\documentstyle [12pt] {article}
\renewcommand{\baselinestretch}{1}
\newcommand{\xo}{\mbox{$x_{0}$}}
\newcommand{\xu}{\mbox{$x_{1}$}}
\newcommand{\ds}{\mbox{$\not\!\partial$}}
\newcommand{\gc}{\mbox{$\gamma_{5}$}}
\newcommand{\fimu}{\mbox{$\Phi_{AB}(\mu)$}}
\newcommand{\fioi}{\mbox{$\Phi_{AB}^{-1}(0)$}}
\newcommand{\pa}{\mbox{$P_{A}(\mu)$}}
\newcommand{\pao}{\mbox{$P_{A}(0)$}}
\newcommand{\pbe}{\mbox{$P_{B}(\mu)$}}
\newcommand{\pbep}{\mbox{$P'_{B}(\mu)$}}
\newcommand{\pbo}{\mbox{$P_{B}(0)$}}
\newcommand{\be}{\mbox{$|_{B}$}}
\newcommand{\dmu}{\mbox{$d \over d\mu$}}
\newcommand{\la}{\mbox{$L_{A}(\mu)$}}
\newcommand{\las}{\mbox{$L_{A}$}}
\newcommand{\lao}{\mbox{$L_{A}(0)$}}
\newcommand{\lb}{\mbox{$L_{B}(\mu)$}}
\newcommand{\lbs}{\mbox{$L_{B}$}}
\newcommand{\lbo}{\mbox{$L_{B}(0)$}}
\newcommand{\lai}{\mbox{$L_{A}^{-1}(\mu)$}}
\newcommand{\lasi}{\mbox{$L_{A}^{-1}$}}
\newcommand{\laoi}{\mbox{$L_{A}^{-1}(0)$}}
\newcommand{\lbi}{\mbox{$L_{B}^{-1}(\mu)$}}
\newcommand{\lbsi}{\mbox{$L_{B}^{-1}$}}
\newcommand{\lboi}{\mbox{$L_{B}^{-1}(0)$}}
\newcommand{\lp}{\mbox{$L'(\mu)$}}
\newcommand{\detp}{\mbox{$det_{p}$}}
\newcommand{\ud}{\mbox{${\cal U}$}}
\newcommand{\ui}{\mbox{${\cal U}^{-1}$}}
\newcommand{\lau}{\mbox{$L_{A{\cal U}^{-1}}$}}
\newcommand{\lbu}{\mbox{$L_{B{\cal U}^{-1}}$}}
\newcommand{\laui}{\mbox{$L_{A{\cal U}^{-1}}^{-1}$}}
\newcommand{\lbui}{\mbox{$L_{B{\cal U}^{-1}}^{-1}$}}
\newcommand{\ik}{\mbox{$I_{k}(\lambda R)$}}
\newcommand{\ikp}{\mbox{$I'_{k}(\lambda R)$}}
\newcommand{\iip}{\mbox{$a \lambda {{\ikp}\over{\ik}}$}}
\newcommand{\iipp}{\mbox{$a \lambda {{\ikpp}\over{\ipk}}$}}
\newcommand{\ipk}{\mbox{$I_{k'}(\lambda R)$}}
\newcommand{\ikpp}{\mbox{$I'_{k'}(\lambda R)$}}
\newcommand{\nik}{\mbox{$I_{k}(\lambda_{n} R)$}}
\newcommand{\nikp}{\mbox{$I'_{k}(\lambda_{n} R)$}}
\newcommand{\niip}{\mbox{$a \lambda_{n} {{\nikp}\over{\nik}}$}}
\newcommand{\niipp}{\mbox{$a \lambda_{n} {{\nikpp}\over{\nipk}}$}}
\newcommand{\nipk}{\mbox{$I_{k'}(\lambda_{n} R)$}}
\newcommand{\nikpp}{\mbox{$I'_{k'}(\lambda_{n} R)$}}
\title{P-determinants and boundary values
\thanks{\it{This work was partially supported by CONICET(Argentina).}}}
\author{O.Barraza, H.Falomir, R.E.Gamboa
Sarav\'{\i}\\and E.M.Santangelo\\ \\
Departamento de F\'{\i}sica\\Fac. de
Ciencias Exactas, U.\ N.\ L.\ P.\\c.\ c.\ 67, 1900 La Plata, Argentina.}
\date{may 1991}
\begin{document}
\maketitle
\begin{abstract}
We show that a regularized determinant based on Hilbert's approach (wich we
call the "p-determinant") of a quotient of elliptic operators defined on a
manifold with boundary is equal to the "p-determinant" of a quotient of
pseudodifferential operators. The last ones are entirely expressible in terms
of boundary values of solutions of the original differential operators. We
argue that, in the context of Quantum Field Theory, this boundary values also
determine the subtractions (i.e., the counterterms) to which this
regularization scheme gives rise.
{\em Pacs}: 11.10 Ef, 12.40 Aa.
\end{abstract}
\newpage
\section {- Introduction}
In a recent paper\cite{r1}, R. Forman has established a relation between the
quotient of $\zeta$-function regularized determinants of elliptic operators
defined on a manifold with boundary and the boundary values of their solutions.
In fact, under certain conditions\cite{r1}, such quotient is equal to the
determinant of a quotient of pseudodifferential operators, entirely expressible
in terms of boundary values of the solution space of the original elliptic
operators. Thus, this approach has the advantage that, if one has a good
understanding of this solution space (which is independent of the boundary
conditions), one can study how the determinant varies as a function of the
operators and of the boundary conditions. In such way, all considerations can
be restricted to the boundary, which is a manifold of dimension one less than
the original one, without requiring the knowledge of eigenvalues, which are
global quantities of difficult handling.
Even though succesfully employed in some applications of physical \linebreak
interest\cite{r1,r2}, the conditions under which Forman's result is valid
appear to be too restrictive. In fact, they can be addressed to the existence
of the derivative of the Fredholm determinant of the quotient of elliptic
operators. This clearly excludes those situations in which this Fredholm
determinant does not exist, but a more general regularized determinant (which
we will call "p-determinant"), based on Hilbert's approach , can be defined
\cite{r3,r4}.
The aim of this paper is to generalize Forman's result, without loosing
its forementioned advantages.
In section 2 we establish our main result:\ Under certain conditions to be
specified later, the p-determinant \detp(\lbi\la\laoi\lbo) (where \la\ (\lb) is
an elliptic differential operator depending on a parameter $\mu$, $L(\mu)$,
restricted to functions $f$ defined in a region $\Omega \subset {\bf R}^{n}$
and satisfying $Af=0$ ($Bf=0$) at the boundary $\partial\Omega$) equals the
p-determinant \detp(\fioi\fimu) (where \fimu, acting on functions living on
$\partial\Omega$, is Forman's map defined in reference \cite{r1}).
We also argue that the application of this method in the context of Quantum
Field Theory allows to determine the subtractions which occur in this
regularization scheme (i.e., the pieces requiring renormalization) also from
boundary values of solutions of the operators.
In section 3 we apply our result to some physically interesting problems. In
particular, for a bosonic field at finite temperature, the difference of free
energies corresponding to two boundary conditions is evaluated, carefully
studying the subtractions occuring and the related problem of renormalization.
Finally, section 4 contains some comments and conclusions.
\section {- Generalization of Forman's result}
In what follows, we will be interested in elliptic linear differential
operators $L(\mu)$, depending on a parameter $\mu$ and restricted to functions
defined in a region $\Omega \subset {\bf R}^{n}$ and satisfying elliptic
boundary conditions corresponding to the projection $Bf=0$ at $\partial\Omega$.
We will suppose that the homogeneous problem:
\[ L f(x) = 0 , \ x\in \Omega \]
\begin{equation}
B f(x) = 0 , \ x \in \partial\Omega
\label{e1}
\end{equation}
has no nontrivial solution. So, following reference \cite{r1}, we introduce
the Poisson map for the operator $L$ and the boundary condition $B$,\ \pbe,
such that the unique solution of
\[ L f(x) = 0 , \ x \in \Omega \]
\begin{equation}
B f(x) = h , \ x \in \partial\Omega
\label{e2}
\end{equation}
is given by
\begin{equation}
f(x)=\pbe h(x).
\label{e2p}
\end{equation}
As shown in Lemma 1.4 in the same reference, the
Poisson map satisfies:
\begin{equation}
\dmu \pbe |_{B} = -L_{B}^{-1}(\mu) L'(\mu) \pbe |_{B} ,
\label{e3}
\end{equation}
where $L'(\mu)=\dmu L(\mu)$ and \lbi\ is the Green's function of $L$, satisfying
$B\lbi=0$ at $\partial\Omega$, and \be\ means acting on functions projected by
$B$ at the boundary.
Let us consider a second elliptic boundary condition, defined by the operator
$A$, and its corresponding Green's function \lai\ ($A\lai=0$ at
$\partial\Omega$). The second result quoted in the abovementioned Lemma is:
\begin{equation}
P_{A}(\mu) A \left( -L_{B}^{-1}(\mu)\right) =
L_{A}^{-1}(\mu) - L_{B}^{-1}(\mu) ,
\label{e4}
\end{equation}
since the right hand side is the unique smooth solution of $L$ with the
same $A$-boundary value as (-\lbi):
\begin{equation}
A \left( L_{A}^{-1}(\mu) - L_{B}^{-1}(\mu) \right) =
A \left( -L_{B}^{-1}(\mu) \right) .
\label{e5}
\end{equation}
\bigskip
Now, the p-determinant of $(1-M)$, where $M\in{\cal I}_{p}$ (i.e., $M^{p}$
is trace class), is defined as\cite{r4}:
\begin{equation}
\ln \detp ( 1 - M ) = - \int_{\gamma} dz z^{p - 1}
Tr \left\{ M^{p} \left( 1 - z M \right)^{-1} \right\} ,
\label{e6}
\end{equation}
where $\gamma$ is any differentiable path,
\[ \gamma : [0,1] \rightarrow {\bf C},\]
\[ \gamma(0)=0 , \gamma(1)=1,\]
such that $\left( \gamma (t) \right)^{-1}$ does not belong to the spectrum
of $M$.
If $M$ is a differentiable function of a parameter $\mu$, from equation
(\ref{e6}), one gets:
\begin{equation}
\dmu \ln \detp \left( 1 - M \right) =
Tr \left\{ M^{p-1} \left( 1 - M \right)^{-1}
\dmu M \right\} .
\label{e7}
\end{equation}
\bigskip
We will now suppose that \la\ and \lb \ are such that the operator\linebreak[4]
[\lbi \la \laoi \lbo] has a p-determinant, and
\begin{equation}
Tr \left\{ \left( \lai - \lbi \right)
\left( 1 - \la \laoi \lbo \lbi \right)^{p-1} \lp |_{ker L} \right\} < \infty,
\label{e8p}
\end{equation}
where $L'$ acts on functions in the kernel of the differential operator
$L$.
Then, from equation (\ref{e7}), it is easy to get:
\[ \dmu\ln\detp \left(\lbi \la \laoi \lbo \right) = \]
\begin{equation}
Tr \left\{ \left( \lai - \lbi \right)
\left( 1 - \la \laoi \lbo \lbi \right)^{p-1} \lp |_{ker L} \right\} .
\label{e8}
\end{equation}
By repeated application of equations (\ref{e4}) and (\ref{e5}), we will
prove the following
\bigskip
LEMMA
\begin{eqnarray}
\left( \lai - \lbi \right) \la \laoi \lbo \lbi \nonumber \\
= \pbe B \pao A \left( \lai
- \lbi \right)
\label{e9}
\end{eqnarray}
\bigskip
PROOF
\begin{eqnarray}
\left( \lai - \lbi \right) \la \laoi \lbo \lbi \nonumber \\
= \left( \pbe B \lai \right) \la \laoi \lbo \lbi \nonumber\\
= \pbe B \left( \laoi - \lboi \right) \lbo \lbi \nonumber\\
= \pbe B \left( - \pao A \lboi \right) \lbo \lbi \nonumber\\
= \pbe B \pao A \left( \lai - \lbi \right) .
\label{e9p}
\end{eqnarray}
When replaced in equation (\ref{e8}), this result allows us to write:
\[ \dmu \ln \detp \left( \lbi \la \laoi \lbo \right) \]
\[ = Tr \left\{ \left( 1 - \pbe B \pao A \right)^{p-1}
\left( \lai - \lbi \right) \lp |_{ker L} \right\}\]
\begin{equation}
= Tr \left\{ \pbe \left( 1 - B \pao A \pbe \right)^{p-1} B
\left( \lai - \lbi \right) \lp |_{ker L} \right\} .
\label{e10}
\end{equation}
Notice that, using equation (\ref{e3}), the last factors in the trace can
be rewritten as:
\[ \left( \lai - \lbi \right) \lp \pbe B |_{ker L}= \pa A \left( - \lbi \right)
\lp \pbe B |_{ker L} \]
\begin{equation}
= \pa A \pbep B |_{ker L}.
\label{e11}
\end{equation}
Therefore:
\[\dmu \ln \detp \left( \lbi \la \laoi \lbo \right) \]
\[ = Tr \left\{ \pbe \left( 1 - B \pao A \pbe \right)^{p-1} B \pa A \pbep B
|_{ker L}
\right\} \]
\[ = Tr \left\{ \left( 1 - B \pao A \pbe \right)^{p-1} \left( B \pa \right)
\left( A \pbe \right) ' \be \right\} \]
\begin{equation}
= \dmu \ln \detp \left( \fioi\fimu \right) ,
\label{e12}
\end{equation}
where \fimu\ is the operator introduced by R. Forman in reference \cite{r1}:
\begin{equation}
\fimu = A \pbe .
\label{e13}
\end{equation}
So, we have proved the following
\bigskip
THEOREM
\begin{equation}
\detp \left( \lbi \la \laoi \lbo \right) = \detp \left( \fioi\fimu \right) ,
\label{e14}
\end{equation}
as long as these p-determinants exist and equation (\ref{e8p}) is satisfied
(since both sides equal 1 at $\mu=0$).
\bigskip
Notice that this is a generalization of Forman's result. Indeed,
if\- \la\ and \lb\ are such that \lbo\lbi\ and \la\laoi\ have a Fredholm
determinant
(1-determinant), by the well-known properties of 1-determinants\cite{r3}
and their relation with $\zeta$-function regularized determinants\cite{r1},
one can write:
\begin{eqnarray}
{{Det_{\zeta}\la}\over{Det_{\zeta}\lb}}
{{Det_{\zeta}\lbo}\over{Det_{\zeta}\lao}}
= det_{1}\left(\lbo \lbi\right) det_{1} \left(\la \laoi\right) \nonumber \\
= det_{1} \left(\lbo \lbi \la \laoi \right) = det_{1} \left(\fioi\fimu
\right) ,
\label{e15}
\end{eqnarray}
which is the integrated version of Theorem 1 in reference \cite{r1}.
In this case ($p=1$) one could also follow reference \cite{r1} and collect
all $\mu$-independent factors into an integration constant (see equations
(\ref{e8}) and (\ref{e12})) which, under particularly lucky circumstances,
can be evaluated\cite{r1,r2}. In the general case ($p>1$), no such
factorization takes place.
\bigskip
Equation (\ref{e14}) in the Theorem above states that the p-determinant
on the left can be entirely expressed in terms of objects acting on functions
defined just on the boundary, which can be evaluated from the boundary values
of solutions of the elliptic operators\cite{r1,r2} (see section 3 later
in this paper for an explicit example).
We wish to stress that, in the context of Quantum Field Theory, the problem
of analyzing the subtractions taking place in this regularization scheme can
also be reduced to the consideration of objects at the boundary. In fact,
let us consider for a while that the (p-1)-determinants of the operators in
equation (\ref{e14}) also exist. Then, from the following property of
p-determinants\cite{r3}:
\begin{equation}
\detp \left( 1 - M \right) =
\exp\left({1 \over p-1} Tr M^{p-1}\right) det_{p-1}( 1 - M ),
\label{e16}
\end{equation}
we conclude that
\[ Tr \left\{ \left( 1 - \lbi \la \laoi \lbo \right)^{p-1} \right\} \]
\begin{equation}
= Tr\left\{ \left( 1 - \fioi\fimu \right)^{p-1} \right\}
\label{e17}
\end{equation}
When the (p-1)-determinant does not exist, equation (\ref{e17}) can be
considered to be a formal identity, which can be extended to all integer
powers from 1 to (p-1). In such a way, the (divergent) subtractions in the
present regularization scheme (and, therefore, the finite counterterms
required) can be expressed entirely in terms of the operator \fimu. We will
make explicit application of these arguments in the third example of Section
3.
\bigskip
Our Theorem concerns variations of the elliptic operator $L$. To end this
section, a comment on variations of the boundary conditions is in place:
Let us consider a situation in which boundary conditions, rather than
operators, depend on $\mu$ through a regular transformation ${\cal U}(\mu)$
such that the relevant boundary values of a solution of $L$, $f$, are given
by $A{\cal U}^{-1}(\mu)f$ ($B{\cal U}^{-1}(\mu)f)$. Again, we will suppose
that the boundary value problem is nonsingular.
We can translate the $\mu$-dependence to the differential operator by noticing
that:
\begin{equation}
\left(
\begin{array}{c}
L \Psi = 0 \\
A \ui \Psi = h
\end{array}
\right)
\Rightarrow
\left(
\begin{array}{c}
\ui L \ud \chi = 0 \\
A \chi = h
\end{array}
\right), for\ \chi = {\cal U}^{-1} \Psi,
\label{e18}
\end{equation}
and therefore apply our previous result, equation \ref{e14}, to
$\la \equiv \left( {\cal U}^{-1}L{\cal U} \right)_{A}$.
Since
\begin{equation}
\left( {\cal U}^{-1} L {\cal U}\right)^{-1}_{A} = {\cal U}^{-1} \laui {\cal
U} ,
\label{e19}
\end{equation}
equation (\ref{e14}) implies:
\[ \detp \left\{ \left( \ui L \ud\right)_{B}^{-1} \left( \ui L \ud\right)_{A}
\lasi \lbs \right\} \]
\[ = \detp \left\{ \ui \lbui \lau \ud\lasi \lbs \right\} \]
\begin{equation}
= \detp \left\{ \fioi\fimu \right\} ,
\label{e20}
\end{equation}
where
\begin{equation}
\fimu B \ui \Psi = A \ui \Psi ,
\label{e21}
\end{equation}
for any solution of $L\Psi=0$. In the next section we apply this result
to some examples of physical interest.
\section{- Some applications}
\subsection{- The Laplacian on the disc}
In this section, we will explicitly construct the operator \fimu\ for the
case in which the differential operator is:
\begin{equation}
L = - \bigtriangleup + \lambda ^{2} ,
\label{e22}
\end{equation}
acting on functions $f(r,\theta)$ defined on a disc of radius $R$, and the
boundary conditions correspond to the projections:
\begin{equation}
\begin{array}{lll}
Af & = & a \partial_{r} f(R,\theta) + f(R,\theta)\\
Bf & = & f(R,\theta),
\end{array}
\label{e23}
\end{equation}
We will introduce a smooth function ${\cal U}^{-1}(r)$, such that
${\cal U}^{-1}(R)=1$ and $\partial_{r}{\cal U}^{-1}(R)=-\mu$. Then, we have:
\begin{equation}
\begin{array}{lll}
A\ui f & = & a \partial_{r} f(R,\theta) + \left( 1 - \mu a \right)
f(R,\theta) \\
B\ui f & = & f(R,\theta) .
\end{array}
\label{e24}
\end{equation}
In order to determine \fimu, we choose as a basis in the kernel of $L$
the system:
\begin{equation}
\left\{ \Psi_{k}(r,\theta) = I_{k}\left(\lambda r\right) \exp \{ ik\theta \}
, for\ k \in {\bf Z}\right\}
\label{e25}
\end{equation}
where $I_{k}(z)$ is the modified Bessel function. The boundary values of
those functions are given by:
\begin{equation}
\begin{array}{lllll}
h'_{k}(\theta) & = & A \ui \Psi_{k} & = & \left\{ a \lambda \ikp +
\left( 1 - \mu a \right) \ik \right\} \exp \{ ik\theta \} \\
h_{k}(\theta) & = & B \ui \Psi_{k} & = & \ik \exp \{ ik\theta \} .
\end{array}
\label{e26}
\end{equation}
Since, for all k:
\begin{equation}
h'_{k}(\theta) = \fimu h_{k}(\theta) ,
\label{e27}
\end{equation}
we conclude that, referred to the basis $\left\{\exp \{ik\theta\},
k\in{\bf Z}\right\}$,
\fimu\ is diagonal and given by:
\begin{equation}
\left( \fimu \right)_{k',k} =
\left\{ \left( 1 - \mu a \right) + \iip \right\} \delta_{k',k}.
\label{e28}
\end{equation}
So:
\begin{equation}
\left( \fioi\fimu \right)_{k',k} =
\left\{ 1 - \frac{\mu a}{1 + \iip} \right\} \delta_{k',k}.
\label{e29}
\end{equation}
From equation (\ref{e6}) and the asymptotic behavior of $I_{k}(z)$ for
large $k$, which gives:
\begin{equation}
\left( 1 - \fioi\fimu \right)_{k',k} {\approx}\
{{\mu R}\over{|k|}} \delta_{k',k} ,\ |k|\gg 1,
\label{e30}
\end{equation}
one can see that the Fredholm determinant of the operator in equation
(\ref{e29})
doesn't exist, but its p-determinant for $p\geq 2$ does.
Then, the Theorem in section 2 implies that:
\[ det_{2} \left( \ui \lbui \lau \ud \lasi \lbs \right) = \]
\begin{equation}
\prod_{k=-\infty}^{\infty}
\left\{ 1 - {{\mu a}\over{1 + \iip}} \right\}
\exp \left\{ \frac{\mu a}{1 + \iip} \right\} .
\label{e31}
\end{equation}
Notice that, in the present case, $L_{B{\cal U}^{-1}}^{-1}=\lbsi$, which
follows from equations (\ref{e22}), (\ref{e23}) and (\ref{e24})).
\bigskip
If $\lambda=0$, one must consider the basis of functions in the kernel
of $L=-\bigtriangleup$ given by $\{ \Psi_{k} = r^{|k|} \exp \{ik\theta\},$
$ k\in {\bf Z}\}$.
One finally gets:
\begin{equation}
det_{2} \left( \fioi\fimu \right) = \prod_{k=-\infty}^{\infty}
\left\{ 1 - {{\mu a}\over{1 + |k| a / R}} \right\}
\exp \left\{ \frac{\mu a}{1 + |k|a/R} \right\} .
\label{e32}
\end{equation}
Notice that choosing $\mu=1/a$ the $k=0$ factor vanishes, thus
showing that $L_{A{\cal U}^{-1}}$, the Laplacian with Neumann boundary
conditions, has a null eigenvalue. This is related to the property of
p-determinants that\linebreak[4]
$\detp(1-M)\ne 0$ if and only if $(1-M)$ is invertible\cite{r3}.
\subsection{- Bosonic field at temperature $\frac{1}{\beta}>0$}
In this section, we study a massive scalar field confined to a circle of
radius $R$, at finite temperature $T=\beta^{-1}>0$ and subject, at the spatial
boundary, to the conditions defined in equation (\ref{e23}) of the previous
example.
As is well-known, the partition function of such a system leads to the
consideration of the fuctional deteminant of the differential operator:
\begin{equation}
L = -\bigtriangleup^{2} - \partial_{t}^{2} + m^{2} ,
\label{e33}
\end{equation}
where $t\in [0,\beta]$ is the "temporal" coordinate, and $L$ acts on functions
periodic in the $t$-direction and
satisfying $A{\cal U}^{-1}f = 0$ ($B{\cal U}^{-1}f = 0$) at $r=R$.
A basis in the kernel of $L$ is given by:
\begin{equation}
\left\{ \Psi_{n,k}(r,\theta,t) =
I_{k}(\lambda_{n}r) \exp \{ik\theta + i\omega_{n}t\},\
for\ n,k \in {\bf Z} \right\} ,
\label{e34}
\end{equation}
where $\lambda_{n}=\sqrt(\omega_{n}^{2}+m^{2})$ and
$\omega_{n}=\frac{2n\pi}{\beta}$.
To determine Forman's map, it is enough to consider projections on the
spatial boundary (since $\Psi_{nk}$ is periodic in the $t$-direction):
\[ h_{n,k}(\theta,t) = A \ui \Psi_{n,k} \]
\[ = \left\{(1-\mu a)\nik + a \lambda_{n}\nikp\right\}
\exp \{ik\theta+i\omega_{n}t\} \]
\begin{equation}
h'_{n,k}(\theta,t)=B\ui \Psi_{n,k} = \nik \exp \{ik\theta + i\omega_{n}t\}
\label{e35}
\end{equation}
Once more, \fimu\ is diagonal when referred to the basis of functions at
the boundary
\begin{equation}
\left\{ \exp \{ik\theta+i\omega_{n}t\}, for\ n,k \in {\bf Z} \right\},
\label{e36}
\end{equation}
so:
\begin{equation}
\left( \fioi\fimu \right)^{n',n}_{k',k} =
\left\{ 1 - {{\mu a}\over{1+\niip}} \right\} \delta_{k',k} \delta^{n',n} .
\label{e37}
\end{equation}
Now, the asymptotic Debye expansion for Bessel functions\cite{r5,r6} at
large order and/or argument:
\begin{equation}
z {{I'_{k}(z)}\over{I_{k}(z)}} \approx \rho -
{{z^{2}}\over{2\rho^{2}}}+O(1/\rho),\ for \ \rho=\sqrt{k^{2}+z^{2}} ,
\label{e38}
\end{equation}
shows, when replaced in equation (\ref{e6}) that the p-determinant of the
operator in equation (\ref{e37}) exists only for $p\geq 3$.
So, from our main result, it turns that:
\[ det_{3}( \ui \lbui \lau \ud \lasi \lbs ) =
\prod_{n=-\infty}^{\infty}\prod_{k=-\infty}^{\infty} \]
\begin{equation}
\left\{1-{{\mu a}\over{1+\niip}}\right\}
\exp\left\{{{\mu a}\over{1+\niip}} + {1 \over 2} \left( {{\mu a}\over{1+\niip}}
\right)^{2} \right\} .
\label{e39}
\end{equation}
In order to define a quotient of
partition functions of the system (subject to the different boundary conditions)
in the regularization
scheme provided by this approach, one should study the subtractions which
take place. As pointed in section 2, this can be reduced to the consideration
of the two terms subtracted in equation (\ref{e39}) (corresponding to the
argument of the exponential in each factor of the double product). These
are two (divergent) terms, which must be readded after being renormalized
\cite{r7}. Notice that they are
linear and quadratic in $\mu$ respectively, which
suggests the introduction of two finite counterterms, given by the integral
over the boundary of densities linear and quadratic in ${\cal U}\partial_{r}
{\cal U}^{-1}$. This point will be more explicitly considered
in our last example, where a variable external field will be introduced.
\subsection{- Variable external field}
In order to analyze the renormalization scheme to which the approach developed
in this paper leads, we now treat the situation in which a more general
external $\cal{U}$ field appears. We consider ${\cal U}(r,\theta)$ such that:
\[ \ud (R,\theta) = 1 \]
\begin{equation}
\ud \partial_{r} \ui (R,\theta) = - \mu \sum_{l=-\infty}^{\infty} C_{l}
\exp \{ il\theta \} .
\label{e40}
\end{equation}
Since only boundary conditions have changed, we take the same basis for
the kernel of $L$ as in the previous example. So, the boundary values are
now given by:
\[ h'_{n,k}(\theta,t) = \sum_{l=-\infty}^{\infty}
\left\{ \left( 1 + \niip \right)
\delta_{l,0} - \mu a C_{l} \right\}
\nik \exp \{i(k+l)\theta +i\omega_{n} t \} \]
\begin{equation}
h_{n,k}(\theta,t)=\nik \exp \{ ik\theta +i\omega_{n} t \}.
\label{e41}
\end{equation}
In the present case, \fimu\ is diagonal only in $n$ when referred to the
basis in equation (\ref{e36}). After a direct algebra, one gets:
\begin{equation}
\left( \fioi\fimu \right)^{n',n}_{k',k} = \left\{ \delta_{k',k} -
{{\mu a C_{k'-k}}\over {1+\niip}} \right\} \delta^{n',n} .
\label{e42}
\end{equation}
It can be seen that the p-determinant of the operator in equation (\ref{e42})
exists only for $p\geq 3$. This means that, in order to obtain a finite
\linebreak 3-determinant, two (divergent) traces,
\begin{equation}
{{1}\over{q}} Tr \left\{ \left(1 - \fioi\fimu \right)^{q} \right\} ,
\ for\ q=1,2\
(=p-1),
\label{e43}
\end{equation}
are subtracted (see equation(\ref{e16})). To define (a difference of) free
energies, this traces must be readded after being renormalized. So, we will
write (see equation (\ref{e20})):
\[ 2\beta \left\{ [F_{A}(\mu) - F_{B}(\mu)] - [F_{A}(0) - F_{B}(0)] \right\} \]
\begin{equation}
\equiv \ln det_{3} \left( \fioi\fimu \right) + \sum_{q=1}^{2} {1\over q}
Tr \left\{ \left(1 - \fioi\fimu \right)^{q} \right\}|_{ren.}.
\label{e43p}
\end{equation}
From equation (\ref{e42}) and the development in equation (\ref{e38}),
it can be seen that, for $q=1$, this trace is proportional to $C_{0}$ and
can be regularized as follows:
\[G_{1} = Tr \left\{ 1 - \fioi\fimu \right\}|_{reg.} =
\]
\[\mu a C_{0}\sum_{k,n=-\infty}^{\infty} \left\{{1\over {1+\niip}} -
{R\over {a\rho}} + {R^{2}\over {a^{2}\rho^{2}}}\left(1 -
{{aR\lambda_{n}^{2}}\over{2\rho^{2}}}\right)\right\} +
\]
\begin{equation}
\mu R C_{0} \ F.P.|_{s=0} \sum_{k,n=-\infty}^{\infty} {1\over {\rho^{s}}}
\left\{{1\over {\rho}} - {R\over {a\rho^{2}}}\left(1 -
{{aR\lambda_{n}^{2}}\over{2\rho^{2}}}\right)\right\},
\label{et1}
\end{equation}
where $\rho = \sqrt{k^{2}+(R\lambda_{n})^{2}}$. In equation (\ref{et1})
the first series is absolutely convergent, while the second is analytically
regularized through the introduction of a factor $1/\rho^{s}$, and only
its finite part at $s=0$ is retained. It can be verified that the residue
at $s=0$ is linear in $\beta$, so that only temperature-independent
divergencies must be removed from the free energy\cite{r8}.
This clearly requires the introduction of a (finite) local counterterm
of the form:
\begin{equation}
K_{1} = \beta\alpha_{1} C_{0} =
\beta\alpha_{1} \int_{0}^{2\pi} \ud \partial_{r}
\ui(R,\theta) d\theta .
\label{e44}
\end{equation}
For $q=2$, the situation is more involved, since one can distinguish a finite
nonlocal contribution plus a divergent local term. Indeed, one can write:
\[ \left([ 1-\fioi\fimu]^{2} \right)_{k,k}^{n,n} =
\sum_{k'} {{(\mu a)^{2} |C_{k-k'}|^{2}}\over{1+\niip}} \]
\begin{equation}
\left\{ {{1}\over{1+\niipp}} - {{1}\over{1+\niip}} \right\} +
\sum_{k'} {{(\mu a)^{2} |C_{k-k'}|^{2}}\over{\left(1+\niip\right)^{2}}} .
\label{e45}
\end{equation}
The first term in the right hand side gives a finite nonlocal contribution:
\begin{equation}
G_{2} =
{{-1}\over{2}} (\mu a)^{2} \sum_{n,k,k'} {{|C_{k-k'}|^{2}}\over{1+\niip}}
\left\{ {{1}\over{1+\niipp}} - {{1}\over{1+\niip}} \right\}.
\label{e47}
\end{equation}
The trace of the second one can be regularized to give:
\[G_{3} = {{(a\mu)^{2}}\over 2} \sum_{{\cal K}=-\infty}^{\infty}
|C_{{\cal K}}|^{2}\sum_{k,n=-\infty}^{\infty}\left\{
{1\over{\left(1+\niip\right)^{2}}} - \left({R\over {a\rho}}\right)^{2}\right\}
\]
\begin{equation}
+ {{(R\mu)^{2}}\over 2} \sum_{{\cal K}=-\infty}^{\infty}
|C_{{\cal K}}|^{2} \ F.P.|_{s=0} \sum_{k,n=-\infty}^{\infty}
{1\over {\rho^{2+s}}}.
\label{eg3}
\end{equation}
Again, the residue is linear in $\beta$. Also in this case, the introduction
of a (finite) local counterterm is needed:
\begin{equation}
K_{2} = \beta\alpha_{2} \sum_{\cal K} |C_{\cal K}|^{2} = \beta
\alpha_{2} \int_{0}^{2\pi} \left(\ud \partial_{r} \ui (R,\theta) \right)^{2}
d\theta .
\label{e46}
\end{equation}
Moreover, it is obvious from equations (\ref{e23}) and (\ref{e24}) that
$F_{B}(\mu)$ is \linebreak $\mu$-independent (since energy eigenvalues are so).
Thus we finally define the difference of renormalized free energies of this
bosonic system subject to the boundary condition $A$ with and without the
presence of the external field as:
\[ 2\beta [F_{A}(\mu) - F_{A}(0)] \equiv
\]
\begin{equation}
\ln det_{3} \left( \fioi\fimu \right) + G_{1} + G_{2} + G_{3} + K_{1} + K_{2},
\label{e48}
\end{equation}
where the last two terms on the right hand side contain the
undetermined constants $\alpha_{1}$ and $\alpha_{2}$.
\section{- Conclusions}
To summarize, in Section 2 we have presented a generalization of the results
in reference \cite{r1} for the determinant of the quotient \lbi\la\laoi\lbo,
where \la\ (\lb) are elliptic differential operators
depending on a parameter $\mu$ ($L(\mu)$ restricted to functions $f$ satisfying
$Af=0$ ($Bf=0$) at the boundary). Such generalization can be applied
to situations in which Forman's map fails to have a Fredholm determinant,
but a regularized \linebreak p-determinant can be defined.
Although we have not attempted to establish the most general conditions
under which our result holds, we have proved that, when equation
(\ref{e8p}) is satisfied, \detp(\lbi\la\laoi\lbo) equals \detp(\fioi\fimu)
(where \fimu\ is Forman's map defined in reference \cite{r1}), provided
that both p-determinants exist. Notice that \fimu\ acts on functions defined
on the boundary of the manifold. In principle, it can be determined just
from boundary values of solutions of the homogeneous equation $Lf = 0$,
regardless of boundary conditions.
By making use of well-known properties of p-determinants, we have argued
that the subtractions to which our regularization scheme gives rise can
also be determined from boundary values, which is an important point when
applying our results in the context of Quantum Field Theory. Indeed, these
are the divergent pieces which require to be renormalized.
In Section 3, we have studied some models of interest to Physics, which clearly
show the need for this generalization as well as the great simplification due
to the reduction to the boundary. In particular, in the third one, we have
discussed the correction to the free energy of confined free bosons as a
function of a variable external field coupled through boundary conditions. The
study of the discarded traces, once regularized, allowed us not only to
determine the required counterterms, but also to recover a nonlocal finite
contribution as well as local finite temperature-dependent pieces thrown away
by the definition of the p-determinant.
Notice that, even though this approach requires the introduction of two
different boundary conditions, in the examples treated here, a convenient
selection of one of the boundary projection operators (such that the free
energy $F_{B}(\mu)$ it determines is $\mu$-independent) allowed us to obtain
the difference of free energies with and without external field for a unique
boundary condition. Another [Afreedom this method allows is the possibility of
choosing also a particular value of the parameter $\mu$. This idea has been
applied to evaluate the chiral correction to the free energy of a
four-dimensional chiral bag, and will be presented somewhere else\cite{r9}.
Finally, we wish to mention that the relationship between this regularization
with $p>1$ and the $\zeta$-function one is rather more involved than in the p=1
case, and is at present under study.
\newpage
\begin{thebibliography}{9}
\bibitem{r1} R.\ Forman, { Invent.\ Math.\ {\bf 88}, 447 (1987).}
\bibitem{r2} H.\ Falomir and E.\ M.\ Santangelo, { Phys.\ Rev.\ {\bf D42},
590 (1990)}; {Phys.\ Rev.\ {\bf D}, in press.}
\bibitem{r3} B.\ Simon, "Trace ideals and their applications", { London Math.\
Soc.\ lecture note series 35, Cambridge University Press (1979).}
\bibitem{r4} I.\ C.\ Gohberg and M.\ G.\ Krein, "Introduction to the theory
of linear nonselfadjoint operators", Vol.\ 18 (Translation of Mathematical
Monographs), Providence, Rhode Island, American Mathematical Society (1969).
\bibitem{r5} M.\ Abramowitz and I.\ A.\ Stegun (eds.), "Handbook of Mathematical
Functions", { Dover Publications, New York (1972).}
\bibitem{r6} I.\ Zahed, A.\ Wirzba and Ulf-G.\ Meissner, { Ann.\ of Phys.\
{\bf 165}, 406 (1985).}
\bibitem{r7} E.\ Seiler, { Phys.\ Rev.\
{\bf D22}, 2412 (1980).}
\bibitem{r8} N.\ P.\ Landsman and Ch.\ G.\ van Weert, { Phys.\ Rep.\
{\bf 145}, 141 (1987).}
\bibitem{r9} M.\ De Francia, H.\ Falomir and E.\ M.\ Santangelo,
submitted to Phys. Rev. D.
\end{thebibliography}
\end{document}