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\begin{document}
\title{Scattering theory of infrared divergent\\ Pauli-Fierz Hamiltonians}
%
\author{J. Derezi\'{n}ski
\\ Department of Mathematical Methods in Physics\\
Warsaw University\\ Ho\.{z}a 74, 00-682 Warszawa Poland,\\C. G\'erard \\ D\'epartement de Math\'ematiques \\Universit\'e de
Paris Sud\\ 91405 Orsay Cedex France.}\date{July 2003}
\maketitle
\begin{abstract}
We consider in this paper the scattering theory of infrared divergent
massless Pauli-Fierz Hamiltonians. We show that the CCR
representations obtained from the asymptotic field contain so-called
{\em coherent sectors} describing an infinite number of asymptotically
free bosons. We formulate some conjectures leading to mathematically
well defined notion of {\em inclusive and non-inclusive scattering
cross-sections} for Pauli-Fierz Hamiltonians. Finally we give a general
description of the scattering theory of QFT models in the presence of
coherent sectors for the asymptotic CCR representations.
\end{abstract}
\noindent{\small {\bf Acknowledgments.}
Both authors were partly supported by the NATO
Grant PST.CLG.979341 and by the
Postdoctoral Training Program HPRN-CT-2002-0277.
The research of J. D. was also partly supported
by the Komitet Bada\'{n} Naukowych (the grants SPUB127 and 2 P03A 027 25).
A part of this work was done during his visit
to Aarhus University supported by MaPhySto funded
by the Danish National Research Foundation.
}
\section{Introduction}
\init
The {\em infrared problem} in QED consists in the divergence at small
momenta of some
integrals used to compute scattering amplitudes.
Mathematical interpretation of this problem is made
difficult by the fact that QED does not have a satisfactory
definition in terms of a unitary dynamics in a Hilbert space.
The lack of a satisfactory definition of QED
seems, however, to be mostly due to the ultraviolet
problem.
Infrared divergences persist even for various simplified models of
QED that have ultraviolet cutoffs or treat a part of the system in a
classical way---models that can be rigorously defined.
Infrared divergences arise mostly if we try to compute
scattering amplitudes. They seem to be a symptom of a pathological scattering
theory.
QED is a theory of charged particles interacting with
photons. Correspondingly, one has two distinct aspects of the
infrared problem in QED: the first aspect
involves the dynamics of charged particles and the second involves photons.
\subsection{ Infrared problem for charged particles}
Let us shortly discuss
the first aspect. Scattering of charged particles is made difficult by the
long-range nature of their interaction. To partly understand this
phenomenon, let us fix the Coulomb gauge in QED, drop photons and use
the non-relativistic approximation. Then QED becomes a theory of charged
particles whose dynamics is described by the many body Schr\"odinger
Hamiltonian with Coulomb interactions. As is well known, the usual
scattering theory breaks down for such systems.
Naive rules for computing scattering amplitudes in terms of Feynman
diagrams presuppose that we want to construct the usual wave and
scattering operators, which do not exist because Coulomb potentials
are long-range. Therefore, we get meaningless divergent expressions.
It is well understood how to cure this problem, at least in the
context of many body Schr\"odinger Hamiltonians. Two approaches are
possible:
\ben\item One can compute only scattering cross-sections, staying away
from ill-defined wave and scattering operators. The standard way
is to approximate Coulomb interaction by the Yukawa interaction of
mass $m>0$, which is short range, compute the cross-sections and take
the $m\to0$ limit. This is the approach found in most textbooks on
quantum mechanics.
\item One can introduce {\em modified wave and scattering operators}. From
the conceptual point of view it is a more satisfactory approach---it
gives deeper insight into the problem. The mathematics of this
approach is very interesting and nowadays well understood (see
e.g. \cite{DG0}). On the other hand, it is more
complicated computationally than the first approach
and uses non-canonical objects: the modified wave and scattering
operators depend on the choice of the so-called modifier.
\een
Apart from the remarks above, in our paper we will not touch this aspect of
the infrared problem.
\subsection{Infrared problem for photons}
Let us now discuss the photonic aspect of the infrared problem. In our
discussion we
will consider both the perturbative QED and various simplified
models such as Pauli-Fierz Hamiltonians.
If
one tries to compute scattering cross-sections involving
states with a finite number of asymptotic photons, one often
obtains infrared divergent integrals.
After an appropriate
renormalization, one obtains scattering cross-sections equal to
zero.
This is usually interpreted by saying that ``the vacuum escapes from
the physical Hilbert space'' and that ``all states contain an infinite number
of soft photons''.
In the literature one can find 4 approaches to cure this
problem in QED-like theories that make possible
computing physically meaningful cross-sections.
\ben\item
One can restrict oneself to
the so-called {\em inclusive cross-sections}, which
take into account all possible ``soft photon states'' below a certain
energy $\epsilon>0$. The philosophy behind this prescription is: do
not attempt to compute or even ask about the existence of the wave and
scattering operators---try to compute
scattering cross-sections relevant for realistic experiments.
This point of view is most common in standard textbooks \cite{JR} and can
be traced back to \cite{BN} (see also \cite{YFS}).
\item Naive rules for computing scattering amplitudes in terms of
Feynman diagrams presuppose that the asymptotic fields
form a Fock CCR representation. This assumption can be wrong because of the
infrared problem. To eliminate this difficulty, one can treat
seriously non-Fock representations. One class of non-Fock
representations is especially easy to handle---the so-called {\em coherent
representations}. One can define wave and scattering
operators between coherent sectors, and also asymptotic
Hamiltonians. Scattering theory is somewhat less intuitive than in the
case of Fock representations, but it is naturally defined and not
much more difficult.
This approach can be traced back to Kibble \cite{Ki}.
We regard it as the
most satisfactory approach to the infrared problem.
It provides an appropriate framework for the infrared problem in the
case of exactly solvable
van Hove Hamiltonians \cite{De}. In our
paper we will argue that this approach works also well
in the case of Pauli-Fierz
Hamiltonians, although one can not rule out the appearance of other
types of CCR representations besides the coherent ones.
In order for this approach to be meaningful, one needs to use a
certain version of the so-called
{\em LSZ approach}, that means, one needs to construct
the {\em asymptotic fields}. This requires some, usually mild,
assumptions on the interaction of the ``short range'' type.
This is the main weakness of this approach.
\item One can keep the formal expression for the Hamiltonian and
change the CCR representation. This amounts to a change of the
underlying Hilbert space and of the Hamiltonian. The new Hamiltonian
is sometimes called the renormalized Hamiltonian. The main requirement
for the renormalized Hamiltonian is to have a ground state, which
implies that
the representation of its asymptotic fields is of Fock type.
Shifting the asymptotic CCR representations can always be done in the case of
exactly solvable van Hove Hamiltonians. In the case of Pauli-Fierz
Hamiltonians it seems possible only under some special assumptions on
the interaction, such as Assumption \ref{main4}, (the
possibility to split the interaction into a scalar part and a regular part).
One can criticize this approach in two separate points.
First of all, as we mentioned above, we need special assumptions to
make this approach work. One can argue that Approach (2) is more general
and does not need these assumptions.
Secondly, in general there is a large degree of arbitrariness in how to
shift the Hamiltonian. Therefore, the renormalized Hamiltonian is not
a canonical object.
One can try to give a justification of this approach by using
$C^*$-algebras---Approach (4). The passage from the
initial to a renormalized Hamiltonian would correspond to a change of
a representation of the given $C^*$-algebraic system.
If applicable, this approach is very useful. In
recent literature it was applied in \cite{A} and \cite{HHS}. It will
be also an important tool in our paper.
\item
Sometimes one can describe a quantum system in terms of a dynamics
on a $C^*$-algebra \cite{FNV,BR}. This algebra may have many inequivalent
representations. In some of them the dynamics may be generated by
a Hamiltonian with a ground state, so that the infrared problem disappears.
This approach can be used to justify Approach (3). One can say that
the initial Hilbert space is just one of many representations of the
$C^*$-algebra and one needs to go to a different representation, where
the representation of asymptotic fields is of Fock type.
It seems that this approach is inadequate for Pauli-Fierz systems
unless one makes some very special assumptions on the interaction. In
general it is difficult (probably impossible) to find a physically motivated
$C^*$-algebra which is preserved by the dynamics.
\een
In our paper we will discuss in detail approach (2) to the
infrared problem in the context of Pauli-Fierz Hamiltonians.
Approach (3) will play an important role, but it will be
treated as a tool in the study of Approach (2). We will
also discuss approach (1).
\subsection{Scattering theory for Pauli-Fierz Hamiltonians}
There exist a number of simplified models that can be used
to test some of the photonic aspects of QED.
Probably the simplest are quadratic bosonic
Hamiltonians with a linear perturbation. In \cite {Sch} such Hamiltonians are called {\em van Hove
Hamiltonians}, and we will use this name. They are exactly solvable
and one can study their scattering theory in full detail \cite{De}.
A typical van Hove Hamiltonian
can be written in the form:
\beq
\int\left(a^*(k)+\frac{z(k)}{\omega(k)}\right)
\omega(k)\left(a(k)+\frac{\bar z(k)}{\omega(k)}\right) \d
k,\label{kaka0}\eeq
where $z(k)$ is some given function and $\omega(k)$ is the dispersion
relation, e.g. $\omega(k)=|k|$.
Note that if we consider QED with prescribed classical charges, then
we obtain a van Hove Hamiltonian
In our paper we consider the so-called
{\em abstract Pauli-Fierz Hamiltonians}.
They can also be used
to understand interaction of photons with matter, but are more
difficult and rich than the van Hove Hamiltonians.
They are not exactly solvable and their mathematical understanding
is far from complete. They are a caricature of
QED with charged particles confined in an infinite well.
Consider the Hilbert space
$\cK\otimes\Gamma_\s(L^2(\rr^d))$, where the Hilbert space $\cK$ describes the confined
charged particles and $\Gamma_\s(L^2(\rr^d))$ is a bosonic Fock
space. Following the terminology of \cite{DG1,DJ,Ge1}, an operator of the form
\[\begin{array}{rl}
H&:=K\otimes\one+\one\otimes
\int \omega(k)a^*(k)
a(k) \d k\\[3mm]&
+\int v(k)\otimes a^{*}(k)\d k+\int v^{*}(k)\otimes
a(k)\d k\end{array}\]
will be called a Pauli-Fierz Hamiltonian.
For
simplicity, in our paper
charged particles are described by an
abstract Hamiltonian $K$ and their confinement is expressed by the
condition that $K$ has a compact resolvent.
Let us sketch the main ideas of scattering theory for Pauli-Fierz
Hamiltonians. We follow the formalism of \cite{DG1,DG2}, which can
be traced back to much earlier work, such as \cite{HK}. In the
introduction
we will not aim
at the mathematical precision, for instance we will freely use the
operator valued measures $a^{(*)}(k)$ and we will not precise the type
of the limits involved in our statements. All the rigorous details
will be provided in next sections.
Under appropriate assumptions one can show the existence of the
following limits:
\[\begin{array}{l}
a^{*\pm}(k):=\lim\limits_{t\to\infty}\e^{\i tH}\e^{-\i
t\omega(k)}a^*(k)\e^{-\i tH},\ \ \
a^{\pm}(k):=\lim\limits_{t\to\infty}\e^{\i tH}\e^{\i
t\omega(k)}a(k)\e^{-\i tH}.\end{array}\]We will call $a^{*\pm}(k)$ and
$a^\pm(k)$ the asymptotic creation/annihilation
operators. (If we want to be more precise, then we will say
outgoing/incoming creation/annihilation
operators).
Note that they form covariant CCR representations:
\[\begin{array}{l}[a^\pm(k_1),a^\pm(k_2)]=0,\ \ \
[a^{\pm*}(k_1),a^{\pm*}(k_2)]=0,\ \ \
[a^\pm(k_1),a^{\pm*}(k_2)]=\delta(k_1-k_2),\\[3mm]
\e^{\i tH} a^{*\pm}(k)\e^{-\i tH}=\e^{\i t\omega(k)}a^{\pm*}(k),\ \ \
\e^{\i tH} a^{\pm}(k)\e^{-\i tH}=\e^{-\i
t\omega(k)}a^{\pm*}(k).\end{array}
\]
We define $\cK_0^\pm$ to be the space of $\Psi\in\cH$ satisfying
\[a^\pm(k)\Psi=0, \ \ \ k\in \rr^{d}.\]
Elements of $\cK_0^\pm$ will be called asymptotic
vacua. The Fock sectors of the
asymptotic space are defined as
\[\cH_0^\pm:=\cK_0^\pm\otimes\Gamma_\s(L^2(\rr^d)).\]
The wave operators in the Fock sector are defined as linear
maps $\Omega_0^\pm:\cH_0^\pm\to\cH$ satisfying
\[\Omega_0^\pm\ \Psi\otimes a^*(k_1)\cdots a^*(k_n)\Omega:=
a^{\pm*}(k_1)\cdots a^{\pm*}(k_n)\Psi,\ \ \ \Psi\in\cK_0^\pm\]
($\Omega$ denotes the vacuum in the Fock space.
The same letter decorated by the superscript
$+$ or $-$ denotes the appropriate wave operator).
We also introduce the Hamiltonian of the asymptotic vacua
\[K_0^\pm:=H\Big|_{\cK_0^\pm},\]
and
the full asymptotic Hamiltonian:
\[H_0^\pm:=K_0^\pm\otimes\one+\one\otimes\int\omega(k)a^*(k)a(k)\d
k.\]
Now the following is true:
\ben
\item
$\Omega_0^\pm$ are isometric;
\item $\Omega_0^\pm\ \one\otimes a(k)=a^\pm(k)\Omega_0^\pm$,
\newline$\Omega_0^\pm\ \one\otimes a^*(k)=a^{\pm*}(k)\Omega_0^\pm$;
\item
$\cK_0^\pm$ contains all eigenvectors of $H$;
\item $\Omega_0^\pm H_0^\pm=H\Omega_0^\pm$.
\een
One can formulate two desirable properties,
called sometimes jointly
the asymptotic completeness
\cite{DG1,DG2}:
\begin{itemize}
\item {\bf The operators $\Omega_0^\pm$ are unitary}, in other words,
{\bf the asymptotic CCR representations are Fock}.
\item
{\bf All asymptotic vacua are linear combinations of bound states of $H$}.
\end{itemize}
For massive
bosons, (e.g. if
$\omega(k)=\sqrt{k^2+m^2}$ with $m>0$), under quite weak assumptions
one can show that both above properties
are true \cite{HK,DG1,DG2}.
If $m=0$, little is known about these two properties except for
the case of van Hove Hamiltonians \cite{De}.
Typically, the breakdown of the above properties is closely
related to the infrared problem.
Note that
the conventional scattering theory starts from a given pair of
operators: the full Hamiltonian $H$ and the free Hamiltonian $H_0$ and
then proceeds to construct wave operators by considering the limit (in
appropriate sense) of $\e^{\i tH}\e^{-\i tH_0}$ as $t$ goes to
$\pm\infty$. The formalism of scattering theory that we described above
differs substantially from the conventional one.
Instead of the ``free Hamiltonian'' we have the
asymptotic Hamiltonians $H_0^\pm$. The Hamiltonians $H_0^\pm$ are
simpler than the
full Hamiltonian $H$: they have the form of a ``free Pauli-Fierz Hamiltonian''.
Nevertheless, they are not given a priori---they are constructed from $H$.
If $\Omega_0^\pm$ is not unitary, then the asymptotic fields have some
non-Fock sectors.
It may even happen that there are no nonzero asymptotic vacua, so that
there are no asymptotic Fock sectors at all. This motivates us to give
a description of scattering theory in the presence of non-Fock
sectors.
Among non-Fock sectors the most manageable ones are the
so-called coherent sectors. Our paper is to a large extent devoted to
the description of scattering theory in their presence.
Let $\rr^d\ni k\mapsto g(k)$ be a complex function. We define
$\cK_g^\pm$ to be the space of $\Psi\in\cH$ satisfying
\[a^{\pm}(k)\Psi=\sqrt{2}g(k)\Psi,\ \ \ k\in\rr^d.\]
The elements of $\cK_g^\pm$ will be called asymptotic
$g$-coherent vectors.
The asymptotic $g$-coherent space is defined as
\[\cH_g^\pm:=\cK_g^\pm\otimes\Gamma_\s(L^2(\rr^d)).\]
The $g$-coherent wave operator is the linear map
$\Omega_g^\pm:\cH_g\to\cH$ defined as
\[\Omega_g^\pm\ \Psi\otimes a^*(k_1)\cdots a^*(k_n)\Omega:=
(a^{\pm*}(k_1)-\sqrt 2\bar g(k_1))\cdots (a^{\pm*}(k_n)-\sqrt 2\bar g(k_n))
\Psi,\ \ \ \Psi\in\cK_0^\pm.\]
We define the asymptotic Hamiltonian in the $g$-coherent sector
as $H_g^\pm:=\Omega_g^{\pm*}H\Omega_g^\pm$.
The following can be easily shown:
\ben\item
$\Omega_g^\pm$ are isometric;
\item $\Omega_g^\pm\ \one\otimes a(k)=(a^\pm(k)- \sqrt 2g(k))\Omega_g^\pm$, \\
$\Omega_g^\pm\ \one\otimes a^*(k)=(a^{\pm*}(k)-\sqrt 2g(k))\Omega_g^\pm$;
\item $\Omega_g^\pm H_g^\pm=H\Omega_g^\pm$.
\item
There exists a decomposition
\beq H_g^\pm=K_g^\pm\otimes\one
+\one\otimes\int\left(a^*(k)+\sqrt 2g(k)\right)
\omega(k)\left(a(k)+\sqrt 2g(k)\right) \d k\label{qsz}\eeq
\item If $g_1$ and $g_2$ differ by a square integrable function, then
the ranges of $\Omega_{g_1}^\pm$ and $\Omega_{g_2}^\pm$ coincide.
\een
Note that the second term on the right of (\ref{qsz}) is a van Hove
Hamiltonian. If $g$ is not square integrable then
the asymptotic CCR representations on the range of
$\Omega_g^\pm$ are non-Fock and the
asymptotic Hamiltonians do not have a ground state---nevertheless,
we have well defined wave operators that can be used to compute
scattering cross-sections.
We are not aware of a full description of the above formalism in the
literature, although some of its elements may belong to the so-called
folklore. In particular, the fact that the asymptotic Hamiltonians
have the form given in the equation (\ref{qsz}) is quite interesting and
not obvious.
\subsection{Renormalized Hamiltonian and dressing operator}
%The result about (\ref{qsz}) is purely algebraic.
The main new `analytical'
result of the paper is the proof of the existence of
a nontrivial non-Fock coherent sector for asymptotic fields
in a certain nontrivial class of Pauli-Fierz
Hamiltonians. The
most important additional assumption that we need to get this
result is the possibility to split the interaction into two parts:
an infrared divergent scalar part and an infrared convergent matrix
part.
Using this assumption we can define the {\em renormalized
Hamiltonian } $H_\ren$.
On the formal level the so-called renormalized Hamiltonian
is unitarily equivalent to the initial Hamiltonian $H$:
\[H_\ren=\one{\otimes}W(-\i g)\, H_\ren\,\one{\otimes}W(\i g),\]
where $W(\i g)$ is formally a Weyl operator. Strictly speaking,
however, $W(\i g)$ is not well defined, since $g$ is not square
integrable. Still, $H_\ren$ turns out to be a correctly defined
Pauli-Fierz operator.
Moreover, with an appropriate choice of $g$, $H_\ren$ has a mild
infrared singularity, so that one can apply the results of \cite{Ge1},
which imply that $H_\ren$ possesses a ground state.
Under appropriate assumptions, one can show that for both $H$ and
$H_\ren$ one can define asymptotic fields. Besides, one can define the
so-called {\em dressing operators }$U^\pm$. The dressing operators
are some kind of unitary intertwiners between the objects related to
$H_\ren$ and $H$. They do not intertwine, however, in the usual
meaning of this word: it is not true that $HU^\pm=U^\pm H_\ren$. The
action of $U^\pm$ gives some sort of a translation in phase space by
$g$. In particular, $U^\pm$ map coherent sectors of the asymptotic
fields of $H_\ren$ onto the coherent asymptotic sectors of
$H$ shifting them by $g$. In particular, they map the Fock sector of
the asymptotic
CCR representation for $H_\ren$ onto the $g$-coherent sector of the
CCR representation for $H$,
which is non-Fock. But we know that $H_\ren$ has a ground state. Hence
it has nontrivial Fock asymptotic sectors. Therefore, $H$ has nontrivial
$g$-coherent asymptotic sectors.
Let us note that the use of $H_\ren$ is well known from the
literature, eg. \cite{A} and \cite{HHS} for recent applications.
According to
Approach (3) described above one could
discard $H$ in favor of $H_\ren$ and treat $H_\ren$ as the physical
Hamiltonian. After this replacement, the asymptotic fields
have Fock sectors,
where the infrared problem is avoided.
We, however, prefer the (more canonical and general) Approach (2), which treats
$H$ as the basic physical
Hamiltonian and $H_\ren$ as a technical tool used to prove certain
properties of scattering for $H$.
\subsection{Comparison with literature}
It is difficult to compare our results with the literature, since a
large part of it
is non-rigorous and devoted to different models.
Our class of models has one simplifying feature---they are not
translation invariant and the small system is ``confined''. In the
literature one can find attempts to understand the infrared problem for
translation invariant Hamiltonians.
Let us mention an interesting attempt to define
wave and scattering operators for the full QED taking into account both the
long-range nature of the interaction between charged particles and the
emergence of non-Fock representations of photons, due to Faddeev and
Kulish \cite{KF}.
In \cite{FMS} infrared features of full
QED were studied on basis of appropriately chosen axioms.
The infrared problem for the
so-called {\em Nelson model }in the one-electron sector
was studied by Fr\"ohlich in \cite{F}, and more recently by Pizzo
\cite{Pi}.
In these papers one can find an
operator essentially equivalent to our dressing operators
$U^\pm$. Fr\"ohlich and Pizzo consider translation invariant models,
which introduces additional complications in their analysis.
A complete construction of dressed one
electron states is not achieved in \cite{F}, (some parts of the
construction relied on physically reasonable but conjectural
assumptions). A complete construction was recently given by Pizzo \cite{Pi}.
Note that the fact that we restrict ourselves to
a confined system without translation invariance
enables us to give a more transparent and thorough analysis of the
scattering theory in presence of the infrared divergences.
\subsection{Organization of the paper}
Our paper can be divided into two parts. The first consists of
Section \ref{main}, where we describe the main results of our paper.
We introduce a certain class of abstract Pauli-Fierz Hamiltonians. We recall
and partly extend basic results on the existence of asymptotic fields
\cite{DG1}, \cite{Ge2} and on the existence and non-existence of ground states
\cite{Ge1}. The asymptotic fields may have non-Fock sectors. We
concentrate our attention on the so-called coherent sectors. We show
how to define wave operators, scattering operators and asymptotic
Hamiltonians for coherent sectors. We demonstrate that they are not
much more difficult than the usual Fock sectors, and thus we explain
how one can overcome the conceptual problems caused by the infrared
problem.
We show the existence of non-Fock sectors for a class of Pauli-Fierz
Hamiltonians, that includes a certain class of Nelson Hamiltonians.
We end Section \ref{main} with a discussion of inclusive cross-sections
in our model. Let us stress that, in principle, by using our formalism
one can describe predictions for experiments that measure ``soft
components of the system'' and
one does not need to restrict oneself to inclusive
cross-sections. One can argue, however, that in realistic experiments
the soft background should be irrelevant and measurable quantities
should depend only on the ``hard components''.
We discuss how to define such inclusive cross-sections and state some
physically motivated conjectures about them.
The remaining part of our paper is somewhat more
mathematical. It contains a systematic exposition of various elements
of mathematical formalism used in Section \ref{main}.
Some of them are
presented in a more general context and proved in bigger generality.
Let us stress that Sections \ref{s2}, \ref{s3}, \ref{s4}, \ref{s4a}, \ref{s5}
and the Appendix can be read independently of Section \ref{main}.
In Section \ref{s2} we study general CCR representations. A
particular attention is devoted to the so-called
{\em coherent representations}. These representations
are obtained by
translating the Fock representation by an antilinear functional. If
the functional is not continuous, then this representation is not
unitarily equivalent to the Fock representation.
In Section \ref{s3} we study the so-called {\em covariant CCR
representations}.
They are CCR representations equipped with a dynamics, which
is implemented both on the level of the full space and of the 1-particle
space.
We show how to describe covariant representations in a coherent
sector. It turns out that in every coherent sector the dynamics has a
certain natural decomposition, one part of which is given by a
quadratic Hamiltonian perturbed by a linear one (a van Hove
Hamiltonian). In our opinion this is quite an interesting and hitherto
unknown fact.
Covariant CCR representations arise naturally in scattering theory
of certain quantum systems. Based on the ideas of the LSZ formalism,
such representations were constructed and studied e.g. in \cite{HK}, and
more recently in \cite{DG1}, \cite{DG2} and \cite{Ge2}.
In Section \ref{s4} we study such
representations in an abstract context. One of them describes the
observables in the distant past---the {\em incoming representation}
$W^-(\cdot)$, the other describes the observables in the distant
future---the {\em outgoing representation} $W^+(\cdot)$. Collectively, they
are called {\em asymptotic representations}. We show in particular that
eigenvectors of the Hamiltonian are always vacua of both asymptotic
representations and thus give rise to nontrivial Fock sectors.
Note that the material of Sections \ref{s2}, \ref{s3} and \ref{s4} is
rather basic and mostly belongs to the folklore (although our
presentation has some points which are new).
Section \ref{s4a} is more special: here
we introduce the so-called {\em dressing operator}
between two CCR representations.
In Section \ref{s5} we introduce a relatively general class of
Pauli-Fierz Hamiltonians. For these Hamiltonians, under some
relatively mild assumptions on the interaction, asymptotic CCR
representations exist and one can apply the formalism developed
in the previous sections. One can also introduce the renormalized
Hamiltonian and the dressing operators.
\section{Overview of main results and some open problems}
\label{main}
\init
In this section we describe most of main results of our paper in a
somewhat simplified form. We also discuss some aspects of
the physical content of our mathematical constructions.
We formulate some open mathematical
problems inspired by physical considerations.
\subsection{Pauli-Fierz Hamiltonians}
Suppose that $\cK$ is
a separable Hilbert space representing the
degrees of freedom of the atomic system.
Let $K$ be a positive operator on $\cK$---the Hamiltonian of
the atomic system. We will sometimes use
\bea
\[
(K+\i)^{-1} \hbox{ is compact on }\cK.
\]
\label{main1}\eea
The physical interpretation of this assumption
is that the small system is confined.
Let $\fh=L^{2}(\rr^{d}, \d k)$ be the $1-$particle
Hilbert space in the momentum representation
and let $\Gamma_\s(\fh)$ be the bosonic Fock space over $\fh$,
representing the field degrees of freedom.
$\Omega$ will stand for the vacuum in $\Gamma_\s(\fh)$.
We will denote by $k$ the
momentum operator of multiplication by
$k$ on $L^{2}(\rr^{d}, \d k)$.
Let \[\omega:=|k|\]
be the dispersion
relation.
For $f\in\fh$ the operators of creation and annihilation of $f$
are denoted by
\[
\int f(k)a^*(k)\d k,\ \ \ \ \int \bar{f}(k)a(k)\d k.\]
The Hamiltonian describing the field is equal
to \[\d\Gamma(\omega)=\int\omega(k)a^*(k)a(k)\d k.\]
(See e.g. \cite[vol. II]{BR} or \cite{DG1,DG2} for basic concepts
related to the second quantization).
\bea
The interaction between the atom and the boson
field is described with a coupling function $v$
\[
\rr^{d}\ni k\mapsto v(k),
\]
such that for a.e. $k\in \rr^{d}$, $v(k)$ is a bounded operator
from $\Dom(K^{\12})$ into $\cK$.
We will assume:
\[
\begin{array}{l}\hbox{ for a.e. }\ \ k\in \rr^{d},\ \ \:
v(k)(K+1)^{-\12}\in B(\cK),\\[3mm]
\forall\: \Psi_{1}, \Psi_{2}\in \cK,\ \ \:k\mapsto (\Psi_{2},
v(k)(K+1)^{-\12}\Psi_{1})
\hbox{ is measurable,}\\[3mm]
\lim\limits_{R\to\infty}\int
(1+\omega(k)^{-1})\|v(k)(K+R)^{-\12}\|^{2}
\d k<1/2.
\end{array}
\]\label{main2}\eea
Note that the functions
$k\mapsto \|v(k)(K+R)^{-\12}\|$
is measurable (see for example \cite[Appendix]{Ge2}), and hence the last condition in Assumption \ref{main2} has a
meaning.
We set
\[ H_0=K{\otimes}\one+\one{\otimes}\int\omega(k)a^*(k)a(k)\ k,\]
\[
H=H_0+\int v(k)\otimes a^{*}(k)\d k+\int v^{*}(k)\otimes
a(k)\d k.
\]
$H_0$ is called the {\em free Pauli-Fierz Hamiltonian} and $H$ {\em the full
Pauli-Fierz Hamiltonian}.
One can easily show that
\bet Under Assumptions \ref{main1} and \ref{main2},
the operator $H$ is self-adjoint and bounded from
below with the form domain $\Dom(H_0^{1/2})$.
\eet
\subsection{The confined massless Nelson model}
\def\rx{{\rm x}}
In this subsection we describe one of the main examples of Pauli-Fierz
Hamiltonians.
It is a model describing a confined atom
interacting with a field of scalar bosons.
A similar model (without the ultraviolet cut-off) was studied in a
well known paper by Nelson \cite{Ne}. Hence, in a part of the
mathematical literature it is called
the {\em Nelson model} (see \cite{A},
\cite{Ar}, \cite{LMS}).
To be more precise, the model that we will consider can be called
the {\em confined massless ultraviolet cut-off Nelson model}.
We will prove that a large class of such models
satisfies all the assumptions of this section.
Thus Lemma \ref{nelson} means that all the results presented in
in Sections \ref{main} and \ref{s5} apply to this class.
In particular, their asymptotic CCR representations contain a
non-Fock coherent sector.
The atom is described with the Hilbert space
\[
\cK:=L^{2}(\rr^{3P}, \d \rx),
\]
where $\rx= (\rx_{1}, \dots, \rx_{P})$, $\rx_{i}$ is the position of
particle $i$, and the Hamiltonian:
\[
K:= \sum_{i=1}^{P}\frac{-1}{2m_{i}}\Delta_{i}+
\sum_{i0,\: \alpha>0.
\end{array}
\]
It follows from {\it (H0)} that $K$ is symmetric and bounded below on
$\coinf(\rr^{3P})$. We still denote by $K$ its Friedrichs
extension. Moreover we have $\Dom((K+b)^{\12})\subset
H^{1}(\rr^{3P})\cap \Dom(|\rx|^{\alpha})$, which implies that
\beq
|\rx|^{\alpha}(K+b)^{-\12}\hbox{ is bounded}.
\label{nelson.e0}
\eeq
Note also that {\it (H0)} implies that $K$ has compact resolvent on
$L^{2}(\rr^{3P})$.
The one-particle space for bosons is
\[
\ch:=L^{2}(\rr^{3}, \d k),
\]
where the observable $k$ is the boson momentum. and the one-particle
energy is $\omega(k)=|k|$.
The interaction is given by the operator $\rr^3\ni k\mapsto v(k)\in \cB(\cK)$, where $v(k)$ is a multiplication operator on $L^{2}(\rr^{3P},
\d \rx)$
equal to
\[
v(k, \rx)=
\frac{1}{\sqrt{2}}\sum_{j=1}^{P}\frac{\chi(|k|)}{|k|^{\12}}\e^{-\i
k\cdot
\rx_{j}}
\]
where $\chi\in \coinf(\rr)$ is a real, even function such that
$\chi\equiv 1$ near $0$. The function $\chi$ plays the role of an
ultraviolet cutoff.
\begin{lemma}\label{nelson}
If hypothesis {\it (H0)} holds for $\alpha>1$, the confined
Nelson model satisfies assumptions \ref{main1}, \ref{main2} and
\ref{qea4}, \ref{main4}, \ref{main5}, \ref{main6} below, where in
Assumptions \ref{main4} and \ref{main6} we set
\[
z(k)=\frac{P}{\sqrt{2}}\frac{\chi(|k|)}{|k|^{\12}},\ \ \ \:v_\ren(k, \rx)=
\frac{1}{\sqrt{2}}\sum_{j=1}^{P}\frac{\chi(|k|)}{|k|^{\12}}(\e^{-\i
k\cdot\rx_{j}}-1).
\]
\end{lemma}
\proof
We already know that Assumption \ref{main1} is true.
We have $|v(k,x)|\leq C|k|^{-1/2}$. Therefore,
\[\|v(k)\|\in L^2(\rr^3,(1+|k|^{-1})\d k),\]
and hence Assumptions \ref{main2} and \ref{main5} are satisfied.
We will now show that Assumption \ref{qea4} holds with $\fg:=
C_0^\infty(\rr^3\backslash\{0\})$. Let $h\in \fg$. Define
\[\begin{array}{rl}
m_{j,t}(x)&:=
\int \bar h(k)\e^{\i t|k|} v(k,\rx_j)\d k+{\rm cc}\\[3mm]
&=\int\e^{\i (t|k|-\rx_j \cdot k)}\frac{\bar
h(k)\chi(k)}{|k|^{1/2}}\d k+{\rm cc}.\end{array}\]
(The symbol cc denotes the complex conjugate). We can write
\beq\begin{array}{rl}
m_{j,t}(x)(1+K)^{-1/2}&=m_{j,t}(x)1_{[0,\frac{t}{2}]}(|x|)(1+K)^{-1/2}\\[3mm]
&+m_{j,t}(x)1_{]\frac{t}{2},\infty[}(|x|)\langle \rx\rangle^{-\alpha}
\langle \rx\rangle^\alpha(1+K)^{-1/2}.
\end{array}\label{lala}\eeq
Since by (\ref{nelson.e0}) $|\rx|^{\alpha}(K+1)^{-\12}$ is bounded,
the second term is $O(t^{-\alpha})$, hence integrable.
To deal with the first term,
note that the function $\overline{h}(k)\frac{\chi(|k|)}{|k|^{\12}}$ is
in $\coinf(\rr^{3}\backslash\{0\})$. Because of the cutoff function,
the phase $t|k|-k\cdot\rx_{j}$ is smooth without stationary points on
$|\rx|1$, Assumption \ref{qea4} is satisfied.
Consider now Assumption \ref{main4}. We note that
\beq\label{stup}
|\e^{-\i k\cdot\rx_{j}}-1|\leq|k|\:|\rx_j|.
\eeq
Hence
\[\|v_\ren(k)\langle \rx\rangle^{-1}\|\leq C|k|^{1/2},\]
which implies $\|v_\ren(k)(1+K)^{-1/2}\|\in L^2(\rr^3,|k|^{-2}\d k)$. This proves
Assumption \ref{main4}.
Finally, we prove Assumption \ref{main6}.
We set
\[\begin{array}{rl}
m_{j,t}(\rx_j)&:=
\int g(k)\e^{\i t|k|} v_\ren(k,\rx_j)\d k+{\rm cc}\\[3mm]
&=P\sqrt 2\int \frac{\chi(|k|)^2}{k^2}\bigl(\cos(t|k|-\rx_j \cdot k)-\cos
t|k|\bigr)
\d k.
\end{array}\]
We go to spherical coordinates $(r,\theta,\phi)$,
$r\in\rr^{+}$, $\theta\in[0,\pi]$, $\phi\in[0,2\pi]$, and get:
\[\begin{array}{rl}
m_{j,t}(\rx_j):=
&P\sqrt 2\int_0^\infty\int_0^{\pi}\int_0^{2\pi}
\chi(r)^2\bigl(\cos(tr-|\rx_j|\cos\theta r)-\cos
tr\bigr)\d r\d\cos\theta\d \phi\\[3mm]
&=P\sqrt 22\pi\int_0^\infty
\chi(r)^2\bigl(
\frac{\sin(tr +|\rx_j| r)-\sin(tr-|\rx_j|r)}{|\rx_j|r}-2\cos tr\bigr)\d
r\\[3mm]
&=P\sqrt 22\pi\int_{-\infty}^\infty
\chi(r)^2\cos tr\bigl(
\frac{\sin(|\rx_j| |r|)}{|\rx_j||r|}-1\bigr)\d r
\\[3mm]&=O(t^{-n}\langle \rx_j\rangle^n).
\end{array}\]
for any $n\in\nn$, where in the last step we integrated by parts. By
interpolation we actually can replace $n$ with any positive real
$\alpha$.
Thus we see that
\[\begin{array}{l}m_{j,t}(\rx_j)(1+K)^{-1/2}
\\[3mm]=m_{j,t}(\rx_j)\langle \rx \rangle^{-\alpha}\langle
\rx
\rangle^{\alpha}(1+K)^{-1/2}
=O(t^{-\alpha}),\end{array}\]
which ends the proof of Assumption \ref{main6}.
\qed
\subsection{Asymptotic fields}
For $h\in\fh$ we define the field and the Weyl operators
\[\phi(h):=\frac{1}{\sqrt 2}\int\bigl( h(k)a^*(k)+\bar h(k)
a(k)\bigr)\d k,\ \ \ W(f):=e^{\i\phi(h)}.\]
Let
\[\fh_1:=\left\{h\in\fh\ | \ \int(1+\omega(k)^{-1})|h(k)|^2\d
k<\infty\right\}=\Dom (\omega^{-1/2}),\]
with the norm $\|h\|_{\fh_1}:=\|(1+\omega^{-1})^{1/2}h\|_\fh$.
The following assumption can be called the {\em short range condition}.
\bea There exists a dense subspace
$\fg\subset\fh_1\cap\Dom(\omega^{1/2}) $ such that for $h\in\fg$,
\[\int_0^\infty\left
\|\int\bigl(\e^{\i t\omega(k)}\bar h(k)v(k)
+v^*(k)\e^{-\i t\omega(k)}h(k)\bigr)(1{+}K)^{-1/2}\d k\right\|_{B(\cK)}\d t<\infty.
\]
\label{qea4}\eea
\bet Suppose that assumptions \ref{main2} and \ref{qea4} hold. Then:
\ben\item
For all $h\in \fh_1$, there exist
\beq
W^{\pm}(h):= \slim_{t\fld \pm\infty}\e^{\i tH}\one_{\cK}{\otimes}W (\e^{-\i t\omega}h)
\e^{-\i tH}.
\eeq
\item
\[W^\pm(h_1)W^\pm(h_2)=\e^{-\frac\i2\Im(h_1|h_2)}W^\pm
(h_1+h_2),\ \ \ h_1,h_2\in\fh_1,\]
\[\rr\ni t\mapsto W^\pm(th) \ \ \hbox{is strongly continuous}
,\ \ h\in\fh_1;\]
in other words,
\beq\fh_1\ni h\mapsto W^\pm(h)\label{piu}\eeq
are regular CCR representations (see Section \ref{s2}).
\item
\[\e^{\i tH}W^\pm(h)\e^{-\i tH}=W^\pm(\e^{\i t\omega}h),\ \ \
h\in\fh_1,\]
in other words, $(W^\pm,\omega,H)$ are covariant CCR representations
(see Section \ref{s3}).
\item
If $H\Psi=E\Psi$, then
\beq(\Psi|W^\pm(h)\Psi)=\e^{-\|h\|^2/4}\|\Psi\|^2,\label{vac}\eeq
in other words, eigenvectors of $H$ are vacua for (\ref{piu}) (see Theorem
\ref{eig1}).\een\label{scatt}\eet
The above theorem is a simplified version of Theorem \ref{scac} proved
later in our paper.
It is convenient to introduce the following notation.
$\cH_\p(H)$ will denote the closure of the span of eigenvectors of
$H$.
The set of vacua
for (\ref{piu}), i.e. the set of
$\Phi\in\cH$ satisfying (\ref{vac}) is denoted by $\cK_0^\pm$. Note
that $\cK_0^\pm$ is a closed subspace of $\cH$. By Theorem \ref{scatt}
(4), $\cK_0^\pm$ contains
$\cH_\p(H)$.
The closure of the span of vectors $W(h)\Phi$ with $h\in\fh_1$,
$\Phi\in\cK_0^\pm$ will be denoted by $\cH_{[0]}^\pm$. It is the
largest subspace of $\cH$ on which (\ref{piu}) is equivalent to the
Fock representation.
Let us state the following conjecture:
\begin{conjecture}
Suppose Assumptions \ref{main1}, \ref{main2} and \ref{qea4}
hold. Assume also
\beq
\int \frac{\|v(k)\|^2}{k^2}\d k<\infty. \label{sdf}\eeq
Then
\ben\item $\cH_{[0]}^\pm=\cH$, in other words, the asymptotic
representations are multiples of the Fock representation.
\item
$\cK_0^\pm=\cH_\pp(H)$, in
other words, all the asymptotic vacua are linear combinations of
eigenstates of $H$.
\een\label{conj}\end{conjecture}
There are two situations when we can prove the above conjecture.
If $\dim\cK=1$, then the Hamiltonian $H$ is the exactly solvable van
Hove Hamiltonian and the
conjecture follows by explicit computations, see e.g. \cite{De}.
If $v(k)=0$ in a neighborhood of zero, then the problem reduces to the
case with a positive mass. Conjecture \ref{conj} (1) follows then from the
arguments due to Hoegh-Krohn \cite{HK} described in \cite{DG1}, see also a
different proof in \cite{DG2}. Conjecture
\ref{conj} (2)
follows then
from \cite{DG1}, see also a somewhat simpler proof given in \cite{DG2}
Note that the power $|k|^{-2}$ in (\ref{sdf}) is natural, since it is
suggested by the exactly solvable case. However, we do not know how to
prove our conjecture under much stronger assumptions, e.g. if for any $N$
\[\int \frac{\|v(k)\|^2}{k^N}\d k<\infty. \]
\subsection{Existence and nonexistence of a ground state}
The following assumption will be very important in the sequel:
\bea $v(k)$ can be split as\[
\:\begin{array}{l}
v(k)= z(k)\one_{\cK}+ v_{\ren}(k),\hbox{ where}\\[3mm]
z(k)\in \cc, \ \ \ \ \ v_\ren(k)\in B(\Dom(K^{\12}), \cK),\\[3mm]
\int (1+\omega(k)^{-1})|z(k)|^{2}\d k<\infty,\\[3mm]
\int \omega(k)^{-2}\|v_\ren(k)(K+1)^{-1/2}\|^{2}\d k<\infty.
\end{array}
\]\label{main4}\eea
In order to use the results of \cite{Ge1}
we will also need the following (probably
unnecessary) assumption, which is stronger than Assumption \ref{main2}:
\bea
\[
\begin{array}{l}\hbox{ for a.e. }\ \ k\in \rr^{d},\ \ \:
v(k)(K+1)^{-\12},\:(K+1)^{-\12}v(k)\in B(\cK),\\[3mm]
\forall \Psi_{1}, \Psi_{2}\in \cK,\ \ \:k\mapsto (\Psi_{2},
(K+1)^{-\12}v(k)\Psi_{1}) \hbox{ and } k\mapsto (\Psi_{2},v(k)(K+1)^{-\12}\Psi_{1})
\hbox{ are measurable,}\\[3mm]
\lim\limits_{R\to\infty}\int
(1+\omega(k)^{-1})\bigl(\|v(k)(K+R)^{-\12}\|^{2}
+\|(K+R)^{-\12}v(k)\|^2\bigr)\d k=0.
\end{array}
\]\label{main5}\eea
\bet
Assume Hypotheses \ref{main4} and \ref{main5}. Then:
\ben\item
if Assumption \ref{main1} holds and $\int \omega(k)^{-2}|z(k)|^{2}\d
k<\infty$, then $\inf\sp(H)$ is an eigenvalue.
\item
if $\inf\sp(H)$ is an eigenvalue,
then $\int \omega(k)^{-2}|z(k)|^{2}\d
k<\infty$.
\een\label{2.1}
\eet
In particular under Assumption \ref{main1},
the existence of a ground state is {\em
equivalent} to the condition
\[
\int \omega(k)^{-2}|z(k)|^{2}\d
k<\infty.
\]
\proof
Part (1) has been shown in \cite[Thm. 1]{Ge1}. Let us prove part
(2) by contradiction. Assume that
\[
\int \omega(k)^{-2}|z(k)|^{2}\d
k=\infty,
\]
and let $\Psi_{0}\in \cH$ be a ground state of $H$. The following
pull-through formula is valid (see e.g. \cite[Sect. III.4]{Ge1}):
\beq
\begin{array}{rl}
&(H+\omega(k)-z)^{-1}a(k)\Psi\\[3mm]
=& a(k)(H-z)^{-1}\Psi+
(H+\omega(k)-z)^{-1}v(k)(H-z)^{-1}\Psi,\: \Psi\in \cH,
\end{array}
\label{e2.1} \eeq
as an identity on $L^{2}_{\rm loc}(\rr^{d}\backslash\{0\}, \d k ;\cH)$. Applying this identity
to $\Psi_{0}$, we obtain
\[
a(k)\Psi_{0}= (E-H-\omega(k))^{-1}v(k)\Psi_{0},
\]
as an identity on $L^{2}_{\rm loc}(\rr^{d}\backslash\{0\}, \d k ;\cH)$.
Hence
\[
a(k)\Psi_{0}= \frac{z(k)}{\omega(k)}\Psi_{0}+
(E-H-\omega(k))^{-1}v_\ren(k)\Psi_{0}.
\]
Let
\[
r(k):=
a(k)\Psi_{0}-\frac{z(k)}{\omega(k)}\Psi_{0}=(E-H-\omega(k))^{-1}v_\ren(k)
\Psi_0.
\]
We have
\[
\|r(k)\|\leq
c\frac{1}{\omega(k)}\|v_\ren(k)(K+1)^{-\12}\|.
\] Hence, by the last condition of Assumption \ref{main4},
$r\in L^{2}(\rr^{d}, \d
k ;\cH)$. Since $\frac{z}{\omega}\not\in \ch$, applying Lemma
\ref{2.2} below we obtain $\Psi_{0}=0$, which is a contradiction.
\qed
\begin{lemma}
Let $\Psi\in \Gamma_\s(L^{2}(\rr^{d}))$ such that
\beq
\int \|(a(k)- g(k))\Psi\|^{2}\d k<\infty,
\label{e2.2}
\eeq
where $k\mapsto g(k)\in \cc$ is measurable and
\[
\int |g(k)|^{2}\d k=\infty.
\]
Then $\Psi=0$.
\label{2.2}
\end{lemma}
\proof
We write
\[
\Psi=(\Psi_{0}, \Psi_{1}, \cdots, \Psi_{n},\cdots)
\]
where $\Psi_{n}\in \otimes_{\rm s}^{n}\ch$. From (\ref{e2.2}) we
obtain
\[
\Psi_{1}(k)- g(k)\Psi_{0}\in L^{2}(\rr^{d}),
\]
which implies that $\Psi_{0}=0$, since $g\not\in L^{2}(\rr^{d})$.
%Assume by induction that we have shown that $\Psi_{l}=0$ for $l\leq
%n-1$. Then
>From (\ref{e2.2}) we obtain
\[
(n+1)^{\12}\Psi_{n+1}(k, k_{1}, \dots, k_{n})-g(k)\Psi_{n}(k_{1},
\dots, k_{n})\in \otimes^{n+1}L^{2}(\rr^{d}),
\]
which implies that $\Psi_{n}=0$. Hence $\Psi=0$. \qed
\subsection{Existence of non-Fock sectors for asymptotic fields}
Set
\beq g(k):=\sqrt{2}\omega^{-1}(k)z(k).\label{gz}\eeq
Let us introduce the following assumption:
\bea
\[\int_0^\infty\left\|
\int\bigl(\e^{\i t\omega(k)}\bar g(k)v_\ren(k)+
v_\ren^*(k)\e^{-\i t\omega(k)}g(k)\bigr)(1{+}K)^{-1/2}\d k\right\|\d t<\infty.
\]\label{main6}\eea
\bet Assume Hypotheses \ref{main1},% \ref{main2},
\ref{qea4},
\ref{main4}, \ref{main5} and \ref{main6}.
Then there exists a nonzero vector
$\Phi\in\cH$ such that
\beq (\Phi|W^\pm(h)\Phi)=\|\phi\|^2\e^{\i\Re(h|g)}\e^{-\|h\|^2/4}.
\label{cohos}\eeq
%(a $g$-coherent vector for (\ref{piu})---see Section \ref{s2}).
In particular, if $g\not\in
L^2$, then the CCR representations (\ref{piu}) have non-Fock coherent sectors.
\label{qwo}\eet
Let us introduce the following notation. The set of vectors $\Phi$
satisfying (\ref{cohos}) will be denoted $\cK_g^\pm$. Such vectors will
be called $g$-coherent vectors for (\ref{piu}) (see Section \ref{s2}).
They form a closed
subspace of $\cH$.
The closure of the span of vectors $W(h)\Phi$ with $h\in\fh_1$,
$\Phi\in\cK_g^\pm$, will be denoted by $\cH_{[g]}^\pm$. It is the
largest subspace of $\cH$ on which (\ref{piu}) is equivalent to the
so-called $g$-{\em coherent representation}.
\begin{conjecture}
Under the hypotheses of Theorem \ref{qwo}, $\cH_{[g]}=\cH$, in other
words, the representation of asymptotic fields is equivalent to a
multiple of the $g$-coherent representation.
\label{pno}\end{conjecture}
Note that given the methods of the proof of Theorem \ref{qwo},
Conjecture \ref{pno} essentially follows from Conjecture \ref{conj} (1).
\subsection{Renormalized Hamiltonian}
In this and the next subsection we will describe the main ideas of the
proof of Theorem \ref{qwo}.
One of them is the use of the so-called renormalized Hamiltonian. It is
defined as
\[H_\ren:=K_\ren\otimes\one+\one\otimes\d\Gamma(\omega)+\int (v_\ren(k)a^*(k)
+v_\ren^*(k)a(k))\d k,\]
where
\[\begin{array}{rl}
K_\ren&:=K-\int\Big(\frac{|z(k)|^2}{\omega(k)}+\frac{\bar
z(k)v_\ren(k)}{\omega(k)}+
\frac{v_\ren^*(k)z(k)}{\omega(k)}\Big)\d k.\end{array}\]
Note that Assumptions
\ref{main1},
\ref{main4}, \ref{main5} for $H$ imply
Assumptions \ref{main1}, \ref{main4}
and \ref{main5} for $H_\ren$
with $z_\ren=0$. Therefore, by the result of \cite{Ge1} quoted
in Theorem
\ref{2.1} (1), $H_\ren$ has a ground state.
Suppose Assumptions \ref{qea4} and \ref{main6} hold as well. Then, by
Theorem \ref{dressa},
we can define
asymptotic fields for $H_\ren$
\[W_\ren^{\pm}(h):= \slim_{t\fld \pm\infty}\e^{\i tH_\ren}\one{\otimes}W
(\e^{-\i t\omega}h)
\e^{-\i tH_\ren}.
\]
Clearly, $W_\ren^\pm$ satisfy the obvious analog of Theorem
\ref{scatt}. The ground state of $H_\ren$ is a vacuum for the renormalized
asymptotic fields.
\ber
Note that if $g\in\fh$, then
\beq H=W(\i g)H_\ren W(-\i g).\label{qgq}\eeq
If $g\not\in\fh$, then $W(\pm\i g)$ is not well defined. Still, we can use
(\ref{qgq}) on a formal level. To make it rigorous we can proceed in a
variety of ways. We can choose a sequence of approximations of $g$
\[g_\sigma:=g\one_{[\sigma,\infty[}(\omega),\ \ \ 0<\sigma<1.\]
Then it is easy to show that
\[(\i+H_\ren)^{-1}=\slim_{\sigma\searrow0}\bigl(\i +W(\i g_\sigma) HW(\i
g_\sigma)\bigr)^{-1}. \]
\eer
\subsection{Dressing operators}
Clearly, $\Im
(g|\e^{-\i t\omega}g)$ is well defined and $(1-\e^{-\i t\omega})g\in
\fh$. Therefore the following definition makes sense:
\[
U(t):= \e^{\frac\i2\Im
(g|\e^{-\i t\omega}g)}\e^{\i tH}W(\i(1-\e^{-\i t\omega})g)\e^{-\i tH_\ren}.
\]
\bet Under Assumptions \ref{main2},
\ref{main4} and \ref{main6}, there exists
$U^\pm:=
\slim_{t\to \pm\infty}U(t)$. $
\slim_{t\to \pm\infty}U(t)^*$ also exists and equals $U^{\pm*}$.
\label{assa1}
\eet
The above theorem will be proved under more general conditions later
as Theorem \ref{dressa}.
The operators $U^\pm$ will be called the {\em dressing operators}.
They have the following properties:
\bet Suppose Assumptions \ref{main2},
\ref{qea4}, \ref{main4} and \ref{main6} are true. Then
For $h\in \fh_1$, we have
\[\begin{array}{rl}
W^{\pm}(h)U^{\pm}&= U^{\pm}W_{\ren}^{\pm}(h)\e^{\i\Re(h, g)},\\[3mm]
\e^{\i tH}U^{\pm}\e^{-\i t H_\ren}&=
U^{\pm}W_{\ren}^{\pm}(\i(1-\e^{\i t\omega})g)
\e^{-\frac{\i}{2}\Im(g|\e^{-\i t\omega}g)}
\\[3mm]
&=W^\pm(\i(1-\e^{\i t\omega})g)U^\pm
\e^{\frac{\i}{2}{\rm Im }(g|\e^{-\i t\omega}
g)}.
\end{array}
\]
Therefore, $U^\pm$ maps $\cK_{0,\ren}^\pm$ onto $\cK_g^\pm$.
\label{5.2b1}
\eet
The above properties of dressing operators are proved in Section \ref{s4a}.
\subsection{Wave operators}
We define the $g$-{\em coherent asymptotic space} as
\[\cH_g^\pm:=\cK_g^\pm\otimes\Gamma_\s(\ch).\]
It is easy to show that there exists a unique linear operator
$\Omega_g^\pm:\cH_g^\pm\to\cH$ such that
\[\Omega_g^\pm\
\Phi{\otimes}W(h)\Omega=\e^{-\i\Re(h|g)}W^\pm(h)\Phi,\ \ \
\Phi\in\cK_g^\pm,\ \ \ h\in\fh_1.\]
The operator $\Omega_g^\pm$ is isometric and its range equals
$\cH_{[g]}^\pm$. It will be called the
$g$-{\em coherent wave operator}. (Note that $\Omega$, without any
superscripts,
still denotes the
vacuum in a Fock space).
The $g$-{\em coherent asymptotic Hamiltonian} is defined as
\[H_g^\pm:=\Omega_g^{\pm*} H\Omega_g^{\pm}.\]
Clearly, $H_g^\pm$ is a self-adjoint operator on $\cH_g^\pm$
satisfying
\[\Omega_g^\pm H_g^\pm=H\Omega_g^\pm.\]
What is a little less obvious is the following decomposition of
$H_g^\pm$, proved in
Theorem \ref{kaka}:
\beq H_g^\pm=K_g^\pm\otimes\one+\one\otimes\int\left(a^*(k)+\frac{z(k)}{\omega(k)}\right)
\omega(k)\left(a(k)+\frac{\bar {z(k)}}{\omega(k)}\right) \d k\label{kaka1}\eeq
Thus, in particular,
the asymptotic Hamiltonians $H_g^\pm$ do not have ground states.
Note that the subspaces $\cH_{[g]}^\pm$ are invariant with respect to
$W^\pm(h)$ and $H$. They depend only on the equivalence class $[g]$ of
$g$ in $\fh_1^*/\fh$,
where $\fh_1^*$ denotes the space of all antilinear functionals on
$\fh_1$ (see Theorem \ref{pas} (1)).
If one introduces
\[H_{[g]}^\pm=H\Big|_{\cH_{[g]}^\pm}=\Omega_g^\pm H_g^{\pm}\Omega_g^{\pm*},\]
then again $H_{[g]}^\pm$ depends only on $[g]$.
If Conjecture \ref{pno}
is true then $\cH=\cH_{[g]}^\pm$ and $H=H_{[g]}^\pm$.
\subsection{Scattering operator}
We can define the {\em scattering operator} for the $g-g$ channel as
\[S_{gg}:=\Omega_g^{+*}\Omega_g^-.\]
It is unitary iff $\cH_{[g]}^-=\cH_{[g]}^+$.
Suppose that we prepare a state in a distant past
inside the incoming $g$-coherent
sector. We can describe it by a density matrix (a positive operator of
trace 1) $\rho$ on $\cH_g^-$.
Suppose that we make a measurement in a distant future in the outgoing
$g$-coherent sector. We can describe it by an observable (a
self-adjoint operator) $A$ on $\cH_g^+$.
The expectation value of the experiment is given by
\beq\Tr S_{gg}\rho S_{gg}^*A.\label{expo}\eeq
Note that there is no infrared problem in the formula above. In
principle, one has a well defined procedure to compute the expectation
value of an experiment involving any initial state and any final
observable---there is no need to restrict oneself to ``inclusive
cross-sections''.
The infrared problem manifests itself in the non-canonical choice of the
functional $g$. In fact, $g$ is not determined by the Hamiltonian $H$
itself. One can argue that in a realistic experiment all the
quantities depending on the choice of $g$ are unmeasurable (or at
least are much more
difficult to measure).
This is quite similar to long-range scattering for Schr\"odinger
operators, where the modified scattering operator depends on a
non-canonical modifier and one usually assumes that measurable
quantities are independent of its choice. In the remaining part of this
section we will analyze scattering of infrared singular Pauli-Fierz
Hamiltonians and point out quantities that are likely to be physically
relevant.
Let us note a certain discrepancy between mathematics and physics of
the problem. In the construction of wave and scattering operators the
past is treated in the same way as the future. Thus mathematics of
scattering theory is in some sense symmetric with respect to time
reversal.
This is not the case for the formula (\ref{expo}), which gives
physical interpretation of the scattering operator: the past is
represented by a density matrix whereas the future by an arbitrary
selfadjoint operator. This asymmetry between past and future will be
even more pronounced in the next subsections, where we discuss
inclusive cross-sections. It will be clear which observables should be
considered in the future, it will be less clear which
initial states should be taken into account.
\subsection{Soft and hard photons}
Let $\epsilon\geq0$. Define
\[\fh_{\leq\epsilon}:=\Ran\one_{[0,\epsilon]}(\omega),\ \ \
\fh_{>\epsilon}:=\Ran\one _{]\epsilon,\infty[}(\omega),\]
so that $\fh=\fh_{\leq\epsilon}\oplus\fh_{>\epsilon}$. Clearly, we can
make the identification
\beq
\cH_g^\pm\simeq\cH_{g,\leq\epsilon}^\pm\otimes\Gamma_\s(\fh_{>\epsilon}),
\label{hsa}\end{equation}
where $\cH_{g,\leq\epsilon}^\pm:=\cK_g^\pm\otimes
\Gamma_\s(\fh_{\leq\epsilon})$.
Let us make an additional assumption
\beq \one_{]\epsilon, +\infty[}(\omega)g=0.\label{gdh}\eeq
Since $g$ is given in terms of $z$ by the equality (\ref{gz}), this is
equivalent to $\one_{]\epsilon, +\infty[}(\omega)z=0$, which we can
always assume. By this assumption, the asymptotic Hamiltonian can be
written as
\[\begin{array}{rl}
H_g^\pm&=K_g^\pm\otimes\one+\one\otimes\int\limits_{\omega<\epsilon}
\left(a^*(k)+\frac{z(k)}{\omega(k)}\right)
\omega(k)\left(a(k)+\frac{\bar {z(k)}}{\omega(k)}\right) \d k\\[5mm]&
+\one\otimes\int\limits_{\omega\geq\epsilon}
\omega(k)a^*(k)a(k) \d k.\end{array}
\]
Therefore, with respect to the decomposition (\ref{hsa}), the asymptotic
Hamiltonians can be written as
\[H_g^\pm=H_{g,\leq\epsilon}^\pm\otimes\one+\one\otimes\d\Gamma(\omega_{>\epsilon}),\]
where $\omega_{>\epsilon}=\omega \one_{]\epsilon,\infty[}(\omega)$.
One can ask whether the decomposition into soft and hard components is
sensitive to the choice of $g$.
Introduce the soft Hamiltonian
\[H_{[g],\leq\epsilon}^\pm:=\Omega_g^\pm \
H_{g,\leq\epsilon}^{\pm}{\otimes}\one\ \Omega_g^{\pm*},\]
and the hard Hamiltonian
\[H_{[g],>\epsilon}^\pm:=\Omega_g^\pm \
\one{\otimes}\d\Gamma(\omega_{>\epsilon})\ \Omega_g^{\pm*}.\]
We have
\beq H_{[g]}^\pm=H_{[g],\leq\epsilon}^\pm+H_{[g],>\epsilon}^\pm
\label{deco}\eeq
and the Hamiltonians in (\ref{deco}) depend only on $[g]$.
A similar question can be asked concerning the observables.
On the level of asymptotic spaces we have clearly
\beq B(\cH_g^\pm)\simeq B(\cH_{g,\leq\epsilon}^\pm)\otimes B(
\cH_{g,>\epsilon}^\pm) .\label{paf}\eeq
Denote the range of the homomorphism
\[B(\cH_g^\pm)\ni A\mapsto \Omega_g^\pm\
A \Omega_g^{\pm*}\in B(\cH)\] by $\fA_{[g]}$. $\fA_{[g]}$
depends only on $[g]$ and is equal to
$B(\cH_{[g]}^\pm)$. Inside $\fA_{[g]}$ we
can distinguish the ``algebra of soft observables''
\beq \fA_{[g],\leq\epsilon}^\pm:=\Omega_g^\pm\
B(\cH_{g,\leq\epsilon}^\pm){\otimes}\one\ \Omega_g^{\pm*},\label{ga1}\eeq
and the ``algebra of hard observables''
\beq \fA_{[g],>\epsilon}^\pm:=\Omega_g^\pm\
\one{\otimes}B(\cH_{g,>\epsilon}^\pm)\ \Omega_g^{\pm*}.\label{ga2}\eeq
(\ref{ga1})
and (\ref{ga2}) depend only on $[g]$. The hard observables are even more
independent of $g$.
The automorphism
\[B(\Gamma_\s(\fh_{>\epsilon}))\ni A_{>\epsilon}
\mapsto \Omega_g^\pm \ \one{\otimes}A_{>\epsilon}\ \Omega_g^{\pm*}\in
\fA_{[g],>\epsilon}^\pm\] depends only on $[g]$ if we assume (\ref{gdh}).
\subsection{Inclusive cross-sections}
To simplify the discussion, we will assume in what follows that
\beq\cH=\cH_{[g]}^+=\cH_{[g]}^-.\label{hax}\end{equation}
In what follows we will drop the subscripts $g$ wherever possible,
thus we will write $\Omega^\pm$, $\cH^\pm$, $H^\pm$,
$H_{\leq\epsilon}^\pm$,
etc. instead of $\Omega_g^\pm$, $\cH_g^\pm$, $H_g^\pm$,
$H_{g,\leq\epsilon}^\pm$, etc.
Set $E:=\inf H$. Clearly,
\[E=\inf H^-=\inf H^+=\inf H_{\leq\gamma}^-=\inf H_{\leq\gamma}^+\]
for any $\gamma>0$.
Note that by the assumption (\ref{hax}), the wave operators
$\Omega^\pm$ are unitary from $\cH^\pm$ to $\cH$
and the scattering operator $S=\Omega^{+*}\Omega^-$ is unitary from
$\cH^-$ to $\cH^+$.
Suppose now that the experimentalist can only control
the components of the system above the threshold $\epsilon$.
In particular, since the functional $g$ depends on the soft
components, the quantities that depend on $g$ are not measurable.
The quantum description of an experiment has two aspects:
preparation of the incoming
state and measurement of the observable. It is easy to say which
observables can in principle be measured by the experimentalist. They
are the observables in the hard algebra $\fA_{>\epsilon}^+$, that
means the observables of the form $\Omega^+\ \one{\otimes} A_{>\epsilon}\
\Omega^{+*}$, where $A_{>\epsilon}\in B(\Gamma_\s(\fh_{>\epsilon}))$.
It is more difficult to say which incoming states the experimentalist can
prepare.
Recall that
$\cH^-=\cH_{\leq\epsilon}^-\otimes\cH_{>\epsilon}^-$. Thus we can
introduce the partial trace wrt $\cH_{\leq\epsilon}^-$, denoted
\[l^1(\cH^-)\ni\rho\mapsto\Tr_{\leq\epsilon}^-\ \rho\in
l^1(\cH_{>\epsilon}^-),\]
where $l^1(\cH^-)$ denotes the space of trace class operators on the
Hilbert space $\cH^-$.
In particular, if $\rho$ is a density matrix on $\cH^-$, then
$\Tr_{\leq\epsilon}^-\ \rho$
is a density matrix on $\cH_{>\epsilon}^-$.
We assume that the initial state of the system
is described
by a density matrix $\rho$ on $\cH^-$.
We also suppose that the experimentalist does not have full
information about $\rho$ and is able
to control only $\Tr_{\leq\epsilon}^-\rho$. More precisely,
for a given density matrix $\rho_{>\epsilon}$ on
$\cH_{>\epsilon}^-$, while preparing his experiment,
he can make sure that
\beq\Tr_{\leq\epsilon}^-\ \rho=\rho_{>\epsilon}.\label{faf}\eeq
Of course, there are many density matrices $\rho$ satisfying
(\ref{faf}). The choice of $\rho$ should be determined by physics.
Let us suppose that the experiment is conducted at a low temperature,
so that everything tends to have the lowest possible energy.
Suppose for a moment that the infrared problem is absent in the sense
that the Hamiltonian $H$, hence also $H^\pm$ and $H_{\leq\epsilon}^\pm$,
has a non-degenerate ground state. Then it is natural to assume that
the incoming density matrix equals
\[\one_E(H_{\leq \epsilon}^-)\otimes\rho_{>\epsilon}.\]
(recall that $\inf\sp(H_{\leq\epsilon}^-)=E$, and hence
$\one_E(H_{leq \epsilon}^-)$ denotes the spectral projection onto the
ground state of $H_{\leq\epsilon}^-$).
Thus one can argue that if the experimentalist prepared the hard part
of the incoming state as $\rho_{>\epsilon}$ and measures the
observable $A$, then
the expectation value of the
measurement (which we will somewhat imprecisely call the
cross-section)
will be
\beq\Tr\ S\ \one_E(H_{\leq\epsilon}^-){\otimes}\rho_{>\epsilon}\ S^*\
A .\label{nonin}\eeq
If we have an infra-red problem---if $H$ has no ground state at all or
even if its ground state is degenerate---then it is not clear
which $\rho$ satisfying (\ref{faf}) should be taken. We can argue that
$\rho$ should satisfy
\[\one_{[E,E+\delta]}(H_{\leq\epsilon}^-){\otimes}\one\ \rho=\rho\]for some
small $\delta>0$. Of course this does not fix the choice of $\rho$
either.
Motivated by these considerations,
if $\delta>0$, $\rho_{>\epsilon}$ is
a density matrix on $\cH_{>\epsilon}^-$ and
$A$ is observable on $\cH^+$, we define
\[\begin{array}{rl}
\Cross
_\delta(\rho_{>\epsilon},A)&:=\bigl\{\Tr\rho S^*AS\ :\rho\ \hbox{ is a
density matrix on }\cH^-,\\[3mm]
&\one_{[E,E+\delta]}(H_{\leq\epsilon}^-)\otimes\one\ \rho=\rho, \Tr_{\leq\epsilon}^-\ \rho=\rho_{>\epsilon}\bigr\}.
\end{array}\]
This is the set of
all possible cross-sections compatible with the pair $(\rho_{>\epsilon},A)$
under the assumption that the soft part of the initial state has the
excess energy below $\delta$.
Clearly, $\Cross_\delta(\rho_{>\epsilon},A)$ is a family of nonempty
intervals in $[-\|A\|,\|A\|]$ decreasing as $\delta\searrow0$.
It would be interesting to investigate whether a large class of
Pauli-Fierz Hamiltonians has the following property:
\begin{property}
The Pauli-Fierz Hamiltonian $H$
has the property of
the {\bf continuity of cross-sections at the
bottom of spectrum} iff for
any $\rho_{>\epsilon}$ and $A$,
\beq\bigcap_{0<\delta<\epsilon}\Cross_\delta(\rho_{>\epsilon},A)^\cl\label{cap1}
\eeq
is a single point. (The superscript $\rm cl$ denotes the closure of a set).
\label{con1}\end{property}
If
Property \ref{con1} holds, then
the
number given by (\ref{cap1}) can be viewed as the cross-section for
the experiment described by $\rho_{>\epsilon}$ and $A$.
Note that if $H$ has a non-degenerate
ground state, then (\ref{cap1}) contains
the number (\ref{nonin}).
Clearly (\ref{cap1}) depends on the choice of $g$ within its
equivalence class, hence one can
argue that in such a case
it does not correspond to a physical experiment. If one
assumes that the observable is of the form $A=\one{\otimes}A_{>\epsilon}$
with $A_{>\epsilon}$ an observable on $\Gamma_\s(\fh_{>\epsilon})$, then
\beq\Cross_\delta(\rho_{>\epsilon},\one\otimes A_{>\epsilon})\label{faf2}\eeq
does not depend on the choice of $g$ satisfying (\ref{gdh}), using the
covariance properties shown in Subsection \ref{cova}.
(\ref{faf2}) is the set of
possible inclusive cross-sections compatible with the pair $(
\rho_{>\epsilon},A_{>\epsilon})$.
One can introduce a property weaker
than (\ref{con1}):
\begin{property} The Pauli-Fierz Hamiltonian $H$ has the property of
the {\bf continuity of inclusive cross-sections at the
bottom of spectrum}
iff
for
any $\rho_{>\epsilon}$ and $A_{>\epsilon}$,
\[\bigcap_{0<\delta<\epsilon}\Cross_\delta(
\rho_{>\epsilon},\one{\otimes}A_{>\epsilon}
)^\cl\]
is a single point.\label{con2}\end{property}
If Property \ref{con1} is true
then the theory based on the Pauli-Fierz Hamiltonian $H$ has quite a
strong
predictive power. The experimentalist does not
have to worry about preparing precisely the soft part of the initial state;
it is enough if its soft part is sufficiently low energetic. Then he
can measure all observables he likes, even those involving soft modes.
The theory will give well defined
cross-sections for his experiments.
If the experimentalist measures only hard components of the final state,
then it is sufficient that Property \ref{con2} holds to have well defined
cross-sections for all experiments.
Note that the stronger Property \ref{con1} is true in the case of the
exactly solvable van Hove Hamiltonian, where the scattering operator
is equal to identity.
\subsection{Insensitivity to soft background}
One could argue, however, that Properties \ref{con1} and \ref{con2}
are too modest and do not correspond to realistic physical
situations. It may be unjustified to expect that the soft modes of the
radiation will dissipate their energy while the experimentalist
prepares the experiment. Nevertheless, one can hope that soft modes
should not influence the outcome of measurement too much provided that
their energy is reasonably bounded. This intuition leads to yet
another conjecture.
In order to state it, we introduce a new
definition. Let $\delta>0$ and $0<\gamma\leq\epsilon$. Suppose that the
experimentalist can control the incoming states up to the modes of
energy $\gamma$. He can make sure that there are no photons of energy
in $[\gamma,\epsilon]$---the system is in the lowest possible
energetic state for the modes of energy in this energy range.
This means that
\beq \Tr_{\leq\gamma}^-\ \rho=|W(-\i g_{[\gamma,\epsilon]})\Omega)
(W(-\i g_{[\gamma,\epsilon]})\Omega|\otimes
\rho_{>\epsilon}.\label{fga}\eeq
Here
\[
g_{[\gamma,\epsilon]}=\one_{[\gamma,\epsilon]}(\omega)g,
\]
and
$|W(-\i g_{[\gamma,\epsilon]})\Omega)
(W(-\i g_{[\gamma,\epsilon]})\Omega|$ denotes the orthogonal
projection onto the coherent vector
$W(-\i g_{[\gamma,\epsilon]})\Omega$.
Suppose also that the experimentalist can
guarantee that the soft modes
have the excess of the energy below $\delta>0$, which however does not
have to be very small.
This means that
\[\one_{[E,E+\delta]}(H_{\leq\gamma}^-){\otimes}\one\ \rho
= \rho
.\]
Note that by (\ref{fga})
this is equivalent to
\[\one_{[E,E+\delta]}(H_{\leq\epsilon}^-){\otimes}\one\ \rho
= \rho
.\]
Cross-sections compatible
with this information are given by the set
\[\begin{array}{rl}
\Cross
_{\delta,\gamma}(\rho_{>\epsilon},A)&:=\bigl\{\Tr\rho S^*AS\ :\rho\ \hbox{ is a
density matrix on }\cH^-,\\[3mm]
&\one_{[E,E+\delta]}(H_{\leq\gamma}^-){\otimes}\one\ \rho=\rho,
\Tr_{\leq\gamma}^-\ \rho=|W(-\i g_{[\gamma,\epsilon]})\Omega)
(W(-\i g_{[\gamma,\epsilon]})\Omega|{\otimes}
\rho_{>\epsilon}\bigr\}.
\end{array}\]
Clearly $\Cross
_{\delta,\gamma}(\rho_{>\epsilon},A)$ decrease if $\delta$ or $\gamma$
decrease.
Moreover if $\delta<\gamma$, then
\[\Cross
_{\delta,\gamma}(\rho_{>\epsilon},A)=
\Cross
_{\delta}(\rho_{>\epsilon},A).\]
If $A_{>\epsilon}$ is as above, then
\beq\Cross_{\delta,\gamma}
(\rho_{>\epsilon},\one\otimes A_{>\epsilon})\label{faf1}\eeq
does not depend on the choice of $g$ satisfying the condition (\ref{gdh}).
Let $[0,\epsilon]\ni\gamma\mapsto\delta(\gamma)$ be a
function with values in positive real numbers.
One could expect that a large class of Pauli-Fierz Hamiltonians
satisfy the following property for $\delta(\gamma)$ much larger than
$\gamma$:
\begin{property}
A Pauli-Fierz Hamiltonian $H$ has the property of
{\bf $\delta$-insensitivity of inclusive cross-sections to soft
background}
iff the following is
true. Let $\rho_{>\epsilon}$ and $A_{>\epsilon}$ be as above. Then
\[\bigcap_{0<\gamma<\epsilon}\Cross_{\delta(\gamma),\gamma}
(\rho_{>\epsilon},\one{\otimes}A_{>\epsilon}
)^\cl\]
is a single point.\label{con3}\end{property}
Note that the van Hove Hamiltonians have the Property \ref{con3} with
$\delta(\gamma)=\infty$---soft modes and hard modes are completely decoupled.
\section{Canonical commutation relations}
\label{s2}
\init
Here begins the second part of this paper, consisting of Sections
\ref{s2}-\ref{s5} and Appendix, which is more
mathematical than the previous section.
In this part we develop systematically various elements
of mathematical
formalism useful in the study of infrared problem. In particular we
prove most of the statements described in Section \ref{main}.
Let us stress that
this and the following sections can be read independently
of Section \ref{main} and of the introduction.
In this section we collect basic constructions and facts concerning
CCR representations \cite{BR}, \cite{BSZ}, \cite{DG2},
concentrating especially on the so-called {\em coherent representations}.
The notation that we develop here will be used
throughout the paper. Note in particular that in the applications
that will start with Section \ref{s4}, the superscript $\pi$ will be
replaced by the superscript $-$ or $+$ corresponding to the incoming
or outgoing representation.
\subsection{CCR Representations}
Let $\fg$ be a complex vector space with a scalar product
$(\cdot|\cdot)$ antilinear wrt the first argument. Let $\cH$ be
a Hilbert space. Let $U(\cH)$ denote the set of unitary operators on
$\cH$.
Recall
that
\beq \fg\ni h\mapsto W^\pi(h)\in U(\cH)\label{regu}\end{equation}
is a {\em CCR representation} over $\fg$ in $\cH$
if
\[W^\pi(h_1)W^\pi(h_2)=\e^{-\frac\i2\Im(h_1|h_2)}W^\pi
(h_1+h_2),\ \ \ h_1,h_2\in\fg.\]
We say that a vector $\Psi\in\cH$ is {\em regular} if
\[\rr\ni t\mapsto W^\pi(th)\Psi,\ \ h\in\fg\]
is continuous. Let $\cH_\reg^\pi$ be the set
of regular vectors---the regular sector of (\ref{regu}).
It is easy to see that $\cH_\reg^\pi$ is a closed subspace of $\cH$ invariant under
(\ref{regu}). We say that (\ref{regu}) is {\em regular }if
$\cH=\cH_\reg^\pi$.
The field operator associated to the representation $\pi$ and $h\in\fg$
is the self-adjoint operator defined as follows:
$\Psi\in\Dom(\phi^\pi(h))$ iff there exists
\[\phi^\pi(h)\Psi=\frac{\d}{\i\d t}W^\pi(th)\Psi\Big|_{t=0}.\]
Clearly, $\Dom (\phi^\pi(h))$
is contained and dense in $\cH_\reg^\pi$.
The creation and annihilation operators associated to the
representation $\pi$ are defined as
\[a^\pi(h):=\frac1{\sqrt2}(\phi^\pi(h)+\i\phi^\pi(\i
h)),
\ \ \ a^{\pi*}(h):=\frac1{\sqrt2}(\phi^\pi(h)-\i\phi^\pi(\i
h)).\]
For further reference let us note the identities
%\[W^\pi(g)h(a^{\pi*})W^\pi(-g)=h(a^{\pi*})+\frac\i{\sqrt 2}(g|h),\ \
%W^\pi(g)\bar h(a^\pi)W^\pi(-g)=\bar h(a^\pi)-\frac\i{\sqrt 2}(h|g);\]
\[W^\pi(\i g)a^{\pi*}(h)W^\pi(-\i g)=a^{\pi*}(h)+\frac1{\sqrt 2}(g|h),\ \
W^\pi(\i g)a^\pi(h)W^\pi(-\i g)=a^\pi(h)+\frac1{\sqrt 2}(h|g).\]
\subsection{The Fock representation}
\label{focko}
Let $\fh$ be a Hilbert space.
$\Gamma_\s(\fh)$ will denote the symmetric
Fock space over $\fh$. $\Omega$ will
denote the corresponding vacuum vector and $N$ the number
operator.
If $h\in\fh$, then $a^*(h)$ denotes the corresponding creation
operator, that is the operator defined on finite particle vectors $\Phi$ as
\[a^*(h)\Phi:=h\otimes_\s\sqrt{N+1}\Phi,\ \ \ \]
The same symbol $a^*(h)$ denotes the closure of this operator.
The annihilation operator is defined as $a(h):=h(a^*)^*$ and the
field and Weyl operators are
\[\phi(h):=\frac{1}{\sqrt2}(a^*(h)+a(h)),\ \ \ \
W(h):=\e^{\i\phi(h)}.\]
It is well known that
\beq \fh\ni h\mapsto W(h)\in
U(\Gamma_\s(\fh)),\label{regu1}\end{equation}
is
a regular CCR representation. It is
called the Fock representation.
(See \cite{BR}, \cite{BSZ}).
If $f\in \fh$, then $W(-\i f)\Omega$ is called the {\em coherent vector
centered at }$f$. Note that it satisfies
\[\sqrt{2}\,a(h)W(-\i f)\Omega=(h|f)W(-\i f)\Omega.\]
This property characterizes coherent vectors, as is seen from Theorem
\ref{coho}.
In the remaining part of this section, $\fg$ will be
a dense subspace of $\fh$ and $f$ will be an
antilinear functional on $\fg$. The action of $f$ on $h\in\fg$ will be
denoted by $(h|f)$, as in the scalar product.
The following theorem is well known, for the proof see eg. \cite{De}.
\bet
Let
$\Psi\in\Gamma_\s(\fh)$. Suppose that for any $h\in\fg$ we have
\[
\Psi\in\Dom(a(h)),\ \ \ \
\sqrt{2}a(h)\Psi=(h|f)\Psi.\]
Then the following is true:
\ben\item If $f\in\fh$, then $\Psi$ is proportional to $W(-\i f)\Omega$.
\item If $f\not\in\fh$, then $\Psi=0$.
\een\label{coho}\eet
%\proof By induction we show that for $h_1,\dots,h_n\in\fg$,
%$\bar h_{n-1}(a)\cdots \bar h_1(a)\Psi\in\Dom (\bar h_n(a))$ and
%\[2^{n/2}\bar h_{n}(a)\cdots \bar h_1(a)\Psi=(h_1|f)\cdots(h_n|f)\Psi.\]
%This implies
%\beq2^{n/2}(h_{1}(a^*)\cdots
%h_n(a^*)\Omega|\Psi)=(h_1|f)\cdots(h_n|f)(\Omega|\Psi) .\label{asas}
%\end{equation}
%In particular, \[\sqrt2(h|\Psi)=
%\sqrt2(h(a^*)\Omega|\Psi)=(h|f)(\Omega|\Psi),\ \ h\in\fg.\]
%Using the fact that $\fg$ is dense in $\fh$ we see that $(\Omega|\Psi)f$
%is a bounded functional on $\fh$, hence it belongs to $\fh$. Thus
%either $f\in\fh$ or $(\Omega|\Psi)=0$. In the latter case, (\ref{asas})
%implies that $\Psi=0$. \qed
\subsection{Coherent representations}
Note that
\beq \fg\ni h\mapsto W^f(h):=W(h)\e^{\i\Re(f|h)}
\in U(\Gamma_\s(\fh))\label{coh}\end{equation}
is a regular CCR representation in $\Gamma_\s(\fh)$.
We will call (\ref{coh}) the $f$-{\em coherent representation}.
The corresponding field, creation and annihilation operators will be
denoted
$\phi^f(h)$, $a^{f*}(h)$, $a^f(h)$. Clearly,
\beq\begin{array}{l}
\phi^f(h)=\phi(h)+\Re(h|f),\\[3mm]
a^{f*}(h)=a^*(h)+\frac1{\sqrt2}(f|h),\\[3mm]
a^{f}(h)=a(h)+\frac1{\sqrt2}(h|f).\end{array}\label{qaq}
\end{equation}
Note that the vacuum satisfies for $h\in\fg$:
\[\sqrt{2}\,a^f(h)\Omega=(h|f)\Omega.\]
%\[(\Omega|W^f(h)\Omega)=\e^{-\frac{1}{4}\|h\|^2+\i\Re(f|h)}.\]
\bet
\ben\item If $f\in\fh$, then $W^f(h)=W(\i f)W(h)W(-\i f)$, $h\in\fg$.
\item
If $f\not\in\fh$, then there is no operator $U$ such that
\beq W^f(h)=UW(h)U^*,\ \ h\in\fg.\label{raf}\end{equation}
\een\eet
\proof (1) is immediate. To prove (2),
suppose that $U$ satisfies
(\ref{raf}). Then
$a^f(h)=Ua(h)U^*$.
Using $a(h)\Omega=0$ and the last identity of (\ref{qaq}) we see
that
\[\sqrt{2}\,a(h)U^*\Omega=(h|f)U^*\Omega,\]
which means that $U^*\Omega$ satisfies the assumptions of Theorem
\ref{coho}.
But $U^*\Omega\neq0$. Hence $f\in\fh$. \qed
\subsection{Coherent sectors}
In this and the following subsection we consider an arbitrary
CCR representation
\beq \fg\ni h\mapsto W^\pi(h)\in U(\cH).\label{regu5}\end{equation}
We are going to describe how to extract $f$-coherent
sub-representations of (\ref{regu5}).
A vector $\Psi\in\cH$ is called an $f$-{\em coherent vector } for (\ref{regu5})
if for any $h\in\fg$ we have
\[\Psi\in\Dom(a^\pi(h)),\ \ \ \ \
\sqrt2a^\pi(h)\Psi=(h|f)\Psi.\]
Let $\cK_f^\pi$ be the set of $f$-coherent vectors for (\ref{regu5}). Elements of $\cK_0^\pi$
will be called {\em vacua }for (\ref{regu5}).
\bet\ben
\item\label{z4}
$\cK_f^\pi$ is a closed linear subspace.
\item\label{z3} $\Psi\in\cK_f^\pi$ iff \[(\Psi|W^\pi(h)\Psi)=\|\Psi\|^2
\e^{-\frac{1}{4}\|h\|^2+\i\Re(f|h)}.\]
\item \label{z2}
All vectors in $\cK_f^\pi$ are analytic for
$\phi^\pi(h)$, $h\in\fg$.
\item\label{z5}
If $\Psi_1,\Psi_2\in\cK_f^\pi$, then
\[(\Psi_1|W^\pi(h)\Psi_2)=(\Psi_1|\Psi_2)
\e^{-\frac{1}{4}\|h\|^2+\i\Re(f|h)}.\]
\een\label{qsq}\eet
\proof
\arabicref{z4} is obvious, since $a^\pi(h)$ are closed operators.
Let us prove \arabicref{z3} $\Leftarrow$. Let
$\Psi\in\cK_f^\pi$ and $\|\Psi\|=1$. Taking the first two terms of the Taylor
expansion of
\[t\mapsto(\Psi|W^\pi(th)\Psi)=\|\Psi\|^2
\e^{-\frac{1}{4}t^2\|h\|^2+\i t\Re(f|h)},\]
we obtain
\[(\Psi|\phi^\pi(h)\Psi)=\Re(f|h),\
\ \ \ (\Psi|\phi^\pi(h)^2\Psi)=\frac12\|h\|^2+(\Re(f|h))^2.\]
Similarly,
\[(\Psi|\phi^\pi(\i h)\Psi)=-\Im(f|h),\ \ \ \ \
(\Psi|\phi^\pi(\i h)^2\Psi)=\frac12\|h\|^2+(\Im(f| h))^2.\]
Clearly,
\[[\phi^\pi(h),\phi^\pi(\i h)]=\i\|h\|^2.\]
Therefore,
\[\begin{array}{rl}
\|(\sqrt2\,a^\pi(h)-(h|f)
)\Psi\|^2=&
\|(\phi^\pi(h)+\i\phi^\pi(\i h)-(h|f)
)\Psi\|^2\\[3mm]
=&\Big(\Psi|\bigl(\phi^\pi(h)^2+\phi(\i h)^2
+\i[\phi^\pi(h),\phi^\pi(\i h)]\\[3mm]&-2\phi^\pi
(h)\Re(f|h)-2\phi^\pi(\i h)\Im(f|h)
+|(f|h)|^2\bigr)\Psi\Big)=0.\end{array}
\]
To prove \arabicref{z3} $\Rightarrow$, note that
$\Dom(a^\pi(h))=\Dom(\phi^\pi(h))\cap\Dom(\phi^\pi(\i h))$.
Hence if $\Psi\in
\Dom(a^\pi(h))$, then the function
\[\rr\ni t\mapsto F(t):=(\Psi|W^\pi(th)\Psi)\]
is $C^1$. Now
\[\begin{array}{rl}
\frac{\d}{\d t}F(t)&=\frac{\i}{\sqrt2}(a^\pi(h)\Psi|W^\pi(th)\Psi)+
\frac{\i}{\sqrt2}(\Psi|W^\pi(th)a^\pi(h)\Psi)-\frac{t}{2}\|h\|^2F(t)\\[3mm]
&=(\i\Re(f|h)-\frac{t}{2}\|h\|^2)F(t).\end{array}\]
This implies that $F(t)=
\|\Psi\|^2
\e^{-\frac{1}{4}t^2\|h\|^2+\i t\Re(f|h)}$.
\arabicref{z2} follows immediately from \arabicref{z3}.
\arabicref{z5} follows from \arabicref{z3} by polarization.
In fact, let $\Psi_1,\Psi_2\in\cK_f^\pi$. By \arabicref{z4},
$\Psi_1+\i^j \Psi_2\in\cK_f^\pi$. Hence
\[\begin{array}{rl}
(\Psi_1|W^\pi(h)\Psi_2)&=
\sum\limits_{j=0}^3\frac{\i^{-j}}{4}\left((\Psi_1+\i^j\Psi_2)|W^\pi(h)
(\Psi_1+\i^j\Psi_2)\right)\\[3mm]&=
\sum\limits_{j=0}^3\frac{\i^{-j}}{4}\|\Psi_1+\i^j\Psi_2\|^2
\exp(-\frac14\|h\|^2+\i\Re(f|h))\\[3mm]
&=(\Psi_1|\Psi_2)\exp(-\frac14\|h\|^2+\i\Re(f|h)).\:\: \Box\end{array}\]
\medskip
Set \[\cH_{[f]}^\pi:=\Span^\cl\{W^\pi(h)\Psi\ :\
\Psi\in\cK_f^\pi,\ \ h\in\fg\},\]
where $\Span^\cl \cA$ denotes the
closure of the span of the set $\cA\subset\cH$. Let $P_{[f]}^\pi$ be
the orthogonal projection
onto $\cH_{[f]}^\pi$. We will call $\cH_{[f]}^\pi$ the
$f$-{\em coherent sector }of (\ref{regu5}).
Set \[\cH_f^\pi:=\cK_f^\pi\otimes\Gamma_\s(\fh).\]
$\cH_{[0]}^\pi$ will be called the {\em Fock sector }of $\pi$.
If $\cH_{[f]}^{\pi}= \cH$ (resp. $\cH_{[0]}^{\pi}=\cH$) we will say
that the representation $W^{\pi}$ is of $f-${\em coherent type} (resp.
of {\em Fock type}).
%We will write $\cH_{[]}^\pi$ for $\cH_{[0]}^\pi$ and $P_{[]}^\pi$ for
%$P_{[0]}^\pi$ and call it the vacuum sector. Likewise, we will write
%$\cH^\pi$ for $\cK^\pi\otimes\Gamma_\s(\fh)$.
\bet
\ben \item\label{z1}
$\cH_{[f]}^\pi$ is an invariant subspace of (\ref{regu5})
contained in $\cH_\reg^\pi$.
\item\label{z6}
There exists a unique operator
$\Omega_f^\pi:\cH_f^\pi\to\cH_{[f]}^\pi$ satisfying
\[\Omega_f^\pi\ \Psi{\otimes}
W(h)\Omega=\e^{-\i\Re(h|f)}W^\pi(h)\Psi,\ \ \ \Psi\in\cK_f^\pi,\ \ h\in\fg.\]
The operator $\Omega_f^\pi$ is unitary.
\item\label{z7}
\beq\Omega_f^\pi\ \one{\otimes}
W(g)=\e^{-\i\Re(g|f)}W^\pi(g)\Omega_f^\pi, \ \ g\in\fg.\label{pio}\end{equation}
\een\eet
\proof \arabicref{z1} is obvious.
Let us prove \arabicref{z6}.
Let $\Psi_1,\Psi_2\in\cK_f^\pi$, $h_1,h_2\in\fg$.
Then, by Theorem \ref{qsq} \arabicref{z5},
\[\begin{array}{rl}
(\e^{-\i\Re(h_1|f)}W^\pi(h_1)\Psi_1|
\e^{-\i\Re(h_2|f)}W^\pi(h_2)\Psi_2)&=
(\Psi_1|\Psi_2)\e^{\frac\i2\Im(h_1|h_2)-\frac{1}{4}\|h_1-h_2\|^2}\\[3mm]&=
(\Psi_1|\Psi_2)(W(h_1)\Omega|W(h_2)\Omega)
\end{array}\]
Hence for $\alpha_j\in\cc$, $\Psi_j\in\cK_f^\pi$, $h_j\in\fg$.
\[\Big\|\sum_j\alpha_j\e^{-\i\Re(h_j|f)}W^\pi(h_j)\Psi_j\big\|^2
=\Big\|\sum_j\alpha_j\Psi_j{\otimes}W(h_j)\Omega\big\|^2.\]
Therefore, $\Omega_f^\pi$ is well defined and isometric. It is
obvious that its range equals $\cH_{[f]}^\pi$.
To show \arabicref{z7}, we note:
\[\begin{array}{rl}
&\Omega_f^\pi\ \one{\otimes} W(g)\ \Psi{\otimes} W(h)\Omega\\[3mm]
=&\e^{-\frac\i2\Im(g|h)}
\Omega_f^\pi\ \Psi{\otimes} W(g+h)\Omega\\[3mm]
=&\e^{-\frac\i2\Im(g|h)}\e^{-\i\Re(g+h|f)}
W^\pi(g+h)\Psi\\[3mm]
=&\e^{-\i\Re(g+h|f)}
W^\pi(g)W^\pi(h)\Psi\\[3mm]
=&\e^{-\i\Re(g|f)}
W^\pi(g)\ \Omega_f^\pi\ \Psi{\otimes} W(h)\Omega.\:\: \Box
\end{array}\]
\subsection{Comparison of coherent sectors}
For $h\in\fh$ we set
\[W_f^\pi(h):=\Omega_f^\pi\,\one{\otimes}W(h)\,
\Omega_f^{\pi*}.\]
\bet
\ben
\item\label{z8}
The map
\[\fh\ni h\mapsto W_f^\pi(h)\in U(\cH_{[f]}^\pi)\]
is a regular CCR representation
\item\label{z8a}
\[\Omega_f^\pi\ \Psi{\otimes} W(h)\Omega=W_f^\pi(h)\Psi,\ \
\ \Psi\in\cK_f^\pi,\ \ h\in\fh.\]
\item\label{z9}
For $h\in\fg$ we have
\[\begin{array}{l}
W_f^\pi(h)=\e^{-\i\Re(f|h)}P_{[f]}^\pi W^\pi(h),\\[3mm]
\phi_f^\pi(h)=P_{[f]}^\pi(\phi^\pi(h)-\Re(f|h)),\\[3mm]
a_f^{\pi*}(h)=P_{[f]}^\pi\bigl(a^{\pi*}(h)-\frac1{\sqrt2}(h|f)\bigr),\\[3mm]
a_f^\pi(h)=P_{[f]}^\pi\bigl(a^\pi(h)-\frac1{\sqrt2}(f|h)\bigr).\end{array}\]
\een\label{ffg}\eet
\arabicref{z8} and (2) are immediate.
If we multiply
(\ref{pio}) from the right by $\Omega_f^{\pi*}$, use
$P_{[f]}^\pi =\Omega_f^\pi\Omega_f^{\pi*}$ and the fact
that $P_{[f]}^\pi$ commutes with $W^\pi(h)$, we obtain
the first identity of \arabicref{z9}. The other follow immediately.
\qed
\ber
Let us make a comment on the purpose of introducing the operators
$W_f^\pi(h)$. As we see from Theorem \ref{ffg} (3), for various
applications, as long as $h\in\fg$
we could use $W^\pi(h)$ instead of $W_f^\pi(h)$. The advantage of the
operators $W_f^\pi(h)$, however, lies in the fact that they are
defined for any $h\in\fh$.
Note also that $W_f^\pi(h)$ is a different object from the
$f$-coherent representation $W^f(h)$ introduced earlier.
\eer
\bet
Let $f,g$ be antilinear functionals on $\fg$.
\ben\item
Assume that $g\in\fh$. Then
\begin{romanenumerate}\item
\label{d1}$\cK_{g+f}^\pi=W_f^\pi(-\i g)\cK_f^\pi$.\\
\item \label{d2}
$\cH_{[f]}^\pi=\cH_{[f+g]}^\pi$ and $P_{[f]}^\pi=P_{[f+g]}^\pi$.
Consequently the $f-$coherent sector $\cH_{[f]}^{\pi}$ depends only on
the class $[f]$ of $f$ in $\fg^{*}/\ch$.
\item\label{d2a}
Set
$W_{\coh,f}^\pi(-\i g):= W_f^\pi(-\i g)\Big|_{\cK_f^\pi}$. Then
$W_{\coh,f}^\pi(-\i g)$
is is a unitary map from
$\cK_f^\pi$ to $\cK_{f+g}^\pi$.
\item\label{d2b} We have $W_{\coh,f}^\pi(-\i g)= W_{f+g}^\pi(-\i
g)\Big|_{\cK_f^\pi}$ and
$W_{\coh,f+g}^\pi(\i g)=W_{\coh,f}^\pi(-\i g)^*$.
\item \label{d3}
$\begin{array}{rl}
\Omega_{f}^\pi
&=\Omega_{f+g}^\pi\
W_{\coh,f}^\pi(-\i g){\otimes} W(\i g)\end{array}$.
\end{romanenumerate}
\item
If $g\not\in\fh$, then $\cH_{[f]}^\pi\perp\cH_{[f+
g]}^\pi$.
\een
\label{pas}\eet
\proof Let us
first prove (1.{\rm i}). $W_f^\pi$ is a CCR representation, hence for
$h\in\fg$
\[\begin{array}{l}
\bar h(a_f^\pi)W_f^\pi(-\i g)=W_f^\pi(-\i g)(
a_f^\pi(h)+\frac1{\sqrt2}(h|g)) .\end{array}\]
Therefore,
\[\begin{array}{l}
a^\pi(h)W_f^\pi(-\i g)=W_f^\pi(-\i g)(
a^\pi(h)+\frac1{\sqrt2}(h|g)) .\end{array}\]
This implies $W_f^\pi(-\i g)\cK_f^\pi\subset\cK_{f+g}^\pi$.
An analogous reasoning shows the converse inclusion.
(1.ii) and (1.iii) follow immediately from (1.i).
To prove (1.iv) note that
$W_f^\pi(-\i g)=W_{f+g}^\pi(-\i g)$, which
follows from $\Re(g|\i g)=0$.
Let us prove (1.v).
Let $\Psi\in\cK_f^\pi$ and
$h\in\fg$.
\[\begin{array}{l}
\Omega_{f+g}^\pi\
W_{\coh,f}^\pi(-\i g){\otimes} W(\i g)
\ \Psi{\otimes}W(h)\Omega\\[3mm]
=W_{f+g}^\pi(\i g)\Omega_{f+g}^\pi\
W_{f}^\pi(-\i g)\Psi{\otimes}W(h)\Omega\\[3mm]
=W_{f+g}^\pi(\i g)W_{f+g}^\pi(h)
W_{f}^\pi(-\i g)\Psi\\[3mm]=W_{f+g}^\pi(\i g)W_{f+g}^\pi(h)
W_{f+g}^\pi(-\i g)\Psi\\[3mm]
=W_{f}^\pi(h)\Psi\\[3mm]
=\Omega_{f}^\pi\ \Psi\otimes W(h)\Omega.
\end{array}\]
Let us prove (2).
Let us first show that
\beq
0\neq\Phi\in\cK_{f+g}^\pi\cap\cH_{[f]}^\pi\ \Rightarrow\
g\in\fh.\label{rer}\end{equation}
In fact, for $h\in\fg$ we have
\[
%\begin{array}{rl}
\one{\otimes}a(h)\,\Omega_f^{\pi*}\Phi=\Omega_f^{\pi*}\bigl(a^\pi(h)
-\frac1{\sqrt2} (h|f)\bigr)\Phi
=\frac1{\sqrt2} (h|g)\Omega_f^{\pi*}\Phi.
%\end{array}
\]
But $\Ran\Omega_f^\pi=\cH_{[f]}^\pi$, hence
$\Omega_f^{\pi*}\Phi\neq0$. By Theorem \ref{coho}, this implies
$g\in\fh$.
Now suppose that $\cH_{[f]}^\pi$ is not perpendicular to
$\cH_{[f+g]}^\pi$.
Then there exist vectors
$\Psi_1\in\cK_f^\pi$, $\Psi_2\in\cK_{f+ g}^\pi$, $h_1,
h_2\in\fg$ such that
\beq(W^\pi(h_1)\Psi_1|W^\pi(h_2)\Psi_2)\neq0.\label{asq}\end{equation}
Set $\Phi:=P_{[f]}^{\pi}\Psi_2$.
Clearly, $\Phi\in\cH_{[f]}^\pi$.
Note that $P_{[f]}^\pi$ commutes with $a^\pi(h)$. Hence
$\Phi\in\cK_{f+g}^\pi$. Clearly,
$(W^\pi(-h_2)W^\pi(h_1)\Psi_1|\Phi)$ equals the left hand side of
(\ref{asq}), hence is nonzero. Therefore, $\Phi\neq0$.
By (\ref{rer}), this implies $g\in\fh$. \qed
\section{Covariant CCR representations }
\label{s3}
\init
\subsection{Definition of a covariant CCR representation }
In this section we describe properties of a CCR representation
equipped with a dynamics.
Let $\fh$ and $\cH$ be Hilbert spaces.
Let $\fg$ be a dense subspace of $\fh$. Let
\beq
\fg\ni h\mapsto W^\pi(h)\in U(\cH)\label{repi}\end{equation} be
a CCR representation.
Let $\omega$ be a self-adjoint operator on $\fh$ and $H$ a
self-adjoint operator on $\cH$. We say that the triple
$(W^\pi
,\omega,H)$ is a
covariant CCR representation iff
$\fg$ is invariant w.r.t. $\e^{\i t\omega}$
and
\[\e^{\i tH}W^\pi(h)\e^{-\i tH}=W^\pi(\e^{\i t\omega}h),\:t\in \rr, \: h\in\fg.\]
\subsection{Operators $\d\Gamma(\cdot)$}
Let $\d\Gamma(\omega)$ be defined in the usual way as a self-adjoint
operator on $\Gamma_\s(\ch)$. Recall that $W(h)$ denote the Weyl
operators on $\Gamma_\s(\fh)$. It is well known that
\[\e^{\i t\d\Gamma(\omega)}W(h)\e^{-\i t\d\Gamma(\omega)}=W(\e^{\i
t\omega}h).\]
Therefore,
the triple $(W,\omega,\d\Gamma(\omega))$
is a covariant CCR
representation (by $W$ we mean
the Fock representation over $\fh$ recalled in Subsection \ref{focko}).
For further reference let us note the following identities, where we set
$z=\frac1{\sqrt 2}\omega g$:
%\[W(g)\d\Gamma(\omega)W(-g)=\d\Gamma(\omega)-\i a^*(z)
%+\i a(z)+(z|\omega^{-1} z),\]
\[W(\i g)\d\Gamma(\omega)W(-\i g)=\d\Gamma(\omega)+ a^*(z)+a(z)
+(z|\omega^{-1} z),\]
\[[W(g),\d\Gamma(\omega)]=-\i a^*(z)W(g)+\i W(g)a(z).\]
\subsection{Van Hove Hamiltonians}
Let $\fh_n$ be the scale of Hilbert spaces associated with the
operator $\omega^{-1}$. This means that for $n\geq0$,
$\fh_n=\Dom(\omega^{-n/2})$,
$\fh_{-n}$ is the space of continuous antilinear functionals on
$\fh_n$. (An alternative notation for $\fh_{-n}$ is $(|\omega|^{-n/2}+1)\fh$).
Let $f\in\fh_{-1}$. Set
\[z:=\frac1{\sqrt2}\omega f\in (\omega^{1/2}+\omega)\fh.\]
It is easy to see that
\beq\rr\ni t\mapsto\e^{\frac{\i}{2}\Im(f|\e^{\i t\omega}f)}
W\left(\i(1-\e^{\i t\omega})f\right)\Gamma(\e^{\i t\omega})
\in U(\Gamma_\s(\fh)).\label{vanh}\eeq
is a strongly continuous unitary group.
Therefore there exists a unique self-adjoint operator
$\d\Gamma_f(\omega)$, that we will call the {\em Van Hove Hamiltonian}, such
that (\ref{vanh}) equals $\e^{\i t\d\Gamma_f(\omega)}$.
Formally, the van Hove Hamiltonian is given by the following expression:
\[\d\Gamma_f(\omega):=\d\Gamma(\omega)+a^*(z)+a(z)+(z|\omega^{-1}z).\]
(In \cite{De} it is called a van Hove Hamiltonian
of the second kind).
Note that the infimum of the spectrum of $\d\Gamma_f(\omega)$ equals $0$
and
\[\begin{array}{l}
\e^{\i t\d\Gamma_f(\omega)}W(h)\e^{-\i t\d\Gamma_f(\omega)}
=\exp\left(\i\Re(f|(\e^{\i t\omega}-1)h)\right)
W(\e^{\i t\omega}h)
.\end{array}\]
If $f\in\fh$, then $\d\Gamma_f(\omega)=W(\i f)\d\Gamma(\omega)W(-\i f)$.
\bet
\ben\item
If $f,g\in\fh_{-1}$, then
the following identities holds:
\[\e^{\i t\d\Gamma_{f+g}(\omega)}
=\e^{\frac{\i}{2}\Im(g|\e^{\i t\omega}g)+\i\Im(f|(\e^{\i
t\omega}-1)g)}
W(\i(1-\e^{\i t\omega})g)\e^{\i t\d\Gamma_{f}(\omega)}.\]
\item If moreover $g\in\fh$, then:
\[\d\Gamma_{f+g}(\omega)=W(\i g)\d\Gamma_f(\omega)W(-\i g)\]
\een\label{vah}\eet
Note that if we consider the
$f$-coherent representation
\beq\fh_1\ni h\mapsto W^f(h):=W(h)\e^{\i\Re(f|h)},\label{coho2}\end{equation}
then it satisfies
\[\e^{\i t\d\Gamma_f(\omega)}W^f(h)\e^{-\i t\d\Gamma_f(\omega)}
=W^f(\e^{\i t\omega}h).\]
Thus, the triple $(W^f,\omega,\d\Gamma_f(\omega)\bigr)$
is a covariant CCR representation.
%\ber If $g\in\omega^{-1/2}\fh+\omega^{-1}\fh$, then it is possible to
%define a self-adjoint operator formally equal to
%\[\d\Gamma(\omega)+a^*( z)+a( z),\]
%In \cite{} it is called a van Hove Hamiltonian of the first kind. It
%differs from the corresponding van Hove Hamiltonian of the second
%kind by a constant (sometimes infinite). \eer
\subsection{Hamiltonian in the Fock sector}
In the remaining part of this section we consider a covariant
representation $\bigl(W^\pi,\omega,H\bigr)$, as at the
beginning of this section.
The following facts are immediate \cite{DG2}:
\bet \ben
\item
The space of vacua $\cK_0^\pi$ is $\e^{\i tH}$ invariant.
\item
The Fock sector $\cH_{[0]}^\pi$ is $\e^{\i tH}$ invariant.\een
\eet
On $\cH_0^\pi=\cK_0^\pi\otimes\Gamma_\s(\fh)$ we define the operator
\[H_0^\pi:=\Omega_0^{\pi*}H\Omega_0^\pi.\]
\bet
We have
\[H_0^\pi:=K_0^\pi\otimes \one+\one\otimes\d\Gamma(\omega),\]
where $K_0^\pi:=H\Big|_{\cK_0^\pi}$. Moreover,
\[H\Omega_0^\pi=\Omega_0^\pi H^\pi.\]
\eet
\subsection{Hamiltonian in a coherent sector}
One can generalize the constructions described in the previous
subsection to the case of coherent sectors.
\bet\ben
\item Let $\fg$ be a dense subspace of $\fh$ and
let $f$ be an antilinear
functional on $\fg$. Then $\e^{\i tH}\cK_f^\pi=\cK_{\e^{\i t\omega}f}^\pi$.
\item
If in addition $f\in\fh_{-2}$,
then $\cH_{[f]}^\pi$ is $\e^{\i tH}$-invariant.
\een\eet
\proof (1)
Let $\Psi\in\cK_f^\pi$. Then
\[\begin{array}{rl}
(\e^{\i tH}\Psi|W(h)\e^{\i tH}\Psi)&=(\Psi|W(\e^{-\i
t\omega}h)\Psi)\\[3mm]
&=\|\Psi\|^2\e^{-\frac14\|h\|^2+\i\Re(f|\e^{-\i t\omega}h)}\\[3mm]
&=\|\Psi\|^2\e^{-\frac14\|h\|^2+\i\Re(\e^{\i
t\omega}f|h)}.\end{array}\]
(2) Since $f\in\fh_{-2}$, we have $(\e^{\i t\omega}-1)f\in\fh$. Hence by Theorem \ref{pas},
$\cH_{[f]}^\pi=\cH_{[\e^{\i t\omega}f]}^\pi$. Thus it
suffices to apply (1). \qed
Set
\[H_f^\pi:=\Omega_f^{\pi*}H\Omega_f^{\pi}.\]
\bet
Suppose that $\fg=\fh_1$ and $f\in\fh_{-1}$.
\ben\item
There exists a unique operator $K_f^\pi$ on $\cK_f^\pi$
such that
\[H_f^\pi:=K_f^\pi\otimes \one+\one\otimes\d\Gamma_f(\omega),\]
\item
$\Omega_f^\pi H_f^\pi=H\Omega_f^\pi$.
\een\label{kaka}
\eet
\proof
Let $h\in\fh_1$.
We first check that
\[\begin{array}{rl}
&\e^{\i tH_f^\pi}
\,\one\otimes W(h)\,\e^{-\i tH_f^\pi}\\[3mm]=&
\Omega_f^{\pi*}\e^{\i tH}
\,W^\pi(h)\e^{-\i\Re(f|h)}\, \e^{-\i
tH}\Omega_f^{\pi}\\[3mm]=&
\Omega_f^{\pi*}
\,W^\pi(\e^{\i t\omega}h)\e^{-\i\Re(f|h)}\Omega_f^{\pi}
\\[3mm]=&
\one\otimes W(\e^{\i t\omega}h)\e^{-\i\Re(f|h)+\i\Re(f|\e^{\i t\omega}h)}
\\[3mm]=&
\one\otimes\e^{\i t\d\Gamma_f(\omega)} W(h)
\e^{-\i t\d\Gamma_f(\omega)}.\end{array}\]
Since linear combinations of
$W(h)$, $h\in\fh_1$, are weakly dense in $B(\Gamma_\s(\fh))$,
for $B\in B(\Gamma_\s(\fh))$ we have
\[
%\begin{array}{l}
\e^{\i tH_f^\pi}\ \one{\otimes } B\ \e^{-\i tH_f^\pi}%\\[3mm]
=\e^{\i t\one{\otimes}\d\Gamma_f(\omega)}\ \one{\otimes} B\
\e^{-\i t\one{\otimes}\d\Gamma_f(\omega)}.%\end{array}
\]
By Lemma \ref{aut}, this implies that $H_f^\pi-\one{\otimes}\d\Gamma_f(\omega)$ is of
the form
$K_f^\pi{\otimes}\one$ for some self-adjoint operator $K_f^\pi$
on $\cK_f^\pi$.
\qed
\subsection{Comparison of coherent sectors of a covariant representation}
In this subsection we assume that
$\fg=\fh_1$ and $f\in\fh_{-1}$.
\bet Let $g\in\fh$.
Then
\[H_{g+f}^\pi=W_{\coh,f}^\pi(-\i g) {\otimes} W(\i g)\,
H_f^\pi \, W_{\coh,f}^{\pi*}(-\i g) {\otimes}W^*(\i g).\]
\[K_{g+f}^\pi=W_{\coh,f}^\pi(-\i g)
K_f^\pi W_{\coh,f}^{\pi*}(-\i g) .\]
\eet
\proof This follows immediately from Theorem \ref{pas}. \qed
Other natural objects that can be introduced in the context of
coherent sectors are the following self-adjoint operators:
\[\begin{array}{l}
K_{[f]}^\pi:=\Omega_f^\pi K_f^\pi{\otimes}\one\Omega_f^{\pi*},\\[3mm]
\d\Gamma_{[f]}^\pi(\omega):=
\Omega_f^\pi \one{\otimes}\d\Gamma_{f}(\omega)\Omega_f^{\pi*}.\end{array}\]
Clearly, they give a natural decomposition of the operator $H$ on the
sector $\cH_{[f]}^\pi$:
\beq HP_{[f]}^\pi=K_{[f]}^\pi+\d\Gamma_{[f]}^\pi(\omega).\label{qhq}\eeq
The decomposition (\ref{qhq}) depends only on the class $[f]$ of $f$
in $\fg^{*}/\ch$.
\bet
\label{stip} If $g\in\fh$, then $K_{[f+g]}^\pi=K_{[f]}^\pi$ and
$\d\Gamma_{[f+g]}^\pi(\omega)=\d\Gamma_{[f]}^\pi(\omega)$.
\eet
Theorem \ref{stip} follows from Theorem \ref{pas} {\rm (iii)} and
Theorem \ref{vah} (2).
\section{Asymptotic CCR representations}
\label{s4}\init
\subsection{Construction of asymptotic CCR representations }
Suppose that $\omega$ is a self-adjoint operator with an absolutely
continuous spectrum
on a Hilbert space
$\fh$.
Let $\cK$ be an additional Hilbert space and $H$ a self-adjoint
operator on $\cH:=\cK\otimes\Gamma_\s(\fh)$. Let $\fg$ be a subspace of
$\fh$ invariant w.r.t. $\e^{\i t\omega}$.
Throughout this section we make the following assumption:
\bea
For any $h\in \fg$, there exists
\[\slim_{t\to\pm\infty}\e^{\i tH} \one{\otimes}W(\e^{-\i t\omega} h)\e^{-\i tH}
=:W^\pm(h).\]
\label{as1}\eea
It is easy to see that the above assumption implies the following theorem:
\bet \ben
\item We have
\[W^\pm(h_1)W^\pm(h_2)=\e^{-\frac\i2\Im(h_1|h_2)}W^\pm
(h_1+h_2),\ \ \ h_1,h_2\in\fg.\]
In other words,
\beq\fg\ni h\mapsto W^\pm(h) \in U(\cK\otimes\Gamma_\s(\fh)),
\label{asu}\end{equation}
are CCR representations.
\item
\[\e^{\i tH}W^\pm(h)\e^{-\i tH}=W^\pm(\e^{\i t\omega}h),\ \ h\in\fg.\]
In other words, $(W^\pm,\omega,H)$ are
covariant CCR representations .\een
\eet
We will call (\ref{asu}) the {\em asymptotic CCR representations}.
Let $\phi^\pm(h)$, $a^\pm(h)$, $a^{\pm*}(h)$,
etc, denote the field, annihilation, creation operators,
etc. associated with the
representations (\ref{asu}). All these objects will be called
``asymptotic'' (or, if there will be a need for a greater precision,
``outgoing/incoming'').
\subsection{Wave and scattering operators}
For any antilinear functional $f$ on $\fg$
we can define the space of asymptotic $f$-coherent vectors
$\cK_f^\pm$,
the asymptotic spaces $\cH_f^\pm$,
the asymptotic Hamiltonian in the
$f$-coherent sector $H_f^\pm$, etc.
The intertwining operators $\Omega_f^\pm$ will be called {\em
$f$-coherent wave
operators}.
In the physical interpretation of these concepts an important role is
played by the so-called {\em scattering operators}:
\[S_{g,f}:=\Omega_g^{+*}\Omega_f^-.\]
Note that they satisfy
\[S_{g,f}H_f^-=H_g^+S_{g,f}.\]
Suppose that we prepare a state in the $f$-coherent sector. It is
natural to describe it by a density matrix $\rho$, which is a
positive trace 1 operator on $\cH_f^-$.
Suppose that we measure an observable within the sector $g$. We can
describe it by a self-adjoint operator $A\in B(\cH_g^+)$.
Then according to the standard rules of quantum mechanics, the
expectation value of the measurement is given by
\[\Tr S_{g,f} \rho S_{g,f}^* A.\]
\subsection{Fock sector of asymptotic representations}
\bet
Eigenvectors of $H$ are contained in the Fock sector $\cK_0^\pm$.
\label{eig1}\eet
\proof We
will show first the following property of Weyl operators on the Fock space:
\beq
\wlim_{t\to\infty} W(\e^{\i t\omega}h)=\exp(-{\scriptstyle\frac{1}{4}}\|h\|^2)
.\label{wlim}\end{equation}
Let $\Psi_1,\Psi_2$ be vectors with a finite number of
particles. Then, by the absolute continuity of $\omega$, $
a(\e^{\i t\omega}h)^n\Psi_i\to0$ when $t\to \infty$. Hence
\[(\Psi_1|W(\e^{\i t\omega}h)\Psi_2)=
\exp(-{\scriptstyle\frac{1}{4}}\|h\|^2)
(\e^{-\frac{\i}{\sqrt2}a(\e^{\i
t\omega}h)}\Psi_1|\e^{\frac{\i}{\sqrt2}
a(\e^{\i t\omega}h)}\Psi_2)\to
\exp(-{\scriptstyle\frac{1}{4}}\|h\|^2)(\Psi_1|\Psi_2).\]
Since $W(\e^{\i t\omega}h)$ is uniformly bounded, this proves (\ref{wlim}).
Assume that $H\Psi=\lambda\Psi$. Then
\[\begin{array}{rl}
(\Psi|W^\pm(h)\Psi)&
=\lim\limits_{t\to\pm\infty}
(\Psi|\e^{\i tH}\one{\otimes}W(\e^{-\i t\omega}h)\e^{-\i tH}\Psi)\\[3mm]&
=\lim\limits_{t\to\pm\infty}
(\Psi|\one{\otimes}W(\e^{-\i
t\omega}h)\Psi)=\|\Psi\|^2\exp(-\frac{1}{4}\|h\|^2).\:\: \Box\end{array}
\]
\section{Dressing operators}
\label{s4a}\init
\subsection{Dressing operator for a pair of CCR representations }
Suppose that $\fh$, $\cH$ are Hilbert spaces and $\fg$ is a dense
subspace of $\fh$.
Consider two CCR representations
\beq \fg\ni h\mapsto W^\pi(h)\in U(\cH),\label{regu6}\end{equation}
\beq \fg\ni h\mapsto W_\ren^\pi(h)\in U(\cH).\label{reg2}\end{equation}
For the representation (\ref{regu6})
we use the notation described in the previous
three sections.
All the objects constructed from (\ref{reg2}) will have an additional
subscript $\ren$ (for ``renormalized'').
For instance, $\phi_\ren^\pi$, $a_\ren^\pi$ and
$a_\ren^{\pi*}$ will denote the field, annihilation and creation
operators for (\ref{reg2}).
Let $g$ be an antilinear functional on $\fg$.
We say that $U^\pi\in U(\cH)$ is a $g$-{\em dressing operator }between
(\ref{reg2}) and (\ref{regu6}) if
for $h\in\fg$, we have
\[
W^{\pi}(h)U^{\pi}= U^{\pi}W_{\ren}^{\pi}(h)\e^{\i\Re(h|g)}.\]
\bet
\ben
\item \label{j1} If $h\in\fg$, then
\[\begin{array}{l}
\phi^{\pi}(h)U^{\pi}= U^{\pi}(\phi^{\pi}_{\ren}(h)+\Re(g| h)),
\\[3mm]
a^{\pi*}(h)U^{\pi}= U^{\pi}(a^{\pi*}_{\ren}(h)+\frac{1}{\sqrt{2}}(g| h)),
\\[3mm]
a^{\pi}(h)U^{\pi}= U^{\pi}(a^{\pi}_{\ren}(h)
+\frac{1}{\sqrt{2}}(h|g)).
\end{array}
\]
\item \label{j2} Let $f$ be an antilinear functional on $\fg$. Then
$\cK_{g+f}^{\pi}=U^\pi\cK_{\ren,f}^{\pi}$.
\item\label{j2a}
Set $U_{\coh,f}^\pi:=U^\pi\Big|_{\cK_{\ren,f}^\pi}$. Then
$U_{\coh,f}^\pi$ is a unitary operator from $\cK_{\ren,f}^\pi$ to
$\cK_{f+g}^\pi$.
\item \label{j3}
$\cH_{[g+f]}^{\pi}=U^\pi\cH_{\ren,[f]}^{\pi}$.
\item \label{j4} $\Omega_{g+f}^\pi=U^\pi\Omega_{\ren,f}^\pi\
U_{\coh,f}^{\pi*}{\otimes}\one$
\een\eet
\proof
\arabicref{j1} is immediate.
Consider $\Psi\in\cK_{\ren,f}^\pi$. Then
\[
%\begin{array}{rl}
(U^\pi\Psi|W^\pi(h)U^\pi\Psi)
=\e^{\i\Re(h|g)}(\Psi|W_\ren^\pi(h)\Psi)%\\[3mm]
=\e^{-\frac14\|h\|^2+\i\Re(h|f+g)}\|\Psi\|^2.
%\end{array}
\]
This proves \arabicref{j2}, which implies
\arabicref{j3} and \arabicref{j2a}.
To show \arabicref{j4}, we compute for $h\in\fg$, $\Psi\in\cK_{\ren,f}$,
\[
\begin{array}{rl}
&U^\pi\Omega_{\ren,f}^\pi\ U_{\coh,f}^{\pi*}{\otimes}\one\
\Psi{\otimes} W(h)\Omega=
U^\pi\Omega_{\ren,f}^\pi\
U^{\pi*}\Psi{\otimes} W(h)\Omega\\[3mm]
=&\e^{-\i\Re(h|f)}U^\pi W_\ren^\pi(h)U^{\pi*}\Psi
=\e^{-\i\Re(h|f+g)}W^\pi(h)\Psi
=\Omega_{f+g}^\pi
\ \Psi{\otimes} W^\pi(h)\Omega.\:\:\Box
\end{array}\]
\subsection{Dressing operators for a pair of covariant representations}
Suppose that $H$ and $H^\ren$ are self-adjoint operators on $\cH$ and
$\omega$ is a self-adjoint operator on $\fh$. We assume that $\fg=\fh_{-2}$.
Consider two covariant CCR representations $(W^\pi,\omega,H)$ and
$(W_\ren^\pi,\omega,H_\ren)$. Recall that this
means that the representations of
CCR (\ref{regu6}) and (\ref{reg2}) satisfy
\beq \begin{array}{l}
\e^{\i tH}W^\pi(h)\e^{-\i tH}=W^\pi(\e^{\i t\omega}h).\end{array}
\label{reg2a}\end{equation}
\beq \begin{array}{l}
\e^{\i tH_\ren}W_\ren^\pi(h)\e^{-\i tH_\ren}=W_\ren^\pi(\e^{\i
t\omega}h)
.\end{array}\label{reg2b}\end{equation}
Let $g\in\fh_{-2}$ and let
$U^\pi\in U(\cH)$ be a $g$-dressing operator between
(\ref{regu6}) and (\ref{reg2}). We say that it is a {\em covariant
$g$-dressing operator} between the covariant representations
(\ref{reg2a}) and (\ref{reg2b})
if
\[\begin{array}{lll}
\e^{\i tH}U^{\pi}\e^{-\i t H_\ren}&=
U^{\pi}W_{\ren}^{\pi}(\i(1-\e^{\i t\omega})g)
\e^{-\frac{\i}{2}\Im(g|\e^{-\i t\omega}g)}
&\\[3mm]
&=W^\pi(\i(1-\e^{\i t\omega})g)U^\pi
\e^{\frac{\i}{2}{\rm Im }(g|\e^{-\i t\omega}
g)},&\ t\in\rr.
\end{array}
\]
\bet Suppose that $\fg=\fh_1$ and $f,g\in\fh_{-1}$. Then
\[K_{g+f}^{\pi}=U_{\coh,f}^\pi K_{\ren,f}^\pi U_{\coh,f}^{\pi *}.\]
\eet
\proof Recall that
\[\begin{array}{lll}
\Omega_{\ren,f}^{\pi*}H_\ren\Omega_{\ren,f}^{\pi}&=
H_{\ren,f}^\pi&=K_{\ren,f}^\pi\otimes\one+\one\otimes\d\Gamma_{f}(\omega),\\[3mm]
\Omega_{g+f}^{\pi*}H\Omega_{g+f}^{\pi}&=
H_{g+f}^\pi&=K_{g+f}^\pi\otimes\one+\one\otimes\d\Gamma_{g+f}(\omega).
\end{array}\]
Hence
\[\begin{array}{rl}
&\e^{\i t U_{\coh,f}^{\pi*}K_{g+f}^\pi U_{\coh,f}^{\pi}}
\otimes\e^{\i t\d\Gamma_{g+f}(\omega)}\\[3mm]
= &U_{\coh,f}^{\pi*}{\otimes}\one\ \e^{\i tH_{g+f}^\pi}\
U_{\coh,f}^{\pi}{\otimes}\one \\[3mm]
=& U_{\coh,f}^{\pi*}{\otimes}\one\
\Omega_{g+f}^{\pi*}
\e^{\i tH}\Omega_{g+f}^{\pi}\
U_{\coh,f}^{\pi}{\otimes}\one \\[3mm]
=&\Omega_{\ren,f}^{\pi*}U^{\pi*}
\e^{\i tH}U^\pi
\Omega_{\ren,f}^{\pi}
\\[3mm]=&\e^{-\frac{\i}{2}\Im(g|\e^{-\i
t\omega}g)}
\Omega_{\ren,f}^{\pi*}W_\ren^\pi(\i(1-\e^{\i t\omega})g)\e^{\i t H_\ren}
\Omega_{\ren,f}^{\pi}
\\[3mm]
=&
\e^{-\frac{\i}{2}\Im(g|\e^{-\i
t\omega}g)+\i\Re(f|\i(1-\e^{\i t\omega})g)}
\ \one{\otimes}W(\i(1-\e^{\i t\omega})g)\ \e^{\i t H_{\ren,f}^\pi}
\\[3mm]
=&\e^{\i tK_{\ren,f}^\pi}\otimes
\e^{\frac{\i}{2}\Im(g|\e^{\i
t\omega}g)+\i\Im(f|(\e^{\i t\omega}-1)g)}
W(\i(1-\e^{\i t\omega})g)\e^{\i t \d\Gamma_f(\omega)}
\\[3mm]
=&\e^{\i tK_{\ren,f}^\pi}\otimes \e^{\i t\d\Gamma_{g+f}(\omega)}.\:\:
\Box
\end{array}\]
\subsection{Coherent asymptotic renormalization}
Let $g\in\fh_{-1}$.
Suppose $H_\ren$ is a self-adjoint operator on $\cH$.
Set
\beq\begin{array}{rl}
U(t)&= \e^{\frac\i2\Im
(g|\e^{-\i t\omega}g)}\e^{\i tH}W(\i(1-\e^{-\i t\omega})g)\e^{-\i
tH_\ren}
\\[3mm]&=\e^{\i tH}\e^{-\i t \one{\otimes}\d\Gamma_g(\omega)}
\e^{\i t \one{\otimes}\d\Gamma(\omega)}\e^{-\i
tH_\ren}.\end{array}\label{fak}\eeq
Clearly, $\Im
(g|\e^{-\i t\omega}g)$ is well defined and $(1-\e^{-\i t\omega})g\in
\fh$, therefore $U(t)$ is well defined.
Moreover, in (\ref{fak}) we used the identity from Theorem \ref{vah}.
Suppose the
following assumption holds:
\bea
$
\slim_{t\to \pm\infty}U(t)$ and $
\slim_{t\to \pm\infty}U^*(t)$ exist.
\label{assa}
\eea
Under Assumption \ref{assa} we set $U^\pm:=
\slim_{t\to \pm\infty}U(t)$. Clearly,
$ \slim_{t\to\pm \infty}U^{*}(t)= U^{\pm *}.$
\bet Suppose Assumption \ref{as1} holds for the Hamiltonian $H$ and
the space $\fg=\fh_1$. Suppose also that
Assumption \ref{assa} is satisfied.
Then the following is true:
\ben
\item
Assumption \ref{as1} holds for the operator $H_\ren$ with $\fg=\fh_1$, that means,
for any $h\in \fh_1$, there exists
\[\slim_{t\to\pm\infty}\e^{\i tH_\ren} \one{\otimes}W(\e^{-\i t\omega} h)
\e^{-\i tH_\ren}
=:W_\ren^\pm(h).\]
\item
\[W_\ren^\pm(h_1)W_\ren^\pm(h_2)=\e^{-\frac\i2\Im(h_1|h_2)}W_\ren^\pm
(h_1+h_2),\ \ \ h_1,h_2\in\fh_1,\]
\[\e^{\i tH_\ren}W_\ren^\pm(h)\e^{-\i tH_\ren}=W_\ren^\pm(\e^{\i t\omega}h),\ \ h\in\fh_1.\]
In other words, the triples
%\beq\fh_1\ni h\mapsto W_\ren^\pm(h) \in U(\cK\otimes\Gamma_\s(\fh)),
%\label{asu2}\end{equation}
$(W_\ren^\pm,\omega,H_\ren)$
are covariant CCR representations .
\item
For $h\in \fh_1$, we have
\[\begin{array}{rl}
W^{\pm}(h)U^{\pm}&= U^{\pm}W_{\ren}^{\pm}(h)\e^{\i\Re(h, g)},\\[3mm]
\e^{\i tH}U^{\pm}\e^{-\i t H_\ren}&=
U^{\pm}W_{\ren}^{\pm}(\i(1-\e^{\i t\omega})g)
\e^{-\frac{\i}{2}\Im(g|\e^{-\i t\omega}g)}
\\[3mm]
&=W^\pm(\i(1-\e^{\i t\omega})g)U^\pm
\e^{\frac{\i}{2}{\rm Im }(g|\e^{-\i t\omega}
g)}.
\end{array}
\]
Therefore, $U^\pm$ are covariant $g$-dressing operators between the
covariant CCR representations $(W_\ren^\pm, \omega, H_\ren)$ and
$(W^\pm, \omega, H)$.
\een
\label{5.2b}
\eet
\noindent
\proof We have
\[\begin{array}{rl}
&\e^{\i tH_\ren}\one{\otimes}W(\e^{-\i t\omega}h)\e^{-\i tH_\ren}\\[3mm]
=&\e^{\i\Re((1-\e^{-\i t\omega})g|\e^{-\i t\omega}h)}
\e^{\i tH_\ren}\ \one{\otimes}W(-\i(1-\e^{-\i t\omega})g)
W(\e^{-\i t\omega}h)
W(\i(1-\e^{-\i t\omega})g)\ \e^{-\i tH_\ren}
\\[3mm]=&\e^{\i\Re((1-\e^{-\i t\omega})g|\e^{-\i t\omega}h)}
U(t)^*\e^{\i tH}\one{\otimes}W(\e^{-\i t\omega}h)\e^{-\i tH}
U(t)\\[3mm]
\to &\e^{-\i\Re(g|h)}U^{\pm*}W^\pm(h)U^\pm,
\end{array}\]
where we used $\lim_{t\to\infty}(g|\e^{-\i t\omega}h)=0$, which follows
from the Riemann-Lebesgue lemma.
This proves (1), (2) and the first identity of (3).
Let us now prove the second identity of (3). We compute:
\[
\begin{array}{rl}
&\e^{\i tH}U(s)\e^{-\i t\Hren}\\[3mm]
=&\e^{\frac\i2\Im(g|\e^{-\i s\omega}g)}
\e^{\i (t+s)H}W(\i(1-\e^{-\i s\omega})g)
\e^{-\i (s+t)H_\ren}\\[3mm]
=&\e^{\frac{\i}{2}\Im(g|\e^{-\i s\omega}g)}
\e^{\frac\i2\Im((\e^{-\i(s+t)\omega}-\e^{-\i
s\omega})g|(1-\e^{-\i(s+t)\omega})g)} \\[3mm]&\times
\e^{\i(s+t)H}W(\i(\e^{-\i(s+t)\omega}-\e^{-\i s\omega})g)
W(\i(1-\e^{-\i(s+t)\omega})g)
\e^{-\i(s+t)\Hren}
\\[3mm]
=&\e^{\frac{\i}{2}\Im(g|\e^{-\i t\omega}g)}
\e^{\i\Im(g|\e^{-\i s\omega}(1-\e^{-\i
t\omega})g)}
\e^{\i(s+t)H}W(\i(\e^{-\i(s+t)\omega}-\e^{-\i s\omega})g)\e^{-\i
(s+t)H}\\[3mm]
&\times\e^{\frac{\i}{2}\Im(g|\e^{-\i(s+t)\omega}g)}
\e^{\i
(s+t)H}W(\i(1-\e^{-\i(s+t)\omega})g)
\e^{-\i(s+t)\Hren}\\[3mm]
\to&\e^{\frac{\i}{2}\Im(g|\e^{-\i t\omega}g)}
W^\pm(\i(1-\e^{\i t\omega})g)U^\pm,
\end{array}
\]
where we used the Riemann-Lebesgue lemma to show that
$\lim\limits_{s\to\infty}\Im(g|\e^{-\i s\omega}(1-\e^{-\i
t\omega})g)=0$.
\qed
\section{Pauli-Fierz Hamiltonians}
\label{s5}
\init
In this section we apply the abstract formalism developed in Sections
\ref{s2}-\ref{s4} to a class of Pauli-Fierz Hamiltonians. We will
formulate a set of assumptions that will guarantee a satisfactory
scattering theory and the existence of a dressing operator.
\subsection{Coupling Fock space}
Let $\fh$, $\cK$ be Hilbert spaces. Let $\fh_1$ and $\cK_1$ be dense
subspaces of $\fh$ and $\cK$.
Let $\otimesal$ denote the algebraic tensor product. Let
\[(\cK_1\otimesal\fh_1)\times\fh_1\ni(\Psi_1,\Psi_2)\mapsto
(\Psi_1|v\Psi_2)\in\cc\]
be a sesquilinear form.
Let $\Gammal_\s(\fh_1)$ denote the algebraic Fock space over the vector
space $\fh_1$.
We define the annihilation form and creation forms $ \Wick(v^*)$ and
$\Wick(v)$
as the forms on
$\cK\otimes \Gamma_\s(\fh)$ with the domain $\cK_1\otimesal
\Gammal_\s(\fh_1)\subset\cK\otimes \Gamma_\s(\fh)$
as follows:
if $h_1,h_2\in\ch_1$ and $\Psi_1,\Psi_2\in\cK_1$, then
\[(\Psi_2{\otimes}h_2^{\otimes m}|
\Wick(v)\Psi_1{\otimes}h_1^{\otimes n})=\left\{\begin{array}{ll}\sqrt{m}
(\Psi_2\otimes h_2|v h_1)(h_2|h_1)^n,& m=n+1\\[3mm]
0,&m\neq n+1;\end{array}\right.\]
\[(\Psi_2{\otimes}h_2^{\otimes m}|
\Wick(v^*)\Psi_1{\otimes}h_1^{\otimes n})=\left\{\begin{array}{ll}
\sqrt{n}
(\Psi_2|v^*\Psi_1{\otimes } h_1)(h_2|h_1)^{m}& m=n-1,\\ 0& m\neq
n-1\end{array}\right.\]
Note that if $v$ is bounded, then $\Wick(v)$ and $\Wick(v^*)$ extend to
closed operators adjoint to one another. We will write
$\Wick(v_1+v_2^*)$ for $\Wick(v_1)+\Wick(v_2^*)$.
For a vector $z\in\fh$ the operators
$|z)\in B(\cc,\fh)$ and its adjoint
$(z|\in B(\fh,\cc)$ are defined in the
in the usual way:
\[\cc\ni \lambda\mapsto |z)\lambda:=\lambda z\in\fh,\ \ \ \
\fh\ni h\mapsto (z|h:=(z|h)\in\cc.\]
Note that the usual creation and annihilation operators correspond to
the case $\cK=\cc$: if $z\in\fh$, then
\[\Wick(|z))=a^*(z),\ \ \ \ \Wick((z|)=a(z).\]
For further reference let us note the identities
\[
\begin{array}{l}
W(\i g)\Wick(v)W(-\i g)=\Wick(v)+\frac1{\sqrt 2}\one{\otimes}(g|v,
\\[2mm]
W(\i g)\Wick(v^*)W(-\i g)=\Wick(v^*)+\frac1{\sqrt 2}v^*\one{\otimes}|g).
\end{array}
\]
(In the above identities we dropped
the factors ${\otimes}\one_{\Gamma_\s(\fh)}$).
%{\it Comment: I replaced the operator inequality in your version by a
%form inequality, it seems needed if one wants to use the resolvent
%$(K+1)^{-\12}$.}
Let us note the following inequalities:
\bel
For $\Psi\in\Gamma_\s(\fh)$, $R>0$ and a positive operator $\omega$ on $\fh$ we have
\beq\|\Wick(v^*)\Psi\|^2\leq(\Psi|\one{\otimes}\d\Gamma(\omega)\Psi)\|v^*\
\one{\otimes}
\omega^{-1}\ v\|;\label{dfg1}\eeq
\beq
\label{dfg}
%\begin{array}{rl}
|(\Psi|\Wick(v^*)\Psi)|%\\
\leq \|\one\otimes
\omega^{-\12}v(K+R)^{-\12}\|_{\cB(\cK, \cK\otimes\ch)}\|(K+R)^{\12}\otimes\one
\Psi\|\|\one\otimes \d\Gamma(\omega)^{\12}\Psi\|.
%\end{array}
\eeq
\eel
\proof The proof of the first inequality can be found e.g
in \cite{DJ} and \cite{GGM}.
The second inequality is proved e.g in
\cite[Corollary 3.10]{GGM}).
For the reader's convenience we will show how the first inequality
implies the second.
Set $\tilde v:=v(K+R)^{-1/2}$. Now
\[\begin{array}{rl}
|(\Psi|\Wick(v^*)\Psi)|&=
|\bigl((R+K)^{1/2}{\otimes}\one\ \Psi\ |\
\Wick(\tilde v^*)\Psi\bigr)|\\[3mm]
&\leq\|(R+K)^{1/2}{\otimes}\one\ \Psi\|\|\Wick(\tilde v^*)\Psi\|\\[3mm]
&\leq\|(R+K)^{1/2}{\otimes}\one\ \Psi\|\|1{\otimes}\omega^{-1/2}\ \tilde
v\|\|\one\otimes\d\Gamma(\omega)^{1/2} \Psi\|.\end{array}\]
\qed
\subsection{Pauli-Fierz Hamiltonians}
Consider a positive operator $K$ on $\cK$ and a positive operator
$\omega$ on $\fh$.
The operator
\[H_0:=K\otimes\one+\one\otimes\d\Gamma(\omega)\]
will be called a free Pauli-Fierz Hamiltonian.
The following assumption is weaker than Assumption \ref{main2}:
\bea
$v$ is a form on $\cK_{1}{\otimes}\ch_{1}\times\cK_{1}$ such that
$\limsup\limits_{R\to\infty}\|\omega^{-1/2}v(K+R)^{-1/2}\|<1/2$
\label{pf1}\eea
>From the inequality (\ref{dfg}) one deduces the following theorem:
\bet Under Assumption \ref{pf1}, the quadratic form
\[\Wick(v+v^*)\]
is form bounded wrt $H_0$ with the bound less than $1$.
Therefore, by the KLMN theorem, we can define the Pauli-Fierz
Hamiltonian as the self-adjoint operator
\[H:=H_0+\Wick(v+v^*),\]
with the same form domain as $H_0$.
\eet
\subsection{Asymptotic CCR representations for Pauli-Fierz Hamiltonians}
As before, let $\fh_n$ be the scale of Hilbert spaces associated with
$\omega^{-1}$.
The following assumption can be called the {\em short range condition} and
is the equivalent of Assumption \ref{qea4}:
\bea There exists a subspace $\fg\subset\fh_1\cap\Dom(\omega^{1/2}) $
dense in $\fh_1$ in the topology of $\fh_1$ such that
for $h\in\fg$ and almost all
$t\in\rr$, the operator \beq B(t):=
\bigl(\one_\cK{\otimes}(\e^{-\i t\omega}h|\ v
+\hc\bigr)(1{+}K)^{-1/2}\label{gag}\eeq is bounded and
\[\int_0^\infty\|B(t)\|\d t<\infty.
\]
\label{qea}\eea
\ber
Note that in (\ref{gag})
$\one_\cK{\otimes}(\e^{-\i t\omega}h|\ v
$ denotes an operator in $B(\cK)$
and $\hc$ stands for its hermitian conjugate, that is the operator
$v^*\ \one_\cK{\otimes}|\e^{-\i t\omega}h)$. \eer
\bet Suppose Assumptions \ref{pf1} and \ref{qea} hold. Then
\ben\item
For all $h\in \fh_1$ there exist
\beq
W^{\pm}(h):= \slim_{t\fld \pm\infty}\e^{\i tH}\one{\otimes}W (\e^{-\i t\omega}h)
\e^{-\i tH}.
\label{eas.3bis}
\end{equation}
\item
The map
\beq\fh_1\ni h\mapsto W^\pm(h)\label{kwer.1}\end{equation}
is strongly continuous.
Consequently $(W^{\pm}, \omega, H)$ are two regular covariant CCR
representations.
\item
For all $h\in\fh_1$
\beq
W^{\pm}(h)(\i+H)^{-1/2}= \lim_{t\fld \pm\infty}\e^{\i tH}\one{\otimes}W
(\e^{-\i t\omega}h)
(\i+H)^{-1/2}\e^{-\i tH}.
\label{eas.3ter}
\end{equation}
\item
The map
\beq
\fh_1\ni h\mapsto W^\pm(h)(\i+H)^{-1/2}\label{kwer.2}
\end{equation}
is norm continuous.
\item for all $h\in \ch_{1}$, $\Dom(H+c)^{\12}\subset
\Dom(\phi^{\pm}(h))$ and
\[
\phi^{\pm}(h)(H+c)^{-\12}= \slim_{t\to\pm \infty}\e^{\i tH}\phi(\e^{-\i
t\omega}h)\e^{\i tH}(H+c)^{-\12}.
\]
\item for any $\epsilon>0$ the CCR representations
$W^{\pm}$ are of Fock type when restricted to $\one_{[\epsilon,
+\infty[}(\omega)\ch$.
\een\label{scac}\eet
\proof For shortness, we drop $\one_\cK\otimes$ in the formulas below.
We have
\[
W(\e^{-\i t\omega}h)= \e^{-\i tH_{0}}W(h)\e^{\i tH_{0}},
\]
which implies that $t\mapsto (1+H_0)^{-1}W(\e^{-\i
t\omega}h)(1+H_0)^{-1}$ is $C^1$ and
\beq
\begin{array}{l}
\partial_{t}(1+H_0)^{-1}W (\e^{-\i t\omega} h)(1+H_0)^{-1}\\[3mm]=
-(1+H_0)^{-1}[H_{0}, \i W (\e^{-\i t\omega}h)](1+H_0)^{-1}\\[3mm]=
-\frac{1}{\sqrt 2}(1+H_0)^{-1}\left(a^*(\e^{-\i t\omega}h)
W (\e^{-\i t\omega}h)- W (\e^{-\i t\omega}h)a(\e^{-\i t\omega}
h)\right)
(1+H_0)^{-1}.\end{array}
\label{eas.2}\end{equation}
Using
the fact that $\e^{\i t\omega}h\in\Dom(\omega^{-1/2})$ we see that
$(1+H_0)^{-1/2}a^*(\e^{-\i t\omega}h)$ is bounded. Therefore, from
(\ref{eas.2}) and Lemma \ref{diffo} we can actually conclude that
\[t\mapsto(1+H_0)^{-1/2}W(\e^{-\i
t\omega}h)(1+H_0)^{-1/2}\] is $C^1$ and
\beq
\begin{array}{l}
\partial_{t}(1+H_0)^{-1/2}W (\e^{-\i t\omega} h)(1+H_0)^{-1/2}\\[3mm]=
-\frac{1}{\sqrt 2}(1+H_0)^{-1/2}\left(a^*(\e^{-\i t\omega}h)
W (\e^{-\i t\omega}h)- W (\e^{-\i t\omega}h)a(\e^{-\i t\omega}
h)\right)
(1+H_0)^{-1/2}.\end{array}
\label{eas.2a}\end{equation}
But $(1+H_0)^{-1/2}(c+H)^{1/2}$ is bounded, so
$t\mapsto(c+H)^{-1/2}W(\e^{-\i
t\omega}h)(c+H)^{-1/2}$ is $C^1$ and
we can replace $(1+H_0)^{-1/2}$ with $(c+H)^{-1/2}$ in (\ref{eas.2a}).
Now, $t\mapsto(c+H)^{-1}\e^{\i tH}W (\e^{-\i t\omega}h)\e^{-\i
tH}(c+H)^{-1}$ is $C^1$ and we have
\[\begin{array}{l}
\partial_{t}(c+H)^{-1}\e^{\i tH}W(\e^{-\i t\omega}h)\e^{-\i
tH}(c+H)^{-1}\\[3mm]=(c+H)^{-1}\e^{\i tH}\i[H,W(\e^{-\i t\omega}h)]\e^{-\i
tH}(c+H)^{-1}\\[3mm]
+\e^{\i tH}\left(\partial_{t}(c+H)^{-1}W(\e^{-\i
t\omega}h)(c+H)^{-1}\right) \e^{-\i
tH}\\[3mm]
=\e^{\i tH}(c+H)^{-1}\i[\Wick(v+v^*),W(\e^{-\i
t\omega}h)](c+H)^{-1} \e^{-\i
tH}\\[3mm]
= \frac1{\sqrt2}(c+H)^{-1}\e^{\i tH}W(\e^{-\i t\omega}h)\left(
(\e^{-\i t\omega}h|v
-v^*|\e^{-\i t\omega}h)\right)
\e^{-\i tH}(c+H)^{-1}.\end{array}
\]
Eventually, using again Lemma \ref{diffo}, we can write
\beq\begin{array}{l}
\partial_{t}\e^{\i tH}W(\e^{-\i t\omega}h)\e^{-\i
tH}(c+H)^{-1/2}\\[3mm]
= \frac1{\sqrt2}\e^{\i tH}W(\e^{-\i t\omega}h)\left(
(\e^{-\i t\omega}h|v
-v^*|\e^{-\i t\omega}h)\right)
\e^{-\i tH}(c+H)^{-1/2}.\end{array}
\label{qrw}\end{equation}
The norm of (\ref{qrw}) can be estimated by
\[c\left\|\left((\e^{-\i t\omega}h|v
-v^*|\e^{-\i t\omega}h)\right)(1{+}K)^{-1/2}\right\|.\]
By Assumption
\ref{qea}, if $h\in\fg$,
this is integrable. Therefore, by the Cook method there
exists
\beq
\lim_{t\fld \pm\infty}\e^{\i tH}W
(\e^{-\i t\omega}h)
(\i+H)^{-1/2}\e^{-\i tH}.\label{qaf}\end{equation}
If $h\in\fh_1$, then we will find a sequence $(h_n)$ in $\fg$ such
that $h_n\to h$ in the norm of $\fh_1$. Clearly, $\|h_n\|_{\fh_1}$
is uniformly
bounded.
Now, using Lemma \ref{diffo} and estimate (\ref{dfg}) we get
\[\begin{array}{l}
\sup\limits_t
\|\e^{\i tH}W(\e^{-\i t\omega}h)\e^{-\i tH}(c+H)^{-1/2}-
\e^{\i tH}W(\e^{-\i t\omega}h_n)\e^{-\i tH}(c+H)^{-1/2}\|\\[3mm]
\leq
\sup\limits_t c\|(W(\e^{-\i t\omega}h)-
W(\e^{-\i t\omega}h_n))(1+\d\Gamma(\omega))^{-1/2}\|
\\[3mm]\leq c_1(\|h-h_n\|+\|\phi(h-h_n)(1+\d\Gamma(\omega))^{-1/2}\|\\[3mm]
\leq c_2\left(\|h-h_n\|+\|\omega^{-1/2}(h-h_n)\|\right).\end{array}\]
This proves the existence of the norm limit (\ref{qaf}) for an
arbitrary $h\in\fh_1$, and also shows
\[\lim_{n\to\infty} W^\pm(h_n)(c+H)^{-1/2}=W^\pm (h)(c+H)^{-1/2}.\]
This proves (3) and (4). Now (1) and (2) follow
by a simple density argument. The proof of (5) can be done as in eg
\cite[Thm. 8.2]{Ge2}. It remains to prove (6). We will use the notion
of the {\em number quadratic form }associated to a regular CCR
representation (see eg \cite[Sect. 4.2]{DG2}). Let us fix $\epsilon>0$
and let $\ff$ be
a finite dimensional subspace in $\ch_{\epsilon}:=\one_{[\epsilon,
+\infty[}(\omega)\ch$. Let $n_{\ff}^{\pm}$ be the quadratic form
equal to:
\[
n_{\ff}^{\pm}(u,u)=\sum_{i=1}^{n}\|a^{\pm}(f_{i})u\|^{2}, \hbox{
with domain }\bigcap_{i=1}^{n}\Dom(a^{\pm}(f_{i})).
\]
where $(f_{1}, \dots, f_{n})$ is an orthonormal basis of ${\ff}$.
It is easy to see that $n_{\ff}^{\pm}$ does not depend on the
choice of the o.n.b. of ${\ff}$. One can then define the {\em number
quadratic forms} $n^{\pm}$ as:
\[
n^{\pm}:=\sup_{{f}\subset \ch_{\epsilon}, \: {\rm dim}{\ff}
<\infty }n_{\ff}^{\pm}.
\]
Then (see e.g. \cite[Thm. 4.3]{DG2}) the CCR representations $W^{\pm}$
are of Fock type iff $n^{\pm}$ are densely defined.
We claim that there exist a
constant $C$, independent of ${\ff}\subset \ch_{\epsilon}$ such that:
\beq
\label{yrch}
n_{\ff}^{\pm}(u,u)\leq C(u, (H+c)u), \: u\in \Dom((H+c)^{\12}),
\eeq
which implies that $\Dom((H+c)^{\12})\subset \Dom(n^{\pm})$ and hence
completes the proof of (6). In fact using (5), we obtain for $u\in
\Dom((H+c)^{\12})$:
\beq
\label{elf}
n^{\pm}_{\ff}(u,u)=\lim_{t\to\pm \infty}(\e^{-\i tH}u,
\d\Gamma(\e^{\i t\omega}\pi_{\ff}\e^{-\i t\omega})\e^{-\i tH}u),
\eeq
where $\pi_{\ff}$ is the orthogonal projection on ${\ff}$. Next
we have $\e^{\i t\omega}\pi_{\ff}\e^{-\i t\omega}\leq
\one_{[\epsilon, +\infty[}(\omega)\leq \epsilon^{-1}\omega$, and
hence $\d\Gamma(\e^{\i t\omega}\pi_{\ff}\e^{-\i t\omega})\leq
\epsilon^{-1}\d \Gamma(\omega)\leq C(H+c)$, uniformly w.r.t. $\ff$. by
(\ref{elf}) this implies (\ref{yrch}) and completes the proof of the
theorem. \qed
\subsection{Renormalized Pauli-Fierz Hamiltonian}
\label{normre}
The following assumption is weaker than Assumption \ref{main4}:
\bea
We assume that
\[\begin{array}{l}
v=|z)\otimes \one_\cK+v_\ren,\\[3mm]
z\in\fh,\ \ \ v_\ren\in B(\Dom(K^{\12}),\cK\otimes\fh),\\[3mm]
(z|(1+\omega^{-1})z)<\infty,\\[3mm]
\|\omega^{-1}v_\ren (1{+}K)^{-1/2}\|<\infty.
\end{array}\]
\label{pf2}\eea
Set \[
g:=\sqrt 2\omega^{-1}z.
\]
Note that $g\in\omega^{-1/2}\fh$.
The assumption below is the equivalent of Assumption \ref{main6}:
\bea For almost all $t\in\rr$, the operator
\[C(t):=\bigl(\one_\cK{\otimes}(\e^{-\i t\omega}g|\ v_\ren+\hc
\bigr)(1{+}K)^{-1/2}\] is bounded and
\[\int_0^\infty\|C(t)\|\d t<\infty.
\]
\label{pf3}\eea
Introduce the renormalized Hamiltonian
\[H_\ren:=K_\ren\otimes\one+\one\otimes\d\Gamma(\omega)+\Wick(v_\ren+
v_\ren^*),\]
where
\[v_\ren:=v-\one_\cK\otimes|z),\]
\[\begin{array}{rl}
K_\ren&:=K+(z|\omega^{-1}z)-\one{\otimes}(\omega^{-1}z|\ v\
-\ v^*\ \one{\otimes}|\omega^{-1}z)\\[3mm]
&
=K-(z|\omega^{-1}z)-\one{\otimes}(\omega^{-1}z\ |v_\ren\ -\
v_\ren^*\one{\otimes}\ |\omega^{-1}z).\end{array}\]
Note that if $g\in\fh$, then
\[H=W(\i g)H_\ren W(-\i g).\]
As in (\ref{fak}), set
\[
U(t)= \e^{\frac\i2\Im
(g|\e^{-\i t\omega}g)}\e^{\i tH}W(\i(1-\e^{-\i t\omega})g)\e^{-\i tH_\ren}.
\]
\bet \ben\item
Suppose Assumptions \ref{pf1}, \ref{pf2} and \ref{pf3} hold. Then
there exist
\[U^\pm:=\slim_{t\to\pm\infty}U(t).\]
Moreover,
\[U^{\pm*}=\slim_{t\to\pm\infty}U^*(t).\]
\item Suppose in addition Assumption \ref{qea}. Then there exist the
limits
\[\slim_{t\to\pm\infty}\e^{\i tH_\ren} \one{\otimes}W(\e^{-\i t\omega} h)
\e^{-\i tH_\ren}
=:W_\ren^\pm(h).\]
Moreover, $U^\pm$ are covariant $g$-dressing operators
between the
representations $(W_\ren^\pm,\omega,H_\ren)$ and $(W^\pm,\omega,H)$
and
satisfy all the properties described in Section $\ref{s4a}$.\een
\label{dressa}\eet
\proof We have
\[\begin{array}{rl}
&\frac{\d}{\d t}\e^{\i tH}\e^{-\i t \one\otimes\d\Gamma_g(\omega)}
\\[3mm]=&\i\e^{\i tH}\left(\Wick(v+v^*)+K\otimes\one-\one\otimes
a^*(z)-\one\otimes a(z)-(z|\omega z)\right)\e^{-\i
t\one\otimes\d\Gamma_g(\omega)}\\[3mm]
=&\i\e^{\i tH}\left(\Wick(v_\ren+v_\ren^*)+K_\ren\otimes\one
+(\omega^{-1}z|v_\ren+v_\ren^*|\omega^{-1}z)\right)\e^{-\i
t\one\otimes\d\Gamma_g(\omega)}
\end{array}
\]
and
\[\begin{array}{l}
\frac{\d}{\d t}\e^{\i t\one\otimes\d\Gamma(\omega)}
\e^{-\i tH_\ren}=-\i\e^{\i t\one\otimes\d\Gamma(\omega)}
\left(K_\ren\otimes\one+\Wick(v_\ren+v_\ren^*)\right)\e^{-\i tH_\ren}
.\end{array}\]
Hence,
\[\begin{array}{rl}\frac{\d}{\d t}U(t)&=
\Big(\frac{\d}{\d t}\e^{\i tH}\e^{-\i t \one\otimes\d\Gamma_g(\omega)}\Big)
\e^{\i t\one\otimes\d\Gamma(\omega)}
\e^{-\i tH_\ren}+
\e^{\i tH}\e^{-\i t \one\otimes\d\Gamma_g(\omega)}
\frac{\d}{\d t}\e^{\i t\one\otimes\d\Gamma(\omega)}
\e^{-\i tH_\ren}\\[4mm]
&=
\i\e^{\frac\i2\Im
(g|\e^{-\i t\omega}g)}\e^{\i
tH}\Big(\Wick(v_\ren+v_\ren^*)+(\omega^{-1}z|v_\ren+
v_\ren^*|\omega^{-1}z)\\[4mm]
&-
W(\i(1-\e^{-\i t\omega})g)\Wick(v_\ren+v_\ren^*)W(-\i(1-\e^{-\i
t\omega})g)\Big)W(\i(1-\e^{-\i t\omega})g)
\e^{-\i tH_\ren}\\[4mm]
&=\i \e^{\frac\i2\Im
(g|\e^{-\i t\omega}g)}\e^{\i
tH}\Big((\e^{-\i t\omega}g|v_\ren+
v_\ren^*|\e^{-\i t\omega}g)\Big)
W(\i(1-\e^{-\i t\omega})g)
\e^{-\i tH_\ren}.
\end{array}\]
Therefore,
\[\begin{array}{l}
\int_0^\infty\|\frac{\d}{\d t}U(t)(\i +H_\ren)^{-1/2}\|\d
t<\infty.\end{array} \]
This means that Assumption \ref{assa} holds and we can apply the
results of Section \ref{s4a}.
\qed
\subsection{Covariance of renormalized objects}
\label{cova}
The renormalization depends on the splitting of $v$ given in
Assumption \ref{pf2} into a singular scalar part and the regular
part. This splitting is to some extent arbitrary. In this subsection
we study how various ``renormalized'' objects depend on this splitting.
We will replace $g$ by $\tilde{g}= g+h$, for
$h\in \ch$ and denote with tildes the new objects obtained with the
function $\tilde{g}$.
\bet Suppose Assumptions \ref{pf1}, \ref{qea}, \ref{pf2} and \ref{pf3} hold.
Let $f\in\fh_{-1}$ and $h_1\in\fh_1$. Then
\ben
\item
$\tilde H_\ren=W(-\i h)H_\ren W^*(\i h)$.
\item
$\tilde W_\ren^\pm(h_1)=W(-\i h)W_\ren^\pm(h_1) W^*(\i h)$, $h_1\in\fh_1$.
\item
$\tilde \cK_{\ren,f}^\pm=W(-\i h)\cK_{\ren,f}^\pm$.
\item
$\tilde \cH_{\ren,[f]}^{\pm}=W(-\i h)\cH_{\ren,[f]}^{\pm}$.
\item
$\tilde \cH_{\ren,f}^{\pm}=W(-\i h)\cH_{\ren,f}^{\pm}$.
\item
$\tilde K_{\ren,f}^\pm=W(-\i h)K_{\ren,f}^\pm W(\i h)$.
\item
$\tilde H_{\ren,f}^{\pm}=W(-\i h){\otimes}\one \,H_{\ren,f}^{\pm}\,W(\i
h){\otimes} \one$.
\item
If in addition $h\in\fh_1$, then there exists $\tilde U^\pm$ and
$\tilde{U}^{\pm}= W^{\pm}(-\i h)U^{\pm}\ \one{\otimes}W(\i h)$.
\een\eet
\proof Direct computation proves (1). To prove (2) we compute
for $h_1\in\fh_1$\[
\begin{array}{rl}
&\tilde{W}_{\ren}^{\pm}(h_1)\\[3mm]
=&\slim_{t\to\pm\infty}W(-\i h)\e^{\i t\Hren}W(\i h)
W(\e^{-\i\omega t}h_1)W(-\i h)\e^{-\i t\Hren}W(\i h)\\[3mm]
=&\slim_{t\to\pm\infty}\e^{\i \Re(h|\e^{-\i t\omega}h_1)}
W(-\i h)\e^{\i t\Hren}
W(\e^{-\i\omega t}h_1)\e^{-\i t\Hren}W(\i h)
\\[3mm]
= &W(-\i h)W_{\ren}^{\pm}(h_1)W(\i h),
\end{array}
\]
since $(h| \e^{-\i t\omega}h_1)\to 0$ when $t\to\pm\infty$ by the
Riemann-Lebesgue
lemma.
(1) and (2) directly imply all the statements but (8), which we prove below:
\[\begin{array}{rl}
&U^{*}(t)\tilde U(t) W(-\i h)\\[3mm]
=&\e^{\i tH_\ren}\e^{-\i t\d\Gamma(\omega)}\e^{\i t\d\Gamma_g(\omega)}
\e^{-\i t\d\Gamma_{\tilde g}(\omega)}\e^{\i t\d\Gamma(\omega)}W(-\i
h)\e^{-\i tH_\ren}\\[3mm]
=&
\e^{-\frac\i2\Im(h|\e^{\i t\omega}h)-\i \Im(g|(\e^{\i t\omega}-1)h)}
\e^{\i tH_\ren}\e^{-\i t\d\Gamma(\omega)}W(-\i(1-\e^{\i t\omega})h)
\e^{\i t\d\Gamma(\omega)}W(-\i
h)\e^{-\i tH_\ren}\\[3mm]
=&
\e^{-\frac\i2\Im(h|\e^{\i t\omega}h)-\i \Im(g|(\e^{\i t\omega}-1)h)}
\e^{\i tH_\ren}W(\i(1-\e^{-\i t\omega})h)
W(-\i
h)\e^{-\i tH_\ren}\\[3mm] =&
\e^{-\i\Im(\tilde g|\e^{\i t\omega}h)+\i\Im(g|h)}
\e^{\i tH_\ren}W(-\i\e^{-\i t\omega}h)
\e^{-\i tH_\ren}\\[3mm]
\underset{\rm strongly}{\to}&\e^{\i\Im(g|h)}W_\ren^\pm(-\i h),\end{array}\]
where we used the Riemann-Lebesgue lemma, and the fact that
$h\in\fh_1$, $\tilde g\in\fh_{-1}$
to show that $\lim_{t\to\infty}(\tilde g|\e^{\i t\omega}h)=0$.
Therefore
\[
%\begin{array}{rl}
\tilde U^\pm=\e^{\i\Im(g| h)}U^\pm W_\ren^\pm(-\i h)W(\i h)%\\[3mm]
=W^\pm(-\i h)U^\pm W(\i h).\:\:\Box
%\end{array}
\]
\appendix
\section{Appendix}
\init
In the appendix we prove a number of technical lemmas needed in
Section \ref{s5}.
\subsection{Differentiability of operator valued functions}
\bel Consider a function \beq]-\epsilon,\epsilon[\ni t\mapsto
C(t)\in B(\cH).\label{ava1}\end{equation}Suppose that for some dense
subspaces $\cB$, $\cD$ and $\Phi\in\cB$, $\Psi\in\cD$ the derivative
\beq\frac{\d}{\d t}(\Phi|C(t)\Psi)\label{ava}\end{equation}
exists. Suppose that $]-\epsilon,\epsilon[\ni t\mapsto C'(t)\in B(\cH)$ is a
continuous function and (\ref{ava})
equals $(\Phi|C'(t)\Psi)$. Then (\ref{ava1}) is norm differentiable
and its derivative equals $C'(t)$, that means
\beq\lim_{s\to0}\frac{C(t+s)-C(t)}{s}=C'(t).\label{difu}\eeq
\label{diffo}\eel
\proof It suffices to prove (\ref{difu}) for $t=0$.
For $\Phi\in\cB$ and $\Psi\in\cD$,
\[\begin{array}{rl}
\Big(\Phi|\bigl(s^{-1}(C(s)-C(0)-C'(0)\bigr)\Psi\Big)&=s^{-1}\int_0^s
(\Phi|\bigl(C'(s_1)-C'(0)\bigr)\Psi\Big)\d s_1
.\end{array}\]
Hence
\[\begin{array}{rl}
\Big(\Phi|\bigl(s^{-1}(C(s)-C(0)-C'(0)\bigr)\Psi\Big)&\leq
\sup\{\|C'(s_1)-C'(0)\|\ :\ |s_1|<|s|\}\|\Phi\|\|\Psi\|.
\end{array}\]
Thus
\[\|s^{-1}\bigl(C(s)-C(0)\bigr)-C'(0)\|\leq\sup\{\|C'(s_1)-C'(0)\|\ :\
|s_1|