``bounded''
in space-like directions. We consider here that
a constantly spreading wave-packet cannot represent the
state of a stable free particle.
We show in this paragraph that soliton solutions
to the Klein-Gordon equation cannot have a finite energy,
as a particular case of a stronger result concerning
periodic-in-time solutions. The free scalar lagrangian is:
\be
{\cal L}= \partial_\mu\Phi^* \partial^\mu\Phi -m^2 \Phi^*\Phi \label{sl04}
\ee
The energy-momentum tensor and the corresponding conserved
total momentum are:
\bea
T_{\mu\nu} &=& \partial_\mu\Phi^*\partial_\nu\Phi+
\partial_\nu\Phi^* \partial_\mu\Phi -{\cal L}g_{\mu\nu}
\label{sl08} \\
P_\nu &=& \int_{\Sigma} d\sigma^\mu T_{\mu\nu} \label{sl09}
\ena
The linearity of the equations of motion allows us to
expand the field in a Fourier serie:
\bea
\phi(t,\vec x) &=& \Sigma_{n=-\infty}^\infty a_n(\vec x)
e^{in\omega t} \label{pKG01} \\
&=& \Sigma_{n=-\infty}^\infty a_n(r)
e^{in\omega t} \label{pKG02} \\
\ena
We first look for solutions of the form
$\phi = \exp(i\eta p\cdot x)g(\sqrt{u})$,
where $\eta$ is a real parameter. We have:
\bea
\partial^\mu \phi &=& e^{i\eta p\cdot x}\left(i\eta g(\sqrt{u})p^\mu
+\frac{\lambda^\mu}{\sqrt{u}}g'(\sqrt{u}) \right) \label{pKG03} \\
\Box \phi &=& e^{i\eta p\cdot x}\left(-m_0^2\eta^2 g(\sqrt{u})
-\frac{2m_0^2}{\sqrt{u}}g'(\sqrt{u})-m_0^2g''(\sqrt{u})\right)
\nonumber \\
&=& -\frac{m_0^2}{y} e^{i\eta p\cdot x}\left(\eta^2 t(y)+t''(y)
\right)~~~~~~(g(y) = t(y)/y) \label{pKG04} \\
0 &=& (\Box+m^2)\Phi \nonumber \\
\Leftrightarrow 0 &=& t''(y) + t(y)\left(\eta^2
-\frac{m^2}{m_0^2}\right)
\label{pKG05} \\
\Rightarrow t(y) &=& A e^{-\sqrt{\frac{m^2}{m_0^2}-\eta^2}\,y}
\label{pKG06}
\ena
We have not considered the other solution that increases as
$y$ (or $r$) increases, because we look for normalized solutions.
Thus, the general solution is:
\bea
\phi(x) &=& \frac{1}{y}\sum_{|n|\leq \left[\frac{m}{m_0}\right]}
A_n e^{in p\cdot x}e^{-\sqrt{\frac{m^2}{m_0^2}-n^2}y}
\label{pKG07} \\
&=& \frac{1}{m_0 r}\sum_{|n|\leq \left[\frac{m}{m_0}\right]}
A_n e^{in m_0 t}e^{-\sqrt{m^2-n^2 m_0^2}r} \label{pKG08}
\ena
The sum has a finite number of terms because we limit
ourselves to exponentially
decreasing terms. It will be clear in the following that the
oscillating solutions for $|n|> \left[\frac{m}{m_0}\right]$
will not provide normalizable solutions.
Contrary to the claim of Hormuzdiar and Hsu in \cite{breather}
which considered only the large r behaviour, the solutions are
not normalizable. This is due to their small $r$ behaviour. This
can be shown by computing the conserved momentum:
\\
\bea
P_0 &=& \int_{t=0} d\vec x\,\left[2\partial_0\phi^*
\partial_0\phi-g_{00}(\partial_0\phi^*\partial_0\phi
-\vec\nabla\phi^*\vec\nabla\phi-m^2\phi^*\phi)\right] \label{normal01} \\
&=& \int_{t=0} d\vec x\,\left[\partial_0\phi^*
\partial_0\phi +\vec\nabla\phi^*\vec\nabla\phi+m^2\phi^*\phi)\right]~~
(\geq 0) \label{normal02} \\
&=& 4\pi\int_0^\infty r^2dr\, \frac{1}{r^2}\left[
\left|\sum_n inm_0A_ne^{-\sqrt{m^2-n^2m_0^2}r} \right|^2
+\left|\sum_n A_ne^{-\sqrt{m^2-n^2m_0^2}r}\left(\sqrt{m^2-n^2m_0^2}
+\frac{1}{r}\right)\right|^2 \right. \nonumber \\
&&\left. +m^2\left|\sum_n A_ne^{-\sqrt{m^2-n^2m_0^2}r} \right|^2 \right]
\ena
and the $1/r$ term in the second squared term makes the
integral divergent. The integral converges if $\sum A_n=0$
but the computation on another space-like hypersurface $t=t_0\ne 0$
would be still divergent, which is an indication that the computation
at $t=0$ is meaningless, even if it can be accidentally convergent. \\
\subsection{Solitons for the coupled SQED equations}
The field $v^\mu$ may also be written in a simple
generic form if we suppose that it
obeys the spherically-symmetric soliton condition.
The most general form compatible with the symmetries
of the solution is given by:
\bea
v^\mu(x) &=& a(u)\lambda^\mu + b(u)p^\mu \label{vsoliton} \\
v^2 &=& m_0^2(b^2-a^2 u) \label{vdeux}
\ena
The first term of $v^\mu$ does not contribute to the
field strength tensor because if we set $A=\int_0^u a(s)ds$,
then $a(u)\lambda^\mu = \partial^\mu(A(u)/2)$
which is a pure gauge term. And we will further demonstrate
that this term must vanish. However, we will see in the
next sections that for periodic solutions, this term is
important. \\
We will also need to comply with the classical asymptotic
conditions at infinity in space-like directions. One
must therefore have $A^\mu$ decreasing as $1/r$ at infinity, and
thus $b(u)\simeq C/\sqrt{u}$ when $u\rightarrow \infty$. \\
Then we can substitute $v^\mu$ and \fbox{$z(x)=f(u)$} in the equations
of motion Eq.~\ref{sqed-gauge-invar1} and
Eq.~\ref{sqed-gauge-invar2}~:
\bea
(\Box+(m^2-e^2v^2))z &=& 0 \label{sqed-gauge-invar1b} \\
\Box(v_\nu) -\partial_\nu (\partial\cdot v) &=&
- 2e^2 z^2 v_\nu \label{sqed-gauge-invar2b}
\ena
Using the parameterization of $v^\mu$ given in Eq.~\ref{vsoliton}
one gets:
\bea
\partial_\alpha v_\beta &=&
2\frac{d(a)}{du}\lambda_\alpha\lambda_\beta
+2\frac{d(b)}{du}\lambda_\alpha p_\beta+ a\tau_{\alpha\beta} \\
F_{\mu\nu} = \partial_\mu v_\nu -\partial_\nu v_\mu
&=& 2 \frac{d(b)}{du}(\lambda\wedge p)^{\mu\nu} = -2m_0^2 \frac{d(b)}{du}
(x\wedge p)^{\mu\nu} \\
F_{\mu\lambda}F_\nu^{~\lambda} &=& 4m_0^4 {\frac{d(b)}{du}}^2
(x_\mu x_\nu m_0^2 -(p\cdot x)(p_\mu x_\nu+p_\nu x_\mu)
+x^2 p_\mu p_\nu) \\
F^2 &=& -8m_0^4{\frac{d(b)}{du}}^2 u \\
\Box v_\nu -\partial_\nu(\partial\cdot v)
&=& -4m_0^2u\frac{d^2(b)}{du^2}p_\nu
-6m_0^2\frac{d(b)}{du}p_\nu
\ena
Thus Eq.~\ref{sqed-gauge-invar2b} yields:
\bea
-4m_0^2u\frac{d^2(b)}{du^2}p_\nu
-6m_0^2\frac{d(b)}{du}p_\nu
&=& -2e^2f^2(a\lambda_\nu+bp_\nu) \\
\Rightarrow~~~~ a&=& 0 \\
and~~~~~~~ 4u\frac{d^2(b)}{du^2}
+ 6\frac{d(b)}{du}
&=& \frac{2e^2}{m_0^2}f^2b \\
\fbox{$\ds b(u) = \frac{\tilde b(\sqrt{u})}{\sqrt{u}}$}
\Rightarrow~~~~ \tilde b'' &=&
\frac{2e^2}{m_0^2}f^2\tilde b \label{eqb}
\ena
Similarly, we will use the change of variable \fbox{$\ds f(u)=
\frac{m_0 t(\sqrt{u})}{\sqrt{u}}$} in Eq.~\ref{sqed-gauge-invar1b}
and in Eq.~\ref{eqb}. We finally obtain this system of coupled
differential equations:
\bea
t''(y) - \left(\frac{m^2}{m_0^2}
-e^2\frac{\tilde b(y)^2}{y^2}\right)t
&=& 0 \label{eqt} \\
\tilde b''(y) -2e^2\frac{t(y)^2}{y^2}\tilde b(y)
&=& 0 \label{eqb2}
\ena
\subsubsection{Normalization of the solutions}
In this paragraph, we compute the conserved momentum
of spherically symmetric solitons. We will
show that the solutions cannot be normalized. Considering
the energy-momentum tensor of Eq.~\ref{em-tensor2}, we get
for a soliton:
\bea
\vec P &=& \vec 0 \\
F_{0\lambda}F_0^{~\lambda}
&=& -4m_0^6 \left(\frac{db}{du}\right)^2
r^2~~~~(rest~frame,~t=0,~r=|\vec x| ) \\
v^0 &=& m_0 b(u)~~~;~~~\partial_0 z =0~~~;~~~
(\partial_\mu z)^2 = -4m_0^2 u \left(\frac{df}{du}\right)^2 \\
T_{00} &=& e^2z^2(2m_0^2b^2-m_0^2b^2+m_0^2a^2u)
+4m_0^2 u \left(\frac{df}{du}\right)^2
+m^2 f^2 +4m_0^6\left(\frac{db}{du}\right)^2r^2
-2m_0^4\left(\frac{db}{du}\right)^2u \\
&=& m_0^2 e^2 f^2 b^2 + 4m_0^2 u \left(\frac{df}{du}\right)^2 +m^2 f^2
+2m_0^6 \left(\frac{db}{du}\right)^2 r^2 \\
&=& e^2m_0^4 \frac{t^2{\tilde b}^2}{y^4} +m^2\frac{t^2}{y^2}
+4m_0^6 r^2\left(\frac{1}{2y}\frac{d}{dy}\left(\frac{t(y)}{y}
\right) \right)^2 +2m_0^6 r^2\left(\frac{1}{2y}\frac{d}{dy}
\left(\frac{\tilde b(y)}{y} \right) \right)^2 \\
&=& e^2m_0^4 \frac{t^2{\tilde b}^2}{y^4} +m^2\frac{t^2}{y^2}
+m_0^4\left(\frac{d}{dy}\left(\frac{t(y)}{y}
\right) \right)^2 +\frac{m_0^4}{2}\left(\frac{d}{dy}
\left(\frac{\tilde b(y)}{y} \right) \right)^2 \\
P_0 &=& 4\pi\int_0^\infty r^2\, dr\, T_{00} \\
P_0 &=& \frac{4\pi}{m_0^3}\int_0^\infty y^2\, dy
\left[ e^2m_0^4 \frac{t^2{\tilde b}^2}{y^4} +m^2\frac{t^2}{y^2}
+m_0^4\left(\frac{d}{dy}\left(\frac{t(y)}{y}
\right) \right)^2 +\frac{m_0^4}{2}\left(\frac{d}{dy}
\left(\frac{\tilde b(y)}{y} \right) \right)^2 \right]
\nonumber \\
&=& 4\pi m_0\int_0^\infty dy\left[ e^2 \frac{t^2{\tilde b}^2}{y^2}
+\left(\frac{m}{m_0}\right)^2 t^2
+y^2\left(\frac{d}{dy}\left(\frac{t(y)}{y}
\right)\right)^2
+\frac{y^2}{2}\left(\frac{d}{dy}\left(\frac{\tilde b(y)}{y}
\right)\right)^2
\right] \label{solit-norm}
\ena
We have seen that $\tilde b$ must tend to a
non-vanishing constant
at infinity in space-like directions ($A^\mu \simeq 1/r$),
but from Eq.~\ref{eqb2}
we can conclude that $\tilde b$ is a convex
function when $\tilde b>0$ and
the converse for the other sign. From the last term
in Eq.~\ref{solit-norm}, we get that $\tilde b$ cannot tend
to a non-vanishing value in $y=0$ (otherwise the integral is
divergent). Thus if $\tilde b$ vanish
in $y=0$, it cannot tend to a non-vanishing constant at
infinity because it is a convex function if $\tilde b>0$ or
the converse if $\tilde b<0$. The only possibility
is $\tilde b=0$, and we are then back to the free case, which
we have previously rejected.
\subsection{Is there some periodic solutions to the coupled equations?}
\label{periodic}
Now we introduce a ``time'' variable $\tau = p\cdot x$ which
is dimensionless and $y=\sqrt{u}$ like in the soliton case. We have:
\bea
z(x) &=& f(\tau,y) = \frac{t(\tau,y)}{y}\\
v^\mu &=& a(\tau,y)\lambda^\mu +b(\tau,y)p^\mu \\
\Rightarrow \partial_\mu z &=& p_\mu \partial_0 f
+\frac{\lambda^\mu}{y}\partial_1 f \\
\Rightarrow \Box z &=& m_0^2\left(\partial_0^2 f -\frac{2}{y}\partial_1 f
-\partial_1^2 f \right)
= \frac{m_0^2}{y}\left(\partial_0^2 t -\partial_1^2 t \right) \\
\Box v_\mu -\partial_\mu(\partial\cdot v) &=&
m_0^2 p_\mu \left[\rho\partial_0\partial_1 a +3\partial_0 a
-\partial_1^2 b -\frac{2}{y}\partial_1 b \right]
+m_0^2 \lambda_\mu\left[\partial_0^2 a
-\frac{\partial_0\partial_1 b}{y} \right]
\ena
From these basic calculations we get for the equations of motion:
\bea
\partial_0^2 t -\partial_1^2 t
+\left(\frac{m^2}{m_0^2}-e^2(b^2-y^2 a^2)\right)t &=& 0 \label{eq1}\\
\partial_0^2 a
-\frac{\partial_0\partial_1 b}{y} &=& -2\frac{e^2 t^2}{m_0^2 y^2}a
\label{eq2} \\
y\partial_0\partial_1 a +3\partial_0 a
-\partial_1^2 b -\frac{2}{y}\partial_1 b &=&
-2\frac{e^2 t^2}{m_0^2 y^2}b \label{eq3}
\ena
These equations are much more complicated than in the case of
solitons and the fundamental structure of the solutions,
even periodic in time is not clear so far. We will restrict
ourselves in this paragraph to a description of what is
really different in this case and why we conjecture
the existence of some normalized periodic solutions. \\
The conservation of the electromagnetic current leads
to the emergence of a kind of pre-potential:
\bea
\partial_\mu(z^2 v^\mu) &=& 0 \\
\Leftrightarrow \partial_1 (y t^2 a) &=& \partial_0 (t^2 b)
\label{conserved-c} \\
\Rightarrow y t^2 a &=& \partial_0\vp(\tau,y) \label{vpa} \\
and~~~~~~~ t^2 b &=& \partial_1\vp(\tau,y) \label{vpb}
\ena
Introducing this potential in the equations for the
electromagnetic field we get:
\bea
\partial_0\left( \partial_0 a
-\frac{\partial_1 b}{y} \right) &=& -2\frac{e^2}{m_0^2}
\partial_0\left(\frac{\vp}{y^3}\right)
\label{eq2b} \\
\partial_1\left(y^2\partial_1 b\right)
-\partial_1\partial_0 (y^3a) &=&
2\frac{e^2}{m_0^2}\partial_1\vp \label{eq3b}
\ena
These equations can be partially integrated, and
we obtain:
\bea
\partial_0 a
-\frac{\partial_1 b}{y} &=& -2\frac{e^2}{m_0^2}
\frac{\vp}{y^3} +A(y)
\label{eq2c} \\
y^2\partial_1 b -\partial_0 (y^3a) &=&
2\frac{e^2}{m_0^2}\vp +B(\tau) \label{eq3c}
\ena
The presence of these two functions A and B
enlarges significantly the set of possibilities for the
solutions. We therefore hope that some of
these might be normalizable, as we shall discuss further.
\subsubsection{Normalization of the time-dependent solutions}
In order to normalize these periodic solutions, the
computation of the conserved momentum gives for
Eq.~\ref{em-tensor2}:
\bea
F_{\mu\nu}&=& (p\wedge\lambda)_{\mu\nu}\left(\partial_0 a -
\frac{\partial_1 b}{y}\right) \\
F_{\mu\alpha}F_\nu^{~\alpha} &=& -m_0^2\left(\partial_0 a -
\frac{\partial_1 b}{y}\right)^2(y^2 p_\mu p_\nu
-\lambda_\mu\lambda_\nu) \Rightarrow F_{0\alpha}F_\mu^{~\alpha}
= -m_0^3y^2\left(\partial_0 a -
\frac{\partial_1 b}{y}\right)^2 p_\mu \\
F_{\beta\alpha}F^{\beta\alpha} &=& -2m_0^4 y^2\left(\partial_0 a -
\frac{\partial_1 b}{y}\right)^2 \\
T_{0i} &=& 2e^2f^2m_0ab\lambda^i
+2m_0\partial_0f \frac{\lambda^i}{y} \\
\Rightarrow P_i &=& 0 \\
T_{00} &=& e^2m_0^2f^2(b^2+a^2y^2)+m^2f^2
+m_0^2\left( (\partial_0f)^2+(\partial_1f)^2 \right)
+\frac{y^2}{2}\left(\partial_0 a -
\frac{\partial_1 b}{y}\right)^2 \\
P_0 &=& \frac{4\pi}{m_0^3}\int_0^\infty y^2dy\, T_{00} \\
&=& \frac{4\pi}{m_0}\int_0^\infty y^2dy\,
\left( e^2f^2(b^2+a^2y^2)+\frac{m^2}{m_0^2}f^2
+\left( (\partial_0f)^2+(\partial_1f)^2 \right)
+\frac{y^2}{2m_0^2}\left(\partial_0 a -
\frac{\partial_1 b}{y}\right)^2
\right) \label{periodic-energy}
\ena
The last term in Eq.~\ref{periodic-energy}
also appears in Eq.~\ref{eq2c}, equation
that was absent when we considered soliton solutions. In this
equation, the function A is
undetermined but if $\frac{\vp}{y^3}$
is sufficiently singular at 0, the function A will certainly not
compensate the singularity because it is time-independent, and
$\vp$ is periodic. Thus, if A accidentally compensate
$\frac{\vp}{y^3}$ at $y=0$ for $t=0$, it may not be the case at a
different time. As a consequence, $\vp$ must certainly vanish at
$y=0$ if one wants the integral to be convergent.
We still have in this case some dramatic constraints
on the behaviour of the solutions at $y=0$. However,
what prevented us from finding normalized periodic solutions
in the free case was the finite value of t at $y=0$.
In the free case, solutions are only composed of exponentially
decreasing functions. Here we have another ``mass'' term
in the equation of motion for t. If $b^2-a^2y^2$ becomes large
in the vicinity of the origin, one may obtain solutions
that are spatially oscillating (and only near $y=0$).
Such a possibility allows to have a t function that vanishes
at $y=0$, while still featuring an exponentially
decreasing behaviour at infinity.
We expect soon to be able to confirm this conjecture by numerical
simulations, before we can get more rigorous answers
to this problem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTION 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The non-abelian case}
\subsection{The standard equations of motion}
In this case we consider a scalar field $\Phi$ lying in an
N-dimensional vector space of representation of the
Lie group ${\cal G}$ (a subgroup of $U(N)$). The results
presented in this section will not work for all the possible
gauge groups,
yet our method is valid for $U(N)$ or $SU(N)$.
There are very few constraints that may
be imposed on a generic gauge group.
Probably the most important one
is that there must exist a scalar product on the Lie algebra
which is invariant under an inner automorphism. One can then
demonstrate that the solvable part (in the Levi decomposition
of the group) must be abelian. Thus if we also
constrain the group to be compact, it is relevant to consider
gauge groups as being a sum of $U(1)$ terms, plus
any semi-simple part like $SU(N)$. Since the $U(1)$ case
has been previously solved, we focus here on $SU(N)$ groups.
Actually, we will see that
our formalism works if the orbit of any vector $\Phi$ under
the gauge group is $C^N$, which is the case for $SU(N)$. \\
We will denote by $i{\cal A}$ the real Lie algebra, such
that the matrices lying in ${\cal A}$ are hermitian.
If $\rho$ stands for
the representation of the Lie algebra,
we can endow the algebra with the following
scalar product for the computation of the
Yang-Mills part of the lagrangian:
$(A,B)_\rho=\tr[\rho(A)^\dagger \rho(B)]$.
The scalar product generally used with a semi-simple group
is the Killing form applied to the field strength tensor.
Since this scalar product is proportional to any
scalar product of the form $(A,B)_\rho$ (the coefficient
being the Dynkin index of $\rho$), we will simply use
this scalar product $(A,B)=\tr[A^\dagger B]= \tr[AB]$.
In the following, we give the
lagrangian and the corresponding equations of motion, using
$\Phi$ and $W_\mu$ as variables.
\bea
{\cal L} &=& {\cal L}_0 + {\cal L}_{YM} \nonumber \\
{\cal L}_0 &=& (D_\mu\Phi)^\dagger D^\mu\Phi - m^2\Phi^\dagger\Phi
\nonumber \\
&=& \partial_\mu\Phi^\dagger \partial^\mu\Phi
-ig\Phi^\dagger W_\mu\partial^\mu\Phi
+ig\partial_\mu\Phi^\dagger W^\mu\Phi +g^2\Phi^\dagger W_\mu W^\mu\Phi
- m^2\Phi^\dagger\Phi \\
{\cal L}_{YM} &=& -\frac{1}{4}(F_{\mu\nu},F^{\mu\nu}) =
-\frac{1}{4}\Big\{\tr\left[(\partial_\mu W_\nu-\partial_\nu W_\mu)
(\partial^\mu W^\nu-\partial^\nu W^\mu)\right] \nonumber \\
&& -g^2\tr\left[[W_\mu,W_\nu][W^\mu,W^\nu]\right]
+2ig\tr\left[(\partial_\mu W_\nu-\partial_\nu W_\mu)
[W^\mu,W^\nu]\right]\Big\} \\
\frac{\partial{\cal L}}{\partial(\partial_\mu W_\nu)}(\Omega_\nu)
&=& -\,\tr[\Omega_\nu G^{\mu\nu}] \\
\partial_\mu \frac{\partial{\cal L}}{\partial(\partial_\mu W_\nu)}
(\Omega_\nu) - \frac{\partial{\cal L}}{\partial(W_\nu)}
(\Omega_\nu) &=& -\,\tr[\Omega_\nu \partial_\mu G^{\mu\nu}]
-\left(-ig\Phi^\dagger \Omega_\mu\partial^\mu\Phi
+ig\partial_\mu\Phi^\dagger \Omega^\mu\Phi +g^2\Phi^\dagger
\{\Omega_\mu, W^\mu\}\Phi
\right) \nonumber \\
&& +ig\,\tr\left[[W_\mu,\Omega_\nu]G^{\mu\nu}
\right] \\
&=& -\,\tr[\Omega_\nu \partial_\mu G^{\mu\nu}]
+ig\tr\left[\Omega_\nu\left(D^\nu\Phi \Phi^\dagger
-\Phi (D^\nu\Phi)^\dagger
\right)\right] \nonumber \\
&& +ig\,\tr\left[\Omega_\nu[G^{\mu\nu},W_\mu]
\right] \\
\Rightarrow 0&=& \tr\left[\Omega_\nu \left(-{\cal D}_\mu G^{\mu\nu} +
ig(D^\nu\Phi\Phi^\dagger -\Phi (D^\nu\Phi)^\dagger) \right)
\right]~~~~~(\forall~{\Omega_\nu}~\in~{\cal A}) \label{gf-na} \\
{\cal D}^\mu (G_{\mu\nu})
&=& \Pi_{\cal A}\left[ig\left(D_\nu\Phi\Phi^\dagger
-\Phi(D_\nu\Phi)^\dagger \right)\right]= \Pi_{\cal A}(J_\nu)
\label{gf-na1} \\
J_\mu &=& ig\left(\partial_\mu \Phi \Phi^\dagger -\Phi\partial_\mu
\Phi^\dagger +ig\{ W_\mu, \Phi\Phi^\dagger\}\right) \label{curr} \\
\partial_\mu \frac{\partial{\cal L}}{\partial(\partial_\mu \Phi^\dagger)}
- \frac{\partial{\cal L}}{\partial(\Phi^\dagger)}
=0\Rightarrow 0 &=& (D_\mu D^\mu +m^2)\Phi \nonumber \\
&=& (\Box+m^2)\Phi + 2igW_\alpha\partial^\alpha\Phi
+ig(\partial_\alpha W^\alpha)\Phi - g^2W_\mu W^\mu\Phi
\label{scalar-na}
\ena
We shall note that in Eq.~(\ref{gf-na}), the
fact that the equation is only valid for $\Omega_\nu~\in~{\cal A}$ is very
important. It comes from the fact that in the variationnal
principle leading to the Euler-Lagrange equations, the variation
of the gauge field ($\Omega_\nu$) must lie in the Lie algebra also.
If this equation were valid for any matrix $\Omega_\nu$, then
we would have ${\cal D}^\alpha (G_{\alpha\beta})$ (which is
in ${\cal A}$) equal to $ig\left(D_\beta\Phi\Phi^\dagger
- \Phi(D_\beta\Phi)^\dagger \right)$, which is not necessarily
in ${\cal A}$, and this is why there is this projection
operator on the Lie algebra $\Pi_{\cal A}$ in Eq.~\ref{gf-na1}.
For $su(N)$ algebras, this
projection is simply $M\mapsto M -\tr(M)\frac{I}{N}$, where
$I$ stands for the identity matrix.
Eq.~\ref{gf-na1} and Eq.~\ref{scalar-na} are the equations of motion
respectively for the gauge fields and for the scalar fields, which
can be related to the abelian equations of Eq.~\ref{gf}
and Eq.~\ref{scalar}. The non-abelian equivalent of the
current is now extracted from Eq.~\ref{gf-na1} and is
given by the matrix: $J_\nu = ig\left(D_\nu\Phi\Phi^\dagger
-\Phi(D_\nu\Phi)^\dagger \right)$ \\
Contrary to the abelian case, this current is
not gauge invariant anymaore but rather gauge covariant, that is
$J_\nu =U{J'}_\nu U^{-1}$ under a gauge transformation. \\
We now operate as in the previous section, and
observe that if we compute $(\ref{scalar-na})\Phi^\dagger -
\Phi(\ref{scalar-na})^\dagger$, we get:
\bea
0 &=& (D_\mu D^\mu\Phi) \Phi^\dagger-\Phi(D_\mu D^\mu\Phi)^\dagger
\Phi \nonumber \\
&=& {\cal D}_\mu \left( (D^\mu\Phi) \Phi^\dagger
-\Phi(D^\mu\Phi)^\dagger \right) \\
\Rightarrow 0 &=& {\cal D}_\mu \left( J^\mu \right) \label{non-ab-cons-c}
\ena
If one projects this equation on the Lie algebra, the
resulting equation is redundant with Eq.~\ref{gf-na1} on which
we apply the operator ${\cal D}^\nu$. Like in the abelian case,
we find a redundancy, but it is important to note at this stage
that eq.~\ref{non-ab-cons-c} is a stronger condition than if
we just applied ${\cal D}^\nu$ on Eq.~\ref{gf-na1}. It seems
that we missed some degrees of freedom in Eq.~\ref{gf-na1}.
The fundamental
structure of the gauge group is responsible for this fact.
For instance, in the case of a $u(N)$ algebra,
$\Pi_{\cal A}(M)=M$ if
M is hermitian, and all the ``degrees of freedom'' of $J_\mu$
are concerned with this redundancy between the equation
for the matter and the equation for the gauge field. \\
We therefore have too much information in the set of equations
of the matter field and one should replace eq.~(\ref{scalar-na}) by
$(\ref{scalar-na})\Phi^\dagger +
\Phi(\ref{scalar-na})^\dagger$, i.e.:
\bea
0 &=& (D_\mu D^\mu\Phi)\Phi^\dagger +
\Phi(D_\mu D^\mu\Phi)^\dagger +2m^2 \Phi\Phi^\dagger \\
&=& {\cal D}_\mu \left((D^\mu\Phi) \Phi^\dagger
+\Phi(D^\mu\Phi)^\dagger \right)
-2D_\mu\Phi(D^\mu\Phi)^\dagger+2m^2\Phi\Phi^\dagger\\
&=& \left({\cal D}_\mu{\cal D}^\mu(\Phi\Phi^\dagger)
-2(D_\mu\Phi)(D^\mu\Phi)^\dagger+2m^2\Phi\Phi^\dagger\right)
\ena
\subsection{The gauge invariant variables}
The procedure used to obtain Eq.~\ref{ym-from-c} consists in
eliminating the two first terms of $\frac{J_\mu}{ie} =
\vp^*\partial_\mu\vp-\partial_\mu\vp^*\vp +2ie\rho A_\mu$ in order
to extract the gauge field. We have $\frac{J_\mu}{ie\rho}=2ie
A_\mu +\partial_\mu \Lambda$ and the pure gauge term disappears in
$F_{\mu\nu}$. But in our case we have a matrix and this procedure
does not work. However, the extraction of $A_\mu$ can be seen in
another way. In the abelian case, we could also have taken a
unitary gauge, that is to say a gauge in which $\vp$ is real. This
automatically eliminates the desired terms. We may proceed here in
a similar way. The essential hypothesis is that any two scalar
fields $\Phi$ and $\Psi_0$ can be related by an element of the
gauge group. It is the case for $U(N)$ or $SU(N)$. Thus, the
central point of the method is to choose a constant unitary vector
$\Psi_0$, and therefore one can find $U$ in the gauge group such
that:
\bea
\Phi &=& zU\Psi_0 \\
z &=& \sqrt{\Phi^\dagger\Phi} = \sqrt{\rho}
\ena
A consequence is that if ${W'}_\mu$ is the gauge field in the
``unitary'' gauge obtained by the matrix $U$ we have
from Eq.~\ref{curr}:
\bea
{J'}_\mu = U^{-1}J_\mu U &=& ig\left( z\partial_\mu(z)
\Psi_0\Psi_0^\dagger - \Psi_0\Psi_0^\dagger z\partial_\mu(z)
+ig\rho\{{W'}_\mu,\Psi_0\Psi_0^\dagger\} \right)\\
&=& (ig)^2\rho \{{W'}_\mu,\Psi_0\Psi_0^\dagger\}
\ena
However, it is in general impossible to reconstruct the
entire gauge field ${W'}_\mu$ from this equation,
except for the $SU(2)$ case because of the relation
$\{\sigma^i,\sigma^j\}=2\delta^{ij}$ (and this
anticommutator has no residue
lying in the Lie algebra). Using this property, the traceless
part of ${J'}_\mu$ gives $(ig)^2\rho {W'}_\mu$. Since it works
only for $SU(2)$, we need to find a way to get the missing
degrees of freedom of the gauge field. The method consists
in constructing an orthonormal basis of $C^N$, starting
from $\Psi_0$: $(\Psi_0,\Psi_1,...,\Psi_{N-1})$, which does not
depend on space-time coordinates. If we set $\Phi_k= zU\Psi_k$
($k\geq 1$), then $(\rho^{-1}\Phi,\rho^{-1}\Phi_1,...,\rho^{-1}\Phi_{N-1})$
forms also an orthonormal basis. A gauge transformation will
naturally apply also to these new scalar fields, and
we consider the gauge invariant variables:
\bea
J_{mn\mu} &=& ig\left(\Phi_m^\dagger D_\mu\Phi_n
- (D_\mu\Phi_m)^\dagger\Phi_n\right)
\ena
The simple reason why we do not consider some other
gauge invariant variables, by taking the sum of the two terms
above instead of their difference is that
$\Phi_m^\dagger D_\mu\Phi_n + (D_\mu\Phi_m)^\dagger\Phi_n= \partial_\mu
(\Phi_m^\dagger\Phi_n) =\partial_\mu(\rho\delta_{m,n})$
and thus they can be expressed using the gauge invariant
variable $z=\sqrt{\rho}$. In the unitary gauge, these gauge
invariant variables allow to reconstruct the
gauge field completely:
\bea
J_{mn\mu} &=& 2(ig)^2\rho\Psi_m^\dagger{W'}_\mu\Psi_n
\quad\quad\quad (J_{nm\mu} = J_{mn\mu}^*) \\
v_{mn\mu} &=& \frac{-1}{2g^2\rho}J_{mn\mu} \\
\Rightarrow {W'}_\mu &=& \sum_{m,n} v_{mn\mu}\Psi_n\Psi_m^\dagger
\ena
The equations of motion for the gauge field in Eq.~\ref{gf-na1} can
then be rewritten in the unitary gauge (note that
$\Pi_{\cal A}(U^{-1}JU)= U^{-1}\Pi_{\cal A}(J)U$)
\bea
{G'}_{\mu\nu} &=& \sum_{m,n} \left(
\partial_\mu\left(v_{mn\nu}\right) -
\partial_\nu\left(v_{mn\mu}\right)
+ig \sum_k (v_{mk\mu}v_{kn\nu} - v_{mk\nu}v_{kn\mu})
\right)\Psi_n\Psi_m^\dagger \label{curvature} \\
{\cal D'}^\mu({G'}_{\mu\nu}) &=& \partial^\mu{G'}_{\mu\nu}
+ig[{W'}_\mu,{G'}_{\mu\nu}] \\
&=&\sum_{m,n} \left(
\Box \left(v_{mn\nu}\right) -
\partial_\nu\left(\partial\cdot v_{mn}\right)
+ig \sum_k \partial^\mu (v_{mk\mu}v_{kn\nu} - v_{mk\nu}v_{kn\mu})
\right)\Psi_n\Psi_m^\dagger \nonumber \\
&&+ig\sum_{m,n} \Psi_n\Psi_m^\dagger \left[\sum_l v_{ml}^{~~\mu}
\left(
\partial_\mu\left(v_{ln\nu}\right) -
\partial_\nu\left(v_{ln\mu}\right)
+ig \sum_k (v_{lk\mu}v_{kn\nu} - v_{lk\nu}v_{kn\mu})
\right)\right. \nonumber \\
&&\left. -\left(
\partial_\mu\left(v_{ml\nu}\right) -
\partial_\nu\left(v_{ml\mu}\right)
+ig \sum_k (v_{mk\mu}v_{kl\nu} - v_{mk\nu}v_{kl\mu})
\right)
v_{ln}^{~~\mu}\right] \\
&=& -g^2z^2 \sum_m \Pi_{\cal A}\left(v_{0m\mu}
\Psi_m\Psi_0^\dagger +v_{m0\mu}\Psi_0\Psi_m^\dagger
\right) \\
&=& -g^2z^2 \sum_m \left(v_{0m\mu}
\Psi_m\Psi_0^\dagger +v_{m0\mu}\Psi_0\Psi_m^\dagger
-2\frac{v_{00\mu}}{N}\Psi_m\Psi_m^\dagger\right)
\ena
The last equality is only valid for $SU(N)$. One must adapt this
formula for another gauge group. Projecting these equations on the
basis of matrices $\Psi_m\Psi_n^\dagger$ leads to a large set of
$N^2$ equations in which only gauge invariant variables are
present. In the $SU(N)$ case we can also separate these
equations into four different classes depending on the indices m
and n, because of the specific form of the current matrix
projected on the Lie algebra. The four cases correspond to the
diagonal case with indices in the form $(m,m)$ ($m>0$), the case
with indices in the form $(0,m)$ or $(m,0)$ ($m>0$), and finally
the case where $m=n=0$. The projection on these different cases
can be easily done and we will not present them here. It is clear
that the gauge fields ${W'}_\mu$ expressed in the basis of the
$\Psi_k$'s is nothing but the matrix composed of the gauge
invariant coefficients $(v_{m,n\mu})$. Of course, these
coefficients depend on the constant basis we choose, but physical
solutions must be independent of this choice.
It remains to demonstrate that these equations of motion can be
re-expressed using only variables that are also independent from
the constant basis chosen: we can consider some objects of the
form $\tr[({W'}_\mu)^n]$, or equivalently the characteristic
polynomial of ${W'}_\mu$. We expect to have new results in the
near future.
We may conclude this
last section with the equation of motion for the
matter fields. The simplest way is to look at the lagrangian
and to use the following equality:
\be
\frac{1}{-g^2}\sum_{m,n}J_{mn\mu}J_{nm}^{~~\mu} = N\partial_\mu \rho
\partial^\mu \rho -4\rho (D_\mu \Phi)^\dagger (D^\mu\Phi)
\ee
The matter part of the lagrangian can then be written:
\be
{\cal L}_0 = N\partial_\mu z \partial^\mu z -m^2z^2
+g^2z^2\sum_{m,n}v_{mn\mu}v_{nm}^{~~\mu}
\ee
And the equation of motion for the scalar field is finally:
\be
\left(\Box+\frac{m^2}{N}\right)z =
\frac{g^2}{N}z \sum_{m,n}v_{mn\mu}v_{nm}^{~~\mu}
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% CONCLUSION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
In this paper, we give a certain number of results which
are really encouraging
for the purpose of reformulating gauge theories using only
gauge-invariant variables.
Within the prospects of this work, a short-term project would
naturally be to find an equivalent formulation when fermions are
involved. Then, the quantization of the theory has to be
constructed. Within this subtopic, it would be interesting to
revisit the general formalism of quantization in QFT. An equation
like $\frac{dA}{dt}= i[H,A]$ is a very old non-relativistic
formula which is surprisingly still used in textbooks about
relativistic quantum field theory. Instead of the Hamiltonian, one
would naturally consider an operator of the form $\int_\Sigma
d\sigma^\mu T_{\mu\nu}$ in order to quantize a theory. This has
not been done yet and one of the possible reasons is that there is no
unique expression for the energy-momentum tensor $T_{\mu\nu}$.
There are some current research activities on this
topic~\cite{gotay1}, in order to find the ``best'' criteria to
define uniquely $T_{\mu\nu}$. So far, it seems that the Belinfante
tensor is a good candidate, since it is gauge-invariant. Therefore
it can be naturally inserted in the formalism presented in this
paper. Finally, in the long-term we hope
to be able to compute some scattering
cross sections using directly gauge invariant variables, and
also to provide a revised version of Quantum Field Theory which
would apply to unstable particles and more generally, to physical
systems that evolve on a ``long-time'' scale,
(CP violation, neutrino oscillations,...) as mentionned in the
introduction. \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendices}
\subsection{Review of basic cohomological formulas}
As noted in this paper, one of the main problems regarding
gauge independence is to have a method to find the set of
gauge fields with a given Field-Strength tensor $F^{\mu\nu}$.
We will separate the abelian case
from the non-abelian one, because the curvature tensor
depends linearly on the gauge field in the abelian case,
quadratically in the latter
case. Linearity is lost in the non-abelian case, which renders
the problem much more complicated. \\
The problem can be summarized as follows: if one has a specific
tensor F of rank $n$, we look for another tensor of rank $n-1$
such that $F=dA$ where $d$ represents the
exterior derivative.
The tensor $F$ must obey $dF=0$ because of the property $d^2=0$.
So we want to find $A$ from a given $F$,
assumed that $F$ is a closed form (i.e.
$dF=0$). Given a solution $A$, one can find another solution
$A'$ by adding to $A$ any term of the form $d\Lambda$, again
because $d^2=0$. Therefore, we will say that two tensors
of rank $n-1$ are co-homologous if there exists $\Lambda$ such
that $A-A' = d\Lambda$. It is an equivalence relation
and the equivalence classes are called
cohomology classes
(for the de-Rahm cohomology, and we will
further explain why it is important to make this
distinction when the non-abelian case is involved).
\subsection{Abelian gauge fields}
Let $M=R^4$ be the Minkowsky space-time, and consider
$X^\mu(u,x)$ an application from $[0,1]\times M$ into $M$
such that:
\bea
\forall~x~\in~M,~X^\mu(0,x) &=& x_0^\mu \label{contraction00a} \\
\forall~x~\in~M,~X^\mu(1,x) &=& x^\mu \label{contraction01a}
\ena
We also assume that $X^\mu$ is infinitely smooth. It is
then called a ``contraction''. The reader will recover the
standard Poincar\'e formula by taking $X^\mu(u,x)=ux^\mu$.
Suppose $A^\mu(x)$ is a vector field with vanishing curvature,
then if we define $V(x)$ as follows:
\bea
\label{de-rahm0}
V(x) &=& \int_0^1 du \frac{\partial X^\mu}{\partial u}
A_\mu(X(u,x)) \\
Then~~~~~~~~~~~\partial_\mu V &=& A_\mu(x) -\int_0^1 du
\frac{\partial X^\alpha}{\partial u}\frac{\partial X^\beta}
{\partial x^\mu}F_{\alpha\beta}(X)\label{de-rahm00}
\ena
Therefore, if the curvature
of $A$ vanishes, $V(x)$ is a possible solution for the potential.
Also, if one replaces explicitly $A^\mu$ by $\partial^\mu V'$ in
eq.~\ref{de-rahm0}, one gets $V'(x)-V'(x_0)$, and not $V'(x)$.
$V(x)$ is therefore not a ``fixed point solution'' of an integral
equation, but can be defined as the solution for which
$V(x_0) = 0$. The rest in the expression of $\partial_\mu V$
vanishes explicitly for a vanishing curvature, but when the
curvature is not $0$, this formula provides us with an explicit
expression for $A^\mu$ as a function of $F^{\mu\nu}$ up to a gauge
transformation by $\partial^\mu V$. Thus we have already the
next step, and if we consider a given field-strength tensor
$F^{\mu\nu}$, we can define the following vector field:
\be
\label{de-rahm1}
A^\mu(x) = \int_0^1 du \frac{\partial X^\alpha}{\partial u}
\frac{\partial X^\beta}{\partial x^\mu} F_{\alpha\beta}(X(u,x))
\ee
Then, with this definition we have:
\be
\partial_\mu A_\nu - \partial_\nu A_\mu = F_{\mu\nu}(x)
-\int_0^1 du \frac{\partial X^\alpha}{\partial u}
\frac{\partial X^\beta}{\partial x^\mu}\frac{\partial X^\gamma}
{\partial x^\nu}(\partial_\alpha F_{\beta\gamma} +
\partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta})
\ee
The last term vanishes if $dF=0$, and we recognize here the
homogeneous Maxwell equations.
In this case, the expression we have
chosen for $A^\mu$ is a possible gauge field, and this
formula is of course very important because it allows us
to ``parameterize'' the orbits of gauge fields.
It is possible to go on with this scheme, and for a given
3-form $\omega_{\alpha\beta\gamma}$ we can define $F^{\mu\nu}$
using:
\be
F_{\mu\nu} = \int_0^1 \frac{\partial X^\alpha}{\partial u}
\frac{\partial X^\beta}{\partial x^\mu}\frac{\partial X^\gamma}
{\partial x^\nu}\omega_{\alpha\beta\gamma}(X(u,x))
\ee
and when $d\omega =0$, we have $\partial_\alpha F_{\beta\gamma} +
\partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta}
= \omega_{\alpha\beta\gamma}(x)$, and so on (but there is actually
only one next step because we have assumed here that we are in
four space-time dimensions and any four form is proportional
to the Levi-Civita pseudo-tensor\index{Levi-Civita pseudo-tensor}).
To summarize, given an $n$ form $F$ such that $dF=0$, we have been
able to exhibit a $n-1$ form $A$ such that $F=dA$. This element
$A$ can be interpreted as an element of an equivalent class of
cohomology with a given curvature. In other words, we have
``computed'' the cohomology. Expressed this way, it looks simple
but hides the real difficulties, which are of a topological nature.
In all these calculations, we have assumed the existence of $X^\mu$,
which imposes some constraints on the topology of the four dimensional
space-time. If the whole Minkowsky space is
taken under consideration, no topological problem occurs,
and more generally, this is true if we consider a
simply connected space. Then, one
can find $X^\mu$ and proceed to the previous calculations.
\subsection{Conventions for the Non-abelian case}
$i{A}$ and $iB$ are supposed to lie in the real Lie
algebra corresponding
to the Lie Group\index{Lie Group} ${\cal G}$,
which is a subgroup\index{Subgroup} of $U(N)$ here.
Therefore $A$ and $B$ are hermitian. We set ${A}= U{A}'U^{-1}$.
$X$ and $Y$ are vectors lying in the same representation
as the matter field $\Phi$.
\bea
\Phi &=& U\Phi' = e^{iT}\Phi'~~~(T~~small)\label{gt01} \\
W_\mu &=& U{W'}_\mu U^{-1} +\frac{i}{g}\partial_\mu (U) U^{-1}
~~~~~~{W'}_\mu = U^{-1}W_\mu U-\frac{i}{g} U^{-1}\partial_\mu (U)
\label{gt02} \\
\delta W_\mu = {W'}_\mu-W_\mu &=& -\frac{i}{g}{\cal D'}_\mu(U) U^{-1}
= -\frac{i}{g}U^{-1}{\cal D}_\mu(U) \label{gt03} \\
U{W'}_\nu U^{-1}-{W'}_\nu +\frac{i}{g}(\partial_\nu U)U^{-1}
&=& e^{iT}{W'}_\nu e^{-iT}-{W'}_\nu+\frac{i}{g}(\partial_\nu e^{iT})
e^{-iT} \\
&\simeq & [iT,{W'}_\nu] -\frac{1}{g}(\partial_\nu T) = -\frac{1}{g}
{\cal D}_\nu ({A}) \label{gt03b}\\
D_\mu \Phi &=& (\partial_\mu +igW_\mu)\Phi \Rightarrow
D_\mu \Phi = D_\mu (U\Phi') = U {D'}_\mu \Phi' \label{gt04} \\
{\cal D}_\mu ({A}) &=& \partial_\mu{A} +ig[W_\mu,{A}]
\Rightarrow {\cal D}_\mu (U{A}'U^{-1}) =
U ({\cal D'}_\mu {A}')U^{-1} \label{gt05} \\
{\cal D}_\mu (AB) &=& {\cal D}_\mu (A)B + A{\cal D}_\mu (B) \\
{\cal D}_\mu (XY^\dagger) &=& {D}_\mu (X)Y^\dagger
+ X({D}_\mu (Y))^\dagger \label{gt08}\\
D_\mu (AX) &=& {\cal D}_\mu (A)X + A{D}_\mu (X)\label{gt09} \\
{[D_\mu,D_\nu]}\Phi &=& ig(\partial_\mu W_\nu - \partial_\nu W_\mu
+ig[W_\mu,W_\nu])\Phi = igG_{\mu\nu}\Phi \label{gt06} \\
G_{\mu\nu} &=& U{G'}_{\mu\nu}U^{-1} \label{gt07} \\
\ [{\cal D}_\alpha,{\cal D}_\beta](A)
&=& ig[G_{\alpha\beta},A] \label{gt14} \\
0 &=& [D_\nu,[D_\rho,D_\sigma]]\Phi+[D_\rho,[D_\sigma,D_\nu]]\Phi+
[D_\sigma,[D_\nu,D_\rho]]\Phi \label{gt15} \\
\Leftrightarrow 0 &=& \eps^{\mu\nu\rho\sigma}[D_\nu,G_{\rho\sigma}](\Phi)
~~~~~~~(\forall\, \Phi) \\
\Leftrightarrow 0 &=& {\cal D}_\nu
(\tilde G^{\mu\nu})~~~~~(Bianchi) \label{gt16}
\ena
For $SU(N)$ gauge groups, it may be useful to use the relation:
\be
\Phi\Phi^\dagger = \Phi^\dagger\Phi\frac{1}{N} I + A_\phi
\label{alg-decomp}
\ee
where $I$ stands for the identity matrix in $N$ dimensions,
$A_\Phi$ lies therefore in the Lie algebra
$su(N)$, and we will conveniently denote by
$\rho_\Phi= \Phi^\dagger\Phi$ the probability density
of $\Phi$. \\
\subsection{Non abelian case and the Path Ordered Exponential}
If $A$ is an operator valued function of the real variable
$\lambda$, a solution to the differential equation
$f'(\lambda) = A(\lambda)f(\lambda)$ is given by (see~\cite{karp}):
\bea
f(x) &=& \left[1+\int_0^x d\lambda\, A(\lambda)
+\int_0^x d\lambda_1\, A(\lambda_1)
\int_0^{\lambda_1} d\lambda_2\, A(\lambda_2)
+\ldots \right. \nonumber \\
&&\left. + \int_0^x d\lambda_1\, A(\lambda_1)
\cdot\ldots\cdot
\int_0^{\lambda_{n-1}} d\lambda_n\, A(\lambda_n) \right]f(0) \\
&=& \expf^{\int_0^x d\lambda\, A(\lambda)}f(0) =
\lim_{ds\rightarrow 0}\prod_{k=n}^1 e^{A(s_k)ds}f(0)\quad
\quad with \quad s_k=\frac{k\times x}{n}\label{eproduct} \\
\expf^{\int_0^x d\lambda\, A(\lambda)} &=&
\expf^{\int_y^x d\lambda\, A(\lambda)}
\expf^{\int_0^y d\lambda\, A(\lambda)} \\
\frac{d}{ds} \expf^{\int_0^s F(v)dv} &=&
F(s) \expf^{\int_0^s F(v)dv} \quad \quad
\frac{d}{ds} \expf^{\int_s^1 F(v)dv} =
-\expf^{\int_s^1 F(v)dv}F(s) \label{expfprop01}
\ena
Note that the product in Eq.~\ref{eproduct}
is done ``from right to left''. In the following, we list
a few properties of the path order exponential:
\bea
\left(\expf^{\int A}\right)^{-1} = \expb^{-\int A}
&=& 1-\int_0^1 A(u)du +\int_0^1 du_1\,\int_0^{u_1} du_2 A(u_2)A(u_1)
+\ldots \\
\expf^{\int_a^x A'(s)A^{-1}(s)ds} &=& A(x)A^{-1}(a) \label{o-exp-t01a} \\
P(x) = \expf^{\int_a^x A(s)ds}\Rightarrow
\expf^{\int_a^x A(s)+B(s)ds} &=&
P(x)\expf^{\int_a^x P^{-1}(s)B(s)P(s)ds} \label{o-exp-t02a} \\
A(x) \expf^{\int_a^x B(s)ds}A^{-1}(a) &=&
\expf^{\int_a^x \left(A'(s)A^{-1}(s)+A(s)B(s)A^{-1}(s)\right)ds}
\label{o-exp-t03a} \\
\frac{\partial}{\partial\lambda}\expf^{\int_a^b A(u,\lambda)du}
&=& \int_a^b ds\, \expf^{\int_s^b A(u,\lambda)du}\,
\frac{\partial A(s,\lambda)}{\partial\lambda}
\,\expf^{\int_a^s A(u,\lambda)du}\label{o-exp-t04a}
\ena
The last formula can be demonstrated easily if one uses
the product form of the ordered exponential (Eq.~\ref{eproduct})
\subsubsection{Introduction of a space-time contraction}
If we now consider a contraction $X_\mu(u,x)$ where
$X_\mu(0,x)=x_0$ and $X_\mu(1,x)=x$ (see Eq.~\ref{contraction01a}),
we obtain the following definition:
\bea
F(u,x) &=& \frac{\partial X_\mu}{\partial u}(u,x)
A^\mu(X_\mu(u,x)) \label{Fdef} \\
\partial_\mu X_\alpha A^\alpha(X_\mu(u,x))|_{u=1}&=& A_\mu(x)
\label{Fdefb} \\
f(x)= \expf^{ig\int_\gamma dl^\mu A_\mu}(x) &=&
1+(ig)\int_0^1 du\, F(u,x)
+(ig)^2\int_0^1 du_1\, F(u_1,x)
\int_0^{u_1} du_2 F(u_2,x)
+\ldots \nonumber \\
&& + (ig)^n\int_0^1 du_1\, F(u_1,x)
\cdot\ldots\cdot
\int_0^{u_{n-1}} du_n\, F(u_n,x)
+\ldots \label{ordered-exp02a} \\
&=& \sum_k (ig)^k\int_{[0;1]^k} du_1...du_k \theta(u_1,\ldots,u_k)
F(u_1,x)\ldots F(u_k,x) \label{ordered-exp02b} \\
&=& \exp\left( ig\int_0^1du\, F(u,x)\right)~~~~(if~[A(x),A(x')]=0)
\label{ordered-exp02c} \\
\theta(u_1,\ldots,u_k) &=& 1~~~iff~~~~u_1\geq u_2 \ldots \geq u_k,
~~0~~if~~not \label{ordered-exp02d} \\
&=& H(u_1-u_2)H(u_2-u_3)...H(u_{k-1}-u_k) \label{ordered-exp02e}
\ena
Each term in the sum can be obtain by the following recursion:
\bea
J_0(a,b,x)&=&1 \label{recurr01a}\\
J_n(a,b,x)&=&\int_a^b ds\,
F(s,x)J_{n-1}(a,s) \label{recurr02a} \\
J_n(a,a,x)&=& 0 ~~~~\forall~n,x \label{recurr02b} \\
J_n(a,b,x) &=&\int_a^b ds\, \partial_s X_\mu
A^\mu(X_\mu(s,x)) J_{n-1}(a,s) \label{recurr02c} \\
\ena
Let $\Phi$ be a solution (if it exists) to the system of PDE
$\partial_\mu\Phi=-igW_\mu(x)\Phi$, then:
\bea
\frac{\partial X^\mu}{\partial u}\partial_\mu\Phi&=&-ig
\frac{\partial X^\mu}{\partial u}W_\mu(x)\Phi \\
\frac{d}{du} \Phi(X(u,x)) &=& F(u,x)\Phi(X(u,x)) \\
\Rightarrow \Phi(X(u,x)) &=& \expf^{\int_0^u dv\,F(v,x)}\Phi_0 \\
\Rightarrow \Phi(x) =\Phi(X(1,x)) &=&
\expf^{\int_0^1 dv\,F(v,x)}\Phi_0 \\
&=&
\expf^{-ig\int_0^1 du\,\frac{\partial X^\mu}
{\partial u}W_\mu(X(u,x))}\Phi_0 \label{covar-int}
\ena
If $\Phi$ is a square matrix and $\Phi_0=I$, then $\Phi$
is invertible because $\det(\Phi)=e^{-ig\int \tr F}\neq 0$, thus
$W_\mu = \frac{i}{g}\partial_\mu\Phi \Phi^{-1}$ which is
a right invariant form, the curvature of which vanishes. It is not
surprising to get such a constraint. Already in the
abelian case, if $\phi = e^{-ig\int A}$ then
$\partial_\mu \phi = -ig\left(A_\mu+\int \partial_u X^\alpha
\partial_\mu X^\beta F_{\alpha\beta}\right)\phi$ (see Eq.~\ref{de-rahm00})
and we explicitly show the presence of a curvature term
as an obstacle to solve the system of differential equations.
To obtain a similar formula in the non-abelian case, let us take
the partial derivatives of Eq.~\ref{covar-int}. We get:
\bea
\frac{i}{g}\partial_\mu \Phi &=& \int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\partial_\mu\left(\frac{\partial X^\nu}
{\partial s}W_\nu(X(s,x))\right) \expf^{-ig\int_0^s W_x} \\
&=& \int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\left(\partial_\mu\partial_s X^\nu W_\nu(X(s,x))
+\partial_s X^\nu \partial_\mu X^\rho
\partial_\rho W_\nu(X(s,x))
\right) \expf^{-ig\int_0^s W_x} \\
&=& \int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\{\partial_\mu\partial_s X^\nu W_\nu(X)
+\partial_u X^\nu \partial_\mu X^\rho G_{\rho\nu}(X) \nonumber \\
&& +\partial_s X^\nu \partial_\mu X^\rho
(\partial_\nu W_\rho(X)-ig[W_\rho,W_\nu])
\} \expf^{-ig\int_0^s W_x} \\
&=& \int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\partial_s \left(\partial_\mu X^\nu W_\nu(X)\right)
\expf^{-ig\int_0^s W_x} \nonumber \\
&&+\int_0^1 ds\, \expf^{-ig\int_s^1 W_x} \{
-ig\partial_s X^\nu \partial_\mu X^\rho[W_\rho,W_\nu]
+\partial_s X^\nu \partial_\mu X^\rho G_{\rho\nu}
\} \expf^{-ig\int_0^s W_x} \\
&=& \int_0^1 ds\,\partial_s\left( \expf^{-ig\int_s^1 W_x}
\partial_\mu X^\nu W_\nu(X)
\expf^{-ig\int_0^s W_x} \right) \nonumber \\
&& -\int_0^1 ds\, \expf^{-ig\int_s^1 W_x}(ig \partial_s
X^\nu W_\nu(X))\partial_\mu X^\rho W_\rho(X)
\expf^{-ig\int_0^s W_x} \label{residue01} \\
&& +\int_0^1 ds\, \expf^{-ig\int_s^1 W_x}\partial_\mu X^\rho W_\rho(X)
(ig\partial_s
X^\nu W_\nu(X))\expf^{-ig\int_0^s W_x}
\label{residue02} \\
&&+\int_0^1 ds\, \expf^{-ig\int_s^1 W_x} \{
-ig\partial_s X^\nu \partial_\mu X^\rho[W_\rho,W_\nu]
+\partial_s X^\nu \partial_\mu X^\rho G_{\rho\nu}
\} \expf^{-ig\int_0^s W_x} \\
&=& W_\mu(x)\expf^{-ig\int_0^1 W_x}- 0
+\int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\partial_s X^\nu \partial_\mu X^\rho G_{\rho\nu}
\expf^{-ig\int_0^s W_x} \\
&=& W_\mu(x)\expf^{-ig\int_0^1 W_x}
-\int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\partial_s X^\nu \partial_\mu X^\rho G_{\nu\rho}
\expf^{-ig\int_0^s W_x} \label{cohom-rel01}
\ena
where Eq.~\ref{residue01} and Eq.~\ref{residue02} make use of
Eq.~\ref{expfprop01}. The result
of Eq.~\ref{cohom-rel01} is nothing but the non-abelian equivalent
of Eq.~\ref{de-rahm00}, and it can be interesting to
rewrite it as follows:
\bea
W_\mu(x) &=& \int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\partial_s X^\nu \partial_\mu X^\rho G_{\nu\rho}
\expf^{-ig\int_0^s W_x}
\left(\expf^{-ig\int_0^1 W_x}\right)^{-1}
+\frac{i}{g}\partial_\mu \Phi \Phi^{-1}
\ena
This expression gives $W_\mu(x)$ as a gauge equivalent
of (see Eq.~\ref{gt02}):
\bea
{W'}_\mu &=& \left(
\expf^{-ig\int_0^1 W_x}\right)^{-1}
\int_0^1 ds\, \expf^{-ig\int_s^1 W_x}
\partial_s X^\nu \partial_\mu X^\rho G_{\nu\rho}
\expf^{-ig\int_0^s W_x} \nonumber \\
&=& \int_0^1 ds\, \left(\expf^{-ig\int_0^s W_x}\right)^{-1}
\partial_s X^\nu \partial_\mu X^\rho G_{\nu\rho}
\left( \expf^{-ig\int_0^s W_x}\right)
\ena
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\end{document}