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\begin{document}

\title{\textbf{On the Meaning of \ the String-Inspired Noncommutativity and its
Implications}}
\author{\ \textbf{\ \ \ G. Dourado Barbosa} \\
%EndAName
\\
{\small \ \ Centro Brasileiro de Pesquisas F\'{i}sicas, CBPF-CCP}\\
{\small Rua Dr. Xavier Sigaud 150 \ , 22290-180, Rio de Janeiro, Brazil}}
\maketitle

\begin{abstract}
We propose an alternative interpretation for the meaning of noncommutativity
of the string inspired field theories and quantum mechanics. Arguments are
presented to show that the noncommutativity generated in the stringy context
ought be assumed to be only between the particle coordinate observables, and
not of the spacetime coordinates. Some implications of this fact for
noncomutative field theories and quantum mechanics are discussed. In
particular, a consistent interpretation is given for the wavefunction in
quantum mechanics.\ An analysis of the noncommutative theories\ in the
Schr\"{o}dinger formulation employing a generalized quantum Hamilton-Jacobi
formalism originated from the polar decomposition of the wavefunction
revealed a formal structure for noncommutative quantum mechanics richer than
the one of noncommutative quantum field theory. Conditions for the classical
and commutative limits of these theories were also determined and applied in
some examples.\vspace{2cm}

email: gbarbosa@cbpf.br

Keywords: string theory, noncommutative field theory, Schr\"{o}dinger
formulation,

Hamilton-Jacobi formalism, quantum mechanics, classical and commutative
limits

PACS numbers :11.10.Lm; 11.10.Ef; 03.65.-w; 03.65.Bz

\pagebreak
\end{abstract}




\section{Introduction}

Recently there has been a great interest in noncommutative theories of
the\ canonical type. They are characterized by the following commutation
relation
\begin{equation}
\lbrack\widehat{X}^{\mu},\widehat{X}^{\nu}]=i\theta^{\mu\nu}\;,\label{1}%
\end{equation}
where $\theta^{\mu\nu}$ is an antisymmetric constant tensor function of the coordinates.

During the latest years, a great deal of work and effort has been done in the
direction of understanding this kind of noncommutativity and its implications
in context of quantum\ field theory and also quantum mechanics (for good
reviews see \cite{1,1.5}). Part of the interest in these theories happens
because noncommutativity is present in certain models of string theory
\cite{2,3}, and M-theory \cite{4,5}. Another motivation comes from field
theory itself. Semi-classical arguments combining General Relativity and the
Heisenberg uncertainty principle also lead to noncommutative field theory
\cite{6}. The study of these theories\ gives us the opportunity to understand
interesting phenomena, like nonlocality and IR/UV mixing \cite{7}, new physics
at very short distances \cite{1.5,8}, and possible implications of Lorentz
violation \cite{9}.

But the interest in noncommutativity goes beyond the formal manipulations.
There are several lines of investigation about the possible phenomenology
associated with the canonical noncommutativity, from cosmology and high energy
scattering experiments to low energy precision tests\ \cite{10}.

The articles in this emerging branch of physics are growing up very fast.
Analyzing carefully the literature in the area it is possible to identify two
primary research lines in canonical noncommutative quantum field theory. The
first one presupposes a noncommutative spacetime from the beginning,\ being an
intrinsic approach. It is closely related with the ideas of Connes
noncommutative geometry \cite{12}. This is the line followed, for example, by
Chaichian et all \cite{13}, Cho et all \cite{14} and\ Wulkenhaar \cite{15}.

The second research line has its roots in a low-energy approximation
of\ string theory in\ the presence of a strong background\ $B$\ field
\cite{3}. This is the approach assumed, for example, in \cite{7} and
followers, and is the one adopted in this work. Therefore we will refer to the
noncommutative quantum field theory (NCQFT) and\ the associated noncommutative
quantum mechanics (NCQM) always presuming this point of view.

The implications of considering noncommutativity as property of the spacetime
has been a subject under discussion. As it\ can be seen in \cite{7}, NCQFT is
supported by the use of the Feynman graph approach for its calculations. As
was argued in \cite{15}, for example, that such kind of procedure would not be
expected from a theory defined on a true noncommutative spacetime, which not
a\ manifold, at least in the usual sense. At this point, it is important to
take into account that\ NCQFT came from string theory, which is based on a
continuous manifold structure. Therefore it would be quite reasonable to
interpret\ it\ as commutative in what concerns the spacetime. This naturally
leads to the formulation of one question: what kind of noncommutativity can be
behind NCQFT, if not of the spacetime?

The first goal of this work is to present an answer for the question above,
proposing an alternative interpretation for the string inspired
noncommutativity and discussing its implications for NCQFT and NCQM. As we
shall see later, one of these implications is the necessity of the
reinterpretation of the meaning of the wavefunction. Since the interpretation
for the noncommutativity proposed in this work is entirely new, we feel the
necessity to analyze more carefully the NCQFT and NCQM formulations until now
available in the literature. This is quite related to our second goal, which
is to explore the Schr\"{o}dinger picture of NCQFT and NCQM, whose advantages
to discuss some fundamental questions, like the quantum/classical and
noncommutative/commutative passages, will be underlined in our exposition.

The organization of the work is the following. In Section 2, we give a new
interpretation for the meaning of the noncommutativity originated in the
string theory context by tracing an intuitive parallel with the Landau
problem. The Third Section is about NCQFT in the Schr\"{o}dinger picture and
the discussion about the nontriviality to achieve its commutative and
classical limits. The remaining of the work is about NCQM. In the Fourth
Section, we show how to construct a consistent interpretation for NCQM in the
Schr\"{o}dinger formulation, a problem still now open in the literature. We
also apply the method proposed in Section 3 to carry out an analysis on the
formal structure of quantum mechanics and determine the conditions for the
achievement of its classical and commutative limits. Two simple examples of
application to illustrate how the ideas proposed work in concrete models are
presented in Section 5.\ We finish the work with a concluding section that
discuss the principal results and signalizes for\ some questions that deserve investigation.

\section{ The Meaning of Stringy\ Noncommutativity}

Let us discuss the origin of the noncommutativity relation (\ref{1}) and its
interpretation. For simplicity, we will restrict our work to the case where
$\theta^{oi}=0$. To grasp some intuition we will appeal for the similarity
between the derivation of the canonical noncommutativity in the string context
and the one present in the Landau problem when a system is projected onto its
lowest Landau level \cite{1,3}. Though being exhaustively explored in the
literature, this analogy has been always centered in what is the origin of the
noncommutativity,\ and not in what it is about.

We start analyzing the Landau problem. Consider a nonrelativistic particle
moving on the $x-y$ plane in the presence of a constant magnetic$\ B$ field
pointing in the $z$-direction. The classical Lagrangian of the system is
\begin{equation}
L=\frac{1}{2}m\overset{\cdot}{\vec{x}}^{2}+e\overset{\cdot}{\vec{x}}\cdot
\vec{A}\text{ ,}\label{2}%
\end{equation}
where $e$ is the particle charge, $\vec{A}$ is the electromagnetic vector
potential and $m$ its mass. We are considering units where $c=1.$ The quantum
Hamiltonian is
\begin{equation}
\widehat{H}=\frac{1}{2m}\hat{\pi}^{i}\hat{\pi}^{i},\label{2.5}%
\end{equation}
where $\hat{\pi}^{i}=m\overset{\cdot}{\hat{x}^{i}}=\hat{p}^{i}-e\hat{A}^{i}$
are the physical momenta and $\hat{p}^{i}$ are\ the canonical momenta. Notice
that the canonical momenta commute, while the physical momenta satisfy the
commutation relation
\begin{equation}
\lbrack\hat{\pi}^{i},\hat{\pi}^{j}]=i\hbar eB\epsilon^{ij}.\label{2.7}%
\end{equation}
To understand how the noncommutativity (\ref{1}) arises in the model\ it is
useful to define, in analogy with the classical case, the center-of-orbit
operator, whose components are given by
\begin{equation}
\widehat{X}^{i}=\hat{x}^{i}+\frac{i}{eB}\hat{\pi}^{i}.\label{2.8}%
\end{equation}
These components can be shown to satisfy the commutation relation
\begin{equation}
\lbrack\widehat{X}^{i},\widehat{X}^{j}]=i\frac{\hbar}{eB}\epsilon^{ij}%
=i\theta^{ij},\label{2.9}%
\end{equation}
where $\theta^{ij}=\left(  \hbar/eB\right)  \epsilon^{ij}$. Notice that while
$[\hat{x}^{i},\hat{x}^{j}]=0$, the $\widehat{X}^{i}$ are not allowed to
commute due to the presence of the term containing the magnetic field.\ The
spacetime, on the other hand, is the same ordinary commutative one. In this
scenery, the\ uncertainty relation
\begin{equation}
\Delta X^{i}\Delta X^{j}\geq\frac{\hbar}{2eB}\left|  \epsilon^{ij}\right|
,\label{2.95}%
\end{equation}
\ introduced by (\ref{2.9}), must be understood as just being a consequence of
the\ limitation on the information available about the $X^{i}$ coordinates\ in
a process of measurement.

Now we consider\ the strong magnetic field limit. In this case, the system is
projected onto the lowest Landau level. A rigorous prescription of how to work
in this limit, which is achieved by solving the constraint $\hat{\pi}%
^{i}\approx0$\ (using a projection technique) can be found in \cite{15.3}. On
heuristic grounds, one can understand the projection onto the lowest Landau
level as a process where the particles have their kinetic degrees of liberty
frozen and are confined into their respective orbit centers \cite{15.4}. The
particle coordinate observables in this limit clearly satisfy (\ref{2.9}) as a
consequence of the coincidence between $\widehat{X}^{i}$ and $\hat{x}^{i}$,
but there is no fundamental reason to believe that the spacetime coordinates
will become noncommutative in this limit as a consequence of the projection.
In reality the spacetime should be assumed as being unaltered, do not becoming
fuzzy or pointless in any sense.

Usually the relation (\ref{2.9}) is achieved in the literature by dropping the
kinetic term directly from the Lagrangian (\ref{2}).\ If we write the vector
potential as $\overrightarrow{A}=\left(  0,Bx,0\right)  $ and consider the
$B\rightarrow\infty$ or $m\rightarrow0$ limits, we can discard the kinetic
term and write the Lagrangian as
\begin{equation}
L=eBx\dot{y}.\label{3}%
\end{equation}
In this Lagrangian the $x$ and $y$ variables are canonically conjugate and
their respective quantum operators satisfy a commutation relation identical
to\ (\ref{2.9}),
\begin{equation}
\left[  \hat{x}^{i},\hat{x}^{j}\right]  =\frac{i\hbar}{eB}\epsilon
^{ij}.\label{4}%
\end{equation}

Notice that, when the discussion is directed from this point of view, the
derivation of the canonical noncommutativity deviates attention from the
fundamental point that the noncommutativity obtained as a consequence of
projection onto the lowest Landau level, which enforces the identification
between $\widehat{X}^{i}$ and $\hat{x}^{i}$, ought in reality be assumed to
be\ of the particle coordinate observables, and not of the spacetime
coordinates. This is the source of some confusion in the literature on the
Landau problem and also in the high energy physics articles on NCQFT that
refer to it. For instance, take for example the following assertion found in
\cite{15.5} : ``An example of a system where space-time coordinates do not
commute is that of a particle in a strong magnetic field''. How could a strong
magnetic field in a laboratory turn the spacetime noncommutative and thus
destroy its pointwise structure ? It seems to be a preposterous conclusion to
believe in this as a serious possibility!

The canonical noncommutativity originated from string theory in \cite{3} is
based on the same heuristic approximation of the lowest Landau level just
described.\ Consider bosonic strings moving in a flat euclidean space with
metric $g_{\mu\nu}$ in the presence of a constant Neveu-Schwarz $B$-field and
with $Dp$ branes. The $B$-field is equivalent to a constant magnetic field on
the branes, and it can be gauged away in the directions transverse to the $Dp$
brane worldvolume. The worldsheet action is
\begin{align}
S &  =\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}\left(  g_{\mu\nu}%
\partial_{\eta}X^{\mu}\partial^{\eta}X^{\nu}-2\pi i\alpha^{\prime}B_{\mu\nu
}\epsilon^{\eta\lambda}\partial_{\eta}X^{\mu}\partial_{\lambda}X^{\nu}\right)
\label{5}\\
&  =\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}g_{\mu\nu}\partial_{\eta}X^{\mu
}\partial^{\eta}X^{\nu}-\frac{i}{2}\int_{\partial\Sigma}B_{\mu\nu}X^{\mu
}\partial_{t}X^{\nu}\text{ \ ,}\nonumber
\end{align}
where $\alpha^{\prime}=l_{s}^{2}$, $\Sigma$ is the string worldsheet and
$\partial_{t}$ is a tangential derivative along the worldsheet boundary
$\partial\Sigma$. Now consider the limit $g_{\mu\nu}\sim\left(  \alpha
^{\prime}\right)  ^{2}\rightarrow0$ keeping $B_{\mu\nu}$\ fixed \cite{3}. In
this limit the bulk kinetic terms vanish and the theory is topological. All
that remains are the boundary degrees of freedom, which are governed by the action%

\begin{equation}
S=-\frac{i}{2}\int_{\partial\Sigma}B_{\mu\nu}X^{\mu}\partial_{t}X^{\nu
}.\label{6}%
\end{equation}
If we regard (\ref{6}) as a one dimensional action and ignore the fact that
the $X^{\mu}\left(  t\right)  $ are the endpoints of a string, it can be
considered as analogous to the action corresponding to the Lagrangian of the
Landau problem (\ref{3}). Under the approximation being considered\ the
$X^{\mu}\left(  t\right)  $ can be regarded as operators satisfying the
canonical commutation relation
\begin{equation}
\lbrack\widehat{X}^{\mu},\widehat{X}^{\nu}]=\left(  \frac{i}{B}\right)
^{\mu\nu},\label{7}%
\end{equation}
which is identical to the one of (\ref{1}) by defining\ $\theta^{\mu\nu
}=\left(  1/B\right)  ^{\mu\nu}$.

This heuristic derivation (here abbreviated to go direct to the point) has the
same lack present on the one of the Landau problem where the kinetic term is
discarded. It deviates attention from the fact that the noncommutativity
achieved by the approximation under consideration should be assumed to be of
the particle coordinate observables, the spacetime coordinates remaining commutative.

Although the great majority of papers about NCQFT consider the spacetime as
being\ ``pointless'', they use the basis of plane wave for their calculations
in the momentum space (see for example \cite{1,1.5,3,7} and references
therein) and interpret the space of Weyl symbols \cite{13} as the physical
position space. This is equivalent to allow the localization of information at
individual points on the spacetime. The apparent contradiction in this
procedure is clearly shown to be absent if we interpret the stringy spacetime
that they are considering as commutative. It is important to keep in mind that
the distinction between the intrinsic spacetime noncommutativity and
the\ coordinate observable noncommutativity is not a question of metaphysics.
Since the calculation of the Green functions in the first case must employ the
procedure of averaging over localized states in the space of Weyl
symbols\ \cite{13} (which smears the pointwise information), the finiteness of
the theories, as well as the measurable physical quantities, will differ
according to the approach adopted for the noncomutativity.

Let us review the Weyl quantization procedure considering stringy
noncommutativity from the new point of view.\ For the case under
consideration, it consists in the establishment of a map from the
noncommutative space of the particle coordinate observables to the space of
commutative coordinates by the use of the Moyal\ star product
\begin{align}
\left(  f\star g\right)  (x) &  =\frac{1}{(2\pi)^{n}}%
%TCIMACRO{\dint }%
%BeginExpansion
{\displaystyle\int}
%EndExpansion
d^{n}kd^{n}pe^{i(k_{\mu}+p_{\mu})x^{\mu}-\frac{i}{2}k_{\mu}p_{\nu}\theta
^{\mu\nu}}f(k)g(p)\nonumber\\
&  =\left.  e^{\frac{i}{2}\theta^{\mu\nu}\frac{\partial}{\partial\xi^{\mu}%
}\frac{\partial}{\partial\eta^{\nu}}}f(x+\xi)g(x+\eta)\right|  _{\xi=\eta
=0}.\label{8}%
\end{align}
According to this technique, the representation of a given NCQFT\ originally
valued on the space of noncommutative variables on the space of ordinary
commutative functions is done by the replacement of the noncommuting variables
in the arguments of the fields in the action by the commutative ones and the
ordinary product of the fields by the Moyal product. Since the
noncommutativity is assumed to be only of the particle coordinate observables,
what the Weyl procedure does is just to represent the NCQFT originally valued
on this noncommutative variables in function of the spacetime coordinates and
canonical momenta. This will be particularly clear later when we will consider
the NCQM, which is the fixed particle sector of NCQFT.

Due to the property $\int d^{4}xA\star B=\int d^{4}x$ $AB$, the Hilbert (Fock)
space of a noncommutative field theory can be chosen to be the same as the one
of its commutative counterpart at the perturbative level \cite{26.5}, and the
noncommutativity is manifest only through the interaction terms of the action.
This gives us no surprise, since the difference of a commutative quantum field
theory and its noncommutative counterpart is only the presence of the\ strong
Neveau-Schwartz $B$ field in the background acting to forbid the simultaneous
measurement of all the position coordinates of the particles. Of course the
$B$ field does this by interacting with the quantum fields, and thus its
presence must be encoded in the interaction terms of the action.

\section{NCQFT and the Hamilton-Jacobi Theory}

Here, we discuss some aspects of the Schr\"{o}dinger representation for NCQFT
based on the canonical deformation of the algebra of the coordinate
observables. For simplicity, we will consider scalar field theories with
action
\begin{equation}
S=\int d^{4}x\left\{  \frac{1}{2}\left(  \partial^{\mu}\varphi\partial_{\mu
}\varphi-\frac{m^{2}}{\hbar^{2}}\varphi^{2}\right)  -V_{\star}\left(
\varphi\right)  \right\}  \text{ \ ,}\label{10}%
\end{equation}
where $V_{\star}\left(  \varphi\right)  $ is the usual commutative interaction
part of the potential with the star product replacing the ordinary one. Since
the discussion\ here is general and the procedure presented can be in
principle applied to other dimensions than four, we will keep the potential as
a generic real analytic one. Of course, the question of renormalizability must
be taken into account when choosing the acceptable potentials \cite{30}. All
the quantization procedures of the usual formalism developed for the
commutative quantum field theory in the Schr\"{o}dinger representation
\cite{31} are similar in the noncommutative case under consideration, which
presupposes that $\theta^{0i}=0$. We briefly present the aspects of the
procedure that will be useful in this work.

The Hamiltonian corresponding to (\ref{10})\ is%

\begin{equation}
H=\int d^{4}x\left\{  \frac{1}{2}\left(  \pi^{2}+\left|  \nabla\varphi\right|
^{2}+\frac{m^{2}}{\hbar^{2}}\varphi^{2}\right)  +V_{\star}\left(
\varphi\right)  \right\}  ,\label{11}%
\end{equation}
where $\pi\left(  x\right)  =\dot{\varphi}\left(  x\right)  $ is the conjugate
field momentum. The operators $\widehat{\varphi}\left(  x\right)  $ and
$\widehat{\pi}\left(  x\right)  $ satisfy the canonical equal-time
commutators,
\begin{align}
\left[  \widehat{\varphi}\left(  \vec{x},t\right)  ,\widehat{\pi}\left(
\vec{y},t\right)  \right]   &  =i\hbar\delta\left(  \vec{x}-\vec{y}\right)
,\nonumber\\
\left[  \widehat{\varphi}\left(  \vec{x},t\right)  ,\widehat{\varphi}\left(
\vec{y},t\right)  \right]   &  =\left[  \widehat{\pi}\left(  \vec{x},t\right)
,\widehat{\pi}\left(  \vec{y},t\right)  \right]  =0\text{ .}\label{12}%
\end{align}

To work in a coordinate field representation,\ we shall\ consider a basis for
the Fock space where the operator $\widehat{\varphi}\left(  \vec{x}\right)  $
is time independent and diagonal $\widehat{\varphi}\left(  \vec{x}\right)
\mid\phi\rangle=\phi\left(  \vec{x}\right)  \mid\phi\rangle$. In this basis
the state $\mid\Psi\rangle$ is represented by\ the time dependent wave
functional $\Psi\left[  \phi\right]  =\langle\phi\mid\Psi\rangle$, and the
momentum field operator $\hat{\pi}\left(  \vec{x}\right)  $ by $-i\hbar
\delta/\delta\phi\left(  \vec{x}\right)  $. The Schr\"{o}dinger equation is
written as
\begin{equation}
i\hbar\frac{\partial}{\partial t}\Psi\left[  \phi,t\right]  =\int
d^{3}x\left\{  \frac{1}{2}\left(  -\hbar^{2}\frac{\delta^{2}}{\delta\phi
^{2}\left(  \vec{x}\right)  }+\left|  \nabla\phi\right|  ^{2}+\frac{m^{2}%
}{\hbar^{2}}\phi^{2}\right)  +V_{\star}\left(  \phi\right)  \right\}
\Psi\left[  \phi,t\right]  .\label{18}%
\end{equation}

If we write the wavefunctional in its polar form $\Psi=R$ $\exp\left(
iS/\hbar\right)  $ and substitute it into (\ref{18}), we obtain
\begin{align}
\left[  i\hbar\frac{\partial R}{\partial t}-R\frac{\partial S}{\partial
t}\right]  e^{iS/\hbar} &  =\int d^{3}x\left\{  -\frac{\hbar^{2}}{2}\left[
\frac{\delta^{2}R}{\delta\phi^{2}}-\frac{R}{\hbar^{2}}\left(  \frac{\delta
S}{\delta\phi}\right)  ^{2}\right.  \right. \nonumber\\
&  +\left.  \left.  i\left(  \frac{2}{\hbar}\frac{\delta R}{\delta\phi}%
\frac{\delta S}{\delta\phi}+\frac{R}{\hbar}\frac{\delta^{2}S}{\delta\phi^{2}%
}\right)  \right]  \right\}  e^{\frac{i}{\hbar}S}+\left[  \frac{1}{2}\left|
\nabla\phi\right|  ^{2}+\frac{m^{2}}{2\hbar^{2}}\phi^{2}+V_{\star}\left(
\phi\right)  \right]  \text{ }Re^{iS/\hbar}\label{18.5}%
\end{align}
Dividing (\ref{18.5})\ by $\Psi$ and\ separating the real part we find
\begin{equation}
\frac{\partial S}{\partial t}+\int d^{3}x\left\{  \frac{1}{2}\left[  \left(
\frac{\delta S}{\delta\phi}\right)  ^{2}+\left|  \nabla\phi\right|  ^{2}%
+\frac{m^{2}}{\hbar^{2}}\phi^{2}\right]  +V\left(  \phi\right)  \right\}
+V_{nc}[\phi]+Q_{K}[\phi]=0,\label{19}%
\end{equation}
where
\begin{equation}
V_{nc}[\phi]=\int d^{3}x\left\{  \operatorname{Re}\left[  V_{\star}\left(
\phi\right)  \right]  -V\left(  \phi\right)  \right\}  ,\label{19.2}%
\end{equation}
and
\begin{equation}
Q_{K}[\phi]=-\frac{\hbar^{2}}{2R}\int d^{3}x\frac{\delta^{2}R}{\delta\phi
^{2}\left(  \vec{x}\right)  }.\label{19.4}%
\end{equation}
Equation (\ref{19}) is the ordinary Hamilton-Jacobi equation for the scalar
field with two additional potentials $V_{nc}$ and $Q_{K}.$

The complete information contained in the functional Schr\"{o}dinger equation
is extracted only if we consider (\ref{19}) and the other coupled equation
that is obtained from the decomposition of (\ref{18}) into its real and
imaginary parts. Multiplying (\ref{18.5})\ by\ $2R/\hbar\exp\left(
-iS/\hbar\right)  $,\ taking the imaginary part and doing\ some simplification
using the properties of the star product, the equation obtained reads
\begin{equation}
\frac{\partial R^{2}}{\partial t}+\int d^{3}x\frac{\delta}{\delta\phi}\left(
R^{2}\frac{\delta S}{\delta\phi}\right)  =0.\label{21}%
\end{equation}
This equation must be interpreted as a continuity equation for the probability
density $R^{2}[\phi(\vec{x}),t]$ that the field configuration be $\phi(\vec
{x})$ at time $t$. Notice that (\ref{19}) and (\ref{21}) constitute a set of
nonlinear\ coupled equations. In practice it is convenient to solve directly
(\ref{18})\ and then obtain $R$ and $S$ from the wavefunctional $\Psi$.\ The
advantage of decomposing the functional Schr\"{o}dinger equation in a more
complex system of two coupled equations is for the sake of its physical interpretation.

In what follows, we shall use the generalized Hamilton-Jacobi formalism
introduced to perform an analysis of\ the conditions for a system achieve its
classical and commutative limits. Before going on, however, it is necessary to
define precisely what we mean for achieving these limits.

Finding the classical or commutative limit of a system is to establish the
conditions that must be imposed on the environment, quantum numbers or
physical constants in order to enforce the system to assume the behavior
identical to the one of a classical or commutative analog. These conditions
can be determined by analyzing equations (\ref{19}) and (\ref{21}). When we
consider a solution of the equations (\ref{19}) and (\ref{21}) and imagine it
as being substituted on then, the original set of equations must now be
understood as a set of identities that are being trivially satisfied. The
classical or commutative limits are achieved when the conditions are such that
this set of equations (seen as a set of identities)\ assume the classical form
or the quantum commutative form. Notice that, according to this criterion, the
establishment of the adequate prescription for the achievement of the limits
of each individual system must be done by accounting for the properties that
characterize its physical state. The procedure proposed will thus lead to the
determination of state dependent criteria, thought they can be written in an
universal and compact form, as will be shown later.

The first implication of\ the adoption of the prescription suggested here is
that the achievement of the classical limit will in most cases happen under
conditions that differ form the naive one, $\hbar\rightarrow0,$ usually
employed in many textbooks. Here, and in what follows,\ the symbol
$``A\rightarrow B"$ must be understood as taking a limit such that $A-B$ is
sufficiently small to be neglected in comparison with the other quantities
under consideration. If the scalar field is assumed to be massive, in
order\ that the limit $\hbar\rightarrow0$ have sense on (\ref{19}) it would be
necessary to impose that $m\rightarrow0$, which may not be a desired
condition. Moreover, no matter if the field is massive or not, in\ the
achievement of the classical limit starting from (\ref{19}) and (\ref{21}) we
must\ take into account the fact that $R$ (and$\ S$) in principle depends on
$\hbar$, and hence in general $Q_{K}$ does not go to zero when $\hbar$ goes. A
detailed discussion about the fallacy in adopting $\hbar\rightarrow0$ as the
universal criterion for the classical limit\ in the context of ordinary
quantum mechanics and field theory containing examples\ is found in \cite{18}
and references therein.

An analogous reasoning can be applied when we consider the commutative limit.
Although $V_{nc}\rightarrow0$ when $\theta^{ij}\rightarrow0$, as can be seen
from (\ref{8}) and (\ref{19.2}), this criterion may not be valid for the
achievement of the commutative limit. This happens because$\ Q_{K}$ can
contain contributions generated by the noncommutativity that do not depend on
$\theta^{ij}$. In this case, taking $\theta^{ij}\rightarrow0$ will not vanish
them, do not conducting to the commutative limit. Moreover, since in principle
the dependence of $Q_{K}$ (and $S$)\ on $\theta^{ij}$ is arbitrary and totally
state dependent, the possibility of $Q_{K}$ blowing-up when $\theta
^{ij}\rightarrow0$ must not be discarded. In reality this is expected to occur
in some models.

When we consider a differential equation which is deformed depending on a set
of parameters (for example $\theta^{ij}$), the set of admissible solutions is
richer than the original one corresponding to the undeformed equation. If we
undo the deformation smoothly,\ part of the expanded set of solutions will of
course suffer a process of homotopy, and in the end of it will be contained in
the set of solutions of the original undeformed equation. But part of the same
set will not present the same behavior, and can become unaltered, go to
another limit outside the set of undeformed solutions, or blow-up. In fact, it
has been verified that in many of the perturbative loopwise calculations of
NCQFT, some peculiar phenomena happens when $\theta^{ij}\rightarrow0$, like
UV/IR mixing \cite{7}, which is responsible for the non-analytic behavior in
the $\theta^{ij}\rightarrow0$ limit, the blowing-up of the self-energy in some
models \cite{32.7}, among others \cite{1.5}.

In what follows we will turn operational the technique proposed for the
determination of the classical and commutative limits by showing how it can be
applied to perform an analysis of NCQFT. In Section 4 we will do the same
analysis for the most complex case of\ NCQM and also complement it with
illustrative examples involving concrete models.

The origin of the quantum contributions on (\ref{19}) is traced back to\ the
action of the field momentum in its differential representation, $\hat{\pi
}\left(  \vec{x}\right)  =-i\hbar\delta/\delta\phi\left(  \vec{x}\right)  ,$
on the wavefunctional $\Psi=R\exp\left(  iS/\hbar\right)  $ in\ (\ref{18}).
When this equation is divided by $\Psi$ and the real part is taken,\ the
quantum Hamilton-Jacobi equation (\ref{19}) is originated containing terms
that can not appear on the classical Hamilton-Jacobi equation, where the field
momentum is represented by $\pi\left(  \vec{x}\right)  =\delta S/\delta
\phi\left(  \vec{x}\right)  $. These terms are nonclassical, and thus must be
responsible by the quantum effects. This is a\ simple and secure criterion for
the identification of the quantum contributions.

In practice, it is not necessary to do all the steps and write the
Hamilton-Jacobi equation containing all its terms.\ When one wants to identify
and compute\ the quantum effects corresponding to an specific part of the
Schr\"{o}dinger Hamiltonian, the procedure\ can be\ applied\ directly to the
term under consideration to extract its quantum contributions, which are
grouped to constitute its associated quantum potential. In equation
(\ref{18}), for example, the only term that has derivatives (here functional)
acting on $\Psi$ is kinetic one. Thus, there will be only one quantum
potential, which must be associated with this term and is given by
\begin{align}
Q_{K} &  =\operatorname{Re}\left(  -\frac{\hbar^{2}}{2}\frac{1}{\Psi}\int
d^{3}x\frac{\delta^{2}\Psi}{\delta\phi^{2}}\right)  -\left(  \frac{\hbar^{2}%
}{2}\int d^{3}x\left(  \frac{\delta S}{\delta\phi}\right)  ^{2}\right)
\nonumber\\
&  =-\frac{\hbar^{2}}{2R}\int d^{3}x\frac{\delta^{2}R}{\delta\phi^{2}\left(
\vec{x}\right)  }.\label{22}%
\end{align}
This reproduces (\ref{19.4}), and is the expression used to define
the\ quantum potential in the de Broglie-Bohm interpretation of\ quantum
theory \cite{32}. However, this must not be cause of confusion. Usually, the
polar decomposition of the wavefunctional and the associated Hamilton-Jacobi
formalism are explored by the Bohmian community in the direction of
establishing an ontological meaning for quantum field theory, with an
additional assumption about the objective existence of quantum fields
independent of the act of observation \cite{18,32}. However, this assumption
is not being\ considered in this work. What is present in the Bohmian
literature is in reality a procedure that is a particular case of the one
presented here in the sense that it considers only the kinetic quantum
potential. This will be clear in the section about quantum mechanics. The most
general case where we intend that this method could be applied may not yet
admit an ontological interpretation.

It is possible to associate a quantum potential with each term or, according
to the necessity, a group of terms involving derivatives in the Hamiltonian.
To suppress its corresponding quantum effects, it is sufficient to impose the
cancelation of its corresponding quantum potential as an equation. The
classical limit of the theory is found when the sum of\ all the quantum
potentials vanish. In this case the Hamilton-Jacobi equation loses its
dependence on $R$ and\ on second and higher order derivatives of $S$,
decouples from (\ref{21}) and\ assumes its classical form, which is equation
exclusively for $S$. This is a general result, totally independent of the
specific form of equation\ (\ref{19}). By definition, the classical
contributions are the ones obtained by the replacement $\pi\left(  \vec
{x}\right)  =\delta S/\delta\phi\left(  \vec{x}\right)  $, and thus do not
contain any derivative of $R$ that could spoil the decoupling process of the
Hamilton-Jacobi equation. Once (\ref{19}) and (\ref{21}) assume a form
characteristic of the classical physics, the physical content of the system is
the same of the classical field theory, and therefore the observable
quantities, as well as the equations of evolution of their averages, will be
indistinguishable of the ones of classical physics.

Having determined the criterion for the classical limit as $Q_{K}\rightarrow
0$, let us concentrate our attention on the criterion for the commutative one.
Since equation (\ref{21}) is still in the ordinary commutative classical form
the discussion about how to achieve the commutative limit reduces to the
analysis of the potentials $Q_{K}$ and $V_{nc}$ of (\ref{19}). One secure way
for the identification of the noncommutative effects present in $Q_{K}$ is to
calculate the same quantum potential with respect to the analog wavefunctional
corresponding to the associated commutative field theory and compare with it.
This is done by defining the commutative quantum potential as
\begin{equation}
Q_{c}=-\frac{\hbar^{2}}{2R}\int d^{3}x\frac{\delta^{2}R_{c}}{\delta\phi
^{2}\left(  \vec{x}\right)  },\label{23}%
\end{equation}
where $R_{c}=\sqrt{\Psi_{c}^{\ast}\Psi_{c}}$ and $\Psi_{c}$ is the solution of
(\ref{18}) with $\theta^{ij}=0$, that is , the solution of the commutative
counterpart. The noncommutative contributions contained in $Q_{K} $ can thus
be attributed to the functional
\begin{equation}
Q_{nc}=Q_{K}-Q_{c},\label{24}%
\end{equation}
which we define as the noncommutative quantum potential.

When $V_{nc}+Q_{nc}\rightarrow0$,\ (\ref{19}) becomes similar to the
Hamilton-Jacobi equation corresponding to the conventional quantum field
theory, assuring the noncommutative/commutative passage. This can happen in
two ways:

1) $V_{nc}+Q_{nc}\rightarrow0$\ is achieved by taking the limit $\theta
^{ij}\rightarrow0$ directly without any obstruction caused by the quantum
effects represented by $Q_{nc}$. In this case the $\theta^{ij}$ disappear
completely from all terms of the Hamilton-Jacobi\ equation. This is the most
common approach and easy to understand in comparison with classical
noncommutative field theory, where $Q_{nc}$\ is absent the commutative limit
is usually achieved\ by taking $\theta^{ij}\rightarrow0$.

2) $V_{nc}+Q_{nc}\rightarrow0$ is achieved by varying a physical constant or
parameter of the system, keeping $\theta^{ij}$ unaltered. This approach is
less common, but can be useful specially in cases where the first is not
possible to be realized. Two examples in the quantum mechanical context are
presented to illustrate how this can be done in the end of the article.

\section{ Noncommutative Quantum Mechanics}

Here we discuss the implications of considering the coordinate observables of
the particles as operators satisfying (\ref{1}) for quantum mechanics. We
shall show how to work in the Schr\"{o}dinger formulation\ with a consistent
interpretation for the wavefunction and how the Hamilton-Jacobi formalism of
the previous section can be applied to give information about the formal
structure of NCQM in comparison with the one of NCQFT.\ 

\subsection{Establishing the Foundations}

As in the commutative theories, NCQM can be derived from NCQFT in its\ the low
energy limit. A rigorous treatment for this derivation was done P. M. Ho and
H. C. Kao in \cite{33}. For chargeless particles their noncommutative
Schr\"{o}dinger\ equation reduces to
\begin{equation}
i\hbar\frac{\partial\Psi(x^{i},t)}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla
^{2}\Psi(x^{i},t)+V(x^{i})\star\Psi(x^{i},t).\label{26}%
\end{equation}
By using the properties of the star product the potential term of the
Schr\"{o}dinger equation can also be written as
\begin{equation}
V(x^{i})\star\Psi(x^{i},t)=V\left(  x^{i}+\frac{\theta^{ij}\partial_{j}}%
{2}\right)  \Psi(x^{i},t).\label{27}%
\end{equation}
The Hamiltonian is then%

\begin{equation}
H=\frac{-\hbar^{2}}{2m}\nabla^{2}+V\left(  x^{i}+\frac{\theta^{ij}\partial
_{j}}{2}\right)  .\label{28}%
\end{equation}
Thus, the noncommutative quantum mechanics is physically equivalent to a
commutative quantum mechanics with new momentum dependent interactions,
dictated by the replacement $x^{i}\rightarrow x^{i}-\theta^{ij}\hat{p}%
_{j}/2\hbar$ in the potential term. This can be assumed to be the Weyl
correspondence for quantum mechanics, and has been widely used in the
literature (see, for example \cite{27}). However, the complete investigation
of all of its consequences has not ever been done.

To understand physics that lies behind the correspondence above we again
appeal to the Landau problem, this time do not considering the projection onto
the lowest Landau level. An analogy between the physical variables suggests
that the shift $x^{i}\rightarrow x^{i}-\theta^{ij}\hat{p}_{j}/2\hbar$ caused
by the external Neveu-Schwartz $B$ field is similar to the shift $\hat{p}%
^{i}\rightarrow\hat{p}^{i}-e\hat{A}^{i}/c$, which is\ caused by an external
magnetic field in ordinary quantum mechanics. Since in the Landau problem the
physical momentum is $\hat{\pi}^{i}=\hat{p}^{i}-e\hat{A}^{i}/c,$ according to
the analogy the physical position must be $\widehat{X}^{i}=x^{i}-\theta
^{ij}\hat{p}_{j}/2\hbar$. This intuitively explain why the position
observables must remain noncommuting after the Weyl correspondence is applied.
This map is no more than a representation of the noncommuting position
observables in terms of the canonical momentum $\hat{p}^{i}$ and canonical
coordinates $x^{i}$, the last having the meaning of being points of the real
physical space. The association of the $\widehat{X}^{i}$ with the observables
corresponding to the\ physical coordinates of the particles was also\ proposed
in \cite{28}, but following the same line of \cite{27}, which interpret the
canonical positions $x^{i}$ just as auxiliary variables, do not having any
physical interpretation.

Some physical intuition about the meaning of the canonical position variables
can be gained by the fictitious dipole picture \cite{35}. For the case of the
NCQM under consideration, this consists of imagining that, instead of
a\ particle, the elementary object of the theory is a half dipole whose extent
is proportional to its momentum, $\Delta x^{i}=\theta^{ij}p_{j}/2\hbar$. One
of the endpoints of the fictitious\ dipole carries its mass and is responsible
for its interactions. The other extreme is empty. According to this intuitive
view, the change of variables $X^{i}=x^{i}-\theta^{ij}p_{j}/2\hbar$
corresponds to a change of coordinates of the interacting extreme of the
dipole $X^{i}$,\ where the corresponding physical particle is located, to
its\ empty one $x^{i}$. Therefore the effects of the background field, which
acts to\ forbid the coordinates to commute, is compensated in this new
commutative\ coordinate system obtained by change of variable that is
dependent on the momentum and background intensity (remember $\theta
^{ij}=(1/B)^{ij}$). Although\ the simultaneous measurement of the observables
$\hat{x}^{i}$ is in principle possible, it is not sufficient for the
determination of the physical position of the associated particle. Because the
$\hat{x}^{i}$ and the $\hat{p}^{i}$ do not commute, the knowledge of the three
$x^{i}$ forbids the simultaneous determination of the three $p^{i}$ and thus
the physical location of the particle.

The Hilbert space of states of noncommutative quantum mechanics is assumed to
be the same of the commutative one \cite{27}, which is a remnant from a result
of field theory discussed in the Second Section. Having the Hilbert space of
states and the Schr\"{o}dinger evolution equation for the wavefunction, it
rests to show that this wavefunction admits the definition of a density of
probability. The usual definition of probability density%

\begin{equation}
\rho=\Psi^{\ast}\Psi=\left|  \Psi\right|  ^{2}\label{29}%
\end{equation}
can be employed with some care in interpreting its meaning. The wavefunction
here is valued on the canonical coordinates $x^{i}$, rather than on
eigenvalues of the physical position observables $\widehat{X}^{i}$, which do
not commute. Thus $\rho(\vec{x},t)d^{3}x$ must be interpreted as the
probability of finding the canonical coordinate of the particle in the volume
$d^{3}x$ around the point $\vec{x}$ at time $t$.

To extract information about\ the physical coordinate position of the
particles from the wavefunction it is necessary to expand it in eigenfunctions
of the position operators $\widehat{X}^{i}=x^{i}+i\theta^{ij}\partial_{j}/2$.
Of course, since the $\widehat{X}^{i}$ do not commute, it is impossible to
find a base for all the spatial directions simultaneously. This is the simple
manifestation of the fact that, although the spacetime is endowed with a
pointwise structure, it is impossible to localize a particle on a given point
of it in a measurement. The background Neveau-Schwartz field always acts
conspiring against any attempt in this direction, enforcing the results of the
measurements to obey the uncertainty relation
\begin{equation}
\Delta X^{i}\Delta X^{j}\geq\left|  \theta^{ij}\right|  /2.\label{29.5}%
\end{equation}
This solves the open question about the possibility of the existence of the
wavefunction and how to interpret it \cite{27,27.5} in the context of the
string inspired noncommutativity. Had we done the wrong interpretation of the
canonical position observables $\hat{x}^{i}$ as the physical ones, the
uncertainty relation (\ref{29.5}) would be lost in the Schr\"{o}dinger
formulation. Of course, due to the smallness expected for $\theta^{ij},$ in
practice\ the noncommutative effects are not possible to be perceived by a
direct verification of (\ref{29.5}), which would demand an energy to work on a
length scale far beyond the limits of validity of nonrelativistic quantum
mechanics, but through its indirect consequences on other observable
quantities, like atomic energy spectrum \cite{27,27.3}, phase shifts on the
Aharonov -Bohm effect \cite{40}, etc.

By differentiating (\ref{29}) and using (\ref{26}) we obtain
\begin{equation}
\frac{\partial\rho}{\partial t}=\frac{i\hbar}{2m}\nabla\cdot\left(  \Psi
\nabla\Psi^{\ast}-\Psi^{\ast}\nabla\Psi\right)  -\frac{i}{\hbar}\left(
\Psi^{\ast}V\star\Psi-\Psi\left(  V\star\Psi\right)  ^{\ast}\right)
,\label{30}%
\end{equation}
where the term$\ i\hbar\left(  \Psi\nabla\Psi^{\ast}-\Psi^{\ast}\nabla
\Psi\right)  /2m$ is well known from ordinary quantum mechanics and
corresponds in that case to the current density $\vec{J}$. Notice that since
the canonical coordinates $x^{i}$ do not represent the physical coordinates of
the particles, the fact that the usual continuity equation$\ \partial
\rho/\partial t+\nabla\cdot\vec{J}=0$ is not satisfied\ does not mean that
there is no local conservation law of probability. Such form of continuity
equation would be expected only if $\left|  \Psi(x^{i},t)\right|  ^{2}d^{3}x$
had the meaning of probability of finding the particle in a volume $d^{3}x$
around the point $\vec{x}$ at time $t$.

Actually, the extra term that appeared in (\ref{30}) due to the
noncommutativity is not responsible for any inconsistency of the theory, but
its role must be understood. One way to justify its presence is to perceive
that it is essential in order to assure that the \textit{equivariance}
property \cite{22.5} is satisfied by $\rho=\left|  \Psi\right|  ^{2}$. This
means that if $\rho(x^{i},t_{0})=\left|  \Psi(x^{i},t_{0})\right|  ^{2}$ at
some time $t_{0}$, then $\rho(x^{i},t)=\left|  \Psi(x^{i},t)\right|  ^{2}$ for
all $t$. In other words, $\rho(t)$ preserves its form as a functional of
$\Psi(t)$ for all times. This property is trivially satisfied because the
point of departure for the derivation of the equation (\ref{30}) was exactly
the definition of $\rho(x^{i},t)=\left|  \Psi(x^{i},t)\right|  ^{2}$ as the
density of probability for an arbitrary time and the noncommutative
Schr\"{o}dinger equation. Thus it is the noncommutative Schr\"{o}dinger
equation itself that\ enforces the appearance of the noncommutative term in
the local probability\ conservation law for a question of consistency. This
term could called ``\textit{equivariance} correction'', and be denoted by
$\Sigma_{\theta}$. It should be expected, however, that when integrated over
all the space $\Sigma_{\theta}$ vanish, as occurs with the one containing the
divergence. This is a\ necessary condition\ for the global conservation of
probability. By integrating the equation (\ref{30}) over the space\ and using
the properties of the star product\ it is easy to verify that
\begin{equation}
\int\left(  \frac{\partial\rho}{\partial t}+\nabla\cdot\frac{i\hbar}%
{2m}\left(  \Psi\nabla\Psi^{\ast}-\Psi^{\ast}\nabla\Psi\right)  \right)
d^{3}x=-\frac{i}{\hbar}\int\left(  \Psi^{\ast}V\star\Psi-\Psi\left(
V\star\Psi\right)  ^{\ast}\right)  d^{3}x=0,\label{32}%
\end{equation}
thanks to the antisymetry of $\theta^{\mu\nu}$.

\subsection{Analyzing From Hamilton-Jacobi Point of View}

The Hamilton-Jacobi formalism associated with the NCQM is found by applying
the same procedure previously discussed when considering the NCQFT. We write
the wavefunction in its polar form $\Psi=R\,e^{iS/\hbar}$, substitute into
equation (\ref{26}), and separate its real and imaginary parts. For the real
part we obtain%

\begin{equation}
\frac{\partial S}{\partial t}+\frac{\left(  \nabla S\right)  ^{2}}%
{2m}+V+V_{nc}+Q_{K}+Q_{I}=0.\label{39}%
\end{equation}
The three new potential terms are defined as
\begin{equation}
V_{nc}=V\left(  x^{i}-\frac{\theta^{ij}\partial_{j}S}{2\hbar}\right)
-V\left(  x^{i}\right)  ,\label{40}%
\end{equation}%
\begin{equation}
Q_{K}=\operatorname{Re}\left(  -\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\psi}%
{\psi}\right)  -\left(  \frac{\hbar^{2}}{2m}\left(  \nabla S\right)
^{2}\right)  =-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R}{R},\label{41}%
\end{equation}
and
\begin{equation}
Q_{I}=\operatorname{Re}\left(  \frac{V\left(  x^{i}+i\frac{\theta^{ij}}%
{2}\partial_{j}\right)  \Psi}{\Psi}\right)  -V\left(  x^{i}-\frac{\theta
^{ij}\partial_{j}S}{2\hbar}\right)  .\label{42}%
\end{equation}
The $V_{nc}$ potential accounts for the noncommutative
classical\ interactions, while $Q_{K}$ and $Q_{I}$ for the quantum effects.
The origin of the $Q_{K}$ potential is from\ the kinetic term of the
Hamiltonian,\ closely related to the case of NCQFT discussed before. Its
definition is identical to the one of the quantum potential in ordinary
Bohmian mechanics \cite{18}, and here we will call it the kinetic quantum
potential. The remaining term, $Q_{I}$, is the potential that will accounts
for the quantum effects that came from the interaction term, and thus should
be called interaction quantum potential.

The imaginary part of the Schr\"{o}dinger equation yields
\begin{equation}
\frac{\partial R^{2}}{\partial t}+\nabla\cdot\left(  R^{2}\frac{\nabla S}%
{m}\right)  +\frac{2R}{\hbar}\operatorname{Im}\left(  e^{-\frac{i}{\hbar}%
S}V\star(Re^{\frac{i}{\hbar}S})\right)  =0,\label{44}%
\end{equation}
which\ is identical to the equation (\ref{30}) for probability conservation,
now written in function of the $R$ and $S$ fields.

Returning to equation (\ref{39}), we see that while$\ V_{nc}$\ and $Q_{K}%
$\ have their partners in NCQFT, $Q_{I}$, on the other hand, has no analog
when a comparison is done with field theory. It could be written as
$Q_{I}(x^{i},S,\partial^{i}S,\partial^{ij}S,\partial^{ijk}S,...,R,\partial
^{i}R,\partial^{ij}R,\partial^{ijk}R,...)$, keeping in mind, of course, that
the higher derivative interactions came from a star product of $V$ with the
$\Psi$ field, and therefore $Q_{I}$ is not exactly an arbitrary function of
$x^{i},S,R$ and their partial derivatives. When $V$ is polynomial, $Q_{I}$
contains a finite number of derivatives, otherwise it will be infinite series
containing derivatives of all orders, which characterizes a interaction
nonlocal in space. This case can still be analyzed, at least perturbatively. A
general\ discussion of how to handle mathematically nonlocal particle
mechanics and field theory can be found in \cite{36,36.5}, and in the context
of noncommutative theories in \cite{37}. The reason for the absence of the
interaction\ quantum potential $Q_{I}$ in the case of NCQFT is that in that
case there\ is no star product between the potential and the wavefunctional in
the Schr\"{o}dinger equation. In the case of\ quantum field theory the
potential term is valued on the star algebra and its product with the
wavefunctional is the ordinary one, and in the quantum mechanical case the
potential is valued on the ordinary algebra and multiplies the wave function
with the star product.

Notice that in case of considering quantum field theory with an external
potential it will be incorporated in the quadratic term of the Hamiltonian
with the star product, but\ its product with the wavefunctional will still\ be
the ordinary one.\ This point marks a difference between the implications of
noncommutativity for high and low energy phenomena. At least in what concerns
the structure of the equations in the Hamilton-Jacobi formalism, the
complexity and richness of the new physics originated from the
noncommutativity may be manifest in a more impressive\ way\ in the low energy processes.

\subsection{The Classical and Commutative Limits}

The most quoted criterion for finding the classical and commutative limits of
a quantum mechanical systems is $\hbar\rightarrow0$. As pointed out when
discussing the NCQFT, the adoption of this criterion\ as universal for the
classical limit\ has\ several problems. For the determination of the
commutative limit the most employed criterion is $\theta^{ij}\rightarrow0$
\cite{1,1.5}. However, arguments for its failure\ in the most general case
were presented in the framework of NCQFT in the Third Section. The arguments
presented there are still valid and reinforced here, where they become yet
stronger by the presence of the additional\ quantum potential $Q_{I}$ and the
\textit{equivariance} correction $\Sigma_{\theta}$. Among other criteria
commonly used for the classical limit are high quantum numbers ($n\rightarrow
\infty$), large mass ($m\rightarrow\infty$), and short de Broglie wavelength.
All of these criteria are not universal, and can be obtained from the
application for particular systems of the most general and systematic method
based on the generalized\ Hamilton-Jacobi formalism. The last,\ according to
the case under consideration, furnishes the best choice as the parameter to vary.

The conditions for the attainment of the classical and commutative limits of
the theory can be obtained easily from the examination of the system of
equations constituted by (\ref{39}) and (\ref{44}). For simplicity,\ we shall
attain our considerations to the case where $\Sigma_{\theta}\rightarrow0$,
which will be the one of interest for the examples that follow.\ According to
the prescription for the identification of the quantum effects proposed in
Section 3 and applied in this subsection to NCQM, all the quantum
manifestations of a given noncommutative quantum mechanical system must be
governed by $Q_{K}$ and $Q_{I}$ Thus, in the limit where $Q_{K}+Q_{I}%
\rightarrow0$ the system is expected to present a classical behavior.

By looking for the criteria for the commutative limit, however, we see that
the conditions that must be imposed on the set $\{V_{nc},Q_{K},Q_{I}\}$\ are
less trivial. While useful for the formal analysis comparing NCQM with NCQFT,
the\ grouping of the quantum contributions in the two potentials $Q_{K}$ and
$Q_{I}$ is not convenient for the identification of the noncommutative quantum
effects. This can be done by following an approach similar to the one adopted
for NCQFT. We define the noncommutative quantum contributions by
\begin{equation}
Q_{nc}=Q_{K}+Q_{I}-Q_{c},\label{44.2}%
\end{equation}
where
\[
Q_{c}=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R_{c}}{R_{c}}\text{ \ ,\ \ \ }%
R_{c}=\sqrt{\psi_{c}^{\ast}\psi_{c}}.
\]
$\psi_{c}$ is the wavefunction obtained from the commutative Schr\"{o}dinger
equation containing the usual potential $V(x^{i})$, that is, the equation
obtained by making $\theta^{ij}=0$ on (\ref{26}) before solving it. The
conditions to assure that (\ref{39}) and (\ref{44}) assume a form identical to
the one corresponding to a commutative system are
\begin{equation}
V_{nc}+Q_{nc}\rightarrow0.\label{44.4}%
\end{equation}
The conditions for the classical limit can now be written as
\begin{equation}
Q_{c}+Q_{nc}\rightarrow0.\label{44.6}%
\end{equation}

\section{Simple Applications}

\subsection{Noncommutative Harmonic Oscillator}

Here, we show a simple application of the ideas presented in the last section.
Consider a\ two dimensional noncommutative harmonic oscillator. In two
dimensions (\ref{1}) can be written as
\begin{equation}
\lbrack\widehat{X}^{\mu},\widehat{X}^{\nu}]=i\theta\epsilon^{\mu\nu
}.\label{44.8}%
\end{equation}
The Hamiltonian of the system is given by
\begin{equation}
H=\frac{1}{2m}\left(  p_{x}^{2}+p_{y}^{2}\right)  +\frac{1}{2}mw^{2}\left[
\left(  x-\frac{\theta}{2\hbar}p_{y}\right)  ^{2}+\left(  y+\frac{\theta
}{2\hbar}p_{x}\right)  ^{2}\right]  ,\label{45}%
\end{equation}
where $m$ is the particle mass and $w$ the frequency of the oscillator.

The corresponding Schr\"{o}dinger equation in polar coordinates is
\begin{align}
i\hbar\frac{\partial\Psi_{\theta}\left(  r,\varphi,t\right)  }{\partial t}  &
=H_{\theta}\Psi_{\theta}\left(  r,\varphi,t\right) \nonumber\\
& =-\frac{\hbar^{2}}{2m}\left(  1+\left(  \frac{mw\theta}{2\hbar}\right)
^{2}\right)  \left(  \partial_{r}^{2}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}%
}\partial_{\varphi}^{2}\right)  \Psi_{\theta}\left(  r,\varphi,t\right)
\nonumber\\
& +\left(  i\frac{m}{2}\theta w^{2}\partial_{\varphi}+\frac{m}{2}w^{2}%
r^{2}\right)  \Psi_{\theta}\left(  r,\varphi,t\right)  .\label{46}%
\end{align}
By plugging $\Psi_{\theta}\left(  r,\varphi,t\right)  =e^{-iEt/\hbar}%
\psi_{\theta}\left(  r,\varphi\right)  $ into (\ref{46}) we obtain the
eigenvalue equation $H_{\theta}\psi_{\theta}\left(  r,\varphi\right)
=E_{\theta}\psi_{\theta}\left(  r,\varphi\right)  $. The solution of this
eigenvalue equation\ is straightforward (see, for example \cite{38}), and
gives
\begin{equation}
\psi_{\theta}\left(  r,\varphi\right)  =\psi_{n,\alpha,\theta}\left(
r,\varphi\right)  =\left(  -1\right)  ^{n}\sqrt{\frac{n!\tilde{\zeta}}%
{\pi\left(  n+\left|  \alpha\right|  \right)  !}}\exp\left(  -\frac
{\tilde{\zeta}r^{2}}{2}\right)  \left(  \sqrt{\tilde{\zeta}}r\right)
^{\left|  \alpha\right|  }L_{n,\theta}^{^{\left|  \alpha\right|  }}\left(
\tilde{\zeta}r^{2}\right)  e^{i\alpha\varphi},\label{47}%
\end{equation}
where
\begin{equation}
L_{n,\theta}^{^{\left|  \alpha\right|  }}\left(  \tilde{\zeta}r^{2}\right)
=\overset{n}{\underset{m=0}{\sum}}\left(  -1\right)  ^{n}\left(
\begin{array}
[c]{c}%
n+\left|  \alpha\right| \\
n-m
\end{array}
\right)  \frac{\left(  \tilde{\zeta}r^{2}\right)  ^{m}}{m!},\text{
\ \ }E_{n,\alpha,\theta}=\hbar w_{nc}\left(  n+\frac{\left|  \alpha\right|
+1}{2}\right)  -\frac{m\theta w^{2}\alpha}{2},\label{48}%
\end{equation}
$n=0,1,2,$...is the principal quantum number, $\alpha=0,\pm1,\pm2$... is the
angular momentum quantum number and
\begin{equation}
\tilde{\zeta}^{2}=\frac{\left(  \frac{mw}{\hbar}\right)  ^{2}}{1+\left(
\frac{mw\theta}{2\hbar}\right)  ^{2}},\text{ \ }w_{nc}=2w\left(  1-\left(
\frac{mw\theta}{2\hbar}\right)  ^{2}\right)  ^{1/2}.\label{49}%
\end{equation}

Let us consider the state where $n=0$, whose wavefunction is
\begin{equation}
\Psi_{\theta}\left(  r,\varphi,t\right)  =e^{-iEt/\hbar}\sqrt{\frac
{\tilde{\zeta}}{\pi\left|  \alpha\right|  !}}\exp\left(  -\frac{\tilde{\zeta
}r^{2}}{2}\right)  \left(  \sqrt{\tilde{\zeta}}r\right)  ^{\left|
\alpha\right|  }e^{i\alpha\varphi}.\label{50}%
\end{equation}
For this physical state the corresponding $V\ ,V_{nc}$, $Q_{c},$ $Q_{nc}$ and
$\Sigma_{\theta}$ are
\begin{align}
V  & =\frac{1}{2}mw^{2}r^{2}\nonumber\\
V_{nc}  & =\left(  \frac{mw\theta}{2\hbar}\right)  ^{2}\frac{\alpha^{2}%
\hbar^{2}}{2mr^{2}}-\frac{m\theta w^{2}\alpha}{2}\nonumber\\
Q_{c}  & =-\frac{mw^{2}r^{2}}{2}+\hbar w\left(  \left|  \alpha\right|
+1\right)  -\frac{\alpha^{2}\hbar^{2}}{2mr^{2}}\label{51}\\
Q_{nc}  & =\left[  \sqrt{1-\left(  \frac{mw\theta}{2\hbar}\right)  ^{2}%
}-1\right]  \hbar w\left(  \left|  \alpha\right|  +1\right)  -\left(
\frac{mw\theta}{2\hbar}\right)  ^{2}\frac{\alpha^{2}\hbar^{2}}{2mr^{2}%
}\nonumber\\
\Sigma_{\theta}  & =0.\nonumber
\end{align}

A first inspection on the potentials shows that in\ the lowest energy state,
characterized by $\alpha=0$ and $r=0,$ the zero point energy is given by
\begin{equation}
E_{0,0,\theta}=Q_{c}+Q_{nc}=\frac{\hbar w_{nc}}{2},\label{52.5}%
\end{equation}
exactly what would be expected from quantum potentials, which must account for
all nonclassical behaviors. In the $\theta\rightarrow0$ limit it will be
reduced to $\hbar w$, the result of commutative quantum mechanics.

According to the criteria proposed in the last subsection, the condition for
the classical limit is $Q_{c}+Q_{nc}\rightarrow0$, while for the commutative
limit the condition is reduced to $V_{nc}+Q_{nc}\rightarrow0$. We start by the
commutative limit. In order that the sum $V_{nc}+Q_{nc}$ be negligible there
are two principal ways.

1) Taking the limit $\theta\rightarrow0$. This is the most obvious choice. In
this case (\ref{53}) can be written as%

\begin{align}
V  & =\frac{1}{2}mw^{2}r^{2}\nonumber\\
V_{nc}  & =0\nonumber\\
Q_{c}  & =-\frac{mw^{2}r^{2}}{2}+\hbar w\left(  \left|  \alpha\right|
+1\right)  -\frac{\alpha^{2}\hbar^{2}}{2mr^{2}}\label{53}\\
Q_{nc}  & =0\nonumber\\
\Sigma_{\theta}  & =0.\nonumber
\end{align}
This is exactly the result that would have been obtained is the
oscillator\ have been considered commutative from the beginning.

2) Modifying the length scale of the system. When $r$ is sufficiently big $V$
and $Q_{c}$ in (\ref{51}) can be perfectly approximated by $V=mw^{2}r^{2}/2$,
$Q_{c}=-mw^{2}r^{2}/2$, and\ the remaining of its constituents is totally
negligible. This is the same result that is obtained from (\ref{53}) in the
limit of large $r$. Of course the observables corresponding to the physical
coordinates will remain satisfying (\ref{29.5}), but, due to the smallness
expected for $\theta$ and\ length scales under consideration, for practical
purposes it will not have any physical implication.

In looking for the classical limit it is easy to see that there is no
parameter available to vary in order that $Q_{c}+Q_{nc}$ become negligible
without cancelling the interaction term, which would be a trivial and
uninteresting choice. This is a manifestation of the fact that the system can
not assume a classical behavior in the state described by (\ref{50}). The
classical limit of the harmonic oscillator is achieved by constructing
coherent states, which are superpositions of states containing different
quantum numbers \cite{38.5}

This simple example of the harmonic oscillator is useful to illustrate the
spirit of the ideas defended here, that the commutative and classical limits
of a system must be considered as realized\ on its physical states, rather
than on the equations of motion.

\subsection{WKB Approximation in the Case of a One-Direction Interacting Potential}

This quite simple example is useful to complement the one presented in the
last subsection in a discussion about some implications of\ noncommutativity
for the quantum/classical passage and how to obtain the commutative limit by
varying other parameters than $\theta.$ Consider a particle in two dimensional
plane under the action of a one-direction interacting potential. The
noncommutative Schr\"{o}dinger equation is given by
\begin{equation}
i\hbar\frac{\partial}{\partial t}\Psi=\left[  \frac{\hat{p}_{x}^{2}}{2m}%
+\frac{\hat{p}_{y}^{2}}{2m}+V\left(  x-\frac{\theta\hat{p}_{y}}{2\hbar
}\right)  \right]  \Psi.\label{59}%
\end{equation}
Writing the wavefunction as
\begin{equation}
\Psi\left(  x,y,t\right)  =e^{-iEt/\hbar+ik_{y}y}\psi(x),\label{60}%
\end{equation}
we separate the variables, obtaining
\begin{align}
-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+V\left(  x-\frac{\theta k_{y}%
}{2}\right)  \psi & =E_{x}\psi\label{61}\\
\frac{\hbar k_{y}}{2m}  & =E_{y}.\label{62}%
\end{align}

Performing a WKB approximation inserting the $ansatz$
\begin{equation}
\psi(x)=e^{i\phi(x)/\hbar},\text{ }\phi=\phi_{0}+\hbar\phi_{1}+...\label{63}%
\end{equation}
on (\ref{61}) we find, after simplification
\begin{equation}
\Psi_{WKB}^{\pm}(x,y,t)=\frac{C_{\pm}}{\left[  2m\left(  E_{x}-V-V_{nc}%
\right)  \right]  ^{1/4}}\exp\left[  \frac{i}{\hbar}\left(  \pm\int2m\left(
E_{x}-V-V_{nc}\right)  ^{1/2}dx-\left(  E_{x}+E_{y}\right)  t+\hbar
k_{y}y\right)  \right]  .\label{64}%
\end{equation}

The conditions for the validity of this approximation are \cite{18}
\begin{align}
m\hbar\left|  \left(  V+V_{nc}\right)  ^{\prime}\right|  /\left[  2m\left(
E_{x}-V-V_{nc}\right)  \right]  ^{3/2}  & \ll1,\nonumber\\
m\hbar^{2}\left|  \left(  V+V_{nc}\right)  ^{\prime\prime}\right|  /\left[
2m\left(  E_{x}-V-V_{nc}\right)  \right]  ^{2}  & \ll1,\nonumber\\
m\hbar^{3}\left|  \left(  V+V_{nc}\right)  ^{\prime\prime\prime}\right|  /
\left[  2m\left(  E_{x}-V-V_{nc}\right)  \right]  ^{5/2}  & \ll1,\label{65}%
\end{align}
and so on for higher derivatives of $\left(  V+V_{n}\right)  ^{\prime
}=d\left(  V+V_{nc}\right)  /dx$. The potentials corresponding to the
wavefunction (\ref{64}) are
\begin{align}
V_{nc}  & =V\left(  x-\frac{\theta k_{y}}{2}\right)  -V(x)\nonumber\\
Q_{c}  & =-\frac{\hbar^{2}}{2m}\left(  E_{x}-V\right)  ^{1/4}\frac{d^{2}%
}{dx^{2}}\left(  E_{x}-V\right)  ^{-1/4}\nonumber\\
Q_{nc}  & =-\frac{\hbar^{2}}{2m}\left(  E_{x}-V-V_{nc}\right)  ^{1/4}%
\frac{d^{2}}{dx^{2}}\left(  E_{x}-V-V_{nc}\right)  ^{-1/4}+\frac{\hbar^{2}%
}{2m}\left(  E_{x}-V\right)  ^{1/4}\frac{d^{2}}{dx^{2}}\left(  E_{x}-V\right)
^{-1/4}\label{66}\\
\Sigma_{\theta}  & =0\nonumber
\end{align}

Notice that $\theta$ appears multiplying$\ k_{y}=k\sin\varphi$, where
$\varphi$ is the angle between the direction of motion and the direction where
the potential acts. It can thus be chosen as our parameter to vary regulating
the degree of deviation of the commutative behavior. If $\varphi$ is small
such that the $V_{nc}+Q_{nc}$ is negligible comparing with the other
potentials,\ the system assumes a commutative behavior. When the speed of the
particle is sufficiently slow, the system can achieve this limit for
relatively\ large angles. Of course here\ considerations similar to the ones
of the previous subsection apply. We are supposing that $\theta$ is small and
considering energy scales where quantum mechanics is valid, that is, on length
scales where the uncertainty relation (\ref{29.5}) is not expected to be
detectable. The only manifestations of noncommutativity\ that could be
relevant would be through\ $V_{nc}$ and $Q_{nc}$, which are suppressed
if$\ k_{y}=k\sin\varphi$ is small.

From the conditions (\ref{65}) one can infer that $\left|  Q_{c}%
+Q_{nc}\right|  \ll E_{x}-V-V_{nc}$, and thus the wavefunction is
semiclassical if they are satisfied. Observe that the turning points, where
$E_{x}-V-V_{nc}=0$ and the approximation fails, are shifted in comparison with
the ones of the analogous commutative problem due to the presence of $V_{nc}.$

\section{Discussion and Outlook}

In this work we presented an alternative interpretation for the meaning of the
canonical noncommutativity obtained from arguments of string theory. In order
to justify our point of view about the validity of interpreting (\ref{1}) as
being realized just by the coordinate observables of the particles, and not by
the spacetime itself, we discussed some of the aspects of the derivation of
the noncommutativity in the Landau problem and in the string context.
Arguments for the existence of a loophole in the usual interpretation of the
results found in the literature were presented.

When we consider the stringy noncommutativity from the new point of view, the
interpretation of the Weyl symbols as points of the spacetime, considered as
senseless in \cite{13,27,27.5}, but adopted by the great majority of the
string inspired articles, like \cite{7}, for example, is perfectly justified.
Although these articles do not presuppose our interpretation for
noncommutativity, their calculations\ are\ in accordance with it. The possible
problem in the legitimacy of the use of the Feynman graph approach on
noncommutative spaces, which was underlined in a discussion carried on in
\cite{15}, is absent here, since NCQFT is assumed as being based on a
commutative spacetime.

In the context of quantum mechanics the implications of adopting\ the new
point of view for noncommutativity are manifest into the foundations. The
common assertion that the concept wavefunction does not make sense in
noncommutative spaces \cite{13,27.5} loses its validity for the stringy
noncommutativity considered because it is not of the spacetime coordinates.
The wavefunction, however, has a different meaning in NCQM than in its
commutative counterpart. It is possible to define a consistent probability
density given by the expression $\rho=\left|  \Psi\right|  ^{2}$, but the
interpretation of $\rho(\vec{x},t)d^{3}x$ is that of being the probability of
finding the canonical coordinate of the particle in the volume $d^{3}x$ around
the point $\vec{x}$ at time $t$.

Other related topic of interest discussed was the use of the Schr\"{o}dinger
formulation and the associated Hamilton-Jacobi formalism to perform an
analysis of the structure of NCQFT and NCQM and their possible limiting
behaviors. A formal analysis comparing the potential terms according to the
derivatives present on their definitions revealed\ that\ NCQM is richer than
NCQFT from this point of view. While the\ complete description of\ NCQFT\ can
be done by considering the properties of the\ three potentials $V,$ $V_{nc}$
and$\ Q_{K}$, the corresponding one for NCQM is more complex, needing the four
potentials $V,$ $V_{nc}$, $Q_{K},$ $Q_{I}$ and the $\Sigma_{\theta}$ term
present in the probability conservation equation.

The conditions for the achievement of the classical and commutative limits of
NCQFT were determined as being $Q_{K}=Q_{c}+Q_{nc}\rightarrow0$ and
$V_{nc}+Q_{nc}\rightarrow0$, respectively. For NCQM our considerations were
restricted to the case where $\Sigma_{\theta}\rightarrow0$, which was the one
of interest for the examples of application presented. The conditions for the
classical limit were found to be $Q_{K}+Q_{I}=Q_{c}+Q_{nc}\rightarrow0$, and
for the commutative one $V_{nc}+Q_{nc}\rightarrow0$.

In the same way that $\hbar$ must not be considered as the agent responsible
for the quantum effects in ordinary and noncommutative field theories and
quantum mechanics, $\theta^{ij}$ must not be considered as the one responsible
for the noncommutative manifestations in NCQFT and NCQM. Although achieving
the commutative limit by setting $\theta^{ij}\rightarrow0$ was possible in the
examples presented, this route for the classical limit will not always be
available when Hamiltonians\ containing more complicated potentials are
considered \cite{40}. In these cases the possibility of finding other routes
in the spirit of the alternative ones discussed here must be analyzed.

There is the immediate possibility to apply the Hamilton-Jacobi formalism
present in this work\ to investigate the properties\ of some simple models of
noncommutative\ scalar field theory or\ noncommutative quantum mechanical
systems whose Hamiltonians possesses higher order derivatives or are nonlocal,
like, for example, the anharmonic oscillator and the hydrogen atom. The
extension of the ideas presented here is in principle possible for other
theories than the scalar field ones, like the noncommutative version of\ QED.

The generalized\ Hamilton-Jacobi formalism is a tool that can be used for
investigation of\ higher order derivative quantum theories in general. The
classical, commutative and any other limit that can be available for a given
model in an arbitrary higher order derivative theory can be found by
performing a polar decomposition of the wavefunction and following the same
steps done here.

\section*{Acknowledgments}

The author is greatly indebted to Nelson Pinto Neto for valuable discussions,
suggestions and for all the\ corrections\ on an earlier manuscript. We also
acknowledge Ronaldo Penna Neves for discussions in the earlier stage of this
work and Jos\'{e} Hel\"{a}yel Neto for discussions, encouragement and
suggestions on this manuscript. This work was financially supported by CAPES.

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