
corresponding classical systems. At $\hbar \rightarrow 0$, the star-product
quantization recovers the classical constrained dynamics.

To illustrate calculation of the quantum corrections to the Dirac bracket,
in Appendix C we derive the analytical expression for the $\hbar ^{2}$
correction.


\section{Quantum spherical pendulum in the phase space}

\setcounter{equation}{0} 

As an application we consider, in the phase space, the evolution of the
Wigner function of a mathematical pendulum on an $S^{n-1}$ sphere of a unit
radius in an $n$-dimensional Euclidean space with the coordinates $\phi
^{\alpha }$. 

At the classical level, there exists, within the generalized Hamiltonian
framework, two constraint functions 
\begin{equation}
{}\mathcal{G}^{1}=\ln \phi,\;\;\;{}\mathcal{G}^{2}=\pi ^{\alpha }\phi
^{\alpha }  \label{OME}
\end{equation}
where $\phi =(\phi ^{\alpha }\phi ^{\alpha })^{1/2}$. The first constraint $%
{}\mathcal{G}^{1}=0$ implies that the particle stays on the sphere $\phi =1$%
, the second constraint ${}\mathcal{G}^{2}=0$ suggests that the radial
component of the momenta vanishes. The constraint functions constitute a
canonical pair (\ref{SB}). One can check that the projected canonical
variables $\xi _{s}(\xi )=(\phi _{s}^{\alpha },\pi _{s}^{\alpha })$, with 
\begin{eqnarray}
\phi _{s}^{\alpha } &=&\phi ^{\alpha }/\phi, \\
\pi _{s}^{\alpha } &=&\phi \pi ^{\alpha }-\phi ^{\alpha }\phi \pi /\phi,
\end{eqnarray}
and the Hamiltonian projected onto the constraint submanifold (see \cite
{Krivoruchenko:2005wb}, Sect. 6), 
\begin{equation}
\mathcal{H}{}_{s}=\frac{1}{2}(\phi ^{2}\delta ^{\alpha \beta }-\phi ^{\alpha
}\phi ^{\beta })\pi ^{\alpha }\pi ^{\beta },  \label{HAMI}
\end{equation}
are identically in involution with ${}\mathcal{G}^{a}$, which is on a par
with Eqs.(\ref{CG}) and (\ref{XO}).

Since ${}\mathcal{G}^{2}$ is a second order polynomial with respect to the
canonical variables, the infinite power series in the Poisson operator in
Eq.(\ref{EA}) is truncated at $O(\hbar ^{0})$. Indeed, all the higher order
terms vanish, so that $\mathcal{G}{}^{1}\wedge {}\mathcal{G}^{2}=\{{}%
\mathcal{G}^{1},{}\mathcal{G}^{2}\}=1$. Assuming ${G}^{a}={}\mathcal{G}^{a}$%
, Eqs.(\ref{SBAS}) hold.

Similarly, Eqs.(\ref{CG3}) hold for $\xi _{t}(\xi )=\xi _{s}(\xi )$: ${}%
\mathcal{G}^{1}$ and $\phi _{s}^{\alpha }$ depend on $\phi ^{\alpha }$, so $%
\phi _{s}^{\alpha }\wedge {}\mathcal{G}^{1}=0$. $\pi _{s}^{\alpha }$ is a
first degree polynomial with respect to the canonical momenta. In the
expression $\pi _{s}^{\alpha }\wedge \mathcal{G}{}^{1}$, the operators ${}%
\mathcal{P}^{2k+1}$ act to the right on the coordinates only, and to the
left on the momenta. The power series terms entering Eq.(\ref{EA}) vanish
starting with $O(\hbar ^{2})$, so $\pi _{s}^{\alpha }\wedge {}\mathcal{G}%
^{1}=\{\pi _{s}^{\alpha },{}\mathcal{G}^{1}\}=0$ holds in virtue of Eqs.(\ref
{CG}). ${}\mathcal{G}^{2}$ is a second degree polynomial with respect to the
canonical variables. The infinite power series in Eq.(\ref{EA}) is truncated
at $O(\hbar ^{0})$ again. Due to Eqs.(\ref{CG}), we have ${}\mathcal{G}%
^{2}\wedge \phi _{s}^{\alpha }=\{{}\mathcal{G}^{2},\phi _{s}^{\alpha }\}=0$
and ${}\mathcal{G}^{2}\wedge \pi _{s}^{\alpha }=\{{}\mathcal{G}^{2},\pi
_{s}^{\alpha }\}=0$.

Finally, one has to check that ${}\mathcal{G}^{a}\wedge {}\mathcal{H}_{s}=0$%
. The first equation, for $a=1$, is valid since ${}\mathcal{G}^{1}$ depends
on $\phi ^{\alpha }$, while $\mathcal{H}_{s}$ depends on $\pi ^{\alpha }$
quadratically. The infinite power series in Eq.(\ref{EA}) is truncated at $%
O(\hbar ^{0})$, so the Moyal bracket can be replaced by the Poisson bracket.
Since $\{{}\mathcal{G}^{1},{}\mathcal{H}_{s}\}=0$, equation ${}\mathcal{G}%
^{1}\wedge {}\mathcal{H}_{s}=0$ holds. The second equation, for $a=2$, is
valid since ${}\mathcal{G}^{2}$ is a second degree polynomial. Consequently, 
$\mathcal{G}^{2}\wedge \mathcal{H}_{s}=\{ \mathcal{G}^{2}, \mathcal{H}_{s}
\}=0$.

The projected canonical variables $\xi _{s}(\xi )$, the constraint functions 
${}\mathcal{G}^{a}$, and the classical Hamiltonian $\mathcal{H}_{s}$
coincide with the projected canonical variables $\xi _{t}(\xi )$, the
constraint functions $G^{a}$, and the Hamiltonian $H_{t}$, respectively.

The evolution equation for the Wigner function has the form (\ref{PEV2})
where the sign of the right side should be changed. The power series
expansion of the Moyal bracket over the Poisson operator ${}$ is truncated
at $O(\hbar ^{2})$, since the Hamiltonian (\ref{HAMI}) is a fourth degree
polynomial of the canonical variables, so we obtain 
\begin{equation}
\frac{\partial }{\partial t}W=-\{W,{}\mathcal{H}_{s}\}+\frac{\hbar ^{2}}{8}(%
\frac{\partial ^{3}W}{\partial \phi ^{\alpha }\partial \phi ^{\beta
}\partial \pi ^{\gamma }}(2\delta ^{\alpha \beta }\phi ^{\gamma }-\delta
^{\alpha \gamma }\phi ^{\beta }-\delta ^{\beta \gamma }\phi ^{\alpha })-%
\frac{\partial ^{3}W}{\partial \pi ^{\alpha }\partial \pi ^{\beta }\partial
\phi ^{\gamma }}(2\delta ^{\alpha \beta }\pi ^{\gamma }-\delta ^{\alpha
\gamma }\pi ^{\beta }-\delta ^{\beta \gamma }\pi ^{\alpha })).  \label{PEV3}
\end{equation}
The first term in the right side is of the classical origin, while the
second term represents a quantum correction to the classical Liouville
equation and there are no other quantum corrections. Given $W(\xi,0)$ in
the unconstrained phase space, $W(\xi,t)$ can be found by solving the
partial differential equation (\ref{PEV3}).

\begin{table}[tbp]
\caption{
Skew-symmetric multiplication operations are listed for functions (second column) and equivalence classes of functions (third column) in the phase space of classical systems (first row) and in the phase space of quantum systems (second row).
}
\label{lab2}
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
Systems: & unconstrained & constrained \\ \hline
classical & $\{f,g\} $ & $\{f,g\}_{D}$ \\ \hline
quantum & $f\wedge g$ & $f_{t}\wedge g_{t} $ \\ \hline
\end{tabular}
\end{center}
\par
\vspace{-2mm}
\end{table}


\section{Conclusion}

\setcounter{equation}{0} 

Real functions in the unconstrained phase space of second-class constraints
systems split into equivalence classes corresponding to different physical
observables. The quantum observables are described by equivalence classes of
operators acting on the Hilbert space of states. The Weyl's association rule
extends the equivalence relations to functions in the phase space of quantum
systems.

The Dirac bracket can be calculated as the Poisson bracket between functions
projected onto the constraint submanifold using the phase flows generated by
the constraint functions. The quantum deformation of the Dirac bracket is
the Moyal bracket calculated for functions projected onto the constraint
submanifold of the phase space.

The skew-symmetric multiplication operations are synthesized in Table I.

\textit{The operation $f_{t}\wedge g_{t} $ designates the quantum
deformation of the Dirac bracket consistent with the canonical quantization
of constraint systems.}

As an application of the general formalism, in Sect. VI we derived with the