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\begin{document}
\title{\textbf{THE GEOMETRICAL BASIS OF THE NON-LINEAR GAUGE}\thanks{to be submitted for publication}}
\author{Jose A. Magpantay \thanks{email: jamag@nip.upd.edu.ph}\\National Institute of Physics\\University of the Philippines,\\Diliman Quezon City, 1101\\ Philippines}
\maketitle
\begin{abstract}
We consider Yang-Mills theory in Euclidean space-time $(R^4)$ and construct its configuration space.  The orbits are first shown to form a congruence set.  Then we show that in spite of the fact that there can be no orthogonal gauge in the non-Abelian theory, the submanifolds defined by the non-linear gauge previously proposed by the author are shown to foliate the configuration space.  This was accomplished by making use of the form version of Frobenius theorem.
\end{abstract}
\clearpage
\section{Introduction}

Today, a mathematical treatment of Yang-Mills theory generally makes use of fiber bundles and topology\cite{top}.  But in spite of the use of such powerful mathematics, we are nowhere near the solution to the problem of confinement.  In fact, however, this statement is not exactly correct.  Physicists do catch a glimpse of confinement by making use of particular gauge conditions.  Some examples are the Abelian\cite{gauge}, center\cite{cond} and non-linear gauges\cite{linear}.  These gauge conditions focus on specific configurations -monople for Abelian, vortex for center and spherically synemetric scalars $(f^a=\partial \cdot{A}^a)$ for the non-linear gauge, which may be responsible for confinement.

Naively, this result is paradoxical because confinement seems to be dependent on the choice of gauge.  Gauge theorists have always assumed that physical phenomena are gauge-independent.  But is this really true?  In electrodynamics and perturbative non-Abelian theory, the equivalence of quantization in various linear gauges can be shown using formal operations on the path-integral.  Alternatively, in a particular gauge, gauge-invariance is guaranteed by the Ward-Takahashi identity for Abelian theory and Lee-Slavnov identities for non-Abelian theory.

However, it is also true that physical states of the gauge fields are more transparent in certain gauges.  For example, in Abelian theory and in the short-distance regime of the non-Abelian theory, the transverse photon and gluons satisfy the Coulomb gauge.  This shows that an appropriate choice of gauge can expose the physical degrees of freedom.  Thus, if confinement is due to a specific gauge field or a class of gauge fields, then choosing a gauge, which highlights the field configuration(s) is absolutely necessary.

The gauge-independence of physical results must only be true then for gauge-fixing conditions that intersect all the orbits.  This will guarantee that all field configurations are represented in the path-integral.  Thus, if certain physical phenomena are transparent in one gauge, the same physical phenomena must also be accounted for, although may not be as transparent, in another gauge as long as the two gauge conditions intersect all the orbits.

In this paper, we will discuss the problem of gauge-fixing by analyzing the configuration space of Yang-Mills theory.  We will be employing concepts used in finite dimensional Euclidean space and extend them in the infinite dimensional configuration space.  To visualize the concepts used, we will naively count the dimension and the number of elements in the gauge parameter and configuration spaces of both the Abelian and non-Abelian theories.  We then show that a global orthogonal gauge can be defined for the Abelian theory but not for the non-Abelian case.  Next we present arguments why non-linear gauge-fixing is natural for the Yang-Mills theory.  Finally, using the form version of Frobenius theorem, we show that the particular non-linear gauge proposed by the author can define submanifolds that foliate the configuration space.  This establishes the geometric basis of the non-linear gauge.\\
  
\section{The Geometry of Configuration Space}

Consider Yang-Mills theory in 4D Euclidean space-time.  The configuration space is an infinite dimensional space where the (Cartesian) axes are $A^a_\mu(x)$, i.e., the components of the gauge field at each point defined by $x_\alpha, \alpha=1,2,3,4$.  Naively, the ``infinite'' dimension of the configuration space is ${\aleph}=3\times 4\times(2\infty)^4$, where 3 comes from the SU(2) index a, 4 from the Lorentz index $\mu$, and $(2\infty)^4$ from the 4D Euclidean space-time coordinates (the number of points in a line that extends to $\infty$ in both directions is $2\infty)$.

In configuration space a gauge field function $A^a_\mu(x)=a^a_\mu(x)$ is just a point.  We can also treat this as a ``vector'', which is pictorially represented by connecting  $A^a_\mu(x)=0$ (the origin) to $a^a_\mu(x)$ by a line directed from the origin to $a^a_\mu(x)$.  This ``vector'' can also be represented by a $(\mathcal{N}\times1)$ column vector $\mathcal{A}$ the components of which are the values of $a^a_\mu(x)$ (all real) for each $a$, $\mu,$ and x.  The configuration space is flat as reflected by the norm
\begin{equation}\label{1}
\Vert\mathcal{A}\Vert^2=\int d^4x  a^a_\mu(x) a^a_\mu(x).
\end{equation}
This means that the ``metric'' in configuration space is $\delta^{ab}\delta_{\mu\nu}\delta^4(x-x')$.\\

The gauge transformation, which leaves the Yang-Mills action
\begin{equation}\label{2}
S=\frac{1}{4}\int{d^4}xF^a_{\mu\nu}(x)F^a_{\mu\nu}(x),
\end{equation}
invariant is
\begin{eqnarray}
A^{'}_{\mu}&=&\Omega A_{\mu}\Omega^{-1}-i(\partial_{\mu}\Omega)\Omega^{-1}\label{3}\\
&=&\Omega[A_{\mu}+i(\partial_{\mu}\Omega^{-1})\Omega]\Omega^{-1}\label{4}
\end{eqnarray}
where
\begin{eqnarray}
\Omega&=&exp[{i\wedge}]\label{5}\\
\wedge &=&\vec{\wedge}\cdot{T}=\wedge^{a}(x)T^{a},\label{6}
\end{eqnarray}
is an element of SU(2).  Using
\begin{eqnarray}
\Omega A_{\mu}\Omega^{-1}&=& A_{\mu}+i[\wedge,A_{\mu}]-\frac{1}{2}[\wedge,[\wedge,A_{\mu}]]+\cdots,\label{7}\\
-i(\partial_{\mu}\Omega)\Omega^{-1}&=&\partial_{\mu}\wedge+\frac{1}{2}[\wedge,\partial_{\mu}\wedge]-\frac{1}{6}[\wedge,[\wedge,\partial_{\mu}\wedge]]+\cdots,
\label{8}
\end{eqnarray}
the gauge transformation can be written in configuration space as
\begin{eqnarray}
\mathcal{A}&=&R_{\Omega}\mathcal{A}+T_{\Omega},\label{9}\\
&=& R_{\Omega}(\mathcal{A}-T_{\Omega^{-1}}).\label{10}
\end{eqnarray}
$R_{\Omega}$ is $(\mathcal{N}\times\mathcal{N})$ and its action on $\mathcal{A}$ is given by
\begin{eqnarray}
[R_{\Omega}\mathcal{A}]^a_{\mu}(x)=2\int{d^{4}}x'\delta_{\nu\mu}\delta^{4}(x-
x')tr\{T^{a}\langle1+i[\wedge, \quad]-\frac{1}{2}[\wedge,[\wedge,    ]]\nonumber\\
-\frac{1}{6}[\wedge,[\wedge,[\wedge,   ]]]+\cdots\rangle{A}_{\nu}(x')\}.\label{11}
\end{eqnarray}
$T_{\Omega}$, on the other hand is $(\mathcal{N}\times 1)$ and its components are read from Equation (\ref{8}).

Since $\parallel\Omega \mathcal{A}\Omega^{-1}\parallel=\parallel \mathcal{A}\parallel$, then
\begin{eqnarray}
R^+_{\Omega} R_{\Omega}&=&\mathbf{1}	\quad(\mathcal{N}\times{\mathcal{N}}\quad identity)\nonumber\\
&=& \mathbf{1}_{SU(2)}\otimes\mathbf{1}_{Lorentz}\otimes\delta^{q}_{4}(x-x')\label{12}.
\end{eqnarray}
We will take $det\ R_{\Omega}=1$. Equations(\ref{9}), (\ref{10}) and (\ref{12}) establish that gauge transformation is a combination of translation and rotation in configuration space.  This makes the configuration space an affine space.

Let us now focus on pure gauge fields
\begin{equation}\label{13}
a_{\mu}=-i(\partial_{\mu}\Omega)\Omega^{-1}.
\end{equation}
with $\Omega  \epsilon {SU(2)}$.  Note that this field configuration has zero field strength; thus it contains the trivial vacuum.  Since $\Omega$ has three parameters $\Lambda^{a}(x)$ at each point in $R^4$, the infinite dimension $\mathcal{D}$ of the space of functions of $\Omega$, which we will designate as $\mathcal{G}$, is naively equal to $3\times(2\infty)^4$. Each point in this space is mapped to a point in configuration space by equation (\ref{13}).  Equivalently, as suggested by equations (\ref{9}) and (\ref{10}), each point in the function space of the gauge parameters is mapped to $T_\Omega$, which belongs to a class of translation group which have vanishing field strengths.  Let us call this particular class of the translation group $\tau_o$.

We will now show that $\tau_o$ forms an orbit that passes through the origin of the configuration space.  To remove the degeneracy of the origin $\mathcal{A}=0$, we will remove the class of constant gauge transformations except $\Omega=\mathbf{1}$ from $\mathcal{G}$.  We need to show that a gauge transform of a pure gauge field is also a pure gauge field.  Let $a_{\mu}=-i(\partial_{\mu}\Omega)\Omega^{-1}$ and consider its gauge transform under $\tilde{\Omega}$, i.e.,
\begin{eqnarray}
a'_{\mu}&=&\tilde{\Omega}i(\partial_{\mu}\Omega)\tilde{\Omega}^{-1}-i(\partial_{\mu}\tilde{\Omega})\tilde{\Omega}^{-1}\nonumber\\
&=& -i[\partial_{\mu}(\tilde{\Omega}\Omega](\tilde{\Omega}\Omega)^{-1}.\label{14}
\end{eqnarray}
This shows that $a'_{\mu}$ is a pure gauge with gauge element $\Omega'=\tilde{\Omega}\Omega$.  In configuration space, equation (\ref{14}) becomes
\begin{equation}\label{15}
T'_{\Omega}=R_{\tilde{\Omega}}T_{\Omega}+T_{\tilde{\Omega}}=T_{\Omega'}
\end{equation}
Equations (\ref{14}) and (\ref{15}) show that we can generate $\tau_o$, the orbit of the pure gauge configurations which has zero field strength, from the origin $\mathcal{A}=0.$

Let the orbit passing through a point $\mathcal{A}$ in configuration space be $\tau_{\mathcal{A}}$.  This can be generated in two ways.  The first is by rotating the vector $\mathcal{A}$ using $R_{\Omega}$ and then translating with $T_{\Omega}$ as presented in equation (\ref{9}).  This should be done on $\mathcal{A}$ using all the operations $\{R_{\Omega},T_{\Omega}\}$.  The second is by first translating $\mathcal{A}$ by $(-)T_{\Omega^{-1}}$ and then rotating by $R_{\Omega}$.  This is prescribed in equation (\ref{10}) and should also be done using all the operations $\{R_{\Omega},T_{\Omega^{-1}}\}$.

The orbits $\tau_{\mathcal{A}}$ and $\tau_o$, no matther how they twist and turn in configuration space, always maintain the distance $\parallel{A}\parallel$ between their corresponding points as shown in Figure \ref{fig1}.
\begin{figure}
\centering
\includegraphics[totalheight=2.5in]{Fig1Geom.EPS}
\caption{The orbits $\tau_{o}$ and $\tau_{A}$ showing three sets of corresponding points connected by steps of a ladder.  The distances between each set of points, which is equal to the length of the ladder steps, are all equal to $\Vert A-0\Vert$.}
\label{fig1}
\end{figure}
This ladder-like structure follows from applying equation (\ref{9}) on both $\mathcal{A}$ and the origin and then subtracting the gauge transformed results.  This will yield $R_{\Omega}(\mathcal{A}-0)$, which has the same norm as $\mathcal{A}$.  Doing this for all $\{R_{\Omega},T_{\Omega}\}$ we generate the ladder-like structure of the two orbits.  If this is true for $\tau_{\mathcal{A}}$ and $\tau_o$, it is also true for $\tau_{\tilde\mathcal{A}}$ and $\tau_o$ and also for $\tau_{\mathcal{A}}$ and $\tau_{\tilde{\mathcal{A}}}$.  The orbits, though twisting and turning in a complicated way, all ``flow'' in the same ``direction'' maintaining the same distance between corresponding points on the two orbits.  The orbits therefore form a congruence set, i.e., they cover the configuration space without intersecting.  This simple observation is significant for the following reasons.  If the gauge-fixing submanifold does not intersect $\tau_o$ uniquely, it will not intersect neighboring orbits uniquely also (see Figure \ref{fig2}).  If we know how $\tau_o$ twists and turns, then we know how all the other orbits twist and turn also.  This will suggest how to choose the gauge-fixing submanifold, if not throughout the entire configuration space, at least in the vicinity of physically interesting field configurations.
\begin{figure}
\centering
\includegraphics[totalheight=2.5 in]{Fig3Geom.EPS}
\caption{The gauge-fixing surface $\mathcal{F}_{o\cdots o}$, defined by the non-linear gauge and the orbit $\tau_{A}$ make an angle $\theta$, which is never equal to $\frac{\pi}{2}$. $\theta$ is the angle between the normal to $\mathcal{F}_{o\cdots o}$ (given by $\frac{\delta\mathcal{F}_{o\cdots o}}{\delta A}$) and the tangent to the orbit (given by $\mathcal{T}$).}
\label{fig2}
\end{figure}

Finally, we note why the path-integral is invariant under gauge transformation.  The path-integral measure
\begin{equation}\label{16}
[dA^{a}_{\mu}(x)]={\prod_{a,\mu,x_{\alpha}}dA^{a}_{\mu}(x_{\alpha})={\prod_{I=1}^{\mathcal{N}}dA_{I}}}
\end{equation}
where the last term is the ``infinitesimal volume'' in configuration space.  This measure is invariant under the affine transformation defined by equation (\ref{9}).

The gaue-invariant action can be written as
\begin{eqnarray}
S=-\frac{1}{2}\int d^{4}x d^{4}x'A^{a}_{\mu}(x')\{\delta^{4}(x-x')\delta^{ab}[\Box^{2}\delta_{\nu\mu}-\partial_{\mu}\partial_{\nu}]\}A^{b}_{\nu}(x)\nonumber\\
+\int d^{4} x d^{4}x'd^{4}x''A^{b}_{\mu}(x')A^{c}_{\nu}(x'')\{\delta^{4}(x-x')\delta^{4}(x'-x'')\epsilon^{bca}\delta_{v\alpha}\partial_{\mu}\}A^{a}_{\alpha}(x)\nonumber\\
+\frac{1}{2}\int d^{4}xd^{4}x'd^{4}x''d^{4}x'''\{[\delta^{ac}\delta^{bd}-\delta^{ad}\delta^{bc}]\delta^{4}(x-x')\delta^{4}(x'-x'')\nonumber\\
\times\delta^{4}(x''-x''')\delta_{\mu\alpha}\delta_{\nu\beta}\}A^{a}_{\mu}(x)A^{b}_{\nu}(x')A^{c}_{\alpha}(x'')A^{d}_{\beta}(x''')\label{17}
\end{eqnarray}

The action forms a quartic hyperplane (we use the convention where hyperplane is a submanifold of dimension one less than the manifold) in configuration space of the general form
\begin{equation}\label{18}
S=\alpha_{IJ}\mathcal{A}_{I}\mathcal{A}_{J}+\beta_{IJK}\mathcal{A}_{I}\mathcal{A}_{J}\mathcal{A}_{K}+\gamma_{IJKL}\mathcal{A}_{I}\mathcal{A}_{J}\mathcal{A}_{K}\mathcal{A}_{L}.
\end{equation}
Although it is not apparent, this form of the action is invariant under the combined operations of rotation and translation as given by equation (\ref{9}).  Equations (\ref{16}) and (\ref{17}) establish the invariance of the path-integral under gauge-transformation.

The normal vector to the hyperplane S=constant at $\mathcal{A}$ is $\mathcal{P}$ with components
\begin{equation}\label{19}
P^{a}_{\mu}(x)=\frac{\delta{S}}{\delta{A^{a}_{\mu}(x)}}=D^{ab}_{\nu}F^{b}_{\nu\mu}(x).
\end{equation}
The tangent to the orbit at $\mathcal{A}$ is $\mathcal{T}$ with components
\begin{equation}\label{20}
T^{a}_{\mu}(x)=D^{ab}_{\mu}\Lambda^{b}.
\end{equation}
The expected orthogonality of $\mathcal{P}$ and $\mathcal{T}$ follow from
\begin{eqnarray}
\mathcal{P}\cdot\mathcal{T}&=&\int{d^{4}}x(D^{ab}_{\nu}F^{b}_{\nu\mu})(D^{ac}_{\mu}\Lambda^{c})\nonumber\\
&=&-\int{d^{4}}x(D^{ca}_{\mu}D^{ab}_{\nu}F^{b}_{\nu\mu})\Lambda^{c}\nonumber\\
&=&0.\label{21}  
\end{eqnarray}

\section{The Orthogonal Gauge Condition}

Gauge fixing is the process of choosing representative field configurations from each orbit.  Ideally, the gauge-fixing should choose only one representative from each orbit and all orbits should be represented.  This condition is equivalent to saying that the gauge-fixing condition is unique and always realizable.

Generally, gauge-fixing is done by imposing a local condition on the potentials, i.e., requiring
\begin{equation}\label{22}
F^{a}[A_{\mu}(x)]=0,\quad{a}=1,2,3.
\end{equation}
Uniqueness requires that if $A_{\mu}(x)$ satisfies equation (\ref{22}), then $F^{a}[A^{\Omega}_{\mu}(x)]\neq{0}$, i.e., all the gauge transformed fields of $A_{\mu}(x)$ must not satisfy the condition.  Realizability requires that for all $A_{\mu}(x)$  which does not satisfy equation (\ref{22}), there must be an $\Omega  \epsilon {SU(2)}$ such that $F^{a}[A^{\Omega}]=0$.

The ideal condition is satisfied in only one case, the Coulomb gauge fixing of an Abelian theory.  In all the other linear gauge-fixing of Abelian and non-Abelian theories, realizability is generally taken for granted while non-uniqueness is rectified through subsidiary conditions (Gupta-Bleuler condition in Lorentz gauge) or Fadeev-Popov determinants.

In configuration space, gauge-fixing is tantamount to choosing a submanifold where all orbits must pass through.  Since one of the four $A^{a}_{\mu}(x)$ for each a and at each x essentially becomes a dependent variable when $F^{a}[A]=0$ is imposed, the submanifold $\mathcal{F}$ defined by the gauge-fixing is naively $3\times3\times(2\infty)^{4}$ dimensional.  The issue now is what geometrical principle should be used in choosing $\mathcal{F}$.  The simplest and most compelling principle is to require global orthogonality of the orbit to $\mathcal{F}$.  This will guarantee uniqueness and realizability.  As stated already this is only achieved in the Coulomb gauge formulation of an Abelian theory.  This will be shown below.

In 4D euclidean space, the Coulomb gauge is given by the local condition
\begin{equation}\label{23}
F[A]=\partial_{\mu}A_{\mu}=0.
\end{equation}
For an Abelian theory, uniqueness and realizability of this condition follow from the positive definitness of the Laplacian operator.  The configuration space, naively, has dimension $\mathcal{N}=4\times(2\infty)^{4}$.  From equation (\ref{23}), there are only three independent potentials thus the submanifold of transverse potentials $\mathcal{F}$ has dimension $\mathcal{M}=3\times(2\infty)^{4}$.  This means that $\mathcal{F}$ is defined by
\begin{equation}\label{24}
\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}=F^{\Lambda_1}\cap{F^{\Lambda_2}}\cap\cdots,F^{\Lambda_\mathcal{H}},
\end{equation}
i.e., it is the intersection of hyperplanes (dimension equal to $4\times(2\infty)^{4}-1$) defined by each $F^{\Lambda_{I}}=c_{I}$, the total of which is $\mathcal{H}=(2\infty)^{4}$.  Each $F^{\Lambda}$ is given by a specific choice of $\Lambda$ in the gauge parameter space.  We can easily choose $(2\infty)^{4}$ functions which makes the integrand in $F^{\Lambda}$ positive-definite because there is a total of $(2\infty)^{(2\infty)^{4}}$ functions in configuration space.  To understand this count, note that the discretized $R^{3}$ has total number of points equal to $(2\infty)^{3}$.  Thus, the infinite dimensional $U(1)$ parameter space with dimension $(2\infty)^{4}$ will have $(2\infty)^{(2\infty)^{4}}$ points with each point representing a particular function $\Lambda(x)$.  The choice of $\Lambda(x)'s$ that will give positive definite integrands in $F^{\Lambda}$ is important so that when we choose $F^{\Lambda}=c$, the hyperplane translates into a local condition on the potential $A_{\mu}(x)$.  

Let us now determine $F^{\Lambda}$ by imposing that the normal to $F^{\Lambda}$ at $\mathcal{A}$ is equal to the tangent to the orbit.  This is equivalent to the orbit being orthogonal to the $F^{\Lambda}$=const. hyperplane.  In component form, this condition means
\begin{equation}\label{25}
\frac{\delta F^{\Lambda}}{\delta A_{\mu}(x)}=\partial_{\mu}\Lambda(x).
\end{equation}

The solution is
\begin{eqnarray}
F^{\Lambda}&=&\int{d^{4}}x(\partial_{\mu}\Lambda(x))A_{\mu}(x)\nonumber\\
&=&(-)\int{d^{4}}x\Lambda(x)(\partial_{\mu}A_{\mu}(x))\label{26}
\end{eqnarray}	
Imposing all $c_{I}$'s equal to zero gives the local condition $\partial_{\mu}A_{\mu}=0$, i.e., the Coulomb gauge, while $c_I\neq 0$ corresponds to $\partial_{\mu}A_{\mu}=f(x)$.

To prove that the submanifold defined by equations (\ref{26}) and (\ref{24}) defines a global orthogonal gauge, we make use of the Froenius theorem.  In the following, we will use the form version.  Define the set of one-forms $((2\infty)^{4}$ in total) in configuration space, which live on the cotangent space, by\\
\begin{equation}\label{27}
w^{\Lambda}=\int{d^{4}}x(\partial_{\mu}\Lambda)dA_{\mu}(x).
\end{equation}
Since $w^{\Lambda}=dF^{\Lambda}$, the set of one forms is a closed set.  The ``new coordinates'' $F^{\Lambda}$ defined by equation (\ref{26}) form a surface $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ given by equation (\ref{24}) when each $F^{\Lambda_{I}} = c_{I}$, for $I = 1,\cdots\mathcal{H}$.

From this construction, it follows that
\begin{equation}\label{28}
w^{\Lambda}\vert\mathcal{F}_{_{c_{I}\cdots c_{\mathcal{H}}}} = 0,
\end{equation}
i.e., on the submanifold $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$, the tangent vectors annul the one forms.  In particular, on the Coulomb gauge submanifold given by $\mathcal{F}_{o \cdots o}$, this follows from

\begin{eqnarray}
w^{\Lambda}\vert\mathcal{F}_{_{o \cdots o}}&=&\int d^{4}x \partial_{\mu}\Lambda dA_{\mu}(x)\vert\mathcal{F}_{_{o \cdots o}}\nonumber\\
&=&(-)\int{d^{4}}x\Lambda(x)\partial_{\mu}dA_{\mu}(x)\vert\mathcal{F}_{_{o\cdots o}}\label{29}
\end{eqnarray}
Since $A_{\mu}(x)$ is transverse on $\mathcal{F}_{o\cdots o}$, we can write
\begin{equation}\label{30}
A_{\mu}(x)\vert\mathcal{F}_{_{o\cdots o}} = (\delta_{\mu\nu}-\partial_{\mu}\frac{1}{\Box^{2}}\partial_{\nu})A_{\nu}
\end{equation}
giving
\begin{equation}\label{31}
dA_{\mu}(x)\vert\mathcal{F}_{_{o\cdots o}} = (\delta_{\mu\nu}-\partial_{\mu}\frac{1}{\Box^{2}}\partial_{\nu})A_{\nu}.
\end{equation}
All these prove that the submanifolds $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ where the orbits are orthogonal, are leaves in the foliation of the entire configuration space.\\

In the non-Abelian case, the tangent to the orbit is the vector $\mathcal{T}$ defined by equation(\ref{20}).  Imposing that this is equal to the normal to the surface defined by $F^{\vec{\Lambda}}$ = constant implies
\begin{equation}\label{32}
\frac{\delta F^{\vec\Lambda}}{\delta A^{a}_{\mu}(x)}=D^{ab}_{\mu}\Lambda^{b}=\partial_{\mu}\Lambda^{a}-\epsilon^{abc}A^{c}_{\mu}\Lambda^{b}.
\end{equation}
Because of the second term, there is no solution to equation (\ref{32}).  Hence, the conclusion that there exists no orthogonal gauge condition, local or global, in the non-Abelian case.  This result had been established by various authors, including Chodos and Moncrief\cite{chodos},
who used the vector version of Frobenius theorem.\\

\section{Geometry of the Non-Linear Gauge}

We will now discuss the non-linear gauge, which was discussed by the author in a series of articles.  Initially, the author's justification for the gauge condition is the fact that there are field configurations missed by the Coulomb gauge\cite{group}. These are the field configurations that are on the Gribov horizon of the $\partial\cdot{A}={\mathit{f}}\neq 0$ surface.  In subsequent papers, the author showed that the gauge condition ``reveals'' the physical degrees of  freedom, which depend on the distance scale, of the non-Abelian theory.  At short distance, i.e., well inside hadrons, transverse gluons exist and interact very weakly with quarks.  This is accounted for by the linear limit of the non-linear gauge.  At large distance scales, the important field configurations are the new scalar fields $\mathit{f}^{a}=\partial\cdot{A}^{a}$, which has an infinitely non-linear effective action.  The classical, stochastic dynamics of spherically symmetric $\mathit{f}^{a}$ leads to the linear potential\cite{linear} while the full quantum dynamics leads to dimensional reduction\cite{hadron}.

What we would like to raise at this point is the question, Is there a geometrical basis for the non-linear gauge?

Before we answer this question, we note that since the orbit through $\mathcal{A}$ twists and turns in configuration space (see discussions in Section II), it is most unlikely that there exists a linear submanifold that intersects all orbits uniquely.  As shown in reference (6), it may also happen that a linear submanifold may not intersect some orbits at all.  For this reason, there are those who proposed covering the submanifold by local patches centered around background gauge fields\cite{chaos}. This gauge fixing is essentially a collection of linear gauges.  However, beyond formal expressions for the path-integral and global expectation values for gauge-invariant quantities, this formalism has not really shown confinement.  Also, a collection of linear gauges actually suggests non-linearity of the entire submanifold.

The non-linear gauge is also hinted by equation (\ref{32}), which states that the orbit is orthogonal to the gauge-fixing surface.  Since the RHS of equation (\ref{32}) is linear in A, the hyperplane $F^{\vec{\Lambda}}$ = constant must be quadratic in A.  Unfortunately, the anti-symmetric $\epsilon^{abc}$ precludes the existence of a solution.

But suppose we modify equation (\ref{32}) to something like
\begin{equation}\label{33}
\frac{\delta F^{\vec\Lambda}}{\delta A^{a}_{\mu}(x)}=\int {d^{4}} x'h^{ab}_{\mu\nu}(x;x')(D^{bc}_{\nu}A^{c})_{x'}
\end{equation}
Equation (\ref{33}) states that the normal to the hyperplane $F^{\vec{\Lambda}}$ = constant is a linear combination of the components of the tangent to the orbit at $\mathcal{A}$ (with components $(D^{ab}_{\mu}\Lambda^{b})_{x}$).  This means that the gauge-fixing submanifold $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ given by the intersections of the hyperplanes $F^{\vec{\Lambda_{I}}}=c_{I}$, i.e.
\begin{equation}\label{34}
\mathcal{F}_{c_{I}}\cdots c_{\mathcal{H}} =F^{\vec{\Lambda}_{1}}\cap F^{\vec{\Lambda}_{2}}\cap F^{\vec{\Lambda}_{3}}\cap\cdots,F^{\Lambda_{\mathcal{H}}}
\end{equation}
intersects the orbit but is not orthogonal to it.  The submanifold $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ is tilted slightly relative to the orbit, with the tilting determined by $h^{ab}_{\mu\nu}(x;x')$ given in equation (\ref{33}).

Before we solve equation (\ref{33}), we give a naive counting of dimensions. Each $F^{\vec{\Lambda}}$ is a hyperplane (dimension equal to $3\times 4\times(2\infty)^{4}-1)$ and we will need a total $\mathcal{H}=3\times(2\infty)^{4}$ specified by choosing an equal number of $\Lambda^{a}(x)$ from a total of $[(2\infty)]^{3\times(2\infty)^{4}}$ functions in the gauge parameter space.  This will make the gauge-fixing submanifold $\mathcal{F}_{c_{1}\cdots c_{\mathcal{H}}}[3\times3\times(2\infty)^{4}]$ dimensional.

Consider the following $h^{ab}_{\mu\nu}$ 
\begin{equation}\label{35}
h^{ab}_{\mu\nu}(x;x')=\delta^{4}(x-x')\delta^{ab}\partial'_{\mu}\partial'_{\nu}+\frac{1}{4}\partial'_{\mu}(\partial\cdot A^{b})_{x'}\frac{\delta}{\delta A^{a}_{\nu}(x)}.
\end{equation}
Substituting in (\ref{33}), we find
\begin{equation}\label{36}
\frac{\delta F^{\vec{\Lambda}}}{\delta A^{a}_{\mu}(x)}=\partial_{\mu}(\partial\cdot D)^{ab}\Lambda^{b}+\epsilon^{abc}\partial_{\mu}(\partial\cdot A^{c})\Lambda^{b}.
\end{equation}
From equation (\ref{36}), we find 
\begin{eqnarray}
F^{\vec\Lambda} &=& -\int d^{4}x\Lambda^{b}(x)[(D\cdot\partial)^{bc}(\partial\cdot A^{c})],\nonumber\\
&=&-\int d^{4}x\Lambda^{b}(x)[(\partial\cdot D)^{bc}(\partial\cdot A^{c})],\nonumber\\
&=&-\frac{1}{2}\int d^{4}x\Lambda^{b}(x)\{[(\partial\cdot D)^{bc}+(D\cdot\partial)^{bc}](\partial\cdot A^{c})\}.\label{37}
\end{eqnarray}
From equation (\ref{37}), we read that the submanifold $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ with all $c_{I} = 0$ defines the non-linear gauge condition
\begin{equation}\label{38}
(\partial\cdot D)^{ab}(\partial\cdot A^{b})=(D\cdot\partial)^{ab}(\partial\cdot A^{b})=\frac{1}{2}[(\partial\cdot D)^{ab}+(D\cdot\partial)^{ab}](\partial\cdot A^{b})= 0.
\end{equation}
And for arbitrary set of constants $c_{I}$, with $I=1,\cdots,\mathcal{H}$; the submanifold $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ is defined by the gauge condition
\begin{equation}\label{39}
(\partial\cdot D)^{ab}(\partial\cdot A^{b})=s^{a}(x),
\end{equation}
with
\begin{equation}\label{40}
F^{\vec{\Lambda}_{i}}=c_{I}=-\int d^{4}x\Lambda^{a}_{I}(x)s^{a}(x).
\end{equation}

Now let us consider the set of one forms
\begin{equation}\label{41}
w^{\vec{\Lambda}}=\int d^{4}x\{\partial_{\mu}(\partial\cdot D)^{ab}\Lambda^{b}+\epsilon^{abc}\partial_{\mu}(\partial\cdot A^{c})\Lambda^{b}\}dA^{a}_{\mu}(x).
\end{equation}
Just like in the Abelian case, since $w^{\vec{\Lambda}} = dF^{\vec{\Lambda}}$, the set of one forms is a closed set.  the ``new coordinates'' $F^{\vec{\Lambda}}$ form a surface $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$ as given in equation (\ref{34}).\\

From this construction it follows that
\begin{equation}\label{42}
w^{\vec{\Lambda}}\vert_{\mathcal{F}_{_{c_{I}}\cdots c_{\mathcal{H}}}}= 0.
\end{equation}
And in the particular case of $\mathcal{F}_{o\cdots o}$, i.e., the submanifold defined by the nonlinear regime of the non-linear gauge condition given by equation (\ref{38}), the result follows from the following arguments.  Starting from a field configuration $A^{a}_{\mu}$ that does not satisfy equation (\ref{38}), we can always gauge transform to one that satisfies the non-linear gauge.  This field configuration is given by
\begin{equation}\label{43}
A^{a}_{\mu}(x)\vert_{\mathcal{F}_{_{o\cdots o}}}=A^{a}_{\mu}(x)-D^{ab}_{\mu}(x)\int d^{4}x'H^{bc}(x,x';A)[(\partial\cdot D)^{cd}(\partial\cdot A^{d})]_{x'},
\end{equation}
where $H^{ab}(x;x'; A)$ is the Greens function of the non-singular operator
\begin{equation}\label{44}
\mathbf{\theta}^{ab}=(D\cdot\partial)^{ac}(\partial\cdot D)^{cb}-\epsilon^{acd}[\partial(\partial\cdot A^{c})]\cdot D^{db}.
\end{equation}
The non-singularness of $\mathbf{\theta}^{ab}$, even if $(\partial\cdot D)$ has a zero mode $(\partial\cdot A^{a}=\mathit{f}^{a}\neq 0)$ is verified in first-order perturbation theory.  Since this is crucial to what follows, we will outline the proof of this claim.

First, $\mathbf{\theta}^{ab}$ is hermitian on the submanifold defined by equation (\ref{36}).  Since the first terms of $\mathbf{\theta}$ is a fourth-order operator (dominant term), with zero mode $\partial\cdot A$, the zero mode of $\mathbf{\theta}$, if it exists, must be of the form
\begin{equation}\label{45}
{z}^{a}=\partial\cdot A^{a}+\lambda^{a}
\end{equation}
with $\lambda^{a}\ll\partial\cdot A^{a}$.  The correction $\lambda^{a}$ must be solved from
\begin{equation}\label{46}
(D\cdot\partial)(\partial\cdot D)\lambda = [\partial(\partial\cdot A)\cdot D](\partial\cdot A)
\end{equation}
The solution to equation (\ref{46}) only exists if the zero mode $\partial\cdot A$ is orthogonal to the source in the above equation, i.e.
\begin{equation}\label{47}
\int d^{4}x(\partial\cdot A^{a})\epsilon^{abc}[\partial(\partial\cdot A^{b})\cdot D^{cd}](\partial\cdot A^{d})= 0.
\end{equation}
But by integration by parts, it is easy to show that the above integral is
\begin{equation}\label{48}
\geq c^{2}\parallel\partial\cdot A^{a}\parallel^{2},
\end{equation}
where $c^{a}$ is the minimum value of $\partial\cdot A^{a}$.  Since equation (\ref{47}) can never be satisfied, $\lambda$ does not exist and $\mathbf{\theta}$ is non-singular.

Going back to equation (\ref{43}), we find that
\begin{eqnarray}
dA^{a}_{\mu}\vert_{\mathcal{F}_{o\cdots o}}&=& dA^{a}_{\mu}(x)-\epsilon^{abe} dA^{e}_{\mu}(x)\int d^{4}y H^{bc}(x;y;A)[(D\cdot\partial)^{cd}(\partial\cdot A^{d})]_{y}\nonumber\\
&-& D^{ab}_{\mu}(x)\int d^{4} y H^{bc}(x,y;A)[(D\cdot\partial)^{cd}\partial_{\alpha}d A^{d}_{\alpha}]_{y}\nonumber\\
&-&D^{ab}_{\mu}(x)\int d^{4}y H^{bc}(x,y;A)[(-)\epsilon^{cde} dA^{e}_{\alpha}\partial_{\alpha}(\partial\cdot A^{d})]_{y}\nonumber\\
&-& D^{ab}_{\mu}(x)\int d^{4} y d^{4} z\frac{\delta H^{bc}}{\delta A^{f}_{\alpha}(z)}(x,y;A)dA^{f}_{\alpha}(z)[(D\cdot\partial)^{cd}(\partial\cdot A^{d})]_{y}\label{49}
\end{eqnarray}
where $H^{ab}$ is the Greens function of $\mathbf\theta$.  The last term is evaluated by using
\begin{equation}\label{50}
\delta H = -\int H\delta\mathbf{\theta} H.
\end{equation}

Substituting equation (\ref{49}), in equations (\ref{41}), we verify equation (\ref{42}) after doing integration by parts.  Equations (\ref{37}), (\ref{41}), (\ref{42}), (\ref{43}) and (\ref{49}) are the analogue of equations (\ref{27}) to (\ref{31}) in the Abelian theory.

All these show that the configuration space can foliated by the submanifolds
$\mathcal{F}_{c_{1}\cdots c_{\mathcal{H}}}$, where each set of values of $c_{1},\cdots, c_{\mathcal{H}}$ is a leaf of the foliation.

Let us compute the angle between the tangent to the orbit at $A^{a}_{\mu}$, which is $\mathcal{T}$ with components $T^{a}_{\mu}=D^{ab}_{\mu}\Lambda^{b}$, and the normal to the $F^{\vec{\Lambda}}= c$ hyperplane (also orthogonal to $\mathcal{F}_{c_{1}\cdots c_{\mathcal{H}}}$ submanifold).  The angle $\theta$ (see Figure \ref{fig3}) is given by
\begin{equation}\label{51}
cos\theta=\frac{\int d^{4}x(D^{ab}_{\mu}\Lambda^{b})\frac{\delta F^{\vec\Lambda}}{\delta A^{a}_{\mu}(x)}}{\parallel T\parallel\parallel\frac{\delta F^{\vec{\Lambda}}}{\delta A}\parallel}
\end{equation}
Substituting equation (\ref{36}) and using integration by parts, we find
\begin{equation}\label{52}
cos\theta\approx\int d^{4}x\Lambda^{a}\mathbf{\theta}^{ab} \Lambda^{b}
\end{equation}

\begin{figure}
\centering
\includegraphics[totalheight=2.5in]{Fig2Geom.EPS}
\caption {The gauge-fixing surface $\mathcal{F}$, which is intersected by the orbits $\tau_{o}$ and $\tau_{A}$ at three points each.  Note the distance between corresponding points on the orbits are all the same}. 
\label{fig3}
\end{figure}

Since $\mathbf{\theta}^{ab}$ is a non-singular operator, it has no zero mode and $cos \theta$ is never zero.  Thus, on the non-linear gauge submanifold, although the orbit is not orthogonal, it is also never tangential to $\mathcal{F}_{o\cdots o}$.  The gauge transform of a field configuration on the non-linear gauge surface is always off the surface.  The non-linear gauge condition is a unique gauge-fixing.

Contrast this with the Coulomb gauge condition.  The angle between the tangent to the orbit and the normal to the surface (see equation (\ref{25})) is
\begin{equation}\label{53}
cos\theta_{c}=\int d^{4}x\Lambda^{a}(\partial\cdot D)^{ab} \Lambda^{b}
\end{equation}
For field configurations $A^{a}_{\mu}$ on the Coulomb surface that have gauge copies, the operator $(\partial\cdot D)$ has $L^{2}$ zero modes.  This yields $\theta_{c}=\frac{\pi}{2}$, i.e., the tangent to the orbit is tangential to the Coulomb surface.\\

\section{Conclusion}

In this paper, we have established the geometrical basis of the non-linear gauge condition.  We have shown that although the orbit is never orthogonal to $\mathcal{F}_{o\cdots o}$, it is also never tangential to the surface $(\theta_{c}\neq\frac{\pi}{2})$.  We have also shown using Frobenius theorem that the configuration space can be foliated by submanifolds $\mathcal{F}_{c_{I}\cdots c_{\mathcal{H}}}$.

\section{Acknowledgement}

This research was supported in part by the National Research Council of the Philippines and the Natural Sciences Research Institute of the University of the Philippines.  The research was started when the author visited the University of Mainz through the financial support of the Alexander von Humboldt Stiftung.  Discussions with Martin Reuter are gratefully acknowledged.	

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\bibitem{chodos} Alan Chodos and Vincent Moncrief, {Journal of Mathematical Physics} \textbf{21} (1980) 364.
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\bibitem{chaos}See for example, H. Huffel and G. Kelnhofer, {Annals of Physics} \textbf {270} (1998) 231.
\end{thebibliography}
\end{document}
 

