text:
\section{Introduction}\label{sec:intro}
Noncommutative quantum mechanics (NCQM) is an active field of research these
days. The starting point here is to alter the canonical commutation relations
(CCR) among the respective positions and momenta coordinates and hence
introduce a new noncommutative Lie algebra structure. Consult
\cite{scholtzt,delducetalt} for a detailed account on this approach.

There is another approach available where one starts with noncommutative field theory (NCFT) studying quantum field theory on noncommutative space-time. An excellent pedagogical treatment to noncommutative quantum field theory can be found in \cite{Szabo}. Refer to the excellent review \cite{Douglasetal} to delve further into the studies of NCFT. A noncommutative quantum field theory is the one where the fields are functions of space-time coordinates with spatial coordinates failing to commute with each other. Among others, Snyder and Yang were the leading proponents to introduce the concept of noncommutative structure of space-time (see \cite{Sny,Yang}). Introduction of such assumption of spatial noncommutativity eliminates ultraviolet (UV) divergences of quantum field theory and runs parallel to the technique of renormalization as a cure to such annoying divergence issues in quantum field theory. NCQM can then be seen as nonrelativistic approximation of NCFT (see \cite{Hoetal,Chaichianetal}).

In yet another approach (see \cite{ncqmjmp},\cite{ncqmjpa}), the authors start from a certain nilpotent Lie group $G_{\NC}$ as the defining group of NCQM for a system of 2 degrees of freedom and compute its unitary dual using the method of orbits introduced by Kirillov (see \cite{Kirillovbook}). Although $G_{\NC}$ does not contain the Weyl-Heisenberg group $\GWH$ as its subgroup, the unitary dual of $\GWH$ is found to be sitting inside that of $G_{\NC}$. Various gauges arising in NCQM are also found to be related to a certain family of unitary irreducible representations (UIRs) of $G_{\NC}$.  The Lie group $G_{\NC}$ was later identified as the kinematical symmetry group of NCQM in \cite {wigjmp} where various Wigner functions for such model of NCQM were computed that were found to be supported on relevant families of coadjoint orbits associated with $G_{\NC}$. The purpose of this paper is to construct coherent states arising from $G_{\NC}$, study their properties and discuss the associated Berezin-Toeplitz quantization of the observables on the 4-dimensional phase space, including the relevant ``semi-classical'' asymptotics.

In~Section~\ref{sec:cs ncqm}, we~review the relevant facts about the UIRs of
the 7-dimensional real Lie group $G_{\NC}$ from \cite{ncqmjpa} and proceed to
construct the associated coherent states and reproducing kernel.
The~UIRs in question are parameterized by triples $(\hbar,\vthta,\cB)\in\RR^3$
satisfying $\hbar^2-\vthta\cB\neq0$;
the ``degenerate'' case $\hbar^2=\vthta\cB$ is then discussed separately in
Section~\ref{sec:degenrt-rep}.
With~the coherents states in hand, one~constructs the Toeplitz operators in
the standard fashion, and we~examine the corresponding ``semiclassical limit''
of the resulting Berezin-Toeplitz quantization in Section~\ref{secTOEP}.
The~main novelty here, of~course, is~the presence of the three deformation
parameters $\hbar,\vthta,\cB$ instead of the sole Planck constant~$\hbar$,
thus making it already somewhat unclear in what manner these three should be
allowed to approach zero in any analogue of the ordinary semi-classical limit
where one just has $\hbar\searrow0$; note that $(\hbar,\vthta,\cB)$ cannot
approach $(0,0,0)$ completely unrestrictedly due to the non-degenaracy
condition $\hbar^2-\vthta\cB\neq0$. It~turns out that the right objects from
this point of view are the renormalized quantities $B:=\cB/\hbar$ and
$T:=\vthta/\hbar$, and nice semiclassical asymptotics are established for
the situation when $\hbar,B,T$ all tend to zero, without any restrictions.
(Note that $B$ has the physical interpretation of the applied magnetic field,
see eqn.~(3.6) in \cite{ncqmjpa}. Note also that in the $B,T$ notation the
non-degeneracy condition becomes simply $BT\neq1$, so~there is no longer
any problem with $B,T,\hbar$ simultaneously approaching zero.)

One~can in principle consider the Berezin-Toeplitz quantization and the
corresponding semi-classical behaviour also in the degenerate case
$\hbar^2-\vthta\cB=0$. In~this case, the~underlying Hilbert space
carrying the coherent states undergoes a dimension reduction from 4 to~2,
and while we still get a unique associated reproducing kernel,
it~turns out that the measure defining the inner product in the reproducing
kernel Hilbert space at hand is no longer uniquely determined.
Furthermore, the~semi-classical asymptotics of the associated Berezin-Toeplitz
operators turn out not to depend at all on the choice of this measure,
nor in fact on the deformation parameter~$\vthta$ (or~on~$\cB=\hbar^2/\vthta$),
and reduce just to the plain Berezin-Toeplitz deformation quantization
(star-product) on the complex plane.
Details are supplied in Section~\ref{secTOEPdeg}.

\section*{Acknowledgements} The~third author (M.E) was supported by GA~\v{C}R
grant no.~201/12/G028 and Institutional funding (RVO) for I\v{C}~47813059.
The~first author (S.~H.~H.~C.) gratefully acknowledges a grant from National
Natural Science Foundation of China (NSFC) under Grant No.~11550110186.

The~middle named author, S.~Twareque~Ali, passed away unexpectedly on
January~25 2016, before this manuscript was finished.
The~other two authors dedicate this paper to~him, in~memory of his scientific
enthusiasm and great and lasting friendship.

\section{Noncommutative coherent states associated to $G_{\NC}$}
\label{sec:cs ncqm}
The geometry of the coadjoint orbits associated with the 7-dimensional real Lie group $G_{\NC}$ is studied in detail in \cite{ncqmjpa}. There the orbits were classified based on the triple $(\rho,\sigma,\tau)\in\mathbb{R}^{3}$. In this paper, we will focus on the family of UIRs corresponding to $(\rho,\sigma,\tau)\in\mathbb{R}^{3}$ such that $\rho\neq 0$, $\sigma\neq 0$ and $\tau\neq 0$ with $\rho^{2}\alpha^{2}-\gamma\beta\sigma\tau\neq 0$. The UIRs are given by
\begin{eqnarray}\label{eq:UIR-gen-NCQM}
\lefteqn{(\hat{U}^{\rho}_{\sigma,\tau}(\theta,\phi,\psi,\mathbf q,\mathbf p)f)(r_{1},s_{2})}\nonumber\\
    &&=e^{i\rho(\theta+\alpha q_{2}s_{2}+\alpha p_{1}r_{1}+\frac{\alpha}{2}q_{1}p_{1}-\frac{\alpha}{2}q_{2}p_{2})}e^{i\sigma(\phi+\beta p_{1}s_{2}-\frac{\beta}{2}p_{1}p_{2})}\nonumber\\
    &&\times e^{i\tau(\psi+\gamma q_{2}r_{1}+\frac{\gamma}{2}q_{2}q_{1})}f(r_{1}+q_{1},s_{2}-p_{2}).
\end{eqnarray}
Note that in (\ref{eq:UIR-gen-NCQM}), $r_{1}$ has the dimension of length and $s_{2}$ has that of momentum. By taking the inverse Fourier transform of (\ref{eq:UIR-gen-NCQM}) in the second coordinate $s_2$ yields the following representation on $L^{2}(\mathbb{R}^{2},d\mathbf r)$:
\begin{eqnarray}\label{eq:UIR-Innv-FT-gen-NCQM}
\lefteqn{(U^{\rho}_{\sigma,\tau}(\theta,\phi,\psi,\mathbf q,\mathbf p)f)(r_{1},r_{2})}\nonumber\\
    &&=e^{i\rho(\theta+\alpha p_{1}r_{1}+\alpha p_{2}r_{2}+\frac{\alpha}{2}q_{1}p_{1}+\frac{\alpha}{2}q_{2}p_{2})}e^{i\sigma(\phi+\frac{\beta}{2}p_{1}p_{2})}\nonumber\\
    &&\times e^{i\tau(\psi+\gamma q_{2}r_{1}+\frac{\gamma}{2}q_{1}q_{2})}f\left(r_{1}+q_{1},r_{2}+q_{2}+\frac{\sigma\beta}{\rho\alpha}p_{1}\right),
\end{eqnarray}
where $f\in L^{2}(\mathbb{R}^{2},d\mathbf r)$.

\begin{defi}\label{cs-def}
For a given fixed vector $\chi\in L^{2}(\mathbb{R}^{2},d\mathbf r)$ and a fixed point $(\mathbf q,\mathbf p)\in\mathbb{R}^{4}$, the underlying phase space for the 2-dimensional system under study, let us define the following vectors in $L^{2}(\mathbb{R}^{2},d\mathbf r)$ as
\begin{equation}\label{def-nc-cs}
\chi_{\mathbf{q},\mathbf{p}}^{\NC}=U^{\rho}_{\sigma,\tau}(0,0,0,-\mathbf q,\mathbf p)\chi.
\end{equation}
Introducing the following change of variables:
\begin{equation*}
\hbar=\frac{1}{\rho\alpha},\;\;\vartheta=-\frac{\sigma\beta}{\rho^{2}\alpha^{2}}\;\hbox{and}\;\mathcal{B}=-\frac{\tau\gamma}{\rho^{2}\alpha^{2}},
\end{equation*}
the vectors $\chi_{\mathbf{q},\mathbf{p}}^{\NC}$ read
\begin{equation*}
\chi_{\mathbf{q},\mathbf{p}}^{\NC}(\mathbf r)=e^{\frac{i}{\hbar}\left(\mathbf r-\frac{1}{2}\mathbf q\right).\mathbf p-\frac{i\vartheta}{2\hbar^{2}}p_{1}p_{2}+\frac{i\mathcal{B}}{\hbar^{2}}\left(q_{2}r_{1}-\frac{1}{2}q_{1}q_{2}\right)}\chi\left(r_{1}-q_{1},r_{2}-q_{2}-\frac{\vartheta}{\hbar}p_{1}\right).
\end{equation*}
Let us now define the noncommutative coherent states using the vectors $\chi_{\mathbf{q},\mathbf{p}}^{\NC}$ as
\begin{equation} \label{normlzd-nc-cs}
\eta_{\mathbf{q},\mathbf{p}}^{\NC}(\mathbf r)=e^{\frac{i}{\hbar}\left(\mathbf r-\frac{1}{2}\mathbf q\right).\mathbf p-\frac{i\vartheta}{2\hbar^{2}}p_{1}p_{2}+\frac{i\mathcal{B}}{\hbar^{2}}\left(q_{2}r_{1}-\frac{1}{2}q_{1}q_{2}\right)}\eta\left(r_{1}-q_{1},r_{2}-q_{2}-\frac{\vartheta}{\hbar}p_{1}\right),
\end{equation}
where $\eta$ is a vector given by $\eta=\frac{\chi}{\vert\vert\chi\vert\vert}$.
\end{defi} \medskip
The phase space for the 2-dimensional noncommutative system is
$\mathbb{R}^{4}$. The underlying observables are functions defined over the
phase space variables $\mathbf q$, $\mathbf p$. These functions are taken to be elements of
the Hilbert space $L^{2}(\mathbb{R}^{4},d\nu(\mathbf q,\mathbf p))$ equipped with the
measure
$d\nu(\mathbf q,\mathbf p)=\frac{\vert\hbar^{2}-\mathcal{B}\vartheta\vert}
 {4\pi^{2}\hbar^{4}}d\mathbf q\;d\mathbf p$.
The noncommutative coherent states $\eta_{\mathbf{q},\mathbf{p}}^{\NC}$, given by (\ref{normlzd-nc-cs}),
satisfy the resolution of identity as stated by the following lemma:
\begin{lem}$\!\!${\bf }\label{lem-res-idnty}
The vectors $\eta_{\mathbf{q},\mathbf{p}}^{\NC}$ defined as the noncommutative coherent states by (\ref{normlzd-nc-cs}) satisfy the following integral relation:
\begin{equation}\label{stmnt-res-idnty}
\int_{\mathbb{R}^{4}}\vert\eta_{\mathbf{q},\mathbf{p}}^{\NC}\rangle\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert\;d\nu(\mathbf q,\mathbf p)=\mathbb{I},
\end{equation}
where $\mathbb{I}$ is the identity operator on $L^{2}(\mathbb{R}^{2},d\mathbf r)$.
\end{lem}
\noindent{\bf{ Proof}\,\,}{.}
Introducing the following changes of variables:
\begin{equation}\label{nc-chng-vrbl}
\begin{aligned}
q_{1}^{\NC}&=q_{1}\\
q_{2}^{\NC}&=q_{2}+\frac{\vartheta}{\hbar}p_{1}\\
p_{1}^{\NC}&=p_{1}+\frac{\mathcal{B}}{\hbar}q_{2}\\
p_{2}^{\NC}&=p_{2},
\end{aligned}
\end{equation}
the noncommutative coherent states $\eta_{\mathbf{q},\mathbf{p}}^{\NC}$, appearing in (\ref{normlzd-nc-cs}), can neatly be written as
\begin{equation}\label{nc-cs-smpfd}
\eta_{\mathbf{q},\mathbf{p}}^{\NC}(\mathbf r)=e^{\frac{i}{\hbar}\left(\mathbf r-\frac{1}{2}\mathbf q^{\NC}\right).\mathbf p^{\NC}}\eta\left(\mathbf r-\mathbf q^{\NC}\right),
\end{equation}
where $(q_{1}^{\NC},q_{2}^{\NC})$ and $(p_{1}^{\NC},p_{2}^{\NC})$ are denoted as $\mathbf q^{\NC}$ and $\mathbf p^{\NC}$, respectively.

The associated measures transform as
\begin{equation}\label{jacbn-ncqm-qm}
d\mathbf q^{\NC}d\mathbf p^{\NC}=\frac{\vert\hbar^{2}-\mathcal{B}\vartheta\vert}{\hbar^{2}}d\mathbf q\;d\mathbf p.
\end{equation}

Let us now choose two compactly supported smooth functions $f$ and $g$ in $L^{2}(\mathbb{R}^{2},d\mathbf r)$. One then obtains,
\begin{eqnarray}\label{prf-res-idnty}
\lefteqn{\int_{\mathbb{R}^{4}}\langle f\vert\eta_{\mathbf{q},\mathbf{p}}^{\NC}\rangle\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert g\rangle d\nu(\mathbf q,\mathbf p)}\nonumber\\
&&=\frac{\vert\hbar^{2}-\mathcal{B}\vartheta\vert}{4\pi^{2}\hbar^{4}}\int_{\mathbb{R}^{4}}\langle f\vert\eta_{\mathbf{q},\mathbf{p}}^{\NC}\rangle\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert g\rangle d\mathbf q\;d\mathbf p\nonumber\\
&&=\frac{1}{4\pi^{2}\hbar^{2}}\int_{\mathbb{R}^{4}}\langle f\vert\eta_{\mathbf{q},\mathbf{p}}^{\NC}\rangle\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert g\rangle d\mathbf q^{\NC}d\mathbf p^{\NC}\nonumber\\
&&=\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{4}}\left[\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}e^{\frac{i}{\hbar}(\mathbf r-\mathbf r^{\prime}).\mathbf p^{\NC}}\overline{f(\mathbf r)}\;\eta(\mathbf r-\mathbf q^{\NC})\overline{\eta(\mathbf r^{\prime}-\mathbf q^{\NC})}g(\mathbf r^{\prime})\;d\mathbf r\;d\mathbf r^{\prime}\right]d\mathbf q^{\NC}d\left(\frac{\mathbf p^{\NC}}{\hbar}\right)\nonumber\\
&&=\int_{\mathbb{R}^{2}}\left[\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\delta^{(2)}(\mathbf r-\mathbf r^{\prime})\overline{f(\mathbf r)}\;\eta(\mathbf r-\mathbf q^{\NC})\overline{\eta(\mathbf r^{\prime}-\mathbf q^{\NC})}g(\mathbf r^{\prime})\;d\mathbf r\;d\mathbf r^{\prime}\right]d\mathbf q^{\NC}\nonumber\\
&&=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\overline{f(\mathbf r)}\;\eta(\mathbf r-\mathbf q^{\NC})\overline{\eta(\mathbf r-\mathbf q^{\NC})}g(\mathbf r)\;d\mathbf r\;d\mathbf q^{\NC}\nonumber\\
&&=\vert\vert\eta\vert\vert^{2}\langle f\vert g\rangle\nonumber\\
&&=\langle f\vert g\rangle.
\end{eqnarray}
Using the continuity of the inner product of the underlying Hilbert space and the fact that the compactly supported smooth functions are dense in $L^{2}(\mathbb{R}^{2},d\mathbf r)$, one can extend the above equality to any pair of functions $f$, $g$ in $L^{2}(\mathbb{R}^{2},d\mathbf r)$ and hence proving the lemma.
\qed

Let us rewrite (\ref{nc-cs-smpfd}) and observe that
\begin{eqnarray}\label{cs-smplfd-oprtorial}
\eta_{\mathbf{q},\mathbf{p}}^{\NC}(\mathbf r)&=&e^{\frac{i}{\hbar}\left(\mathbf r-\frac{1}{2}\mathbf q^{\NC}\right).\mathbf p^{\NC}}\eta\left(\mathbf r-\mathbf q^{\NC}\right)\nonumber\\
&=&e^{\frac{i}{\hbar}\left(\mathbf r-\frac{1}{2}\mathbf q^{\NC}\right).\mathbf p^{\NC}}e^{\left(-q^{\NC}_{1}\frac{\partial}{\partial r_{1}}-q^{\NC}_{2}\frac{\partial}{\partial r_{2}}\right)}\eta(\mathbf r)\nonumber\\
&=&e^{\frac{i}{\hbar}\left[\mathbf r.\mathbf p^{\NC}-q^{\NC}_{1}\left(-i\hbar\frac{\partial}{\partial r_{1}}\right)-q^{\NC}_{2}\left(-i\hbar\frac{\partial}{\partial r_{2}}\right)\right]}\eta(\mathbf r)\nonumber\\
&=&e^{\frac{i}{\hbar}(\mbox{\boldmath $\xi$}^{T}\omega^{\NC}\mathbf X)}\eta(\mathbf r),
\end{eqnarray}
where $\mbox{\boldmath $\xi$}$ and $\mathbf X$ are $4\times 1$ column vectors given by
\begin{equation}\label{nc-dis-op-symbls}
\mbox{\boldmath $\xi$}=\begin{bmatrix}q^{\NC}_{1}\\q^{\NC}_{2}\\p^{\NC}_{1}\\p^{\NC}_{2}\end{bmatrix}=\begin{bmatrix}q_{1}\\q_{2}+\frac{\vartheta}{\hbar}p_{1}\\p_{1}+\frac{\mathcal{B}}{\hbar}q_{2}\\p_{2}\end{bmatrix}\;\;\hbox{and}\;\;
\mathbf X=\begin{bmatrix}\hat{Q}^{\NC}_{1}\\\hat{Q}^{\NC}_{2}\\\hat{P}^{\NC}_{1}\\\hat{P}^{\NC}_{2}\end{bmatrix}=\begin{bmatrix}r_{1}+i\vartheta\frac{\partial}{\partial r_{2}}\\r_{2}\\-i\hbar\frac{\partial}{\partial r_{1}}\\-\frac{\mathcal{B}}{\hbar}r_{1}-i\hbar\frac{\partial}{\partial r_{2}}\end{bmatrix},
\end{equation}
while $\omega^{\NC}$ in (\ref{cs-smplfd-oprtorial}) is given by the following $4\times 4$ matrix:
\begin{equation}\label{nc-anlg-J}
\omega^{\NC}=\begin{bmatrix}0&0&-1&0\\-\frac{\mathcal{B}\hbar}{(\hbar^{2}-\mathcal{B}\vartheta)}&0&0&-\frac{\hbar^{2}}{(\hbar^{2}-\mathcal{B}\vartheta)}\\\frac{\hbar^{2}}{\hbar^{2}-\mathcal{B}\vartheta}&0&0&\frac{\vartheta\hbar}{\hbar^{2}-\mathcal{B}\vartheta}\\0&1&0&0\end{bmatrix}.
\end{equation}
The 4 entries of the column vector $\mathbf X$, i.e. $\hat{Q}^{\NC}_{i}$, $\hat{P}^{\NC}_{i}$ for $i=1,2$,  in (\ref{nc-dis-op-symbls}) represent the non-central generators of $G_{\NC}$ in the Landau gauge representation defined on $L^{2}(\mathbb{R}^{2},d\mathbf r)$ as obtained in (\cite{ncqmjpa}).
\begin{defi}\label{nc-displcmnt-oprt-def}
The unitary operator $\mathcal{D}^{\NC}(\mathbf q,\mathbf p)$ that generates the noncommutative coherent state vectors $\eta_{\mathbf{q},\mathbf{p}}^{\NC}$ by acting upon the normalized ground state vector $\eta$ is defined as the noncommutative displacement operator.
\begin{eqnarray}\label{eq-nc-displc-opt-def}
\eta_{\mathbf{q},\mathbf{p}}^{\NC}(\mathbf r)&=&\mathcal{D}^{\NC}(\mathbf q,\mathbf p)\eta(\mathbf r)\nonumber\\
&=&e^{\frac{i}{\hbar}\left(\mbox{\boldmath $\xi$}^{T}\omega^{\NC}\mathbf X\right)}\eta(\mathbf r),
\end{eqnarray}
where $\mbox{\boldmath $\xi$}$, $\mathbf X$ and $\omega^{\NC}$ are as given in (\ref{nc-dis-op-symbls}) and (\ref{nc-anlg-J}).
\end{defi} \medskip
\begin{remark}
A few remarks on the quantum mechanical limits of the above formulations are in order. From (\ref{nc-dis-op-symbls}) and (\ref{nc-anlg-J}), it can easily be seen that as $\vartheta\rightarrow 0$, $\mathcal{B}\rightarrow 0$, the noncommutative displacement operator $\mathcal{D}^{\NC}(\mathbf q,\mathbf p)$ approaches the canonical displacement operator:
\begin{eqnarray}\label{lim-nc-displcmnt-oprtr}
\mathcal{D}^{\NC}(\mathbf q,\mathbf p)&\xrightarrow{\vartheta\rightarrow 0,\;\vartheta\rightarrow 0}&\exp\frac{i}{\hbar}\left(\begin{bmatrix}q_{1}&q_{2}&p_{1}&p_{2}\end{bmatrix}\begin{bmatrix}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{bmatrix}\begin{bmatrix}\hat{Q}_{1}\\\hat{Q}_{2}\\\hat{P}_{1}\\\hat{P}_{2}\end{bmatrix}\right)\nonumber\\
&=&\exp\frac{i}{\hbar}(\mathbf p.\mathbf{\hat{Q}}-\mathbf q.\mathbf{\hat{P}}),
\end{eqnarray}
where $\mathbf{\hat{Q}}=(\hat{Q}_{1},\hat{Q}_{2})=(r_{1},r_{2})$ and $\mathbf{\hat{P}}=(\hat{P}_{1},\hat{P}_{2})=(-i\hbar\frac{\partial}{\partial r_{1}},-i\hbar\frac{\partial}{\partial r_{2}})$ are the standard quantum mechanical representation of the position and momentum operators of the underlying 2-dimensional system defined on $L^{2}(\mathbb{R}^{2},d\mathbf r)$. The cases $\mathcal{B}\rightarrow 0$ or $\vartheta\rightarrow 0$ can similarly be studied. For example,
\begin{equation}\label{case-B-0}
\omega^{\NC}\xrightarrow{\mathcal{B}\rightarrow 0}\begin{bmatrix}0&0&-1&0\\0&0&0&-1\\1&0&0&\frac{\vartheta}{\hbar}\\0&1&0&0\end{bmatrix},\;\omega^{\NC}\xrightarrow{\vartheta\rightarrow 0}\begin{bmatrix}0&0&-1&0\\-\frac{\mathcal{B}}{\hbar}&0&0&-1\\1&0&0&0\\0&1&0&0\end{bmatrix}.
\end{equation}
\end{remark}

Now in view of (\ref{cs-smplfd-oprtorial}), the noncommutative coherent states $\eta_{\mathbf{q},\mathbf{p}}^{\NC}$ can be read off as
\begin{equation}\label{nc-cs-rep-ker}
\eta_{\mathbf{q},\mathbf{p}}^{\NC}(\mathbf r)=e^{\frac{i}{\hbar}(p_{1}^{\NC}\hat{Q}_{1}+p_{2}^{\NC}\hat{Q}_{2}-q_{1}^{\NC}\hat{P}_{1}-q_{2}^{\NC}\hat{P}_{2})}\eta(\mathbf r).
\end{equation}
Let us choose an element $\phi\in L^{2}(\mathbb{R}^{2},d\mathbf r)$ and consider the map $\Phi:\mathbb{R}^{4}\rightarrow\mathbb{C}$ for a fixed vector $\phi\in L^{2}(\mathbb{R}^{2},d\mathbf r)$ by
\begin{equation}\label{def-map}
\Phi(\mathbf q,\mathbf p)=\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert\phi\rangle.
\end{equation}
It is immediate that $\Phi\in L^{2}(\mathbb{R}^{4},d\nu(\mathbf q,\mathbf p))$. Now consider the isometry map
\begin{equation}\label{iso-map-def}
W:L^{2}(\mathbb{R}^{2},d\mathbf r)\rightarrow L^{2}(\mathbb{R}^{4},d\nu)
\end{equation}
defined by $W\phi=\Phi$. The range $\mathcal W$ of this isometry map, a~closed subspace of $L^{2}(\mathbb{R}^{4},d\nu)$, is~a~reproducing kernel Hilbert space (RKHS) and the function $K^{\NC}:\mathbb{R}^{4}\times\mathbb{R}^{4}\rightarrow \mathbb{C}$ with
\begin{equation}\label{rep-ker-def}
K^{\NC}(\mathbf q,\mathbf p;\mathbf q^{\prime},\mathbf p^{\prime})=\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert\eta_{\mathbf{q}^{\prime},\mathbf{p}^{\prime}}^{\NC}\rangle
\end{equation}
is its reproducing kernel.

Let us now recall a few facts about $G_{\NC}$ from \cite{ncqmjmp}. The group composition rule for $G_{\NC}$ are given by
\begin{eqnarray}\label{grplawtriplyextendedgrp}
\lefteqn{(\theta,\phi,\psi,\mathbf q,\mathbf p)(\theta^{\prime},\phi^{\prime},\psi^{\prime},\mathbf q^{\prime},\mathbf p^{\prime})}\nonumber\\
&&=(\theta+\theta^{\prime}+\alpha\;\xi((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime})),\phi+\phi^{\prime}+\beta\;\xi^{\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))\nonumber\\
&&\;\;\;,\psi+\psi^{\prime}+\gamma\;\xi^{\prime\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime})),\mathbf q+\mathbf q^{\prime},\mathbf p+\mathbf p^{\prime}),
\end{eqnarray}
where the 3-inequivalent local exponents of the abelian group of translations in $\mathbb{R}^{4}$ is given by
\begin{equation}\label{local-expnts}
\begin{aligned}
\xi((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))&=\frac{1}{2}[q_{1}p_{1}^{\prime}+q_{2}p_{2}^{\prime}-p_{1}q_{1}^{\prime}-p_{2}q_{2}^{\prime}],\\
\xi^{\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))&=\frac{1}{2}[p_{1}p_{2}^{\prime}-p_{2}p_{1}^{\prime}],\\
\xi^{\prime\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))&=\frac{1}{2}[q_{1}q_{2}^{\prime}-q_{2}q_{1}^{\prime}].
\end{aligned}
\end{equation}
\begin{Prop}
Provided one chooses the ground state vector $\eta$ in (\ref{nc-cs-rep-ker}) to be the following normalized Gaussian function:
\begin{equation}\label{grnd-state-gaussn}
\eta(\mathbf r)=\frac{1}{\sqrt{\pi}s}e^{-\frac{1}{2s^{2}}\vert\mathbf r\vert^{2}},
\end{equation}
then, the reproducing kernel $K^{\NC}$ (see \ref{rep-ker-def}) associated with the Lie group $G_{\NC}$ reads
\begin{eqnarray}\label{rep-krnl-dfn-grp-ncqm}
\lefteqn{K^{\NC}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))}\nonumber\\
&&=e^{\frac{i}{\hbar}\xi((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))-\frac{s^{2}}{4\hbar^{2}}\vert\mathbf p-\mathbf p^{\prime}\vert^{2}-\frac{1}{4s^{2}}\vert\mathbf q-\mathbf q^{\prime}\vert^{2}}e^{\frac{i\vartheta}{\hbar^{2}}\xi^{\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))}\nonumber\\
&&\times e^{-\frac{1}{4s^{2}}\left[\frac{\vartheta^{2}}{\hbar^{2}}(p_{1}-p_{1}^{\prime})^{2}+2\frac{\vartheta}{\hbar}(q_{2}-q_{2}^{\prime})(p_{1}-p_{1}^{\prime})\right]}e^{\frac{i\mathcal{B}}{\hbar^{2}}\xi^{\prime\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))-\frac{s^{2}}{4\hbar^{2}}\left[\frac{\mathcal{B}^{2}}{\hbar^{2}}(q_{2}-q_{2}^{\prime})^{2}+2\frac{\mathcal{B}}{\hbar}(p_{1}-p_{1}^{\prime})(q_{2}-q_{2}^{\prime})\right]},\nonumber\\
\end{eqnarray}
where $\xi$, $\xi^{\prime}$ and $\xi^{\prime\prime}$ are all given by (\ref{local-expnts}). Also, $s$ stands for the standard deviation associated with the position vector $\mathbf r=(r_1,r_2)$ and hence has the dimension of length.
\end{Prop}
\noindent{\bf{ Proof}\,\,}{.}
Using (\ref{nc-cs-rep-ker}), one finds that
\begin{eqnarray}\label{rep-ker-proof}
\lefteqn{\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert\eta_{\mathbf{q}^{\prime},\mathbf{p}^{\prime}}^{\NC}\rangle}\nonumber\\
&&=\langle\eta\vert e^{-\frac{i}{\hbar}(\mathbf p^{\NC}.\mathbf{\hat{Q}}-\mathbf q^{\NC}.\mathbf{\hat{P}})}e^{\frac{i}{\hbar}(\mathbf p^{\prime\;\NC}.\mathbf{\hat{Q}}-\mathbf q^{\prime\;\NC}.\mathbf{\hat{P}})}\eta\rangle\nonumber\\
&&=e^{-\frac{i}{2\hbar}(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})}\langle\eta\vert e^{\left[-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).\mathbf{\hat{Q}}+\frac{i}{\hbar}(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}).\mathbf{\hat{P}}\right]}\eta\rangle\nonumber\\
&&=e^{-\frac{i}{2\hbar}(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})}\int_{\mathbb{R}^{2}}\overline{\eta(\mathbf r)} e^{\left[-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).\mathbf{\hat{Q}}+\frac{i}{\hbar}(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}).\mathbf{\hat{P}}\right]}\eta(\mathbf r)d\mathbf r\nonumber\\
&&=e^{-\frac{i}{2\hbar}\left[(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})+(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right]}\int_{\mathbb{R}^{2}}\overline{\eta(\mathbf r)} e^{-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).\mathbf{\hat{Q}}}\widetilde{\eta}(\mathbf r)d\mathbf r,\nonumber\\
\end{eqnarray}
where
\begin{equation}\label{def-tilde-eta}
\widetilde{\eta}=e^{\frac{i}{\hbar}(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}).\mathbf{\hat{P}}}\eta.
\end{equation}
Therefore, (\ref{rep-ker-proof}) now reads
\begin{eqnarray}\label{rep-ker-proof-contd}
\lefteqn{\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert\eta_{\mathbf{q}^{\prime},\mathbf{p}^{\prime}}^{\NC}\rangle}\nonumber\\
&&=e^{-\frac{i}{2\hbar}\left[(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})+(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right]}\nonumber\\
&&\times \int_{\mathbb{R}^{2}}\overline{\eta(\mathbf r)}e^{-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\NC}).\mathbf r}e^{\frac{i}{\hbar}(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}).\mathbf{\hat{P}}}\eta(\mathbf r)d\mathbf r\nonumber\\
&&=e^{-\frac{i}{2\hbar}\left[(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})+(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right]}\nonumber\\
&&\times \int_{\mathbb{R}^{2}}\overline{\eta(\mathbf r)}e^{-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\NC}).\mathbf r}\eta(\mathbf r+\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})d\mathbf r\nonumber\\
&&=e^{-\frac{i}{2\hbar}\left[(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})+(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right]}\nonumber\\
&&\times\frac{1}{\pi s^{2}}\int_{\mathbb{R}^{2}}e^{-\frac{1}{2s^{2}}\vert\mathbf r\vert^{2}}e^{-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\NC}).\mathbf r}e^{-\frac{1}{2s^{2}}\vert\mathbf r+\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}\vert^{2}}d\mathbf r\nonumber\\
&&=\frac{1}{\pi s^{2}}e^{-\frac{i}{2\hbar}\left[(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})+(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right]}\nonumber\\
&&\times\int_{\mathbb{R}^{2}}e^{-\frac{i}{\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\NC}).\mathbf r}e^{-\frac{1}{s^{2}}\vert\mathbf r\vert^{2}-\frac{1}{s^{2}}\mathbf r.(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})-\frac{\vert\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}\vert^{2}}{2s^{2}}}d\mathbf r\nonumber\\
&&=\frac{1}{\pi s^{2}}e^{-\frac{i}{2\hbar}\left[(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC})+(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right]}\nonumber\\
&&\times e^{\left[-\frac{s^{2}}{4\hbar^{2}}\vert\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}\vert^{2}-\frac{1}{4s^{2}}\vert\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}\vert^{2}+\frac{i}{2\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}).(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC)}\right]}\nonumber\\
&&\times\int_{\mathbb{R}^{2}}e^{-\frac{1}{s^{2}}\left\vert\mathbf r+i\frac{s^{2}}{2\hbar}(\mathbf p^{\NC}-\mathbf p^{\prime\;\NC})+\frac{1}{2}(\mathbf q^{\NC}-\mathbf q^{\prime\;\NC})\right\vert^{2}}d\mathbf r\nonumber\\
&&=e^{\left[-\frac{i}{2\hbar}\left(\mathbf p^{\NC}.\mathbf q^{\prime\;\NC}-\mathbf p^{\prime\;\NC}.\mathbf q^{\NC}\right)-\frac{s^{2}}{4\hbar^{2}}\vert\mathbf p^{\NC}-\mathbf p^{\prime\;\NC}\vert^{2}-\frac{1}{4s^{2}}\vert\mathbf q^{\NC}-\mathbf q^{\prime\;\NC}\vert^{2}\right]}\nonumber\\
\end{eqnarray}
Now writing $\mathbf q^{\NC}$,$\mathbf p^{\NC}$ in terms of $\mathbf q$ and $\mathbf p$ with the help of (\ref{nc-chng-vrbl}) and subsequently using (\ref{local-expnts}), one finally obtains
\begin{eqnarray}\label{rep-krnl-fin-exprssn}
\lefteqn{\langle\eta_{\mathbf{q},\mathbf{p}}^{\NC}\vert\eta_{\mathbf{q}^{\prime},\mathbf{p}^{\prime}}^{\NC}\rangle}\nonumber\\
&&=e^{\frac{i}{\hbar}\xi((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))-\frac{s^{2}}{4\hbar^{2}}\vert\mathbf p-\mathbf p^{\prime}\vert^{2}-\frac{1}{4s^{2}}\vert\mathbf q-\mathbf q^{\prime}\vert^{2}}e^{\frac{i\vartheta}{\hbar^{2}}\xi^{\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))}\nonumber\\
&&\times e^{-\frac{1}{4s^{2}}\left[\frac{\vartheta^{2}}{\hbar^{2}}(p_{1}-p_{1}^{\prime})^{2}+2\frac{\vartheta}{\hbar}(q_{2}-q_{2}^{\prime})(p_{1}-p_{1}^{\prime})\right]}e^{\frac{i\mathcal{B}}{\hbar^{2}}\xi^{\prime\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))-\frac{s^{2}}{4\hbar^{2}}\left[\frac{\mathcal{B}^{2}}{\hbar^{2}}(q_{2}-q_{2}^{\prime})^{2}+2\frac{\mathcal{B}}{\hbar}(p_{1}-p_{1}^{\prime})(q_{2}-q_{2}^{\prime})\right]}.\nonumber
\end{eqnarray}
\qed
\begin{remark}\label{rep-krnl-nc-qm-limits}
It is worth remarking here that as $\mathcal{B},\vartheta\rightarrow 0$,
\begin{equation}\label{rep-krnl-lim-eqn}
K^{\NC}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))\rightarrow e^{\frac{i}{\hbar}\xi((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))-\frac{s^{2}}{4\hbar^{2}}\vert\mathbf p-\mathbf p^{\prime}\vert^{2}-\frac{1}{4s^{2}}\vert\mathbf q-\mathbf q^{\prime}\vert^{2}},
\end{equation}
which is the canonical reproducing kernel that one obtains for the Weyl-Heisenberg group in 2-dimensions.
\end{remark}

\section{Noncommutative coherent states in the degenerate case}
\label{sec:degenrt-rep}
In the previous section, we have computed the reproducing kernel $K^{\NC}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))$ from the generic unitary irreducible representations of $G_{\NC}$ due to nonzero $\rho$, $\sigma$ and $\tau$ satisfying  $\rho^{2}\alpha^{2}-\gamma\beta\sigma\tau\neq0$. In this section, using similar arguments, we shall compute the reproducing kernel $\tilde{K}^{\NC}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))$ from the unitary irreducible representations of $G_{\NC}$ for which each of $\rho$, $\sigma$ and $\tau$ is nonzero and $\rho^{2}\alpha^{2}-\gamma\beta\sigma\tau=0$ holds. This family of representations has been computed in \cite{ncqmjpa}:
\begin{eqnarray}\label{eq:degenerate-rep-momentumspace}
    \lefteqn{(\hat{U}^{\kappa,\delta}_{\rho,\zeta}(\theta,\phi,\psi,q_{1},q_{2},p_{1},p_{2})\hat{f})(s)}\nonumber\\
    &&=e^{i\kappa q_{1}+i\delta q_{2}+i\rho\left(\theta-\alpha q_{1}s-\frac{\alpha q_{1}p_{1}}{2}-\frac{\zeta\alpha^{2}q_{1}q_{2}}{2\beta}\right)}e^{i\frac{\rho}{\zeta}\left(\phi+\beta p_{2}s+\frac{\zeta\alpha q_{2}p_{2}}{2}+\frac{\beta}{2}p_{1}p_{2}\right)}\nonumber\\
    &&\times e^{i\frac{\zeta\rho\alpha^{2}}{\gamma\beta}\psi}\hat{f}(s+p_{1}+\frac{\zeta\alpha q_{2}}{\beta}),
\end{eqnarray}
where $f\in L^{2}(\hat{\mathbb{R}},ds)$. It is to be noted that given  $\rho\neq 0$, an ordered pair $(\kappa,\delta)$ and $\zeta\in (-\infty, 0)\cup(0,\infty)$ satisfying $\rho=\sigma\zeta=\frac{\gamma\beta\tau}{\zeta\alpha^{2}}$, one precisely obtains a unitary irreducible representation of $G_{\NC}$.

Inverse Fourier transform of (\ref{eq:degenerate-rep-momentumspace}) leads to
\begin{eqnarray}\label{eq:degenerate-rep-momentumspace-inv-trnsfrm}
    \lefteqn{(U^{\kappa,\delta}_{\rho,\zeta}(\theta,\phi,\psi,q_{1},q_{2},p_{1},p_{2})f)(r)}\nonumber\\
    &&=e^{i\rho\left(\theta+\frac{1}{\zeta}\phi+\frac{\zeta\alpha^{2}}{\gamma\beta}\psi\right)+i\kappa q_{1}+i\delta q_{2}-i\rho\alpha rp_{1}-\frac{i\rho\alpha^{2}\zeta}{\beta}rq_{2}+\frac{i\rho\alpha}{2}(q_{1}p_{1}-q_{2}p_{2})}\nonumber\\
    &&\times e^{i\rho\left(\frac{\alpha^{2}\zeta}{2\beta}q_{1}q_{2}-\frac{\beta}{2\zeta}p_{1}p_{2}\right)}f(r-q_{1}+\frac{\beta}{\alpha\zeta}p_{2}),
\end{eqnarray}
where $f\in L^{2}(\mathbb{R},dr)$.

In analogy with section \ref{sec:cs ncqm}, given a fixed vector $\tilde{\chi}\in L^{2}(\mathbb{R},dr)$ and a fixed point $(\mathbf q,\mathbf p)\in\mathbb{R}^{4}$, let us first define the vector in $L^{2}(\mathbb{R},dr)$ as:
\begin{equation}\label{def-nc-cs-deg}
\tilde{\chi}_{\mathbf{q},\mathbf{p}}^{\NC}=U^{\kappa,\delta}_{\rho,\zeta}(0,0,0,-\mathbf q,\mathbf p)\chi.
\end{equation}
With the following change of variables:
\begin{equation*}
\hbar=\frac{1}{\rho\alpha},\;\;\vartheta=-\frac{\sigma\beta}{\rho^{2}\alpha^{2}},
\end{equation*}
and recalling that $\rho=\sigma\zeta=\frac{\gamma\beta\tau}{\zeta\alpha^{2}}$ holds, one can rewrite $\tilde{\chi}_{\mathbf{q},\mathbf{p}}^{\NC}$ suitably as:
\begin{equation}
\tilde{\chi}_{\mathbf{q},\mathbf{p}}^{\NC}(r)=e^{-i\kappa q_{1}-i\delta q_{2}-\frac{ir}{\hbar}\left(p_{1}+\frac{\hbar}{\vartheta}q_{2}\right)-\frac{i}{2\hbar}(q_{1}p_{1}-q_{2}p_{2})+\frac{i}{2}\left(-\frac{q_{1}q_{2}}{\vartheta}+\frac{\vartheta}{\hbar^{2}}p_{1}p_{2}\right)}\tilde{\chi}\left(r+q_{1}-\frac{\vartheta}{\hbar}p_{2}\right).
\end{equation}
\begin{defi}\label{cs-degnrt-def}
Define the degenerate noncommutative coherent states as the following vectors in $L^{2}(\mathbb{R},dr)$:
\begin{equation}\label{eq:normlzd-deg-cohrnt-states}
\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}(r)=e^{-i\kappa q_{1}-i\delta q_{2}-\frac{ir}{\hbar}\left(p_{1}+\frac{\hbar}{\vartheta}q_{2}\right)-\frac{i}{2\hbar}(q_{1}p_{1}-q_{2}p_{2})+\frac{i}{2}\left(-\frac{q_{1}q_{2}}{\vartheta}+\frac{\vartheta}{\hbar^{2}}p_{1}p_{2}\right)}\tilde{\eta}\left(r+q_{1}-\frac{\vartheta}{\hbar}p_{2}\right),
\end{equation}
where $\tilde{\eta}$ is a vector given by $\tilde{\eta}=\frac{\tilde{\chi}}{\vert\vert\tilde{\chi}\vert\vert}$.
\end{defi} \medskip

Now the underlying observables are functions in the Hilbert space
$L^{2}(\mathbb{R}^{4},d\tilde\nu(\mathbf q,\mathbf p))$ equipped with the measure
\begin{equation} \label{tilde-nu}
 d\tilde\nu(\mathbf q,\mathbf p)=\frac{1}{2\pi\hbar}dq_{1}dp_{1}d\mu(q_{2},p_{2}),
\end{equation}
where $d\mu$ is an arbitrary probability measure on~$\mathbb R^2$.

\begin{lem}$\!\!${\bf }\label{lem-res-idnty-deg-cohrnt-states}
The vectors $\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}$ defined as the degenerate noncommutative coherent states by (\ref{eq:normlzd-deg-cohrnt-states}) satisfy the following integral relation:
\begin{equation}\label{stmnt-res-idnty-deg}
\int_{\mathbb{R}^{4}}\vert\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}\rangle\langle\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}\vert\;d\tilde{\nu}(\mathbf q,\mathbf p)=\mathbb{I},
\end{equation}
where $\mathbb{I}$ is the identity operator on $L^{2}(\mathbb{R},dr)$.
\end{lem}
\noindent{\bf{ Proof}\,\,}{.}
Using the following change of variables:
\begin{equation}\label{nc-chng-vrbl-dgnrt-case}
\begin{aligned}
q_{1}^{\NC}&=q_{1}-\frac{\vartheta}{\hbar}p_{2}\\
q_{2}^{\NC}&=q_{2}\\
p_{1}^{\NC}&=p_{1}+\frac{\hbar}{\vartheta}q_{2}+2\kappa\hbar\\
p_{2}^{\NC}&=p_{2},
\end{aligned}
\end{equation}
one can rewrite the degenerate noncommutative coherent states (\ref{eq:normlzd-deg-cohrnt-states}) as:
\begin{equation}\label{nc-cohrnt-states-deg-nc-coordnts}
\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}(r)=e^{-i\delta q_{2}^{\NC}-\frac{ir}{\hbar}(p_{1}^{\NC}-2\kappa\hbar)-\frac{i}{2\hbar}q_{1}^{\NC}p_{1}^{\NC}-\frac{i\vartheta}{\hbar}\kappa p_{2}^{\NC}}\tilde{\eta}(r+q_{1}^{\NC}).
\end{equation}
Observe that $dq_1 dp_1=dq_1^{\NC}dp_1^{\NC}$,
by~(\ref{nc-chng-vrbl-dgnrt-case}).
Thus for any $f,g\in L^{2}(\mathbb{R},dr)$, we obtain
\begin{eqnarray}
\lefteqn{\int_{\mathbb{R}^{4}}\langle f\vert\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}\rangle\langle\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}\vert g\rangle\;d\tilde{\nu}(\mathbf q,\mathbf p)}\nonumber\\
&&=\frac{1}{2\pi\hbar}\int_{\mathbb{R}^{4}}\left[\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{f(r)}e^{-\frac{i}{\hbar}(r-r^{\prime})(p_{1}^{\NC}-2\kappa\hbar)}\tilde{\eta}(r+q_{1}^{\NC})\overline{\tilde{\eta}(r^{\prime}+q_{1}^{\NC})}g(r^{\prime})dr dr^{\prime}\right]\nonumber\\
&&\qquad\qquad\times dq_{1}^{\NC} dp_{1}^{\NC} d\mu(q_{2}^{\NC},p_{2}^{\NC})\nonumber\\
&&=\int_{\mathbb{R}^{3}}\left[\int_{\mathbb{R}}\int_{\mathbb{R}}\delta(r-r^{\prime})\overline{f(r)}\tilde{\eta}(r+q_{1}^{\NC})\overline{\tilde{\eta}(r^{\prime}+q_{1}^{\NC})}g(r^{\prime})dr\;dr^{\prime}\right] dq_{1}^{\NC} d\mu(q_{2}^{\NC},p_{2}^{\NC})\nonumber\\
&&=\vert\vert\tilde{\eta}\vert\vert^{2}\int_{\mathbb{R}^{2}}\left[\int_{\mathbb{R}}\overline{f(r)}g(r)dr\right] d\mu(q_{2}^{\NC},p_{2}^{\NC})\nonumber\\
&&=\langle f\vert g\rangle .
\end{eqnarray}
\qed

The reproducing kernel associated with the degenerate noncommutative coherent states will be given by
\begin{equation}\label{eq:deg-cohrnt-states-rep-kern}
\tilde{K}^{\NC}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))=\langle\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}\vert\tilde{\eta}_{\mathbf{q}^{\prime},\mathbf{p}^{\prime}}^{\NC}\rangle.
\end{equation}
This degenerate reproducing kernel is explicitly computed in the following proposition:
\begin{Prop}
Provided one chooses the ground state vector $\tilde{\eta}\in L^{2}(\mathbb{R},dr)$ in (\ref{eq:normlzd-deg-cohrnt-states}) to be the following normalized Gaussian function:
\begin{equation}\label{grnd-state-gaussn-deg-rep}
\tilde{\eta}(r)=\frac{1}{{\pi}^{\frac14}s^{\frac12}}e^{-\frac{r^{2}}{2s^{2}}},
\end{equation}
then, the reproducing kernel $\tilde{K}^{\NC}$ (see (\ref{eq:deg-cohrnt-states-rep-kern})) associated with the Lie group $G_{\NC}$ reads
\begin{eqnarray}\label{rep-krnl-dfn-grp-ncqm-degnrt-rep}
\lefteqn{\tilde{K}^{\NC}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))}\nonumber\\
&&=e^{\frac{i}{\hbar}\xi((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))+\frac{i}{\vartheta}\xi^{\prime\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))+\frac{i\vartheta}{\hbar^{2}}\xi^{\prime}((\mathbf q,\mathbf p),(\mathbf q^{\prime},\mathbf p^{\prime}))+i\kappa(q_{1}-q_{1}^{\prime})+i\delta(q_{2}-q_{2}^{\prime})-\frac{1}{4s^{2}}(q_{1}-q_{1}^{\prime})^{2}}\nonumber\\
&&\times e^{-\frac{s^{2}}{4\vartheta^{2}}(q_{2}-q_{2}^{\prime})^{2}-\frac{s^{2}}{4\hbar^{2}}(p_{1}-p_{1}^{\prime})^{2}-\frac{\vartheta^{2}}{4s^{2}\hbar^{2}}(p_{2}-p_{2}^{\prime})^{2}+\frac{\vartheta}{2s^{2}\hbar}(q_{1}p_{2}-q_{1}p_{2}^{\prime}-q_{1}^{\prime}p_{2}+q_{1}^{\prime}p_{2}^{\prime})}\nonumber\\
&&\times e^{-\frac{s^{2}}{2\hbar\vartheta}(p_{1}q_{2}-p_{1}q_{2}^{\prime}-p_{1}^{\prime}q_{2}+p_{1}^{\prime}q_{2}^{\prime})},
\end{eqnarray}
where $\xi$, $\xi^{\prime}$ and $\xi^{\prime\prime}$ are all given by (\ref{local-expnts}). Also, $s$ stands for the standard deviation associated with the position coordinate $r$ and hence has the dimension of length.
\end{Prop}

\noindent{\bf{ Proof}\,\,}.
Using (\ref{nc-cohrnt-states-deg-nc-coordnts}) we get
\begin{align*}
\langle\tilde{\eta}_{\mathbf{q},\mathbf{p}}^{\NC}|\tilde{\eta}_{\mathbf{q}^{\prime},\mathbf{p}^{\prime}}^{\NC}\rangle
&= e^{-i\delta(q_2^{\prime\NC}-q_2^{\NC})
 -\frac{i}{2\hbar}
   (q_1^{\prime\NC}p_1^{\prime\NC}-q_1^{\NC}p_1^{\NC})
 -\frac{i\vartheta}{\hbar}\kappa
   (p_2^{\prime\NC}-p_2^{\NC})}   \\
&\qquad\times
 \int_{\mathbb R}
   e^{-\frac{ir}{\hbar}(p_1^{\prime\NC}-p_1^{\NC})}
   \tilde{\eta}(r+q_1^{\prime\NC})
   \overline{\tilde{\eta}(r+q_1^{\NC})} \, dr   \\
&= e^{-i\delta(q_2^{\prime\NC}-q_2^{\NC})
 -\frac{i}{2\hbar}
   (q_1^{\prime\NC}p_1^{\prime\NC}-q_1^{\NC}p_1^{\NC})
 -\frac{i\vartheta}{\hbar}\kappa
   (p_2^{\prime\NC}-p_2^{\NC})}   \\
&\qquad\times
 \frac1{\pi^{1/2}s} \int_{\mathbb R}
   e^{-\frac{ir}{\hbar}(p_1^{\prime\NC}-p_1^{\NC})}
   e^{-\frac{(r+q_1^{\prime\NC})^2}{2s^2}
      -\frac{(r+q_1^{\NC})^2}{2s^2}} \, dr   \\
&= e^{-i\delta(q_2^{\prime\NC}-q_2^{\NC})
 -\frac{i}{2\hbar}
   (q_1^{\prime\NC}p_1^{\prime\NC}-q_1^{\NC}p_1^{\NC})
 -\frac{i\vartheta}{\hbar}\kappa
   (p_2^{\prime\NC}-p_2^{\NC})}   \\
&\qquad\times
    e^{-\frac{\hbar^2(q_1^{\NC}-q_1^{\prime\NC})^2
              +2i\hbar s^2 (p_1^{\NC}-p_1^{\prime\NC})
                      (q_1^{\NC}+q_1^{\prime\NC})
              + s^4 (p_1^{\NC}-p_1^{\prime\NC})^2 } {4h^2s^2}} ,
\end{align*}
by~the familiar formula for Gaussian integrals
$$ \int_{\mathbb R} e^{ar-b-cr^2}\,dr =
 \pi^{1/2} c^{-1/2} e^{\frac{a^2}{4c}-b} ,  \qquad c>0 .    $$
A~routine manipulation gives~(\ref{rep-krnl-dfn-grp-ncqm-degnrt-rep}).
\qed

\section{Toeplitz operators and semiclassical limits}\label{secTOEP}
We~now proceed to consider a variant of the well-known Berezin-Toeplitz
quantization procedure in the context of the coherent states,
and resolution of the identity, from the preceding section.
Our~strategy will be to relate the corresponding Toeplitz operators
(defined in (\ref{defTOEP}) below) to~the analogous operators in
the standard setting.

Consider the Fock space
$$ \FFh:=\{f\in L^2(\CC^2,e^{-|\mathbf z|^2}(\pi\hbar)^{-2}\,dA(\mathbf z)):
f\text{ is holomorphic on }\CC^2\}    $$
(here~$dA$ stands for the Lebesgue area measure), and let $V$ be the map
$$ Vf(\mathbf q,\mathbf p) := e^{-|\bznc|^2/{2\hbar}} f(\bznc),   \qquad f\in\FFh,   $$
where we have introduced the notation
\begin{equation}
 \bznc_j = \sqrt{\frac\hbar2} \frac{\mathbf q_j^{\NC}}s
 - \frac{is\mathbf p_j^{\NC}}{\sqrt{2\hbar}} , \qquad j=1,2.   \label{XXX}
\end{equation}
From the equality
\begin{equation} \label{MEA}
 dA(\bznc) = \frac{d\mathbf p^{\NC} d\mathbf q^{\NC}}4
 = \frac{|\hbar^2-\cB\vthta|}{4\hbar^2} \,d\mathbf p\,d\mathbf q
 = \pi^2 \hbar^2 \,d\nu(\mathbf q,\mathbf p),
\end{equation}
where as before
\begin{equation} \label{MEB}
 d\nu(\mathbf q,\mathbf p) = \frac{|\hbar^2-\cB\vthta|}{4\pi^2\hbar^4}\,d\mathbf q\,d\mathbf p,
\end{equation}
one~verifies that $V$~is an isometry from $\FFh$ into $L^2(\RR^4,d\nu(\mathbf q,\mathbf p))$.
If~$\{e_n\}_n$ is an arbitrary orthonormal basis of~$\FFh$, then $\{Ve_n\}_n$
will be an orthonormal basis of the image $\operatorname{Ran}V=:\WW'$ of~$V$.
Using the standard formula for a reproducing kernel in terms of an orthonormal
basis~\cite{Arns}, and the fact that the reproducing kernel of $\FFh$ is well
known to be given by $K_{\FFh}(\mathbf x,\mathbf y)=
e^{\spr{\mathbf x,\mathbf y}/\hbar}$, we~see that $\WW'$ is a reproducing
kernel Hilbert space (RKHS) with reproducing kernel
\begin{align*}
K_{\WW'}((\mathbf q,\mathbf p),(\mathbf q',\mathbf p')) &= \sum_n Ve_n(\bznc) \overline{Ve_n(\bZnc)} \\
&= e^{-|\bznc|^2/2\hbar-|\bZnc|^2/2\hbar} \sum_n e_n(\bznc) \overline{e_n(\bZnc)}\\
&= e^{-|\bznc|^2/2\hbar-|\bZnc|^2/2\hbar} e^{\spr{\bznc,\bZnc}/\hbar}   \\
&= K^{\NC}((\mathbf q,\mathbf p),(\mathbf q',\mathbf p')),
\end{align*}
upon a short computation (cf.~(\ref{rep-ker-proof-contd})).
Since a RKHS is uniquely determined by its reproducing kernel~\cite{Arns},
it~follows that in fact $\WW'=\WW$.
Thus $V$ is a unitary isomorphism of $\FFh$ onto~$\WW$.

Recall that for $f\in L^\infty(\CC^2)$, the Toeplitz operator $T_f$ on $\FFh$
is given~by
$$ T_f u = P(fu), \qquad u\in\FFh ,    $$
where $P:L^2(e^{-|\mathbf z|^2/\hbar}(\pi\hbar)^{-2}\,dA(\mathbf z))\to\FFh$ is the orthogonal
projection. Alternatively, $T_f$~is determined by the property that
$$ \spr{T_f u,v} = \int_{\CC^2} f(\mathbf z) u(\mathbf z) \overline{v(\mathbf z)}
 e^{-|\mathbf z|^2/\hbar} \, \frac{d\mathbf z}{(\pi\hbar)^2}  \qquad \forall u,v\in\FFh. $$
Similarly, we~have Toeplitz operators~$\cT_F$, $F\in L^\infty(\RR^4)$,
on~$\WW$ defined~by
\begin{equation}
 \cT_F u = \cP(Fu),  \qquad u\in\WW ,    \label{defTOEP}
\end{equation}
where $\cP:L^2(\RR^4,d\nu)\to\WW$ is the orthogonal projection;
alternatively, $\cT_F$~is determined by the property that
$$ \spr{\cT_F u,v} = \int_{\RR^4} F(\mathbf q,\mathbf p) u(\mathbf q,\mathbf p) \overline{v(\mathbf q,\mathbf p)}
 \,d\nu(\mathbf q,\mathbf p)  \qquad \forall u,v\in\WW .   $$
Now~by a simple change of variable (cf.~(\ref{MEA}))
\begin{align*}
\spr{\cT_F Vu,Vv}
&= \int_{\RR^4} F(\mathbf q,\mathbf p) Vu(\mathbf q,\mathbf p) \overline{Vv(\mathbf q,\mathbf p)} \,d\nu(\mathbf q,\mathbf p)\\
&= \int_{\RR^4} F(\mathbf q,\mathbf p) e^{-|\bznc|^2/\hbar} u(\bznc) \overline{v(\bznc)}
 \,d\nu(\mathbf q,\mathbf p)   \\
&= \int_{\CC^2} F(\iota(\bznc)) e^{-|\bznc|^2/\hbar} u(\bznc) \overline{v(\bznc)}
 \,\frac{dA(\bznc)}{(\pi\hbar)^2},
\end{align*}
for~all $u,v\in\FFh$, where $\iota$ is the inverse to the coordinate
transformation (\ref{XXX}):
\begin{equation} \label{IOTA}
\begin{gathered}
 \iota(\bznc) := \Big(q_1^{\NC},
 \frac{\hbar^2 q_2^{\NC}-\hbar\vthta p_1^{\NC}}{\hbar^2-\cB\vthta},
 \frac{\hbar^2 p_1^{\NC}-\hbar\cB q_2^{\NC}}{\hbar^2-\cB\vthta},
 p_2^{\NC}\Big) \in\RR^4,  \\
 \mathbf q^{\NC}=\frac s{\sqrt{2\hbar}}(\bznc+\overline{\bznc}), \quad
 \mathbf p^{\NC}=\frac{\sqrt{\hbar/2}}{is}(\overline{\bznc}-\bznc) .
\end{gathered}
\end{equation}
Consequently,
\begin{equation}
 V^* \cT_F V = T_{F\circ\iota}.   \label{VTV}
\end{equation}

\begin{remark}
From the last formula one can see what are the commutators of the Toeplitz
operators $\cT_{p_j},\cT_{q_k}$, $j=1,2$, on~$L^2(\RR^4,d\nu)$.
Indeed, from the formulas for the Toeplitz operators on~$\FFh$,
$$ T_{z_j}=z_j, \qquad T_{\overline z_j}=h\frac\partial{\partial z_j},  $$
and the resulting commutator identity
$$ [T_{z_j},T_{\overline z_k}] = -\delta_{jk}\hbar {\mb I},   $$
one gets using (\ref{VTV})
\begin{align*}
[\cT_{p_1},\cT_{p_2}] &= -\frac{i\cB\hbar^2}{\hbar^2-\cB\vthta}{\mb I}, \\
[\cT_{p_1},\cT_{q_1}] &= [\cT_{p_2},\cT_{q_2}] =
 -\frac{i\hbar^3}{\hbar^2-\cB\vthta}{\mb I}, \\
[\cT_{q_1},\cT_{q_2}] &= -\frac{i\hbar^2\vthta}{\hbar^2-\cB\vthta}{\mb I} ,
\end{align*}
all remaining commutators being zero.
Note that these differ from the commutator relations for the corresponding
quantum observables (cf.~eqn.~(3.7) in~\cite{ncqmjpa})
$$ [\hat Q_j,\hat P_k]=i\hbar\delta_{jk}{\mb I}, \quad
[\hat Q_1,\hat Q_2]=i\vthta{\mb I}, \quad [\hat P_1,\hat P_2]=i\cB{\mb I}  $$
by a factor of $\frac{\hbar^2}{\cB\vthta-\hbar^2}$.
\end{remark}

Now~from the Berezin-Toeplitz quantization (see~e.g.~\cite{Schl},~\cite{AE}),
it~is known that for $f,g$, say, smooth with compact support,
one~has the asymptotic expansion
$$ T_f T_g \approx \sum_{j=0}^\infty \hbar^j T_{C_j(f,g)}    $$
as $\hbar\searrow0$, in~the sense of operator norms, where
$$ C_j(f,g) = (-1)^j \sum_{|\alpha|=j} \frac1{\alpha!}
 \frac{\partial^\alpha f}{\partial z^\alpha}
 \frac{\partial^\alpha g}{\partial \overline z^\alpha}   $$
(here the summation is over all multiindices $\alpha\in\mathbf N^2$
of length~$j$). From the computation
\begin{align*}
V^*\cT_F\cT_G V &= (V^*\cT_F V)(V^*\cT_G V) = T_{F\circ\iota}T_{G\circ\iota}\\
&\approx \sum_{j=0}^\infty \hbar^j T_{C_j(F\circ\iota,G\circ\iota)}
 = \sum_{j=0}^\infty \hbar^j
 V^* \cT_{\cC_j(F\circ\iota,G\circ\iota)\circ\iota^{-1}}V ,
\end{align*}
we~thus see that we have an asymptotic expansion,
in~the sense of operator norms,
$$ \cT_F\cT_G \approx \sum_{j=0}^\infty \hbar^j \cT_{\cC_j(F,G)}   $$
as $\hbar\searrow0$, with
\begin{equation}
 \cC_j(F,G) := C_j(F\circ\iota,G\circ\iota)\circ\iota^{-1} .  \label{CCJ}
\end{equation}
This gives rise also to the associated star-product
$$ F*G := \sum_{j=0}^\infty \hbar^j \cC_j(F,G) ,   $$
so~that, heuristically, $\cT_F\cT_G\approx\cT_{F*G}$.

In~particular, $\cC_0(F,G)=FG$ (the~pointwise product), while
$$ \cC_1(F,G) =
 \begin{bmatrix}
  \dfrac{\partial F}{\partial p_1},
  \dfrac{\partial F}{\partial p_2},
  \dfrac{\partial F}{\partial q_1},
  \dfrac{\partial F}{\partial q_2}  \end{bmatrix}
  \cdot A \cdot
 \begin{bmatrix} \dfrac{\partial G}{\partial p_1} \\
  \dfrac{\partial G}{\partial p_2} \\
  \dfrac{\partial G}{\partial q_1} \\
  \dfrac{\partial G}{\partial q_2} \end{bmatrix} ,   $$
where
$$ A = \begin{bmatrix}
 -\dfrac{\hbar(\hbar^4+\cB^2 s^4)}{2s^2(\hbar^2-\cB\vthta)^2}  &
 \dfrac{-i\cB \hbar}{2(\hbar^2-\cB\vthta)} &
 \dfrac{-i\hbar^2}  {2(\hbar^2-\cB\vthta)} &
 \dfrac{\hbar^2(\cB s^4+\hbar^2\vthta)}{2s^2(\hbar^2-\cB\vthta)^2}  \\
 \dfrac{i\cB\hbar}{2(\hbar^2-\cB\vthta)}  &
 -\dfrac \hbar{2s^2}  & 0 &
 \dfrac{-i\hbar^2}{2(\hbar^2-\cB\vthta)}  \\
 \dfrac{i\hbar^2}{2(\hbar^2-\cB\vthta)} &
 0 & -\dfrac{s^2}{2\hbar}  &
 \dfrac{-i\hbar\vthta}{2(\hbar^2-\cB\vthta)}  \\
 \dfrac{\hbar^2(\cB s^4+\hbar^2\vthta)}{2s^2(\hbar^2-\cB\vthta)^2}  &
 \dfrac{i\hbar^2}{2(\hbar^2-\cB\vthta)} &
 \dfrac{i\hbar\vthta}{2(\hbar^2-\cB\vthta)} &
 -\dfrac{\hbar^3(s^4+\vthta^2)}{2s^2(\hbar^2-\cB\vthta)^2}
\end{bmatrix} .   $$
The~last matrix looks much nicer in terms of the ``renormalized'' parameters
$$ B=\frac{\cB}\hbar, \quad T:=\frac\vthta\hbar,
 \quad S:=\frac s{\sqrt{\hbar}} ;   $$
namely,
$$ A = \begin{bmatrix}
 -\dfrac{1+B^2S^4}{2S^2(1-BT)^2} & -\dfrac{iB}{2(1-BT)} &
  -\dfrac i{2(1-BT)} & \dfrac{BS^4+T}{2S^2(1-BT)^2} \\
 \dfrac{iB}{2(1-BT)} & -\dfrac1{2S^2} & 0 & -\dfrac i{2(1-BT)} \\
 \dfrac i{2(1-BT)} & 0 & -\dfrac{S^2}2 & -\dfrac{iT}{2(1-BT)}  \\
 \dfrac{BS^4+T}{2S^2(1-BT)^2} & \dfrac i{2(1-BT)} & \dfrac{iT}{2(1-BT)} &
  -\dfrac{S^4+T^2}{2S^2(1-BT)^2}
\end{bmatrix} .   $$
Note that both $A$ and the inverse transform $\iota$ depend only on $B,T$
and~$S$, but not~on~$\hbar$.
The~parameter~$S$, which arises solely from the choice of the vector~$\eta$
in~(\ref{grnd-state-gaussn}), is~in a sense responsible only for re-scaling
the Planck constant~$\hbar$, and we can choose $S=1$.
\footnote{Note that as far as length only is concerned, $s$~has the same
physical dimension as~$\sqrt{\hbar}$.}
It~then follows from (\ref{CCJ}) that all $\cC_j$, $j\ge0$, will be
bidifferential operators with coefficients given by expressions involving
only $B,T$, in~fact, by~rational functions in $B,T$ with powers of $1-BT$
as the denominators. Replacing the latter by their Taylor expansions around
$(B,T)=(0,0)$, we~thus obtain a joint asymptotic expansion for the product
$F*G$ as $(\hbar,B,T)\to(0,0,0)$. Its~beginning looks as follows
\begin{align*}
F*G &\approx FG - \frac\hbar2 \sum_{k=1}^2
  \Big(\frac{\partial F}{\partial p_k}-i\frac{\partial F}{\partial q_k}\Big)
  \Big(\frac{\partial G}{\partial p_k}+i\frac{\partial G}{\partial q_k}\Big) \\
& \qquad\qquad
 + \frac{B\hbar}2 \Big[ i\frac{\partial G}{\partial p_1}
    \Big(\frac{\partial F}{\partial p_2}-i\frac{\partial F}{\partial q_2}\Big)
        -i \frac{\partial F}{\partial p_1}
    \Big(\frac{\partial G}{\partial p_2}+i\frac{\partial G}{\partial q_2}\Big)
   \Big]  \\
& \qquad\qquad
 + \frac{T\hbar}2 \Big[ \frac{\partial F}{\partial q_2}
    \Big(\frac{\partial G}{\partial p_1}+i\frac{\partial G}{\partial q_1}\Big)
        + \frac{\partial G}{\partial q_2}
    \Big(\frac{\partial F}{\partial p_1}-i\frac{\partial F}{\partial q_1}\Big)
   \Big]  \\
& \qquad\qquad
 + \frac{\hbar^2}8 \Big[ \sum_{k=1}^2
    \Big(\frac\partial{\partial p_k}+i\frac\partial{\partial q_k}\Big)^2 G
    \cdot
    \Big(\frac\partial{\partial p_k}-i\frac\partial{\partial q_k}\Big)^2 F \\
& \hskip1em
    + 2 \Big(\frac\partial{\partial p_1}+i\frac\partial{\partial q_1}\Big)
        \Big(\frac\partial{\partial p_2}+i\frac\partial{\partial q_2}\Big) G
    \cdot
        \Big(\frac\partial{\partial p_1}-i\frac\partial{\partial q_1}\Big)
        \Big(\frac\partial{\partial p_2}-i\frac\partial{\partial q_2}\Big) F
    \Big]  \\
& \qquad\qquad
  + O((h+B+T)^3).
\end{align*}
(Here the differentiations stop at each~$\cdot$,
i.e.~$\partial F\cdot\partial G$ means $(\partial F)(\partial G)$.)

For~the corresponding commutator, we~get
\begin{align*}
F*G-G*F &\approx
i\hbar \sum_{k=1}^2  \Big(
  \frac{\partial F}{\partial q_k} \frac{\partial G}{\partial p_k} -
  \frac{\partial F}{\partial p_k} \frac{\partial G}{\partial q_k}  \Big)  \\
& \qquad\qquad
 + i\hbar B  \Big(
  \frac{\partial F}{\partial p_2} \frac{\partial G}{\partial p_1} -
  \frac{\partial F}{\partial p_1} \frac{\partial G}{\partial p_2}  \Big)  \\
& \qquad\qquad
 + i\hbar T  \Big(
  \frac{\partial F}{\partial q_2} \frac{\partial G}{\partial q_1} -
  \frac{\partial F}{\partial q_1} \frac{\partial G}{\partial q_2}  \Big)  \\
& \qquad\qquad
 + \frac{\hbar^2}8  \Big[
     2 (\partial_{+1}\partial_{+2}G) (\partial_{-1}\partial_{-2}F) -
     2 (\partial_{+1}\partial_{+2}F) (\partial_{-1}\partial_{-2}G)   \\
& \hskip6em
     + \sum_{k=1}^2 \Big(
        (\partial_{+k}^2 G)(\partial_{-k}^2 F) -
        (\partial_{+k}^2 F)(\partial_{-k}^2 G)  \Big)  \Big]    \\
& \qquad\qquad
  + O((h+B+T)^3) ,
\end{align*}
where for the sake of brevity, we~have denoted $\partial_{\pm k}=
\frac\partial{\partial p_k}\pm i\frac\partial{\partial q_k}$, $k=1,2$.
The~first term~is $i\hbar\{F,G\}$, the~Poisson bracket of $F$ and~$G$,
which takes care of the correct semi-classical behavior as $\hbar\to0$;
note that it does not contain any $B$ and~$T$, which come only in the
second-order terms.

\section{Toeplitz operators and semiclassical limits: the degenerate case}
\label{secTOEPdeg}

In~a~similar way as in the preceding section, we~can also treat the
``degenerate'' representation and kernel from Section~\ref{sec:degenrt-rep}.
This time, we~need the Fock space just on the complex plane,
$$ \tFFh:=\{f\in L^2(\CC,e^{-|z|^2}(\pi\hbar)^{-1}\,dA(z)):
f\text{ is holomorphic on }\CC\},   $$
and let $\tV$ be the map
$$ \tV f(\mathbf q,\mathbf p) := e^{i\delta q_2+i\kappa\vthta p_2/\hbar}
  e^{-|\znc|^2/{2\hbar}} f(\znc),   \qquad f\in\tFFh,   $$
where
\begin{equation}
 \znc = \sqrt{\dfrac\hbar2} \frac{q_1^{\NC}}s
 - \frac{is p_1^{\NC}}{\sqrt{2\hbar}} ,    \label{defZNC}
\end{equation}
with $p_1^{\NC},q_1^{\NC}$ given by~(\ref{nc-chng-vrbl-dgnrt-case}).
One~verifies that $\tV$~is an isometry from $\tFFh$ into $L^2(\RR^4,d\tnu)$:
\begin{align*}
\int_{\RR^4} |\tV f(\mathbf q,\mathbf p)|^2 \,d\tnu(\mathbf q,\mathbf p)
&= \frac1{2\pi\hbar} \int_{\RR^4} |f(\znc)|^2 e^{-|\znc|^2/\hbar}
 \, dq_1^{\NC}\,dp_1^{\NC}\,d\mu(q_2,p_2)   \\
&= \frac1{2\pi\hbar} \int_{\RR^2} |f(\znc)|^2 e^{-|\znc|^2/\hbar}
 \,dq_1^{\NC}\,dp_1^{\NC}   \\
&= \frac1{\pi\hbar} \int_{\CC} |f(\znc)|^2 e^{-|\znc|^2/\hbar} \,dA(\znc),
\end{align*}
since $dA(\znc)=dp_1^{\NC}dq_1^{\NC}/2$.
Thus if~$\{e_n\}_n$ is an arbitrary orthonormal basis of~$\tFFh$,
then $\{\tV e_n\}_n$ will be an orthonormal basis of the image
$\operatorname{Ran}\tV=:\tWW'$ of~$\tV$.
Using the standard formula for a reproducing kernel in terms of an orthonormal
basis, and the fact that the reproducing kernel of $\tFFh$ is well known to be
given by $K_{\tFFh}(x,y)=e^{x\overline y/\hbar}$,
we~see that $\tWW'$ is a RKHS with reproducing kernel
\begin{align*}
K_{\tWW'}((\mathbf q,\mathbf p),(\mathbf q',\mathbf p'))
&= \sum_n \tV e_n(\znc) \overline{\tV e_n(\Znc)} \\
&= e^{i\delta(q_2-q'_2)+i\kappa\vthta(p_2-p'_2)/\hbar}
   e^{-|\znc|^2/2\hbar-|\Znc|^2/2\hbar}
   \sum_n e_n(\znc) \overline{e_n(\Znc)}  \\
&= e^{i\delta(q_2-q'_2)+i\kappa\vthta(p_2-p'_2)/\hbar}
   e^{-|\znc|^2/2\hbar-|\Znc|^2/2\hbar} e^{\znc\overline{\Znc}/\hbar}   \\
&= \tilde K^{\NC}((\mathbf q,\mathbf p),(\mathbf q',\mathbf p')),
\end{align*}
again upon a short computation. As~before, it~follows that $\tWW'=\tWW$,
the~space for which $\tilde K^{\NC}$ is the reproducing kernel.
Thus $\tV$ is a unitary isomorphism of $\tFFh$ onto~$\tWW$.

The Toeplitz operators~$\tcT_F$, $F\in L^\infty(\RR^4,d\tnu)$,
on~$\tWW$ are now related to the Toeplitz operators $T_f$,
$f\in L^\infty(\CC)$, on~$\tFFh$ via
\begin{align*}
\spr{\tcT_F \tV u,\tV v}
&= \int_{\RR^4} F(\mathbf q,\mathbf p) \tV u(\mathbf q,\mathbf p) \overline{\tV v(\mathbf q,\mathbf p)}
 \,d\tnu(\mathbf q,\mathbf p)  \\
&= \frac1{2\pi\hbar} \int_{\RR^4} F(\mathbf q,\mathbf p) e^{-|\znc|^2/\hbar} u(\znc)
 \overline{v(\znc)} \,dq_1^{\NC}\,dp_1^{\NC}\,d\mu(q_2,p_2)   \\
&= \frac1{2\pi\hbar} \int_{\RR^2}
 \Big(\int_{\RR^2}
 F(q_1^{\NC}+\frac\vthta\hbar p_2,q_2,
 p_1^{\NC}-\frac\hbar\vthta q_2-2\kappa\hbar,p_2) \,d\mu(q_2,p_2) \Big)\\
&\qquad\qquad\times
 u(\znc) \overline{v(\znc)} e^{-|\znc|^2/\hbar}
  \,dq_1^{\NC}\,dp_1^{\NC} \\
&= \int_{\CC} \varrho F(\znc) u(\znc) \overline{v(\znc)}
 e^{-|\znc|^2/\hbar} \,\frac{dA(\znc)}{\pi\hbar}  \\
&= \spr{T_{\varrho F} u,v} ,
\end{align*}
for~all $u,v\in\tFFh$, where $\varrho$ is the mapping
$$ \begin{gathered}
 \varrho F(\znc) := \int_{\RR^2}
 F(q_1^{\NC}+\frac\vthta\hbar p_2,q_2,
   p_1^{\NC}-\frac\hbar\vthta q_2-2\kappa\hbar,p_2) \,d\mu(q_2,p_2), \\
 q_1^{\NC}= \frac s{\sqrt{2\hbar}} (\znc+\overline\znc), \quad
 p_1^{\NC}= \frac{\sqrt{\hbar/2}}{is} (\overline\znc-\znc).
\end{gathered}  $$
Consequently,
\begin{equation}
 \tV^* \tcT_F \tV = T_{\varrho F}.    \label{TDEG}
\end{equation}

\begin{remark}
From the last formula one can again get the commutator relations for the
Toeplitz operators $\tcT_{p_j},\tcT_{q_k}$, $j=1,2$, on~$L^2(\RR^4,d\tnu)$.
This time we get $\tcT_{p_2}=(\int p_2\,d\mu(q_2,p_2)){\mb I}$,
$\tcT_{q_2}=(\int q_2\,d\mu(q_2,p_2)){\mb I}$ and
$$ [\tcT_{q_1},\tcT_{p_1}]=-[\tcT_{p_1},\tcT_{q_1}]=i\hbar{\mb I},   $$
all remaining commutators being zero.
\end{remark}

Using again the asymptotic expansion known from the Berezin-Toeplitz
quantization:
$$ T_f T_g \approx \sum_{j=0}^\infty \hbar^j T_{\tilde C_j(f,g)}    $$
as $\hbar\searrow0$, in~the sense of operator norms, where
$$ \tilde C_j(f,g) = \frac{(-1)^j}{j!}
 \frac{\partial^j f}{\partial z^j}
 \frac{\partial^j g}{\partial \overline z^j} ,   $$
one sees from
$$ \tV^*\tcT_F\tcT_G\tV = T_{\varrho F}T_{\varrho G}
\approx \sum_{j=0}^\infty \hbar^j T_{\tilde C_j(\varrho F,\varrho G)}
 = \sum_{j=0}^\infty \hbar^j
 \tV^* \tcT_{\varrho^* \tilde C_j(\varrho F,\varrho G)}\tV , $$
that there is an asymptotic expansion, in~the sense of operator norms,
$$ \tcT_F\tcT_G \approx \sum_{j=0}^\infty \hbar^j \tcT_{\tcC_j(F,G)}   $$
as $\hbar\searrow0$, with
\begin{equation}
 \tcC_j(F,G) := \varrho^* \tilde C_j(\varrho F,\varrho G) .   \label{TCC}
\end{equation}
Here $\varrho^*$ is in principle any right inverse for $\varrho$, for~instance,
$$ \varrho^* f(\mathbf q,\mathbf p) := f(\znc)    $$
with the notations (\ref{nc-chng-vrbl-dgnrt-case}) and~(\ref{defZNC}).

\begin{remark} In~some sense, the freedom of choice for $\varrho^*$,
as~well as for the probability measure $d\mu$ in~(\ref{tilde-nu}),
reflects the ``degeneracy'' of the representation, as~does the reduction
of the number of variables of~$\eta$.
Note that the above choice for $\varrho^*$ has the virtue that it works
for all probability measures~$d\mu$ and values of $\hbar,\vthta$.
\end{remark}

Not~all choices of $d\mu$ and $\varrho^*$, however, are~physically relevant.
For the associated star-product
$$ F*G := \sum_{j=0}^\infty \hbar^j \tcC_j(F,G) , \qquad \text{i.e.} \quad
 \tcT_F\tcT_G\approx\tcT_{F*G}  ,   $$
we~would like to have the usual requirement that $\tcC_0(F,G)=FG$,
the~pointwise product. Applying $\varrho$ to~(\ref{TCC}), this implies
$$ \varrho(FG) = (\varrho F)(\varrho G)    \qquad \forall F,G .   $$
It~is easily seen that this is only possible when $d\mu$ is a Dirac~mass:
$$ d\mu(q_2,p_2) = \delta(q_2-q_2^*) \delta(p_2-p_2^*) ,  $$
for some fixed $(q_2^*,p_2^*)\in\RR^2$. The~functions $F,G$, being elements
of $L^\infty(\RR^4,d\tnu)$, are~then effectively defined only on the plane
$(q_2,p_2)=(q_2^*,p_2^*)$ (the~complement of this plane has zero measure);
and the right inverse $\varrho^*$ becomes simply the ordinary inverse.
Viewing $F(\mathbf q,\mathbf p)=F(q_1,q_2^*,p_1,p_2^*)$ as a function of $q_1,p_1$ only,
and similarly for~$G$, we~then get as desired $\tcC_0(F,G)=FG$
(the~pointwise product), while
$$ \tcC_1(F,G) = -\frac1{2\hbar s^2}
  \Big(\hbar\frac{\partial F}{\partial p_1} -
       is^2\frac{\partial F}{\partial q_1}\Big)
  \Big(\hbar\frac{\partial G}{\partial p_1} +
       is^2\frac{\partial G}{\partial q_1}\Big) .   $$
Note that this expression does not depend at all on the parameter $\vthta$,
nor~on the choice of the base-point $(q_2^*,p_2^*)$; this~can be shown to
prevail also for $\tcC_j$, $j=2,3,\dots$.
In~fact, setting again $s=\sqrt{\hbar}$, we~arrive via (\ref{TCC}) just at
the formulas for the ordinary Berezin-Toeplitz quantization on~$\CC$ in the
``free'' variables~$q_1,p_1$.

\section{Conclusion and Outlook}\label{sec:conclsn}
In this paper, we have constructed noncommuative coherent states associated with a system of 2 degrees of freedom by means of the continuous families of UIRs of the kinematical symmetry group $G_{\NC}$ of the underlying system. Subsequently, we computed the pertaining reproducing kernels. Since the generic families of UIRs are indexed by 3 nonzero continuous parameters $\hbar$, $\mathcal{B}$ and $\vartheta$ subject to a quadratic constraint $\hbar^{2}-\mathcal{B}\vartheta\neq 0$, it was natural to quest for the Berezin-Toeplitz quantization of the observables on the underlying 4-dimensional Phase space using all 3 deformation parameters instead of the single Planck's constant $\hbar$. In fact, when $\mathcal{B}$ (or $B=\frac{\mathcal{B}}{\hbar}$) and $\vartheta$ (or $T=\frac{\vartheta}{\hbar}$) are both zero, the UIRs of $G_{\NC}$ indexed by this single nonzero Planck's constant $\hbar$ are nothing but the UIRs of the 2-dimensional Weyl-Heisenberg group $\GWH$. But the asymptotic analysis of Berezin-Toeplitz quantization pertaining to the generic sector (where $\hbar^{2}-\mathcal{B}\vartheta\neq 0$) of the unitary dual of $G_{\NC}$ reveals the fact that $\hbar$, $\mathcal{B}$ and $\vartheta$ cannot all approach 0 independently due to the imposed quadratic constraint. But upon ``renormalizing'' the deformation parameters $\mathcal{B}$ and $\vartheta$ to $B$ and $T$, respectively, one achieves the desired semiclassical asymptotics with $B$, $T$ and $\hbar$ simultaneously approaching 0.

We have subsequently handled the degenerate case $\hbar^{2}-\mathcal{B}\vartheta=0$ or $BT=1$ by defining the associated family of noncommutative coherent states on a dimensionally reduced Hilbert space and observed the crucial fact that the deformation parameter $\vartheta=\frac{\hbar^{2}}{\mathcal{B}}$ disappears from the picture completely yielding the standard setting of Berezin-Toeplitz quantization on the complex plane $\mathbb{C}$.
Our analysis of the degenerate noncommutative coherent states is propelled by the family of UIRs (see (\ref{eq:degenerate-rep-momentumspace-inv-trnsfrm})) of $G_{\NC}$. In other words, the group representation theoretic analysis of NCQM conducted in \cite{ncqmjpa} has enabled us to study such degenerate setting in the context of NCQM. As has been insinuated towards the end of Section \ref{sec:intro} that $B$ here stands for the constant magnetic field applied perpendicular to a charged particle constrained to move on a two dimensional plane. The degenerate case that we have studied here is closely tied with the massless limit of a charged particle moving on such a plane subject to a vertical constant magnetic field (p. 213, \cite{Szabo}). Such a model has zero Hamiltonian in the limiting case turning the theory into a topological one. The string theoretic analog of this degenerate case is also discussed there in \cite{Szabo}.

In an earlier paper \cite{nctori} by one of the present authors, noncommutative
4~tori have been constructed explicitly using the unitary dual of $G_{\NC}$.
We~propose to undertake an in-depth study of the pertinent aspects from the
point of view of noncommutative geometry, in~particular, classifying projective
modules over this NC-4~tori, computing connections with constant curvature and
the Chern character on the relevant projective modules in near future.

redpajama_set_name: RedPajamaArXiv


text:
The COVID-19 has pushed tens of thousands of people out of the workforce, which also has a certain impact on the bhutan bookings_News_Robot,Robot Tech,Robot Industry,Environment,Low Ca
The COVID-19 has pushed tens of thousands of people out of the workforce, which also has a certain impact on the bhutan bookings
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The Kingdom of Bhutan and Bhutan Tourism the bhutan bookings
The Kingdom of Bhutan referred to as Bhutan, is a landlocked country located on the southern slope of the eastern part of the Himalayas between China and India. It covers an area of about 47,000 square kilometers and is about the same size as Switzerland. The forest coverage rate is about 70% of the land, and the terrain is mostly mountainous and alpine. The peaks are in the northern and some subtropical southern foothills.
The country name of Bhutan is called "Zhuyu" in the local language, which means the land of thunder and dragon. Bhutan's economy is relatively backward, but in the "Global Happiest Country Ranking" released in 2006, bhutan bookings Bhutan ranked 8th, ranking first in Asia. Rich and unique tourism resources make tourism an important pillar of Bhutan's economy. In 2007, the income was 29.85 million US dollars, and its contribution rate to GDP was 2.5%; in 2008, it was 38.83 million US dollars, and its contribution rate to GDP was 3.1%; 2009 In 2010, it was 31.88 million US dollars, and its contribution rate to GDP was 2.5%; in 2010, it was 35.98 million US dollars, and its contribution rate to GDP was 2.20%; The $14.89 million in tax and is owned by the Bhutanese government.
Statistics from the Bhutan Tourism Bureau show that bhutan bookings since 1974, when Bhutan opened its high-end markets to tourists, the number of visitors in that year was 300. By 2019, inbound tourists had reached 355,600, an increase of 15.1% over 2018. After the border reopens, foreign tourists can freely choose travel agencies and accommodation arrangements.
Timmy, who runs a travel agency in Thimphu, the capital, believes that the increase in environmental protection tax may deter some foreign tourists, but it will not affect tourists with affluent economic conditions: "Bhutan welcomes customers who are willing to spend money. Hope."
Where is Bhutan?
Bhutan is sandwiched between the Indian Ocean and the Himalayas. Due to bhutan bookings the high and low altitude difference and the low latitude, Bhutan has diverse topography, diverse climate types and rich and colorful natural landscapes, forming a very attractive and unique place. There are many high mountains in Bhutan, bhutan bookings the dangerous peaks and glaciers are widely distributed, the colorful prayer flags flutter in the wind, and the silver snow peaks stand tall against the blue sky background, which extremely accurately interprets the "Shangri-La" Tibetan Buddhism's complete overview of the wonderland, and is also a hobby for adventure and mountaineering. the place of desire. Manas Wildlife Sanctuary and Raya Wildlife Sanctuary in Bhutan have many precious wild animals and are also good places for tourists to visit and travel. Bhutan is also one of the most environmentally friendly countries in the world. It is known as "the last Shangri-La on earth", "Xanadu", "Pure Land on Earth", etc. It has the reputation of being "farthest from the world and closest to the blue sky".
Seasonality of tourism in Bhutan
Spring and autumn are the peak tourist seasons, while winter and summer are the off-seasons. The seasonal performance of tourism is outstanding. Due to bhutan bookings the climate of Bhutan, spring and autumn are the peak tourist seasons every year, and a large number of tourists pour in, which has caused huge pressure on Bhutan's hotels and other reception facilities and traffic, the seasonal contradiction between supply and demand is prominent. As the only international airline bhutan bookings in Bhutan, Royal Bhutan Airlines has very limited capacity. It is hard to get a ticket during the peak tourist season, namely spring and autumn, but there are few passengers in the off-season. The same is true for hotels, where rooms are very hard to book during peak season, but there are plenty of vacancies in summer and winter.
Current Situation of Tourism in Bhutan
Due to the special geographical location of Bhutan, the unique natural environment, cultural landscape and cultural customs constitute an important reason bhutan bookings for promoting the development of tourism in Bhutan. The Black-necked Crane Centre at the foot of the Black Mountains, for example, is a biodiversity zone with lush coniferous forests, rhododendrons, magnolias, oaks, banana trees and cacti, and the ridges offer views of distant peaks and forests. Fujica is also home to the rare and endangered black-necked crane, a beloved bird of Bhutanese people. It can bhutan bookings be seen everywhere in the lowlands of the Fujika River Valley. Ancient castles such as Tashi Que Dzong and Punakha Dzong also have new landmarks that reflect the social and cultural construction of the Bhutanese royal family and people since modern times, such as the National Monument and National Museum of Bhutan. Tashi Chodzong is located in the government center of Thimphu, the capital, on the west bank of the Wangchu River.
Globally Customized Bhutan Travel of Asia Odyssey Travel bhutan bookings
Asia Odyssey Tours & Travel Company Limited ( HYPERLINK "https://www.bhutantrip.com/" Bhutantrip.com ) is a leading online travel service focus on Bhutan trips for bhutan bookings about 10 years. Headquartered in Hong Kong, we have branch offices in Tibet, Nepal, India, Malaysia, Taiwan, Chengdu, Guilin, Xian, Beijing, and Guangzhou. Since 2013, we have been arranging both private & customized Bhutan tours for our clients from all over the world. We can also do Nepal tours, Nepal & Bhutan travel, Tibet & Nepal tours, Tibet-Nepal-Bhutan trips, India Tours, Nepal-Bhutan-India trips, and so on if you want to add more charming destinations to your Bhutan tour.
Asia Odyssey Travel to the matching of bhutan bookings our travel consultant who is the person you contact with in the first place, we're here to make a difference to ensure exceptional experiences on your Bhutan trip. Aiming at providing the best tour service at the most reasonable cost, we guarantee that we will present you a worry-free, unforgettable and unique Bhutan journey once in your lifetime. If you have any questions about traveling to Bhutan, please feel free to contact Bhutantrip !
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