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\begin{document}
\title{Block Renormalization Group in a Formalism with Lattice
Wavelets: Correlation Function Formulas for Interacting Fermions}
\author{Emmanuel Pereira$^{1}$ and Aldo Procacci$^{2}$\\
$^{1}$Dep. F\'{\i}sica-ICEx, UFMG,
CP 702, Belo Horizonte MG 30.161-970, Brazil\\
$^{2}$Dep. Matem\'atica-ICEx, UFMG, CP 702,
Belo Horizonte MG 30.161-970, Brazil}
\maketitle
\begin{abstract}
Searching for a general and technically simple multi-scale formalism to treat
interacting fermions, we develop a (Wilson-Kadanoff) block renormalization group
mechanism, which, due to the property of ``orthogonality between scales'',
establishes a trivial link between the correlation functions and the effective
potential flow, leading to simple expressions for the generating and correlation functions. Everything is based on the existence of ``special configurations''
(lattice wavelets) for multi-scale problems: using a simple linear change of
variables relating the initial fields to these configurations we establish the
formalism. The algebraic formulas show a perfect parallel with those obtained
for bosonic problems, considered in previous works.
%\noindent
%{\bf Key Words:}
\end{abstract}
\vskip1.5cm
{\bf n. pages}: 19
\newpage
\noindent
{\bf Running Head}: BLOCK RG FOR INTERACTING FERMIONS
\noindent
{\bf Mailing address}: Emmanuel Pereira, Dep. F\'{\i}sica-ICEx, UFMG,
CP 702, Belo Horizonte MG 30.161-970, Brazil.
{\bf E-mail}: aldo@mat.ufmg.br
%\vskip1.5cm
\section{Introduction}
\zeq
For a long time Renormalization Group (RG) methods have been a successful guide
to the study of problems with many scales of length. Roughly, the rigorous
formalization of the RG ideas leads to the analysis of a Gaussian measure through
some multi-scale structure of its covariance:
the covariance is written as a sum of new ones in distinct scales (now massive
interactions), which automatically leads to a
decomposition of the initial fields; and the procedure
follows with the study of the effective interactions resulting by integrating
out, step by step, the field in each scale, where the massive covariance is
expected to make easier the computation. See the refs. [1-5], and references therein. According to the problem, several techniques may
be used: polymer and tree expansions, small and large field analysis, etc. But, in spite of all this formalism, the resulting mathematical problem generally
involves an entanglement of variables and intricate propositions, becoming quite
difficult to be solved. In short, it is very useful to develop ideas
and structures in order to simplify this mechanism.
Recently [6,7], searching for such simplifications, in the study of the well
known lattice dipole gas (bosonic models with a laplacian plus perturbation as
interaction) it was noted that, specifically in the RG formalism already developed
in ref. [1] (Wilson-Kadanoff transformation: RG with $\delta$ weight function -
details in the next section), there was an orthogonality between several
terms associated to the covariance decomposition into different scales. This
property, named ``orthogonality between scales'', appeared gratuitously due to
lattice wavelets, fortunately implicit into the RG structure used (in the present
article, we make this relation more explicit). The use of such property allowed
establishing exact and simple formulas for the correlation functions (showing
a trivial link between the correlation and effective potential flows), with a
good control of the dominant and subdominant terms. In a few words, the
Wilson-Kadanoff RG appeared as a very good tool for those problems.
As a natural step in order to expand these results one may ask about the
possibility of obtaining a similar lattice formalism for fermionic systems.
Here, and throughout this paper, for such systems we mean fermions with
interaction given by a derivative plus perturbations, such as in a wide class of
models: Gross-Neveu, Luttinger, Fermi liquids.
For many reasons, fermions have been mostly treated in a different approach,
with smooth cutoffs (avoiding the lattice [2], [4], [8]), or even mapped on bosonic
models [9], but working in the lattice, although facing obstacles such as the
doubling of spectrum, one may start from a precise mathematical problem (e.g.,
without worrying about continuous Grassmann algebra) with a simplified solution
once implemented this special orthogonal property. A ``smooth'' RG transformation
for
lattice fermions is presented in ref. [10], but the results show that the
orthogonal property is not there, and its implementation is not a trivial fact,
which draws a pessimistic scenario.
In spite of that, in a successive work [11] useful
structures were established, towards an orthogonal multi-scale formalism
for fermions. The free propagator (corresponding to the Wilson
version of Dirac operator in a lattice) was decomposed into operators associated
to different scales and with orthogonal relations, and the
uniform exponential decay (locality) of the effective actions (actions after $k$
steps of the RG transformation) was shown. For that, an unusual procedure
(complex averages in the position space) was adopted, and the desired effective actions were
obtained as limit of expressions due to smooth RG transformations (with
exponential weight function - details in sec.2). That is, the RG transformation
with the orthogonal property (which shall facilitate the treatment of interacting
systems) was not directly defined.
In the present paper, besides reviewing previous works, now improving some
results in order to make clear the origin and extent of the orthogonal property,
we conclude the formalism for fermions. In other words, we redeem the
Wilson-Kadanoff block RG ($\delta$ type RG transformation) for fermionic
systems. We rigorously establish this multi-scale structure with orthogonality
between scales, and use it to obtain simple formulas for the generating and
correlation functions of interacting fermions. We are guided by the bosonic
results, in particular by the fact that, from a certain covariance decomposition,
we may get ``special field configurations'' related to wavelets (lattice
wavelets). Now even though working with Grassmann variables, we prove that it is
still possible, using a simple linear change of variables, to write the initial
field in terms of those ``special configurations'', which have properties quite
useful for a multi-scale analysis, such as localization, orthogonality between
scales, etc.
The rest of the article is organized as follows: in the section 2, to make the
paper essentially self-contained, we review some results and structures
already known (some propositions are just listed - proofs in the references).
We also introduce some modifications in order to show the generality of the
method and to make clear the connexion with wavelets. Section 3 is devoted to
the derivation of the desired multi-scale formalism for interacting fermions,
and section 4 to concluding remarks.
\section{Previous Results and First Generalizations}
\zeq
\subsection{Bosonic Systems}
Now, reviewing some results [6,7] we describe (without details) how to derive
the bosonic multi-scale formalism, emphasizing (and sometimes improving) the main aspects.
Let us consider scalar field models on unitary finite lattices
$\Lambda_{N} = \left[-\frac{L^{N}}{2}, \frac{L^{N}}{2}\right]^{d}\cap
{\bf Z}^{d}$, $L$ odd, $d \geq 3$, given by interactions such as
\begin{equation}
{\cal H}(\phi) = \frac{1}{2}b_{0}(\phi, \Delta\phi) + V(\phi) ,
\end{equation}
where $b_0$ is a constant, $\phi\in {\bf R}^{|\Lambda_{N}|}$,
$\Delta \equiv \partial^{\dagger}\partial$ (for Dirichlet boundary
conditions, otherwise plus a regularizer), $V$ a function of
$\partial_{\mu}\phi(x)$ (such as in the dipole gas or $(\nabla\phi)^{4}$
models). To obtain the formalism we follow the flow of the generating function
\begin{equation}
Z(h) \equiv \int\exp[-{\cal H}(\phi) + (h, \phi)] D\phi ,~~~~~~~~~ D\phi =
\prod_{x \in \Lambda_{N}} d\phi(x),
\end{equation}
via the Wilson-Kadanoff RG transformation (with $\delta$ weight function)
\begin{equation}
\exp[- H^{1}(\psi)] = \frac{\int\exp[- H(\phi)]\delta(C\phi - \psi)
D\phi}{{\rm numerator \, with} \; h, {\psi} = 0}
\end{equation}
where $H(\phi) = {\cal H}(\phi) - (h,\phi)$, $\psi \in {\bf R}^{|\Lambda_{N-1}|}$,
$\delta(C\phi - \psi) = \prod_{x \in \Lambda_{N-1}} \delta(C\phi(x) - \psi(x))$,
with $C\phi(x)$ meaning the rescaled average (canonical scaling) over blocks
$b_{Lx}^{L}$ of size $L$, centered in $Lx \in \Lambda_{N}$,
\begin{equation}
C\phi(x) = L^{(d-2)/2} L^{-d} \sum_{y \in b_{Lx}^{L}}\phi(y) ,
\end{equation}
(we maintain the notation $C$ for averages from $\Lambda_{N-j}$ to
$\Lambda_{N-j-1}$).
After $n$ steps of the RG transformation ($n \leq N$), minimizing at each
step the effective action (after discarding the perturbative potential $V$
and considering the constraint $\delta(C\phi^{j} - \phi^{j+1})$, where
$\phi^{j}$ means the block field at the $j$th scale), changing the variables
in order to expand around the minima, and also separating the marginal terms
(quadratic part) of the potential, we obtain (all details in ref. [6])
\begin{eqnarray}
\lefteqn{Z(h) = c \exp\left[\frac{1}{2}(h, \tilde{P}_{n}h)\right]} \nonumber \\
& & \times \int\exp\left\{-\tilde{V}^{n}(\partial_{\mu}[M_{n}\phi + \tilde{G}_{n}h]) -
\frac{1}{2}b_{n}(\phi, \Delta_{n}\phi)\right\} D\phi ,
\end{eqnarray}
where $\phi \in {\bf R}^{\Lambda_{N-n}}$;
$c$ does not depend on $h$; $b_{n}$ is the wavefunction renormalization
constant at
step $n$; $\tilde{V}^{n}$ is the $n$th irrelevant perturbative potential (the potential
without its marginal quadratic part); the propagators $\tilde{P}_{n}$ and $\tilde{G}_{n}$
given by
\begin{equation}
\tilde{P}_{n} = P_{n} +
\frac{1}{b_{n}}\tilde{\Delta}_{n}^{-1},~~~~
P_{n}= \sum_{j=0}^{n-1}\left(2 -
\frac{b_{n}}{b_{j}}\right)\frac{1}{b_{j}}\tilde{\Gamma}_{j} ,
\end{equation}
\begin{equation}
\tilde{G}_{n} = G_{n} +
\frac{1}{b_{n}}\tilde{\Delta}_{n}^{-1} ,~~~~
G_{n}=\sum_{j=0}^{n-1}\frac{1}{b_{j}}\tilde{\Gamma}_{j}
\end{equation}
with $\tilde{\Gamma}_{j} = M_{j}\Gamma_{j}M_{j}^{\dagger}$, ~~
$\tilde{\Delta}_{n}^{-1} = M_{n}\Delta_{n}^{-1}M_{n}^{\dagger}$, ~~and
\begin{equation}
\Gamma_{j} = \Delta_{j}^{-1} -
\Delta_{j}^{-1}C^{\dagger}\Delta_{j+1}C\Delta_{j}^{-1} , \; \;
\Delta_{j} = (C_{j}\Delta^{-1}C^{\dagger}_{j})^{-1} , \; \;
M_{j} = \Delta^{-1}C_{j}^{\dagger}\Delta_{j} ,
\end{equation}
where $C_{j}$ is the rescaled average over blocks of side $L^{j}$, given by
(2.4) changing $L$ by $L^{j}$. It is interesting to note that the structure
of the operators $\tilde{P}_{n}$ and $\tilde{G}_{n}$ is directly related to the free propagator
decomposition
\begin{equation}
\Delta^{-1} = \sum_{j=0}^{n-1}\tilde{\Gamma}_{j} + \tilde{\Delta}_{n}^{-1} ,
\end{equation}
which is a decomposition into massive terms
\begin{equation}
|\tilde{\Gamma}_{j}(x, y)| \leq L^{-j(d-2)}\exp[-\alpha'L^{-j}|x
- y|], \; \; \alpha'>0 ,
\end{equation}
($x$ and $y$ in unitary lattices); $\Delta_{j}$ also with exponential
decay, and
$\tilde{\Delta}^{-1}_{n}$ bounded by $c L^{-n(d-2)}$ (vanishing as $n
\rightarrow \infty$). Roughly, $\tilde{\Gamma}_{j}$ describes the interaction
around the momentum scale $L^{-j}$.
In a few words, the expressions above say that, using a properly chosen RG
transformation, it is possible to write the generating function (of several
systems) in terms of a ``local'' effective action $\Delta_{n}$ (which goes,
as $n \rightarrow \infty$, to the Gaussian fixed point), a ``small''
irrelevant perturbative potential ${\tilde V}_{n}$, and two propagators $\tilde{P}_{n}$ (which
shall contain the dominant part of the two-point function) and
$\tilde{G}_{n}$ written in terms of interactions $\tilde{\Gamma}_{j}$, living in
different momentum scales. The long distance behavior of the correlations is
determined by a sequence of wavefunction renormalization constants and by
field derivatives of the effective action at zero field.
We must emphasize the simplicity of the formulas: there is no mix between
different momentum scales in the expressions for $\tilde{P}_{n}$ and $\tilde{G}_{n}$, fact
due to the orthogonal property
\begin{equation}
\tilde{\Gamma}_{j}\Delta\tilde{\Gamma}_{k} = \delta_{ij}\tilde{\Gamma}_{k},
\, \, \, \tilde{\Gamma}_{j}\Delta\tilde{\Delta}_{n}^{-1} =
\tilde{\Delta}_{n}^{-1}\Delta\tilde{\Gamma}_{j} = 0, \, \, \,
\tilde{\Delta}_{n}^{-1}\Delta\tilde{\Delta}_{n}^{-1} =
\tilde{\Delta}_{n}^{-1} .
\end{equation}
It is also important to note that this property is due to the type of the RG
transformation considered here (with $\delta$ weight function - more comments
in the part 2 of this section), which, say, ``abruptly separates the scales'':
RG with exponential weight function, for instance, does not present this
orthogonality.
Now we derive some additional results which will later guide us in the construction of the fermionic formalism. From the free propagator decomposition (2.9) we get
\begin{equation}
I ~=~ \sum_{j=0}^{n-1}~\tilde{\Gamma}_{j}~\Delta ~+~ \tilde{\Delta}_{n}^{-1}~\Delta ~~
\equiv ~\sum_{j=0}^{n}~{\cal P}_{j}
\end{equation}
with, from (2.11), for $j = 0, 1, \ldots, n$
\begin{equation}
{\cal P}_{j}{\cal P}_k =\delta_{jk}{\cal P}_k, \; \; \;
{\cal P}_{j}^{\dagger}\Delta{\cal P}_k =\delta_{jk}\Delta
{\cal P}_k,
\end{equation}
where $\dagger$ means conjugate transposte (i.e. adjoint in
${\bf C}^{|{\Lambda}_{N}|}$ with the canonical scalar product).
In ref. [6], ${\cal P}_j$ and ${\cal P}_n$ (defined there as
$\Delta^{1/2}\tilde{\Gamma}_{j}\Delta^{1/2}$ and
$\Delta^{1/2}\tilde{\Gamma}_{n}\Delta^{1/2}$) are self-adjoint in the
canonical Hilbert space ${\bf R}^{|{\Lambda}_{N}|}$, and useful properties
follows. Here we use the definitions above in order to obtain results
extendable to fermions (see next section). However, defining a new inner
product in ${\bf C}^{|{\Lambda}_{N}|}$: $ = (f,\Delta{g})$,
where $(\cdot ,\cdot )$ indicates the canonical inner product (note that
$\Delta$ is strictly positive),
the adjoints (respect to $<\cdot, \cdot>$) become
\[
{\cal P}^{*}_{j}=\Delta^{-1}{\cal P}^{\dag}_{j}\Delta =
\Delta^{-1}\Delta\tilde{\Gamma}_{j}\Delta={\cal P}_j , \; \; \; \;
{\cal P}^{*}_{n}=\Delta^{-1}{\cal P}^{\dag}_{n}\Delta =
\Delta^{-1}\Delta\tilde{\Delta}_{n}\Delta={\cal P}_n .
\]
Thus we can still view ${\cal P}_j$ as orthogonal projections and,
from (2.12), (2.13), it follows that the eigenfunctions of
${\cal P}_j ~~(j=1,2,\dots ,n)$ furnish a basis for
${\bf C}^{|{\Lambda}_{N}|}$. Let us derive the eigenfunctions. From
${\cal P}f=f$ we get $\tilde{\Gamma}_{j}\Delta f
=M_j \Gamma_j \Delta_j C_j f$, and so, for
\begin{equation}
f=M_j v_j, ~~~~~~~~~~~Cv_{j}=0, ~~~~~~v_{j}\in {\bf C}^{|{\Lambda}_{N-j}|}
\end{equation}
i.e., $f=\Delta^{-1}C_{j}^{\dag}\Delta_{j}$, we have ${\cal P}_{j}f=f$. The
eigenfunctions of ${\cal P}_{n}$ are given by
\begin{equation}
g=M_{n}w~,~~~w\in{\bf C}^{|{\Lambda}_{N-n}|}
\end{equation}
(since ${\cal P}_{n}M_{n}w=M_{n}\Delta^{-1}_{n}M_{n}^{\dag}\Delta M_{n}w=
M_{n}w$, once $M_{n}^{\dag}\Delta M_{n}=\Delta_{n}$). It may be checked that
the total number of eigenfunctions described above is $|\Lambda_{N}|$, and
so we have a basis for ${\bf C}^{|{\Lambda}_{N}|}$.
Suitable translations of eigenfunctions are still eigenfunctions,
but not dilations. However, using operators that go from
$\varepsilon$-lattices to $L\varepsilon$-lattices, we may pass to the
continuum, taking $\varepsilon\rightarrow 0$, and in this limit
the eigenfunctions above, toghether with
their translations and dilations (details in [12]), generate
a basis of continuum wavelets. For this reason we baptize
these eigenfunctions as lattice wavelets.
>From (2.12-15) we may write any field configuration in terms
of lattice wavelets, which we claim to be ``special configurations''
for problems with many scale of length. Actually, for any field
configuration
$\phi\in {\bf C}^{|{\Lambda}_{N}|}$,
\begin{equation}
\phi = \sum_{j=0}^{n-1}M_{j}v_{j}+M_{n}w,
~~~~v_{j}\in{\bf C}^{|{\Lambda}_{N-j}|},~~~Cv_{j}=0,
~~~~w\in{\bf C}^{|{\Lambda}_{N-n}|}
\end{equation}
where the vectors $v_{j}$, $w$ are uniquely determinated by
$\phi$.
The special feature of this decomposition is that lattice-wavelets
bring out the ideas of
multi-scale, orthogonality and localization present in the RG
procedure. These properties are related to algebraic and analytic
behavior of the structures present in the lattice wavelets. The analytic behavior,
for instance, depends sensibly on the choice of the averaging operator $C$.
The decomposition above, interpreted as a linear
change of variables, is a simple way to get the generating
functional formula (2.5) in the multi-scale decomposition.
For example, taking $n=1$, we
pose (introducing a shift to adjust the external field $h$)
$\phi =M_{1}\psi +Q\zeta + b_{0}^{-1}\Gamma_{0}h,$
with $CQ=0$,
$Q$ injection from ${\bf C}^{|{\Lambda}_{N}|-|{\Lambda}_{N-1}|}$
to ${\bf C}^{|{\Lambda}_{N}|}$ (i.e., a parametrization of Ker $C$),
$\psi\in {\bf C}^{|{\Lambda}_{N-1}|}$,
$\zeta\in{\bf C}^{|{\Lambda}_{N}|-|{\Lambda}_{N-1}|}$;
and thus, in terms of the variables \{$\psi$, $\zeta$\},
we may rewrite (2.2) as
\begin{eqnarray*}
Z(h)
% & = & \int \exp[-{1\over 2}b_{0}(\phi ,\Delta\phi )-V(\phi)
%+(h,\phi)] D\phi\\
& = & const. \, \exp[{1\over 2}b_{0}^{-1}(h,\Gamma_{0}h)]\int D\psi \exp[-{1\over 2}b_{0}(\psi ,\Delta_{1} \psi) +
(h,M_{1}\psi)] \\
& & \int D\zeta \;
\exp[-{1\over 2}b_{0}(Q\zeta ,\Delta Q\zeta)-V(M_{1}\psi +Q\zeta +b_{0}^{-1}\Gamma h)] ,
\end{eqnarray*}
where we used $CQ=0$, the definitions (2.8) and the
orthogonal relations (2.11).
>From the effective potential appearing in formula above
$$ \exp[-V_{1}(\chi = M_{1}\psi+b_{0}^{-1}\Gamma h)]=
\frac{\int D\zeta
\exp[-{1\over 2}b_{0}(Q\zeta ,\Delta Q\zeta)-V(M_{1}\psi +Q\zeta+b_{0}^{-1}\Gamma h)]}{{\rm numerator \; with}\; \zeta, h = 0} ,$$
extracting the marginal quadratic part
proportional to $(\chi ,\Delta\chi)$, and using the orthogonal
properties (2.11) we finally obtain
\begin{eqnarray*}
Z(h)
%& = & \int \exp[-{1\over 2}b_{0}(\phi ,\Delta\phi )-V(\phi)
%+(h,\phi)] D\phi\\
& = & const. \exp[{1\over 2}(h,P_{1}h)]\int D\psi \; \exp[-{1\over 2}b_{1}(\psi ,\Delta_{1} \psi) +
(h,M_{1}\psi)-\tilde{V}_{1}(M_{1}\psi +G_{1}h)]
\end{eqnarray*}
where $\tilde{V}_{1}$ is the irrelevant effective
potential (
$\tilde{V}_{1}(\chi)=V_{1}(\chi)-{1\over 2}\delta b_{0}(\chi, \Delta\chi)$),
and $b_{1}=b_{0}+\delta b_{0}$.
The usual definition of the effective potential using
the standard definition of the RG transformation (eq. (2.3)), with
the $\delta$-function made explicit, is given by
$$\exp[-V_{1}(\chi = M_{1}\psi+b_{0}^{-1}\Gamma h)] =
\frac{\int D\eta \delta (C\eta)
exp[-{1\over 2}b_{0}(\eta ,\Delta \eta)-V(M_{1}\psi +\eta+b_{0}^{-1}\Gamma
h)]}{{\rm numerator \, with} \; \psi, \, h = 0}$$
which, up to a constant, is the same as in the formulas above.
As said, the injection $Q$ is just a
parametrization of Ker $C$
with parameters $\zeta$ (a choice of $Q$ is
found in ref. [1]).
As a final comment, note that, specifying the lattice wavelets in
each scale
$$M_{j}Q\eta_{j}(x) =\sum_{i}a_{j}^{i}\eta_{j}^{i},
~~~~~~~~~~a_{j}^{i}\in {\bf R},$$
supposing $\eta_{j}^{i}$ a basis for the
$j$th scale with $(\eta_{j}^{i}, \Delta\eta_{j}^{i}) = \delta_{ij}$,
writing
$\phi = \sum_{(k)}a_{(k)}\eta_{(k)}(x)$,
$a_{(k)}\in {\bf R}$ (where the index $(k)$ include the scale
index $j$ and the internal index $i$ at a fixed scale), one can obtain the
Wilson-Kadanoff block RG (for $h = 0$) in a expression
similar to eq. (0.1) in ref. [13], there the starting point
for a phase
cell cluster expansion for Euclidean field theories. We need also to mention that in ref. [14] wavelets are related to the
Gaussian fixed point of a block spin RG.
\subsection{Fermionic Systems}
Now we recall some results and structures present in fermionic systems in
order to understand their specificness. We will see that it is still
possible, after suitable manipulations, to obtain multi-scale structures with
the orthogonal property which will lead (next section) to simple formulas for
the correlation functions.
In [10] {\it free} lattice Euclidean fermions are carefully studied via an RG
transformation with a Gaussian weight function. The authors consider,
in lattices with spacing $\varepsilon$ (initially), the Wilson version of the
Dirac operator
\begin{equation}
D = \sum_{\mu = 1}^{d} \gamma_{\mu}\left(\frac{\partial_{\mu}^{\varepsilon} -
{\partial_{\mu}^{\varepsilon}}^{\dagger}}{2}\right) -
\frac{1}{2}\varepsilon\Delta^{\varepsilon} , \;\; \Delta^{\varepsilon} =
\sum_{\mu = 1}^{d} \frac{1}{\varepsilon}(\partial_{\mu}^{\varepsilon} +
{\partial_{\mu}^{\varepsilon}}^{\dagger}) ,
\end{equation}
(for finite lattices, depending on boundary conditions, it may be necessary
to introduce another regularizer in order to make $D$ invertible)
where $\partial_{\mu}^{\varepsilon}$ is the $\varepsilon$-lattice forward derivative
(${\partial_{\mu}^{\varepsilon}}^{\dagger}$ the canonical adjoint), and $\gamma_{\mu}$
anti-hermitian Dirac matrices. The extra term breaking chiral symmetry,
introduced to supress the doubling of spectrum, vanishes in the continuous limit
($\varepsilon \rightarrow 0$) and is subdominant in relation to the infrared
behavior of the free propagator. The RG transformation
$T_{a, L}^{\varepsilon}$ with a ``smooth'' Gaussian weight function is defined as
\begin{eqnarray}
\lefteqn{\exp(\bar{\chi}, D_{1}\chi) \equiv [T^{\varepsilon}_{a,L}\exp(\cdot,
D\cdot)](\bar{\chi}, \chi)} \nonumber \\
& = & N\int d\bar{\psi}d\psi
\exp[a(L\varepsilon)^{-1}(\bar{\chi} - C\bar{\psi}, \chi -
C\psi)]\exp(\bar{\psi}, D\psi)
\end{eqnarray}
where the $\varepsilon$ ($L\varepsilon$) lattice fields
$\bar{\psi}$, $\psi$ $(\bar{\chi}, \chi)$ are independent Grassmann algebra
generators (with supressed spinor and lattice indices); $C$ is the usual
arithmetic averaging operator
over a block of side size $L\varepsilon$; $a$ is a real positive parameter;
$N$ a normalization constant such that
\begin{equation}
\int\exp[\bar{\chi}, D_{1}\chi] d\bar{\chi} d\chi =
\int\exp[\bar{\psi}, D\psi] d\bar{\psi} d\psi .
\end{equation}
Successive RG transformations are introduced according to the semi-group
property
\[ T^{L^{k-1}\varepsilon}_{a,L} \; T^{L^{k-2}\varepsilon}_{a,L}\ldots
T^{\varepsilon}_{a,L} =
T^{\varepsilon}_{a_{k},L^{k}} ,\]
$T^{\varepsilon}_{a_{k},L^{k}}$ defined as in (2.18) with $a_{k} =
\frac{1 - L^{-1}}{1 - L^{-k}}a$, $L^{k}$ and
$C_{k}$ (arithmetic averaging operators over blocks of side $L^{k}\varepsilon$)
replacing $a$, $L$ and $C$. The same symbol is used for the arithmetic averages
over $L^{kd}$ points irrespective of the domain lattice (lattice spacement).
After some manipulations, it is obtained the telescopic decomposition of the
free propagator as in the bosonic formula (2.9), now just replacing $\Delta$
by $D$, and taking
\begin{equation}
D_{k} = a_{k}(I + a_{k}C_{k}D^{-1}C_{k}^{\dagger})^{-1} ,
\end{equation}
with the decomposition still involving massive terms. In fact, it is shown that
(Th. III.1 in ref.
[10]), rescaling the operators after $k$ steps to the unitary lattice,
$D_{k}(x-x')$ (and other operators in the decomposition of $D^{-1}$)
admit a bound with uniform exponential decay
(independent on $k$). But the results hold only for $a$ sufficiently small.
%\begin{teorema}
%$\exists \beta>0, \; c>0$ independent of $k$, but for sufficiently small $a$ such that
%\[\noindent |D_{(k)}(x, x')|, \; \; |\Gamma_{(k)}(x, x')| \leq c
%\exp[-\beta|x - x'|] ,\]
%\noindent
%\[|D^{-1}_{(\eta)}C^{\dagger}_{k}D_{k}(y, x)|, \; \;
%|D_{(k)}C_{k}D^{-1}_{(k)}(x, y)| \leq c \exp[-\beta|y - x|] ,\]
%\noindent
%for $y, y' \in L^{-k}{\bf Z}^{d}$, $x, x' \in {\bf Z}^{d}$; $\Gamma_{(k)}$,
%$D_{(k)}$ in the unitary lattice, and $D_{(\eta)}$ the Dirac operator
%in the lattice $\eta = L^{-k}$.
%\end{teorema}
Hence, for the RG with smooth weight (finite $a$) there is no orthogonal
property (2.11), and consequently, no hope to obtain simple correlation
function formulas for interacting systems. In the limit $a \rightarrow \infty$,
$D^{-1}_{k}$ becomes $C_{k}D^{-1}C_{k}^{\dagger}$ (as for bosons) and the
orthogonality is recovered, but the uniform exponential decay of
effective actions is invalidated, which destroys the usefulness of the
multi-scale decomposition. In a few words, ref. [10] says how one may obtain
a decomposition of the free propagator in massive terms without the orthogonal
property, or with orhogonality but non massive terms (without uniform
exponential decay).
In ref. [11] the theorem about uniform exponential decay was extended for infinite $a$ (i.e., when $D^{-1}_{k} =
C_{k}D^{-1}C_{k}^{\dagger}$) by taking suitable complex averages $C$ (the
consideration of complex averages is a must - lemma 3.1 in ref. [11]). Roughly,
this necessity appears because
the main
term in $D$ is proportional to $\partial_{\mu}$, an odd function in the momentum space ($\sin p_{\mu}$), and using real averages the effective actions, due to
cancellations between positive and negative terms, may acquire a nasty behavior
(spoiling the fixed point [10], [11]). In order to differentiate $p_{\mu}$
of $-p_{\mu}$ one is forced to use different weights, which means complex averages in the position space.
It is easy to see that this procedure works also for other fermionic
actions $D$ with main part given by a derivative.
However, in ref. [11] the RG transformation with the orthogonal property
(able to treat a perturbative potential) is not directly presented: all the structures (related
to the free propagator) are studied as limit of those coming from the
smooth Gaussian RG transformation. Surely, it is not a good idea to introduce
a perturbative potential in the Gaussian RG transformation, study the
correlation formulas carrying out enormous expressions, and then take the
limit $a \rightarrow \infty$ in order to recover orthogonality, hoping for
drastic simplifications. So, in the next section, guided by the idea of
``special configurations'' for muti-scale problems, we develop this expected
formalism and show how to get simple formulas for correlation functions of
interacting fermions.
\section{Fermionic RG Formalism}
\zeq
\let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon
\let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda
\let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ph=\varphi
\let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\vartheta
\let\y=\upsilon \let\x=\xi \let\z=\zeta
\let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta
\let\O=\Omega \let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi
\let\Y=\Upsilon
\def\pro{{\cal P}}
\def\psib{\bar{\psi}}
\def\xb{\bar{\xi}}
\def\zb{\bar{\zeta}}
\def\cb{\bar{\chi}}
\def\hb{\bar{h}}
\def\lat{|\Lambda_{N}|}
\def\rightarrow{\to}
\def\Gt{{\tilde \Gamma}}
In this section, we present the general multi-scale structure
(orthogonal RG transformation) which we hope to be useful for a
large class of fermionic systems (in particular, the final formulas are directly applicable to asymptotically free models). We omit some technical details
dependent on the specificness of each model but irrelevant
for the properties of the formalism (details such as regularizers in the
interaction, subdominant terms, etc.).
The results of ref. [11], although carefully proved for a specific model
(Wilson version of the Dirac operator), are still considered since they are
trivially extendable for the class of fermionic systems considered here.
We take models in $\L_N\subset{\bf Z}^{d}$, finite unitary lattice,
$\lat = L^{Nd}$, with the inverse free propagator of the Fermi theory
given by $D$ (always ruled by a derivative), a linear, self-adjoint, inversible operator acting
on ${{\bf C}^{|{\Lambda}_{N}|}}$ (with suitable periodic conditions, regulators, etc.).
We deliberately ignore the internal degree of freedom, since they do not
play any role in the algebraic contruction of our RG. The matricial notation
${\bar{\psi}} D\psi$ will denote the product of a row vector
$\psib$, a matrix $D$ and a column vector $\psi$ (entries of $\psib$,
$\psi$ given by $\psib_{x}$, $\psi_{x}$, $x\in\L_{N}$, independent
generators of a finite Grassmann algebra). Newly introduced fields
in sequel (unless stated otherwise) are Grassmann generators, anticommuting with
all the others.
The generating function now is given by
\begin{equation}
Z(\hb ,h)=\int d\psi d\psib \, \exp[-b_{0}\psib D\psi - V_{0}(\psib ,\psi)
+\hb\psi +\psib h ],
\end{equation}
where $b_{0}$ is a constant, and $V_{0}$, depending on $\psib$, $\psi$,
the perturbative potential.
The multiscale decomposition starts with the
definition of the averaging operator $C$. We consider rescaled operators
$C~:~{{\bf C}^{|{\Lambda}_{N}|}} \to {{\bf C}^{|{\Lambda}_{N-1}|}}$ (or generally from ${{\bf C}^{|{\Lambda}_{N-j}|}}$ to ${{\bf C}^{|{\Lambda}_{N-j-1}|}}$) defined as
\begin{equation}
(C\psi)_{u}=\sum_{x\in\L_{N}}W(x)\psi_{Lu+x},~~~~~~~~~~~~~~~
u\in\L_{N-1}~~~(or~~ x\in\L_{N-j}, u\in\L_{n-j-1}).
\end{equation}
In the bosonic case of sec.2 (where Laplacian $\D$ replaces $D$)
$W(x)$ was taken as the canonical average: constant for $x$ inside a block
of size $L$ centered in $u$, and zero outside it. For $D$ the Wilson-Dirac
operator (properly adjusted to the finite lattice) $W(x)$ may be taken
as in ref. [11], leading to a multi-scale decomposition of $D^{-1}$ with
nice properties (as discussed in last section). Actually, the general
contitions that we need for $C$ are $C^{\dagger}$ injection and $C$ surjection
such that the effetive actions (in the expression related to the orthogonal
property) $D_{j}=(CD^{-1}_{j-1}C^{\dagger})^{-1}$ make sense (i.e. the inverses
exist), and also that these effective actions admit a uniform exponential
bound (theorem (2.1) in [11]).
Admiting that $C$ satisfies the conditions above,
we construct the operators $D_j$, $\G_j$, $M_j$ as in the
bosonic case:
\begin{equation}
D_{j}=(CD_{j-1}^{-1}C^{\dagger})^{-1}=(C_{j}D_{0}^{-1}C_{j}^{\dagger})^{-1},
~~~~~D_{0}=D,~~j\geq 1 ,
\end{equation}
here,
$C_{j}$ is just the composition of $C$ $j$ times,
\begin{eqnarray}
\G_{j} & = & D^{-1}_{j}-D^{-1}_{j}C^{\dagger}D_{j+1}CD^{-1}_{j}, ~~~~j\geq 0\\
M_{j} & = & m_{1}m_{2}\dots m_{j}=D_{0}^{-1}C^{\dagger}D_{j},~~~~~~~
m_{j}=D_{j-1}^{-1}C^{\dagger}D_{j},~~~m_{0}=M_{0}=1 \nonumber
\end{eqnarray}
$D_{j}$ and $\G_{j}$ operators in ${{\bf C}^{|{\Lambda}_{N-j}|}}$; $M_{j}$ from
${{\bf C}^{|{\Lambda}_{N-j}|}}$ to ${{\bf C}^{|{\Lambda}_{N}|}}$; $m_{j}$ from ${{\bf C}^{|{\Lambda}_{N-j}|}}$ to ${{\bf C}^{|{\Lambda}_{N-j+1}|}}$. As in the previous
section, a telescopic decomposition of the free propagator follows
\begin{equation}
D^{-1} = \sum_{j=0}^{n-1}\tilde{\G} _{j}+{\tilde{D}}_{n}^{-1}
= \sum_{j=0}^{n-1}[D^{-1}C_{j}^{\dagger}D_{j}C_{j}D^{-1}-
D^{-1}C_{j+1}^{\dagger}D_{j+1}C_{j+1}D^{-1}] +D^{-1}C^{\dagger}_{n}D_{n}C_{n}D^{-1}
\end{equation}
(just an algebraic fact)
where ${\tilde \G}_{j}\equiv M_{j}\G_{j}M^{\dagger}_{j}$,
${\tilde D}^{-1}_{n}=M_{n}D^{-1}_{n}M^{\dagger}_{n}$; and it is also immediate
the desired orthogonal properties:
\begin{equation}
\Gt_{j}D\Gt_{k}=\d_{jk}\Gt_{k}, ~~~\Gt_{j}D{\tilde D}^{-1}_{n}=
{\tilde D}_{n}^{-1} D\Gt_{j} = 0,~~~
{\tilde D}^{-1}_{n}D{\tilde D}^{-1}_{n}={\tilde D}_{n}^{-1} .
\end{equation}
Our strategy, on a parallel with the derivation of the bosonic formalism
(sec.2), is to exploit the decomposition above in order
to find the ``special configurations'', and use them to contruct
the multi-scale structure, i.e., to define the othogonal RG for fermions.
>From (3.5) we have
\begin{equation}
I=\sum_{j=0}^{n-1}\Gt_{j}D + {\tilde D}^{-1}_{n}D = \sum_{j=0}^{n}\pro_{j},
\end{equation}
and from (3.6),
\begin{equation}
\pro_{j}\pro_{k} =\d_{jk}\pro_{k},~~~~~
\pro_{j}^{\dagger}D\pro_{k}=\d_{jk}D\pro_{k},~~~~~(A^{\dagger}={\bar A}^{T})
\end{equation}
Comparing to the bosonic case of sec. 2, the difference is that $D$
now is not positive, and so we cannot define $(\cdot ,D \cdot)$ as a scalar
product in ${{\bf C}^{|{\Lambda}_{N}|}}$ respect to which $\pro_{j}$ would be self-adjoint and
$\{\pro_j\}$ a set of orthogonal projections. Anyway, we
will see that the eigenfunctions of ${\pro_j}$
are all that we need to reach our aim.
Let us first make some remarks. Considering $C$ as an operator from ${{\bf C}^{|{\Lambda}_{N-j}|}}$ to ${{\bf C}^{|{\Lambda}_{N-j-1}|}}$, since $C$ is a surjection, it
follows that dim $Ker~C=|\L_{N-j}|-|\L_{N-j-1}|$. Thus for $u\in Ker~C$,
we may write $u=Qv$, with $Q$ any injection from ${{\bf C}^{|{\Lambda}_{N-j}|-|{\Lambda}_{N-j-1}|}}$ to ${{\bf C}^{|{\Lambda}_{N-j-1}|}}$,
with $CQ=0$ (i.e., $Q$ is a parametrization of $Ker~C$), and with $v$
uniquely determinated by $u$ (and by the choice of $Q$).
Now turning to the eigenfunctions of $\pro_j$ and
$\pro_{n}$, $j\leq n-1$, we have
\begin{equation}
\pro_{j}M_{j}Qv_{j} = M_{j}Qv_{j},~~~~~~~~~~\pro_{n}M_{n}w_{n} = M_{n}w_{n},
\end{equation}
where $v_{j}\in {{\bf C}^{|{\Lambda}_{N-j}|-|{\Lambda}_{N-j-1}|}}$ and
$w_{n}\in {{\bf C}^{|{\Lambda}_{N-n}|}}$.
%there are many ways to check that these eigenfunctions, when
%$v_{j}$ varies in ${{\bf C}^{|{\Lambda}_{N-j}|-|{\Lambda}_{N-j-1}|}}$ and %$w_{n}$
%varies in ${{\bf C}^{|{\Lambda}_{N-n}|}}$, are linearly
%independent.
Clearly, if $v_j$, $v'_{j}$ are linearly independent in
${{\bf C}^{|{\Lambda}_{N-j}|-|{\Lambda}_{N-j-1}|}}$, then $M_{j}Qv_{j}$, $M_{j}Qv'_{j}$ are linearly independent in
${{\bf C}^{|{\Lambda}_{N}|}}$ (since $M_{j}Q$, as composition of injections, is an injection), and so, the correspondent eigenfunctions are also linearly independent.
For different $j$'s the same follows since $\pro_{j}\pro_{k}=0$ if $j\neq k$.
Thus, any vector $f$ in ${{\bf C}^{|{\Lambda}_{N}|}}$ may be decomposed in a unique way as
\begin{equation}
f=\sum_{j=0}^{n-1}M_{j}Qv_{j} + M_{n}w_{n}~ ,
\end{equation}
where $v_{j}\in {{\bf C}^{|{\Lambda}_{N-j}|-|{\Lambda}_{N-j-1}|}}$ and $w_{n}\in{{\bf C}^{|{\Lambda}_{N-n}|}}$. For bosons, a similar
decomposition (2.16) relates the new variables to wavelets. So,
we maintain here the name
{\it lattice wavelets} for these eigenfunctions (special configurations with properties
of localization, orthogonality, etc.).
Turning to the Grassmann algebra, the procedure above suggests constructing
the block RG transformation via a change of {\it Grassmann} variables following
(3.10). Thus, we take
\begin{equation}
\psi = \sum_{j=0}^{n-1}M_{j}Q\z_{j}+M_{n}\x_{n}~~~~,~~~~
\psib =\sum_{j=0}^{n-1}\zb_{j}Q^{\dagger}M^{\dagger}_{j} + \xb_{n}M_{n}^{\dagger} .
\end{equation}
This is a genuine change of variables: $2|\L_{N}|$ variables $\psi$, $\psib$
related to others $2|\L_{N}|$ given by
$\{\z_{0},\zb_{0},\z_{1},\zb_{1}, \cdots ,\z_{n-1},\zb_{n-1},\x_{n},\xb_{n}\}$.
Now everything follows quite similarly to what we have done for bosons in the
end of sec. 2. To study the flow of the effective potential, all that we
need is (3.11) (perform the change of variables (3.11) in (3.1) with
$h=0$, $\hb =0$, then integrate step by step the fields in each scale).
To study the generating function, it is again necessary to make some shifts in order
to adjust the external fields $h$ and $\hb$.
Let us see the first step of the RG tranformation in details. We replace
in r.h.s. of (3.1)
\[
\psi = M_{1}\x_{1}+Q\z_{0}+b_{0}^{-1}\G_{0}h~~~~,~~~~
\psib =\zb_{0}Q^{\dagger} + \xb_{1}M_{1}^{\dagger}+b_{0}^{-1}\hb\G_{0}
\]
i.e., (3.11) with $n=1$ plus a shift; we use then elementary properties
of Grassmann integrals, in particular: if $\psi =B\x$, $\psib =\xb B^{\dagger}$ such that $\det B \neq 0$, then
$
\int d\psi d\psib \r (\psi ,\psib ) =
\left.\int d\x d\xb ~ \r (B\x ,\xb B^{\dagger} )\right/ {\rm det}[B^{\dagger}B]$ (change of variables);
if $h,~\hb ,~ \psi ,~\psib$ are independent generators, then
$\int d\psi d\psib \r (\psi ,\psib ) =
\int d\psi d\psib \r (\psi + h,\psib +\hb )$
(translation formula).
Also using $CQ=0$ and orthogonal properties, we finally rewrite r.h.s. of (3.1)
as
\begin{eqnarray*}
\lefteqn{Z(h ,\hb) = \exp[b_{0}^{-1}\hb \G_{0} h ]N_{0}
\int d\x_{1} d\xb_{1}
\exp[-b_{0}\xb_{1} D_{1}\x_{1} + \hb m_{1}\x_{1}+\xb_{1}m_{1}^{\dagger}h]} \\
& & \times \int d\z_{0} d\zb_{0} \exp [-b_{0}\zb_{0}Q^{\dagger}D_{0}Q\z_{0}]
\exp[-V(m_{1}\x_{1} + Q\z_{0} + b_{0}^{-1}\G_{0}h ,~~
\xb_{1}m_{1}^{\dagger} + \zb_{0}Q^{\dagger} + b_{0}^{-1}\hb\G_{0})]
\end{eqnarray*}
where $N_{0}$ is a constant, and the last expression in r.h.s. define
the effective potential at scale $1$
\[
\exp[-V_{1}(\c ,\cb )] = \frac{\int d\z_{0} d\zb_{0} \, \exp [-b_{0}\zb_{0}Q^{\dagger}D_{0}Q\z_{0} -
V(\c + Q\z_{0}~,~\cb_ + \zb_{0}Q^{\dagger})]}{{\rm numerator~with} \; \c, \cb =0}
\]
with $\c= m_{1}\x_{1} + b_{0}^{-1}\G_{0}h $,
$\cb =\xb_{1}m_{1}^{\dagger} + b_{0}^{-1}\hb\G_{0}$.
Now we separate out the main part (respect to the scaling properties)
in $V_{1}$ in order to isolate the dominant term. Here we will only
take care of the quadratic marginal term, i.e. the term in $V_{1}(\c ,\cb ) $ which is proportional to $\cb D_{0}\c$ (depending on the specificness of the model a more detailed analysis
in the ``main part'' of the effective potential may be necessary, e.g. massive and quartic terms may be controlled in separate). We
define the ``irrelevant'' potential $\tilde{V}_{1}$ by
$V_{1}(\c ,\cb ) = \tilde{V}_{1}(\c ,\cb ) + \d b_{0}\cb D_{0}\c$,
where, due to orthogonal relations:
$\d b_{0}\cb D_{0}\c= \d b_{0}\xb_{1}D_{1}\x_{1} +{\d b_{0}\over b_{0}^{2}}\hb\G_{0}h$.
So, writing
$b_{1} = b_{0} +\d b_{0}$, $\g_{0}^{(1)} = b_{0}^{-1} - (b_{1} -b_{0})b_{0}^{-2}$, $P_{1}=\g_{0}^{(1)}\G_{0}$ and $G_{1}=b_{0}^{-1}\G_{0}$
we get
\begin{eqnarray*}
\lefteqn{Z(h ,\hb ) = N_{1}~\exp [\hb P_{1}h]} \\
& & \times \int d\x_{1} d\xb_{1}
\exp[-b_{1}\xb_{1} D_{1}\x_{1} + \hb m_{1}\x_{1}+\xb_{1}m_{1}^{\dagger}h]
\exp[-\tilde{V}_{1}(M_{1}\x_{1}+G_{1}h~,~\xb_{1}M_{1}^{\dagger}+\hb G_{1})].
\end{eqnarray*}
The expression above corresponds to the first step of RG tranformation. The second step follows with the change of variables
$
\x_{1}=m_{2}\x_{2}+ Q\z_{1} + b_{1}^{-1}\G_{1}M_{1}^{\dagger}h~~,
~\xb_{1}=\xb_{2}m_{2}^{\dagger}+
\zb_{1}Q^{\dagger} + b_{1}^{-1}\hb M_{1}\G_{1}
$
and similar procedures. Iterating we obtain (as in the bosonic case)
\begin{eqnarray*}
Z(h ,\hb ) & = & N~\exp [\hb (P_{n}+ b_{n}^{-1}M_{n}D^{-1}_{n}M^{\dagger}_{n})h]
\int d\x_{n} d\xb_{n}
\exp[-b_{n}\xb_{n} D_{n}\x_{n}]\\
& & \times
\exp[-\tilde{V}_{n}(M_{n}\x_{n} + b_{n}^{-1}M_{n}D_{n}^{-1}M_{n}^{\dagger}h
+G_{n}h~,~\xb_{n}M_{n}^{\dagger}+ b_{n}^{-1}\hb M_{n}D_{n}^{-1}M_{n}^{\dagger}
+\hb G_{n})]
\end{eqnarray*}
with
\begin{eqnarray}
b_{j} & = & b_{j-1} +\d b_{j-1}, ~~~~ \g_{j}^{(n)} = b_{j}^{-1} - (b_{n} -b_{j})b_{j}^{-2}, \\
G_{n} & = & \sum_{j=0}^{n-1}~b_{j}^{-1}\tilde{\G}_{j}, ~~~~
P_{n} = \sum_{j=0}^{n-1}\g_{j}^{(n)}\tilde{\G}_{j} \nonumber
\end{eqnarray}
Once more we emphasize that the final formulas are simple due
to the orthogonal properties (3.6). Using Grassmann derivative
the correlation function formulas are immediate. For example, for the
two point function $S_{2}(x,y)$
\begin{equation}
S_{2}(x,y) = \tilde{P}_{n}(x,y)
%+b_{n}M_{n}D_{n}^{-1}M_{n}^{\dagger}(x,y)
- \sum_{u,v}\tilde{G}_{n}(x,u)\left[{\partial \over \partial\cb}W_{n}
{\partial \over \partial\c}(u,v)\right]_{\c ,\cb =0}\tilde{G}_{n}(v,y)
\end{equation}
where
\[\tilde{P}_{n}(x,y) = P_{n}(x,y) +
b_{n}^{-1}M_{n}D_{n}^{-1}M_{n}^{\dagger}(x,y), ~~~~
\tilde{G}_{n}(x,y) = G_{n}(x,y) +
b_{n}^{-1}M_{n}D_{n}^{-1}M_{n}^{\dagger}(x,y);\]
\[
\exp[-W_{n}(\c ,\cb)] = \int d\x_{n}d\xb_{n}~\exp[-b_{n}\xb_{n}D_{n}\x_{n}]
\exp[-{\tilde V}_{n}(M_{n}\x_{n} + \c ,\xb_{n}M^{\dagger}_{n}+\cb )].
\]
The usefulness of the representation above is that its analysis poses
no difficulty: the dominant term is isolated in $\tilde{P}_{n}$ and the subdominant
contribution in the correlations are given by the field derivatives of the
``irrelevant'' effective potential at zero field (in a combination with
$\tilde{G}_n$). The behaviour of $\tilde{P}_n$ and $\tilde{G}_n$ depends only on the
running coupling constants $b_{0}, ~b_{1},\cdots ,~b_{n}$. And since this sequence is
given by the effective potential flow (remind that, such as in the bosonic case, the fermionic RG is also expected to present suitable multiscale properties), we see that our formalism
provides a trivial link between the effective potential theory and
the correlation function theory. This triviality is a consequence
of the particular block RG here constructed, which does not mix the scales
(orthogonal property).
We want to remark that the key idea to build up the block RG
transformation is the {\it linear} change of the initial Grassmann variables
by the ``special fields'' configurations (our lattice-wavelets).
We also give emphasis to the perfect parallel between the fermionic and bosonic
RG tranformation, which is not such a surprise, since everything is based on a
linear change of variables, and in linear combinations Grassmann variables
behave just like trivial complex numbers.
\section{Conclusion}
\zeq
Although making only general considerations without using the proposed
orthogonal multi-scale formalism in the study of a particular fermionic
system, that is, without carrying out the calculation of the running coupling constants and the effective potential flow for a precise
model, we strongly believe that now one may be optmistic about the usefulness
of the Wilson-Kadanoff ($\delta$-type) RG transformation in the study of
fermions. Besides the comments at the end of last section concerning the
simplicity of the generating and correlation functions (due to the orthogonal
property) which facilitates their analysis, we point out the ``good'' behavior of
fermionic perturbation to say that the computation of these parameters in the
RG flow may be even easier than for bosonic models (already successfully
considered). Fermions, for instance, are free of the unpleasant large field
regions of bosons, responsible for intrincate technical problems.
We intend to use the RG formalism developed here to study also systems controlled
by a non trivial fixed point: with new rescaling in the averaging operators,
i.e., carefully redefining the effective potential theory, we hope to
adapt the formulas deduced here to face anomalous scaling problems
(see [15] as an example for such systems).
Finally, we also emphasize the promising aspect of the connexion between
multi-scale structures (RG transformations) and the lattice wavelets. It has been
proved [12] that the simplest structure of lattice wavelets (with canonical
averaging and laplacian operator in the expressions) leads to true wavelets in
the continuum limit (in this case, Lemarie functions). Thus, it comes to mind
questions such as what to learn about wavelet construction with RG
techniques, and vice-versa, which other properties of these functions may improve
the multi-scale analysis.
\vskip1.5cm
\noindent
{\bf Acknowledgements}: This work was partially supported by CNPq and
FAPEMIG (Brazil).
\section*{References}
\zeq
\addcontentsline{toc}{section}{References}
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\end{document}
ENDBODY