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\begin{document}


\title{\textbf{A Bohmian Interpretation for Noncommutative Quantum Field Theory and
Quantum Mechanics }}
\author{\ \textbf{\ \ \ G. Dourado Barbosa} \\
%EndAName
{\small \ \ \ }\\
{\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 develop Bohmian interpretations for noncommutative quantum field theory
and quantum mechanics. The conditions for the classical and commutative
limits for both theories are derived and applied for a\ formal discussion of
the quantum/classical and noncommutative/commutative passages. Classical
field theory and Newtonian mechanics are obtained as limit cases of
noncommutative quantum field theory and noncommutative quantum mechanics.
Some advantages of the Bohmian interpretation against the orthodox one are
underlined. An application of the theory for the noncommutative harmonic
oscillator is presented and worked\ out\ in detail. \vspace{5cm}

email: gbarbosa@cbpf.br

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

noncommutative quantum mechanics,\ Bohmian interpretation, classical and

$\left. \text{commutative}\right. $ limits

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

\pagebreak
\end{abstract}

\section{Introduction}

The natural appearance of noncommutativity of the canonical type in string
theory \cite{1}\ has been motivating an intense investigation of its
implications for quantum field theory and quantum mechanics \cite{2,3}. The
theoretical\ relevance of this new and growing-up branch of physics was soon
recognized, since it\ gives us the opportunity to understand very interesting
phenomena. Among them, we quote nonlocality and IR/UV mixing \cite{5}, new
physics at very short distances $\left.  \cite{3,6}\right.  $, and possible
implications of Lorentz violation \cite{7}. From the experimental point of
view, a great deal of\ effort has been devoted in the search for evidences of
possible manifestations of the noncommutative effects in cosmology, high
energy and low energy experiments\ \cite{8}.

In a previous work \cite{9}, we argued that, at least from the stringy point
of view, the canonical commutation relation that characterizes the theories
under consideration,%

\begin{equation}
\lbrack\widehat{X}^{\mu},\widehat{X}^{\nu}]=i\theta^{\mu\nu}\;,\label{1}%
\end{equation}
ought to be assumed as commutation relation of the particle coordinate
observables, instead of the spacetime coordinates. This fact was shown to have
implications in the way of performing the calculations of noncommutative
quantum field theory (NCQFT), and enforced a reinterpretation of the meaning
of the wavefunction in noncommutative quantum mechanics (NCQM).

The aim of this work is to construct consistent Bohmian interpretations
\cite{8.5}\ for NCQFT and NCQM. In doing that, we shall have an opportunity to
reinforce and deepen\ the ideas presented in \cite{9}. We develop ontological
theories, defined on the ordinary commutative spacetime, which reproduce all
the results of NCQFT and NCQM. The idea to combine noncommutative geometry
with a theory of hidden variables was previously proposed by Lee Smolin in
\cite{9.5}, where stochastic quantization was employed.

Part of our interest in the Bohmian approach\ comes from the fact that it is
an object of intense investigation and application in a wide range of branches
of Physics,\ like quantum field theory \cite{11}, phenomenology of high energy
physics \cite{12}, condensed matter \cite{14} and atomic-molecular physics
\cite{15}, among others. The enormous reheating\ of\ interest in the Bohmian
interpretation comes from multiple directions. From the experimental point of
view, there is an indication from condensed matter and
atomic-molecular\ physics for a reassessment of the fundamentals of quantum
theory \cite{16}. In fact, there is a strong effort in\ the direction of
finding experimental tests to distinguish between the ontological and orthodox
interpretations (see, for example \cite{17,17.5}).\ From the theoretical point
of view, on the other hand, there is a large amount of phenomena that do not
fit comfortably within the standard operator quantum formalism of the orthodox
interpretation. Among them, we quote the dwell and tunneling times \cite{18},
escape times and escape positions \cite{19} and scattering theory \cite{20}.
They are easily handled by the Bohm's ontological approach \cite{20.5}.

Beyond the general raising of interest in the physical community on the
Bohmian interpretation, a special motivation for its choice for NCQFT and NCQM
is the variety of evidences indicating that noncommutativity must, in some
way, be related to a quantum theory of gravitation; consequently, it may have
implications for quantum cosmology. First of all, NCQFT was shown to emerge
from a combination of General Relativity and the Heisenberg principle in
\cite{4}, and thus, in that context, it incorporates gravitation in itself.
After this first connection,\ an extension of a model originally proposed by
't Hooft for black hole scattering \cite{21.5} was presented that contains\ an
action analogous to the one of string theory\ with an antisymmetric tensor
background \cite{22.5}. This was the source of a noncommutativity between the
coordinate operators corresponding to\ in-going and out-going scattered
particles in the neighborhood of a black hole. Finally, the fact that NCQFT
appeared naturally in string \cite{1} and M- theory \cite{22.7} was pointed
out as a strong evidence that noncommutativity is a general feature of a
unified theory of quantum gravity \cite{2,3}.

The inadequacy of the application of the Copenhagen interpretation for quantum
cosmology has been stressed for long time by many proeminent physicists, like
Feynman\ \cite{23} (for a review of the subject see, for example,
\cite{23.2}). As an alternative to the Copenhagen interpretation, the Bohmian
one is employed in several works of quantum cosmology (see \cite{23.5} and
references therein). Thus, it is important to show, having in mind future
applications for this area, that the string-inspired noncommutativity is
compatible with the Bohmian interpretation of quantum theory.

The organization of this work is this following. In Section 2, we summarize
the essential concepts of the Bohmian interpretation illustrating their
application for the formulation of the Bohmian noncommutative quantum field
theory (BNCQFT). Section 3 deals with noncommutative Bohmian quantum mechanics
(BNCQM). After an informal presentation of the construction of the theory of
motion, we formalize it in a simple and compact form. In Section 4, we discuss
the conditions for the achievement of the classical and commutative limits of
BNCQM and derive the ordinary Newtonian mechanics as one of its particular
cases. An application of the theory for the noncommutative harmonic oscillator
is presented in Section 5. Finally, in Section 6, we end up with a general
discussion and summary\ of the principal results.

\section{Bohmian Interpretation for NCQFT}

\subsection{Preliminary Considerations}

Here, for the sake of completeness, we present a brief summary of the
background on noncommutativity effects in quantum field theory in the
Schr\"{o}dinger picture necessary for the comprehension of the of this
section. More details may be found in \cite{9}. The construction of NCQFT is
generally done according to the following prescription. To obtain the
noncommutative version of a given field theory, take its action and replace
the product of the fields by the star product. The latter is defined by
\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}\theta^{\mu\nu
}p_{\nu}}f(k)g(p)\nonumber\\
& =e^{\frac{i}{2}\theta^{\mu\nu}\frac{\partial}{\partial\xi^{\mu}}%
\frac{\partial}{\partial\eta^{\nu}}}f(x+\xi)g(x+\eta)\mid_{\xi=\eta
=0}.\label{2}%
\end{align}
The commutative coordinates $x^{i}$ are called Weyl symbols and, at least from
the stringy point of view \cite{9}, they ought to be considered as spacetime
coordinates. In this work, we shall assume that $\theta^{oi}=0$. This makes
the formulation of NCQFT quite simple, the Schr\"{o}dinger functional
differential equation being given by%

\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{3}%
\end{equation}
where $V_{\star}\left(  \phi\right)  $ is the usual commutative potential
$V\left(  \phi\right)  $ with the star product between the fields replacing
the ordinary one. As the discussion\ that follows is intended to be general,
we shall keep the potential $V\left(  \phi\right)  $ as a generic real one. A
detailed study of the renormalizability may be found in \cite{24}.

By writing the wavefunctional in its polar form, $\Psi=R\exp(iS/\hbar),$ the
Schr\"{o}dinger differential equation (\ref{3}) is shown to be equivalent to
the system of coupled equations%

\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]  \right\}  +V_{c}+V_{nc}%
+Q_{K}=0\label{4}%
\end{equation}
and
\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{5}%
\end{equation}
where%

\begin{equation}
V_{nc}=\int d^{3}x\left[  \operatorname{Re}\left(  V_{\star}\right)
-V\right]  ,\label{6}%
\end{equation}%
\begin{equation}
V_{c}=\int d^{3}xV,\label{6.5}%
\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{7}%
\end{equation}

Expression (\ref{4}) is a generalized Hamilton-Jacobi equation, and (\ref{5})
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 $t$. While $Q_{K}$ is responsible for the quantum effects, $V_{nc}$ is
the $\theta^{ij}$-dependent contribution of the real\ part of the classical
noncommutative potential. The splitting of the noncommutative effects
contained in $Q_{K}$ can be done by defining%

\begin{equation}
Q_{nc}=Q_{K}-Q_{c}.\text{ \ \ ,\ \ \ }Q_{c}=-\frac{\hbar^{2}}{2R}\int
d^{3}x\frac{\delta^{2}R_{c}}{\delta\phi^{2}\left(  \vec{x}\right)
},\label{7.3}%
\end{equation}
where $R_{c}=\sqrt{\Psi_{c}^{\ast}\Psi_{c}},$ and $\Psi_{c}$ is the solution
of (\ref{3}) with $\theta^{ij}=0$, that is , the solution of the commutative
associated equation.

\subsection{On the Cornerstones of The Bohmian Philosophy}

The formalism to be\ presented from now on is on the same line of Bohm and
followers (see for example \cite{25,21}). We shall not center our attention
into the fundamentals, since in the case of quantum field theory they are
essentially the same as in the ordinary Bohmian theory \cite{25,22} for the
commutative and noncommutative versions, with a small modification due to the
presence of an extra potential $V_{nc}$ in the Hamilton-Jacobi equation. This
is a consequence of the fact that the noncommutativity (\ref{1}) of the
spatial coordinate observables does not involve the degrees of freedom of
NCQFT. In the case of NCQM, on the other hand, the noncommutativity\ touches
directly in the degrees of freedom of the system \cite{25.5}, which are the
particle coordinates. This is the source of great conceptual differences
between noncommutative and ordinary quantum mechanics, and this will be the
reason for a reformulation of the fundamentals of the theory, as shown in
Section 3.\ Independently of the modifications that will be necessary there,
the essential ideas that lie behind the Bohmian interpretation are always the
same, and we\ summarize them here in the context of field theory.

The Bohmian approach is founded on the assumption that the complete
characterization of a quantum system cannot not be provided by a wavefunction
alone. The description of individual processes, which are not statistical in
character, enforces the adoption of an objective view of matter. In order to
comprise the\ notion of objective reality with the known results from quantum
theory, an individual physical system is assumed to be\ composed by\ a wave
propagating with a field. The field moves under the guidance of the wave,
which satisfies the Schr\"{o}dinger equation (\ref{3}) and\ contains the
information how the energy of the field must be directed.

Since the field $\phi$ has an ontological character, we must assume that it
has a well-defined value for all $\vec{x}$ at each instant $t$, whatever the
state of the wavefunctional $\Psi$ is. The equation proposed for its time
evolution is
\begin{equation}
\frac{\partial\phi(\vec{x},t)}{\partial t}=\left.  \frac{\delta S\left[
\phi(\vec{x}),t\right]  }{\delta\phi(\vec{x})}\right|  _{\phi(\vec{x}%
)=\phi(\vec{x},t)},\label{8}%
\end{equation}
once we have specified the initial configuration of the field $\phi_{0}%
(\vec{x})$. This is the same equation as in\ ordinary classical field theory
in the Hamilton-Jacobi formalism. The crucial difference is that here $S$
plays a fundamental role for the dynamics. In the usual classical scalar field
theory, the field evolution is governed by the Klein-Gordon equation\ and the
dynamics dictated by this equation determines the evolution of the field $S$
in\ configuration space. In other words, the Hamilton-Jacobi equation is just
another way to encode the mathematical information contained in Klein-Gordon
equation. The ontological approach, on the other hand, gives to the field $S$
and the coupled field $R$ a primary role, in the sense that the equation of
motion (\ref{8}) can be solved only after the evolution of the fields
$R(x^{i},t)$ and $S(x^{i},t)$ is known. Since the $S$-field\ with $R$ is the
solution of a system of coupled equations ((\ref{4}) and (\ref{5})), which has
the same physical content as the Schr\"{o}dinger equation (\ref{3}), it
contains encoded in itself the quantum and noncommutative\ effects of the
theory. These will be manifest in the ``field trajectory'', which is the
solution to the equation (\ref{8}).

A good intuitive picture of the quantum and noncommutative\ effects as being
deviations from the ordinary classical behavior in the evolution of the system
can be obtained by writing an equation of motion\ for the field coordinate
similar to the Klein-Gordon one of the usual formalism. This can be done by
applying a functional derivative $\delta/\delta\phi$ to equation (\ref{4}).
The resulting equation is%

\[
\left[  \frac{\partial}{\partial t}+\int d^{3}x\left(  \frac{\delta S}%
{\delta\phi}\frac{\delta}{\delta\phi}\right)  \right]  \frac{\delta S}%
{\delta\phi}-\nabla^{2}\phi(\vec{x},t)+\frac{m^{2}}{\hbar^{2}}\phi(\vec
{x},t)+\hspace{4cm}%
\]%
\begin{equation}
\hspace{4cm}+\left.  \left[  \frac{\delta V_{c}\left[  \phi(\vec{x}),t\right]
}{\delta\phi(\vec{x})}+\frac{\delta V_{nc}\left[  \phi(\vec{x}),t\right]
}{\delta\phi(\vec{x})}+\frac{\delta Q_{K}\left[  \phi(\vec{x}),t\right]
}{\delta\phi(\vec{x})}\right]  \right|  _{\phi(\vec{x})=\phi(\vec{x}%
,t)}=0.\label{9}%
\end{equation}
By identifying $\frac{\delta S}{\delta\phi}=\frac{\partial\phi}{\partial t}$,
using$\ \frac{d}{dt}=\frac{\partial}{\partial t}+\int d^{3}x$\ $\frac
{\partial\phi}{\partial t}\frac{\delta}{\delta\phi}$ and equation (\ref{7.3}),
and by noting that$\frac{d}{dt}\dot{\phi}=\frac{\partial}{\partial t}\dot
{\phi},$ equation (\ref{9}) can be simplified to%

\begin{equation}
\square\phi(\vec{x},t)+\frac{m^{2}}{\hbar^{2}}\phi(\vec{x},t)+\left.  \left[
\frac{\delta V_{c}}{\delta\phi(\vec{x})}+\frac{\delta V_{nc}}{\delta\phi
(\vec{x})}+\frac{\delta Q_{c}}{\delta\phi(\vec{x})}+\frac{\delta Q_{nc}%
}{\delta\phi(\vec{x})}\right]  \right|  _{\phi(\vec{x})=\phi(\vec{x}%
,t)}=0\text{. }\label{10}%
\end{equation}
This is the Klein-Gordon equation with three extra inhomogeneous force terms
$\delta V_{nc}/\delta\phi(\vec{x})$, $\ \delta Q_{c}/\delta\phi(\vec{x})$ and
$\delta Q_{c}/\delta\phi(\vec{x}).$ The reason for calling $V_{nc}$, $Q_{c}$
and $Q_{nc}$ potentials is justified exactly by this fact. They are
responsible for the new quantum and noncommutative force terms that account
for the deviations of the field evolution from the one expected for
an\ ordinary commutative classical field.

Now, let us see how the noncommutative /commutative and quantum/classical
transitions occur. In the commutative limit, we wish that the equations
(\ref{4}) and (\ref{5}) coincide with the well-known results of conventional
de Broglie-Bohm theory (\cite{25,22}), which are obtained whenever
$V_{nc}+Q_{nc}\rightarrow0$. Here and in what follows,\ the symbol
$``A\rightarrow B"$ must be understood as taking a limit such that
$A-B\footnote{Since $S=S(\hbar,\theta,n,..)$ and $R=R(\hbar,\theta,n,..)$, the
limits must be taken under conditions such that all the terms in (\ref{4}) and
(\ref{5}) are well behaviored when they are considered.}$ is sufficiently
small to be neglected in comparison with the other quantities under
consideration. Thus, $V_nc+Q_nc\rightarrow0$, for example, means that
$V_nc+Q_nc$ is negligible when compared to the sum of the other terms of
the\ Hamilton-Jacobi equation. In this case, (\ref{10}) assumes a form
identical to the quantum Klein-Gordon equation of the ordinary commutative
Bohmian theory. The limit of ordinary classical field theory is then attained
if $Q_c\rightarrow0.$

It is also possible to achieve the classical limit without canceling the
noncommutative contributions. If the conditions are such that $Q_{c}%
+Q_{nc}\rightarrow0$, we may write the potential term of (\ref{10}) as $\delta
V_{c}/\delta\phi(\vec{x})+\delta V_{nc}/\delta\phi(\vec{x})=dV_{\star}/d\phi$.
In this case, equation (\ref{10}) can be recognized as\ the noncommutative
classical Klein-Gordon one \cite{25.7}, and the commutative classical field
theory is obtained if $V_{nc}\rightarrow0$, that is, when $V_{\star
}\rightarrow V.$

As it can be seen, in the Bohmian interpretation the conditions for the
achievement of the classical and commutative limits of NCQFT, previously
proposed in \cite{9}, have a deeper physical meaning. They are more than
mathematical conditions that assure that the set of equations (\ref{4}) and
(\ref{5}) take the characteristic form that enforces the physical\ system to
follow a classical or commutative behavior. Here, they\ represent the
conditions under which the deviations from the classical or commutative
behavior in the equation of motion of an objective field are suppressed.

\section{Bohmian Interpretation for NCQM}

\subsection{Background on NCQM}

The results of NCQM discussed in \cite{9} essential for this work can be
summarized as follows. The Hilbert space of states is the same as in the
commutative quantum mechanics, and the noncommutative Schr\"{o}%
dinger\ equation is given by
\begin{align}
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)\;\nonumber\\
& =-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi(x^{i},t)+V\left(  x^{i}+i\frac
{\theta^{ij}\partial_{j}}{2}\right)  \Psi(x^{i},t).\label{14}%
\end{align}
The operators $\widehat{X}^{i}=x^{i}+\theta^{ij}\partial_{j}/2$ are the
observables that correspond to the physical positions of the particles, and
$x^{i}$ are the associated canonical coordinates. Since the $\widehat{X}^{i}$
do not commute and satisfy the relation (\ref{1}), the particles cannot be
localized in a measurement process. The usual expression for the definition of
probability density, $\rho=\Psi^{\ast}\Psi=\left|  \Psi\right|  ^{2}$, is
valid. However, $\rho(\vec{x},t)d^{3}x$ must be interpreted as the probability
that the system is found in a configuration such that the canonical coordinate
of the particle is contained in the volume $d^{3}x$ around the point $\vec{x}$
at time $t$.

The Hamilton-Jacobi formalism for NCQM\ is found by applying the same
procedure discussed previously\ when NCQFT was considered. We write the
wavefunction in its polar form, $\Psi=R\,e^{iS/\hbar}$, substitute it in
equation (\ref{14}), 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{15}%
\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{16}%
\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{17}%
\end{equation}
and
\begin{equation}
Q_{I}=\operatorname{Re}\left(  \frac{V\left(  x^{i}+\frac{i\theta^{ij}}%
{2}\partial_{j}\right)  \Psi}{\Psi}\right)  -V\left(  x^{i}-\frac{\theta
^{ij}\partial_{j}S}{2\hbar}\right)  .\label{18}%
\end{equation}
\ $V_{nc}$ is the\ potential that\ accounts for the noncommutative
classical\ interactions, while $Q_{K}$ and $Q_{I}$ account\ for\ the quantum
effects. The noncommutative contributions contained in the last two can be
split out by defining
\begin{equation}
Q_{nc}=Q_{K}+Q_{I}-Q_{c}\text{,}\label{18.3}%
\end{equation}
where
\begin{equation}
Q_{c}=-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R_{c}}{R_{c}}\text{ \ ,\ \ \ }%
R_{c}=\sqrt{\Psi_{c}^{\ast}\Psi_{c}}.\label{18.6}%
\end{equation}
$\Psi_{c}$ 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 $\theta^{ij}=0$ in (\ref{14}) before solving it.

The imaginary part of the Schr\"{o}dinger equation yields the local
probability conservation law
\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^{-iS/\hbar}%
V\star\left(  R\,e^{iS/\hbar}\right)  \right)  =0.\label{19}%
\end{equation}
The last term on the LHS of (\ref{19}) has its presence justified in order to
assure the \textit{equivariance }property \cite{9,27} of the probability
density, and is denoted by
\begin{equation}
\Sigma_{\theta}=\frac{2R}{\hbar}\operatorname{Im}\left(  e^{-iS/\hbar}%
V\star\left(  R\,e^{iS/\hbar}\right)  \right)  .\label{19.5}%
\end{equation}

\subsection{Constructing the Ontological Theory of Motion}

As previously discussed in Section 2, the ontological interpretation is
founded on the assumption that the complete characterization of a quantum
system cannot be provided by a wavefunction alone. In the quantum mechanical
case, the system is composed by the wavefunction, satisfying equation
(\ref{14}), and a point particle, which is assumed to have an objective meaning.

Having the equation for the evolution of the guiding wave $\Psi$, one still
has to look for an equation of motion for the particle. To find this equation,
it is interesting to consider the Heisenberg formulation and the equations of
motion for the observables \cite{27.01}. For the variables $\widehat{X}^{i},$
$\hat{x}^{i}$ and $\hat{p}^{i}$ they\ are\ given by
\begin{equation}
\frac{d\widehat{X}^{i}}{dt}=\frac{1}{i\hbar}[\widehat{X}^{i},H]=\frac{\hat
{p}^{i}}{m}+\frac{\theta^{ij}}{\hbar}\frac{\partial\widehat{V}(\widehat{X}%
^{i})}{\partial\widehat{X}^{j}},\label{20.5}%
\end{equation}%
\begin{equation}
\frac{d\hat{x}^{i}}{dt}=\frac{1}{i\hbar}[\hat{x}^{i},H]=\frac{\hat{p}^{i}}%
{m}+\frac{\theta^{ij}}{2\hbar}\frac{\partial\widehat{V}(\widehat{X}^{i}%
)}{\partial\widehat{X}^{j}},\label{21}%
\end{equation}
and
\begin{equation}
\frac{d\hat{p}^{i}}{dt}=\frac{1}{i\hbar}[\hat{p}^{i},H]=-\frac{\partial
\widehat{V}(\widehat{X}^{i})}{\partial\widehat{X}^{i}}.\label{22}%
\end{equation}

Since $p^{i}=\partial^{i}S$\ in the classical Hamilton-Jacobi
theory\footnote{It is easy to see from the Hamilton-Jacobi theory
\cite{27.1}\ that the relation $p^{i}=\partial^{i}S$\ is still valid for
Hamiltonians containing higher order powers of the $p^{i}$, like the one of
(\ref{14}). The relation between the canonical momenta and canonical position
coordinates, however, is no more $p^{i}=m\dot{x}^{i}.$}, we can define
\begin{equation}
X^{i}=x^{i}-\frac{\theta^{ij}}{2\hbar}\partial_{j}S(x^{i},t),\label{22.3}%
\end{equation}
inspired on the relation $\widehat{X}^{i}=x^{i}+\theta^{ij}\partial_{j}/2$. A
reasonable equation of motion for the particles is therefore
\begin{equation}
\frac{dX^{i}(t)}{dt}=\left.  \left[  \frac{\partial^{i}S(x^{i},t)}{m}%
+\frac{\theta^{ij}}{\hbar}\frac{\partial V(X^{i})}{\partial X^{j}}\right]
\right|  _{x^{i}=x^{i}(t)},\label{23}%
\end{equation}
where $S(x^{i},t)$ is the phase of $\Psi$.

Notice that the $S$-function on the RHS of (\ref{23}) is obtained directly
from the wavefunction, which is valued on canonical coordinates. The use of
canonical coordinates is, at least for the attainment of the wavefunction and
the $S$ field, unavoidable. Once the $S$-field is known, it is possible to
invert (\ref{22.3}) and express to the RHS of (\ref{23}) as a function of the
$X^{i}$-variables. However, this procedure is not necessary, and in
practice\ it is convenient to find the particle tracks by solving the
differential equation for the canonical coordinate $x^{i}(t)$,
\begin{equation}
\frac{dx^{i}(t)}{dt}=\left.  \left[  \frac{\partial^{i}S(x^{i},t)}{m}%
+\frac{\theta^{ij}}{2\hbar}\frac{\partial V(X^{i})}{\partial X^{j}}\right]
\right|  _{x^{i}=x^{i}(t)},\label{24}%
\end{equation}
and then obtain the $X^{i}(t)$ via (\ref{22.3})$.$ This will be the procedure
adopted as preferential for the problems of application and also for the
formalization of the theory, which will be done in the next subsection.
Although not essential for the foundations, equation (\ref{23}) will be useful
later in an intuitive approach for the classical and commutative limits of the theory.

One important property of (\ref{23}) is that the velocity field of the
particles is not defined on nodal regions of $\Psi$, where $S$ is undefined.
Thus, the particles cannot pass through these regions or occupy them. An
interesting consequence of this property is that, although the wavefunction is
valued in the canonical position variables, its vanishing can be adopted as a
boundary condition when it is desired that the particles do not pass through a
region. This is a non-trivial conclusion, since the wavefunction is valued on
canonical coordinates, and $\left|  \Psi\right|  ^{2}$ $d^{3}x$ does not
represent the probability that the particles are in the volume $d^{3}x$ around
the point $\vec{x}$ at time $t$. Indeed, it must exclusively be attributed to
the fact that the particles in the theory under consideration are objective
and obey (\ref{23}). Had one considered, for example, the problem of how to
apply boundary conditions on NCQM to calculate the energy levels of a particle
in an infinite square well potential from\ the orthodox point of view, there
would be no preferred answer. Thus the ontological interpretation provides an
unambiguous information in a situation where the orthodox one does not say
anything. This fact has also been observed in the context of ordinary
commutative quantum mechanics, for example in \cite{17.5}, where it is
employed as a possible source for an experimental test of the de Broglie-Bohm
interpretation against the orthodox one.

\subsection{The Basic Postulates}

In the last subsection, we have proposed an objective quantum theory of motion
for NCQM. Let us now\ summarize the complete theory in a formal structure.
This can be done with the help of the following postulates:

1) The spacetime is commutative and has a pointwise manifold structure. The
observables corresponding to\ position coordinates of the particles satisfy
the commutation relation
\begin{equation}
\lbrack\widehat{X}^{i},\widehat{X}^{j}]=i\theta^{ij}.\label{28}%
\end{equation}

2) A quantum system is composed by a point particle and a wave $\Psi$. The
particle moves in spacetime under the guidance of the wave, which satisfies
the Schr\"{o}dinger equation
\begin{equation}
i\hbar\frac{\partial\Psi(x^{i},t)}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla
^{2}\Psi(x^{i},t)+V(\widehat{X}^{i})\Psi(x^{i},t)\;,\label{29}%
\end{equation}
where the position observables can be represented in the coordinate space as
$\widehat{X}^{i}=x^{i}+\frac{i}{2}\theta^{ij}\partial_{j}$, and the $x^{i}$
are the canonical coordinates and represent points of\ the physical
space\footnote{In the intrinsic approach for noncommutative theories
considering $[\widehat{X}^{\mu},\widehat{X}^{\nu}]=i\theta^{\mu\nu}$ as a
true\ property of the spacetime,\ the commutative variables $x^{i}$ are known
as Weyl symbols. On that context they do not have any physical meaning, being
just auxiliary tools to perform the calculations \cite{17.7}.}.

3) The particle moves along the trajectory
\begin{equation}
X^{i}(t)=x^{i}(t)-\frac{\theta^{ij}}{2\hbar}\partial_{j}S(x^{i}%
(t),t),\label{29.5}%
\end{equation}
independent of observation, where $x^{i}(t)$ is the trajectory followed by the
canonical position variables. Their equation of motion is%

\begin{equation}
\frac{dx^{i}(t)}{dt}=\left.  \left[  \frac{\partial^{i}S(\vec{x},t)}{m}%
+\frac{\theta^{ij}}{2\hbar}\frac{\partial V(X^{i})}{\partial X^{j}}\right]
\right|  _{x^{i}=x^{i}(t)}\text{ \ ,}\label{30}%
\end{equation}
where $S(\vec{x},t)$ is the phase of $\Psi$. To find the path followed by a
particle, we must specify its initial position $X^{i}\left(  0\right)  $,
obtain the associated canonical one as $x^{i}(0)=X^{i}(0)+\frac{\theta^{ij}%
}{2\hbar}\partial_{j}S(x^{i}(0),0),$ solve (\ref{30}) and then obtain the
physical path via (\ref{29.5}).

The three postulates presented constitute a\ consistent theory of motion.
However, the theory presented is intended to be a finer view of NCQM, which
gives detailed description of the individual\ physical processes and provides
the same statistical predictions.\ In order to reproduce the statistical
predictions of NCQM, we can impose the additional requirement that at a
certain instant of time, which will be taken as the initial one, the\ random
distribution of the canonical position of the particles is given by$\ \left|
\Psi(\vec{x},0)\right|  ^{2}$. This assumption and the property of
\textit{equivariance} of the probability\ distribution, assured by the
evolution law of (\ref{19}) \cite{9}, are sufficient to guarantee that, at any
arbitrary time $t$, the probability of a particle to have its canonical
position inside a volume $d^{3}x$ around the point $\vec{x}$ is given by
$\left|  \Psi(\vec{x},t)\right|  ^{2}d^{3}x.$ Under this condition, all
observable quantities, as well as all the expected values that are computed by
employing the wavefunction, will agree with the ones computed considering the
orthodox interpretation of NCQM.

If we were to obtain information about the physical position of the particles
from the wavefunction, it\ would be necessary to expand it in eigenfunctions
of the observables $\widehat{X}^{i}=x^{i}+\frac{i}{2}\theta^{ij}\partial_{j}$,
which do not commute. Thus, it is impossible to localize a particle by
measuring its position coordinates simultaneously.\ This is an intuitive way
of understanding the origin of the uncertainty relation%

\begin{equation}
\Delta X^{i}\Delta X^{j}\geq\left|  \theta^{ij}\right|  /2\label{31}%
\end{equation}
in the Schr\"{o}dinger formulation \cite{9}. Here, the ontological
interpretation gives an additional intuition on how this uncertainty can be
understood from a finer point of view. In the ordinary de Broglie-Bohm theory,
the impossibility of determining simultaneously the position and momentum of a
particle is attributed to the perturbation introduced on $p^{i}=\partial^{i}S$
by the evolution of the wavefunction during the measurement process. The
uncertainty (\ref{31}) is generated by a similar mechanism, since the $X^{i}$
contain $\partial^{i}S$ in their definition. Notice that, contrary to the
ordinary de Broglie -Bohm theory, where the initial particle positions are not
perfectly known by imprecisions of the measurements, the initial positions of
the particles here are intrinsically indeterminable.

\section{The Classical and Commutative Limits}

Here, we show how to obtain the ordinary classical mechanics starting from
BNCQM. The theory, which was summarized in the postulates of the previous
section, is perfectly consistent for arbitrary values of $\theta^{ij}$. When
considering the attainment of the ordinary quantum mechanical or
classical\ laws as its limiting cases, however, it is important to take into
account the actual values that can be assumed for the noncommutativity
parameters $\theta^{ij}$. Since if non-zero,\ they should be very small, the
verification of the consequences of (\ref{1}) through the direct\ verification
of (\ref{31}), for example, is not expected to be possible, at least in the
energy scale where nonrelativistic quantum mechanics is valid. However, the
detection of its indirect manifestations on other observable quantities, like
the atomic energy spectrum, can in principle be done \cite{27.2,27.3}. Thus,
although very important for the fundamentals of the theory, the distinction
between $X^{i}$ and $x^{i}$ will not be relevant for any\ experiment measuring
the position of the particles. This gives support for the use of the
definition $\rho(\vec{x},t)d^{3}x=$ $\left|  \Psi\right|  ^{2}d^{3}x$
as\ being interpreted approximately\ as the probability of finding a particle
in the volume $d^{3}x$ around the point $\vec{x}$ at time $t$ \cite{27.2}. In
this section, we shall see how this argument can be refined in the context of
the Bohmian interpretation.

Our considerations will be restricted to the case where $\Sigma_{\theta
}\rightarrow0,$ which is the one of interest for the example of application in
the next section. In this case, our discussion will be centered on (\ref{15}),
since (\ref{19}) is assumed to be in the ordinary classical form. The
generalization including $\Sigma_{\theta}$ is straightforward. The conditions
for the commutative and classical limits in this case\ are formally the same
as the ones for field theory. The commutative limit\ is achieved by imposing
that $V_{nc}+Q_{nc}\rightarrow0$, as it\ can be seen from\ (\ref{15}) and
(\ref{18.3}). The classical limit is achieved if the quantum contributions are
dropped from (\ref{15}), that is, when\ $Q_{c}+Q_{nc}\rightarrow0.$

An intuitive view\ of how the quantum effects in (\ref{23}) appear as
deviations from the classical behavior\ can be obtained by the derivation of a
generalized second Newton law. When doing this, it is important to keep in
mind that the actual equation of motion for the particles is (\ref{23}), a
first-order differential equation in time. Differentiating (\ref{23}) with
respect to $t$, we obtain, after simplification using (\ref{15}) and
(\ref{22.3}),
\begin{equation}
m\frac{d^{2}X^{i}}{dt^{2}}=\left.  \left[  -\frac{\partial V\left(
X^{i}\right)  }{\partial X^{i}}+\frac{\theta^{ij}}{\hbar}\frac{\partial
V\left(  X^{i}\right)  }{\partial X^{j}\partial X^{k}}\dot{X}^{k}%
-\frac{\partial}{\partial x^{i}}\left(  Q_{c}(x^{i})+Q_{nc}(x^{i})\right)
\right]  \right|  _{x^{i}=x^{i}(t)},\label{33}%
\end{equation}
which in quantum mechanics plays a role analogous of the one of\ (\ref{10}) in
field theory. We can see the emergence of the ordinary second Newton law as a
particular case of (\ref{33}) by following the path:%

\[
\text{\textbf{NC Quantum Mechanics}}\rightarrow\text{\textbf{NC Classical
Mechanics}}\rightarrow\text{\textbf{C\ Classical Mechanics}\textit{,}}%
\]
where \textbf{NC} accounts for noncommutative and \textbf{C} for commutative.
After eliminating the quantum effects by finding conditions such that
$Q_{c}+Q_{nc}\rightarrow0$, we obtain the noncommutative classical second
Newton law, which can also be found by deforming the classical phase space
\cite{27.01, 28}:
\begin{equation}
m\frac{d^{2}X^{i}}{dt^{2}}=\left.  \left[  -\frac{\partial V\left(
X^{i}\right)  }{\partial X^{i}}+\frac{\theta^{ij}}{\hbar}\frac{\partial
V\left(  X^{i}\right)  }{\partial X^{j}\partial X^{k}}\dot{X}^{k}\right]
\right|  _{X^{i}=X^{i}(t)}.\label{33.3}%
\end{equation}

The commutative limit has to be achieved by imposing that $V_{nc}\rightarrow
0$. Formally, this step necessarily involves the limit $\theta^{ij}%
\rightarrow0$ or the limit of low momenta $p^{i}=\partial^{i}S\rightarrow0.$
The latter must be understood in the following way: for all arbitrary but
fixed values supported for $\theta^{ij},$ the momenta must not be sufficiently
big to justify the consideration of the terms containing $\theta^{ij}%
\partial_{j}S$\ on (\ref{15}). The two criteria will assure that
$V_{nc}\rightarrow0$, but\ the low momenta criterion is
particularly\ interesting for conceiving the classical limit with $\theta
^{ij}$ fixed. This is in the spirit of the ideas previously proposed in
\cite{9}, according to which to explain the noncommutative/commutative
transition one must not necessarily always appeal for the condition
$\theta^{ij}\rightarrow0$. Notice that, according to the reasoning developed,
taking the limit $V_{nc}\rightarrow0$ automatically assures that
$X^{i}\rightarrow x^{i}$, no matter the $\theta^{ij}$ are fixed or not when
considering this limit. Therefore, the potential $V_{nc}$ is, in a certain
sense, a measure of the deviation of the physical positions from the canonical ones.

In practice, since $\partial_{j}S$ and $\theta^{ij}$\ appear multiplying each
other in (\ref{15}), the application of both criteria for the commutative
limit will be equivalent to discard the terms containing $\theta^{ij}$ in
(\ref{15}),\ (\ref{22.3}), (\ref{23}) and (\ref{24}) by making $\theta^{ij}=0$
in these equations$.$ In doing this, (\ref{15}) is reduced to the classical
Hamilton-Jacobi equation, which allows the identification of $S$ with
Hamilton's principal function, and (\ref{33.3}) with the ordinary second
Newton law.

Now, we consider the alternative route:%

\[
\text{\textbf{NC Quantum Mechanics}}\rightarrow\text{\textbf{C Quantum
Mechanics}}\rightarrow\text{\textbf{C\ Classical Mechanics}\textit{.}}%
\]
After the transition for commutative quantum mechanics by the
condition\footnote{We are excluding the umprobable case where $V_{nc}=-Q_{nc}%
$} $V_{nc}+Q_{nc}\rightarrow0$, (\ref{23}) can be written as
\begin{equation}
m\frac{d^{2}x^{i}}{dt^{2}}=\left.  \left[  -\frac{\partial}{\partial x^{i}%
}\left(  V(x^{i})+Q_{c}(x^{i})\right)  \right]  \right|  _{x^{i}=x^{i}%
(t)},\label{33.6}%
\end{equation}
which is the generalized second Newton law of Bohmian mechanics \cite{22}.
When $Q_{c}\rightarrow0,$ the ordinary classical Newton law is attained.

As in the case of field theory, the transition from the noncommutative quantum
world\ to the commutative classical one occurs\ in a continuous and
transparent way, with the objective point particle being present during all
the process. This is regulated by the variation of the three potentials
$V_{nc},Q_{c}$ and $Q_{nc}$ until their contribution is zero or completely
negligible. The restoration of the status of probability of finding a particle
in the volume $d^{3}x$ around the point $\vec{x}$ at time $t$ by $\left|
\Psi(\vec{x},t)\right|  ^{2}d^{3}x$ is gradual and is\ perfectly understood as
a consequence of the identification between $x^{i}$ with $X^{i} $ in the limit
where $V_{nc}\rightarrow0.$

\section{Simple Application: Noncommutative Harmonic Oscillator}

Here, we show a simple application of the BNCQM for the analysis of a model
previously discussed in \cite{9}. Consider a\ two dimensional noncommutative
harmonic oscillator. In two dimensions (\ref{1}) is reduced to
\begin{equation}
\lbrack\widehat{X}^{\mu},\widehat{X}^{\nu}]=i\theta\epsilon^{\mu\nu
}.\label{34}%
\end{equation}
The position observables of the particles can therefore be represented by
$\widehat{X}^{i}=x^{i}-\theta\epsilon^{ij}\hat{p}_{j}/2\hbar$, and the
Hamiltonian written as
\begin{equation}
H=\frac{1}{2m}\left(  \hat{p}_{x}^{2}+\hat{p}_{y}^{2}\right)  +\frac{1}%
{2}mw^{2}\left[  \left(  x-\frac{\theta}{2\hbar}\hat{p}_{y}\right)
^{2}+\left(  y+\frac{\theta}{2\hbar}\hat{p}_{x}\right)  ^{2}\right]
,\label{35}%
\end{equation}
where $m$ and $w$ are the mass and frequency of the associated commutative 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)
\label{36}\\
& +\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)  ,\nonumber
\end{align}
whose solution is \cite{9}%

\begin{equation}
\Psi_{\theta}\left(  r,\varphi,t\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-iEt/\hbar},\label{36.5}%
\end{equation}
where $L_{n,\theta}^{^{\left|  \alpha\right|  }}\left(  \tilde{\zeta}%
r^{2}\right)  $ are the Laguerre polynomials
\begin{equation}
L_{n,\theta}^{^{\left|  \alpha\right|  }}\left(  \tilde{\zeta}r^{2}\right)
=\overset{n}{\underset{l=0}{\sum}}\left(  -1\right)  ^{n}\left(
\begin{array}
[c]{c}%
n+\left|  \alpha\right| \\
n-l
\end{array}
\right)  \frac{\left(  \tilde{\zeta}r^{2}\right)  ^{l}}{l!}\text{
\ \ \ },\text{ \ \ \ }\tilde{\zeta}^{2}=\frac{\left(  \frac{mw}{\hbar}\right)
^{2}}{1+\left(  \frac{mw\theta}{2\hbar}\right)  ^{2}},\label{36.7}%
\end{equation}
$n=0,1,2,$...is the principal quantum number and $\alpha=0,\pm1,\pm2$... is
the angular canonical momentum quantum number.

The energy levels are given by
\begin{equation}
E_{n,\alpha,\theta}=2\hbar w\left(  1-\left(  \frac{mw\theta}{2\hbar}\right)
^{2}\right)  ^{1/2}\left(  n+\frac{\left|  \alpha\right|  +1}{2}\right)
-\frac{m\theta w^{2}\alpha}{2}.\label{36.9}%
\end{equation}
Notice that, due to the noncommutative effects, the degeneracy of the energy
levels corresponding to\ the right-handed and left-handed polarizations for
the same $n$ is removed. When noncommutativity (\ref{1}) is assumed as being
originated from an action of a strong background field, as the Neveu-Schwartz
in the string context \cite{1}, or a magnetic field when a condensed matter
system is projected onto its lowest Landau level, the lifting of the
degeneracy can be \ intuitively understood as consequence of a chirality
introduced by the background field.

For simplicity, let us consider the state where $n=0$. In\ this state,
(\ref{36.5}) is simplified to
\begin{equation}
\Psi_{\theta}\left(  r,\varphi,t\right)  =\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-iEt/\hbar},\label{37}%
\end{equation}
and 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{38}\\
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}

The canonical trajectories are found by solving the equations
\begin{equation}
\frac{dx}{dt}=\frac{1}{m}\frac{\partial S}{\partial x}+\frac{\theta}{\hbar
}mw^{2}\left(  y+\frac{\theta}{2\hbar}\frac{\partial S}{\partial x}\right)
\text{ \ \ \ , \ \ \ }\frac{dy}{dt}=\frac{1}{m}\frac{\partial S}{\partial
y}-\frac{\theta}{\hbar}mw^{2}\left(  x-\frac{\theta}{2\hbar}\frac{\partial
S}{\partial y}\right)  .\label{39}%
\end{equation}
Changing into polar coordinates, and substituting $S=\alpha\hbar\varphi-Et $
in (\ref{39}), we find
\begin{equation}
\frac{dr}{dt}=0\text{ \ \ \ ,\ \ \ \ }\frac{d\varphi}{dt}=\frac{\alpha\hbar
}{r^{2}}-\frac{mw^{2}\theta}{\hbar}+\frac{mw^{2}\alpha\theta^{2}}{2\hbar
r^{2}},\label{40}%
\end{equation}
whose solutions are
\begin{equation}
r=r_{0}\text{ \ \ \ \ , \ \ \ \ \ }\varphi=\varphi_{0}+w_{\theta}t\text{,
\ \ \ }w_{\theta}=\left(  \frac{\alpha\hbar}{r_{0}^{2}}+\frac{mw^{2}%
\alpha\theta^{2}}{2\hbar r_{0}^{2}}\right)  .\label{41}%
\end{equation}
The physical radius and angle are
\begin{align}
R(t)  & =\sqrt{X^{2}(t)+Y^{2}(t)}=\left[  \left(  x-\frac{\theta}{2\hbar}%
\frac{\partial S}{\partial y}\right)  ^{2}+\left(  y+\frac{\theta}{2\hbar
}\frac{\partial S}{\partial x}\right)  ^{2}\right]  ^{1/2}\nonumber\\
& =r_{0}\left(  1-\frac{\alpha\theta}{r_{0}^{2}}+\frac{\alpha^{2}\theta^{2}%
}{4r_{0}^{4}}\right)  ^{1/2}=R_{o}\label{42}%
\end{align}
and
\begin{equation}
\Phi(t)=\arctan\left[  \frac{Y(t)}{X(t)}\right]  =\arctan\left[  \frac
{y(t)}{x(t)}\right]  =\varphi\left(  t\right)  .\label{43}%
\end{equation}
The velocity of the particles is tangential to their circular orbits, being
given by
\begin{equation}
\mathbf{V}\left(  t\right)  =\sqrt{\dot{X}^{2}(t)+\dot{Y}^{2}(t)}=R_{0}%
\dot{\varphi}=R_{0}w_{\theta}.\label{44}%
\end{equation}

As expected, the particle trajectories are circles. Although the absolute
value of the\ angular velocity $w_{\theta}$\ is the same for both the right
and left polarized states, their tangential velocity is not due to a small
difference in the radii of their corresponding orbits. It is this difference
that is responsible for the lifting of the degeneracy of the\ energy spectrum
(\ref{36.9}) of the oscillator. While the orbits corresponding to
the\ right-handed excitations have the associated energy levels shifted
downwards by a diminution of their radii with respect to the commutative one,
the left-handed excitations have their energy levels shifted upwards,
corresponding to motions on larger radii. Notice that when the system is\ in
the low energy state, characterized by $\alpha=0$, the particle is at rest at
$R=r_{0}$, as in the ordinary Bohmian theory. Analyzing (\ref{38}), it is
possible to see that the condition $V_{nc}+Q_{nc}\rightarrow0$\ is satisfied
if $\theta\rightarrow0$. In this case, we have $R(t)=R_{0}=r_{0}$, and
$\Phi(t)=\varphi(t)=\varphi_{0}+\alpha\hbar t/r_{0}^{2},$ exactly the results
corresponding to the conventional Bohmian orbits.

In a previous discussion of NCQM \cite{9}, it was underlined the importance of
adopting the correct interpretation of the $\widehat{X}^{i}$ as the
observables carrying the information about the physical position of the
particles and representing them as $\widehat{X}^{i}=x^{i}+\frac{i}{2}%
\theta^{ij}\partial_{j}$ when considering the Schr\"{o}dinger formulation. Had
one considered the canonical variables $x^{i}$ as the ones containing the
physical information about the position of the particles, the\ uncertainty
relation (\ref{31}), for example, would have been\ lost in the quantum
mechanical context.

Here, the detailed description of the particle motion provided by the Bohmian
approach reinforce the arguments presented in \cite{9}, and renders evident
the preference for the interpretation of $X^{i}$-variables as the true markers
of the physical\ position of the particles rather than the canonical ones. To
see this, observe that the lifting of the degeneracy of energy spectrum, which
was\ easily understood by considering the $X^{i}$ as the physical position
variables, does not have a satisfactory explanation if this role is attributed
to the canonical variables $x^{i}$. This would reduce the description of the
particle motion to the solution\ (\ref{41}), giving identical $r_{0}$ and
$\left|  w_{\theta}\right|  $ (and thus the same energy) for the orbits
corresponding to left-handed and right-handed polarizations of the oscillator.

\section{Discussion and Outlook}

In this work, we proposed ontological interpretations for NCQFT and NCQM. We
have shown how to construct consistent theories for objective\ fields and
point-particles living in the ordinary commutative spacetime and leading to
the same physical predictions of NCQFT and NCQM\ in the orthodox
interpretation. The only noncommutativity that must necessarily be\ present to
reproduce the predictions of NCQFT and NCQM is the one between the particle
coordinate observables. This result reinforces the previous claim \cite{9}
that the noncommutativity derived in the stringy context \cite{1} can (and
ought to) be interpreted as being a property of the particle coordinate
observables induced by the presence of a strong background field.

For field theory, this development of the Bohmian interpretation was
straightforward. The\ mathematical conditions for the classical and
commutative limits previously obtained in an interpretation-independent
discussion \cite{9} here were shown to acquire a precise physical meaning.
They represent the circumstances under which the deviations from the
commutative and classical behaviors on the evolution of the objective system
are negligible or null.

A deep change in the fundamentals of the theory characterized the quantum
mechanical case. The particles were shown to have commutative coordinates and
follow well-defined spacetime tracks. The realization of the noncommutativity
only between their coordinate observables means that, although the particles
are point-like, their complete localization in a process of measurement is
forbidden by the disturbances caused by the apparatus of measurement
interacting with the quantum system. The intrinsic uncertainty for the
localization during a measurement process must be faced on the same foot as
the one that forbids the simultaneous determination of the momentum and the
position of a particle in the ordinary Bohmian\ theory.

Beyond the clarity and elegance with which Bohmian approach explains the
quantum/classical passage there\ is the advantage that it offers unambiguous
predictions in situations where the standard orthodox one is ambiguous. This,
for example occurs when one wishes to know which boundary conditions must be
applied for the simple calculation of the spectrum corresponding to\ an
infinite square well potential, as we discussed in Section 3.

One interesting environment where the predictions of Bohmian interpretation
can be confronted against the predictions of other interpretations\ is the one
of quantum cosmology. Recently, noncommutativity at early times of the
universe was introduced by deforming the commutation relation of the
minisuperspace variables\ in a cosmological model based on the Kentowski-Sachs
metric, originating a noncommutative Wheeler-de Witt equation. Since in the
formalism of minisuperspace the Wheeler-de Witt equation is essentially
quantum mechanical, the application of the Bohmian interpretation developed in
this work for models like the one of \cite{35} is\ almost immediate. In\ case
of conceiving a quantum cosmology based on the canonical noncommutativity of
the spatial coordinates (\ref{1}), for example, the ideas presented in this
work may also be a good departure point.

Since the ontological interpretations have variants and are still under
construction, this work should not be considered a closed structure. Many of
the rules just stated are open and may be subject to reformulation after a
further discussion. There are still many open questions to explore in the
formulation from the theoretical point of view, like the extension of the
theory for many-bodies, where some care must be taken when considering charged
particles \cite{37} , etc. In the case of field theory it should be
interesting to apply the ideas introduced here to the case of noncommutative
QED. The de Broglie-Bohm version for ordinary QED is available in \cite{38}.

\section*{Acknowledgments}

The author is greatly indebted to Nelson Pinto-Neto and Jos\'{e}
Hela\"{y}el-Neto\ for relevant suggestions and for all the\ corrections\ on
this manuscript.\ He also acknowledges Jos\'{e} Acacio de Barros\ for useful
discussions underlining important aspects of the Bohmian interpretation.\ This
work was financially supported by CAPES.

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\end{document}