\documentclass[]{JHEP3}
\usepackage{bm,subequation}
\newcommand{\xbf}{\bm{x}}
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\title{Quantum Theory of Noncommutative Fields}
\author{J.M. Carmona, J.L. Cort\'es\\ Departamento de F\'{\i}sica Te\'orica,
Universidad de Zaragoza, Zaragoza 50009, Spain\\ 
E-mail: \email{jcarmona@posta.unizar.es}, \email{cortes@posta.unizar.es}}
%\author{J.L. Cort\'es\\ Departamento de F\'{\i}sica Te\'orica,
%Universidad de Zaragoza, Zaragoza 50009, Spain\\
%E-mail: \email{cortes@posta.unizar.es}}
\author{J. Gamboa, F. M\'endez\\ Departamento de F\'{\i}sica, 
Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile\\
E-mail: \email{jgamboa@lauca.usach.cl}, \email{fmendez@lauca.usach.cl}}
%\author{F. M\'endez\\ Departamento de F\'{\i}sica, 
%Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile\\
%E-mail: \email{fmendez@lauca.usach.cl}}
\abstract{
Generalizing the  noncommutative harmonic oscillator construction, we propose a
new extension of quantum field theory based on the concept of ``noncommutative 
fields''. Our description permits to break the usual particle-antiparticle 
degeneracy at the dispersion relation level and introduces naturally an
ultraviolet and an infrared cutoff. Phenomenological bounds for these new 
energy scales are given.}
\keywords{Noncommutativity, Lorentz invariance violation, Field theory}
%\pacs{04.60.-m,11.30.Cp}

%\preprint{}

\begin{document}

\section{Introduction}

Relativistic quantum field theory (RQFT) is the general framework of our 
present microscopical theories. Its validity in particle physics has been 
confirmed by experiments covering a very wide range of energies, from the 
eV to the TeV~\cite{cutoffs}. From a modern perspective~\cite{weinberg}, RQFT 
is the necessary form adopted by any low-energy or ``effective'' theory 
satisfying  the following three principles: special relativity, quantum 
mechanics and the cluster decomposition principle, which basically states that
distant experiments yield uncorrelated results. The success of RQFT
is then a confirmation of the validity of its ingredients, which describe
correctly low-energy phenomena.

There seems however to be strong difficulties in obtaining a RQFT containing
gravitation. Indeed there is no \textit{a priori} reason why RQFT should
be the correct framework of a high-energy theory, and even one or more of
its ingredients could fail at these energies. In particular, Lorentz invariance
could not be an exact symmetry at high energies, as recent developments
in quantum gravity suggest~\cite{qugr}. It is clear however that any 
high-energy theory of particle physics should reduce to a RQFT at low energies.

In this paper we propose an extension of the RQFT framework based on the
notion of what we will call ``a noncommutative field''. Motivated by the
appearance of noncommutative spaces in string theory~\cite{string}, 
there has recently been quite a few developments on ``noncommutative quantum
mechanics'' (NCQM), which is an extension of quantum mechanics (QM) 
consisting in the formulation of quantum mechanical systems 
on a noncommutative coordinate space (or even phase-space)~\cite{NCQM,nair}.
``Noncommutative quantum field theories'', meaning quantum field theories 
on such spaces, have also been studied~\cite{NCQFT}. These theories 
violate relativistic invariance~\cite{carroll} and modify in a peculiar
way the short-distance behavior of the theory (although there are still
ultraviolet divergences).

There is however another way of introducing a noncommutativity in quantum
field theory. In NCQM, the coordinates (and momenta), which are the 
degrees of freedom of the system, are made noncommutative.
The degrees of freedom in field theory are the fields at every point of space 
(and their conjugated momenta). Therefore the natural generalization of 
NCQM to field theory leads to a noncommutative field (instead of a field in a 
noncommutative space). In Sec.~\ref{sec:NCF} we show how a quantum theory of 
such a field can be constructed and then in Sec.~\ref{sec:prop} we will
study the properties of the extension of RQFT it provides. Finally in
Sec.~\ref{sec:phen} we will consider its phenomenological implications,
the bounds that present experimental status produce on the size of the
noncommutativity and the possibility to measure effects coming from this 
source.

\section{The noncommutative field}
\label{sec:NCF}

\subsection{Definition of the free noncommutative field theory}

Let us consider a scalar field theory with two fields $(\phi_1,\phi_2)$,
i.e., a complex field $\Phi=(\phi_1+i\phi_2)/\sqrt{2}$, and let us
introduce a noncommutativity between the two fields at every point of space
(for now we work in the Schr\"odinger picture where fields and momenta do
not depend on time)
\begin{equation}
\left[\Phi(\xbf),\Phi^\dagger(\xbf')\right]=\theta \,\delta^3(\xbf-\xbf').
\label{theta}
\end{equation}
We may consider at the same time a noncommutativity in the momenta
$\Pi=(\pi_1+i \pi_2)/\sqrt{2}$
\begin{equation}
\left[\Pi(\xbf),\Pi^\dagger(\xbf')\right]=B \,\delta^3(\xbf-\xbf'),
\label{B}
\end{equation}
where $\theta$ and $B$ are the parameters which parametrize the 
noncommutativity. The fields and their conjugated momenta are related by 
the conventional commutation relations
\begin{subequations}
\begin{eqnarray}
\left[\Phi(\xbf),\Pi^\dagger(\xbf')\right] &=& i \,\delta^3(\xbf-\xbf'), \\
\left[\Phi^\dagger(\xbf),\Pi(\xbf')\right] &=& i \,\delta^3(\xbf-\xbf').
\end{eqnarray}
\label{phipi}
\end{subequations}
The free theory of the complex noncommutative quantum field is completely
defined by the above commutation relations, together with the Hamiltonian
\begin{eqnarray}
H &=& \int d^3 \xbf \,\, \mathcal{H}(\xbf), \nonumber \\
\mathcal{H}(\xbf) &=&
\Pi^{\dagger}({\xbf}) \Pi({\xbf}) \, + \,
\lambda\bm{\nabla}\Phi^{\dagger}({\xbf}) \bm{\nabla}\Phi({\xbf}) \, + \,
m^2 \Phi^{\dagger}({\xbf}) \Phi({\xbf}), \label{H}
\end{eqnarray}
where $\mathcal{H}$ is the Hamiltonian density. Note that there is one
additional dimensionless parameter $\lambda$ in the Hamiltonian as
compared to the canonical theory because we already set the scale of
the fields through the commutation relations~(\ref{phipi}).

The commutation relations~(\ref{theta}) and~(\ref{B}) are not the most
general ones to define a noncommutative field: one could introduce
nonzero commutators between fields (and momenta) at different spatial points,
which would then involve a much more complicated and arbitrary parametrization
than that of Eqs.~(\ref{theta}) and~(\ref{B}). We will examine the properties
and characteristic features of this simple definition for the noncommutative
field in Sec.~\ref{sec:prop}.

In order to solve this theory, it is convenient to recall the method 
followed in standard RQFT ($\theta=B=0$, $\lambda=1$). 
There one identifies the ``field''
as a superposition of an infinite number of decoupled one-dimensional quantum 
harmonic oscillators, each with a frequency $\omega(\pbf)=\sqrt{\pbf^2+m^2}$, 
and then uses the solution of the harmonic oscillator in QM to define the Fock
space of particle-states where the field acts. This suggests that
considering the noncommutative generalization of the harmonic oscillator in 
QM might help us to define a Fock space for the noncommutative field.

\subsection{Harmonic oscillator in noncommutative quantum mechanics}

Let us consider a particle in two noncommuting spatial dimensions 
(the coordinates will be in correspondence with the two noncommuting real 
fields) in the presence of the harmonic oscillator potential 
(in correspondence with the relation between a free field theory and a 
superposition of oscillators) and a constant magnetic field [in correspondence
with the noncommutativity in momenta Eq.~(\ref{B})].

This system is defined by the Hamiltonian
\begin{equation}
H \, = \, \frac{\omega}{2} \left({\hat p}_1^2 + {\hat p}_2^2 + 
{\hat q}_1^2 + {\hat q}_2^2\right)
\label{Hqp}
\end{equation}
\textsl{and} the commutation rules
\begin{equation}
[{\hat q}_1,{\hat q}_2]=i\,{\hat \theta}, \quad
[{\hat p}_1,{\hat p}_2]=i\,{\hat B}, \quad
[{\hat q}_i,{\hat p}_j]=i\,\delta_{ij}\, .
\label{[qp]}
\end{equation}
Note that we have expressed the Hamiltonian and the commutation rules in
terms of adimensional phase-space coordinates (hence the small angles over
them) appropriately rescaled, and omitted $\hbar$ factors.

This problem was recently considered and solved in Ref.~\cite{nair}.
Since we are interested in the case where the noncommutativity 
is going to be a small correction to RQFT, 
we will restrict in the following to the $\hat B\hat\theta<1$ case.
The appropriate way to solve this problem is to identify a linear 
transformation of the phase-space coordinates
\begin{equation}
{{\hat q}_1, {\hat p}_1, {\hat q}_2, {\hat p}_2} \longrightarrow
{{\hat Q}_1, {\hat P}_1, {\hat Q}_2, {\hat P}_2} , 
\label{linear}
\end{equation}
so that the commutation rules in the new variables are
\begin{equation} 
[{\hat Q}_1,{\hat Q}_2] \,=\, [{\hat P}_1,{\hat P}_2] \,=\, 0, \quad
[{\hat Q}_i,{\hat P}_j]=i\,\delta_{ij}, 
\label{[QP]}
\end{equation} 
and the Hamiltonian in the new variables is still diagonal
\begin{equation}
H \, = \, \frac{\omega_1}{2} \left({\hat P}_1^2 + {\hat Q}_1^2\right) \, + \,
\frac{\omega_2}{2} \left({\hat P}_2^2 + {\hat Q}_2^2\right).
\label{HQP}
\end{equation}

However, there is no a unique linear transformation~(\ref{linear}) 
satisfying Eqs.~(\ref{[QP]}) and~(\ref{HQP}). In fact the authors of 
Ref.~\cite{nair} got a very special (and, after having obtained the 
general result, complicated) linear transformation. In order to 
adequately solve our original problem (the free theory of the noncommutative
field) it will prove convenient to work out the most general solution of this
associated quantum mechanical problem. We will give now the final result. 

This system is equivalent to a set of two decoupled one-dimensional 
oscillators of frequencies~\footnote{This result can also be obtained directly 
from the second order equations for the operators ${\hat q}_i$ which one 
gets after eliminating ${\hat p}_i$ in the Hamilton equations 
$i\mathrm{d}\mathcal{O}/\mathrm{d}t=[\mathcal{O},H]$, where $\mathcal{O}$
is a phase-space variable.}
\begin{subequations}
\begin{eqnarray}
\omega_1 &=& \omega \, \left[\sqrt{ 1 + 
\left(\frac{{\hat B} - {\hat \theta}}{2}\right)^2} \,+\,
\left(\frac{{\hat B} + {\hat \theta}}{2}\right)\right] ,
\label{omega1} \\
\omega_2 &=& \omega \, \left[\sqrt{ 1 + 
\left(\frac{{\hat B} - {\hat \theta}}{2}\right)^2} \,-\,
\left(\frac{{\hat B} + {\hat \theta}}{2}\right)\right] .
\label{omega2}
\end{eqnarray}
\label{omegas}
\end{subequations}
If we set ${\hat\theta}={\hat B}=0$ then $\omega_1=\omega_2=\omega$ and
we recover the result of the symmetric bidimensional harmonic oscillator of
frequency $\omega$.

In order to have a simple expression for the most general linear 
transformation~(\ref{linear}) which passes from the Hamiltonian~(\ref{Hqp}) 
and the commutation rules~(\ref{[qp]}) to the Hamiltonian~(\ref{HQP}) and 
the commutation rules~(\ref{[QP]}) it is convenient to use the following 
combination of variables:
\begin{eqnarray}
z \,&=&\, \frac{{\hat q}_1 + i {\hat q}_2}{\sqrt{2}}\, , \quad
w \,=\, \frac{{\hat p}_1 + i {\hat p}_2}{\sqrt{2}}\, , \nonumber \\
{\bar z} \,&=&\, \frac{{\hat q}_1 - i {\hat q}_2}{\sqrt{2}}\, , \quad
{\bar w} \,=\, \frac{{\hat p}_1 - i {\hat p}_2}{\sqrt{2}}\, ,  
\label{zw}
\end{eqnarray} 
instead of the original variables ${\hat q}_1$, ${\hat p}_1$, 
${\hat q}_2$, ${\hat p}_2$. The creation-annihilation operators of the 
one-dimensional oscillators of frequencies $\omega_1$, $\omega_2$ are
\begin{eqnarray}
a \,&=&\, \frac{{\hat Q}_1 + i {\hat P}_1}{\sqrt{2}}\, , \quad
b \,=\, \frac{{\hat Q}_2 + i {\hat P}_2}{\sqrt{2}}\, , \nonumber \\
a^{\dagger} \,&=&\, \frac{{\hat Q}_1 - i {\hat P}_1}{\sqrt{2}}\, , \quad
b^{\dagger} \,=\, \frac{{\hat Q}_2 - i {\hat P}_2}{\sqrt{2}}\, .
\label{ab}
\end{eqnarray}
Then the most general linear transformation that allows to solve the
problem of noncommutative quantum mechanics is 
\begin{subequations}
\begin{eqnarray}
z &=& \eta \epsilon_1 \,e^{i\alpha} a \,+\,
\epsilon_2 \,e^{i\beta} b^{\dagger} , \\
{\bar z} &=& \eta \epsilon_1 \,e^{-i\alpha} a^{\dagger} \,+\, 
\epsilon_2 \,e^{-i\beta} b , \\
w &=& -i \epsilon_1 \,e^{i\alpha} a \,+\,
i \eta \epsilon_2 \,e^{i\beta} b^{\dagger}, \\
{\bar w}&=&i \epsilon_1 \,e^{-i\alpha} a^{\dagger}\,-\,
i \eta \epsilon_2 \,e^{-i\beta} b ,
\end{eqnarray}
\label{zwab}
\end{subequations}
where $\alpha$, $\beta$ are two angles which parametrize the most general
linear transformation. Both angles appear only in exponential factors
accompanying the $a$ and $b$ operators. Then we can use the freedom in 
the phase choice of the particle states to take, without any lost of
generality, $\alpha = \beta = 0$ in Eq.~(\ref{zwab}). The coefficients 
$\eta$, $\epsilon_1$ and $\epsilon_2$ are expressed in terms of the 
noncommutative parameters
\begin{subequations}
\begin{eqnarray}
\eta \,&=&\, \sqrt{ 1 + \left(\frac{{\hat B} - {\hat \theta}}{2}\right)^2} 
\,-\, \left(\frac{{\hat B} - {\hat \theta}}{2}\right), \\
\epsilon_1^2 \,&=&\, \frac{{\hat B} + \eta}{1 + \eta^2} 
             \,=\, \frac{1/\eta +{\hat \theta}}{1 + \eta^2}, \\
\epsilon_2^2 \,&=&\, \frac{\eta -{\hat \theta}}{1 + \eta^2} 
             \,=\, \frac{1/\eta -{\hat B}}{1 + \eta^2}.
\end{eqnarray}
\label{etaepsilon}
\end{subequations}
Since there are only two parameters for the noncommutativity, we have
a relation between these three coefficients
\begin{equation}
\epsilon_1^2 + \epsilon_2^2 \, =\, 1/\eta \, .
\end{equation}
Finally, we can obtain from Eq.~(\ref{etaepsilon}) the following simple 
relations:
\begin{equation}
\epsilon_1^2 - \eta^2 \epsilon_2^2 \,=\,{\hat B}, \quad
\eta^2 \epsilon_1^2 - \epsilon_2^2 \,=\,{\hat \theta}.
\end{equation}

\subsection{Construction of the noncommutative field}

The noncommutative field is constructed from the above solution of the
noncommutative quantum mechanics problem by considering an oscillator for each
value of the momentum $\pbf$ of frequency 
$\omega(\pbf)=\sqrt{\lambda\pbf^2+m^2}$. Then, the extension to the 
complex field $\Phi$ of the expression~(\ref{zwab}) of the $z$ coordinate 
as a function of the creation and annihilation operators is
\begin{equation}
\Phi ({\xbf}) = \int \frac{d^{3}{\pbf}}{(2\pi)^3} 
\frac{1}{\sqrt{ \omega(\pbf)}} \left[\eta(\pbf) \epsilon_1(\pbf) \, a_{\pbf} \,
e^{i{\pbf}\cdot{\xbf}}  
+ \epsilon_2(\pbf) \, b^{\dagger}_{\pbf} \, 
e^{- i{\pbf}\cdot{\xbf}}\right],
\label{Phi}
\end{equation}
which generalizes the conventional expression of the field in RQFT as a 
superposition of infinite oscillators (each for every momentum) at every
space point. Eq.~(\ref{Phi}) includes the plane-wave factors 
$e^{i\pbf\cdot\xbf}$, an explicit momentum-dependence in the coefficients 
$\eta$ and $\epsilon_1$, $\epsilon_2$, and a global factor $1/\sqrt{\omega}$ 
to take into account the rescaling between the adimensional coordinates and 
the field. The momentum has the analogue extension of the expression of $w$ in 
Eq.~(\ref{zwab})
\begin{equation}
\Pi ({\xbf}) = \int \frac{d^{3}{\pbf}}{(2\pi)^3} 
\sqrt{\omega(\pbf)} \left[- i\,\epsilon_1(\pbf) \, a_{\pbf} \, 
e^{i{\pbf}\cdot{\xbf}} + 
i\,\eta(\pbf) \epsilon_2(\pbf) \, b^{\dagger}_{\pbf} \, 
e^{- i{\pbf}\cdot{\xbf}}\right].
\label{Pi}
\end{equation}

Eqs.~(\ref{Phi}) and~(\ref{Pi}) (and their respective conjugated expressions)
give the fields and momenta as a function of creation and annihilation 
operators. If these satisfy the commutation rules
\begin{subequations}
\begin{eqnarray}
[a_{\pbf}, a^{\dagger}_{\pbf'}] 
&=& (2\pi)^3 \delta^{3}({\pbf}- {\pbf'}), \\
\left. [b_{\pbf}, b^{\dagger}_{\pbf'}] \right.
&=& (2\pi)^3 \delta^{3}({\pbf}- {\pbf'}),
\label{[ab]}
\end{eqnarray}
\end{subequations}
and we choose the noncommutative parameters of the system of quantum
mechanics corresponding to each momentum as
\begin{equation}
{\hat \theta} (\pbf) = \theta \,\omega(\pbf), \quad
{\hat B} (\pbf) = \frac{B}{\omega(\pbf)}\, ,
\label{thetaB(p)}
\end{equation}
with $\theta$, $B$ constants which do not depend of the momentum, then 
the calculation of the commutators of fields and momenta give the
commutation rules~(\ref{theta}), (\ref{B}) and~(\ref{phipi}).
This proves that the field constructed as in Eq.~(\ref{Phi}) is a 
representation of the noncommutative field in the Fock space defined by the 
creation and annihilation operators $a_{\pbf}$, $a^\dagger_{\pbf}$, 
$b_{\pbf}$ and $b^\dagger_{\pbf}$.

Moreover using the representation of fields and momenta as a linear 
combination of creation and annihilation operators, Eqs.~(\ref{Phi}) 
and~(\ref{Pi}), one can express the Hamiltonian~(\ref{H}) in the form
\begin{equation}
H = \int \frac{d^{3}{\pbf}}{(2\pi)^3} \left[E_1(\pbf) \left(
a^{\dagger}_{\pbf} a_{\pbf} \,+\,\frac{1}{2}\right)
+ E_2(\pbf) \left(b^{\dagger}_{\pbf} b_{\pbf} 
\,+\,\frac{1}{2}\right)\right],
\label{Hab}
\end{equation}
which shows that the theory of the free complex noncommutative field is
a theory of free particles of two types. $E_1(\pbf)$ and $E_2(\pbf)$ give 
the expressions of the energy of one of these particles with momentum $\pbf$. 
These energies are simply the frequencies in Eq.~(\ref{omegas})
of the two decoupled oscillators appearing in the solution of the 
quantum mechanical system corresponding to each momentum $\pbf$ with the 
parameters of noncommutativity given in Eq.~(\ref{thetaB(p)})
\begin{subequations}
\begin{eqnarray}
E_1(\pbf) &=& \omega(\pbf) \, \left[\sqrt{1 + \frac{1}{4} 
\left(\frac{B}{\omega(\pbf)} - \theta \omega(\pbf)\right)^2}
+ \frac{1}{2}
\left(\frac{B}{\omega(\pbf)} + \theta \omega(\pbf)\right) \right], 
\label{E12p}\\
E_2(\pbf) &=& \omega(\pbf) \, \left[\sqrt{ 1 + \frac{1}{4} 
\left(\frac{B}{\omega(\pbf)} - \theta \omega(\pbf)\right)^2} 
- \frac{1}{2}
\left(\frac{B}{\omega(\pbf)} + \theta \omega(\pbf)\right) \right].
\label{E12a}
\end{eqnarray}
\label{E12}
\end{subequations}

In summary we have seen that the free theory of the scalar noncommutative field
can be solved in a similar way as in the conventional case of RQFT. It is
a theory of free particles. The simplest way to incorporate interactions is
by using the same Hamiltonians as in RQFT, now in terms of noncommutative
fields. Then the solution of the free theory can be taken as a starting point
for a perturbative treatment of interactions analogously to what is done
in RQFT: identification of propagators, Feynman rules, etc. We will sketch
this procedure in the following section, and examine the characteristic
properties of the noncommutative theories defined in such a way.

\section{Properties of the quantum theory of noncommutative fields}
\label{sec:prop}

The free noncommutative field was defined by the commutation 
relations~(\ref{theta}),~(\ref{B}) and by the Hamiltonian~(\ref{H}).
%---that is, the same Hamiltonian of standard RQFT of the complex 
%scalar field. 
This simple implementation of a noncommutativity in field space
has the following properties:
\begin{enumerate} 
\item Standard RQFT is trivially recovered in the $\theta\to 0$,
$B\to 0$, $\lambda\to 1$ limit. As in the standard case, the quantum 
theory is obtained from a classical Hamiltonian which is relativistic 
invariant [the apparent noninvariance of $\lambda\neq 1$ in 
Eq.~(\ref{H}) is fictitious; as we remarked before, it is a consequence of 
the choice of the scale for the fields in order to write the standard
commutation relations between fields and momenta Eqs.~(\ref{phipi})],
but now we follow a quantification procedure, given by the 
new commutation relations~(\ref{theta}) and~(\ref{B}), which 
explicitely violates Lorentz symmetry. This is what we understand by the
``quantum theory of a noncommutative field''.
\item The Hamiltonian density defined in Eq.~(\ref{H}) is made of fields 
which commute at different space points, and therefore satisfies
\begin{equation}
[\mathcal{H}(\xbf),\mathcal{H}(\xbf')]=0\quad \mathrm{for}\, \,
\xbf\neq\xbf'.
\label{density1}
\end{equation}
This property is essential in RQFT to guarantee that the $S$-matrix will
be Lorentz-invariant. More specifically, what is required in RQFT is that
\begin{equation}
[\mathcal{H}(x),\mathcal{H}(x')]=0\quad \mathrm{for}\, \,
(x-x')\geq 0,
\label{density2}
\end{equation}
which is equivalent to Eq.~(\ref{density1}) in a Lorentz-invariant theory.
The two conditions are not equivalent however when relativistic invariance 
is lost. In fact, in the quantum theory of the noncommutative field, 
the Hamiltonian density satisfies Eq.~(\ref{density1}) but not 
Eq.~(\ref{density2}). The preservation of the property~(\ref{density1}) is 
in any case welcome since it allows to speak consistently about the concept 
of a Hamiltonian density. Without this condition, the energy of a closed 
finite system could depend on the energy of another system very far away. 
Commutation relations more general than Eqs.~(\ref{theta}) and~(\ref{B}) 
would violate Eq.~(\ref{density1}).
\item Keeping the property~(\ref{density1}) requires the introduction of
two real fields as the only way to implement a noncommutativity in field space.
This leads to a theory with two types of particles which correspond to
the particle and the antiparticle in the $\theta, B\to 0, \lambda\to 1$ 
limit (conventional RQFT). 
Particle and antiparticle are no longer degenerated in this extension
of RQFT, and their energy is different from the standard expression 
$\sqrt{\pbf^2+m^2}$ by small corrections parametrized by $\theta$, $B$
and $\lambda$. The theory naturally incorporates in this way a 
matter-antimatter asymmetry.
\item We have a new, specific form of the dispersion relation, or relation
between energy and momentum of a particle, Eq.~(\ref{E12}),
which is no longer Lorentz-invariant, while the theory still preserves 
rotational symmetry. Relativistic invariance is therefore
an ingredient which is lost in this extension of RQFT. We will see in 
Sec.~\ref{sec:phen} that this symmetry is violated not only at high energies,
but, surprisingly enough, also at low energies, being still compatible with
phenomenological observations. Relativistic causality is also violated, as
we check later in this section. 
\item An essential property of RQFT, which in principle should hold in any
sensible physical theory, is the cluster decomposition principle: experiments
which are sufficiently separated in space should have unrelated results.
A general theorem states that the $S$-matrix satisfies this crucial requirement
if the Hamiltonian can be expressed as a sum of products of creation and 
annihilation operators, with suitable non-singular 
coefficients~\cite{weinberg}. This theorem garantees that the cluster property
still holds in the noncommutative extension of RQFT.
\end{enumerate}

To discuss causality and the formulation of perturbation theory we have
to consider the field operator in the interaction picture
\begin{eqnarray}
\Phi ({\xbf}, t) &=& e^{i H_0 t}  \,\Phi ({\xbf})\, e^{- i H_0 t} \nonumber \\
&=& \int \frac{d^{3}{\pbf}}{(2\pi)^3} \frac{1}{\sqrt{\omega(\pbf)}}  
\left[\eta(\pbf) \epsilon_1(\pbf) \, a_{\pbf} \, 
e^{-i E_1(\pbf) t} e^{i{\pbf}.{\xbf}}
+ \epsilon_2(\pbf) \, b^{\dagger}_{\pbf} \, e^{i E_2(\pbf) t} 
e^{- i{\pbf}.{\xbf}}\right].
\label{Phi(t)}
\end{eqnarray}

From Eq.~(\ref{Phi(t)}) one can calculate the commutator of operators at
different times
\begin{equation}
\left[\Phi(\xbf, t) , \Phi^{\dagger}(\xbf', t')\right]
\label{[t]}
\end{equation}
and verify that it is different from zero at causally disconnected points
($(\xbf-\xbf')^2>(t-t')$) owing to the noncommutativity. 

The modification to the standard propagator caused by the noncommutativity
is rather simple. One has
\begin{eqnarray}
\langle 0|{\cal T}(\Phi(\xbf,t) \Phi^{\dagger}(\xbf', t')|0\rangle &=& 
\int \frac{d^4 p}{(2\pi)^4} \,e^{-i p^0 (t-t')}
e^{i {\pbf}\cdot(\xbf-\xbf')} \nonumber \\
&\times& \,\frac{i \,(1 - \theta B + \theta p^0)}
{(p^0 - E_1(\pbf) +i\epsilon) (p^0 + E_2(\pbf) -i\epsilon)}\, , \quad 
\label{propagador}
\end{eqnarray}
that is, the effect of the noncommutativity is a displacement in the position
of the poles 
\begin{equation}
\pm\sqrt{\pbf^2+m^2}\to E_1(\pbf), -E_2(\pbf), 
\end{equation}
together with a modification of the residues
\begin{equation}
\pm \frac{1}{2\sqrt{\pbf^2+m^2}}\to 
\frac{(1-\theta B+\theta E_1(\pbf))}{(E_1(\pbf)+E_2(\pbf))},
-\frac{(1-\theta B-\theta E_2(\pbf))}{(E_1(\pbf)+E_2(\pbf))}.
\end{equation}

We will finally note that there is not any obstruction to the introduction
of gauge symmetries in the theory of noncommutative fields. In the free
noncommutative complex scalar field theory we have a global U(1) symmetry
that can be made local in a Hamiltonian with interaction terms containing
so many $\Phi$ fields as $\Phi^\dagger$ fields, if every derivative
of the field appears in the combination $-i\,\bm{\nabla} \Phi + 
\bm{A}\Phi$. We leave further exploration on the dynamics of interactions
in theories of noncommutative fields for future work and consider in the
following section the phenomenological implications coming from the solution
of the free theory.
   
\section{Phenomenological bounds on the parameters of noncommutativity}
\label{sec:phen}

We would like to show in this section how the quantum theory of noncommutative
fields can be a sensible extension of RQFT, in the sense that it does not
contradict the present understanding of low-energy phenomena and, at the
same time, may have observable consequences.

The effect of the noncommutativity at the level of the free theory is
the substitution of the particle or antiparticle states of momentum $\pbf$
and energy $E=\sqrt{\pbf^2+m^2}$ in RQFT by two states of energies 
$E_1(\pbf)$ and $E_2(\pbf)$ given by Eqs.~(\ref{E12}). From these expressions
one sees that if
\begin{equation}
B \ll \sqrt{\lambda\pbf^2 + m^2} \,\ll\, 1/\theta 
\Rightarrow E_1(\pbf) \approx E_2(\pbf) \approx   
\sqrt{\lambda\pbf^2 + m^2}
\end{equation}
and the relativistic dispersion relation is recovered in the 
$\lambda=1$ limit.

We then have two energy scales coming from the noncommutativity: $B$ and
$1/\theta$, and from the dispersion relation we see that these 
two scales are respectively 
the infrared (IR) and ultraviolet (UV) scales which limit the range of 
validity of the relativistic invariant theory.

The noncommutativity induces a violation of relativistic invariance both
at high and low energies. 
A maximum velocity of propagation different from the speed of light 
$(\lambda\neq 1)$ and
high-energy violations coming from the presence of an
UV scale are properties which have been explored in other 
contexts~\cite{cutoffs,qugr,kostel,uv}, especially those trying to incorporate 
effects coming from the Planck length. What is new in the present extension 
of RQFT is the presence of an additional IR scale and a violation of 
relativistic invariance at low energies.

This result allows to explore consequences of the noncommutativity already
at the level of the kinematics, without a necessity of considering in detail
the dynamics. The analysis will alternatively give restrictions on the
values of the parameters of the noncommutativity, based on the success of
the RQFT description of nature. We note here that this kinematics study 
needs a further assumption. The new dispersion relations~(\ref{E12}) were 
obtained for the free theory of the scalar noncommutative field. The extension 
to a free theory of fermions is not however a trivial task, and we leave it
for future work. 
We will now make the phenomenological analysis to obtain bounds on the
parameters of noncommutativity assuming that a similar dispersion relation
holds for fermions. 

The most sensitive experiments to detect a violation of relativistic invariance
at high energies are those involving ultra high-energy cosmic rays (UHECR).
Not only they reach energies as high as $10^{20}$ eV, but they are even 
sensible to effects parametrized by much larger energy scales, such as the 
Planck scale, thanks to amplification mechanisms coming from the presence of 
very different scales. This is what happens at the very end of the cosmic ray
spectrum: relativistic kinematics predicts a cutoff in the spectrum 
for UHECR coming from distant sources (the GZK cutoff)~\cite{gzk}, caused by
the energy loss they experiment in their interaction with the cosmic
background radiation (CBR), which seems to be avoided somehow~\cite{gzkexps}. 
Relativistic invariance violations induced by the Planck scale are,
among others~\cite{othersgzk}, a possible 
explanation for the disappearance of the GZK 
cutoff~\cite{uv,capolavoro,grillo}. 
The sensibility to the Plank scale results in this case from the presence of 
a very small energy scale: the kinetic energy of photons of the CBR,
$10^{-3}$ eV.

In this range of momenta, we expect $|\theta|\,(\pbf^2+m^2)\gg |B|$, and then
$B$ can be neglected in the dispersion relations~(\ref{E12}). Bounds on
$1/\theta$ in the $B\to 0$ limit coming from the physics of the UHECR were
studied in Ref.~\cite{capolavoro}. Taking the energy-momentum 
relation Eq.~(\ref{E12p}) for the particle, and Eq.~(\ref{E12a}) for the
antiparticle, so that the noncommutative field Eq.~(\ref{Phi}) generalizes
the expression of the conventional field in RQFT, as a linear combination
of particle annihilation operators $a_{\pbf}$ and antiparticle
creation operators $b^\dagger_{\pbf}$, then the analysis of 
Ref.\cite{capolavoro} shows that the sign of $\theta$ has to be negative.
This is because $E_1(\pbf)$ with $\theta>0$ would generate a mechanism
of energy loss for particles independent of the CBR: particle
disintegrations prohibited by relativistic kinematics, would be now allowed.
The observation of UHECR excludes then this possibility. With $\theta<0$,
$E_1(\pbf)<\omega(\pbf)$, and the interaction with the CBR is now kinematically
forbidden so that the GZK cutoff no longer exists. Experiments coming
in the near future~\cite{newgzkexps} will clarify the situation with respect
to the GZK cutoff violation, which will then be a stringent test for 
the theory of noncommutative fields. The bounds provided for
the UV energy scale are~\cite{capolavoro}:
$10^{21}\mbox{ eV}\lesssim 1/|\theta| \lesssim 10^{43}\mbox{ eV}$.

Let us now consider the IR corrections to the relativistic dispersion
relation given by Eqs.~(\ref{E12}). In the range of momenta where
$|\theta|\,(\pbf^2+m^2)\ll |B|$, we can neglect in a first approximation
the effects parametrized by the UV scale ($1/\theta$) and then we obtain
\begin{equation}
E_1(\pbf)\approx \sqrt{\pbf^2+m^2+\frac{B^2}{4}}-\frac{B}{2}
\end{equation}
for the energy of the particle, and an analogous expression for the 
antiparticle (replacing $-B/2$ by $+B/2$). We then see that the effect of
the noncommutativity is a constant contribution to the energy (opposite in
sign for particles and antiparticles) and a ``renormalization'' of the
mass $m^2 \to m^2_{\mathrm{eff}}=m^2+B^2/4$. Conservation laws in physical
process will however make invisible the $\pm B/2$ constant contributions 
to the energy coming from the noncommutativity. On the other hand, the
bound $B^2/4\leq m^2_{\mathrm{eff}}$ will restrict the value of $B$ from
the bounds to neutrino masses. The $\beta$ disintegration of tritium, which
is the most sensitive experiment to neutrino mass~\cite{tritioexps}, gives
$B<5$ eV.

Finally, bounds on the adimensional parameter $\lambda$ were considered in 
Ref.~\cite{uv} in different scenarios. The typical bound is 
$|1-\lambda|\leq 10^{-23}$. A more detailed  phenomenological analysis 
including cosmological implications
of the noncommutativity will be given elsewhere.  
  
\acknowledgments
We would like to thank J. Clemente-Gallardo for discussions. 
This work has been partially supported by the
grants 1010596, 7010596 and 3000005 from Fondecyt-Chile,
by M.AA.EE./AECI and 
by MCYT (Spain), grants FPA2000-1252 and FPA2001-1813.

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\end{thebibliography}

\end{document}




