[Ground States of Quantum Electrodynamics with Cutoffs ]{}\
$\;$\
[Toshimitsu Takaesu ]{}\
$\;$\
*Faculty of Science and Technology, Gunma University,\
Gunma, 371-8510, Japan*

> **Abstract** In this paper, we investigate a system of quantum electrodynamics with cutoffs. The total Hamiltonian is defined on a tensor product of a fermion Fock space and a boson Fock. It is shown that, under spatially localized conditions and momentum regularity conditions, the total Hamiltonian has a ground state for all values of coupling constants. In particular, its multiplicity is finite.\
> $\; $\
> [MSC 2010 : 47A10, 81Q10. $\; $\
> key words : Fock spaces, Spectral analysis, Quantum Electrodynamics]{}.

Introduction 
=============

$\;$ This articles is concerned with a system of quantum electrodynamics with cutoffs. In quantum field theory, the interactions of charged particles and photons are described by quantum electrodynamics. We consider the system of a massive Dirac field coupled to a radiation field. The radiation filed is quantized in the Coulomb gauge. In this system, the process of electron-positron pair production and annihilation occurs. We mathematically investigate the spectrum of the total Hamiltonian for the system. The Hilbert space for the system is defined by a tensor product of a fermion Fock space and boson Fock space, which is called a boson-fermion Fock space. The total Hamiltonian is given by $$\begin{aligned}
 {H_{\textrm{QED}}}=   H_{{ \textrm{D} }}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}}}& + {\kappa_{\textrm{I}}}\sum_{j=1}^{3} {\int_{\mathbf{R}^{3}}}\chi_{{\textrm{I}}} ({\ensuremath{\mathbf{x}}}) ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}})  }{\otimes}A_j ({\ensuremath{\mathbf{x}}} ) ) d{\ensuremath{\mathbf{x}}} \notag  \\
  & \; \; + {\kappa_{\textrm{II}}}\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
     \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} 
\left(  {\psi^{\dagger} (\mathbf{x}) }{ \psi ({\ensuremath{\mathbf{x}}})  }{\psi^{\dagger} (\mathbf{y}) }{ \psi ({\ensuremath{\mathbf{y}}})  }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right)
 d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}}  \notag \end{aligned}$$ on the Hilbert space. Here $H_{{ \textrm{D} }}$ and $H_{{\textrm{rad}}} $ denote the energy Hamiltonians of the Dirac field and radiation field, respectively, $\psi ({\ensuremath{\mathbf{x}}})$ the Dirac field operator, ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})=(A_j({\ensuremath{\mathbf{x}}}))_{j=1}^3$ the radiation field operator, ${\ensuremath{\mathbf{\alpha}}}=(\alpha_j)_{j=1}^3$ $4\times 4$ Dirac matrices, and ${\chi_{\textrm{I}} (\mathbf{x}) }$ and ${\chi_{\textrm{II}} (\mathbf{x}) }$ the spatial cutoffs. The constants ${\kappa_{\textrm{I}}}\in {\ensuremath{\mathbf{R}}}$ and ${\kappa_{\textrm{II}}}\in {\ensuremath{\mathbf{R}}}$ are called coupling constants. Ultraviolet cutoffs are imposed on ${ \psi ({\ensuremath{\mathbf{x}}})  }$ and ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})$, respectively.\
By making use of the spacial cutoffs and ultraviolet cutoffs, ${H_{\textrm{QED}}}$ is self-adjoint operator on the Hilbert space, and the spectrum of ${H_{\textrm{QED}}}$ is bounded from below. The main interest in this paper is the lower bound of the spectrum of ${H_{\textrm{QED}}}$. If the infimum of the spectrum of a self-adjoint operator is eigenvalue, the eigenvector is called ground state. The infimum of the spectrum of $H_{0}= H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}}}$ is eigenvalue, but it is embedded in continuous spectrum. This is because the radiation field is a massless field. It is not clear that ${H_{\textrm{QED}}}$ has a ground state since the embedded eigenvalue is not stable when interactions are turned on.\
The ground state of ${H_{\textrm{QED}}}$ for sufficiently small values of coupling constants was proven in [@Ta09]. The aim of this paper is to prove that ${H_{\textrm{QED}}}$ has a ground state for all values of coupling constants. In particular, its multiplicity is finite. For the ground states of other QED models, Dimassi-Guillot [@DiGu03] and Barbaroux-Dimassi-Guillot [@BDG04] investigated the system of the Dirac field in external potential coupled to the radiation field. They proved the existence of the ground state of the total Hamiltonian with generalized interactions for sufficiently small values of coupling constants. As far as we know, the existence of the ground states for the systems of a fermionic field coupled to a massless bosonic field, which include QED models, has not been proven for all values of coupling constants until now.\
To prove the existence of the ground state of ${H_{\textrm{QED}}}$ for all values of coupling constants, we apply the methods for systems of particles coupled bosonic fields. The spectral analysis and scattering theory for these systems, which include the non-relativistic QED models, have been progressed since the middle of ’90s. The existence of the ground states was established by Arai-Hirokawa [@AH97], Bach-Fr[ö]{}hlich-Sigal [@BFS98; @BFS99], G[é]{}rard [@Ge00], Griesemer-Lieb-Loss [@GLL01], Lieb-Loss [@LiLo03], Spohn [@Sp98] and many researchers. The strategy is as follows.

$\; $\
**\[1st Step\]** We introduce approximating Hamiltonians $H_{m}$, $m>0$. Physically, $m>0$ denotes the artificial mass of photon, and we call $H_{m}$ a massive Hamiltonian. To prove the existence of ground states of $H_{m}$, we use partition of unity on Fock space, which was developed by Dereziński-Gérard [@DeGe99]. We especially need the partitions of unity for both Dirac field and radiation field. By the partitions of unity and the Weyl sequence method, we prove that a positive spectral gap above the infimum of the spectrum exists for all values of coupling constants. From this, the existence of the ground states of $H_m$ for all values of coupling constants follows.

$\; $\
**\[2nd Step\]** Let $\Psi_m $ be the ground state of $H_{m}$, $m>0$. Without loss of generality, we may assume that the $\Psi_m$ is normalized. Then, there exists a subsequence of $\{ \Psi_{m_j} \}_{j=1}^\infty$ with $m_{j+1} < m_{j}$, $j\in {\ensuremath{\mathbf{N}}}$, such that the weak limit of $ \{ \Psi_{m_j}\}_{j=1}^{\infty}$ exists. The key point is to show that the the weak limit is non-zero vector. To prove this, we consider a combined method of Gerard [@Ge00] and Griesemer-Lieb-Loss in [@GLL01]. We use the electron positron derivative bounds and photon derivative bounds. To derive these bounds, the argument of the spatially localization is needed. For the spatially localized conditions, we suppose $$\int_{{\mathbf{R}^{3} }}  |{\ensuremath{\mathbf{x}}}| \, | \chi_{{\textrm{I}}}({\ensuremath{\mathbf{x}}})  | dx\, < \, \infty  , \qquad 
 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }}  
\frac{| \chi_{{\textrm{II}}}({\ensuremath{\mathbf{x}}}) \chi_{{\textrm{II}}}({\ensuremath{\mathbf{y}}}) |}{|{\ensuremath{\mathbf{x}}}- {\ensuremath{\mathbf{y}}}|} |{\ensuremath{\mathbf{x}}}| {d \mathbf{x} }{d \mathbf{y} }<  \infty.$$ I addition, We imposed momentum regularity conditions on the Dirac field and radiation field, which include the infrared regularity condition $$\int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{| {\ensuremath{\mathbf{k}}}  |^{5} } {d \mathbf{k} }< \infty .$$ $\;$\
$\; $\
We briefly review the results for the systems of fermionic fields coupled bosonic fields. For QED models, the Gell-Mann - Low formula of ${H_{\textrm{QED}}}$ was obtained by Futakuchi-Usui [@FuUs14]. For the Yukawa model, which is the system for a massive Dirac field interacting with a massive Klein-Gordon field, the existence of the ground state was proven in [@Ta11]. The spectaral analysis for the the weak interaction models has been analyzed, and refer to Barbaroux-Faupin-Guillot [@BFG14], Guillot [@Gu15] and the reference therein.

$\;$\
This paper is organized as follows. In section 2, full Fock spaces, fermion Fock spaces and boson Fock spaces are introduced, and Dirac field operators and radiation field operators are defined on a Fermion Fock space and boson Fock space, respectively. The total Hamiltonian is defined on a boson-fermion Fock space and the main theorem is stated. In Section 3, partitions of unity for the Dirac field and radiation field are investigated. Then the existence of the ground state of $H_{m}$ is proven. In section 4, the derivative bounds for electrons-positrons and photons are derived. In Section 5, we give the proof of the main theorem.\

Notations and Main Results
==========================

Fock Spaces
-----------

**(i) Full Fock Space**\
The full Fock space over a complex Hilbert space ${\ensuremath{\mathscr{Z}}}$ is defined by $ {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Z}}})= \oplus_{n=0}^\infty ( {\otimes^{n}}{\ensuremath{\mathscr{Z}}})$ where ${\otimes^{n}}{\ensuremath{\mathscr{Z}}}$ is the $n$ fold tensor product of $Z$. The Fock vacuum is defined by $\Omega = \{1, 0,0, \cdots  \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Z}}})$. Let $ {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}})$ be the set which consists of all linear operators on ${\ensuremath{\mathscr{Z}}}$. The functor of $Q\in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$ is defined by $\Gamma (Q) = \oplus_{n=0}^{\infty} \left( {\otimes^{n}}Q \right)$ and the second quantization of $T\in{\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$ is given by $ {\ensuremath{d\Gamma({T}) }} = \oplus_{n=0}^{\infty}  \tilde{T}^{(n)} $ with $ \tilde{T}^{(n)} =\sum\limits_{j=1}^{n} ( ({\otimes}^{j-1} {{\small \text{1}}\hspace{-0.32em}1}) {\otimes}T {\otimes}( {\otimes}^{n-j} {{\small \text{1}}\hspace{-0.32em}1})  )$. The number operator is defined by $N= {\ensuremath{d\Gamma({{{\small \text{1}}\hspace{-0.32em}1}}) }} $.\
$\;$\
**(ii) Fermion Fock Space**\
The fermion Fock space over a complex Hilbert space ${\ensuremath{\mathscr{X}}}$ is defined by $ {\mathscr{F}_{\textrm{f}}  }( {\ensuremath{\mathscr{X}}})= \oplus_{n=0}^\infty ( {\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}})$ where $ {\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}}$ denotes the $n$-fold anti-symmetric tensor product of ${\ensuremath{\mathscr{X}}}$. The Fock vacuum is defined by $\Omega_{{\textrm{f}}} = \{1, 0, 0, \cdots  \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{X}}})$. Let $T_{{\textrm{f}}}$ and $Q_{{\textrm{f}}}$ be linear operators on ${\ensuremath{\mathscr{X}}}$. We set ${\ensuremath{d\Gamma_{\textrm{f}}({T_{{\textrm{f}}}}) }}= {\ensuremath{d\Gamma({T_{{\textrm{f}}}}) }}_{\upharpoonright {\mathscr{F}_{\textrm{f}}  }( {\ensuremath{\mathscr{X}}})}$ and $\Gamma_{{\textrm{f}}}( Q_{{\textrm{f}}})= \Gamma (Q_{{\textrm{f}}})_{\upharpoonright {\mathscr{F}_{\textrm{f}}  }( {\ensuremath{\mathscr{X}}})}$ where $ X_{\upharpoonright {\ensuremath{\mathscr{M}}}} $ is the restriction of the operator $X$ to the subspace ${\ensuremath{\mathscr{M}}}  $. The number operator is defined by $N_{{\textrm{f}}}= {\ensuremath{d\Gamma_{\textrm{f}}({{{\small \text{1}}\hspace{-0.32em}1}}) }} $. The creation operator $C^{\,  \dagger}(f)$, $f \in {\ensuremath{\mathscr{X}}}$, is defined by $(C^{\, \dagger}(f) \Psi )^{(n)} = \sqrt{n} \, U_{\textrm{a}}^{ n}(f {\otimes}\Psi^{(n-1)})$, $n \geq 1$, and $(C^{\, \dagger }(f) \Psi )^{(0)} = 0 $ where $ U_{\textrm{a}}^{ n} $ is the projection from $ {\otimes^{n}}{\ensuremath{\mathscr{X}}} $ to ${\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}} $. The annihilation operator $C(f)$ is defined by $C(f)=(C^{\, \dagger}(f))^{\ast}$ where $X^{\ast} $ denotes the adjoint of the operator $X$. For each subspace ${\ensuremath{\mathscr{M}}} \subset {\ensuremath{\mathscr{X}}}$, the finite particle space $ {\ensuremath{\mathscr{F}}}_{{\textrm{f}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{M}}}) $ is defined by the linear hull of $ \Omega_{{\textrm{f}}} $ and $C^{\, \dagger}(f_{1}) , \cdots C^{\, \dagger}(f_{n}) \Omega_{{\textrm{f}}}  $, $j=1 , \cdots , n$, $n \in {\ensuremath{\mathbf{N}}}$. The creation and annihilation operators satisfy the canonical anti-commutation relations on ${\mathscr{F}_{\textrm{f}}  }({\ensuremath{\mathscr{X}}})$ : $$\qquad 
\{ C(f) , \, C^{ \, \dagger}(f')  \} = (f , f' ) , \quad   \{ C^{ \, \dagger }(f) ,
 C^{ \, \dagger }(f')  \} 
= \{ C(f) , C (f')  \} =0 , \quad f, f' \in {\ensuremath{\mathscr{X}}},$$ where $\{ X ,Y \} = XY +YX $.

$\;$\
**(ii) Boson Fock Space**\
The boson Fock space over a complex Hilbert space ${\ensuremath{\mathscr{Y}}}$ is defined by $ {\mathscr{F}_{\textrm{b}}  }( {\ensuremath{\mathscr{Y}}})= \oplus_{n=0}^\infty ( {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}})$ where $ {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}}$ denotes the $ n$-fold symmetric tensor product of ${\ensuremath{\mathscr{Y}}}$. The Fock vacuum is given by $\Omega_{{\textrm{b}}} = \{1, 0, 0,   \cdots  \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Y}}})$ Let $T_{{\textrm{b}}}$ and $Q_{{\textrm{b}}}$ be linear operators on ${\ensuremath{\mathscr{Y}}}$. Then we define ${\ensuremath{d\Gamma_{\textrm{b}}({T_{{\textrm{b}}}}) }}= {\ensuremath{d\Gamma({T_{{\textrm{b}}} }) }}_{\upharpoonright {\mathscr{F}_{\textrm{b}}  }( {\ensuremath{\mathscr{Y}}})}$ and $\Gamma_{{\textrm{b}}}( Q)= \Gamma (Q_{{\textrm{b}}})_{\upharpoonright {\mathscr{F}_{\textrm{b}}  }( {\ensuremath{\mathscr{Y}}})}$. The number operator is defined by $N_{{\textrm{f}}}= {\ensuremath{d\Gamma_{\textrm{f}}({{{\small \text{1}}\hspace{-0.32em}1}}) }} $. The creation operator $A^{\dagger}(g)$, $g \in {\ensuremath{\mathscr{Y}}}$, is defined by $(A^{\dagger}(g) \Phi )^{(n)} = \sqrt{n} \, U_{\textrm{s}}^{ n}(f {\otimes}\Phi^{(n-1)})$, $n \geq 1$, and $(A^{\dagger}(g) \Phi )^{(0)} = 0 $ where $ U_{\textrm{s}}^{ n} $ is the projection from $ {\otimes^{n}}{\ensuremath{\mathscr{Y}}} $ to $ {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}}  $. The annihilation operator $A(f)$ is defined by $A(g)= ( A^{\dagger}(g) )^{\ast}  $. The finite particle space $ {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{N}}}) $ on the subspace ${\ensuremath{\mathscr{N}}} \subset {\ensuremath{\mathscr{Y}}}$ defined by the linear hull of $ \Omega_{{\textrm{f}}} $ and $A^{\, \dagger}(g_{1}) , \cdots A^{\, \dagger}(g_{n}) \Omega_{{\textrm{f}}}  $, $j=1 , \cdots , n$, $n \in {\ensuremath{\mathbf{N}}}$. The creation and annihilation operators satisfy the canonical commutation relations on $ {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{N}}}) $: $$\qquad
[  A(g),A^{ \dagger }(g ')   ] = (g , g' ) , \quad   [ A^{  \dagger }(g) , A^{ \dagger }(g')  ] 
= [ A(g) , A (g') ] =0 , \quad g, g' \in {\ensuremath{\mathscr{Y}}} ,$$ where $[X ,Y]=XY-YX $.\

Dirac field
-----------

Let ${\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }}  = {\mathscr{F}_{\textrm{f}}  }(L^2( {\mathbf{R}^{3} };{\ensuremath{\mathbf{C}}}^4))$. The energy Hamiltonian of the Dirac field is defined by $$H_{{ \textrm{D} }} = {\ensuremath{d\Gamma_{\textrm{f}}({\omega_{\,M} }) }}$$ where $\omega_M ({\ensuremath{\mathbf{p}}}) = \sqrt{| {\ensuremath{\mathbf{p}}}|^2 + M^2 }$, ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, and $M>0$. Let $C^{\, \dagger } (^{t}(f_{1} , \cdots , f_{4}))$, $f_{l} \in L^2( {\mathbf{R}^{3} })$, $l=1, \cdots ,4$, be the creation operator on ${\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}$. For each $f \in L^2( {\mathbf{R}^{3} })$, we set $$\begin{aligned}
&b_{1/2}^{\dagger}(f) = C^{\, \dagger}({}^{t} (f,0,0,0) ), \quad \quad  
b_{-1/2}^{\dagger}(f) = C^{\, \dagger}({}^{t}(0,f,0,0) ), \\
&d_{1/2}^{\, \dagger}(f) = C^{\, \dagger}({}^{t}(0,0,f,0) ), \quad \quad  
d_{-1/2}^{\, \dagger}(f) = C^{\, \dagger}({}^{t}(0,0,0,f) ) . \end{aligned}$$ We define $b_s ( f)$ and $d_{s} ( g)$ by the conjugate of $b_s^{\dagger} ( f)$ and $d_{s}^{ \, \dagger } ( g)$, respectively. Then the canonical anti-commutation relations $$\begin{aligned}
& \{ b_s (f) ,b_{s'}^{\dagger} (g ) \} = \{ d_s (f) ,d_{s'}^{\, \dagger } (g ) \}= \delta_{s, s'}
 (f,g  ),  \quad  \label{ACR1} \\
& \{ b_s (f) ,b_{s'} (g) \} =\{ d_s (f) ,d_{s'} (g) \}=  \{ b_s (g) ,d_{s'}^{\, \dagger} (g) \} = 0 ,  \quad  \label{ACR2}\end{aligned}$$ are satisfied and it holds that $$\|b_s (f)  \| = \|b_s^{\dagger }(f)  \| =  \| f \| , \qquad \|d_s(g)  \| = \|d_s^{\, \dagger }(g)  \|=  \|  g \|  . \label{bdNorm}$$ Let $h_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}})={\ensuremath{\mathbf{\alpha}}} \cdot {\ensuremath{\mathbf{p}}}+ M\beta $ be the Fourier transformed Dirac operator with $4 \times 4$ Dirac matrices $\boldsymbol{\alpha} = (\alpha^j)_{j=1}^3$ and $\beta $. Let ${\ensuremath{\mathbf{S}}}({\ensuremath{\mathbf{p}}})= {\ensuremath{\mathbf{S}}} \cdot {\ensuremath{\mathbf{p}}}$, ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, where ${\ensuremath{\mathbf{S}}}= -\frac{i}{4} \boldsymbol{\alpha} \wedge \boldsymbol{\alpha} $ is the spin angular momentum. The spinors $ u_{s}({\ensuremath{\mathbf{p}}} ) = 
(u_{s}^{l} ({\ensuremath{\mathbf{p}}}) )_{l=1}^{4} \; $ and $v_{s}({\ensuremath{\mathbf{p}}} ) = (v_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) )_{l=1}^{4}  \; $ are function which satisfy the following : $$\begin{aligned}
\textbf{(D.1)}\;  \; \;  &h_{D} ({\ensuremath{\mathbf{p}}}) u_{s} ({\ensuremath{\mathbf{p}}}) = E_{M} ({\ensuremath{\mathbf{p}}}) u_{s} ({\ensuremath{\mathbf{p}}}), 
\quad h_{D} ({\ensuremath{\mathbf{p}}}) v_{s} ({\ensuremath{\mathbf{p}}}) = 
-E_{M} ({\ensuremath{\mathbf{p}}}) v_{s} ({\ensuremath{\mathbf{p}}}),    \\ 
\textbf{(D.2)}  \; \;  \; & S({\ensuremath{\mathbf{p}}} ) u_{s} ({\ensuremath{\mathbf{p}}}) = s | {\ensuremath{\mathbf{p}}} | u_{s} ({\ensuremath{\mathbf{p}}}),
\quad  S({\ensuremath{\mathbf{p}}} ) v_{s} ({\ensuremath{\mathbf{p}}}) = s | {\ensuremath{\mathbf{p}}} | v_{s} ({\ensuremath{\mathbf{p}}}) , \\
 \textbf{(D.3)} \; \; \; & \sum_{l=1}^4 u_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast}    u_{s'}^{\,l} ({\ensuremath{\mathbf{p}}}' ) 
=   \sum_{l=1}^4 v_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast}    v_{s'}^{\, l} ({\ensuremath{\mathbf{p}}}' ) 
=\delta_{s,s'}   ,\quad  
\sum_{l=1}^4 u_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast}    v_{s'}^{\, l} ({\ensuremath{\mathbf{p}}}' )  = 0 . \end{aligned}$$

\[exa1\] We review the example of spinors in the standard representation (see [@Tha] ; Section 1). The Pauli matrices are defined by $ \sigma_{1} = \left( 
\begin{array}{cc}
0 & 1 \\
1  & 0
\end{array}
\right)  $, $ \sigma_{2} = \left( 
\begin{array}{cc}
0 & -i \\
i  & 0
\end{array}
\right)
$ and $ \sigma_{3} = \left( 
\begin{array}{cc}
1 & 0 \\
0  & -1
\end{array}
\right)  $. Then, the Dirac marices are $
  \alpha =  \left( 
\begin{array}{cc}
O & \sigma \\
\sigma  & O
\end{array}
\right) $, $
 \beta = 
 \left( 
\begin{array}{cc}
{{\small \text{1}}\hspace{-0.32em}1}& O \\
O  & -{{\small \text{1}}\hspace{-0.32em}1}\end{array}
\right) $, and the spin angular momentum is ${\ensuremath{\mathbf{S}}} = \frac{1}{2} \left( \begin{array}{cc}
\ \sigma & O \\
O  & \sigma
\end{array} \right)
 $. Let $ O_{\textrm{SR}} = \{  {\ensuremath{\mathbf{p}}}  = (p^{1} , p^{2}, p^{3} )  \in {\mathbf{R}^{3} }\; \left| \frac{}{} 
 \right. |{\ensuremath{\mathbf{p}}}| -p^3 = 0 \}  $. We see that the Lebesgue measure of $O_{\textrm{SR} }$ is zero. We set $$\eta_{+} ({\ensuremath{\mathbf{p}}} ) = \left\{
\begin{array}{c}
\frac{1}{\sqrt{ 2 | {\ensuremath{\mathbf{p}}} | (  | {\ensuremath{\mathbf{p}}} | -p^{3} ) } }
 \begin{pmatrix}
  p^{1} - i p^{2}  \\  | {\ensuremath{\mathbf{p}}} |  - p^{3}
\end{pmatrix} , \;  {\ensuremath{\mathbf{p}}}  \notin O_{\textrm{SR} }    ,  \\ 
 \qquad \qquad
 \begin{pmatrix} 
  1 \\ 0   \end{pmatrix} ,  \quad  \quad \qquad 
    {\ensuremath{\mathbf{p}}}  \in O_{\textrm{SR} }   ,
 \end{array} 
 \right.
\; \;   
 \eta_{-} ({\ensuremath{\mathbf{p}}} ) = 
 \left\{
 \begin{array}{c}
\frac{1}{\sqrt{ 2 | {\ensuremath{\mathbf{p}}} | ( | {\ensuremath{\mathbf{p}}} | - p^{3}   ) } }
    \begin{pmatrix}
p^{3} - | {\ensuremath{\mathbf{p}}} |  \\  p^{1} + i p^{2} 
\end{pmatrix}  , \,
    {\ensuremath{\mathbf{p}}}  \notin O_{\textrm{SR} }    ,  \\ 
\qquad \quad   \begin{pmatrix}   0  \\  1 \end{pmatrix} ,
 \quad  \qquad \qquad {\ensuremath{\mathbf{p}}}  \in O_{\textrm{SR} } .
  \end{array}  
\right.$$ Let $$u_{\pm1/2} ({\ensuremath{\mathbf{p}}}) = 
 \begin{pmatrix} \lambda_{+} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) 
 \\  \pm \lambda_{-} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}})   \end{pmatrix} ,  \qquad
 v_{\pm 1/2} ({\ensuremath{\mathbf{p}}}) = 
 \begin{pmatrix} \mp \lambda_{-} ({\ensuremath{\mathbf{p}}} ) \phi_{\pm} ({\ensuremath{\mathbf{p}}}) 
 \\  \pm \lambda_{+} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}})   \end{pmatrix} ,$$ with $ \lambda_{\pm} ({\ensuremath{\mathbf{p}}}) = 
\frac{1}{\sqrt{2}} \sqrt{  1 \pm  M \, E_{M} ({\ensuremath{\mathbf{p}}})^{-1 }}$. Here note that $ u_{s} $ and $  v_{s} $ satisfy $  u_{s} , v_s  \in 
 \oplus^4 (  C^{1} ( {\mathbf{R}^{3} }\backslash   O_{\textrm{SR}} ) ) $.\
$\;$

$\;$\
The Dirac field operator $  \psi({\ensuremath{\mathbf{x}}}) =  {}^{t} (\psi_{1}({\ensuremath{\mathbf{x}}}) ,  \cdots ,  \psi_{4}({\ensuremath{\mathbf{x}}}))$ is defined by $$\qquad \qquad \quad \psi_{l}({\ensuremath{\mathbf{x}}}) = \sum_{s=\pm 1/2} \left( b_s ( f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ) +  
 d^{\, \dagger}_s ( g_{s , {\ensuremath{\mathbf{x}}}}^l) \frac{}{} \right) , \quad \qquad l=1, \cdots , 4 ,$$ where $f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}})= f_{s }^{\, l} ({\ensuremath{\mathbf{p}}})e^{-{\ensuremath{\mathbf{p}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $f_{s }^{\, l} ({\ensuremath{\mathbf{p}}})= \frac{1}{\sqrt{ (2 \pi )^3 }} \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) u_s^{l}({\ensuremath{\mathbf{p}}})$ and $g_{s , {\ensuremath{\mathbf{x}}}}^l ({\ensuremath{\mathbf{p}}})= g_{s }^l ({\ensuremath{\mathbf{p}}})e^{-{\ensuremath{\mathbf{p}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $g_{s }^l ({\ensuremath{\mathbf{p}}})= \frac{1}{\sqrt{ (2 \pi )^3 }}\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \tilde{v}_s^{l}({\ensuremath{\mathbf{p}}})$ and $\tilde{v}_s^{\, l}({\ensuremath{\mathbf{p}}})= v_{s}^{\, l} (-{\ensuremath{\mathbf{p}}})$. Here $\chi_{{ \textrm{D} }} $ satisfy the following condition.\

> **(A.1 ; Ultraviolet Cutoff for Dirac Field)** $$\int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}})  |^2 d {\ensuremath{\mathbf{p}}} 
> < \infty  .$$

Then it holds that $$\qquad 
\|\psi_l ({\ensuremath{\mathbf{x}}}) \| \leq  c_{\, { \textrm{D} }}^{ \, l } ,  \label{psiBound}$$ where $ c_{\, { \textrm{D} }}^{\, l } = \frac{1}{\sqrt{(2\pi)^3}} \sum\limits_{s= \pm 1/2} 
\left( \| \chi_{{ \textrm{D} }} \, u_{s}^{\,l}  \| +  \| \chi_{{ \textrm{D} }} \,  \tilde{v}_{s}^{\, l} \|  \right)$, $l=1 , \cdots , 4.$\

Radiation Field in the Coulomb Gauge
------------------------------------

Let ${\mathscr{F}_{\textrm{rad}}}= {\mathscr{F}_{\textrm{b}}  }( \oplus_{r= 1,2} L^2( {\mathbf{R}^{3} }))$. The free Hamiltonian is defined by $${H_{\textrm{rad}}}= {\ensuremath{d\Gamma_{\textrm{b}}({\omega }) }}$$ where $\omega ({\ensuremath{\mathbf{k}}}) = |{\ensuremath{\mathbf{k}}}|$, ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$. Let $A^{\ast }(h_1 , h_2 )$, $h_{r} \in L^2( {\mathbf{R}^{3} })$, $r=1,2$, be the creation operators on ${\mathscr{F}_{\textrm{rad}}}$. Let $$\qquad 
a^{\dagger }_{1} (h ) = A (( h, 0)) , \quad a^{\dagger }_{2} (h )= A ((  0, h )) , \qquad h \in L^2( {\mathbf{R}^{3} }) ,$$ and $a_{r}(h')=(a^{\dagger} (h'))^{\ast}$, $h' \in L^2( {\mathbf{R}^{3} }) $, $r=1,2$. The creation operators and annihilation operators satisfy the canonical commutation relations $$\begin{aligned}
& [a_r (h) ,a_{r'}^{\dagger} (h') ] = \delta_{r, r'} (h, \, h' ), \; \; 
 \label{radCCR1}  \\
& [a_r (h) ,a_{r'} ( h' ) ] = [a_r^{\dagger } (h' ) ,a_{r'}^{\dagger} (h') ] = 0 , 
 \label{radCCR2}\end{aligned}$$ on ${\mathscr{F}_{\textrm{rad}}}^{\, {\textrm{fin}}}({\ensuremath{\mathscr{M}}})$ where ${\ensuremath{\mathscr{M}}} $ is a subspace of $\oplus_{r= 1,2} L^2( {\mathbf{R}^{3} })   $. For all $h \in {\ensuremath{\mathscr{D}}}(\omega^{-1/2}) $, it follows that $$\quad
\| a_r (h )  ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq \| \frac{h}{\sqrt{\omega }} \|  , \quad
 \| a^{\dagger }_r (h )  ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq  \| \frac{h}{\sqrt{\omega }}    \|  + \| h \|  . \label{radfiedBound}$$ The polarization vectors ${\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}}) = (e_r^{j} ({\ensuremath{\mathbf{k}}}))$, $r=1,2$, satisfy the following relations. $$\textbf{(R.1)} \; \qquad  {\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}}) \cdot {\ensuremath{\mathbf{e}}}_{r'} ({\ensuremath{\mathbf{k}}}) =0 , \quad 
{\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}})\cdot {\ensuremath{\mathbf{k}}} = 0 , \quad \quad {\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} . \notag$$ $\;$

\[exa2\] We check the example of the polarization vectors. For all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash    \{ {\ensuremath{\mathbf{0}}} \} $, we set $$\qquad 
{\ensuremath{\mathbf{e}}}_1({\ensuremath{\mathbf{k}}}) = \frac{1}{\sqrt{(k^1)^2 + (k^2) ^2}} \left( \begin{array}{c} - k^{2} \\ k^{1} \\ 0 \end{array} \right)  , \; \; \;  {\ensuremath{\mathbf{e}}}_2({\ensuremath{\mathbf{k}}})   =\frac{1}{|{\ensuremath{\mathbf{k}}}|\sqrt{k_1^2 +k_2^2}}
 \left( \begin{array}{c}  k^{1} k^3 \\ k^{2} k^3 \\ - (k^1)^2 - (k^2)^2 \end{array} \right)  . \quad$$ Then **(R.1)** is satisfied. Here it is noted that ${\ensuremath{\mathbf{e}}}_{r} \in \oplus^3 \, ( C^{1}({\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} ))$, $r=1,2$.

$\;$\
The radiation field operator ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})= ( A_j ({\ensuremath{\mathbf{x}}}))_{j=1}^3$ is defined by $$A_j({\ensuremath{\mathbf{x}}}) = \sum_{r=1,2} \left( a_r (h_{r , {\ensuremath{\mathbf{x}}}}^j) +   a^{\dagger}_r (h_{r , {\ensuremath{\mathbf{x}}}}^j) \frac{}{} \right)$$ where $h_{r , {\ensuremath{\mathbf{x}}}}^j ({\ensuremath{\mathbf{k}}})= h_{r }^j ({\ensuremath{\mathbf{k}}})e^{-{\ensuremath{\mathbf{k}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $h_{r }^j ({\ensuremath{\mathbf{k}}})= \frac{1}{\sqrt{(2 \pi )^3}}  \frac{\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) e_r^{j}({\ensuremath{\mathbf{k}}})}{\sqrt{2  \omega({\ensuremath{\mathbf{k}}})}}$, and $\chi_{{\textrm{rad}}}$ satisfy the following condition.\

> **(A.2 : Ultraviolet Cutoff for Radiation Field)** $$\qquad 
> \int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ \omega({\ensuremath{\mathbf{k}}})^{l}} {d \mathbf{k} }< \infty , \quad l= 1,2 .$$

Then $$\|A_j ({\ensuremath{\mathbf{x}}}) ( {H_{\textrm{rad}}}+1 )^{-1/2}  \| \leq  c^{\, j}_{{\textrm{rad}}}$$ where $c^{\,j}_{{\textrm{rad}}}=  \frac{1}{\sqrt{(2\pi )^3}}\sum\limits_{r=1,2}\left( \sqrt{2} \| \frac{\chi_{{\textrm{rad}}} e_r^j }{ \omega} \|  +   \|  \frac{\chi_{{\textrm{rad}}} e_r^j }{\sqrt{2 \omega}}  \| \right) $.\

Total Hamiltonian and Main Theorem
----------------------------------

We define the system of the Dirac field interacting with the radiation field. The Hilbert space for the system is defined by ${\mathscr{F}_{\textrm{QED}}}= {\mathscr{F}_{\textrm{Dirac}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$. The free Hamiltonian is defined by $$H_0= H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}}}\notag$$ on the domain ${\ensuremath{\mathscr{D}}}(H_{0}) = {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )  \cap {\ensuremath{\mathscr{D}}}({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}}})$. To define the interactions, we introduce spatial cutoff $\chi_{{\textrm{I}}}$ and $\chi_{{\textrm{II}}}$, which satisfy the condition below.\

> **(A.3 : Spatial Cutoff )** $$\int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }< \infty , \qquad   \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
>      \frac{|{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} {d \mathbf{x} }{d \mathbf{y} }< \infty .$$

First we define a functional on $  {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}  $ by $$\qquad \qquad 
\ell_{{\textrm{I}}}(\Phi , \Psi ) = \sum_{j=1}^3   \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , ({\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}})  }{\otimes}A_j ({\ensuremath{\mathbf{x}}}) ) \Psi \right) ,
\; \; \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} , \, \Psi \in {\ensuremath{\mathscr{D}}}(H_0 ) ,$$ where $ {\psi^{\dagger} (\mathbf{x}) }= ( \psi_{1}({\ensuremath{\mathbf{x}}})^{\ast} , \cdots , \psi_{4}({\ensuremath{\mathbf{x}}})^{\ast} ) $. We see that $$| \ell_{{\textrm{I}}}(\Phi , \Psi ) |   \leq  \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }\right) \,
\sum\limits_{j=1}^3 \, \sum_{l,l'=1}^4 |\alpha_{l,l'}^j | c_{\, { \textrm{D} }}^{\, l } c_{\, { \textrm{D} }}^{ \, l' } c_{{\textrm{rad}}}^{\,  j } \,
 \|  \Phi \| \,  \| {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}({H_{\textrm{rad}}}+1)^{1/2} \Psi \| .  \notag$$ By the Riesz representation theorem, we can define the operator ${H_{\textrm{I}}}$ which satisfy $  (\Phi , {H_{\textrm{I}}}\Psi ) 
= \ell_{{\textrm{I}}}(\Phi , \Psi )   $ and $$\|  {H_{\textrm{I}}}\Psi \| \leq c_{\, {\textrm{I}}} \,   \| {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}({H_{\textrm{rad}}}+1)^{1/2} \Psi \|   , 
  \label{HIbound}$$ where $ c_{\, {\textrm{I}}} = \| \chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \, \sum\limits_{l,l'=1}^4 |\alpha_{l,l'}^j | c_{\, { \textrm{D} }}^{\, l } c_{\, { \textrm{D} }}^{ \, l' } c_{{\textrm{rad}}}^{\,  j }$. By the spectral decomposition theorem, it is proven that for all $\epsilon > 0$, $$\|{H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \|H_{0} \Psi \| +c_{\textrm{I}}\left( \frac{1 }{2 \epsilon}  +1 \right)  \| \Psi \|  .
 \label{HIbound'}$$ Next we define a functional on $ {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} $ by $$\ell_{{\textrm{II}}}(\Phi , \Psi ) =    \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
     \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} 
\left( \Phi , \left(  {\psi^{\dagger} (\mathbf{x}) }{ \psi ({\ensuremath{\mathbf{x}}})  }{\psi^{\dagger} (\mathbf{y}) }{ \psi ({\ensuremath{\mathbf{y}}})  }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  \right) \Psi \right) 
 d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}}  , \; \; \Phi , \, \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}  . \\$$ We see that $$|\ell_{{\textrm{II}}}(\Phi , \Psi ) |   \leq 
 \left( \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \left| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{ |{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \right| {d \mathbf{x} }{d \mathbf{y} }\right)
 \,  \sum_{l , l' =1}^4
 (c_{\, { \textrm{D} }}^{ \, l} c_{\, { \textrm{D} }}^{\, l'})^2 \| \Phi  \| \, \| \Psi \| .  \notag$$ Then, by the Riesz representation theorem, we can define an operator ${H_{\textrm{II}}}$ satisfying $  (\Phi , {H_{\textrm{II}}}\Psi ) = \ell_{{\textrm{II}}}(\Phi , \Psi )$ and $$\|{H_{\textrm{II}}}\| \leq  c_{{\textrm{II}}} ,   \label{HIIbound}$$ where $ c_{{\textrm{II}}} = \left\| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{ |{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \right\|_{L^1} \,  \sum\limits_{l , l' =1}^4  (c_{\, { \textrm{D} }}^{ \, l} c_{\, { \textrm{D} }}^{\, l'})^2$. By (\[HIbound’\]) and (\[HIIbound\]), it holds that $$\| ({\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}) \Psi  \| 
\leq    c_{{\textrm{I}}} {\kappa_{\textrm{I}}}\epsilon  \| H_0 \Psi \| + 
\left(  c_{{\textrm{I}}} {\kappa_{\textrm{I}}}\left(  \frac{1 }{2 \epsilon}  +1 \right)  +  c_{{\textrm{II}}} {\kappa_{\textrm{II}}}\right) \| \Psi \|
. \notag$$ Then the Kato-Rellich theorem yields that that ${H_{\textrm{QED}}}$ is self-adjoint on ${\ensuremath{\mathscr{D}}}(H_{0})$ and essentially self-djoint on any core of $H_{0}$. Hence, in particular, ${H_{\textrm{QED}}}$ is essentially self-adjoin on $${\ensuremath{\mathscr{D}}}_{0} = {\mathscr{F}_{\textrm{Dir}}}^{\, {\textrm{fin}}}{{\ensuremath{\mathscr{D}}}(\omega_{\, M})} \hat{{\otimes}}{\mathscr{F}_{\textrm{rad}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}}(\omega ))$$ where $\hat{{\otimes}}$ denotes the algebraic tensor product.

$\; $\
To prove the existence of the ground state of ${H_{\textrm{QED}}}$, we suppose additional conditions below.

> **(A.4 : Spatial Localization)** $$\int_{{\mathbf{R}^{3} }} |{\ensuremath{\mathbf{x}}}|| {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }< \infty ,  \quad  \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
>      \frac{|{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|}|{\ensuremath{\mathbf{x}}}| {d \mathbf{x} }{d \mathbf{y} }< \infty       .$$

> **(A.5 : Momentum Regularity Condition for Dirac Field)**\
> There exists a subset $O_{{ \textrm{D} }} \subset {\mathbf{R}^{3} }$ with Lebesgue measure zero such that $u_{s}, v_{s} \in \oplus^4 \, ( C^1 ({\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }})) $, $s= \pm 1/2$. $\chi_{{ \textrm{D} }} \in  C^{1}({\mathbf{R}^{3} }) $, and it satisfies that $$\int_{{\mathbf{R}^{3} }}  | \partial_{p^{\nu}}\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}}
> < \infty ,
>  \;   \int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p }}}) \partial_{p^{\nu }} u_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}}
> < \infty , \;  \int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p }}}) \partial_{p^{\nu }} v_{s}^{\, l} (-{\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}}
> < \infty ,$$ for all $\nu =1, 2,3$, $ l = 1 , \cdots ,4$, $  s= \pm 1/2 $.

> **(A.6 : Momentum Regularity Condition for Radiation Field )**\
> There exists a subset $O_{{\textrm{rad}}} \subset {\mathbf{R}^{3} }$ with Lebesgue measure zero such that ${\ensuremath{\mathbf{e}}}_{r} \in \oplus^3 \, ( C^{1}({\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}} ))$, $r=1,2$, where $O_{{\textrm{rad}}} $. $\chi_{{\textrm{rad}}} \in C^{1}({\mathbf{R}^{3} })$ and it satisfies that $$\int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ | {\ensuremath{\mathbf{k}}}|^5} {d \mathbf{k} }< \infty , \quad 
> \int_{{\mathbf{R}^{3} }} \frac{ | \partial_{k^\nu }\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ | {\ensuremath{\mathbf{k}}} |^3} {d \mathbf{k} }< \infty , \quad  \int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu } e_{r}^{j} ({\ensuremath{\mathbf{k}}}) |^2}{ |{\ensuremath{\mathbf{k}}}|^3} {d \mathbf{k} }< \infty ,$$ for all $\nu =1, 2, 3$, $j=1, 2, 3$, $r=1,2$.

\[exa3\] Examples of $O_{{ \textrm{D} }}$ and $O_{{\textrm{rad}}}$ in **(A.5)** and **(A.6)** are as follows. In the case of the standard representation, $O_{{ \textrm{D} }} = O_{\textrm{SR}}$ where $ O_{\textrm{SR}} $ is defined in Remark \[exa1\]. For the polarization vectors considered in Remark \[exa2\], $O_{{\textrm{rad}}} = \{ {\ensuremath{\mathbf{0}}} \}$.

$\; $\
The main theorem in this paper is as follows.

\[Main-Theorem\] **(Existence of a Ground State)**\
Suppose **(A.1)** - **(A.6)**. Then $ {H_{\textrm{QED}}}$ has a ground state for all values of coupling constants. In particular, its multiplicity is finite.

Ground States of Massive case
=============================

In this section, we consider a massive Hamiltonian defined by $$H_m = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}+ {\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}},  \notag$$ where ${H_{\textrm{rad}, \, m}}= {\ensuremath{d\Gamma_{\textrm{b}}({\omega_{m}}) }}$ with $\omega_m ({\ensuremath{\mathbf{k}}})=\sqrt{{\ensuremath{\mathbf{k}}}^2 + m^2} $, $m>0$.\

Fock Spaces on Direct Sum of Hilbert Spaces
-------------------------------------------

$\, $ We review basic properties of Fock spaces on direct sum of Hilbert spaces. These are useful for constructing partitions of unity on Fock spaces (see, Dereziński-Gérard [@DeGe99]).

$\, $\
**(i) Full Fock Space on** ${\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$\
Let $ Z = \left[ \begin{array}{c} Z_0 \\ Z_\infty \end{array} \right] $, $ Z_0 , Z_\infty \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $, where ${\ensuremath{\mathscr{Z}}}$ is a complex Hilbert space. We consider $ Z = \left[ \begin{array}{c} Z_0 \\ Z_\infty \end{array} \right] $ is an operator $ {\ensuremath{\mathscr{Z}}} \to {\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$ which acts for $$\qquad 
 \qquad Z h = \left[ \begin{array}{c} Z_0 \,  h \\ Z_\infty \, h \end{array} \right] , \quad \; h \in  {\ensuremath{\mathscr{D}}} (Z_{0}) \cap {\ensuremath{\mathscr{D}}} (Z_{\infty}) .$$ Let $J = \left[ \begin{array}{c} J_0 \,   \\ J_\infty \,  \end{array} \right]$, $ J_0 , J_\infty \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $ and $ B = \left[ \begin{array}{c} B_0 \,   \\ B_\infty \,  \end{array} \right] $, $ B_0 , B_\infty \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $. We define $d \Gamma ( J , B) : {\ensuremath{\mathscr{F}}}({\ensuremath{\mathscr{Z}}}) \to {\ensuremath{\mathscr{F}}} ({\ensuremath{\mathscr{Z}}}) \oplus  {\ensuremath{\mathscr{F}}}({\ensuremath{\mathscr{Z}}})$ by $$d \Gamma (J , B ) = 
\oplus_{n=0}^{\infty} \left( \sum\limits_{j=1}^n 
 ( {\otimes}^{j-1} J )   {\otimes}B
  {\otimes}( {\otimes}^{n-j} J ) \right) .$$ If $B_{0}$ and $B_{\infty}$ are bounded, and $J_{0}^{\ast}J_0 + J_{\infty}^{\ast}J_{\infty} \leq 1 $, it holds that $$\|   d \Gamma ( J, B ) (N +1)^{-1} \| \leq  \sqrt{ \|B_{0} \|^2 + \|  B_{\infty}  \|^2 } .  
    \label{3.1.1}$$ Let $T \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$. Then it holds that $$\Gamma (J ) {\ensuremath{d\Gamma({T}) }} =  d \Gamma \left( \left[ 
\begin{array}{cc} T & 0 \\ 
 0 & T  \end{array} \right]   \right)
\Gamma ( J ) + d \Gamma ( J ,\tilde{\text{ad}}_{T}( J )  ) ,
  \label{3.1.2}$$ where $ \tilde{\text{ad}}_{T}(J)  :  {\ensuremath{\mathscr{Z}}} \to {\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$ is defined by $$\qquad  \qquad \quad 
\tilde{\text{ad}}_{T}(J) h=\left[ 
\begin{array}{c} [T ,J_0 ] h \\ 
 {[} T , J_\infty {]} h  \end{array} \right] , \quad h \in {\ensuremath{\mathscr{D}}}([T ,J_0 ]) 
\cap {\ensuremath{\mathscr{D}}}([T ,J_\infty ] ) .$$

$\;$\
**(ii) Fermion Fock Space on** ${\ensuremath{\mathscr{X}}} \oplus {\ensuremath{\mathscr{X}}}$\
Let ${\ensuremath{\mathscr{X}}}$ be a complex Hilbert space. Let $ J_{{\textrm{f}}}  = \left[ \begin{array}{c} J_{{\textrm{f}}}^{\, 0 } \,   \\ J_{{\textrm{f}}}^{\, \infty} \,  \end{array} \right] $, $ J_{{\textrm{f}}}^{\, 0 } ,  J_{{\textrm{f}}}^{\, \infty} \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{X}}}) $ and $  B_{{\textrm{f}}} = \left[ \begin{array}{c}
 B_{{\textrm{f}}}^{\, 0 } \,   \\   B_{{\textrm{f}}}^{\, \infty}\,  \end{array} \right] $, $ B_{{\textrm{f}}}^{\, 0 }, B^{\, \infty}_{{\textrm{f}}} \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{X}}}) $. We set ${\ensuremath{d\Gamma_{\textrm{f}}({ J_{{\textrm{f}}} , B_{{\textrm{f}}} }) }} = 
{\ensuremath{d\Gamma({ J_{{\textrm{f}}} , B_{{\textrm{f}}}  }) }}_{{\upharpoonright}{\mathscr{F}_{\textrm{f}}  }({\ensuremath{\mathscr{X}}})} $. Suppose that $B_{{\textrm{f}}}^{\, 0 } $ and $B^{\, \infty}_{{\textrm{f}}} $ are bounded, and $ (J_{{\textrm{f}}}^{\, 0 })^\ast J_{{\textrm{f}}}^{\, 0 } +   (J_{{\textrm{f}}}^{\, \infty})^{\ast} J_{{\textrm{f}}}^{\, \infty}  \leq 1 $. By (\[3.1.1\]), it holds that $$\|   d {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} , B_{{\textrm{f}}} ) (N_{{\textrm{f}}} +1)^{-1} \| \leq 
 \sqrt{ \| B_{{\textrm{f}}}^{\, 0 } \|^2 + \|   B_{{\textrm{f}}}^{\, \infty}  \|^2 } . 
\label{3.1.3}$$ Let $T_{{\textrm{f}}}  \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{X}}})$. From (\[3.1.2\]), it holds that $${\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ) {\ensuremath{d\Gamma_{\textrm{f}}({T_{{\textrm{f}}} }) }} =  d {\Gamma_{\textrm{f}}}\left( \left[ 
\begin{array}{cc} T_{{\textrm{f}}}  & 0 \\ 
 0 & T_{{\textrm{f}}}  \end{array} \right]   \right)
{\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ) + d {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}}  ,\tilde{\text{ad}}_{T_{{\textrm{f}}} }(
 J_{{\textrm{f}}} )  ).  \label{3.1.4}$$ Let $ C(f)$ and $C^{ \, \dagger} (f)$, $f \in {\ensuremath{\mathscr{X}}}$, be the annihilation and creation operators on ${\mathscr{F}_{\textrm{f}}  }({\ensuremath{\mathscr{X}}})$, respectively. Then it follows that $$\begin{aligned}
& \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) C (f) = C \left( 
\left[  \begin{array}{c} f  \\   0  \end{array} \right]  
 \right)  \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) + \Gamma_{{\textrm{f}}} 
( J_{{\textrm{f}}} ) \,  C \left( (1-(  J_{{\textrm{f}}}^{\, 0 })^{\ast})f \frac{}{}\right) ,
 \label{3.1.5}
 \\
 & \Gamma ( J_{{\textrm{f}}} ) C^{\, \dagger} (f) = C^{\, \dagger} \left( 
\left[  \begin{array}{c} f  \\   0  \end{array} \right]  
 \right)  \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) +  C^{\, \dagger} \left( 
\left[  \begin{array}{c} J_{{\textrm{f}}}^{\, 0 } -1  \\      J_{{\textrm{f}}}^{\, \infty}   \end{array} \right]  
 f \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} )  . 
\label{3.1.6}\end{aligned}$$ $\;$\
$\; $\
**(iii) Boson Fock Space on** ${\ensuremath{\mathscr{Y}}} \oplus {\ensuremath{\mathscr{Y}}}$\
Let ${\ensuremath{\mathscr{Y}}}$ be a complex Hilbert space. Let $ J_{{\textrm{b}}}  = \left[ \begin{array}{c} J_{{\textrm{b}}}^{\, 0 } \,   \\ J_{{\textrm{b}}}^{\, \infty} \,  \end{array} \right] $, $ J_{{\textrm{b}}}^{\, 0 } ,  J_{{\textrm{b}}}^{\, \infty} \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Y}}}) $ and $  B_{{\textrm{b}}} = \left[ \begin{array}{c}
 B_{{\textrm{b}}}^{\, 0 } \,   \\   B_{{\textrm{b}}}^{\, \infty} \,  \end{array} \right] $, $ B_{{\textrm{b}}}^{\, 0 },  B_{{\textrm{b}}}^{\, \infty} \in   {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Y}}}) $. We define ${\ensuremath{d\Gamma_{\textrm{b}}({ J_{{\textrm{b}}} , B_{{\textrm{b}}} }) }} = 
{\ensuremath{d\Gamma({ J_{{\textrm{b}}} , B_{{\textrm{b}}}  }) }}_{{\upharpoonright}{\mathscr{F}_{\textrm{b}}  }({\ensuremath{\mathscr{Y}}})} $. Assume that $B_{{\textrm{b}}}^{\, 0 } $ and $ B_{{\textrm{b}}}^{\, \infty} $ are bounded, and $ (J_{{\textrm{b}}}^{\, 0 })^\ast J_{{\textrm{b}}}^{\, 0 } +   (J_{{\textrm{b}}}^{\, \infty})^{\ast} J_{{\textrm{b}}}^{\, \infty}  \leq 1 $. By (\[3.1.1\]), it follows that $$\|   d {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} , B_{{\textrm{b}}} ) (N_{{\textrm{b}}} +1)^{-1} \| \leq 
 \sqrt{ \| B_{{\textrm{b}}}^{\, 0 } \|^2 + \|   B_{{\textrm{b}}}^{\, \infty}  \|^2 } . 
\label{3.1.7}$$ Let $ T_{{\textrm{b}}} \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Y}}})$. Then (\[3.1.2\]) yields that $${\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) {\ensuremath{d\Gamma_{\textrm{b}}({T_{{\textrm{b}}} }) }} =  d {\Gamma_{\textrm{b}}}\left( \left[ 
\begin{array}{cc} T_{{\textrm{b}}}  & 0 \\ 
 0 & T_{{\textrm{b}}}  \end{array} \right]   \right)
{\Gamma_{\textrm{b}}}( J_{{\textrm{f}}} ) + d {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}}  ,\tilde{\text{ad}}_{T_{{\textrm{b}}} }(
 J_{{\textrm{b}}} )  ).   \label{3.1.8}$$ Let $ A(g)$ and $A^{\dagger} (g)$, $g \in {\ensuremath{\mathscr{Y}}}$, be the annihilation and creation operators on ${\mathscr{F}_{\textrm{b}}  }({\ensuremath{\mathscr{Y}}})$, respectively. Then it follows that $$\begin{aligned}
& \Gamma_{{\textrm{b}}} ( J_{{\textrm{b}}} ) A (g) = A \left( 
\left[  \begin{array}{c} g  \\   0  \end{array} \right]  
 \right)  {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) + {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) \, A \left( (1-( J_{{\textrm{b}}}^{\, 0 } )^{\ast}) g \frac{}{}\right) , 
 \label{3.1.9} \\
 & {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) A^{\dagger } (g) = A^{\dagger} \left( 
\left[  \begin{array}{c} g  \\   0  \end{array} \right]  
 \right)  \Gamma_{{\textrm{b}}} ( J_{{\textrm{b}}} ) +  A^{\dagger } \left( 
\left[  \begin{array}{c}   J_{{\textrm{b}}}^{\, 0 } -1 \\    J_{{\textrm{b}}}^{\, \infty }  \end{array} \right]   g \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} )  .
\label{3.1.10}\end{aligned}$$

Partition of Unity for the Dirac Field
--------------------------------------

We construct a partition of unity for the Dirac field. For general properties of partition of unity for fermionic fields, refer to Ammari [@Am04].

$\;$\
Let $$\quad 
c_{ \tau , s} (f) = \left\{ \begin{array}{c} b_{s}(f), \; \; \tau = + , \\
 d_{s}(f) , \; \;\tau = - . \end{array}  \right.$$ Let $U_{{\textrm{f}}}:   {\mathscr{F}_{\textrm{f}}  }\left( L^2( {\mathbf{R}^{3} }_{\bf{p}} ; {\ensuremath{\mathbf{C}}}^4  ) \oplus L^2( {\mathbf{R}^{3} }_{\bf{p}} ; {\ensuremath{\mathbf{C}}}^4  ) \right)  \to {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} $ be an isometric operator which satisfy $U_{{\textrm{f}}} \,  \Omega_{{ \textrm{D} }} =\Omega_{{ \textrm{D} }} {\otimes}\Omega_{{ \textrm{D} }} $ and $$\begin{aligned}
&U_{{\textrm{f}}}  \;  c^{\dagger }_{\tau_1, s_1} \left(\left[ \begin{array}{c} f_1 \\ g_1 \end{array} \right]\right) \cdots  c^{\dagger }_{\tau_1 , s_1} \left(\left[ \begin{array}{c} f_1 \\ g_n \end{array} \right]\right) \Omega_{{ \textrm{D} }}   \notag \\
&  =  \left( c^{\dagger}_{\tau_1 , s_1} (f_1) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}}
 {\otimes}c^{\dagger}_{\tau_1 ,  s_1}(g_1) \frac{}{}\right)
\cdots  \left( c^{\dagger }_{\tau_n, s_n} (f_n) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger}_{\tau_n , r_n}(g_n) \frac{}{} \right)
 \Omega_{{ \textrm{D} }} {\otimes}\Omega_{{ \textrm{D} }} . \notag\end{aligned}$$ Here note that $(-1)^{N_{{ \textrm{D} }}}\Psi = (-1)^n \Psi  $ for the vector of the form $ \Psi = c^\dagger_{\tau_1 ,  s_1}(f_1)  \cdots   c^\dagger_{\tau_n, s_n} (f_n) \Omega_{{ \textrm{D} }} $, $f_{j} \in L^2 ({\mathbf{R}^{3} }) $, $j= 1, \cdots ,n$, $n \in {\ensuremath{\mathbf{N}}}$. Let $j_{0} , j_{\infty} \in C^{\, \infty} ({\ensuremath{\mathbf{R}}})$. We assume that $j_0 \geq 0 $, $j_{\infty} \geq 0$, $j_{0} ({\ensuremath{\mathbf{x}}})^2 +  j_{\infty} ({\ensuremath{\mathbf{x}}})^2 =1$, $j_{0}({\ensuremath{\mathbf{x}}})=1 $ for $|{\ensuremath{\mathbf{x}}}| \leq 1 $ and $j_{0}({\ensuremath{\mathbf{x}}})=0 $ for $|{\ensuremath{\mathbf{x}}}| \geq 2 $. Let $j_{{\textrm{f}}, R } = \left[ \begin{array}{c} j_{{\textrm{f}}, R }^{\, 0}   \\ 
  j_{{\textrm{f}}, R }^{\, \infty}  \end{array} \right]$ where $ j_{{\textrm{f}}, R }^{\, 0}= j_{0} (\frac{-i \, {\ensuremath{\mathbf{\nabla_{{\ensuremath{\mathbf{p}}}}}}}}{R})$ and $ j_{{\textrm{f}}, R }^{\, \infty}=  j_{\infty}
 (\frac{-i \, {\ensuremath{\mathbf{ \nabla_{{\ensuremath{\mathbf{p}}}}}}}}{R})    $ with $\nabla_{{\ensuremath{\mathbf{p}}}}= (\partial_{p^1}, \partial_{p^2},\partial_{p^3} ) $.\
$\;$\
Let $X_{{\textrm{f}}, R} : {\mathscr{F}_{\textrm{Dir}}}\to  {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }}  $ defined by $$\quad 
 X_{{\textrm{f}}, R}  = \,    U_{{\textrm{f}}} \, \Gamma_{{\textrm{f}}}(j_{{\textrm{f}}, R}) . \notag$$ From (\[3.1.4\])-(\[3.1.6\]), it holds that $$\begin{aligned}
 &  X_{{\textrm{f}}, R} \, H_{{ \textrm{D} }} = 
\left( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} \frac{}{} \right) 
  X_{{\textrm{f}}, R} \, +   U_{{\textrm{f}}} \,  d \Gamma_{{\textrm{f}}} (j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{M} }(j_{{\textrm{f}},R})) , \label{3.2.1}  \\
& X_{{\textrm{f}}, R} \,  c_{\tau , s}(f)
=  (c_{\tau, s } (f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  X_{{\textrm{f}}, R} \, +
 X_{{\textrm{f}}, R} \,
 c_{\tau , s}((1-j_{{\textrm{f}}, R}^{\, 0} ) f) , \label{3.2.2} \\
& X_{{\textrm{f}}, R} \, c^{\dagger}_{\tau , s }(f) 
= ( c^\ast_{\tau , s } (f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} 
+    \left( c^{\dagger}_{\tau , s } ((j_{{\textrm{f}}, R}^{\, 0} -1 )f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+  (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger }_{\tau , s }(j_{{\textrm{f}}, R}^{\, \infty} f ) \right)
X_{{\textrm{f}}, R}  . \label{3.2.3}\end{aligned}$$

\[lemma32a\] Assume **(A.1)**. Then, $$\begin{aligned}
& \textbf{(i)}
\left\| \left(  X_{{\textrm{f}}, R} \,  H_{{ \textrm{D} }}
-   (      H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} ) \right) 
  X_{{\textrm{f}}, R}  ( N_{{ \textrm{D} }} + 1 )^{-1} \right\|  \leq  \frac{c_{ \, {\textrm{f}}}}{R} ,  \\
 & \textbf{(ii)} 
\left\|   X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})
-   (       \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R}    
 \right\|  \leq \delta^{1,l}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) ,\quad l= 1, \cdots 4,  \\
&\textbf{(iii)} 
\left\|   X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}
-   (       \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R}     \right\|  \leq
 \delta^{2 , l}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) , \quad  l= 1, \cdots 4 . \end{aligned}$$ Here $c_{ \, {\textrm{f}}} \geq 0$ is a constant, and $\delta^{i, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) \geq0  $, $l=1 ,\cdots ,4$, $i= 1, 2$, are error terms which satisfy $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{ i , l }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{i,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0  $ for all ${\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$.

$\;$\
**(Proof)** **(i)** By (\[3.2.1\]), we have $$\left\| \left(   X_{{\textrm{f}}, R}  H_{{ \textrm{D} }}
-   (     H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }}  ) X_{{\textrm{f}}, R} \right) 
   (N_{{ \textrm{D} }} +1)^{-1}  \right\| 
   \leq   \|  d \Gamma_{{\textrm{f}}} ( j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{\, M}}(j_{{\textrm{f}},R})) (N_{{ \textrm{D} }} +1)^{-1}   \|  , \notag$$ and (\[3.1.3\]) yields that $$\|  d \Gamma_{{\textrm{f}}} ( j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{\, M}}(j_{{\textrm{f}},R})) (N_{{ \textrm{D} }} +1)^{-1}   \|
\leq  \sqrt{ \| [\omega_{\, M},j^{\, 0}_{{\textrm{f}}, R} ]  \|^2_{B(L^2({\mathbf{R}^{3} }))} 
+ \| [\omega_{\, M},j^{\, \infty}_{{\textrm{f}}, R} ]  \|^2_{B(L^2({\mathbf{R}^{3} }) )} }  \notag$$ By pseudo-differential calculus (e.g., [@FGS02] ; Appendix A, [@Hida11] ; Section IV), it follows that $\;$ $\| [\omega_{M},j^{\, \sharp}_{{\textrm{f}}, R} ]  \|_{B(L^2({\mathbf{R}^{3} }) )} \leq$ $ \frac{ c_{\sharp }}{R}$, $\sharp = 0, \infty $, where $c_{\sharp} \geq 0$ are constants. Thus **(i)** is proven.\
**(ii)** By the definition of $\psi_{l} ({\ensuremath{\mathbf{x}}})= \sum\limits_{s= \pm 1/2}(b_{s} (f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + d^{\, \dagger }_{s} (g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) )$, we have from (\[3.2.2\]) and (\[3.2.3\]) that $$\begin{aligned}
& X_{{\textrm{f}}, R} \,  \psi_{l}({\ensuremath{\mathbf{x}}}) -  ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  X_{{\textrm{f}}, R}   \notag \\
& =\sum_{s= \pm 1/2} \left( X_{{\textrm{f}}, R} \,  b_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s, {\ensuremath{\mathbf{x}}}}^{\,l})
 + \left(     d^{\, \dagger }_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s,{\ensuremath{\mathbf{x}}}}^{\, l}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}+  (-1)^{N_{{ \textrm{D} }}} {\otimes}d_{s}^{\, \dagger } (j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^{l\, } )\right)   X_{{\textrm{f}}, R} \right) . \end{aligned}$$ Then we have $$\begin{aligned}
& \left\|  X_{{\textrm{f}}, R} \,  \psi_{l}({\ensuremath{\mathbf{x}}})
-   (      \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  X_{{\textrm{f}}, R}      \right\|   \notag \\
 & \leq  \sum_{s= \pm 1/2 }  \left( 
\|  b_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s ,{\ensuremath{\mathbf{x}}}}^{\, l})   \|
+ \| (   d^{\, \dagger}_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s ,{\ensuremath{\mathbf{x}}}}^l){\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R}  \| + \|  ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}d_{s}^{\, \dagger } (j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l )  X_{{\textrm{f}}, R}   \|   \right)   \notag \\
  & \leq \sum_{s= \pm 1/2}  \left( \| ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s ,{\ensuremath{\mathbf{x}}}}^{\, l})  \|
+  \| ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s,{\ensuremath{\mathbf{x}}}}^l \|
+ \| j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l   \| \frac{}{}   \right)  .\end{aligned}$$ Let $\delta^{1,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{s= \pm 1/2 } \left( \| ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s,{\ensuremath{\mathbf{x}}}}^{\, l})  \|
+  \| ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s,{\ensuremath{\mathbf{x}}}}^l) \|
+ \| j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l   \| \frac{}{}   \right) $. We see that $  \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} |
\delta^{1, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}})| \leq   \sum\limits_{s= \pm 1/2 } \left(   \|f_{s}^{\,l} \| + 2 \|g_{s}^l \|\right) $ and $\lim\limits_{R \to \infty}
\delta^{1,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Hence **(ii)** follows.\
$\; $\
**(iii)** From the definition of $\psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}= \sum\limits_{s= \pm 1/2}(b^{\dagger }_{s} (f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + d_{s} (g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) )$, (\[3.2.2\]) and (\[3.2.3\]) yield that $$\begin{aligned}
& X_{{\textrm{f}}, R} \,  \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} -  ( \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  X_{{\textrm{f}}, R}   \notag \\
&  = \sum_{s= \pm 1/2} \left( 
  \left(     b^{\dagger }_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) f_{s,{\ensuremath{\mathbf{x}}}}^{\, l}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}+  (-1)^{N_{{ \textrm{D} }}} {\otimes}b_{s}^{\dagger } (j_{{\textrm{f}}, R}^{\, \infty} f_{s,{\ensuremath{\mathbf{x}}}}^{\, l } )
\right)   X_{{\textrm{f}}, R} +
X_{{\textrm{f}}, R} \,  d_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) 
\right) . \end{aligned}$$ Then it follows that $$\| X_{{\textrm{f}}, R} \,  \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} -  ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  X_{{\textrm{f}}, R}   \| \leq 
\delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) ,
\notag$$ where $\delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{s= \pm 1/2 } \left( 
  \| ((j_{{\textrm{f}}, R}^{\, 0} -1) f_{s,{\ensuremath{\mathbf{x}}}}^l) \|
+ \| j_{{\textrm{f}}, R}^{\, \infty} f_{s,{\ensuremath{\mathbf{x}}}}^l   \|
+ \| ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s,{\ensuremath{\mathbf{x}}}}^{\, l})  \|
\,   \right) $. It is seen that $  \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} |
\delta^{2, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}})| \leq   \sum\limits_{s= \pm 1/2 } \left(    2 \|f_{s}^{\,l} \| +  \|g_{s}^l \|\right) $ and $\lim\limits_{R \to \infty}
\delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Thus we obtain **(iii)**. $\blacksquare $\

\[coro32a\] Assume **(A.1)**. Then, for all $l ,l' = 1, \cdots 4$, $$\begin{aligned}
\textbf{(i)} \, & \left\|   X_{{\textrm{f}}, R} \,  \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) 
-   (       \psi_{l}  ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R}     \right\|  \leq \delta^{3, l,l'}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) , 
\\
\textbf{(ii)}  
& \left\|   X_{{\textrm{f}}, R} \,   \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l}({\ensuremath{\mathbf{x}}})   \psi_{l'}({\ensuremath{\mathbf{y}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{y}}})   
-   (    \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l}({\ensuremath{\mathbf{x}}})   \psi_{l'}({\ensuremath{\mathbf{y}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{y}}})    {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R}     \right\|  \leq \delta^{ \, 4, l, l'  }_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}, {\ensuremath{\mathbf{y}}}) .\notag\end{aligned}$$ Here $\delta^{ 3 , l, l' }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) \geq 0  $ satisfies $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{ 3, l,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{ 3, l ,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0  $ for all ${\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$, and $\delta^{ 4 , l, l' }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}} ) \geq0 $ satisfies $ \sup\limits_{ ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}}) \in {\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} | \delta^{ \, 4, l,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{ \, 4, l ,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) =0  $ for all $ {\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}} \in  {\mathbf{R}^{3} }$.

**(Proof)** **(i)** By Lemma \[lemma32a\] **(ii)** and **(iii)**, it is seen that $$\begin{aligned}
& \left\|   X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}})
-   (       \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R}  \right\|
\notag \\
&\leq \left\|  \left(  X_{{\textrm{f}}, R} \, \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}  
 -    \left( (       \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right) \right) \psi_{l'}({\ensuremath{\mathbf{x}}}) \right\|  \notag \\
& \qquad \qquad \qquad + \left\| (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \left(  X_{{\textrm{f}}, R} \,   \psi_{l'}({\ensuremath{\mathbf{x}}})   -   (       \psi_{l'}({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right)\right\| \notag \\
& \leq \delta_{{\textrm{f}}, R}^{2,l} \|\psi_{l'}({\ensuremath{\mathbf{x}}})  \| 
+ \delta_{{\textrm{f}}, R}^{1,l'} \| \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}  \|  . \notag \end{aligned}$$ Note that $  \|   \psi_{\, l'}({\ensuremath{\mathbf{x}}})  \| \leq c_{\, { \textrm{D} }}^{\, l'}$ and $ \| \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}  \| \leq c_{\, { \textrm{D} }}^{l} $. Hence **(i)** is obtained. Similarly, we can prove **(ii)** by using **(i)**. $\blacksquare$\

Partition of Unity for Radiation Field 
---------------------------------------

Let $U_{{\textrm{b}}}:   {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}( L^2( {\mathbf{R}^{3} }_{{\ensuremath{\mathbf{k}}}} \times \{ 1,2  \}  )  \oplus  L^2( {\mathbf{R}^{3} }_{{\ensuremath{\mathbf{k}}}} \times \{ 1,2  \}  ) ) \to  {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} $ an isometric operator satisfying $U_{{\textrm{b}}} \,  {\Omega_{\textrm{rad}}}= {\Omega_{\textrm{rad}}}{\otimes}{\Omega_{\textrm{rad}}}$ and $$\begin{aligned}
&U_{{\textrm{b}}} \, a_{r_1}^{\dagger } \left(\left[ \begin{array}{c} f_1 \\ g_1 \end{array} \right]\right) \cdots  a_{r_1}^{\dagger } \left(\left[ \begin{array}{c} f_1 \\ g_n \end{array} \right]\right) {\Omega_{\textrm{rad}}}\\
= & \,  \left( a^{\dagger }_{r_1} (f_1) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a^{\dagger }_{r_1}(g_1) \frac{}{}\right)
\cdots  \left( a^{\dagger}_{r_n} (f_n) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a^{\dagger}_{r_n}(g_n) \frac{}{} \right)
 {\Omega_{\textrm{rad}}}{\otimes}{\Omega_{\textrm{rad}}}.  \end{aligned}$$ Let $j_{0} , j_{\, \infty} \in C^{\infty} ({\ensuremath{\mathbf{R}}})$. We suppose that $j_0 \geq 0$, $j_{\infty} \geq0 $, $j_{0}^2 +  j_{\infty}^2 =1$, $j_{0}({\ensuremath{\mathbf{y}}})=1 $ if $|{\ensuremath{\mathbf{y}}}| \leq 1 $ and $j_{0}({\ensuremath{\mathbf{y}}})=0 $ if $|{\ensuremath{\mathbf{y}}}| \geq 2 $. We set $j_{{\textrm{b}}, R } =\left[ \begin{array}{c} j_{{\textrm{b}}, R }^{\, 0} \\ j_{{\textrm{b}}, R }^{\, \infty}  \end{array} \right] $ where $ j_{{\textrm{b}}, R }^{\, 0}= j_{0} (\frac{-i {\ensuremath{\mathbf{\nabla_{{\ensuremath{\mathbf{k}}}}}}}}{R})$ and $ j_{{\textrm{b}}, R }^{\, \infty}=  j_{\infty}
 (\frac{-i {\ensuremath{\mathbf{ \nabla_{{\ensuremath{\mathbf{k}}}}}}}}{R})    $ with $\nabla_{{\ensuremath{\mathbf{k}}}}= (\partial_{k^1}, \partial_{k^2},\partial_{k^3} ) $.\
$\;$\
Let $Y_{{\textrm{b}}, R} : {\mathscr{F}_{\textrm{rad}}}\to {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} $ defined by $$Y_{{\textrm{b}}, R} = U_{{\textrm{b}}} \, \Gamma_{{\textrm{b}}}(j_{{\textrm{b}}, R})  .
\notag$$ From (\[3.1.8\]) - (\[3.1.10\]), it follows that $$\begin{aligned}
& Y_{{\textrm{b}}, R} \, {H_{\textrm{rad}, \, m}}= 
\left( {H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}\frac{}{} \right) 
Y_{{\textrm{b}}, R}
-  U_{{\textrm{b}}} \, d \Gamma_{{\textrm{b}}} ( j_{{\textrm{b}}, R} , \tilde{\text{ad}}_{\omega_{m} }(j_{{\textrm{b}},R})) ,  \label{3.3.1} \\
& Y_{{\textrm{b}}, R} \,  a_{r}(h)
= (a_r (h) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R}
+ Y_{{\textrm{b}}, R} \,
 a_{r}((1-j_{{\textrm{b}}, R}^{\, 0} )h) ,   \label{3.3.2} \\
&Y_{{\textrm{b}}, R} \, a^{\dagger}_{r}(h)
= (a^\ast_r (h) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R}
+ \left(a^{\dagger}_r ((j_{{\textrm{b}}, R}^{\, 0}-1) h ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger }(j_{{\textrm{b}}, R}^{\, \infty} h) \right) Y_{{\textrm{b}}, R}  .
  \label{3.3.3}\end{aligned}$$

\[lemma33a\] Assume **(A.2)**. Then $$\begin{aligned}
& \textbf{(i)}
\left\| \left(  Y_{{\textrm{b}}, R} \,  {H_{\textrm{rad}, \, m}}-   (      {H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}) \, Y_{{\textrm{b}}, R} \frac{}{} \right) 
    (N_{{\textrm{rad}}} +1)^{-1} \right\|  \leq  \frac{c_{ \, {\textrm{b}}}}{R}  , \\   
 & \textbf{(ii)} 
\left\| \left(  Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}})
-   (      A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \frac{}{} \right) (N_{{\textrm{rad}}}+1)^{-1/2} \right\|  \leq \delta^j_{ \, {\textrm{b}},R } ({\ensuremath{\mathbf{x}}}) .\end{aligned}$$ Here $c_{ \, {\textrm{b}}} \geq 0$ is a constant and $\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) \geq0  $, $j=1, 2,3$, are error terms which satisfy $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) =0  $ for all $ {\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$.

**(Proof)** **(i)** It is proven in a similar way to Lemma \[lemma32a\] **(i)**.\
**(ii)** By the definition of $ A_{j}({\ensuremath{\mathbf{x}}}) = \sum\limits_{r=1,2} \left( a_{r}(h^j_{r , {\ensuremath{\mathbf{x}}}}) +   a^{\dagger}_{r}(h^j_{r , {\ensuremath{\mathbf{x}}}}) \right)$, it follows from (\[3.3.2\]) and (\[3.3.3\]) that $$\begin{aligned}
&  Y_{{\textrm{b}}, R} \,  A_{j}({\ensuremath{\mathbf{x}}})
-   (      A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  Y_{{\textrm{b}}, R}    \notag \\
&= \sum_{r=1,2} \left( Y_{{\textrm{b}}, R} \,
 a_{r}((1-j_{{\textrm{b}}, R}^{\, 0} )h_{r, {\ensuremath{\mathbf{x}}}}^j)
+ \left(a^{\dagger}_r ((j_{{\textrm{b}}, R}^{\, 0}-1) h_{r, {\ensuremath{\mathbf{x}}}}^j ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger }(j_{{\textrm{b}}, R}^{\, \infty} h_{r, {\ensuremath{\mathbf{x}}}}^j ) \right) Y_{{\textrm{b}}, R} \right)
 .   \notag \end{aligned}$$ Since $\|a_{r}(h) (N_{{\textrm{rad}}} + 1)^{-1/2}  \| \leq \| h \|$ and $\|a^{\dagger }_{r}(h) (N_{{\textrm{rad}}} + 1)^{-1/2}  \| \leq 2 \| h \|$, we have $$\begin{aligned}
& \left\| \left( Y_{{\textrm{b}}, R} \,  A_{j}({\ensuremath{\mathbf{x}}})
-   (      A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  Y_{{\textrm{b}}, R}  \right)  (N_{{\textrm{rad}}} +1)^{-1/2}
 \right\|   \notag \\
 & \leq  \sum_{r=1,2}  \left( 
\| ( a_{r} (1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j)   (N_{{\textrm{rad}}} +1)^{-1/2} \|
 \right. \notag  \\
 & \qquad \qquad \left.  
+ \|  (a^{\dagger}_{r} ((j_{{\textrm{b}}, R}^{\, 0}-1) h_{r,{\ensuremath{\mathbf{x}}}}^j)(N_{{\textrm{rad}}} +1)^{-1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  ( (N_{{\textrm{rad}}} +1)^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})Y_{{\textrm{b}}, R}    (N_{{\textrm{rad}}} +1)^{-1/2}
 \| \right. \notag  \\
&\qquad \qquad \qquad \left.  + \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger} (j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j ) (N_{{\textrm{rad}}} +1)^{-1/2} ) 
( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}(N_{{\textrm{rad}}} +1)^{1/2}  )
 Y_{{\textrm{b}}, R}    (N_{{\textrm{rad}}} +1)^{-1/2}  \|  \frac{}{} \right) \notag \\
& \leq  \sum\limits_{r=1,2}  \left( \| (1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j  \|
+2  \| (j_{{\textrm{b}}, R}^{\, 0}-1)h_{r,{\ensuremath{\mathbf{x}}}}^j \|
+ 2 \| j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j   \| \frac{}{}   \right) . 
  \notag \end{aligned}$$ Let $\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{r=1,2}  \left( 3 \| ((1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j)  \|
+ 2 \| j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j   \| \frac{}{}   \right) $, $j=1,2  ,3$. We see that $  \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} |\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}})| 
\leq 5  \left(   \|h_{1}^j \| + \|h_{2}^j \|\right) $ and $\lim\limits_{R \to \infty}\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Thus we obtain the proof. $\blacksquare$\

Existence of Ground State of $H_{m}$
------------------------------------

We recall that the massive Hamiltonian is defined by $$H_{m} = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}+ {\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}. \notag$$ Throughout this subsection, we do not omit the subscripts of the identities ${{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$ and ${{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}$.\
$\;$\
Since $\frac{1}{\omega_m({\ensuremath{\mathbf{k}}})^{\lambda}} \leq \frac{1}{\omega ({\ensuremath{\mathbf{k}}})^{\lambda}}  $, $\lambda>0$, it holds that $$\|A_j ({\ensuremath{\mathbf{x}}}) ( {H_{\textrm{rad}, \, m}}+1 )^{-1/2}  \| 
\leq \sum_{r=1,2}  \left( 
2 \| \frac{ \chi_{{\textrm{rad}}} e_{r}^{\, j}}{ \omega_{m}} \|  +   \| \frac{ \chi_{{\textrm{rad}}} e_{r}^{\, j}}{ \sqrt{\omega_m}} \| \right) 
\leq  c_{{\textrm{rad}}}^{\, j} . \label{9/1.1}$$ Then, we have $$\| {H_{\textrm{I}}}\Psi \| \leq  c_{\,{\textrm{I}}} \,   \| {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} \Psi \| , 
  \quad  \label{HImbound}$$ and it holds that for all $\epsilon > 0$, $$\|{H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \|H_{0 , m} \Psi \| + c_{\textrm{I}}\left( \frac{1 }{2 \epsilon}  +1 \right)  \| \Psi \|  .
  . \label{HImbound'}$$ From (\[HImbound’\]) and $\| H_{{\textrm{II}}}\| < \infty $, it is proven that $H_{m}$ is self-adjoint and essentially self adjoint on any core of $H_{0,m}$.\

\[Massive-Case\] **(Existence of a Ground State of $H_m $)**\
Suppose **(A.1)** - **(A.3)**. Let $m <M$. Then $H_m $ has purely discrete spectrum in $[ E_{0} (H_m ) , E_{0} (H_m ) + m )$. In particular, $ {H_{m}}$ has a ground state.

$\; $\
To prove Theorem \[Massive-Case\], we need some preparations. We define $ \tilde{X}_{{\textrm{f}}, R } : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ by $$\tilde{X}_{{\textrm{f}}, R } =   X_{{\textrm{f}}, R} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} .$$ We introduce Hamiltonian $\tilde{H}_{m} : {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}\to {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ defined by $$\tilde{H}_{m} = \tilde{H}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\tilde{H}_{{\textrm{rad}}}  + {\kappa_{\textrm{I}}}\tilde{H}_{{\textrm{I}}} + {\kappa_{\textrm{II}}}\tilde{H}_{{\textrm{II}}}  , \notag$$ where $\tilde{H}_{{ \textrm{D} }} =  H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} $, $ \tilde{H}_{{\textrm{rad}}} =  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}  {\otimes}{H_{\textrm{rad}, \, m}}$ and $$\begin{aligned}
&\tilde{H}_{{\textrm{I}}} = \sum_{j=1}^{3} {\int_{\mathbf{R}^{3}}}\chi_{{\textrm{I}}} ({\ensuremath{\mathbf{x}}}) (\tilde{\psi}^{\dagger}({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi}({\ensuremath{\mathbf{x}}} )  {\otimes}A_j ({\ensuremath{\mathbf{x}}} )) d{\ensuremath{\mathbf{x}}} , \\
& \tilde{H}_{{\textrm{II}}} = 
\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
     \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} 
\left( \tilde{\psi}^{\dagger }  ({\ensuremath{\mathbf{x}}}) \tilde{\psi}  ({\ensuremath{\mathbf{x}}}) \tilde{\psi}^{\dagger}   ({\ensuremath{\mathbf{y}}}) \tilde{\psi} ({\ensuremath{\mathbf{y}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right)
 d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}}. 
 \end{aligned}$$ with $\tilde{\psi}({\ensuremath{\mathbf{x}}})=\psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$ and $\tilde{\alpha}^{j} = \alpha^j  {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$, $j=1, \cdots 3$. $\; $\

\[9/9.a\] Assume **(A.1)** - **(A.3)**. Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_{m})$. Then, it holds that $$\begin{aligned}
& \textbf{(i)} \; \;
\left\|  \left(   \tilde{X}_{{\textrm{f}}, R }   ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  )
-   (    \tilde{H}_{{ \textrm{D} }}   {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}   +  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{X}_{{\textrm{f}}, R }  \right) \Psi  \right\|    \notag \\
& \quad \;  \; \; \; \leq  \frac{ c_{ \, {\textrm{f}}}}{R} \, 
 \left( \left\|        (   N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}   ) \Psi \right\|  + \left\| \Psi \right\| \right) , \\
 & \textbf{(ii)} \;  \, \,
 \left\|  \left(  \tilde{X}_{{\textrm{f}}, R }  {H_{\textrm{I}}}-         \tilde{{H_{\textrm{I}}}}    \tilde{X}_{{\textrm{f}}, R } \right) \Psi      \right\| 
 \leq \delta_{\, {\textrm{f}}, {\textrm{I}}} (R )  \,   
 \left( \|  (  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2}  ) \Psi \|
+  \left\|  \Psi \right\| \right), \\
& \textbf{(iii)} \;
\left\|   \left(  \tilde{X}_{{\textrm{f}}, R }  {H_{\textrm{II}}}-         \tilde{{H_{\textrm{II}}}}    \tilde{X}_{{\textrm{f}}, R } \right) \Psi   \right\|
  \leq \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \,  \| \Psi \| .\end{aligned}$$ Here $c_{{\textrm{f}}} \geq 0$ is the constant in Lemma \[lemma32a\]**(i)**, and $\delta_{\, {\textrm{f}}, {\textrm{I}}}(R )  \geq 0 $ and $\delta_{\, {\textrm{f}}, {\textrm{II}}} (R)  \geq 0 $ are error terms satisfying that $\lim\limits_{R \to \infty }\delta_{\, {\textrm{f}}, {\textrm{I}}} (R)  =0  $ and $\lim\limits_{R \to \infty }\delta_{\, {\textrm{f}}, {\textrm{II}}} (R) =0  $, respectively.

**(Proof)**\
**(i)** It directly follows from Lemma \[lemma32a\] (**i**).\
**(ii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m ) $ and $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $. Then, $$\begin{aligned}
& \left( \tilde{\Phi} ,  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, 
-   \tilde{H}_{{\textrm{I}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right)    \notag \\
 & = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left(   \Phi , 
 \left( \left( X_{{\textrm{f}}, R} {\psi^{\dagger} (\mathbf{x}) }({\ensuremath{\mathbf{x}}})  \alpha^j  \psi  ({\ensuremath{\mathbf{x}}}) 
-  \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{x}}})  \tilde{\alpha}^j  \tilde{\psi} ({\ensuremath{\mathbf{x}}}) 
  X_{{\textrm{f}}, R} \right) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})
  \right) \Psi \right) {d \mathbf{x} }. \notag   \end{aligned}$$ Then we have $$\begin{aligned}
& \left| \left( \tilde{\Phi} ,  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, 
-   \tilde{H}_{{\textrm{I}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right)  \right|  \notag \\
 & \leq   \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }}  \left| {\chi_{\textrm{I}} (\mathbf{x}) }\right| \, 
  \left\|  X_{{\textrm{f}}, R} \psi^{\dagger} ({\ensuremath{\mathbf{x}}})  \alpha^j  \psi  ({\ensuremath{\mathbf{x}}}) 
-  \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{x}}})  \tilde{\alpha}^j  \tilde{\psi} ({\ensuremath{\mathbf{x}}}) 
  X_{{\textrm{f}}, R} \right\| \,  \left\| \left(  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})  \right) \Psi \right\| {d \mathbf{x} }\notag \\
& \leq   \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha_{l , l'}^j | \int_{{\mathbf{R}^{3} }}  \left| {\chi_{\textrm{I}} (\mathbf{x}) }\right| \, 
  \|  \left( X_{{\textrm{f}}, R} \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}   \psi_{l'} ({\ensuremath{\mathbf{x}}})
-  \tilde{\psi}_{l} ({\ensuremath{\mathbf{x}}})^{\ast}  \tilde{\psi}_{l'} ({\ensuremath{\mathbf{x}}}) 
  X_{{\textrm{f}}, R} \right) \|  \left\| \left(  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})  \right) \Psi \right\| {d \mathbf{x} }.  \notag\end{aligned}$$ By Corollary \[coro32a\] **(i)**, we have $ \|  \left( 
X_{{\textrm{f}}, R} \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}  \psi_{l'} ({\ensuremath{\mathbf{x}}}) 
-  \tilde{\psi}_{l} ({\ensuremath{\mathbf{x}}})^{\ast}   \tilde{\psi}_{l'} ({\ensuremath{\mathbf{x}}}) 
  X_{{\textrm{f}}, R} 
\right) \| \leq  \delta_{\, {\textrm{f}}, R}^{3, l, l'} ({\ensuremath{\mathbf{x}}}) $. We also see that $ \| A_{j} ({\ensuremath{\mathbf{x}}}) (N_{{\textrm{rad}}} +1)^{-1/2}\| \leq 3 \sum\limits_{r=1,2}\| h_{r}^j\|  $. Then it follows that $$\begin{aligned}
 & \left|  \left( \tilde{\Phi} ,  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, 
-   \tilde{H}_{{\textrm{I}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right)  \right|  \notag \\
&   \quad \qquad \leq     \sum_{r=1,2} \sum_{j=1}^3 \sum_{l,l'=1}^4    |\alpha_{l , l'}^j | \| h_{r}^j  \|
 \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }|
   \delta_{\, {\textrm{f}}, R}^{3, l, l'} ({\ensuremath{\mathbf{x}}}) {d \mathbf{x} }\frac{}{} \right)    \,
\|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}} +1 )^{1/2}) \Psi \|  .  \label{9/9.2}\end{aligned}$$ Since (\[9/9.2\]) holds for all $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $, it follows that $$\left\|  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, 
-   \tilde{H}_{{\textrm{I}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi   \right\|    \leq    \delta_{\, {\textrm{f}}, {\textrm{I}}} (R) \,  
\|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}} +1 )^{1/2}) \Psi \|  ,$$ where $\delta_{\, {\textrm{f}}, {\textrm{I}}} (R)  =  3 \sum\limits_{r=1,2} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 \, |\alpha_{l , l'}^j | \, \| h_{r}^j  \| \, 
  \|  \chi_{{\textrm{I}}}  \, 
   \delta_{\, {\textrm{f}}, R}^{3, l, l'} \|_{L^1} $. We see that $ \lim\limits_{R \to \infty}
 \delta_{\, {\textrm{f}},{\textrm{I}}} (R) =0 $, and hence **(ii)** follows.\
**(iii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$. We set $Q_{ l  }({\ensuremath{\mathbf{x}}})= \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}  \psi_{l} ({\ensuremath{\mathbf{x}}}) $. Then for all $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $, $$\begin{aligned}
& \left( \tilde{\Phi} ,  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, 
-   \tilde{H}_{{\textrm{II}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right)    \notag \\
 & = \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}  \left(   \tilde{\Phi} , 
 \left( ( X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}})  Q_{l' } ({\ensuremath{\mathbf{y}}} )
-   (( {Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l'} ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}   ) 
  X_{{\textrm{f}}, R} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}
  \right) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }. \notag\end{aligned}$$ Then we have $$\begin{aligned}
& \left|  \left( \tilde{\Phi} ,  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, 
-   \tilde{H}_{{\textrm{II}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right)   \right|  \notag \\
 & \leq    \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }}
 \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }| }{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}   
\left\|  \left( X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}})  Q_{l' } ({\ensuremath{\mathbf{y}}} )
-   ( {Q}_{ l }({\ensuremath{\mathbf{x}}})  {Q}_{l' } ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}   ) 
  X_{{\textrm{f}}, R} \right)  \Psi  \right\| 
  {d \mathbf{x} }{d \mathbf{y} }.  \notag \end{aligned}$$ From Corollary \[coro32a\] **(ii)**, it holds that $\left\|   X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}})  Q_{l' } ({\ensuremath{\mathbf{y}}} )
-   ({Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l' } ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}   ) 
  X_{{\textrm{f}}, R}  \right\|  \leq \delta_{{\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}})$. Then we have $$\left|  \left( \tilde{\Phi} ,  \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, 
-   \tilde{H}_{{\textrm{II}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right)   \right|   \leq \left( \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
\frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }| }{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}   \delta_{{\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}})   {d \mathbf{x} }{d \mathbf{y} }\right) \,  \left\|  \Psi  \right\|    . \notag$$ This implies that $$\left\|   \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, 
-   \tilde{H}_{{\textrm{II}}}    \tilde{X}_{{\textrm{f}}, R } 
 \right)  \Psi \right\|   \leq  \delta_{\, {\textrm{f}}, {\textrm{II}}} (R)  \, \left\|  \Psi  \right\| , \notag$$ where $ \delta_{ \, {\textrm{f}}, {\textrm{II}}} (R)  = \sum\limits_{l, l'=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}   \delta_{\, {\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}})   {d \mathbf{x} }{d \mathbf{y} }$. We see that $\lim\limits_{R \to \infty }
\delta_{\, {\textrm{f}}, {\textrm{II}}} (R) =0$, and thus the proof is obtained. $\blacksquare$\
$\;$\
We define $\tilde{Y}_{{\textrm{b}}, R } : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ by $$\tilde{Y}_{{\textrm{b}}, R } = {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}Y_{{\textrm{b}}, R}  .$$

\[9/9.b\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$\begin{aligned}
& \textbf{(i)}
\left\|  \left( \tilde{Y}_{{\textrm{b}}, R } ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}})
-  (  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}   +  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{QED} }} {\otimes}{H_{\textrm{rad}, \, m}})
 \tilde{Y}_{{\textrm{b}}, R }  \right) \Psi   \right\|  \notag \\
 & \qquad \qquad   \leq  \frac{ c_{ \, {\textrm{b}}}}{R}  \,  
  \left( \left\|    ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}   )   \Psi \right\|   + 
 \left\| \Psi \right\|  \right) , \\
 & \textbf{(ii)} 
 \left\|   \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}-   (      {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \tilde{Y}_{{\textrm{b}}, R } 
 \right)  \Psi \right\|  \leq 
  \delta_{ \, {\textrm{b}}, {\textrm{I}}} (R )  \left( 
  \|  ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2}  )  \Psi  \| + \|  \Psi  \|  \right)  ,\end{aligned}$$ where $c_{ \, {\textrm{b}}} \geq 0 $ and $\delta_{ \, {\textrm{b}},{\textrm{I}}} (R)  \geq 0 $ satisfying $\lim\limits_{R \to \infty } 
 \delta_{ \, {\textrm{b}}, {\textrm{I}}} (R)  =0 $.

**(Proof)** **(i)** It immediately follows from Lemma \[lemma33a\] **(i)**.\
**(ii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$ and $\tilde{\Xi} \in  {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Xi} \| =1$. We see that $$\begin{aligned}
& \left( \tilde{\Xi} ,  \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}-   (      {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \tilde{Y}_{{\textrm{rad}}, R } 
 \right)  \Psi \right)    \notag \\
 & = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left(   \tilde{\Xi} , 
\left( {\psi^{\dagger} (\mathbf{x}) }\alpha^j  \psi ({\ensuremath{\mathbf{x}}}) \right)
 {\otimes}\left( Y_{{\textrm{b}}, R }  A_{j} ({\ensuremath{\mathbf{x}}}) \, -   (   A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  )
 Y_{{\textrm{b}}, R } )
 \right)  \Psi \right) {d \mathbf{x} }, \notag .\end{aligned}$$ Then, $$\begin{aligned}
& \left| \left( \tilde{\Xi} ,  \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}-   (      {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \tilde{Y}_{{\textrm{b}}, R } 
 \right)  \Psi \right)  \right|    \notag \\
 & \leq  \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \,   
\left\| (  {\psi^{\dagger} (\mathbf{x}) }\alpha^j  \psi ({\ensuremath{\mathbf{x}}}) {\otimes}\left( Y_{{\textrm{b}}, R }  A_{j}  \, -   (   A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  )
 Y_{{\textrm{b}}, R } )  \right)  \Psi  \right\| {d \mathbf{x} }\notag \\
&  \leq \left(  \sum_{j=1}^3 \sum_{l,l'=1 }^4 |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l} c_{\, { \textrm{D} }}^{\, l'} \right)
\int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }|  \left\|
 \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\left( Y_{{\textrm{b}}, R }  A_{j}  \, -   (   A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  )
 Y_{{\textrm{b}}, R } ) \right)  \right)  
\Psi  \right\| {d \mathbf{x} }.  \notag \end{aligned}$$ From Lemma \[lemma33a\] **(ii)**, it holds that $$\left\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\left( Y_{{\textrm{b}}, R }  A_{j}  \, -   (   A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  )  Y_{{\textrm{b}}, R } )  \right)  \Psi  \right\|
\leq  \delta_{\, {\textrm{b}},R}^j ({\ensuremath{\mathbf{x}}}) \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2})  \Psi   \| , \notag$$ where $ \delta_{{\textrm{b}}, R}^j ({\ensuremath{\mathbf{x}}}) \geq 0 $ is the error term, and hence, $$\left| \left( \tilde{\Xi} ,  \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}-   (      {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \tilde{Y}_{{\textrm{b}}, R } 
 \right)  \Psi \right)  \right| \leq  \delta_{\, {\textrm{b}}, {\textrm{I}}} (R) \, \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2})  \Psi   \| , \label{9/9.3}$$ where $\delta_{\, {\textrm{b}},{\textrm{I}}} (R)   = \sum\limits_{l,l'=1 }^4 |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l} c_{\, { \textrm{D} }}^{\, l'} \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }|  \delta^j_{\, {\textrm{b}}} ({\ensuremath{\mathbf{x}}})  dx  $. Since (\[9/9.3\]) holds for all $\tilde{\Xi} \in  {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Xi} \| =1$, we have $$\left\|   \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}-   (      {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \tilde{Y}_{{\textrm{rad}}, R } 
 \right)  \Psi  \right\| \leq  \delta_{\, {\textrm{b}}, {\textrm{I}}} (R) \, \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2})  \Psi   \| . \notag$$ Since $\lim\limits_{R \to \infty }\delta_{\, {\textrm{b}}, {\textrm{I}}}(R)=0$, the proof is obtained. $\blacksquare$\
$\; $\
Here we introduce a new norm defined by $$\qquad 
 \|  \Psi \|_{ \lambda  , \, \lambda  '  } \; = \; \|(  N_{{ \textrm{D} }}^{\, \lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi  \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{ \, \lambda ' /2}  )\Psi  \| + \| \Psi \| , \quad \Psi \in {\ensuremath{\mathscr{D}}} (N_{{ \textrm{D} }}^{\lambda  /2 } {\otimes}N_{{\textrm{rad}}}^{ \lambda  ' /2}  ) .$$

$\;$\
From Proposition \[9/9.a\] and Proposition \[9/9.b\], the next corollary follows. $\;$\

\[9/9.c\] Assume **(A.1)** - **(A.3)**. Then for all $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$, $$\begin{aligned}
&\textbf{(i)} \;  \left\|  \left( \tilde{X}_{{\textrm{f}}, R } H_{m}
-  ( \tilde{H}_{m}   +  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )
 \tilde{X}_{{\textrm{f}}, R}   \right) \Psi  \right\| 
\leq \delta_{ \, {\textrm{f}}} (R)  \| \Psi \|_{2,1 } ,  \\
&\textbf{(ii)} \; \left\|  \left( \tilde{Y}_{{\textrm{b}}, R } H_{m}
-  ( H_{m} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}   +  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{H_{\textrm{rad}, \, m}})
 \tilde{Y}_{{\textrm{b}}, R }  \right) \Psi \right\| \leq \delta_{ \,{\textrm{b}}} (R)
  \| \Psi \|_{0, 2} \, .\end{aligned}$$ Here $ \delta_{ \, {\textrm{f}}} (R) \geq 0 $ and $ \delta_{ \, {\textrm{b}}} (R) \geq 0 $ are error terms which satisfy that $ \lim\limits_{R \to \infty} \delta_{ \, {\textrm{f}}} (R)=0 $ and $ \lim\limits_{R \to \infty} \delta_{ \, {\textrm{b}}} (R) =0 $, respectively.\

\[9/9.d\] \[LformboundHm\] Assume **(A.1)** - **(A.3)**. Let $q_{\, {\textrm{f}}, R} = (j^{\, 0}_{\, {\textrm{f}}, R})^2$ and $q_{\, {\textrm{b}}, R} = (j^{\, 0}_{\, {\textrm{b}}, R})^2$. Then, for all $\Psi \in 
{\ensuremath{\mathscr{ D}}} (H_m ) $ with $\| \Psi \| =1$, $$\begin{aligned}
 (\Psi, H_m \Psi )     \geq & E_{0} (H_m)  + \, m  \,  +  \,   (M-m)  
 \left( \Psi,   ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\Gamma_{{\textrm{b}}} (q_{\, {\textrm{b}}, R}) )  \Psi \right) \\
&    - M \left( \Psi ,  
 \left( \Gamma_{{\textrm{f}}} (q_{\, {\textrm{f}}, R}) {\otimes}\Gamma_{{\textrm{b}}} (q_{\, {\textrm{b}}, R}) \right) \Psi \right) +  \left(  \delta_{\, {\textrm{f}}} (R ) \|  \Psi  \|_{2,1 } + \delta_{\, {\textrm{b}}} (R )  \|  \Psi \|_{0,2 }  \right) .
 \end{aligned}$$

**(Proof)** Let $\Psi \in {\ensuremath{\mathscr{ D}}} (H_m ) $ with $\| \Psi \| =1$. By Lemma Corollary \[9/9.c\] **(ii)**, $$\begin{aligned}
 (\Psi , H_m \Psi )
& = \left(  \Psi , 
 \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{Y}_{\, {\textrm{b}},R} H_m  \Psi \right) \notag \\
&  \geq ( \Psi ,  \tilde{Y}_{\, {\textrm{b}},R}^{\ast}  (  H_m  {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}})
   \tilde{Y}_{\, {\textrm{b}},R}   \Psi )  + ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast}
 ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  {\otimes}{H_{\textrm{rad}, \, m}})
   \tilde{Y}_{\, {\textrm{b}},R}   \Psi )  - \delta_{\, {\textrm{b}}} (R)  \| \Psi \|_{0,2} . \notag   
   \end{aligned}$$ We see that $ {H_{\textrm{rad}, \, m}}\geq m (  {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {P_{\textrm{rad}}}) $ with $ {P_{\textrm{rad}}}= E_{N_{{\textrm{rad}}}}(\{ 0\})$ where $E_{X}(J)$ denotes the spectral projection on a Borel set $J \in {\ensuremath{\mathscr{B}}}({\ensuremath{\mathbf{R}}})$ for a self-adjoint operator $X$. Then $$\begin{aligned}
(\Psi , H_m \Psi ) &  \geq 
(   \Psi ,  \tilde{Y}_{\, {\textrm{rad}},R}^{\ast}  (  H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  )
   \tilde{Y}_{\, {\textrm{rad}},R}  \Psi )  + m   \notag \\
&   \qquad \qquad \quad  - m (  \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^\ast ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  {\otimes}{P_{\textrm{rad}}})
  \tilde{Y}_{\, {\textrm{rad}},R}  \Psi )  - \delta_{\, {\textrm{b}}} (R)  \| \Psi \|_{0,2 }   \notag \\
&  \geq (  \Psi ,   \tilde{Y}_{\, {\textrm{b}},R}^{\ast} (  H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )
   \tilde{Y}_{\, {\textrm{b}},R}   \Psi )  + m-m (\Psi , ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}( q_{\, {\textrm{b}}, R})  \Psi ) - \delta_{\, {\textrm{b}}} (R)  \| \Psi \|_{0,2 }  . \label{9/10.1}\end{aligned}$$ Here we used $ 
 {Y}_{\, {\textrm{b}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{P_{\textrm{rad}}})
   {Y}_{\, {\textrm{b}},R}  =    {\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) $ in the last line. We evaluate the first term in the right hand side of (\[9/10.1\]). Let $\tilde{\tilde{X}}_{{\textrm{f}},R}= \tilde{X}_{\, {\textrm{f}},R} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}$. By Corollary \[9/9.c\] **(i)**, $$\begin{aligned}
 & (  \Psi ,  \tilde{Y}_{\, {\textrm{b}},R}^{\ast}   (  H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )
  \tilde{Y}_{\, {\textrm{b}},R}   \Psi ) \notag \\
& = \left(   \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast}
    ( ( \tilde{X}_{\, {\textrm{f}},R}^{\ast} \tilde{X}_{\, {\textrm{f}},R} H_m)    {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )  
  \tilde{Y}_{\, {\textrm{b}},R}   \Psi ) \right)  \notag \\
&\geq    \left(    \Psi , 
\tilde{Y}_{\, {\textrm{rad}},R}^{\ast} 
 \tilde{\tilde{X}}^{\ast}_{\, {\textrm{f}},R} (    \tilde{H}_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}})
 \tilde{\tilde{X}}_{\, {\textrm{f}},R}  \tilde{Y}_{\, {\textrm{b}},R}   \Psi \right)  \notag  \\
& \qquad  \qquad \qquad  \quad + \left(  \Psi_n ,  
 \tilde{Y}_{\, {\textrm{b}},R}^{\ast} 
 \tilde{\tilde{X}}^{\ast}_{\, {\textrm{f}},R} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )
  \tilde{\tilde{X}}_{\, {\textrm{b}},R}  \tilde{Y}_{ {\textrm{b}},R} \Psi \right) 
 - \delta_{\, {\textrm{f}}} (R) \| \tilde{Y}_{  {\textrm{b}},R} \Psi \|_{2,1}^{\sim }, \label{9/9.3}\end{aligned}$$ where we set $$\| \tilde{ \Phi}  \|_{ \lambda , \lambda '  }^{\sim} \; = \; \|(  N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )   \tilde{ \Phi} \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{ \Phi} \| + \| \tilde{ \Phi} \| ,$$ for $  \tilde{ \Phi} \in {\ensuremath{\mathscr{D}}} ( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}})  \cap {\ensuremath{\mathscr{D}}}({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) $. We see that $$\begin{aligned}
\| \tilde{Y}_{  {\textrm{b}},R} \Psi  \|_{ 2,1 }^{\sim} & = 
 \|  \tilde{Y}_{  {\textrm{b}},R} ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )  \Psi  \|
+ \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{  {\textrm{b}},R} \Psi \| + \| \tilde{Y}_{  {\textrm{b}},R} \Psi  \| \\
 &  \leq \|  \tilde{Y}_{  {\textrm{b}},R} ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )  \Psi  \|  + \|\tilde{Y}_{  {\textrm{b}},R} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2}  )  \Psi \|+ \| \tilde{Y}_{  {\textrm{b}},R} \Psi  \|
=  \| \Psi \|_{2,1} ,\end{aligned}$$ and $ H_{{ \textrm{D} }} \geq M (  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}- P_{{ \textrm{D} }}) $ with $ P_{{ \textrm{D} }} =  E_{N_{{ \textrm{D} }}}(\{ 0 \})$. Then we have $$\begin{aligned}
(\ref{9/9.3})& 
\geq  E_{0} (\tilde{H}_m ) +   M   - M 
 \left(    \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{\tilde{X}}_{\, {\textrm{f}},R}^{\ast}  ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{P_{\Omega_{\textrm{D}}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} )    \tilde{\tilde{X}}_{\, {\textrm{f}},R}  \tilde{Y}_{\, {\textrm{b}},R}   \Psi \right)  - \delta_{\, {\textrm{f}}} (R) \|  \Psi \|_{ 2,1} 
  \notag \\
  & \geq E_{0} (H_m ) +   M - M ( \Psi , ( {\Gamma_{\textrm{f}}}( q_{ \, {\textrm{f}}, R} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \Psi )
  - \delta_{\, {\textrm{f}}, m } (R) \|  \Psi \|_{2,1} . \notag\end{aligned}$$ Here we used $  E_{0} (\tilde{H}_m)= E_{0} (H_m)$ and $   {X}_{\, {\textrm{f}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}P_{{ \textrm{D} }} )
   {X}_{\, {\textrm{f}},R}  =    {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}}, R} ) $ in the last line. Thus we have $$\begin{aligned}
(\Psi,  H_m \Psi )   & \geq E_{0} (H_m ) + m + M 
-  M \left( \Psi , ( {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}  ) \Psi \right) \notag  \\
& \qquad \qquad  -m \left( \Psi , \left(  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R}  \right) \right)  \Psi \right)
 -\delta_{\, {\textrm{b}}} (R) \|  \Psi \|_{0,2} -
 \delta_{\, {\textrm{f}}} (R) \|  \Psi \|_{2,1} .\end{aligned}$$ Note that $$\begin{aligned}
 {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}    & \geq {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}+  ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -  {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R}  \right)  )   {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right)  \notag \\
& = {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}+ {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right)
  -  {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R}  \right)     {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right)  . \notag\end{aligned}$$ Then we have $$\begin{aligned}
 (\Psi, H_m \Psi )    &  \geq E_{0} (H_m)  + \, m  \,  +  \,  
(M-m)   \left( \Psi,   ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}   {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R}  \right) ) \Psi \right)  \\
& \quad- M \left( \Psi ,  ({\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R}  \right) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R}  \right) ) \Psi \right) -\delta_{\, {\textrm{b}}} (R) \|  \Psi \|_{0,2} -
 \delta_{\, {\textrm{f}}} (R) \|  \Psi \|_{2,1} .\end{aligned}$$ Thus the proof is obtained. $\blacksquare$\

\[9/9.e\] Assume **(A.1)** - **(A.3)**. Then for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\qquad  \qquad 
\| H_{0, m} \Psi \| \leq  L_{\epsilon} \| H_{m}  \Psi \|  +   R_{\epsilon} \|  \Psi \|  , \qquad  \Psi \in {\ensuremath{\mathscr{D}}}(H_m )  ,  \notag$$ where $L_{\epsilon} = \frac{1}{1- c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|\epsilon} $ and $R_{\epsilon}= \frac{1}{1- c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|\epsilon}\left( c_{\textrm{I}}|{\kappa_{\textrm{I}}}| \, ( \frac{1 }{2 \epsilon} +1 )  + |{\kappa_{\textrm{II}}}| \, \|  {H_{\textrm{II}}}\| \right)$.

**(Proof)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$. Since $H_{0,m} = H_m - {\kappa_{\textrm{I}}}{H_{\textrm{I}}}- {\kappa_{\textrm{II}}}{H_{\textrm{II}}}$, we see that $$\|H_{0,m} \Psi  \|  \leq  \| H_m \Psi \| + \left| {\kappa_{\textrm{I}}}\right| \, \left|  {H_{\textrm{I}}}\Psi \right\|
 + \left| {\kappa_{\textrm{II}}}\right| \, \,  \| {H_{\textrm{II}}}\| \, \| \Psi \| . \notag$$ From (\[HImbound’\]), it holds that $ \| {H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon   \| H_{0 , m } \Psi  \|  + c_{{\textrm{I}}} ( 
\frac{1 }{2 \epsilon} +1 ) \| \Psi \| $ for all $\epsilon > 0$. Hence $$(1- c_{\, {\textrm{I}}} |{\kappa_{\textrm{I}}}| \epsilon  )\| H_{0,m} \Psi  \|  \leq \| H_m \Psi   \|
+ \left( c_{\textrm{I}}|{\kappa_{\textrm{I}}}| \, \left( \frac{1 }{2 \epsilon} +1 \right)   +  | {\kappa_{\textrm{II}}}| \, \| {H_{\textrm{II}}}\|  \,  \right)
\| \Psi \| .$$ Taking $\epsilon >0$ such that $ \epsilon < \frac{1}{c_{\, {\textrm{I}}}|{\kappa_{\textrm{I}}}|}$, we obtain the proof. $\blacksquare $

$\;$\
Since $\| N_{{ \textrm{D} }} \Psi \| \leq  \frac{1}{M} \| H_{{ \textrm{D} }} \Psi  \|$, $\Psi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$, and $\| N_{{\textrm{rad}}} \Phi \| \leq  \frac{1}{m} \| H_{{\textrm{rad}}} \Phi  \|$, $\Phi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}}})$, the next corollary follows from Lemma \[9/9.e\].\

\[9/9.f\] Assume **(A.1)** - **(A.3)**. Then for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$\begin{aligned}
& \textbf{(i)} \; \| (N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| \leq  
\frac{L_{\epsilon}}{M} \|H_{m} \Psi  \|  + \frac{R_\epsilon}{M} \| \Psi \| ,     \\
& \textbf{(ii)} \;\| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi \| \leq 
 \frac{L_{\epsilon }}{m} \|H_{m} \Psi  \|  + \frac{R_\epsilon}{m} \| \Psi \|  . \end{aligned}$$

$\;$\
[**(Proof of Theorem \[Massive-Case\] )**]{}\
It is enough to show that $ \sigma_{{ \textrm{ess} }} (H_m )  \subset [E_{0} (H_m ) + m , \infty )$. Let $\lambda \in \sigma_{{ \textrm{ess} }} (H_m )$. Then by the Weyl’s theorem, there exists a sequence $\{ \Psi_n \}_{n=1}^{\infty}$ of ${\ensuremath{\mathscr{D}}}(H_m )$ such that (i) $\| \Psi_n \| =1$, $n \in {\ensuremath{\mathbf{N}}}$, (ii) s-$\lim\limits_{ n \to \infty} (H_m -\lambda ) \Psi_n =0 $, and (iii) w-$\lim\limits_{n \to \infty} \Psi_n = 0$. Since $ | \lambda- (\Psi_n , H_m \Psi_n ) | \leq |  ( \Psi_n , ( H_m- \lambda  )\Psi_n |
 \leq \|  ( H_m- \lambda  )\Psi_n \| $, it holds that $  \lambda = \lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) $. Here we show that $$\lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) \geq  E_{0} ({H_{m}}) + m  , \notag$$ and then, the proof is obtained. Let $m \leq M$. From Lemma \[LformboundHm\], $$\begin{aligned}
 ( \Psi_{n} ,H_m \Psi_{n} ) \geq  & E_{0} (H_m)  + m - M  (\Psi_n , ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) ,\Psi_n  )  \notag \\
 &  \qquad \qquad \qquad \quad  \;\;- \delta_{\, {\textrm{f}}} (R) \|  \Psi_n \|_{2,1} -\delta_{\, {\textrm{b}}, m } (R) \|  \Psi_n \|_{0,2 }. \notag
 \end{aligned}$$ Since s-$\lim\limits_{ n \to \infty} (H_m -\lambda ) \Psi_n =0 $, we can set $$E_{m} = \sup_{n \in {\ensuremath{\mathbf{N}}}} \| H_m \Psi_n  \| < \infty .$$ Let $0 \leq  \lambda \leq 2 $ and $0 \leq  \lambda' \leq 2$. From Corollary \[9/9.f\], it is seen that for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\begin{aligned}
\| \Psi_{n} \|_{\lambda ,\lambda ' }
& = \|( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_n \|    +  \|(  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' } ) \Psi_n \| + \| \Psi_n \| \notag  \\
& \leq \|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_n \|    +  \|(  {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi_n \| + 3
\| \Psi_n \| \notag \\
& \leq (\frac{1}{M} + \frac{1}{m} ) \left( L_{\epsilon }  \|H_{m} \Psi_n \|  + 2R_{\epsilon} \right) +3\| \Psi_n \|  \notag \\ 
& \leq E_{m} L_{\epsilon} (\frac{1}{M} + \frac{1}{m} ) +   2 (\frac{1}{M} + \frac{1}{m} )R_{\epsilon} +3  . \notag\end{aligned}$$ Then we have $$( \Psi_{n} ,H_m \Psi_{n} ) \geq   E_{0} (H_m)  + m - M  (\Psi_n , ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) ,\Psi_n  )  - \delta_{m, \epsilon} (R) , \label{9/9.6}$$ where $\delta_{m , \epsilon} (R) =  c_{\, m , \epsilon}  ( \delta_{\, {\textrm{b}}} (R) +  \delta_{ \, {\textrm{f}}} (R) ) $ with $c_{\, m , \epsilon }  = E_{m} L_{\epsilon} (\frac{1}{M} + \frac{1}{m} ) +   2 (\frac{1}{M} + \frac{1}{m} )R_{\epsilon} +3$. It is seen that $$\begin{aligned}
 &\left| ( \Psi_n , ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n  ) \right| \notag \\
& \qquad \qquad  \leq \| (H_{0,m} +1)^{1/2} \Psi_n  \| \,  \|
(H_{0,m} +1)^{-1/2}  ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n   \| .  \label{9/16.1}\end{aligned}$$ From Lemma \[9/9.e\], we see that $$\| (H_{0,m} +1)^{1/2} \Psi_n \| \leq \| H_{0,m} \Psi_n \| + \|\Psi_n  \|  \leq  L_\epsilon \| H_{m} \Psi_n \| + ( R_\epsilon +1)  \|\Psi_n  \| = E_{0} (H_m) L_{\epsilon} + R_\epsilon +1 ,$$ and hence, $$\sup\limits_{n \in {\ensuremath{\mathbf{N}}}} \| (H_{0,m} +1)^{1/2} \Psi_n \| \leq E_{m} L_\epsilon  +  R_\epsilon +1 .  \label{9/16.2}$$ It holds that $$\begin{aligned}
   (H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) = 
  & (H_{0,m} +1)^{-1/2} ( (H_{{ \textrm{D} }} + 1)^{1/2} {\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} ) \notag \\
  & \;  \times ( (H_{{ \textrm{D} }} + 1)^{-1/2}  {\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R} )  ) {\otimes}( ({H_{\textrm{rad}, \, m}}+1)^{-1/2} {\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}},R} ) )  ) ,
    \notag\end{aligned}$$ and hence, $(H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) )$ is compact, since $\| (H_{0,m} +1)^{-1/2} ( (H_{{ \textrm{D} }} + 1)^{1/2} {\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} )\| \leq 1$ and $ \left(  (H_{{ \textrm{D} }} + 1)^{-1/2}{\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R})  \right) {\otimes}\left(   ({H_{\textrm{rad}, \, m}}+1)^{-1/2} {\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}}, R}) \right)$ is compact. Therefore it holds that $$\lim_{n \to \infty}  \left\| (H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n \right\|=0. \label{9/16.3}$$ From (\[9/16.1\]) - (\[9/16.3\]) we have $
\lim\limits_{n \to \infty}\left| \left( \Psi_n , ({\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R}) {\otimes}{\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}},R}) ) ,\Psi_n  \right) \right|
 =0 $. Then by taking the limit of (\[9/9.6\]) as $R \to \infty$, we have $ \lim\limits_{n \to \infty} (\Psi_n , H_m  \Psi_n ) \geq  E_{0}(H_m ) + m $. $\blacksquare$\

Derivative Bounds
=================

From Theorem \[Massive-Case\], $H_{m}$ has the ground state. Let $\Psi_{m}$ be the normalized ground state of $H_{m}$, i.e. $$\qquad  \qquad H_{m} \Psi_m = E_{0}(H_m ) \Psi_m , \quad \| \Psi_m  \| = 1.  \notag$$

Electron-Positron Derivative Bounds
-----------------------------------

We introduce the distribution kernel of the annihilation operator for the Dirac field. For all $ \Psi = \left\{  \Psi^{(n)} = {}^{t}\left( \Psi^{(n)}_1 , \cdots ,  \Psi^{(n)}_4  \right)  \right\}_{n=0}^{\infty}
\in {\ensuremath{\mathscr{D}}} ( H_{{ \textrm{D} }} )$, we set $$\qquad 
C_{l}({\ensuremath{\mathbf{p}}})\Psi^{(n , \nu )}( {\ensuremath{\mathbf{p}}}_{1} , \cdots ,  {\ensuremath{\mathbf{p}}}_{n} )
= \delta_{\, l , \nu } \sqrt{n+1} \Psi^{(n+1 , \nu )}( {\ensuremath{\mathbf{p}}} , {\ensuremath{\mathbf{p}}}_{1} , \cdots ,  {\ensuremath{\mathbf{p}}}_{n} ) . \quad l=1 ,\cdots 4 .$$ Let $$b_{1/2} ({\ensuremath{\mathbf{p}}}) = C_{1} ({\ensuremath{\mathbf{p}}}), \; \; b_{-1/2} ({\ensuremath{\mathbf{p}}}) = C_{2} ({\ensuremath{\mathbf{p}}}) , \;
  \;   d_{1/2} ({\ensuremath{\mathbf{p}}}) = C_{3} ({\ensuremath{\mathbf{p}}}), \; \;   d_{-1/2} ({\ensuremath{\mathbf{p}}}) = C_{4} ({\ensuremath{\mathbf{p}}})  .$$ $\;$\
Then it follows that for all $\Phi \in {\mathscr{F}_{\textrm{Dirac}}}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} )$, $$\begin{aligned}
\qquad \quad & ( \Phi ,  b_{s} (f)  \Psi ) \, = \,  \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} ( \Phi ,  b_{s} ({\ensuremath{\mathbf{p}}})  \Psi ) d {\ensuremath{\mathbf{p}}}  , \quad  \quad   \; f  \in {\ensuremath{\mathscr{D}}}(\omega_{\, M}) , \\
& ( \Phi ,  d_{s} (g)  \Psi ) \, = \,  \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} ( \Phi ,  d_{s} ({\ensuremath{\mathbf{p}}})  \Psi ) d {\ensuremath{\mathbf{p}}} , \quad  \quad   \;  g \in {\ensuremath{\mathscr{D}}}(\omega_{\, M})  .
 \end{aligned}$$ The number operator for electrons and positrons are defined by $$N_{{ \textrm{D} }}^{+} = d {\Gamma_{\textrm{f}}}\left( \left( \begin{array}{cc} {{\small \text{1}}\hspace{-0.32em}1}& O \\ O& O \end{array} \right) \right) , \qquad N_{{ \textrm{D} }}^{-} = d {\Gamma_{\textrm{f}}}\left( \left( \begin{array}{cc} O & O \\ O& {{\small \text{1}}\hspace{-0.32em}1}\end{array} \right) \right) ,$$ respectively. It holds that for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} )$, $$\begin{aligned}
&(\Phi , N_{{ \textrm{D} }}^{+}  \Psi ) = \sum_{\pm1/2}\int_{{\mathbf{R}^{3} }}( b_{s}({\ensuremath{\mathbf{p}}})  \Phi ,  b_{s}({\ensuremath{\mathbf{p}}}) \Psi) d {\ensuremath{\mathbf{p}}}  , \notag \\
&(\Phi , N_{{ \textrm{D} }}^{-}  \Psi ) = \sum_{\pm1/2} \int_{{\mathbf{R}^{3} }}( d_{s}({\ensuremath{\mathbf{p}}})  \Phi ,  d_{s}({\ensuremath{\mathbf{p}}}) \Psi) d {\ensuremath{\mathbf{p}}} \notag  .\end{aligned}$$

$\; $\
By the canonical anti-commutation relation, it is proven in ([@Ta09] ; Section III) that $$\begin{aligned}
&[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}})  }, b_{s}(f ) ] = - \sum_{l,l'=1}^4 \alpha_{l,l'}^{j}
( f , f_{s,{\ensuremath{\mathbf{x}}}}^{l}  ) \;  \psi_{l'}({\ensuremath{\mathbf{x}}}) ,  \label{ori1}
\\
&[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}})  }, d_{s}(g ) ] = \sum_{l,l'=1}^4 \alpha_{l,l'}^{j}
( g ,g_{s ,{\ensuremath{\mathbf{x}}}}^{l'}  ) \; \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} ,  \label{ori2}
 \end{aligned}$$ and for $\rho({\ensuremath{\mathbf{x}}})={\psi^{\dagger} (\mathbf{x}) }\psi ({\ensuremath{\mathbf{x}}})$, $$\begin{aligned}
&[ \rho({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}), \; b_{s}(f  ) ] 
= -\sum_{l=1}^4 \left( ( f  , f_{s, {\ensuremath{\mathbf{y}}}}^{l}  )
\rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) 
+ ( f , f_{s, {\ensuremath{\mathbf{x}}}}^{l}  ) \; \psi_{l}({\ensuremath{\mathbf{x}}})
 \rho ({\ensuremath{\mathbf{y}}}) \right) ,   \label{ori3}  \\
&[ \rho ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}), \; d_{s}( g ) ] 
= \sum_{l=1}^4 \left( ( g , g_{s, {\ensuremath{\mathbf{y}}}}^{l}  )
\rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) ^{\ast}
+ (g , g_{s, {\ensuremath{\mathbf{x}}}}^{l}  ) \;  \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}
 \rho ({\ensuremath{\mathbf{y}}}) \right) . \label{ori4}\end{aligned}$$

$\; $\
Let $X$ and $Y$ be operators on a Hilbert space. The weak commutator is defined by $$[X, \,Y ]^0 (\Phi , \Psi ) = (X^{\ast}\Phi , Y \Psi  ) - (Y^{\ast}\Phi , X \Psi  ) ,$$ where $ \Psi \in {\ensuremath{\mathscr{D}}}(X) \cap {\ensuremath{\mathscr{D}}} ( Y) $ and $ \Phi \in {\ensuremath{\mathscr{D}}}(X^{\ast}) \cap {\ensuremath{\mathscr{D}}} ( Y^{\ast}) $.\

\[9/12.a\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $f \in L^2 ({\mathbf{R}^{3} })$, $$\begin{aligned}
\textbf{(i)} & \; \;  [{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi )  \,
= \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{+} ({\ensuremath{\mathbf{p}}})   \Psi  \right)
 d {\ensuremath{\mathbf{p}}} ,  \quad \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} ,\; \Psi \in {\ensuremath{\mathscr{D}}}(H_m ) ,  \\
 \textbf{(ii)} & \; \;
[{H_{\textrm{II}}}, b_{s}(f){\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi)  \,
= \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , (S_s^{\, +} ({\ensuremath{\mathbf{p}}}) + T_s^{+} ({\ensuremath{\mathbf{p}}}) )    \Psi  \right) d {\ensuremath{\mathbf{p}}} ,  \quad \Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}  . \end{aligned}$$ Here $ K_s^{+}({\ensuremath{\mathbf{p}}})$, $S_s^{\, +} ({\ensuremath{\mathbf{p}}})   $ and $T_s^{+} ({\ensuremath{\mathbf{p}}})   $ are operators which satisfy $$\begin{aligned}
& (\Phi , K_s^{+}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, 
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) 
\left( \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi \right) {d \mathbf{x} }, \notag \\
 & (\Phi , S^{+}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{l=1}^4  \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
\frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} f_{s , {\ensuremath{\mathbf{y}}} }^{\, l}({\ensuremath{\mathbf{p}}}) 
\left( \Phi , \,  ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi \right)  {d \mathbf{x} }{d \mathbf{y} }, 
\notag \\ 
&(\Phi , T^{+}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
\frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}  f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}})  
\left( \Phi , \,  (\psi_{l}({\ensuremath{\mathbf{x}}})  \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi \right)  {d \mathbf{x} }{d \mathbf{y} }.
\notag\end{aligned}$$

**(Proof)**\
**(i)** Let $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$. By (\[ori1\]), we have $$\begin{aligned}
[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi)  
& =\sum_{j=1}^3 \int_{{\mathbf{R}^{3} }}
 {\chi_{\textrm{I}} (\mathbf{x}) }[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) , b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 
( \Phi , \Psi )   {d \mathbf{x} }\notag \\
 & = \sum_{j=1}^3\int_{{\mathbf{R}^{3} }}
{\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , \left( [ {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}} ), b_{s}(f) ]   {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right)   \Psi \right) {d \mathbf{x} }\notag  \\
& = -  \sum_{ j=1}^3  \sum_{ l, l'=1}^4 \alpha_{l , l'}^j \int_{{\mathbf{R}^{3} }}  {\chi_{\textrm{I}} (\mathbf{x}) }\, (f,  f_{s, {\ensuremath{\mathbf{x}}}}^{\, l})  \left( \Phi , ( \psi_{l'} ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})) \Psi \right)  {d \mathbf{x} }.  \notag \end{aligned}$$ Let $\ell_{s, {\ensuremath{\mathbf{p}}}} : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times  {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\ensuremath{\mathbf{C}}}  $ be a functional defined by $$\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ') = 
 - \sum_{j=1}^3 \sum_{l, l'=1}^4 \alpha_{l , l'}^j  \int_{{\mathbf{R}^{3} }}  {\chi_{\textrm{I}} (\mathbf{x}) }\,   f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}})  \left( \Phi ' , ( \psi_{l'} ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})) \Psi '\right)  {d \mathbf{x} },$$ for $  \Phi ' \in {\mathscr{F}_{\textrm{QED}}}, \Psi ' \in {\ensuremath{\mathscr{D}}}(H_{0, m} )$. We see that $$\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' )  \leq 
c_{ \, {\textrm{I}}, \, s, {\ensuremath{\mathbf{p}}}}  \| \Phi ' \|  \,  \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}( {H_{\textrm{rad}, \, m}}+1)^{1/2}) \Psi ' \| ,$$ where $c_{\, {\textrm{I}}, \, s, {\ensuremath{\mathbf{p}}}} = \sum\limits_{j=1}^3 \sum\limits_{l, l'=1}^4 | \alpha_{l , l'}^j| \, \| \chi_{\, {\textrm{I}}}  \|_{L^1} \, |f_{s}^{\, l}({\ensuremath{\mathbf{p}}})|c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} $. Then from the Riesz Representation theorem, we can define an operator $ K_{s}^{\, +} ({\ensuremath{\mathbf{p}}})$ which satisfy $\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ', \Psi ' ) = (\Phi ' , K_{s}^{\, +} ({\ensuremath{\mathbf{p}}}) \Psi' )   $. Then it holds that $$[{H_{\textrm{I}}}, b_{s}(f) ]^0 (\Phi , \Psi) =  \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast}
 \ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) 
 d {\ensuremath{\mathbf{p}}} = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{+} ({\ensuremath{\mathbf{p}}})   \Psi  \right)  d {\ensuremath{\mathbf{p}}} .$$ **(ii)** From (\[ori3\]), we see that for all $\Phi , \Psi  \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$, $$\begin{aligned}
[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi)  & = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } [ \rho ({\ensuremath{\mathbf{x}}} ) 
\rho ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi ,\Psi ) {d \mathbf{x} }{d \mathbf{y} }\notag \\
& =\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | }
\left( \Phi , \left( [  \rho ({\ensuremath{\mathbf{x}}} )  \rho ({\ensuremath{\mathbf{y}}} ) , b_{s}(f)  ]  {\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right)  \Psi \right) {d \mathbf{x} }{d \mathbf{y} }\notag  \\ 
& = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } \left\{ 
( f  , f_{s, {\ensuremath{\mathbf{y}}}}^{\, l}  ) ( \Phi , 
( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l} ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) \right. \notag \\
& \qquad \qquad \qquad \qquad   \left. + ( f , f_{s, {\ensuremath{\mathbf{x}}}}^{\, l}  ) (\Phi ,  \, 
 ( \psi_{l} ({\ensuremath{\mathbf{x}}})  \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi ) \right\} {d \mathbf{x} }{d \mathbf{y} }.  \notag \end{aligned}$$ We set functionals $q_{s, {\ensuremath{\mathbf{p}}}} $ and $r_{s,{\ensuremath{\mathbf{p}}}}  $ on ${\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times
 {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ by $$\begin{aligned}
&\quad  q_{s, {\ensuremath{\mathbf{p}}}} (\Phi '  , \Psi ' ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } f_{s, {\ensuremath{\mathbf{y}}}}^{\, l}  ({\ensuremath{\mathbf{p}}}) ( \Phi ' , 
( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l} ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ' ) {d \mathbf{x} }{d \mathbf{y} }, \quad \Phi ', \, \Psi ' \in {\mathscr{F}_{\textrm{QED}}},  \\
&\quad   r_{s, {\ensuremath{\mathbf{p}}}} (\Phi '', \Psi '') =  -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } f_{s, {\ensuremath{\mathbf{x}}}}^{\, l}  ({\ensuremath{\mathbf{p}}}) ( \Phi '' , 
(\psi_{l} ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}})  {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi '' ) {d \mathbf{x} }{d \mathbf{y} },  \quad \Phi '' , \, \Psi '' \in {\mathscr{F}_{\textrm{QED}}}.\end{aligned}$$ We see that $$\begin{aligned}
& q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' )  \leq c_{ \, {\textrm{II}},  \, s, {\ensuremath{\mathbf{p}}}}  \| \Phi ' \| \, \| \Psi '\| ,  \notag \\
& r_{s, {\ensuremath{\mathbf{p}}}} (\Phi '' , \Psi '' )  \leq  c_{ \, {\textrm{II}},  \, s, {\ensuremath{\mathbf{p}}}}  \| \Phi '' \| \, \| \Psi '' \| ,\end{aligned}$$ where $c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} = \sum\limits_{l, l'=1}^4 
 \left\| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-  {\ensuremath{\mathbf{y}}}| }  \right\|_{L^1} \,
 |f_{s}^{\, l}({\ensuremath{\mathbf{p}}})|(c_{\, { \textrm{D} }}^{\, l'})^2  c_{\, { \textrm{D} }}^{\, l} $. Then from Riesz Representation theorem, we can define operators $ S_s^{+} ({\ensuremath{\mathbf{p}}})$ and $ T_s^{+} ({\ensuremath{\mathbf{p}}})$ such that $q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) = (\Phi ', S_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi ' )   $ and $r_{{\ensuremath{\mathbf{p}}}} (\Phi '' , \Psi '' ) = (\Phi '' , T_s^{+} ({\ensuremath{\mathbf{p}}}) \Psi '' )   $, respectively. Then it holds that $$\begin{aligned}
[{H_{\textrm{II}}}, b_{s}(f) ]^0 (\Phi , \Psi)  
& =  \int_{{\mathbf{R}^{3} }} \overline{f({\ensuremath{\mathbf{p}}})} \left( q_{s,  {\ensuremath{\mathbf{p}}}}  (\Phi , \Psi )
+ r_{ s, {\ensuremath{\mathbf{p}}}}  (\Phi , \Psi ) \right)
 d {\ensuremath{\mathbf{p}}} \\ 
&=  \int_{{\mathbf{R}^{3} }} \overline{f({\ensuremath{\mathbf{p}}})} \left( \Phi , (S_s^{+} ({\ensuremath{\mathbf{p}}}) + T_s^{+} ({\ensuremath{\mathbf{p}}}) )    \Psi  \right) d {\ensuremath{\mathbf{p}}}.\end{aligned}$$ Thus proof is obtained. $\blacksquare $

$\; $\
In a similar way to Lemma \[9/12.a\], the following lemma is also proven.\

\[9/12.b\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $g \in L^2 ({\mathbf{R}^{3} })$, $$\begin{aligned}
\textbf{(i)} \; & \; \;  [{H_{\textrm{I}}}, d_{s}(g) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi )  \,
= \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{\,-} ({\ensuremath{\mathbf{p}}})   \Psi  \right)
 d {\ensuremath{\mathbf{p}}} ,  \qquad \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} ,\Psi \in {\ensuremath{\mathscr{D}}}(H_m ) ,  \\
 \textbf{(ii)} & \; \;
[{H_{\textrm{II}}}, d_{s}(g){\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi)  \,
= \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , (S_s^{\, -} ({\ensuremath{\mathbf{p}}}) + T_s^{-} ({\ensuremath{\mathbf{p}}}) )    \Psi  \right) d {\ensuremath{\mathbf{p}}} ,  \qquad \Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}  . \end{aligned}$$ Here $ K_s^{\, -}({\ensuremath{\mathbf{p}}})$, $S_s^{\, -} ({\ensuremath{\mathbf{p}}})   $ and $T_s^{-} ({\ensuremath{\mathbf{p}}})   $ are operators which satisfy $$\begin{aligned}
& (\Phi , K_s^{\, -}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, 
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }g_{s , {\ensuremath{\mathbf{x}}}}^{\, l'} ({\ensuremath{\mathbf{p}}}) 
\left( \Phi , \,  (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi \right) {d \mathbf{x} }, \notag  \\
 & (\Phi , S^{-}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{l=1}^4  \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
\frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} g_{s , {\ensuremath{\mathbf{y}}} }^{\, l}({\ensuremath{\mathbf{p}}}) 
\left( \Phi , \,  ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi \right)  {d \mathbf{x} }{d \mathbf{y} }, \notag \\ 
&(\Phi , T^{-}_{-}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} 
\frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}  g_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}})  
\left( \Phi , \,  (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}  \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi \right)  {d \mathbf{x} }{d \mathbf{y} },
\notag\end{aligned}$$ $\; $

\[9/12.c\] Assume **(A.1)** - **(A.5)**. Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$. Then, $K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$, $S_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$ and $T_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$, $s= \pm 1/2$, are strongly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$.

**(Proof)** We show that $K_{s}^{+}({\ensuremath{\mathbf{p}}}) \Psi$ is strongly differentiable. Let $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $ \| \Phi \| =1$. From **(A.4)**, $K_{s}^{+}({\ensuremath{\mathbf{p}}}) \Psi$ is weakly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$, and we have $$\partial_{p^{\nu}} (\Phi , K_{s}^{+ }({\ensuremath{\mathbf{p}}}) \Psi )
= -\sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, 
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\partial_{p^\nu}f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) 
\left( \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi \right) {d \mathbf{x} },$$ and $ 
 |\partial_{p^{\nu}} (\Phi , K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi )| \leq \sum\limits_{j=1}^3 \sum\limits_{l , l'=1}^4  | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} \left( \int_{{\mathbf{R}^{3} }}|  \partial_{p^\nu}f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}})  | {d \mathbf{x} }\, \right) \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}^{1/2} ) \Psi \| $. Then the Riesz representation theorem shows that there exists a vector $\Xi_{\Psi} ({\ensuremath{\mathbf{p}}}) \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ such that $( \Phi , \Xi_{\Psi} ({\ensuremath{\mathbf{p}}}) )= \partial_{p^{\nu}} (\Phi , K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi )$. Let ${\ensuremath{\mathbf{e}}}_{\nu} = (\delta_{\nu , j})_{j=1}^3 $. It is seen that $$\begin{aligned}
&(\Phi ,  \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - 
 K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon}  \Psi) -  
( \Phi , \Xi ({\ensuremath{\mathbf{p}}}) ) \notag \\
& = -\sum_{j=1}^3 \sum_{l , l'=1}^4  \alpha^{j}_{l, l'} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - 
 f_{s , {\ensuremath{\mathbf{x}}}}^{\,l } ({\ensuremath{\mathbf{p}}})}{\epsilon}  - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{l }  ({\ensuremath{\mathbf{p}}})\right)  
(\Phi , ( \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) )  \Psi ) {d \mathbf{x} }, \notag \end{aligned}$$ and hence, $$\begin{aligned}
&| (\Phi , (  \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) -  K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon}  \Psi - \Xi ({\ensuremath{\mathbf{p}}}))  |  \notag \\
& \leq  \sum_{j=1}^3 \sum_{l , l'=1}^4  | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j'} \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }|
\left| \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - 
 f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}}) }{\epsilon}  - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }  ({\ensuremath{\mathbf{p}}})\right|  
  {d \mathbf{x} }\right)  \|    \Psi \| .  \label{9/13.1}\end{aligned}$$ Since (\[9/13.1\]) holds for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $ \| \Phi \| =1$, we have $$\begin{aligned}
& \left\|  \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) -  K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon}  \Psi - \Xi ({\ensuremath{\mathbf{p}}}) \right\|  \notag \\
&   \leq 
\sum_{j=1}^3 \sum_{l , l'=1}^4  | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j'} \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }|
\left| \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - 
 f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}}) }{\epsilon}  - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }  ({\ensuremath{\mathbf{p}}})\right|  
  {d \mathbf{x} }\right)  \|    \Psi \|  \to 0 ,   \notag\end{aligned}$$ as $\epsilon \to 0$. Thus $K_{s}^{+ }({\ensuremath{\mathbf{p}}} ) \Psi $ is strongly differentiable. Similarly, it is proven that $ K_{s}^{\, -}({\ensuremath{\mathbf{p}}} ) \Psi$, $ S_{s}^{\pm  }({\ensuremath{\mathbf{p}}} ) \Psi$ and $ T_{s}^{\pm  }({\ensuremath{\mathbf{p}}} ) \Psi$ are strongly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$. $\blacksquare $\

\[9/12.d\] For all $\Phi , \Psi  \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$, it holds that $$\begin{aligned}
&\textbf{(i)} \; \; [H_{{ \textrm{D} }}  , b_{s}(f) ]^0 (\Phi , \Psi )=-  \left(  \Phi , b_{s}( 
\omega_{\, M} f )\Psi  
\right)  , \\
& \textbf{(ii)} \; \;
[H_{{ \textrm{D} }}  , d_{s}(f) ]^0 (\Phi , \Psi )=-  \left(  \Phi , d_{s}( \omega_{\, M} f )\Psi  
\right) .\end{aligned}$$

**(Proof)** It holds that for all $\Phi  \in {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, M} )$, $$[H_{{ \textrm{D} }}, b^{\dagger}_{s}(f)   ]\Phi = b^{\dagger}_{s}( \omega_{\, M} f)  \Phi .$$ Let $\Psi \in {\ensuremath{\mathscr{D}}} (H_m )$. Then $$(H_{{ \textrm{D} }} \Phi,    b_{s}(f)   \Psi ) -  ( b_{s}(f)   \Phi , H_{{ \textrm{D} }} \Psi )
= ( [ b^{\dagger}_{s}(f) ,  H_{{ \textrm{D} }} ] \Phi , \Psi  ) = (- b^{\dagger}_{s}( \omega_{\, M} f) \Phi ,  \Psi) ,$$ and hence, $$(H_{{ \textrm{D} }} \Phi,    b_{s}(f)   \Psi ) -  ( b_{s}(f)   \Phi , H_{{ \textrm{D} }} \Psi ) =  -(  \Phi , 
b_{s}( \omega_{\, M} f) \Psi) .  \label{9/11.1}$$ Since $ {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, M} ))$ is a core of $H_{{ \textrm{D} }}$ and $ b_{s}(f)$ is bounded, (\[9/11.1\]) holds for all $\Phi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$. Hence **(i)** follows. Similarly, we can also prove **(ii)**. $\blacksquare $\

\[EP-Pullth\] **(Electron-Positron Pull-Through Formula)**\
Assume **(A.1)** - **(A.3)**. Then that $$\begin{aligned}
& \textbf{(i)} \; \; \;  (b_s ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  = 
 (H_m- E_0 (H_m) + \omega_{M}({\ensuremath{\mathbf{p}}}))^{-1} \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) +
 {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m ,
 \\ 
& \textbf{(ii)} \; \; (d_s ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  
 =   (H_m- E_0 (H_m) + \omega_{M}({\ensuremath{\mathbf{p}}}))^{-1} \left( {\kappa_{\textrm{I}}}K^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) +
 {\kappa_{\textrm{II}}}( S^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m ,
 \end{aligned}$$ for almost everywhere ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$.

**(Proof)** Let $\Phi, \in {\ensuremath{\mathscr{D}}} (H_m )$. By Lemma \[9/12.d\] **(i)**, we have $$\begin{aligned}
& [{H_{m}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m )  \notag \\
&=  -  \left(  \Phi , ( b_{s}( \omega_M f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  \right) + {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) +
{\kappa_{\textrm{II}}}[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) .  \notag\end{aligned}$$ On the other hand, $H_m \Psi_m=E_{0}(H_m) \Psi_m$ yields that $$[{H_{m}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) = \left(   (  {H_{m}}- E_{0}({H_{m}})  ) \Phi , ( b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) . \notag$$ Then, we have $$\begin{aligned}
& (    ({H_{m}}- E_{0}({H_{m}}))  \Phi , ( b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  )   + ( \Phi , ( b_{s}( \omega_M f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_m ) \notag \\
&\qquad \qquad \qquad   =  {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) +
{\kappa_{\textrm{II}}}[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) . \notag\end{aligned}$$ By Lemma \[9/12.a\], it follows that $$\begin{aligned}
& \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left(    ({H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}}))
\Phi , ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  \frac{}{} \right)  d {\ensuremath{\mathbf{p}}} \notag \\
&\quad  \qquad \quad 
= \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi ,  \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) +
 {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m  \right) d {\ensuremath{\mathbf{p}}}.
 \label{9/12.3}\end{aligned}$$ Since (\[9/12.3\]) holds for all $f  \in L^2 ({{\mathbf{R}^{3} }} )$, it follows that $$(    ({H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}}))  \Phi , ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  )   
=( \Phi ,  \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) +
 {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m ) 
    , \notag$$ for almost everywhere ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$. This implies that $ ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \in  {\ensuremath{\mathscr{D}}} ({H_{m}}) $ and $$( {H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m   
= \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) +
 {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m   . \label{9/12.7}$$ From (\[9/12.7\]), we obtain **(i)**. Similarly, **(ii)** is also proven. $\blacksquare $\

\[EP-DB\] (**Electron-Positron Derivative Bounds**)\
Assume **(A.1)** - **(A.5)**. Then, it holds that for all ${\ensuremath{\mathbf{p}}} \in {\ensuremath{\mathbf{R}}}^3  \backslash O_{{ \textrm{D} }}$ and $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\begin{aligned}
&\textbf{(i)} \; \;  \left\| \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m}  \right\| 
\leq \left( ( L_{\epsilon} E_{0}(H_{m} )  +  R_\epsilon + 1 \,)
|{\kappa_{\textrm{I}}}|+ 2|{\kappa_{\textrm{II}}}| \frac{}{}
\right) F_{s , +}^{\nu}({\ensuremath{\mathbf{p}}})  , \\
&\textbf{(ii)} \; \; \left\| \partial_{p^{\nu}} (d_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m}  \right\| 
\leq  \left( ( L_{\epsilon} E_{0}(H_{m} )  +  R_\epsilon + 1 \,)
|{\kappa_{\textrm{I}}}|+ 2|{\kappa_{\textrm{II}}}| \frac{}{}
\right)  F_{s , -}^{\nu}({\ensuremath{\mathbf{p}}}) . \end{aligned}$$ Here $  F_{s ,\pm }^{\nu}  $ are functions satisfying $ F_{s , \pm }^{\nu} \in L^{2} ({\mathbf{R}^{3} }) $, $s= \pm  1/2$, $\nu=1 , \cdots , 3$.

**(Proof)** Let $R_{m,M}({\ensuremath{\mathbf{p}}}) =(H_m- E_0 (H_m) + \omega_{\, M}({\ensuremath{\mathbf{p}}}))^{-1}  $. From Proposition \[EP-Pullth\] it holds that for all $\Phi \in {\mathscr{F}_{\textrm{QED}}}$ with $ \| \Phi \|=1$, $$\begin{aligned}
( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) 
& =   {\kappa_{\textrm{I}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}})  K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right)  
 +  {\kappa_{\textrm{II}}}\left( \Phi,  \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) 
 S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right) \notag \\
& \qquad \qquad  + {\kappa_{\textrm{II}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) 
 T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right) . \label{9/13.3} \end{aligned}$$ Here we evaluate the three terms in the right-hand side of (\[9/13.3\]) as follows.\
(First term) We see that $$\begin{aligned}
&  \left( \Phi,\partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) 
 K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right)  \notag \\
& =- \sum_{j=1}^{3} \sum_{l,l'=1}^{4} \alpha^{j}_{l,l'}\, \partial_{p^{\nu}} \left(   f_{s}^{\, l} ({\ensuremath{\mathbf{p}}})
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right) {d \mathbf{x} }\right) , \notag  \\
& = -\sum_{j=1}^{3} \sum_{l,l'=1}^{4} \alpha^{j}_{l,l'}\, \left\{  ( \partial_{p^{\nu}}  f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}))
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right) {d \mathbf{x} }\right. , \notag  \\
& \qquad \qquad \qquad  -i  f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right) {d \mathbf{x} }\notag \\
& \qquad \qquad \qquad  \left.  - \frac{ f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})}
   \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left(  R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right)
{d \mathbf{x} }\right\} \notag  .\end{aligned}$$ Since $ \| R_{m,M}({\ensuremath{\mathbf{p}}})\|  \leq \frac{1}{\omega_{\, M}({\ensuremath{\mathbf{p}}})} \leq  \frac{1}{M}$ and $\|\Phi \|=1$, we have [$$\begin{aligned}
& \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right) {d \mathbf{x} }\right|  \leq  \frac{ c_{\, { \textrm{D} }}^{\, l'}  c_{{\textrm{rad}}}^{\, j}}{ M}
 \| \chi_{{\textrm{I}}} \|_{L^1}   \|    ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \|,  \\
 & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right)
 {d \mathbf{x} }\right|   \leq \frac{ c_{\, { \textrm{D} }}^{\, l'}  c_{{\textrm{rad}}}^{\, j}}{ M} \, 
\| |{\ensuremath{\mathbf{x}}}| \chi_{{\textrm{I}}} \|_{L^1}  \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} )   \Psi_{m} \|, \\
& \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}}
\left(  R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \,  (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}}))  \Psi_m \right)
{d \mathbf{x} }\right|   \leq \frac{c_{\, { \textrm{D} }}^{\, l'}  c_{{\textrm{rad}}}^{\, j}}{ M^2} \,
\| \chi_{{\textrm{I}}} \|_{L^1}   \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \|  .\end{aligned}$$ ]{} It is seen that $ \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}^{1/2} +1)^{1/2}) \Psi_m \| \leq 
  \|  H_{0,m} \Psi_{m} \| +  \| \Psi_m\| = \|  H_{0,m} \Psi_{m} \| +1 $, and hence, $$\begin{aligned}
& \left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) K_{s}^{\, +} ({\ensuremath{\mathbf{p}}})\Psi_m \right) \right|  \notag \\
& \leq   \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{j=1}^{3} \sum_{l,l'=1}^{4} c_{\, { \textrm{D} }}^{\, l'}  c_{{\textrm{rad}}}^{\, j}
 \left( 
 \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} 
+ \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{M} 
+  \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{M^2}
\right)   \, \left( \|   H_{0,m}  \Psi_m \| +1 \frac{}{}  \right) .
\label{9/13.I}
 \end{aligned}$$ (Second term) It is seen that $$\begin{aligned}
&  \left( \Phi, \partial_{p^{\nu}}R_{m,M}({\ensuremath{\mathbf{p}}}) 
 S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right)  \notag \\
& =- \sum_{l=1}^4 \partial_{p^{\nu}} \left( f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m \right)  {d \mathbf{x} }{d \mathbf{y} }\right) \notag   \\
&  =- \sum_{l=1}^4 \left\{ ( \partial_{p^{\nu}}  f_{s }^{\, l}({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m  \right)  {d \mathbf{x} }{d \mathbf{y} }\right. \notag \\
&\qquad \quad - i  f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} y^{\nu}  e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m  \right)  {d \mathbf{x} }{d \mathbf{y} }\notag \\
& \qquad \quad \left. - \frac{f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}   e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \,  ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))
  \Psi_m  \right)  {d \mathbf{x} }{d \mathbf{y} }\right) . \label{9/13.9}\end{aligned}$$ By evaluating the right-hand side of (\[9/13.9\]), we have $$\left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) 
 S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right)  \right|
   \leq  \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{l,l'=1}^{4} ( c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l}\left( 
 \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} 
+ \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{ M} 
+  \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{M^2}
\right)     . \label{9/13.II}$$ (Third term) We see that $$\begin{aligned}
&  \left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) 
 T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right)  \notag \\
& =-\sum_{l=1}^4 \partial_{p^{\nu}} \left( f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  ( \psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}})  {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m  \right)  {d \mathbf{x} }{d \mathbf{y} }\right) \notag   \\
&  =- \sum_{l=1}^4 \left\{ ( \partial_{p^{\nu}}  f_{s }^{\, l}({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}})  ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m  \right)  {d \mathbf{x} }{d \mathbf{y} }\right. \notag \\
&\qquad \quad - i  f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} y^{\nu}  e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}}
\left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \,  (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}})  ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m \right)  {d \mathbf{x} }{d \mathbf{y} }\notag \\
& \qquad \quad \left. - \frac{f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}}  (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|}   e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \,  (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}})  ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}))  \Psi_m  \right)  {d \mathbf{x} }{d \mathbf{y} }\right) . \label{9/13.10}\end{aligned}$$ We estimate the right-hand side of the absolute value of (\[9/13.10\]), and then, $$\left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) 
 T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} )    \Psi_m \right)  \right|   \leq 
  \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{l,l'=1}^{4}c_{\, { \textrm{D} }}^{\, l} ( c_{\, { \textrm{D} }}^{\, l'})^2 
 \left( 
 \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{M} 
+ \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{ M} 
+  \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{M^2}
\right)  . \label{9/13.III}$$ From (\[9/13.I\]), (\[9/13.II\]) and (\[9/13.III\]), we have $$\begin{aligned}
& \left| ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right| 
 \notag \\
&  \leq    \sum_{l=1}^4 c_{+}^{\,l}  \left( 
 \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} 
+ \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{M} 
+  \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \,   }{ M^2}
\right)   \left(   |\kappa_{\, {\textrm{I}}}| \,  \| H_{0,m} \Psi_m \|  + |{\kappa_{\textrm{I}}}| + 2 |\kappa_{\, {\textrm{II}}}|    \frac{}{} \right) ,  \notag \end{aligned}$$ where $c_{+}^{\,l} =  \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \, \times  \max \left\{ 
\sum\limits_{j=1}^{3}\sum\limits_{l'=1}^{4} |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} , \;   \sum\limits_{l'=1}^{4}  ( c_{\, { \textrm{D} }}^{\, l'})^2  c_{\, { \textrm{D} }}^{\, l}   \right\}$. By the definition of $f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) = \frac{\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \,  u_{s}^{\,l}({\ensuremath{\mathbf{p}}})}{
\sqrt{(2 \pi )^3  } }$, we have $$\left| ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right| 
  \leq   F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}})
   \, \left(   |\kappa_{\, {\textrm{I}}}| \, \| H_{0,m} \Psi_m \| + |{\kappa_{\textrm{I}}}|+ 2 |\kappa_{\, {\textrm{II}}}|    \frac{}{} \right) , \label{9/13.11}$$ where $$F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) = \frac{1}{\sqrt{(2 \pi )^3}} \sum_{l=1}^4 c_{+}^{\, l} \left(
  \frac{| \partial_{p^{\nu}} \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \,   |}{ M} 
+ \frac{ | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \partial_{p^{\nu}} u_{s}^{\,l}({\ensuremath{\mathbf{p}}}) | \,   }{M} + \frac{ |\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}})  | \,   }{ M}
+  \frac{ |\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}})  | \,   }{ M^2}
\right)  .$$ We see that (\[9/13.11\]) holds for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $\| \Phi \| =1$, and this implies that $$\left\|  \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right\| 
  \leq    F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}})
   \, \left(   |\kappa_{\, {\textrm{I}}}| \|  H_{0,m}  \Psi_m \| +| {\kappa_{\textrm{I}}}| + 2 |\kappa_{\, {\textrm{II}}}|   \frac{}{} \right)  . \notag$$ From Lemma \[9/9.e\], it holds that for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\|  H_{0,m} \Psi_{m} \|   \leq 
 \, L_{\epsilon} \| H_{m} \Psi_m \| +  R_{\epsilon}  \| \Psi_m \| 
=  \, L_{\epsilon} E_{0}(H_m)  + R_{\epsilon}    .$$ Thus **(i)** is obtained. Similarly, **(ii)** is also proven in a same way as **(i)**. $\blacksquare $\

Photon Derivative Bound
-----------------------

In a similar to the Dirac field, we introduce the distribution kernel of the annihilation operator for the radiation field. For all $ \Psi = \left\{  \Psi^{(n)} = \left( \Psi^{(n)}_1 ,   \Psi^{(n)}_2  \right)  \right\}_{n=0}^{\infty}
\in {\ensuremath{\mathscr{D}}} ( H_{{\textrm{rad}}, m } )$, we define $a_{r}({\ensuremath{\mathbf{k}}})$, by $$a_{r}({\ensuremath{\mathbf{k}}})\Psi^{(n)}_{\varrho} ( {\ensuremath{\mathbf{k}}}_{1} , \cdots ,  {\ensuremath{\mathbf{k}}}_{n} )
= \delta_{\, r , \varrho  } \sqrt{n+1} \Psi^{(n+1 )}_{\varrho }( {\ensuremath{\mathbf{k}}} , {\ensuremath{\mathbf{k}}}_{1} , \cdots ,  {\ensuremath{\mathbf{k}}}_{n} ) , \qquad \varrho = 1,2.$$ It holds that $$\qquad \qquad 
(\Phi ,  a_{r}(h)  \Psi )=
\int_{{\mathbf{R}^{3} }}h({\ensuremath{\mathbf{k}}})^{\ast}(\Phi ,  a_{r}({\ensuremath{\mathbf{k}}}) \Psi) d {\ensuremath{\mathbf{k}}}, 
\quad \Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} , \; \Psi \in  {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}}, m})  .$$

\[9/13.c\]Assume (**A.2**). Then for all $\Phi , \Psi  \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}},m })$, $$\begin{aligned}
& \textbf{(i)} \; \;
[H_{{\textrm{rad}}, m }  \, , a_{r} (h)]^0 (\Phi , \Psi )=  \left(  \Phi , a_{r}( \omega_m h )\Psi
\right) , \\
&\textbf{(ii)} \; \; [A_{j}({\ensuremath{\mathbf{x}}} )  , a_{r}(h) ]^0 (\Phi , \Psi )=
-  (h, h_{r , {\ensuremath{\mathbf{x}}}}^j  )\left(  \Phi , \Psi \right)  .
 \end{aligned}$$

**(Proof)** It holds that for all $\Phi  \in {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, m} ))$, $$\begin{aligned}
& [ H_{{\textrm{rad}}, m}, a^{\dagger}_{r}(h)   ]\Phi =  -  a^{\dagger}_{r}(\omega_{\,m }h)\Phi ,
\label{9/14.1}  \\
&[ A_{j} ({\ensuremath{\mathbf{x}}}), a^{\dagger}_{r}(h)   ]\Phi =  ( h_{r , {\ensuremath{\mathbf{x}}}}^j ,h  ) \Phi .\label{9/14.2}\end{aligned}$$ In a similar way to Lemma \[9/12.d\], we can prove **(i)** by (\[9/14.1\]) and **(ii)** by (\[9/14.2\]). $\blacksquare $\

\[9/13.d\] Assume **(A.1)** - **(A.3)**. Then\
**(i)** it holds that for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$[{H_{\textrm{I}}},  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ( h )]^0 (\Phi , \Psi)  \,
= \, \int_{{\mathbf{R}^{3} }} h({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi , Q_{r} ({\ensuremath{\mathbf{k}}})   \Psi  \right)
 d {\ensuremath{\mathbf{k}}} . \notag$$ Here $ Q_r ({\ensuremath{\mathbf{k}}})$ is an operator which satisfy $$(\Phi , Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi ) = - \sum_{j=1}^3  \, 
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }h_{r, {\ensuremath{\mathbf{x}}}}^{\, j} ({\ensuremath{\mathbf{k}}})
\left( \Phi , \,  ({\psi^{\dagger} (\mathbf{x}) }\alpha^j  { \psi ({\ensuremath{\mathbf{x}}})  }{\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi \right) {d \mathbf{x} }$$ with $ \| Q_{r} ({\ensuremath{\mathbf{k}}}) \| \leq \|\chi_{{\textrm{I}}} \|_{L^1}  \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |h_{r}^j ({\ensuremath{\mathbf{k}}})| \, |\alpha^{j}_{l,l'}| \, |c_{\, { \textrm{D} }}^{\, l}| \, |c_{\, { \textrm{D} }}^{\, l'}|$.\
**(ii)** Additionally assume **(A.4)** and **(A.6)**. Then, $Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi$ is strongly differential for all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}}$.

**(Proof)** **(i)** Let $\Phi \in {\ensuremath{\mathscr{D}}}(H_m)$ From Lemma \[9/13.c\], $$\begin{aligned}
[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h ) ]^0 (\Phi , \Psi ) &=  
\sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi  ({\ensuremath{\mathbf{x}}}) {\otimes}A_{j}({\ensuremath{\mathbf{x}}}), {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)]^0( \Phi , \Psi ) {d \mathbf{x} }\notag \\
&= \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }}  {\chi_{\textrm{I}} (\mathbf{x}) }[ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}A_{j}({\ensuremath{\mathbf{x}}}) , {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)  ]^0 
(  \Phi ,( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi  ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi   )   {d \mathbf{x} }\notag \\
& =- \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }}  {\chi_{\textrm{I}} (\mathbf{x}) }( h, h_{r ,{\ensuremath{\mathbf{x}}}}^j  )(  \Phi ,( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi  ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi   )   {d \mathbf{x} }.\end{aligned}$$ We define $\ell_{r, {\ensuremath{\mathbf{k}}}}:{\mathscr{F}_{\textrm{QED}}}{\otimes}{\mathscr{F}_{\textrm{QED}}}\to {\ensuremath{\mathbf{C}}}$ by $$\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ') = - \sum_{j=1}^3  h_{r}^j  ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}}  \,  (  ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi  ({\ensuremath{\mathbf{x}}})   {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Phi ', \Psi  ' ) {d \mathbf{x} }$$ We see that $|\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ' ) | \leq 
\|\chi_{{\textrm{I}}} \|_{L^1}  \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |h_{r}^j ({\ensuremath{\mathbf{k}}})| \, |\alpha^{j}_{l,l'}| \, |c_{\, { \textrm{D} }}^{\, l}| \, |c_{\, { \textrm{D} }}^{\, l'}|\, \| \Phi ' \| \, \| \Psi' \| $. By Riesz representation theorem, we can define an operator $Q_{r} ({\ensuremath{\mathbf{k}}})$ such that $\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ')  = (\Phi ', Q_{r}({\ensuremath{\mathbf{k}}}) \Psi ')$. Then we have $$[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(f) ]^0 (\Phi , \Psi )
= \int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi , \Psi ) {d \mathbf{k} }=   \int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left(  \Phi ,Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi    \right)   {d \mathbf{k} }.
\notag$$ Then **(i)** is obtained.\
**(ii)** The strong differentiability of $ Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi $ is proven by **(A.4)** and **(A.6)** in a similar way to Lemma \[9/12.c\], and the proof is omitted. $\blacksquare $.\

\[PF-P\] ()\
Assume **(A.1)** - **(A.3)**. Then it holds that for almost everywhere ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$, $$({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m ={\kappa_{\textrm{I}}}(H_m- E_0 (H_m) + \omega_{m}({\ensuremath{\mathbf{k}}}))^{-1} 
Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m  .$$

**(Proof)** Let $\Phi \in {\ensuremath{\mathscr{D}}}(H_m) $. By Lemma \[9/13.c\] **(i)**, $$[{H_{m}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h ) ]^0 (\Phi , \Psi_m )= - \left( \Phi , ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ( \omega_m h)) \Psi_m \right)
 +   {\kappa_{\textrm{I}}}\, [{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 (\Phi , \Psi_m )  . \notag$$ It also holds that $$[{H_{m}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 (\Phi , \Psi_m ) =
 \left( (H_{m}-E_{0}({H_{m}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)) \Psi_m \right) . \notag$$ Then we have $$\left( (H_{m}-E_{0}({H_{m}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)) \Psi_m \right) 
+ \left( \Phi, ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(\omega_m h)) \Psi_m \right) = {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h )]( \Phi, \Psi_{m} ).  \notag$$ By Lemma \[9/13.d\], $$\int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast}
\left(  (H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m    \right) {d \mathbf{k} }=  {\kappa_{\textrm{I}}}\int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left(  \Phi ,Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m    \right)   {d \mathbf{k} }. \label{9/13.14}$$ Note that (\[9/13.14\]) holds for all $h \in L^2 ({\mathbf{R}^{3} })$. Then we have $$\left(  (H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m    \right) =    \left(  \Phi ,\,  {\kappa_{\textrm{I}}}Q_r ({\ensuremath{\mathbf{k}}}) \Psi_m    \right) ,  \label{9/13.15}$$ for almost everywhere $ {\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$. In addition, (\[9/13.15\]) yields that $ ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \in {\ensuremath{\mathscr{D}}} (H_m )$ and $$(H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m  =   {\kappa_{\textrm{I}}}Q_r ({\ensuremath{\mathbf{k}}}) \Psi_m    . \notag$$ Thus the proof is obtained. $\blacksquare $

\[P-DB\] (**Photon Derivative Bounds**)\
Assume **(A.1)**-**(A.4)** and **(A.6)**. Then it holds that for all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}}$, $$\left\| \partial_{k^\nu} ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m  \right\|     \leq |{\kappa_{\textrm{I}}}|  F_{r}^{\, \nu} ({\ensuremath{\mathbf{k}}}) \notag$$ where $  F_{r}^{\, \nu} $ is a function which satisfy $ F_{r}^{\, \nu} \in L^2 ({\mathbf{R}^{3} }) $.

**(Proof)** $\; $\
Let $ R_{m}({\ensuremath{\mathbf{k}}}) = (H_m- E_0 (H_m) + \omega_{\, m}({\ensuremath{\mathbf{k}}}))^{-1} $. From Proposition \[PF-P\], it holds that ${{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m =  R_{m}({\ensuremath{\mathbf{k}}}) 
  Q_{r} ({\ensuremath{\mathbf{k}}})   \Psi_m   $. Then for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$, $$\begin{aligned}
& = ( \Phi ,  \partial_{k^\nu } ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m  )  \notag \\
& =- {\kappa_{\textrm{I}}}\sum_{j=1}^3  \, \partial_{k^{\nu}} \left(   h_{r}^{\, j} ({\ensuremath{\mathbf{k}}})
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \,  ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j  \psi({\ensuremath{\mathbf{x}}})
  {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_m  \right) {d \mathbf{x} }\right) \notag  \\
& =- {\kappa_{\textrm{I}}}\sum_{j=1}^3  \, \left\{  ( \partial_{k^{\nu}}  h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}))
 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \,  ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j  \psi({\ensuremath{\mathbf{x}}}) 
{\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_m \right) {d \mathbf{x} }\right. \notag  \\
& \qquad \qquad \qquad  -i  h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}}
\left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \,  ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j  \psi({\ensuremath{\mathbf{x}}})
 {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_m  \right) {d \mathbf{x} }\notag \\
& \qquad \qquad \qquad  \left.  - \frac{ h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) k^{\nu} }{\omega_{\, m}({\ensuremath{\mathbf{k}}})}
   \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}}
\left(  R_{m}({\ensuremath{\mathbf{k}}})^2 \Phi , \,  ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j  \psi({\ensuremath{\mathbf{x}}})
 {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_m \right)
{d \mathbf{x} }\right\}  . \label{9/13.16}\end{aligned}$$ By estimating the absolute value of the right-hand side of (\[9/13.16\]), we have $$\begin{aligned}
& \left| \partial_{k^{\nu}} \left( \Phi, R_{m}({\ensuremath{\mathbf{k}}})Q ({\ensuremath{\mathbf{k}}})\Psi_m \right) \right| 
\notag \\
& \leq  \| (1+ |{\ensuremath{\mathbf{x}}}| ) \chi_{{\textrm{I}}} \|_{L^1}|{\kappa_{\textrm{I}}}|
\sum_{j=1}^3 \sum_{l,l'=1}^4 \, | \alpha^j_{l,l'} | \, | c_{\, { \textrm{D} }}^{\,l}| \, |  c_{\, { \textrm{D} }}^{\,l'}| \, 
 \left( 
 \frac{| \partial_{k^{\nu}} h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) |}{\omega_{\, m}({\ensuremath{\mathbf{k}}})} 
+ \frac{ | h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) | \,   }{\omega_{\, m}({\ensuremath{\mathbf{k}}})} 
+  \frac{ | h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) | \,   }{\omega_{\, m}({\ensuremath{\mathbf{k}}})^2}
\right) . \notag
 \end{aligned}$$ From the definition of $h_{r}^{\, j}({\ensuremath{\mathbf{k}}})=\frac{\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) 
e_{r}^{\,j}({\ensuremath{\mathbf{k}}}))}{\sqrt{2 (2 \pi )^3 \omega ({\ensuremath{\mathbf{k}}}) }}$, we have $$\partial_{k^\nu} h_{r}^{\, j}({\ensuremath{\mathbf{k}}}) = \frac{1}{\sqrt{2 (2 \pi )^3 }} \left( 
\frac{ ( \partial_{k^\nu} \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) )  e_{r}^{\,j}({\ensuremath{\mathbf{k}}}) }{\omega({\ensuremath{\mathbf{k}}})^{1/2}}
+ \frac{ \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu} e_{r}^{\,j}({\ensuremath{\mathbf{k}}})}{\omega ({\ensuremath{\mathbf{k}}})^{1/2}}
- \frac{1}{2} \frac{  \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) k^{\nu}}{ \omega ({\ensuremath{\mathbf{k}}})^{5/2}} \right) . \notag$$ Hence, it holds that $$\left| ( \Phi ,  \partial_{k^\nu } ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_{m} ) \right|
  \leq    | {\kappa_{\textrm{I}}}| \, F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) ,  \label{9/13.17}$$ where $$\begin{aligned}
  F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) =\frac{\| (1+ |{\ensuremath{\mathbf{x}}}| ) \chi_{{\textrm{I}}} \|_{L^1}}{\sqrt{2 (2 \pi )^3 }}   & \sum_{j=1}^3 \sum_{l,l'=1}^4 \, | \left\{  \alpha^j_{l,l'} | \, | c_{\, { \textrm{D} }}^{\,l}| \, |  c_{\, { \textrm{D} }}^{\,l'}| \, \frac{}{} \right.\notag \\
  &   \times \left.
\left( \frac{| \partial_{k^\nu} \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |+| \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu} e_{r}^{\,j}({\ensuremath{\mathbf{k}}})| 
 + | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) | }{  \omega ({\ensuremath{\mathbf{k}}})^{3/2}}
+ \frac{3}{2} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |}{ \omega ({\ensuremath{\mathbf{k}}})^{5/2}}  \right) \right\} . \notag 
 \end{aligned}$$ Since (\[9/13.17\]) holds for all $\Phi \in {\mathscr{F}_{\textrm{QED}}}$, we have $$\| \partial_{k^{\nu}} ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_{m}  \| \leq  | {\kappa_{\textrm{I}}}| F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) .$$ The condition **(A.6)** yields that $ F_{r}^{\nu} \in L^2 ({\mathbf{R}^{3} }) $, and hence the proof is obtained. $\blacksquare $

Proof of Theorem \[Main-Theorem\] 
==================================

Let $\{ \Psi_m\}_{m>0}$ be the sequence of the normalized ground state of $H_m$, $m>0$. Then there exists a subsequence of $\{ \Psi_{m_{j}} \}_{j=1}^{\infty}$ with $m_{j+1} < m_{j} $, $j \in {\ensuremath{\mathbf{N}}}$, such that the weak limit $\Psi_{0} := $w-$\lim\limits_{j \to \infty } \Psi_{m_{j}} $ exists.

\[9/16.a\] Suppose **(A.1)** - **(A.3)**. Then,\
$\qquad $ **(i)** ${\ensuremath{\mathscr{D}}}_{0}$ is a common core of $H_{{ \textrm{QED} }}$ and $H_{m}$, $m>0$, and $H_{m}$ strongly converges to $H_{{ \textrm{QED} }}$ on ${\ensuremath{\mathscr{D}}}_{0}$\
$\qquad $ **(ii)** $  \lim\limits_{m \to \infty} E_{0}(H_m) = E_{0}(H_{{ \textrm{QED} }} ) $.

**(i)** Since ${\ensuremath{\mathscr{D}}}_{0}$ is a core of $H_{0,m}$, ${\ensuremath{\mathscr{D}}}_{0}$ is also a core of $H_{m}$. It is directly proven that $ \lim\limits_{m \to 0} H_{m} \Psi = H_{{ \textrm{QED} }} \Psi $ for all $\Psi \in {\ensuremath{\mathscr{D}}}_0$.\
**(ii)** We see that $(\Psi , H_{m} \Psi )  \geq ( \Psi, H_{{ \textrm{QED} }} \Psi ) \geq {E_{0}(H_{{ \textrm{QED} }})}$, for all $\Psi \in {\ensuremath{\mathscr{D}}}_{0}$. Hence $\inf\limits_{m>0}E_{0}(H_{m}) \geq E_{0}(H_m )$. From **(i)**, it follows that $H_{m}$ converges to $H_{{ \textrm{QED} }}$ as $m \to 0$ in the strong resolvent sense, and this yields that $\limsup\limits_{m \to 0}E_{0}(H_{m}) \leq E_{0}({H_{\textrm{QED}}})$. Hence **(ii)** follows. $\blacksquare $\
$\;$\
From Lemma \[9/16.a\] **(ii)**, we can set $$E_{\infty} = \sup\limits_{j\in {\ensuremath{\mathbf{N}}}} |E_{0}(H_{m_j })|  \; < \infty . \\$$

\[9/14.a\] **(Number Operator Bounds)**\
Suppose **(A.1)** - **(A.6)**. Then, for all $
 0 < \epsilon < \frac{1}{c_{\textrm{I}}| {\kappa_{\textrm{I}}}|} ,$ $$\begin{aligned}
 \quad {\ensuremath{\mathbf{(i)}}} \; \; & 
\sup_{j\in {\ensuremath{\mathbf{N}}} } \| (N_{{ \textrm{D} }}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m_j } \|
 \leq \left( \frac{L_{\epsilon}}{M}E_{\infty} + \frac{R_{\epsilon} }{M}\right)^{1/2} , \notag \\
{\ensuremath{\mathbf{(ii)}}} \; \;&  \sup_{j\in {\ensuremath{\mathbf{N}}} } \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j } \|
 \leq c_0 |{\kappa_{\textrm{I}}}|  \, \left\|  \frac{ \chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\|   , \notag \end{aligned}$$ where $c_{0} = \sqrt{\frac{11}{2(2\pi)^3}} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |\alpha^{j}_{l,l'}| c_{{ \textrm{D} }}^{\,l} c_{{ \textrm{D} }}^{\,l'} $.

**(Proof)** **(i)** We see that $ \|( N_{{ \textrm{D} }}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2  
 = (\Psi_m , ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} )\leq \|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|$, and Corollary \[9/9.f\] yields that for all $0 < \epsilon < \frac{1}{c_{\textrm{I}}| {\kappa_{\textrm{I}}}| }$, $$\|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|
 \leq \frac{L_{\epsilon}}{M} \| H_m \Psi_m  \| + \frac{ R_{\epsilon} }{M}\|\Psi_m  \| = 
 \frac{L_{\epsilon}}{M}E_{0}(H_m ) + \frac{R_{\epsilon}}{M} .
 \notag$$ Hence **(i)** follows.\
**(ii)** From the photon pull-through formula in Proposition \[PF-P\], it follows that $$\begin{aligned}
 (\Psi_m , ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}) \Psi_m ) 
 & = \sum_{r=1,2} \int_{{\mathbf{R}^{3} }} \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\
 & = |{\kappa_{\textrm{I}}}|^2 \sum_{r=1,2} \int_{{\mathbf{R}^{3} }}\| (H_m -E_{0}(H_m ) + \omega_{m}({\ensuremath{\mathbf{k}}})) Q_{r}({\ensuremath{\mathbf{k}}}) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\
 & \leq |{\kappa_{\textrm{I}}}|^2 \frac{11}{{2(2\pi)^3}} \sum_{r=1,2} \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha^{j}_{l,l'}|^2 ( c_{{ \textrm{D} }}^{\,l} c_{{ \textrm{D} }}^{\,l'} )^2 \left( \int_{{\mathbf{R}^{3} }}   \frac{| \chi_{{\textrm{rad}}}|^2}{|{\ensuremath{\mathbf{k}}}|^3}  d {\ensuremath{\mathbf{k}}} \right) . \label{9/15.1}\end{aligned}$$ From (\[9/15.1\]), we obtain **(ii)**. $\blacksquare $\

\[9/15.a\] Assume **(A.1)**-**(A.6)**. Let $ F \in C_0^{\, \infty} ({\ensuremath{\mathbf{R}}}^3)$ which satisfy $0 \leq F \leq 1$ and $F({\ensuremath{\mathbf{x}}})=1$ for $|{\ensuremath{\mathbf{x}}}| \leq 1$, and set $ F_{ R}({\ensuremath{\mathbf{x}}}) = F (\frac{  {\ensuremath{\mathbf{x}}} }{R} ) $. Let $\hat{{\ensuremath{\mathbf{p}}}} = - i \nabla_{{\ensuremath{\mathbf{p}}}}  $ and $\hat{{\ensuremath{\mathbf{k}}}} = - i \nabla_{{\ensuremath{\mathbf{k}}}}  $. Then for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} {\kappa_{\textrm{I}}}}$, $R \geq 1$ and $R' \geq 1$, $$\begin{aligned}
&\textbf{(i)} \; \;  \,   \sup_{j\in {\ensuremath{\mathbf{N}}} } \|  (  {{\small \text{1}}\hspace{-0.32em}1}- 
 {\Gamma_{\textrm{f}}}( F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_{m_j } \|)  \leq  
\frac{c_{1,\epsilon  }}{\sqrt{R}} , \\
&\textbf{(ii)} \; \;  \,    \sup_{  j\in {\ensuremath{\mathbf{N}}}} \|  ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}( {{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{b}}}( 
F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ))   \Psi_{m_j } \| )  \leq   \frac{c_2}{\sqrt{R'}}  , \notag \end{aligned}$$ where $$c_{1, \epsilon } =   \left( 4 \frac{L_{\epsilon} E_{\infty} +  R_{\epsilon}}{M}\right)^{1/4}
\left(   \left( \frac{L_{\epsilon} E_{\infty} +  R_{\epsilon}}{M}\right)^{1/2}
+  ( L_{\epsilon} E_{\infty} + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 2|{\kappa_{\textrm{II}}}|  \sum_{s=\pm 1/2} \sum_{\nu =1}^3  \sum_{\tau= \pm } \|F_{s ,\tau}^{\nu} \| 
\right)^{1/2}  \notag$$ and $$c_2 =    |{\kappa_{\textrm{I}}}|^{1/2} \left(  c_0  \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| \,\right)^{1/2}  \, \left(
c_0  \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\|
+ \, \sum_{r=1,2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}\right)^{1/2} . \notag$$

**(Proof)** It follows that $ ({{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )^2 \leq  {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \leq 
 {\ensuremath{d\Gamma_{\textrm{f}}({1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})}) }}  $, and then, $$\begin{aligned}
 \|  ( ( {{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_{m} \|^2 
&\leq ( \Psi_m ,\left(    {d \Gamma_{\textrm{f}}}( 1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) \Psi_{m} ) \notag \\
& = \sum_{s= \pm 1/2}
\left(  \int_{{\mathbf{R}^{3} }} \left( (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m}   , 
(1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  )
(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right)  d {\ensuremath{\mathbf{p}}}  \right. \notag \\
& \qquad \quad  \left. +  \int_{{\mathbf{R}^{3} }} \left( (d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m}   ,
 (1-    F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m } \right)  d {\ensuremath{\mathbf{p}}}  \right) . \label{9/15.2}\end{aligned}$$ We evaluate the two terms in the right-hand side of (\[9/15.2\]). The first term is estimated as $$\begin{aligned}
&\left| \int_{{\mathbf{R}^{3} }} \left( (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m}   ,
 (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right)  d {\ensuremath{\mathbf{p}}} \right| \notag \\
& \leq \left( \int_{{\mathbf{R}^{3} }} \left\| (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m  \right\|^2 d {\ensuremath{\mathbf{p}}}  
 \right)^{1/2} \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2  d {\ensuremath{\mathbf{p}}} \right)^{1/2} \notag \\
 & =  \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times  \left( \int_{{\mathbf{R}^{3} }}
 \left\| (1-F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2  d {\ensuremath{\mathbf{p}}} \right)^{1/2}. \notag\end{aligned}$$ It is seen that $$\begin{aligned}
 & \int_{{\mathbf{R}^{3} }} \| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  \|^2  d {\ensuremath{\mathbf{p}}} 
\notag \\
&  \leq 4  \int_{{\mathbf{R}^{3} }} \| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \, \frac{1}{1+ \hat{{\ensuremath{\mathbf{p}}}}^2}  (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  \|^2  d {\ensuremath{\mathbf{p}}}   \notag \\
& \qquad \qquad \qquad  \qquad    + 4 \sum_{\nu=1}^3
 \int_{{\mathbf{R}^{3} }} \| (1-F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})  ) \, \frac{(\hat{p}^\nu )^2}{1+ \hat{{\ensuremath{\mathbf{p}}}}^2} 
 \, (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  \|^2  d {\ensuremath{\mathbf{p}}}  . \notag\end{aligned}$$ Note that for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, $$\sup_{{\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }} \left|
\left(  1- F_{ R} ({\ensuremath{\mathbf{p}}})\right) \, \frac{1}{ 1+ {\ensuremath{\mathbf{p}}}^2} \right| \leq  \frac{1}{R^2} \; , \quad  \; \;  \; \; \sup_{{\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }} \left|
 \left( 1- F_{ R} ({\ensuremath{\mathbf{p}}})\right) \, \frac{p^{\nu}}{ 1+ {\ensuremath{\mathbf{p}}}^2} \right| \leq  \frac{1}{R} .$$ Then by the electron derivative bounds in Theorem \[EP-DB\] **(i)** and the spectral decomposition theorem, we have $$\begin{aligned}
& \int_{{\mathbf{R}^{3} }} \left\| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}))(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2  d {\ensuremath{\mathbf{p}}} \ \notag \\
& \leq  \frac{4}{R^4} 
  \int_{{\mathbf{R}^{3} }} \left\| (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2  d {\ensuremath{\mathbf{p}}}
+  \frac{4}{R^2} \sum_{\nu =1}^3  \int_{{\mathbf{R}^{3} }} \left\| \partial_{p^{\nu}}(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2  d {\ensuremath{\mathbf{p}}}   \notag \\
& \leq  \frac{4}{R^4} 
  \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \|^2 +   \frac{ c_m( \epsilon )^2}{R^2} \sum_{\nu =1}^3
   \int_{{\mathbf{R}^{3} }}|F_{s, +}^{\nu}({\ensuremath{\mathbf{p}}})|^2 d {\ensuremath{\mathbf{p}}} , \notag 
  \end{aligned}$$ where $c_m ( \epsilon ) =  2 ( L_{\epsilon} E_{0} (H_m ) + R_{\epsilon} +1) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|$. Therefore, $$\begin{aligned}
& \left| \int_{{\mathbf{R}^{3} }} (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m  , (1-  F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  ) d {\ensuremath{\mathbf{p}}} \right| \notag \\
& \leq \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left(
\frac{2}{R^2} \| ( N_{{ \textrm{D} }}^+ {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| +  \frac{c_m ( \epsilon )}{R} \sum_{\nu =1}^3 \|F_{s ,+}^{\nu} \|_{L^2} \right) . \label{9/15.3}\end{aligned}$$ In a same way as the first term, we can estimate the second term in the right-hand side of (\[9/15.2\]) by the positron derivative bounds in Theorem \[EP-DB\] **(ii)**, and then, $$\begin{aligned}
& \left| \int_{{\mathbf{R}^{3} }} (d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m  , (1-   F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m  ) d {\ensuremath{\mathbf{p}}}  \right| \notag \\
& \leq \| ( N_{{ \textrm{D} }}^{-} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left(
\frac{2}{R^2} \| ( N_{{ \textrm{D} }}^- {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| +  \frac{c_m ( \epsilon )}{R} \sum_{\nu =1}^3  \|F_{s , -}^{\nu} \|_{L^2} \right) . \label{9/15.4}\end{aligned}$$ From (\[9/15.3\]) and (\[9/15.4\]), we have for all $R>1$,\
$$\begin{aligned}
& \|  ( ( {{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})  \Psi_{m} \|^2  \notag  \\
&\leq \frac{1}{R}\sum_{\tau = \pm} \| ( N_{{ \textrm{D} }}^{\tau} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \|  \left( 
2 \| ( N_{{ \textrm{D} }}^{\tau} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| +  c_m ( \epsilon )
\sum_{s=\pm 1/2} \sum_{\nu =1}^3  \|F_{s ,\tau}^{\nu} \|_{L^2} \right) . \label{9/15.7}\end{aligned}$$ From Lemma \[9/14.a\] **(i)**, $$\sup_{j \in {\ensuremath{\mathbf{N}}}} \| ( N_{{ \textrm{D} }}^{\pm} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_{m_j} \| \leq 
\sup_{j \in {\ensuremath{\mathbf{N}}}} \| ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_{m_j} \|  \leq 
  \left( \frac{L_{\epsilon}}{M} E_{\infty} +\frac{R_{\epsilon}}{M} \right)^{1/2}  ,$$ and we see that $$\sup_{j \in {\ensuremath{\mathbf{N}}}} c_{m_{j}} ( \epsilon ) 
= \sup_{j \in {\ensuremath{\mathbf{N}}}} \left(2 ( L_{\epsilon} E_{0}(H_{m_j}) + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|  \right)
\; \leq  \;
 2 ( L_{\epsilon} E_{\infty} + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|.$$ Hence **(i)** follows.\
**(ii)** In a similar way to the proof of **(i)**, it follows that $ ({{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})))^2 \leq  {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}(F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})) \leq 
 {\ensuremath{d\Gamma_{\textrm{b}}({1-F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) }) }}  $, and hence, $$\begin{aligned}
 \|   ( {{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{b}}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}))  )  \Psi_{m} \|^2 
&\leq ( \Psi_m ,\left(    {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{d \Gamma_{\textrm{b}}}( 1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )   \right) \Psi_{m} ) \notag \\
& = \sum_{r= 1,2}
  \int_{{\mathbf{R}^{3} }} \left(  ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m}   , 
(1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )
({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right)  d {\ensuremath{\mathbf{k}}}  .
 \label{9/15.5}\end{aligned}$$ We see that $$\begin{aligned}
&\left| \int_{{\mathbf{R}^{3} }} \left(  ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m}   , 
(1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )
({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right)  d {\ensuremath{\mathbf{k}}}  \right| \notag \\
& \leq \left( \int_{{\mathbf{R}^{3} }} \left\| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m  \right\|^2 d {\ensuremath{\mathbf{p}}}  
 \right)^{1/2} \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )
(  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right\|^2  d {\ensuremath{\mathbf{p}}} \right)^{1/2} \notag \\
 & =  \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times  \left( \int_{{\mathbf{R}^{3} }} \left\| (1-F_{{\textrm{b}}, R'} )( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})   ) \Psi_m \right\|^2  d {\ensuremath{\mathbf{k}}} \right)^{1/2} . \notag\end{aligned}$$ By the photon derivative bounds in Theorem \[P-DB\] and the spectral decomposition theorem, $$\begin{aligned}
 & \int_{{\mathbf{R}^{3} }} \| (1-F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )(  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m  \|^2  d {\ensuremath{\mathbf{k}}}
 \notag \\
 & \leq 4 \int_{{\mathbf{R}^{3} }} \| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \, \frac{1}{1+ \hat{{\ensuremath{\mathbf{k}}}}^2} 
 ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m  \|^2  d {\ensuremath{\mathbf{k}}} \notag \\
& \qquad \qquad + 4 \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }} \|
 (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \, \frac{(\hat{k}^\nu )^2}{1+ \hat{{\ensuremath{\mathbf{k}}}}^2} 
 \, ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m  \|^2  d {\ensuremath{\mathbf{k}}} \notag  \\
 &  \leq \frac{4}{R'^4} 
  \int_{{\mathbf{R}^{3} }} \left\| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2  d {\ensuremath{\mathbf{k}}}
+ \frac{4}{R'^2}  \sum_{\nu=1}^3\int_{{\mathbf{R}^{3} }}  \left\| \partial_{k^{\nu}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2  d {\ensuremath{\mathbf{k}}} \notag \\
& \leq \frac{4}{R'^4}  \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|^2 + \frac{ 4 |{\kappa_{\textrm{I}}}|^2}{R'^2}\sum_{\nu =1}^3
   \int_{{\mathbf{R}^{3} }}|F_{r}^{\nu}({\ensuremath{\mathbf{k}}})|^2 d {\ensuremath{\mathbf{k}}} . \notag \end{aligned}$$ Then we have $$\begin{aligned}
& \left| \int_{{\mathbf{R}^{3} }} \left(  ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m}   , 
(1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )
({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right)  d {\ensuremath{\mathbf{k}}}  \right| \notag \\
& \leq  \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left(  \frac{4}{{R'}^4} \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|^2
+ \frac{ 4|{\kappa_{\textrm{I}}}|^2 }{{R'}^2} \sum_{\nu =1}^3
   \| F_{r}^{\nu} \|_{L^2}^2 \right)^{1/2} \notag , \end{aligned}$$ and hence, for all $R'>1$, $$\begin{aligned}
&  \|  ( ( {{\small \text{1}}\hspace{-0.32em}1}-  {\Gamma_{\textrm{b}}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ))  )  \Psi_{m} \|^2   \notag \\
&\leq  \frac{2}{R'}   \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left(  \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|
+|{\kappa_{\textrm{I}}}|\, \sum_{r=1,2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}\right) .\end{aligned}$$ From Lemma \[9/14.a\] (**ii**), it holds that $ \sup\limits_{j \in {\ensuremath{\mathbf{N}}}} \| (  {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j} \| < c_{0} |{\kappa_{\textrm{I}}}|
\left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} , \right\| $. Therefore the proof is obtained. $\blacksquare$\
$\;$\
[**(Proof of Theorem \[Main-Theorem\])**]{}\
From Proposition \[9/16.a\] and a general theorem ([@AH97] ; Lemmma 4.9), it is enough to show that w-$ \lim\limits_{j \to \infty} \Psi_{m_j} \neq 0$. We see that $$\begin{aligned}
 {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} & =   ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -{\Gamma_{\textrm{f}}}(F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}   +   {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} -  {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ))     \notag \\
& \quad + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([0, n] )) + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}}  )) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )  ) E_{N_{{\textrm{rad}}}}([ n+1, \infty) )) . \notag \end{aligned}$$ Then by Proposition \[9/15.a\], we have for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $R>1$ and $R'> 1 $, $$\begin{aligned}
& \left\| \left( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}}  )) {\otimes}( {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([0, n ] ) )\right) \Psi_{m_j}  \right\| 
\notag \\
& \geq 1-  \left(  \frac{}{} \| (( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}}  ))) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_{m_j}  \|  
  \right. \notag \\
& \qquad \qquad \qquad  \left. + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} -  
{\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) )  \Psi_{m_j}  \|
+ \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j}  \| \frac{}{}
 \right) . \notag \\
 & \geq 1-  \left( \frac{c_{1, \epsilon }}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}}  + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j}  \|  \right) , \notag\end{aligned}$$ It is seen that $$\sqrt{n+1 }\|  ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j}  \| \leq
 \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j}  \|  \leq c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}}\right\| .$$ Then from Lemma \[9/14.a\] **(ii)**, we have $$\sup\limits_{j \geq 1}\|  ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j}  \| \leq \frac{c_3 }{(n+1)^{1/2}}. \notag$$ where $c_{3}= c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}}\right\| $. Then it follows that $$\| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j}  \|
\geq 1 - \left( \frac{c_{1 , \epsilon}}{R} + \frac{c_2}{R'}  + \frac{c_3}{(n+1)^{1/2}} \right) .
\label{9/15.9}$$ We also see that $$\begin{aligned}
& \| {\Gamma_{\textrm{f}}}(  F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}(   F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j}  \|^2 
\notag \\ 
& = ((H_0 +1) E_{N_{{\textrm{rad}}}}([0, n ] ) \Psi_{m_j} ,(H_0 +1)^{-1} 
( {\Gamma_{\textrm{f}}}(F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] )) ) \Psi_{m_j}  ) \notag \\
& \leq \| (H_0 +1)\Psi_{m_j} \| \, \| (H_0 +1)^{-1} \, ( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2 ) {\otimes}(
 {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] )))  \, \Psi_{m_j}  \| . \notag\end{aligned}$$ We see that $\| (H_0 +1)\Psi_{m_j} \| \leq \| H_0 \Psi_{m_j} \| +1 \leq   \| H_{0,m_{j}} \Psi_{m_j} \| +1 $ and Lemma \[9/9.e\] yields that $$\| H_{0,m_{j}} \Psi_{m_j} \|
\leq \frac{L_{\epsilon } }{M}  \| H_{m_j}  \Psi_{m_j}  \| + \frac{R_{\epsilon }}{M} 
\leq \frac{L_{\epsilon}}{M} E_{\infty} + \frac{R_{\epsilon}}{M}  .$$ Then we have $$\begin{aligned}
&\| {\Gamma_{\textrm{f}}}(  F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}(   F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j}  \| \notag   \\
& \qquad \qquad \qquad \leq  c_{4 ,\epsilon}\| (H_0 +1)^{-1} \,
\left( ( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}(  {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))) \right) \, \Psi_{m_j}  \|^{1/2}, \label{9/15.10}\end{aligned}$$ where $c_{4 , \epsilon }= (\frac{L_{\epsilon}E_{\infty} + R_{\epsilon} }{M} +1)^{1/2}$. From (\[9/15.9\]) and (\[9/15.10\]) $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}(  F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2  E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j}   \| \geq  \frac{1}{c_{4, \epsilon }}^2 \left( 1 - \left( \frac{c_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}}  + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 \notag$$ Since $(H_0 +1)^{-1} \left( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}(   F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2) E_{N_{{\textrm{rad}}}}([0, n ] ))) \right) $ is compact, we have $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}(  F_{R}(\hat{{\ensuremath{\mathbf{p}}}})^2 ) {\otimes}( {\Gamma_{\textrm{b}}}(  F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{0}   \| \geq  \frac{1}{c_{4, \epsilon }^2}\left( 1 - \left( \frac{c_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}}  + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2  ,
 \label{9/15.20}$$ where we set $\Psi_{0} =$ w-$ \lim\limits_{j \to \infty} \Psi_{m_j} $. Then for sufficiently large $R>0$, $R'>0 $ and $n >0$, the right-hand side of (\[9/15.20\]) is greater than zero, and hence $  \Psi_{0}  \ne 0 $.\
$\;$\
**(Multiplicity)** Assume dim ker $(H -E_{0}(H_{{ \textrm{QED} }})) = \infty $. Let $ \Psi_{l} $, $l \in {\ensuremath{\mathbf{N}}}$, be the ground states. Let ${\ensuremath{\mathscr{M}}} $ be the closure of the linear hull of $\{ \Psi_l  \}_{l=0}^{\infty}$. Then ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }}$ is decomposed as ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }} = {\ensuremath{\mathscr{M}}} \oplus {\ensuremath{\mathscr{M}}}^{\bot} $. Let $\{ \Phi_l \}_{l=0}^{\infty}$ be a complete orthogonal system of ${\ensuremath{\mathscr{M}}}^{\bot}$. We can set a complete orthonormal system $\{ \Xi_l \}_{l=0}^{\infty}$ of ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }}$ by $  \Xi_{2 l-1 } = \Psi_{l} $ and $  \Xi_{2 l} = \Phi_{l} $ for all $l \in {\ensuremath{\mathbf{N}}}$. Since $\{ \Xi_l \}_{l=0}^{\infty}$ is a complete orthonormal system , w-$\lim\limits_{l \to \infty}\Xi_l =0$. On the other hand, $\Xi_{2 l-1 }$ is ground state for all $l \in {\ensuremath{\mathbf{N}}}$, and hence $H_{{ \textrm{QED} }}\Xi_{2 l-1 }=E_{0}({H_{\textrm{QED}}}) \Xi_{2 l-1 } $. In a same argument of the proof of the existence of the ground state, we have $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}(  F_{R}(\hat{{\ensuremath{\mathbf{p}}}})^2 ) {\otimes}( {\Gamma_{\textrm{b}}}(  
F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Xi_{2l-1}   \| \geq  \frac{1}{\tilde{c}_{4, \epsilon }^2}\left( 1 - \left( \frac{ \tilde{c}_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}}
  + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 ,
 \label{9/15.15}$$ where $\tilde{c}_{1, \epsilon} $ and $\tilde{c}_{4, \epsilon} $ are the constants $ c_{1 , \epsilon}$ and $ c_{1 , \epsilon}$ replacing $E_{\infty}$ with $E_{0}({H_{\textrm{QED}}})$. Then by taking sufficiently large $R>0$, $R'>0 $ and $n >0$, we have w-$\lim\limits_{l \to \infty}\Xi_{2l-1} \ne 0$, but this is contradict to w-$\lim\limits_{l \to \infty}\Xi_l = 0$. Hence dim ker $(H_{{ \textrm{QED} }} -E_{0}(H_{{ \textrm{QED} }})) < \infty $. $\blacksquare $\
$\;$\
$\;$\
[**\[Concluding remarks\]**]{}\
**(1) The case of Massless Dirac field**\
It is not realistic model, but we can consider the system of a massless Dirac field coupled to the radiation field. In such a case, by replacing (**A.5**) with similar conditions to (**A.6**), we can also prove the existence of the ground state in a same ways as ${H_{\textrm{QED}}}$. $\; $\
**(2) Infrared divergent problem**\
For some systems of particles coupled to massless Bose fields, the existence of the ground states without infrared regularity conditions was obtained (refer to e.g., Bach-Fröhlich-Sigal [@BFS99], Griesemer-Lieb-Loss [@GLL01] and Hasler-Herbst [@HaHe11]), and non-existence of the ground states for other other systems was also investigated (see e.g., Arai-Hirokawa-Hiroshima [@AHH99]). To prove the existence or non-existence of the ground state of ${H_{\textrm{QED}}}$ without infrared regularity conditions is left for future study.

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