%\documentstyle[preprint,aps]{revtex}
\documentstyle[prl,aps]{revtex}
\begin{document}
\title{Coarse graining and a new strategy for
renormalization\footnote{Work supported in part by the National
Nature Science Foundation of China under Grant No. 10205004.}}
\draft
\author{Ji-Feng Yang}
\address{Department of Physics, East China Normal University,
Shanghai 200062, P. R. China}
%\date{Dec 15 2002}
\maketitle

\begin{abstract}
We present the natural arguments for the rationality of a recently
proposed simple approach for renormalization. The renormalization
group equation is also derived in a natural way and recognized as
a decoupling theorem of the UV modes that underlie a QFT. This new
strategy has direct implications to the scheme dependence problem.
\end{abstract}

\pacs{PACS Number(s): 11.10.Gh; 11.10.Hi; 02.30.Jr} Recently, we
proposed a new approach for calculating radiative corrections
without introducing any form of regulator and any form of removal
of UV divergence\cite{YYY,PRD65,CTP38}, which is a differential
equation approach with ambiguities to be fixed through rational
boundary conditions. In this simple approach, many complicated
aspects associated with conventional regularizations (e.g., the
definition of $\{\gamma^5, \gamma^{\mu}\}$ and $g^{\mu\nu}$ in
dimensional regularization; the notorious power law divergences in
cutoff regularization, and so on) simply do not show
up\cite{CTP38}. It is especially efficient in the nonperturbative
contexts where conventional regularization and/or subtraction
schemes often make it very hard to extract physical information
from the calculated quantities, as the proposed approach can make
the issue dramatically easy\cite{LOU}. It is also applied in
Ref.\cite{PRD65} to massless $\lambda\phi^4$ to discuss the
problem of nontrivial symmetry breaking solution in various
regularization and renormalization prescriptions.

In this short report, we shall: (1) to present the natural
rationality of the simple approach in Sec. II; (2) sketch a simple
derivation of the renormalization group equation as a natural
decoupling theorem of the underlying short distance modes in Sec.
III. The final section is devoted to discussion and summary.

Our starting point is the well known point of view that the
conventional QFT should be replaced by a complete quantum theory
of everything (QTOE) with correct high energy details. The low
energy physics are defined by the coarse grained low energy
sectors of QTOE with the high energy details integrated
out\cite{Coarse}. The high energy processes' contributions are
physically suppressed by certain mechanism in QTOE (unknown to us)
rather than 'cut off' by hand. This understanding naturally
motivates the presence of a set of parameters(denoted as $\left\{
\sigma \right\} $) to characterize the high energy modes'
contributions in the coarse grained objects. For this coarse
graining or emergence scenario to be effective, the magnitude of
the parameters in energy unit must be {\bf such that} $\sup
\left\{ \Lambda _{QFT} \right\} \ll \inf \left\{ \sigma \right\} $
with $\Lambda _{QFT}$ representing a general dimensional parameter
(momenta or masses) in the QFT in under consideration.

The preceding magnitude order analysis automatically activates a
limit operation with respect to $\left\{ \sigma \right\} $ on the
coarse grained amplitudes for describing 'low' energy processes,
which will be denoted as $L_{\left\{ \sigma \right\} }$($ \equiv
\lim_{\left\{ \sigma \right\} \rightarrow 0}$ in length unit).
Then the finite coarse grained vacuum functional in the presence
of the external sources for low energy processes reads \FL
\begin{eqnarray}
&&Z\left( J\left( x\right) |\{\bar{c}\}\right) \equiv L_{\left\{
\sigma \right\} }Z\left( J\left( x\right) |\{\sigma
\}\right)\equiv L_{\left\{ \sigma \right\} }\int D\Phi \left(
x{\bf |}\left\{ \sigma \right\} \right)
\exp \left[ \frac i\hbar S\left( \Phi \left( x%
{\bf |}\left\{ \sigma \right\} \right) ;\left\{ \sigma \right\}
\Vert J\right) \right] ,
\end{eqnarray}
where the objects with $\left\{ \sigma \right\} $ dependence being
coarse grained and well defined in QTOE. It is the presence of
$\left\{ \sigma \right\}$ defined by QTOE that makes the
functional integration over the coarse grained degrees UV
finite\footnote{This corresponds to the loop integration in QFT or
equivalently the low energy sectors of QTOE, which should be
clearly discriminated from the 'integrating out of high energy
modes' that in our presentation {\em refers to} the coarse
graining operation.}, or the UV regions contribute a finite amount
at most. Then in Eq.(1) the order of functional integration and
$L_{\left\{ \sigma \right\} }$ can not be trivially exchanged,
otherwise we would get the ill defined QFT's or divergences, i.e.,
\begin{eqnarray}
L_{\left\{ \sigma \right\} }\int D\Phi \left( x{\bf |}\left\{
\sigma \right\} \right) \exp \left[ \frac i\hbar S\left( \Phi
\left( x{\bf |}\left\{ \sigma \right\} \right) ;\left\{ \sigma
\right\} \Vert J\right) \right] \neq \int D\Phi \left( x\right)
\exp \left[ \frac i\hbar S\left( \Phi \left( x\right) \Vert
J\right) \right] ,
\end{eqnarray}
with $ S\left( \Phi \left( x\right) \Vert J\right) \equiv
L_{\left\{ \sigma \right\} }S\left( \Phi \left( x{\bf |}\left\{
\sigma \right\} \right) ;\left\{ \sigma \right\} \Vert J\right) $
and $ \Phi \left( x\right)\equiv L_{\left\{\sigma
\right\}}\Phi\left( x{\bf |} \left\{ \sigma \right\} \right) $
being well defined. But the path integral $\int D\Phi \left(
x\right) \exp \left[ \frac i\hbar S\left( \Phi \left( x\right)
\Vert J\right) \right]$ does not exist, which is just the 'formal'
QFT definition that requires extra cutoff or other artificial
regularization in conventional programs.

In terms of an ill-defined one-loop Feynman diagram, this is,\FL
\begin{eqnarray}
L_{\left\{ \sigma \right\} }\Gamma \left( \left( p\right) ,\left(
m\right) ;\{\sigma \}\right) \equiv L_{\left\{ \sigma \right\}
}\int d^DQ\bar{f}_\Gamma \left( Q,\left( p\right) ,\left( m\right)
;\{\sigma \}\right) \neq \int d^DQf_\Gamma \left( Q,\left(
p\right) ,\left( m\right) \right),
\end{eqnarray}
with $f_\Gamma (Q,\left( p\right) ,\left( m\right) )$ being the
integrand of this diagram defined in conventional QFT. The loop
momentum, external momenta and masses are denoted respectively by
$Q,\left( p\right) $ and $\left( m\right)$. Again, the presence of
$\{\sigma \}$ makes the loop integration finite or the UV
contribution suppressed, as the propagators and vertices defined
by $S\left( \Phi \left( x{\bf |}\left\{ \sigma \right\} \right)
;\left\{ \sigma \right\} \Vert J\right)$ must be strikingly
different from the ones given by $S\left( \Phi \left( x\right)
\Vert J\right)$ {\em in UV region.}

In principle we could not evaluate the generating functional or
the Feynman amplitudes without knowing the exact dependence upon
$\{\sigma \}$. However, we can determine each one loop amplitude
(ill defined in QFT) $ L_{\left\{ \sigma \right\} }\Gamma \left(
\left( p\right) ,\left( m\right) ;\{\sigma \}\right) $ up to an
appropriate polynomial of momenta and masses with finite but
undetermined coefficients {\em merely from the existence} of the
QTOE.

{\bf THEOREM}. \emph{A one loop amplitude $\Gamma $ defined in
QTOE (ill defined in the conventional QFTs) satisfies the
following kind of natural differential equation, \FL
\begin{eqnarray}
\label{diffeq} \left( {\partial }_p\right) ^{\omega _\Gamma
+1}L_{\left\{ \sigma \right\} } \bar{\Gamma}\left( \left( p\right)
,\left( m\right);\left\{ \sigma \right\} \right) =\int d^DQ\left(
{\partial }_p\right) ^{\omega _\Gamma +1}f_\Gamma (Q,\left(
p\right) ,\left( m\right) )\ \ \left(\equiv \Gamma ^{\left( \omega
_\Gamma \right) }\left( ( p) ,( m) \right)\right)
\end{eqnarray}
with $\omega _\Gamma $ being the superficial divergence degree or
scaling dimension of such a diagram}.

${\sl Proof}$: Since QTOE is completely well defined, then in any
dimension $D$ of spacetime we have\cite{YYY} $$ \begin{array}{l}
 \left( {\partial }_p\right)^{\omega _\Gamma +1}
 L_{\left\{ \sigma \right\}}\bar{\Gamma}\left(
 \left( p\right) , \left( m\right);\left\{\sigma \right\}
 \right)  =L_{\left\{ \sigma \right\}}\int d^DQ
 \left( {\partial }_p\right) ^{\omega _\Gamma
+1}\bar{f}_{\bar{\Gamma}}(Q,\left( p\right) , \left(
m\right);\left\{ \sigma \right\} )\\ =\int
d^DQ\left({\partial}_p\right) ^{\omega _\Gamma +1}L_{\left\{
\sigma \right\} }\bar{f}_{\bar{\Gamma} }(Q,\left( p\right) ,\left(
p\right) ;\left\{ \sigma \right\} ) =\int d^DQ\left( {\partial
}_p\right) ^{\omega _\Gamma +1}f_\Gamma (Q,\left( p\right) ,\left(
m\right) )\ \ \ \ Q.E.D.
\end{array}$$
Similar differential equations also hold with ${\partial }_p$
replaced by $\partial _m$.

The solutions to such differential equations are easy to obtain as
%%eq.(5)
\FL
\begin{eqnarray}
\label{solution} &&\Gamma \left( ( p) ,( m);\{\bar{c}\}
\right)\equiv L_{\left\{ \sigma \right\} }\bar{\Gamma}\left(
\left( p\right) ,\left( m\right);\left\{ \sigma \right\}
\right)\doteq \left( \int_p\right) ^{\omega _\Gamma +1} \Gamma
^{\left( \omega _\Gamma \right) }\left( ( p) ,( m)
\right)\nonumber \\
&&=\left( \int_p\right) ^{\omega _\Gamma +1}\int d^DQ\left(
{\partial }_p\right) ^{\omega _\Gamma +1}f_\Gamma (Q,\left(
p\right) ,\left( m\right) )
\end{eqnarray}
with the symbol '$\doteq $' indicating that the two sides are
equal up to certain integration constants in a polynomial of
momenta and masses of power $\omega _\Gamma $. To determine the
integration constants (which will be denoted as $\left\{ C\right\}
$ in contrast to $ \left\{ \bar{c}\right\} $ that are defined in
QTOE from the operation $L_{\left\{ \sigma \right\} }$) we need
'boundary conditions' like symmetries, sum rules and finally
experimental data, a natural procedure that surpasses the
renormalization conditions\footnote{Conventionally, after the
renormalized amplitudes are defined under certain renormalization
conditions, one still have to resort to physical inputs or
experiments, i.e., to employ boundary conditions to pin down the
physical scales and parameters\cite{Sterman}. In our strategy the
problem is simply to fix the integration constants via certain
physical boundary conditions. No intermediate renormalization
condition is needed.}, just like what we do in quantum mechanics
and classical electrodynamics. We note that there must be a
dimensional constant $\bar{\mu}$ in $\{\bar{c}\}$ characterizing
the typical scale of the QFT under consideration (the one in
$\left\{ C\right\} $ will be denoted as $ \mu _{int}$).
Eq.(~\ref{diffeq}) or (~\ref{solution}) is just our general
strategy for evaluating the Feynman amplitudes that dispenses
artificial regularizations and the manipulation of divergences. It
works in the same way for multiloop diagrams\cite {YYY}, i.e., one
simply inserts a pair of $\left( \int_p\right) ^{\omega _\Gamma
+1}$ and $\left( {\partial }_p\right) ^{\omega _\Gamma +1}$ to the
two sides of each divergent loop integration as $L_{\left\{ \sigma
\right\} } $ crosses the loop integration from the left until the
$L_{\left\{ \sigma \right\} }$ is finally removed from all loops
in the diagram. For convergent loops $L_{\left\{ \sigma \right\}
}$ can trivially cross the loop integrations. However, by defining
that $\left( \partial \right) ^n\equiv {\left( \int \right) }^{
|n|},\ {\left( \int \right) }^n\equiv \left( \partial \right)
^{|n|},\ $for$\ n<0,{\left( \int \right) }^n=\left( \partial
\right) ^n=1,\ $ for$\ n=0$ and noting that $\left( \int \right)
^n\times \left(
\partial \right) ^{n}=\left( \partial \right) ^{|n|}\times \left(
\int \right) ^{|n|}=1 $ for $n<0$ we can also put a convergent
loop into the form of Eq.~(\ref{solution}).

We emphasize that the above expressions are correct provided the
magnitude order $\sup \left\{ |p|,m,\bar{\mu}\right\} \ll \inf
\left\{ \sigma \right\} $ is satisfied. It is clear that no
infinite counterterms and bare parameters is present except the
finite 'bare' or tree parameters. It is also evident that our
strategy is obviously applicable to any interactions (fields with
any spin) in any spacetime.

The integration constants $ \left\{ C\right\} $ span a space in
which the QTOE prediction $\left\{ \bar{c }\right\} $ just lies on
one point of this space (up to equivalence). Obviously, the QTOE
definition of the Lagrangian constants and $\left\{
\bar{c}\right\} $ should be scheme and scale (SAS)
invariant\cite{scheme}. This observation may help to reduce or
even resolve the scheme dependence problem\cite{scheme} by
adopting our simple strategy, as in this strategy one could start
with finite 'bare' (hence SAS invariant) Lagrangian parameters,
and the problem is to find physical or reasonable boundary
conditions. Thus inequivalent choices of $\left\{ C\right\} $
would correspond to different physics. (Note that here the words
'bare parameters' does not mean no interaction as they are just
the tree parameters that characterize the low energy processes.)

Now we turn to the issue how the renormalization group equation
(RGE) arises in this QTOE scenario.

From the preceding discussions, we can parametrize the constants
$\{\bar{c}\}$ so that $\{\bar{c}\}=\{\bar{\mu},\left[ \bar{c}
^0\right] \}$, $\forall \bar{c}^0,\forall g:$ dim$\left\{
\bar{c}^0\right\} =0$, dim$\left\{ g\right\} \neq 0,\ \partial
_{\bar{\mu}}\bar{c} ^0=\partial _g\bar{c}^0=0,$ , i.e., all the
dimensional constants are factorized as $\{\bar{\mu}{\bar{c}}^0\}$
with $\{\bar{c}^0\}$ dimensionless, as $\{\bar{c}\}$ are all of
the same order as $\{g\}$. Otherwise they should belong to
$\{\sigma\}$. Then the full scaling law for a complete vertex
function $\Gamma ^{\left( n\right) }\left( \left( p\right) ,\left(
g\right) ;\{\bar{c}\}\right) $ (with $ \left( g\right) $ denoting
the 'bare' or tree couplings and masses)reads, \FL
\begin{equation}
\label{CSE} \left\{ s\partial _s+\Sigma d_gg\partial
_g+\bar{\mu}\partial _{\bar{\mu} }-d_{\Gamma ^{(n)}}\right\}
\Gamma ^{(n)}(\left( sp\right) ,\left( g\right) ;\{\bar{c}\})=0.
\end{equation}
with $d_g\equiv$ dim$\{g\}$. Since all the constants $\{\bar{c}\}$
only appear in the local parts of 1PI vertices, then
$\bar{\mu}\partial _{\bar{\mu}} (=\Sigma d_{\bar {c}}{\bar
{c}}\partial_{\bar {c}})$ induces the insertion of the vertex
operators $\{O\}$, i.e., $\bar{\mu}\partial
_{\bar{\mu}}=\Sigma_{\{O\}}\delta _O\hat{I}_O$, or equivalently,
\FL
\begin{eqnarray}
\bar{\mu}\partial _{\bar{\mu}}\Gamma ^{(n)}\left( \left( p\right)
,\left( g\right) ;\{\bar{\mu},[\bar{c}^0] \}\right)
=\Sigma_{\{O\}}\delta _O\hat{I}_O\Gamma ^{(n)}\left( \left(
p\right) ,\left( g\right) ;\{\bar{\mu} ,[\bar{c}^0] \}\right) .
\end{eqnarray}
This is just the general form of RGE for any type of interactions
in our approach. The solutions of Eq.(4) in terms of masses will
show that the anomalous dimension $\delta _O$ of a vertex operator
$O$
must be functions of dimensionless tree couplings $[g^0] $ and $%
[\bar{c}^0] $, i.e., $\delta _O=\delta _O\left( [g^0] ,[\bar{c}^0]
\right) $\cite{YY}.

Some of the vertex operators are defined in the Lagrangian with
couplings $\left( g\right)$, so $\Sigma_{\{O\}}\delta
_O\hat{I}_O=\Sigma \delta _gg\partial _g+\Sigma \delta _\phi
\hat{I}_{\phi;kinetic}+ \Sigma_{\{\bar{O}\}}\delta
_{\bar{O}}\hat{I}_{\bar{O}}$, with $\phi $ and $ \bar{O}$ denoting
respectively the 'elementary' fields in Lagrangian and the
operators not defined in Lagrangian. $\hat{I}_{\phi;kinetic}$
denotes the kinetic vertex for field $\phi$ and the operation
$\delta _\phi \hat{I}_{\phi;kinetic}$ rescales the field operator
$\phi $ by amount $\frac{\delta _\phi }2$. Apparently
$\Sigma_{\{\bar{O}\}}\delta _{ \bar{O}}\hat{I}_{\bar{O}}$ is
absent in renormalizable theories. Thus in renormalizable
theories, we obtain that \FL
\begin{equation}
\label{RG}
\left\{ \bar{\mu}\partial _{\bar{\mu}}-\Sigma \bar{\delta}_gg\partial _g-%
\bar{\delta}_{\Gamma ^{\left( n\right) }}\right\} \Gamma
^{(n)}\left( \left( p\right) ,\left( g\right)
;\{\bar{\mu},[\bar{c}^0] \}\right) =0
\end{equation}
with $\bar{\delta}_g\equiv \delta _g-\Sigma _{\left[ \phi \right] _g}\frac{%
\delta _\phi }2$. Since $\left( g\right) $ and $\{\bar{\mu},\left[ \bar{c}%
^0\right] \}$ should be uniquely determined by QTOE, the variation
in Eq.(~\ref{RG}) should be understood as the change due to the
global rescaling of everything. By introducing a set of scale
co-moving parameters based on Coleman's bacteria
analogue\cite{Cole}, we finally arrive at the standard form of RGE
in place of Eq.(~\ref{RG}) \FL
\begin{equation}
\label{RG2}
\left\{ \mu \partial _\mu -\Sigma \bar{\delta}_{\bar{g}}\bar{g}\partial _{%
\bar{g}}-\bar{\delta}_{\Gamma ^{\left( n\right) }}\right\} \Gamma
^{(n)}\left( \left( p\right) ,\left( \bar{g}\right) ;\{\mu ,[ \bar{c}%
^0] \}\right) =0,\ \ \
\end{equation}
with $\mu \partial _\mu \bar{g}\left( \mu ;\left( g\right) \right) =\bar{g}%
\left( \mu ;\left( g\right) \right) \bar{\delta}_{\bar{g}}\left( \left[ \bar{%
g}^0\left( \mu ;[g^0] \right) \right] ,[\bar{c}^0]
\right) ,$ $\ \bar{g}\left( \mu ;\left( g\right) \right) |_{\mu =\bar{\mu}%
}=g,\ \ \mu \equiv t\bar{\mu},\ \hspace{0.01in}t:\max \left[ \mu
\right] \ll \inf \left\{ \sigma \right\} $. We see that the
'running' of the parameters is nothing but the rescaling of $\bar
{\mu}$ whose existence is naturally guaranteed by QTOE in the low
energy limit, the mystery in the dimensional transmutation
phenomenon is therefore removed.

Inserting Eq.(~\ref{RG2}) back into Eq.(~\ref{CSE}) we will get
the full scaling law defined by the Callan-Symanzik
equation\cite{CS} \FL
\begin{eqnarray}
&&\left\{ s\partial _s+\Sigma
\bar{\delta}_{\bar{g}}\bar{g}\partial_{\bar{g}}+
\bar{\delta}_{\Gamma ^{\left( n\right) }} -d_{\Gamma
^{(n)}}\right\} \Gamma ^{(n)}\left( \left( sp\right) ,\left(
\bar{g}\right) ;\{\bar{\mu},[\bar{c}^0] \}\right) =-i\Gamma
_\Theta ^{(n)}\left( 0,\left( sp\right) ,\left( \bar{g}\right) ;\{
\bar{\mu},[\bar{c}^0] \}\right),
\end{eqnarray}
where
%%EQ(11,12)
\FL
\begin{eqnarray}
&&s\partial _s\bar{g}\left( s\bar{\mu};\left( g\right) \right)
 =\bar{g}\left( s\bar{\mu};\left( g\right) \right)
 \bar{\delta}_{\bar{g}}\left(
[ \bar{g}^0( s\bar{\mu};[g^0]) ] ,[\bar{c}^0] \right),\ \ \
\bar{g}\left(s\bar{\mu};( g) \right) |_{s=1}=g,\\ & & i\Gamma
_\Theta ^{(n)}\left( 0,\left( sp\right) ,\left( \bar{g}\right)
;\{\bar{\mu},[\bar{c}^0] \}\right)\equiv \Sigma
d_{\bar{g}}\bar{g}\partial _{\bar{g}}\Gamma ^{(n)}\left( \left(
sp\right) ,\left( \bar{g}\right) ;\{\bar{\mu},[\bar{c}^0]
\}\right) ,
\end{eqnarray}
with $\Theta $ being the trace of the energy tensor of the theory.
Of course in practice we are forced to use $\{C\}(=\{\mu
_{int},\left[ C^0\right] \})$ instead of $\{\bar{c%
}\}$ (Here $\mu _{int},\left[ C^0\right]$ parallel
$\bar{\mu},[\bar{c}^0]$).

Next Let us show why RGE can be interpreted as a decoupling
theorem of the underlying high energy modes. Since the constants
$\{\bar {c}\}$ arise from the low energy limit operation, then
before taking this limit Eq.(~\ref{CSE}) must be written as \FL
\begin{eqnarray}
&&\left\{ s\partial _s+\Sigma d_gg\partial _g+\Sigma d_\sigma \sigma
\partial _\sigma -d_{\Gamma ^{(n)}}\right\} \bar{%
\Gamma}^{(n)}(\left( sp\right) ,\left( g\right) ;\{\sigma \}) =0
\end{eqnarray}with the $\Sigma d_{\bar {c}}{\bar
{c}}\partial_{\bar {c}}$ or $\bar{\mu}\partial _{\bar{\mu}}$
replaced by $\Sigma d_\sigma \sigma
\partial _\sigma $. This is obviously a normal scaling law in
QTOE. However, as $\left\{\sigma\right\}$ go to zero, though the
high energy modes become 'formally' decoupled, their contributions
in the scaling law persist as the scaling 'anomalies' in terms of
the 'tree' parameters $\{g\}$, which can be subsumed into
$\bar{\mu}\partial _{\bar{\mu}}$, a coarse grained way to
reproduce the underlying structures' contributions, i.e., \FL
\begin{eqnarray}
\label{decoupl}  L_{\left\{ \sigma \right\} }\left(\Sigma d_\sigma
\sigma\partial _\sigma \bar{%
\Gamma}^{(n)}(\cdots)\right)= \left\{\Sigma
\bar{\delta}_{\bar{g}}\bar{g}\partial _{
\bar{g}}+\bar{\delta}_{\Gamma ^{\left( n\right) }}\right\} \Gamma
^{(n)}\left( \cdots\right)= \bar{\mu}\partial _{\bar{\mu}}
\Gamma^{(n)}(\cdots).
\end{eqnarray}Thus it is the
'decoupling effects of high energy modes' that lead to the
violation of naive scaling law, no divergence is involved here.
Moreover, Eqs.~(\ref{RG2}) and (11), (12) tell us that these
'scaling anomalies' from the underlying modes decoupling could
lead to finite 'renormalization' of the tree parameters. One might
gain some intuition about this mechanism from the decoupling
effects of heavy fermions upon the beta function in QCD or QED as
was illustrated in Ref.\cite{PLB393}, where
$\lim_{M\rightarrow\infty}M\partial_{M}\Gamma(=i \Gamma _\Theta)
=\Delta\beta\alpha
\partial_{\alpha} \Gamma$ with $\Theta = M \overline{\psi} \psi $.
Thus the meaning of RGE is deepened as a decoupling theorem
following from the coarse grained formulations of the low energy
sectors of QTOE.

Conventionally, one is forced to use some artificial regulators to
define the UV or high energy ends of a QFT so that the loop
amplitudes could be calculated. Such procedures do not
automatically make the loop amplitudes finite and subsequent
subtraction of divergent pieces is necessary. The subtraction
leads to a residual ambiguity that is in fact encoded in RGE.
While in the QTOE scenario, the amplitudes that are only formally
defined in QFT are well defined as coarse grained objects in QTOE
with high energy details integrated out. In the low energy limit,
the decoupling of the high energy modes give rise to the so-called
scaling 'anomalies' that in turn lead to the freedom of
redefinition encoded in RGE. Neither regularization nor
subtraction is involved here. In a sense, we provide a more
physical and reasonable foundation for RGE and finite
renormalization.

Finally, we stress the importance of the simplicity of our
strategy in obtaining finite loop amplitudes, since conventional
renormalization approaches often make certain nonperturbative
framework very complicated or even useless. The same is true for
the electroweak theory whose renormalization is rather complicated
in conventional approaches\cite{EW}. It is easy to see that our
strategy could be readily applied to the nonperturbative problems
such as multi-nucleon interactions\cite{Nucl} that is currently a
hot topic. Applications of our approach to these problems will be
intensively pursued in the future.

In summary, we provided natural arguments in favor of a recently
proposed simple strategy for renormalization. The key point is the
{\em existence of a complete quantum theory of everything} which
contains full information of high energy physics. From this QTOE
scenario, the Callan-Symanzik equation and RGE can be derived in a
natural way with the RGE recognized as a decoupling theorem of the
high energy modes. The conceptual foundation for renormalization
group is more reasonable in the QTOE scenario and is not entangled
with divergence at all.
\section*{Acknowledgement}
The author is grateful to W. Zhu for helpful conversations and
encouragement.
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\bibitem{Nucl} For recent reviews, see, e.g., P. F. Bedaque, and U. van
Kolck, e-print No.  and references therein.
\end{references}

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