text:
\section{Introduction.}

Generally covariant systems are a particular kind of dynamical systems in which time is included among the canonical variables \cite{Henneaux}. The state of the system evolves along orbits parametrized by an unphysical scalar parameter. The formalism is invariant under reparametrizations of this parameter (for this reason these systems are also called general parametrized systems) leading to a gauge symmetry, which in the Hamiltonian formulation is implemented by a first class (Hamiltonian) constraint. One of the most striking properties of generally covariant theories is that the Hamiltonian is identically zero on the constraint hypersurface. Consequently there is no standard Hamiltonian time evolution (which coincides exactly with the gauge transformation). Instead, dynamics is to be found among the relations between the canonical variables that are determined by the Hamiltonian constraint \cite{Rovelli1,Gambini1,Rovelli2}. In other words, dynamics is the unfolding of the gauge transformation. The most famous example of a generally covariant theory is, of course, general relativity \cite{Kuchar1,Ashtekar1,Rovelli3}.

Upon quantization these systems display a number of technical and conceptual problems which, motivated by the quest to quantize general relativity, have been intensively studied in the literature. Since the publication of Dirac's seminal work on the quantization of constrained systems \cite{Dirac}, several quantization programs have been developed with the aim of refining Dirac's approach and making it suitable to address the generally covariant case \cite{Henneaux,Ashtekar1,Ashtekar2,Marolf,Woodhouse}. In spite of this, several important problems concerning the quantization of these systems (the problem of time, the observables' problem and the measurement problem, just to mention a few) are still lacking a definitive answer \cite{Gambini1,Rovelli2,Marolf}.

One quantization program that has only been scarcely explored in this context is the deformation approach \cite{Weyl}-\cite{Bayen2}. And this regardless of the fact that the deformation quantization method displays some remarkable features that make it especially suitable to address the generally covariant case \cite{Antonsen,Martinez,Takao}.
On the one hand the deformation approach leads to a formulation of quantum mechanics in terms of classical like objects. The theory lives in phase space and mimics the structure of classical statistical mechanics. The state of the system is described by a phase space quasi-distribution and observables are also functionals on the phase space. From the deformation point of view quantization amounts to the substitution of the standard product of functions by a non-commutative star product \cite{Bayen1,Bayen2}. The theory has been proved useful when addressing a wide range of fields of research ranging from topics in non-relativistic quantum mechanics \cite{Lee1}, \cite{Lee2}-\cite{Shin} to some current developments in $M$-theory \cite{Fairlie2,Fairlie3,Seiberg}.
In the context of generally covariant systems it has been advocated that the relation between classical statistical and standard operator quantum mechanics  should be emphasized when approaching the quantization of these systems \cite{Kuchar2}. The point of view is that the problems besetting their quantization, being to a large extend conceptual ones, may receive key physical insights from the more intuitive classical statistical analysis. From this perspective the deformation methods seem to be especially suited.

On the technical side the deformation approach acquired the status of a powerful mathematical theory not displaying any of the subtleties of the path integral or canonical quantization prescriptions. Most remarkable is the fact that the theory is able to generate the quantum version of a classical system living on a generic (possibly curved) Poisson or sympletic manifold \cite{Kontsevich,Fedosov,Batalin,Bordemann}.
Of special relevance for generally covariant systems is that the deformation approach displays some powerful tools to address the quantum eigenvalue problem, the most meaningful of which being a covariant formalism unifying all distinct phase space representations of the eigenvalue problem \cite{Bayen1,Nuno6,Nuno7}, a formal solution of a generic stargenvalue equation \cite{Nuno1,Dahl} and a set of efficient methods to determine the semiclassical expansion of the stargenfunctions \cite{Almeida1,Almeida2}.

In this paper we aim at providing a pedagogical incursion into some of these methods by studying the deformation quantization of a special simple kind of generally covariant systems: those obtained by parametrizing an originally non-covariant version of the system. The parametrization procedure leads to a new formulation of the dynamics living on an enlarged phase space where the physical time is incorporated as a canonical variable \cite{Henneaux}.
The existence of such a natural time variable considerably simplifies the (technical and conceptual) analysis of the dynamical structure of the system. For this reason these models have been used \cite{Henneaux,Gambini1,Rovelli2} to test a set of quantization techniques and interpretative prescriptions aiming at addressing the far more difficult problem of quantizing "already parametrized systems" of which the most significant example is general relativity.

The Hamiltonians of these models (as in the general case) are identically zero on the constraint hypersurface. At the quantum level the imposition of this constraint determines the physical states of the system.
 In the deformation context the imposition of first class constraints yields a stargenvalue equation \cite{Nuno1,Fairlie1} and so the physical states are the zero stargenfunctions of the Hamiltonian symbol. We are then able to apply the powerful tools of the deformation approach to study the quantum eigenvalue problem. A complete characterization of the physical states will be provided in three different phase space representations. In particular, we obtain a representation where the dynamics of the quasidistribution (with respect to the physical time variable) is dictated by the classical Liouville equation. This is an interesting side result that shows that through general parametrization a causal formulation for the distributional sector of quantum mechanics is made possible.
More generally we find that the remarkable parallelism between classical statistical mechanics and phase space quantum mechanics carries on intact to the generally covariant setting, a property that establishes a promising starting point for the future research on the deformation quantization of generally covariant systems.

\section{Covariant Wigner quantum mechanics}

The aim of this section is to review the main structures of the covariant formulation of the deformation quantization procedure. Our analysis will be restricted to the case of flat phase space as we will be dealing only with systems of this sort. The reader should refer to \cite{Batalin,Nuno6} for a more detailed presentation of these results and to \cite{Fedosov,Bordemann} for the generalization of the formalism to the non-flat case. An important part of this section (eq.(9) to eq.(15)) focuses on the covariant generalization of the $*$-genvalue equation, once again, just for the flat case. This is a topic that will play an important role latter on.

Before proceeding and to avoid future misunderstandings let us make the following remark: Throughout the paper the word {\it covariant} will be used in two different contexts: to designate the invariance under time reparametrizations of the original classical system and to designate the invariance under general coordinate transformations of the Moyal plane. The aim of this paper is to present the deformation quantization of a generally covariant (parametrized) classical system. We use the covariant version of the deformation quantization procedure (a subject where this paper makes no original contribution) because this leads to a larger set of possible quantum phase space representations, among which we will find the causal representation presented in section 4.

Let us then settle down the preliminaries: we consider a $N$-dimensional dynamical system, its classical formulation living on the flat phase space $T^*M$. A set of global canonical coordinates ($\{q_i,p_i,i=1..N\}$) can then be defined on $T^*M$ in terms of which the sympletic structure reads $w=dq_i \wedge dp_i$.
Upon quantization the set $\{\hat q_i \} $ constitutes a complete set of commuting observables. Let us introduce the vector notation $\hat{\vec q}=(\hat q_1,...,\hat q_N)$ and designate by $|\vec q>$ the general eigenstate of $\hat{\vec q}$ associated to the array of eigenvalues $\vec q$ and spanning the Hilbert space ${\cal H}$ of the system. Let also ${\cal A}({\cal H})$ be the algebra of quantum observables over ${\cal H}$ and ${\cal A}(T^*M)$ the algebra of classical functions over the classical phase space $T^*M$.  Let $(\vec Q,\vec P)$ be a second set of phase space coordinates, related to $(\vec q,\vec p)$ by a generic phase space diffeomorphism (not necessarily canonical): $\vec q=\vec q(\vec Q,\vec P)$ and $\vec p=\vec p(\vec Q,\vec P)$. The covariant Weyl transform \cite{Nuno6}:
\begin{eqnarray}
W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}: {\cal A}({\cal H}) & \longrightarrow & {\cal A}(T^*M);  \\
\hat A & \longrightarrow & W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat A) = \hbar^N \int d^N\vec x \int d^N\vec y \, e^{-i\vec p(\vec Q,\vec P) \cdot \vec y}
\delta (\vec x-\vec q(\vec Q,\vec P))
<\vec x+\frac{\hbar}{2} \vec y| \hat A |\vec x-\frac{\hbar}{2} \vec y> \nonumber
\end{eqnarray}
where $|\vec x \pm \frac{\hbar}{2} \vec y>$ are eigenstates of $\hat{\vec q}$,
yields the entire structure of covariant phase space quantum mechanics.
The map $W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}$ can be applied both to an observable $\hat A$ as well as to the density matrix $|\psi(t)><\psi(t)|$. In the first case it yields the $^{(\vec q,\vec p)}_{(\vec Q,\vec P)}$-Weyl symbol $A'(\vec Q,\vec P)=W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat A)$ of the original quantum operator and, in the second case, the celebrated Wigner function $f'_W(\vec Q,\vec P,t)=\frac{1}{(2\pi \hbar)^N}W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(|\psi(t)><\psi(t)|)$. Notice that if $(\vec Q,\vec P)=(\vec q,\vec p)$ then the covariant map $W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}$ reduces to the standard Weyl transform in the variables $(\vec q,\vec p)$. In this case we will use the notation $W_{(\vec q,\vec p)}$ to designate $W_{(\vec q,\vec p)}^{(\vec q,\vec p)}$. Notice also that $A'(\vec Q,\vec P)=A(\vec q(\vec Q,\vec P),\vec p(\vec Q,\vec P))$ where $A(\vec q,\vec p)=W_{(\vec q,\vec p)}(\hat A)$ and the same relation is valid for the Wigner function.
The covariant Weyl map implements the transformation $(\vec q,\vec p) \to (\vec Q,\vec P)$ as a coordinate transformation in quantum phase space and defines a covariant star-product $*'_{(\vec Q,\vec P)}$ and Moyal bracket $[,]_{M'_{(\vec Q,\vec P)}}$ \cite{Bayen1,Nuno6} ($\hat A,\hat B \in {\cal A}({\cal H})$):
\begin{eqnarray}
W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat A\hat B) & = & W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat A)*'_{(\vec Q,\vec P)} W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat B)\nonumber \\
W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\frac{1}{i\hbar}[\hat A,\hat B]) &=& [W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat A),W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat B)]_{M'_{(\vec Q,\vec P)}}
\end{eqnarray}
which display the functional form:
\begin{eqnarray}
A'( \vec Q, \vec P) *'_{(\vec Q,\vec P)} B'( \vec Q, \vec P)& = & A'( \vec Q, \vec P) e^{\frac{i \hbar}{2} {\buildrel { \leftarrow}\over\nabla'}_i  J'^{ij}_{(\vec Q,\vec P)}  {\buildrel
{ \rightarrow}\over\nabla'}_j} B' (\vec Q, \vec P) \nonumber \\
\left[ A'(\vec Q,\vec P), B'(\vec Q,\vec P) \right]_{M'_{(\vec Q,\vec P)}} & = & \frac{2}{\hbar} A'(\vec Q,\vec P) \sin \left(\frac{\hbar}{2}
{\buildrel { \leftarrow}\over\nabla'}_i  J'^{ij}_{(\vec Q,\vec P)}  {\buildrel
{ \rightarrow}\over\nabla'}_j \right) B'(\vec Q,\vec P),
\end{eqnarray}
where the covariant derivative is given by (let $O'^i = P_i, O^i=p_i, i=1, \cdots ,N$; $O'^i = Q_{i-N}, O^i=q_{i-N}, i=N+1, \cdots, 2N$):
\begin{equation}
\nabla'_i A'  = \partial'_i A', \quad
\nabla'_i \nabla'_j A'  = \partial'_i \partial'_j A' - \Gamma'^k_{ij} \partial'_k A', \quad \partial'_i= \partial /\partial {O'}^{i};
\quad i,j,k= 1, \cdots, 2N,
\end{equation}
and
\begin{equation}
J'^{ij}_{(\vec Q,\vec P)}(\vec Q,\vec P) = \frac{\partial O'^i}{\partial O^k} \frac{\partial O'^j}{\partial O^l} J^{kl}_{(\vec q,\vec p)}, \quad
\Gamma'^i_{jk}(\vec Q,\vec P) =  \frac{\partial O'^i}{\partial O^b} \frac{\partial^2 O^b}{\partial O'^j \partial O'^k},
\end{equation}
are the new symplectic structure and Poisson connection associated with the coordinates $(\vec Q,\vec P)$, respectively. Finally, notice that in eq.(5) we explicitly took into account the phase space flat structure.

When formulated in terms of these structures Wigner mechanics becomes fully invariant under the action of general phase space diffeomorphisms:
\begin{equation}
A'(\vec Q,\vec P) *^{\prime}_{(\vec Q,\vec P)} B'(\vec Q,\vec P) = A(\vec q(\vec Q,\vec P),\vec p(\vec Q,\vec P))*_{(\vec q,\vec p)} B(\vec q(\vec Q,\vec P),\vec p(\vec Q,\vec P)) \qquad \forall_{A,B \in {\cal A}(T^{\ast}M)},
\end{equation}
the covariant generalization of the Moyal and stargenvalue equations reading:
\begin{eqnarray}
& \frac{\partial f'_W}{\partial t}  = [H',f'_W]_{M'_{(\vec Q,\vec P)}} & \nonumber \\
& \left\{ \begin{array}{lll}
A'(\vec Q,\vec P) *'_{(\vec Q,\vec P)} \rho'_{a;b}(\vec Q,\vec P) & = &  a \rho'_{a;b}(\vec Q,\vec P) \\
\rho'_{a;b}(\vec Q,\vec P) *'_{(\vec Q,\vec P)} A'(\vec Q,\vec P) & = &  b \rho'_{a;b}(\vec Q,\vec P)
\end{array} \right. &
\end{eqnarray}
where $\rho'_{a;b}(\vec Q,\vec P)$ is the $a$-left and $b$-right $*'_{(\vec Q,\vec P)}$-genfunction of $A'(\vec Q,\vec P)$.
These equations transform covariantly under arbitrary phase space diffeomorphisms yielding, in any coordinates, identical mathematical solutions and thus identical physical predictions:
\begin{equation}
P(A'(\vec Q,\vec P;t)=a)=  \int d^N\vec Q \int d^N\vec P (\mbox{det} J'^{ij}_{(\vec Q,\vec P)})^{-1/2}
\rho'_{a;a}(\vec Q,\vec P)  f'_W(\vec Q,\vec P;t).
\end{equation}

Let us consider the stargenvalue equation (7) in more detail. Let $\{\hat A_1,...,\hat A_N\}$ and $\{\hat B_1,...,\hat B_N\}$ be two sets of commuting observables satisfying $[\hat A_i,\hat B_j]=i \hbar \delta_{ij}$. Let $A'_i(\vec Q,\vec P)=W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat A_i)$ and $B'_i(\vec Q,\vec P)=W^{(\vec q,\vec p)}_{(\vec Q,\vec P)}(\hat B_i)$. Then the simultaneous solution of:
\begin{equation}
\left\{ \begin{array}{lll}
A'_i(\vec Q,\vec P) *'_{(\vec Q,\vec P)} \rho'_{a_1,..,a_N;b_1,..,b_N}(\vec Q,\vec P)& = & a_i \rho_{a_1,..,a_N;b_1,..,b_N}(\vec Q,\vec P) \\
\rho'_{a_1,..,a_N;b_1,..,b_N}(\vec Q,\vec P) *'_{(\vec Q,\vec P)} A'_i(\vec Q,\vec P)
& = & b_i \rho_{a_1,..,a_N;b_1,..,b_N}(\vec Q,\vec P) \quad , \quad i=1..N
\end{array} \right.
\end{equation}
is given by \cite{Nuno1}:
\begin{eqnarray}
\rho'_{a_1,..,a_N;b_1,..,b_N}(\vec Q,\vec P) & \equiv & \Delta_{*'_{(\vec Q,\vec P)}}(A'_1,...,A'_N;a_1,...,a_N;b_1,...,b_N)   ={\prod_{i=1}^{N}}{}_{*'_{(\vec Q,\vec P)}}  \rho'_{a_i;b_i}(\vec Q,\vec P), \\
\rho'_{a_i;b_i}(\vec Q,\vec P) & \equiv & \Delta_{*'_{(\vec Q,\vec P)}}(A'_i;a_i;b_i) = \frac{1}{2\pi}
e_{*'_{(\vec Q,\vec P)}}^{\frac{i}{\hbar}(b_i-a_i)B'_i(\vec Q,\vec P)}*'_{(\vec Q,\vec P)}
\int \,dk e_{*'_{(\vec Q,\vec P)}}^{ik(A'_i(\vec Q,\vec P)-b_i)},
\end{eqnarray}
where ${\prod_{i=1}^{N}}{}_{*'_{(\vec Q,\vec P)}} $ is the $N$-fold starproduct and the starexponential $e_{*'_{(\vec Q,\vec P)}}^{ikA'_i(\vec Q,\vec P)} \equiv E_{*'_{(\vec Q,\vec P)}}(k,\vec Q,\vec P)$ is defined as the solution of the differential problem:
$ \frac{\partial}{\partial k} E_{*'_{(\vec Q,\vec P)}}(k,\vec Q,\vec P)=i A_i'(\vec Q,\vec P) *'_{(\vec Q,\vec P)} E_{*'_{(\vec Q,\vec P)}}(k,\vec Q,\vec P)$ and $ E_{*'_{(\vec Q,\vec P)}}(0,\vec Q,\vec P)=1$, which is unique  for the Weyl symbol $A'_i(\vec Q,\vec P)$ of a generic self-adjoint operator $\hat A_i$ (a result that is just the Weyl-Wigner translation of an equivalent result in operator quantum mechanics \cite{Reed}). Furthermore $ E_{*'_{(\vec Q,\vec P)}}(k,\vec Q,\vec P)= \sum_{n=1}^{+\infty} \frac{(ik)^n}{n!} {\prod_{j=1}^{N}}{}_{*'_{(\vec Q,\vec P)}} A'_i(\vec Q,\vec P)$ whenever $k$ is in the radius of convergence of the series, which justifies the notation used in eq.(11). We also introduced the $*$-delta notation
$\Delta_{*'_{(\vec Q,\vec P)}}(A'_1,...,A'_N;a_1,...,a_N;b_1,...,b_N) $ that will be extensively used later on. Notice that the stargenfunctions $\rho'_{a_1,..,a_N;b_1,..,b_N}(\vec Q,\vec P)$ are the $_{(\vec Q,\vec P)}^{(\vec q,\vec p)}$-Weyl transform of the projectors $|a_1,..,a_N><b_1,..,b_N|$ where $|a_1,..,a_N>$ and $|b_1,..,b_N>$ are simultaneous eigenvectors of $\hat A_i$, $i=1..N$. Moreover the stargenfunctions $\rho'_{a_i;b_i}(\vec Q,\vec P)$ are the $_{(\vec Q,\vec P)}^{(\vec q,\vec p)}$-Weyl transform of the projector:
\begin{equation}
|a_i><b_i|=\int\, d\vec z\, |a_i, \vec z><b_i,\vec z|= \frac{1}{2\pi} \int \, dk e^{\frac{i}{\hbar}(b_i-a_i)\hat B_i} e^{ik(\hat A_i-b_i)},
\end{equation}
where $\vec z=(a_1,...,a_{i-1},a_{i+1},...,a_N)$ is the $N-1$ array of degeneracy indices. The Weyl transform of the first identity in eq.(12) yields the inverse relation of eq.(10):
\begin{equation}
\rho'_{a_i;b_i}(\vec Q,\vec P)= \int \, da_1..da_{i-1}da_{i+1}..da_N \rho'_{a_1,..,a_i,..,a_N;a_1,..,b_i,..,a_N}(\vec Q,\vec P).
\end{equation}
Finally, the diagonal elements $\rho'_{a_i;a_i}(\vec Q,\vec P)$ are given by:
\begin{equation}
\rho'_{a_i;a_i}(\vec Q,\vec P)= \frac{1}{2\pi} \int \, dk e_{*'_{(\vec Q,\vec P)}}^{ik (A'_i(\vec Q,\vec P)-a_i)}=
\Delta_{*'_{(\vec Q,\vec P)}}(A'_i;a_i;a_i) \equiv  \Delta_{*'_{(\vec Q,\vec P)}}(A'_i-a_i),
\end{equation}
this being the object that enters the probability functional (8).
The stargenfunctions (10,11) transform as scalars under arbitrary phase space coordinate transformations. For instance $
\rho'_{a_i;a_i}(\vec Q,\vec P)$ satisfies:
\begin{equation}
\rho'_{a_i;a_i}(\vec Q,\vec P)=\Delta_{*'_{(\vec Q,\vec P)}}(A'_i(\vec Q,\vec P)-a_i)=\Delta_{*_{(\vec q,\vec p)}}(A_i(\vec q(\vec Q,\vec P),\vec p(\vec Q,\vec P))-a_i)=\rho_{a_i;a_i}(\vec q(\vec Q,\vec P),\vec p(\vec Q,\vec P))
\end{equation}
where $\rho_{a_i;a_i}(\vec q,\vec p)$ is the diagonal solution of the stargenvalue equation in the $(\vec q,\vec p)$-representation.

Lastly, we should point out that the former results (in the presented form) are valid whenever we are able to provide the complete sets of commuting observables $\{\hat A_i\}$ and $\{\hat B_i\}$. In fact this restriction can be considerably weakened. It is crucial to the overall approach that the complete set $\{\hat A_i\}$ exists, but the requirement on the existence of the set $\{\hat B_i\}$ can be discarded, while preserving the validity of eqs.(8,10,13,14) exactly and that of eqs.(11,12) under slight modifications. This is the case, for instance, when $\hat A_i$ displays a discrete spectrum. The reader should refer to \cite{Nuno1} for a detailed presentation of these results.

\section{Deformation quantization of a parametrized non-relativistic system}

Let us consider an arbitrary $N$-dimensional dynamical system with configuration variables $\vec q=(q_1,...,q_N)$, described by the Lagrangian $L_0(\vec q,\frac{d\vec q}{dt})$.  Starting from the standard action $S=\int dt L_0(\vec q,\frac{d\vec q}{dt})$ we impose the time reparametrization invariance by introducing an unphysical time $\tau$ and promoting $t$ to a configuration variable. The action is now re-written as \cite{Henneaux}:
\begin{equation}
S=\int d \tau \, {\dot t} L_0 \left( \vec q,\frac{\dot{\vec q}}{\dot t} \right)
\end{equation}
where the dot represents the derivative with respect to $\tau$. The Legendre transform yields the Hamiltonian formulation living in a $(2N+2)$-dimensional phase space spanned by the canonical variables $t,P_t,\vec q,\vec p$ where $P_t=\partial L/ \partial \dot{t}$, $p_i=\partial L/ \partial \dot{q}_i$, $i=1..N$ and $L(t,\dot t,\vec q,\dot{\vec q})={\dot t} L_0 \left( \vec q,\frac{\dot{\vec q}}{\dot t} \right)$. This, as expected, is a "zero-Hamiltonian" system:
\begin{equation}
H=\lambda \phi \quad , \quad \phi = P_t+ H_0(\vec q,\vec p)
\end{equation}
where $\phi$ is a first class constraint, $\lambda $ is a Lagrange multiplier and $H_0$ is the Hamiltonian associated to the original Lagrangian $L_0$.

Dirac's quantization procedure imposes the constraint as a restriction on the space of physical states: $ \hat{\phi} |\psi> =0$. In the density matrix formulation this equation reads:
\begin{equation}
\hat{\phi} |\psi><\psi| = |\psi><\psi| \hat{\phi}= 0.
\end{equation}
To find the complete specification of its solutions, and according to the general method of section 2, the first step is to introduce the complete set of observables $\{\hat{\phi},\hat{\vec A}=(\hat A_1,...,\hat A_N) \}$ and the set of operators $\hat{\vec B}=(\hat B_1,...,\hat B_N)$, generators of  translations in the spectrum of  $\hat{\vec A}$, satisfying $[\hat A_j,\hat B_k]=i\hbar \delta_{jk} $ and $[\hat{\phi},\hat B_k ]=0$, $j,k=1..N$. We easily find that:
\begin{equation}
\hat A_j=\hat F_j(\hat{\vec q},\hat{\vec p},-\hat t) \quad , \quad \hat B_k=\hat G_k(\hat{\vec q},\hat{\vec p},-\hat t)
\end{equation}
where $\hat F_j(\hat{\vec q},\hat{\vec p},t)$ and $\hat G_k(\hat{\vec q},\hat{\vec p},t)$ are the solutions of the original Heisenberg equations:
\begin{equation}
\frac{\partial \hat F_j}{\partial t}=[\hat F_j,\hat H_0] \quad , \quad  \frac{\partial \hat G_k}{\partial t}=[\hat G_k,\hat H_0] \quad, \quad j,k=1..N
\end{equation}
together with the initial conditions $\hat F_j(\hat{\vec q},\hat{\vec p},0)=\hat q_j$ and $\hat G_k(\hat{\vec q},\hat{\vec p},0)=\hat p_k$, satisfy the aforementioned requirements. $\hat{\vec A}$ and $\hat{\vec B}$ are the quantum histories of the system.

The phase space representation of eq.(18) is determined by the Weyl transform. We may write:
\begin{equation}
\phi *_{(t,\phi,{\vec A},{\vec B})} f_W (t,\phi,{\vec A},{\vec B}) = f_W(t,\phi,{\vec A},{\vec B}) *_{(t,\phi,{\vec A},{\vec B})} \phi =0
\end{equation}
by using the map $W_{(t,\phi,{\vec A},{\vec B})}$ or equivalently:
\begin{equation}
\phi(t,P_t,{\vec q},{\vec p}) *_{(t,P_t,{\vec q},{\vec p})} f_W (t,P_t,{\vec q},{\vec p}) = f_W(t,P_t,{\vec q},{\vec p}) *_{(t,P_t,{\vec q},{\vec p})} \phi(t,P_t,{\vec q},{\vec p}) =0
\end{equation}
by using the map $W_{(t,P_t,{\vec q},{\vec p})}$. We start by considering the simplest representation, which is provided by eq.(21). In this case the Wigner function is a left and right zero $(t,\phi,{\vec A},{\vec B})$-stargenfunction of the constraint symbol $\phi$. To determine $f_W$ explicitly we will follow the procedure described in section 2. From eqs.(10,11,14), the fundamental zero stargenfunctions of the Hamiltonian constraint (which will be designated by $\rho_{{\vec a},{\vec b}} \def \rho_{h=0,{\vec a};h=0,{\vec b}}$) are given by:
\begin{eqnarray}
\rho_{{\vec a},{\vec b}}(t,\phi,{\vec A},{\vec B}) & = & \Delta_{*_{(t,\phi,{\vec A},{\vec B})}}(\phi,{\vec A};h=0,{\vec a};h=0,{\vec b}) = \Delta_{*_{(t,\phi,{\vec A},{\vec B})}}(\phi) *_{(t,\phi,{\vec A},{\vec B})} \Delta_{*_{(t,\phi,{\vec A},{\vec B})}}({\vec A};{\vec a};{\vec b}) =\nonumber \\
& = & \Delta _{*_{(t,\phi,{\vec A},{\vec B})}}(\phi ){\prod_{j=1}^{N}}{}_{
{*_{(t,\phi,{\vec A},{\vec B})}}}\left\{ e_{*_{(t,\phi,{\vec A},{\vec B})}}^{\frac{i}{\hbar} (b_j-a_j)B_j} *_{(t,\phi,{\vec A},{\vec B})} \Delta _{*_{(t,\phi,{\vec A},{\vec B})}}(A_j-b_j) \right\}\nonumber \\
& = & \delta(\phi) {*_{(t,\phi,{\vec A},{\vec B})}}\prod_{j=1}^{N} \left\{ e^{\frac{i}{\hbar} (b_j-a_j)B_j} *_{(t,\phi,{\vec A},{\vec B})} \delta(A_j-b_j) \right\}\nonumber \\
& = & \delta (\phi) \prod_{j=1}^N \left\{ \frac{1}{2\pi} \sum_{n=0}^{+\infty} \frac{1}{n!} \left( \frac{-i\hbar}{2} \right)^n e^{\frac{i}{\hbar} (b_j-a_j)B_j} \left[\frac{i}{\hbar} (b_j-a_j) \frac{\partial }{\partial A_j} \right]^n \int dk \, e^{ik(A_j-b_j)} \right\} \nonumber \\
& = & \frac{\delta(\phi)}{(2\pi)^N}\prod_{j=1}^N \left\{ e^{\frac{i}{\hbar} (b_j-a_j)B_j} \int \, dk \sum_{n=0}^{+\infty} \frac{1}{n!} \left[\frac{i}{2}k(b_j-a_j)\right]^n e^{ik(A_j-b_j)} \right\} \nonumber \\
& = & \delta(\phi) \prod_{j=1}^N \left\{ e^{\frac{i}{\hbar} (b_j-a_j)B_j} \frac{1}{2\pi} \int \, dk  e^{ik(A_j-b_j+\frac{b_j}{2}-\frac{a_j}{2})}\right\} \nonumber \\
& = & \delta(\phi) \prod_{j=1}^N \left\{ e^{\frac{i}{\hbar} (b_j-a_j)B_j} \delta(A_j-\frac{a_j}{2}-\frac{b_j}{2}) \right\}
 =  \delta(\phi) e^{\frac{i}{\hbar} (\vec b-\vec a)\cdot \vec B} \delta(\vec A-\frac{\vec a+\vec b}{2})
\end{eqnarray}
where ${\prod_{j=1}^{N}}{}_{
{*_{(t,\phi,{\vec A},{\vec B})}}}$ stands for the $N$-fold starproduct and $\prod_{j=1}^{N}$ for the standard $N$-fold product of functions. The stargenfunctions $\rho_{{\vec a},{\vec b}}(t,\phi,{\vec A},{\vec B})$ are simultaneously solutions of eq.(21) and:
\begin{equation}
\left\{ \begin{array}{lll}
A_j*_{(t,\phi,{\vec A},{\vec B})}\rho_{{\vec a},{\vec b}}(t,\phi,{\vec A},{\vec B}) & = & a_j \rho_{{\vec a},{\vec b}}(t,\phi,{\vec A},{\vec B}) \\
\rho_{{\vec a},{\vec b}}(t,\phi,{\vec A},{\vec B}) *_{(t,\phi,{\vec A},{\vec B})} A_j & = & b_j \rho_{{\vec a},{\vec b}}(t,\phi,{\vec A},{\vec B}) \quad , j=1..N
\end{array} \right.
\end{equation}
Notice that the diagonal elements ${\vec a}={\vec b}$ fully identify a history of the system: $\rho_{{\vec a},{\vec a}}(t,\phi,{\vec A},{\vec B})=\delta(\phi)\prod_{j=1}^N \delta(A_j-a_j)$.
It is also straightforward to realize that the fundamental stargenfunctions (23) are the $(t,\phi,{\vec A},{\vec B})$-Weyl transform of the projectors:
\begin{equation}
|h=0,{\vec a}><h=0,{\vec b}| = \hat{\Delta}(\hat{\phi})\prod_{j=1}^N\left\{ e^{\frac{i}{\hbar} (b_j-a_j) \hat B_j}  \hat{\Delta} (\hat A_j-b_j) \right\}
\end{equation}
where the general ket satisfies:
\begin{equation}
\hat{\phi} |h,{\vec x}> =h|h,{\vec x}> \quad \mbox{and} \quad \hat A_j |h,{\vec x}> = x_j |h,{\vec x}>
\end{equation}
and
\begin{equation}
\hat{\Delta}(\hat{\phi})= \int d{\vec x} |h=0,{\vec x}><h=0,{\vec x}| = \frac{1}{2\pi} \int dk e^{ik \hat{\phi}},
\end{equation}
is the operator analogue of (14).
The most general solution of eq.(21) is a linear combination of the fundamental solutions (23):
\begin{equation}
f_W(t,\phi,{\vec A},{\vec B}) =\frac{1}{(2\pi \hbar)^N}  \int d{\vec a} \int d{\vec b} C({\vec a}) C^* ({\vec b}) \rho_{{\vec a},{\vec b}} (t,\phi,{\vec A},{\vec B})
\end{equation}
where $C({\vec a})$ obeys to the normalization condition that is induced by the normalization of the Wigner function. Let us then calculate its norm:
\begin{eqnarray}
&& \int \int \int \int dtd\phi d{\vec A}d{\vec B} f_W (t,\phi,{\vec A},{\vec B})
\nonumber \\
&=&  \frac{1}{(2\pi \hbar)^N} \int \int \int dt d{\vec A}d{\vec B} \int d{\vec a} \int d{\vec b}
C({\vec a}) C^* ({\vec b})\prod_{j=1}^N e^{\frac{i}{\hbar} (b_j-a_j)B_j}  \delta (A_j-\frac{a_j+b_j}{2}) = \nonumber \\
&=& \int dt \int d{\vec a} \int d{\vec b} C({\vec a})C^*({\vec b})  \delta (\vec b-\vec a) =
\int dt \int d{\vec a} C({\vec a})C^*({\vec a})
\end{eqnarray}
The divergence of the previous integral indicates that we are integrating over the gauge orbits thus spoiling the normalization of the Wigner function. In this case this is quite simple to correct. In fact we just have to introduce the phase space measure $d\mu = d\phi d{\vec A}d{\vec B}$ and use it from now on whenever we have to integrate the Wigner function. The procedure corresponds to cutting the gauge orbits through a single time hypersurface. With this measure the Wigner function satisfies:
\begin{equation}
\int d \mu f_W = 1 \quad, \forall t
\end{equation}
provided the parameters $C({\vec a})$ satisfy $\int d{\vec a} | C({\vec a})|^2 =1$. The proper normalization of the Wigner function determines a new phase space measure and a restriction on the factors $C({\vec a})$. These parameters display a natural physical interpretation. To see this explicitly let us calculate the probabilities for the output of a measurement of ${\vec A}$. The general ${\vec x}=(x_1,...,x_N)$-left and -right stargenfunction of ${\vec A}$ (and simultaneously $y$-left and -right stargenfunction of the constraint) is:
\begin{eqnarray}
 \rho_{y,{\vec x};y,{\vec x}}  & = & \Delta_{*_{(t,\phi,{\vec A},{\vec B})}}(\phi,{\vec A};y,{\vec x};y,{\vec x}) = \Delta_{*_{(t,\phi,{\vec A},{\vec B})}}(\phi -y) {\prod_{j=1}^N}{}_{*_{(t,\phi,{\vec A},{\vec B})}} \Delta_{*_{(t,\phi,{\vec A},{\vec B})}} (A_j-x_j)=\nonumber \\
& = & \delta (\phi -y)\prod_{j=1}^N \delta (A_j-x_j) =\delta (\phi -y) \delta (\vec A-\vec x)
\end{eqnarray}
Therefore, the probability density for ${\vec A}={\vec x}$ and $\phi=y$ is given by:
\begin{eqnarray}
&& {\cal P}(\phi=y, {\vec A}= {\vec x} )  =  \int d \mu \, f_W(t,\phi,{\vec A},{\vec B}) \delta (\phi-y) \delta ({\vec A}-{\vec x}) = \int \, d{\vec B} f_W(t,y,{\vec x},{\vec B}) \nonumber \\
& = &   \frac{1}{(2\pi \hbar)^N} \int \, d{\vec B} \int \int d{\vec a} d{\vec b}  C({\vec a}) C^* ({\vec b}) \delta (y) \prod_{j=1}^N \left\{ e^{\frac{i}{\hbar} (b_j-a_j)B_j}  \delta (x_j-\frac{a_j+b_j}{2}) \right\}\nonumber \\
& = &  \int \int d{\vec a} d{\vec b} \, C({\vec a})C^*({\vec b})  \delta({\vec b}-{\vec a}) \delta ({\vec x}-\frac{{\vec a}+{\vec b}}{2}) \delta (y) = |C({\vec x})|^2 \delta (y)
\end{eqnarray}
from where it follows:
\begin{equation}
P(\phi =y , {\vec A}={\vec x} ) = \lim_{\epsilon \to 0^+} \int_{y-\epsilon}^{y + \epsilon} dy' {\cal P} (\phi = y', {\vec A}={\vec x} ) = |C(\vec x)|^2 \delta_{y,0}.
\end{equation}
It is clear from the previous equation that the term $|C({\vec x})|^2$ represents the probability for the system to be found in the history ${\vec A}={\vec x}$. Under the measurement of ${\vec A}$ with output ${\vec x}$ the Wigner function will indeed collapse to the state:
\begin{equation}
\rho_{0,{\vec x};0,{\vec x} } = \delta (\phi ) \delta (\vec A-\vec x)
\end{equation}
Let us also point out that in the $(t,\phi,{\vec A},{\vec B})$-representation the Wigner function is static both with respect to the external time $\tau$ and to the physical time $t$. In fact from eqs.(23,28) we see that $\frac{\partial }{\partial t} f_W=0$.

The former results can now be easily translated to the $(t,P_t,{\vec q},{\vec p})$ representation where the intention is to solve eq.(22). The two representations are related by the unitary transformation:
\begin{eqnarray}
t(t,P_t,{\vec q},{\vec p}) & = & U^{-1}*_{(t,P_t,{\vec q},{\vec p})}t*_{(t,P_t,{\vec q},{\vec p})}U=t \nonumber \\
\phi (t,P_t,{\vec q},{\vec p}) & = & W_{(t,P_t,{\vec q},{\vec p})}(\hat{\phi}) =U^{-1}*_{(t,P_t,{\vec q},{\vec p})}P_t*_{(t,P_t,{\vec q},{\vec p})}U=P_t+H_0({\vec q},{\vec p}) \nonumber \\
A_j (t,{\vec q},{\vec p}) & = & W_{(t,P_t,{\vec q},{\vec p})}(\hat{A_j}) =U^{-1}*_{(t,P_t,{\vec q},{\vec p})}q_j*_{(t,P_t,{\vec q},{\vec p})}U \quad, \quad j=1..N \nonumber \\
 B_j (t,{\vec q},{\vec p})& = & W_{(t,P_t,{\vec q},{\vec p})}(\hat{B_j}) =U^{-1}*_{(t,P_t,{\vec q},{\vec p})}p_j*_{(t,P_t,{\vec q},{\vec p})}U \quad , \quad j=1..N
\end{eqnarray}
where $U=e_{*_{(t,P_t,{\vec q},{\vec p})}}^{\frac{i}{\hbar}H_0({\vec q},{\vec p})t}$.The fundamental stargenfunctions are:
\begin{eqnarray}
&& \rho_{{\vec a},{\vec b}}(t,P_t,{\vec q},{\vec p})   =  \Delta_{*_{(t,P_t,{\vec q},{\vec p})}}(\phi,{\vec A};h=0,{\vec a};h=0,{\vec b}) \\
& = & \Delta_{*_{(t,P_t,{\vec q},{\vec p})}}(P_t+H_0({\vec q},{\vec p})) {\prod_{j=1}^N}{}_{*_{{(t,P_t,{\vec q},{\vec p})}}}\left\{
e_{*_{(t,P_t,{\vec q},{\vec p})}}^{\frac{i}{\hbar} (b_j-a_j)B_j(t,{\vec q},{\vec p})} {*_{(t,P_t,{\vec q},{\vec p})}} \Delta _{*_{(t,P_t,{\vec q},{\vec p})}}(A_j(t,{\vec q},{\vec p})-b_j) \right\} \nonumber
\end{eqnarray}
which in general do not simplify any further as in equation (23). These stargenfunctions are the ${(t,P_t,{\vec q},{\vec p})}$-Weyl transform of the projector (25). The most general solution of (22) is then:
\begin{equation}
f_W(t,P_t,{\vec q},{\vec p})=\frac{1}{(2\pi \hbar)^N} \int d{\vec a} d{\vec b} \, C({\vec a})C^*({\vec b}) \rho_{{\vec a},{\vec b}}(t,P_t,{\vec q},{\vec p})
\end{equation}
and can be obtained directly from the Wigner function (28) by applying the unitary transformation (35). Hence, $f_W(t,P_t,{\vec q},{\vec p})$
is properly normalized for the phase space measure $d\mu = dP_t d{\vec q}d{\vec p} $ provided the coefficients $C({\vec a})$ satisfy the normalization $\int d{\vec a} \, C({\vec a})C^*({\vec a})=1$. Notice that in general the new Wigner function does not have support exclusively on the classical constraint hypersurface. This is due to the non-local character of the $*$-delta functions in eq.(36). Notice also that $f_W$ does not
evolve with respect to the external time $\tau$ but it displays the standard time evolution with respect to the canonical time $t$:
\begin{eqnarray}
\frac{\partial f_W(t,P_t,{\vec q},{\vec p})}{\partial \tau} & = & [H(t,P_t,{\vec q},{\vec p}),f_W(t,P_t,{\vec q},{\vec p})]_{M_{(t,P_t,{\vec q},{\vec p})}}=0 \nonumber \\
& \Longleftrightarrow &
\frac{\partial f_W(t,P_t,{\vec q},{\vec p})}{\partial t}=[H_0({\vec q},{\vec p}),f_W(t,P_t,{\vec q},{\vec p})]_{M_{({\vec q},{\vec p})}}
\end{eqnarray}
We conclude that the $(t,P_t,{\vec q},{\vec p})$-representation yields the extended phase space Schr\"odinger picture for the quantum generally covariant system. Likewise the $(t, \phi, {\vec A},{\vec B})$-representation provides the phase space Heisenberg picture.

\section{Causal representation}

We now study another possible phase space representation of the system. The first step is to specify a new set of classical phase space coordinates $(t,P_t,{\vec Q},{\vec P})$. Let us define the phase space diffeomorphism by:
\begin{equation}
t=t \quad ,\quad P_t=\phi-H_0({\vec A},{\vec B}) \quad , \quad {\vec Q}={\vec Q}(t,{\vec A},{\vec B}) \quad , \quad {\vec P}={\vec P}(t,{\vec A},{\vec B})
\end{equation}
where $H_0({\vec A},{\vec B})=W_{({\vec A},{\vec B})}(\hat H_0)=W_{({\vec q},{\vec p})}(\hat H_0)|_{{\vec q}={\vec A} \wedge {\vec p}={\vec B}}$ and ${\vec Q}(t,{\vec A},{\vec B})$, ${\vec P}(t,{\vec A},{\vec B})$ satisfy:
\begin{equation}
\frac{\partial {\vec Q}(t,{\vec A},{\vec B})}{\partial t}=\{{\vec Q}(t,{\vec A},{\vec B}),H_0({\vec A},{\vec B})\}_{({\vec A},{\vec B})} \quad , \quad
\frac{\partial {\vec P}(t,{\vec A},{\vec B})}{\partial t}=\{{\vec P}(t,{\vec A},{\vec B}),H_0({\vec A},{\vec B})\}_{({\vec A},{\vec B})}
\end{equation}
together with the initial conditions: ${\vec Q}(0,{\vec A},{\vec B})={\vec A}$ and ${\vec P}(0,{\vec A},{\vec B})={\vec B}$. That is ${\vec Q}(t,{\vec A},{\vec B})$ and ${\vec P}(t,{\vec A},{\vec B})$ constitute the {\it classical} time evolution of the deparametrized system. The transformation (39) is canonical and can be easily inverted. It yields:
\begin{equation}
t=t \quad ,\quad \phi=\phi'(t,P_t,{\vec Q},{\vec P})=P_t+H_0({\vec Q},{\vec P}) \quad , \quad {\vec A}={\vec A}'(t,{\vec Q},{\vec P}) \quad , \quad {\vec B}={\vec B}'(t,{\vec Q},{\vec P})
\end{equation}
where this time $H_0({\vec Q},{\vec P})=W_{({\vec q},{\vec p})}(\hat H_0)|_{{\vec q}={\vec Q} \wedge {\vec p}={\vec P}}$ and ${\vec A}'(t,{\vec Q},{\vec P})$, ${\vec B}'(t,{\vec Q},{\vec P})$ satisfy:
\begin{equation}
\frac{\partial {\vec A}'(t,{\vec Q},{\vec P})}{\partial t}=\{H_0({\vec Q},{\vec P}),{\vec A}'(t,{\vec Q},{\vec P})\}_{({\vec Q},{\vec P})} \quad , \quad
\frac{\partial {\vec B}'(t,{\vec Q},{\vec P})}{\partial t}=\{H_0({\vec Q},{\vec P}),B'(t,{\vec Q},{\vec P})\}_{({\vec Q},{\vec P})}
\end{equation}
together with the initial conditions: ${\vec A}'(0,{\vec Q},{\vec P})={\vec Q}$ and ${\vec B}'(0,{\vec Q},{\vec P})={\vec P}$. This means that the {\it functions} ${\vec A}'(t,{\vec Q},{\vec P})$ and ${\vec B}'(t,{\vec Q},{\vec P})$ are the classical histories of the system. We should point out that in general the two functions ${\vec A}'(t,{\vec Q},{\vec P})$ and ${\vec A}(t,{\vec q},{\vec p})$ given by eqs.(41,35), respectively, as well as ${\vec B}'(t,{\vec Q},{\vec P})$ and ${\vec B}(t,{\vec q},{\vec p})$, display different functional forms (they are respectively the classical and the quantum Weyl-Wigner histories of the system). To make this point explicit, we introduced the prime notation which is in agreement with the fact that ${\vec A}'(t,{\vec Q},{\vec P})=W_{(t,P_t,{\vec Q},{\vec P})}^{(t,\phi,{\vec A},{\vec B})}(\hat {\vec A})$, ${\vec B}'(t,{\vec Q},{\vec P})=W_{(t,P_t,{\vec Q},{\vec P})}^{(t,\phi,{\vec A},{\vec B})}(\hat {\vec B})$ and $\phi'(t,P_t,{\vec Q},{\vec P})=W_{(t,P_t,{\vec Q},{\vec P})}^{(t,\phi,{\vec A},{\vec B})}(\hat{\phi} )$. Finally, let us concisely designate the transformation (41) by $\vec T: \R^{(2N+2)} \longrightarrow \R^{(2N+2)};\; (t,P_t,{\vec Q},{\vec P}) \longrightarrow (t,\phi,{\vec A},{\vec B})=\vec T(t,P_t,{\vec Q},{\vec P})$.

The transformation $\vec T$ is canonical but in general it does not preserve neither the starproduct nor the Moyal bracket.
Using the generalized Weyl transform $W_{(t,P_t,{\vec Q},{\vec P})}^{(t,\phi,{\vec A},{\vec B})}$ we find a new phase space representation of the eigenvalue equation (18):
\begin{equation}
\phi'(t,P_t,{\vec Q},{\vec P}) *'_{(t,P_t,{\vec Q},{\vec P})} f'_W (t,P_t,{\vec Q},{\vec P}) = f'_W(t,P_t,{\vec Q},{\vec P}) *'_{(t,P_t,{\vec Q},{\vec P})} \phi'(t,P_t,{\vec Q},{\vec P}) =0
\end{equation}
where the $*'_{(t,P_t,{\vec Q},{\vec P})}$ is the covariant starproduct given by (3) with:
\begin{equation}
J'^{ij}_{(t,P_t,{\vec Q},{\vec P})}=J^{ij}_{(t,\phi ,{\vec A},{\vec B})} \quad \mbox{and} \quad \Gamma'^i_{jk} =  \frac{\partial O'^i}{\partial O^b} \frac{\partial^2 O^b}{\partial O'^j \partial O'^k},
\end{equation}
where $O^b \in \{t,\phi ,{\vec A},{\vec B}\}$ and $O'^i \in \{t, P_t,{\vec Q},{\vec P}\}$.
The solutions of eq.(43) can be read from eq.(23,28):
\begin{equation}
f'_W(t,P_t,{\vec Q},{\vec P}) =\frac{1}{(2\pi \hbar)^N} W_{(t,P_t,{\vec Q},{\vec P})}^{(t,\phi,{\vec A},{\vec B})}(|\psi><\psi|)= \frac{1}{(2\pi \hbar)^N} \int d{\vec a} \int d{\vec b} C({\vec a}) C^* ({\vec b}) \rho'_{{\vec a},{\vec b}} (t,P_t,{\vec Q},{\vec P}),
\end{equation}
where:
\begin{eqnarray}
&&\rho'_{{\vec a},{\vec b}}(t,P_t,{\vec Q},{\vec P}) = \Delta_{*'_{(t,P_t,{\vec Q},{\vec P})}}(\phi'(t,P_t,{\vec Q},{\vec P}),{\vec A}'(t,{\vec Q},{\vec P});h=0,{\vec a};h=0,{\vec b})  \nonumber \\
&=& \Delta_{*'_{(t,P_t,{\vec Q},{\vec P})}}(\phi'(t,P_t,{\vec Q},{\vec P})) {\prod_{j=1}^N}{}_{*'_{(t,P_t,{\vec Q},{\vec P})}} \Delta_{*'_{(t,P_t,{\vec Q},{\vec P})}}(A_j'(t,{\vec Q},{\vec P}),a_j,b_j) =\nonumber \\
& = & \Delta _{*_{(t,\phi,{\vec A},{\vec B})}}(\phi'(t,P_t,{\vec Q},{\vec P}) ){\prod_{j=1}^N}{}_{*_{(t,\phi,{\vec A},{\vec B})}}\left\{ e_{*_{(t,\phi,{\vec A},{\vec B})}}^{\frac{i}{\hbar} (b_j-a_j)
B'_j(t,{\vec Q},{\vec P})} *_{(t,\phi,{\vec A},{\vec B})} \Delta _{*_{(t,\phi,{\vec A},{\vec B})}}(A'_j(t,{\vec Q},{\vec P})-b_j) \right\}\nonumber \\
& = & \delta (\phi' (t,P_t,{\vec Q},{\vec P}) )\prod_{j=1}^N \left\{e^{\frac{i}{\hbar} (b_j-a_j)B'_j(t,{\vec Q},{\vec P})}  \delta (A_j'(t,{\vec Q},{\vec P})-\frac{a_j+b_j}{2})\right\} \nonumber \\
& = & \delta (\phi' (t,P_t,{\vec Q},{\vec P}) ) e^{\frac{i}{\hbar} (\vec b-\vec a)\cdot \vec B'(t,{\vec Q},{\vec P})}  \delta (\vec A'(t,{\vec Q},{\vec P})-\frac{\vec a+\vec b}{2})
\end{eqnarray}
The new Wigner function satisfies:
\begin{equation}
f_W'(t,P_t,{\vec Q},{\vec P})=f_W(t,\phi '(t,P_t,{\vec Q},{\vec P}),\vec A'(t,{\vec Q},{\vec P}),\vec B'(t,{\vec Q},{\vec P}))
\end{equation}
where $f_W$ is given by eq.(28).
Furthermore and since the coordinate transformation is canonical we have:
\begin{equation}
d\mu=d\phi d{\vec A}d{\vec B}=  \delta (t) dt  d\phi d{\vec A}d{\vec B}= \delta (t) dt dP_td{\vec Q}d{\vec P}=dP_td{\vec Q}d{\vec P},
\end{equation}
and the new Wigner function obeys the proper normalization: $\int d \mu f'_W = 1$.
Also notice that both the diagonal and the non-diagonal stargenfunctions (and thus also the Wigner function) have support only on the classical constraint hypersurface a property that it is not shared by the previous representation (36,37).

Finally, let us briefly elaborate on the dynamical structure of the system in this representation.
It is clear from eq.(43) that the Wigner function is static:
\begin{eqnarray}
\frac{\partial}{\partial \tau} f'_W(t,P_t,{\vec Q},{\vec P};\tau) & = & [H'(t,P_t,{\vec Q},{\vec P}),f'_W(t,P_t,{\vec Q},{\vec P};\tau)]_{M'_{(t,P_t,{\vec Q},{\vec P})}} \nonumber \\
& = & \lambda [\phi' (t,P_t,{\vec Q},{\vec P}),f'_W(t,P_t,{\vec Q},{\vec P};\tau)]_{M'_{(t,P_t,{\vec Q},{\vec P})}} =0,
\end{eqnarray}
confirming the typical picture of frozen dynamics. However, the Wigner function does evolve with respect to the physical time $t$. From eqs.(42,47) we have:
\begin{eqnarray}
\frac{\partial}{\partial t} f'_W(t,P_t,{\vec Q},{\vec P}) & = & \frac{\partial f_W}{\partial \phi}(\vec T(t,P_t,{\vec Q},{\vec P}))\frac{\partial \phi'}{\partial t}+\sum_{j=1}^N \frac{\partial f_W}{\partial A_j}(\vec T(t,P_t,{\vec Q},{\vec P}))\frac{\partial A'_j}{\partial t} + \sum_{j=1}^N \frac{\partial f_W}{\partial B_j}(\vec T(t,P_t,{\vec Q},{\vec P}))\frac{\partial B'_j}{\partial t}  \nonumber \\
&=& \sum_{j=1}^N \frac{\partial f_W}{\partial A_j} (\vec T(t,P_t,{\vec Q},{\vec P}))\{H_0,A'_j\}_{(t,P_t,{\vec Q},{\vec P})} + \sum_{j=1}^N \frac{\partial f_W}{\partial B_j}(\vec T(t,P_t,{\vec Q},{\vec P})) \{H_0,B'_j\}_{(t,P_t,{\vec Q},{\vec P})} \nonumber \\
&=& \{H_0,f'_W\}_{(t,P_t,{\vec Q},{\vec P})},
\end{eqnarray}
reproducing the classical Liouville equation $\{H',f'_W\}_{(t,P_t,{\vec Q},{\vec P})} =0$.

We see that the $^{(t,\phi , {\vec A},{\vec B})}_{(t,P_t,{\vec Q},{\vec P})}$-representation of the system leads to an interesting mathematical picture: the Hamiltonian vector field lives on the extended phase space and is given by:
\begin{equation}
\xi_H = \lambda \left\{ \frac{\partial}{\partial t} + \sum_{j=1}^N \frac{\partial H_0}{\partial P_j} \frac{\partial}{\partial Q_j} -\sum_{j=1}^N
\frac{\partial H_0}{\partial Q_j} \frac{\partial}{\partial P_j} \right\}.
\end{equation}
The flows of this vector field define lines in phase space. These lines cross each time hypersurface once and only once: they are the histories of the system. These histories are one-dimensional hypersurfaces that can be identified by the values of the $1+2N$ constants of motion: $\phi, {\vec A}, {\vec B}$. Along these lines the Wigner distribution function is constant. However, the correlations between the canonical variables, namely between ${\vec Q}$ and $t$, do change and time evolution is imprint on these correlations. The interesting point is that this picture is not of the classical description but of the quantum mechanical instead.

A natural question is then: what happened to the quantum content of the theory? What happened to the interfering trajectories and to the non-local behavior? The answer is that the quantum features have been completely removed from the distributional sector and are now exclusively placed in the observables' sector of the theory. This can be checked explicitly by applying the proper generalized Weyl transform to one of the fundamental operators of the system. We have for instance:
\begin{equation}
W_{(t,P_t,{\vec Q},{\vec P})}^{(t,\phi,{\vec A},{\vec B})} (\hat q_1)=q_1(t,\phi,{\vec A},{\vec B})|_{(t,\phi,{\vec A},{\vec B})=\vec T(t,P_t,{\vec Q},{\vec P})}=q_1(t,\phi' (t,P_t,{\vec Q},{\vec P}),A'(t,{\vec Q},{\vec P}),B'(t,{\vec Q},{\vec P})),
\end{equation}
leading to the time evolution:
\begin{eqnarray}
\frac{dq_1}{dt}& = & \frac{\partial q_1}{\partial t}+ \frac{\partial q_1}{\partial \phi}\frac{\partial \phi'}{\partial t} +\sum_{j=1}^N
\frac{\partial q_1}{\partial A_j}\frac{\partial A'_j}{\partial t} + \sum_{j=1}^N\frac{\partial q_1}{\partial B_j} \frac{\partial B'_j}{\partial t} \nonumber \\
& = & [q_1,H_0]_{M_{({\vec A},{\vec B})}}-\sum_{j=1}^N\frac{\partial q_1}{\partial A_j} \{A'_j,H_0 \}_{({\vec Q},{\vec P})} -\sum_{j=1}^N \frac{\partial q_1}{\partial B_j}
\{B'_j,H_0 \}_{({\vec Q},{\vec P})} \nonumber \\
&=&   [q_1,H_0]_{M_{({\vec A},{\vec B})}}-\{q_1,H_0 \}_{({\vec A},{\vec B})},
\end{eqnarray}
where the identity $\frac{\partial q_1}{\partial t}= [q_1,H_0]_{M_{({\vec A},{\vec B})}}$ follows from eq.(20) (or alternatively from eq.(35)), we used the fact that $\phi '$ is time independent and that the transformation $({\vec Q},{\vec P}) \to ({\vec A},{\vec B})$ is canonical.  Equation (53) does in fact constitute a quantum correction to the classical statistical description where we have:
\begin{equation}
\frac{\partial \rho}{\partial t}= \{H_0,\rho \} \quad \mbox{and} \quad \frac{d q_1}{d t}=0 .
\end{equation}
Notice that in the classical description ${\vec Q}={\vec q}$,  ${\vec P}={\vec p}$ and ${\vec q}(t,\phi' (t,P_t,{\vec q},{\vec p}),{\vec A}'(t,{\vec q},{\vec p}),{\vec B}'(t,{\vec q},{\vec p}))={\vec q}$.

We finally conclude by pointing out that a causal representation can also be obtained for deparametrized systems by using an explicit "time dependent phase space representation" where the starproduct and Moyal bracket are themselves (scalar) time dependent \cite{Nuno7}. In this approach we apply a time dependent generalization of the Weyl map to the density matrix formulation of the deparametrized system and obtain a phase space causal representation of the Wigner function, which is formally identical to the one described by eqs.(45,46,52), although $t$ is a canonical variable in the former equations and an external scalar parameter in the approach of \cite{Nuno7}. Accordingly, the two resulting quasidistributions live on different phase spaces.
In spite of this the two formulations are consistent with each other leading to the single conclusion that a phase space formulation of quantum mechanics where the distributional sector displays a fully classical causal structure is made possible through a suitable choice of representation. This may either be a (scalar) time dependent representation in the deparametrized phase space or a history representation in the generally covariant setting.

\section{Example}

To illustrate our previous results let us consider the simple system compose by two coupled particles and described by the Hamiltonian:
\begin{equation}
H_0= \frac{ p_1^2}{2M} + \frac{ p_2^2}{2m} + k  q_1 p_2^2
\end{equation}
where $( q_1, p_1)$ are the canonical variables of the particle of mass $M$, $(q_2,p_2)$ those of the particle of mass $m$ and $k$ is a coupling constant.

The generally covariant version of this system is obtained by promoting $t$ to a canonical variable and imposing the time reparametrization invariance. The
extended Hamiltonian formulation of the system lives on a $6$-dimensional phase space spanned by the canonical variables $t,P_t,\vec q=(q_1,q_2),\vec p=(p_1,p_2)$ which satisfy the commutation relations $\{t,P_t\}=1$, $\{q_i,p_j\}=\delta_{ij}$, $i,j=1,2$, all others being zero. Upon quantization the
system is described by the "zero Hamiltonian":
\begin{equation}
\hat H=\lambda \hat{\phi} \quad , \quad \hat{\phi}=\hat{P}_t+\hat H_0
=\hat{P}_t+\frac{\hat p_1^2}{2M} + \frac{\hat p_2^2}{2m} + k  \hat q_1\hat p_2^2
\end{equation}
where $\hat{\phi}$ is the first class Hamiltonian constraint and $\lambda$ is a Lagrange multiplier. In standard the Dirac formulation the physical states of this system are the wave functions $\psi$ solutions of the constraint equation:
\begin{equation}
\hat{\phi} |\psi>=0
\end{equation}

We now address the deformation quantization of the system. As in the main text, three distinct quantum phase space representations will be presented.
\\

{\bf 1.} {\it The map $W_{(t,P_t,\vec q,\vec p)}$ and the generally covariant Schr\"odinger picture}.

The Weyl map $W_{(t,P_t,\vec q,\vec p)}$ yields the $(t,P_t,\vec q,\vec p)$-constraint symbol:
\begin{equation}
\phi= W_{(t,P_t,\vec q,\vec p)}(\hat{\phi})=P_t+\frac{ p_1^2}{2M} + \frac{ p_2^2}{2m} + k q_1 p_2^2
\end{equation}
and the quantum phase space version of the Hamiltonian constraint (57) is given by:
\begin{equation}
\phi (t,P_t,\vec q,\vec p) *_{(t,P_t,\vec q,\vec p)}f_W(t,P_t,\vec q,\vec p) =f_W(t,P_t,\vec q,\vec p) *_{(t,P_t,\vec q,\vec p)} \phi (t,P_t,\vec q,\vec p)=0
\end{equation}
The solutions of this right and left stargenvalue equation are given by eqs.(36,37)
and they do not easily simplify any further. It is worth noticing that due to the non-local character of the $*$-product the phase space quasidistribution $f_W$ will have support on phase-space regions which are not classically allowed. One can easily calculate the evolution of $f_W$ with respect to the physical time $t$. From eq.(59) we have:
\begin{eqnarray}
&&\frac{\partial f_W}{\partial \tau}=\lambda [\phi,f_W]_{M_{(t,P_t,\vec q,\vec p)}}=0 \Longleftrightarrow
\frac{\partial f_W}{\partial t}=[H_0,f_W]_{M_{(\vec q,\vec p)}}\nonumber \\ &\Longleftrightarrow &\frac{\partial f_W}{\partial t}=\{H_0,f_W\}_{{(\vec q,\vec p)}}
+\frac{\hbar^2}{24} \left[2\left\{2kp_2,\frac{\partial^2 f_W}{\partial q_2 \partial p_1}\right\}_{(\vec q,\vec p)} -
\left\{2kq_1,\frac{\partial^2 f_W}{\partial q_2^2}\right\}_{(\vec q,\vec p)} \right]
\end{eqnarray}
which is obviously not of the form of the Liouville equation. Consequently, the Wigner function does not display a classical causal structure.\\

{\bf 2.} {\it The map $W_{(t,\phi,\vec A,\vec B)}$ and the generally covariant Heisenberg picture}.

Following the approach of section 3 our first step is to determine the quantum histories associated to $\hat H$. We easily find that [check eqs.(19,20)]:
\begin{equation}
\left\{ \begin{array}{lll}
\hat A_1 & = & \hat q_1 - \frac{\hat p_1}{M} \hat t - \frac{k}{2M} \hat p_2^2\hat t^2 \\
\hat B_1 & = & \hat p_1+k\hat p_2^2\hat t \\
\hat A_2 & = & \hat q_2 - \left\{\frac{\hat p_2}{m} + 2k\hat q_1\hat p_2 \right\}\hat t + \frac{k}{M} \hat p_1\hat p_2\hat t^2 + \frac{k^2}{3M}\hat p_2^3\hat t^3 \\
\hat B_2 & = &\hat p_2
\end{array} \right.
\end{equation}
satisfy the requisites following eq.(18) i.e. $\hat A_j$, $\hat B_j$, $j=1,2$ commute with the constraint $\hat{\phi}$ and with $\hat t$ and furthermore they satisfy the Heisenberg algebra $[\hat A_1,\hat B_1]=[\hat A_2,\hat B_2]=i\hbar$, all other commutators being zero. Since $[\hat t,\hat{\phi}]=i\hbar$ the set
$\{\hat t,\hat{\phi},\hat{\vec A}=(\hat A_1,\hat A_2),\hat{\vec B}=(\hat B_1,\hat B_2)\}$ is a complete set of fundamental operators for this system.

Using the Weyl map $W_{(t,\phi,\vec A,\vec B)}$ we find the Heisenberg phase space representation of the constraint equation (57):
\begin{equation}
\phi *_{(t,\phi,\vec A,\vec B)} f_W(t,\phi,\vec A,\vec B)=f_W(t,\phi,\vec A,\vec B)*_{(t,\phi,\vec A,\vec B)}\phi=0
\end{equation}
The fundamental solutions of this equation $\rho_{\vec a,\vec b}$ (where $\vec a=(a_1,a_2)$ and $\vec b=(b_1,b_2)$), and also satisfying:
\begin{equation}
A_j *_{(t,\phi,\vec A,\vec B)} \rho_{\vec a,\vec b}= a_j \rho_{\vec a,\vec b} \quad , \quad
\rho_{\vec a,\vec b}*_{(t,\phi,\vec A,\vec B)} A_j=b_j \rho_{\vec a,\vec b}
\quad, \quad j=1,2
\end{equation}
are given by:
\begin{equation}
\rho_{\vec a,\vec b}(t,\phi,\vec A,\vec B)=\delta(\phi) e^{\frac{i}{\hbar}\{ (b_1- a_1) B_1+(b_2-a_2)B_2 \}} \delta( A_1-\frac{ a_1+ b_1}{2})\delta( A_2-\frac{ a_2+ b_2}{2})
\end{equation}
And the Wigner function, solution of (62), is just a hermitian combination of the fundamental solutions. We have:
\begin{equation}
f_W(t,\phi,\vec A,\vec B) =\frac{1}{2\pi \hbar}  \int d\vec a \int d\vec b C(\vec a) C^* (\vec b) \rho_{\vec a,\vec b} (t,\phi,\vec A,\vec B)
\end{equation}
where $C(\vec a)$ obeys to the normalization condition that is induced by the normalization of the Wigner function: $\int d\vec a |C(\vec a)|^2=1$.
The Wigner function satisfies eq.(62) which implies that $\frac{\partial f_W}{\partial \tau}=0$. We also have from eq.(64) that $\frac{\partial f_W}{\partial t}=0$. That is, the Wigner function is static both with respect to the unphysical scalar parameter as well as to the physical time. This is an expected result since, in this representation, the Wigner quasidistribution is an exclusive function of observables of the system (the histories) i.e. of quantities that commute with the Hamiltonian constraint.

On the other hand, in this representation, the stargenfunctions of the fundamental variables $\hat t,\hat{\vec q}$ and $\hat{\vec p}$ do evolve with respect to the physical time. For instance (let $|x>$ be the general eigenket of $\hat q_1$ with associated eigenvalue $x$):
\begin{eqnarray}
g_{x}(t,\phi,\vec A,\vec B) & = & W_{(t,\phi,\vec A,\vec B)} (|x><x|)  =  \Delta_{*_{(t,\phi,\vec A,\vec B)}} (q_1(t,\vec A,\vec B)-x)  \\
&=& \frac{1}{2\pi} \int d k \, e_{*_{(t,\phi,\vec A,\vec B)}}^{ik(q_1(t,\vec A,\vec B)-x)}=\frac{1}{2\pi} \int d k \, e^{ik(q_1(t,\vec A,\vec B)-x)}=\delta (q_1(t,\vec A,\vec B)-x) \nonumber
\end{eqnarray}
where $q_1(t,\vec A,\vec B)$ can be easily derived from eq.(61): $q_1(t,\vec A,\vec B)=W_{(t,\phi,\vec A,\vec B)}(\hat q_1(\hat t,\hat{\vec A},\hat{\vec B}))= A_1 + \frac{ B_1}{M} t - \frac{k}{2M}  B_2^2 t^2$. Hence, $g_{x}(t,\phi ,\vec A,\vec B)$ satisfies:
\begin{equation}
\frac{\partial }{\partial t} g_{x}(t,\phi ,\vec A,\vec B) =  [g_{x}(t,\phi ,\vec A,\vec B),H_0(\vec A,\vec B)]_{M_{(\vec A,\vec B)}}=
\{g_{x}(t,\phi ,\vec A,\vec B),H_0(\vec A,\vec B)\}_{(\vec A,\vec B)},
\end{equation}
where $H_0(\vec A,\vec B)=W_{(\vec A,\vec B)}(\hat H_0)=
 \frac{ B_1^2}{2M} + \frac{ B_2^2}{2m} + k  A_1 B_2^2$.
We conclude that the Weyl transform $W_{(t,\phi,\vec A,\vec B)}$ casts the phase space dynamics in the Heisenberg picture. Accordingly, the time dependence is exclusively displayed by the observable (stargenfunction) sector of the theory.\\

{\bf 3.} {\it The map $W_{(t,P_t,\vec q,\vec p)}^{(t,\phi,\vec A,\vec B)}$ and the generally covariant causal picture}.

Following the prescription of section 4 let us introduce a new set of  phase space coordinates $(t,P_t,\vec Q=(Q_1,Q_2),\vec P=(P_1,P_2))$ defined by eqs.(39,40). We get: $t=t$, $P_t=\phi-H_0(\vec A,\vec B)$ and,
\begin{equation}
\left\{ \begin{array}{lll}
Q_1 & = & A_1 + \frac{B_1}{M} t - \frac{k}{2M}  B_2^2 t^2 \\
 P_1 & = &  B_1-k B_2^2 t \\
 Q_2 & = &  A_2 + \left\{\frac{ B_2}{m} + 2k A_1 B_2 \right\} t \\
&& + \frac{k}{M}  B_1 B_2 t^2 - \frac{k^2}{3M} B_2^3 t^3 \\
 P_2 & = & B_2
\end{array} \right.
\Longleftrightarrow
\left\{ \begin{array}{lll}
A_1 & = & Q_1 - \frac{P_1}{M} t - \frac{k}{2M}  P_2^2 t^2 \\
 B_1 & = &  P_1+k P_2^2 t \\
 A_2 & = &  Q_2 - \left\{\frac{ P_2}{m} + 2k Q_1 P_2 \right\} t \\
&& + \frac{k}{M}  P_1 P_2 t^2 + \frac{k^2}{3M} P_2^3 t^3 \\
 B_2 & = & P_2
\end{array} \right.
\end{equation}
We notice that $\vec Q(t,\vec A,\vec B)$ and $\vec P(t,\vec A,\vec B)$ coincide with $W_{(t,\phi,\vec A,\vec B)}(\hat{\vec q})$, $W_{(t,\phi,\vec A,\vec B)}(\hat{\vec p})$, respectively (this is an easy result that follows from eq.(61)), i.e the classical and the quantum histories of this system are exactly the same. Indeed, eq.(68) solves both the Moyal and the Hamiltonian equations of motion. In the notation of section 4 we have $\vec A'=\vec A$ and  $\vec B'=\vec B$ (one should notice that this result is not valid in general). Hence, for this system, we are not required to introduce a second set of "classical coordinates" and can simplify the notation by making:  $Q_j=q_j$ and $P_j=p_j$, $j=1,2$.

We now consider
the action of the generalized Weyl map $W_{(t,P_t,\vec q,\vec p)}^{(t,\phi,\vec A,\vec B)}$. The associated covariant starproduct $*'_{(t,P_t,\vec q,\vec p)}$ and Moyal bracket $[\, , \, ]_{M_{(t,P_t,\vec q,\vec p)}}$ are characterized by (let $O'^1=P_t,\,O'^2=p_1,\, O'^3=p_2,\,O'^4=t,\, O'^5=q_1,\,O'^6=q_2$,
$O^1=\phi ,\,O^2=B_1,\, O^3=B_2,\,O^4=t,\, O^5=A_1,\,O^6=A_2$ and $i,j=1..6$) [check eq.(5)]:
\begin{eqnarray}
& J'^{ij}_{(\vec q,\vec p)} =  J^{ij}_{(\vec q,\vec p)} & \\
& \Gamma'^1_{22}=-\Gamma'^5_{24}=-\Gamma'^5_{42}=\frac{1}{M}, \quad \Gamma'^1_{24}=\Gamma'^1_{42}=-\Gamma'^5_{44}=\frac{k}{M}p_2^2, \quad \Gamma'^1_{33}=2kA_1(t,\vec q,\vec p), & \nonumber \\
& \Gamma'^1_{34}= \Gamma'^1_{43}=-\Gamma'^6_{44}=-\frac{2k}{M}p_1p_2,
\quad \Gamma'^1_{35}=\Gamma'^1_{53}=\Gamma'^2_{34}=\Gamma'^2_{43}=-\Gamma'^6_{45}=-\Gamma'^6_{54}=2kp_2,& \nonumber \\
& \Gamma'^1_{44}= \frac{k^2}{M}p_2^4, \quad \Gamma'^2_{33}=-\Gamma'^6_{35}=-\Gamma'^6_{53}= 2kt, \quad
 \Gamma'^5_{33}=\Gamma'^6_{23}=\Gamma'^6_{32}=\frac{k}{M}t^2 ,& \nonumber \\
& \Gamma'^5_{34}=\Gamma'^5_{43}=\frac{k}{M}p_2t ,\quad
\Gamma'^6_{33}=\frac{2k^2}{M}p_2t^3 ,\quad \Gamma'^6_{34}=\Gamma'^6_{43}=-\frac{1}{m}-2kA_1(t,\vec q,\vec p) & \nonumber
\end{eqnarray}
all other Christoffel symbols being zero.

In this representation the constraint equation reads:
\begin{equation}
(P_t+H_0(\vec q,\vec p)) *'_{(t,P_t,\vec q,\vec p)} f'_W (t,P_t,\vec q,\vec p) = f'_W(t,P_t,\vec q,\vec p) *'_{(t,P_t,\vec q,\vec p)}  (P_t+H_0(\vec q,\vec p))=0
\end{equation}
where $H_0(\vec q,\vec p)=W_{(t,P_t,\vec q,\vec p)}^{(t,\phi,\vec A,\vec B)}(\hat H_0)$ is given by eq.(54). The fundamental solutions of eq.(70) are given by:
\begin{eqnarray}
&&\rho'_{\vec a,\vec b}(t,P_t,\vec q,\vec p) = \Delta_{*'_{(t,P_t,\vec q,\vec p)}}(\phi(t,P_t,\vec q,\vec p),\vec A(t,\vec q,\vec p);h=0,\vec a;h=0,\vec b)  \nonumber \\
& = & \delta (\phi (t,P_t,\vec q,\vec p) ) e^{\frac{i}{\hbar} (\vec b-\vec a)\cdot\vec B(t,\vec q,\vec p)}  \delta (\vec A(t,\vec q,\vec p)-\frac{\vec a+\vec b}{2}).  \nonumber \\
&=& \delta (P_t+H_0(\vec q,\vec p) ) e^{\frac{i}{\hbar} \{(b_1- a_1) (p_1+kp_2^2t) + (b_2-a_2) p_2 \}}  \delta \left(q_1 - \frac{p_1}{M} t - \frac{k}{2M}  p_2^2 t^2 -\frac{a_1+b_1}{2}\right)\nonumber \\
&& \cdot \delta \left( q_2 - \left\{\frac{ p_2}{m} + 2k q_1 p_2 \right\} t
 + \frac{k}{M}  p_1 p_2 t^2 + \frac{k^2}{3M} p_2^3 t^3
-\frac{a_2+b_2}{2}\right)
\end{eqnarray}
and the Wigner function is once again a hermitian combination of the fundamental solutions $\rho'_{\vec a,\vec b}$:
\begin{equation}
f'_W(t,P_t,\vec q,\vec p) =\frac{1}{(2\pi \hbar)^N} W_{(t,P_t,\vec q,\vec p)}^{(t,\phi,\vec A,\vec B)}(|\psi><\psi|)= \frac{1}{(2\pi \hbar)^N} \int d\vec a \int d\vec b C(\vec a) C^* (\vec b) \rho'_{\vec a,\vec b} (t,P_t,\vec q,\vec p),
\end{equation}
It is related with the $(t,\phi,\vec A,\vec B)$-representation by:
\begin{equation}
f'_W(t,P_t,\vec q,\vec p)=f_W(0,\phi(t,P_t,\vec q,\vec p),\vec A(t,\vec q,\vec p),\vec B(t,\vec q,\vec p))
\end{equation}
where $f_W$ is given by eqs.(64,65). We conclude that: 1) The support of $f_W'$ is confined to the classically allowed regions [check eq.(71)] and 2) its evolution with respect to the physical time satisfies
the Liouville equation:
\begin{equation}
\frac{\partial f'_W}{\partial t}=\sum_{j=1}^2 \frac{\partial f_W}{\partial A_j}  \frac{\partial A_j}{\partial t} +\sum_{j=1}^2
\frac{\partial f_W}{\partial B_j}  \frac{\partial  B_j}{\partial t}
=\sum_{j=1}^2 \frac{\partial f_W}{\partial A_j} \{H_0,A_j\}_{(\vec q,\vec p)}+\sum_{j=1}^2\frac{\partial f_W}{\partial B_j} \{H_0,B_j\}_{(\vec q,\vec p)} =\{H_0,f'_W\}_{(\vec q,\vec p)}
\end{equation}
Hence, in this representation the quantum behavior is displayed by the stargenfunction sector alone.
However, for this system, we also have (let $z=q_1,p_1 \vee p_2$ and $|z_0>$ be a generic eigenket of $\hat z$ with associated eigenvalue $z_0$):
\begin{eqnarray}
W_{(t,P_t,\vec q,\vec p)}^{(t,\phi,\vec A,\vec B)}(|z_0><z_0|) & = & \Delta_{*'(t,P_t,\vec q,\vec p)}(z-z_0)=\Delta_{*(t,\phi,\vec A,\vec B)}(z(t,\vec A,\vec B)-z_0)|_{\vec A=\vec A(t,\vec q,\vec p) \wedge \vec B=\vec B(t,\vec q,\vec p)} \nonumber \\
& = & \delta(z(t,\vec A,\vec B)-z_0)|_{\vec A=\vec A(t,\vec q,\vec p) \wedge \vec B=\vec B(t,\vec q,\vec p)}= \delta(z-z_0)
\end{eqnarray}
where in the third step we used the fact that  $e_{*_{(t,\phi,A,B)}}^{ik(z(t,\vec A,\vec B)-z_0)}= e^{ik(z(t,\vec A,\vec B)-z_0)}$ for $z=q_1,p_1 \vee p_2$, (a simple result that follows from eq.(61)).
Hence, the former three fundamental stargenfunctions display a classical structure and satisfy $\frac{\partial}{\partial t}\Delta_{*'(t,P_t,q,p)}(z-z_0)=0$. We conclude that for this system, in this representation, the non-trivial (quantum) behavior is displayed by the stargenfunction $z=q_2$ alone.

\section{Conclusions}

We addressed the deformation quantization of the non-relativistic particle in the generally parametrized form and provided the complete specification of its physical quantum states in three different phase space representations.
We proved that in one of these representations the distributional sector of the system displays a classical causal structure. This result confirms a similar conclusion that has been recently obtained for a generic non-relativistic deparametrized system and reinforces the point of view that the De Broglie-Bohm formulation is not the unique possible causal formulation of quantum mechanics.
In the De Broglie-Bohm theory \cite{Holland,Nuno8,Nuno9} the source of quantum behavior is the quantum potential determining a causal (although not fully classical) dynamics for the quasidistribution. Furthermore, the theory also displays a non-trivial quantum correction to the momentum stargenfunction. On the other hand, in the "causal covariant formulation" presented here the quantum effects have been completely removed from the distributional sector (which now displays a fully classical causal structure) and the price to pay was the appearance of some further (quantum) corrections on the observable's sector of the theory.

We finish by recalling the point of view according to which the relation between the classical statistical and the quantum mechanical formulations of generally covariant systems should be further explored as it may provide key physical insights into some of the conceptual problems displayed by the quantum version of these systems.
From this perspective the deformation methods seem to be especially suited. We proved that (at least for the simplest case of the parametrized non-relativistic particle) the formal similarities between classical statistical and phase space quantum mechanics carry on intact to the generally covariant context. This close analogy is obviously superior to the one displayed by the standard operator formulation, and supports the point of view that the deformation methods should be further explored as an alternative, conceptually simpler approach to the quantization of generally covariant systems.

\paragraph*{Acknowledgments.}

 This work was partially supported by the grant POCTI/MAT/45306/2002.

redpajama_set_name: RedPajamaArXiv


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text:
Home/Music/T.I.'s Lawsuit Against Toymaker Ends in Mistrial Following Testimony Alleging Racism
T.I.'s Lawsuit Against Toymaker Ends in Mistrial Following Testimony Alleging Racism
A federal judge on Wednesday (Jan. 25) declared a mistrial in the high-profile courtroom battle pitting T.I. and wife Tameka "Tiny" Harris against toymaker MGA over a line of dolls, ending the proceedings after jurors heard inadmissible testimony claiming the company "steals from African Americans."
A day after attorneys for MGA argued that the "inflammatory" testimony about cultural appropriation had ruined their chances of a fair trial, Judge James V. Selna agreed, granting a mistrial. That means the case will need to be re-tried in front of a new jury at some point in the future.
Following the mistrial, MGA told Billboard that "diversity has always been a key value" at the company: "We are disappointed that the trial was cut short, but look forward to vindicating our rights in the next trial." An attorney for T.I. and Tiny did not immediately return a request for comment.
The ruling marks an abrupt end for the closely-watched intellectual property trial, in which T.I. and Tiny were trying to persuade a jury that MGA's line of "OMG" dolls stole their look and name from the OMG Girlz, a defunct teen pop trio created by Tiny and starring her daughter Zonnique Pullins.
In their 2021 complaint, T.I. and Tiny alleged that MGA had committed both "cultural appropriation and outright theft of the intellectual property," stealing the look of a group of "young multicultural women." The lawsuit included side-by-side images, aiming to show how each doll was directly based on a particular member of the OMG Girlz, who disbanded in 2015.
On the fifth day of the trial, jurors heard videotaped deposition testimony from a woman named Moneice Campbell, a former MGA customer. According to court documents, Campbell testified that she would no longer purchase the company's products because MGA "steals from African Americans and their ideas and profit off of it." She also said that "hundreds" of social media users had agreed with the accusations, citing the fact that "people often steal from the black community and make money off of it."
Earlier in the case, Judge Selna had already expressly prohibited such testimony from the trial. In one such order, he ruled that statements about "cultural appropriation" were "immaterial and impertinent" to the actual legal issues at play in the case and could not be made in front of jurors.
In a written motion filed after Tuesday's courtroom proceedings had concluded, MGA's lawyers demanded an immediate mistrial, arguing that the impact of the inadmissible testimony on the fairness of the case "cannot be understated."
"There is no way to unring the bell of the jury's hearing Ms. Campbell's emotionally charged accusations that MGA has been 'stealing' from the African-American community," the MGA attorneys wrote. "Her improper testimony cannot be challenged, rebutted or cured without drawing further attention to it."

redpajama_set_name: RedPajamaCommonCrawl


KEEDIO has several ideas in common with the Red Hat philosophy about open source and the technologies for enterprise.
For KEEDIO it is a huge pleasure, because from now on we will collaborate closely in our developments, projects and strategies to make our customers improve their businesses with the best support that we could have: the community and the company that is at the forefront of open source software development.
Red Hat has been for years the reference for the open source community and has become a strong leader developing open source software for the enterprise world.
Red Hat has found in KEEDIO a strong player, as a specialist in Big Data and Cloud Computing technologies that is aligned with its open source vision and IT services.


JAB is acquiring the UK food outlet for $2 billion. After spending roughly $30 bln on coffee and food chains over the past two years, the German conglomerate is a growing threat to larger rivals Nestle and Starbucks. An agile and ambitious model gives it a good shot at success.
JAB Holding Company, the conglomerate owned by Germany's billionaire Reimann family, said on May 29 that it will buy British sandwich and coffee shop chain Pret A Manger from Bridgepoint. The purchase price including debt is approximately 1.5 billion pounds, a source close to the deal said.
Private equity firm Bridgepoint bought Pret in 2008 for about 345 million pounds with the aim of increasing the number of stores by 15 percent each year. The food chain, which had sales of 223 million pounds and 200 shops in 2007, now has 530 stores worldwide and group revenue of 879 million pounds in the financial year that ended in December 2017.


Do I need to book a pre wedding day makeup application?
I like doing a trial run with all of my brides! I love making sure I can get to know your personal style and bring that into your Makeup look. We can go through your inspiration images, discuss details of your wedding day, such as: your dress, colors, flowers etc. and decide on a look that we both feel is fitting. Once I've completed your application, I will record all of the details and snap an image of the final product to have on hand for the wedding day. This will ensure that once you get to my chair on your wedding day, you can relax and know exactly what you're getting.
Do I Need more than one trial?
No, not at all. Even if there are a couple things you'd like to change after our initial meeting, I can just make note of it on your chart. The only time an additional trial might be recommended would be if you decided on a completely different look from what we had already done.
I consider 3 months prior to the wedding date to be the "Sweet Spot". Its not so far in advance that you forget what it looks like or what we discussed, but its not so close to the wedding day that it feels rushed. Doing it closer to 2 or 3 months in advance should give you confidence that you know exactly what you like and wont be changing your mind.
How do I communicate the look I want for my wedding day?
Every time you come across a makeup look you like, Pin It, Snap a screenshot or tear it out of a magazine. Even if you don't like a look in its entirety but you like one aspect of it, such as the lips or the eyes, just save it! Compile a few of these. I like flipping through all the ideas you have and being able to pull inspiration from each of them to create something that will suit your face and highlight your best features.
99% of my items are MAC Cosmetics. Through 10 years of experience MAC has proven to be the longest lasting and highest quality for color payoff. They are a cruelty free company as well as hypoallergenic.
Come in a neutral, solid top or tee. You don't want colors that will clash or throw off the coloring of the makeup.
Because I'm booked on the date of receiving a signed contract, I always recommend my clients to sign on the date of their trial. This is a way for you to ensure someone else won't book me on your wedding date. It's always good to have something in writing between us, that confirms date, time of service and rates.
What if I'm in another state or having a destination Wedding?
If budget allows, then consider bringing me with you! Traveling to the destination is optional, provided that travel expenses are covered. We would set up the trial prior to your trip. Another option is for us to have a private makeup lesson before your destination wedding. This would be a detailed lesson with step by step directions that you can recreate or give to an artist on site. If you're out of state and are hiring me to travel to you, we can set up your trial at your earliest convenience once I've arrived.
Should I do Makeup or hair first?
Makeup should be the very last thing you do prior to putting on your dress and starting photos. Fresh Makeup is the best makeup.
Should I do my makeup first or last?
It's best to do your makeup last so it's the freshest before photos.
How many appointments can you take on the wedding day?
I typically take between 5-10 people total for a bridal party. All bridal party appointments will be discussed and booked at the time of the trial session. If more than 10 appointments are needed, I have the option of bringing an assistant for an additional cost.
Can you stay through the wedding for touchups?
Absolutely! Many of my clients prefer to have me stay through the ceremony for touchups prior to the reception. I have some brides who even prefer to slightly change their look from the ceremony to the reception. I use all the products necessary to keep your makeup lasting all night. Touch ups are most frequently powder and lips, eye touch ups for those who may shed a few tears during the ceremony and toasts at the reception, and brides who wish to do a second look.
Gratuity is not expected but is greatly appreciated!


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A Guide To Birthstones Pt 1
Posted on March 7, 2018 by Mitchell Perry
The idea of birthstones dates back to ancient times when the Greeks, Egyptians, Romans, Persians and Indians all assigned different gemstones to each month of the year. According to the stories behind birthstones, legend has it that wearing a gemstone during its assigned month boosts its healing and therapeutic powers. For the full effect, individuals needed to own all twelve and alternate them monthly. The birthstone system we use today is believed to originate from the biblical system created around 40 AD. Jewish historian Titus Flavius Josephus declared that the 12 stones in Aaron's breastplate had a special connection to the months of the year and the signs of the Zodiac. In 1912, the National Association of Jewellers in America drew up a list that is still in use today, although there have been a few modifications along the way, most notably in 2002 with the addition of tanzanite as a birthstone for December.
Birthstones are a fantastic way to add a special meaning to jewellery and make great gifts to mark a birthday, anniversary, graduation or other memorable occasion. They can be incorporated into rings, necklaces, pendants, bracelets and more. Paired with diamonds, they can add a splash of colour and variety to your jewellery piece.
This guide will be separated into four parts, each giving an overview of three birthstones:
Birthstone for January, the word "garnet" comes from the 14th Century Middle English word "gernet" meaning dark red. The word is derived from Latin "granatum" which means seed, and is called so because of the gemstone's resemblance to the beautifully red seeds of the pomegranate. Garnet is actually the name of a group of minerals that comes in a rainbow of colours. Some rare garnets are even blue, colorless, or—most rare of all—change colors in different lights. But the most common colour is a beautiful range of reds, from rust coloured to deep violet-red. The folklore on garnet is extensive. Legend has it that the garnet can bring peace, prosperity and good health to the home. Some even called it the "Gem of Faith," and it's believed that to those who wear it and do good, more good will come. (Conversely, it was also said to bring very bad fortune to those who commit bad acts while wearing it.)
The garnet is a very durable gemstone (6.5-7.5 on the Mohs scale) found all over the world and thus is a firm favourite throughout the ages. In particular, the Victorians made garnets very popular during that time period and so is common amongst vintage jewellery pieces. Some of the loveliest garnet jewellery from that era mimics its pomegranate namesake, with clusters of tiny red gems forming a larger statement piece. Today, the garnet can be found in a range of jewellery pieces and styles, from beautiful rings to stunning tiaras. Since the garnet can come in a range of colours, rare garnets in green or blue make breathtaking pieces, especially in pendants or drop earrings.
When buying garnets, it should be noted that some garnets have inclusions that are part of the beauty of the overall stone (like "horsetails" in Demantoid garnets, or Hessonite garnets which sometimes have a "turbulent" look). So you may discover that you like the unique look these inclusions bring to the piece. No matter why or how you wear garnet, this beautiful gemstone is perfect for those who share January birthdays and get to start each New Year with a sense of goodwill, happiness, and purpose.
February – Amethyst
Often associated with qualities of peace, courage and stability, Amethyst is purple quartz and is a beautiful blend of violet and red that can be found in every corner of the earth. The name comes from the Ancient Greek, derived from the word "methustos," which means "intoxicated." Ancient wearers believed the gemstone could protect them from drunkenness. For many years, amethyst was held to be one of the most precious gemstones, often favoured by royalty or exclusively by the clergy as a symbol for the deity of Christ. It was even held for many years in the same regard as the diamond. It wasn't until the discovery of more abundant supplies of amethyst that it became a gemstone enjoyed by more than just the wealthiest buyers.
It is a crystalline quartz that sits at a relatively hard 7-8 on the (1-10) Mohs scale and as such it is a fairly easy stone to look after. Higher grade examples can be facetted into a jewellery gem, whereas lower-end material tends to be carved into beads and ornamental pieces. when buying, check the colour of the stone. Amethyst often has "stripes" or layers of colour from how and when it was formed, so it takes a skilled gemologist to cut and polish the stone to show the overall colour of the stone evenly. Avoid brownish or rust coloured tints to the stone and be careful the colour is not too deep or it can appear black in some lights. So, whether you're purchasing an amethyst for yourself or a loved one, you're making an investment in beauty that will stand the test of time. Amethyst makes a great gift for individuals born in February, or as a celebratory milestone for a 6th or 17th wedding anniversary.
March – Aquamarine and Bloodstone
Not only is aquamarine one of the March birthstones, it's also used to celebrate 19th wedding anniversaries. It's a beautiful stone with little or no yellow in it, so it looks great in many settings with different coloured metals and gemstones. The colour ranges from nearly clear to a strong dark blue and is a perfect way to communicate affection, tranquility, and peace. The second birthstone for March is bloodstone, a dark-green stone flecked with vivid red spots of iron oxide. Generally found embedded in rocks or in riverbeds as pebbles, primary sources for this stone are India, Brazil, and Australia.
For those looking for a good quality bloodstone, it is generally considered that a solid green colour with visible veins of red is best. It also comes in many shapes and cuts including traditional cuts like emerald, oval, and cushion. In regards to Aquamarine, it's generally accepted that lighter coloured aquamarines are less valuable than the stronger, deeper hues of blue or blue green. However, both stones make for a lovely purchase for any occasion, but especially for someone with a March birthday or to celebrate or re-kindle romantic love.
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Mitchell Perry
Gold Four Pronged Diamond Bracelet
Guide To Birthstones Pt 2
Tips for Choosing the Right Wedding Bands
The Facets of Buying Diamond Earrings in Hatton Garden
Different Types Of Diamond Cuts
What to look for in a Luxury Diamond Jeweller
Jewellery Spotlight
News (Demo)


Kohlberg believed that individuals progress through stages of moral reasoning in one specific order, without skipping any in between. Such perspectives should be mutually respected as unique to each person or community. Although Kohlberg insisted that stage six exists, he found it difficult to identify individuals who consistently operated at that level. Kohlberg divided this thought process into 6 stages, which he based on the findings of an experiment. For the stage two theorist, the world's perspective is often seen as. Just as every individual person is different, every child is different. He brought forth the idea that children learn moral sense from the interactions between the social world as well as their inner thoughts.
That is to say, most people take their moral views from those around them and only a minority think through ethical principles for themselves. Consider implementing a system to reward students who follow the rules and who exhibit helpful behaviors in the classroom. Social and Personality Development 5th ed. They involve the rare ability to simultaneously grasp several abstract principles when considering moral behavior. In this sense, each new stage is more cognitively adequate than the prior stage. Stage 5 emphasizes democratic processes, and stage 6 considers how all parties take one another's perspectives according to the principles of justice.
The individual attempts to take the perspective of all individuals. Justice is not black and white, but the circumstances of each situation are considered individually. In response to the Heinz story, many subjects say they understand that Heinz's motives were good, but they cannot condone the theft. Children develop not because they are shaped through external reinforcements but because their curiosity is aroused. Children, regardless of their beliefs, will always move to stage 4 thinking some time after stage 1 thinking because it is cognitively so much more sophisticated. What they preferred was the highest stage they heard, whether they fully understood it or not.
What Kohlberg was mainly interested in was not whether the boys judged the action right or wrong, but the reasons given for the decision. With law-and-order morality, they fulfill duties to which they have agreed to carry out. Harvard Educational Review, 47 4 , 481-517. Stage 3 is related to social roles. Consistent with this observation, Gilligan's theory of moral development does not value justice above other considerations.
Gilligan has made an initial effort to trace women's moral development. Recent research in moral psychology has brought back the value of witnessing moral exemplars in action or learning about their stories. Student C pointed out that the stranger violated no law. There are many more details to Piaget's work on moral judgment, but he essentially found a series of changes that occur between the ages of 10 and 12, just when the child begins to enter the general stage of formal operations. Kohlberg claims that there are, but the evidence does not always support this conclusion. However, Bee 1994 suggests that we also need to take account of: a habits that people have developed over time.
The individual views laws and rules as flexible tools for improving human purposes. He found that these reasons tended to change as the children got older. It is a behaviour that one has been accustomed to due to the laws and customs in a particular society. After Kohlberg's body was recovered and his death confirmed, former students and colleagues published special issues of scholarly journals to commemorate his contribution to developmental psychology. Democratic government is theoretically based on stage five reasoning.
They understood lower-stage reasoning, but they disliked it. Kohlberg insists, however, that we do not judge people to be at stage 5 merely from their verbal labels. Heavily influenced by the six-stage model, it made efforts to improve the criteria by using a test, the , to rate moral dilemmas similar to Kohlberg's. In addition, Kohlberg and his co-workers Colby et al. The ideal is formal justice, in which all parties evaluate one another's claims in an impartial manner. Presumably, if their group valued property over life, they would too.
The first criticism that I would like to address is that of his research methods. Dilemma discussions in schools was another method proposed by Kohlberg to increase moral reasoning. Blatt found that over half the students moved up one full stage after the 12 weeks. What would happen if we all started breaking the laws whenever we felt we had a good reason? He neglects the feminine voice of compassion, love, and non-violence, which is associated with the socialization of girls. Moral becomes more than consideration of close ties to others.
It is also to do with social factors. Laws are valid only insofar as they are grounded in justice, and a commitment to justice carries with it an obligation to disobey unjust laws. If a person is punished, they must have done wrong. Use our sample or order a custom written research paper from Paper Masters. Stage 6: Universal ethical principle orientation According to Kohlberg, this is the highest stage of functioning.