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{\it Keywords}: quantum electrodynamics, polaron,
Dirac-Maxwell operator, non-relativistic
limit, scalig limit, Fock space
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\begin{document}
\title{\bf Non-relativistic Limit
of a Dirac Polaron
in Relativistic Quantum Electrodynamics}
\author{ Asao Arai\thanks{
Supported
by the Grant-in-Aid No.17340032 for Scientific Research from
the JSPS.} \\
{Department of Mathematics, Hokkaido University} \\
{Sapporo 060-0810, Japan} \\
{E-mail: arai@math.sci.hokudai.ac.jp}
}
\maketitle
\bigskip
\begin{abstract}
A quantum system of a Dirac particle interacting with the quantum radiation field
is considered in the case where no external potentials exist. Then
the total momentum of the system is conserved and the total Hamiltonian
is unitarily equivalent to
the direct integral $\int_{{\bf R}^3}^\oplus\overline{H({\bf p})}d{\bf p}$
of a family of self-adjoint operators
$\overline{H({\bf p})}$ acting in the Hilbert space $\oplus^4{\cal F}_{\rm rad}$,
where ${\cal F}_{\rm rad}$ is the Hilbert space of
the quantum radiation field. The fibre operator $\overline{H({\bf p})}$ is
called the Hamiltonian of the Dirac polaron with total momentum ${\bf p}
\in {\bf R}^3$. The main result of this paper is concerned with
the non-relativistic (scaling) limit of
$\overline{H({\bf p})}$. It is proven that the non-relativistic limit
of $\overline{H({\bf p})}$ yields
a self-adjoint extension of
a Hamiltonian of a polaron with spin $1/2$
in non-relativistic quantum electrodynamics.
\end{abstract}
\bigskip
\noindent
{\it Keywords}: quantum electrodynamics, polaron,
Dirac-Maxwell operator, non-relativistic
limit, scalig limit, Fock space
\bigskip
\section{Introduction}
In a previous paper \cite{Ar03}, the author
discussed the non-relativistic (scaling) limit
of a particle-field Hamiltonian $H$, called a Dirac-Maxwell (DM)
operator or a DM Hamiltonian, in relativistic quantum electrodynamics (QED).
The Hamiltonian $H$ describes a Dirac particle
---a relativistic charged particle with spin $1/2$---
in an external potential $V$ and interacting with the quantum
radiation field. In the case where $V=0$,
the total momentum of the quantum system is conserved,
implying that
$H$ is unitarily equivalent to a self-adjoint
operator $\widetilde H$ which has a direct integral decomposition $\widetilde H=\int_{\R^3}^{\oplus}
\overline{H({\bf p})}d{\bf p}$ with $H({\bf p})$ being an
essentially
self-adjoint operator acting in the Hilbert
space
\begin{equation}
{\cal H}:=\C^4\otimes {\cal F}_{\rm rad}=\oplus^4{\cal F}_{\rm rad}
\end{equation}
($\overline{H({\bf p})}$
denotes the closure of $H({\bf p})$),
where
\begin{equation}
{\cal F}_{\rm rad}:=\oplus_{n=0}^{\infty}
\otimes_{\rm s}^n[L^2(\R^3)\oplus L^2(\R^3)]
\end{equation}
is the Fock space over
$L^2(\R^3)\oplus L^2(\R^3)$ ($\otimes_{\rm s}^n$ denotes
$n$-fold symmetric tensor prodct and
$\otimes_{\rm s}^0[L^2(\R^3)\oplus L^2(\R^3)]:=\C$), that is, the Hilbert
space of state vectors for
the quantum radiation field. The 3-dimensional vector ${\bf p}\in
\R^3$ physically
means a point in the spectrum of the total momentum
\cite[Theorem 1.6]{Ar00}.
The operator $H({\bf p})$ describes
a {\it Dirac polaron}, namely, the Dirac particle
coupled to the quantum radiation field
with the total momentum equal to ${\bf p}$.
Based on this picture, we call the
fiber operator $H({\bf p})$
the Hamiltonian of the Dirac polaron with total momentum ${\bf p}$.
In this paper we consider a partial non-relativistic limit
of $H(\p)$, where \lq\lq{partial}" means
that non-relativistic limit is taken
only with respect to the Dirac particle.
For this purpose, in a way similar to
that done in \cite{Ar03},
we put a scaling
parameter $\kappa >0$ into $H(\p)$
and make an energy cutoff for the free Hamiltonian
of the quantum radiation field as well as an energy
renormalization to obtain a natural candidate
$H_{\kappa}(\p)$ on which
the non-relativistic limit $\kappa\to\infty$
should be considered (the parameter $\kappa$ plays a role of the speed of light).
We show that the resolvent $(\overline{H_{\kappa}(\p)}-z)^{-1}$
($z\in\C\setminus\R$) strongly converges to
an operator matrix, one of the components
of which is the resolvent
of a self-adjoint extension
of
the Hamiltonian of the non-relativistic polaron
with spin $1/2$ in non-relativistic QED.
This is the main result of this paper.
In Section 2 we describe some basic facts
on both relativistic and non-relativistic
polarons in QED. In Section 3 we precisely formulate the main result and
prove it.
\section{Preliminaries}
\subsection{The explicit form of the operator
$H(\p)$}
We first recall some objects
in relativistic QED. We use a unit system in which
$\hbar$ (the Planck constant divided by $2\pi$)
and $c$ (the speed of light) are equal to $1$ respectively.
We denote by $\alpha_j,\beta$ ($j=1,2,3$)
the Dirac matrices, i.e.,
$4\times 4$-Hermitian matrices satisfying the anticommutation
relations
\begin{eqnarray}
&&\{\alpha_j,\alpha_l\}=2\delta_{jl}1_4, \quad \{\alpha_j,\beta\}=0,
\quad j,l=1,2,3,
\quad \beta^2=1_4,
\end{eqnarray}
where $\{X,Y\}:=XY+YX$ and $1_4$ is the $4\times 4$-identity matrix.
We set $\balpha:=(\alpha_1,\alpha_2,\alpha_3)$.
In the present paper we take the following representations of
$\alpha_j$ and $\beta$, called the standard representation \cite[p.36]{Thaller}:
\begin{equation}
\alpha_j=\left(
\begin{array}{cc}
0_2 & \sigma_j\\
\sigma_j & 0_2
\end{array}
\right), \quad
\beta=\left(
\begin{array}{cc}
1_2 & 0_2\\
0_2 & -1_2
\end{array}
\right), \quad j=1,2,3, \label{DPR}
\end{equation}
where $1_2$ (resp. $0_2$) is the $2\times 2$-identity
(resp. zero) matrix and $\sigma_j$'s are
the Pauli matrices:
$$
\sigma_1:=\left(
\begin{array}{cc}
0 & 1\\
1& 0
\end{array}
\right), \quad\sigma_2:=\left(
\begin{array}{cc}
0 & -i\\
i& 0
\end{array}
\right), \quad
\sigma_3:=\left(
\begin{array}{cc}
1 & 0\\
0& -1
\end{array}
\right).
$$
Let $\omega:\R^3 \to [0,\infty)$ be a Borel measurable function,
strictly positive a.e. (almost everywhere) with respect to the Lebesgue
measure. For mathematical generality,
we take one-particle (one-photon) Hamiltonian to
be the multiplication operator on $L^2(\R^3)\oplus L^2(\R^3)$
by the function $\omega$
; we denote it by the same symbol $\omega$.
Then the free Hamiltonian of the quantum radiation field
is defined by
\begin{equation}
H_{\rm rad}:=d\Gamma(\omega),
\end{equation}
the second quantization of $\omega$
(\cite[p.302]{RS1}, \cite[\S X.7]{RS2}).
The momentum operator
\begin{equation}
\P_{\rm rad}=(P_{{\rm rad},1},
P_{{\rm rad},2},P_{{\rm rad},3})
\end{equation}
of the quantum radiation field is given by
\begin{equation}
P_{{\rm rad},j}:=d\Gamma(k_j), \quad j=1,2,3,
\end{equation}
the second quantization of $k_j$ as the
multiplication operator by the $j$-th coordinate
function $k_j$ in the momentum space $\R^3$ of a photon.
We denote by $a(f)$ ($f\in L^2(\R^3)\oplus L^2(\R^3)$)
the annihilation operator acting in ${\cal F}_{\rm rad}$
and define
\begin{equation}
\phi(f):=\frac{a(f)+a(f)^*}{\sqrt{2}},
\end{equation}
the Segal field operator on ${\cal F}_{\rm rad}$ \cite[\S X.7]{RS2}.
There exist $\R^3$-valued continuous functions
$\e^{(r)}, r=1,2$, on
the nonsimply connected space
$\M_0:=\R^3\setminus\{(0,0,k_3)|k_3\in \R\}$ such that
$$
\e^{(r)}(\k)\cdot \e^{(s)}(\k)=\delta_{rs}, \quad
\e^{(r)}(\k)\cdot \k=0, \quad r,s=1,2, \ \k \in \M_0.
$$
We set $\e^{(r)}(0,0,k_3)=0, \ k_3\in \R$.
The vector-valued functions $\e^{(r)}, r=1,2,$ describe the polarization
of one photon.
Let $g \in L^2(\R^3)$ and
\begin{equation}
g_j:=(ge_j^{(1)}, ge_j^{(2)})\in L^2(\R^3)\oplus L^2(\R^3),
\quad j=1,2,3.
\end{equation}
We define a smeared quantum radiation field
$\A:=(A_1,A_2,A_3)$ by
\begin{equation}
A_j:=\phi(g_j), \quad j=1,2,3
\end{equation}
We assume the following:
\begin{list}{}{}
\item{(g.1)} $\omega g,\, g/\sqrt{\omega}, \,|\k|g, \,
|\k|g/\sqrt{\omega} \in L^2(\R^3)$.
\item{(g.2)} For all $R>0$, $\int_{|\k|\leq R}\omega(\k)^2d\k <\infty$
(i.e., $\omega \in L^2_{\rm loc}(\R^3)$).
\end{list}
For objects ${\bf a}=(a_1,a_2,a_3)$ and ${\bf b}=(b_1,b_2,b_3)$
such that the product $a_jb_j$ and the sum $\sum_{j=1}^3a_jb_j$
are defined, we write ${\bf a}\cdot {\bf b}:=\sum_{j=1}^3a_jb_j$.
Now we can give the explicit form of the
polaron Hamiltonian $H(\p)$ ($\p\in \R^3$) \cite{Ar00}:
\begin{equation}
H(\p)=
\balpha \cdot (\p-\P_{{\rm rad}}-q\A)
+m\beta +H_{\rm rad},
\end{equation}
where
$q\in \R$ (resp. $m >0$) is a constant, physically
denoting the charge (resp. bare mass) of the Dirac particle
under consideration.
\subsection{Essential self-adjointness}
Let $\Omega_0:=\{1,0,0,\cdots\}$ be the Fock vacuum in ${\cal F}_{\rm rad}$
and ${\cal F}_{{\rm rad},0}^{\infty}$ be the subspace algebraically
spanned by the vectors $\Omega_0,
a(f_1)^*\cdots a(f_n)^*\Omega_0, n\in \N, f_j\in
C_0^{\infty}(\R^3)\oplus C_0^{\infty}(\R^3),j=1,\cdots,n$.
Then ${\cal F}_{{\rm rad},0}^{\infty}$ is dense in ${\cal F}_{\rm rad}$.
We introduce
\begin{equation}
{\cal D}:=\oplus^4{\cal F}_{{\rm rad},0}^{\infty},
\end{equation}
which is dense in ${\cal H}$.
The following fact is proven \cite[Theorem 1.5]{Ar00}:
\begin{th} Assume (g.1) and (g.2). Then, for all $\p\in \R^3$,
$H(\p)$ is essentially self-adjoint on ${\cal D}$.
\end{th}
\begin{rem}{\rm
It is a highly non-trivial problem to
make it clear whether or not $H(\p)$ is bounded below,
because $H({\bf p})$ contains the term $-\balpha \cdot \P_{\rm rad}$
which is unbounded both below and above and
\lq\lq{comparable}" with $H_{\rm rad}$.
Recently Sasaki \cite{Sa} gave a positive solution
to this problem, proving that,
under a suitable condition for $g$,
$H(\p)$ is bounded below for all $\p\in \R^3$ and $q\in \R$.
}
\end{rem}
\subsection{A scaled Hamiltonian}
To consider the non-relativistic limit of $H(\p)$,
we put a scaling parameter $\kappa >0$ in $H(\p)$
and make an energy cutoff for $H_{\rm rad}$
as well as an energy renormalization.
Let $\Lambda:[0,\infty)\to [0,\infty)$
be an increasing measurable function
such that $\lim_{\kappa\to \infty}\Lambda(\kappa)=\infty$
and
\begin{equation}
H_{\rm rad}^{(\kappa)}
:=E_{\rm rad}([0,\Lambda(\kappa)])
H_{\rm rad} E_{\rm rad}([0,\Lambda(\kappa)]),
\end{equation}
where $E_{\rm rad}(\cdot)$ is the spectral measure
of $H_{\rm rad}$. The operator $H_{\rm rad}^{(\kappa)}$
may be interpreted as the free Hamiltonian of the quantum radiation field
with an energy cutoff.
Then the scaled, renormalized polaron Hamiltonian
for which we consider a scaling limit
is defined as follows:
\begin{equation}
H_{\kappa}(\p)
:=\kappa\balpha\cdot (\p-\P_{{\rm rad}}-q\A)
+m\kappa^2\beta-m\kappa^2 +H_{\rm rad}^{(\kappa)}.
\end{equation}
The non-relativistic limit by which we mean in the present
paper
is to take the limit $\kappa \to \infty$
of $H_{\kappa}(\p)$ in a suitable sense, where
the scaling parameter $\kappa$ plays a role of the speed of light.
As remarked
in \cite{Ar03}, this non-relativistic limit
is with respect to the Dirac particle only
and, hence, strictly speaking,
it is a {\it partial} non-relativistic limit.
\subsection{A Dirac type operator }
In connection with the non-relativistic limit of $H_{\kappa}
(\p)$, we find it convenient to introduce a Dirac type operator
defined by
\begin{equation}
{\D}_{\A}(\p)
:=\balpha\cdot (\p-\P_{{\rm rad}}-q\A).
\end{equation}
(For a general background, see \cite[Section 4]{Ar03}.)
\begin{th}\label{DA} Assume (g.1) and (g.2). Then,
for all $\p\in \R^3$, ${\D}_{\A}(\p)$
is essentially self-adjoint on ${\cal D}$.
\end{th}
\noindent
{\bf Proof}. It is sufficient to show that
$-\balpha\cdot (\P_{{\rm rad}}+q\A)$ is essentially self-adjoint,
since $\balpha\cdot {\bf p}$ is a bounded self-adjoint operator.
This is easily done by applying the commutator theorem
\cite[Theorem X.37]{RS2}
(see the proof of
\cite[Theorem 1.5]{Ar00}). \hfill \qed
\medskip
As a corollary to this theorem, we have
the following fact:
\begin{cor}\label{H-sa} Assume (g.1) and (g.2). Then,
for all $\kappa>0$ and $\p\in \R^3$,
$H_{\kappa}(\p)$ is essentially self-adjoint on
${\cal D}$.
\end{cor}
\noindent
{\bf Proof}.
The operator $H_{\kappa}(\p)$ is written as
$H_{\kappa}(\p)=\kappa {\D}_{\A}(\p)
+B$ with $B:=m\kappa^2\beta-m\kappa^2+H_{\rm rad}^{(\kappa)}$.
Note that $H_{\rm rad}^{(\kappa)}$ is a bounded self-adjoint
operator.
Hence $B$ is a bounded
self-adjoint operator. Therefore, by Theorem
\ref{DA} and the Kato-Rellich theorem,
$H_{\kappa}(\p)$ is essentially self-adjoint
on ${\cal D}$.
\hfill \qed
\medskip
Using the representation
(\ref{DPR}), we have for
$\D_{\A}(\p)$ the following
operator matrix representation:
\begin{equation}
\D_{\A}(\p)=\left(
\begin{array}{cc}
0 & P_{\A}(\p)\\
P_{\A}(\p)& 0
\end{array}\right)
\end{equation}
with
\begin{equation}
P_{\A}(\p):=\bsigma\cdot(\p-\P_{\rm rad}-q\A),
\end{equation}
where $\bsigma:=(\sigma_1,\sigma_2,\sigma_3)$.
In the same way as in the case of $\D_{\A}(\p)$,
we can prove the following theorem:
\begin{th}\label{PA} Assume (g.1) and (g.2). Then,
for all $\p\in \R^3$, $P_{\A}(\p)$
is essentially self-adjoint on ${\cal E}:=
\oplus^2{\cal F}_{{\rm rad},0}^{\infty}$.
\end{th}
Theorems \ref{DA} and \ref{PA} imply the following
operator equalities:
\begin{equation}
\overline{\D_{\A}(\p)}=
\left(
\begin{array}{cc}
0 & \overline{P_{\A}(p)}\\
\overline{P_{\A}(\p)}& 0
\end{array}
\right)
\end{equation}
and
\begin{equation}
\overline{\D_{\A}(\p)}^2=
\left(
\begin{array}{cc}
\overline{P_{\A}(p)}^2&0\\
0& \overline{P_{\A}(\p)}^2
\end{array}
\right).
\end{equation}
\subsection{A non-relativistic polaron}
The Hamiltonian (with total momentum $\p\in \R^3$)
of a non-relativistic charged polaron with spin $1/2$
and with bare mass $m>0$ is defined by
\begin{equation}
H_{\rm NR}(\p):=\frac 1{2m}
P_{\A}(\p)^2
+H_{\rm rad}.
\end{equation}
It can be shown that,
for all sufficiently small $|q|$,
$H_{\rm NR}(\p)$ is essentially self-adjoint \cite{HS}.
It seems, however, that the essential self-adjointness
of $H_{\rm NR}(\p)$ for all $q\in \R$ has not been proved
in the literature. In fact, this is a non-trivial problem,
because $H_{\rm NR}(\p)$ contains
the terms $\P_{\rm rad}^2$ and $\A\cdot \P_{\rm rad}$ which
are not relatively bounded with respect to $H_{\rm rad}$.
But here we do not discuss this problem.
Instead we construct a self-adjoint extension
of $H_{\rm NR}(\p)$.
For each $\kappa >0$ we introduce
\begin{equation}
H_{\rm NR}^{(\kappa)}(\p):=
\frac 1{2m}\overline{P_{\A}(\p)}^2 +H_{\rm rad}^{(\kappa)}.
\end{equation}
Since $H_{\rm rad}^{(\kappa)}$ is bounded
as remarked above,
it follows that
$H_{\rm NR}^{(\kappa)}(\p)$ is
self-adjoint with $D(H_{\rm NR}^{(\kappa)}(\p))=
D(\overline{P_{\A}(\p)}^2)$ and nonnegative.
The following lemma is a key to the main result of this paper:
\begin{lem}\label{keylem} Assume (g.1) and (g.2).
Then there exists a unique self-adjoint extension
$\tilde H_{\rm NR}(\p)$ of $H_{\rm NR}(\p)$ such that
the following hold:
\begin{list}{}{}
\item{(i)} $\tilde H_{\rm NR}(\p)\geq 0$.
\item{(ii)} $D(\tilde H_{\rm NR}(\p)^{1/2})
\subset D(|\overline{P_{\A}(\p)}|)\cap D(H_{\rm rad}^{1/2})$.
\item{(iii)} For all $z\in \C\setminus [0,\infty)$,
$$
\mbox{{\rm s-}}\lim_{\kappa \to\infty}
(H_{\rm NR}^{(\kappa)}(\p)-z)^{-1}=(\tilde H_{\rm NR}(\p)-z)^{-1}.
$$
\item{(iv)} For all $\xi <0$ and $\psi \in
D(\tilde H_{\rm NR}(\p)^{1/2})$,
$$
\mbox{{\rm s-}}\lim_{\kappa \to\infty}
(H_{\rm NR}^{(\kappa)}(\p)-\xi)^{1/2}\psi=
(\tilde H_{\rm NR}(\p)-\xi)^{1/2}\psi.
$$
\end{list}
\end{lem}
\noindent
{\bf Proof}. We need only to apply
\cite[Theorem A.1]{Ar03}
to the following case:
$$
N=1, \ A=\frac 1{2m}\overline{P_{\A}(\p)}^2, \
B_1=H_{\rm rad},
$$
where $N, A, B_j$ ($j=1,\cdots,N$) are the notations used in
\cite[Appendix A]{Ar03}.
\hfill \qed
\medskip
\begin{rem}{\rm Since
$H_{\rm NR}$ is nonnegative, it has a self-adjoint extension
$\hat H_{\rm NR}$ which is defined as the Friedrichs extension of
$H_{\rm NR}$. Another self-adjoint extension
$H_{\rm NR}'$ of $H_{\rm NR}$ can be defined through the sesquilinear
form $s(\Psi,\Phi):=\sum_{j=1}^3\lang \overline{P_{\bf A}(\p)_j}\Psi,
\overline{P_{\bf A}(\p)_j}\Phi\rang/2m \, +
\lang H_{\rm rad}^{1/2}\Psi,H_{\rm rad}^{1/2}\Phi\rang,
\ \Psi,\Phi \in \cap_{j=1}^3D(\overline{P_{\bf A}(\p)_j})\cap D(H_{\rm rad}^{1/2})$.
But it seems to be nontrivial to see if $\hat H_{\rm NR}=H_{\rm NR}'$.
Unfortunately we have not been
able to clarify if
$\tilde H_{\rm NR}=\hat H_{\rm NR}$ or $\tilde H_{\rm NR}=H_{\rm NR}'$.
Also we remark that,
if $H_{\rm NR}$ is not essentially self-adjoint,
then the operator ${\tilde H}_{\rm NR}(\p)$ may depend on
the choice of the energy cutoff parameter $\Lambda(\cdot)$.
}
\end{rem}
\section{The Main Result}
The main result of the present paper is as follows:
\begin{th}\label{mainth} Assume (g.1) and (g.2). Suppose that
\begin{equation}
\lim_{\kappa \to \infty}\frac{\Lambda(\kappa)^2}{\kappa}
=0.\label{L-cond}
\end{equation}
Then, for all $ z\in \C\setminus \R$ and all $\p\in \R^3$,
\begin{equation}
\mbox{{\rm s-}}\lim_{\kappa \to\infty}
(\overline{H_{\kappa}(\p)}-z)^{-1}
=\left(
\begin{array}{cc}
(\tilde H_{\rm NR}(\p)-z)^{-1} & 0\\
0 & 0
\end{array}
\right). \label{limit}
\end{equation}
\end{th}
\noindent
{\bf Proof}. It is easy to see that
$$
\overline{H_{\kappa}(\p)}
=\overline{\D_{\A}(\p)}+m\kappa^2\beta-m\kappa^2 +H_{\rm rad}^{(\kappa)}.
$$
In the same way as in the proof
of \cite[Lemma 5.1]{Ar03}, we can show that
the self-adjoint operator $\overline{\D_{\A}(\p)}$ strongly anticommutes
with $\beta$. Since we assume (\ref{L-cond}) and we have
Lemma \ref{keylem}, we can apply \cite[Theorem 4.3]{Ar03}
to the following case:
$$
A=\overline{\D_{\A}(\p)}, \quad B=m\beta, \quad
C(\kappa)=H_{\rm rad}^{(\kappa)},
$$
where $A, B$ and $C(\kappa)$ are the notations used in
\cite[Section 4]{Ar03}.
Thus we obtain (\ref{limit}).
\hfill \qed
\medskip
Theorem \ref{mainth} establishes
a natural connection of relativistic QED
to non-relativistic QED in the context of
polaron theory.
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\bibitem{Ar00} A.~Arai,
A particle-field Hamiltonian in relativistic quantum
electrodynamics, {\it J. Math. Phys}. {\bf 41} (2000),
4271--4283.
\bibitem{Ar03}A.~Arai, Non-relativistic limit of a Dirac-Maxwell
operator in relativistic quantum electrodynamics,
{\it Rev. Math. Phys.} {\bf 15} (2003), 245--270.
\bibitem{HS}F.~Hiroshima and H.~Spohn, Ground state degeneracy of the Pauli-Fierz model with spin, {\it Adv. Theor. Math. Phys}. {\bf 5}
(2001), 1091--1104.
\bibitem{RS1}M.~Reed and B.~Simon, Methods of
Modern Mathematical Physics I: Functional Analysis,
Academic Press, New York, 1972.
\bibitem{RS2}M.~Reed and B.~Simon, Methods of
Modern Mathematical Physics II: Fourier Analysis, Self-adjointness,
Academic Press, New York, 1975.
\bibitem{Sa}I. Sasaki, Ground state energy of the
polaron in the relativistic quantum electrodynamics,
preprint, 2005, to be published in {\it J. Math. Phys.}
\bibitem{Thaller}B.~Thaller, The Dirac Equation, Springer-Verlag,
Berlin, Heidelberg, 1992.
\end{thebibliography}
\end{document}
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