IRMP-CP3-24-01

Noncommutativity in Configuration Space

Induced by

A Conjugate Magnetic Field in Phase Space

Jan Govaertsa,b,c

aCentre for Cosmology, Particle Physics and Phenomenology (CP3),
Institut de Recherche en Mathématique et Physique (IRMP),
Université catholique de Louvain (UCLouvain),
2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
E-mail<EMAIL_ADDRESS>
ORCID: http://orcid.org/0000-0002-8430-5180
bInternational Chair in Mathematical Physics and Applications (ICMPA–UNESCO
Chair)
University of Abomey-Calavi, 072 B.P. 50, Cotonou, Republic of Benin
cFellow of the Stellenbosch Institute for Advanced Study (STIAS),
Stellenbosch, Republic of South Africa

As is well known, an external magnetic field in configuration space coupled to
a quantum dynamics induces noncommutativity in its velocity momentum space. By
phase space duality, an external vector potential in the conjugate momentum
sector of the system induces noncommutativity in its configuration space. Such
a rationale for noncommutativity is explored herein for an arbitrary
configuration space of Euclidean geometry. Ordinary quantum mechanics with a
commutative configuration space is revisited first. Through the introduction
of an arbitrary positive definite $*$-product, a one-to-one correspondence
between the Hilbert space of abstract quantum states and that of the
enveloping algebra of the position quantum operators is identified. A parallel
discussion is then presented when configuration space is noncommutative, and
thoroughly analysed when the conjugate magnetic field is momentum independent
and nondegenerate. Once again the space of quantum states may be identified
with the enveloping algebra of the noncommutative position quantum operators.
Furthermore when the positive definite $*$-product is chosen in accordance
with the value of the conjugate magnetic field which determines the commutator
algebra of the coordinate operators, these operators span a Fock algebra of
which the canonical coherent states are the localised noncommutative quantum
analogues of the sharp and structureless local points of the associated
commutative configuration space geometry. These results generalise and justify
a posteriori within the context of ordinary canonical quantisation the
heuristic approach to quantum mechanics in the noncommutative Euclidean plane
as constructed and developed by F. G. Scholtz and his collaborators.

##  1 Introduction

It is a widely held conviction that even though conceived as a differentiable
continuum up to the heretofore experimentally accessible distance and energy
scales, in actual fact as may be revealed by quantum fluctuations the space-
time fabric consists of some coarse-grained structure to be physically
manifested at the smallest distance and wavelength, and largest energy scales
possible. Yet the nature and characteristics of this coarse-grained structure
remains at best speculative, presumably to be found among a large ensemble of
possible complementary ideas and concepts.

Among these and by analogy with the coarse-grained properties of the quantised
phase space of a dynamical system, the possibility of a noncommutative
configuration space with its geometry has been explored starting already seven
decades ago along lines largely inspired by the structure of quantum mechanics
[1]. The last twenty-five years or so have seen a very strong resurgence of
interest in such a possibility, from a diversity of vantage points and
complementary approaches all sharing as primary driving motivation nowadays
the quest for a complete quantum unification of all fundamental interactions
inclusive of relativistic gravity [2, 3, 4, 5], beginning with quantum field
theories and even more simply quantum mechanical systems constructed over
noncommutative spaces and geometries [6, 7]. Some recent introductory reviews
may also be found in [8, 9, 10].

Among a large body of rather diversified and most interesting works motivated
by different approaches and points of view, the cases of two and three
dimensional spaces with flat or curved geometries such as the plane or the
sphere have attracted a lot of efforts and developments. Standing out among
the latter is the work and results over the years by F. G. Scholtz and his
collaborators (see for instance [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]).

Based on a heuristic approach conceived within the context of the
noncommutative plane and then extended to three dimensions, a quantum
mechanics on such noncommutative configuration spaces has been constructed.
This construction [9, 13] involves a so-called classical Hilbert space which
provides a representation space for the noncommuting coordinates of
configuration space, to replace and play the role of the ordinary classical
coordinates of a commutative configuration space. Then by analogy with the
ordinary configuration space wave functions of quantum states that are
functions of the commuting configuration space coordinates in the commutative
case, the quantum Hilbert space of a quantum mechanics and its quantum states
defined over the noncommutative configuration space corresponds to the
enveloping algebra of the noncommutative configuration space coordinate
operators, namely composite operators of these coordinates that act on the
classical Hilbert space.

Even though the construction does not rely on Dirac’s correspondence principle
with its classical canonical phase space formulation of a dynamics based on a
minimal action principle which is then quantised to identify a corresponding
quantum dynamical system, Scholtz’s construction of a quantum mechanics over a
noncommutative space and its geometry allows as well for the identification of
momenta canonically conjugate to the noncommuting configuration space
coordinates, whose action on the quantum Hilbert space and its states is
realised in terms of the coordinate operators themselves and their
commutators. Furthermore, a path integral representation of such so-called
noncommutative quantum mechanics also becomes feasible, in terms of an action
principle over configuration space which is then not local in time on account
of space noncommutativity [14]. Yet in the limit of vanishing noncommutativity
parameters in configuration space, the usual results of ordinary or so-called
commutative quantum mechanics are indeed recovered.

Clearly the success of Scholtz’s heuristic construction of a noncommutative
quantum mechanics and of its specific features calls for an independent
understanding of it, and how it may possibly find its justification as well
from within the more traditional and standard approach of Dirac’s canonical
quantisation of a classical phase space realisation of a classical dynamics
over a classical and commutative configuration space. Addressing and
clarifying these questions is the main purpose of the present work.

As is well known at least in the case of the celebrated Landau problem,
coupling a dynamics to a magnetic field in configuration space induces
noncommutativity in the gauge invariant velocity space [22]. Likewise but less
well known than it would deserve, by phase space duality, coupling a dynamics
to a magnetic field in conjugate momentum space induces noncommutativity in
configuration space. This observation serves as a classical starting point for
the exploration of answers to the above questions, leading in fine to the a
posteriori justification through Dirac’s canonical quantisation of Scholtz’s
construction of a quantum mechanics over noncommutative configuration spaces,
at least when their geometry is Euclidean and even dimensional which is the
situation explicitly analysed herein.

The discussion is structured as follows. Section 2 revisits ordinary quantum
mechanics over commutative configuration spaces. A series of considerations
which are usually not to be found in standard quantum mechanics textbooks are
addressed and developed, to anticipate on a parallel ensemble of
considerations which prove to be required in the noncommutative case. In
particular, a specific $*$-product is introduced already for a commutative
quantum mechanics, based on a specific class of localised and normalisable
quantum states that are in one-to-one correspondence with the points in
configuration space, yet that are not sharp position eigenstates. With section
3 the discussion is then extended to include a generic conjugate magnetic
field in the canonical conjugate momentum sector of phase space, which readily
implies a noncommutativity of the configuration space coordinate operators for
the canonically quantised dynamical system. After a series of general
considerations, in section 4 the analysis is restricted to such a conjugate
magnetic field which is simply constant over phase space, thereby leading to
noncommuting coordinate operators whose commutation relations are simply
constants. This analysis is then mostly parallel to that of section 2, to
finally identify, in the case of an even-dimensional configuration space, how
Dirac’s canonical quantisation indeed provides an a posteriori justification
for Scholtz et al.’s heuristic quantum mechanics on noncommutative
configuration spaces. Some concluding considerations are outlined in section
5.

##  2 A “Star” (Product) for Ordinary Quantum Mechanics

###  2.1 A classical formulation

Even though the considerations presented in this work may be extended to more
general geometries, for the purposes of the present discussion let us restrict
to any physical system whose configuration space is specifically an Euclidean
space of finite dimension $N\geq 1$, of metric $\delta_{ij}$, and parametrised
by real cartesian coordinates $x_{i}\in\mathbb{R}$ ($i=1,2,\cdots,N$). Within
its canonical Hamiltonian formulation the system’s phase space is spanned by
coordinates $(x_{i},p_{i})$, with the variables $p_{i}\in\mathbb{R}$
($i=1,2,\cdots,N$) being conjugate momenta canonically conjugate to $x_{i}$.
Their (equal time) canonical Poisson brackets hence determine the phase
space’s symplectic structure, with

$\left\\{x_{i},x_{j}\right\\}=0,\qquad\left\\{x_{i},p_{j}\right\\}=\delta_{ij},\qquad\left\\{p_{i},p_{j}\right\\}=0,\qquad
i,j=1,2,\cdots,N.$ (1)

The system’s time evolution, say relative to a physical time coordinate
$t\in\mathbb{R}$, is thereby generated from some Hamiltonian function
$H(\vec{x},\vec{p}\,)=H(x_{i},p_{i})$ – assumed herein to be time independent,
in order to keep the presentation the simplest possible – with the first-order
Hamiltonian equations of motion

$\dot{x}_{i}=\left\\{x_{i},H\right\\}=\frac{\partial H}{\partial
p_{i}},\qquad\dot{p}_{i}=\left\\{p_{i},H\right\\}=-\frac{\partial H}{\partial
x_{i}},\qquad\dot{}\equiv\frac{d}{dt},\qquad i=1,2,\cdots,N.$ (2)

Within the context of nonrelativistic Newtonian dynamics, for a single massive
particle of mass $\mu$ the Hamiltonian is generally of the form

$H(\vec{x},\vec{p}\,)=\frac{1}{2\mu}\vec{p}\,^{2}+V(\vec{x}\,),$ (3)

where $V(\vec{x}\,)$ is some mechanical potential energy for the particle.

As a matter of fact, these different structures are encoded in the Hamiltonian
first-order action defined over phase space,

$S_{0}[x_{i},p_{i}]=\int
dt\left(\dot{\vec{x}}\cdot\vec{p}-H(\vec{x},\vec{p}\,)\right)=\int
dt\left(\dot{x}_{i}p_{i}-H(x_{i},p_{i})\right)$ (4)

(with the implicit summation convention over repeated indices being understood
throughout). Of course, the variations of this action relative to $x_{i}$ and
$p_{i}$ lead back to the above first-order equations of motion. Note that an
arbitrary total time derivative, or time surface term in the form of
$d\Lambda(\vec{x},\vec{p}\,)/dt$ may be added to this action without affecting
neither these equations of motion nor the phase space symplectic structure.
Finally in the case of a Hamiltonian in the form of (3), the reduction of the
conjugate momenta through the Hamiltonian equations of motion for $x_{i}$
leads to the familiar form of the Lagrangian action for this class of systems,

$S_{0}[x_{i}]=\int
dt\left(\frac{1}{2}\mu\dot{\vec{x}}\,^{2}\,-\,V\left(\vec{x}\,\right)\right).$
(5)

###  2.2 A canonically quantised formulation

Through the correspondence principle, the canonically quantised system is
implicitly specified by the equal time commutation relations which are in
correspondence with the above Poisson brackets. These commutation relations
are to be considered at some reference time $t=t_{0}$ which is not made
explicit hereafter (note that because of the lack of time dependency of the
Hamiltonian, invariance of the dynamics under constant translations in time
implies that one may always take $t_{0}=0$ for convenience). Hence one has the
following tensor product over $i,j=1,2,\cdots,N$ of $N$ copies of the
Heisenberg algebra with generators and quantum phase space observables111Note
that throughout all operators are systematically distinguished by a caret
above their symbol. $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$,

$\left[\hat{x}_{i},\hat{x}_{j}\right]=0,\quad\left[\hat{x}_{i},\hat{p}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I},\quad\left[\hat{p}_{i},\hat{p}_{j}\right]=0,\quad\hat{x}^{\dagger}_{i}=\hat{x}_{i},\quad\hat{p}^{\dagger}_{i}=\hat{p}_{i},\quad
i,j=1,2,\cdots,N.$ (6)

The abstract Hilbert space of quantum states of the system provides a
representation of this Heisenberg algebra. It must be equipped with a positive
definite sesquilinear inner product for which $\hat{x}_{i}$ and $\hat{p}_{i}$
must each be self-adjoint (or at least hermitian), as must be the quantum
Hamiltonian itself as well,

$\hat{H}=\hat{H}(\hat{\vec{x}},\hat{\vec{p}}\,),\qquad\hat{H}^{\dagger}=\hat{H},$
(7)

in order for time evolution of the quantised system as generated from this
quantum operator to be unitary. As is well known from the Stone-von Neumann
theorem, in the case of a configuration space of trivial fundamental group –
such as applies to the space $\mathbb{R}^{N}$ –, up to unitary equivalence
there exists a single unitary representation of the Heisenberg algebra [23],
namely the well known canonical representation and its wave function
realisations whether in configuration or in momentum space.

The construction of these $|\vec{x}\,\rangle$\- and
$|\vec{p}\,\rangle$-representations is standard. Since either all position
operators $\hat{x}_{i}$, or all momentum operators $\hat{p}_{i}$, commute
among themselves there always exists a basis of eigenstates of each of these
two ensembles of operators, $|\vec{x}\,\rangle$ or $|\vec{p}\,\rangle$,
respectively,

$\hat{x}_{i}|\vec{x}\,\rangle=x_{i}|\vec{x}\,\rangle,\qquad\hat{p}_{i}|\vec{p}\,\rangle=p_{i}|\vec{p}\,\rangle,\qquad
x_{i},p_{i}\in\mathbb{R},\quad i=1,2,\cdots,N.$ (8)

Given the Euclidean metric of configuration space, the normalisation of these
states is chosen such that,

$\langle\vec{x}\,|\vec{x}\,^{\prime}\rangle=\delta^{(N)}(\vec{x}-\vec{x}\,^{\prime}),\qquad\langle\vec{p}\,|\vec{p}\,^{\prime}\rangle=\delta^{(N)}(\vec{p}-\vec{p}\,^{\prime}),$
(9)

which in turn implies the corresponding completeness relations or
representations of the unit operator,

$\mathbb{I}=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\langle\vec{x}\,|,\qquad\mathbb{I}=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\langle\vec{p}\,|.$
(10)

Note that since these normalisation choices do not impact the pure imaginary
phase factors of these position and momentum eigenstates, these phase factors,
local in $\vec{x}$ or in $\vec{p}$, are left unspecified [23], with the
understanding that some choice albeit implicit has been effected at this
stage.

Consequently, in view of the above spectral representations of the unit
operator, any abstract quantum state $|\varphi\rangle$ may be decomposed in
either of these two bases in terms of complex valued functions whether in
configuration or in momentum space, namely its configuration or momentum wave
function representations $\varphi(\vec{x}\,)=\langle\vec{x}\,|\varphi\rangle$
or $\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$, respectively,
such that,

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\,\varphi(\vec{x}\,),\quad|\varphi\rangle=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,\tilde{\varphi}(\vec{p}\,),\quad\varphi(\vec{x})=\langle\vec{x}\,|\varphi\rangle,\quad\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle.$
(11)

It may be established [23] as well that the operators $\hat{x}_{i}$ and
$\hat{p}_{i}$ then possess the following differential operator realisations
acting on these wave functions, whether in the
$|\vec{x}\,\rangle$-representation,

$\langle\vec{x}\,|\hat{x}_{i}|\varphi\rangle=x_{i}\,\varphi(\vec{x}\,),\qquad\langle\vec{x}\,|\hat{p}_{i}|\varphi\rangle=-i\hbar\frac{\partial}{\partial
x_{i}}\varphi(\vec{x}\,),\qquad i=1,2,\cdots,N,$ (12)

or in the $|\vec{p}\,\rangle$-representation,

$\langle\vec{p}\,|\hat{x}_{i}|\varphi\rangle=i\hbar\frac{\partial}{\partial
p_{i}}\tilde{\varphi}(\vec{p}\,),\qquad\langle\vec{p}\,|\hat{p}_{i}|\varphi\rangle=p_{i}\,\tilde{\varphi}(\vec{p}\,),\qquad
i=1,2,\cdots,N.$ (13)

Once these choices made for the differential operator representations of
$\hat{p}_{i}$ in the $|\vec{x}\,\rangle$ basis and of $\hat{x}_{i}$ in the
$|\vec{p}\,\rangle$ basis, it follows that the unitary transformations mapping
one of these bases onto the other are realised in terms of the following
matrix elements,

$\langle\vec{x}\,|\vec{p}\,\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}},\qquad\langle\vec{p}\,|\vec{x}\,\rangle=\langle\vec{x}\,|\vec{p}\,\rangle^{*}=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}},$
(14)

where an implicit choice for the relative quantum phase factor between these
two classes of quantum states is made such that no other pure imaginary
constant phase factor is involved. As a consequence for any abstract state
$|\varphi\rangle$ and its position and momentum wave functions one has the
reciprocal (Fourier transformation) relations,

$\varphi(\vec{x}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,\tilde{\varphi}(\vec{p}\,),\qquad\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,\varphi(\vec{x}\,).$
(15)

###  2.3 Alternative quantum operator representations

Motivated by the spectral decompositions or completeness relations of the unit
operator,

$\mathbb{I}=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\langle\vec{x}\,|,\qquad\mathbb{I}=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\langle\vec{p}\,|,$
(16)

it is also possible to establish similar spectral decomposition
representations of the position and momentum operators, $\hat{x}_{i}$ and
$\hat{p}_{i}$, whether in the $|\vec{x}\,\rangle$ or the $|\vec{p}\,\rangle$
basis. Such results can also find their own practical utility, as the case may
be. They also extend to the noncommutative configuration spaces considered
later on.

Obviously in their respective eigenbases one has,

$\hat{x}_{i}=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\,x_{i}\,\langle\vec{x}\,|,\qquad\hat{p}_{i}=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,p_{i}\,\langle\vec{p}\,|,\qquad
i=1,2,\cdots,N.$ (17)

While in the other basis which is not that of their eigenstates, one also
finds for $\hat{p}_{i}$,

$\displaystyle\hat{p}_{i}$ $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\left(-\frac{1}{2}i\hbar\left(\frac{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}{\partial
x_{i}}-\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
x_{i}}\right)\right)\langle\vec{x}\,|$ (18) $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\left(-i\hbar\frac{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}{\partial
x_{i}}\right)\langle\vec{x}\,|=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\left(i\hbar\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
x_{i}}\right)\langle\vec{x}\,|,\qquad i=1,2,\cdots,N,$

and for $\hat{x}_{i}$,

$\displaystyle\hat{x}_{i}$ $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\left(\frac{1}{2}i\hbar\left(\frac{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}{\partial
p_{i}}-\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
p_{i}}\right)\right)\langle\vec{p}\,|$ (19) $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\left(i\hbar\frac{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}{\partial
p_{i}}\right)\langle\vec{p}\,|=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\left(-i\hbar\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
p_{i}}\right)\langle\vec{p}\,|,\qquad i=1,2,\cdots,N.$

These different relations for the operators $\hat{x}_{i}$ and $\hat{p}_{i}$
may easily be confirmed, for example by computing their action on any state
$|\varphi\rangle$ and then projecting onto the basis in which their spectral
representation is considered.

However it should be emphasised that for the last two sets of identities for
$\hat{p}_{i}$ and for $\hat{x}_{i}$ to be valid, it must be understood that
these operators are implicitly assumed to be acting on abstract quantum states
$|\varphi\rangle$ whose wave functions $\varphi(\vec{x}\,)$ and
$\tilde{\varphi}(\vec{p}\,)$ are normalisable over $\mathbb{R}^{N}$, so that
no surface contributions at infinity can be generated through integrations by
parts.

###  2.4 Configuration space localised and normalisable quantum states

Even though the states $|\vec{x}\,\rangle$ and $|\vec{p}\,\rangle$ are sharp
in the sense that they are eigenstates of the position and momentum operators
$\hat{x}_{i}$ and $\hat{p}_{i}$, respectively, they are not normalisable since
the values of their overlaps within each ensemble are given by the Dirac
$\delta^{(N)}$ distribution whether in $\vec{x}$ or in $\vec{p}$ space.

Within the noncommutative configuration space context to be addressed
hereafter, the momentum operators $\hat{p}_{i}$ remain commuting among
themselves. In that case their eigenbasis of non-normalisable though sharp
states $|\vec{p}\,\rangle$ still provides a basis of quantum Hilbert space.
However since in that context the configuration space operators $\hat{x}_{i}$
are no longer all commuting among themselves, they cannot possess a common
basis of sharp eigenstates, as are the present states $|\vec{x}\,\rangle$.
Nonetheless, it then remains possible to identify quantum states that are not
only localisable in configuration space at least in a weak sense, namely in
terms of expectation values, but these states are then also of finite norm.
This is then made possible because of the existence of length scales intrinsic
to the noncommutativity of configuration space.

Likewise within the present context of ordinary quantum physics with its
commuting configuration space degrees of freedom $\hat{x}_{i}$, let us
consider the possibility of constructing similarly localisable quantum states
of finite norm, given specific length squared scale factors $\lambda_{ij}$
with $\lambda_{ij}\in\mathbb{R}$ representing a real positive definite
symmetric matrix of constant coefficients of which the inverse
$(\lambda^{-1})_{ij}$ is thus also well defined. By definition, these states,
to be denoted $|\vec{x};\lambda\rangle$, possess the following representation
in the basis of momentum eigenstates,

$\langle\vec{p}\,|\vec{x};\lambda\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}},\qquad|\vec{x};\lambda\rangle=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,\langle\vec{p}\,|\vec{x};\lambda\rangle.$
(20)

Obviously one has,

$\lim_{\lambda_{ij}\rightarrow 0}|\vec{x};\lambda\rangle=|\vec{x}\,\rangle.$
(21)

Due to the exponential suppression of the large momentum components of these
states $|\vec{x};\lambda\rangle$ induced by the nonvanishing scale factors
$\lambda_{ij}$, the latter quantities are responsible for a spreading or
fuzziness on distance scales of the order of $\sqrt{|\lambda_{ij}|}$ for these
states’ probability density in configuration space, and centered at $\vec{x}$.
This is made explicit by the configuration space wave function of these
states,

$\langle\vec{x}\,|\vec{x}_{0};\lambda\rangle=\frac{1}{(\pi^{N}\,{\rm
det}\,\lambda_{ij})^{1/2}}\,e^{-(x-x_{0})_{i}\,(\lambda^{-1})_{ij}\,(x-x_{0})_{j}}=W_{\lambda}(\vec{x}-\vec{x}_{0})=W_{\lambda}(\vec{x}_{0}-\vec{x}),$
(22)

where the real Weierstrass kernel $W_{\lambda}(\vec{x}\,)$ associated to the
coefficients $\lambda_{ij}$ is defined by,

$W_{\lambda}(\vec{x}\,)\equiv\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}}=\frac{1}{(\pi^{N}\,{\rm
det}\,\lambda_{ij})^{1/2}}\,e^{-x_{i}(\lambda^{-1})_{ij}x_{j}},\qquad
W^{*}_{\lambda}(\vec{x}\,)=W_{\lambda}(\vec{x}\,).$ (23)

Note that one may consider that the parameters $\lambda_{ij}$ jointly with the
choice of localisable states $|\vec{x};\lambda\rangle$ provide a gaussian
regularisation of the sharp but non-normalisable position eigenstates
$|\vec{x}\,\rangle$, that leads as well to an ensemble of localised and
normalisable quantum states attached specifically to all points in
configuration space which ought to share with the states $|\vec{x}\,\rangle$
most of their properties but in a weak sense through matrix elements and
expectation values.

Even though the above notation for the states $|\vec{x};\lambda\rangle$ is
strictly reserved for real values of the parameters $x_{i}\in\mathbb{R}$,
since $\lambda_{ij}$ defines a real positive definite symmetric matrix the
above definition of these states extends to a larger ensemble of quantum
states by allowing the parameters $x_{i}$ to rather take complex values
$z_{i}\in\mathbb{C}$. This extended set of quantum states is then to be
denoted specifically as $|z_{i};\lambda\rangle$ with,

$|z_{i};\lambda\rangle=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,\langle\vec{p}\,|z_{i};\lambda\rangle,\qquad\langle\vec{p}\,|z_{i};\lambda\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}z_{i}p_{i}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}},$
(24)

while obviously
$|z_{i}=x_{i}\in\mathbb{R};\lambda\rangle=|x_{i};\lambda\rangle=|\vec{x};\lambda\rangle$.
Of course, the states $|z_{i};\lambda\rangle$ remain normalisable for
$z_{i}\in\mathbb{C}$, with,

$\langle
z_{1,i};\lambda|z_{2,i};\lambda\rangle=\frac{1}{\left((2\pi)^{N}\,{\rm
det}\,\lambda_{ij}\right)^{1/2}}\,e^{-\frac{1}{2}(z^{*}_{1,i}-z_{2,i})\,\left(\lambda^{-1}\right)_{ij}\,(z^{*}_{1,j}-z_{2,j})},$
(25)

and in particular,

$\langle z_{i};\lambda|z_{i};\lambda\rangle=\frac{1}{\left((2\pi)^{N}\,{\rm
det}\,\lambda_{ij}\right)^{1/2}}\,e^{2\,{\rm
Im}\,z_{i}\,\left(\lambda^{-1}\right)_{ij}\,{\rm Im}\,z_{j}}.$ (26)

###  2.5 Properties of Weierstrass kernels and their $*_{\lambda}$-products

From the above $\vec{p}\,$-integral representation of the Weierstrass kernel
associated to the coefficients $\lambda_{ij}$ one readily sees that,

$\lim_{\lambda_{ij}\rightarrow
0}W_{\lambda}(\vec{x}\,)=\delta^{(N)}(\vec{x}\,),\qquad\int_{(\infty)}d^{N}\vec{x}\,W_{\lambda}(\vec{x}\,)=1,$
(27)

while of course,

$\int_{(\infty)}d^{N}\vec{x}\,\delta^{(N)}(\vec{x}\,)=1.$ (28)

The Weierstrass kernel $W_{\lambda}(\vec{x}\,)$ thus provides a specific
gaussian regularisation of the Dirac $\delta^{(N)}(\vec{x}\,)$ distribution.

By using the Weierstrass kernel as a normalised weight distribution in
configuration space, direct evaluations of some expectation values in
configuration space find,

$\int_{(\infty)}d^{N}\vec{x}\,W_{\lambda}(\vec{x}\,)\,x_{i}=0,\quad\int_{(\infty)}d^{N}\vec{x}\,W_{\lambda}(\vec{x}\,)\,x_{i}x_{j}=\frac{1}{2}\lambda_{ij},\quad\int_{(\infty)}d^{N}\vec{x}\,W_{\lambda}(\vec{x}\,)\,\vec{x}\,^{2}=\frac{1}{2}{\rm
Tr}\,\lambda_{ij}.$ (29)

It may readily be checked that this kernel also obeys the following elliptic
differential equation,

$\left(-\lambda_{ij}\frac{\partial}{\partial x_{i}}\frac{\partial}{\partial
x_{j}}\,+\,4\,x_{i}\,(\lambda^{-1})_{ij}\,x_{j}\,-\,2N\right)\,W_{\lambda}(\vec{x}\,)=0.$
(30)

In order to display one last most noteworthy property of the Weierstrass
kernel $W_{\lambda}(\vec{x}\,)$, let us finally introduce a
$*_{\lambda}$-product which will prove to be most relevant hereafter. This
$*_{\lambda}$-product involves the scale parameters $\lambda_{ij}$, is given
as the exponentiated series of powers of the $\lambda_{ij}$ contracted product
of two configuration space gradients acting towards the left and the right on
functions of $\vec{x}$ that are multiplied through the $*_{\lambda}$-product,
and is defined by,

$*_{\lambda}\equiv
e^{\frac{1}{2}\lambda_{ij}\stackrel{{\scriptstyle\leftarrow}}{{\frac{\partial}{\partial
x_{i}}}}\stackrel{{\scriptstyle\rightarrow}}{{\frac{\partial}{\partial
x_{j}}}}}\equiv*_{\vec{x},\lambda}.$ (31)

The very last notation may be used when the variables on which the
$*_{\lambda}$-product acts are to be made explicit if confusion could arise
otherwise. Note that in the limit of vanishing scale coefficients
$\lambda_{ij}$ the $*_{\lambda}$-product reduces to the ordinary commutative
product of functions over configuration space,

$\lim_{\lambda_{ij}\rightarrow
0}*_{\lambda}=1,\qquad\lim_{\lambda_{ij}\rightarrow
0}\,f(\vec{x}\,)\,*_{\lambda}\,g(\vec{x}\,)=f(\vec{x}\,)\,g(\vec{x}\,).$ (32)

A direct evaluation using for the Weierstrass kernel its $\vec{p}\,$-integral
representation then finds the following identity valid specifically for the
$*_{\lambda}$-product,

$W_{\lambda}(\vec{x}-\vec{a}\,)*_{\lambda}W_{\lambda}(\vec{x}-\vec{b}\,)=W_{\lambda}(\vec{x}-\vec{a}\,)\,\delta^{(N)}(\vec{a}-\vec{b}\,).$
(33)

Once again, this result is to be considered on a par with the similar identity
valid for the Dirac $\delta^{(N)}(\vec{x}\,)$ distribution, to which
furthermore the above relation reduces in the limit $\lambda_{ij}\rightarrow
0$,

$\delta^{(N)}(\vec{x}-\vec{a}\,)\,\delta^{(N)}(\vec{x}-\vec{b}\,)=\delta^{(N)}(\vec{x}-\vec{a}\,)\,\delta^{(N)}(\vec{a}-\vec{b}\,).$
(34)

###  2.6 Properties of the localisable states $|\vec{x};\lambda\rangle$

Given any quantum operator $\hat{\cal O}$ acting on Hilbert space, let us
denote its normalised expectation values for the localised states
$|\vec{x};\lambda\rangle$ as

$\langle\hat{\cal
O}\rangle_{\lambda}(\vec{x}\,)\equiv\frac{\langle\vec{x};\lambda|\hat{\cal
O}|\vec{x};\lambda\rangle}{\langle\vec{x};\lambda|\vec{x};\lambda\rangle}.$
(35)

A simple evaluation finds that the states $|\vec{x};\lambda\rangle$ are indeed
of finite norm for nonvanishing scale factors $\lambda_{ij}$, with a value
independent of $\vec{x}$ given as,

$\langle\vec{x};\lambda|\vec{x};\lambda\rangle=\frac{1}{\left((2\pi)^{N}\,{\rm
det}\,\lambda_{ij}\right)^{1/2}}=W_{2\lambda}(\vec{0}\,),$ (36)

where $W_{2\lambda}(\vec{x}\,)$ denotes the Weierstrass kernel now associated
to the coefficients $2\lambda_{ij}$, namely,

$W_{2\lambda}(\vec{x}\,)\equiv\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N}}\,e^{-\frac{2\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}}=\frac{1}{((2\pi)^{N}\,{\rm
det}\,\lambda_{ij})^{1/2}}\,e^{-\frac{1}{2}x_{i}(\lambda^{-1})_{ij}x_{j}}.$
(37)

Direct evaluations find for the operators $\hat{x}_{i}$,
$\hat{x}_{i}\hat{x}_{j}$ and $\hat{\vec{x}}\,^{2}$ the following expectations
values for the states $|\vec{x};\lambda\rangle$,

$\langle\hat{x}_{i}\rangle_{\lambda}(\vec{x}\,)=x_{i},\qquad\langle\hat{x}_{i}\hat{x}_{j}\rangle_{\lambda}(\vec{x}\,)=x_{i}x_{j}+\frac{1}{4}\lambda_{ij},\qquad\langle\hat{\vec{x}}\,^{2}\rangle_{\lambda}(\vec{x}\,)=\vec{x}\,^{2}+\frac{1}{4}{\rm
Tr}\,\lambda_{ij}.$ (38)

And likewise for the operators $\hat{p}_{i}$, $\hat{p}_{i}\hat{p}_{j}$ and
$\hat{\vec{p}}\,^{2}$,

$\langle\hat{p}_{i}\rangle_{\lambda}(\vec{x}\,)=0,\qquad\langle\hat{p}_{i}\hat{p}_{j}\rangle_{\lambda}(\vec{x}\,)=\hbar^{2}\,\left(\lambda^{-1}\right)_{ij},\qquad\langle\hat{\vec{p}}\,^{2}\rangle_{\lambda}(\vec{x}\,)=\hbar^{2}\,{\rm
Tr}\,\left(\lambda^{-1}\right)_{ij}.$ (39)

Consequently for the associated uncertainty relations, one finds,

$\Delta^{(\hat{\vec{x}})}_{ij}(\vec{x}\,)\equiv\langle\left(\hat{x}_{i}-\langle\hat{x}_{i}\rangle_{\lambda}\right)\left(\hat{x}_{j}-\langle\hat{x}_{j}\rangle_{\lambda}\right)\rangle_{\lambda}(\vec{x}\,)=\frac{1}{4}\lambda_{ij},$
(40)

as well as,

$\Delta^{(\hat{\vec{p}}\,)}_{ij}(\vec{x}\,)\equiv\langle\left(\hat{p}_{i}-\langle\hat{p}_{i}\rangle_{\lambda}\right)\left(\hat{p}_{j}-\langle\hat{p}_{j}\rangle_{\lambda}\right)\rangle_{\lambda}(\vec{x}\,)=\hbar^{2}\,\left(\lambda^{-1}\right)_{ij},$
(41)

so that finally,

$\Delta^{(\hat{\vec{x}})}_{ij}(\vec{x}\,)\,\Delta^{(\hat{\vec{p}}\,)}_{k\ell}(\vec{x}\,)=\frac{1}{4}\hbar^{2}\,\lambda_{ij}\,\left(\lambda^{-1}\right)_{k\ell}.$
(42)

In other words the states $|\vec{x};\lambda\rangle$ are saturating states for
the mixed uncertainty relations of the position and momentum quantum
operators, as is characteristic as well of usual canonical coherent states for
the Heisenberg algebra. Indeed it will be seen hereafter that these localised
and normalised quantum states $|\vec{x};\lambda\rangle$ are indeed a specific
subclass of such coherent quantum states.

This observation also reflects the fact that of course, the states
$|\vec{x};\lambda\rangle$ cannot be sharp position eigenstates of
$\hat{x}_{i}$, of which the action on these states includes as well
contributions from the action of the momentum operators $\hat{p}_{i}$ that
directly involve the coefficients $\lambda_{ij}$. Indeed, by inserting the
spectral decomposition of the unit operator in the $|\vec{p}\,\rangle$
eigenbasis to the left of $\hat{x}_{i}|\vec{x};\lambda\rangle$ and using the
matrix elements $\langle\vec{p}\,|\hat{x}_{i}|\vec{x};\lambda\rangle$ one
establishes the identity,

$\hat{x}_{i}\,|\vec{x};\lambda\rangle=x_{i}\,|\vec{x};\lambda\rangle\,-\,\frac{i}{2\hbar}\lambda_{ij}\,\hat{p}_{j}\,|\vec{x};\lambda\rangle,\qquad\left(\hat{x}_{i}+\frac{i}{2\hbar}\lambda_{ij}\hat{p}_{j}\right)|\vec{x};\lambda\rangle=x_{i}\,|\vec{x};\lambda\rangle.$
(43)

Hence, the localised states $|\vec{x};\lambda\rangle$ are eigenstates of the
“momentum twisted” position operators
$(\hat{x}_{i}+\frac{i\lambda_{ij}}{2\hbar}\hat{p}_{j})$ with eigenvalues
$x_{i}$, rather than of the position operators $\hat{x}_{i}$ themselves.
Further consequences of this observation are to be addressed hereafter.

Similar considerations may be extended to matrix elements of the operators
$\mathbb{I}$, $\hat{x}_{i}$, $\hat{x}_{i}\hat{x}_{j}$, $\hat{\vec{x}}\,^{2}$,
$\hat{p}_{i}$, $\hat{p}_{i}\hat{p}_{j}$ and $\hat{\vec{p}}\,^{2}$ but now for
two distinct external localised states. Through direct evaluations one finds
the following explicit results. First for their overlaps,

$\langle\vec{x}_{1};\lambda|\vec{x}_{2};\lambda\rangle=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N}}\,e^{\frac{i}{\hbar}\vec{p}\cdot(\vec{x}_{1}-\vec{x}_{2})}e^{-\frac{\lambda_{ij}}{2\hbar^{2}}p_{i}p_{j}}=W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$
(44)

an expression which shows that any of these states has a finite nonvanishing
overlap with any other such state. Next for the operators $\hat{x}_{i}$,
$\hat{x}_{i}\hat{x}_{j}$ and $\hat{\vec{x}}\,^{2}$,

$\langle\vec{x}_{1};\lambda|\hat{x}_{i}|\vec{x}_{2};\lambda\rangle=\frac{1}{2}\left(x_{1}+x_{2}\right)_{i}\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$
(45)
$\langle\vec{x}_{1};\lambda|\hat{x}_{i}\hat{x}_{j}|\vec{x}_{2};\lambda\rangle=\left[\frac{1}{4}\left(x_{1}+x_{2}\right)_{i}\left(x_{1}+x_{2}\right)_{j}+\frac{1}{4}\lambda_{ij}\right]\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$
(46)

and,

$\langle\vec{x}_{1};\lambda|\hat{\vec{x}}\,^{2}|\vec{x}_{2};\lambda\rangle=\left[\left(\frac{\vec{x}_{1}+\vec{x}_{2}}{2}\right)^{2}+\frac{1}{4}{\rm
Tr}\,\lambda_{ij}\right]\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}).$ (47)

And finally for the operators $\hat{p}_{i}$, $\hat{p}_{i}\hat{p}_{j}$ and
$\hat{\vec{p}}\,^{2}$ as well,

$\langle\vec{x}_{1};\lambda|\hat{p}_{i}|\vec{x}_{2};\lambda\rangle=-i\hbar\frac{\partial}{\partial
x_{1i}}\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$ (48)
$\langle\vec{x}_{1};\lambda|\hat{p}_{i}\hat{p}_{j}|\vec{x}_{2};\lambda\rangle=\left(-i\hbar\frac{\partial}{\partial
x_{1i}}\right)\left(-i\hbar\frac{\partial}{\partial
x_{1j}}\right)\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$ (49)

and,

$\langle\vec{x}_{1};\lambda|\hat{\vec{p}}\,^{2}|\vec{x}_{2};\lambda\rangle=\left(-i\hbar\vec{\nabla}_{\vec{x}_{1}}\right)^{2}\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}).$
(50)

###  2.7 Displacement operators and canonical quantum coherent states

As is well known the momentum operators $\hat{p}_{i}$ are the generators of
displacements in configuration space, with a finite displacement by $\vec{a}$
being generated through the action of the unitary operator
$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}$. Explicitly one finds222By
using the Baker-Campbell-Haussdorff (BCH) formula
$e^{\hat{A}}\,e^{\hat{B}}=e^{\hat{A}+\hat{B}+\frac{1}{2}[\hat{A},\hat{B}]}$
which is valid for any two operators $\hat{A}$ and $\hat{B}$ that commute with
their commutator $[\hat{A},\hat{B}]$.,

$e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,\hat{x}_{i}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}=\hat{x}_{i}\,+\,a_{i}\,\mathbb{I},\qquad
e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,\hat{p}_{i}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}=\hat{p}_{i},$
(51)

while,

$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{x}\,\rangle=|\vec{x}+\vec{a}\,\rangle,\qquad
e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{p}\,\rangle=e^{\frac{i}{\hbar}\vec{a}\cdot\vec{p}}\,|\vec{p}\,\rangle.$
(52)

In a likewise manner the position operators $\hat{x}_{i}$ are the generators
of displacements in momentum space, with a finite displacement by
$\vec{p}_{0}$ being generated through the action of the unitary operator
$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}$. One has this time,

$e^{-\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}=\hat{x}_{i},\qquad
e^{-\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,\hat{p}_{i}\,e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}=\hat{p}_{i}\,+\,p_{0i}\,\mathbb{I},$
(53)

while,

$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{x}\,\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,|\vec{x}\,\rangle,\qquad
e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{p}\,\rangle=|\vec{p}+\vec{p}_{0}\rangle.$
(54)

Let us now reconsider the result in (43). Combined with the fact that

$\hat{p}_{i}\,|\vec{x};\lambda\rangle=i\hbar\,\frac{\partial}{\partial
x_{i}}\,|\vec{x};\lambda\rangle,$ (55)

which is readily established from the knowledge of the momentum wave functions
$\langle\vec{p}\,|\vec{x};\lambda\rangle$, one thus also obtains the following
identities,

$\hat{x}_{i}\,|\vec{x};\lambda\rangle=\left(x_{i}+\frac{1}{2}\lambda_{ij}\frac{\partial}{\partial
x_{j}}\right)\,|\vec{x};\lambda\rangle,\qquad\hat{p}_{i}\,|\vec{x};\lambda\rangle=2i\hbar\left(\lambda^{-1}\right)_{ij}\,\left(\hat{x}_{j}-x_{j}\mathbb{I}\right)\,|\vec{x};\lambda\rangle.$
(56)

Given that the localised states $|\vec{x};\lambda\rangle$ are eigenstates with
eigenvalues $x_{i}$ of the momentum twisted coordinates
$\left(\hat{x}_{i}+i\frac{\lambda_{ij}}{2\hbar}\hat{p}_{j}\right)$, for any
conjugate momentum value $\vec{p}_{0}$ one also has the property that,

$e^{\frac{i}{\hbar}p_{0i}\left(\hat{x}_{i}+\frac{i\lambda_{ij}}{2\hbar}\hat{p}_{j}\right)}\,|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,|\vec{x};\lambda\rangle.$
(57)

Using the BCH formula this last identity may also be brought into the
following form,

$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle=e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{0i}p_{0j}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}_{0}}\,e^{\frac{\lambda_{ij}}{2\hbar^{2}}p_{0i}\hat{p}_{j}}\,|\vec{x};\lambda\rangle,$
(58)

a result which is to be compared to that of the action of the same momentum
displacement operator acting on the sharp states $|\vec{x}\,\rangle$ (see
(54)). Note that the momentum wave functions of these transformed localised
states are thus simply,

$\langle\vec{p}\,|e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}\left(p-p_{0}\,\right)_{i}\left(p-p_{0}\right)_{j}}\,e^{-\frac{i}{\hbar}\left(\vec{p}-\vec{p}_{0}\,\right)\cdot\vec{x}}=\langle\vec{p}-\vec{p}_{0}|\vec{x};\lambda\rangle,$
(59)

which display the expected shift $\vec{p}\rightarrow(\vec{p}-\vec{p}_{0})$ in
their momentum dependency as generated by the momentum displacement operator
${\rm exp}(i\vec{p}_{0}\cdot\hat{\vec{x}}/\hbar)$.

Since
$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{x}\,\rangle=|\vec{x}+\vec{a}\,\rangle$,
the result in (58) may also be expressed in terms of the extended quantum
states $|z_{i};\lambda\rangle$, with

$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle=e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{0i}p_{0j}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}_{0}}\,\Big{|}z_{i}=x_{i}+\frac{1}{2}i\lambda_{ij}p_{0j};\lambda\Big{\rangle}.$
(60)

Incidentally the matrix elements (59) also imply the following simple
representation of momentum eigenstates in terms of a superposition of the
transformed localised states, whatever the values for $\lambda_{ij}$. Indeed
one has, using (59),

$\int_{(\infty)}d^{N}\vec{x}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle=\int_{(\infty)}d^{N}\vec{x}\int_{(\infty)}d^{N}\vec{p}_{1}\,|\vec{p}_{1}\rangle\,\langle\vec{p}_{1}|e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}|\vec{x};\lambda\rangle=(2\pi\hbar)^{N/2}\,|\vec{p}\,\rangle,$
(61)

so that,

$|\vec{p}\,\rangle=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle.$
(62)

This observation remains valid as well for $\lambda_{ij}=0$ and the sharp
position eigenstates $|\vec{x}\,\rangle$, since

$\int_{(\infty)}d^{N}\vec{x}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x}\,\rangle=\int_{(\infty)}d^{N}\vec{x}\,e^{\frac{i}{\hbar}\vec{p}\cdot\vec{x}}\int_{(\infty)}d^{N}\vec{p}_{1}\,|\vec{p}_{1}\rangle\langle\vec{p}_{1}|\vec{x}\,\rangle=(2\pi\hbar)^{N/2}\,|\vec{p}\,\rangle,$
(63)

where in the last equality the value $\langle\vec{p}_{1}|\vec{x}\,\rangle={\rm
exp}(-i\vec{p}_{1}\cdot\vec{x}/\hbar)/(2\pi\hbar)^{N/2}$ is applied. So that
indeed,

$|\vec{p}\,\rangle=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x}\,\rangle.$
(64)

Finally, let us revisit the issue of the coherent state status of the states
$|\vec{x};\lambda\rangle$. First consider the momentum twisted coordinates in
the following form,

$\hat{a}_{i}=\hat{x}_{i}+\frac{i}{2\hbar}\lambda_{ij}\,\hat{p}_{j},\qquad\hat{a}^{\dagger}_{i}=\hat{x}_{i}-\frac{i}{2\hbar}\lambda_{ij}\,\hat{p}_{j},$
(65)

to which one may associate the complex valued phase space quantities

$z_{i}=x_{i}+\frac{i}{2\hbar}\,\lambda_{ij}\,p_{j},\qquad
z^{*}_{i}=x_{i}-\frac{i}{2\hbar}\,\lambda_{ij}\,p_{j}.$ (66)

A direct evaluation then finds for their algebra of commutation relations,

$\left[\hat{a}_{i},\hat{a}_{j}\right]=0,\qquad\left[\hat{a}_{i},\hat{a}^{\dagger}_{j}\right]=\lambda_{ij}\,\mathbb{I},\qquad\left[\hat{a}^{\dagger}_{i},\hat{a}^{\dagger}_{j}\right]=0.$
(67)

Since $\lambda_{ij}$ is real, symmetric and definite positive, these operators
$(\hat{a}_{i},\hat{a}^{\dagger}_{i})$ thus in fact span a $N$-dimensional Fock
algebra of which the Fock vacuum $|\Omega_{\lambda}\rangle$ is such that

$\hat{a}_{i}\,|\Omega_{\lambda}\rangle=0,$ (68)

with a non standard choice of normalisation adapted to the present context,
namely,

$\langle\Omega_{\lambda}|\Omega_{\lambda}\rangle=W_{2\lambda}(\vec{0}\,)=\frac{1}{\left((2\pi)^{N}{\rm
det}\,\lambda_{ij}\right)^{1/2}}.$ (69)

It follows that the momentum wave function representation of the Fock vacuum,
together with a trivial choice of phase factor, is given as

$\langle\vec{p}\,|\Omega_{\lambda}\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}.$
(70)

Consequently,

$e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\Omega_{\lambda}\rangle=\int_{(\infty)}d^{N}\vec{p}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\vec{p}\,\rangle\,\langle\vec{p}\,|\Omega_{\lambda}\rangle=|\vec{x};\lambda\rangle,$
(71)

thus proving that the localised and normalised state $|\vec{x};\lambda\rangle$
is simply the Fock vacuum $|\Omega_{\lambda}\rangle$ displaced in
configuration space by the translation vector $\vec{x}$, with in particular
the identification

$|\Omega_{\lambda}\rangle=|\vec{x}=\vec{0};\lambda\rangle.$ (72)

The states $|\vec{x};\lambda\rangle$ are thus indeed a particular class of
canonical quantum coherent states for the Fock algebra
$(\hat{a}_{i},\hat{a}^{\dagger}_{i})$, but specifically associated to
configuration space alone and to the choice of symmetric definite positive
matrix $\lambda_{ij}$. Indeed given the complex parameters $(z_{i},z^{*}_{i})$
defined in (66), the complete set of canonical quantum phase space coherent
states associated to that Fock algebra are constructed as

$|z_{i};\Omega_{\lambda}\rangle\equiv
e^{z_{i}(\lambda^{-1})_{ij}\hat{a}^{\dagger}_{j}-z^{*}_{i}(\lambda^{-1})_{ij}\hat{a}_{j}}\,|\Omega_{\lambda}\rangle=e^{-\frac{i}{\hbar}\left(x_{i}\hat{p}_{i}-p_{i}\hat{x}_{i}\right)}\,|\Omega_{\lambda}\rangle\equiv|(x_{i},p_{i});\Omega_{\lambda}\rangle,$
(73)

with thus in particular,

$|\vec{x};\lambda\rangle=|z_{i}=x_{i};\Omega_{\lambda}\rangle=|(x_{i},p_{i}=0);\Omega_{\lambda}\rangle=e^{-\frac{i}{\hbar}x_{i}\hat{p}_{i}}\,|\Omega_{\lambda}\rangle,$
(74)

namely with $p_{i}=0$.

###  2.8 The completeness relation of localised states and the
$*_{\lambda}$-representation

Even though the localised states $|\vec{x};\lambda\rangle$ cannot provide a
basis in quantum Hilbert space for having nonvanishing overlaps among
themselves and thus not being linearly independent, they do generate the
entire quantum Hilbert space and are therefore overcomplete. Indeed, while one
has the identity

$\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\lambda\rangle\,\langle\vec{x};\lambda|=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,e^{-\frac{\lambda_{ij}}{2\hbar^{2}}p_{i}p_{j}}\,\langle\vec{p}\,|=e^{-\frac{\lambda_{ij}}{2\hbar^{2}}\,\hat{p}_{i}\hat{p}_{j}},$
(75)

which is not of much relevance as such except for the charateristic property
of the states $|\vec{x};\lambda\rangle$ that it represents (namely a
suppression of their components at large momenta), given the
$*_{\lambda}$-product defined in (31) one has the following actual
completeness relation this time in terms of the $*_{\lambda}$-product,

$\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\lambda\rangle*_{\lambda}\,\langle\vec{x};\lambda|=\mathbb{I}.$
(76)

Both these identifies may readily be established by computing the matrix
elements of the operator on the l.h.s. of each expression whether in the
$|\vec{x}\,\rangle$ or in the $|\vec{p}\,\rangle$ basis, to check that the
considered operator indeed provides yet another integral representation of the
operator on the r.h.s., in particular in the second case the unit operator for
the quantum Hilbert space representation of the Heisenberg algebra generated
by the operators $\mathbb{I}$, $\hat{x}_{i}$ and $\hat{p}_{i}$
($i=1,2,\cdots,N$).

Consequently, besides the configuration and momentum space wave functions
representations of any abstract quantum state $|\varphi\rangle$, namely
$\varphi(\vec{x}\,)=\langle\vec{x}\,|\varphi\rangle$ and
$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$, respectively, a
third type of overcomplete wave function representation in $\vec{x}$ -space
arises whatever the values of the spatial scale parameters $\lambda_{ij}\neq
0$, namely the complex valued function of $\vec{x}$ defined by
$\langle\vec{x};\lambda|\varphi\rangle\equiv\varphi_{\lambda}(\vec{x}\,)$. Let
us call it the $*_{\lambda}$-representation (in configuration space333Note
that this $*_{\lambda}$-representation is simply the ordinary canonical
coherent state wave function representation of the Heisenberg algebra for the
choice of coefficients $\lambda_{ij}$, restricted however to configuration
space as is warranted by the completeness relation (76) that applies thanks to
the existence of the $*_{\lambda}$-product and its specific properties.) of
the abstract quantum Hilbert space that defines a representation of the
Heisenberg algebra $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$.

More explicitly, in terms of the $*_{\lambda}$-product any abstract quantum
state $|\varphi\rangle$ is decomposed as a linear combination of the localised
states $|\vec{x};\lambda\rangle$ according to the relation,

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\lambda\rangle*_{\lambda}\varphi_{\lambda}(\vec{x}\,),\qquad\varphi_{\lambda}(\vec{x}\,)\equiv\langle\vec{x};\lambda|\varphi\rangle.$
(77)

In turn the wave function $\varphi_{\lambda}(\vec{x}\,)$ possesses an integral
representation in terms either of the configuration space or of the momentum
wave function of the state $|\varphi\rangle$,

$\varphi_{\lambda}(\vec{x}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}\,\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}d^{N}\vec{x}\,^{\prime}\,W_{\lambda}(\vec{x}-\vec{x}\,^{\prime})\,\varphi(\vec{x}\,^{\prime}).$
(78)

Inverse relations are also available for $\varphi(\vec{x}\,)$ and
$\tilde{\varphi}(\vec{p}\,)$ in terms of $\varphi_{\lambda}(\vec{x}\,)$, but
involving now the $*_{\lambda}$-product, in the form of,

$\varphi(\vec{x}_{1})=\int_{(\infty)}d^{N}\vec{x}\
W_{\lambda}(\vec{x}-\vec{x}_{1})*_{\lambda}\varphi_{\lambda}(\vec{x}\,),$ (79)

as well as,

$\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}*_{\lambda}\varphi_{\lambda}(\vec{x}\,).$
(80)

In contradistinction to the operators $|\vec{x}\,\rangle\langle\vec{x}\,|$
which are projection operators (more correctly, a projection operator density)
acting on quantum Hilbert space, the operators
$|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|$ are not such a
projection operator density. Yet since they are the operator density of the
completeness relation integrated over configuration space, they arise
continuously when working in the $*_{\lambda}$-representation, in particular
through their commutators with the position and momentum operators
$\hat{x}_{i}$ and $\hat{p}_{i}$. Thus one may also establish the following
fundamental commutation relations, beginning with the operators $\hat{x}_{i}$,

$\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{x}_{i}\,\Big{]}=0,$
(81)

next with the operators $\hat{p}_{i}$,

$\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{p}_{i}\,\Big{]}=-i\hbar\frac{\partial}{\partial
x_{i}}\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,\Big{]},$
(82)

and with the operators $\hat{p}_{i}\hat{p}_{j}$,

$\displaystyle\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{p}_{i}\hat{p}_{j}\,\Big{]}$
$\displaystyle=$ $\displaystyle\ \ \ \ \left(-i\hbar\frac{\partial}{\partial
x_{j}}\right)\left(\,|\vec{x};\lambda\rangle*_{\lambda}\left(-i\hbar\frac{\partial}{\partial
x_{i}}\langle\vec{x};\lambda|\right)\,\right)$ (83)
$\displaystyle\,-\,\left(-i\hbar\frac{\partial}{\partial
x_{i}}\right)\left(\,\left(-i\hbar\frac{\partial}{\partial
x_{j}}|\vec{x};\lambda\rangle\right)*_{\lambda}\langle\vec{x};\lambda|\right),$

and finally with the operator $\hat{\vec{p}}\,^{2}$,

$\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{\vec{p}}\,^{2}\,\Big{]}=-i\hbar\vec{\nabla}_{\vec{x}}\cdot\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\left(-i\hbar\vec{\nabla}_{\vec{x}}\langle\vec{x};\lambda|\right)\,-\,\left(-i\hbar\vec{\nabla}_{\vec{x}}|\vec{x};\lambda\rangle\right)*_{\lambda}\langle\vec{x};\lambda|\,\Big{]}.$
(84)

###  2.9 Quantum probability conservation in the $*_{\lambda}$-representation

The abstract Schrödinger equation of ordinary quantum dynamics in the
Schrödinger picture for a single nonrelativistic particle reads,

$i\hbar\frac{d}{dt}|\psi,t\rangle=\hat{H}\,|\psi,t\rangle=\left[\frac{1}{2\mu}\hat{\vec{p}}\,^{2}+\hat{V}(\hat{\vec{x}}\,)\right]\,|\psi,t\rangle.$
(85)

Hence in the $|\vec{x}\,\rangle$-representation one has the ordinary
Schrödinger wave equation for
$\psi(\vec{x},t)\equiv\langle\vec{x}\,|\psi,t\rangle$,

$i\hbar\frac{\partial}{\partial
t}\psi(\vec{x},t)=\left[-\frac{\hbar^{2}}{2\mu}\vec{\nabla}^{2}_{\vec{x}}+V(\vec{x}\,)\right]\,\psi(\vec{x},t).$
(86)

Since quantum evolution is unitary (for $\hat{H}$ self-adjoint, or at least
hermitian), the normalisation of any initial state is constant under time
evolution, reflecting quantum probability conservation. Thus for an initial
state normalised to unity, one has,

$1=\langle\psi,t|\psi,t\rangle=\int_{(\infty)}d^{N}\vec{x}\,\psi^{*}(\vec{x},t)\,\psi(\vec{x},t)=\int_{(\infty)}d^{N}\vec{x}\,\rho(\vec{x},t),\qquad\rho(\vec{x},t)\equiv\psi^{*}(\vec{x},t)\,\psi(\vec{x},t),$
(87)

a property which indicates that the density $\rho(\vec{x},t)$ represents a
probability density. As is well known, this probability density obeys the
following continuity equation for quantum probability conservation,

$\frac{\partial}{\partial
t}\rho+\vec{\nabla}_{\vec{x}}\cdot\vec{J}=0,\qquad\vec{J}=\frac{\hbar}{2i\mu}\left(\psi^{*}\,(\vec{\nabla}_{\vec{x}}\,\psi)\,-\,(\vec{\nabla}_{\vec{x}}\,\psi^{*})\,\psi\right),$
(88)

as may easily be established based on the above Schrödinger wave equation,
while $\vec{J}(\vec{x},t)$ thus represents the probability current density.

Let us develop the same considerations now in the
$*_{\lambda}$-representation, in which case the Schrödinger equation reads as
follows for the wave function
$\psi_{\lambda}(\vec{x},t)\equiv\langle\vec{x};\lambda|\psi,t\rangle$,

$i\hbar\frac{\partial}{\partial
t}\psi_{\lambda}(\vec{x},t)=\langle\vec{x};\lambda|\left[\frac{1}{2\mu}\hat{\vec{p}}\,^{2}+\hat{V}(\hat{\vec{x}}\,)\right]|\psi,t\rangle.$
(89)

Note how the r.h.s. of this equation may be expressed in terms of the wave
function $\psi_{\lambda}(\vec{x},t)$, a convolution integral, the Weierstrass
kernel $W_{2\lambda}(\vec{x}\,)$, and the $*_{\lambda}$-product. First insert
to the left of $|\psi,t\rangle$ the completeness relation for the unit
operator in terms of the $*_{\lambda}$-product and the states
$|\vec{x};\lambda\rangle$. And then use the matrix elements in (45) and (50)
given in terms of the Weierstrass kernel
$W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2})$.

Once again since the normalisation of any quantum state is conserved under
quantum time evolution, when the initial quantum state $|\psi,t\rangle$ is
normalised to unity one has for $\psi_{\lambda}(\vec{x},t)$,

$1=\langle\psi,t|\psi,t\rangle=\int_{(\infty)}d^{N}\vec{x}\,\psi^{*}_{\lambda}(\vec{x},t)*_{\lambda}\psi_{\lambda}(\vec{x},t)=\int_{(\infty)}d^{N}\vec{x}\,\rho_{\lambda}(\vec{x},t),$
(90)

a property which indicates that in the $*_{\lambda}$-representation the
probability density in configuration space, $\rho_{\lambda}(\vec{x},t)$, is
given this time by the $*_{\lambda}$-product of the wave function
$\psi_{\lambda}(\vec{x},t)$ with its complex conjugate,

$\rho_{\lambda}(\vec{x},t)\equiv\langle\psi,t|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\psi,t\rangle=\psi^{*}_{\lambda}(\vec{x},t)*_{\lambda}\psi_{\lambda}(\vec{x},t).$
(91)

Using now the Schrödinger wave equation in the $*_{\lambda}$-representation
(89), it readily follows that the time evolution of this density is governed
by the equation,

$\frac{\partial}{\partial
t}\rho_{\lambda}(\vec{x},t)=-\frac{i}{\hbar}\langle\psi,t|\,\left[\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{H}\,\right]\,|\psi,t\rangle.$
(92)

The detailed evaluation of the commutator
$\left[|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|,\hat{H}\right]$
directly derives from (81-84). This calculation leads to the following
continuity equation for probability conservation in the
$*_{\lambda}$-representation,

$\frac{\partial}{\partial
t}\rho_{\lambda}(\vec{x},t)+\vec{\nabla}_{\vec{x}}\cdot\vec{J}_{\lambda}(\vec{x},t)=0,$
(93)

with the following probability current density $\vec{J}_{\lambda}(\vec{x},t)$,
itself also given in terms of the $*_{\lambda}$-product of the wave function
and its configuration space gradient in the $*_{\lambda}$-representation,

$\vec{J}_{\lambda}(\vec{x},t)=\frac{\hbar}{2i\mu}\left(\psi^{*}_{\lambda}(\vec{x},t)*_{\lambda}(\vec{\nabla}_{\vec{x}}\,\psi_{\lambda}(\vec{x},t))\,-\,(\vec{\nabla}_{\vec{x}}\,\psi^{*}_{\lambda}(\vec{x},t))*_{\lambda}\psi_{\lambda}(\vec{x},t)\right).$
(94)

###  2.10 An enveloping algebra representation in configuration space

With this last subsection dedicated to ordinary quantum mechanics, one last
perspective complementary to the usual discussion of representations of the
Heisenberg algebra is to be addressed. Even though this perspective does not
actually provide new useful insight into that issue in the case of commuting
configuration space coordinate operators, this discussion has its exact
parallel in the context of the noncommutative configuration spaces to be
considered hereafter where it does indeed provide new powerful and useful
insight.

Given an abstract quantum Hilbert space representation of the Heisenberg
algebra, quantum states of the physical system are represented by abstract
states in Hilbert space, $|\varphi\rangle$. By identifying a basis or an
overcomplete basis in Hilbert space, as are the states $|\vec{x}\,\rangle$,
$|\vec{p}\,\rangle$ or $|\vec{x};\lambda\rangle$, the abstract states are
represented by complex quantities that are functions of the discrete or
continuous labels that distinguish the basis or overcomplete basis vectors,
viz, the wave functions $\varphi(\vec{x}\,)$, $\tilde{\varphi}(\vec{p}\,)$ or
$\varphi_{\lambda}(\vec{x}\,)$ considered above.

The construction to be described presently provides a representation of
physical quantum states not in terms of vectors in the abstract Hilbert space,
i.e., the abstract states $|\varphi\rangle$, but rather in terms of operators
that belong to the enveloping algebra of the configuration space coordinate
operators $\hat{x}_{i}$ ($i=1,2,\cdots,N)$, namely all the composite operators
that may be built out of the $\hat{x}_{i}$’s, i.e.,
$\hat{\varphi}(\hat{\vec{x}}\,)$. To any given abstract state
$|\varphi\rangle$ there corresponds a unique element
$\hat{\varphi}(\hat{\vec{x}}\,)$ of that enveloping algebra, and vice-versa,
This correspondence defines a one-to-one map between states in the abstract
quantum Hilbert space and operators in the enveloping algebra of the
$\hat{x}_{i}$’s that act on that same quantum Hilbert space which spans all
possible abstract quantum physical states of the considered physical system.

The construction relies on the existence of the momentum eigenbasis
$|\vec{p}\,\rangle$ of the commuting conjugate momentum operators
$\hat{p}_{i}$ ($i=1,2\cdots,N$). When the position operators $\hat{x}_{i}$
($i=1,2,\cdots,N$) all commute, as is the case in this subsection, the
construction also goes through by using the position eigenbasis
$|\vec{x}\,\rangle$. However, when configuration space is noncommutative only
the conjugate momentum path remains available with commuting momentum
operators $\hat{p}_{i}$, as is to be discussed when considering noncommutative
configuration spaces hereafter. This is the reason why the conjugate momentum
path is taken here as well.

The existence of such a one-to-one map follows from the representations of
momentum eigenstates expressed in (62) and (64),

$|\vec{p}\,\rangle=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle,\qquad|\vec{p}\,\rangle=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x}\,\rangle.$
(95)

Note that in particular for $\vec{p}=\vec{0}$ these relations imply,

$\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\lambda\rangle=(2\pi\hbar)^{N/2}\,|\vec{p}=\vec{0}\rangle,\qquad\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle=(2\pi\hbar)^{N/2}\,|\vec{p}=\vec{0}\rangle.$
(96)

Hence any given abstract quantum state $|\varphi\rangle$ in Hilbert space may
be represented according to, with
$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$,

$\displaystyle|\varphi\rangle$ $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}d^{N}\vec{p}\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle\,\tilde{\varphi}(\vec{p}\,)$
(97) $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{x}\left(\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\tilde{\varphi}(\vec{p}\,)\right)\,|\vec{x};\lambda\rangle=\int_{(\infty)}d^{N}\vec{x}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x};\lambda\rangle,$

or equivalently, through a series of identical manipulations in the case that
one uses the above second representation of $|\vec{p}\,\rangle$ with
$\lambda_{ij}=0$,

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{x}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x}\,\rangle,$
(98)

where in each case the operator $\hat{\varphi}(\hat{\vec{x}}\,)$ is defined by

$\hat{\varphi}(\hat{\vec{x}}\,)\equiv\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\hat{\vec{x}}\cdot\vec{p}}\,\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\langle\vec{p}\,|\varphi\rangle.$
(99)

In other words, any abstract quantum state $|\varphi\rangle$ defines a unique
operator $\hat{\varphi}(\hat{\vec{x}}\,)$ in the enveloping algebra of the
coordinate operators $\hat{x}_{i}$ through the latter correspondence. And
conversely, any operator $\hat{\varphi}(\hat{\vec{x}}\,)$ in that enveloping
algebra defines a unique abstract quantum state $|\varphi\rangle$ through

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{x}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x};\lambda\rangle=\int_{(\infty)}d^{N}\vec{x}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x}\,\rangle=(2\pi\hbar)^{N/2}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{p}=\vec{0}\,\rangle.$
(100)

Note that when the coordinate operators all commute among themselves the
position eigenbasis $|\vec{x}\,\rangle$ exists, as is the case in the present
subsection, so that one has right away for any such operator in the enveloping
algebra,

$\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x}\,\rangle=\varphi(\vec{x}\,)|\vec{x}\,\rangle=|\vec{x}\,\rangle\,\langle\vec{x}\,|\varphi\rangle,$
(101)

hence leading once again to the result that,

$\int_{(\infty)}d^{N}\vec{x}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x}\,\rangle=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\,\langle\vec{x}\,|\varphi\rangle=|\varphi\rangle.$
(102)

There also exists the following alternative representation of the operator
$\hat{\varphi}(\hat{\vec{x}}\,)$ associated to the quantum state
$|\varphi\rangle$, valid again when the coordinates $\hat{x}_{i}$
($i=1,2,\cdots,N$) are all commuting among themselves,

$\hat{\varphi}(\hat{\vec{x}}\,)=\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,|\vec{x}_{1}\rangle\langle\vec{x}_{1}|\hat{\varphi}(\hat{\vec{x}}\,)|\vec{x}_{2}\rangle\langle\vec{x}_{2}|=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x}\,\rangle\,\varphi(\vec{x}\,)\,\langle\vec{x}\,|.$
(103)

Under such a circumstance this representation of
$\hat{\varphi}(\hat{\vec{x}}\,)$ could be a most natural alternative
definition of it as well.

The enveloping algebra representation of quantum states in terms of the
operators $\hat{\varphi}(\hat{\vec{x}}\,)$ is also equipped with a
sesquilinear positive definite inner product as is required of a quantum
mechanical representation of the abstract Heisenberg algebra. This inner
product over the enveloping algebra of the position operators $\hat{x}_{i}$ is
simply identified from the abstract inner product
$\langle\varphi_{1}|\varphi_{2}\rangle$ of any two abstract quantum states
$|\varphi_{1}\rangle$ and $|\varphi_{2}\rangle$, using the above one-to-one
map between $|\varphi\rangle$ and $\hat{\varphi}(\hat{\vec{x}}\,)$ involving
the states $|\vec{x};\lambda\rangle$, namely,

$\langle\varphi_{1}|\varphi_{2}\rangle=\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1};\lambda|\hat{\varphi}^{\dagger}_{1}(\hat{\vec{x}}\,)\,\hat{\varphi}_{2}(\hat{\vec{x}}\,)|\vec{x}_{2};\lambda\rangle.$
(104)

It is obvious that the r.h.s. of this relation defines a sesquilinear inner
product over the enveloping algebra, which is furthermore positive definite
since

$\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1};\lambda|\,\hat{\varphi}^{\dagger}(\hat{\vec{x}}\,)\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x}_{2};\lambda\rangle=\langle\varphi|\varphi\rangle\geq
0,$ (105)

for any operator $\hat{\varphi}(\hat{\vec{x}}\,)$ and the corresponding
abstract quantum state $|\varphi\rangle$.

By considering rather the one-to-one map (98) between $|\varphi\rangle$ and
$\hat{\varphi}(\hat{\vec{x}}\,)$ involving this time the sharp position
eigenstates $|\vec{x}\,\rangle$, one realises that the relation (104) extends
to the localised and normalisable states $|\vec{x};\lambda\rangle$ the usual
definition of the inner product for the abstract Hilbert space representation,
since,

$\displaystyle\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1}|\hat{\varphi}^{\dagger}_{1}(\hat{\vec{x}}\,)\,\hat{\varphi}_{2}(\hat{\vec{x}}\,)|\vec{x}_{2}\rangle$
$\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\varphi^{*}_{1}(\vec{x}_{1})\,\varphi_{2}(\vec{x}_{2})\,\langle\vec{x}_{1}|\vec{x}_{2}\rangle$
(106) $\displaystyle=$
$\displaystyle\int_{(\infty)}d^{N}\vec{x}\,\varphi^{*}_{1}(\vec{x}\,)\,\varphi_{2}(\vec{x}\,)=\langle\varphi_{1}|\varphi_{2}\rangle.$

Now that the one-to-one map between the abstract quantum Hilbert space and the
enveloping algebra of the coordinate operators has been identified, one may
enquire how the action on that Hilbert space of the phase space operators
$(\hat{x}_{i},\hat{p}_{i})$ ($i=1,2,\cdots,N$) is represented as linear
operators acting on that enveloping algebra, to be denoted
$(\hat{X}_{i},\hat{P}_{i}$) ($i=1,2,\cdots,N$), respectively. Working once
again in the momentum eigenbasis the action of the operators
$(\hat{x}_{i},\hat{p}_{i})$ on Hilbert space is represented as, respectively,

$\langle\vec{p}\,|\hat{x}_{i}|\varphi\rangle=i\hbar\frac{\partial}{\partial
p_{i}}\,\tilde{\varphi}(\vec{p}\,),\qquad\langle\vec{p}\,|\hat{p}_{i}|\varphi\rangle=p_{i}\,\tilde{\varphi}(\vec{p}\,).$
(107)

Hence under the map to the enveloping algebra, these operators are now
realised as,

$\hat{X}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\hat{\vec{x}}\cdot\vec{p}}\,i\hbar\frac{\partial}{\partial
p_{i}}\tilde{\varphi}(\vec{p}\,)=\hat{x}_{i}\,\hat{\varphi}(\hat{\vec{x}}\,),$
(108)
$\hat{P}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\hat{\vec{x}}\cdot\vec{p}}p_{i}\tilde{\varphi}(\vec{p}\,)=-i\hbar\frac{\partial}{\partial\hat{x}_{i}}\,\hat{\varphi}(\hat{\vec{x}}\,).$
(109)

Note that since the coordinate operators $\hat{x}_{i}$ all commute with one
another, whether the multiplication by $\hat{x}_{i}$ or the derivative
relative to $\hat{x}_{i}$ is taken from the left or from the right is
irrelevant in the present situation. No ordering issue arises. In particular,
one thus has indeed as operators acting on the considered enveloping algebra
the following commutation relations,

$\left[\hat{X}_{i},\hat{X}_{j}\right]=0,\qquad\left[\hat{X}_{i},\hat{P}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I}_{q},\qquad\left[\hat{P}_{i},\hat{P}_{j}\right]=0,$
(110)

which is indeed once again the Heisenberg algebra defining the quantised
dynamical system (with $\mathbb{I}_{q}$ standing for the identity operator on
the enveloping algebra). Note as well that for the inner product over the
enveloping algebra of the coordinate operators $\hat{x}_{i}$ defined in (104),
the operators $\hat{X}_{i}$ and $\hat{P}_{i}$ are indeed at least hermitean if
not formally self-adjoint, as it should be.

In particular given now the enveloping algebra representation of the abstract
quantum system, the Schrödinger equation then reads,

$i\hbar\frac{\partial}{\partial
t}\hat{\psi}(\hat{\vec{x}},t)=\hat{H}\left(\hat{\vec{x}},-i\hbar\vec{\nabla}_{\hat{\vec{x}}}\right)\,\hat{\psi}(\hat{\vec{x}},t),$
(111)

with for instance,

$\hat{H}\left(\hat{\vec{x}},-i\hbar\vec{\nabla}_{\hat{\vec{x}}}\right)=-\frac{\hbar^{2}}{2\mu}\vec{\nabla}_{\hat{\vec{x}}}\cdot\vec{\nabla}_{\hat{\vec{x}}}\,+\,\hat{V}(\hat{\vec{x}}\,).$
(112)

Clearly, whether working in terms of the commuting coordinate operators
$\hat{x}_{i}$ and the enveloping algebra operators
$\hat{\psi}(\hat{\vec{x}},t)$, or in terms of their eigenvalues $x_{i}$ in the
position eigenbasis $|\vec{x}\,\rangle$ and thus the wave functions
$\psi(\vec{x},t)$, is totally irrelevant from the algebraic point of view. The
same differential equations ensue. One is simply dealing once more with
different realisations of a same and unique abstract Hilbert space in which
the same abstract Heisenberg algebra is represented.

When extending hereafter the discussion now to noncommutative configuration
spaces of a certain general class, all the above considerations will find
their rightful place in an entirely parallel analysis, based once again on a
Hamiltonian formulation and its canonical quantisation.

##  3 A Conjugate Magnetic Field in Phase Space

###  3.1 Classical dynamics considerations

Let us now deform the action of the system given in the form of (4) together
with its implicit symplectic structure by adding to it a real non-exact 1-form
in phase space, specifically restricted to the conjugate momentum sector of
phase space coordinates $p_{i}$ in the form of a local vector field
$\bar{A}_{i}(p_{i})$ ($i=1,2,\cdots,N$) and the 1-form
$dp_{i}\bar{A}_{i}(\vec{p}\,)$. One then has a new Hamiltonian first-order
action which reads,

$S_{1}[x_{i},p_{i}]=\int
dt\left(\dot{\vec{x}}\cdot\vec{p}+\dot{\vec{p}}\cdot\vec{\bar{A}}(\vec{p}\,)-H(\vec{x},\vec{p}\,)\right)=\int
dt\left(\dot{x}_{i}p_{i}+\dot{p}_{i}\bar{A}_{i}(p_{i})-H(x_{i},p_{i})\right).$
(113)

The Hamiltonian equations of motion are then deformed into,

$\dot{x}_{i}=\frac{\partial H(\vec{x},\vec{p}\,)}{\partial
p_{i}}+\bar{A}_{ij}(\vec{p}\,)\frac{\partial H(\vec{x},\vec{p}\,)}{\partial
x_{j}},\qquad\dot{p}_{i}=-\frac{\partial H(\vec{x},\vec{p}\,)}{\partial
x_{i}},\qquad i,j=1,2,\cdots,N,$ (114)

where $\bar{A}_{ij}(\vec{p}\,)$ is the nonvanishing exterior derivative or
curl of the vector field $\bar{A}_{i}(\vec{p}\,)$ in momentum space,

$\bar{A}_{ij}(\vec{p}\,)=\frac{\partial\bar{A}_{j}(\vec{p}\,)}{\partial
p_{i}}\,-\,\frac{\partial\bar{A}_{i}(\vec{p}\,)}{\partial p_{j}},\qquad
i,j=1,2,\cdots,N.$ (115)

Consequently, in presence of the non integrable vector field
$\bar{A}_{i}(\vec{p}\,)$ the symplectic structure of phase space is deformed
in such a way that the phase space parametrisation in terms of the variables
$(x_{i},p_{i})$ now possesses the following Poisson brackets,

$\left\\{x_{i},x_{j}\right\\}=\bar{A}_{ij}(\vec{p}\,),\qquad\left\\{x_{i},p_{j}\right\\}=\delta_{ij},\qquad\left\\{p_{i},p_{j}\right\\}=0,\qquad
i,j=1,2,\cdots,N.$ (116)

Note how the Poisson brackets of the configuration space coordinates are no
longer vanishing. For the canonically quantised system, its configuration
space coordinate operators have become noncommutative, with
$\left[\hat{x}_{i},\hat{x}_{j}\right]=i\hbar\,\hat{\bar{A}}_{ij}(\hat{\vec{p}}\,)$.

Clearly, by analogy with what happens for genuine physical magnetic fields and
their interactions in configuration space, the vector field
$\bar{A}_{i}(\vec{p}\,)$ may be given the interpretation of a conjugate
magnetic vector potential with its conjugate magnetic field
$\bar{A}_{ij}(\vec{p}\,)$ present in the conjugate momentum sector of phase
space. Thus, any nontrivial conjugate magnetic field in conjugate momentum
space induces noncommutativity in configuration space.

This interpretation, which resonates with a physicist’s intuition, is
corroborated as well by the following fact. Any dynamical action is defined up
to a total derivative in time, without affecting its equations of motion. By
redefining the vector field through what is in effect a local gauge
transformation in conjugate momentum space,

$\bar{A}_{i}(\vec{p}\,)\longrightarrow\bar{A}_{i}(\vec{p}\,)+\frac{\partial\chi(\vec{p}\,)}{\partial
p_{i}},$ (117)

where $\chi(\vec{p}\,)$ is an arbitrary function over conjugate momentum
space, the action (113) sees its value modified by a total time derivative
contribution, thereby not affecting the equations of motion. Incidentally,
this observation also explains why the equations of motion must involve the
field strength $\bar{A}_{ij}(\vec{p}\,)$ rather than directly the vector
potential $\bar{A}_{i}(\vec{p}\,)$, namely the conjugate magnetic field only.
Indeed, this magnetic field is the local gauge invariant field that may be
constructed out of the vector potential $\bar{A}_{i}(\vec{p}\,)$.

Even though such gauge transformations do not modify the classical equations
of motion, they may affect the identification of canonically conjugate phase
space coordinates, with still further more subtle consequences for the
canonically quantised system in terms of induced passive unitary
transformations of the unitary representation of the Heisenberg algebra used
to identify the actual quantum physical system [22]. Since no such
transformation is to be contemplated in this work, we shall not have to
concern ourselves with these subtle issues here, even though they could be of
interest in their own right.

Finally, the following phase space reparametrisation in terms of the new set
of variables $(y_{i},p_{i})$ ($i=1,2,\cdots,N$) defined as,

$y_{i}=x_{i}-\bar{A}_{i}(\vec{p}\,),\qquad p_{i}=p_{i},\qquad i=1,2,\cdots,N,$
(118)

consists in a Darboux transformation to canonically conjugate phase space
coordinates, with the then canonical Poisson brackets,

$\left\\{y_{i},y_{j}\right\\}=0,\qquad\left\\{y_{i},p_{j}\right\\}=\delta_{ij},\qquad\left\\{p_{i},p_{j}\right\\}=0,\qquad
i,j=1,2,\cdots,N.$ (119)

Note that in contradistinction to $(x_{i},p_{i})$ that are invariant under the
above gauge transformations of $A_{i}(\vec{p}\,)$, the coordinates $y_{i}$
canonically conjugate to $p_{i}$ are not gauge invariant under such
transformations [22].

###  3.2 Quantum dynamics considerations

The canonically quantised system is thus characterised by the (equal time)
commutation relations,

$\left[\hat{x}_{i},\hat{x}_{j}\right]=i\hbar\,\hat{\bar{A}}_{ij}(\hat{\vec{p}}\,),\qquad\left[\hat{x}_{i},\hat{p}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I},\qquad\left[\hat{p}_{i},\hat{p}_{j}\right]=0,\qquad
i,j=1,2,\cdots,N,$ (120)

or equivalently,

$\left[\hat{y}_{i},\hat{y}_{j}\right]=0,\qquad\left[\hat{y}_{i},\hat{p}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I},\qquad\left[\hat{p}_{i},\hat{p}_{j}\right]=0,\qquad
i,j=1,2,\cdots,N,$ (121)

where

$\hat{x}_{i}=\hat{y}_{i}+\hat{\bar{A}}_{i}(\hat{\vec{p}}\,),\qquad
i=1,2,\cdots,N.$ (122)

Hence the operators $(\hat{y}_{i},\hat{p}_{i},\mathbb{I})$ ($i=1,2,\cdots,N$)
generate a Heisenberg algebra, of which an abstract Hilbert space
representation provides an abstract Hilbert space representation of the
algebra of the original operators $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$
($i=1,2,\cdots,N$), which include the operators $\hat{x}_{i}$ that are
identified with the cartesian coordinates of the noncommutative configuration
space. Note that all the operators $\hat{x}_{i}$, $\hat{p}_{i}$ and
$\hat{y}_{i}$ need to be self-adjoint (or at least hermitian) as well.

Since all conjugate momentum operators $\hat{p}_{i}$ commute among themselves,
one may choose as basis of Hilbert space all sharp momentum eigenstates
$|\vec{p}\,\rangle$, with precisely all the same properties and conventions as
discussed already in the absence of the vector potential
$\vec{\bar{A}}(\vec{p}\,)$. Hence in particular in that
$|\vec{p}\,\rangle$-representation, we have for the operators $\hat{y}_{i}$
and $\hat{x}_{i}$, given any abstract quantum state $|\varphi\rangle$ and its
momentum space wave function representation
$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$,

$\langle\vec{p}\,|\hat{y}_{i}|\varphi\rangle=i\hbar\frac{\partial}{\partial
p_{i}}\tilde{\varphi}(\vec{p}\,),\qquad\langle\vec{p}\,|\hat{x}_{i}|\varphi\rangle=\left(i\hbar\frac{\partial}{\partial
p_{i}}+\bar{A}_{i}(\vec{p}\,)\right)\,\tilde{\varphi}(\vec{p}\,),\qquad
i=1,2,\cdots,N.$ (123)

In the last of these two expressions of course one recognises the covariant
derivative for the conjugate vector potential $\vec{\bar{A}}(\vec{p}\,)$ in
conjugate momentum space.

However, since the configuration space coordinate operators $\hat{x}_{i}$ do
not commute among themselves, there does not exist a basis of sharp position
eigenstates for the complete set of these operators, that could have been
labelled by all values of $x_{i}\in\mathbb{R}$. Nevertheless, there exists a
basis of eigenstates $|\vec{y}\,\rangle$ for the twisted configuration space
operators $\hat{y}_{i}$, since these operators all commute with one another,
$\hat{y}_{i}|\vec{y}\,\rangle=y_{i}|\vec{y}\,\rangle$. The states
$|\vec{y}\,\rangle$ possess all the same properties as the position
eigenstates $|\vec{x}\,\rangle$ considered above in the commutative case, with
for instance the change of basis matrix elements,

$\langle\vec{p}\,|\vec{y}\,\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{y}\cdot\vec{p}}.$
(124)

The existence of the basis $|\vec{y}\,\rangle$ and different available
representations for the twisted configuration space operators $\hat{y}_{i}$
may sometimes be put to good use in intermediate steps of calculations for a
noncommutative configuration space. For example as an illustration, one has,

$\hat{x}_{i}=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\left(\frac{1}{2}i\hbar\left(\frac{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}{\partial
p_{i}}-\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
p_{i}}+\bar{A}_{i}(\vec{p}\,)\right)\right)\langle\vec{p}\,|,\qquad
i=1,2,\cdots,N.$ (125)

By analogy with the previous construction of localised but not sharp quantum
states associated to positions $\vec{x}$ in configuration space in the
commutative case, one could contemplate as well now in the present
noncommutative context the study of localised quantum states in configuration
space labelled by all points $\vec{x}$ in configuration space, in the form of,

$\langle\vec{p}\,|\vec{x};B\rangle\equiv\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-B(\vec{p}\,)},\qquad|\vec{x};B\rangle=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\langle\vec{p}\,|\vec{x};B\rangle,$
(126)

where $B(\vec{p}\,)$ is some specifically chosen function of $\vec{p}$ such
that $\lim_{|\vec{p}\,|\rightarrow\infty}B(\vec{p}\,)=+\infty$, so that these
states would be normalisable and with an exponential suppression in their
large momentum components. Possible choices could be once again
$B(\vec{p}\,)=\lambda_{ij}p_{i}p_{j}/(4\hbar^{2})$ with $\lambda_{ij}$ a
symmetric positive definite matrix of real spatial scale factors, or also for
instance $B(\vec{p}\,)=\alpha\vec{\bar{A}}^{2}(\vec{p}\,)/2$ or
$B(\vec{p}\,)=\beta\bar{A}_{ij}(\vec{p}\,)\bar{A}_{ij}(\vec{p}\,)/4$ with
$\alpha,\beta>0$ and of the proper physical dimension, provided of course
these fields grow infinite for infinite $|\vec{p}\,|$.

Some properties valid for the states $|\vec{x};B\rangle$ analogous to those
presented in the commutative case for the states $|\vec{x};\lambda\rangle$ may
then be established here as well, but not to the same degree of completeness
nor generality unless the choice for $B(\vec{p}\,)$ is quadratic in $\vec{p}$
and the conjugate magnetic field $\bar{A}_{ij}(\vec{p}\,)$ is constant.
Indeed, when $\bar{A}_{ij}(\vec{p}\,)$ is independent of $\vec{p}$, the
nonvanishing noncommutativity of the configuration space operators
$\hat{x}_{i}$ is then decoupled from the conjugate momentum operators
$\hat{p}_{i}$, which themselves commute among one another and have constant
Heisenberg-like commutators with the $\hat{x}_{i}$. Rather than try to pursue
a general analysis for an arbitrary choice of vector potential
$\vec{\bar{A}}(\vec{p}\,)$ with its conjugate magnetic field
$\bar{A}_{ij}(\vec{p}\,)$, the remainder of the discussion henceforth focusses
now on the specific case of a constant conjugate magnetic field
$\bar{A}_{ij}(\vec{p}\,)$.

##  4 A Uniform Conjugate Magnetic Field in Phase Space

###  4.1 A canonical quantum representation

From now on, let us thus assume that the conjugate magnetic field
$\bar{A}_{ij}(\vec{p}\,)$ is constant over phase space,
$\bar{A}_{ij}(\vec{p}\,)=\bar{A}_{ij}$ with $\bar{A}_{ji}=-\bar{A}_{ij}$ being
the constant entries of a real antisymmetric matrix. A convenient choice of
gauge for the vector potential $\bar{A}_{i}(\vec{p}\,)$ is then that of the
so-called symmetric gauge with

$\bar{A}_{i}(\vec{p}\,)=-\frac{1}{2}\bar{A}_{ij}\,p_{j}.$ (127)

For ease of notation for the quantised system, let us also introduce the
quantities $A_{i}(\vec{p}\,)$ and $A_{ij}$ in the form of,

$A_{ij}=\hbar\,\bar{A}_{ij},\qquad
A_{i}(\vec{p}\,)=\hbar\,\bar{A}_{i}(\vec{p}\,)=-\frac{1}{2}A_{ij}\,p_{j},\qquad\bar{A}_{i}(\vec{p}\,)=\frac{1}{\hbar}A_{i}(\vec{p}\,)=-\frac{1}{2\hbar}\,A_{ij}\,p_{j}.$
(128)

Consequently the quantised system is now characterised by the commutation
relations

$\left[\hat{x}_{i},\hat{x}_{j}\right]=i\,A_{ij}\,\mathbb{I},\qquad\left[\hat{x}_{i},\hat{p}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I},\qquad\left[\hat{p}_{i},\hat{p}_{j}\right]=0,$
(129)

with in particular noncommutative configuration space coordinate operators of
constant commutation relations specified through the real antisymmetric matrix
$A_{ij}$.

The canonical parametrisation of classical phase space in terms of the
variables $(y_{i},p_{i})$ is then provided by the following changes of
variables

$y_{i}=x_{i}+\frac{1}{2\hbar}\,A_{ij}\,p_{j},\qquad
x_{i}=y_{i}-\frac{1}{2\hbar}\,A_{ij}\,p_{j},$ (130)

while for the canonically quantised system one has the following operator
redefinitions

$\hat{y}_{i}=\hat{x}_{i}-\hat{A}_{i}(\hat{\vec{p}}\,)=\hat{x}_{i}+\frac{1}{2\hbar}A_{ij}\,\hat{p}_{j},\qquad\hat{x}_{i}=\hat{y}_{i}-\frac{1}{2\hbar}A_{ij}\,\hat{p}_{j}.$
(131)

A direct evaluation then finds that these operators
$(\hat{y}_{i},\hat{p}_{i},\mathbb{I})$ indeed generate a Heisenberg algebra
for $N$-dimensional Euclidean space,

$\left[\hat{y}_{i},\hat{y}_{j}\right]=0,\qquad\left[\hat{y}_{i},\hat{p}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I},\qquad\left[\hat{p}_{i},\hat{p}_{j}\right]=0.$
(132)

Since all conjugate momenta operators $\hat{p}_{i}$ do commute among
themselves, the $|\vec{p}\,\rangle$ -representation certainly still exists
with its basis of sharp and non-normalisable momentum eigenstates
$|\vec{p}\,\rangle$, and all the same properties as those recalled above in
the commutative configuration space case. Thus abstract quantum states
$|\varphi\rangle$ may be represented in terms of their momentum wave function
$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$, while the
operators $\hat{y}_{i}$, $\hat{x}_{i}$ and $\hat{p}_{i}$ then have the
following differential operator representations,

$\langle\vec{p}\,|\hat{y}_{i}|\varphi\rangle=i\hbar\frac{\partial}{\partial
p_{i}}\,\tilde{\varphi}(\vec{p}\,),\qquad\langle\vec{p}\,|\hat{x}_{i}|\varphi\rangle=\left(i\hbar\frac{\partial}{\partial
p_{i}}\,-\,\frac{1}{2\hbar}A_{ij}\,p_{j}\right)\tilde{\varphi}(\vec{p}\,),\qquad\langle\vec{p}\,|\hat{p}_{i}|\varphi\rangle=p_{i}\,\tilde{\varphi}(\vec{p}\,).$
(133)

###  4.2 Configuration space localised and normalisable quantum states

In the same way as in the previous discussion of the commutative Euclidean
configuration space, let us reconsider the length-squared scale factors
defined by some real positive definite symmetric matrix of entries
$\lambda_{ij}$, and introduce once again the localised and normalisable
quantum states $|\vec{x};\lambda\rangle$ labelled by all possible vector
values for $\vec{x}$ parametrising now the noncommutative configuration space.
These states coincide with those in the commutative case, and are defined
again in the form of,

$\langle\vec{p}\,|\vec{x};\lambda\rangle=\frac{1}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}\,p_{i}p_{j}},\qquad|\vec{x};\lambda\rangle=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,\langle\vec{p}\,|\vec{x};\lambda\rangle.$
(134)

Related to this choice of states, one has again the same definition of the
$W_{\lambda}(\vec{x}\,)$ and $W_{2\lambda}(\vec{x}\,)$ Weierstrass kernels
with all their same properties as discussed previously already, say in the
case of $W_{\lambda}(\vec{x}\,)$,

$W_{\lambda}(\vec{x}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}\,p_{i}p_{j}}=\frac{1}{\left(\pi^{N}{\rm
det}\,\lambda_{ij}\right)^{1/2}}\,e^{-x_{i}(\lambda^{-1})_{ij}x_{j}},$ (135)

and the same definition of a $*_{\lambda}$-product acting on pairs of
functions of configuration space parametrised by $\vec{x}$,

$*_{\lambda}\equiv
e^{\frac{1}{2}\stackrel{{\scriptstyle\leftarrow}}{{\frac{\partial}{\partial
x_{i}}}}\,\lambda_{ij}\,\stackrel{{\scriptstyle\rightarrow}}{{\frac{\partial}{\partial
x_{j}}}}}.$ (136)

Hence for instance one finds again for the finite normalisation of these
states $|\vec{x};\lambda\rangle$,

$\langle\vec{x};\lambda|\vec{x};\lambda\rangle=\frac{1}{\left((2\pi)^{N}{\rm
det}\,\lambda_{ij}\right)^{1/2}}=W_{2\lambda}(\vec{x}=\vec{0}\,).$ (137)

Furthermore, note that like in the commutative case, it is possible to extend
the ensemble of states $|\vec{x};\lambda\rangle=|x_{i};\lambda\rangle$ with
real values for the parameters $x_{i}$, to the larger set of states
$|z_{i};\lambda\rangle$ with $z_{i}\in\mathbb{C}$ defined in the present
noncommutative case exactly as in (24).

More specifically, as a matter of fact it should be remarked that the states
$|\vec{x};\lambda\rangle$ and $|z_{i};\lambda\rangle$ introduced here in the
case of the noncommutative configuration space coincide exactly still with
those introduced above in the case of a commutative configuration space, then
in the absence of the conjugate magnetic field.

In contradistinction however, the expectation values for these states of
operators involving the configuration space coordinate operators $\hat{x}_{i}$
(rather than $\hat{y}_{i}$) take different values now, on account of
additional contributions due to the conjugate magnetic field $A_{ij}$. Thus
ready evaluations find the following results, for $\hat{x}_{i}$,

$\langle\hat{x}_{i}\rangle_{\lambda}(\vec{x}\,)=x_{i},$ (138)

and for $\hat{x}_{i}\,\hat{x}_{j}$,

$\langle\hat{x}_{i}\hat{x}_{j}\rangle_{\lambda}(\vec{x}\,)=x_{i}x_{j}+\frac{1}{4}\lambda_{ij}+\frac{1}{2}iA_{ij}-\frac{1}{4}A_{ik}(\lambda^{-1})_{k\ell}A_{\ell
j}.$ (139)

On the other hand, operators only involving the momentum operators
$\hat{p}_{i}$ retain the same expectation values as in the absence of the
conjugate magnetic field, such as $\hat{p}_{i}$ and
$\hat{p}_{i}\,\hat{p}_{j}$,

$\langle\hat{p}_{i}\rangle_{\lambda}(\vec{x}\,)=0,\qquad\langle\hat{p}_{i}\hat{p}_{j}\rangle_{\lambda}(\vec{x}\,)=\hbar^{2}\,(\lambda^{-1})_{ij}.$
(140)

As a consequence, in the noncommutative case the uncertainties in position and
momentum for the states $|\vec{x};\lambda\rangle$ now read, respectively,

$\Delta^{(\hat{\vec{x}})}_{ij}(\vec{x}\,)\equiv\langle\left(\hat{x}_{i}-\langle\hat{x}_{i}\rangle_{\lambda}\right)\left(\hat{x}_{j}-\langle\hat{x}_{j}\rangle_{\lambda}\right)\rangle_{\lambda}(\vec{x}\,)=\frac{1}{4}\lambda_{ij}+\frac{1}{2}iA_{ij}-\frac{1}{4}A_{ik}(\lambda^{-1})_{k\ell}A_{\ell
j},$ (141)
$\Delta^{(\hat{\vec{p}}\,)}_{ij}(\vec{x}\,)\equiv\langle\left(\hat{p}_{i}-\langle\hat{p}_{i}\rangle_{\lambda}\right)\left(\hat{p}_{j}-\langle\hat{p}_{j}\rangle_{\lambda}\right)\rangle_{\lambda}(\vec{x}\,)=\hbar^{2}\,(\lambda^{-1})_{ij},$
(142)

thus leading to the following uncertainly relations,

$\Delta^{(\hat{\vec{x}})}_{ij}(\vec{x}\,)\,\Delta^{(\hat{\vec{p}}\,)}_{k\ell}(\vec{x}\,)=\frac{1}{4}\hbar^{2}\,(\lambda^{-1})_{k\ell}\left(\lambda_{ij}+2iA_{ij}-A_{im}(\lambda^{-1})_{mn}A_{nj}\right).$
(143)

Hence once again the states $|\vec{x};\lambda\rangle$ are saturating states,
in the character of quantum coherent states.

For completeness, let us also list matrix elements of some operators involving
$\hat{x}_{i}$ and $\hat{p}_{i}$ for two distinct external states belonging to
the class $|\vec{x};\lambda\rangle$, beginning with the overlaps of such
states,

$\langle\vec{x}_{1};\lambda|\vec{x}_{2};\lambda\rangle=W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}).$
(144)

For the operators $\hat{x}_{i}$ and $\hat{x}_{i}\hat{x}_{j}$ one finds,
respectively,

$\langle\vec{x}_{1};\lambda|\hat{x}_{i}|\vec{x}_{2};\lambda\rangle=\left(\frac{1}{2}(x_{1}+x_{2})_{i}-\frac{1}{2}iA_{ij}(\lambda^{-1})_{jk}(x_{1}-x_{2})_{k}\right)\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$
(145)

and,

$\displaystyle\langle\vec{x}_{1};\lambda|\hat{x}_{i}\hat{x}_{j}|\vec{x}_{2};\lambda\rangle=$
(146) $\displaystyle=$
$\displaystyle\left\\{\frac{1}{4}\left[\left(x_{1}+x_{2}\right)_{i}-i\left(A\cdot\lambda^{-1}\right)_{ik}\left(x_{1}-x_{2}\right)_{k}\right]\left[\left(x_{1}+x_{2}\right)_{j}-i\left(A\cdot\lambda^{-1}\right)_{j\ell}\left(x_{1}-x_{2}\right)_{\ell}\right]\right.$
$\displaystyle\left.\ \ \
+\frac{1}{4}\lambda_{ij}+\frac{1}{2}iA_{ij}-\frac{1}{4}\left(A\cdot\lambda^{-1}\cdot
A\right)_{ij}\right\\}\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}).$

While for operators linear and quadratic in $\hat{p}_{i}$,

$\langle\vec{x}_{1};\lambda|\hat{p}_{i}|\vec{x}_{2};\lambda\rangle=-i\hbar\frac{\partial}{\partial
x_{1i}}\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$ (147)
$\langle\vec{x}_{1};\lambda|\hat{p}_{i}\hat{p}_{j}|\vec{x}_{2};\lambda\rangle=\left(-i\hbar\frac{\partial}{\partial
x_{1i}}\right)\left(-i\hbar\frac{\partial}{\partial
x_{1j}}\right)\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}),$ (148)

and finally,

$\langle\vec{x}_{1};\lambda|\hat{\vec{p}}\,^{2}|\vec{x}_{2};\lambda\rangle=-\hbar^{2}\,\vec{\nabla}^{2}_{\vec{x}_{1}}\,W_{2\lambda}(\vec{x}_{1}-\vec{x}_{2}).$
(149)

###  4.3 Displacement operators and canonical quantum coherent states

Since the commutation relations of the momentum operators $\hat{p}_{i}$ with
either the configuration space or momentum operators, $\hat{x}_{i}$ and
$\hat{p}_{i}$, respectively, have not been modified in the noncommutative
configuration space case as compared to the commutative one, the status of the
momentum operators $\hat{p}_{i}$ as generators of displacements in
configuration space through the unitary transformations
$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}$ has not be modified either.
One still has of course,

$e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,\hat{x}_{i}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}=\hat{x}_{i}\,+\,a_{i}\,\mathbb{I},\qquad
e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,\hat{p}_{i}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}=\hat{p}_{i}.$
(150)

When acting with that operator on the states $|\vec{x};\lambda\rangle$ and
$|\vec{p}\,\rangle$ and by inserting then to the left of it the completeness
relation of the unit operator in the $|\vec{p}\,\rangle$-representation, one
readily finds again for the states $|\vec{x};\lambda\rangle$ and
$|\vec{p}\,\rangle$,

$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{x};\lambda\rangle=|\vec{x}+\vec{a};\lambda\rangle,\qquad
e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{p}\,\rangle=e^{-\frac{i}{\hbar}\vec{a}\cdot\vec{p}}\,|\vec{p}\,\rangle.$
(151)

In particular, note the following simple realisation of any state
$|\vec{x};\lambda\rangle$ in terms of the displacement of the state at the
origin of configuration space, namely $|\vec{x}=\vec{0};\lambda\rangle$,

$|\vec{x};\lambda\rangle=e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\vec{x}=\vec{0};\lambda\rangle.$
(152)

We shall come back to this observation from the perspective of quantum
coherent states once again.

In contradistinction, the situation regarding the position operators
$\hat{x}_{i}$ is certainly much different, on account of the now nonvanishing
commutation relations of these operators among themselves determined by the
conjugate magnetic field $A_{ij}$. In other words, which types of
transformation in phase space, and of quantum states do the unitary operators
$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}$ generate in the
noncommutative case?

Using the BCH formula and the relevant commutation relations, it is
straightforward to establish that one has,

$e^{-\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}=\hat{x}_{i}\,-\,\frac{1}{\hbar}A_{ij}\,p_{0j}\,\mathbb{I},\qquad
e^{-\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,\hat{p}_{i}\,e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}=\hat{p}_{i}\,+\,p_{0i}\,\mathbb{I},$
(153)

thus showing that the operators
$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}$ not only generate
displacements in momentum space as they do in the commutative case, but that
because of the noncommutativity of the $\hat{x}_{i}$ operators among
themselves, the same unitary transformations do also generate as well
displacements in configuration space by a quantity dependent both on the
conjugate magnetic field and the transformation parameters $p_{0i}$. In other
words, in the case of configuration space noncommutativity, displacements in
configuration space may be generated by the coordinate operators as well, and
not only by the momentum operators as is usually the case. This specific
feature is to be explored further hereafter.

However, determining now the action of these unitary operators on the quantum
states $|\vec{x};\lambda\rangle$ and $|\vec{p}\,\rangle$ is another matter
altogether. And this even by attempting to use the completeness relation of
the unit operator whether in the $|\vec{p}\,\rangle$ or in the
$|\vec{y}\,\rangle$ eigenbases, that correspond to the canonical phase space
parametrisation in terms of the variables $(y_{i},p_{i})$. Another path is to
be followed.

First, by considering in the $|\vec{p}\,\rangle$-representation the action of
the operators $\hat{x}_{i}$ and $\hat{p}_{i}$ on the states
$|\vec{x};\lambda\rangle$ one finds the representations,

$\hat{x}_{i}\,|\vec{x};\lambda\rangle=x_{i}\,|\vec{x};\lambda\rangle\,-\,\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j}\,|\vec{x};\lambda\rangle,\qquad\hat{p}_{i}\,|\vec{x};\lambda\rangle=i\hbar\frac{\partial}{\partial
x_{i}}\,|\vec{x};\lambda\rangle.$ (154)

Hence one may write for the action of $\hat{x}_{i}$ on the states
$|\vec{x};\lambda\rangle$,

$\hat{x}_{i}\,|\vec{x};\lambda\rangle=\left(x_{i}+\frac{1}{2}\left(\lambda_{ij}-iA_{ij}\right)\frac{\partial}{\partial
x_{j}}\right)\,|\vec{x};\lambda\rangle,$ (155)

which includes an explicit contribution of the conjugate magnetic field
$A_{ij}$. On the other hand, since one also observes that444Note that
$\hat{x}_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j}=\hat{y}_{i}+\frac{i}{2\hbar}\lambda_{ij}\,\hat{p}_{j}$,
to be compared to the Fock algebra operators $(a_{i},a^{\dagger}_{i})$
introduced in (65) in the commutative case.

$\left(\hat{x}_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j}\right)\,|\vec{x};\lambda\rangle=x_{i}\,|\vec{x};\lambda\rangle,$
(156)

it follows that the states $|\vec{x};\lambda\rangle$ are eigenstates with
eigenvalues $x_{i}$ of the momentum twisted coordinate operators now defined
by

$\hat{x}_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j},$
(157)

thus extending in the presence of the conjugate magnetic field $A_{ij}$ a
parallel result already observed for the same states $|\vec{x};\lambda\rangle$
in the case of a commutative configuration space.

Consequently, one finds that

$e^{\frac{i}{\hbar}p_{0i}\left(\hat{x}_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j}\right)}\,|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,|\vec{x};\lambda\rangle.$
(158)

Using once again the BCH formula in the l.h.s. of this identity, at this
intermediary step the action of the unitary operator
$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}$ on the states
$|\vec{x};\lambda\rangle$ may thus be represented in the form of,

$e^{\frac{i}{\hbar}p_{0i}\hat{x}_{i}}\,|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{0i}p_{0j}}\,e^{\frac{1}{2\hbar^{2}}\left(\lambda_{ij}-iA_{ij}\right)p_{0i}\hat{p}_{j}}\,|\vec{x};\lambda\rangle.$
(159)

One first conclusion that may be drawn from this result is the value of the
following matrix element,

$\langle\vec{p}\,|e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}|\vec{x};\lambda\rangle=e^{-\frac{i}{2\hbar^{2}}A_{ij}p_{0i}p_{j}}\,\langle\vec{p}-\vec{p}_{0}|\vec{x};\lambda\rangle.$
(160)

This expression confirms that the unitary operator
$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}$ indeed generates a
displacement in momentum space as already established above in operator form.

Furthermore from the knowledge of the result for the action of
$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}$ on the states
$|\vec{x};\lambda\rangle$, in terms of the states
$|z_{i}\in\mathbb{C};\lambda\rangle$ the result in (159) may also be given in
the form of,

$e^{\frac{i}{\hbar}p_{0i}\hat{x}_{i}}\,|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{0i}p_{0j}}\,\Big{|}z_{i}=x_{i}-\frac{1}{2\hbar}A_{ij}\,p_{0j}+\frac{i}{2\hbar}\lambda_{ij}\,p_{0j};\lambda\Big{\rangle}.$
(161)

Or else in the following form still in terms of the states
$|\vec{x};\lambda\rangle$,

$e^{\frac{i}{\hbar}p_{0i}\hat{x}_{i}}\,|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{0i}p_{0j}}\,e^{\frac{\lambda_{ij}}{2\hbar^{2}}p_{0i}\hat{p}_{j}}\,\Big{|}x_{i}-\frac{1}{2\hbar}A_{ij}\,p_{0j};\lambda\Big{\rangle},$
(162)

thus showing that $e^{\frac{i}{\hbar}p_{0i}\hat{x}_{i}}$ indeed generates as
well displacements in configuration space. Alternatively by using the fact
that
$|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{x}+\vec{a};\lambda\rangle$
and that the momentum operators $\hat{p}_{i}$ all commute with one another,
one also establishes that

$e^{\frac{i}{\hbar}p_{0i}\hat{x}_{i}}\,|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\vec{x}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{0i}p_{0j}}\,e^{\frac{1}{2\hbar^{2}}\left(\lambda_{ij}+iA_{ij}\right)p_{0i}\hat{p}_{j}}\,\Big{|}x_{i}-\frac{1}{\hbar}\,A_{ij}\,p_{0j};\lambda\Big{\rangle},$
(163)

where the choice of $|\vec{x};\lambda\rangle$ state on the r.h.s. of this
identity is motivated by the result in (153) for the transformation under the
operator $e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}$ of the coordinate
operators $\hat{x}_{i}$.

By inserting to the left of (159) the completeness relation for the unit
operator in the $|\vec{p}\,\rangle$ eigenbasis, another conclusion that may be
drawn from that relation is the following representation of momentum
eigenstates in terms of the localised and normalisable states
$|\vec{x};\lambda\rangle$,

$\int_{(\infty)}d^{N}\vec{x}\,e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle=\left(2\pi\hbar\right)^{N/2}\,|\vec{p}_{0}\rangle,\qquad|\vec{p}\,\rangle=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle.$
(164)

Hence finally based on this result as well as the identity (159), one finds
for the action of the operators
$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot{\hat{\vec{x}}}}$ on the momentum
eigenstates,

$e^{\frac{i}{\hbar}\vec{p}_{0}\cdot\hat{\vec{x}}}\,|\vec{p}\,\rangle=e^{-\frac{i}{2\hbar^{2}}A_{ij}p_{0i}p_{j}}\,|\vec{p}+\vec{p}_{0}\rangle.$
(165)

It is to be pointed out that this result is confirmed by inserting to the left
of the l.h.s. of this identity the completeness relation for the unit operator
in the $|\vec{y}\,\rangle$ eigenbasis of the coordinate twisted operators
$\hat{y}_{i}$ and then using the values for the matrix elements
$\langle\vec{y}\,|\vec{p}\,\rangle$.

In the case that the configuration space is even dimensional with $N=2n$ and
$n\geq 1$, and that the real antisymmetric matrix is invertible (and thus with
no vanishing eigenvalue), it is also possible to express the results for the
action of the exponentiated coordinate operators $\hat{x}_{i}$ on the states
$|\vec{x};\lambda\rangle$ in a form which makes more explicit their character
as generators of displacements in noncommutative configuration space. For this
purpose it suffices to consider the correspondence (see (153)),

$a_{i}=-\frac{1}{\hbar}A_{ij}\,p_{0j},\qquad
p_{0i}=-\hbar\,(A^{-1})_{ij}\,a_{j}=\hbar\,a_{j}\left(A^{-1}\right)_{ji}.$
(166)

One thus finds for the transformation of the phase space operators,

$e^{-ia_{i}(A^{-1})_{ij}\hat{x}_{j}}\,\hat{x}_{i}\,e^{ia_{i}(A^{-1})_{ij}\hat{x}_{j}}=\hat{x}_{i}+a_{i}\,\mathbb{I},\qquad
e^{-ia_{i}(A^{-1})_{ij}\hat{x}_{j}}\,\hat{p}_{i}\,e^{ia_{i}(A^{-1})_{ij}\hat{x}_{j}}=\hat{p}_{i}-\hbar(A^{-1})_{ij}a_{j}\,\mathbb{I},$
(167)

while for the quantum states $|\vec{x};\lambda\rangle$, by using the fact that
$|\vec{x};\lambda\rangle=e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}}\,|\vec{x}+\vec{a};\lambda\rangle$,

$e^{ia_{i}(A^{-1})_{ij}\hat{x}_{j}}\,|\vec{x};\lambda\rangle=e^{ia_{i}(A^{-1})_{ij}x_{j}}\,e^{-\frac{\lambda_{ij}}{4}(A^{-1})_{ik}a_{k}(A^{-1})_{j\ell}a_{\ell}}\,e^{\frac{i}{2\hbar}a_{i}\left(\delta_{ij}-i(A^{-1})_{ik}\lambda_{kj}\right)\hat{p}_{j}}\,|\vec{x}+\vec{a};\lambda\rangle,$
(168)

and finally for the momentum eigenstates $|\vec{p}\,\rangle$,

$e^{ia_{i}(A^{-1})_{ij}\hat{x}_{j}}\,|\vec{p}\,\rangle=e^{-\frac{i}{2\hbar}a_{i}p_{i}}\,\Big{|}p_{i}-\hbar\left(A^{-1}\right)_{ij}\,a_{j}\Big{\rangle}.$
(169)

Let us now come back to the issue of the states $|\vec{x};\lambda\rangle$ as
quantum coherent states, along lines parallel to those discussed in the case
of a commutative configuration space, and this again independently of whether
$N$ is even or odd. Thus consider the momentum twisted coordinate operators
defined this time by,

$\hat{A}_{i}=\hat{x}_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,\hat{p}_{j},\qquad\hat{A}^{\dagger}_{i}=\hat{x}_{i}-\frac{i}{2\hbar}\left(\lambda_{ij}+iA_{ij}\right)\,\hat{p}_{j},$
(170)

jointly with the associated complex valued phase space variables,

$Z_{i}=x_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,p_{j},\qquad
Z^{*}_{i}=x_{i}-\frac{i}{2\hbar}\left(\lambda_{ij}+iA_{ij}\right)\,p_{j}.$
(171)

By construction the states $|\vec{x};\lambda\rangle$ are eigenstates with
eigenvalues $x_{i}$ of the operators $\hat{A}_{i}$,

$\hat{A}_{i}\,|\vec{x};\lambda\rangle=x_{i}\,|\vec{x};\lambda\rangle.$ (172)

Note as well that one has,

$\hat{A}_{i}=\hat{y}_{i}+\frac{i}{2\hbar}\,\lambda_{ij}\,\hat{p}_{j},\qquad\hat{A}^{\dagger}_{i}=\hat{y}_{i}-\frac{i}{2\hbar}\,\lambda_{ij}\,\hat{p}_{j},$
(173)

together with,

$Z_{i}=y_{i}+\frac{i}{2\hbar}\,\lambda_{ij}\,p_{j},\qquad
Z^{*}_{i}=y_{i}-\frac{i}{2\hbar}\,\lambda_{ij}\,p_{j},\qquad
y_{i}=x_{i}+\frac{1}{2\hbar}\,A_{ij}\,p_{j}.$ (174)

A direct evaluation finds that the operators
$(\hat{A}_{i},\hat{A}^{\dagger}_{i})$ again span a $N$ dimensional Fock
algebra since the matrix $\lambda_{ij}$ is real, symmetric and positive
definite, with the commutation relations,

$\Big{[}\,\hat{A}_{i},\hat{A}_{j}\,\Big{]}=0,\qquad\Big{[}\,\hat{A}_{i},\hat{A}^{\dagger}_{j}\,\Big{]}=\lambda_{ij}\,\mathbb{I},\qquad\Big{[}\,\hat{A}^{\dagger}_{i},\hat{A}^{\dagger}_{j}\,\Big{]}=0.$
(175)

By substitution one also checks that the displacement operator in phase space
for the associated canonical quantum coherent states is given as,

$Z_{i}\left(\lambda^{-1}\right)_{ij}\,\hat{A}^{\dagger}_{j}\,-\,Z^{*}_{i}\left(\lambda^{-1}\right)_{ij}\,\hat{A}_{j}=-\frac{i}{\hbar}x_{i}\hat{p}_{i}+\frac{i}{\hbar}p_{i}\hat{x}_{i}+\frac{i}{\hbar^{2}}p_{i}\,A_{ij}\,\hat{p}_{j}=-\frac{i}{\hbar}y_{i}\hat{p}_{i}+\frac{i}{\hbar}p_{i}\hat{y}_{i},$
(176)

as may have been expected for its very last expression in terms of the
momentum twisted operators $\hat{y}_{i}$ and their eigenvalues $y_{i}$.

Consider then the associated Fock vacuum $|\Omega_{\lambda}\rangle$ again with
a non standard choice of normalisation, such that,

$\hat{A}_{i}\,|\Omega_{\lambda}\rangle=0,\qquad\langle\Omega_{\lambda}\,|\,\Omega_{\lambda}\rangle=\frac{1}{\left((2\pi)^{N}\,{\rm
det}\,\lambda_{ij}\right)^{1/2}}.$ (177)

Since in view of the above expression for $\hat{A}_{i}$ in terms of
$\hat{y}_{i}$, the $|\vec{p}\,\rangle$ momentum space wave function of this
Fock vacuum $|\Omega_{\lambda}\rangle$ is such that,

$\left(i\hbar\frac{\partial}{\partial
p_{i}}+\frac{i}{2\hbar}\,\lambda_{ij}\,p_{j}\right)\,\langle\vec{p}\,|\Omega_{\lambda}\rangle=0,$
(178)

this wave function $\langle\vec{p}\,|\Omega_{\lambda}\rangle$ is identical to
that of the Fock vacuum $|\Omega_{\lambda}\rangle$ introduced earlier already
in (70) in the case of a commutative configuration space. Hence these two Fock
states are a same and identical quantum state irrespective of the conjugate
magnetic field $A_{ij}$, which justifies the same notation used here for both
without any risk of confusion. And one has as well that
$|\Omega_{\lambda}\rangle=|\vec{x}=\vec{0};\lambda\rangle$.

The canonical phase space coherent states related to the Fock algebra
$(\hat{A}_{i},\hat{A}^{\dagger}_{i})$ are thus constructed as,

$\displaystyle\Big{|}Z_{i};\Omega_{\lambda}\Big{\rangle}$
$\displaystyle\equiv$ $\displaystyle
e^{Z_{i}\left(\lambda^{-1}\right)_{ij}\,\hat{A}^{\dagger}_{j}\,-\,Z^{*}_{i}\left(\lambda^{-1}\right)_{ij}\,\hat{A}_{j}}\,\Big{|}\Omega_{\lambda}\Big{\rangle}$
(179) $\displaystyle=$ $\displaystyle
e^{-\frac{i}{\hbar}\left(x_{i}\hat{p}_{i}-p_{i}\hat{x}_{i}\right)+\frac{i}{\hbar^{2}}p_{i}A_{ij}\hat{p}_{j}}\,\Big{|}\Omega_{\lambda}\Big{\rangle}\equiv\Big{|}(x_{i},p_{i};A_{ij});\Omega_{\lambda}\Big{\rangle},$

and are the eigenstates of the Fock operators $\hat{A}_{i}$ with complex
eigenvalues $Z_{i}$,

$\hat{A}_{i}\,\Big{|}Z_{i};\Omega_{\lambda}\Big{\rangle}=Z_{i}\,\Big{|}Z_{i};\Omega_{\lambda}\Big{\rangle},\qquad
Z_{i}=y_{i}+\frac{i}{2\hbar}\,\lambda_{ij}\,p_{j}=x_{i}+\frac{i}{2\hbar}\left(\lambda_{ij}-iA_{ij}\right)\,p_{j}.$
(180)

Hence the localised and normalisable quantum states $|\vec{x};\lambda\rangle$
belong to the ensemble of such canonical coherent states for the
$(\hat{A}_{i},\hat{A}^{\dagger}_{i})$ Fock algebra, restricted however solely
to configuration space with $Z_{i}=x_{i}$ or $p_{i}=0$, namely,

$|\vec{x};\lambda\rangle=\Big{|}(x_{i},p_{i}=0;A_{ij});\Omega_{\lambda}\Big{\rangle}=\Big{|}Z_{i}=x_{i};\Omega_{\lambda}\Big{\rangle}.$
(181)

With the obvious identification now that

$|\Omega_{\lambda}\rangle=|\vec{x}=\vec{0};\lambda\rangle,$ (182)

one thus also recovers the general relation observed earlier already, namely,

$|\vec{x};\lambda\rangle=e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\Omega_{\lambda}\rangle=e^{-\frac{i}{\hbar}\vec{x}\cdot\hat{\vec{p}}}\,|\vec{x}=\vec{0};\lambda\rangle.$
(183)

Based on this observation, together with the facts that the momentum operators
$\hat{p}_{i}$ and Fock vacua $|\Omega_{\lambda}\rangle$ for the Fock algebras
$(\hat{a}_{i},\hat{a}^{\dagger}_{i})$ and
$(\hat{A}_{i},\hat{A}^{\dagger}_{i})$ remain identical irrespective of whether
configuration space is commutative or noncommutative, one also concludes that
the entire ensemble of localised and normalisable quantum states
$|\vec{x};\lambda\rangle$ labelled by all points $\vec{x}$ in configuration
space are as well the same and identical quantum states irrespective of the
value of the conjugate magnetic field $A_{ij}$, for any given choice of
gaussian length squared scale parameters $\lambda_{ij}$.

###  4.4 The $*_{\lambda}$-representation

Consequently, the completeness relation (76) established in section 2.8
remains valid, namely

$\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|=\mathbb{I},$
(184)

thereby allowing for the $*_{\lambda}$-representation of any abstract quantum
state $|\varphi\rangle$ in terms of a $*_{\lambda}$-wave function
$\varphi_{\lambda}(\vec{x}\,)$, such that,

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{x}\,|\vec{x};\lambda\rangle*_{\lambda}\varphi_{\lambda}(\vec{x}\,),\qquad\varphi_{\lambda}(\vec{x}\,)=\langle\vec{x};\lambda|\varphi\rangle,$
(185)

while this $*_{\lambda}$-wave function $\varphi_{\lambda}(\vec{x}\,)$ and the
momentum wave function $\tilde{\varphi}(\vec{p}\,)$ of the same state are
transformed into one another according to the following expressions involving
the $*_{\lambda}$-product,

$\varphi_{\lambda}(\vec{x}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}\tilde{\varphi}(\vec{p}\,),$
(186)

and

$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle=\int_{(\infty)}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,e^{-\frac{\lambda_{ij}}{4\hbar^{2}}p_{i}p_{j}}*_{\lambda}\varphi_{\lambda}(\vec{x}\,).$
(187)

All other similar considerations as they have already been discussed in
sections 2.8 and 2.9 in the commutative configuration space case apply here as
well without modification, except of course for those that involve the now
noncommuting configuration space coordinates $\hat{x}_{i}$, and the
configuration space wave functions $\varphi(\vec{x}\,)$ which only exist in
the commutative case. Thus for instance the commutators of the operator
$|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|$ with the operators
$\hat{p}_{i}$, $\hat{p}_{i}\hat{p}_{j}$, and $\hat{\vec{p}}\,^{2}$ remain as
given in (82), (83) and (84), respectively.

However, the commutators of that same operator
$|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|$ with the
coordinate operators $\hat{x}_{i}$ are now dependent on the conjugate magnetic
field $A_{ij}$, with values given according to,

$\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|,\hat{x}_{i}\,\Big{]}=\frac{1}{2}iA_{ij}\frac{\partial}{\partial
x_{j}}\Big{(}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,\Big{)}.$
(188)

This important distinctive feature of the noncommutative situation thus
impacts, for instance, the continuity equation for the probability current
density as expressed in the $*_{\lambda}$-representation.

Consider the abstract time dependent Schrödinger equation,

$i\hbar\frac{d}{dt}|\psi,t\rangle=\hat{H}\,|\psi,t\rangle,\qquad-i\hbar\langle\psi,t|\frac{\stackrel{{\scriptstyle\leftarrow}}{{d}}}{dt}=\langle\psi,t|\,\hat{H},$
(189)

with the quantum Hamiltonian operator, say in the typical nonrelativistic
situation of a single particle of mass $\mu$, expressed as,

$\hat{H}=\frac{1}{2\mu}\,\hat{\vec{p}}\,^{2}\,+\,\hat{V}_{q}\left(\hat{x}_{i}\right),\qquad\hat{V}_{q}(\hat{x}_{i})=\frac{1}{2}\left(\hat{V}(\hat{x}_{i})+\hat{V}^{\dagger}(\hat{x}_{i})\right).$
(190)

Note that because of the noncommutativity of the coordinate operators
$\hat{x}_{i}$ and the requirement that the Hamiltonian be self-adjoint (or at
least hermitian), the quantum potential energy now needs to be chosen in terms
of $\hat{V}_{q}(\hat{x}_{i})$ as defined above, rather than simply by
$\hat{V}(\hat{x}_{i})$ which generally is such that
$\hat{V}^{\dagger}(\hat{x}_{i})\neq\hat{V}(\hat{x}_{i})$ because of
configuration space noncommutativity.

For a unitary quantum time evolution generated by $\hat{H}$ the normalisation
of any time dependent normalised abstract quantum state remains conserved,
namely,

$1=\langle\psi,t|\psi,t\rangle=\int_{(\infty)}d^{N}\vec{x}\,\rho_{\lambda}(\vec{x},t),\quad\rho_{\lambda}(\vec{x},t)=\psi^{*}_{\lambda}(\vec{x},t)*_{\lambda}\psi_{\lambda}(\vec{x},t),\quad\psi_{\lambda}(\vec{x},t)\equiv\langle\vec{x};\lambda|\psi,t\rangle.$
(191)

Hence,
$\rho_{\lambda}(\vec{x},t)=\psi^{*}_{\lambda}(\vec{x},t)\,*_{\lambda}\,\psi_{\lambda}(\vec{x},t)$
still represents the probability density of the state $|\psi,t\rangle$. As
follows from the Schrödinger equation, its local conservation equation thus
reads,

$\frac{\partial}{\partial
t}\rho_{\lambda}(\vec{x},t)\,+\,\frac{i}{\hbar}\langle\psi,t|\,\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\,\langle\vec{x};\lambda|\,,\,\hat{H}\,\Big{]}\,|\psi,t\rangle=0.$
(192)

The kinetic energy term of the Hamiltonian provides the following contribution
to the second term on the l.h.s. of this identity,

$\frac{i}{\hbar}\langle\psi,t|\,\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\frac{1}{2\mu}\hat{\vec{p}}\,^{2}\,\Big{]}\,|\psi,t\rangle=\vec{\nabla}_{\vec{x}}\cdot\vec{J}_{\lambda,K}(\vec{x},t),$
(193)

where the kinetic probability current density
$\vec{J}_{\lambda,K}(\vec{x},t)$, which coincides with its expression in the
commutative configuration space case, is defined by,

$\vec{J}_{\lambda,K}(\vec{x},t)=\frac{\hbar}{2i\mu}\,\psi^{*}_{\lambda}(\vec{x},t)\,*_{\lambda}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\vec{x}}\psi_{\lambda}(\vec{x},t),\qquad\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\vec{x}}\right)_{i}=\left(\frac{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}{\partial
x_{i}}\right)\,-\,\left(\frac{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}{\partial
x_{i}}\right).$ (194)

However in contradistinction to the commutative case, the quantum potential
energy term of the Hamiltonian now also provides an additional nonvanishing
contribution to the total probability current density, since one now has,

$\frac{i}{\hbar}\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{x}_{i}\,\Big{]}=-\frac{1}{2\hbar}\,A_{ij}\,\frac{\partial}{\partial
x_{j}}\Big{(}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,\Big{)}.$
(195)

By extension, and through a power series expansion representation of
$\hat{V}_{q}(\hat{x}_{i})$, it thus follows that the contribution to the
second term on the l.h.s. of the identity (193) of the quantum potential
energy is given again by the divergence of a current density, in the form of,

$\langle\psi,t|\,\frac{i}{\hbar}\,\Big{[}\,|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,,\,\hat{V}_{q}(\hat{x}_{i})\,\Big{]}\,|\psi,t\rangle=\frac{\partial}{\partial
x_{i}}\,J_{\lambda,V,i}(\vec{x},t),$ (196)

with,

$\frac{\partial}{\partial
x_{i}}\,J_{\lambda,V,i}(\vec{x};t)=\oint_{0}\frac{du}{2i\pi}\,\frac{1}{u^{2}}\,\langle\psi,t|\,\hat{V}_{q}\left(\hat{x}_{i}+u\frac{\partial}{\partial
x_{j}}\left(-\frac{A_{ij}}{2\hbar}|\vec{x};\lambda\rangle*_{\lambda}\langle\vec{x};\lambda|\,\right)\right)\,|\psi,t\rangle.$
(197)

Here in accordance with the residues theorem, $u$ is a parameter in the
complex plane and $\oint_{0}$ denotes an arbitrary closed contour in the
complex plane encircling once the origin with the trigonometric orientation.
Quite obviously, the potential energy probability current density
$\vec{J}_{\lambda,V}(\vec{x},t)$, being directly proportional to $A_{ij}$,
vanishes identically in the commutative limit of a vanishing conjugate
magnetic field $A_{ij}$, as it should.

In conclusion in the case of noncommutativity in configuration space induced
by a nonvanishing conjugate magnetic field $A_{ij}$, in the
$*_{\lambda}$-representation the conservation of quantum probability is
expressed in terms of the continuity equation,

$\frac{\partial}{\partial
t}\rho_{\lambda}(\vec{x},t)+\vec{\nabla}_{\vec{x}}\cdot\vec{J}_{\lambda}(\vec{x},t)=0.$
(198)

This time, the total probability current density
$\vec{J}_{\lambda}(\vec{x},t)$ is comprised of two contributions, namely the
usual kinetic energy one, together now with a potential energy one on account
of configuration space noncommutativity which induces a Lorentz-like force
generated by the conjugate magnetic field $A_{ij}$, thus,

$\vec{J}_{\lambda}(\vec{x},t)=\vec{J}_{\lambda,K}(\vec{x},t)+\vec{J}_{\lambda,V}(\vec{x},t).$
(199)

###  4.5 The noncommutative configuration space enveloping algebra
representation

Following the same line of argumentation as that presented in section 2.10 in
the commutative case based on the momentum space
$|\vec{p}\,\rangle$-representation of the Heisenberg algebra, in the
noncommutative case it readily follows once again that there exists a one-to-
one map between abstract quantum states $|\varphi\rangle$ in the abstract
quantum Hilbert space and the operators $\hat{\varphi}(\hat{\vec{x}}\,)$ in
the enveloping algebra of the now noncommuting coordinate operators
$\hat{x}_{i}$, constructed in terms of the momentum wave representation
$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$of the state
$|\varphi\rangle$ and the following direct correspondence,

$\hat{\varphi}(\hat{\vec{x}}\,)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\tilde{\varphi}(\vec{p}\,),\qquad\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle.$
(200)

Conversely the abstract state $|\varphi\rangle$ is reconstructed through the
following representations,

$|\varphi\rangle=\int_{(\infty)}d^{N}\vec{p}\,|\vec{p}\,\rangle\,\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}d^{N}\vec{p}\frac{d^{N}\vec{x}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,|\vec{x};\lambda\rangle\,\tilde{\varphi}(\vec{p}\,)=\int_{(\infty)}d^{N}\vec{x}\,\hat{\varphi}(\hat{\vec{x}}\,)\,|\vec{x};\lambda\rangle.$
(201)

This enveloping algebra representation of quantum states relies on the
identity (164) which allows to represent momentum eigenstates
$|\vec{p}\,\rangle$ in terms of configuration space integrals over the
localised and normalisable states $|\vec{x};\lambda\rangle$ in the
noncommutative case as well.

Furthermore the positive definite inner product over the enveloping algebra
that represents the inner product of abstract quantum states
$|\varphi_{1}\rangle$ and $|\varphi_{2}\rangle$ is thus also given by the
configuration space doubly integrated matrix element for the
$|\vec{x};\lambda\rangle$ states of the $\hat{x}_{i}$-enveloping algebra
composite operator
$\hat{\varphi}^{\dagger}_{1}(\hat{\vec{x}}\,)\,\hat{\varphi}_{2}(\hat{\vec{x}}\,)$,

$\langle\varphi_{1}|\varphi_{2}\rangle=\int_{(\infty)}d^{N}\vec{x}_{1}\,d^{N}\vec{x}_{2}\,\langle\vec{x}_{1};\lambda|\,\hat{\varphi}^{\dagger}_{1}(\hat{\vec{x}}\,)\,\hat{\varphi}_{2}(\hat{\vec{x}}\,)\,|\vec{x}_{2};\lambda\rangle,$
(202)

as is the case for the commutative situation (see (104)).

However there remains still to establish under which conditions this
enveloping algebra representation of quantum states also provides a
representation of the abstract commutation relations of the operators
$(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$. In the commutative case this
representation is provided by the expressions in (108) and (109). In
particular that for the operators $\hat{P}_{i}$ involves a partial derivative
relative to the coordinate operator $\hat{x}_{i}$ of the operator
$\hat{\varphi}(\hat{\vec{x}}\,)$ that represents the state $|\varphi\rangle$.
However in the noncommutative case such a representation is not well defined
anymore since the operators $\hat{x}_{i}$ then do not commute among
themselves. Hence the situation has to be considered anew, based on the
momentum space representations of the abstract operators $\hat{x}_{i}$ and
$\hat{p}_{i}$ acting on abstract quantum states.

As it turns out (see hereafter), the existence of a representation of the
abstract momentum operators $\hat{p}_{i}$ for the enveloping algebra
representation requires that the real antisymmetric matrix $A_{ij}$
representing the conjugate magnetic field be invertible. Hence the
noncommutative configuration space enveloping algebra representation of the
quantum system is available only provided that the configuration space is even
dimensional with $N=2n$ and $n\geq 1$ and that none of the (complex)
eigenvalues of $A_{ij}$ is vanishing. Let us henceforth assume to be working
under such specific conditions.

Consider first the situation for the abstract momentum operators
$\hat{p}_{i}$, for which
$\langle\vec{p}\,|\hat{p}_{i}|\varphi\rangle=p_{i}\,\tilde{\varphi}(\vec{p}\,)$.
Hence one has,

$\hat{P}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,p_{i}\,\tilde{\varphi}(\vec{p}\,).$
(203)

In order to express this linear operator acting within the
$\hat{x}_{i}$-enveloping algebra solely in terms of the noncommuting
coordinate operators, let us revisit the identity,

$e^{-\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=\hat{x}_{i}\,-\,\frac{1}{\hbar}A_{ij}\,p_{j},\qquad\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\hat{x}_{i}\,-\,\frac{1}{\hbar}\,A_{ij}\,p_{j}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}},$
(204)

which thus implies,

$-\frac{1}{\hbar}A_{ij}\,p_{j}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=\Big{[}\,\hat{x}_{i}\,,\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\Big{]}.$
(205)

Therefore provided the matrix $A_{ij}$ is invertible one has,

$p_{i}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=\Big{[}-\hbar\left(A^{-1}\right)_{ij}\hat{x}_{j}\,,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\Big{]}.$
(206)

Consequently one finally finds, under that specific condition, for the
representation of the abstract momentum operators in the
$\hat{x}_{i}$-enveloping algebra representation,

$\hat{P}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,\Big{[}-\hbar\left(A^{-1}\right)_{ij}\hat{x}_{j}\,,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\Big{]}\tilde{\varphi}(\vec{p}\,)=-\hbar(A^{-1})_{ij}\left[\hat{x}_{j},\hat{\varphi}(\hat{\vec{x}}\,)\right].$
(207)

Let us now turn to the situation for the noncommutative coordinate operators
$\hat{x}_{i}$, for which
$\langle\vec{p}\,|\hat{x}_{i}|\varphi\rangle=\left(i\hbar\frac{\partial}{\partial
p_{i}}-\frac{1}{2\hbar}A_{ij}p_{j}\right)\,\tilde{\varphi}(\vec{p}\,)$. Hence
one has,

$\hat{X}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\left(i\hbar\frac{\partial}{\partial
p_{i}}-\frac{1}{2\hbar}A_{ij}p_{j}\right)\,\tilde{\varphi}(\vec{p}\,).$ (208)

In order to complete this evaluation by integration by parts for the term in
$\partial/\partial p_{i}$, for a fixed value of the index $i$ let us denote by
$j\neq i$ the other values of the running index $j$, so that (no summation
over $i$ in this expression),

$\vec{p}\cdot\hat{\vec{x}}=p_{i}\hat{x}_{i}\,+\,\sum_{j\neq
i}p_{j}\hat{x}_{j}.$ (209)

Then using the BCH formula one finds (in this expression the index $i$ is not
summed over),

$e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=e^{\frac{i}{2\hbar^{2}}\sum_{j\neq
i}A_{ij}p_{i}p_{j}}\,e^{\frac{i}{\hbar}p_{i}\hat{x}_{i}}\,e^{\frac{i}{\hbar}\sum_{j\neq
i}p_{j}\hat{x}_{j}}.$ (210)

Therefore one has,

$-i\hbar\frac{\partial}{\partial
p_{i}}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=\left(\hat{x}_{i}+\frac{1}{2\hbar}\sum_{j\neq
i}A_{ij}p_{j}\right)\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}},\quad\left(-i\hbar\frac{\partial}{\partial
p_{i}}\,-\,\frac{1}{2\hbar}A_{ij}p_{j}\right)\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}.$
(211)

Thus finally,

$\hat{X}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\int_{(\infty)}\frac{d^{N}\vec{p}}{(2\pi\hbar)^{N/2}}\,\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\tilde{\varphi}(\vec{p}\,)=\hat{x}_{i}\,\hat{\varphi}(\hat{\vec{x}}\,).$
(212)

Note well that the above analysis implies that this simple representation of
the noncommuting coordinate operators on the $\hat{x}_{i}$-enveloping algebra
is such that $\hat{X}_{i}$ acts by left multiplication by the operator
$\hat{x}_{i}$ of the operator $\hat{\varphi}(\hat{\vec{x}}\,)$ representing
the abstract quantum state $|\varphi\rangle$, rather than by right
multiplication. One could expect that an alternative representation where
$\hat{x}_{i}$ would be acting by right multiplication may be achieved as well,
using the second relation in (204), but then accompanied by an explicit
contribution in the conjugate magnetic field $A_{ij}$, since,

$\left(-i\hbar\frac{\partial}{\partial
p_{i}}\,-\,\frac{1}{2\hbar}A_{ij}p_{j}\right)\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=\hat{x}_{i}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}=e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}\,\hat{x}_{i}\,-\,\frac{1}{\hbar}A_{ij}\,p_{j}\,e^{\frac{i}{\hbar}\vec{p}\cdot\hat{\vec{x}}}.$
(213)

However based on that identity one then finds again for the operator
$\hat{X}_{i}$,

$\hat{X}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\hat{\varphi}(\hat{\vec{x}}\,)\,\hat{x}_{i}\,-\frac{1}{\hbar}\,A_{ij}\,\hat{P}_{j}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\hat{\varphi}(\hat{\vec{x}}\,)\hat{x}_{i}\,+\,\left[\hat{x}_{i},\hat{\varphi}(\hat{\vec{x}}\,)\right]=\hat{x}_{i}\,\hat{\varphi}(\hat{\vec{x}}\,).$
(214)

In conclusion, on the $\hat{x}_{i}$-enveloping algebra representation of
operators $\hat{\varphi}(\hat{\vec{x}}\,)$ the abstract algebra of operators
$(\hat{x}_{i},\hat{p}_{i})$ is realised by the following linear operators only
involving the noncommuting coordinate operators $\hat{x}_{i}$,

$\hat{X}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\hat{x}_{i}\,\hat{\varphi}(\hat{\vec{x}}\,),\qquad\hat{P}_{i}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=-\hbar(A^{-1})_{ij}\left[\hat{x}_{j},\hat{\varphi}(\hat{\vec{x}}\,)\right].$
(215)

Furthermore and as is to be expected, it may readily be checked that these
operators do indeed generate the same algebra as that of the abstract
operators $(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$, with,

$\left[\hat{X}_{i},\hat{X}_{j}\right]=iA_{ij}\,\mathbb{I}_{q},\qquad\left[\hat{X}_{i},\hat{P}_{j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I}_{q},\qquad\left[\hat{P}_{i},\hat{P}_{j}\right]=0,$
(216)

where $\mathbb{I}_{q}$ denotes the identity operator on the enveloping
algebra,
$\mathbb{I}_{q}\left(\hat{\varphi}(\hat{\vec{x}}\,)\right)=\hat{\varphi}(\hat{\vec{x}}\,)$.

With the above choices of inner product in (202) and of these actions of the
operators $\hat{X}_{i}$ and $\hat{P}_{i}$, its has thus been established that
the enveloping algebra of the noncommuting coordinate operators $\hat{x}_{i}$
with operators $\hat{\varphi}(\hat{\vec{x}}\,)$ provides an explicit
representation of the abstract algebra of the operators
$(\hat{x}_{i},\hat{p}_{i},\mathbb{I})$ and its Hilbert space of abstract
quantum states $|\varphi\rangle$.

Furthermore, the reconstruction of the abstract quantum state
$|\varphi\rangle$ represented by the operator $\hat{\varphi}(\hat{\vec{x}}\,)$