\section{Introduction}
Monopoles in Yang-Mills-Higgs theories are known to posses a spectrum of excitations called dyons. In the
simplest of these theories the gauge group is ${\cal G}=SU(2)$ spontaneously broken to ${\cal H}=U(1)$ by a non-zero vacuum
expectation value of the Higgs field. Gauss' Law implies that the system is invariant under small gauge
transformations. Large gauge transformations with respect to the unbroken gauge group $U(1)$ induce an electric field
and therefore do have an observable effect; the monopole acquires electric charge and is then called a dyon. The gauge
parameter of large $U(1)$ gauge transformations is thus interpreted as a collective coordinate of the monopole.

The simple picture described in the preceding paragraphs is still valid when the gauge group $G$ is arbitrary but the Higgs
field enforces maximal symmetry breaking. The unbroken gauge group is of the form $U(1)\times\ldots\times U(1)$ and for
each $U(1)$ factor there will be an electric charge. In this paper we shall generalize the previous ideas to
larger gauge groups and general symmetry breaking, paying special attention to transformation properties of physical states
under large gauge transformations. From the point of view of semiclassical
quantization, the most important question is to identify the internal degrees of freedom of a monopole when
the unbroken gauge group is non-abelian or, in more technical words, what is the moduli space of monopole solutions.
The parameters of the moduli space are identified with collective coordinates, and quantization of ``motion"
in moduli space yields a tower of dyon states.

The example of maximal symmetry breaking suggests that the group parameters of the unbroken gauge group are collective
coordinates of the monopole, and that motion in the moduli space corresponds to large gauge transformations. However, it
has been known for a long time that in a spontaneously broken Yang-Mills-Higgs theory not all generators of the unbroken
gauge group ${\cal H}$ correspond to collective coordinates in semiclassical quantization of monopoles If ${\cal H}$ is non-abelian
\cite{abou, nelson, cole, bala}. Problems arise when we try to define the action of the generators of ${\cal H}$ that do not
commute with the magnetic field of the monopole. We shall denote the set of these generators by ${\cal H}'$. These problems
can be exposed through semiclassical quantization \cite{abou} of monopole solutions, through topological considerations
\cite{nelson}, or by studying the quantum mechanics of a test particle in the presence of a non-abelian monopole \cite{bala}.
One of the aims of this paper is to analyze these problems from the point of view of physical states

It was shown in \cite{abou} that the momenta of inertia corresponding to ${\cal H}'$ are vanishing, and that therefore the
collective coordinates associated to those gauge generators somehow decouple from the theory. This fact suggests that
it should be possible to formulate the theory of non-abelian dyons in such a way that the
decoupling of ${\cal H}'$ is proven from first principles. In this paper we shall present a Hamiltonian formulation of non-abelian
Yang-Mills-Higgs dyons that meets that
demand. The crucial feature of a gauge theory is the appearance of Gauss' Law and the necessity of ``choosing a gauge" in
order to obtain a non-singular symplectic structure acting on its phase space. Once the asymptotic values of the group
parameters are given, the gauge condition fixes their value for all points. Therefore only the boundary values
of the group parameters remain undetermined and can be considered dynamical variables \cite{wadia}. The extension of the
Hamiltonian formalism to Yang-Mills-Higgs theories with dyons requires, therefore, the introduction of boundary terms that
act as dynamical variables. These new variables enforce a new, extended Gauss' Law that will be shown to restrict physical
states to singlets under ${\cal H}'$. Therefore, there are no collective coordinates for the excitation of the ${\cal H}'$ modes. The
effect of the inclusion of a vacuum angle $\vartheta$ will also be considered.

The rest of this paper is organized as follows. Section 2 introduces the Hamiltonian formalism for Yang-Mills-Higgs theories,
including the boundary terms that arise in the presence of dyons. Section 3 discusses the consequences of the extended
Gauss' Law on physical states. In Section 4 we present our conclusions.

\section{Boundary Terms in Yang-Mills-Higgs theories}
Let s consider a Yang-Mills-Higgs theory with simply-connected gauge group ${\cal G}$ spontaneously broken to a subgroup
${\cal H}$ by a Higgs field in the adjoint representation of ${\cal G}$. We shall consider the theory to be defined on a large sphere
${\cal S}$ with boundary ${\cal B}$ in order to exhibit the importance of boundary terms in the action when the sphere contains
magnetic monopoles. The lagrangean density of this theory is
\begin{equation}
{\cal L}={\rm Tr}\left(-{1\over 4} F_{\mu\nu}\,F^{\mu\nu}-\half D_{\mu}\phi\, D^{\mu}\phi\right) -V(\phi),
\label{lagr}
\end{equation}
where the potential $V(\phi)$ must ensure spontaneous symmetry breaking. The classical action of this theory is, in
first-order formalism,
\begin{eqnarray}
S&=&\int_{t_1}^{t_2} dt\,\left[\int\,dx\,{\rm Tr}\left(E_i \dot{A}_i + \pi\dot{\phi}\right)-H(E_i, A_i, \pi, \phi, A_0)\right],\nonumber\\
H&=&\int\,dx\,{\rm Tr}\left[\half \,E_i\,E_i+\half \,\pi^2 +{1\over 4}\,F_{ij}\,F_{ij}+\half\,D_i\phi\,D_i\phi+V(\phi)
-A_0\,\left(D_i\,E_i+e\,[\phi,\pi]\right)\right]\nonumber\\
&+&\int_{{\cal B}} d\sigma_i\,{\rm Tr}\,A_0\,E_i.
\label{act}
\end{eqnarray}
where the momenta are $E_i=F_{0i}$ and $\pi=D_0\phi$. Integrations in $x$ extend over the whole sphere
${\cal S}$. The surface element in ${\cal B}$ is defined as $d\sigma_i=r^2\,\hat{r}_i\, d\omega$ with $\hat{r}_i$ a unit vector normal to
${\cal B}$ and $d\omega$ the solid angle element ($\omega$ is a shorthand for angular coordinates on ${\cal B}$). The hamiltonian
$H$ depends not only on the canonical coordinates and momenta, but also on $A_0$, the time
component of the gauge potential. In standard expositions of gauge theories in the hamiltonian formalism it is assumed
that the asymptotic value of $A_0$ is zero, so that when the radius of ${\cal S}$ tends to infinity the last term in $H$
vanishes. Of course it is always possible to choose a gauge where $A_0$ vanishes at infinity, or even everywhere as in
the temporal gauge $A_0=0$. This possibility will be considered below, but for the sake of generality we shall keep the
boundary value of $A_0$ arbitrary and possibly time-dependent \cite{wadia}. The spatial components of the
gauge field $A_i^a$ are such that the monopole is a source of a magnetic field that, at large distances from the central
region of the monopole, takes the form
\begin{equation}
B_i={G(\omega)\over r^2}\,\hat{r}_i,
\label{B}
\end{equation}
with $G(\omega)=G^a(\omega)\,T_a$ an element of the unbroken gauge algebra ${\cal H}$. The magnetic field must satisfy
the Bianchi identity, which implies that $G$ is covariantly constant, $D_i\,G=0$. At this point it is convenient to choose a
Cartan basis for the generators of ${\cal H}$,
\begin{eqnarray}
{\cal H}&=&\left\{T_1, \ldots, T_l, T_{l+1},\ldots,T_r, E_{\alpha_1}, E_{-\alpha_1}, E_{\alpha_2}, E_{-\alpha_2},\ldots\right\}
\nonumber\\
\left[ T_I, E_{\alpha}\right] &=& \alpha_I\,E_{\alpha}, \\
\left[ E_{\alpha}, E_{-\alpha} \right] &=& \sum_{I=1}^l\alpha_I\, T_I , \nonumber\\
\left[ T_I, T_J\right]&=&0.
\label{alg}
\end{eqnarray}
The last $r-l$ generators $T_I$ correspond to possible abelian factors in ${\cal H}$, while the first $l$ generate its maximal
torus. The roots $\alpha$ are all non-zero, distinct, have non-vanishing components for $I=1,\ldots,l$, and span the
$l$-dimensional space \cite{gno}. It is always possible to choose the generators of ${\cal H}$ in such a way that
\begin{eqnarray}
{\rm Tr}(T_I\,T_J)&=&\delta_{IJ},\quad I,J=1,\ldots,l\nonumber\\
{\rm Tr}(E_{\alpha}\,E_{-\alpha})&=&1,
\label{traces}
\end{eqnarray}
and the remaining traces of the products of two generators are zero. It will be convenient to work in the Wu-Yang formalism
and use the freedom present in the definition of the maximal torus of ${\cal H}$ to have
\begin{equation}
G(\omega)=\sum_{I=1}^r g_I(\omega)\,T_I,
\label{Q}
\end{equation}
with $g_i$ the magnetic weights introduced in \cite{gno}. In this formalism the Higgs field is constant on ${\cal B}$ and the
unbroken gauge group ${\cal H}$ lies inside ${\cal G}$ in a position independent way. For future reference we will now give a
criterion for the
commutativity of a general element $X$ of ${\cal H}$ with $G$. Let us write $X$ is the Cartan basis as
\begin{equation}
X=X_I\,T_I +X_{\alpha}\,E_{\alpha},
\label{x}
\end{equation}
where repeated indices $I$ are summed over. The commutator of $X$ with $G$ is easily found to be
\begin{equation}
[X, G]=\sum_{\alpha}(g\cdot \alpha)\,X_{\alpha}\,E_{\alpha}.
\label{xg}
\end{equation}
Therefore $X$ commutes with $G$ if and only if all the roots $\alpha$ included in the decomposition (\ref{x}) satisfy
$g\cdot \alpha =0$. We shall denote those roots by $\alpha^{\perp}$:
\begin{equation}
[X, G(P)]=0 \quad \Longleftrightarrow\quad X=X_I\,T_I + X_{\alpha^{\perp}}\,E_{\alpha^{\perp}}
\label{condition}
\end{equation}
Group elements will be represented by the parameters $\theta^a(x,t)$ that appear in the exponential map of the
gauge group, $g=\exp(\theta^a\,T_a)\in{\cal H}$. The asymptotic values
of the group parameters $\theta^a(\omega,t)$ and their canonical momenta $P_a(\omega,t)$ will be included in the phase
space of the theory. The motivation for this extension of the
phase space is that we shall impose Gauss' Law on physical states; this requirement, together with a choice of
gauge, eliminates gauge transformations that vanish at infinity as symmetries of the theory but leaves the asymptotic
values of the group parameters undetermined. These asymptotic values are candidates for collective coordinates
of the dyon. As we want the dynamics of these degrees of freedom to appear as
equations of motion of the theory, we have to extend the phase space and the hamiltonian (\ref{act}) to accommodate the
new variables. We shall assume standard Poisson brackets for $\theta^a$ and $P_a$,
\begin{eqnarray}
\{\theta^a(\omega,t), P_b(\omega',t)\}&=&\delta_b^{\,\,a}\delta(\omega-\omega')\nonumber\\
\{\theta^a(\omega,t), \theta^b(\omega',t)\}&=&\{P_a(\omega,t), P_b(\omega',t)\}=0.
\label{poi}
\end{eqnarray}
We shall consider physically static field configurations, that is those whose time dependence is given
by a gauge transformation:
\begin{eqnarray}
\partial_0\,A_i&=&D_i\,\Lambda,\nonumber\\
\partial_0\,\phi&=&\left[ \phi, \Lambda\right],
\label{static}
\end{eqnarray}
where $\Lambda=\Lambda^a\,T_a$ is a gauge parameter that, at long distance from the monopole core, lies in the unbroken
gauge algebra. It is always possible to write the asymptotic values of $\Lambda$ at any point $\omega$ in the boundary
${\cal B}$ as
\begin{equation}
\Lambda(\omega)=\Lambda_I(\omega)\,T_I+\lambda_{\alpha}(\omega)\,E_{\alpha}.
\label{lamb}
\end{equation}
We must also specify the boundary conditions that $A_i$ is assumed to obey. At large distance $A_i$ and $F_{\mu\nu}$
should vanish at least as fast as $r^{-1}$ and $r^{-2}$ respectively:
\begin{equation}
\lim_{r\to\infty}\,A_i = O(r^{-1}), \quad\quad\quad \lim_{r\to\infty}\,F_{\mu\nu}=O(r^{-2}).
\label{cond}
\end{equation}
In addition we require that the radial component of $A_i$ decreases for large $r$ as $r^{-2}$,
\begin{equation}
\lim_{r\to\infty}\, \hat{r}^i\,A_i = O(r^{-2}).
\label{radial}
\end{equation}
These boundary conditions imply that $\hat{r}^i\,D_iA_0$ should decrease as $r^{-2}$. This
fact does not imply that $A_0$ decreases as $r^{-1}$; it may behave at long distances like the
Higgs field in the BPS limit, which has a non-vanishing limit ar $r$ goes to infinity and at the same time satisfies
$D_i\phi=B_i=O(r^{-2})$.

We still have to determine what is the time evolution of the asymptotic group parameters
$\theta^a(\omega,t)$. We shall follow the assumption that their time evolution is a gauge transformation of parameter
$\Lambda=\Lambda^a\,T_a$, as in (\ref{static}). In order to express this idea we introduce of the Racah function
$\Phi^a(\theta,\eta)$, which defines the product of two group elements:
\begin{equation}
\exp\left( i\,T_a\eta^a\right)\,\exp\left( i\,T_a\theta^a\right)=\exp\left( i\,T_a\Phi^a(\theta,\eta)\right).
\label{racah1}
\end{equation}
The derivative of this function with respect to its second variable acts as a vierbein in the sense that
gives the directions of small fluctuations about a certain point in the group manifold
\begin{equation}
E_b^{\,\,a}(\theta)={\partial\over\partial\eta^b}\,\Phi^a(\theta,\eta)\Big|_{\eta=0}.
\label{racah2}
\end{equation}
This vierbein relates elements of the Lie algebra to tangent vectors of the group manifold at a generic group element
$g$,
\begin{equation}
i\,T_a\,g=E_a^{\,\,b}\,\partial_b\,g,\quad\quad g=\exp(i\,\theta^a\,T_a)
\label{vier}
\end{equation}
where $\partial_a$ is the derivative with respect to $\theta^a$. The integrability condition of the Lie group requires
$E_a^{\,\,b}$ to satisfy the following relationship
\begin{equation}
E_a^{\,\,c}\,\partial_c E_b^{\,\,d}-E_b^{\,\,c}\,\partial_c E_a^{\,\,d}=f_{ab}^{\,\,\,c}\,E_c^{\,\,d}.
\label{inte}
\end{equation}
The assumed time evolution of the group parameters of a general gauge group is therefore
\begin{eqnarray}
g(t+\delta t)&=&\exp\left\{i\Lambda\,\delta t\right\}\,g=\exp\left\{i\Lambda\,\delta t\right\}\,\exp\left\{i\,\theta^a\,T_a\right\}
=\exp\left\{i\,\Phi^a(\theta, \Lambda\,\delta t)\right\}\nonumber\\
&=&\exp\left\{i\,\left(\Phi^a(\theta,0)+E_b^{\,\,a}\,\Lambda^b\,\delta t\right)\right\}=
\exp\left\{i\,\left(\theta^a+E_b^{\,\,a}\,\Lambda^b\,\delta t\right)\right\}
\label{asu}
\end{eqnarray}
Writing the group element $g(t+\delta t)$ as $\exp(i\,\theta^a(t+\delta t)\,T_a)$ we conclude that the time derivative of
the parameter $\theta$ is
\begin{equation}
{d\over dt}\theta^a(\omega,t)= E_b^{\,\,a}(\theta)\,\Lambda^b(\omega,t),
\label{evol}
\end{equation}
Once we accept that the boundary values of the gauge
parameters must evolve in time according to (\ref{evol}) and that the phase space of the theory must include the
group parameters and their canonical momenta, it is necessary to extend the hamiltonian shown in (\ref{act}) with a new
term that reproduces the equation of motion (\ref{evol}):
\begin{eqnarray}
H'&=&\int\,dx\,{\rm Tr}\left[\half \,E_i\,E_i+\half \,\pi^2 +{1\over 4}\,F_{ij}\,F_{ij}+\half\,D_i\phi\,D_i\phi+V(\phi)
-A_0\,\left(D_i\,E_i+e\,[\phi,\pi]\right)\right]\nonumber\\
&+&\int_{{\cal B}} d\sigma_i\,{\rm Tr}\,(A_0\,E_i) +\int_{{\cal B}} d\omega\, {\rm Tr}\,(\Lambda\,J),
\label{ham}
\end{eqnarray}
where we have introduced the intrinsic momentum $J_a=E_a^{\,\,b}\/P_b$. The Poisson bracket of two $J$ reproduces the
Lie algebra of the unbroken gauge group ${\cal H}$,
\begin{equation}
\{J_a(\omega), J_b(\omega')\}=if_{ab}^{\,\,\,c}\,J_c(\omega)\,\delta(\omega-\omega'),
\label{lie}
\end{equation}
where $f_{ab}^{\,\,\,c}$ are the structure constants of ${\cal H}$. We can think of the momenta $J_a$ as generators of ${\cal H}$ and,
after quantization, as the operators that implement large gauge transformations on physical states.

\section{Generalized Gauss' Law}
The last term of the extended hamiltonian $H'$ generates the desired time evolution for the group parameters through the
Poisson brackets (\ref{poi}). We are
interested only in the equations of motion that, after quantization, are first order constraints on physical states of the
theory. These constraints should appear as stationary points of the action for variations of $A_0(x,t)$ and its boundary
values $A_0(\omega,t)$. The difficulty is that the variations of a field are not entirely independent of the variations of its
boundary values and therefore we cannot vary $A_0(x,t)$ and $A_0(\omega,t)$ independently. The way out of this
problem is to restrict the phase space to a subspace where Gauss' Law is satisfied:
\begin{equation}
\left(D_i\,E_i +\left[ \phi, D_0\,\phi\right]\right)\,|\Psi\rangle =0.
\label{ga}
\end{equation}
where $|\Psi\rangle$ is a physical state. This eliminates $A_0(x,t)$ from the variational problem and leaves its boundary
value $A_0(\omega,t)$ as the only lagrange multiplier. This is not the only effect of Gauss' Law; inserting the conditions
(\ref{static}) into (\ref{ga}) leads to a link between $A_0$ and the gauge parameter that will be
most relevant to our discussion:
\begin{equation}
\left( D_i\,D_i + \left[\phi,\left[\phi,\quad\right]\right]\right)(\Lambda-A_0)=0.
\label{link}
\end{equation}
An obvious solution is $\Lambda=A_0$, which implies that the electric field is zero. It is therefore clear that
$\Lambda-A_0$ must not vanish if we want to turn a monopole into a dyon. An important non-trivial solution is
$\Lambda-A_0=C\,\phi$ with $C$ a constant; this solution, however, corresponds to picking the $U(1)$ direction in the
unbroken gauge group defined by the Higgs field. It is easy to show that this would reproduce the usual charge quantization
condition on the Abelian electric charge of a dyon \cite{witten}. For that reason we will concentrate on the semisimple
part of the unbroken gauge group, that is we shall ignore possible $U(1)$ factors in ${\cal H}$.

Far from the monopole core, (\ref{link}) is the Schr{\"o}dinger equation of a zero-energy adjoint particle in the presence of
the magnetic field of the monopole. The long-distance behaviour of the solutions depends on whether the ``wave function''
commutes with $G$ or not \cite{abou}. We should therefore consider three cases:
\begin{enumerate}
\item Components of $\Lambda-A_0$ within the maximal torus clearly commute with $G$ as defined in (\ref{Q}). Equation
(\ref{link}) reduces to Laplace's far from the monopole core, and thus the solution is of the form $(\Lambda-A_0)_I \sim
Q_I\,r^{-1}$ with $Q\in{\cal H}$ some $r$-independent operator.
\item Components of $\Lambda-A_0$ corresponding to roots $\alpha^{\perp}$ defined by $\alpha^{\perp}_{\,I}\,g_{I}=0$ also
commute with $G$ and thus behave as $(\Lambda-A_0)_{\alpha^{\perp}} \sim Q_{\alpha^{\perp}}\,r^{-1}$.
\item Components corresponding to the rest of the generators do not commute with $G$ and thus decrease faster,
$(\Lambda-A_0)_{\alpha} \sim Q_{\alpha}\,r^{-n}$ with $n>1$.
\end{enumerate}
From now on we shall understand that roots $\alpha$ are not orthogonal to the vector whose components are the magnetic
weights $g_I$ unless explicitly indicated by the index $\perp$. The general solution for $\Lambda-A_0$ is then
\begin{equation}
\Lambda-A_0 \sim {1\over r}\,Q_I\,T_I + {1\over r}\,Q_{\alpha^{\perp}}\,E_{\alpha^{\perp}}+{1\over r^n}\,Q_{\alpha}\,E_{\alpha},
\quad{\rm with}\quad n>1.
\label{so}
\end{equation}
It is clear that the asymptotic values $\Lambda(\omega,t)$ and $A_0(\omega,t)$ must coincide. This important fact
implies that variations of the boundary value of $A_0$ produce a new constraint that includes $J$:
\begin{equation}
\lim_{r\to\infty} r^2\hat{r}_i\,E_a^i+J_a=0.
\label{cons}
\end{equation}
The meaning of this new constraint is that large gauge transformations induce an electric field, thus giving electric charge to
the monopole. Including the parameter $\Lambda$ of the gauge transformation, the quantum version of this constraint is
\begin{equation}
\left[\int_{{\cal B}}d\sigma_i\,{\rm Tr}\left(\Lambda\,E_i \right)+\int_{{\cal B}}d\omega\,{\rm Tr}(\Lambda\,J)\right]\,|\Psi\rangle=0.
\label{cons2}
\end{equation}
Together with (\ref{ga}), this constitutes a generalized Gauss Law \cite{wadia}. It is important to note that the radial part of
the electric field should decrease exactly as $r^{-2}$ in order to contribute to the constraint (\ref{cons2}). Taking into
account the general solution (\ref{so}), together with the fact that the radial component of $A_i$ already decreases as
$r^{-2}$, we find that the radial part of the electric field behaves as
\begin{equation}
\hat{r}^i\,E_i =\hat{r}^i \,D_i\,(\Lambda-A_0) \sim {\partial\over\partial r}\,\left( {1\over r}\,Q_I\,T_I + {1\over
r}\,Q_{\alpha^{\perp}}\,E_{\alpha^{\perp}}\right) +O(r^{-m}), \quad{\rm with}\quad m>2.
\label{oo}
\end{equation}
The implication of Eq. (\ref{oo}) is that the generators $E_{\alpha}$ disappear from the constraint (\ref{cons2}), due to the
fact that the trace of $E_{\alpha}$ with $T_I$ or  $E_{\alpha^{\perp}}$ vanishes:
\begin{equation}
\int_{{\cal B}}d\sigma_i\,{\rm Tr}(\Lambda\,E_i)\sim -\int_{{\cal B}}d\omega \left(\lambda_I\,Q_I +
\lambda_{\alpha^{\perp}}\,Q_{-\alpha^{\perp}}\right).
\label{hot}
\end{equation}
We have used the traces given in (\ref{traces}). Before inserting the result (\ref{hot}) into the constraint (\ref{cons2}) we
will introduce a by now obvious decomposition of the generators $J$:
\begin{equation}
J(\omega)= J_I(\omega)\,T_I+J_{\alpha^{\perp}}(\omega)\,E_{\alpha^{\perp}}+J_{\alpha}(\omega)\,E_{\alpha}.
\label{jota}
\end{equation}
Using (\ref{lamb}), (\ref{hot}) and (\ref{jota}) and comparing the terms in $\lambda_I$, $\lambda_{\alpha^{\perp}}$ and
$\lambda_{\alpha}$ we finally find the action of the different components of $J$ on physical states:
\begin{eqnarray}
J_I\,|\Psi\rangle &=& Q_I\,|\Psi\rangle,\nonumber\\
J_{\alpha^{\perp}}\,|\Psi\rangle &=&Q_{\alpha^{\perp}}\,|\Psi\rangle,\\
J_{\alpha}\,|\Psi\rangle &=&0.\nonumber
\label{result1}
\end{eqnarray}
We can summarize these results as follows: physical states transform under all the $T_I$ and under the generators
$E_{\alpha^{\perp}}$. Large gauge transformations generated by the $E_{\alpha}$ leave physical states invariant and
therefore do not correspond to collective coordinates. This phenomenon is due to the  anomalous behaviour of the radial
part of the electric field in the isodirections $E_{\alpha}$.

\subsection{Inclusion of a vacuum angle}
The discussion of the preceding Section can be readily generalized to include a vacuum angle $\vartheta$. There is no need
to redo the calculations since the effect of the vacuum angle is essentially a shift in the electric field:
\begin{equation}
E_i^a\rightarrow E_i^a-\vartheta\,B_i^a,
\label{angle}
\end{equation}
where $\vartheta$ absorbs some inessential constants. This can be plugged directly into Eq. (\ref{cons2}) with the following
results
\begin{eqnarray}
J_I\,|\Psi\rangle &=& \left (Q_I+\vartheta\,g_I\right)\,|\Psi\rangle,\nonumber\\
J_{\alpha^{\perp}}\,|\Psi\rangle &=&Q_{\alpha^{\perp}}\,|\Psi\rangle,\\
J_{\alpha}\,|\Psi\rangle &=&0.\nonumber
\label{result2}
\end{eqnarray}
We see that physical states are invariant under the $J_{\alpha}$ even when a vacuum angle is introduced. The effect of
the vacuum angle is limited to transformations under the commuting generators $J_I$. Equations (\ref{result1}) or
(\ref{result2}) can be interpreted as a symmetry breaking induced by the non-abelian monopole whereby the unbroken
gauge algebra ${\cal H}$ is broken further down to its generators $T_I$ and $E_{\alpha^{\perp}}$.

\section{Conclusions}
We have demonstrated that physical states of non-abelian dyons are invariant under large gauge transformations that do
not commute with the magnetic field of the dyon. In the language of root systems those gauge transformations correspond
to generators $E_{\alpha}$ of the unbroken gauge group with the root $\alpha$ not orthogonal to the magnetic weights. This
result is still true if a vacuum angle $\vartheta$ is included. The only effect of the vacuum angle is an additional term in the
transformation properties of physical states under the generators of the maximal torus of ${\cal H}$. This additional term is
proportional to the magnetic weights, which are weights of ${\cal H}^v$, the dual of the unbroken gauge group \cite{gno};
physical states of dyons therefore carry a representation of ${\cal H}^v$. This may be relevant to a better understanding of the
Montonen-Olive conjecture \cite{mo}.

Excitation of internal degrees of freedom of the monopole corresponding to the roots $\alpha$ not orthogonal to the
magnetic weights induce no electric charge in the monopole and hence have no observable effects far from the monopole
core. From the point of view of collective coordinate quantization, this fact implies that there are no collective coordinates
associated to those degrees of freedom. This result complements and clarify the results of references \cite{abou} to
\cite{bala}.

\vskip 2cm
\begin{center}
\bf{Acknowledgements}
\end{center}
I would like to thank Professor T.J. Hollowood for many useful discussions.  This
research has been supported by the British Engineering and Physical Sciences Research Council (EPSRC).

\vskip 2cm