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\begin{document}
\title{
Ground State
of the Massless Nelson Model Without
Infrared Cutoff in a Non-Fock Representation}
\author{Asao Arai
\thanks{Supported by the Grant-In-Aid No.
11440036 for Scientific  Research  from the Ministry of Education,
Science, Sports and Culture,
Japan.}\\
Department  of Mathematics\\
Hokkaido University \\
Sapporo 060-0810\\
Japan
}
\maketitle

\begin{abstract}
We consider a model of quantum particles
coupled to a  massless  quantum scalar
field, called the massless Nelson model,
{\it in a non-Fock representation} of the
time-zero fields which satisfy the canonical commutation
relations.
We show that the model has
a ground  state
for {\it all} values of the coupling
constant {\it even in the case where
no infrared cutoff
is made}. The non-Fock representation used
is inequivalent to the Fock one
if no infrared cutoff is made.
\end{abstract}

\noindent
{\bf Key words}:
Nelson's model, massless quantum field,
infrared divergence, ground state, canonical
commutation relations, non-Fock representation
\bigskip
\section{Introduction}
We consider a  system of $N$ quantum particles
($N\in \N$)
moving on the $d$-dimensional Euclidean space
$\R^d$  ($d\in \N$) under the influence of
an external potential $V: \R^{dN} \to \R$ (Borel measurable)
and coupled to a massless quantum
scalar field. The model we discuss  here is the
so-called {\it massless Nelson model} \cite{Ne}.
The problem to which we address ourselves in this paper is
that of existence of a ground state
of  the model {\it without infrared cutoff}.
It has been shown  that
the massless Nelson model {\it with infrared cutoff}
has a  ground state \cite{Sp2,Ger}.
A natural  question to be asked next is
if the  model {\it without infrared cutoff}
has a ground state or not.

We remark that, generally speaking,
there is some subtlety
on the existence of ground states of  massless quantum field models,
which is due to possible infrared divergences.
Indeed, in a class of  models which describe
interactions of particles and massless quantum fields,
it is proved or suggested
that, if the time-zero fields are
given by the Fock representation and no infrared cutoff is made
in the interactions, then
the models have no ground states,
althogh it may depend on
the strength of the parameters
contained in the Hamiltonians \cite{Fro, Sp1, AHH, AH}.
On the other hand, the Pauli-Fierz model
in non-relativistic quantum electrodynamics
has a ground state in the Fock representation of time-zero fields
even if no infrared cutoff  is made.
\cite{BFS, GLL}.

As for the massless
Nelson model without infrared cutoff,
L\"orinczi, Minlos and Spohn \cite{LMS} recently  proved
that,
in the case $d=3$ and $N=1$, it
has no ground state within the Fock space
where the time-zero fields are given by the usual Fock representation
of the canonical commutation relations
(CCR) indexed by ${\cal S}_{\rm real}(\R^d)$, the
space of  real-valued, rapidly decreasing $C^{\infty}$-functions
on $\R^d$.

The purpose  of this paper is to point out  that,
if we consider the massless Nelson model
in a non-Fock representation  of the CCR for time-zero fields, then
it has a ground state
even in the case
where no infrared cutoff is made.
This new representation of
the massless Nelson model
is inequivalent to the Fock one
if no infrared cutoff is made.


This paper is organized as follows.
In Section 2 we first briefly review the  Nelson model
in the standard form in which
the time-zero fields  are given by
the Fock representation of the CCR
indexed by ${\cal S}_{\rm real}(\R^d)$. We call it
the \lq\lq{standard Nelson model}"(SNM).
Then we give an algebraic characterization of the
Nelson model in a way independent of the choice of  representations
of the time-zero fields and derive rigorously  field equations
in a weak sense. To the author's best knowledge,
field equations of the Nelson model
has not  been rigorously established so far
(only formal or classical ones are available).
In Section 3 we define the  Nelson model in a non-Fock
representation of the time-zero fields.
In the last section, we
prove that, under suitable hypotheses, the massless
Nelson model introduced in Section 3 has a ground state
even in the case where
no infrared cutoff is made.

\section{
Algebraic characterization  and
field equations of the Nelson model}


In this section we first recall
the SNM. Then
we present an algebraic characterization
of the Nelson model in an abstract manner.
Finally we derive rigorously  field equations of the abstract
Nelson model.


\subsection{The SNM}

The coordinate of the
configuration space $\R^{dN}$  of the $N$ particles is denoted
$q=(q_1, \cdots,$ $q_N) \in \R^{dN}$ with
$q_j:=(q_{j1}, \cdots, q_{jd}) \in \R^d$ ($j=1, \cdots,N$).
Let $D_{j\mu}$ ($j=1,\cdots,N, \, \mu=1,\cdots,d$)
be the generalized partial differential operator in the variable
$q_{j\mu}$, so that the momentum
operator of the $j$-th particle is given by
$p_j:=(p_{j1}, \cdots, p_{jd})$ with
$p_{j\mu}:=-iD_{j\mu}$.
The Hamiltonian of the particle system is then
given by the Schr\"odinger operator
\begin{equation}
H_{\rm p}:=\sum_{j=1}^N \frac {p_j^2}{2m_j}
+V,
\end{equation}
acting on $L^2(\R^{dN})$,
where $m_j > 0$ is the mass of the $j$-th particle and
$p_j^2=\sum_{\mu=1}^dp_{j\mu}^2=-\Delta_{q_j}$ is the $d$-dimensional
generalized Laplacian in the variable
$q_j$.
For a linear operator $T$, we denote its domain by $D(T)$.
We assume the following:
\begin{description}
\item[(H.1)] The operator
$H_{\rm p}$ is self-adjoint on its natural domain
$D(H_{\rm p})=\cap_{j=1}^ND(\Delta_{q_j})\cap D(V)$ and
bounded from below.
\end{description}

The Hilbert space for state vectors of the quantum scalar field
of the SNM
is given by
\begin{equation}
{\cal F}_{\rm b}
:=\oplus_{n=0}^{\infty}
\otimes_{\rm s}^nL^2(\R^d),
\end{equation}
the Boson Fock space
over $L^2(\R^d)$, where
$\otimes_{\rm s}^nL^2(\R^d)$ is the symmetric tensor product
of $L^2(\R^d)$ ($\otimes_{\rm s}^0L^2(\R^d):=\C$).
Let $\omega$ be a nonnegative Borel measurable
function on $\R^d$
such that $0 < \omega(k) < \infty$
for almost everywehre (a.e.) $k\in\R^d$ with respect to
the $d$-dimensional
Lebesgue measure and
\begin{equation}
\omega(k)=\omega(-k) \quad {\rm a.e.}k.  \label{rinv}
\end{equation}
For a.e. $k\in \R^d$, $\omega(k)$ physically means
the energy of one free boson with momentum $k$.
The function $\omega$ defines a nonnegative
self-adjoint multiplication operator on $L^2(\R^d)$
which is injective.
\begin{rem}{\rm A physical example for $\omega$  is given
by
\begin{equation}
\omega_m(k):=\sqrt{k^2+m^2}, \quad k \in \R^d, \label{omega}
\end{equation}
with $m \geq 0$ a constant denoting the mass of
one boson. We are interested  in the massless case $m=0$:
$\omega_0(k)=|k|$}
\end{rem}


The free Hamiltonian of the quantum scalar field is defined by
\begin{equation}
H_{\rm b}:=d\Gamma(\omega),
\end{equation}
the second quantization of $\omega$ \cite[\S X.7]{RS2}.

We denote the annihilation operators on ${\cal F}_{\rm b}$
by $a(f)=\int_{\R^d}a(k)f(k)^*dk$ ($f \in L^2(\R^d)$)
\cite[\S X.7]{RS2}.
The symmetric operator
\begin{equation}
\Phi_{\rm S}(f):=\frac {1}{\sqrt{2}}(a(f)^*+a(f)),
\end{equation}
called the Segal field operator,
is essentially self-adjoint \cite[\S X.7]{RS2}.  We denote its closure
by the same symbol.

Let
${\cal S}_{\rm real}'(\R^d)$
be the set of real tempered distributions on $\R^d$
 and, for
each $s \in \R$,
\begin{equation}
H_{\omega}^s:=
\{f\in {\cal S}_{\rm real}'(\R^d)|
\omega^s\hat f \in L^2(\R^d)\},
\end{equation}
where
\begin{equation}
\hat f(k):=\frac 1{(2\pi)^{d/2}}\int_{\R^d}f(x)e^{-ikx}dx
\end{equation}
is the Fourier transform of $f$.
For  $f \in H_{\omega}^{-1/2}$ and $g\in
H_{\omega}^{1/2}$, we can define
\begin{equation}
\phi_{\rm F}(f):=
\Phi_{\rm S}\left(\frac{\hat f}{\sqrt{\omega}}\right), \quad
\pi_{\rm F}(g):=
\Phi_{\rm S}\left(i\sqrt{\omega}\hat g\right).
\end{equation}
Let
\begin{equation}
{\cal F}_0:=\{\psi=\{\psi^{(n)}\}_{n=0}^{\infty}
\in {\cal F}_{\rm b}|\mbox{$\psi^{(n)}=0$ for all but finitely
many $n$'s}\},
\end{equation}
called the subspace of finite-particle vectors.
Then, for all $f\in H_{\omega}^{-1/2}$ and $g \in H_{\omega}^{1/2}$,
$\phi_{\rm F}(f)$ and $\pi_{\rm F}(g)$ leave
${\cal F}_0$ invariant
satisfying the CCR
\begin{eqnarray}
&& [\phi_{\rm F}(f), \pi_{\rm F}(g)]
=i\int_{\R^d}\hat f(k)^*\hat g(k)dk, \label{CCR1}\\
&&
[\phi_{\rm F}(f), \phi_{\rm F}(f')]=0,
\quad
[\pi_{\rm F}(g), \pi_{\rm F}(g')]=0, \ f,f'\in H_{\omega}^{-1/2},
g,g'\in H_{\omega}^{1/2}, \label{CCR2}
\end{eqnarray}
on ${\cal F}_0$.
Namely,
$\{\phi_{\rm F}(f),
\pi_{\rm F}(g)|
f\in H_{\omega}^{-1/2}, g\in H_{\omega}^{1/2}
\}$ gives a representation of the CCR.
This representation  is called
the Fock representation of the CCR.
In the SNM, $\phi_{\rm F}(f)$ and $\pi_{\rm F}(g)$
are taken to be the time-zero fields.

The Hilbert space for
 the SNM is
\begin{equation}
{\cal H}:=L^2(\R^{dN})\otimes {\cal F}_{\rm b}.
\end{equation}
As usual,
we freely use the natural identification
of ${\cal H}$ with
$\int_{\R^{dN}}^{\oplus}{\cal F}_{\rm b}dq$,
the constant fibre direct integral  with base space $(\R^{dN},dq)$ and
fibre ${\cal F}_{\rm b}$ \cite[\S XIII.16]{RS4}.
In what follows, for notational simplicity,
a decomposable operator $A=\int_{\R^{dN}}^{\oplus}
A(q)dq$ on ${\cal H}$ with fibre $A(q)$ (which is an operator
on ${\cal F}_{\rm b}$ for each $q \in \R^{dN}$) is
denoted $A(q)$ also.

To describe an interaction between the particles and
the quantum scalar field, we fix distributions
$\rho_j \in {\cal S}_{\rm real}'(\R^d), j=1,\cdots,N$,
which  satisfy the following:
\begin{description}
\item[(H.2)]
For $j=1, \cdots, N$,
\begin{equation}
\rho_j\in H_{\omega}^s, \quad s=-1/2,\  -1.
\end{equation}
\end{description}
The interaction of the particles  and the quantum  field
in the SNM is given by the operator
\begin{equation}
H_{I}^{\rm F}
:=
\sum_{j=1}^N\Phi_{\rm S}\left(e^{-ikq_j}\frac{{\hat \rho_j}{}^*}
{\sqrt{\omega}}\right) \label{HI}
\end{equation}
acting in ${\cal H}$.
Formally we have
$H_I^{\rm F}=\sum_{j=1}^N\int_{\R^d}\phi_{\rm F}(q_j-x)\rho_j(x)dx$,
where
$$
\phi_{\rm F}(x):=\int_{\R^d}
\frac 1{\sqrt{2(2\pi)^d\omega(k)}}\{a(k)^*e^{-ikx}+a(k)e^{ikx}\}dk.
$$

The total Hamiltonian of the SNM is defined by
\begin{equation}
H_{\rm SNM}:=H_0+\lambda H_I^{\rm F},
\end{equation}
where
\begin{equation}
H_0:=H_{\rm p}+
H_{\rm b}
\end{equation}
and $\lambda  \in \R\setminus\{0\}$ denotes the coupling constant
of the model.


\begin{pr}\label{pr2-1} Assume (H.1) and (H.2). Then
$H_{\rm SNM}$ is self-adjoint
with $D(H_{\rm SNM})=D(H_0)$  and bounded from below.
Moreover,
$H_{\rm SNM}$ is essentially self-adjoint on each core
of $H_0$
\end{pr}

{\it Proof}. A simple application of
\cite[Proposition B.2]{Ar89} which employs
the Kato-Rellich theorem (cf. \cite{Fro}).
\hfill \qed
\medskip

In summary, the SNM is characterized in terms of
the Hamiltonian $H_{\rm SNM}$ and the time-zero-fields
$\{\phi_{\rm F}(f), \pi_{\rm F}(g)
|f\in H_{\omega}^{-1/2}, g \in H_{\omega}^{1/2}\}$.





\subsection{An abstract  definition of the Nelson
model}

As is seen above, the SNM uses the Fock representation
of the CCR to give its time-zero fields and its Hamiltonain.
We want to define
the Nelson model in a way independent of the choice
of representations of the CCR for time-zero fields.
A natural manner for this  is to use commutation relations
fulfilled by observables, i.e., to find a possible Lie algebraic
structure.  We shall take
commutation relations in a weak sense.

We denote the inner product and the norm of a Hilbert space ${\cal X}$
by
$\lb \,\cdot\, , \,\cdot\, \rb_{\cal X}$ and $\|\,\cdot\,\|_{\cal X}$
respectively. But, if there is  no danger of confusion,
then we simply write them as
$\lb \,\cdot\, , \,\cdot\, \rb$ and $\|\,\cdot\,\|$.

\begin{df}\label{df2-2}{\rm Let $A$ and $B$ be densely defined
linear operators on a Hilbert space ${\cal X}$
and ${\cal D}$ be a subspace of ${\cal X}$ such that
${\cal D}\subset D(A)\cap D(B)\cap D(A^*)\cap D(B^*)$.
Then we define a quadratic form $[A,B]_{\rm w}^{\cal D}$
by
$$
[A,B]_{\rm w}^{\cal D}(\psi,\phi):=\lb A^*\psi, B\phi\rb-\lb B^*\psi,
A\phi\rb,
\quad
\psi,\phi\in {\cal D}.
$$
(\lq\lq{w}" means \lq\lq{weak}".)}
\end{df}

\begin{rem}{\rm If $A$ and $B$ are bounded
on ${\cal X}$ with $D(A)=D(B)={\cal X}$,
then, for all dense subspaces ${\cal D}$
of ${\cal X}$,
$[A,B]_{\rm w}^{\cal D}(\psi,\phi)=\lb \psi,[A,B]\phi\rb$
for all $\psi,\phi \in {\cal D}$,
where $[A,B]:=AB-BA$ (the usual commutator).
}
\end{rem}


Let ${\cal F}$ be a Hilbert space
and
\begin{equation}
{\cal K}:=L^2(\R^{dN})\otimes {\cal F}=\int_{\R^{dN}}^{\oplus}
{\cal F}dq.
\end{equation}


\begin{df}\label{df2-3}{\rm  Assume (H.1) and (H.2).
Let $s_0$ and $s_1$ be real constants.
A Nelson model is a set
${\M}_{\rm Nelson}:=\{{\cal K}, {\cal D},
L_{\lambda},  \{\phi(f),\pi(g)|f\in H_{\omega}^{s_0},
g\in H_{\omega}^{s_1}\}\}$
having  the following properties:
\begin{description}
\item[{\rm (N.1)}]
${\cal D}$ is a dense subspace of ${\cal K}$.
\item[{\rm (N.2)}]  $L_{\lambda}$ is a symmetric operator on ${\cal K}$
and the operator
\begin{equation}
H_{\rm NM}:=H_{\rm p}+L_{\lambda}
\end{equation}
is self-adjoint ($D(H_{\rm NM}):=D(H_{\rm p})\cap
D(L_{\lambda})$). We call $H_{\rm NM}$
the Hamiltonian of the Nelson model ${\M}_{\rm Nelson}$.
\item[{\rm (N.3)}] The function $\omega$ is such that
\begin{equation}
{\cal S}_{\rm real}(\R^d) \subset
H_{\omega}^{s_0}\cap H_{\omega}^{s_0+2}\cap H_{\omega}^{s_1}
\cap H_{\omega}^1
\end{equation}
and, for all $j=1,\cdots,N, \mu=1,\cdots,d$,
\begin{equation}
D_{\mu}\rho_j \in H_{\omega}^{s_0},
\end{equation}
where $D_{\mu}$ ($\mu=1,\cdots,d$) is the generalized partial
differential operator in the $\mu$-th variable $x_{\mu}$
in $x=(x_1,\cdots, x_d)\in \R^d$.

\item[{\rm (N.4)}]
$\{{\cal K}, {\cal D},
 \{\phi(f),\pi(f)|f\in{\cal S}_{\rm real}(\R^d)\}\}$
is a representation of the CCR indexed by
${\cal S}_{\rm real}(\R^d)$. Namely, for all $f,g \in
{\cal S}_{\rm real}(\R^d)$,
$\phi(f)$ and $\pi(f)$ are self-adjoint operators on ${\cal K}$
with
${\cal D}\subset
D(\phi(f)\pi(g))\cap
D(\pi(f)\phi(g))\cap
D(\phi(f)\phi(g))\cap
D(\pi(f)\pi(g))$
satisfying the CCR
\begin{eqnarray}
&&[\phi(f),\pi(g)]=i\int_{\R^d}f(x)g(x)dx, \\
&&[\phi(f),\phi(g)]=0, \quad
[\pi(f),\pi(g)]=0,
\end{eqnarray}
on ${\cal D}$ and the linearity:
$$
\phi(af+bg)=a\phi(f)+b\phi(g), \quad
\pi(af+bg)=a\pi(f)+b\pi(g), \quad a,b\in\R,
$$
on ${\cal D}$.
\item[{\rm (N.5)}] Let
$X=q_{j\mu}, p_{j\mu}, Y=\phi(f),\pi(f)$,
$j=1, \cdots,N, \mu=1,\cdots,d, \
f \in {\cal S}_{\rm real}(\R^d)$.
There exists a dense subspace ${\cal E}$ of ${\cal K}$ such that
\begin{equation}
{\cal E} \subset D(X) \cap D(Y) \cap D(L_{\lambda})\cap D(H_{\rm p})
\label{E}
\end{equation}
and
the following relations hold in the sense of quadratic
form on ${\cal E}$:
\begin{eqnarray*}
&& [L_{\lambda},q_{j\mu}]_{\rm w}^{\cal E}=0, \quad
[L_{\lambda},p_{j\mu}]_{\rm w}^{\cal E}
=i\lambda
\phi(D_{\mu}\rho_j(q_j-\cdot)), \\
&& [L_{\lambda},\phi(f)]_{\rm w}^{\cal E}=-i\pi(f), \\
&& [L_{\lambda}, \pi(f)]_{\rm w}^{\cal E}=i\phi(
\omega(-i\nabla)^2f)+i\lambda\sum_{j=1}^N
(\rho_j*f)(q_j), \\
&&[X,Y]_{\rm w}^{\cal E} =0,\quad [H_{\rm p},Y]_{\rm w}^{\cal E}=0,
\end{eqnarray*}
where
$\nabla=(D_1,\cdots, D_d)$
and $\rho_j*f$
is the convolution of $\rho_j$ and $f$:
$(\rho_j*f)(x):=\int_{\R^d}\rho_j(x-y)f(y)dy$.
\end{description}
}
\end{df}

\begin{rem}
{\rm (i) We have
$$
(\omega(-i\nabla)^2f)
(x)=\frac 1{(2\pi)^{d/2}}\int_{\R^d}
\omega(k)^2\hat f(k)e^{ikx}dk, \quad f\in {\cal S}(\R^d).
$$
Hence (N.3) implies that
$$
\omega(-i\nabla)^2f \in H_{\omega}^{s_0},
$$
so that $\phi(\omega(-i\nabla)^2f)$ is defined.
(ii) It follows from (H.2) and (N.3) that,
for all $f \in {\cal S}_{\rm real}(\R^d)$,
$\rho_j*f$
 is a bounded continuous function on $\R^d$,
so that
$\rho_j*f(q_j)$ is a bounded self-adjoint multiplication
operator with $D(\rho_j*f(q_j))={\cal K}$.
}
\end{rem}
Let us show that the SNM is indeed a Nelson model in the above sense.
We introduce
\begin{equation}
{\cal F}_{\omega,{\rm fin}}:={\cal L}
\{\Omega_0, \
a(f_1)^*\cdots a(f_n)^*\Omega_0|
n\in\ \N, f_j \in D(\omega), j=1, \cdots,n\},
\end{equation}
where
$\Omega_0:=\{1,0,0, \cdots\}\in {\cal F}_{\rm b}$ is the
Fock vacuum and ${\cal L}\{\cdots\}$ means the subspace
algebraically spanned by all the vectors in
the set $\{\cdots\}$,
and
\begin{equation}
{\cal D}_0:=
C_0^{\infty}(\R^{dN})\otimes_{\rm alg} {\cal F}_{\omega,{\rm fin}},
\end{equation}
where
$\otimes_{\rm alg}$ denotes algebraic tensor product.


\begin{pr}\label{pr2-4}  Assume (H.1) and (H.2).
Suppose that
\begin{eqnarray}
&&
\int_{|q| \leq R}|V(q)|^2dq < \infty,
\forall R > 0, \label{V}\\
&&{\cal S}_{\rm real}(\R^d)
\subset
H_{\omega}^{3/2}\cap
H_{\omega}^{-1/2}, \label{SH}
\end{eqnarray}
and, for $j=1,\cdots, N, \mu=1,\cdots,d$,
\begin{equation}
D_{\mu}\rho_j \in H_{\omega}^{-1/2}. \label{Drho}
\end{equation}
Let
\begin{equation}
L_{\lambda}^{\rm SNM}:=
H_{\rm b}+\lambda H_I^{\rm F}.
\end{equation}
Then
$$
{\M}_{\rm SNM}:=\{{\cal H},
{\cal D}_0, L_{\lambda}^{\rm SNM},  \{\phi_{\rm F}(f),\pi_{\rm F}(g)
|f\in H_{\omega}^{-1/2}, g\in H_{\omega}^{1/2}\}\}
$$
is a Nelson model.
\end{pr}
{\it Proof}. (N.1) is obvious.
(N.2) follows from Proposition \ref{pr2-1}.
(N.3) with $s_0=-1/2, s_1=1/2$
follows from the present assumption.
(N.4) follows from (\ref{CCR1}) and (\ref{CCR2}).
For (N.5), we take ${\cal E}={\cal D}_0$.
By condition (\ref{V}),
$C_0^{\infty}(\R^{dN}) \subset
D(V)$ so that
$C_0^{\infty}(\R^{dN}) \subset D(H_{\rm p})$.
Hence (\ref{E}) follows. The commutation relations
in (N.5) follow from direct computations using
(2.10), (2.11) and
\begin{equation}
[H_{\rm b},a(f)]=-a(\omega f), \quad
[H_{\rm b},a(f)^*]=a(\omega f),
\end{equation}
on ${\cal F}_{\omega,{\rm fin}}$ ($f, \omega f \in L^2(\R^d)$).
\hfill \qed

\subsection{Field equations}

We derive field equations for the Nelson model
${\M}_{\rm Nelson}$ in a weak sense.

\begin{df}{\rm
Let ${\cal X}$ be a Hilbert space
and $A(t)$ ($t\in \R$)  be a linear operator
on ${\cal X}$. Suppose that
there exists a dense subspace ${\cal D}$ of ${\cal X}$
such that ${\cal D} \subset D(A(t))$ for all $t \in \R$.
We say that $A(t)$ is weakly differentiable
on ${\cal D}$ if, for all $\psi, \phi \in {\cal D}$,
the function
$\lb \psi, A(t)\phi\rb $ is differentiable
in $t\in \R$. In that case we define a quadratic
form ${\mbox{\rm w-}}dA(t)/dt|_{\cal D}$ on ${\cal D}$ by
$$
{\mbox{\rm w-}}\frac{dA(t)}{dt}\bigg|_{\cal D}(\psi,\phi):=
\frac {d}{dt}\lb \psi, A(t)\phi\rb, \quad \psi,\phi \in {\cal D}.
$$
}
\end{df}

\begin{df}\label{df2-6}{\rm
Let ${\cal X}$ be a Hilbert space and $H$ be a self-adjoint
operator on ${\cal H}$.
Let $A$ be a delnsely defined linear operator
on ${\cal X}$.
We say that $A$ is in the set ${\sf A}_H$ if it
satisfies the following (i) and (ii):
\begin{description}
\item[{\rm (i)}]
There exists a dense subspace $D_A\subset D(H)$
such that, for all $s \in \R$, $e^{isH}D_A \subset D(A)\cap D(A^*)$.
\item[{\rm (ii)}] For all $\psi \in D_A$,
the ${\cal X}$-valued functions:
$s \to Ae^{isH}\psi $ and $s\to A^*e^{isH}\psi$ are
strongly continuous on $\R$.
\end{description}
For $A\in {\sf A}_H$, we set
\begin{equation}
D_{A,H}:={\cal L}\{e^{isH}\psi|\psi \in D_A, s\in \R\}.
\end{equation}
}
\end{df}


\begin{lem}
\label{lem2-7}
Let ${\cal X}$ be a Hilbert space and $H$ be a self-adjoint
operator on ${\cal X}$.
Let $A$ be a densely defined linear operator
on ${\cal X}$ and set
$$
A(t):=e^{itH}Ae^{-itH}, \quad t\in \R,
$$
Assume that $A\in {\sf A}_H$.
Then $A(t)$ is weakly differentiable
on $D_{A,H}$ and
\begin{eqnarray}
{\mbox{\rm w-}}
\frac{dA(t)}{dt}\bigg|_{D_{A,H}}=i[H,A(t)]_{\rm w}^{D_{A,H}}.
\end{eqnarray}
\end{lem}
{\it Proof}. Let $\psi,\phi \in D_{A,H}$.
Then $\psi,\phi \in D(A(t))\cap D(A(t)^*)$ for all $t \in \R$.
For $h\in \R\setminus\{0\}$,
\begin{eqnarray*}
\lb \psi, \frac{A(t+h)-A(t)}{h}\phi\rb &=&
\lb e^{-itH}\frac{e^{-ihH}-1}{h}\psi,
Ae^{-i(t+h)H}\phi\rb \\
&&\quad
+\lb A(t)^*\psi, \frac{e^{-ihH}-1}{h}\phi\rb.
\end{eqnarray*}
Since $\psi \in D(H)$, it follows that
$$
\lim_{h\to 0}
\frac{e^{-ihH}-1}{h}\psi=-iH\psi.
$$
It follows from Definition \ref{df2-6}(ii)
that  $Ae^{isH}\phi$ is strongly continuous in $s$. Hence
$$
\lim_{h\to 0}Ae^{-i(t+h)H}\phi=Ae^{-itH}\phi.
$$
Hence
$$
\lim_{h\to 0}\lb \psi, \frac{A(t+h)-A(t)}{h}\phi\rb
= i
\lb e^{-itH}H\psi,Ae^{-itH}\phi\rb -
i
\lb A(t)^*\psi,H\phi\rb.
$$
Thus the desired result follows. \hfill\qed
\medskip

The canonical  Heisenberg operators of the Neson model
${\M}_{\rm Nelson}$
(Definition \ref{df2-3})
are defined as follows:
\begin{eqnarray}
&&
q_{j\mu}(t)
:=e^{itH_{\rm NM}}q_{j\mu}e^{-itH_{\rm NM}}, \quad
p_{j\mu}(t):=e^{itH_{\rm NM}}p_{j\mu}e^{-itH_{\rm NM}}, \\
&&\phi(t,f):=e^{itH_{\rm NM}}\phi(f)e^{-itH_{\rm NM}},\quad
\pi(t,f):=e^{itH_{\rm NM}}\pi(t,f)e^{-itH_{\rm NM}}, \\
&&j=1,\cdots,N,\ \mu=1,\cdots,d,
\ f\in
{\cal S}_{\rm real}(\R^d), \ t\in \R. \nonumber
\end{eqnarray}
For a self-adjoint operator $T$, $Q(T)$ denotes the form domain
of the quadratic form associated with $T$.
\begin{th}\label{FE} Consider the Nelson model ${\M}_{\rm Nelson}$
(Definition \ref{df2-3}). Assume (H.1), (H.2) and
the following (A.1)--(A.5):
\begin{description}
\item[{\rm (A.1)}] (\ref{V}) holds and $H_{\rm p}$ is
essentially self-adjoint on $C_0^{\infty}(\R^{dN})$.
\item[{\rm (A.2)}]
For all $j=1,\cdots,N,\
\mu=1,\cdots,d$, the distributional partial
derivative $D_{j\mu}V$ in the variable $q_{j\mu}$
is a Borel measurable function
on $\R^{dN}$ which is a.e. finite with respect to the Lebesgue
measure. Moreover, $C_0^{\infty}(\R^{dN})$
is a form core of the self-adjoint multiplication operator $D_{j\mu}V$
and $D(H_{\rm p})\subset \cap_{j=1}^N\cap_{\mu=1}^dQ(D_{j\mu}V)$.
\item[{\rm (A.3)}]  In addition to (\ref{E}),
$$
{\cal E}\subset D(H_{\rm p})\cap
\left\{\cap_{j,l=1}^N\cap_{\mu,\nu=1}^d\left[D(p_{j\mu}q_{l\nu})\cap
D(q_{l\nu}p_{j\mu})\right]\right\}\cap
\left[\cap_{j=1}^N\cap_{\mu=1}^dQ(D_{j\mu}V)\right].
$$
\item[{\rm (A.4)}] For all $s \in \R$, $e^{isH_{\rm NM}}$ leaves
${\cal E}$ invariant:
$e^{isH_{\rm NM}}{\cal E}\subset {\cal E}$
\item[{\rm (A.5)}]
For all $j=1,\cdots,N, \ \mu=1,\cdots,d$ and $\psi\in{\cal E}$,
the ${\cal K}$-valued function
$q_{j\mu}e^{isH_{\rm NM}}\psi$
is strongly continuous in $s\in \R$.
\item[{\rm (A.6)}]
For all $f \in {\cal S}_{\rm real}(\R^d)$,
the ${\cal K}$-valued functions
$\phi(f)e^{isH_{\rm NM}}\psi$ and
$\pi(f)e^{isH_{\rm NM}}\psi$
are strongly continuous in $s\in \R$.
\end{description}
Then the Heisenberg operators
$q_{j\mu}(t),
p_{j\mu}(t), \phi(t,f)$ and $\pi(t,f)$ ($f \in {\cal S}_{\rm real}
(\R^d)$) are weakly differentiable on ${\cal E}$
and
\begin{eqnarray}
&&{\mbox{\rm w-}}\frac {dq_{j\mu}(t)}{dt}\bigg|_{\cal E}=
\frac {p_{j\mu}(t)}{m_j}, \label{dq(t)}\\
&&
{\mbox{\rm w-}}\frac {dp_{j\mu}(t)}{dt}\bigg|_{\cal E}=
-D_{j\mu}V(q(t))-\lambda
e^{itH_{\rm NM}}\phi(D_{\mu}\rho_j(q_j-\cdot))e^{-itH_{\rm NM}},
\label{dp}\\
&&
{\mbox{\rm w-}}\frac {d\phi(t,f)}{dt}\bigg|_{\cal E}=
\pi(t,f),\label{dphi} \\
&&
{\mbox{\rm w-}}\frac {d\pi(t,f)}{dt}\bigg|_{\cal E}=
-\phi(t,\omega(-i\nabla)^2f)-
\lambda\sum_{j=1}^N(\rho_j*f)(q_j(t)),\label{dpi}
\end{eqnarray}
where $q_j(t):=(q_{j1}(t), \cdots, q_{jd}(t)),
q(t):=(q_1(t),\cdots, q_N(t))$.
\end{th}
{\it Proof}. By (A.3) and the CCR
$$
[p_{j\mu}, q_{l\nu}]=-i\delta_{jl}\delta_{\mu\nu}
\quad \mbox{on $
D(p_{j\mu}q_{l\nu})\cap
D(q_{l\nu}p_{j\mu})$},
$$
we have
$$
[H_{\rm p},q_{j\mu}]_{\rm w}^{\cal E}=
-i\frac{p_{j\mu}}{m_j},
$$
which, together with (N.5), implies that
\begin{equation}
[H_{\rm NM},q_{j\mu}]_{\rm w}^{\cal E}=
-i\frac{p_{j\mu}}{m_j}.
\end{equation}
By this fact, (A.4) and (A.5), we can apply  Lemma \ref{lem2-7}
to obtain (\ref{dq(t)}).

Let
\begin{equation}
h_0:=-\sum_{j=1}^N\frac {\Delta_{q_j}}{2m_j}.
\end{equation}
For all $\psi \in D(h_0)$, we have
\begin{eqnarray}
\|p_{j\mu}\psi\|^2
&\leq& 2m_j\lb \psi,h_0\psi\rb
\leq 2m_j\|\psi\|\|h_0\psi\| \nonumber\\
& \leq &\varepsilon \|h_0\psi\|^2 +\frac {m_j^2}{\varepsilon}\|\psi\|^2,
\label{ineq1}
\end{eqnarray}
where $\varepsilon > 0$ is arbitrary.
By the self-adjointness of $H_{\rm p}$ and the closed graph theorem,
there exist constants $a\geq 0,b \geq 0$ such that, for all
$\psi \in D(H_{\rm p})=D(h_0)\cap D(V)$,
\begin{equation}
\|h_0\psi\|+\|V\psi\| \leq a\|H_{\rm p}\psi\|+b\|\psi\|. \label{hV}
\end{equation}
Hence, by (\ref{ineq1}),
\begin{equation}
\|p_{j\mu}\psi\|
\leq a\sqrt{\varepsilon}\|H_{\rm p}\psi\|
+\left(b\sqrt{\varepsilon}+\frac{m_j}{\sqrt{\varepsilon}}\right)\|\psi\|.
\label{ineq2}
\end{equation}
Let $\psi,\phi \in C_0^{\infty}(\R^{dN})$. Then we have
$$
\lb H_{\rm p}\psi, p_{j\mu}\phi\rb -
\lb p_{j\mu}\psi, H_{\rm p}\phi\rb=\lb \psi, i(D_{j\mu}V)\phi\rb.
$$
By (A.1), (A.2) and (\ref{ineq2}), we can extend this equality
to all $\psi,\phi \in D(H_{\rm p})$.
Hence, in particular, we have
\begin{equation}
[H_{\rm p}, p_{j\mu}]_{\rm w}^{\cal E}=iD_{j\mu}V
\end{equation}
in the sense of quadratic form on ${\cal E}$.
Hence, by (N.5), we have
\begin{equation}
[H_{\rm NM}, p_{j\mu}]_{\rm w}^{\cal E}=iD_{j\mu}V
+i\lambda \phi(D_{\mu}\rho_j(q_j-\cdot)).
\end{equation}
in the sense of quadratic form on ${\cal E}$.
By this fact, (A.4) and (A.6), we can apply  Lemma \ref{lem2-7}
to obtain (\ref{dp}).
Similarly (more easily) we can prove (\ref{dphi}) and (\ref{dpi}).
\hfill \qed
\medskip
Now we show that Theorem \ref{FE} can be applied to
the SNM.


\begin{lem}\label{lem-cont}
Let $S$ be a self-adjoint operator on a Hilbert space ${\cal X}$
and $A$ be a linear operator on ${\cal X}$ such that,
for some $\alpha > 0$, $A$ is $|S|^{\alpha}$-bounded, i.e.,
$D(|S|^{\alpha}) \subset  D(A)$  and
$$
\|A\psi\| \leq a\||S|^{\alpha}\psi\|+b\|\psi\|, \quad
\psi \in D(|S|^{\alpha}),
$$
where $a,b \geq 0$ are constants.
Then, for all $\psi \in D(|S|^{\alpha})$,
the function: $t \to
Ae^{itS}\psi$ is strongly continuous.
\end{lem}
{\it Proof}. We have for all $h \in \R$
\begin{eqnarray*}
\|Ae^{i(t+h)S}\psi
-Ae^{itS}\psi\|
&\leq&
a\||S|^{\alpha}
(e^{ihS}-1)e^{itS}\psi\|
+b\|(e^{ihS}-1)\psi\| \\
&=& a\|
(e^{ihS}-1)|S|^{\alpha}\psi\|
+b\|(e^{ihS}-1)\psi\|,
\end{eqnarray*}
which implies the strong continuity of
the function:$t \to
Ae^{itS}\psi$.
\hfill \qed
\medskip

For a self-adjoint operator $T$ on a Hilbert space which is bounded from
below, we set
\begin{equation}
E_0(T):=\inf\sigma(T),
\end{equation}
the ground state energy of $T$,
where $\sigma(T)$ denotes the spectrum of $T$.
Under hypotheses (H.1) and (H.2), we set
\begin{equation}
\hat H_{\rm SNM}:=H_{\rm SNM}-E_0(H_{\rm SNM}),
\end{equation}
which is nonnegative.

\begin{lem}\label{lem-qV} Assume (H.1) and (H.2). Suppose that
$Q(V)\subset D(|q|)$ and
there exist constants
$c_1,c_2\geq 0$ such that
for all $u \in Q(V)$
\begin{equation}
\||q|u\|\leq c_1\||V|^{1/2}u\|+c_2\|u\|.\label{qV}
\end{equation}
Then,
for all $j=1,\cdots,N, \ \mu=1,\cdots,d$ and $\psi\in Q(H_{\rm SNM})$,
the ${\cal H}$-valued function
$q_{j\mu}e^{isH_{\rm SNM}}\psi$
is strongly continuous in $s\in \R$.
\end{lem}
{\it Proof}. By Proposition \ref{pr2-1} and
the closed graph theorem,
there exist constants
$a_1,b_1\geq 0$ such that,
for all $\psi \in D(H_{\rm SNM})=D(H_0)$,
\begin{equation}
\|H_{\rm p}\psi\|
+\|H_{\rm b}\psi\| \leq
a_1\|H_{\rm SNM}\psi\| +b_1\|\psi\|.
\label{HHH}
\end{equation}
By this inequality and (\ref{hV}), we have
$$
\|V\psi\| \leq
aa_1\|H_{\rm SNM}\psi\| +(ab_1+b)\|\psi\|,
$$
which, via
 an application \cite[Theorem X.18]{RS2},
implies that $Q(H_{\rm SNM})\subset Q(V)$ and
$$
\||V|^{1/2}\phi\| \leq
a_2\|
\hat H_{\rm SNM}^{1/2}\phi\| +b_2\|\phi\|, \quad \phi \in Q(H_{\rm SNM}),
$$
with $a_2, b_2 \geq 0$ being constants.
Hence, by (\ref{qV}), we obtain
$$
\|q_{j\mu}\psi\|
\leq c_1a_2\|\hat H_{\rm SNM}^{1/2}\psi\| +
(c_1b_2+c_2)\|\psi\|, \quad \psi \in Q(H_{\rm SNM}).
$$
Hence we can apply Lemma \ref{lem-cont} to
obtain the desired result.
\hfill \qed
\begin{lem}\label{lem-pH} Assume (H.1) and (H.2). Suppose that
\begin{equation}
{\cal S}_{\rm real}(\R^d) \subset
H_{\omega}^{-1}\cap
H_{\omega}^{1/2}.
\end{equation}
Let $\psi \in Q(H_{\rm SNM})$.
Then, for all $f \in {\cal S}_{\rm real}(\R^d)$,
the ${\cal H}$-valued functions
$\phi_{\rm F}(f)e^{isH_{\rm SNM}}\psi$ and
$\pi_{\rm F}(f)e^{isH_{\rm SNM}}\psi$
are strongly continuous in $s\in \R$.
\end{lem}
{\it Proof}.
Using the well-known estimates
\begin{eqnarray}
\|a(f)\psi\| &\leq& \|\omega^{-1/2}f\|_{L^2(\R^d)}
\|H_{\rm b}^{1/2}\psi\|,  \\
\|a(f)^*\psi\| &\leq& \|\omega^{-1/2}f\|_{L^2(\R^d)}
\|H_{\rm b}^{1/2}\psi\| +\|f\|_{L^2(\R^d)},  \\
&& \qquad\psi
\in D(H_{\rm b}^{1/2}), \ f,\omega^{-1/2}f\in L^2(\R^d),
\nonumber
\end{eqnarray}
we have
\begin{eqnarray}
\|\phi_{\rm F}(f)\psi\|
&\leq &\sqrt{2}\|\omega^{-1}\hat f\|_{L^2(\R^d)}
\|H_{\rm b}^{1/2}\psi\|
+\frac 1{\sqrt{2}}\|\omega^{-1/2}
\hat f\|_{L^2(\R^d)}\|\psi\|, \nonumber\\
&& \qquad \hskip 3cm f \in H_{\omega}^{-1}\cap
H_{\omega}^{-1/2},
 \\
\|\pi_{\rm F}(g)\psi\|
&\leq &\sqrt{2}\|\hat g\|_{L^2(\R^d)}
\|H_{\rm b}^{1/2}\psi\|
+\frac 1{\sqrt{2}}\|\omega^{1/2}
\hat g\|_{L^2(\R^d)}\|\psi\|,\nonumber \\
&& \qquad \hskip 3cm
g \in L^2(\R^d)\cap
H_{\omega}^{1/2}.
\end{eqnarray}
By (\ref{HHH}) and
 an application \cite[Theorem X.18]{RS2}, we have
\begin{equation}
\|H_{\rm b}^{1/2}\psi\|
\leq a_3\|\hat H_{\rm SNM}^{1/2}\psi\|+b_3\|\psi\|, \quad
\psi \in Q(H_{\rm SNM}),
\end{equation}
with $a_3,b_3 \geq 0$ being constants.
It follows that, for all $f\in{\cal S}_{\rm real}
(\R^d)$,  $\phi_{\rm F}(f)$ and
$\pi_{\rm F}(f)$ are $\hat H_{\rm SNM}^{1/2}$-bounded.
Hence an application of Lemma \ref{lem-cont} yields
the desired result.
\hfill \qed
\medskip
We shall need  the following condition too.
\begin{description}
\item[(H.3)]  The function $\omega$ is such that
\begin{equation}
{\cal S}_{\rm real}(\R^d) \subset H_{\omega}^{-1}\cap
H_{\omega}^{3/2}.
\end{equation}
\end{description}
\begin{th} Assume (H.1)--(H.3), (A.1), (A.2),  (\ref{Drho})
and (\ref{qV}).
Suppose that
\begin{equation}
D(H_{\rm p})\subset
\cap_{j,l=1}^N\cap_{\mu,\nu=1}^d\left[D(p_{j\mu}q_{l\nu})
\cap
D(q_{l\nu}p_{j\mu})\right]\cap \left(\cap_{j=1}^N\cap_{\mu=1}^d
Q(D_{j\mu}V)\right)
\label{DHp}
\end{equation}
Then the conclusion of Theorem \ref{FE} holds
with
$H_{\rm NM}=H_{\rm SNM}, \phi(f)=\phi_{\rm F}(f),
\pi(f)=\pi_{\rm F}(f)$ and ${\cal E}=D(H_{\rm SNM})$.
\end{th}
{\it Proof}. We need only check that the assumption
of Theorem \ref{FE} is satisfied with
$H_{\rm NM}=H_{\rm SNM}, \phi(f)=\phi_{\rm F}(f),
\pi(f)=\pi_{\rm F}(f)$ and ${\cal E}=D(H_{\rm SNM})$.
(A.3) and (A.4) follow from  (\ref{DHp}).
(A.5) and (A.6) follow from
Lemmas \ref{lem-qV} and \ref{lem-pH} respectively.
\hfill \qed
\medskip



\section{A Nelson model in a non-Fock representation}
In this section we define a Nelson model
whose time-zero fields are given
by a non-Fock representation of the CCR and
show that it has a ground state
even if no infrared cutoff is made.

Under hypotheses (H.2) and (H.3),
we can define for each $f\in {\cal S}_{\rm real}(\R^d)$
\begin{equation}
\widetilde \phi(f):=\phi_{\rm F}(f)+\lambda\sum_{j=1}^N
\int_{\R^d}\frac {\hat\rho_j(k)\hat f(k)}{\omega(k)^2}dk.
\end{equation}
We set
\begin{equation}
\widetilde \pi(f):=\pi_{\rm F}(f).
\end{equation}
\begin{pr}\label{pr3-1} Assume (H.2) and (H.3). Then:
\begin{description}
\item[{\rm (i)}] $\{{\cal F}_{\rm b}, {\cal F}_0,
\{\widetilde \phi(f),\widetilde \pi(f)|
{\cal S}_{\rm real}(\R^d)\}\}$
is a representation of the CCR indexed
by ${\cal S}_{\rm real}(\R^d)$.
\item[{\rm (ii)}] The representation
$\{\widetilde \phi(f),\widetilde \pi(f)|
{\cal S}_{\rm real}(\R^d)\}\}$
is unitarily equivalent to the Fock representation
$\{\phi_{\rm F}(f),\pi_{\rm F}(f)|
{\cal S}_{\rm real}(\R^d)\}\}$
if and only if
$\sum_{j=1}^N\rho_j \in H_{\omega}^{-3/2}$.
\end{description}
\end{pr}
{\it Proof}. Part (i) is obvious.
Part (ii) follows from an argument similar to
that in \cite[Chapter 1, \S 1-e]{Emch}.
\hfill \qed
\medskip

We now consider a Nelson model
whose time-zero fields are given by
$\{\widetilde \phi(f), \widetilde
\pi(f)| f\in {\cal S}_{\rm real}(\R^d)$.
We introduce an $L^2(\R^d)$-valued function $G$ on $\R^{dN}$
($G:\R^{dN} \to L^2(\R^d)$,
$G(q) \in L^2(\R^d), \ q\in \R^{dN}$) by
\begin{equation}
G(q)(k):=\sum_{j=1}^N\frac {{\hat \rho_j(k)}^*}{\sqrt{\omega(k)}}
\left(e^{-iq_jk}-1\right) \label{G(q)}
\end{equation}
and a function
\begin{equation}
W(q):=\sum_{j,l=1}^N
\int_{\R^d}\frac{\hat \rho_j(k)\hat\rho_l(k)^*}{\omega(k)^2}
e^{-iq_lk}dk, \quad q\in \R^{dN}.
\end{equation}
By reality of $\rho_j$ and (\ref{rinv}),
$W$ is real-valued.
We define a Hamiltonian by
\begin{equation}
H:=H_0+ \lambda \Phi_{\rm S}(G(q))
-\lambda^2W
+c_0\lambda^2,
\end{equation}
where
\begin{equation}
c_0:=\frac 12\left\|\frac{\sum_{j=1}^N\hat \rho_j}{\omega}
\right\|_{L^2(\R^d)}^2.
\end{equation}
\begin{lem} Assume  (H.1)--(H.3).
Then $H$ is self-adjoint with $D(H)=D(H_0)$ and bounded from below.
Moreover, $H$ is essentially self-adjoint on each core of $H_0$.
\end{lem}
{\it Proof}. In the same way as in the case of  $H_{\rm SNM}$,
we can show that
$H':=H_0+\lambda \Phi_{\rm S}(G(q))$ is self-adjoint
with $D(H')=D(H_0)$, bounded from below,
and essentially self-adjoint on each core of $H_0$.
Since $-\lambda^2W+c_0\lambda^2$ is a bounded self-adjoint operator,
the Kato-Rellich theorem yields the desired result.
\hfill \qed

\medskip
Let
\begin{equation}
L_{\lambda}^{\rm NF}:=H_{\rm b}+\lambda \Phi_{\rm S}(G(q))-\lambda^2W
+c_0\lambda^2,
\end{equation}
so that
\begin{equation}
H=H_{\rm p}+L_{\lambda}^{\rm NF}.
\end{equation}
\begin{pr}
Assume
(H.1)--(H.3). Then
\begin{equation}
\M_{\rm NF}:=\{{\cal H}, {\cal D}_0, L_{\lambda}^{\rm NF},
\{\widetilde \phi(f),\widetilde \pi(g)
|f\in H_{\omega}^{-1}, g\in H_{\omega}^{1/2}\}\}
\end{equation}
is a Nelson model.
\end{pr}
{\it Proof}. Similar to the proof of Proposition \ref{pr2-4}.
\hfill \qed
\medskip

\begin{pr}\label{equiv}
Assume
(H.1)--(H.3).
Then
$\{H, \{
\widetilde \phi(f),\widetilde \pi(f)
|f\in {\cal S}_{\rm real}(\R^d)\}\}$
is unitarily equivalent to
$\{H_{\rm SNM}, \{\phi_{\rm F}(f),\pi_{\rm F}(f)
|f\in {\cal S}_{\rm real}(\R^d)\}\}$
if and only if
\begin{equation}
\sum_{j=1}^N\rho_j \in H_{\omega}^{-3/2}. \label{irc}
\end{equation}
\end{pr}
{\it Proof}.
Suppose that there exists a unitary
operator $Y$ on ${\cal H}$ such that
\begin{eqnarray}
&& YHY^{-1}=H_{\rm SNM}, \label{YH}\\
&&Y\widetilde \phi(f)Y^{-1}=\phi_{\rm F}(f), \
Y\widetilde \pi(f)Y^{-1}=\pi_{\rm F}(f), \quad
f\in {\cal S}_{\rm real}(\R^d).\label{Yp}
\end{eqnarray}
Then, by Proposition \ref{pr3-1}, (\ref{irc}) holds.
Conversely, suppose that (\ref{irc}) holds.
Then we can define a unitary operator $U$ by
\begin{equation}
U:=\exp\left(-i\lambda\Phi_{\rm S}\left(i
\frac{\sum_{j=1}^N{\hat \rho_j}{}^*}{\omega^{3/2}}\right)\right). \label{U}
\end{equation}
It is easy to check  that (\ref{YH}) and (\ref{Yp}) hold with
$Y=U$.
\hfill \qed
\medskip
Proposition \ref{equiv} shows that,
under (H.1)--(H.3) and the condition that
$\sum_{j=1}^N\rho_j \in H_{\omega}^{-3/2}$,
the Nelson model $\M_{\rm NF}$ is
equivalent to
the SNM. In this case, under suitable additional conditions,
$H_{\rm SNM}$ has a ground state \cite{Ger} and so does $H$.
Thus we are interested in
the case
\begin{equation}
\sum_{j=1}^N\rho_j \not\in H_{\omega}^{-3/2}. \label{isc}
\end{equation}
This condition is called an {\it infrared singularity
condition}. In this paper,
we say
that {\it the Nelson model $\M_{\rm Nelson}$ has no infrared cutoff}
if (\ref{isc}) holds.
Under condition (\ref{isc}),
the Nelson model $\M_{\rm NF}$ is not unitarily equivalent to
the SNM.
Note that (\ref{isc}) and the natural condition
$\sum_{j=1}^N\rho_j/\sqrt{\omega} \in L^2(\R^d)$ imply
\begin{equation}
{\rm ess.inf}_{k\in \R^d}\,\omega(k)=0 \label{infomega}
\end{equation}
(\lq\lq{ess.inf}" means essential infimum),
i.e., the quantum scalar field under consideration
is \lq\lq{massless}".


\section{Existence of a ground state of the Nelson
model $\M_{\rm NF}$ without infrared cutoff}
Let $T$ be a self-adjoint operator on a Hilbert space and
bounded from below.
We say that $T$ has a ground state
if there exists a non-zero vector $\psi \in D(T)$ such that
$T\psi=E_0(T)\psi$. In this case $\psi$ is called a ground state
of $T$.
We show below that, under suitable conditions, the Hamiltonian
$H$ of the Nelson model $\M_{\rm NF}$ has a ground state
{\it even
in the case where it has no infrared cutoff (i.e.,
(\ref{isc}) holds).}
We formulate some additional conditions:
\begin{description}
\item[(H.4)]
The function
$\omega$ is continuous on $\R^d$ satisfying (\ref{infomega})
and the following conditions:
\begin{eqnarray*}
&&D_{\mu}\omega \in L^{\infty}(\R^d), \
\mu=1, \cdots,d, \\
&&\lim_{|k| \to \infty} \omega(k)=\infty,
\end{eqnarray*}
\item[(H.5)]
For $j=1, \cdots,N$,
\begin{equation}
\int_{\R^d}\frac{|k|^2|\hat\rho_j(k)|^2}{\omega(k)^3}dk < \infty.
\end{equation}
\item[(H.6)] For all $R > 0$,
$\int_{|q| \leq R} |V(q)|dq < \infty$ and
there exist constants $c_1, c_2 \geq 0$ such that
\begin{equation}
|q|^2 \leq c_1 V(q)+c_2, \quad {\rm a.e.}q\in \R^{dN}.
\end{equation}
\end{description}


An infrared-cutoff Hamiltonian
of the SNM is defined by
\begin{equation}
H_{{\rm SNM},\sigma}
:=H_0+\lambda
\sum_{j=1}^N\Phi_{\rm S}\left(e^{-ikq_j}\frac{\chi_{\omega \geq \sigma}
{\hat \rho_j}{}^*}
{\sqrt{\omega}}\right),
\end{equation}
where $\sigma > 0$ is an infrared cutoff parameter and
$\chi_S$ is a characteristic function of the set $S$.


Under (H.2), we can define for all $\sigma > 0$ a unitary operator
\begin{equation}
U_{\sigma}:=\exp\left(-i\lambda\Phi_{\rm S}\left(i\frac{
\sum_{j=1}^N\chi_{\omega \geq \sigma}
\hat \rho_j}{\omega^{3/2}}\right)\right).
\end{equation}

\begin{th}\label{mainth} Assume (H.1), (H.2) and (H.4)--(H.6).
Then $H$ has a ground state $\psi_0$ which has the following property:
there exists a sequence $\{\phi_{\sigma_n}\}_{n=1}^{\infty}$
of unit vectors in $D(H_0)$
such that $\sigma_n > 0, \, n\in \N$,
$\lim_{n\to \infty}\sigma_n=0$,
each $\phi_{\sigma_n}$ is a ground state of
$H_{{\rm SNM}, \sigma_n}$ and
\begin{equation}
\mbox{{\rm w-}}
\lim_{n\to \infty}U_{\sigma_n}^{-1}\phi_{\sigma_n}=\psi_0, \label{w-lim}
\end{equation}
where \lq\lq{\rm w-lim}" means weak limit.
\end{th}

\begin{rem}{\rm Consider the physical case
$\omega=\omega_0$ ((\ref{omega}) with $m=0$). Hence
(H.4) holds. Assume (H.2)
with $\omega=\omega_0$
and that
$$
\sum_{j=1}^N\frac {\hat \rho_j(\cdot)}{|k|^{3/2}}\not\in L^2(\R^d).
$$
Then (\ref{isc}) and (H.5) hold with  $\omega=\omega_0$.
Hence Theorem \ref{mainth} holds in the physical case
without infrared cutoff.
}
\end{rem}

\begin{rem}{\rm Assume   (\ref{isc}).
Then we have
\begin{equation}
\mbox{{\rm w-}}\lim_{\sigma\to 0}U_{\sigma}=0.
\end{equation}
This is an expression of infrared divergence.
Hence the relation
w-$\lim_{n\to \infty}U_{\sigma_n}^{-1}\phi_{\sigma_n}=\psi_0$
in Theorem \ref{mainth} suggests that w-$\lim_{n\to \infty}
\phi_{\sigma_n}=0$.
}
\end{rem}


The idea of proof of Theorem \ref{mainth} is to apply
\cite[Theorem 1]{Ger}.

Let
$$
K:=H_{\rm p}-\lambda^2W+c_0\lambda^2.
$$

\begin{lem}\label{lem1}
Assume (H.1) and (H.2).
Then $K$ is self-adjoint and bounded from below.
\end{lem}
{\it Proof}. This follows from the boundedness of $\lambda^2W$
and a simple application of  the Kato-Rellich theorem.
\hfill \qed
\medskip
Let
$$
\hat K:=K-E_0(K).
$$
\begin{lem}
Assume (H.1), (H.2) and (H.6).
Then
$(\hat K+i)^{-1}$
is compact.
\end{lem}
{\it Proof}.  By (H.6) and the boundedness of $W$,
we have
$$
\lim_{|q| \to \infty}\{V(q)-\lambda^2W(q)+c_0\lambda^2\}=\infty.
$$
Hence, applying \cite[p.249, Theorem XIII.67]{RS4},
we obtain the desired result.
\hfill \qed

\medskip

We denote by
${\sf B}(L^2(\R^{dN}))$
the set of bounded linear
operators on $L^2(\R^{dN})$.
For each $k \in \R^d$, we define a linear operator
on $L^2(\R^{dN})$ as follows:
\begin{equation}
v(k):=\left\{
\begin{array}{ll}
G(\cdot)(k) & \qquad \omega(k) > 0 \\
0 & \qquad \omega(k)=0
\end{array}\right..
\end{equation}
\begin{rem}{\rm
By the original assumption for $\omega$,
the Lebesgue measure of the set $\{k\in \R^d
|\omega(k)=0\}$ is equal to zero. Hence,
$v(k)=G(\cdot)(k)$ a.e. $k\in \R^d$.
}
\end{rem}
\begin{lem}\label{lem3-5}  Assume (H.2). Then,
for a.e. $k\in \R^{d}$, $v(k)
\in {\sf B}(L^2(\R^{dN}))$
with
$$
\|v(k)\| \leq
2\sum_{j=1}^N\frac {|\hat \rho_j(k)|}{\sqrt{\omega(k)}}.
$$
\end{lem}
{\it Proof}. This easily follows from (\ref{G(q)}).
\hfill \qed
\medskip

The following lemma immediately follows from
Lemma \ref{lem3-5}.
\begin{lem} Assume ( H.1) and (H.2). Then,
for a.e. $k\in \R^{d}$,
$$
v(k)(\hat K+1)^{-1/2} \in {\sf B}(L^2(\R^{dN}),
\quad (\hat K+1)^{-1/2}v(k)
\in
{\sf B}(L^2(\R^{dN})).
$$
\end{lem}
\begin{lem} Assume (H.1) and (H.2). Then,
for all $u_1,u_2 \in L^2(\R^{dN})$,
the functions
$$
(u_2, v(\cdot)(\hat K+1)^{-1/2}u_1),
(u_2, (\hat K+1)^{-1/2}v(\cdot)u_1),
$$
on $\R^d$ are measurable. Moreover,
$$
C(R):=\int_{\R^d}\frac 1{\omega(k)}\left(
\|v(k)(\hat K+R)^{-1/2}\|^2+
\|(\hat K+R)^{-1/2}v(k)\|^2\right)dk < \infty
$$
and
\begin{equation}
\lim_{R\to \infty}C(R)=0. \label{C(R)}
\end{equation}
\end{lem}
{\it Proof}. We have
$$
(u_2, v(k)
(\hat K+1)^{-1/2}u_1)
=\sum_{j=1}^N
\frac {{\hat \rho_j(k)}^*}{\sqrt{\omega
(k)}}g_j(k)
$$
with
$$
g_j(k)=
\int_{\R^d}u_2(q)^*
\left(e^{-iq_jk}-1\right)
\left((\hat K+1)^{-1/2}u_1\right)(q)dq.
$$
It is easy to see that $g_j$ is continuous on $\R^d$.
Hence
$(u_2, v(\cdot)
(\hat K+1)^{-1/2}u_1)$
is measurable.
Similarly we can show that
$(u_2, (\hat K+1)^{-1/2}v(\cdot)u_1)$
is measurable.

We have
$$
C(R)\leq
\frac 2{R}\int_{\R^d}\frac 1{\omega(k)}
\|v(k)\|^2dk
\leq
\frac {8N}{R}
\sum_{j=1}^N
\int_{\R^d}
\frac {|\hat \rho_j(k)|^2}{\omega(k)^2}
dk.
$$
Hence $C(R) < \infty$ and (\ref{C(R)}) holds.
\hfill \qed

\begin{lem} Assume (H.1) and (H.6). Let $\psi \in D(\hat K^{1/2})$. Then
$\psi \in D(|q_j|)$ and,
for all $j=1,\cdots,N$,
\begin{equation}
\||q_{j}|\psi\| \leq c_1\|\hat K^{1/2}\psi\|+(c_1\lambda^2\|W\|
+c_1c_0\lambda^2+c_1E_0(K)+c_2)\|\psi\|^2. \label{qj}
\end{equation}
In particular,
\begin{equation}
Q_j:=|q_j|(\hat K+1)^{-1/2}
\end{equation}
is bounded.
\end{lem}
{\it Proof}. Let $\psi \in D(K)$.
Then, using  (H.6), one can easily prove
(\ref{qj}). Since $D(K)$ is a core of $\hat K^{1/2}$,
it follows from a limiting argument
that $D(|q_j|) \supset D(\hat K^{1/2})$ and
(\ref{qj}) holds for all $\psi \in D(\hat K^{1/2})$.
\hfill \qed
\medskip

\begin{lem}\label{lemlast} Assume (H.1), (H.2), (H.5) and (H.6).
Then
\begin{equation}
\int_{\R^d}\frac 1{\omega(k)^2}\|v(k)(\hat K+1)^{-1/2}\|^2dk < \infty.
\label{vK}
\end{equation}
\end{lem}
{\it Proof}. By the elementary inequality
$$
|e^{-iq_jk}-1| \leq |q_j|\,|k|
$$
we have for all $u \in L^2(\R^{dN})$
$$
\|v(k)u\| \leq
\sum_{j=1}^N\frac {|\hat\rho_j(k)| \,|k|}{\sqrt{\omega(k)}}
\||q_j|u\|.
$$
Hence
$$
\|v(k)(\hat K+1)^{-1/2}\| \leq
\sum_{j=1}^N\frac {|\hat\rho_j(k)|\,|k|}{\sqrt{\omega(k)}}
\|Q_j\|,
$$
which, by (H.5),  implies (\ref{vK}). \hfill \qed

\bigskip
\noindent{\bf Proof of Theorem \ref{mainth}}
\medskip
We have
$$
H=K\otimes I +I\otimes H_{\rm b}+H_I(v)
$$
with
$$
H_I(v):=\frac {\lambda}{\sqrt{2}}
\int_{\R^d}(v(k)a(k)^*+v(k)^*a(k))dk.
$$
Hence $H$ is exactly of the same form
as the Hamiltonians considered in \cite{Ger}.
Lemmas \ref{lem1}--\ref{lemlast} show
that $H$ satisfies all the assumptions
of  \cite[Theorem 1]{Ger}.
Thus we can apply
\cite[Theorem 1]{Ger}
to conclude that
$H$ has a ground state $\psi_0$.
>From the proof of
\cite[Theorem 1]{Ger}, we see that
$\psi_0$ is a weak limit of a sequence
$\{\psi_{\sigma_n}\}_{n=1}^{\infty}$
of unit vectors in $D(H_0)$
such that $\sigma_n > 0, \, \lim_{n \to \infty}\sigma_n=0$ and
each $\psi_{\sigma_n}$ is a ground state
of the infrared-cutoff version  $H_{\sigma_n}$ of $H$:
$$
H_{\sigma}:=
K\otimes I +I\otimes H_{\rm b}+H_I(v_{\sigma}), \quad \sigma > 0,
$$
where $v_{\sigma}(k):=v(k)\chi_{\omega\geq \sigma}(k)$.
By Proposition \ref{equiv}, $\phi_{\sigma_n}:=U_{\sigma_n}
\psi_{\sigma_n}$ is a ground state of $H_{{\rm SNM},\sigma_n}$
with $\|\phi_{\sigma_n}\|=1$.
Hence (\ref{w-lim}) holds.
\hfill \qed

\begin{thebibliography}{99}

\bibitem{Ar89}A. Arai, Scaling limit for quantum systems of
nonrelativistic  particles interacting with a Bose field,
Hokkaido University Preprint Series in Mathematics \#59, 1989.

\bibitem{AHH}A. Arai,  M. Hirokawa and F. Hiroshima,
On the absence of eigenvectors of Hamiltonians in a class
of massless quantum field models without infrared cutoff,
J. Funct. Anal. {\bf 168}(1999), 470--497.
\bibitem{AH}A. Arai and M. Hirokawa,
Ground states of a general class of quantum field Hamiltonians,
Rev. Math. Phys. {\bf 12} (2000), 1085--1135.

\bibitem{BFS}V. Bach, J. Fr\"ohlich and I. M. Sigal,
Spectral analysis for systems of atoms and molecules coupled
to the quantized radiation  field, Commun. Math. Phys.
{\bf 207}(1998), 249--290.

\bibitem{Emch}G. G. Emch,
Algebraic Methods in Statistical Mechanics and Quantum
Field Theory, John Wiley \& Sons, 1972.
\bibitem{Fro}J. Fr\"ohlich, On the infrared
problem in a model of scalar electrons and massless scalar bosons,
Ann. Inst. Henri Poincar\'e {\bf 19}(1973), 1--103.


\bibitem{Ger}C. G\'erard, On the existence
of ground states for massless Pauli-Fierz Hamiltonians,
Ann. Henri Poincar\'e {\bf 1}(2000), 443--459.

\bibitem{GLL}M. Griesemer, E. H. Lieb and
M. Loss, Ground states in non-relativistic  quantum electrodynamics,
preprint, 2000.

\bibitem{LMS}J. L\"orinczi,
R. A. Minlos, and
H. Spohn,
The infrared behaviour in Nelson's model
of a quantum particle coupled to a massless scalar field,
preprint, 2000.

\bibitem{Ne}E. Nelson, Interaction of nonrelativistic
particles with a quantized scalar field,
J. Math. Phys. {\bf 5}(1964), 1190--1197.

\bibitem{RS2} M. Reed and B. Simon,
Methods of Modern Mathematical Physics Vol. II:
Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
\bibitem{RS4} M. Reed and B. Simon,
Methods of Modern Mathematical Physics Vol. IV:
Analysis of Operators, Academic Press, New York, 1978.
\bibitem{Sp1}H. Spohn, Ground state(s) of the spin-boson Hamiltonian,
Commun. Math. Phys. {\bf 123} (1989), 277--304.
\bibitem{Sp2}H. Spohn, Ground state of a quantum particle
coupled to a scalar Bose field, Lett. Math. Phys. {\bf 44}
(1998), 9--16.
\end{thebibliography}
\end{document}