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\begin{document}
\title{Bosonization of Massive Fermions}
\author{
J. Dimock\thanks{Research supported by NSF Grant PHY9400626}\\
Dept. of Mathematics \\
SUNY at Buffalo \\
Buffalo, NY 14214 }
\maketitle
\begin{abstract}
We study the Euclidean sine-Gordon field theory on the plane with
$ \beta < 16\pi/3 $ and
an interaction density confined to a finite square.
For $\beta = 4\pi$ we construct correlation functions for the field
$:\sin \f:$ and show that they are equal
to the pseudoscalar $\psib \Gamma \psi $ correlation functions for a free
fermion theory with mass term confined to the finite square.
\end{abstract}
\newpage
\tableofcontents
\section{Introduction}
On a two dimensional space time certain boson and fermion quantum field theories are
equivalent, a phenomenon discovered by Coleman \cite{Col75}. This
equivalence has found applications to statistical mechanics and string theory.
The simplest case is the equivalence of massless free fermions and
massless free bosons on the plane with a Euclidean metric.
The massless free fermion theory is given formally by a (fermionic) functional measure
\be \exp \left ( \int (\bar \psi i \pa \diracslash \psi)(x)dx \right)
d\psib d \psi \ee
The massless free boson theory is given formally by the functional measure
\be \exp \left( -{1 \over 2 \beta }\int |\pa \f(x)|^2 dx \right) d \phi \ee
The equivalence is the statement that all correlation functions (moments) are equal
if $\beta = 4 \pi$ and we make the correspondance
\bea
c:\cos ( \f (x)):\ \ \ && \Longleftrightarrow
\ \ \ \ (\psib \psi )(x) \nn \\
ic:\sin ( \f (x)):\ \ \ && \Longleftrightarrow
\ \ \ \ (\psib \G \psi )(x)
\eea
where $\G = \g_5 = i\g_0\g_1$ and $c$ is a positive constant.
This massless equivalence is now well-understood. There is a large literature,
but for our purposes the best references are \cite{FrSe76}, \cite{FrMa88}.
Equivalence has been found on other two dimensional
manifolds as well, see for example \cite{ABM87}.
The correspondance becomes increasingly complicated
with the topology of the manifold. Already on the torus it seems
that one must take a circle valued field to make the correspondance.
In any case the massless equivalence suggests the equivalence
of the massive free fermions with mass $\mu$ and measure
\be \exp \left(\int ((\bar \psi i \pa \diracslash \psi)(x) +
\mu (\bar \psi \psi )(x))dx \right)
d\psib d \psi \ee
and sine-Gordon bosons at $\beta = 4 \pi$ with coupling constant $\z$
and measure
\be \exp \left( \int (-{1 \over 2 \beta} |\pa \f(x)|^2 + \z :\cos(\f(x)):) dx
\right) d \phi \ee
provided $\z = \mu c$.
Indeed the two theories are formally equivalent order by order in perturbation theory.
The result is remarkable because it is an equivalence between a linear
and a non-linear field theory. The sine-Gordon theory is of particular
interest because it describes the classical statistical mechanics of
a Coulomb gas at temperature $\beta^{-1}$ and activity $\z/2$.
The purpose of this paper is to give a rigorous proof of this widely conjectured
equivalence. What has been
missing till now is a proof that perturbation theory for the sine-Gordon
model converges to the actual model at $\beta = 4 \pi$.
This involves showing that the model exists and is analytic for $\z$ in a neighborhood
of the origin and explicitly computing the derivatives at $\z=0$.
Both steps are non-trivial due to the fact that at this value of $\beta$
the model requires an infinite energy renormalization.
The sine-Gordon model on the torus for $\beta < 8\pi$ was studied
by Dimock and Hurd \cite{DiHu93}.
(See however the remarks in section \ref{overview}).
Here we give a similar, but simpler, treatment on the plane.
Working with cutoffs we study correlation functions and prove analyticity
in $\z$ and bounds
uniform in the cutoffs for $\beta < 16 \pi/3$.
Then for $\beta = 4 \pi$ we prove existence and analyticity of the
theory with no cutoffs, and compute the derivatives at $\z=0$.
A parallel and much more elementary analysis is given for fermions.
Comparing derivatives at $\z=\mu=0$ we conclude that correlation
functions at non-coinciding points are the same in the two models.
The following qualifying remarks need to be made. First,
although the free action is taken on the whole plane,
the interaction/mass term is confined to a finite volume.
Taking the infinite volume limit becomes a separate issue
which is not addressed here. Secondly we only show equality
between correlation functions for $:\sin \f :$ and $ \psib \G \psi $.
The ordinary correlation functions for $:\cos \f :$ and $ \psib \psi $
do not exist. ( Truncated functions are a different story).
An earlier result is due to Fr\"{o}hlich and Seiler \cite{FrSe76} who consider
the sine-Gordon theory for $\beta < 4 \pi$. In this case no energy
renormalization is required and the convergence of perturbation theory
is simpler. They prove equivalence in perturbation
theory with a more complicated
Thirring-Schwinger model.
Our use of the free massless action for the whole plane
introduces some special infrared divergences not present or
not important for compact manifolds like the torus, and also
not present for the massive sine-Gordon model (Yukawa gas). The massive
sine-Gordon model is studied in \cite{Fro76},\cite{BGN82}, \cite{Ben85},
\cite{NRS86}, \cite{BrKe87}. We particularly note the
paper of Brydges and Kennedy \cite{BrKe87}, where
the analyticity of the renormalized partition function is established
for $\beta < 16 \pi/3$ by a precursor of the method used in the present
paper.
The organization of the paper is as follows. In section 2.1 we review some material
about exponential functions of the free boson field, and prove a new result about
truncated correlation functions. In section 2.2 we prove several forms
of the equivalence with the free fermi field.
In section 2.3 we define the sine-Gordon theory in a box,
quote basic existence and analyticity results for $\beta < 16\pi/3 $
and then prove the equivalence at $\beta = 4 \pi$
with massive free fermions in a box.
In section 3 we use a renormalization group technique to prove
the facts about sine-Gordon needed in section 2.
\section{Boson-fermion equivalence}
\subsection{Exponentials of the free boson field}
Our treatment follows Fr\"{o}hlich-Seiler \cite{FrSe76} , and Fr\"{o}hlich-
Marchetti \cite{FrMa88}, but with some innovations
(lemmas \ref{det2}, \ref{truncated}).
The massless boson field theory on ${\bf R}^2$ is defined
as the family of Gaussian random variables $\f(x)$
with covariance which is the inverse Laplacian.
We start with a regularized version of this adding a mass $r$ to
remove the infrared difficulties and an ultraviolet cutoff at $L^N$
(for fixed large $L$) to get a more regular theory. Thus consider
the operator $v_{r,N}$ whose kernel is
\be v_{r,N}(x-y) = (2\pi )^{-2} \int
{e^{ip(x-y)} \over (p^2+ r^2) }e^{-p^2L^{-2N}} dp \label{vrn} \ee
and consider the Gaussian random variables $\f(x)$
with covariance $\beta v_{r,N}$.
We can realize it concretely
by taking the Gaussian measure $\mu_{\beta,r,N}$ on $\cS'(\bf R^2)$ with
covariance $\beta v_{r,N}$
Because of the smoothness of the kernel the measure is actually supported
on smooth functions and we take $\f(x)$ to be evaluation at $x$.
We have the expectation:
\bea < \exp (i \sum_i s_i \f(x_i)) >_{ \beta, r,N} &=&
\int \exp (i \sum_i s_i \f(x_i)) d \mu_{\beta,r,N}(\f) \nn \\
&=& \exp \left( - \half \sum_{i,j}
s_is_j \beta v_{r,N}(x_i - x_j) \right) \eea
We are interested in taking the limits $N \to \infty$ and $ r \to 0$.
To get something non-trivial we need to make some adjustments.
First define Wick-ordered exponentials by
\be
:e^{i\f (x)}:_{r,N} = \exp (\beta v_{r,N}(0)/2) \ e^{i \f (x)}
\ee
($\beta$ dependence is supressed from the notation here).
These normalize the exponential so that
\be < :e^{i \f (x)}:_{r,N}>_{\beta,r,N} =1 \ee
We consider the case where there is a missmatch between the Wick ordering,
taken to be $:e^{i \f (x)}:_{1,N}$, and the measure $\mu_{\beta, r,N}$.
\blem
For non-coinciding points $x_1,...,x_n,y_1,...,y_m$
the limit
\[ < \prod_{i=1}^n :e^{i \f (x_i) }:\
\prod_{j=1}^m :e^{-i \f (y_j)}:>_{\beta} \]
\be = \lim_{r \to 0} \lim_{N \to \infty}
< \prod_{i=1}^n :e^{i \f (x_i)}:_{1,N}\
\prod_{j=1}^m :e^{i \f (y_j) }:_{1,N}>_{\beta,r,N}
\ee
exists. The limit is zero if $n \neq m$,
and if $n =m$ is given by
\be c_0^{-n-m} \left ( \frac{ \prod_{1 \leq i < i' \leq n}|x_i-x_{i'}|
\prod_{1 \leq j < j' \leq m} |y_j-y_{j'}|}
{ \prod_{i=1}^n \prod_{j=1}^m |x_i-y_j|} \right ) ^{\beta / 2 \pi}
\ee
for some positive constant $c_0$.
\elem
\bigskip
\pr The expectation with $r>0, N < \infty$ is evaluated as
\be \exp \left ( {\beta \over 2}\left [ v_{1,N}(0)(n+m)
- \sum_{i,i'}v_{r,N}(x_i - x_i')- \sum_{j,j'}v_{r,N}(y_j - y_j')
+ 2 \sum_{i,j} v_{r,N}(x_i - y_j) \right ] \right ) \ee
The terms $i=i'$ and $j=j'$ contribute $-v_{r,N}(0)(n+m)$ to the bracket.
combining this with the first term and using
\be \lim_{N \to \infty}(v_{1,N}(0) - v_{r,N}(0)) = (2 \pi)^{-1}\log r \ee
we get a contribution $(r^{ \beta / 4 \pi})^{(n+m)}$ as $N \to \infty$. We also have for $x \neq 0$ the existence of
\be v_r(x) = \lim_{N \to \infty} v_{r,N} (x)
= (2\pi )^{-1} \int_{- \infty}^{ \infty}{e^{- \sqrt{p^2 +r^2} |x|} \over
2 \sqrt{p^2 +r^2} } dp \ee
Thus we find that the $N \to \infty$ limit of the expectation is
\be
(r^{ \beta / 4 \pi})^{(n+m)}
\exp \left ( {\beta}\left [
- \sum_{i < i'}v_{r}(x_i - x_i')- \sum_{j < j'}v_{r}(y_j - y_j')
+ \sum_{i,j} v_{r}(x_i - y_j) \right ] \right ) \ee
Now as $r \to 0$ we have
\be v_{r}(x)= v_{1}(rx) =
- (2 \pi)^{-1}\log (r|x|) + \g + h(rx) \ee
where $h(x)$ is bounded on compact subsets of $\bR^2$ and
goes to 0 as $x \to 0$. With $c_0=e^{-\g}$ we find that
our expression can be written
\be
\label{corr10}
(r^{ \beta / 4 \pi})^{ (n-m)^2} c_0^{-n-m+(n-m)^2}
\left ( \frac{ \prod_{1 \leq i < i' \leq n}|x_i-x_{i'}|
\prod_{1 \leq j < j' \leq m} |y_j-y_{j'}|}
{ \prod_{i=1}^n \prod_{j=1}^m |x_i-y_j|} \right ) ^{\beta / 2 \pi}
H( rx, ry) \ee
where $H( x, y)$ is bounded on compact
subsets of $\bR^{2(n+m)}$ and goes to one as $(x,y) \to 0$.
The $r \to 0$ limit gives $0$ if $n \neq m$ and the stated result if $n=m$.
\bigskip
\re It is useful to introduce complex variables
$z_i = x_i^1 + i x_i^0$ and $w_j = y_j^1 +iy_j^0$. Then $|x_i-y_j| = |z_i-w_j|$
and so forth.
Then use the Cauchy lemma (see for example \cite{Hua63}) to obtain
\be \label{det1}
\frac{ \prod_{1 \leq i < i' \leq n}
(z_i-z_j) \prod_{1 \leq j < j' \leq n} (w_i-w_j)}
{ \prod_{i=1}^n \prod_{j=1}^n (z_i-w_j)}= \det \{ ( z_i-w_j)^{-1} \}
\ee
Now we can write our result for $n=m$ as
\be \label{phidet} c_0^{2n} < \prod_{i=1}^n :e^{i \f (x_i)}:\
\prod_{j=1}^n :e^{-i \f (y_j)}:>_{\beta}
= | \det \{ ( z_i-w_j)^{-1} \} | ^{\beta / 2 \pi}
\ee
This is useful for estimates and essential for the identification with
fermions.
More generally we have the following result which is valid for $n \neq m$.
Note that when $n=0$ it is the standard expansion of the
Vandermode determinant.
\blem For $n \leq m$ \label{det2}
\be
\frac{ \prod_{1 \leq i < i' \leq n}
(z_i-z_{i'}) \prod_{1 \leq j < j' \leq m} (w_j-w_{j'})}
{ \prod_{i=1}^n \prod_{j=1}^m (z_i-w_j)} = \det \{ a_{ij} \}
\ee
where $\{ a_{ij} \}$ is the $m \times m $ matrix with entries
\be a_{ij} = \left \{ \barr{rl} ( z_i-w_j)^{-1} & \ \ \ 1 \leq i \leq n \\
(w_j)^{m-i} & \ \ \ n < i \leq m \earr \right. \ee \elem
\pr The proof is by induction on $n$. If $n=m$ it is the Cauchy formula.
We assume it is true for $n$ and prove it for $n-1$. On the left side
of the identity for $n$ the factor $z_n$ occurs $n-1$ times in the numerator
and $m$ times in the denominator. If we let $k=m-n+1$ multiply by $z_n^k $ and
then let $z_n \to \infty$ we get the left side of the identity for $n-1$. We
have to check that the same operation applied to the right side
for $n$ gives the right side for $n-1$.
The first $n-1$ rows of the determinant are independent of $z_n$ and
expanding out these rows we see that the problem is reduced to an
evaluation of the limit of certain determinants of $k \times k$ matrices. Relabeling
the $w_j$'s the required limit is
\be \lim_{z \to \infty} z^k \det \left( \barr{lll}
( z-w_1)^{-1} & ... & ( z-w_k)^{-1} \\
w_1^{k-2} & ... & w_k^{k-2} \\
... & ... & ... \\
w_1 & ... & w_k \\
1 & ... & 1 \earr \right) =
\det \left( \barr{rcl}
w_1^{k-1} & ... & w_k^{k-1} \\
... & ... & ... \\
w_1 & ... & w_k \\
1 & ... & 1 \earr \right)
\ee
But the first row on the left can be expanded as
\be z^{-1}(1,...,1) + z^{-2}(w_1, ...,w_k) + ... +z^{-k}(w_1^{k-1}, ...,w_k^{k-1})
+ \cO ( z^{-k-1})\ee
The first $k-1$ terms do not contribute to the determinant, the $k^{th}$
term gives the answer, and the remainder goes to zero.
\bigskip
Next we investigate the integrability of these correlation functions.
\blem \label{lemma1} For $\beta < 4 \pi$
the expectation
\[ < \prod_{i=1}^n :e^{i \f (x_i)}:\
\prod_{j=1}^m :e^{-i \f (y_j)}:>_{\beta}
\]
is locally integrable. \elem
\pr First consider the case $n=m$
and $2\pi \leq \beta \leq 4\pi$.
Then
\bea \label{basic}
| \det \{ ( z_i-w_j)^{-1} \} | ^{\beta / 2 \pi}
&\leq & (\sum_{\pi} \prod_{i=1}^n| z_i-w_{\pi(i)}|^{-1})^{\beta / 2 \pi} \nn \\
&\leq & \const \sum_{\pi} \prod_{i=1}^n| z_i-w_{\pi(i)}|^{-\beta / 2 \pi}
\eea
where the sum is over permutations $\pi$ of $(1,...,n)$ and in the last step
we have used the inequality $(a+b)^p \leq 2^{p-1}(a^p +b^p)$ valid for $p \geq 1$.
This shows the local integrability. If $\beta < 2\pi$ take out the region where the
determinant is less than one and then increase $\beta$ to $2\pi$ to get
an integrable function. If $n \neq m$ we use (\ref{det2}) to get the result
by a similar argument.
\bigskip
Now define
\be : e^{i\f} :_{1,N}(f) = \int : e^{i\f(x) } :_{1,N} f(x) dx \ee
where the test functions $f$ are
taken to be bounded measurable functions
of compact support.
\blem \label{limit} For $\beta < 4 \pi$ and test
functions $f_i, g_j$ the limit
\[ < \prod_{i=1}^n :e^{i \f }:(f_i)\
\prod_{j=1}^m :e^{-i \f }:(g_j)>_{\beta} \]
\be = \lim_{r \to 0} \lim_{N \to \infty}
< \prod_{i=1}^n :e^{i \f }:_{1,N}(f_i)\
\prod_{j=1}^m :e^{-i \f }:_{1,N}(g_j)>_{\beta, r,N}
\ee
exists and is equal to
\be = \int
< \prod_{i=1}^n :e^{i \f (x_i) }:\
\prod_{j=1}^m :e^{-i \f (y_j) }:>_{\beta} \prod_i f_i(x_i) \prod_j g_j (y_j) dx dy
\ee
\elem
\pr With
the cutoffs in place it is easy to change the order of intergration and obtain
\[ < \prod_{i=1}^n :e^{i \f }:_{1,N}(f_i)\
\prod_{j=1}^m :e^{-i \f }:_{1,N}(g_j)>_{\beta,r,N} \]
\be = \int
< \prod_{i=1}^n :e^{i \f (x_i) }:_{1,N}\
\prod_{j=1}^m :e^{-i \f (y_j) }:_{1,N}>_{\beta,r,N} \prod_i f_i(x_i) \prod_j g_j (y_j) dx dy
\ee
Thus to prove the result one has to show that the right side of
this equation converges to the same thing without cutoffs.
However it is difficult to find a dominating function.
since one does not exactly have a determinant representation for $N < \infty$.
The actual proof for the $N \to \infty$ limit requires
some trickery, see Fr\"{o}hlich \cite{Fro76}, Theorem 3.7., for the details.
The proof is given for $n=m$ and does use the determinant representation (\ref{det1})
at $N= \infty$.
It also works for $n \neq m$ if we use
the representation (\ref{det2}) at $N= \infty$.
The $r \to 0$ limit is easier. One uses
(\ref{corr10}), (\ref{det1}) and dominated convergence .
\bigskip
We also need to discuss truncated correlation functions .
For these we will be able to integrate over coinciding points even at
$\beta = 4 \pi$.
Truncated correlations are defined for any random variables $\{ \chi_i \}$ and any
expectation $<...>$ (including our singular limits) inductively by
\be < \prod_{i=1}^n \chi_i >^T =< \prod_{i=1}^n \chi_i >-
\sum_{\Pi} \prod_j <\prod_{i \in \Pi_j}\chi_i>^T \ee
where the sum is over the proper partitions $\Pi = \{ \Pi_j \}$ of $(1,...,n)$.
The truncated functions are sums of products of ordinary correlation
functions.
For $\beta < 4 \pi$ we have by the previous lemma that
\bea \label{this} && < \prod_{i=1}^n :e^{i \f }:(f_i)\
\prod_{j=1}^m :e^{-i \f }:(g_j)>_{\beta}^T \nn \\
& =& \int
< \prod_{i=1}^n :e^{i \f (x_i) }:\
\prod_{j=1}^m :e^{-i \f (y_j) }:>_{\beta}^T \prod_i f_i(x_i) \prod_j g_j (y_j) dx dy
\eea
Note that these expectations are zero unless $n=m$.
We want to take the limit $\beta \nearrow 4 \pi$
\blem \label{truncated} Except for $n=m=1$
\be \label{that}
< \prod_{i=1}^n :e^{i \f }:(f_i)\
\prod_{j=1}^m :e^{-i \f }:(g_j)>_{4 \pi}^T
= \lim _{ \beta \nearrow 4 \pi}
< \prod_{i=1}^n :e^{i \f }:(f_i)\
\prod_{j=1}^m :e^{-i \f }:(g_j)>_{\beta}^T \ee
exists and is equal to
\be \label{more} \int
< \prod_{i=1}^n :e^{i \f (x_i) }:\
\prod_{j=1}^m :e^{-i \f (y_j) }:>_{4 \pi}^T \prod_i f_i(x_i) \prod_j g_j (y_j) dx dy
\ee
\elem
\pr It suffices to take $n=m \geq 2$. Let $x=(x_1, ...,x_n) $ and
$y = (y_1, ...,y_n)$ and define
\be \cG_{\beta}(x,y) = < \prod_{i=1}^n :e^{i \f (x_i) }:\
\prod_{j=1}^n :e^{-i \f (y_j) }:>_{\beta} \ee
and let $\cG_{\beta}^T(x,y)$ be the truncated function.
We claim that for each $n$ there is a locally integrable function
$h(x,y)$ such that
\be |\cG^T_{\beta}(x,y)| \leq h(x,y) \label{dom} \ee
for all $ \beta \leq 4 \pi$.
Then the integral in (\ref{more}) exists. Furthermore we can take the limit
$\beta \nearrow 4 \pi$ in (\ref{this})
by the dominated convergence theorem and obtain (\ref{that}).
The proof of (\ref{dom}) is by induction on n. We suppose the result is
true for all smaller n (and make no assumption in case n=2).
It suffices to find a dominating function in the sector
\be |x_1 - y_1 | \leq |x_i - y_j | \label{sector}\ee
We write $x=(x_1,\tilde x)$ and
have
\be \cG_{\beta}^T(x,y) \label{split}
=\cG_{\beta}(x,y) - \cG_{\beta}(x_1,y_1) \cG_{\beta}(\tilde x, \tilde y)
-\sum'_{\{(\Pi_j, \Pi_{j'})\} } \prod_i \cG_{\beta}^T(x_{\Pi_j},y_{\Pi_{j'}})
\ee
where the sum is over partitions $\{ (\Pi_j, \Pi_{j'}) \}$ of two copies of $(1,2,...n)$
except that partitions containing $(1,1)$ are excluded.
The first two terms in (\ref{split}) can be written as a constant times
\be |\det(z_i -w_j)^{-1}|^{\beta /2\pi}
- |(z_1 -w_1)^{-1}\det (\tilde z_i - \tilde w_j)^{-1}|^{\beta /2\pi}
\ee
We use the identity for $\al \geq 1$
\be | |a|^{\al} - |b|^{\al}| \leq |a-b| \al ( |a|^{\al-1}+|b|^{\al-1} ) \ee
to bound this by a constant times:
\bea && |\det (z_i -w_j)^{-1}
-(z_1 -w_1)^{-1}\det (\tilde z_i - \tilde w_j)^{-1}| \nn \\
&\times & (|\det(z_i -w_j)^{-1}|^{\beta /2\pi-1} +
|(z_1 -w_1)^{-1}\det (\tilde z_i - \tilde w_j)^{-1}|^{\beta /2\pi}) \eea
which in turn is bounded by
\be
\left (\sum_{\pi : \pi(1) \neq 1} \prod_i |z_i - w_{\pi(i)} |^{-1} \right )
\left( 2 +|\det(z_i -w_j)^{-1}|
+ |(z_1 -w_1)^{-1}\det (\tilde z_i - \tilde w_j)^{-1}| \right) \ee
This function is locally integrable since
the second factor
is locally in $L^{2- \ep}$ for any $\ep >0$ as in (\ref{basic}) and
the first factor is locally $L^{2+ \ep}$
for $\ep$ sufficiently small. To see this last statement
let $B_{2n}$ be a ball in $\bR^{2n}$ and let $B_{2n}'$ be the intersection
with the sector (\ref{sector}).
For a permutation $\pi$ with $\pi(1) \neq 1$ the corresponding term in the
first factor has an $L^{2+\ep}$ norm on $ B_{2n}'$ which is
\bea \label{estimate}
&&\int_{B_{2n}'}|\prod_i|x_i- y_{\pi(i)}|^{-2-\ep}d {x} d {y} \nn \\
&\leq& \const \int_{B_n} |x_1-y_1|^{-n\ep} dx_1 dy_1 ... (dy_{\pi(1)})^{\wedge}... dy_n
\eea
and this is finite if $n\ep < 2$
Now consider the last term in (\ref{split}). Each term in the sum will
be a product of some two-point truncated functions (n=m=1), and some truncated
functions which are dominated by integrable functions by the inductive
hypothesis. The truncated two point functions equal the
ordinary two point functions which are a constant times $|x-y|^{-\beta/ 2\pi}$.
Thus we have to dominate a terms of the form
\be \left( \prod_{i \in I} |x_i- y_{\pi(i)}| \right) ^{-\beta / 2 \pi}
\ee
where $I$ is a subset of $(1,...,n)$ and $\pi$ is a permutation with $\pi(1) \neq 1$.
This is dominated by one on the set where the product is greater than one
and by its value at $\beta = 4\pi$ on the complement.
The latter is locally integrable as in (\ref{estimate}).
\subsection{Massless fermions and equivalence}
Now we turn to massless fermions, mostly following \cite{FrMa88}.
The massless free fermion field theory on the plane is defined
by the two point function which is a fundamental solution $S(x-y)$ for
the Dirac operator $i\pa \diracslash $.
Here $\pa \diracslash = \g_{\mu}\pa_{\mu}$ where $\g_{\mu}$
are the gamma matrices satisfying
$\{ \g_{\mu}, \g_{\nu} \} = - 2 \de_{\mu \nu}$. We choose a
skew adjoint representation in which $\g_0 =-i \si_2$, $\g_1 = -i \si_1$ and
$\G=i\g_0\g_1= \si_3$
where the $\si_i$ are the Pauli matrices.
One can write down $S(x-y)$ directly, but we prefer to
obtain it as a limit of regularized quantities.
We define
\be \ S_{MN}(x) = (2\pi )^{-2} \int_{-M}^{M} dp_0 \int_{-N}^{N} dp_1
e^{ipx}\frac{ p \diracslash}{p^2}
\ee
In the limit $M \to \infty$ one can do the $p_0$ integral by
closing the contour. For $x_0 > 0$ one finds
\be \label{sreg} S_{N}(x) = (2\pi )^{-1} \int_{-N}^{N}
e^{(-|p_1|x_0 +i p_1x_1)}\frac{ (i\g_0|p_1|+\g_1p_1)}{2|p_1|} dp_1
\ee
which is a bounded function.
\blem \ \label{fermion1}
\benum \item For $x \neq 0$,
$ S(x) = \lim_{M,N \to \infty} S_{MN}(x)$ exists,
is continuous, and is a fundamental solution for the Dirac operator.
\item
If $z =ix_0+x_1$ then for $x \neq 0$
\be S(x) = \lim_{N \to \infty} S_N(x) = (2 \pi )^{-1} \left( \barr{lr} 0 & {\bar z} ^{-1} \\
z^{-1} & 0 \earr \right)
\ee
\item Uniformly in $N$:
\be |S_N(x)| \leq \one |x|^{-1} \ee
\eenum
\elem
\pr The first part follows by integrating by parts twice.
This argument also shows that one can take the limit in either order.
Thus for $x_0 >0$
\be S(x) = \lim_{N \to \infty} S_N(x) = (2\pi )^{-1} \int
\frac{e^{(-|p_1|x_0 +i p_1x_1)}}{2|p_1|} (i\g_0|p_1|+\g_1p_1) dp_1 \ee
Then one can compute matrix elements
\bea S_ {21}(x) &=& -i (2\pi )^{-1} \int
\frac{e^{(-|p_1|x_0 +i p_1x_1)}}{2|p_1|} (|p_1|+p_1) dp_1 \nn \\
&=& -i (2\pi )^{-1} \int_0^{\infty} e^{ip_1z} dp_1 \nn \\
&=& (2\pi z)^{-1} \eea
Similarly
$ S_{12}(x) = (2\pi \bar z)^{-1}, S_{11}(x) = 0 , S_{22}(x) = 0$
The same formulas hold for $x_0 <0$, and by continuity for all $x \neq 0$
With a cutoff we have instead
\be (S_N )_{21}(x) =
-i(2\pi )^{-1} \int_0^{N} e^{ip_1z} dp_1 = \cO (|z|^{-1}) \ee
This completes the proof.
\bigskip
The regularized theory is defined by
a fermionic functional integral with respect to the Gaussian
measure with covariance $S_N$. We have for correlation functions
\bea < \prod_{i=1} ^n \psi_{\al_i}(x_i) \psib_{\beta_i}(y_i) >_N
&=& \int \prod_{i=1} ^n \psi_{\al_i}(x_i) \psib_{\beta_i}(y_i)
d \mu_{S_N} \nn \\
&=& \det \{ (S_N)_{\al_i\beta_i}(x_i-y_j) \} \eea
with all other correlation functions equal to zero.
For non-coinciding points we can take the limit $N \to \infty$ and
obtain
\be < \prod_{i=1} ^n \psi_{\al_i}(x_i) \psib_{\beta_i}(y_i) >
= \det \{ S_{\al_i\beta_i}(x_i-y_j) \} \ee
We introduce chiral fermions $b,c$ by
\bea b&=& (1+\G)\psi/2 = \psi_1 \nn \\
\bar c&=& (1-\G)\psi/2 = \psi_2 \eea
Then our results so far can be characterized by the statement that
we have a free fermion field theory with two point functions
\bea <\bar b(x) \bar c(y) > &=& (2\pi \bar z)^{-1} \nn \\
&=& (2\pi z)^{-1} \eea
and all others equal to zero.
\blem For non-coinciding points
\be \label{psidet} < \prod_{i=1}^n \bar b b(x_i)
\ \prod_{j=1}^m c \bar c (y_j)>
= \left \{ \barr{cc} 0 & n \neq m \\
(2 \pi)^{-2n} |\det \{ \frac{1}{ z_i-w_j} \}|^2 & n=m \earr \right \} \ee \elem
\bigskip
\re Since $<\bar b (x)b(x)>_N =0$ we can allow $ \bar b, b$ to be taken
at the same point and still have an $N \to \infty $ limit.
No Wick ordering is required. \bigskip
\pr
The expectation can be explicitly computed by Wicks theorem.
We have two independent field theories (namely $(b,c)$ and $(\bar b , \bar c)$)
and they give complex conjugate
expectations whence
\be < \prod_{i=1}^n \bar b b(x_i)\ \prod_{j=1}^m c \bar c (y_j)>
= |< \prod_{i=1}^n b(x_i)\ \prod_{j=1}^m c(y_j)>|^2
\ee
If $n \neq m$ the expectation is zero, while if $n=m$ we have
\be < \prod_{i=1}^n b(x_i)\ \prod_{j=1}^n c(y_j)> =
(2 \pi )^{-n} \det \{ \frac{1}{ z_i-w_j} \} \ee
This completes the proof
\bigskip
Now by comparing (\ref{phidet}) and (\ref{psidet}) we have:
\bthm \label{equiv1} (massless equivalence).
For non coinciding points and $\beta = 4 \pi$
\bea &&c_0^{n+m} < \prod_{i=1}^n :\exp ( i\f (x_i)):\
\prod_{j=1}^m :\exp (- i \f (y_j)):>_{\beta} \nn \\
&=& (2\pi)^{n+m}< \prod_{i=1}^n :\bar b b:(x_i)
\ \prod_{j=1}^m :c \bar c: (y_j)> \ \eea \ethm
This shows that the massless free theories are equivalent if we take
$c = c_0/\pi$ and make the identification
\bea {c \over 2} :\exp ( i\f ):\ \ && \Longleftrightarrow
\ \ \ \bar b b \nn \\
{c \over 2}:\exp ( - i\f ):\ \ && \Longleftrightarrow
\ \ \ c\bar c \eea
We can also write this as
\bea c:\cos ( \f ):\ \ \ && \Longleftrightarrow
\ \ \ \psib \psi = \bar b b + c \bar c
\nn \\
ic:\sin ( \f ):\ \ \ && \Longleftrightarrow
\ \ \ \ \psib \G \psi = \bar b b - c \bar c
\eea
\bigskip
Next we smear with test functions, still taken to be bounded measurable
functions with compact support. We study scalars formally given by
\be (\psib \psi ) (f) = \int (\psib \psi ) (x) f(x) dx \ee
These are not defined directly but by their expectations given by
\be <(\psib \psi )(f_1)...(\psib \psi)(f_n)>_N =
\int <(\psib \psi )(x_1)...(\psib \psi)(x_n)>_N f(x_1) ...f(x_n) dx \ee
Similarly we define expectations with pseudo-scalars $\psib \G \psi$
We cannot take the limit $N \to \infty$ here unless the test functions
have non-overlapping supports. However for truncated
correlation functions like
\be \label{psitrun} <(\psib \psi )(f_1)...(\psib \psi)(f_n)>^T_N =
\int <(\psib \psi )(x_1)...(\psib \psi)(x_n)>^T_N f(x_1) ...f(x_n) dx \ee
( with $\psib \psi$ treated as a unit) we can take the limit.
\blem \label{ftrun} For $n+m \geq 3$
\be <\prod_{i=1}^{n} (\psib \psi )(x_i)
\prod_{j=1}^m(\psib \G \psi)(y_j)>^T \ee
is locally integrable and
\be <\prod_{i=1}^{n} (\psib \psi )(f_i)
\prod_{j=1}^m (\psib \G \psi)(g_j)>^T =
\lim_{N \to \infty}<\prod_{i=1}^{n} (\psib \psi ) (f_i)
\prod_{j=1}^m\psib \G \psi(g_j)>^T_N \ee
exists and equals
\be \int<\prod_{i=1}^{n} (\psib \psi )(x_i)
\prod_{j=1}^m(\psib \G \psi)(y_j)>^T \prod_{i=1}^n f_i(x_i) \prod_{j=1}^m g_j(y_j) dx dy
\ee \elem
\bigskip
\pr Suppose that $m=0$ and $n \geq 3$. The general case would
just involve inserting $\G's$ at appropriate places.
It suffices to bound $<\psib \psi (x_1)...\psib \psi(x_n)>^T_N$
by a locally integrable function independent of $N$. The the integrability
follows and one can use dominated convergence to take the limit in (\ref{psitrun}).
Truncated correlations for free field theories
are given as sums over connected Feymann diagrams. In this case the only
connected diagram is a circle and we have
\be \label{ft} <(\psib \psi )(x_1)...(\psib \psi )(x_n)>^T_N
= \sum_{\pi} tr (S_N(x_{1} - x_{\pi(1)}) ... S_N(x_{\pi^n(1)} - x_{1}) ) \ee
where the sum is over the cyclic permutations of $(1,2,...,n)$.
Each term has the form
\be tr (S_N(x_1-x_2)S_N(x_2-x_3)...S_N(x_n-x_1)) \ee
On any ball $B$ use lemma \ref{fermion1} to bound this by
\be h(x_1, ...x_n) =\cO(1) g(x_1-x_2)g(x_2-x_3) ...g(x_{n}-x_1) \ee
where
\be g(x) =\chi_{2B}(x)|x|^{-1} \ee
Then by Young's inequality
\bea
\int_{B} h & \leq & |B| \| g *g* ...*g\|_{\infty} \nn \\
& \leq & |B| \| g \|_{3/2} \|g* ...*g\|_3 \nn \\
& \leq & |B| \| g \|^2_{3/2} \|g* ...*g\|_{3/2} \nn \\
& \leq & |B| \| g \|^3_{3/2} \|g* ...*g\|_1 \nn \\
& \leq & |B| \| g \|^3_{3/2} \|g\|^{n-3}_1 \eea
and this is finite since $g \in L_p$ for $p <2$.
\bigskip
\re Note that since the number of permutations in bounded by $n!$,
it follows from our proof that for any $f$ there are constants $A,B$
such that
\be \label{factorialbound} < (\psib \psi )(f)^n >^T_N \leq AB^nn! \ee
for all $N$.
\bigskip
\bthm \label{masslessequiv} For $n+m \geq 3$
\be <\prod_{i=1}^{n} (\psib \psi )(f_i)
\prod_{j=1}^m (\psib \G \psi)(g_j)>^T
= (c)^n(ic)^m <\prod_{i=1}^{n} :\cos \f :(f_i)
\prod_{j=1}^m :\sin \f :(g_j)>_{4\pi }^T \ee
\ethm
\bigskip
\pr The pointwise identity follows from theorem \ref{equiv1}. The
correlations smeared with test functions are given as integrals over
the pointwise correlation functions by lemma \ref{truncated} and lemma
\ref{ftrun}. Hence the result.
\subsection{The main result}
\subsubsection{The sine-Gordon model} \label{sgsection}
We consider the the sine-Gordon model on the plane
with an interaction density supported on a square $\La$. The model
is defined by the interaction
\be \z :\cos \phi :_{1,N} (\chi_{\La}) =
\z \int_{\La}:\cos \phi (x) :_{1,N} dx \ee
By scaling we can always arrange that $\La = \De$ = a unit square, and we make this
choice for the rest of the paper.
Define normalized expectations by
\be < ... >_{\z, \beta,r,N} = { \int [. . .] \exp (\z :\cos \phi :_{1,N} (\chi_{\De}) )
d \mu_{\beta,r,N}(\f) \over
\int \exp (\z :\cos \phi :_{1,N} (\chi_{\De}) )d \mu_{\beta,r,N}(\f)}
\ee
We study expectations of products of the fields $\sin \f$.
The main technical result is:
\bthm Let $f_1, ...,f_n$ be test functions in $\cC^{\infty}_0(\bR^2)$
with disjoint supports in $\De$. Then
for all $r,N$ and for $\beta$ in a compact subset of $(0,16\pi/3)$
the functions
\be \cW_{ \beta, r,N}(\z;f_1,...,f_n)
= \z^n< \prod_{i=1}^n : \sin \f:_{1,N}(f_i) >_{\z, \beta,r,N} \label{early} \ee
are analytic in $\z$ and uniformly bounded on a fixed a neighborhood of the origin.
The same is true for the truncated functions $\cW^T_{ \beta, r,N}(\z;f_1,...,f_n)$.
\label{tech}
\ethm
The proof is given in subsequent sections and involves taking some shortcuts.
This accounts for
the presence of the $\z^n$ which is not really necessary.
Assuming the theorem we can prove:
\blem \label{bosonderiv} Under the hypotheses of the theorem,
\be \cW(\z;f_1,...,f_n) = \z^n< \prod_{i=1}^n : \sin \f:(f_i): >_{\z, 4 \pi} \ee
defined by
\be \cW(\z;f_1,...,f_n) = \lim _{\beta \nearrow 4\pi} \lim_{r \to 0} \lim_{N\to \infty}
\cW_{ \beta, r,N}(\z;f_1,...,f_n) \ee
exists and is analytic in a neighborhood of the origin.
The same is true for truncated functions
\be \cW^T (\z;f_1,...,f_n) = \z^n< \prod_{i=1}^n : \sin \f:(f_i): >^T_{\z, 4 \pi} \ee
and for these we have the derivatives at zero
\bea \label{formula} &&(\cW^T)^{(k)}(0;f_1,...,f_n) = \nn \\
&& \ \ \ \left\{ \barr{ll} 0 & k_{4\pi}^T & k \geq n
\earr \right. \eea \elem
\re
We are only asserting that the $N \to \infty, r \to 0$ limits exist for
$\beta < 4\pi$ and then defining the theory for $\beta = 4\pi$ as the limit
from below. In fact the $N \to \infty, r \to 0$ limits no doubt exist
for all $\beta < 16\pi/3$
and give the same answer at $\beta = 4 \pi$. This should follow from our subsequent
analysis as in \cite{DiHu93} \bigskip
\pr It suffices to prove that the limit exists for the truncated functions.
By a standard computation we have the derivatives
\bea &&(\cW^T)^{(k)}_{ \beta, r,N}(0;f_1,...,f_n) = \nn \\
&& \ \ \ \left\{ \barr{ll} 0 & k_{ \beta, r,N}^T
& k \geq n
\earr \right. \eea
By the theorem we have Cauchy bounds $| (\cW^T)^{(k)}_{\beta, r,N}(0) |
\leq AB^kk!$ uniformly in $\beta,r,N$.
The derivatives also have a limits as $N \to \infty $ given by (\ref{formula}).
For $n+k \geq 3$ this follows by
lemma \ref{limit} and and lemma \ref{truncated}.
For $n=1,k-n=0$ and $n=1,k-n=1$ the
limit vanishes, in the latter case by $\f \to -\f$ symmetry. For $n=2,k-n=0$
the limit exists by the pointwise limit and the assumption of
non-overlapping supports.
Thus in in the Taylor series expansion
\be (\cW^T)_{\beta, r,N}(\z;f_1,...,f_n) = \sum_{k=0}^{\infty}
(\cW^T)^{(k)}_{\beta, r,N}(0;f_1,...,f_n)\z^k / k! \ee
we can take the various limits and all the claimed results follow.
\bigskip
\subsubsection{Massive fermions}
Now consider massive fermions.
We again start with a regularized theory
defining
\be < ...>_{\mu, N} = {<[...]e^{\mu (\psib \psi)(\chi_{\La})}>_N \over
_N } \ee
Again by scaling it suffices to consider $\La = \De=$ a unit square.
We study particularly expectations of products of $\psib \psi $ and $\psi \G \psi$.
With the cutoff in place an expression like
\[ <\prod_{i=1}^n (\psib \psi )(f_i) e^{\mu (\psib \psi )(\chi_{\De})}>_N \]
can be defined by a power series in $\mu$.
The coefficient of $\mu^k/k!$ is an $(n+k) \times (n+k)$ determinant with
bounded entries. For fixed $n$ it can be estimated by $\one ^k k!$ and hence
the series converges for $|\mu| $ small.
(Actually it converges for all $\mu $). These considerations
show that $ <\prod_{i=1}^n (\psib \psi )(f_i) >_{\mu,N}$ is well-defined and
analytic for for $|\mu|$ small. The same is true with the fields $\psib \G \psi$.
\blem \label{fermionderiv} For test functions with disjoint supports the limit
\be <\prod_{i=1}^n (\psib \G\psi )(f_i) >_\mu = \lim_{N \to \infty}
<\prod_{i=1}^n (\psib \G\psi )(f_i) >_{\mu, N}
\ee
exists and is analytic in $\mu$ in a neighborhood of the origin.
The same is true for the truncated functions and these satisfy:
\be \label{psider1} {d^k \over d \mu^k }
<\prod_{i=1}^n (\psib \G\psi )(f_i)>^T_{\mu} |_{\mu =0}
=<\prod_{i=1}^n (\psib \G\psi) (f_i) ((\psib \psi)(\chi_{\De}))^k:>^T
\ee
\elem
\bigskip
\pr It suffices to consider the truncated functions.
The derivatives at zero can be computed as
\be \label{psider2} {d^k \over d \mu^k }
<\prod_{i=1}^n (\psib \G\psi )(f_i)>^T_{\mu,N} |_{\mu =0}
=<\prod_{i=1}^n (\psib \G\psi )(f_i) ((\psib \psi)(\chi_{\De}))^k>_N^T
\ee
The derivatives have bounds of the form $AB^k k!$ uniformly in $N$ by
estimates like (\ref{factorialbound}).
The derivatives have a limits as $N \to \infty $ given by (\ref{psider1}).
For $n+k \geq 3$ this follows by lemma \ref{ftrun}. For $n=1,k=0$ and $n=1,k=1$ the
expression vanishes, in the latter case by reflection invariance. For $n=2,k=0$
the limit exists by the assumption of
non-overlapping supports.
Now in in the Taylor series expansion for
$<\prod_{i=1}^n (\psib \G\psi )(f_i) >_\mu $
we can take the limit $N \to \infty$ and all the results follow.
\bigskip
\subsubsection{Massive equivalence}
Our main result is the following
\bthm Let $f_1, ...f_n$ be test functions with disjoint support in $\De$.
If $\z=\mu c$ is sufficiently small
\be (ic)^n< \prod_{i=1}^n : \sin \f:(f_i): >_{\z, 4 \pi}
=<\prod_{i=1}^n (\psib \G\psi )(f_i) >_\mu \ee \label{main} \ethm
\bigskip
\pr It suffices to show that the truncated functions
are equal. If $\z=\mu c= 0$ the result follow from theorem \ref{masslessequiv}.
If $\z = \mu c \neq 0$
it suffices to establish:
\be (i\z)^n < \prod_{i=1}^n : \sin \f:(f_i): >^T_{\z, 4 \pi}
=\mu^n <\prod_{i=1}^n (\psib \G\psi )(f_i) >^T_\mu \ee
Both sides are now analytic
functions of $\mu$ on a neighborhood of the origin. To establish the identity it suffices
to show that the derivatives at zero agree.
The derivatives are computed in lemma \ref{bosonderiv} and
lemma \ref{fermionderiv}.
To prove the identity we need that
\be
(ic)^n c^k< \prod_{i=1}^n : \sin \f:(f_i)
(: \cos \f:(\chi_{\De}))^k >_{4\pi}^T
=<\prod_{i=1}^n (\psib \G\psi )(f_i)((\psib \psi )(\chi_{\De}))^k>^T
\ee
But this follows from theorem \ref{masslessequiv}. This completes the proof of equivalence.
\bigskip
\re If we try to include terms like $:\cos \f:$ and $\psib \psi$ in the statement of
the theorem we run into the
difficulty that divergent terms like $<(\psib \psi)(f) (\psib \psi) (\chi_{\De})>_N$
occur in the perturbation expansion.
\section{ The sine-Gordon model}
\subsection{Overview} \label{overview}
We now study the sine-Gordon model on the plane
with an interaction density supported on a square.
The analysis uses the renormalization group, and is based on an earlier treatment of
the sine-Gordon model due to Dimock and Hurd \cite{DiHu91},
\cite{DiHu93}.
Unfortunately at the time of this writing there are some problems
with the proofs in \cite{DiHu91}, \cite{DiHu93}, and it is not yet
clear whether they can be resolved. Thus we do not follow the proof too
closely. Instead
we adopt a simpler strategy in which the leading contributions in
perturbation theory play a prominent role. This makes it possible to
treat remainder terms more crudely. In particular we do not make
any extractions from the remainders, which is problematic for dimensionless
fields.
The closer one is to $8 \pi$ the
more orders of pertubation theory one has to take to render the
remainder irrelevant. For this paper we only go to second order perturbation
theory , and the proof works for $\beta < 16\pi/3$ which is sufficient for our
purposes. Possibly something similar will
work all the way up to $\beta = 8 \pi$.
The present model also differs from
from \cite{DiHu93}
in that the base manifold is the plane and not the torus,
and because there are now genuine infrared divergences
in the problem (the $r \to 0$ limit).
Our version of the renormalization group
follows the general framework of Brydges and Yau \cite{BrYa90}.
We also adopt some recent innovations in this technique due to Brydges, Dimock,
and Hurd \cite{BDH95}, \cite{BDH96a}, \cite{BDH96b}.
In particular we use open polymers and background potentials.
\bigskip
\subsection{The Renormalization Group}
The model is defined in section \ref{sgsection}.
Consider expressions of the form
\bea \Xi_{r,N}(\z,\f) &=& \mu_{\beta,r,N}* \ e^{ \tilde V_{N}(\z,\La)}(\f)
= \int e^{ \tilde V_{N}(\z,\La,\f + \f')}d \mu_{\beta,r,N}(\f') \nn \\
\tilde V_N(\z,\La,\f) &=& \z \int_{\La} : \cos ( \f(x)):_{1,N}dx
\label{xi}
\eea
and $\mu_{\beta,r,N}$ has covariance $\beta v_{r,N}$ defined in (\ref{vrn}).
Then $\Xi_{r,N}(\z,0)$ is the partition function, and derivatives of
$\Xi_{r,N}(\z,\f)$ yield correlations functions.
We want to obtain control over $\Xi_{r,N}$ uniformly $r,N$ as $N \to \infty$
and $r \to 0$.
Scaling up by a factor $L^N$ we find
\bea \Xi_{r,N}(\z,\f) &=& (\mu_{\beta,r_N,0}*
\ e^{ V(\z_N, \La_N)})(\f (\cdot / L^N) ) \nn \\
V(\z, X, \f) &=& \z \int_X \cos \f (x) dx \label{vdef} \eea
where
\bea
\La_N &=& L^N \La \nn \\
\z_N &=& L^{-2N} e^{ \beta v_{1,N}(0)/2} \z \nn \\
r_N &=& L^{-N}r
\eea
Note that the Wick ordering constant $\exp ( \beta v_{1, N}(0)/2) $ has been
absorbed into the coupling constant.
The measure $\mu_{\beta,r_N,0}$ has the covariance $\beta v_{r_N,0}$
and it is convenient to write this as $\beta v_{r_N,0}= \beta ' \hat v_N$
where
\bea \beta' &=& e^{r_N^2} \beta \nn \\
\hat v_N (x) &=& e^{-r_N^2} v_{r_N,0}(x) \nn \\
&=& (2\pi)^{-2} \int {e^{ipx} \over p^2 +r_N^2}
e^{-( p^2 +r_N^2)} dp \eea
We seek a sequence of densities
$\cZ^{r,N}_i(\f)$ for $ i = N, N-1, ...,1,0$ localized in $\La_i$ such that
\be \label{zrni} \Xi_{r,N}(\f)= (\mu_{\beta '\hat v_i }*\cZ^{r,N}_i)(\f(\cdot / L^i)) \ee
Initially this holds with $\cZ^{r,N}_N = e^{V(\z_N ,\La_N)} $.
Given such a density for $i$ we define the density for $i-1$ as follows.
First
define $C_i$ by
\be C_i(x) = \hat v_i(x) - \hat v_{i-1}(x/L) \ee
whence
\be \mu_{\beta '\hat v_i } = \mu_{\beta '\hat v_{i-1 }(\cdot/L)} * \mu_{\beta' C_i} \ee
Then scaling down by $L$ changes the first factor to
$ \mu_{\beta '\hat v_{i-1 }}$
and gives the expression for $i-1$ with
\be \label{flow} (\cZ^{r,N}_{i-1})(\f) =
(\mu_{\beta' C_i} * \cZ^{r,N}_{i})(\f (\cdot/L) ) \ee
This is the basic renormalization group transformation.
Since $C_i$ is only weakly dependent on $i$ the study of $\cZ^{r,N}_i$
is essentially the study of iterations of this mapping.
Following the flow down to $i=0$ gives $\cZ^{r,N}_0$,
and for this functional it will be possible to analyze the $r,N$ dependence.
\subsection{Polymer expansions}
Control over the densities $\cZ^{r,N}_i$ depends on keeping careful
track of the localization of the functional. This is accomplished by polymer expansions.
We only sketch the formalism here, referring to \cite{BrYa90}, \cite{BDH94a},
\cite{BDH95} for more details
A (open) polymer $X$ is a union of open unit squares centered on points of the integer lattice
in $\bR^2$. A polymer activity is a function $K(X, \f)$ depending on polymers $X$
and fields $\f$ with the property that the dependence on $\f$ is localized in $X$.
One can define a product on polymer activities and also an exponential
function $\cE xp$. If $\Box$
is the characteristic function of unit blocks, then we have
for an additive polymer activity $V$ and a general polymer activity $K$
\be \cE xp ( \Box e^V+ K)(X, \f) = \sum_{\{X_i\} } \prod_i K(X_i, \f)
e^{V(X \backslash \cup_i X_i, \f)} \ee
where the sum is over collections of disjoint polymers $\{ X_i\}$ in $X$.
The densities
will all be represented in this form. In particular the intial density has this
form with $K=0$:
\be \cZ^{r,N}_N(\f) = e^{V(\z_N,\La_N,\f)} =\cE xp ( \Box e^{V(\z_N)}) (\La_N, \f) \ee
It is useful to keep track
of the dependence on fields and their derivatives separately. Thus
we consider modified polymer activities $K(X, \psi)$ defined on pairs $\psi = (\psi_0, \psi_1)$
consisting of a continuous function $\psi_0$ and
a continuous vector field $\psi_1$. A basic
polymer activity $K^0(X, \f)$ is represented as a modified polymer activity $K(X, \psi)$ if
$ K(X, \psi_{\f})=K^0(X, \f) $ where $\psi_{\f} = (\f ,\pa \f) $.
Modified polymer activities $K,K'$ are equivalent, written $ K \approx K'$, if
$K(X, \psi_{\f})=K'(X, \psi_{\f})$.
Norms are defined on the modified polymer activities. Let $K_n(X,\psi)$ be the
$n^{th}$ derivative
of $K(X,\psi)$ with respect to $\psi$.
This is a multilinear functional on pairs $f=(f_0,f_1)$
of continuous functions, and we assume that restricted
to components it is given by
by a measure of bounded variation supported on $X$. We let $\|K_n(X, \psi) \|$
denote total variation norm (with a sum over components)
and let $\|K_n(X, \psi) \|_{\bf\De}$ denote this
norm restricted to a product of unit blocks
$ {\bf\De}=\De_1\times\dots\times\De_{n}$.
Next we define a large field
regulator
\be \label{Gdef} G(\k, X, \f )=
\exp \left( \k \sum_{1 \leq |\al| \leq 3}\int_X | \pa^{\al} \f |^2 \right) \ee
with constant $0 \leq \k \leq 1$.
Then define
\be
\|K_{n}(X)\|_G= \sum_{\bf\De}
\sup_{\f \in \cC^{3}}\|K_{n}(X,\psi_{\f}) \|_{\bf\De}G(X,\f)^{-1}.
\ee
Dependence on the set $X$ is controlled by a large set regulator $\G(X)$ which
is such that $\G(X)^{-1}$ has polynomial tree decay in $X$. We define
\be \label{gamma}
\|K_{n}\|_{G,\G}= \sup_{\De} \sum_{X\supset
\De}\G(X)\|K_{n}(X)\|_G \ee
Finally, for $h>0$ we define
\be
\|K\|_{G,\G,h}=\sum_{n}\left(h^n/n!\right)
\|K_{n}\|_{G,\G}
\ee
Our polymer expansions will always involve modified
polymer activities $K(X,\psi)$ for which this norm
is finite. Such activities are analytic in $\psi$ in a strip of width $h$.
Sometimes we do that last two sums in the opposite order and call the norm
$ \| K \|_{G, h, \G}$. If $K$ is translation invariant then one can drop
the supremum over $\De$ in (\ref{gamma}) and so
$ \| K \|_{G, h, \G} = \| K \|_{G, \G, h}$. Our $K$ 's are generally
translation invariant, although they appear in expressions like
$ \cE xp (\Box e^V + K)(\La, \psi_{\f})$ which are not.
Another variation is that sometimes we replace $\G$ by
\be \G_n (X) = 2^{n|X|} \G(X) \ee
\subsection{The RG flow to second order} \label{theflow}
As a preliminary we study some aspects of the RG flow to second order in $\z$.
Suppose that after some number of steps we have we have
basic polymer activities of the form $K = Qe^V + \cO(\z^3)$
where $V= \cO(\z), Q = \cO(\z^2)$. Then our density has the form
\be \cE xp ((\Box + Q)e^V)(\La, \f) + \cO(\z^3) \ee
We want to find new densities $Q_1, V_1$ and more generally
continuous functions
$Q_t,V_t, t \in [0,1]$ such that $Q_0=Q,V_0=V$
and such that for $ \beta = \beta', C=C_i$
\be \mu_{t\beta C} * \cE xp ((\Box + Q)e^V)
= \cE xp ((\Box + Q_t)e^{V_t}) + \cO(\z^3) \ee
In general if one defines $A_t$ by
\be \mu_{t\beta C} * \cE xp (A)
= \cE xp (A_t) \label{firstone} \ee
then one finds the $A_t$ satisfies
the differential equation
\be \label{secondone}
E(A_t) \equiv ({\pa \over \pa t} -\De_{\beta C} ) A_t
- \cB_{\beta C}( A_t, A_t ) =0
\ee
where
\bea \De_C A(\f) &=& \frac{1}{2}\int \ d\mu_C(\z)\ A_2(\f;\z,\z)
\nn \\
\cB_C(A,B)(\f) &=& \frac{1}{2} \int \ d\mu_C(\z)
\ A_1(\f;\z)\circ B_1(\f;\z) \eea
and where $A_n(\f)$ is the $n^{th}$ derivative of $A(\f)$.
Futhermore approximate
solutions to (\ref{secondone}) give approximate solutions
to (\ref{firstone} ).
This is the approach we take. Starting with $A=(\Box + Q)e^{V}$ we ask for
$A_t=(\Box + Q_t)e^{V_t}$
so that $E(A_t ) = \cO(\z^3)$.
Associated with $V(\z) = V(\z, X, \f)$ defined in (\ref{vdef}),
we define polymer activities $\cQ^0(\z , W)$
for $W=(W_1,W_2)$ and $|X| \leq 2$ by
\bea \cQ^0(\z , W; X,\f) &=& {\z^2 \over 4} \int_{\tilde X}
\cos (\f(x) + \f(y) ) W_{1}(x-y) dx dy \nn \\
&+& {\z^2 \over 4} \int_{\tilde X} \cos (\f(x) - \f(y) ) W_{2}(x-y)
dx dy \eea
Here for $X=\De \cup \De'$
we define \be \tilde X = \De \times \De' \cup \De' \times \De \ee
If $|X| \geq 3$ then $\cQ^0(\z , W; X,\f)=0$
\blem Let $ V_t = V(\z_t) $ and
$ Q^0_t = \cQ^0(\z_t, W_t)$. Then $A_t=(\Box + Q^0_t)e^{V_t}$
satisfies $E(A_t ) = \cO(\z^3)$ provided
\bea \z_t &=& e^{-t \beta C(0)/2}\z \nn \\
W_{1,t}(x)+1 &=& e^{-t\beta C(x)}(W_{1,0}(x)+1) \nn \\
W_{2,t}(x)+1 &=& e^{+t\beta C(x)}(W_{2,0}(x)+1) \label{before}
\eea
\elem
\pr
For the first order terms in $E$ to vanish we need
\be
\frac{\partial V_{t} }{\partial t} - \Delta_{ \beta C} V_t
=0
\ee
This follows by our choice of $\z_t$.
For the second order terms in $E$ to vanish it turns out we need:
\be ({\pa \over \pa t} - \De_{ \beta C}) Q^0_t - J_t =0
\label{vanish} \ee
where for $|X| \leq 2$
\bea \label{Jdef}
J_t(X,\f) & = &{1 \over 2}
\int_{\tilde X} \beta C(x-y) (V_t)_1(\f,x) (V_t)_1(\f,y) dx dy \nn \\
&=& -{\z_t^2 \over 8}
\sum_{p,q = \pm 1} \int_{\tilde X} e^{ip\f(x)} pq \beta C(x-y) e^{iq\f(y)} dx dy
\eea
If $Q^0_t$ has the general form
\be Q^0_t(X,\f) = {\z_t^2 \over 8}
\sum_{p,q = \pm 1} \int_{\tilde X}
e^{ip\f(x)} W_{pq,t}(x-y) e^{iq\f(y)} dx dy
\ee
Then for (\ref{vanish}) to vanish the kernel must satisfy
\be ({\pa \over \pa t} +pq \beta C(x)) W_{pq,t}(x) = -pq\beta C(x) \ee
which can also be written
\be ({\pa \over \pa t} +pq \beta C(x)) ( W_{pq,t}(x)+1) = 0 \ee
This has the solution
\be W_{pq, t}(x)+1 = e^{-tpq \beta C(x)}(W_{pq,0}(x)+1) \label{2flowprime}
\ee
The theorem is the special case $W_{1,1} = W_{-1,-1}$ and $W_{1,-1} = W_{-1,1}$.
If it is true at $t=0$ then it is true for all $t$. Denoting the first pair
by $W_1$ and the second pair by $W_2$ we have $Q^0_t = \cQ^0(\z_t, W_t)$
and (\ref{2flowprime}) becomes (\ref{before}).
\bigskip
Now we modify our definition of $\cQ$ by defining $\cQ(\z,W; X, \psi)$.
We say that a polymer with $|X| \leq 2$ is a small set if
$\bar X$ is connected. Otherwise it is a large set.
We define
\be \cQ(\z,W; X, \psi) =
\cQ_1(\z,W_1; X, \psi) + \cQ_2(\z,W_2; X, \psi) \ee
where
\bea \cQ_1(\z,W; X, \psi) &=&
{\z^2 \over 4} \int_{\tilde X}
\cos \left( \psi_0 (x) + \psi_0 (y) \right) W(x-y) dx dy \nn \\
\cQ_2(\z,W; X, \psi) &=&
\left\{ \barr{ll}
{\z^2 \over 4} \int_{\tilde X}
\cos \left( \psi_0 (x) - \psi_0 (y) \right) W(x-y) dx dy & {\rm X \ large} \\
{\z^2 \over 4} \int_{\tilde X}
\cos \left( \int_y^x \psi_1 \right) W(x-y) dx dy & {\rm X \ small} \earr
\right.
\eea
and the integral $\int_y^x $ is over the shortest path in $\bar X$.
We recover our earlier definition when we restrict:
\be \cQ (\z, W; X, \psi_{\f}) = \cQ^0(\z, W; X, \f) \label{restrict} \ee
Now define $Q_t(X, \psi) = \cQ(\z_t, W_t; X, \psi)$.
We do not have (\ref{vanish}) for all $\psi$, but (\ref{vanish})
does imply
\be \left( ({\pa \over \pa t} - \De_{ \beta C}) Q_t - J_t
\right) (X, \psi_\f) =0 \label{bigzero} \ee
The result follows from the identity
$(\De_{ \beta C} Q_t)(X,\psi_{\f}) =
(\De_{ \beta C} Q^0_t)(X,\f) $ where
$\De_{ \beta C}$
is the Laplacian in $\psi$; see the Appendix.
\subsection{Estimates}
In estimates we make the convention that constants denoted $\one$ are
independent of $L$,
while constants denoted $C$ may depend on $L$. The value of $C$ may
vary from line to line.
We start with some estimates on $C_i(x)$ which is given by
\bea
C_i(x)
&=&
(2\pi)^{-2} \int {e^{ipx} \over p^2 +r_i^2}
(e^{-(p^2+ r_i^2)} - e^{-L^2(p^2+r_i^2)} ) dp
\nn \\
&=&
(2\pi)^{-2} \int e^{ipx} \int_{1}^{L^2}
e^{-\al (p^2 + r_i^2)} d\al dp
\nn \\
&=&
(4\pi)^{-1}
\int_{1}^{L^2} \al^{-1} e^{- x^2/4\al} e^{-\al r_i^2} d \al
\eea
Then we have the following uniform bounds:
\blem For any multi-index $\beta \neq 0$
\bea 0 < C_i(x) &\leq& \cO(\log L)e^{-|x|/L} \nn \\
|\pa^{\beta} C_i(x)| &\leq& \cO(1)e^{-|x|/2L} \eea \elem
\pr For $\al \leq L^2$ we have
\be {|x|^2 \over 4\al} \geq { |x| \over \sqrt{\al}} -1 \geq {|x| \over L} -1 \ee
This gives the first bound. The second bound is similar. We always
get at least one extra power of $\al^{-1}$ so the $\al$ integral has a
bound independent of $L$.
\bigskip
Next we study the evolution of the kernel $W=(W_1,W_2)$
under the transformation (\ref{before}) at $t=1$, with $\beta = \beta'$,
followed by a scaling
operation. Starting with $W_N = (W_{1,N},W_{2,N})$ define inductively
\bea W_{1,i-1}(x)+1 &=& e^{ - \beta' C_i(Lx)} (W_{1,i}(Lx) +1) \nn \\
W_{2,i-1}(x)+1 &=& e^{ + \beta' C_i(Lx)} (W_{2,i}(Lx) +1) \eea
This is equivalent to
\bea
W_{1,i}(x) + 1 & =& e^{- \beta' w_i(x)} \nn \\
W_{2,i}(x) + 1 & =& e^{+ \beta' w_i(x)}
\eea
where
\bea w_i(x) &=& \sum_{k=i+1}^{N} C_k(L^{k-i} x) \nn \\
&=& (2\pi)^{-2} \int {e^{ipx} \over p^2 +r_i^2}
(e^{-(p^2 + r_i^2)L^{-2(N-i)}} - e^{-(p^2 + r_i^2)} ) dp \nn \\
&=& (4\pi)^{-1}
\int_{L^{-2(N-i)}}^1 \al^{-1} e^{- x^2/4\al} e^{-\al r_i^2} d \al \label{w} \eea
\blem \ \ \label{wbd}
\benum
\item for $|x| \leq 1$
\bea 0 < w_i(x) &\leq & {1 \over 2\pi}\log |x|^{-1} + \one \nn \\
|W_{1,i}(x)|, \
|x|^{ \beta'/2\pi} |W_{2,i}(x)| &\leq& \one
\eea
\item for $|x| \geq 1$
\bea 0 < w_i(x) &\leq & \one e^{-|x|/2 } \nn \\
|W_{1,i}(x)| , \ |W_{2,i}(x)| & \leq & \one e^{-|x|/2 }
\eea
\eenum
\elem
\pr For $|x| \leq 1$
\bea w_i(x) &\leq & (4\pi)^{-1}
\int_0^1 \al^{-1} e^{- x^2/4\al} d \al \nn \\
& = & (4\pi)^{-1}
\int_0^{|x|^{-2}} \al^{-1} e^{- 1/4\al} d \al \nn \\
&\leq & \one + (4\pi)^{-1}
\int_1^{|x|^{-2}} \al^{-1} d \al \nn \\
& = & \one + {1 \over 2\pi}\log |x|^{-1}
\eea
This gives the first
bound on $w_i$ and the bounds on $W_{1,i}, W_{2,i}$ follow.
For $|x| \geq 1 $ and $\al \leq 1$ we have
\be {|x|^2 \over 4\al} \geq { |x| \over \sqrt{\al}} -1 \geq {|x| \over 2}
+ {1 \over 2 \sqrt{\al}} -1 \ee
This gives the second
bound on $w_i$ and the bounds on $W_{1,i}, W_{2,i}$ follow.
\bigskip
Now we turn to some bounds on polymer activities. . We start with
bounds on $V(X) = V(\z,X, \f)$ defined in (\ref{vdef}).
The following hold for any $G=G(\k, X, \f)$.
\blem For a unit block $\De$:
\be \|V(\De) \|_{G, h} \leq |\z|e^h \ee
\elem
\pr The $n^{th}$ derivative of $V(\De, \f)$ with respect to $\f$
is the multilinear function on continuous $f_j =(f_j,0)$
given by
\be V_n( \De , \f; f_1, ...f_n)
= \z \int_{\De} \cos^{(n)}(\f(x)) f_1(x) ...f_n(x) dx \ee
Then $ V_n( \De , \f)$ is given by a measure satisfying
\be \|V_n(\De, \f) \| \leq |\z| \label{vbd} \ee
Similarly $ \|V_n(\z,\De) \|_G \leq |\z| $.
Multiplying by $h^n/n!$ and summing over $n$ gives the result.
\blem \label{evbd} For
$|\z|e^{4h}$ sufficiently small
\bea \|e^{V( \De)} \|_{G, h} &\leq & 2 \nn \\
\|e^{V( \De)}-1 \|_{G, h} &\leq & 2|\z|e^h \nn \\
\|e^{V( X)} \|_{G, h} &\leq & 2^{|X|} \eea
\elem
\pr
First suppose that $X=\De$. Then after some combinatorics (see \cite{BDH95})
one finds that
\be {h^n \over n!} \|(e^{V(\De, \f)})_n\| \leq \exp ( \sum_{n=0}^{\infty}
{h^n \over n!} \|V_n(\De, \f) \| ) \leq \exp(|\z|e^h) \ee
Similarly
\be {h^n \over n!} \|(e^{V(\De)})_n\|_G \leq \exp ( |\z|e^h ) \ee
Now replace $h$ by $4h$, multiply by $4^{-n}$, and sum over
$n$ to obtain the first bound for $|\z|e^{4h}$ small.
The second bound follows from the first and the representation
\be e^{V(\De)} - 1 = \int_0^1 V(\De) e^{sV(\De)} ds \ee
The last bound follows by
\be \|e^{V(X)} \|_{G,h} \leq \prod_{\De \subset X} \|e^{V(\De)} \|_{G,h}
\ee
\blem Let $|\z|e^{4h}$ be sufficiently small, and let $Q$ be a
translation invariant polymer activity. Then
\be \|Qe^V \|_{G, \G, h} \leq \|Q\|_{G, \G_1, h}
\ee \elem
\pr Using lemma \ref{evbd} we have
\be \|Q(X)e^{V(X)} \|_{G,h} \leq \|Q(X) \|_{G,h}
\|e^{V(X)} \|_{1,h} \leq 2^{|X|}\|Q(X) \|_{G,h} \ee
Then multiplying by $\G(X)$ and summing gives the result with
norm $ \| \cdot \|_{G ,h ,\G} = \| \cdot \|_{G, \G, h}$
\bigskip
Next we give some bounds on $\cQ(\z, W)$ .
For good bounds one would like $W_1,W_2$ to be locally integrable and
exponentially decaying. Refering to lemma \ref{wbd} we see that this
is true for $W_1$, but not for $W_2$ if $\beta' \geq 4\pi$. Thus we only assume the weaker
bounds
\bea \label{newwbd}
\int _{\De \times \De'} |W_1(x-y)| dx dy &\leq& C e^{- d(\De, \De')/2L} \nn \\
\int _{\De \times \De'}|x-y||W_2(x-y)| dx dy &\leq& C e^{- d(\De, \De')/2L }
\eea
Note that $W_{1,i}, W_{2,i}$ satisfy this, even with stronger decay
$e^{- d(\De, \De')/2} $. For $W_{2,i}$ we need that $|x|^{1- \beta'/ 2\pi}$
is locally integrable and this is true for $\beta' < 6 \pi$.
For a good uniform bound on $\cQ_2(\z, W_2)$ we subtract the most
singular part which is the constant
\bea \cQ_{sing} (\z,W_2; X, \psi ) &=& \cQ_2 (\z,W_2; X, 0) \nn \\
& =& {\z^2 \over 4} \int_{\tilde X} W_2 (x-y) dx dy \eea
For the difference the bound (\ref{newwbd}) will be adequate.
\blem \label{seven} Let $W_1, W_2$ satisfy (\ref{newwbd}).
Then for some constant $C$:
\bea \|\cQ_1(\z, W_1)\|_{G,\G,h} & \leq & C |\z|^2 e^{\one h} \nn \\
\|\cQ_2(\z, W_2) - \cQ_{sing} (\z,W_2)\|_{G,\G,h}
& \leq & C |\z|^2 e^{\one h} \k^{-1/2} \nn \\
\|\cQ(\z, W) - \cQ_{sing}(\z,W_2)\|_{G,\G,h}
& \leq & C |\z|^2 e^{\one h} \k^{-1/2} \nn \\
\eea
\elem
\pr The last follows from the first two.
For the first bound we compute for $f_j = (f_j,0)$
\[ (\cQ_1)_{n}(X, \psi_{\f}; f_1, ...f_{n}) \]
\be = {\z^2 \over 4} \int_{\tilde X} \cos ^{(n)}(\f(x) + \f(y)) W_1(x-y)
\prod_{j=1}^{n} (f_j(x) + f_j(y) ) dx dy \ee
This leads to
\bea \| (\cQ_1 )_{n}(\De \cup \De', \psi_{\f}) \|
&\leq & \one^{n} |\z|^2 \int_{\De \times \De'} |W_1(x-y)| dx dy \nn \\
&\leq & C\one^{n} |\z|^2 e^{-d(\De, \De')/2L} \eea
The norm restricted to $\bf \De$ has the same bound,
and since there are $2^{n}$ terms in the sum over $\bf \De$
it follows that
\be \| (\cQ_1 )_{n}(\De \cup \De') \|_G
\leq C\one ^{n} |\z|^2 e^{-d(\De, \De')/2L} \ee
Now multiply by $\G(X)$ and sum over $X$, then
multiply by $h^n/n!$ and sum over $n$, and get the result.
The second bound proceeds in just the same way for large
sets. We avoid the singularity at $x=y $ by the large
set hypothesis and the subtraction has the same bound as the function.
For small sets $X$ we compute for continuous vector-valued $f_j =(0,f_j)$
\bea && (\cQ_2- \cQ_{sing})_{n}(X, \psi_{\f}; f_1, ...f_{n}) \nn \\
&&= { \z^2 \over 4} \int_{\tilde X}
(\cos ^{(n)} (\int_y^x \pa \f )- \de_{n,0})
W_2(x-y) \prod_{j=1}^{n} ( \int_y^x f_j) dx dy \eea
It follows that
\bea && \|(\cQ_2- \cQ_{sing})_{n}(X, \psi_{\f}) \| \nn \\
&& \leq \one |\z|^2 \int_{\tilde X}
|(\cos ^{(n)} (\int_y^x \pa \f )- \de_{n,0})|
|W_2(x-y)| |x-y|^{n} dx dy \eea
For $n = 0$ a Sobolev inequality gives
\be | \cos ( \int_y^x \pa \f ) - 1 |
\leq |x-y|\sup _{x \in X}|\pa \f(x)|
\leq |x-y|\k^{-1/2} G(\k, X, \f) \ee
Thus we always have a factor $|x-y|$, and so
\bea && \| (\cQ_2- \cQ_{sing})_{n}(\De \cup \De', \psi_{\f}) \| \nn \\
&& \leq
\one ^{n} |\z|^2 \k^{-1/2} G(\k, X, \f) \int_{\De \times \De'} |x-y|
| W_2(x-y) | dx dy \nn \\
&& \leq
C\one ^{n} |\z|^2 \k^{-1/2} G(\k, X, \f) \eea
The restriction to $\bf \De$ has the same bound. Now multiply by
$G^{-1}$, taking the supremum over $\f$, and sum over $2^{n}$
terms $\bf \De$
to obtain
\be \| (\cQ_2- \cQ_{sing})_{n}(\De \cup \De') \|_G
\leq C\one ^{n} |\z|^2 \k^{-1/2}
\ee
The result now follows as before.
\subsection{The full RG flow}
We now study the full RG flow. The building blocks are the polymer
activities
\bea V^{r,N}_i &=& V(\z_i^{r,N}) \nn \\
Q^{r,N}_i &=& \cQ(\z^{r,N}_i, W_i) - \cQ_{sing}(\z^{r,N}_i, W_{2,i})
\eea
where
\be
\z^{r,N}_i = L^{-2i} \exp ({\beta \over 2}(v_{1,N}(0)-v_{r,N}(0))
+{\beta ' \over 2}\hat v_{i}(0)) \z
\ee
Note that initially $\z_N^{r,N} = \z_N$.
Also note using
$\beta' \hat v_i(0) \leq \beta v_{r_i,0}(0)$
and
$v_{r,N}(0) = v_{r_N,0}(0)$ and
\be v_{r,0}(0) = -\log r / 2 \pi + \one \ee
we have that
\bea |\z^{r,N}_i| &\leq & \one |\z|_i \label{zbd} \nn \\
|\z|_i &=& L^{- (2- \beta / 4\pi )i}|\z| \eea
We define
\be h_i = 3 - \sum_{j=i}^{\infty} 2^{-j} \ee
so that $ 1 \leq h_i \leq 3$.
We also define $G_i(X,\f) = G(\k_i, X, \f)$ where
\be \k_i = \k_0 2^{-i} \ee
for some constant $\k_0$.
For the main theorem we have the following restrictions on the
parameter values. First fix $\beta_0 < 16 \pi/3$ and allow $\beta \leq \beta_0$.
We assume $r_N$ is small enough so $\beta' = e^{r_N^2} \beta \leq \beta_1 < 16 \pi /3$.
Then choose $\ep$ sufficiently small ( depending on $\beta_0$), $L$
sufficiently large (depending on $\beta_0, \ep$), $\k_0$
sufficiently small (depending on $\beta_0, \ep, L$), and finally $|\z|$ sufficiently
small (depending on $\beta_0, \ep, L, \k_0$).
By the previous lemmas we have
\bea \| \Box e^{ V^{r,N}_i} \|_{G_i, \G, h_i} &\leq & C \nn \\
\|\Box ( e^{ V^{r,N}_i}-1) \|_{G_i, \G, h_i} &\leq & C|\z|_i \leq |\z|_i^{1-\ep} \nn \\
\| Q^{r,N}_i e^{ V^{r,N}_i}\|_{G_i, \G, h_i} & \leq&
C |\z|_i^2 \k_i^{-1/2} \leq
|\z|_i^{2-\ep} \label{prelimbds}
\eea
Here we have used that that $L^{-\ep (2 - \beta / 4 \pi)i} 2^{i/2} \leq 1$
for $L$ sufficiently large, and that $C|\z|^{\ep}\k_0^{-1/2} \leq 1 $ for $|\z|$
sufficiently small.
\bthm \label{bigthm} For $\beta < 16 \pi/3$
and $|\z|$ sufficiently small as above,
there exist polymer activities $K^{r,N}_i(X, \psi)$ and constants $ \Om^{r,N}_i$
such that
\be \cZ^{r,N}_i(\f) = e^{\Om^{r,N}_i}
\cE xp ( \Box e^{V^{r,N}_i} + K_i^{r,N})(\La_i,\psi_{\f})
\label{zrep} \ee
The $K_i^{r,N}$ are
analytic in $\z$ and translation invariant.
They have the form
\be K^{r,N}_i = Q_i^{r,N}e^{V^{r,N}_i} + R^{r,N}_i \ee
where for all $\beta,r,N,i$:
\be
\|R^{r,N}_i \|_{G_i, \G, h_i} \leq |\z|_i^{3-\ep} \label{rbd} \ee
and hence
\be \|K^{r,N}_i \|_{G_i, \G, h_i} \leq 2 |\z|_i^{2-\ep} \ee
\ethm
\bigskip
\re The term $\Om^{r,N}_i$ corresponds to the vacuum energy
in second order perturbation theory. It does not have a uniform
bound for $\beta \geq 4\pi$
\bigskip
\pr
We suppress $r,N$ from the notation.
The proof that $\Om_i,R_i$ exist and that $R_i$ satisfies the bounds
is by induction on $i$, starting with $i=N$ and working down to $i=0$.
We assume the result is true for $i$ and prove it for $i-1$.
For i=N the result holds with $W_N =0,Q_N=0, R_N=0,\Om_N=0$.
Each step preserves the analyticity in $\z$.
\bigskip
\noindent{\bf Step 1}.
For the fluctuation integral we have in general that for any polymer
activity $M$
\be \mu_{tC} * \cE xp ( \Box + M) = \cE xp(\Box + \cF_t(M)) \ee
where $\cF_t(M)$ is a certain analytic function of $M$. (See the
Appendix for more details).
For a family $G_t(X,\f)$, $0 \leq t \leq 1$, of large field regulators
satisfying for $t >s$
\be \mu_{(t-s)C} * G_s \leq G_t
\label{hom} \ee
we have the general bound \cite{BrYa90}
\be \|\cF_t(M) \|_{G_t, \G, h'} \leq \| M \|_{G_0, \G, h} \label{fbd} \ee
provided $h' < h$ and
\be \| M \|_{G, \G, h} \leq (h-h')^2/16 \|C\|_{\theta} \label{fcond} \ee
where $ \|C\|_{\theta}$ is a certain norm on $C$.
We study
\be \cZ_{i-1}(\f) = e^{\Om_i} \mu_{\beta ' C_i} *
\cE xp ( \Box e^{V_i} + K_i)(\La_i,\psi_{\f,L^{-1}})
\ee
where $\psi_{L^{-1}}(x) = (\psi_0(x/L), L^{-1}\psi_1(x/L)) $.
Before the scaling the polymer exponential is rewritten
\bea \mu_{t\beta' C_i} * \cE xp ( \Box e^{V_i} + K_i)(\La_i,\psi_{\f})
&=& \mu_{t\beta' C_i} * \cE xp ( \Box + M_i)(\La_i,\psi_{\f}) \nn \\
&=& \cE xp ( \Box + \cF_t(M_i))(\La_i,\psi_{\f}) \label{start} \eea
where
\be M_i = \Box(e^{V_i} -1) + K_i \ee
Now define
\be G_t(X, \f) = (2^{|X|} G_{i-1}(X, \f) )^t G_i(X,\f)^{1-t} \ee
This satisfies (\ref{hom}) as in \cite{BDH96a}. We use that $\k_{i-1}
= 2\k_i$ and that $\k_0$ is sufficiently small in order that
$\k_i \| \beta ' C_i\|$ be sufficiently small for various Sobolev
norms.
We conclude from (\ref{fbd}) that
\be \| \cF_t (M_i) \|_{G_t, \G, h_{i-1}} \leq
\| M_i \|_{G_i, \G, h_i} \leq 3|\z|_i^{1-\ep} \ee
The condition (\ref{fcond}) is satisfied since for $L$ sufficiently large and
$|\z|$ sufficiently small
\be 16\|M_i\|_{G_i, \G, h_i}\|\beta ' C_i\|_{\theta} \leq C |\z|_i^{1-\ep} \leq 2^{-2i+2}
= (h_i-h_{i-1})^2 \ee
Next we isolate the second order contribution. Write $M_i = N_i + R_i$ where
\be
N_i = \Box(e^{V_i} -1) +Q_ie^{V_i}
\ee
As the leading term in $\cF_t(M_i) = \cF_t(N_i +R_i)$
we take
\be N_t = \Box(e^{V_t} -1) +Q_te^{V_t} \ee
where with $ \z_t = e^{-t\beta ' C_i(0)/2} \zeta_i$
\bea
V_t &=& V(\z_t) \nn \\
Q_t &=& \cQ(\z_t, W_t) - \cQ_{sing}(\z_i, W_{2,i}) \eea
and $W_t$ is defined by (\ref{before}) with $\beta = \beta', C=C_i$, and $W_0=W_i$.
This is consistent with the discussion in section \ref{theflow};
the subtraction of the constant $\cQ_{sing}$ will have no effect to second
order.
Now provided that
$ R_t \approx \cF_t(N_i + R_i) - N_t $
we can rewrite (\ref{start}) as \be
\cE xp( \Box +R_t + N_t)(\La_i,\psi_{\f}) =
\cE xp(\Box e^{V_t} + K_t)(\La_i,\psi_{\f}) \ee
where $ K_t = Q_t e^{V_t} + R_t $
To define $R_t$ we write
\bea
&& { \cF}_t(N_i+R_i)-N_t \nn \\
&=& ({ \cF}_t(N_i+R_i)-\cF_t(N_i) )
+ ({ \cF}_t(N_i)-N_t ) \nn \\
&=& \int^1_0({ \cF}_t)_1(N_i+sR_i;R_i)ds
-\int^t_0 \left({\cF}_{t-s}\right)_1(N_s;E_s)ds
\eea
where $(\cF_t)_1$ is the derivative of $\cF_t$ and $E_t = E(N_t) = E(\Box + N_t)$.
An explicit
computation \cite{BDH95}
shows that
\bea
E_t
&=& \left( ( {\pa \over
\pa t} - \Delta_{\beta'C_i} ) Q_t - J_t \right) e^{V_t}
+ \beta'C_i \left( (V_t)_1,(V_t)_1 \right)Q_t e^{V_t} \nn \\
&-&
\beta' C_i \left( (Q_t)_1,(V_t)_1 \right) e^{V_t}
- 2 \cB_{\beta' C_i} \left( \Box e^{V_t}, Q_t e^{V_t} \right)
- \cB_{\beta' C_i}\left(
Q_te^{V_t}, Q_te^{V_t}\right)
\eea
In the first term $\cQ_{sing}$ does not contribute and so the term is
equivalent to zero by (\ref{bigzero}).
Define $E'_t$ to be equal to $E_t$ but with the first term
removed so $ E'_t \approx E_t$.
Now define
\be R_t = \int^1_0({ \cF}_t)_1(N_i+sR_i;R_i)ds
-\int^t_0 \left({\cF}_{t-s}\right)_1(N_s;E'_s)ds \label{rt}
\ee
and we have $ R_t \approx \cF_t(N_i + R_i) - N_t$ as required (see the Appendix).
Now we give some bounds. First write
\bea Q_t &=& \cQ(\z_t, W_t) - \cQ_{sing}(\z_t, W_{2,t}) \nn \\
&+& \cQ_{sing}(\z_t, W_{2,t}) - \cQ_{sing}(\z_i, W_{2,i}) \eea
The kernels $W_t$ still satisfy the bounds (\ref{newwbd}) and we apply
lemma \ref{seven} with $\k = t \k_{i-1} + (1-t)\k_i$. Just as in
the bound on $\cQ_i$ we find that
\be \|\cQ(\z_t, W_t) - \cQ_{sing}(\z_t, W_{2,t}) \|_{G_t,\G, h_i}
\leq |\z|_i^{2-\ep} \label{qtbd1} \ee
For the second term we write
\bea
&& \cQ_{sing}(\z_t, W_{2,t},X) - \cQ_{sing}(\z_i, W_{2,i},X)
\nn \\
&& = {\z_i^2 \over 4} \int_{\tilde X} (e^{t\beta ' (C_i(x-y) -C_i(0))} - 1)
W_{2,i}(x-y) dx dy \nn \\
&& +{\z_i^2 \over 4} \int_{\tilde X} e^{-t\beta' C_i(0)}
(e^{t\beta ' C_i(x-y)} - 1) dx dy
\eea
This also satisfies
\be \| \cQ_{sing}(\z_t, W_{2,t}) - \cQ_{sing}(\z_i, W_{2,i}) \|_{G_t,\G, h_i}
\leq |\z|_i^{2-\ep} \label{qtbd2} \ee
Indeed in the first term
the $(C_i(x-y) -C_i(0))$
supplies the factor $|x-y|$ needed for a good bound on $W_{2,i}(x-y)$,
and in the second term the $C_i(x-y)$ gives the
needed exponential decay.
We conclude that
$ \|Q_t \|_{G_t,\G,h_{i-1}} \leq \one |\z|_i^{2-\ep} $.
The same bound holds with $\G_1$ so
$ \|Q_t e^{V_t}\|_{G_t,\G,h_{i-1}} \leq \one |\z|_i^{2-\ep} $.
A similar first order bound holds for $N_t$.
Estimating $(\cF_t)_1$ and $(\cF_{t-s})_1$ by Cauchy bounds we have from (\ref{rt})
\be \| R_t \|_{G_t, \G, h_{i-1}} \leq
2 (\|R_i\|_{G, \G, h_i} +
\sup_{ s \leq t}\| E'_s \|_{G_s, \G, h_i}) \ee
All the terms in $E'_s$ are $\cO(\z^3)$ and by the techniques we have been using
one can show that $\| E'_s \|_{G_s, \G, h_i} \leq |\z|_i^{3-\ep}$. It
follows that $\| R_t \|_{G_t, \G, h_{i-1}}\leq \one |\z|_i^{3-\ep}$
Now let $\z^\#,W^\#, V^\#, Q^\#, K^\#,R^\#$ be the values of
$\z_t,W_t, V_t, Q_t, K_t,R_t$ at $t=1$.
Then we have
\be \cZ_{i-1}(\f) = e^{\Om_i} \cE xp(\Box e^{V^\#} + K^\#) (\La_i,\psi_{\f,L^{-1}})
\ee
where $ K^\# = Q^\# e^{V^\#}+ R^\# $. After switching a factor
$2^{-|X|}$ from $G_1^{-1} $ to $\G$ we have
\bea
\| Q^\#e^{V^\#} \|_{G_{i-1}, \G_{-1}, h_{i-1}} &\leq & \cO(1) | \z|_i^{2 - \ep} \nn \\
\| R^\# \|_{G_{i-1}, \G_{-1}, h_{i-1}} &\leq & \cO(1) | \z|_i^{3 - \ep}
\label{starbd} \eea
and hence a bound on $K^\#$.
\bigskip
{\bf Step 2}. We extract a constant $F_i(X)$ which is the last two terms
in $Q^\#$:
\be F_i(X) =
\cQ_{sing} (\z^\#, W^\#, X) -\cQ_{sing} (\z_i, W_{2,i}, X) \ee
>From (\ref{qtbd2}) we have
$ \| F_i \|_{G_{i-1}, \G_{-1},h_{i-1}} \leq |\z|_i^{2-\ep} $.
The extraction is a rearrangement \cite{BDH95}
\footnote{In \cite{BDH95}, $\La_i$ was a torus, not a rectangle in the
plane. Nevertheless this formula holds with the same $\cE$ and the same $\de \Om$}
\be \cE xp(\Box e^{V^\#} + K^\#)(\La_i, \psi) = e^{\de \Om_i}
\cE xp(\Box e^{V^*} + K^*)(\La_i, \psi) \ee
where $V^* = V^\#$ and
\bea K^* &=& \cE(K^\#, F_i) \nn \\
& =& (K^\#-F_ie^{V^\#}) + \cE_{\geq 2 } (K^\#,F_i) \nn \\
\de \Omega_i & = & \sum_{X \subset \La_i}F_i(X) \eea
Now we can write
$K^* = Q^*e^{V^*} + R^*$
where
\bea Q^* &=& \cQ(\z^\#, W^\#) - \cQ_{sing}(\z^\#, W_2^\#) \nn \\
R^* &=& R^\# +\cE_{\geq 2 } (K^\#,F_i)
\eea
A general bound on $\cE_{\geq 2}$ is
\be
\| \cE_{\geq 2}(K,F) \|_{G, \G, h} \leq \cO(1) \| K \|_{G, \G_1, h}\|F\|_{G,\G, h})
\label{extractbound }
\ee
Using this and (\ref{starbd}) to bound $R^*$, and (\ref{qtbd1})
to bound $Q^*$ we have:
\bea
\|Q^*e^{V^*}\|_{G_{i-1}, \G_{-2}, h_{i-1}} &\leq & \cO(1)|\z|_i^{2-\ep} \nn \\
\|R^*\|_{G_{i-1}, \G_{-2}, h_{i-1}} &\leq & \cO(1)|\z|_i^{3-\ep} \eea
and hence a bound on $K^*$.
Now we have
\bea \label{one} \cZ_{i-1} (\f)
&=& e^{\Om_{i-1}} \cE xp ( \Box e^{V^*} + K^*)(\La_i,\psi_{\f,L^{-1}}) \nn \\
\Om_{i-1} &=& \Om_{i} + \de \Om_{i} \eea
\bigskip
{\bf Step 3}.
Finally we scale down. The scaling transformation is \cite{BDH95}:
\be \cE xp ( \Box e^{V^*} + K^*))(\La_i,\psi_{\f,{L^{-1}}})
= \cE xp(\Box e^{V_{i-1}}+ K')(\La_{i-1},\psi_{\f})
\label{kprime}
\ee
where $K'= \cS(K^*)$ is a certain function of $K^*$.
We have used
\be L^2 \z^\# = L^2 e^{- \beta'/ 2\pi C_i(0)} \z_i = \z_{i-1} \label{newz} \ee
to identify $V^*(LX, \f(\cdot/L)) = V_{i-1}(X,\f)$.
Now write $ \cS(K^*) =\cS_1(K^*) +\cS_{\geq 2}(K^*)$ where $\cS_1$ is the
linearization. Then we have
$ K' = \cS_1(Q^* e^{V^*}) + R_{i-1} $
where
\be R_{i-1} = \cS_1(R^*) + \cS_{\geq 2}(K^*) \ee
However an explicit calculation using (\ref{restrict}), (\ref{newz}),
and $W^\#(Lx)=W_{i-1}(x)$
shows that
\be \cS_1 (Q^*e^{V^*}) \approx Q_{i-1}e^{V_{i-1}} \ee
Thus if we define $K_{i-1} =Q_{i-1}e^{V_{i-1}} + R_{i-1}$, then
$K_{i-1} \approx K'$, we can replace $K'$ by $K_{i-1}$ in (\ref{kprime}),
and obtain the required
\be \cZ_{i-1} (\f)
= e^{\Om_{i-1}} \cE xp ( \Box e^{V_{i-1}} + K_{i-1})(\La_{i-1},\psi_{\f})
\ee
A basic $\cS$ bound is
\be \| \cS (K) \|_{G, \G,h} \leq
\one L^2 \| K \|_{G,\G_{-2},h}
\ee
The same bound holds for $\cS_1$ and $\cS_{ \geq 2}$ has a bound proportional to $\|K\|^2$.
We conclude
for $\beta \leq \beta_0 < 16 \pi /3$ , $\ep$ sufficiently small, and $L$ sufficiently
large
\bea \| R_{i-1} \|_{G_{i-1}, \G, h_{i-1}}
&\leq & \one L^2 |\z|_i^{3-\ep} \nn \\
& = & \one L^2 L^{-(2- \beta/ 4\pi)(3-\ep)}|\z|_{i-1}^{3-\ep} \nn \\
& \leq & |\z|_{i-1}^{3-\ep} \eea
This completes the proof.
\subsection{Correlation functions}
We now study the correlation functions, see also Hurd \cite{Hur95}.
As in (\ref{early}) we consider:
\be \cW_{\beta, r,N}(\z;f_1,...,f_n)= \z^n
< : \sin \f :_{1,N}(f_1) ... : \sin \f :_{1,N}(f_n)>_{\z,\beta,r,N}
\ee
with test functions $f_i \in \cC^{\infty}_0(\bR^2)$ with non-overlapping
supports in $\De$ .
These have a nice representation
as functional derivatives of $\Xi_{ r,N}(\z, \f)$ as defined in (\ref{xi}).
Indeed we have
\be \label{corr1} (-1)^n \cW_{\beta,r,N}(\z, f_1,...,f_n)=
{ (\Xi_{r,N})_n(\z,0; f_1....,f_n) \over
\Xi_{r,N}(\z,0) }
\ee
Equations (\ref{zrni}) and (\ref{zrep}) for $i=0$ give the representation
for the $\Xi_{r,N}$.
We have
\bea \Xi_{r,N}(\z,\f) &=& e^{\Om^{r,N}_0}
\mu_{\beta' \hat v_0} * \cE xp ( A^{r,N}_0)(\De,\psi_{\f}) \nn \\
&=& e^{\Om^{r,N}_0}
\mu_{\beta' \hat v_0} * A^{r,N}_0(\De,\psi_{\f}) \eea where
\be A^{r,N}_0 =\Box e^{V^{r,N}_0} + K_0^{r,N} \ee
The divergent constants $\exp (\Om^{r,N}_0)$
cancel in the expression (\ref{corr1}) and
the functional derivatives can be taken inside the integral
to obtain
\be \label{corr2} (-1)^n \cW_{\beta, r,N}(\z,f_1,....,f_n)
=
{ \int ( A^{r,N}_0)_n( \De, \psi_{\f};\psi_{ f_1},...,\psi_{ f_n }) d \mu_{\beta' \hat v_0}(\f)
\over \int A^{r,N}_0(\De,\psi_{\f})d \mu_{\beta' \hat v_0}(\f) }
\ee
We need uniform bounds on this function. In prepraration we have
\blem For $\k_0$ sufficiently small
$\int G(\k_0,\De, \f) d \mu_{\beta' \hat v_0}(\f) $
is bounded by a constant independent of $r$. \elem
\pr
Let $B$ be the quadratic form
\be (\f, B\f) = \sum_{1 \leq |\al| \leq 3}\int_{\De} | \pa^{\al} \f |^2
\ee
Then we have
\bea \int G(\k_0,\De, \f) d \mu_{\beta' \hat v_0}(\f) & =&
\int e^{\k_0 (\f, B\f)} d \mu_{\beta' \hat v_0}(\f) \nn \\
&=& (\det(1- \k_0 \beta' \hat v_0^{1/2}B \hat v_0^{1/2}) )^{-1/2} \eea
provided $\hat v_{0}^{1/2}B \hat v_{0}^{1/2}$ is trace class and $\k_0$ is
sufficiently small. We give a bound on the trace norm which is uniform in $r$
and hence get a uniform bound on the integral for $\k_0$ sufficiently
small.
We have $ \hat v_0^{1/2}B\hat v_0^{1/2}= \sum_{\al} A_{\al}^* A_{\al}$
where $A_{\al} = \chi_{\De} \pa^{\al} \hat v_0^{1/2}$. We
show that $A_{\al}$ is Hilbert-Schmidt with a uniformly bounded Hilbert-
Schmidt norm to obtain the result. We have
\bea \|A_{\al} \|_{HS}^2
&=& \int_{\De} dx \int dy |(\pa^{\al}\hat v_{0}^{1/2})(x-y) |^2 \nn \\
&=& \|\pa^{\al} \hat v_0^{1/2}\|_2^2 \nn \\
&=& \one \int { |p^{\al}|^2 \over p^2 +r^2} e^{-(p^2+r^2)}dp
\eea
and this is bounded uniformly in $r$ since $|\al| \geq 1$.
This completes the proof.
\bigskip
We paraphrase theorem \ref{tech} as:
\bthm Under the hypothese of theorem \ref{bigthm}, $ \cW_{\beta, r,N}(\z,f_1,....,f_n)$
is analytic in $\z$ in a neighborhood of the origin
and is bounded there uniformly in $\beta, r,N$. \ethm
\pr If suffices to assume $\|\psi_{f_i}\| \leq 1$.
By the definition of the norm
\be | (A_0^{r,N})_n(\De , \psi_{\f};\psi_{f_1},...,\psi_{f_n})|
\leq n! h_0^{-n} G(\k_0,\De, \f)\G(\De)^{-1}\| A_0^{r,N} \|_{G_0,\G, h_0}
\ee
Using this and the previous lemma we get for each $n$
\be | \int ( A^{r,N}_0)_n( \De, \psi_{\f}; \psi_{f_1},...,\psi_{f_n} )
d \mu_{\beta' \hat v_0}(\f) |
\leq \one \| A_0^{r,N} \|_{G_0,\G, h_0} \ee
By theorem \ref{bigthm}, $\| A_0^{r,N} \|_{G_0,\G, h_0}
\leq C $.
Thus the numerator in (\ref{corr2}) has a uniform bound and the analyticity
follows from the anlayticity of $ A_0^{r,N}$.
The same holds for the denominator, but we also have using (\ref{prelimbds})
\be |1- \int A^{r,N}_0(\De,\psi_{\f}) \ d \mu_{\beta' \hat v_0}(\f)|
\leq \one \|\Box - A^{r,N}_0 \|_{G_0, \G, h_0} \leq \one |\z|^{1-\ep} \ee
so the denominator is bounded away from zero. This completes the proof.
\section{Appendix}
We collect some notes about the fluctuation step. First suppose
we have basic polymer activities $K = K (X, \f)$.
We have the definitions
\bea (\mu_C * K)( \f)
& = & \int K ( \f + \z) d\mu_C(\z) \nn \\
(\De_C K)(\f) &=& {1 \over 2} \int
\ K_2(\f;\z, \z) \ d\mu_C(\z) \nn \\
\cB_C(K,K)(\f) &=& \frac{1}{2} \int \
\ K_1(\f;\z)\circ K_1(\f;\z ) d\mu_C(\z) \eea
We define $K_t(X,\f) $ by
\be \mu_{t C} * \cE xp (\Box + K)
= \cE xp (\Box + K_t) \label{first} \label{a1} \ee
This determines $K_t$ since the exponential function $\cE xp $ is invertible.
Then one finds the $K_t$ satisfy the
the differential equation $E(K_t) = E(\Box + K_t) =0$
or
\be \label{second}
({\pa \over \pa t} -\De_{ C} ) K_t
- \cB_C ( K_t, K_t ) =0
\label{a2}\ee
with the intial condition $K_0 = K$.
This can be integrated to give an integral equation
\be K_t = \mu_{tC} * K + \int_0^t \mu_{(t-s)C}
* \cB_C(K_s,K_s) ds
\label{a3} \ee
A finite iteration gives an explicit formula for $K_t(X,\f)$,
also written $ (\cF_t K)(X,\f) $.
Now suppose we have modified activities $ K = K(X,\psi)$. Now define
\bea (\mu_C * K)( \psi)
& = & \int K ( \psi + \psi_{\z}) \ d\mu_C(\z) \nn \\
(\De_C K)(\psi) &=& {1 \over 2} \int
\ K_2(\psi;\psi_{\z}, \psi_{\z}) \ d\mu_C(\z) \nn \\
\cB_C(K,K)(\psi) &=& \frac{1}{2} \int
\ K_1(\psi;\psi_{\z})\circ K_1(\psi;\psi_{\z}) \ d\mu_C(\z)\eea
where we recall $\psi_{\z} = (\z, \pa \z )$.
Again define $K_t(X,\psi) = (\cF_t K)(X,\psi) $ by
(\ref{a1}), and again the equations (\ref{a2}), (\ref{a3}) hold.
Now consider the relation between the two formulations. Suppose that
$K(\psi_{\f}) = K^0(\f)$.
Directly from the definition we have
$ (\mu_C * K)( \psi_{\f}) = (\mu_C * K^0)(\f)$ and it follows that
$(\cF_t K)(\psi_{\f}) = (\cF_t K^0 )(\f)$.
A corollary is that
If $ K \approx K'$ then
$ \cF_t ( K) \approx \cF_t ( K') $.
If also $A \approx A'$ then
$(\cF_t )_1( K; A) \approx (\cF_t)_1 ( K'; A') $
We also note explicitly that if $K(\psi_{\f}) = K^0(\f)$ then
\bea
(\De_C K)(\psi_{\f}) &=& (\De_C K^0)(\f) \nn \\
\cB_C(K,K)(\psi_{\f} ) &=& \cB_C(K^0,K^0)(\f)
\eea
This follows from
\bea K_1(\psi_{\f}; \psi_f )
&=& {d \over ds} K(\psi_{\f } +s \psi_{f})|_{s=0} \nn \\
&=& {d \over ds} K^0(\f +sf )|_{s=0} \nn \\
&=& (K^0)_1(\f;f) \eea
and similarly for the second derivative.
\bigskip
{\bf Acknowledgement} I would like to thank T. Hurd and D. Brydges for helpful
remarks, and A. Wightman for insisting that somebody should establish
results of this type.
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