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QED, relativistic particles,
self-adjointness, ground state
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\documentclass[a4paper]{amsart}
\usepackage{amssymb}
\newtheorem{definition}{Definition}
\newtheorem{lem}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}
\newcommand{\R}{{\mathbb R}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\D}{{\mathcal D}}
\newcommand{\un}{{\mathbf 1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\title{Quantum electrodynamics of relativistic bound
states with cutoffs. II}
\author{Jean-Marie Barbaroux}
\address{Universit\'e de Nantes, UMR 6629,
2 rue de la Houssini\`ere, 44322 Nantes Cedex, France}
\curraddr{CPT-CNRS, Luminy Case 907, 13288 Marseille Cedex 9, France}
\email{barbarou@cpt.univ-mrs.fr}
\author{Mouez Dimassi}
\address{CNRS-UMR 7539, Universit\'e de Paris-Nord,
Av. J. B. Cl\'ement 93430 Villetaneuse, France}
\email{dimassi@math.univ-paris13.fr}
\author{Jean-Claude Guillot}
\address{CNRS-UMR 7539, Universit\'e de Paris-Nord,
Av. J. B. Cl\'ement 93430 Villetaneuse, France}
\email{guillot@math.univ-paris13.fr}
\thanks{The first author was supported in part by Minist\`ere de
l'Education Nationale, de la Recherche et de la Technologie via
ACI Blanche project.}
%\subjclass[2000]{}
\date{March 4, 2002} \keywords{QED, relativistic particles,
self-adjointness, ground state}
\subjclass{Primary 81Q10, 81V10; Secondary 81V45, 47B25, 47A55}
\date{February 28, 2002.}
\begin{abstract}
In this note we consider an Hamiltonian, with ultraviolet and infrared
cutoffs, describing the interaction of relativistic electrons and
positrons in a Coulomb potential with photons in Coulomb gauge. The
interaction includes both interaction of the current-density with
transversal photons and the Wick ordering corrected Coulomb
interaction of charge-density with itself.
We prove that the Hamiltonian is self-adjoint and has a ground state
for a sufficiently small coupling constant.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% SECTION 1 %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% Introduction %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{s1}
In \cite{ref1} an Hamiltonian with cuttofs describing relativistic
electrons and po\-si\-trons in a Coulomb potential interacting with
transversal photons in Coulomb gauge is studied. In this note we add
the Wick ordering corrected Coulomb interaction of the charge-density
with itself to the Hamiltonian described in \cite{ref1}.
We are studying an Hamiltonian with ultraviolet and infrared cutoff
functions with respect to the momenta of photons, but also with
respect to the momenta of electrons and positrons. This ensure that
the total Hamiltonian in the Fock space of electrons, positrons and
photons is well defined in the Furry picture.
In this note we announce results concerning self-adjointness of
this Hamiltonian and the existence of a ground state when the coupling
constant is sufficiently small. In \cite{ref3}, Bach, Fr\"ohlich and
Sigal proved the existence of a ground state for the Pauli-Fierz
Hamiltonian with an ultraviolet cutoff for photons, and for
sufficiently small values of the fine structure constant, without
introducing an infrared cutoff. Their result has been extended by
Griesemer, Lieb and Loss \cite{ref11} under the binding condition. For
related results see \cite{ref12, ref13, ref14, ref15, ref16}.
No-pair Hamiltonians for relativistic electrons in QED have been
recently considered in \cite{ref17, ref18, ref19}.
The case of relativistic electrons in classical magnetic fields was
studied earlier in \cite{LSS} and \cite{GT}. There, it was proven
instability for the Brown and Ravenhall model in the free picture. In
\cite{GT} it is even deduced from this result that instability also
holds in QED context, i.e. for the Brown and Ravenhall model in the
free picture, coupled to the second quantized radiation field with or
without cutoff.
In our case, working in the Furry picture and imposing both electronic
and photonic cutoff prevents from instability.
Our methods follow those of \cite{ref2, ref3, ref4, ref5}, in which
the spectral theory of the spin-boson and Pauli-Fierz Hamiltonians is
studied. \medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% SECTION 2 %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% HAMILTONIAN %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Hamiltonian}\label{s2}
From now on we use the same notations as in \cite{ref1}. Let
${\mathcal F}_D$ be the Fock space of relativistic electrons and
positrons in a Coulomb potential. We work here in the Furry picture,
i.e., the one-electron space (resp. the one-positron space) is the
positive (resp. negative) spectral subspace of the Dirac-Coulomb
operator. Let $b_{\gamma,+}(p)$ (resp. $b_{\gamma,n}$,
$b_{\gamma,-}(p)$ ) be the annihilation operators corresponding to the
electrons in the continuum (resp. the bound states, the
positrons). See \cite{ref1}, \cite{BDG}, \cite{ref8} and \cite{ref10} for
details. Here $\gamma$ is a triplet of angular momentum and spin-orbit
quantum numbers. Moreover $n\in {\N}$ and $p\in {\R}_+$. The
associated creation operators are denoted by $b_{\gamma,+}^*(p) $,
$b_{\gamma,n}^* $ and $b_{\gamma,-}^*(p)$.
\medskip
The Hamiltonian of the quantized Dirac-Coulomb field is given by:
$$
d\Gamma(H_D) = \sum_{\gamma,n} E_{\gamma,n}\, b_{\gamma,n}^*\,
b_{\gamma,n} + \sum_{\epsilon =+,-}\sum_\gamma \int_{\R^+} dp\,
\omega(p)\, b_{\gamma,\epsilon}^*(p)\, b_{\gamma,\epsilon}(p)
$$
Here $(E_{\gamma,n})_{\gamma,n}$ are the eigenvalues of the Dirac
operator $H_D$ for a relativistic electron in the Coulomb potential
generated by a nucleus of charge $Z\leq 118$. We have $0< E_0={\rm
inf} E_{\gamma,n}$ and $\omega(p)=(p^2+m^2c^4)^{\frac{1}{2}}$,
$p\in\R_+$. Here $m$ is the mass of the electrons and $c$ is the speed
of light. The operator $d\Gamma(H_D) $ is a self-adjoint operator in
${\mathcal F}_D$ and its point spectrum contains the one of $H_D$.
Let ${\mathcal F}_{\rm ph}$ be the Fock space of photons in Coulomb
gauge. Let $a_\mu(k)$ (resp. $a_\mu^*(k)$) denotes the annihilation
(resp. creation) operator associated with a field
$\varepsilon_\mu(k)$, $\mu=1,2$, of transversal polarizations for the
photons (see \cite{ref6}). The Hamiltonian of the quantized
electromagnetic field, denoted by $H_{\rm ph}$, is
$$
H_{\rm ph} = \sum_{\mu =1,2} \int_{{\R}^3} d^3 k\ \omega(k)\, a_\mu^*(k)\,
a_\mu(k)\ ,
$$
where $\omega(k) = c|k|$. The operator $H_{\rm ph}$ is self-adjoint in
${\mathcal F}_{\rm ph}$ and its spectrum is $\lbrack 0,\infty)$.
%(see\cite{ref7}).
The Fock space for electrons, positrons and photons is the following
Hilbert space
$$
{\mathcal F} = {\mathcal F}_D \otimes {\mathcal F}_{\rm ph}.$$ The free
Hamiltonian for electrons, positrons and photons, denoted by $H_0$, is
the following operator in ${\mathcal F}$
$$
H_0 = d\Gamma(H_D) \otimes \un_{\rm ph} + \un_D \otimes H_{\rm ph}.$$ defined
on
$$
{\mathcal D}\left(d\Gamma(H_D) \otimes \un_{\rm ph}\right) \cap {\mathcal
D}\left(\un_D \otimes H_{\rm ph}\right).
$$
$H_0$ is a positive self-adjoint operator which has the same point
spectrum as $d\Gamma(H_D)$. Its continuous spectrum covers the half
line.
The interaction between electrons, positrons and photons that we
consider includes two terms. The first one is the interaction of the
current-density with transversal photons already considered in
\cite{ref1}. The second one is the Coulomb interaction of the
charge-density with itself, with a Wick ordering correction.
We have to introduce several cutoff functions in the Dirac-Coulomb
field and the electromagnetic vector potential in order to get a well
defined total Hamiltonian in the Fock space.
The first term in the interaction, denoted by $H_I^{(1)}$, is given
by:
\begin{eqnarray*}
\lefteqn{H_I ^{(1)} =} & & \\
& & \sum_{\gamma,\gamma',n,\ell} \sum_{\mu =1,2} \int
d^3 k\, \bigg(G_{d,\gamma,\gamma',n,\ell}^\mu(k) b_{\gamma,n}^*
b_{\gamma',\ell} a^*_\mu(k) + {\rm h.c.}\bigg)\\
& & + \sum_{\epsilon=+,-}\sum_{\gamma,\gamma',n} \sum_{\mu =1,2}
\int d^3 k\,dp\,
\bigg(G_{d,\epsilon,\gamma,\gamma',n}^\mu(p;k) \Big(b_{\gamma,n}^*
b_{\gamma',\epsilon}(p) \\
& & \hskip 4.6cm +
b_{\gamma,\epsilon}^*(p)b_{\gamma',n} \Big)a^*_\mu(k) + {\rm h.c.}\bigg)\\
& & + \sum_{\gamma,\gamma'} \sum_{\mu =1,2} \int d^3 k\,dp\,dp'
\bigg(G_{+,-,\gamma,\gamma'}^\mu(p,p';k) \Big(b_{\gamma,+}^*(p)
b_{\gamma',-}^*(p') \\
& & \hskip 3.9cm + b_{\gamma,-}(p)b_{\gamma',+}(p')
\Big) a^*_\mu(k) + {\rm h.c.}\bigg)\\
& & + \sum_{\epsilon=+,-}\sum_{\gamma,\gamma'} \sum_{\mu =1,2}
\int d^3 k\,dp\,dp'
\bigg(G_{\epsilon,\epsilon,\gamma,\gamma'}^\mu(p,p';k)
b_{\gamma,\epsilon}^*(p) b_{\gamma',\epsilon}(p') a^*_\mu(k) + {\rm h.c.}\bigg)
\end{eqnarray*}
For the second term let us introduce some notations. In the case of
electrons, $\xi$ will be equal to $(\gamma,p)$ and $(\gamma,n)$ with
$\int d\xi=\sum_\gamma \int dp + \sum_{\gamma,n}$. In the case of
positrons, $\xi$ will be equal to $(\gamma, p)$ with $\int
d\xi=\sum_\gamma \int dp$. The second term of the interaction,
denoted by $H_I^{(2)}$, is then given by
\begin{eqnarray*}
\lefteqn{H_I^{(2)} =} & & \\
& & \int d\xi_1 d\xi_2d\xi_3d\xi_4\, F^{(1)}(\xi_1,\xi_2,\xi_3,\xi_4)
b_+^*(\xi_1) b_-^*(\xi_2) b_+(\xi_3)b_-(\xi_4)\\
& & + \sum_{\epsilon=+,-} \int d\xi_1 d\xi_2d\xi_3d\xi_4\,
F^{(2)}_\epsilon(\xi_1,\xi_2,\xi_3,\xi_4) b_\epsilon^*(\xi_1)
b_\epsilon^*(\xi_2) b_\epsilon(\xi_3)b_\epsilon(\xi_4)\\
& & +\sum_{\epsilon,\epsilon'=+,-,\atop \epsilon+\epsilon'=0} \int d\xi_1
d\xi_2d\xi_3d\xi_4\, \Big(F^{(3)}_{\epsilon,\epsilon'}
(\xi_1,\xi_2,\xi_3,\xi_4) b_\epsilon^*(\xi_1) b_\epsilon(\xi_2)
b_{\epsilon'}(\xi_3)b_{\epsilon}(\xi_4) \\
& & \hskip4.2cm - F^{(3)}_{\epsilon,\epsilon'} (\xi_4,\xi_2,\xi_3,\xi_1)
b_\epsilon^*(\xi_1) b_\epsilon^*(\xi_2)
b_{\epsilon'}^*(\xi_3)b_{\epsilon}(\xi_4)\Big)\\
& & +\sum_{\epsilon,\epsilon'=+,-,\atop \epsilon+\epsilon'=0} \int d\xi_1
d\xi_2d\xi_3d\xi_4\, \Big(F^{(4)}_{\epsilon,\epsilon'}
(\xi_1,\xi_2,\xi_3,\xi_4) b_\epsilon(\xi_1) b_\epsilon(\xi_2)
b_{\epsilon'}(\xi_3)b_{\epsilon'}(\xi_4)\\
& & \hskip4.2cm + F^{(4)}_{\epsilon,\epsilon'} (\xi_4,\xi_2,\xi_3,\xi_1)
b_\epsilon^*(\xi_1) b_\epsilon^*(\xi_2)
b_{\epsilon'}^*(\xi_3)b_{\epsilon'}^*(\xi_4)\Big)\ .
\end{eqnarray*}
Furthermore, we suppose that
$$
F^{(1)}(\xi_1,\xi_2,\xi_3,\xi_4) =
\overline{F^{(1)}(\xi_3,\xi_4,\xi_1,\xi_2)}
$$
$$
F^{(2)}_\epsilon(\xi_1,\xi_2,\xi_3,\xi_4) =
\overline{F^{(2)}_\epsilon(\xi_4,\xi_3,\xi_2,\xi_1)}
$$
Our Hamiltonian for relativistic electrons and positrons in a Coulomb
potential interacting with photons in Coulomb gauge is finally given
by
$$
H(g) = H_0 + g\, H_I^{(1)}+g^2 H_I^{(2)},
$$
where $g$ is a real coupling constant.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% SECTION 3 %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% MAIN RESULTS %%%%%%%%%%%%%%%%%%
%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}
It is easy to show that $H(g)$ is a symmetric operator in ${\mathcal
F}$ as soon as the kernels $F^{(i)}$'s and $G^{(i)}$'s are square
integrable. With stronger conditions on the $F^{(i)}$'s and
$G^{(i)}$'s, we recover a self-adjoint operator with a ground state,
as stated in Theorem~\ref{theorem1}~and~\ref{theorem2} below.
Let, for $\beta = 0,1,2$,
$$
C_\beta = \sum_{\mu =1,2} \left( \sum_{\gamma,\gamma',n,\ell}
\int_{{\R}^3} |G_{d,\gamma,\gamma',n,\ell}^\mu(k)|^2
\omega(k)^{-\beta} d^3k\right)^{1/2}
$$
$$
+ \sum_{\epsilon = +,-} \sum_{\mu =1,2} \left(
\sum_{\gamma,\gamma',n} \int_{{\R}^3 \times {\R}^+}
|G_{d,\epsilon,\gamma,\gamma',n}^\mu(p;k)|^2 \omega(k)^{-\beta}
dp\,d^3k \right)^{1/2} $$
$$
+ \sum_{r} \sum_{\mu =1,2} \left( \sum_{\gamma,\gamma'} \int_{{\R}^3
\times {\R}^+ \times {\R}^+} |G_{r,\gamma,\gamma'}^\mu(p,p';k)|^2
\omega(k)^{-\beta} dp\,dp'\,d^3k \right)^{1/2},$$ where $r = \{+,+\},
\{+,-\}, \{-,-\}$.
We now state our main results :
\begin{theorem}\label{theorem1} We assume that $F^{(j)}\in L^2$,
$j=1,2,3,4$. Furthermore, we suppose that $C_0 < \infty$ and
$\displaystyle {\frac{|g|}{E_0}} C_1 +{\frac{g^2}{E_0}} \Vert
F^{(1)}\Vert_{L^2}< 1$. Then, $H(g)$ is self-adjoint on the domain
$D(H_0)$.
\end{theorem}
\begin{theorem}\label{theorem2} We assume that $F^{(j)}\in L^2$,
$j=1,2,3,4$. Furthermore, we suppose that $C_0 < \infty,\
\displaystyle {\frac{|g|}{E_0}} C_1 +{\frac{g^2}{E_0}} \Vert
F^{(1)}\Vert_{L^2}< 1$ and $C_2 < \infty$. Then, there exists $g_0 >
0$ such that, for every $g \in [-g_0,g_0]$, the self-adjoint operator
$H(g)$ has a ground state.
\end{theorem}
Recall that $E_0$ is the ground state energy of the Dirac-Coulomb
operator $H_D$.
\medskip
The proofs are partly based on the following lemmas (details can be
found in\cite{ref1} and \cite{BDG}). For a sake of simplicity, we
reduce the discussion here to the case of electrons in the continuum
and positrons only.
Set $\xi=(\gamma,p)$, with $\int d\xi=\sum_\gamma \int dp$. Let
$G(\xi,\xi')=G(\gamma,p,\gamma',p')$, with
$$ \Vert G(\cdot,\cdot)\Vert^2_{L^2}=\sum_{\gamma,\gamma'}\int dp\, dp' \vert
G(\gamma,p,\gamma',p')\vert^2.$$
For $G$ such that $\Vert G\Vert_{L^2}<\infty$ we define
\begin{eqnarray*}
A_\pm & = & \iint d\xi d\xi'\, \overline {G(\xi,\xi')}
b_\pm(\xi)b_\pm(\xi')\ ,\\
A_{+-} & = & \iint d\xi d\xi'\, \overline {G(\xi,\xi')}
b_+(\xi)b_-(\xi')\ ,\\
A_{-+} & = & \iint d\xi d\xi'\, \overline {G(\xi,\xi')}
b_-(\xi)b_+(\xi')\ ,\\
B_\pm & = & \iint d\xi d\xi'\, \overline {G(\xi,\xi')}
b_\pm^*(\xi)b_\pm(\xi')\ .
\end{eqnarray*}
\begin{lem} We have the following norm estimates
\noindent
$$
\begin{array}{lllll}
\Vert A_\pm\Vert& =&\Vert A_\pm^*\Vert &\le & \Vert G_a\Vert_{L^2}\ ,\\
\Vert A_{+-}\Vert &=&\Vert A_{+-}^*\Vert &\le & \Vert G_a\Vert_{L^2}\ ,\\
\Vert A_{-+}\Vert& =&\Vert A_{-+}^*\Vert &\le & \Vert G_a\Vert_{L^2}\ ,
\end{array}
$$
where $G_a(\xi,\xi')=G(\xi,\xi')-G(\xi',\xi)$, and
\noindent
\begin{eqnarray*}
\Vert B_\pm(N_D+1)^{-1/2}\Vert & \le &\Vert G\Vert_{L^2}\ ,\\
\Vert (N_D+1)^{-1/2}B_\pm\Vert & \le &\Vert G\Vert_{L^2} \ ,
\end{eqnarray*}
where $N_D=\sum_{\gamma,n} b^*_{\gamma,n}b_{\gamma,n}
+\sum_{\epsilon=+,-}\int d\xi\, b_\epsilon^*(\xi)b_\epsilon(\xi)$ is the
number operator for electrons and positrons.
\end{lem}
For the quartic terms in $b^\#$ we have the following results
%only consider here the most relevant one associated with
%$F^{(1)}$. All other terms give rise to operators which are relatively
%bounded with respect to $H_0$ with zero relative bound.
\begin{lem} For every $\psi\in {\mathcal D}(N_D)$ we have
\begin{eqnarray*}
\lefteqn{\Vert \int d\xi_1 d\xi_2d\xi_3d\xi_4\,
F^{(1)}(\xi_1,\xi_2,\xi_3,\xi_4) b_+^*(\xi_1) b_-^*(\xi_2)
b_+(\xi_3)b_-(\xi_4) \psi\Vert^2 }
& & \\ & \hskip2.5cm \le & \Vert
F^{(1)}\Vert_{L^2}^2\Vert (N_D+1)\psi\Vert^2+C\Vert
(N_D+1)^{1/2}\psi\Vert^2+ C\Vert \psi\Vert^2
\end{eqnarray*}
where $C$ is a nonnegative constant.
\end{lem}
\begin{lem} For every $\psi\in {\mathcal D}(N_D)$ we have
\begin{eqnarray*}
\lefteqn{\|\sum_{\epsilon=+,-}\!\!\int\!\! d\xi_1 d\xi_2d\xi_3d\xi_4\,
F^{(2)}_\epsilon(\xi_1,\xi_2,\xi_3,\xi_4) b_\epsilon^*(\xi_1)
b_\epsilon^*(\xi_2) b_\epsilon(\xi_3)b_\epsilon(\xi_4)\psi\|^2
\hskip2.5cm \ } & & \\ &\hskip2.5cm \leq & C \left(\sum_{\epsilon=
+,-}\|F^{(2)}_\epsilon\|_{L^2}\right) \|(N_D+1)^{\frac12}\psi\|^2 ,
\end{eqnarray*}
where $C$ is a nonnegative constant.
Similar estimates hold for the term including
$F^{(3)}_{\epsilon,\epsilon'}$ in $H_I^{(2)}$.
\end{lem}
\begin{lem} The operator
\begin{eqnarray*}
\sum_{\epsilon,\epsilon'=+,-,\atop \epsilon+\epsilon'=0} \int d\xi_1
d\xi_2d\xi_3d\xi_4\, \Big(F^{(4)}_{\epsilon,\epsilon'}
(\xi_1,\xi_2,\xi_3,\xi_4) b_\epsilon(\xi_1) b_\epsilon(\xi_2)
b_{\epsilon'}(\xi_3)b_{\epsilon'}(\xi_4)\\
\hskip4.2cm + F^{(4)}_{\epsilon,\epsilon'} (\xi_4,\xi_2,\xi_3,\xi_1)
b_\epsilon^*(\xi_1) b_\epsilon^*(\xi_2)
b_{\epsilon'}^*(\xi_3)b_{\epsilon'}^*(\xi_4)\Big)\ ,
\end{eqnarray*}
is bounded.
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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