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nonrenormalizability
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% The Taming of Nonr3enormalizability
% by John R. Klauder
% November 8, 2008
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%\title{Nonrenormalizability Tamed!}
\title{Taming Nonrenormalizability}
\author{John R. Klauder\footnote{klauder@phys.ufl.edu}
\\Department of Physics and \\Department of Mathematics\\
University of Florida\\
% P.O. Box 118440\\
Gainesville, FL 32611-8440}
\date{ }
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\begin{document}
\maketitle
\begin{abstract}
Unlike asymptotically free quantum field theories, asymptotically
nonfree theories have not been well served by conventional
perturbation analysis. In contrast, the goal of a proper
renormalization scheme for perturbatively nonrenormalizable scalar
quantum field theories should be to neutralize the source of the
divergences. Achieving this goal requires an entirely different kind
of counterterm than those typically employed, and, as we shall
argue, including the appropriate counterterm as part of a pseudofree
model about which to expand leads to an alternative perturbative
formulation that is term-by-term finite.
\end{abstract}
\section*{Introduction}
Nonrenormalizable quantum field models, such as $\varphi^4_n$, with
a spacetime dimension $n\ge5$, require the introduction of
nonclassical, nontrivial counterterms \cite{AF}. A conventional
regularized perturbation theory is {\it re-active} in the sense that
it cancels divergences one-by-one as they are encountered in a
perturbation analysis. Rather than following this traditional
procedure, suppose one could find a {\it pro-active} procedure that
removes the {\it cause} of the divergences in the first place. In a
nutshell, that is our procedure; namely, we choose our counterterm
to remove the {\it cause of the divergences at the outset}, rather
than adopt a series of counterterms that deal with divergences as
they arise.
Initially, we choose an $n$-dimensional, periodic, hypercubic,
Euclidean spacetime lattice with
a lattice spacing $a$, $L$ lattice points on each side, and lattice points
labeled by multi-integers $k=(k_0,k_1,\ldots,k_s)\in {\mathbb Z}^n$,
where $s=n-1$ is the spatial dimension, and $k_0$ refers to a future
time direction. The lattice-regularized functional integral for the
Schwinger function generating functional is given by
\b &&S(h)\equiv M\s\int e^{\t\s Z^{-1/2}\s\Sigma_k h_k\s\p_k\s a^n/\hbar
-I_n(\p,a,N)/\hbar\s-C(\p,a,\hbar)/\hbar}\;\Pi_k\s d\p_k\no\\
&&\hskip.96cm\equiv\\;, \label{t3}\e
where $\{h_k\}$ determines an appropriate test sequence, and
the normalization factor $M$ ensures that $S(0)=1$. The
continuum limit is taken in two steps: (i) The number of
lattice sites on an edge $L\ra\infty$ and the lattice spacing
$a\ra0$ so that $La$ remains constant and finite. Thus the
spacetime volume $V=(La)^n$ as well as the spatial volume (at fixed
Euclidean time) $V'=(La)^s$ are both finite; (ii) The final step involves $V\ra\infty$ and
$V'\ra\infty$. In this article we focus on just the first step in the continuum limit and
assume that both $V$ and $V'$ are sufficiently large. Notationally, we also
introduce $N=L^n$ and $N'=L^s$, and note that
sums (and products) such as $\Sigma_k (\Pi_k)$ are over all spacetime,
while $\Sigma'_k (\Pi'_k)$ are over all space alone at some fixed $k_0$.
In (\ref{t3}), $Z$ denotes the field strength renormalization factor and
$I_n(\p,a,N)$ is the naive lattice action,
\b \hskip.2cm I_n(\p,a,N)\equiv\half{\t\sum_k} {\t\sum_{k^*}}\,(\phi_{k^*}-\phi_k)^2\,
a^{n-2}+\half m_0^2{\t\sum_k} \phi_k^2\, a^n
+ \l_0{\t\sum_k}\phi^4_k\,a^n\,,\label{t4}\e
where $k^*$ denotes any one of the $n$ nearest neighbors to $k$ in
the positive sense, i.e.,
$k^*\in\{\,(k_0+1,k_1,\ldots,k_s)\s,\ldots,\s(k_0,k_1,\dots,k_s+1)\,\}$.
Also in (\ref{t3}) the term
\b C(\p,a,\hbar)\equiv\half\s\hbar^2\s{\t\sum}_k {\F}_k(\p)\s a^n \e
represents the
still-to-be-chosen counterterm.
Assuming that the spacetime volume $V<\infty$, it is clear that full
spacetime averages such as $\<\s[{\t\sum}_k \p_k^r\s a^n]^p\s\>$ are
finite, for all positive integers $r$ and $p$, provided that all the
corresponding sharp-time, spatial averages $\<[\s{\t\sum}'_k
\p_k^r\s a^s]^p\>$ are finite; for a proof, see \cite{kla}. In turn,
the latter expression can be represented as
\b \<\s[\s{\t\sum}'_k \p_k^r\s a^s]^p\s\>=\int [\s{\t\sum}'_k \p_k^r\s
a^s\s]^p\,\Psi(\p)^2\,\Pi'_k\s d\p_k\;, \e
where $\Psi(\p)$ denotes the ground state of the system.
\subsection*{Choice of Counterterm}
As an idealized example of a Gaussian ground-state distribution
characteristic of a typical free theory, let $\Psi_G(\p)^2\equiv R\s
\exp[-A\Sigma'_k \p_k^2\s a^s]$ and, for integral $p\ge0$, consider
the integrals
\b I_p(A)\equiv R\int [\Sigma'_k\p_k^2\s a^s]^p\s e^{\t-A\Sigma'_k
\p_k^2\s a^s}\s\Pi'_kd\p_k\;.\e
Such integrals can be evaluated exactly, but we prefer to study them
in an approximate sense by steepest descent methods. To that end we
introduce hyper-spherical coordinates \cite{klau2} defined by
\b \p_k\equiv\kappa\s\eta_k\hskip.2cm,\hskip.3cm\Sigma'_k\p_k^2=
\kappa^2\hskip.2cm,\hskip.3cm
\Sigma'_k\eta_k^2=1\hskip.2cm,\hskip.3cm
0\le\kappa<\infty\hskip.2cm,\hskip.3cm\
-1\le\eta_k\le1\;, \e
and it follows that
\b I_p(A)=2\s R\int \kappa^{2p}\s a^{sp} e^{\t-A\s\kappa^2\s a^s}\,\kappa^{(N'-1)}d\kappa
\s\s\delta(1-\Sigma'_k\eta_k^2)\s\Pi'_kd\eta_k\;. \e
A steepest descent argument leads to
\b I_p(A)=O((N'/A)^p)\,I_0(A)\;, \e
and a perturbation series for $I_1(A)$ about $I_1(1)$ is given by
\b I_1(A)=I_1(1)-\Delta\s
I_2(1)+\half\s\Delta^2\s I_3(1)-\cdots\;, \label{w17}\e
where $\Delta=A-1$. As $N'\ra\infty$, such a series has
higher-order, term-by-term
divergences because the support of the ground-state
distribution is concentrated on {\it disjoint sets for
distinct $A$ values} due, essentially, to the factor
$\kappa^{(N'-1)}$ in the integrand.
Our goal is to introduce a counterterm that effectively cancels
the factor $\kappa^{(N'-1)}$, and this can be accomplished, loosely speaking, by
choosing an idealized example of a {\it pseudofree model} about which to expand with
a ground-state distribution such that
\b \Psi_I(\p)^2\propto \kappa^{-(N'-1)}\,e^{\t-A\Sigma'_k\p^2_k\s a^s}\;.\label{w6} \e
Observe that the use of the distribution $\Psi_I(\p)^2$
in place of $\Psi_G(\p)^2$ above leads to a series analogous to (\ref{w17})
that is term-by-term finite.
More precisely, to deal with realistic models,
we focus on a
ground state for the pseudofree {\it (pf)} model given by
\b \Psi_{pf}(\p)= K \s\s\frac{e^{\t-\Sigma'_{k,l}\p_k\s A_{k-l}\s\p_l\s
a^{2s}/2\hbar
-W(\p\s\s a^{(s-1)/2}/\hbar^{1/2})/2}}
{\Pi'_k[\Sigma'_lJ_{k,l}\s\p_l^2]^{(N'-1)/4N'}}\;;\label{w7}\e
we discuss the constants $A_{k-l}$ and $J_{k,l}$ and the function $W$ below.
This form for the ground state is ensured if we define the
pseudofree theory -- the theory about which a perturbation
expansion is to take place -- as
\b S_{pf}(h)=M_{pf}\int e^{\t-\half{\t\sum_k}
{\t\sum_{k^*}}\,(\phi_{k^*}-\phi_k)^2\s a^{n-2}/\hbar-\half\s\hbar\s{\t\sum_k}{\F}_k(\p)\s a^n}
\,\Pi_k\s d\p_k\,, \e
and choose ${\F}_k(\p)$ to yield the denominator in (\ref{w7}).
Based on the lattice Hamiltonian for the pseudofree model,
\b &&\H_{pf}= -\half\s{\hbar^2}\, a^{-s}\s{\t\sum_k}'\frac{\d^2}{\d
\phi_k^2}
+\half{\t\sum'_k}{\t\sum'_{k^*}}\,(\p_{k^*}-\p_k)^2a^{s-2}\no\\
&&\hskip1.27cm +\half\s\hbar^2{\t\sum_k}'\F_k(\p)\,a^s -E_0 \;, \e
this connection leads to
\b &&\F_k(\p)
\equiv\frac{1}{4}\s\bigg(\frac{N'-1}{N'}\bigg)^2\s
a^{-2s}\s{\t\sum'_{\s r,\s t}}\s\frac{J_{r,\s k}\s
J_{t,\s k}\s \p_k^2}{[\S_l\s J_{r,\s l}\s\p^2_l]\s[\S_m\s
J_{t,\s m}\s\p_m^2]} \no\\
&&\hskip2.5cm-\frac{1}{2}\s\bigg(\frac{N'-1}{N'}\bigg)
\s a^{-2s}\s{\t\sum'_{\s t}}\s\frac{J_{t,\s k}}{[\S_m\s
J_{t,\s m}\s\p^2_m]} \no\\
&&\hskip2.5cm+\bigg(\frac{N'-1}{N'}\bigg)
\s a^{-2s}\s{\t\sum'_{\s t}}\s\frac{J_{t,\s k}^2\s\p_k^2}{[\S_m\s
J_{t,\s m}\s\p^2_m]^2}\;. \label{w19}\e
Irrespective of the choice for $J_{k,l}$,
we note that: (i) the denominator in the expression for the pseudofree
ground state specifically leads to the counterterm in the Hamiltonian; (ii) the term in the
exponent quadratic in $\p$
is chosen to yield the spatial-gradient term in the
Hamiltonian (and possibly part of $E_0$), and this requires that $A_{k-l}
\propto a^{-(s+1)}$; and (iii) the unspecified term $W$ ensures that no
additional terms (other than the rest of $E_0$) appear in the Hamiltonian. The
functional form of the argument in $W$ follows from the manner in
which both $\hbar$ and $a$ appear in the Hamiltonian. In addition,
note that the quadratic and denominator terms in $\Psi_{pf}(\p)$
are correct for very large and very small field values, respectively; hence $W$
is relatively most effective for intermediate field values.
The choice $J_{k,l}=\delta_{k,l}$ leads to a local covariant
potential for which $\F_k(\p)\propto 1/\p_k^2$, but it also gives rise to an unsuitable form
for the ground-state distribution with incipient normalization divergences at
$\p_k=0$, for each $k$,
as $N'\ra\infty$.
To eliminate that feature, we choose the factors $J_{k,l}$ to provide
a minimally regularized, lattice-symmetric,
local spatial averaging in the form
\b J_{k,\s
l}\equiv\frac{1}{2s+1}\s\delta_{\s k,\s l\in\{k\s\cup \s
k_{nn}\}}\;, \e where $\delta_{k,l}$ is a Kronecker delta. This
notation means that an equal weight of $1/(2s+1)$ is given to the
$2s+1$ points in the set composed of $k$ and its $2s$ nearest
neighbors in the spatial sense only; $J_{k,\s l}=0$ for all other
points in that spatial slice. {\bf [}Specifically, we define
$J_{k,\s l}=1/(2s+1)$ for the points $l=k=(k_0,k_1,k_2,\ldots,k_s)$,
$l=(k_0,k_1\pm1,k_2,\ldots,k_s)$,
$l=(k_0,k_1,k_2\pm1,\s\ldots,k_s)$,\ldots,
$l=(k_0,k_1,k_2,\ldots,k_s\pm1)$.{\bf ]} This definition implies
that $\Sigma'_l\s J_{k,\s l}=1$.
In the continuum limit, it is important to observe that the form of
the counterterm given by (\ref{w19}) leads to a local covariant
potential.
\section*{The Continuum Limit, and Term-by-term \\Finiteness of a Perturbation Analysis}
Before focusing on the limit $a\ra0$ and $L\ra\infty$, we
note several important facts about ground-state averages of the direction
field variables $\{\eta_k\}$. First, we assume that such averages
have two important symmetries: (i) averages of an odd number
of $\eta_k$ variables vanish, i.e.,
\b \<\eta_{k_1}\cdots\eta_{k_{2p+1}}\>=0\;, \e
and (ii) such averages are invariant under any spacetime translation, i.e.,
\b
\<\eta_{k_1}\cdots\eta_{k_{2p}}\>=\<\eta_{k_1+l}\cdots\eta_{k_{2p}+l}\>\;\e
for any $l\in{\mathbb Z}^n$ due to a similar translational
invariance of the lattice Hamiltonian. Second, we note that
for any ground-state distribution, it is
necessary that $\<\s\eta_k^2\s\>=1/N'$
for the simple reason that $\Sigma'_k\s\eta_k^2=1$. Hence,
$|\<\eta_k\s\eta_l\>|\le1/N'$ as follows from the Schwarz
inequality. Since $\<\s[\s\Sigma'_k\s\eta_k^2\s]^2\>=1$, it
follows that $\<\s\eta_k^2\s\eta_l^2\s\>=O(1/N'^{2})$.
Similar arguments show that for any ground-state distribution
\b \<\eta_{k_1}\cdots\eta_{k_{2p}}\>=O(1/N'^{p})\;, \e
which will be useful in the sequel.
\subsubsection*{Field strength renormalization}
For $\{h_k\}$ a suitable spatial test sequence, we insist
that expressions such as
\b \int Z^{-p}\,[\Sigma'_k h_k\s\p_k\,a^s]^{2p}\,\Psi_{pf}(\p)^2\,\Pi'_k\s
d\p_k \label{w20}\e
are finite in the continuum limit. Due to the intermediate
field relevance of the factor $W$ in the pseudofree ground state,
an approximate evaluation of the integral (\ref{w20}) will
be adequate for our purposes.
Thus, we are led to consider
\b &&\hskip-.8cm K\int Z^{-p}\,[\Sigma'_k
h_k\s\p_k\,a^s]^{2p}\,\frac{e^{\t-\Sigma'_{k,l}\s\p_k\s A_{k-l}\s\p_l\,
a^{2s}/\hbar-W}}{\Pi'_k[\s\S_lJ_{k,l}\p_l^2\s]^{(N'-1)/2N'}}\,\Pi'_k\s
d\p_k\\
&&\hskip-.7cm\simeq 2\s K_0\int Z^{-p}\s\k^{2p}\,[\Sigma'_k h_k\s\eta_k\,a^s]^{2p}\,
\frac{e^{\t-\k^2\s\Sigma'_{k,l}\s\eta_k\s A_{k-l}\s\eta_l\,a^{2s}/\hbar}}
{\Pi'_k[\s\S_l J_{k,l}\s\eta^2_l\s]^{(N'-1)/2N'}}\,d\k\,\delta(1-\Sigma'_k\eta_k^2)
\,\Pi'_k\s d\eta_k\;, \label{f5} \no\e
where $K_0$ is the normalization factor when $W$ is dropped.
Our goal is to use this integral to determine a value for
the field strength renormalization constant $Z$.
To estimate this integral we first replace two
factors with $\eta$ variables by their appropriate
averages. In particular, the quadratic expression in the exponent
is estimated by
\b \k^2\s\Sigma'_{k,l}\s\eta_k\s
A_{k-l}\s\eta_l\,a^{2s}\simeq\k^2\s\Sigma'_{k,l}\s N'^{\,-1}
A_{k-l}\,a^{2s}\propto \k^2\s N'\s a^{2s}\s a^{-(s+1)}\;, \e
and the expression in the integrand is estimated by
\b [\Sigma'_k h_k\s\eta_k\,a^s]^{2p}\simeq\s
N'^{\,-p}\,[\Sigma'_k h_k\,a^{s}]^{2p}\;. \e
The integral over $\k$ is then estimated by first rescaling the variable
$\k^2\ra\k^2/(N'\s a^{s-1}/\hbar)$, which then leads to an overall integral estimate proportional
to
\b Z^{-p}\,[N'\s a^{s-1}]^{\,-p}\,N'^{-p}\,[\Sigma'_k h_k\,a^{s}]^{2p}\;;\e
at this point, all factors of $a$ are now outside the
integral.
For this result to be meaningful in the continuum limit,
we are led to choose $Z=N'^{\,-2}\s a^{-(s-1)}$. However, $Z$ must
be dimensionless, so we introduce a fixed positive quantity
$q$ with dimensions of an inverse length, which allows us to
set
\b Z=N'^{\,-2}\s (q\s a)^{-(s-1)}\;. \e
\subsubsection*{Mass and coupling constant renormalization}
A power series expansion of the mass and coupling constant terms
lead to the expressions
$ \<\s [\s m_0^2\,\Sigma_k \p_k^2 a^n\s]^p\s\> $ and
$ \<\s [\s \l_0\,\Sigma_k \p_k^4 a^n\s]^p\s\> $ for $p\ge1$, which we treat
together as part of the larger family governed by
$\<\s [\s g_{0,r}\,\Sigma_k \p_k^{2r} a^n\s]^p\s\>$ for integral $r\ge1$.
Thus we consider
\b &&\hskip-.8cm K\int [\s g_{0,r}\Sigma'_k
\p_k^{2\s r}\,a^s]^{p}\,\frac{e^{\t-\Sigma'_{k,l}\s\p_k\s A_{k-l}\s\p_l\,
a^{2s}/\hbar-W}}{\Pi'_k[\s\S_lJ_{k,l}\p_l^2\s]^{(N'-1)/2N'}}\,\Pi'_k\s
d\p_k\\
&&\hskip-.7cm\simeq 2\s K_0\int g_{0,r}^p\s\k^{2 r p}\,[\Sigma'_k\s\eta_k^{2 r}\,a^s]^{p}\,
\frac{e^{\t-\k^2\s\Sigma'_{k,l}\s\eta_k\s A_{k-l}\s\eta_l\,a^{2s}/\hbar}}
{\Pi'_k[\s\S_l J_{k,l}\s\eta^2_l\s]^{(N'-1)/2N'}}\,d\k\,\delta(1-\Sigma'_k\eta_k^2)
\,\Pi'_k\s d\eta_k\;. \label{ff5} \no\e
The quadratic exponent is again estimated as
\b \k^2\s\Sigma'_{k,l}\s\eta_k\s
A_{k-l}\s\eta_l\,a^{2s}\propto \k^2\s N'\s a^{2s}\s a^{-(s+1)}\;, \e
while the integrand factor
\b [\Sigma'_k\eta_k^{2r}]^p\simeq N'^p\s N'^{-rp}\;. \e
The same transformation of variables used above precedes the integral
over $\k$, and the result is an integral, no longer depending on $a$, that is proportional to
\b g_{0,r}^p \s N'^{-(r-1)p}\s a^{sp}/N'^{rp}\s a^{(s-1)rp}\;.\e
To have an acceptable continuum limit, it suffices that
\b g_{0,r}=N'^{(2r-1)}\,(q\s a)^{(s-1)r-s}\,g_r \;,\e
where $g_r$ may be called the physical coupling factor.
Moreover, it is noteworthy that
$Z^r\,g_{0,r}=[N'\s (q\s a)^s]^{-1}\,g_r $,
for all values of $r$, which for a finite spatial volume $V'=N'\s a^s$ leads to a
finite nonzero result for $Z^r\s g_{0,r}$. It should not be a surprise that there are no
divergences for all such interactions because the source of all divergences has been
neutralized!
We may specialize the general result established above to the two cases of
interest to us. Namely, when $r=1$ this last
relation implies that $m_0^2=N'\s(q\s a)^{-1}\,m^2$, while when
$r=2$, it follows that $\l_0=N'^{\,3}\s(q\s a)^{s-2}\s\l$. In
these cases it also follows that $Z\s m_0^2=[\s N'\s (q\s a)^s\s]^{-1}\s m^2$ and
$Z^2\s \l_0=[\s N'\s (q\s a)^s\s]^{-1}\s \l$,
which for a finite spatial volume $V'=N'\s a^s$ leads to a
finite nonzero results for $Z\s m_0^2$ and $Z^2\s \l_0$, respectively.
\section*{Conclusion}
For scalar nonrenormalizable quantum field models, we
have shown that the choice of a nonconventional counterterm, but one
that is
still nonclassical, leads to a formulation for
which a perturbation analysis of both the mass term and
the nonlinear interaction term, expanded about the appropriate pseudofree model,
are term-by-term finite.
Alternative insight into such models may possibly be
obtained by Monte Carlo studies of the full,
nonperturbative model including the special counterterm; for
a preliminary discussion of such an approach, see \cite{kl}.
\begin{thebibliography}{99}
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$\varphi^4_d$ Field Theory and Some Mean-Field Features of Ising
Models for $d>4$'', {\it Phys. Rev. Lett.} {\bf 47}, 1-4, E-886
(1981); J. Fr\"ohlich, ``On the Triviality of $\l\varphi^4_d$
Theories and the Approach to the Critical Point in $d\ge 4$
Dimensions'', {\it Nuclear Physics B} {\bf 200}, 281-296 (1982).
\bibitem{kla} J.R. Klauder, ``A New Approach to Nonrenormalizable Models'',
{\it Ann. Phys.} {\bf 322}, 2569-2602 (2007).
\bibitem{klau2} J.R. Klauder, ``Poisson Distributions for Sharp-Time
Fields: Antidote for Triviality'', hep-th/9511202, pp 22-28; J.R.
Klauder, ``Isolation and Expulsion of Divergences in Quantum Field
Theory'', {\it Int. J. Mod. Phys. B} {\bf 10}, 1473-1483 (1996).
\bibitem{kl} J.R. Klauder, ``Divergnece-free Nonrenormalizable
Models'', {\it J. Phys. A: Math. Theor.} {\bf 41}, 335206 (15pp) (2008).
\end{thebibliography}
\end{document}
\bibitem{book} J.R. Klauder, Beyond Conventional Quantization, (Cambridge University Press,
Cambridge, 2000 \& 2005).
\bibitem{hida} T. Hida and M. Hitsuda, Gaussian Processes, (American Mathematical
Society, Providence, 2007).
\bibitem{kk6} J.R. Klauder, ``Field Structure through Model
Studies: Aspects of Nonenormalizable Theories'', {\it Acta. Phys.
Austr. Suppl. XI}, 341-387 (1973).
\end{thebibliography}
\end{document}
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