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\begin{center}
{\Large {\bf Conformal Invariance\\ and Quantum Nature of
Particles\\}} \vspace{2cm} ${\bf Hadi~Salehi}$
\footnote{e-mail:~h-salehi@cc.sbu.ac.ir.} \\
\vspace{0.5cm} {\small {Department of Physics, Shahid Beheshti
University, Evin,
Tehran 19839,  Iran.}}\\
\vspace{1cm} ${\bf Yousef~
Bisabr}$\footnote{e-mail:~y-bisabr@myself.com.}\\
\vspace{.5cm} {\small{Department of Physics, Shahid Rajaee
University,
Lavizan, Tehran 16788, Iran.}}\\
\end{center}
\vspace{1cm}
\begin{abstract}
We investigate a gravitational model whose vacuum sector is invariant
under conformal transformations.  In this model matter
is taken to be coupled with a metric which is different
but conformally
related to the metric appearing explicitly in the vacuum sector.  It is then
shown that the effect of a conformal symmetry breaking would lead to a
particle concept.  In particular, a correspondence
between quantum nature of the particles and the gravitational interaction
of matter is established.
\end{abstract}
\vspace{3cm}
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The principle of conformal invariance requires that all the
fundamental equations of physics should be invariant under local
(spacetime dependent) changes of units of length and time
\cite{bm}.  This requirement poses a fundamental symmetry  in
physical theories which is of ever-increasing interest.  The
establishment of such a symmetry in the vacuum sector of a
gravitational model leads one to face with a problem concerning
the incorporation of matter to such models.  In fact, once the
vacuum sector is taken to be conformally invariant, all the
conformal frames should be considered as dynamically equivalent.
By implication there is no $a~ priori$ evidence determining to
which of these frames the matter should be coupled.  From a
classical point of view one may appeal to the weak equivalence
principle which forces us to consider the coupling of matter to
the metric which enters explicitly the action describing the
vacuum sector. This is what is generally known as the normal
coupling. At the quantum level, however, there is not a direct
evidence demonstrating the validity of the weak equivalence
principle. Thus questions such as what is the nature of
gravitational coupling of matter at this level and what are the
physical consequences of this coupling are of obvious theoretical
significance.\\ In this paper we deal with questions of this type
in the framework of a conformal invariant gravitational model in
which the gravitational interaction of matter can be described
both by normal coupling and anomalous coupling.  The anomalous
coupling means that the metric tensor entering the gravitational
part and that entering the matter part of the action correspond to
different conformal frames. We shall establish such a distinction
by introducing a nontrivial conformal factor which in our model is
considered profoundly as a dynamical field. As a basic result we
show that the emergence of this field and the corresponding
gravitational coupling can be used to model a relativistic
particle concept which shares the essential characteristics of the
particle concept in the casual interpretation of quantum mechanics
\cite{bh}. This deduction of the particle concept is carried out
on the basis of a conformal symmetry breaking. The role of such a
symmetry breaking has been previously stressed for the deduction
of a particle concept in general relativity \cite{ssd}. The
present work extends the previous work and establishes a
connection between this effect and the nature of gravitational
coupling of matter.\\  Throughout the following we shall use units
in which $\hbar=c=1$ and the signature is (-+++).\\ Consider the
conformal invariant gravitational action functional
\begin{equation}
S=\frac{1}{2} \int d^{4}x \sqrt{-g}~ (g^{\mu\nu} \nabla_{\mu}\phi
\nabla_{\nu}\phi+\frac{1}{6} R \phi^{2})~,
\label{a1}\end{equation} where $\phi$ is a scalar field, $R$ is
the curvature scalar associated with the metric tensor $g_{\mu
\nu}$ and $\nabla_{\mu }$ denotes a covariant differentiation.\\
The conformal invariance of the action (\ref{a1}) means that it is
invariant under conformal transformations
\begin{equation}
\bar{g}_{\mu\nu} =\Omega^2  g_{\mu\nu}~, \label{2}\end{equation}
\begin{equation}
\bar{\phi}(x) = \Omega^{-1}  \phi(x)~, \label{3}\end{equation}
where $\Omega$ is a smooth dimensionless spacetime function. The
conformal invariance of (\ref{a1}) implies that one may consider
this theory in different conformal frames which are dynamically
equivalent.  This feature leads to an ambiguity concerning the
incorporation of matter to such theories.   Before adding a
matter field action to (\ref{a1}), one has to decide to which of
these frames the matter should be coupled.  In the following we
need to consider besides the normal coupling an anomalous
coupling in which case the matter is coupled to a metric which is
conformally related to $g_{\mu \nu}$. In explicit terms we intend
to investigate the consequences of the action functional
\begin{equation}
S^{'}=\frac{1}{2} \int d^{4}x \sqrt{-g}~ \{g^{\mu\nu}
\nabla_{\mu}\phi \nabla_{\nu}\phi+(g^{\mu\nu}\nabla_{\mu}\sigma
\nabla_{\nu}\sigma +\frac{1}{6} R)\phi^{2} +\frac{1}{2}\lambda
\sigma \phi^4 \}+S_{m}[e^{\sigma}g_{\mu \nu}]~,
\label{4}\end{equation} in which $S_{m}[e^{\sigma}g_{\mu \nu}]$
stands for the matter action which contains some matter field
variables with anomalous coupling.  We have also taken the above
action to involve a kinetic term for $\sigma$ to account for its
dynamical contribution.  By considering $\sigma$ as a dynamical
field we let the action have an interaction term $\sim \lambda
\sigma \phi^4$ in which $\lambda$ is a dimensionless coupling
constant. Note that these changes preserve the conformal
symmetry  of (\ref{a1}) since $\sigma$, as a dimensionless
function, does not transform under a conformal transformation.
The presence of such a dynamical field emphasizes that there is a
dynamical distinction between the two unit systems describing the
gravitational and the matter parts in the action (\ref{4}).  The
choice of a configuration for $\sigma$ is a matter of convenience
and depends on the boundary conditions one wishes to apply in the
theory.  In the following we shall
show that different boundary conditions lead to different representations of the theory.\\
Varying $S^{'}$ with respect to $g^{\mu \nu}$, $\phi$ and $\sigma$
yields, respectively,
\begin{equation}
G_{\mu \nu}-\frac{3}{2}\lambda \sigma \phi^2 g_{\mu\nu}
=6\phi^{-2}(\Sigma_{\mu \nu}(e^{\sigma}g_{\mu
\nu})+\tau_{\mu\nu})+ 6t_{\mu\nu}~, \label{5}\end{equation}
\begin{equation}
\Box \phi-\frac{1}{6}R \phi-\lambda \sigma \phi^3- \phi
\nabla_{\alpha }\sigma \nabla^{\alpha  }\sigma =0~,
\label{6}\end{equation}
\begin{equation}
\nabla_{\mu}J^{\mu} =\frac{1}{2}\sqrt{-g}(\frac{1}{2}\lambda
\phi^4 +g^{\mu\nu}\Sigma_{\mu\nu}(e^{\sigma}g_{\mu \nu}))~,
\label{6a}\end{equation} where
\begin{equation}
\Sigma_{\mu \nu}(e^{\sigma}g_{\mu \nu})=\frac{-2}{\sqrt{-g}}
\frac{\delta}{\delta g^{\mu\nu} } S_{m}[e^{\sigma}g_{\mu \nu}]~,
\end{equation}
\begin{equation}
J^{\mu}=\sqrt{-g}\phi^2 g^{\mu\nu}\nabla_{\nu}\sigma
\label{6aa}\end{equation} and
\begin{equation}
\tau_{\mu \nu}= -(\nabla_{\mu }
\phi \nabla_{\nu}\phi-\frac{1}{2}g_{\mu \nu} \nabla_{\alpha  }\phi
\nabla^{\alpha  }\phi)-\frac{1}{6}(g_{\mu \nu} \Box-\nabla_{\mu }
\nabla_{\nu})\phi^2~,
\end{equation}
\begin{equation}
t_{\mu \nu}=-\nabla_{\mu }\sigma \nabla_{\nu}\sigma +\frac{1}{2}
g_{\mu \nu} \nabla_{\alpha  }\sigma \nabla^{\alpha }\sigma~.
\label{7}\end{equation} Here $\Box \equiv g^{\mu \nu} \nabla_{\mu
} \nabla_{\nu}$ and $G_{\mu \nu}$ is the Einstein tensor of the
metric $g_{\mu \nu}$.  Taking the four-divergence of (\ref{5}) and
combining the result with (\ref{6}) and (\ref{6a}) leads to
\begin{equation}
\nabla^{\mu }\Sigma_{\mu \nu}(e^{\sigma}g_{\mu \nu})
=\frac{1}{2}\nabla_{\nu} \sigma ~g^{\mu\nu}
\Sigma_{\mu\nu}(e^{\sigma}g_{\mu \nu})~, \label{7a}\end{equation}
which indicates that $\Sigma_{\mu \nu}(e^{\sigma}g_{\mu \nu})$ can
not be conserved due to the presence of $\sigma$.  We also note
that in the absence of matter the trace of (\ref{5}) would lead
directly to the equation (\ref{6}) and, therefore, the latter
would not contain any new information. This is a direct
consequence of conformal invariance of the vacuum sector of
(\ref{4}).  In the presence of matter, on the other hand, we
obtain the following condition
\begin{equation}
g^{\mu\nu}\Sigma_{\mu\nu}(e^{\sigma}g_{\mu\nu})=0~,
\label{ab}\end{equation} by comparing the trace of (\ref{5}) with
the equation (\ref{6}). This is a dynamical consistency relation
on the behavior of matter systems in this model. In fact only
traceless matter systems are allowed by (\ref{ab}) to be coupled
to (\ref{a1}).  To consider matter systems with nonvanishing
trace of the corresponding stress-tensor one may introduce a term
such as $\frac{1}{2} \int d^{4} x \sqrt{-g} \mu^2 \phi^2$ to the
action (\ref{4}) with $\mu$ being a constant mass parameter which
is characteristic to the matter system at hand. In this case the
analog of the equations (\ref{5}) and (\ref{6}) are
\begin{equation}
G_{\mu \nu}-3\mu^2 g_{\mu\nu}-\frac{3}{2}\lambda \sigma \phi^2
g_{\mu\nu} =6\phi^{-2}(\Sigma_{\mu \nu}(e^{\sigma}g_{\mu
\nu})+\tau_{\mu\nu})+ 6t_{\mu\nu}~, \label{5-a}\end{equation}
\begin{equation}
\Box \phi-\frac{1}{6}R \phi-\lambda \sigma \phi^3- \mu^2 \phi-\phi
\nabla_{\alpha }\sigma \nabla^{\alpha  }\sigma =0~,
\label{6-a}\end{equation} and the equation (\ref{6a}) remains
unchanged. The relation (\ref{ab}) takes then the form
\begin{equation}
g^{\mu\nu}\Sigma_{\mu\nu}(e^{\sigma}g_{\mu\nu})=-\mu^2 \phi^2~.
\label{ac}\end{equation} This ensures the coupling of matter
systems with nonvanishing trace of the corresponding
stress-tensor to be dynamically
consistent. \\
The appearance of a constant dimensional parameter such as $\mu$
clearly breaks the conformal symmetry. We investigate the effect
of a cosmological symmetry breaking emerging from the
gravitational coupling of the large scale distribution of matter
in the universe. In this case the length scale $\mu^{-1}$ should
be related to the typical size of the universe $R_0$, namely
$\mu^{-1}\sim R_0$. We first consider the normal coupling of the
large scale distribution of matter, demanding a vanishing average
value for $\sigma$ at large scale, namely $\bar{\sigma}=0$. It
this case it is possible to obtain an estimation for the constant
background average value of $\phi$ which provides the strength of
the gravitational coupling. In fact, the trace of
$\Sigma_{\mu\nu}(g_{\mu\nu})$ can be measured in terms of the
average density of the large scale distribution of matter, i.e.,
$\Sigma_{\mu}^{\mu}(g_{\mu\nu}) \sim -M/R_{0}^3$ which $M$
denotes the mass of the universe.  If one uses the empirical fact
that the radius of the universe coincides with its schwarzschild
radius $2GM$, one then gets an estimation of the constant
background value of $\phi$
\begin{equation}
\phi^{-2} \sim  G \sim m_{p}^{-2}~, \label{G}\end{equation} where
$G$ and $m_{p}$ are the gravitational constant and the Planck
mass, respectively. In this case, the gravitational equation
(\ref{5-a}) reduces to the Einstein field equations with a small
cosmological constant.  It is important to note that the use of
$\lambda \sigma \phi^4$, instead of the usual $\lambda \phi^4$
term, allows
to avoid the cosmological constant problem in a cosmological context.\\
Thus it is possible to use a constant configuration of $\phi$ to
measure a constant strength of gravitational coupling of the
large scale distribution of matter in the normal coupling. It
should be emphasized that this feature seems to be characteristic
only to the normal coupling. In the case of anomalous coupling
the conformal symmetry breaking can not be used to achieve a
precise estimation of the energy density of
$\Sigma_{\mu\nu}(e^{\sigma}g_{\mu\nu})$.  This means that we may
not obtain any knowledge of the configuration of the scalar
field $\phi$ as a measure for the strength of the anomalous coupling.\\
We proceed now to the consideration of the anomalous coupling of
the large scale distribution of matter. The corresponding
symmetry breaking is characterized by the relation (\ref{ac}) in
which the length scale $\mu^{-1}$ again measures the typical size
of the universe. But (\ref{ac}) implies that the local variations
of $\sigma$ lead to a variable configuration of $\phi$. As a
consequence, the strength of the anomalous gravitational coupling
of the large scale distribution of matter should not be
considered as constant. It is precisely these local variations
that can be used to derive a particle concept. From (\ref{6-a})
we obtain
\begin{equation}
g^{\mu\nu}\nabla_{\mu}\sigma \nabla_{\nu}\sigma =-m^2-\frac{1}{6}R
+\frac{\Box \phi}{\phi}~, \label{14}\end{equation} which may be
interpreted as a generalized Hamilton-Jacobi equation of a
relativistic particle with a variable mass $m=(\lambda \sigma
\phi^2+ \mu^2)^{1/2}$. The major characteristic of (\ref{14})
lies in the term $\frac{\Box \phi}{\phi}$.  It should be
recognized as the so-called quantum potential in the context of
causal interpretation of relativistic quantum mechanics
\cite{bh}.  In present context, this term emerges because the
anomalous gravitational coupling of large scale matter systems
does not provide us a constant strength for the gravitational
coupling.\\
In order for the particle interpretation of (\ref{14}) to be
valid two general characteristics should be present. Firstly the
particle trajectories must be timelike. Secondly the variable
mass scale $m$ must remain constant along the particle
trajectories. To establish the first characteristic one may
impose the condition $\nabla_{\alpha }\sigma \nabla^{\alpha
}\sigma < 0$ to ensure that the particles move along timelike
trajectories. Looking at (\ref{7}) we see that this acts as an
energy condition on the stress tensor $t_{\mu\nu}$. It selects,
namely, those configurations of $\sigma$ for which the trace of
$t_{\mu\nu}$ is negative. For the stress tensors of matter
systems this assumption is dynamically consistent, as one may
infer from (\ref{ac}). Thus the restriction to timelike
trajectories means that the stress tensor $t_{\mu\nu}$ which
appears on the right hand side of (\ref{5-a}) follows the general
behavior of dynamically allowed matter systems with respect to
the sign of the trace. The establishment of the second
characteristic needs careful considerations. The constant mass
scale of a particle seems to be a feature which is valid if a
particle could ideally be considered as a quasi-isolated object.
To perform the corresponding limiting procedure we propose to
consider the scaling limit $\mu\rightarrow 0$, i.e., the
idealized limit of infinite size of the universe. In this limit
the conformal invariance is restored. Taking this limit as
reasonable we may then select a conformal frame (a particle
frame) by the condition
\begin{equation}
\nabla_{\alpha}\sigma \nabla^{\alpha }m=0~,
\label{9}\end{equation} which ensures that the mass scale $m$
remains constant along the particle trajectories\footnote{The
masses of the particles contribute to vacuum energy density and
appear as a large cosmological constant in the gravitational
equation (\ref{5-a}).  It should however be noted that this would
not lead to the cosmological constant problem.  This problem
arises if no dynamical distinction is allowed to exist between the
two unit systems used in cosmology and particle physics
\cite{bs}.}.\\ Finally we turn to the general particle concept
described by (\ref{14}) and remark that the motion of the
particles can be regarded as influenced by a guidance wave, or a
pilot wave, defined by
\begin{equation}
\psi=\phi e^{i\sigma }~. \label{15}\end{equation} Using
(\ref{6aa}) and (\ref{14}), one can easily show that $\psi$
satisfies the wave equation
\begin{equation}
(\Box-m^2-\frac{1}{6}R)\psi=\frac{i\phi^{-2}}{\sqrt{-g}}~\nabla_{\mu}J^{\mu}~~\psi~.
\end{equation}
As a consequence of (\ref{6a}) and in the scaling limit $\mu
\rightarrow 0$, this reduces to
\begin{equation}
(\Box-m^2-\frac{1}{6}R)\psi=i\frac{m^2}{4\sigma}\psi~.
\label{cc1}\end{equation} The right hand side of this equation
indicates that there is an imaginary contribution to the squared
mass of the particles. However one expects that such a
contribution to be vanishingly small since in order to achieve a
particle interpretation, there should exist a precise distinction
between the two unit systems describing the gravitational and the
matter part of the action (\ref{4}) by requiring that $\sigma$
takes sufficiently large values. In this case the right hand side
of the equation (\ref{cc1}) drops out and this equation takes the
form of the usual field equation in a gravitational background.\\
We summarize our observations as follows: A breakdown of conformal
invariance of the gravitational model (\ref{a1}) provides a
theoretical framework by which one may construct a particle
concept.  This construction entails the existence of a dynamical
field which describes the anomalous coupling of large scale matter
systems, namely the scalar function $\sigma$. This scalar function
makes it possible to make a distinction between the two unit
systems used in the description of the large scale matter systems
and that used in the description of the geometry of spacetime.  In
the absence  of this distinction (the normal coupling) the theory
describes just the usual gravitational coupling of large scale
distribution of matter (cosmological representation).  In the
presence of this distinction (anomalous coupling), on the other
hand, the theory can be used to model local particle systems.  In
this case it is essentially the local variations of the scalar
function $\sigma$ that leads one to a particle interpretation in
which $\sigma$ plays the role of the Hamilton-Jacobi function.
This demonstrates the nontrivial role of absolute value of the
phase in a gravitational context. The whole construction
emphasizes the indispensable role of the anomalous coupling of
large scale distribution of matter in the geometric interpretation
of causal framework of quantum mechanics.
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\begin{thebibliography}{99}
\bibitem{bm} J. D. Bekenstein, A. Meisels, Phys. Rev. D {\bf 22}, 1313 (1980)
\bibitem{bh} D. Bohm, J. Hiley, The Undivided Universe, ( Routledge 1993 )\\
P. R. Holland, The Quantum Theory of Motion, (Cambridge University Press 1993)
\bibitem{ssd} H. Salehi, H. Sepangi, F. Darabi, Found. Phys. Lett.
{\bf 13}, 297 (2000)
\bibitem{bs} Y. Bisabr, H. Salehi, Class. Quantum Grav. {\bf 19}, 2369 (2002)


\end{thebibliography}


\end{document}

