is achieved in terms of the overcomplete ensemble of localised and
normalisable abstract quantum states $|\vec{x};\lambda\rangle$ labelled by all
points of the corresponding classical and commutative configuration space even
though the coordinate operators are now noncommuting and thus not
diagonalisable, whatever the choice of real positive definite symmetric matrix
of length squared coefficients $\lambda_{ij}$. As a matter of fact these
states $|\vec{x};\lambda\rangle$ are a specific subclass of canonical quantum
phase space coherent states whose definition involves the coefficients
$\lambda_{ij}$, thus restricted to configuration space but still generating
through the $*_{\lambda}$-product the whole abstract Hilbert space on which
the algebra of abstract operators $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$ is
realised with a positive definite sesquilinear inner product. This one-to-one
map between abstract quantum states $|\varphi\rangle$ and
$\hat{x}_{i}$-enveloping algebra quantum operators
$\hat{\varphi}(\hat{\vec{x}}\,)$ is provided by the configuration space
integral representations,

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{x}\
\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x};\lambda\rangle,\qquad\langle\varphi_{1}|\varphi_{2}\rangle=\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1};\lambda|\,\hat{\varphi}^{\dagger}_{1}(\hat{\vec{x}}\,)\,\hat{\varphi}_{2}(\hat{\vec{x}}\,)\,|\vec{x}_{2};\lambda\rangle,$
(217)

while the inverse relation involves the momentum eigenbasis
$|\vec{p}\,\rangle$,

$\hat{\varphi}(\hat{\vec{x}}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\langle\vec{p}\,|\varphi\rangle.$
(218)

Finally let us briefly consider the time dependent Schrödinger equation with
an abstract Hamiltonian operator, say once again of the nonrelativistic form,

$\hat{H}=\frac{1}{2\mu}\hat{\vec{p}}\,^{2}\,+\,\hat{V}_{q}(\hat{\vec{x}}\,).$
(219)

On the $\hat{x}_{i}$-enveloping algebra this operator is thus represented as,

$\displaystyle\hat{\mathbb{H}}\left(\hat{\varphi}(\hat{\vec{x}})\,\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\mu}\hat{P}_{i}\left(\hat{P}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)\right)\,+\,\hat{V}_{q}\left(\hat{X}_{i}\right)\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)$
(220) $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\mu}\left(A^{-1}\right)_{ij}\left(A^{-1}\right)_{ik}\Big{[}\hat{x}_{j},\Big{[}\hat{x}_{k},\hat{\varphi}(\hat{\vec{x}}\,)\Big{]}\Big{]}\,+\,\hat{V}_{q}(\hat{\vec{x}}\,)\,\hat{\varphi}(\hat{\vec{x}}\,).$

Hence in this representation the time dependent Schrödinger equation is
expressed as, for any time dependent abstract quantum state $|\psi,t\rangle$
and its operator representation $\hat{\psi}(\hat{\vec{x}},t)$,

$i\hbar\frac{d}{dt}\hat{\psi}(\hat{\vec{x}},t)=\frac{\hbar^{2}}{2\mu}\left(A^{-1}\right)_{ij}\left(A^{-1}\right)_{ik}\Big{[}\hat{x}_{j},\Big{[}\hat{x}_{k},\hat{\psi}(\hat{\vec{x}},t)\Big{]}\Big{]}\,+\,\hat{V}_{q}(\hat{\vec{x}}\,)\,\hat{\psi}(\hat{\vec{x}},t).$
(221)

Given that the conserved inner product of a normalised quantum state
$|\psi,t\rangle$ is expressed as follows in terms of its
$\hat{x}_{i}$-enveloping algebra representation,

$1=\langle\psi,t|\psi,t\rangle=\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1};\lambda|\,\hat{\psi}^{\dagger}(\hat{\vec{x}},t)\,\hat{\psi}(\hat{\vec{x}},t)\,|\vec{x}_{2};\lambda\rangle,$
(222)

indicates that the probability density of the quantum system is now
represented by the following normalised bi-local density in configuration
space,

$\rho_{\lambda}(\vec{x}_{1},\vec{x}_{2},t)=\langle\vec{x}_{1};\lambda|\,\hat{\psi}^{\dagger}(\hat{\vec{x}},t)\,\hat{\psi}(\hat{\vec{x}},t)\,|\vec{x}_{2};\lambda\rangle,\qquad\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\rho_{\lambda}(\vec{x}_{1},\vec{x}_{2},t)=1,$
(223)

or even by the normalised probability density operator
$\hat{\rho}(\hat{\vec{x}},t)$ defined as,

$\hat{\rho}(\hat{\vec{x}},t)=\hat{\psi}^{\dagger}(\hat{\vec{x}},t)\,\hat{\psi}(\hat{\vec{x}},t),\qquad{\rm
with}\quad\rho_{\lambda}(\vec{x}_{1},\vec{x}_{2},t)=\langle\vec{x}_{1};\lambda|\,\hat{\rho}(\hat{\vec{x}},t)\,|\vec{x}_{2};\lambda\rangle.$
(224)

The conservation of probability under a unitary quantum time evolution then
translates into specific equations for either
$d\hat{\rho}(\hat{\vec{x}},t)/dt$ or
$\partial\rho_{\lambda}(\vec{x}_{1},\vec{x}_{2},t)\partial t$, realising in
the enveloping algebra representation the equivalent of an ordinary continuity
equation, of which the consideration and details are not included herein.

###  4.6 Quantum and classical Hilbert spaces

To proceed further with the above general analysis, let us now specifically
restrict to a configuration space of even dimension, $N=2n$ with $n\geq 1$,
and to a situation when the conjugate magnetic field $A_{ij}$ defines an
invertible antisymmetric matrix, ${\rm det}\,A_{ij}\neq 0$. Then in order to
connect with the heuristic construction by F. Scholtz and his collaborators of
quantum mechanics over the noncommutative euclidean plane, it proves useful to
consider an alternative coordinate basis in the configuration and momentum
spaces such that the conjugate magnetic field is then represented by a
$2\times 2$ block diagonalised antisymmetric $2n\times 2n$ matrix.

As is well known, any $2n\times 2n$ real antisymmetric matrix may be block
diagonalised through some O($2n$) orthogonal transformation into $2\times 2$
antisymmetric blocks. When the matrix is non-degenerate as is now the assumed
situation, the off-diagonal entries of these $2\times 2$ blocks are
nonvanishing.

Thus given the constant coefficients $A_{ij}$, there exists a O($2n$)
transformation $R_{(\alpha,a),i}$ with $\alpha,\beta=1,2,\cdots,n$ and
$a,b=1,2$, such that $R^{-1}=R^{T}$, namely,

$R_{(\alpha,a),i}\,R_{(\beta,b),i}=\delta_{\alpha\beta}\,\delta_{ab},\qquad
R_{(\alpha,a),i}\,R_{(\alpha,a),j}=\delta_{ij},\quad{\rm
det}\,R_{(\alpha,a),i}=\pm 1,$ (225)

and leading to the $2\times 2$ block diagonalisation of the matrix $A_{ij}$ in
the form of,

$\Theta_{(\alpha,a),(\beta,b)}=R_{(\alpha,a),i}\,A_{ij}\,R_{(\beta,b),j}=\theta_{\alpha}\,\delta_{\alpha\beta}\,\epsilon_{ab},\quad\left(\Theta^{-1}\right)_{(\alpha,a),(\beta,b)}=-\frac{1}{\theta_{\alpha}}\,\delta_{\alpha\beta}\,\epsilon_{ab},\quad\theta_{\alpha}>0,$
(226)

where the entries $\theta_{\alpha}>0$ are strictly positive, while
$\epsilon_{ab}$ is the 2-index antisymmetric symbol with $\epsilon_{12}=+1$.
Furthermore when requiring that the values $\theta_{\alpha}$ are ordered in
increasing or decreasing order relative to $\alpha=1,2,\cdots,n$, the O($2n$)
orthogonal transformation $R_{(\alpha,a),i}$ is then uniquely determined.

Applying the same linear transformation to the coefficients $\lambda_{ij}$
defining the quantum coherent states $|\vec{x};\lambda\rangle$, one has the
real positive definite symmetric matrix,

$\Lambda_{(\alpha,a),(\beta,b)}=R_{(\alpha,a),i}\,\lambda_{ij}\,R_{(\beta,b),j}.$
(227)

Likewise for the coordinate and momentum operators $\hat{x}_{i}$ and
$\hat{p}_{i}$, and the associated commuting variables $x_{i}$ and $p_{i}$ one
has the quantum operators $(\hat{x}_{\alpha,a},\hat{p}_{\alpha,a})$ and
variables $(x_{\alpha,a},p_{\alpha,a})$,

$\hat{x}_{\alpha,a}=R_{(\alpha,a),i}\,\hat{x}_{i},\qquad\hat{p}_{\alpha,a}=R_{(\alpha,a),i}\,\hat{p}_{i},\qquad
x_{\alpha,a}=R_{(\alpha,a),i}\,x_{i},\qquad
p_{\alpha,a}=R_{(\alpha,a),i}\,p_{i},$ (228)

of which the commutation relations are,

$\left[\,\hat{x}_{\alpha,a},\,\hat{x}_{\beta,b}\,\right]=i\,\theta_{\alpha}\,\delta_{\alpha\beta}\,\epsilon_{ab}\,\mathbb{I},\qquad\left[\,\hat{x}_{\alpha,a},\,\hat{p}_{\beta,b}\,\right]=i\hbar\,\delta_{\alpha\beta}\,\delta_{ab}\,\mathbb{I},\qquad\left[\,\hat{p}_{\alpha,a},\,\hat{p}_{\beta,b}\,\right]=0.$
(229)

Similarly for the Fock algebra generators
$(\hat{A}_{i},\hat{A}^{\dagger}_{i})$ (see (170)) in the presence of the
conjugate magnetic field which is now represented by the quantities
$\theta_{\alpha}>0$, one has the transformed operators,

$\hat{a}_{\alpha,a}=R_{(\alpha,a),i}\,\hat{A}_{i},\qquad\hat{a}^{\dagger}_{\alpha,a}=R_{(\alpha,a),i}\,\hat{A}^{\dagger}_{i},$
(230)

with the following expressions,

$\displaystyle\hat{a}_{\alpha,a}$ $\displaystyle=$
$\displaystyle\hat{x}_{\alpha,a}+\frac{i}{2\hbar}\left(\Lambda_{(\alpha,a),(\beta,b)}-i\Theta_{(\alpha,a),(\beta,b)}\right)\,\hat{p}_{\beta,b},$
$\displaystyle\hat{a}^{\dagger}_{\alpha,a}$ $\displaystyle=$
$\displaystyle\hat{x}_{\alpha,a}-\frac{i}{2\hbar}\left(\Lambda_{(\alpha,a),(\beta,b)}+i\Theta_{(\alpha,a),(\beta,b)}\right)\,\hat{p}_{\beta,b},$
(231)

and their general Fock algebra,

$\Big{[}\hat{a}_{\alpha,a},\hat{a}_{\beta,b}\Big{]}=0,\qquad\left[\hat{a}_{\alpha,a},\hat{a}^{\dagger}_{\beta,b}\right]=\Lambda_{(\alpha,a),(\beta,b)}\,\mathbb{I},\qquad\left[\hat{a}^{\dagger}_{\alpha,a},a^{\dagger}_{\beta,b}\right]=0.$
(232)

Associated to this $N=2n$ dimensional Fock algebra and the complex variables
$Z_{i}=x_{i}+i(\lambda_{ij}-iA_{ij})p_{j}/(2\hbar)$ introduced in (171), the
transformed complex variables thus now read as well,

$\displaystyle z_{\alpha,a}$ $\displaystyle=$ $\displaystyle
R_{(\alpha,a),i}\,Z_{i}=x_{\alpha,a}+\frac{i}{2\hbar}\left(\Lambda_{(\alpha,a),(\beta,b)}-i\Theta_{(\alpha,a),(\beta,b)}\right)\,p_{\beta,b},$
$\displaystyle z^{*}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
R_{(\alpha,a),i}\,Z^{*}_{i}=x_{\alpha,a}-\frac{i}{2\hbar}\left(\Lambda_{(\alpha,a),(\beta,b)}+i\Theta_{(\alpha,a),(\beta,b)}\right)\,p_{\beta,b}.$
(233)

Hence the canonical coherent states $|Z_{i};\Omega_{\lambda}\rangle$ related
to the initial algebra $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$ and the choice
of coefficients $\lambda_{ij}$ in the presence of the conjugate magnetic field
$A_{ij}$ are now realised as

$\displaystyle|Z_{i};\Omega_{\lambda}\rangle$ $\displaystyle\equiv$
$\displaystyle|z_{\alpha,a};\Omega_{\lambda}\rangle$ (234) $\displaystyle=$
$\displaystyle{\rm
exp}\left(z_{\alpha,a}\left(\Lambda^{-1}\right)_{(\alpha,a),(\beta,b)}\,\hat{a}^{\dagger}_{\beta,b}\,-\,z^{*}_{\alpha,a}\left(\Lambda^{-1}\right)_{(\alpha,a),(\beta,b)}\,\hat{a}_{\beta,b}\right)\,|\Omega_{\lambda}\rangle,$

with in particular

$|\vec{x};\lambda\rangle=|z_{\alpha,a}=x_{\alpha,a};\Omega_{\lambda}\rangle=e^{-\frac{i}{\hbar}x_{\alpha,a}\,\hat{p}_{\alpha,a}}\,|\Omega_{\lambda}\rangle=e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\Omega_{\lambda}\rangle.$
(235)

Consequently the fact that the localised and normalisable quantum states
$|\vec{x};\lambda\rangle$ are eigenstates with eigenvalues $x_{i}$ of the Fock
operators $\hat{A}_{i}$,

$\hat{A}_{i}|\vec{x};\lambda\rangle=\left(\hat{x}_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j}\right)\,|\vec{x};\lambda\rangle=x_{i}\,|\vec{x};\lambda\rangle,$
(236)

translates into the fact that these states are eigenstates with eigenvalues
$x_{\alpha,a}$ of the Fock operators $\hat{a}_{\alpha,a}$,

$\hat{a}_{\alpha,a}|\vec{x};\lambda\rangle=\left(\hat{x}_{\alpha,a}+\frac{i}{2\hbar}\left(\Lambda_{(\alpha,a),(\beta,b)}-i\Theta_{(\alpha,a),(\beta,b)}\right)\,\hat{p}_{\beta,b}\right)\,|\vec{x};\lambda\rangle=x_{\alpha,a}\,|\vec{x};\lambda\rangle.$
(237)

In order to consider a specific choice of coefficients $\lambda_{ij}$ or
$\Lambda_{(\alpha,a),(\beta,b)}$ for the construction of the states
$|\vec{x};\lambda\rangle$ such that the above property may be brought into a
form which no longer involves a contribution of the action of the momentum
operators $\hat{p}_{\alpha,a}$ on these states, let us consider the $2\times
2$ block antisymmetric symbol represented by the $2n\times 2n$ matrix $E$ of
which the entries take the values

$E_{(\alpha,a),(\beta,b)}=\delta_{\alpha\beta}\,\epsilon_{ab},$ (238)

and which is such that

$\left(E^{2}\right)_{(\alpha,a),(\beta,b)}=-\delta_{\alpha\beta}\,\delta_{ab}.$
(239)

Note that $E$ and $\Theta$ are two $2n\times 2n$ block diagonal matrices that
commute with one another, $E\Theta=\Theta E$. Let us then introduce the
associated complex valued $2n\times 2n$ projection matrices in the
$(\alpha,a)$ basis,

$\mathbb{P}_{\pm}=\frac{1}{2}\left(\mathbb{I}_{2n}\pm
iE\right),\qquad\left(\mathbb{P}_{\pm}\right)_{(\alpha,a),(\beta,b)}=\frac{1}{2}\delta_{\alpha\beta}\left(\delta_{ab}\pm
i\epsilon_{ab}\right),$ (240)

where $\mathbb{I}_{2n}$ denotes the $2n\times 2n$ identity matrix in the
$(\alpha,a)$ basis,
$\left(\mathbb{I}_{2n}\right)_{(\alpha,a),(\beta,b)}=\delta_{\alpha\beta}\,\delta_{ab}$.
Quite obviously one has,

$\mathbb{P}_{+}+\mathbb{P}_{-}=\mathbb{I}_{2n},\qquad\mathbb{P}_{+}-\mathbb{P}_{-}=iE,\qquad\mathbb{P}^{*}_{\pm}=\mathbb{P}_{\mp},$
(241)

as well as,

$\mathbb{P}^{2}_{\pm}=\mathbb{P}_{\pm},\qquad\mathbb{P}_{+}\mathbb{P}_{-}=0=\mathbb{P}_{-}\mathbb{P}_{+}.$
(242)

In order to exploit these distinctive features in relation to the observation
in (237) aiming to project out the contribution of the action of
$\hat{p}_{\alpha,a}$, let us henceforth now restrict the choice for the
coefficients $\lambda_{ij}$, namely $\Lambda_{(\alpha,a),(\beta,b)}$,
specifically to the values given by,

$\Lambda=-E\,\Theta=-\Theta\,E,\qquad\Lambda_{(\alpha,a),(\beta,b)}=\theta_{\alpha}\,\delta_{\alpha\beta}\,\delta_{ab}.$
(243)

To emphasize this specific choice of values for $\lambda_{ij}$ or
$\Lambda_{(\alpha,a),(\beta,b)}$, henceforth the corresponding localised and
normalisable quantum states are to be denoted as
$|\vec{x};\theta_{\alpha}\rangle$ rather than as $|\vec{x};\lambda\rangle$
which applies to a generic and general arbitrary choice for the coefficients
$\lambda_{ij}$. And likewise now for the generic $*_{\lambda}$-product to be
denoted henceforth as $*_{\theta}$, as well as the generic Fock vacuum
$|\Omega_{\lambda}\rangle$ to be denoted $|\Omega_{\theta}\rangle$.

Given that specific choice for $\lambda_{ij}$ one has now explicitly,

$\lambda_{ij}-iA_{ij}=\sum_{\alpha=1}^{n}\sum_{a,b=1,2}\theta_{\alpha}\left(\delta_{ab}-i\epsilon_{ab}\right)\,R_{(\alpha,a),i}\,R_{(\alpha,a),j},$
(244)

as well as,

$\left(2\mathbb{P}_{\pm}\Theta\right)_{(\alpha,a),(\beta,b)}=\mp\,i\,\theta_{\alpha}\,\delta_{\alpha\beta}\left(\delta_{ab}\,\pm\,i\epsilon_{ab}\right),$
(245)

while,

$\displaystyle\hat{a}_{\alpha,a}$ $\displaystyle=$
$\displaystyle\hat{x}_{\alpha,a}+\frac{1}{\hbar}\,\left(\mathbb{P}_{-}\Theta\right)_{(\alpha,a),(\beta,b)}\,\hat{p}_{\beta,b}=\hat{x}_{\alpha,a}+\frac{i}{2\hbar}\theta_{\alpha}\left(\delta_{ab}-i\epsilon_{ab}\right)\,\hat{p}_{\beta,b},$
$\displaystyle\hat{a}^{\dagger}_{\alpha,a}$ $\displaystyle=$
$\displaystyle\hat{x}_{\alpha,a}+\frac{1}{\hbar}\,\left(\mathbb{P}_{+}\Theta\right)_{(\alpha,a),(\beta,b)}\,\hat{p}_{\beta,b}=\hat{x}_{\alpha,a}-\frac{i}{2\hbar}\theta_{\alpha}\left(\delta_{ab}+i\epsilon_{ab}\right)\,\hat{p}_{\beta,b},$
(246)

with in particular,

$\hat{a}_{\alpha,a}|\vec{x};\theta_{\alpha}\rangle=\left(\hat{x}_{\alpha,a}+\frac{1}{\hbar}\,\left(\mathbb{P}_{-}\Theta\right)_{(\alpha,a),(\beta,b)}\,\hat{p}_{\beta,b}\right)\,|\vec{x};\theta_{\alpha}\rangle=x_{\alpha,a}\,|\vec{x};\theta_{\alpha}\rangle.$
(247)

Note as well that,

$z_{\alpha,a}=x_{\alpha,a}+\frac{i}{2\hbar}\theta_{\alpha}\left(\delta_{ab}-i\epsilon_{ab}\right)\,p_{\beta,b},\qquad
z^{*}_{\alpha,a}=x_{\alpha,a}-\frac{i}{2\hbar}\theta_{\alpha}\left(\delta_{ab}+i\epsilon_{ab}\right)\,p_{\beta,b}.$
(248)

Furthermore the canonical coherent states
$|z_{\alpha,a};\Omega_{\theta}\rangle$ are now realised in the form of,

$|z_{\alpha,a};\Omega_{\theta}\rangle={\rm
exp}\left(\sum_{\alpha=1}^{n}\sum_{a=1,2}\frac{1}{\theta_{\alpha}}\left(z_{\alpha,a}\,\hat{a}^{\dagger}_{\alpha,a}\,-\,z^{*}_{\alpha,a}\,\hat{a}_{\alpha,a}\right)\right)\,|\Omega_{\theta}\rangle,$
(249)

with for the configuration space restricted canonical coherent states for the
algebra $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$,

$|\vec{x};\theta_{\alpha}\rangle=|z_{\alpha,a}=x_{\alpha,a};\Omega_{\theta}\rangle=e^{-\frac{i}{\hbar}x_{\alpha,a}\hat{p}_{\alpha,a}}\,|\Omega_{\theta}\rangle.$
(250)

To project out from the identity (247) the contributions of the momentum
operators $\hat{p}_{\alpha,a}$, it now suffices to multiply it from the left
with the matrix projector $\mathbb{P}_{+}$. Consider thus the following
projected quantum operators, which solely involve the noncommuting
configuration space coordinate operators,

$\displaystyle\hat{u}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{+}\right)_{(\alpha,a),(\beta,b)}\,\hat{a}_{\beta,b}=2\left(\mathbb{P}_{+}\right)_{(\alpha,a),(\beta,b)}\,\hat{x}_{\beta,b}=\hat{x}_{\alpha,a}+i\,\epsilon_{ab}\,\hat{x}_{\alpha,b},$
$\displaystyle\hat{u}^{\dagger}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{-}\right)_{(\alpha,a),(\beta,b)}\,\hat{a}^{\dagger}_{\beta,b}=2\left(\mathbb{P}_{-}\right)_{(\alpha,a),(\beta,b)}\,\hat{x}_{\beta,b}=\hat{x}_{\alpha,a}-i\,\epsilon_{ab}\,\hat{x}_{\alpha,b},$
(251)

as well as,

$\displaystyle\hat{v}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{-}\right)_{(\alpha,a),(\beta,b)}\,\hat{a}_{\beta,b}=2\left(\mathbb{P}_{-}\right)_{(\alpha,a),(\beta,b)}\,\hat{x}_{\beta,b}+\frac{2}{\hbar}\left(\mathbb{P}_{-}\Theta\right)_{(\alpha,a),(\beta,b)}\,\hat{p}_{\beta,b}$
$\displaystyle=$
$\displaystyle\left(\delta_{ab}-i\epsilon_{ab}\right)\left(\hat{x}_{\alpha,b}+\frac{i}{\hbar}\theta_{\alpha}\,\hat{p}_{\alpha,b}\right)=\hat{u}^{\dagger}_{\alpha,a}+\frac{i}{\hbar}\theta_{\alpha}\left(\hat{p}_{\alpha,a}-i\epsilon_{ab}\,\hat{p}_{\alpha,b}\right),$
$\displaystyle\hat{v}^{\dagger}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}+\right)_{(\alpha,a),(\beta,b)}\,\hat{a}^{\dagger}_{\beta,b}=2\left(\mathbb{P}_{+}\right)_{(\alpha,a),(\beta,b)}\,\hat{x}_{\beta,b}+\frac{2}{\hbar}\left(\mathbb{P}_{+}\Theta\right)_{(\alpha,a),(\beta,b)}\,\hat{p}_{\beta,b}$
(252) $\displaystyle=$
$\displaystyle\left(\delta_{ab}+i\epsilon_{ab}\right)\left(\hat{x}_{\alpha,b}-\frac{i}{\hbar}\theta_{\alpha}\,\hat{p}_{\alpha,b}\right)=\hat{u}_{\alpha,a}-\frac{i}{\hbar}\theta_{\alpha}\left(\hat{p}_{\alpha,a}+i\epsilon_{ab}\,\hat{p}_{\alpha,b}\right),$

jointly with the corresponding complex valued commuting variables,

$\displaystyle u_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{+}\right)_{(\alpha,a),(\beta,b)}\,x_{\beta,b}=x_{\alpha,a}+i\,\epsilon_{ab}\,x_{\alpha,b},$
$\displaystyle u^{*}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{-}\right)_{(\alpha,a),(\beta,b)}\,x_{\beta,b}=x_{\alpha,a}-i\,\epsilon_{ab}\,x_{\alpha,b},$
$\displaystyle v_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{-}\right)_{(\alpha,a),(\beta,b)}\,z_{\beta,b}=u^{*}_{\alpha,a}\,+\,\frac{i}{\hbar}\theta_{\alpha}\left(p_{\alpha,a}-i\epsilon_{ab}\,p_{\alpha,b}\right),$
$\displaystyle v^{*}_{\alpha,a}$ $\displaystyle=$ $\displaystyle
2\left(\mathbb{P}_{+}\right)_{(\alpha,a),(\beta,b)}\,z_{\beta,b}=u_{\alpha,a}\,-\,\frac{i}{\hbar}\theta_{\alpha}\left(p_{\alpha,a}+i\epsilon_{ab}\,p_{\alpha,b}\right).$
(253)

Of course one also has,

$\hat{a}_{\alpha,a}=\frac{1}{2}\left(\hat{u}_{\alpha,a}+\hat{v}_{\alpha,a}\right),\qquad\hat{a}^{\dagger}_{\alpha,a}=\frac{1}{2}\left(\hat{u}^{\dagger}_{\alpha,a}+\hat{v}^{\dagger}_{\alpha,a}\right),$
(254)

with analogous relation between $u_{\alpha,a}$, $v_{\alpha,a}$,
$u^{*}_{\alpha,a}$ and $v^{*}_{\alpha,a}$, and $z_{\alpha,a}$ and
$z^{*}_{\alpha,a}$.

For the commutation algebra of the operators $\hat{p}_{\alpha,a}$,
$\hat{u}_{\alpha,a}$, $\hat{u}^{\dagger}_{\alpha,a}$, $\hat{v}_{\alpha,a}$ and
$\hat{v}^{\dagger}_{\alpha,a}$, first one readily finds,

$\left[\hat{u}_{\alpha,a},\hat{p}_{\beta,b}\right]=i\hbar\delta_{\alpha\beta}\left(\delta_{ab}+i\epsilon_{ab}\right)\mathbb{I},\qquad\left[\hat{u}^{\dagger}_{\alpha,a},\hat{p}_{\beta,b}\right]=i\hbar\delta_{\alpha\beta}\left(\delta_{ab}-i\epsilon_{ab}\right)\mathbb{I},$
(255)

so that,

$\displaystyle\left[\hat{u}_{\alpha,a},\left(\delta_{bc}-i\epsilon_{bc}\right)\hat{p}_{\beta,c}\right]$
$\displaystyle=$ $\displaystyle
2i\hbar\delta_{\alpha\beta}\left(\delta_{ab}+i\epsilon_{ab}\right)\mathbb{I},\qquad\left[\hat{u}_{\alpha,a},\left(\delta_{bc}+i\epsilon_{bc}\right)\hat{p}_{\beta,c}\right]=0,$
$\displaystyle\left[\hat{u}^{\dagger}_{\alpha,a},\left(\delta_{bc}-i\epsilon_{bc}\right)\hat{p}_{\beta,c}\right]$
$\displaystyle=$ $\displaystyle
0,\qquad\left[\hat{u}^{\dagger}_{\alpha,a},\left(\delta_{bc}+i\epsilon_{bc}\right)\hat{p}_{\beta,c}\right]=2i\hbar\delta_{\alpha\beta}\left(\delta_{ab}-i\epsilon_{ab}\right)\mathbb{I}.$
(256)

One then establishes that,

$\left[\hat{u}_{\alpha,a},\hat{u}_{\beta,b}\right]=0,\quad\left[\hat{u}_{\alpha,a},\hat{u}^{\dagger}_{\beta,b}\right]=2\theta_{\alpha}\,\delta_{\alpha\beta}\left(\delta_{ab}+i\epsilon_{ab}\right)\,\mathbb{I},\quad\left[\hat{u}^{\dagger}_{\alpha,a},\hat{u}^{\dagger}_{\beta,b}\right]=0,$
(257)

as well as,

$\left[\hat{v}_{\alpha,a},\hat{v}_{\beta,b}\right]=0,\quad\left[\hat{v}_{\alpha,a},\hat{v}^{\dagger}_{\beta,b}\right]=2\theta_{\alpha}\,\delta_{\alpha\beta}\left(\delta_{ab}-i\epsilon_{ab}\right)\,\mathbb{I},\quad\left[\hat{v}^{\dagger}_{\alpha,a},\hat{v}^{\dagger}_{\beta,b}\right]=0,$
(258)

and finally,

$\Big{[}\hat{u}_{\alpha,a},\hat{v}_{\beta,b}\Big{]}=0,\qquad\left[\hat{u}_{\alpha,a},\hat{v}^{\dagger}_{\beta,b}\right]=0,\qquad\left[\hat{u}^{\dagger}_{\alpha,a},\hat{v}_{\beta,b}\right]=0,\qquad\left[\hat{u}^{\dagger}_{\alpha,a},\hat{v}^{\dagger}_{\beta,b}\right]=0.$
(259)

Consequently all the operators
$(\hat{u}_{\alpha,a},\hat{u}^{\dagger}_{\alpha,a})$ commute with all the
operators $(\hat{v}_{\alpha,a},\hat{v}^{\dagger}_{\alpha,a})$.

However the operators $\hat{u}_{\alpha,a}$ on the one hand, and
$\hat{v}_{\alpha,a}$ on the other hand, are not all independent, since one
has,

$i\epsilon_{ab}\,\hat{u}_{\alpha,b}=\hat{u}_{\alpha,a},\qquad
i\epsilon_{ab}\hat{v}_{\alpha,a}=-\hat{v}_{\alpha,a},$ (260)

namely,

$\hat{u}_{\alpha,2}=-i\hat{u}_{\alpha,1},\qquad\hat{u}^{\dagger}_{\alpha,2}=i\hat{u}^{\dagger}_{\alpha,1},\qquad\hat{v}_{\alpha,2}=i\hat{v}_{\alpha,1},\qquad\hat{v}^{\dagger}_{\alpha,2}=-i\hat{v}^{\dagger}_{\alpha,1},$
(261)

and likewise for the associated complex variables $u_{\alpha,a}$,
$u^{*}_{\alpha,a}$, and $v_{\alpha,a}$ and $v^{*}_{\alpha,a}$. Yet, the states
$|\vec{x};\theta_{\alpha}\rangle$ are the eigenstates of the quantum operators
$\hat{u}_{\alpha,a}$ with eigenvalues $u_{\alpha,a}$,

$\hat{u}_{\alpha,a}\,|\vec{x};\theta_{\alpha}\rangle=u_{\alpha,a}\,|\vec{x};\theta_{\alpha}\rangle,\qquad
u_{\alpha,a}=x_{\alpha,a}+i\epsilon_{ab}\,x_{\alpha,b}.$ (262)

Since the operators $\hat{u}_{\alpha,a}$ only involve the configuration space
coordinate operators $\hat{x}_{\alpha,a}$ (or $\hat{x}_{i}$) but not the
momentum ones $\hat{p}_{\alpha,a}$ (or $\hat{p}_{i}$), clearly this last
observation strongly suggests that for the specific choice made for the
coefficients $\lambda_{ij}$ the states $|\vec{x};\theta_{\alpha}\rangle$ are
as well some form of canonical coherent states for the restricted subalgebra
of noncommuting configuration space coordinate operators $\hat{u}_{\alpha,a}$,
in tensor product with the subalgebra of noncommuting operators
$\hat{v}_{\alpha,a}$, and this not only as a configuration space restricted
class of canonical coherent states for the complete initial quantum algebra
$(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$.

To make this structure manifest, finally let us consider the independent
operators $(\hat{b}_{\alpha},\hat{b}^{\dagger}_{\alpha})$ and
$(\hat{d}_{\alpha},\hat{d}^{\dagger}_{\alpha})$, defined by,

$\hat{b}_{\alpha}=\hat{u}_{\alpha,1}=\hat{x}_{\alpha,1}+i\hat{x}_{\alpha,2},\qquad\hat{b}^{\dagger}_{\alpha}=\hat{u}^{\dagger}_{\alpha,1}=\hat{x}_{\alpha,1}-i\hat{x}_{\alpha,2},$
(263)

and,

$\displaystyle\hat{d}_{\alpha}$ $\displaystyle=$
$\displaystyle\hat{v}_{\alpha,1}=(\hat{x}_{\alpha,1}-i\hat{x}_{\alpha,2})+\frac{i}{\hbar}\theta_{\alpha}\left(\hat{p}_{\alpha,1}-i\hat{p}_{\alpha,2}\right)=\hat{b}^{\dagger}_{\alpha}+\frac{i}{\hbar}\theta_{\alpha}\,\hat{p}_{\alpha,-},$
$\displaystyle\hat{d}^{\dagger}_{\alpha}$ $\displaystyle=$
$\displaystyle\hat{v}^{\dagger}_{\alpha,1}=(\hat{x}_{\alpha,1}+i\hat{x}_{\alpha,2})-\frac{i}{\hbar}\theta_{\alpha}\left(\hat{p}_{\alpha,1}+i\hat{p}_{\alpha,2}\right)=\hat{b}_{\alpha}-\frac{i}{\hbar}\theta_{\alpha}\,\hat{p}_{\alpha,+},$
(264)

where,

$\hat{p}_{\alpha,+}=\hat{p}_{\alpha,1}\,+\,i\,\hat{p}_{\alpha,2}=-\frac{i\hbar}{\theta_{\alpha}}\left(\hat{b}_{\alpha}-\hat{d}^{\dagger}_{\alpha}\right),\qquad\hat{p}_{\alpha,-}=\hat{p}_{\alpha,1}\,-\,\hat{p}_{\alpha,2}=\frac{i\hbar}{\theta_{\alpha}}\left(\hat{b}^{\dagger}_{\alpha}-\hat{d}_{\alpha}\right).$
(265)

The complex variables corresponding to $\hat{b}_{\alpha}$ and
$\hat{b}^{\dagger}_{\alpha}$ are to be denoted as, respectively,

$z_{\alpha}=x_{\alpha,1}+ix_{\alpha,2},\qquad
z^{*}_{\alpha}=x_{\alpha,1}-ix_{\alpha,2},$ (266)

while for $\hat{p}_{\alpha,\pm}$ one has,

$p_{\alpha,\pm}=p_{\alpha,1}\pm ip_{\alpha,2},\qquad
p^{*}_{\alpha,\pm}=p_{\alpha,\mp}.$ (267)

The inverse relations are such that,

$\hat{a}_{\alpha,1}=\frac{1}{2}\left(\hat{b}_{\alpha}+\hat{d}_{\alpha}\right),\qquad\hat{a}_{\alpha,2}=-\frac{1}{2}i\left(\hat{b}_{\alpha}-\hat{d}_{\alpha}\right),$
(268)
$\hat{a}^{\dagger}_{\alpha,1}=\frac{1}{2}\left(\hat{b}^{\dagger}_{\alpha}+\hat{d}^{\dagger}_{\alpha}\right),\qquad\hat{a}^{\dagger}_{\alpha,2}=\frac{1}{2}i\left(\hat{b}^{\dagger}_{\alpha}-\hat{d}^{\dagger}_{\alpha}\right).$
(269)

Likewise,

$\hat{x}_{\alpha,1}=\frac{1}{2}\left(\hat{b}_{\alpha}+\hat{b}^{\dagger}_{\alpha}\right),\qquad\hat{x}_{\alpha,2}=-\frac{1}{2}i\left(\hat{b}_{\alpha}-\hat{b}^{\dagger}_{\alpha}\right),$
(270)
$\hat{p}_{\alpha,1}=\frac{i\hbar}{2\theta_{\alpha}}\left(\hat{b}^{\dagger}_{\alpha}-\hat{b}_{\alpha}+\hat{d}^{\dagger}_{\alpha}-\hat{d}_{\alpha}\right),\qquad\hat{p}_{\alpha,2}=-\frac{\hbar}{2\theta_{\alpha}}\left(\hat{b}^{\dagger}_{\alpha}+\hat{b}_{\alpha}-\hat{d}^{\dagger}_{\alpha}-\hat{d}_{\alpha}\right),$
(271)
$\hat{p}_{\alpha,+}=\hat{p}_{\alpha,1}\,+\,i\,\hat{p}_{\alpha,2}=-\frac{i\hbar}{\theta_{\alpha}}\left(\hat{b}_{\alpha}-\hat{d}^{\dagger}_{\alpha}\right),\qquad\hat{p}_{\alpha,-}=\hat{p}_{\alpha,1}\,-\,\hat{p}_{\alpha,2}=\frac{i\hbar}{\theta_{\alpha}}\left(\hat{b}^{\dagger}_{\alpha}-\hat{d}_{\alpha}\right).$
(272)

Obviously, analogous relations apply to the associated complex variables
$z_{\alpha}$, $p_{\alpha,\pm}$, $z_{\alpha,a}$, $x_{\alpha,a}$ and
$p_{\alpha,a}$.

The commutation relations for the operators
$(\hat{b}_{\alpha},\hat{b}^{\dagger}_{\alpha})$ and
$(\hat{d}_{\alpha},\hat{d}^{\dagger}_{\alpha})$ are then obtained as,

$\displaystyle\left[\hat{b}_{\alpha},\hat{b}_{\beta}\right]$ $\displaystyle=$
$\displaystyle
0,\qquad\left[\hat{b}_{\alpha},\hat{b}^{\dagger}_{\beta}\right]=2\theta_{\alpha}\,\delta_{\alpha\beta}\,\mathbb{I},\qquad\left[\hat{b}^{\dagger}_{\alpha},\hat{b}^{\dagger}_{\beta}\right]=0,$
$\displaystyle\left[\hat{d}_{\alpha},\hat{d}_{\beta}\right]$ $\displaystyle=$
$\displaystyle
0,\qquad\left[\hat{d}_{\alpha},\hat{d}^{\dagger}_{\beta}\right]=2\theta_{\alpha}\,\delta_{\alpha\beta}\,\mathbb{I},\qquad\left[\hat{d}^{\dagger}_{\alpha},\hat{d}^{\dagger}_{\beta}\right]=0,$
$\displaystyle\left[\hat{b}_{\alpha},\hat{d}_{\beta}\right]$ $\displaystyle=$
$\displaystyle
0,\qquad\left[\hat{b}_{\alpha},\hat{d}^{\dagger}_{\beta}\right]=0,\qquad\left[\hat{b}^{\dagger}_{\alpha},\hat{d}_{\beta}\right]=0,\qquad\left[\hat{b}^{\dagger}_{\alpha},\hat{d}^{\dagger}_{\beta}\right]=0,$
(273)

while for those involving the operators $\hat{p}_{\alpha,\pm}$ are,

$\displaystyle\left[\hat{b}_{\alpha},\hat{p}_{\beta,-}\right]$
$\displaystyle=$ $\displaystyle
i\hbar\delta_{\alpha\beta}\mathbb{I},\quad\left[\hat{b}^{\dagger}_{\alpha},\hat{p}_{\beta,-}\right]=0,\quad\left[\hat{b}_{\alpha},\hat{p}_{\beta,+}\right]=0,\quad\left[\hat{b}^{\dagger}_{\alpha},\hat{p}_{\beta,+}\right]=i\hbar\delta_{\alpha\beta}\mathbb{I},$
$\displaystyle\left[\hat{d}_{\alpha},\hat{p}_{\beta,-}\right]$
$\displaystyle=$ $\displaystyle
0,\quad\left[\hat{d}^{\dagger}_{\alpha},\hat{p}_{\beta,-}\right]=i\hbar\delta_{\alpha\beta}\mathbb{I},\quad\left[\hat{d}_{\alpha},\hat{p}_{\beta,+}\right]=i\hbar\delta_{\alpha\beta}\mathbb{I},\quad\left[\hat{d}^{\dagger}_{\alpha},\hat{p}_{\beta,+}\right]=0.$
(274)

Hence the operator algebra defining the quantised system is that of the tensor
product of $2n=N$ commuting Fock algebras, namely, for each value of
$\alpha=1,2,\cdots,n$, the two commuting Fock algebras generated, on the one
hand by the operators $(\hat{b}_{\alpha},\hat{b}^{\dagger}_{\alpha})$, and on
the other hand by the operators
$(\hat{d}_{\alpha},\hat{d}^{\dagger}_{\alpha})$, with each of these sectors
characterised by the noncommutativity real parameter $\theta_{\alpha}>0$ as a
normalisation of the Fock algebra nonvanishing commutation relations
$[\hat{b}_{\alpha},\hat{b}^{\dagger}_{\alpha}]=2\theta_{\alpha}\mathbb{I}$ and
$[\hat{d}_{\alpha},\hat{d}^{\dagger}_{\alpha}]=2\theta_{\alpha}\mathbb{I}$ (no
summation over $\alpha$).

Consequently, the structure of the full quantum Hilbert space for the
noncommutative Euclidean configuration space consists of the tensor product of
a $b$-sector and a $d$-sector, each of these two sectors being itself the
$n$-fold tensor product over $\alpha=1,2,\cdots,n$ of the irreducible Fock
spaces associated to the $(\hat{b}_{\alpha},\hat{b}^{\dagger}_{\alpha})$ or to
the $(\hat{d}_{\alpha},\hat{d}^{\dagger}_{\alpha})$ Fock algebras,
respectively.

With this understanding at hand, let us now reconsider the status of the
states $|\vec{x};\theta_{\alpha}\rangle$ which generate the entire quantum
Hilbert space through the $*_{\theta}$-product, and are characterised by the
following properties,

$\hat{a}_{\alpha,a}\,|\vec{x};\theta_{\alpha}\rangle=x_{\alpha,a}\,|\vec{x};\theta_{\alpha}\rangle,\qquad|\vec{x};\theta_{\alpha}\rangle=e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\Omega_{\theta}\rangle,\qquad\langle\Omega_{\theta}|\Omega_{\theta}\rangle=\prod_{\alpha=1}^{n}\frac{1}{2\pi\theta_{\alpha}},$
(275)

with thus in particular the fact that
$|\Omega_{\theta}\rangle=|\vec{x}=\vec{0};\theta_{\alpha}\rangle$. Given that
one has $\hat{a}_{\alpha,1}=(\hat{b}_{\alpha}+\hat{d}_{\alpha})/2$ and
$\hat{a}_{\alpha,2}=-i(\hat{b}_{\alpha}-\hat{d}_{\alpha})/2$, while
$\hat{a}_{\alpha,a}|\Omega_{\theta}\rangle=0$, it follows that,

$\hat{b}_{\alpha}\,|\Omega_{\theta}\rangle=0,\qquad\hat{d}_{\alpha}\,|\Omega_{\theta}\rangle=0.$
(276)

In other words, the state
$|\Omega_{\theta}\rangle=|\vec{x}=\vec{0};\theta_{\alpha}\rangle$ is as well a
Fock vacuum for all Fock algebras in both the $b$\- and the $d$-sectors of the
full quantum Hilbert space. Let us denote as $|\Omega_{b}\rangle$ and
$|\Omega_{d}\rangle$ the Fock vacua for all Fock algebras in the $b$\- and in
the $d$-sectors, respectively, with a nonstandard normalisation chosen to be
identical in both sectors and each equal to
$\sqrt{\langle\Omega_{\theta}|\Omega_{\theta}\rangle}$, so that,

$\hat{b}_{\alpha}|\Omega_{b}\rangle=0,\qquad\hat{d}_{\alpha}|\Omega_{d}\rangle=0,\qquad\langle\Omega_{b}|\Omega_{b}\rangle=\prod_{\alpha=1}^{n}\frac{1}{\sqrt{2\pi\theta_{\alpha}}}=\langle\Omega_{d}|\Omega_{d}\rangle.$
(277)

In terms of these notations, one thus has for these abstract quantum states,

$|\Omega_{\theta}\rangle=|\Omega_{b}\rangle\otimes|\Omega_{d}\rangle.$ (278)

The normalised Fock states for each of the $b$\- or $d$-sectors are therefore
to be denoted as, with $k_{\alpha},\ell_{\alpha}\in\mathbb{N}$,

$|k_{\alpha};\Omega_{b}\rangle=\frac{1}{\sqrt{\langle\Omega_{b}|\Omega_{b}\rangle}}\prod_{\alpha=1}^{n}\frac{1}{\sqrt{(2\theta)^{k_{\alpha}}\,k_{\alpha}!}}\left(\hat{b}^{\dagger}_{\alpha}\right)^{k_{\alpha}}|\Omega_{b}\rangle,\qquad\langle
k_{\alpha};\Omega_{b}|k^{\prime}_{\alpha};\Omega_{b}\rangle=\prod_{\alpha=1}^{n}\delta_{k_{\alpha}k^{\prime}_{\alpha}},$
(279)
$|\ell_{\alpha};\Omega_{d}\rangle=\frac{1}{\sqrt{\langle\Omega_{d}|\Omega_{d}\rangle}}\prod_{\alpha=1}^{n}\frac{1}{\sqrt{(2\theta)^{\ell_{\alpha}}\,\ell_{\alpha}!}}\left(\hat{d}^{\dagger}_{\alpha}\right)^{\ell_{\alpha}}|\Omega_{d}\rangle,\qquad\langle\ell_{\alpha};\Omega_{d}|\ell^{\prime}_{\alpha};\Omega_{d}\rangle=\prod_{\alpha=1}^{n}\delta_{\ell_{\alpha}\ell^{\prime}_{\alpha}},$
(280)

so that their tensor product provides an orthonormalised Fock basis for the
entire quantum Hilbert space with the abstract quantum states,

$|k_{\alpha},\ell_{\alpha};\Omega_{\theta}\rangle\equiv|k_{\alpha};\Omega_{b}\rangle\otimes|\ell_{\alpha};\Omega_{d}\rangle.$
(281)

Correspondingly one also has canonical quantum coherent states in each of the
$b$\- and $d$-sectors. Given complex variables $w_{b,\alpha}$ and
$w_{d,\alpha}$, these states are defined according to, respectively,

$|w_{b,\alpha};\Omega_{b}\rangle=\prod_{\alpha=1}^{n}{\rm
exp}\left(\frac{1}{2\theta_{\alpha}}\left(w_{b,\alpha}\hat{b}^{\dagger}_{\alpha}-w^{*}_{b,\alpha}\hat{b}_{\alpha}\right)\right)|\Omega_{b}\rangle,$
(282) $|w_{d,\alpha};\Omega_{d}\rangle=\prod_{\alpha=1}^{n}{\rm
exp}\left(\frac{1}{2\theta_{\alpha}}\left(w_{d,\alpha}\hat{d}^{\dagger}_{\alpha}-w^{*}_{d,\alpha}\hat{d}_{\alpha}\right)\right)|\Omega_{d}\rangle,$
(283)

whose tensor product defines in the $(\alpha,a)$ parametrisation the canonical
quantum coherent states that generate as an overcomplete ensemble the entire
quantum Hilbert space, to be denoted as,

$|w_{b,\alpha},w_{d,\alpha};\Omega_{\theta}\rangle\equiv|w_{b,\alpha};\Omega_{b}\rangle\otimes|w_{d,\alpha};\Omega_{d}\rangle.$
(284)

Finally, in view of the fact that $|\vec{x};\theta_{\alpha}\rangle={\rm
exp}\,(-i\vec{x}\cdot\hat{\vec{p}}/\hbar)|\Omega_{\theta}\rangle$, and in
terms of the different quantities introduced above, one works out that,

$-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}=-\frac{i}{\hbar}x_{\alpha,a}\hat{p}_{\alpha,a}=\sum_{\alpha=1}^{n}\frac{1}{2\theta_{\alpha}}\left(\left(z_{\alpha}\hat{b}^{\dagger}_{\alpha}-z^{*}_{\alpha}\hat{b}_{\alpha}\right)+\left(z^{*}_{\alpha}\hat{d}^{\dagger}_{\alpha}-z_{\alpha}\hat{d}_{\alpha}\right)\right),$
(285)

so that,

$|\vec{x};\theta_{\alpha}\rangle={\rm
exp}\left\\{\sum_{\alpha=1}^{n}\frac{1}{2\theta_{\alpha}}\left(\left(z_{\alpha}\hat{b}^{\dagger}_{\alpha}-z^{*}_{\alpha}\hat{b}_{\alpha}\right)+\left(z^{*}_{\alpha}\hat{d}^{\dagger}_{\alpha}-z_{\alpha}\hat{d}_{\alpha}\right)\right)\right\\}\,|\Omega_{\theta}\rangle.$
(286)

Hence the localised and normalisable states $|\vec{x};\theta_{\alpha}\rangle$
provide a specific subset of all canonical coherent states
$|w_{b,\alpha},w_{d,\alpha};\Omega_{\theta}\rangle$ restricted such that
$w_{d,\alpha}=w^{*}_{b,\alpha}$ with
$w_{b,\alpha}=z_{\alpha}=x_{\alpha,1}+ix_{\alpha,2}$, namely,

$|\vec{x};\theta_{\alpha}\rangle=|z_{\alpha},z^{*}_{\alpha};\Omega_{\theta}\rangle=|z_{\alpha};\Omega_{b}\rangle\otimes|z^{*}_{\alpha};\Omega_{d}\rangle,\qquad{\rm
with}\ z_{\alpha}=x_{\alpha,1}+ix_{\alpha,2}.$ (287)

This final conclusion thus establishes that indeed as expected, the states
$|\vec{x};\theta_{\alpha}\rangle$ labelled by all points in (the classical and
commutative) Euclidean configuration space are a specific class of canonical
quantum coherent states which, through the $*_{\theta}$-product, generate the
entire quantum Hilbert space.

Having identified these specific structures and characteristics, and by
borrowing and extending from the work by F. G. Scholtz et al. a particular
vocabulary introduced first in the context of quantum mechanics on the
noncommutative two dimensional Euclidean plane, let us denote first the entire
quantum Hilbert space with its two commuting Fock $b$\- and $d$-sectors as
${\cal H}_{q}$. The Hilbert space associated to the $b$-sector Fock algebra
and its irreducible representation spanned by the Fock states
$|k_{\alpha};\Omega_{b}\rangle$, its Fock vacuum $|\Omega_{b}\rangle$ and its
coherent states $|w_{b,\alpha};\Omega_{b}\rangle$ is to be called “the
classical Hilbert space” and denoted as ${\cal H}_{c}$. While the Hilbert
space associate to the $d$-sector Fock algebra and its irreducible
representation spanned by the Fock states $|\ell_{\alpha};\Omega_{d}\rangle$,
its Fock vacuum $|\Omega_{d}\rangle$ and its coherent states
$|w_{d,\alpha};\Omega_{d}\rangle$ is isomorphic to the dual (or adjoint) of
the classical Hilbert space ${\cal H}_{c}$, and thus to be called “the dual
classical Hilbert space” and denoted as ${\cal H}^{\dagger}_{c}$. The above
analysis has thus established that one has the correspondence,

${\cal H}_{q}={\cal H}_{c}\otimes{\cal H}^{\dagger}_{c}.$ (288)

Hence any given value for $\vec{x}$ parametrising the Euclidean configuration
space determines uniquely the specific quantum coherent state
$|\vec{x};\theta_{\alpha}\rangle$ in the quantum Hilbert space ${\cal H}_{q}$,
and thereby uniquely as well the specific coherent state
$|z_{\alpha};\Omega_{b}\rangle$ in the classical Hilbert space ${\cal H}_{c}$,
and the specific coherent state $|z^{*}_{\alpha};\Omega_{d}\rangle$ in the
dual classical Hilbert space ${\cal H}^{\dagger}_{c}$. Conversely any coherent
state $|w_{b,\alpha};\Omega_{b}\rangle$ in the classical Hilbert space
determines uniquely the state $|w_{d,\alpha};\Omega_{d}\rangle$ with
$w_{d,\alpha}=w^{*}_{b,\alpha}$ in the dual classical Hilbert space, and
therefore uniquely as well the quantum coherent state
$|\vec{x};\theta_{\alpha}\rangle=|w_{b,\alpha};\Omega_{b}\rangle\otimes|w^{*}_{b,\alpha};\Omega_{d}\rangle$
in the quantum Hilbert ${\cal H}_{q}$ which itself is uniquely associated to
the coordinate values $\vec{x}$ such that,

$x_{i}=x_{\alpha,a}\,R_{(\alpha,a),i},\qquad
x_{\alpha,1}=\frac{1}{2}(w_{b,\alpha}+w^{*}_{b,\alpha}),\qquad
x_{\alpha,2}=-\frac{1}{2}i(w_{b,\alpha}-w^{*}_{b,\alpha}),$ (289)

since,

$\hat{a}_{\alpha,1}=\frac{1}{2}(\hat{b}_{\alpha}+\hat{d}_{\alpha}),\qquad\hat{a}_{\alpha,2}=-\frac{1}{2}i(\hat{b}_{\alpha}-\hat{d}_{\alpha}),$
(290)

while,

$\displaystyle\hat{b}_{\alpha}|w_{b,\alpha};\Omega_{b}\rangle\otimes|w^{*}_{b,\alpha};\Omega_{d}\rangle$
$\displaystyle=$ $\displaystyle
w_{b,\alpha}|w_{b,\alpha};\Omega_{b}\rangle\otimes|w^{*}_{b,\alpha};\Omega_{d}\rangle,$
$\displaystyle\hat{d}_{\alpha}|w_{b,\alpha};\Omega_{b}\rangle\otimes|w^{*}_{b,\alpha};\Omega_{d}\rangle$
$\displaystyle=$ $\displaystyle
w^{*}_{b,\alpha}|w_{b,\alpha};\Omega_{b}\rangle\otimes|w^{*}_{b,\alpha};\Omega_{d}\rangle.$
(291)

These identifications thus define a one-to-one map between the coherent states
$|z_{\alpha};\Omega_{b}\rangle$ in the classical Hilbert space ${\cal H}_{c}$
and the coherent states $|\vec{x};\theta_{\alpha}\rangle$ in the quantum
Hilbert space ${\cal H}_{q}$, with $z_{\alpha}=x_{\alpha,1}+ix_{\alpha,2}$ and
$x_{\alpha,a}=R_{(\alpha,a),i}\,x_{i}$. While each of these ensembles of
quantum states generates the entire Hilbert space to which they belong, either
through the projection operator density $|z_{\alpha};\Omega_{b}\rangle\langle
z_{\alpha};\Omega_{b}|$ integrated over configuration space in the first case,
or through the $*_{\theta}$-products
$|\vec{x};\theta_{\alpha}\rangle*_{\theta}\langle\vec{x};\theta_{\alpha}|$
likewise integrated over configuration space in the second case.

As a final consideration, let us now turn to the enveloping algebra
representation of the abstract quantum states $|\varphi\rangle$ in the quantum
Hilbert space ${\cal H}_{q}$. As discussed earlier any such state is
represented by an operator in that algebra through the correspondence,

$\hat{\varphi}(\hat{\vec{x}}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\langle\vec{p}\,|\varphi\rangle.$
(292)

Through the operator correspondences
$\hat{x}_{i}\leftrightarrow\hat{x}_{\alpha,a}\leftrightarrow(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})$,
any abstract quantum state $|\varphi\rangle$ is thus represented, in the
enveloping algebra of the coordinate operators $\hat{x}_{i}$ or equivalently
eventually of the Fock algebra operators
$(\hat{b}_{\alpha},\hat{b}^{\dagger}_{\alpha})$, in terms of an operator
$\hat{\varphi}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})$, possibly written
in normal ordered form relative to the Fock vacuum $|\Omega_{b}\rangle$ in the
$b$-sector hence the choice of notation,

$|\varphi\rangle\longleftrightarrow\hat{\varphi}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha}).$
(293)

As established earlier, in the enveloping algebra the inner product of
abstract quantum states $|\varphi_{1}\rangle$ and $|\varphi_{2}\rangle$ is
represented as,

$\langle\varphi_{1}|\varphi_{2}\rangle=\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1};\theta_{\alpha}|\,\hat{\varphi}^{\dagger}_{1}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\,\hat{\varphi}_{2}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\,|\vec{x}_{2};\theta_{\alpha}\rangle.$
(294)

By implementing the following changes of variables to lead to a polar
parametrisation $(\rho_{\alpha},\phi_{\alpha})$ of configuration space in each
of its $\alpha$-sectors,

$d^{N}\vec{x}=\prod_{i=1}^{2n}dx_{i}=\prod_{\alpha=1}^{n}\prod_{a=1,2}dx_{\alpha,a}=\prod_{\alpha=1}^{n}d\rho_{\alpha}\,\rho_{\alpha}\,d\phi_{\alpha},\quad
x_{\alpha,1}=\rho_{\alpha}\cos\phi_{\alpha},\quad
x_{\alpha,2}=\rho_{\alpha}\sin\phi_{\alpha},$ (295)

one may establish that,

$\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\theta_{\alpha}\rangle=\sum_{\\{k_{\alpha}\\}=0}^{\infty}\left(\prod_{\alpha=1}^{n}\sqrt{2\pi\theta_{\alpha}}\right)\,|k_{\alpha};\Omega_{b}\rangle\otimes|k_{\alpha};\Omega_{d}\rangle,$
(296)

where,

$\sum_{\\{k_{\alpha}\\}=0}^{\infty}\equiv\sum_{k_{1},k_{2},\cdots,k_{n}=0}^{\infty}.$
(297)

Consequently the above representation of the inner product in the enveloping
algebra also acquires the following expression as a trace only over the Fock
$b$-sector, namely within the classical Hilbert space ${\cal H}_{c}$, through,

$\displaystyle\langle\varphi_{1}|\varphi_{2}\rangle$ $\displaystyle=$
$\displaystyle\left(\prod_{\alpha=1}^{n}\,2\pi\theta_{\alpha}\right)\sum_{\\{k_{\alpha},\ell_{\alpha}\\}=0}^{\infty}\,\langle
k_{\alpha};\Omega_{b}|\otimes\langle
k_{\alpha};\Omega_{d}|\left(\hat{\varphi}^{\dagger}_{1}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\,\hat{\varphi}_{2}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\right)\,|\ell_{\alpha};\Omega_{b}\rangle\otimes|\ell_{\alpha};\Omega_{d}\rangle$
(298) $\displaystyle=$
$\displaystyle\left(\prod_{\alpha=1}^{n}2\pi\theta_{\alpha}\right)\sum_{\\{k_{\alpha}\\}=0}^{\infty}\langle
k_{\alpha};\Omega_{b}|\,\hat{\varphi}^{\dagger}_{1}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\,\hat{\varphi}_{2}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\,|k_{\alpha};\Omega_{b}\rangle$
$\displaystyle=$
$\displaystyle\left(\prod_{\alpha=1}^{n}2\pi\theta_{\alpha}\right)\,{\rm
Tr}_{c}\left(\hat{\varphi}^{\dagger}_{1}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\,\hat{\varphi}_{2}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})\right),$

where the trace operation ${\rm Tr}_{c}$ in the classical Hilbert space ${\cal
H}_{c}$ of any operator ${\cal
O}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})$ acting on that Fock space is
defined as, since the Fock states $|k_{\alpha};\Omega_{b}\rangle$ are
orthonormalised,

${\rm Tr}_{c}\,\hat{\cal
O}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})=\sum_{\\{k_{\alpha}\\}=0}^{\infty}\langle
k_{\alpha};\Omega_{b}|\hat{\cal
O}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})|k_{\alpha};\Omega_{b}\rangle.$
(299)

As already established earlier, the abstract operators
$(\hat{x}_{i},\hat{p}_{i})$ acting on the abstract quantum Hilbert space
${\cal H}_{q}$ are represented as follows in the enveloping algebra
representation, for any state $\hat{\varphi}(\hat{\vec{x}}\,)$,

$\hat{X}_{i}(\hat{\varphi})=\hat{x}_{i}\,\hat{\varphi},\qquad\hat{P}_{i}(\hat{\varphi})=-\hbar\left(A^{-1}\right)_{ij}\,\left[\hat{x}_{j},\hat{\varphi}\right].$
(300)

Correspondingly in the $(\alpha,a)$ parametrisation, one has for the same set
of operators,

$\hat{X}_{\alpha,a}(\hat{\varphi})=\hat{x}_{\alpha,a}\,\hat{\varphi},\qquad\hat{P}_{\alpha,a}(\hat{\varphi})=\frac{\hbar}{\theta_{\alpha}}\,\epsilon_{ab}\,\left[\hat{x}_{\alpha,b},\hat{\varphi}\right].$
(301)

Furthermore, given the relations
$\hat{b}_{\alpha}=\hat{x}_{\alpha,1}+i\hat{x}_{\alpha,2}$,
$\hat{b}^{\dagger}_{\alpha}=\hat{x}_{\alpha,1}-i\hat{x}_{\alpha,2}$ and
$\hat{p}_{\alpha,\pm}=\hat{p}_{\alpha,1}\pm i\hat{p}_{\alpha,,2}$, and thus
the definitions,

$\hat{B}_{\alpha}=\hat{X}_{\alpha,1}+i\hat{X}_{\alpha,2},\qquad\hat{B}^{\ddagger}_{\alpha}=\hat{X}_{\alpha,1}-i\hat{X}_{\alpha,2},\qquad\hat{P}_{\alpha,\pm}=\hat{P}_{\alpha,1}\pm
i\hat{P}_{\alpha,2},$ (302)

where $\hat{F}^{\ddagger}$ denotes the adjoint of an operator
$\hat{F}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})$ in the enveloping
algebra relative to the above inner product defined for the operators
$\hat{\varphi}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})$, one finds,

$\hat{B}_{\alpha}(\hat{\varphi})=\hat{b}_{\alpha}\,\hat{\varphi},\qquad\hat{B}^{\ddagger}_{\alpha}(\hat{\varphi})=\hat{b}^{\dagger}_{\alpha}\,\hat{\varphi},\quad\hat{P}_{\alpha,+}(\hat{\varphi})=-\frac{i\hbar}{\theta_{\alpha}}\,\left[\hat{b}_{\alpha},\hat{\varphi}\right],\quad\hat{P}_{\alpha,-}(\hat{\varphi})=\frac{i\hbar}{\theta_{\alpha}}\,\left[\hat{b}^{\dagger}_{\alpha},\hat{\varphi}\right],$
(303)

and thus as well,

$\hat{P}_{\alpha,1}\left(\hat{\varphi}\right)=\frac{\hbar}{\theta_{\alpha}}\left[\hat{x}_{\alpha,2},\hat{\varphi}\right],\qquad\hat{P}_{\alpha,2}\left(\hat{\varphi}\right)=-\frac{\hbar}{\theta_{\alpha}}\left[\hat{x}_{\alpha,1},\hat{\varphi}\right],\qquad\hat{P}_{\alpha,a}\left(\hat{\varphi}\right)=\frac{\hbar}{\theta_{\alpha}}\,\epsilon_{ab}\,\left[\hat{x}_{\alpha,b},\hat{\varphi}\right].$
(304)

Finally, in view of the relations
$\hat{d}_{\alpha}=\hat{b}^{\dagger}_{\alpha}+i\theta_{\alpha}\hat{p}_{\alpha,-}/\hbar$
and
$\hat{d}^{\dagger}_{\alpha}=\hat{b}_{\alpha}-i\theta_{\alpha}\hat{p}_{\alpha,+}/\hbar$,
and thus the following definitions for their representatives in the enveloping
algebra representation,

$\hat{D}_{\alpha}=\hat{B}^{\ddagger}_{\alpha}+i\frac{\theta_{\alpha}}{\hbar}\hat{P}_{\alpha,-},\qquad\hat{D}^{\ddagger}_{\alpha}=\hat{B}_{\alpha}-i\frac{\theta_{\alpha}}{\hbar}\hat{P}_{\alpha,+},$
(305)

one finds,

$\hat{D}_{\alpha}(\hat{\varphi})=\hat{\varphi}\,\hat{b}^{\dagger}_{\alpha},\qquad\hat{D}^{\ddagger}_{\alpha}(\hat{\varphi})=\hat{\varphi}\,\hat{b}_{\alpha}.$
(306)

Note how for these operators $(\hat{D}_{\alpha},\hat{D}^{\ddagger}_{\alpha})$
the multiplications of $\hat{\varphi}$ by $\hat{b}^{\dagger}_{\alpha}$ and
$\hat{b}_{\alpha}$ are from the right, to be contrasted with these
multiplications from the left in the case of the operators
$(\hat{B}_{\alpha},\hat{B}^{\ddagger}_{\alpha})$, thereby manifesting again
the adjoint and dual character of the $d$-Fock sector or dual classical
Hilbert space ${\cal H}^{\dagger}_{c}$.

Most obviously, any given $\alpha$ sector of the above structure for a fixed
value of $\alpha=1,2\cdots,n$ and thus fixed value for the noncommutativity
parameter $\theta_{\alpha}>0$ corresponds to a single noncommutative Euclidean
plane and a conjugate magnetic field whose value is set by that parameter
$\theta_{\alpha}$. In this case the analysis of this work based on the
ordinary principles of canonical quantisation and having led to conclusions
that coincide precisely and exactly with it in the $\hat{x}_{i}$-enveloping
algebra representation, justify a posteriori the heuristic quantum mechanics
of the noncommutative Euclidean plane as constructed by Scholtz et al. based
solely on the defining configuration space commutation relation
$[\hat{x}_{1},\hat{x}_{2}]=i\theta\,\mathbb{I}_{c}$ with $\theta>0$ (and
$\mathbb{I}_{c}$ being the unit operator on the associated classical Hilbert
space).

That construction [11, 12, 13] relies directly on the fact that this
commutation relation defines a Fock algebra with
$\hat{b}=\hat{x}_{1}+i\hat{x}_{2}$ and
$[\hat{b},\hat{b}^{\dagger}]=2\theta\,\mathbb{I}_{c}$, of which the canonical
coherent states correspond to the states $|z_{\alpha};\Omega_{b}\rangle$ of
the present discussion, with $z_{\alpha}=x_{1}+ix_{2}$ (and due account of
different choices of normalisation conventions). These localised and
normalisable states play the role of the sharp and non-normalisable position
eigenstates in the commutative case, replace the classical coordinates of the
classical commutative configuration space, and generate the entire so-called
classical Hilbert space through the configuration space integration of the
projection operator density $|z_{\alpha};\Omega_{b}\rangle\langle
z_{\alpha};\Omega_{b}|$.

The quantum Hilbert space of quantum states is then identified with the
enveloping algebra of the coordinate operators $(\hat{x}_{1},\hat{x}_{2})$,
namely $\hat{\varphi}(\hat{x}_{1},\hat{x}_{2})$ or
$\hat{\varphi}(\hat{b}^{\dagger}_{\alpha},\hat{b}_{\alpha})$ in our notations,
which is generated by the configuration space integration of the operators
$|z_{\alpha};\Omega_{b}\rangle*_{\theta}\langle z_{\alpha};\Omega_{b}|$ that
act on the classical Hilbert space spanned by the states
$|z_{\alpha};\Omega_{b}\rangle$, with $*_{\theta}$ being the star product
associated to the parameter $\theta$. Furthermore on the quantum Hilbert space
realised as the enveloping algebra of the coordinate operators
$(\hat{x}_{1},\hat{x}_{2})$, the position operators are realised as the left
multiplication of $\hat{\varphi}$ by the operators
$(\hat{x}_{1},\hat{x}_{2})$, while canonically conjugate momentum operators
may be identified to arise and to be realised in terms of the commutators of
$\hat{\varphi}$ with the coordinate operators $(\hat{x}_{1},\hat{x}_{2})$ in
precisely and exactly the same way as established above. Finally the inner
product of abstract quantum states represented as operators
$\hat{\varphi}_{1}$ and $\hat{\varphi}_{2}$ within the
$(\hat{x}_{1},\hat{x}_{2})$-enveloping algebra is chosen to be defined through
the trace over the classical Hilbert space—the $b_{\alpha}$-sector in our
notations—of the product $\hat{\varphi}^{\dagger}_{1}\hat{\varphi}_{2}$ of the
two operators representing the two quantum states, which is indeed once again
in complete agreement with the above conclusion based on standard canonical
quantisation (and due account of different choices of normalisation
conventions).

This heuristic construction of a quantum mechanics on a noncommutative
Euclidean plane coincides thus exactly and precisely with the above
observations and conclusions that directly derive from a standard canonical
quantisation in the presence of a magnetic field in the phase space sector of
the canonical conjugate momenta of a classical dynamics, specifically however,
in the configuration space enveloping algebra representation of the quantum
system.

As the analysis of this work has explicitly demonstrated based on a standard
canonical quantisation approach, the same structure extends to any even
dimensional Euclidian configuration space in the presence of a nondegenerate
conjugate magnetic field in the canonical conjugate momentum sector of phase
space, provided the quantised system and its states is represented through the
enveloping algebra of the noncommuting configuration space position operators
that act on a classical Hilbert space spanned by states that play a role akin
to position eigenstates without being in fact sharp states, but are specific
finite norm canonical coherent Fock states whose definition is aligned with
the values of the conjugate magnetic field.

When represented in the $(\alpha,a)$ parametrisation of the configuration and
phase spaces which block diagonalises the conjugate magnetic field, and
factorises the classical and quantum Hilbert spaces into commuting and
decoupled sectors for each value of $\alpha$, each $\alpha$ sector then
coincides with the quantum mechanics of the two-dimensional noncommutative
Euclidean plane as constructed by Scholtz et al. Yet, based on the canonical
quantisation approach, other realisations and representations of the same
quantum mechanics on such a noncommutative Euclidean configuration space of
even dimension are also available, some familiar, other less familiar, thereby
offering the potential of complementary methods for explicit calculations
relevant to the dynamics of quantum systems over such noncommutative
configuration spaces.

##  5 Conclusions

As would deserve to be more widely known, a conjugate or dual magnetic field
in the canonically conjugate momentum sector of a classical phase space
induces noncommutativity in configuration space, whether in terms of
nonvanishing classical Poisson brackets, or nonvanishing quantum commutation
relations for the position coordinates or operators of the canonically
quantised system. This observation is the basis for the canonical quantisation
approach to a formulation of a quantum mechanics over a noncommutative
configuration space that is pursued in the present work. More specifically,
this is done herein for an Euclidean configuration space of even dimension,
and a constant and nondegenerate conjugate magnetic field.

As a preliminary, the quantum mechanics of a commutative configuration space
has been revisited first, in order to set up considerations that are directly
relevant to the noncommutative case, but not readily available from standard
treatises of quantum mechanics. In particular through the choice of some real,
symmetric and positive definite matrix of constant coefficients of length-
squared dimension, a specific class of localised and normalisable quantum
states parametrised by all points in configuration space is introduced in
terms of which it is possible to define a specific and related $*$-product
even in the commutative case. These distinguished states turn out to
correspond to a specific subclass of canonical quantum coherent states but
restricted to configuration space. Yet through the associated $*$-product and
by being overcomplete these states generate the entire abstract quantum
Hilbert space of the system. The consideration of these states allows for the
construction of a specific wave function representation in configuration space
of quantum states different from the ordinary position or momentum wave
function ones. Finally, a representation of abstract quantum states and the
abstract Heisenberg algebra in terms of the enveloping algebra of the
commuting position operators is thereby identified, with the specific class of
distinguished quantum states playing the role of ordinary configuration space
coordinates in the commutative case.

When a constant conjugate magnetic field is introduced, a parallel analysis is
developed, to lead to new insight into the understanding of the quantum
mechanics on a noncommutative configuration space. By adjusting the ensemble
of distinguished quantum states introduced in the commutative case to the
choice of conjugate magnetic field, canonical quantisation produces in a
nontrivial manner an appealing organisation of the quantum Hilbert space of
quantum states as a tensor product of two ensembles of Fock algebras and Fock
Hilbert spaces, one associated to the initially distinguished quantum states
that then span a so-called classical Hilbert space, and the other being
isomorphic to the dual of that classical Hilbert space. While in a most
natural way, a representation of the quantum states is achieved in terms of
operators in the enveloping algebra of the noncommuting configuration space
coordinate operators acting on the states of the classical Hilbert space, the
latter thus playing the role of what are in the commutative case position
eigenstates but now being normalisable. However besides this representation,
other representations also arise, of possible practical relevance as well.

By having followed this path towards a quantum mechanics of noncommutative
configuration spaces, has also produced and thus justified a posteriori
precisely the very heuristic construction of a quantum mechanics on the
noncommutative Euclidean plane that F. G. Scholtz and his collaborators have
developed over the years. Combining Dirac’s standard principles of canonical
quantisation with a conjugate magnetic field in phase space thus totally
justifies that work and its results from this complementary point of view, and
provides further understanding of its particular features, as for instance the
apparent emergence of canonically conjugate momenta out of an algebraic
structure only based on the noncommuting quantum position operators in
configuration space of which the abstract representation is that of the
classical Hilbert space.

The specific structures that organise the quantum mechanics of such
noncommutative Euclidean configuration spaces, as they were first identified
by Scholtz et al. by pursuing analogies with quantum mechanics on commutative
configuration spaces, may suggest pathways towards extensions to other
topologies and geometries than those of Euclidean configuration spaces.
Indeed, the doubled tensor structure in terms of a classical Hilbert space and
of its dual is a tantalizing possible avenue, whose exploration is also part
of F. G. Scholtz’s on-going programme. One may also wonder how other choices
for a conjugate magnetic field, now dependent whether on configuration or
momentum space, or both, would produce specific classes of quantum mechanics
on other types of noncommutative structures inclusive of noncommutative
configuration spaces. Or conversely, for instance, which classes of well known
quantum mechanics on other types of noncommutative configuration spaces, say
for example noncommutative spheres (see for instance [24]), could also be
understood from a similar approach combining Dirac’s ordinary canonical
quantisation based on the correspondence principle and a conjugate magnetic
field in phase space.

## Acknowledgements

The author is most grateful to Professor Frederik Scholtz (University of
Stellenbosch, South Africa) for the many, plenty and inspiring discussions
that we have had over the years on the topics of the present paper and on many
other subjects. And even more so, for his always truly most constant welcoming
and warm hospitality at the Department of Physics of the University of
Stellenbosch and at the then National Institute for Theoretical Physics
(NITheP), now the National Institute for Theoretical & Computational Sciences
(NITheCS) of South Africa. This work is supported in part by the Institut
Interuniversitaire des Sciences Nucléaires (IISN, Belgium).

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