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\begin{document}


\centerline{\textbf{\Large}}

\vskip 0.8truecm

\centerline{\textbf{\Large A VARIATIONAL CALCULATION OF POSITRONIUM}}

\vskip 0.3truecm \centerline{\textbf{\Large IN QED}}

\vskip 0.6truecm \centerline{\large Andrei G. Terekidi and Jurij W. Darewych}

\vskip 0.5truecm 
\centerline{\footnotesize \emph{Department of Physics and
Astronomy, York University, Toronto, Ontario, M3J 1P3, Canada}}

\vskip 0.6truecm

%\maketitle

\noindent \textbf{Abstract} We consider a reformulation of QED in which
covariant Green functions are used to solve for the electromagnetic field in
terms of the fermion fields. The resulting modified Hamiltonian contains the
photon propagator directly. A simple Fock-state variational trial function
is used to derive relativistic fermion-antifermion equations variationally
from the expectation value of the Hamiltonian of the field theory. The
interaction kernel of the equations are shown to involve the 
lowest order invariant $\mathcal{M\,}\ $matrices. The two-body eigenenergies are
shown to agree with known result for positronium-like systems to order $%
O\left( \alpha ^{4}\right) $\ in the coupling constant. %\end{abstract}

\vskip 0.8truecm {\normalsize \centerline{\textbf{\large Introduction}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\section{Introduction}
}It has been pointed out in previous publications that various models in
Quantum Field Theory (QFT), including QED, can be reformulated, using
mediating-field Green functions, into a form that is particularly convenient
for variational calculations of relativistic few-body bound states [1,2].
This approach was applied recently to the study of relativistic two-body
eigenstates in the scalar Yukawa (Wick-Cutkosky) theory [3,4,5]. We shall
implement such an approach to two-fermion states in QED in this paper, in
order to demonstrate its utility in the case of a well studied theory.

The Lagrangian of QED is $\left( \hbar =c=1\right) $%
\begin{eqnarray}
\mathcal{L} &=&\overline{\psi }(x)\left( i\gamma ^{\mu }\partial _{\mu
}-m-e\gamma ^{\mu }A_{\mu }(x)\right) \psi (x)  \notag \\
&&-\frac{1}{4}\left( \partial _{\alpha }A_{\beta }(x)-\partial _{\beta
}A_{\alpha }(x)\right) \left( \partial ^{\alpha }A^{\beta }(x)-\partial
^{\beta }A^{\alpha }(x)\right) .
\end{eqnarray}
The corresponding Euler-Lagrange equations of motion are the coupled
Dirac-Maxwell equations, 
\begin{equation}
\left( i\gamma ^{\mu }\partial _{\mu }-m\right) \psi (x)=e\gamma ^{\mu
}A_{\mu }(x)\psi (x),
\end{equation}
and 
\begin{equation}
\partial _{\mu }\partial ^{\mu }A^{\nu }(x)-\partial ^{\nu }\partial _{\mu
}A^{\mu }(x)=j^{\nu }(x)
\end{equation}
where 
\begin{equation}
j^{\nu }(x)=e\overline{\psi }(x)\gamma ^{\nu }\psi (x).
\end{equation}
The equations (2)-(3) can be decoupled in part by using the well-known
formal solution [6,7] of the Maxwell equation (3), namely 
\begin{equation}
A_{\mu }(x)=A_{\mu }^{0}(x)+\int d^{4}x^{\prime }D_{\mu \nu }(x-x^{\prime
})j^{\nu }(x^{\prime }),
\end{equation}
where $D_{\mu \nu }(x-x^{\prime })$ is a Green function (or photon
propagator in QFT terminology), defined by 
\begin{equation}
\partial _{\alpha }\partial ^{\alpha }D_{\mu \nu }(x-x^{\prime })-\partial
_{\mu }\partial ^{\alpha }D_{\alpha \nu }(x-x^{\prime })=g_{\mu \nu }\delta
^{4}(x-x^{\prime }),
\end{equation}
and $A_{\mu }^{0}(x)$ is a solution of the homogeneous (or ``free field'')
equation (3) with $j^{\mu }(x)=0.$

We recall, in passing, that equation (6) does not define the covariant Green
function $D_{\mu \nu }(x-x^{\prime })$ uniquely. For one thing, one can
always add a solution of the homogeneous equation (eq. (6) with $g_{\mu \nu
}\rightarrow 0$). This allows for a certain freedom in the choice of $D_{\mu
\nu }$, as is discussed in standard texts (e.g. ref. [6,7]). In practice,
the solution of eq. (6), like that of eq. (3), requires a choice of gauge.
However, we do not need to specify one at this stage.

Substitution of the formal solution (5) into equation (2) yields the
``partly reduced'' equations,

\begin{equation}
\left( i\gamma ^{\mu }\partial _{\mu }-m\right) \psi (x)=e\gamma ^{\mu
}\left( A_{\mu }^{0}(x)+\int d^{4}x^{\prime }D_{\mu \nu }(x-x^{\prime
})j^{\nu }(x^{\prime })\right) \psi (x),
\end{equation}
which is a nonlinear Dirac equation. To our knowledge no exact (analytic or
numeric) solution of equation (7) for classical fields have been reported in
the literature. However, approximate solutions have been discussed by
various authors, particularly Barut and his co-workers (see ref. [8,9] and
citations therein). In any case, our interest here is in the quantized 
field theory.

The partially reduced equation (7) is derivable from the stationary action
principle 
\begin{equation}
\delta S\left[ \psi \right] =\delta \int d^{4}x\mathcal{L}_{R}=0
\end{equation}
with the Lagrangian density 
\begin{equation}
\mathcal{L}_{R}=\overline{\psi }(x)\left( i\gamma ^{\mu }\partial _{\mu
}-m-e\gamma _{\mu }A_{0}^{\mu }(x)\right) \psi (x)-\frac{1}{2}\int
d^{4}x^{\prime }j^{\mu }(x^{\prime })D_{\mu \nu }(x-x^{\prime })j^{\nu }(x)
\end{equation}
provided that the Green function is symmetric in the sense that 
\begin{equation}
D_{\mu \nu }(x-x^{\prime })=D_{\mu \nu }(x^{\prime }-x)\;\;\;\;\;and\;\;\
\;\ D_{\mu \nu }(x-x^{\prime })=D_{\nu \mu }(x-x^{\prime })
\end{equation}

One can proceed to do conventional covariant perturbation theory using the
reformulated QED Lagrangian (9). The interaction part of (9) has a somewhat
modified structure from that of the usual formulation of QED. Thus, there
are two interaction terms. The last term of (9) is a ``current-current''
interaction which contains the photon propagator sandwiched between the
fermionic currents. As such, it corresponds to Feynman diagrams without
external photon lines. The term containing $A_{0}^{\mu }$ corresponds to
diagrams that cannot be generated by the term containing $D_{\mu \nu }$,
particularly diagrams involving external photon lines (care would have to be
taken not to double count physical effects). However, we shall not pursue
covariant perturbation theory in this work. Rather, we shall consider a
variational approach that allows one to derive relativistic few-fermion
equations, and to study their bound and scattering solutions.

%------------------------- Hamiltonian -------------------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Hamiltonian in the
canonical, equal-time formalism}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\section{Hamiltonian in the canonical, equal-time formalism}
}We consider this theory in the canonical, equal-time formalism. To this end
we write down the Hamiltonian density corresponding to the Lagrangian (9),
with the terms for the free $A_{0}^{\mu }(x)$ field suppressed since it will
not contribute to the results presented in this paper. The relevant
expression is: 
\begin{equation}
\mathcal{H}_{R}=\mathcal{H}_{0}\mathcal{+H}_{I},
\end{equation}
where

\begin{eqnarray}
\mathcal{H}_{0} &=&\psi ^{\dagger }(x)\left( -i\overrightarrow{\mathbf{%
\alpha }}\cdot \nabla +m\beta \right) \psi (x) \\
\mathcal{H}_{I} &=&\frac{1}{2}\int d^{4}x^{\prime }j^{\mu }(x^{\prime
})D_{\mu \nu }(x-x^{\prime })j^{\nu }(x),
\end{eqnarray}

Equal-time quantization corresponds to the imposition of anticommutation
rules for the fermion fields, namely 
\begin{equation}
\left\{ \psi _{\alpha }(\mathbf{x},t),\psi _{\beta }^{\dagger }(\mathbf{y}%
,t)\right\} =\delta _{\alpha \beta }\delta ^{3}\left( \mathbf{x}-\mathbf{y}%
\right) ,
\end{equation}
and all other vanish. In addition, if $A_{0}^{\mu }\neq 0,$ there are the
usual commutation rules for the $A_{0}^{\mu }$ field, and commutation of the 
$A_{0}^{\mu }$ field operators with the $\psi $ field operators.

To specify our notation, we quote the usual Fourier decomposition of the
field operators, namely 
\begin{equation}
\psi (x)=\sum_{s}\int \frac{d^{3}p}{\left( 2\pi \right) ^{3/2}}\left( \frac{m%
}{\omega _{p}}\right) ^{1/2}\left[ b_{\mathbf{p}s}u\left( \mathbf{p}%
,s\right) e^{-ip\cdot x}+d_{\mathbf{p}s}^{\dagger }v\left( \mathbf{p}%
,s\right) e^{ip\cdot x}\right] ,
\end{equation}
with $p=p^{\mu }=\left( \omega _{p},\mathbf{p}\right) $, and $\omega _{p}=%
\sqrt{m^{2}+\mathbf{p}^{2}}$. The mass-$m$ free-particle Dirac spinors $u$\
and $v$,\ where $\left( \gamma ^{\mu }{p}_{\mu }-m\right) u\left( 
\mathbf{p},s\right) =0$,\ $\left( \gamma ^{\mu }{p}_{\mu }+m\right)
v\left( \mathbf{p},s\right) =0$, are normalized such that 
\begin{eqnarray}
u^{\dagger }\left( \mathbf{p},s\right) u\left( \mathbf{p},\sigma \right) 
&=&v^{\dagger }\left( \mathbf{p},s\right) v\left( \mathbf{p},\sigma \right) =%
\frac{\omega _{p}}{m}\delta _{s\sigma } \\
u^{\dagger }\left( \mathbf{p},s\right) v\left( \mathbf{p},\sigma \right) 
&=&v^{\dagger }\left( \mathbf{p},s\right) u\left( \mathbf{p},\sigma \right)
=0
\end{eqnarray}

The creation and annihilation operators $b^{\dagger }$, $b$ of the (free)
fermions of mass $m$, and $d^{\dagger }$, $d$ for the corresponding
antifermions, satisfy the usual anticommutation relations. The
non-vanishing ones are 
\begin{equation}
\left\{ b_{\mathbf{p}s},b_{\mathbf{q}\sigma }^{\dagger }\right\} =\left\{ d_{%
\mathbf{p}s},d_{\mathbf{q}\sigma }^{\dagger }\right\} =\delta _{s\sigma
}\delta ^{3}\left( \mathbf{p}-\mathbf{q}\right) .
\end{equation}

%---------------------------   Var. principle ...  -----------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Variational
principle and fermion-antifermion trial states}} }

{\normalsize \vskip 0.4truecm }

{\normalsize 
%\section{Variational principle and fermion-antifermion trial states}
}Unfortunately we do not know how to obtain exact eigenstates of the
Hamiltonian (11). Therefore we shall resort to a variational approximation,
based on the variational principle 
\begin{equation}
\delta \langle \psi \mid \widehat{H}-E\mid \psi \rangle _{t=0}=0 .
\end{equation}
For a system like $e^{+}e^{-}$, the simplest Fock-space trial state that can
be written down in the rest frame is

\begin{equation}
\mid e^{+}e^{-}\rangle =\underset{s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}%
F_{s_{1}s_{2}}(\mathbf{p})b_{\mathbf{p}s_{1}}^{\dagger }d_{-\mathbf{p}%
s_{2}}^{\dagger }\mid 0\rangle ,
\end{equation}
where $F_{s_{1}s_{2}}$ are four adjustable functions. We use this trial
state to evaluate the matrix elements needed to implement the variational
principle (19), namely 
\begin{equation}
\langle e^{+}e^{-}\mid :\widehat{H}_{0}-E:\mid e^{+}e^{-}\rangle =\underset{%
s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}F_{s_{1}s_{2}}^{\ast }(\mathbf{p}%
)F_{s_{1}s_{2}}(\mathbf{p})\left( 2\omega _{p}-E\right)
\end{equation}
and

\begin{equation*}
\langle e^{+}e^{-}\mid :\widehat{H}_{I}:\mid e^{+}e^{-}\rangle =\frac{%
e^{2}m^{2}}{\left( 2\pi \right) ^{3}}\underset{s_{1}^{\prime }s_{2}^{\prime
}s_{1}s_{2}}{\sum }\int \frac{d^{3}\mathbf{p}d^{3}\mathbf{p}^{\prime }}{%
\omega _{p}\omega _{p^{\prime }}}F_{s_{1}s_{2}}^{\ast }(\mathbf{p}%
)F_{s_{1}^{\prime }s_{2}^{\prime }}(\mathbf{p}^{\prime })\times
\end{equation*}

\begin{equation}
\times \left( 
\begin{array}{c}
\overline{u}\left( \mathbf{p},s_{1}\right) \gamma ^{\mu }u\left( \mathbf{p}%
^{\prime },s_{1}^{\prime }\right) D_{\mu \nu }(p-p^{\prime })\overline{v}%
\left( -\mathbf{p}^{\prime },s_{2}^{\prime }\right) \gamma ^{\nu }v\left( -%
\mathbf{p,}s_{2}\right)  \\ 
-\overline{u}\left( \mathbf{p},s_{1}\right) \gamma ^{\mu }v\left( -\mathbf{p,%
}s_{2}\right) D_{\mu \nu }\left( p+p^{\prime }\right) \overline{v}\left( -\mathbf{p}%
^{\prime },s_{2}^{\prime }\right) \gamma ^{\nu }u\left( \mathbf{p}^{\prime
},s_{1}^{\prime }\right)  
\end{array}
\right)\, , 
\end{equation}
where $p=(\omega_p, {\mathbf p})$, $p^\prime =(\omega_{p^\prime}
, {\mathbf p}^\prime)$, with ${\mathbf p} +
 {\mathbf p}^\prime = 0$ ({\it i.e.} $p+p^\prime = (2 \omega_{\mathbf p}, 0)$) in the rest frame, and  
\begin{equation}
D_{\mu \nu }(x-x^{\prime })=\int \frac{d^{4}k}{(2\pi )^{4}}D_{\mu \nu
}(k)e^{-ik\cdot (x-x^{\prime })}.
\end{equation}
\par
We have normal-order the entire Hamiltonian, since this circumvents the need
for mass renormalization which would otherwise arise. Not that there is a
difficulty with handling mass renormalization in the present formalism (as
shown in various earlier papers; see, for example, ref. [10] and citations
therein). It is simply that we are not interested in mass renormalization
here, since it has no effect on the two-body bound state energies that we
obtain in this paper. Furthermore, the approximate trial state (20), which
we use in this work, is incapable of sampling loop effects. Thus, the
normal-ordering of the entire Hamiltonian does not ``sweep under the
carpet'' loop effects, since none arise at the present level of
approximation, that is with the trial state $\mid e^{+}e^{-}\rangle $ 
specified in eq. (20) .

The variational principle (19) leads to the following equation 
\begin{eqnarray}
&&\sum_{s_{1}s_{2}}\int d^{3}\mathbf{p}\left( 2\omega _{p}-E\right)
F_{s_{1}s_{2}}(\mathbf{p})\delta F_{s_{1}s_{2}}^{\ast }(\mathbf{p})  \notag
\\
&&-\frac{m^{2}}{\left( 2\pi \right) ^{3}}\underset{\sigma _{1}\sigma
_{2}s_{1}s_{2}}{\sum }\int \frac{d^{3}\mathbf{p}d^{3}\mathbf{q}}{\omega
_{p}\omega _{q}}F_{\sigma _{1}\sigma _{2}}(\mathbf{q})\left( -i\right) 
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right)
\delta F_{s_{1}s_{2}}^{\ast }(\mathbf{p})=0,
\end{eqnarray}
where $\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}%
\right) $ is an invariant ``matrix element", which contains two terms:
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right) =%
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope}\left( \mathbf{p,q}%
\right) +\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ann}\left( \mathbf{%
p,q}\right) ,
\end{equation}
where
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope}\left( \mathbf{p,q}%
\right) =\overline{u}\left( \mathbf{p},s_{1}\right) \left( -ie\gamma ^{\mu
}\right) u\left( \mathbf{q},\sigma _{1}\right) iD_{\mu \nu }(p-q)\overline{v}%
\left( -\mathbf{q},\sigma _{2}\right) \left( -ie\gamma ^{\nu }\right)
v\left( -\mathbf{p},s_{2}\right) ,
\end{equation}
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ann}\left( \mathbf{p,q}%
\right) =-\overline{u}\left( \mathbf{p},s_{1}\right) \left( -ie\gamma ^{\mu
}\right) v\left( -\mathbf{p},s_{2}\right) iD_{\mu \nu }\left( p+q\right) 
\overline{v}\left( -\mathbf{q},\sigma _{2}\right) \left( -ie\gamma ^{\nu
}\right) u\left( \mathbf{q},\sigma _{1}\right)\, ,
\end{equation}
correspond to the usual one-photon exchange and virtual annihilation Feynman diagrams. 

We stress that these $\mathcal{M}$-matrices appear naturally in this formalism, that
is the $\mathcal{M}$'s are not put in by hand, nor does their 
derivation require
additional Fock-space terms in the variational trial state (20), as is the
case in traditional formulations (e.g. [11]-[13]).

In the non-relativistic limit, the functions $F_{s_{1}s_{2}}$ can be written
as 
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=F(\mathbf{p})\Lambda _{s_{1}s_{2}},
\end{equation}
where the non-zero elements of $\Lambda _{ij}$ for total spin singlet ($S=0$%
) states are $\Lambda _{12}=-\Lambda _{21}=\frac{1}{\sqrt{2}}$, while for
the spin triplet ($S=1$)\ states the non-zero elements are $\Lambda _{11}=1$
for $m_{s}=+1,$ $\Lambda _{12}=\Lambda _{21}=\frac{1}{\sqrt{2}}$ for $%
m_{s}=0 $, and $\Lambda _{22}=1$ for $m_{s}=-1$. We use the notation that
the subscripts $1$ and 2 of $\Lambda $ correspond to $m_{s}=1/2$ and $%
m_{s}=-1/2$ (or $\uparrow $\ and\ $\downarrow $) respectively. Substituting
(28) into (24), the variational procedure, after multiplying the result by $%
\Lambda _{s_{1}s_{2}}$ and summing over $s_{1}$ and $s_{2}$, gives the
equation 
\begin{equation}
(2\omega _{p}-E)F(\mathbf{p})=\frac{1}{(2\pi )^{3}}\int d^{3}\mathbf{q}%
\,\mathcal{K}(\mathbf{p},\mathbf{q})F(\mathbf{q}),
\end{equation}
where 
\begin{equation}
\mathcal{K}(\mathbf{p},\mathbf{q})=-i\frac{m^{2}}{\omega _{p}\omega _{q}}%
\sum_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\Lambda _{s_{1}s_{2}}\mathcal{M}%
_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right) \Lambda
_{\sigma _{1}\sigma _{2}}.
\end{equation}
To lowest-order in $|{\bf p}|/m$ (i.e. in the non-relativistic limit), the kernel
(30) reduces to $\mathcal{K}=e^{2}/\left| \mathbf{p-q}\right| ^{2}$, and so (29)
reduces to the (momentum-space) Schr\"{o}dinger equation 
\begin{equation}
\left( \frac{\mathbf{p}^{2}}{2\mu }-\varepsilon \right) F(\mathbf{p})=\frac{1%
}{(2\pi )^{3}}\int d^{3} \, \mathbf{q}\frac{e^{2}}{\left| \mathbf{p-q}\right|
^{2}}F(\mathbf{q}),
\end{equation}
where $\varepsilon =E-2m$ and $\mu =m/2$. This verifies that the
relativistic two-fermion equation (24) has the expected non-relativistic
limit.

In the relativistic case we do not complete the variational procedure in
(24) at this stage to get equations for the four adjustable functions $%
F_{s_{1}s_{2}}$, because they are not independent in general. Indeed we
require that the trial state be an eigenstate of the total angular momentum
operator (in relativistic form), its projection, parity and charge
conjugation, namely that 
\begin{equation}
\left[ 
\begin{array}{c}
\widehat{\mathbf{J}}^{2} \\ 
\widehat{J}_{3} \\ 
\widehat{\mathcal{P}} \\ 
\widehat{\mathcal{C}}
\end{array}
\right] \mid e^{+}e^{-}\rangle =\left[ 
\begin{array}{c}
J\left( J+1\right) \\ 
m_{J} \\ 
P \\ 
C
\end{array}
\right] \mid e^{+}e^{-}\rangle \, ,
\end{equation}
where $m_J = J, J-1, ..., -J$ as usual.
Explicit forms for the operators $\widehat{\mathbf{J}}^{2}$, $\widehat{J}_{3}$
are given in Appendix A. 
\par
The functions $F_{s_{1}s_{2}}(\mathbf{p})$ can be
written in the general form 
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=\sum_{\ell
_{s_{1}s_{2}}}\sum_{m_{s_{1}s_{2}}}f_{s_{1}s_{2}}^{\ell
_{s_{1}s_{2}}m_{s_{1}s_{2}}}\left( p\right) Y_{\ell
_{s_{1}s_{2}}}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}}),
\end{equation}
where $Y_{\ell _{s_{1}s_{2}}}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}})$\ are
the usual spherical harmonics. Here and henceforth we will use the notation $%
p=\left| \mathbf{p}\right| $ etc. (four-vectors will be written as $p^{\mu }$%
). The orbital indexes $\ell _{s_{1}s_{2}}$and $m_{s_{1}s_{2}}$ depend on
the spin indexes $s_{1}$ and $s_{2}$\ and are specified by equations (32).
The radial coefficients $f_{s_{1}s_{2}}^{\ell
_{s_{1}s_{2}}m_{s_{1}s_{2}}}\left( p\right) $ in the expansion (33)  also depend on the spin variables. 
\par
Substitution of (33) into (20) and then into (32) leads to two
categories of relations among the adjustable functions, as shown 
in Appendices A and B. It follows from
this solution that the total spin of the system is a good quantum number, and the
states of the system separate into singlet states with the total spin $S=0$
(parastates) and into triplet states with $S=1$ (orthostates). We should point out that this phenomenon is characteristic of the fermion antifermion systems, which are charge conjugation eigenstates, and does not arise for systems like $\mu^+ e^-$.

\vskip .2cm
\noindent \textbf{The singlet states}

In this case $\ell _{s_{1}s_{2}}\equiv \ell =J,\;m_{11}=m_{22}=0\ $and$%
\;m_{12}=m_{21}=m_{J}$.\ The nonzero components of $F_{s_{1}s_{2}}(\mathbf{p}%
)$ are $F_{\uparrow \downarrow }(\mathbf{p})\equiv F_{12}(\mathbf{p}%
),\;F_{\downarrow \uparrow }(\mathbf{p})\equiv F_{21}(\mathbf{p})$\ and have
the form 
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=f_{s_{1}s_{2}}^{\left( sgl\right)J
}(p)Y_{J }^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}}),
\end{equation}
where the relations between \ $f_{12}^{\left( sgl\right) J}(p)\;$and$%
\;\,f_{21}^{\left( sgl\right) J}(p)$ involve the Clebsch-Gordan (C-G)
coefficients $C_{Jm_{J}}^{\left( sgl\right) Jm_{s_{1}s_{2}}}$, that is 
\begin{equation}
f_{s_{1}s_{2}}^{\left( sgl\right) J}(p)=C_{Jm_{J}}^{\left( sgl\right)
Jm_{s_{1}s_{2}}}f^{J}(p),
\end{equation}
as it shown in Appendix A. We see that the spin and radial variables
separate for the singlet states in the sense that the factors $f_{s_{1}s_{2}}^{\left( sgl\right)J}(p)$ have a
common radial function $f^{J}(p)$. Thus, for the singlet states we obtain 
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=C_{Jm_{J}}^{\left( sgl\right)
Jm_{s_{1}s_{2}}}f^{J}(p)Y_{J}^{m_{J}}(\hat{\mathbf{p}}).
\end{equation}
The C-G coefficients $C_{Jm_{J}}^{\left( sgl\right) Jm_{s_{1}s_{2}}}$\ have
a simple form: $C_{Jm_{J}}^{\left( sgl\right) Jm_{11}}=C_{Jm_{J}}^{\left(
sgl\right) Jm_{22}}=0$,\ $C_{Jm_{J}}^{\left( sgl\right)
Jm_{12}}=-C_{Jm_{J}}^{\left( sgl\right) Jm_{21}}=1$ (see Appendix A).\
Therefore for the singlet states we can write expression (20) in the
explicit form 
\begin{equation}
\mid e^{+}e^{-}\rangle =\int d^{3}\mathbf{p}f^{J}(p)Y_{J}^{m_{J}}(\hat{%
\mathbf{p}})\left( b_{\mathbf{p}\uparrow }^{\dagger }d_{-\mathbf{p}%
\downarrow }^{\dagger }-b_{\mathbf{p}\downarrow }^{\dagger }d_{-\mathbf{p}%
\uparrow }^{\dagger }\right) \mid 0\rangle .
\end{equation}
These states are characterized by the quantum numbers $J,m_{J}$ parity $%
P=(-1)^{J+1}$ and charge conjugation $C=\left( -1\right) ^{J}$. As we can
see, the quantum numbers $\ell $ (orbital angular momentum), and total spin $%
S$ are good quantum numbers for the singlet states as well. The
spectroscopical notation is $^{1}J_{J}$.

\vskip .2cm
\noindent  \textbf{The triplet states}

The solution of the system (32) for $S=1$  leads to two cases (Appendix A), namely
$\ell _{s_{1}s_{2}}\equiv \ell =J$, for which

\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=f_{s_{1}s_{2}}^{\left( tr\right)
J}(p)Y_{J}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}}),
\end{equation}
and $\ell _{s_{1}s_{2}}\equiv \ell =J\mp 1$, for which

\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=f_{s_{1}s_{2}}^{J-1}(p)Y_{J-1}^{m_{s_{1}s_{2}}}(%
\hat{\mathbf{p}})+f_{s_{1}s_{2}}^{J+1}(p)Y_{J+1}^{m_{s_{1}s_{2}}}(%
\hat{\mathbf{p}}),
\end{equation}
where 
\begin{equation}
m_{11}=m_{J}-1,\;\ \ \ m_{12}=m_{21}=m_{J},\;\ \ \ m_{22}=m_{J}+1
\end{equation}
The expressions for $f_{s_{1}s_{2}}^{\ell }(p)$ in both cases involve the
C-G coefficients $C_{Jm_{J}}^{\left( tr\right) \ell m_{s}}$ for $S=1$ listed
in Appendix A, that is 
\begin{equation}
f_{s_{1}s_{2}}^{\left( tr\right) \ell }(p)=C_{Jm_{J}}^{\left( tr\right) \ell
m_{s}}f^{\ell }(p),
\end{equation}
where the index $m_{s}$ is defined as 
\begin{eqnarray}
m_{s} &=&+1,\;\ \ when\;\ \ m_{s_{1}s_{2}}=m_{11},  \notag \\
m_{s} &=&0,\;\ \ \ \ when\;\ \ \ m_{s_{1}s_{2}}=m_{12}=m_{21}, \\
m_{s} &=&-1,\;\ \ when\;\ \ m_{s_{1}s_{2}}=m_{22}  \notag
\end{eqnarray}
\par
Thus, for the triplet states with $\ell =J$%
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=C_{Jm_{J}}^{\left( tr\right)
Jm_{s}}f^{J}(p)Y_{J}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}})
\end{equation}
These functions correspond to states, which can be characterized by the 
quantum numbers $J,m_{J}$, parity $P=(-1)^{J+1}$ and charge conjugation 
$C=\left( -1\right) ^{J+1}$. The orbital angular momentum $\ell $, as well as the 
 total spin $S=1$, are good quantum numbers in this case. 
The spectroscopic notation for these states is $^{3}J_{J}$.
\par
For the triplet states with $\ell =J\mp 1$ we obtain the result 
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=C_{Jm_{J}}^{\left( tr\right)
(J-1)m_{s}}f^{J-1}(p)Y_{J-1}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}}%
)+C_{Jm_{J}}^{\left( tr\right) (J+1)m_{s}}f^{J+1}(p)Y_{J+1}^{m_{s_{1}s_{2}}}(%
\hat{\mathbf{p}}),
\end{equation}
which involves two radial functions $f^{J-1}(p)$ and $f^{J+1}(p)$
corresponding to $\ell =J-1$ and $\ell =J+1$. This means that $\ell $ is not
a good quantum number. Such states are characterized by quantum numbers $J,$
$m_{J},$ $P=(-1)^{J}$, charge conjugation $C=\left( -1\right) ^{J}$ and spin 
$S=1$. In spectroscopic notation, these states are a mixture of $^{3}\left(
J-1\right) _{J}$ and $^{3}\left( J+1\right) _{J}$\ states.
\par
The requirement that the states be charge conjugation eigenstates 
(the last equation of (32)) is intimately tied to the conservation of total spin.
 Indeed, a linear combination of singlet and triplet states like 
\begin{equation}
F_{s_{1}s_{2}}(\mathbf{p})=C_{1}f_{s_{1}s_{2}}^{\left( sgl\right)
J}(p)Y_{J}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}})+C_{2}f_{s_{1}s_{2}}^{%
\left( tr\right) J}(p)Y_{J}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}}),
\end{equation}
satisfies the first tree equations of (32). However, it is unacceptable for 
describing a fermion-antifermion system  because
the first and the second terms in (45) have different charge conjugation.
For a system of two particles of different mass (such as $\mu^+ e^-$) charge 
conjugation is not
applicable, so that the total spin would not be  conserved. 

%---------------------------- Radial equations------------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large The relativistic
radial equations for positronium-like systems}} }

{\normalsize \vskip 0.4truecm }

{\normalsize 
%\section{The bound-state relativistic radial equations for $e^{+}e^{-}$ system}
 We return to equation (24) and replace the functions $F_{s_{1}s_{2}}(%
\mathbf{p})$ by the expression (36) for singlet states and by (43) and (44)
for triplet states. The variational procedure then leads to the following results:

For the singlet states $\ell =J$, $P=(-1)^{J+1}$, $C=(-1)^{J}$, the radial
equations are

\begin{equation}
\left( 2\omega _{p}-E\right) f^{J}(p)=\frac{m^{2}}{\left( 2\pi \right) ^{3}}%
\int \frac{q^{2}dq}{\omega _{p}\omega _{q}}\mathcal{K}^{\left( sgl\right)
}\left( p,q\right) f^{J}(q),
\end{equation}
where the kernel 
\begin{equation}
\mathcal{K}^{\left( sgl\right) }\left( p,q\right) =\underset{%
s_{1}s_{2}\sigma _{1}\sigma _{2}m_{J}}{-i\sum }\int d\hat{\mathbf{p}}\,d%
\hat{\mathbf{q}}\,C_{Jm_{J}}^{\left( sgl\right) s_{1}s_{2}\sigma
_{1}\sigma _{2}}\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{%
p,q}\right) Y_{J}^{m_{J}\ast }(\hat{\mathbf{p}})Y_{J}^{m_{J}}(\hat{%
\mathbf{q}}),
\end{equation}
is defined by the invariant $\mathcal{M}$-matrix and the coefficients 
\begin{equation}
C_{Jm_{J}}^{\left( sgl\right) s_{1}s_{2}\sigma _{1}\sigma _{2}}\equiv
C_{Jm_{J}}^{\left( sgl\right) Jm_{\sigma }}C_{Jm_{J}}^{\left( sgl\right)
Jm_{s}}/\sum_{\nu _{1}\nu _{2}m_{J}}\left( C_{Jm_{J}}^{\left( sgl\right)
Jm_{\nu }}\right) ^{2}.
\end{equation}
Here we have summed over $m_{J}$, because of the $(2J+1)$-fold energy degeneracy.

For the triplet states, we obtain different equations for the $\ell
=J$, and $\ell =J\mp 1$ cases.
Thus for the states with $\ell =J$, $P=(-1)^{J+1}$, $C=(-1)^{J+1}$ the
result is
\begin{equation}
\left( 2\omega _{p}-E\right) f^{J}(p)=\frac{m^{2}}{\left( 2\pi \right) ^{3}}%
\int \frac{q^{2}dq}{\omega _{p}\omega _{q}}\mathcal{K}^{\left( tr\right)
}(p,q)f^{J}(q),
\end{equation}
where the kernel $\mathcal{K}^{\left( tr\right) }$\ is formally like that of
(47), namely,

\begin{equation}
\mathcal{K}^{\left( tr\right) }(p,q)=\underset{s_{1}s_{2}\sigma _{1}\sigma
_{2}m_{J}}{-i\sum }C_{Jm_{J}}^{\left( tr\right) s_{1}s_{2}\sigma _{1}\sigma
_{2}}\int d\hat{\mathbf{p}}\,d\hat{\mathbf{q}}\,\mathcal{M}%
_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right)
Y_{J}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{p}})Y_{J}^{m_{\sigma _{1}\sigma
_{2}}}(\hat{\mathbf{q}}).
\end{equation}
However it involves different C-G coefficients, namely

\begin{equation}
C_{Jm_{J}}^{\left( tr\right) s_{1}s_{2}\sigma _{1}\sigma
_{2}}=C_{Jm_{J}}^{\left( tr\right) Jm_{\sigma }}C_{Jm_{J}}^{\left( tr\right)
Jm_{s}}/\sum_{\nu _{1}\nu _{2}m_{J}}\left( C_{Jm_{J}}^{\left( tr\right)
Jm_{\nu }}\right) ^{2}.
\end{equation}

For the triplet states with $\ell =J\mp 1$, we have two independent radial
functions $f^{J-1}(p)\;$and$\;f^{J+1}(p)$.\ Thus the variational equation
(24) leads to a system of coupled equations for $f^{J-1}(p)\;$and$%
\;f^{J+1}(p)$. It is convenient to write them in matrix form,

\begin{equation}
\left( 2\omega _{p}-E\right) \mathbb{F}\left( p\right) =\frac{m^{2}}{\left(
2\pi \right) ^{3}}\int \frac{q^{2}dq}{\omega _{p}\omega _{q}}\mathbb{K}%
\left( p,q\right) \mathbb{F}\left( q\right) ,
\end{equation}
where 
\begin{equation}
\mathbb{F}\left( p\right) =\left[ 
\begin{array}{c}
f^{J-1}(p) \\ 
f^{J+1}(p)
\end{array}
\right] ,
\end{equation}
and 
\begin{equation}
\mathbb{K}\left( p,q\right) =\left[ 
\begin{array}{cc}
\mathcal{K}_{11}\left( p,q\right) & \mathcal{K}_{12}\left( p,q\right) \\ 
\mathcal{K}_{21}\left( p,q\right) & \mathcal{K}_{22}\left( p,q\right)
\end{array}
\right] .
\end{equation}
The kernels $\mathcal{K}_{ij}$ are similar in form to (47) and (50), that is 
\begin{equation}
\mathcal{K}_{ij}\left( p,q\right) =\underset{\sigma _{1}\sigma
_{2}s_{1}s_{2}m_{J}}{-i\sum }C_{Jm_{J}ij}^{s_{1}s_{2}\sigma _{1}\sigma
_{2}}\int d\hat{\mathbf{p}}\,d\hat{\mathbf{q}}\,\mathcal{M}%
_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right) Y_{\ell
_{j}}^{m_{\sigma _{1}\sigma _{2}}}(\hat{\mathbf{q}})Y_{\ell
_{i}}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{p}}).
\end{equation}
However the coefficients $C_{Jm_{J}ij}^{s_{1}s_{2}\sigma _{1}\sigma _{2}}$
are defined by expression

\begin{equation}
C_{Jm_{J}ij}^{s_{1}s_{2}\sigma _{1}\sigma _{2}}=C_{Jm_{J}}^{\left( tr\right)
\ell _{j}m_{\sigma }}C_{Jm_{J}}^{\left( tr\right) \ell
_{i}m_{s}}/\sum_{s_{1}s_{2}m_{J}}\left( C_{Jm_{J}}^{\left( tr\right) \ell
_{i}m_{s}}\right) ^{2},
\end{equation}
where $\ell _{1}=J-1,\;\ell _{2}=J+1$ and $m_S$ is as defined in Eq. (42).
 The system (52) reduses to a single
equation for $J=0$ since $f^{J-1}(p)=0$ in that case.

To our knowledge, it is not possible to obtain analytic solution of the
relativistic radial momentum-space equations (46), (49) and (52). Thus one
must resort to numerical or other approximation methods. Numerical solutions
of such equations are discussed, for example, in [10], while a variational
approximation has been employed in [5]. However, in this paper we shall
resort to perturbative approximations, in order to verify that our\ equations
agree with known result for positronium to $O(\alpha ^{4})$.\ We expect that
this must be so given that the interaction kernels (i.e. momentum-space
potentials) of our equations involve both ``tree-level'' Feynman diagrams.

\par

 Our equations will yield energies which are evidently incomplete 
beyond $O(\alpha ^{4})$. One could,
of course, augment them by the addition of invariant matrix elements
corresponding to higher-order Feynman diagrams to the existing $\mathcal{M}$%
-matrices in the kernels of our equations (as is done in the Bethe-Salpeter formalism).
 Indeed, such an approach has been
used in a similar, though not variational, 
treatment of positronium by Zhang
and Koniuk [14]. These authors show that the inclusion of invariant 
matrix elements corresponding to single-loop
diagrams yields positronium energy eigenstates which are accurate to $%
O\left( \alpha ^{5},\alpha ^{5}\ln \alpha \right) $. However such ad-hoc
augmentation of the kernels would be contrary to the spirit of the present
variational treatment, and we shall not pursue it in this work.

%------------------------------------  Semi-relativistic ---------------------------------

\vskip 0.8truecm {\normalsize 
\centerline{\textbf{\large Semi-relativistic
expansions and the non-relativistic limit}} }

{\normalsize \vskip 0.4truecm }

For perturbative solutions of our radial equations, it is necessary to work
out expansions of the relevant expressions to first order beyond the
non-relativistic limit. This shall be summarized in the present section. We
perform the calculation in the Coulomb gauge, in which the photon
propagator has the form [15] 
\begin{equation}
D_{00}\left( \mathbf{k}\right) =-\frac{1}{\mathbf{k}^{2}},\;\;D_{0l}\left( 
\mathbf{k}\right) =0,\;\;D_{Al}\left( k^{\mu }\right) =-\frac{1}{k^{\mu
}k_{\mu }}\left( \delta _{Al}-\frac{k_{k}k_{l}}{\mathbf{k}^{2}}\right) ,
\end{equation}
where $k^{\mu }=\left( \omega _{p}-\omega _{q},\mathbf{p-q}\right) $.

To expand the amplitudes $\mathcal{M}$\ of (26) and (27)\  to 
one order of $\left( p/m\right) ^{2}$ beyond the non-relativistic 
limit, we take the free-particle spinors to be 
\begin{equation}
u(\mathbf{p,}i)=\left[ 
\begin{array}{c}
\left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)  \\ 
\frac{(\overrightarrow{\mathbf{\sigma }}\cdot \mathbf{p})}{2m}
\end{array}
\right] \varphi _{i},\;\ \ \ \ \ \ \ v(\mathbf{p,}i)=\left[ 
\begin{array}{c}
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{2m} \\ 
\left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right) 
\end{array}
\right] \chi _{i},
\end{equation}
as discussed in Appendix C. In this approximation the photon 
propagator takes on the form 

\begin{equation}
D_{00}\left( \mathbf{p-q}\right) =-\frac{1}{\left( \mathbf{p-q}\right) ^{2}}%
,\;\;\;D_{kl}\left( \mathbf{p-q}\right) \simeq \frac{1}{\left( \mathbf{p-q}%
\right) ^{2}}\left( \delta _{kl}-\frac{\left( p-q\right) _{k}\left(
p-q\right) _{l}}{\left( \mathbf{p-q}\right) ^{2}}\right) .
\end{equation}
Corresponding calculations give for the orbital part of $\mathcal{M}$-matrix 
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope\left( orb\right) }(%
\mathbf{p},\mathbf{q})=ie^{2}\left\{ \frac{1}{\left( \mathbf{p-q}\right) ^{2}%
}+\frac{1}{m^{2}}\left( \frac{1}{4}+\frac{\mathbf{q\cdot p}}{\left( \mathbf{%
p-q}\right) ^{2}}+\frac{\left( \mathbf{p}\times \mathbf{q}\right) ^{2}}{%
\left( \mathbf{p}-\mathbf{q}\right) ^{4}}\right) \right\} \delta
_{s_{1}\sigma _{1}}\delta _{s_{2}\sigma _{2}}.
\end{equation}
The  terms of the expansion linear in spin correspond to the spin-orbital
interaction 
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope\left( s-o\right) }(%
\mathbf{p},\mathbf{q})=\frac{3ie^{2}}{4m^{2}}\varphi _{s_{1}}^{\dagger }\chi
_{\sigma _{2}}^{\dagger }\frac{\left( \overrightarrow{\mathbf{\sigma }}%
^{\left( +\right) }-\overrightarrow{\mathbf{\sigma }}^{\left( -\right)
}\right) \cdot \left( \mathbf{p\times q}\right) }{\left( \mathbf{p-q}\right)
^{2}}\varphi _{\sigma _{1}}\chi _{s_{2}}.
\end{equation}
Here $\overrightarrow{\mathbf{\sigma }}^{\left( +\right) }$ and $%
\overrightarrow{\mathbf{\sigma }}^{\left( -\right) }$\ are positron and
electron\ spin matrices respectively, defined as follow:
 $\overrightarrow{\mathbf{\sigma }}^{\left( +\right) }\varphi
_{\sigma _{1}}\chi _{s_{2}}=\left( \overrightarrow{\mathbf{\sigma }}^{\left(
+\right) }\varphi _{\sigma _{1}}\right) \chi _{s_{2}}$, $\overrightarrow{%
\mathbf{\sigma }}^{\left( -\right) }\varphi _{\sigma _{1}}\chi
_{s_{2}}=\varphi _{\sigma _{1}}\left( \overrightarrow{\mathbf{\sigma }}%
^{\left( -\right) }\chi _{s_{2}}\right) $. The quadratic spin terms or
spin-spin interaction terms are 
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope\left( s-s\right) }(%
\mathbf{p},\mathbf{q})=\frac{ie^{2}}{4m^{2}}\varphi _{s_{1}}^{\dagger }\chi
_{\sigma _{2}}^{\dagger }\left\{ -\frac{\left( \overrightarrow{\mathbf{%
\sigma }}^{\left( +\right) }\cdot \left( \mathbf{p-q}\right) \right) \left( 
\overrightarrow{\mathbf{\sigma }}^{\left( -\right) }\cdot \left( \mathbf{p-q}%
\right) \right) }{\left( \mathbf{p-q}\right) ^{2}}+\overrightarrow{\mathbf{%
\sigma }}^{\left( +\right) }\cdot \overrightarrow{\mathbf{\sigma }}^{\left(
-\right) }\right\} \varphi _{\sigma _{1}}\chi _{s_{2}}.
\end{equation}
Lastly, the virtual annihilation contribution is given by

\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ann}(\mathbf{p},\mathbf{q})=-
\frac{ie^{2}}{4m^{2}}\varphi _{s_{1}}^{\dagger }\chi _{\sigma _{2}}^{\dagger
}\left\{ \overrightarrow{\mathbf{\sigma }}^{\left( +\right) }\cdot 
\overrightarrow{\mathbf{\sigma }}^{\left( -\right) }\right\} \varphi
_{\sigma _{1}}\chi _{s_{2}}.
\end{equation}
\par 
We have used these expressions to obtain the corresponding radial kernels.
Details of calculations can be found in Appendix D. We use the\
notation: $z=\left( p^{2}+q^{2}\right) /2pq$, and\ $Q_{\lambda }(z)$ \ is
the Legendre function of the second kind [16]. The contributions of the
various terms to the kernel are as follows:

\par
\noindent \textbf{The singlet states} $\left( \ell =J\ (J\geq
0),\;P=(-1)^{J+1},\;C=\left( -1\right) ^{J}\right) $:

\textit{Orbital term} 
\begin{equation}
\mathcal{K}^{\left( sgl\right) \left( o\right) }\left( p,q\right) =\frac{%
2\pi e^{2}}{pq}Q_{J}(z)+\frac{\pi e^{2}}{m^{2}}\left( -\frac{J-3}{2}\left( 
\frac{p}{q}+\frac{q}{p}\right) Q_{J}(z)+\left( J+1\right) Q_{J+1}(z)-2\delta
_{J,0}\right) ,
\end{equation}

\textit{Spin-orbital interaction}

\begin{equation}
\mathcal{K}^{\left( sgl\right) \left( s-o\right) }\left( p,q\right) =0
\end{equation}

\textit{Spin-spin interaction}

\begin{equation}
\mathcal{K}^{\left( sgl\right) \left( s-s\right) }\left( p,q\right) =\frac{%
2\pi e^{2}}{m^{2}}\delta _{J,0}
\end{equation}

\par \noindent
\textbf{The triplet states with }  $ \ell =J\ (J\geq
1),\;P=(-1)^{J+1},\;C=\left( -1\right) ^{J+1} $:

\textit{Orbital term}

\begin{equation}
\mathcal{K}^{\left( tr\right) \left( o\right) }(p,q)=\frac{2\pi e^{2}}{pq}%
Q_{J}\left( z\right) +\frac{\pi e^{2}}{m^{2}}\left( -\frac{J-3}{2}\left( 
\frac{p}{q}+\frac{q}{p}\right) Q_{J}\left( z\right) +\left( J+1\right)
Q_{J+1}\left( z\right) \right) ,
\end{equation}

\textit{Spin-Orbital interaction}

\begin{equation}
\mathcal{K}^{\left( tr\right) \left( s-o\right) }(p,q)=-\frac{3\pi e^{2}}{%
m^{2}}\frac{1}{2J+1}\left\{ Q_{J+1}\left( z\right) -Q_{J-1}\left( z\right)
\right\}
\end{equation}

\textit{Spin-Spin interaction}

\begin{equation}
\mathcal{K}^{\left( tr\right) \left( s-s\right) }(p,q)=\frac{\pi e^{2}}{%
2m^{2}}\left( \frac{p}{q}+\frac{q}{p}\right) Q_{J}\left( z\right) -\frac{\pi
e^{2}}{m^{2}}\frac{1}{2J+1}\left\{ JQ_{J+1}\left( z\right) +\left(
J+1\right) Q_{J-1}\left( z\right) \right\}
\end{equation}

\par \noindent
\textbf{The triplet states with} $ \ell =J-1\ (J\geq 1),\;\ell =J+1\ 
(J\geq 0),\;P=(-1)^{J},\;C=\left( -1\right) ^{J} $:

The off-diagonal elements of the kernel matrix (cf. Eqs.. (52)-(54)),
  $\mathcal{K}_{12}$ and $\mathcal{K}%
_{21}$\, which are responsible for mixing of states with $\ell =J-1$ and $\ell =J+1$,
get a non-zero contribution from the spin-spin interactions only: 
\begin{equation}
\mathcal{K}_{12}\left( p,q\right) =\mathcal{K}_{21}\left( p,q\right) =\frac{%
\pi e^{2}}{5m^{2}}\frac{\sqrt{J\left( J+1\right) }}{\left( 2J+1\right) }%
\left( \frac{p}{q}Q_{J+1}\left( z\right) +\frac{q}{p}Q_{J-1}\left( z\right)
-2Q_{J}\left( z\right) \right) .
\end{equation}
The contributions to the diagonal elements of the kernel matrix are the following:
\par
\textit{Orbital terms} 
\begin{equation}
\mathcal{K}_{11}^{\left( o\right) }\left( p,q\right) =\frac{2\pi e^{2}}{pq}%
Q_{J-1}\left( z\right) +\frac{\pi e^{2}}{m^{2}}\left( -\frac{J-4}{2}\left( 
\frac{p}{q}+\frac{q}{p}\right) Q_{J-1}\left( z\right) +JQ_{J}\left( z\right)
-2\delta _{J-1,0}\right) ,
\end{equation}
\begin{equation}
\mathcal{K}_{22}^{\left( o\right) }\left( p,q\right) =\frac{2\pi e^{2}}{pq}%
Q_{J+1}\left( z\right) -\frac{\pi e^{2}}{m^{2}}\left( -\frac{J-2}{2}\left( 
\frac{p}{q}+\frac{q}{p}\right) Q_{J+1}+\left( J+2\right) Q_{J+2}\right)
\end{equation}

\textit{Spin-orbital interaction} 
\begin{equation}
\mathcal{K}_{11}^{\left( s-o\right) }\left( p,q\right) =\frac{3\pi e^{2}}{%
m^{2}}\frac{J-1}{2J-1}\left( Q_{J}\left( z\right) -Q_{J-2}\left( z\right)
\right) ,
\end{equation}
\begin{equation}
\mathcal{K}_{22}^{\left( s-o\right) }\left( p,q\right) =-\frac{3\pi e^{2}}{%
m^{2}c^{2}}\frac{J+2}{2J+3}\left( Q_{J+2}\left( z\right) -Q_{J}\left(
z\right) \right)
\end{equation}

\textit{Spin-spin interaction} 
\begin{equation}
\mathcal{K}_{11}^{\left( s-s\right) }\left( p,q\right) =\frac{\pi e^{2}}{%
2m^{2}}\frac{1}{2J+1}\left( \left( \frac{p}{q}+\frac{q}{p}\right)
Q_{J-1}\left( z\right) -2Q_{J}\left( z\right) \right) ,
\end{equation}
\begin{equation}
\mathcal{K}_{22}^{\left( s-s\right) }\left( p,q\right) =\frac{\pi e^{2}}{%
2m^{2}}\frac{1}{2J+3}\left( \left( \frac{p}{q}+\frac{q}{p}\right)
Q_{J+1}\left( z\right) -2Q_{J+2}\left( z\right) \right)
\end{equation}

\textit{Annihilation term } 
\begin{equation}
\mathcal{K}^{ann}\left( p,q\right) =-\frac{2\pi e^{2}}{m^{2}}\delta _{J-1,0}
\end{equation}

We note that in the non-relativistic limit only the first terms of orbital
part of the kernels survive. They have the common form $2\pi ie^{2}Q_{\ell
}(z)/pq$, hence all radial equations reduce to the form 
\begin{equation}
\left( 2\omega _{p}-E\right) f^{\ell }(p)=\frac{m^{2}e^{2}}{\pi \omega
_{p} p }\int_{0}^{\infty }dq\frac{q}{\omega _{q}}%
Q_{\ell }(z)f^{\ell }(q).
\end{equation}
Recalling, also, that
\begin{equation}
\omega _{p}=\sqrt{m^{2}+\mathbf{p}^{2}}=m\left( 1+\frac{1}{2}\left( \frac{%
\mathbf{p}}{m}\right) ^{2}\right) ,
\end{equation}
we obtain, in the 
non relativistic limit, 
 the momentum-space Schr\"{o}dinger radial equations 
\begin{equation}
\left( \frac{\mathbf{p}^{2}}{2\mu }-\varepsilon \right) f^{\ell }(p)=\frac{%
\alpha }{\pi }\frac{1}{p}\int_{0}^{\infty }dq\, q\, Q_{J}(z)f^{\ell }(q),
\end{equation}
where $\alpha =e^{2}/4\pi $, $\mu =\frac{m}{2}$, $\varepsilon =E-2m$.

% ---------------------------    Fine Structure -----------------------

\vskip 0.8truecm {\normalsize 
\centerline{\textbf{\large Energy levels and
fine structure}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\subsection{Energy levels and fine structure}
}

The relativistic energy eigenvalues $E_{n,J}$ can be calculated 
from the expression 
\begin{eqnarray}
E\int_{0}^{\infty }dp\,p^{2}f^{J}(p)f^{J}(p) &=&\int_{0}^{\infty
}dp\,p^{2}\,2\omega _{p}f^{J}(p)f^{J}(p)  \notag \\
&&-\,\frac{m^{2}}{\left( 2\pi \right) ^{3}}
\int_{0}^{\infty }\frac{dpp^{2}}{%
\omega _{p}}\int_{0}^{\infty }dq\frac{q^{2}}{\omega _{q}}\mathcal{K}^{\left(
sgl,tr\right) }(p\mathbf{,}q)f^{J}(p)f^{J}(q)
\end{eqnarray}
for the singlet and $\ell =J$ triplet states.
\par
For the $\ell =J\mp 1$ triplet states the corresponding result is

\begin{eqnarray}
E\int_{0}^{\infty }dp\,p^{2}\mathbb{F}^{\dagger }(p)\mathbb{F}(p)
&=&\int_{0}^{\infty }dp\,p^{2}\,2\omega _{p}\mathbb{F}^{\dagger }(p)\mathbb{F%
}(p)  \notag \\
&&-\frac{m^{2}}{\left( 2\pi \right) ^{3}}\int_{0}^{\infty }\frac{dpp^{2}}{%
\omega _{p}}\int_{0}^{\infty }dq\frac{q^{2}}{\omega _{q}}\mathbb{K}(p\mathbf{%
,}q)\mathbb{F}^{\dagger }(p)\mathbb{F}(q)\, .
\end{eqnarray}

To obtain results for $E$ to $O\left( \alpha ^{4}\right) $ we use the forms
of the kernels expanded to $O\left( p^{2}/m^{2}\right) $ (eqns. 
(64)-(77)) and replace $f^{\ell }(p)$ by their non-relativistic
(Schr\"{o}dinger) form  (see (146), Appendix D). The most important integrals that
we used for calculating (81) and (82), are given in Appendix D. In Appendix
E we show that the contribution of kernels $\mathcal{K}_{12}$ 
and $\mathcal{K}_{21}$ in (82),  is
zero at $O(\alpha^4)$. Thus, the energy
corrections for the triplet states with $\ell =J-1$ and $\ell =J+1$ can be
calculated independently. 
\par \noindent
The results will be presented in the form $\Delta
\varepsilon =E- 2m + {\alpha ^{2}m}/{2 n^{2}}$.

\textbf{Singlet states} $\left( \ell =J\ (J\geq 0),\;P=(-1)^{J+1},\;C=\left(
-1\right) ^{J}\right) $
\par \noindent
The kinetic energy corrections:
\begin{equation}
\Delta \varepsilon _{K}^{(sgl)}=-\frac{\alpha ^{4}m}{8}\left( \frac{1}{2J+1}%
\frac{1}{n^{3}}-\frac{3}{8}\frac{1}{n^{4}}\right) .
\end{equation}
The potential energy corrections:

\begin{equation}
\Delta \varepsilon _{P}^{(sgl)\left( o\right) }=-\frac{\alpha ^{4}m}{8}%
\left( \left( \frac{3}{2J+1}-2\delta _{J,0}\right) \frac{1}{n^{3}}-\frac{1}{%
n^{4}}\right)
\end{equation}

\begin{equation}
\Delta \varepsilon _{P}^{(sgl)\left( s-o\right) }=0
\end{equation}

\begin{equation}
\Delta \varepsilon _{P}^{(sgl)\left( s-s\right) }=-\frac{\alpha ^{4}m}{4}%
\frac{\delta _{J,0}}{n^{3}}
\end{equation}
The total energy corrections: 
\begin{equation}
\Delta \varepsilon ^{(sgl)}=-\frac{\alpha ^{4}m}{8}\left( \frac{4}{2J+1}%
\frac{1}{n^{3}}-\frac{11}{8n^{4}}\right) .
\end{equation}

\par
\textbf{Triplet states} $\left( \ell =J\ (J\geq 1),\;P=(-1)^{J+1},\;C=\left(
-1\right) ^{J+1}\right) $
\par \noindent
The kinetic energy corrections:

\begin{equation}
\Delta \varepsilon _{K}^{(tr)}=-\frac{\alpha ^{4}m}{8}\left( \frac{1}{2J+1}%
\frac{1}{n^{3}}-\frac{3}{8}\frac{1}{n^{4}}\right) .
\end{equation}
The potential energy corrections:

\begin{equation}
\Delta \varepsilon _{P}^{\left( tr\right) \left( o\right) }=-\frac{\alpha
^{4}m}{8}\left( \left( \frac{3}{2J+1}-2\delta _{J,0}\right) \frac{1}{n^{3}}-%
\frac{1}{n^{4}}\right) ,
\end{equation}
\begin{equation}
\Delta \varepsilon _{P}^{\left( tr\right) \left( s-o\right) }=-\frac{\alpha
^{4}m}{8}\frac{3}{J(J+1)\left( 2J+1\right) }\frac{1}{n^{3}},
\end{equation}
\begin{equation}
\Delta \varepsilon _{P}^{\left( s-s\right) }=\frac{\alpha ^{4}m}{8}\frac{1}{%
J\left( J+1\right) \left( 2J+1\right) }\frac{1}{n^{3}},
\end{equation}
The total energy corrections:

\begin{equation}
\Delta \varepsilon ^{(tr)}=-\frac{\alpha ^{4}m}{8}\left( \left( \frac{4}{2J+1%
}+\frac{2}{J\left( J+1\right) \left( 2J+1\right) }\right) \frac{1}{n^{3}}-%
\frac{11}{8}\frac{1}{n^{4}}\right) .
\end{equation}

\par
\textbf{Triplet states} $\left( \ell =J-1\ (J\geq
1),\;\;P=(-1)^{J},\;C=\left( -1\right) ^{J}\right) $
\par \noindent
The kinetic energy corrections:

\begin{equation}
\Delta \varepsilon _{K}^{(tr)(J-1)}=-\frac{\alpha ^{4}m}{8}\left( \frac{1}{%
2J-1}\frac{1}{n^{3}}-\frac{3}{8}\frac{1}{n^{4}}\right) .
\end{equation}
The potential energy corrections: 
\begin{equation}
\Delta \varepsilon _{P}^{\left( tr\right) \left( o\right) (J-1)}=-\frac{%
\alpha ^{4}m}{8}\left[ \left( \frac{3}{2J-1}-2\delta _{J,1}\right) \frac{1}{%
n^{3}}-\frac{1}{n^{4}}\right] ,
\end{equation}
\begin{equation}
\Delta \varepsilon _{P}^{tr\left( s-o\right) (J-1)}=\frac{\alpha ^{4}m}{8}%
\frac{3\left( 1-\delta _{J,1}\right) }{J\left( 2J-1\right) }\frac{1}{n^{3}},
\end{equation}
\begin{equation}
\Delta \varepsilon _{P}^{tr\left( s-s\right) (J-1)}=-\frac{\alpha ^{4}m}{8}%
\frac{1-\delta _{J,1}}{J\left( 2J+1\right) \left( 2J-1\right) }\frac{1}{n^{3}%
}.
\end{equation}

\begin{equation}
\Delta \varepsilon ^{\left( ann\right) }=\frac{\alpha ^{4}m}{4}\frac{1}{n^{3}%
}\delta _{J,1}
\end{equation}

\par \noindent
The total energy corrections:

\begin{equation}
\Delta \varepsilon ^{tr(J-1)}=-\frac{\alpha ^{4}m}{8}\left( \left( \frac{4}{%
2J-1}-\frac{2\left( 3J+1\right) \left( 1-\delta _{J,1}\right) }{J\left(
2J+1\right) \left( 2J-1\right) }-4\delta _{J,1}\right) \frac{1}{n^{3}}-\frac{%
11}{8}\frac{1}{n^{4}}\right) .
\end{equation}

\par
\textbf{Triplet states }$\left( \ell =J+1\ (J\geq
0),\;\;P=(-1)^{J},\;C=\left( -1\right) ^{J}\right) $
\par \noindent
The kinetic energy corrections: 
\begin{equation}
\Delta \varepsilon _{K}^{(tr)(J+1)}=-\frac{\alpha ^{4}m}{8}\left( \frac{1}{%
2J+3}\frac{1}{n^{3}}-\frac{3}{8}\frac{1}{n^{4}}\right) .
\end{equation}
The potential energy corrections: 
\begin{equation}
\Delta \varepsilon _{P}^{\left( tr\right) \left( o\right) (J+1)}=-\frac{%
\alpha ^{4}m}{8}\left[ \frac{3}{2J+3}\frac{1}{n^{3}}-\frac{1}{n^{4}}\right] ,
\end{equation}
\begin{equation}
\Delta \varepsilon _{P}^{tr\left( s-o\right) (J+1)}=-\frac{\alpha ^{4}m}{8}%
\frac{3}{\left( J+1\right) \left( 2J+3\right) }\frac{1}{n^{3}},
\end{equation}
\begin{equation}
\Delta \varepsilon _{P}^{tr\left( s-s\right) (J+1)}=-\frac{\alpha ^{4}m}{8}%
\frac{1}{\left( J+1\right) \left( 2J+3\right) \left( 2J+1\right) }\frac{1}{%
n^{3}}.
\end{equation}
The total energy corrections: 
\begin{equation}
\Delta \varepsilon ^{tr(J+1)}=-\frac{\alpha ^{4}m}{8}\left( \frac{2}{2J+3}%
\left( 2+\frac{3J+2}{\left( J+1\right) \left( 2J+1\right) }\right) \frac{1}{%
n^{3}}-\frac{11}{8}\frac{1}{n^{4}}\right) .
\end{equation}

These results are in agreement with the well-known positronium fine
structure results [17], [18].

{\normalsize \vskip 0.8truecm \centerline{\textbf{\large Concluding remarks}}
}

%-------------------------------  Conclusions --------------------------

{\normalsize \vskip 0.4truecm }

{\normalsize %\section{Concluding remarks}}
We have considered a reformulation of QED, in which covariant Green functions are used to solve the QED field equations for the mediating electromagnetic field  in terms of the fermion field. 
This leads to a reformulated Hamiltonian with an interaction term 
in which the photon propagator appears sandwiched between fermionic 
currents.  
\par
The variational method within the Hamiltonian formalism of
quantum field theory is used to determine approximate eigensolutions of 
bound relativistic fermion-antifermion states. The reformulation enables us to use the simplest possible trial state to derive a relativistic 
momentum-space Salpeter-like equation for a positronium-like system.
The invariant $\cal M$ matrices corresponding to one-photon exchange and virtual annihilation Feynman diagrams arise directly in the interaction kernel of this equation.
\par
The trial
states are chosen to be eigenstates of the total angular momentum operator $\widehat{\mathbf{J}}^{2}$ and $\widehat{J}_{3}$ of QED, along with parity
and charge conjugation. A general relativistic reduction of the wave
equations to radial form is given. For given $J$ there is a single radial
equation for total spin zero singlet states , but for spin triplet states
there are, in general coupled equations. We show how the classification of
 states follows naturally from the system of eigenvalue equations obtained 
with our trial state. 
\par
We have solved the integral equations perturbatively
to obtain the relativistic two-particle binding energy corrections 
to order $\alpha ^{4}$. The results agree with well known results for
positronium, obtained on the basis of the Bethe-Salpeter equation [18].
\par
It would be of interest to consider more elaborate variational calculations 
within this formalism, using trial states that accommodate virtual pairs.
\par
The financial support of the Natural Sciences and Engineering Research 
Council of Canada for this work is gratefully acknowledged.

{\normalsize %\end{enumerate}}

\vfill \eject

%----------------------------   App. A --------------------------------

%{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Appendix A:
Total angular momentum operator in relativistic
form}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\section{Appendix}
%\subsection{Quantization of total angular momentum operator in relativistic
%reduction}
}

The total angular momentum operator is defined by expression 
\begin{equation}
\widehat{\mathbf{J}}=\int d^{3}\mathbf{x}:\psi ^{\dagger }\left( x\right) 
\mathbf{(}\widehat{\mathbf{L}}+\widehat{\mathbf{S}})\psi \left( x\right): ,
\end{equation}
where $\widehat{\mathbf{L}}=\widehat{\mathbf{x}}\times \widehat{\mathbf{p}}$
and $\widehat{\mathbf{S}}=\frac{1}{2}\widehat{\overrightarrow{\mathbf{\sigma 
}}}$ are the orbital angular momentum and spin operators. We use the
standard representation for the Pauli matrices 
\begin{equation}
\widehat{\overrightarrow{\mathbf{\sigma }}}=\left[ 
\begin{array}{cc}
\overrightarrow{\mathbf{\sigma }} & 0 \\ 
0 & \overrightarrow{\mathbf{\sigma }}
\end{array}
\right]
\end{equation}
\begin{equation}
\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] ,\;\;\;\sigma _{3}=\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & -1
\end{array}
\right]
\end{equation}
Using the field operator $\psi \left( x\right) $ in the form (15), after
tedious calculations we obtain 
\begin{equation*}
\widehat{J}_{1}=\int d^{3}\mathbf{q}\left( 
\begin{array}{c}
\widehat{L}_{q1}\left( b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}%
\uparrow }+b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\downarrow }+d_{%
\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }+d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }\right) \\ 
+\frac{1}{2}\left( b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}\downarrow
}+b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\uparrow }+d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\uparrow }+d_{\mathbf{q}\uparrow
}^{\dagger }d_{\mathbf{q}\downarrow }\right)
\end{array}
\right) ,
\end{equation*}
\begin{equation}
\widehat{J}_{2}=\int d^{3}\mathbf{q}\left( 
\begin{array}{c}
\widehat{L}_{q2}\left( b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}%
\uparrow }+b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\downarrow }+d_{%
\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }+d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }\right) \\ 
+\frac{i}{2}\left( -b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}%
\downarrow }+b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\uparrow }-d_{%
\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\downarrow }+d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\uparrow }\right)
\end{array}
\right) ,
\end{equation}
\begin{equation*}
\widehat{J}_{3}=\int d^{3}\mathbf{q}\left( 
\begin{array}{c}
\widehat{L}_{q3}\left( b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}%
\uparrow }+b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\downarrow }+d_{%
\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }+d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }\right) \\ 
+\frac{1}{2}\left( b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}\uparrow
}-b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\downarrow }+d_{\mathbf{q}%
\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }-d_{\mathbf{q}\downarrow
}^{\dagger }d_{\mathbf{q}\downarrow }\right)
\end{array}
\right) .
\end{equation*}
Here $\widehat{\mathbf{L}}_{q}$ is the orbital angular momentum operator in
momentum representation:

\begin{equation}
(\widehat{\mathbf{L}}_{q})_{i}\equiv \widehat{L}_{qi}=-i\left( \mathbf{%
q\times }\nabla _{q}\right) _{i}
\end{equation}
Note that these expressions are valid for any $t$, since the time-dependent
phase factors of the form $e^{i\omega _{q}t}$\ cancel out.

\par
For the operator $\widehat{\mathbf{J}}^{2}=\widehat{J}_{1}^{2}+\widehat{J}%
_{2}^{2}+\widehat{J}_{3}^{2}$ we have 
\begin{eqnarray}
\widehat{\mathbf{J}}^{2} &=&\int d^{3}\mathbf{q}\left( 
\begin{array}{c}
\left( \widehat{\mathbf{L}}_{q}^{2}+\frac{3}{4}\right) \left( b_{\mathbf{q}%
\uparrow }^{\dagger }b_{\mathbf{q}\uparrow }+b_{\mathbf{q}\downarrow
}^{\dagger }b_{\mathbf{q}\downarrow }+d_{\mathbf{q}\uparrow }^{\dagger }d_{%
\mathbf{q}\uparrow }+d_{\mathbf{q}\downarrow }^{\dagger }d_{\mathbf{q}%
\downarrow }\right) \\ 
+\widehat{L}_{q-}b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}\downarrow }+%
\overset{\symbol{94}}{L}_{q+}b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q%
}\uparrow }+\overset{\symbol{94}}{L}_{q-}d_{\mathbf{q}\uparrow }^{\dagger
}d_{\mathbf{q}\downarrow }+\overset{\symbol{94}}{L}_{q+}d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\uparrow } \\ 
+\widehat{L}_{q3}\left( b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}%
\uparrow }-b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\downarrow }+d_{%
\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }-d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }\right)
\end{array}
\right)  \notag \\
&&+\frac{1}{2}\int d^{3}\mathbf{q}^{\prime }d^{3}\mathbf{q}\left( 
\begin{array}{c}
2\widehat{\mathbf{L}}_{q^{\prime }}\cdot \widehat{\mathbf{L}}_{q}\left( 
\begin{array}{c}
b_{\mathbf{q}^{\prime }\uparrow }^{\dagger }b_{\mathbf{q}^{\prime }\uparrow
}d_{\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }+b_{\mathbf{q}%
^{\prime }\uparrow }^{\dagger }b_{\mathbf{q}^{\prime }\uparrow }d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\downarrow } \\ 
+b_{\mathbf{q}^{\prime }\downarrow }^{\dagger }b_{\mathbf{q}^{\prime
}\downarrow }d_{\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }+b_{%
\mathbf{q}^{\prime }\downarrow }^{\dagger }b_{\mathbf{q}^{\prime }\downarrow
}d_{\mathbf{q}\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }
\end{array}
\right) \\ 
+\frac{1}{2}\left( b_{q^{\prime }\uparrow }^{\dagger }b_{q^{\prime }\uparrow
}d_{q\uparrow }^{\dagger }d_{q\uparrow }-b_{q^{\prime }\uparrow }^{\dagger
}b_{q^{\prime }\uparrow }d_{q\downarrow }^{\dagger }d_{q\downarrow }\right)
\\ 
-\frac{1}{2}\left( b_{q^{\prime }\downarrow }^{\dagger }b_{q^{\prime
}\downarrow }d_{q\uparrow }^{\dagger }d_{q\uparrow }-b_{q^{\prime
}\downarrow }^{\dagger }b_{q^{\prime }\downarrow }d_{q\downarrow }^{\dagger
}d_{q\downarrow }\right) \\ 
+b_{q^{\prime }\uparrow }^{\dagger }b_{q^{\prime }\downarrow }d_{q\downarrow
}^{\dagger }d_{q\uparrow }+b_{q^{\prime }\downarrow }^{\dagger }b_{q^{\prime
}\uparrow }d_{q\uparrow }^{\dagger }d_{q\downarrow } \\ 
+\widehat{L}_{q^{\prime }+}\left( 
\begin{array}{c}
b_{\mathbf{q}^{\prime }\uparrow }^{\dagger }b_{\mathbf{q}^{\prime }\uparrow
}d_{\mathbf{q}\downarrow }^{\dagger }d_{\mathbf{q}\uparrow }+b_{\mathbf{q}%
^{\prime }\downarrow }^{\dagger }b_{\mathbf{q}^{\prime }\downarrow }d_{%
\mathbf{q}\downarrow }^{\dagger }d_{\mathbf{q}\uparrow } \\ 
+b_{\mathbf{q}\downarrow }^{\dagger }b_{\mathbf{q}\uparrow }d_{\mathbf{q}%
^{\prime }\uparrow }^{\dagger }d_{\mathbf{q}^{\prime }\uparrow }+b_{\mathbf{q%
}\downarrow }^{\dagger }b_{\mathbf{q}\uparrow }d_{\mathbf{q}^{\prime
}\downarrow }^{\dagger }d_{\mathbf{q}^{\prime }\downarrow }
\end{array}
\right) \\ 
+\widehat{L}_{q^{\prime }-}\left( 
\begin{array}{c}
b_{\mathbf{q}^{\prime }\uparrow }^{\dagger }b_{\mathbf{q}^{\prime }\uparrow
}d_{\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\downarrow }+b_{\mathbf{q}%
^{\prime }\downarrow }^{\dagger }b_{\mathbf{q}^{\prime }\downarrow }d_{%
\mathbf{q}\uparrow }^{\dagger }d_{\mathbf{q}\downarrow } \\ 
+b_{\mathbf{q}\uparrow }^{\dagger }b_{\mathbf{q}\downarrow }d_{\mathbf{q}%
^{\prime }\uparrow }^{\dagger }d_{\mathbf{q}^{\prime }\uparrow }+b_{\mathbf{q%
}\uparrow }^{\dagger }b_{\mathbf{q}\downarrow }d_{\mathbf{q}^{\prime
}\downarrow }^{\dagger }d_{\mathbf{q}^{\prime }\downarrow }
\end{array}
\right) \\ 
+\left( \widehat{L}_{q^{\prime }3}+\widehat{L}_{q3}\right) \left( b_{\mathbf{%
q}^{\prime }\uparrow }^{\dagger }b_{\mathbf{q}^{\prime }\uparrow }d_{\mathbf{%
q}\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }-b_{\mathbf{q}^{\prime
}\downarrow }^{\dagger }b_{\mathbf{q}^{\prime }\downarrow }d_{\mathbf{q}%
\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }\right) \\ 
-\left( \widehat{L}_{q^{\prime }3}-\widehat{L}_{q3}\right) \left( b_{\mathbf{%
q}^{\prime }\uparrow }^{\dagger }b_{\mathbf{q}^{\prime }\uparrow }d_{\mathbf{%
q}\downarrow }^{\dagger }d_{\mathbf{q}\downarrow }-b_{\mathbf{q}^{\prime
}\downarrow }^{\dagger }b_{\mathbf{q}^{\prime }\downarrow }d_{\mathbf{q}%
\uparrow }^{\dagger }d_{\mathbf{q}\uparrow }\right)
\end{array}
\right) ,
\end{eqnarray}
where 
\begin{equation}
\widehat{L}_{q+}=\widehat{L}_{q1}+i\widehat{L}_{q2},\;\;\;\;\;\;\;\widehat{L}%
_{q-}=\widehat{L}_{q1}-i\widehat{L}_{q2} \, .
\end{equation}

The requirement that the trial state (20) be an eigenstate of $\widehat 
{\bf J}^2$ and $\widehat J_z$ leads to the system of equations

\begin{eqnarray}
\left( \widehat{L}_{3}+1\right) F_{11} &=&m_{J}F_{11}  \notag \\
\widehat{L}_{3}F_{12} &=&m_{J}F_{12}  \notag \\
\widehat{L}_{3}F_{21} &=&m_{J}F_{21} \\
\left( \widehat{L}_{3}-1\right) F_{22} &=&m_{J}F_{22}  \notag
\end{eqnarray}

\begin{eqnarray}
\left( J(J+1)-\widehat{\mathbf{L}}^{2}-2-2\widehat{L}_{3}\right) F_{11} &=&%
\widehat{L}_{-}\left( F_{12}+F_{21}\right)  \notag \\
\left( J(J+1)-\widehat{\mathbf{L}}^{2}-1\right) F_{12} &=&F_{21}+\widehat{L}%
_{+}F_{11}+\widehat{L}_{-}F_{22}  \notag \\
\left( J(J+1)-\widehat{\mathbf{L}}^{2}-1\right) F_{21} &=&F_{12}+\widehat{L}%
_{+}F_{11}+\widehat{L}_{-}F_{22} \\
\left( J(J+1)-\widehat{\mathbf{L}}^{2}-2+2\widehat{L}_{3}\right) F_{22} &=&%
\widehat{L}_{+}\left( F_{12}+F_{21}\right)  \notag
\end{eqnarray}
Substitution of the expressions (33) for $F_{s_1 s_2}$ and use 
of eq. (112)\ gives 
\begin{equation}
m_{12}=m_{21}=m_{J},\;\ \ \ \ m_{11}=m_{J}-1,\;\ \ \ \ \ \ m_{22}=m_{J}+1,
\end{equation}
\begin{equation}
\ell _{11}=\ell _{22}=\ell _{12}=\ell _{21}=\ell ,
\end{equation}
and 
\begin{eqnarray}
\left( J(J+1)-\ell (\ell +1)-2m_{J}\right) f_{11}^{\ell }(p) &=&\sqrt{(\ell
-m_{J}+1)(\ell +m_{J})}f_{12}^{\ell }(p)  \notag \\
&&+\sqrt{(\ell -m_{J}+1)(\ell +m_{J})}f_{21}^{\ell }(p)  \notag \\
\left( J(J+1)-\ell (\ell +1)-1\right) f_{12}^{\ell }(p) &=&f_{21}^{\ell }(p)
\notag \\
&&+\sqrt{(\ell +m_{J})(\ell -m_{J}+1)}f_{11}^{\ell }(p)  \notag \\
&&+\sqrt{(\ell -m_{J})(\ell +m_{J}+1)}f_{22}^{\ell }(p) \\
\left( J(J+1)-\ell (\ell +1)-1\right) f_{21}^{\ell }(p) &=&f_{12}^{\ell }(p)
\notag \\
&&+\sqrt{(\ell +m_{J})(\ell -m_{11}+1)}f_{11}^{\ell }(p)  \notag \\
&&+\sqrt{(\ell -m_{J})(\ell +m_{J}+1)}f_{22}^{\ell }(p)  \notag \\
\left( J(J+1)-\ell (\ell +1)+2m_{J}\right) f_{22}^{\ell }(p) &=&\sqrt{(\ell
+m_{J}+1)(\ell -m_{J})}f_{12}^{\ell }(p)  \notag \\
&&+\sqrt{(\ell +m_{J}+1)(\ell -m_{J})}f_{21}^{\ell }(p)  \notag
\end{eqnarray}

The singlet states correspond to the solution $f_{11}^{\ell }\left( p\right)
=f_{22}^{\ell }\left( p\right) =0$, $f_{12}^{\ell }\left( p\right) =-f_{21}^{\ell }\left( p\right) $%
\ of this system with $\ell =J$ $(J\geq0)$.

For the triplet states the solutions are $f_{12}^{\ell }\left( p\right)
=f_{21}^{\ell }\left( p\right) \equiv f^{\ell }\left( p\right) $, and, 
\par \noindent 
for \ $\ell =J-1$: 
\begin{eqnarray}
\left( J-m_{J}\right) f_{11}^{J-1}(p) &=&\sqrt{\left( J-m_{J}\right)
(J+m_{J}-1)}f^{J-1}(p) \\
\left( J+m_{J}\right) f_{22}^{J-1}(p) &=&\sqrt{\left( J+m_{J}\right)
(J-m_{J}-1)}f^{J-1}(p),
\end{eqnarray}
for\ $\ell =J$:
\begin{eqnarray}
m_{J}f_{11}^{J}(p) &=&-\sqrt{\left( J+m_{J}\right) (J-m_{J}+1)}f^{J}(p) \\
m_{J}f_{22}^{J}(p) &=&\sqrt{\left( J-m_{J}\right) (J+m_{J}+1)}f^{J}(p),
\end{eqnarray}
Equations (116)-(119) make since only for $J\geq1$
\par \noindent 
for\ $\ell =J+1$:
\begin{eqnarray}
\left( J+1+m_{J}\right) f_{11}^{J+1}(p) &=&-\sqrt{\left( J-m_{J}+2\right)
\left( J+m_{J}+1\right) }f^{J+1}(p) \\
\left( J+1-m_{J}\right) f_{22}^{J+1}(p) &=&-\sqrt{\left( J-m_{J}+1\right)
\left( J+m_{J}+2\right) }f^{J+1}(p)
\end{eqnarray}

It is convenient to introduce the table of coefficients $C_{Jm_{J}}^{\left(
tr\right) \ell m_{s}}$ :
\begin{equation*}
\begin{array}{cccc}
J & m_{s}=+1 & m_{s}=0 & m_{s}=-1 \\ 
\ell -1 & \sqrt{\frac{\left( J+m_{J}-1\right) (J+m_{J})}{J\left( 2J-1\right) 
}} & \sqrt{\frac{\left( J-m_{J}\right) \left( J+m_{J}\right) }{J\left(
2J-1\right) }} & \sqrt{\frac{\left( J-m_{J}-1\right) (J-m_{J})}{J\left(
2J-1\right) }} \\ 
\ell & -\sqrt{\frac{\left( J+m_{J}\right) (J-m_{J}+1)}{J\left( J+1\right) }}
& \frac{m_{J}}{\sqrt{J\left( J+1\right) }} & \sqrt{\frac{\left(
J-m_{J}\right) (J+m_{J}+1)}{J\left( J+1\right) }} \\ 
\ell +1 & \sqrt{\frac{\left( J-m_{J}+1\right) \left( J-m_{J}+2\right) }{%
\left( J+1\right) \left( 2J+3\right) }} & -\sqrt{\frac{\left(
J-m_{J}+1\right) \left( J+m_{J}+1\right) }{\left( J+1\right) \left(
2J+3\right) }} & \sqrt{\frac{\left( J+m_{J}+2\right) \left( J+m_{J}+1\right) 
}{\left( J+1\right) \left( 2J+3\right) }}
\end{array}
\end{equation*}
These coefficients coincide with the usual Clebsch-Gordan 
coefficients for $S=1$ except for a factor $2$ in the denominator, 
which we absorb into the  normalization constant.

%-------------------------  Appendix B ------------------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Appendix B: Parity
and charge conjugation}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\subsection{Parity and charge conjugation}
}

We consider the application of the parity operator to the trial state (20): 
\begin{equation}
\widehat{\mathcal{P}}\mid e^{+}e^{-}\rangle =\underset{s_{1}s_{2}}{\sum }%
\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p})\widehat{\mathcal{P}}%
b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle =\underset{%
s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p})\widehat{%
\mathcal{P}}b_{ps_{1}}^{\dagger }\widehat{\mathcal{P}}^{-1}\widehat{\mathcal{%
P}}d_{-ps_{2}}^{\dagger }\widehat{\mathcal{P}}^{-1}\widehat{\mathcal{P}}\mid
0\rangle .
\end{equation}
Making use of the properties 
\begin{equation}
\widehat{\mathcal{P}}b_{ps_{1}}^{\dagger }\widehat{\mathcal{P}}^{-1}=\eta
^{P}b_{-ps_{1}}^{\dagger },\;\;\;\;\;\widehat{\mathcal{P}}%
d_{-ps_{2}}^{\dagger }\widehat{\mathcal{P}}^{-1}=-\eta
^{P}d_{ps_{2}}^{\dagger },\;\;\;\;\;\widehat{\mathcal{P}}\mid 0\rangle =\mid
0\rangle ,
\end{equation}
where $\eta ^{P}$ is the intrinsic parity ($\left( \eta ^{P}\right) ^{2}=1$%
), it follows that 
\begin{equation*}
\widehat{\mathcal{P}}\mid e^{+}e^{-}\rangle =\underset{s_{1}s_{2}}{\sum }%
\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p})\widehat{\mathcal{P}}%
b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle =\underset{%
s_{1}s_{2}}{-\sum }\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(-\mathbf{p}%
)b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle
\end{equation*}
\begin{equation}
=P\underset{s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p}%
)b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle ,
\end{equation}
where the parity eigenvalue $P$ depends on the symmetry of $F_{s_{1}s_{2}}
\mathbf{p})$ in different states:
\par \noindent
 For the singlet states\textbf{\ }$\left(
\ell =J\right) $ we get from (36) $F_{s_{1}s_{2}}(-\mathbf{p})=\left(
-1\right) ^{J}F_{s_{1}s_{2}}(\mathbf{p})$, so that 
$P=\left( -1\right) ^{J+1}$.
\par \noindent
For the triplet states with\textbf{\ }$\ell =J$ we get from (38) $
F_{s_{1}s_{2}}(-\mathbf{p})=\left( -1\right) ^{J}F_{s_{1}s_{2}}(\mathbf{p})$%
, hence $P=\left( -1\right) ^{J+1}$.
\par \noindent For the triplet states with\textbf{\ }$%
\ell =J\pm 1$ we get from (39) $F_{s_{1}s_{2}}(-\mathbf{p})=\left( -1\right)
^{J+1}F_{s_{1}s_{2}}(\mathbf{p})$, therefore $P=\left( -1\right) ^{J}$.

Charge conjugation is associated with the interchange of the particle and
antiparticle. Applying the charge conjugation operator to the trial state
(20) we get 
\begin{equation}
\widehat{\mathcal{C}}\mid e^{+}e^{-}\rangle =\underset{s_{1}s_{2}}{\sum }%
\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p})\widehat{\mathcal{C}}%
b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle =\underset{%
s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p})\widehat{%
\mathcal{C}}b_{ps_{1}}^{\dagger }\widehat{\mathcal{C}}^{-1}\widehat{\mathcal{%
C}}d_{-ps_{2}}^{\dagger }\widehat{\mathcal{C}}^{-1}\widehat{\mathcal{C}}\mid
0\rangle .
\end{equation}
Using the relations 
\begin{equation}
\widehat{\mathcal{C}}b_{ps_{1}}^{\dagger }\widehat{\mathcal{C}}^{-1}=\eta
^{C}d_{ps_{1}}^{\dagger },\;\;\;\;\;\widehat{\mathcal{C}}d_{-ps_{2}}^{%
\dagger }\widehat{\mathcal{C}}^{-1}=\eta ^{C}b_{-ps_{2}}^{\dagger
},\;\;\;\;\;\widehat{\mathcal{C}}\mid 0\rangle =\mid 0\rangle ,
\end{equation}
where $\left( \eta ^{C}\right) ^{2}=1$, we obtain 
\begin{equation*}
\widehat{\mathcal{C}}\mid e^{+}e^{-}\rangle =\underset{s_{1}s_{2}}{\sum }%
\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p})\widehat{\mathcal{C}}%
b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle =-\underset{%
s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}F_{s_{2}s_{1}}(\mathbf{p}%
)b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle
\end{equation*}
\begin{equation}
=C\underset{s_{1}s_{2}}{\sum }\int d^{3}\mathbf{p}F_{s_{1}s_{2}}(\mathbf{p}%
)b_{ps_{1}}^{\dagger }d_{-ps_{2}}^{\dagger }\mid 0\rangle ,
\end{equation}
where the charge conjugation quantum number $C$ depends on the symmetry of $%
F_{s_{1}s_{2}}(\mathbf{p})$ in different states:
\par \noindent
For the singlet states\textbf{\ }$\left( \ell =J\right) $ we get from (36) $%
F_{s_{1}s_{2}}(-\mathbf{p})=\left( -1\right) ^{J+1}F_{s_{1}s_{2}}(\mathbf{p}%
) $, hence $C=\left( -1\right) ^{J}$.
\par \noindent
For the triplet states with\textbf{\ }$\ell =J$ we get from (38) $%
F_{s_{1}s_{2}}(-\mathbf{p})=\left( -1\right) ^{J}F_{s_{1}s_{2}}(\mathbf{p})$%
, therefore $C=\left( -1\right) ^{J+1}$.
\par \noindent
For the triplet states with\textbf{\ }$\ell =J\pm 1$ we get from (39) $%
F_{s_{1}s_{2}}(-\mathbf{p})=\left( -1\right) ^{J+1}F_{s_{1}s_{2}}(\mathbf{p}%
) $, so that $C=\left( -1\right) ^{J}$.


%-----------------------------  App. C ---------------------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Appendix C:
Expansion of the spinors}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\subsection{Expansion of the bispinors}
}

We recall the form of the particle spinors: 
\begin{equation}
u(\mathbf{p,}i)=N_{\mathbf{p}}\left[ 
\begin{array}{c}
1 \\ 
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{\omega _{p}+m}
\end{array}
\right] \varphi _{i},
\end{equation}
where 
\begin{equation}
\varphi _{1}=\left[ 
\begin{array}{c}
1 \\ 
0
\end{array}
\right] ,\;\;\;\varphi _{2}=\left[ 
\begin{array}{c}
0 \\ 
1
\end{array}
\right] ,\;\;\;\;\;N_{\mathbf{p}}=\sqrt{\frac{\omega _{p}+m}{2m}}
\end{equation}
The antiparticle or ``positron'' representation for the $v_{i}(\mathbf{p})$
spinors has the form

\begin{equation}
v(\mathbf{p,}i)=N_{\mathbf{p}}\left[ 
\begin{array}{c}
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{\omega _{p}+m} \\ 
1
\end{array}
\right] \chi _{i},
\end{equation}
where 
\begin{equation}
\chi _{1}=\left[ 
\begin{array}{c}
0 \\ 
1
\end{array}
\right] ,\;\;\;\;\;\chi _{2}=-\left[ 
\begin{array}{c}
1 \\ 
0
\end{array}
\right]
\end{equation}
The normalization is 
\begin{equation}
\overline{u}(\mathbf{p,}i)u(\mathbf{p,}j)=\delta _{ij},\;\;\;\;\;\;\;%
\overline{v}(\mathbf{p,}i)v(\mathbf{p,}j)=-\delta _{ij}
\end{equation}
Expanding in powers of $p/m$ and keeping the lowest non-trivial order terms,

\begin{equation}
\frac{(\overrightarrow{\sigma }\mathbf{\cdot p})}{\omega _{p}+m}\simeq \frac{%
(\overrightarrow{\sigma }\mathbf{\cdot p})}{2m},
\end{equation}
\begin{equation}
N_{\mathbf{p}}=\sqrt{\frac{\omega _{p}+m}{2m}}\simeq 1+\frac{\mathbf{p}^{2}}{%
8m^{2}},
\end{equation}
we obtain the result 
\begin{equation}
u(\mathbf{p,}i)\simeq \left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right) \left[ 
\begin{array}{c}
1 \\ 
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{2m}
\end{array}
\right] \varphi _{i}=\left[ 
\begin{array}{c}
\left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right) \\ 
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{2m}
\end{array}
\right] \varphi _{i}
\end{equation}
\begin{equation}
v(\mathbf{p,}i)\simeq \left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right) \left[ 
\begin{array}{c}
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{2m} \\ 
1
\end{array}
\right] \chi _{i}=\left[ 
\begin{array}{c}
\frac{(\overrightarrow{\mathbf{\sigma }}\mathbf{\cdot p})}{2m} \\ 
\left( 1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)
\end{array}
\right] \chi _{i}.
\end{equation}


%---------------------  App. D -------------------------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Appendix D: Some
useful identities and integrals}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\subsection{Some useful identities and integrals}
}

The following identity is useful for evaluating the $\mathcal{M}$ matrices: 
\begin{equation}
\frac{\left( \left( \mathbf{p}-\mathbf{q}\right) \cdot \mathbf{p}\right) ^{2}%
}{\left( \mathbf{p}-\mathbf{q}\right) ^{4}}=\frac{\mathbf{p}^{2}}{\left( 
\mathbf{p}-\mathbf{q}\right) ^{2}}-\frac{\left( \mathbf{p}\times \mathbf{q}%
\right) ^{2}}{\left( \mathbf{p}-\mathbf{q}\right) ^{4}}.
\end{equation}

The angular integration in (47), (50), (55) involves the following integrals

\begin{equation}
\int d\hat{\mathbf{p}}\,d\hat{\mathbf{q}}\,\digamma \left( \hat{%
\mathbf{p}}\cdot \hat{\mathbf{q}}\right) Y_{J^{\prime }}^{m_{J}^{\prime
}}(\hat{\mathbf{q}})Y_{J}^{m_{J}\ast }(\hat{\mathbf{p}})=2\pi \delta
_{J^{\prime }J}\delta _{m_{J}^{\prime }m_{J}}\int d\left( \hat{\mathbf{p}%
}\cdot \hat{\mathbf{q}}\right) \digamma \left( \hat{\mathbf{p}}\cdot 
\hat{\mathbf{q}}\right) P_{J}\left( \hat{\mathbf{p}}\cdot \hat{%
\mathbf{q}}\right)\, ,
\end{equation}

\begin{equation}
\int d\left( \hat{\mathbf{p}}\cdot \hat{\mathbf{q}}\right) \frac{%
\hat{\mathbf{p}}\cdot \hat{\mathbf{q}}}{\left( \mathbf{p}-\mathbf{q}%
\right) ^{2}}P_{J}\left( \hat{\mathbf{p}}\cdot \hat{\mathbf{q}}%
\right) =\frac{1}{\left| \mathbf{p}\right| \left| \mathbf{q}\right| }\left( 
\frac{J+1}{2J+1}Q_{J+1}\left( z\right) +\frac{J}{2J+1}Q_{J-1}\left( z\right)
\right) \, ,
\end{equation}

\begin{equation}
\int d\left( \hat{\mathbf{p}}\cdot \hat{\mathbf{q}}\right) \frac{%
\left( \hat{\mathbf{p}}\times \hat{\mathbf{q}}\right) ^{2}}{\left( 
\mathbf{p}-\mathbf{q}\right) ^{4}}P_{J}\left( \hat{\mathbf{p}}\cdot 
\hat{\mathbf{q}}\right) =\frac{1}{\left| \mathbf{p}\right| \left| 
\mathbf{q}\right| }\left( \frac{\left( J+1\right) \left( J+2\right) }{%
2\left( 2J+1\right) }Q_{J+1}\left( z\right) -\frac{J\left( J-1\right) }{%
2\left( 2J+1\right) }Q_{J-1}\left( z\right) \right) \, ,
\end{equation}
where $\digamma \left( \hat{\mathbf{p}}\cdot \hat{\mathbf{q}}\right) $
is an arbitrary function of $\hat{\mathbf{p}}\cdot \hat{\mathbf{q}}$%
, $P_{J}\left( x\right) $ is the Legendre polynomial, and $Q_{J}(z)$ is the
Legendre function of the second kind of order $J$.

The following integrals are needed for the calculation of the relativistic
energy corrections. 
\begin{equation}
\int_{0}^{\infty }\int_{0}^{\infty }dp\,dq\,p^{2}q^{2}f^{J}(p)f^{J}(q)=2\pi
\left( \frac{\alpha \mu }{n}\right) ^{3}\delta _{J,0},
\end{equation}
\begin{equation}
\int_{0}^{\infty }\int_{0}^{\infty }dp\,dq\,pqf^{J}(p)f^{J}(q)Q_{J}(z_{1})=%
\frac{\pi \alpha \mu }{n^{2}},
\end{equation}
\begin{eqnarray}
\int_{0}^{\infty }\int_{0}^{\infty
}dp\,dq\,p^{2}q^{2}f^{J}(p)f^{J}(q)Q_{J}(z_{1})&=& \notag \\ 
\int_{0}^{\infty }\int_{0}^{\infty
}dp\,dq\,p^{3}qf^{J}(p)f^{J}(q)Q_{J}(z_{1}) &=&\pi \left( \frac{\alpha \mu }{%
n}\right) ^{3}\left( \frac{4}{2J+1}-\frac{1}{n}\right) ,
\end{eqnarray}
\begin{equation}
\int_{0}^{\infty }\int_{0}^{\infty
}dp\,dq\,p^{2}q^{2}f^{J}(p)f^{J}(q)Q_{J-1}(z_{1})=\pi \left( \frac{\alpha
\mu }{n}\right) ^{3}\left( \frac{2}{J}-\frac{1}{n}\right) ,
\end{equation}
\begin{equation}
\int_{0}^{\infty }\int_{0}^{\infty
}dp\,dq\,p^{2}q^{2}f^{J}(p)f^{J}(q)Q_{J+1}\left( z_{1}\right) =\pi \left( 
\frac{\alpha \mu }{n}\right) ^{3}\left( \frac{2}{J+1}-\frac{1}{n}\right) .
\end{equation}
Here $f^{J}$ is the nonrelativistic hydrogen-like\ radial wave function in
momentum space [18]

\begin{equation}
f^{J}(p)\equiv f_{n}^{J}(p)=\left( \frac{2}{\pi }\frac{\left( n-J-1\right) !%
}{\left( n+J\right) !}\right) ^{1/2}\frac{n^{J+2}p^{J}2^{2\left( J+1\right)
}J!}{\left( n^{2}p^{2}+1\right) ^{J+2}}\mathcal{G}_{n-J-1}^{J+1}\left( \frac{%
n^{2}p^{2}-1}{n^{2}p^{2}+1}\right) ,
\end{equation}
where $\mathcal{G}_{n-J-1}^{J+1}\left( x\right) $ are Gegenbauer functions.


%----------------------------- App. E ------------------------------------

{\normalsize \vskip 0.8truecm 
\centerline{\textbf{\large Appendix E:
$\mathcal{K}_{12}$, $\mathcal{K}_{21}$ kernels for \ $l=J\mp 1$\ states}} }

{\normalsize \vskip 0.4truecm }

{\normalsize 
%\subsection{$\mathcal{K}_{12}$, $\mathcal{K}_{21}$ kernels for \ $l=J\mp 1$\\states}
}

The contribution  of the kernel $\mathcal{K}_{12}$ 
to the energy correction is 
\begin{equation}
\int dp\,dq\,p^{2}q^{2}\mathcal{K}_{12}\left( p,q\right)
f^{J-1}(p)f^{J+1}(q),
\end{equation}
where

\begin{equation}
\mathcal{K}_{12}\left( p,q\right) =\underset{\sigma _{1}\sigma _{2}s_{1}s_{2}%
}{\sum }C_{Jm_{J}12}^{s_{1}s_{2}\sigma _{1}\sigma _{2}}\int d\hat{%
\mathbf{p}}\,d\hat{\mathbf{q}}\,\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma
_{2}}^{ope\left( s-s\right) }\left( \mathbf{p,q}\right) Y_{J+1}^{m_{\sigma
_{1}\sigma _{2}}}(\hat{\mathbf{q}})Y_{J-1}^{m_{s_{1}s_{2}}\ast }(%
\hat{\mathbf{p}}).
\end{equation}
This requires the following integral 
\begin{equation}
\underset{\sigma _{1}\sigma _{2}s_{1}s_{2}}{\sum }C_{Jm_{J}12}^{s_{1}s_{2}%
\sigma _{1}\sigma _{2}}\int d^{3}\mathbf{p\,}d^{3}\mathbf{q\,}%
f^{J-1}(p)Y_{J-1}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{p}})\mathcal{M}%
_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope\left( s-s\right) }\left( \mathbf{p,q%
}\right) f^{J+1}(q)Y_{J+1}^{m_{\sigma _{1}\sigma _{2}}}(\hat{\mathbf{q}}%
).
\end{equation}
We calculate this form in coordinate space. The Fourier transform of $%
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right) \;$%
is

\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{p,q}\right)
=\int d^{3}\mathbf{r} \, d^{3}\mathbf{r}^{\prime }\mathcal{M}_{s_{1}s_{2}\sigma
_{1}\sigma _{2}}\left( \mathbf{r,r}^{\prime }\right) e^{-i\left( \mathbf{p-q}%
\right) \cdot \left( \mathbf{r-r}^{\prime }\right) },
\end{equation}
where the $\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{r,r}%
^{\prime }\right) $ matrix  is a local operator in general [18], 
that is 
\begin{equation}
\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{r,r}^{\prime
}\right) =\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma _{2}}\left( \mathbf{r}%
\right) \delta \left( \mathbf{r-r}^{\prime }\right) .
\end{equation}
We apply this transformation to the  
$\mathcal{M}_{s_{1}s_{2}\sigma _{1}\sigma
_{2}}^{ope\left( s-s\right) }\left( \mathbf{p,q}\right) $ 
 matrix (see eq. (62)).
Because of the angular integration in (129), only the first term in (62)
survives. The\ Fourier transformation of that term is 
\begin{equation}
\frac{\left( \overrightarrow{\mathbf{\sigma }}^{\left( +\right) }\cdot
\left( \mathbf{p-q}\right) \right) \left( \overrightarrow{\mathbf{\sigma }}%
^{\left( -\right) }\cdot \left( \mathbf{p-q}\right) \right) }{4m^{2}\left( 
\mathbf{p-q}\right) ^{2}}\;\;\;\rightarrow \;\;\;3\frac{\left( 
\overrightarrow{\mathbf{\sigma }}^{\left( +\right) }\cdot \mathbf{r}\right)
\left( \overrightarrow{\mathbf{\sigma }}^{\left( -\right) }\cdot \mathbf{r}%
\right) }{16\pi m^{2}\mathbf{r}^{5}}.
\end{equation}
Furthermore,

\begin{equation}
\int d^{3}\mathbf{p}f^{J-1}(p)Y_{J-1}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{%
p}})e^{-i\mathbf{p}\cdot \mathbf{r}}=R_{n}^{J-1}(r)Y_{J-1}^{m_{s_{1}s_{2}}%
\ast }(\hat{\mathbf{r}}),
\end{equation}
\begin{equation}
\int d^{3}\mathbf{q}f^{J+1}(q)Y_{J+1}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{%
q}})e^{-i\mathbf{q}\cdot \mathbf{r}}=
R_{n}^{J+1}(r)Y_{J+1}^{m_{s_{1}s_{2}}}(\hat{\mathbf{r}}),
\end{equation}
where 
\begin{equation}
R_{n}^{\ell }\left( r\right) =-\frac{2}{n^{2}}\sqrt{\frac{\left( n-\ell
-1\right) !}{\left( \left( n+\ell \right) !\right) ^{3}}}e^{-r/n}\left( 
\frac{2r}{n}\right) ^{\ell }L_{n+\ell }^{2\ell +1}\left( \frac{2r}{n}\right)
.
\end{equation}
The associated Laguerre function \ \ $L_{\lambda }^{\mu }\left( \rho \right) 
$\ \ is related to the confluent hypergeometric function by 
\begin{equation}
L_{\lambda }^{\mu }\left( \rho \right) =\left( -1\right) ^{\mu }\frac{\left(
\lambda !\right) ^{2}}{\mu !\left( \lambda -\mu \right) !}F\left( -\lambda
+\mu ,\mu +1;\rho \right) .
\end{equation}
The generating function for the Laguerre function is 
\begin{equation}
U_{\mu }\left( \rho ,u\right) \equiv \left( -1\right) ^{\mu }\frac{u^{\mu }}{%
\left( 1-u\right) ^{\mu +1}}\exp \left( -\frac{u\rho }{1-u}\right)
=\sum_{\lambda =\mu }^{\infty }\frac{L_{\lambda }^{\mu }\left( \rho \right) 
}{\lambda !}u^{\lambda },
\end{equation}
hence 
\begin{eqnarray}
&&\underset{\sigma _{1}\sigma _{2}s_{1}s_{2}}{\sum }C_{Jm_{J}12}^{s_{1}s_{2}%
\sigma _{1}\sigma _{2}}\int d^{3}\mathbf{p\,}d^{3}\mathbf{q\,}%
f^{J-1}(p)Y_{J-1}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{p}})\mathcal{M}%
_{s_{1}s_{2}\sigma _{1}\sigma _{2}}^{ope\left( s-s\right) }\left( \mathbf{p,q%
}\right) f^{J+1}(q)Y_{J+1}^{m_{\sigma _{1}\sigma _{2}}}(\hat{\mathbf{q}})
\notag \\
&=&\underset{\sigma _{1}\sigma _{2}s_{1}s_{2}}{\sum }%
C_{Jm_{J}12}^{s_{1}s_{2}\sigma _{1}\sigma _{2}}\int d^{3}\mathbf{r\,}%
R_{n}^{J-1}\left( r\right) Y_{J-1}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{r}}%
)\left( 3\alpha \frac{\left( \overrightarrow{\mathbf{\sigma }}^{\left(
+\right) }\cdot \mathbf{r}\right) \left( \overrightarrow{\mathbf{\sigma }}%
^{\left( -\right) }\cdot \mathbf{r}\right) }{16\pi m^{2}\mathbf{r}^{5}}%
\right) R_{n}^{J+1}\left( r\right) Y_{J+1}^{m_{s_{1}s_{2}}}(\hat{\mathbf{%
r}}) \\
&=&\frac{3\alpha }{16\pi m^{2}}\int dr\,r^{2}\frac{1}{r^{3}}%
R_{n}^{J-1}\left( r\right) R_{n}^{J+1}\left( r\right) \times   \notag \\
&&\times \underset{\sigma _{1}\sigma _{2}s_{1}s_{2}}{\sum }%
C_{Jm_{J}12}^{s_{1}s_{2}\sigma _{1}\sigma _{2}}\int d\hat{\mathbf{r}}%
\,Y_{J-1}^{m_{s_{1}s_{2}}\ast }(\hat{\mathbf{r}})\left( \overrightarrow{%
\mathbf{\sigma }}^{\left( +\right) }\cdot \hat{\mathbf{r}}\right) \left( 
\overrightarrow{\mathbf{\sigma }}^{\left( -\right) }\cdot \hat{\mathbf{r}%
}\right) Y_{J+1}^{m_{s_{1}s_{2}}}(\hat{\mathbf{r}}).  \notag
\end{eqnarray}
It follows that
\begin{equation}
\underset{\sigma _{1}\sigma _{2}s_{1}s_{2}}{\sum }C_{Jm_{J}12}^{s_{1}s_{2}%
\sigma _{1}\sigma _{2}}\int d\hat{\mathbf{r}}Y_{J-1}^{m_{s_{1}s_{2}}\ast
}(\hat{\mathbf{r}})\left( \overrightarrow{\mathbf{\sigma }}^{\left(
+\right) }\cdot \hat{\mathbf{r}}\right) \left( \overrightarrow{\mathbf{%
\sigma }}^{\left( -\right) }\cdot \hat{\mathbf{r}}\right)
Y_{J+1}^{m_{s_{1}s_{2}}}(\hat{\mathbf{r}})=\frac{1}{15}\frac{\sqrt{%
J\left( J+1\right) }}{2J+1},
\end{equation}
but 
\begin{equation}
\int_{0}^{\infty }dr\,r^{2}\frac{1}{r^{3}}R_{n}^{J-1}\left( r\right)
R_{n}^{J+1}\left( r\right) =0.
\end{equation}
The last expression can be proved in the following way. Let us consider the
more general case 
\begin{equation}
\int_{0}^{\infty }dr\,r^{\beta +2}R_{n}^{\ell }\left( r\right) R_{n}^{\ell
^{\prime }}\left( r\right) .
\end{equation}
The generating function for $R_{n}^{\ell }\left( r\right) $\ is

\begin{equation}
G_{n\ell }\left( r,u\right) =-\frac{2}{n^{2}}\sqrt{\frac{\left( n-\ell
-1\right) !}{\left( \left( n+\ell \right) !\right) ^{3}}}e^{-r/n}\left( 
\frac{2r}{n}\right) ^{\ell }\left( -1\right) ^{2\ell +1}\frac{u^{2\ell +1}}{%
\left( 1-u\right) ^{2\ell +2}}\exp \left\{ -\frac{u}{1-u}\frac{2r}{n}%
\right\} .
\end{equation}
Then we consider the expression
\begin{eqnarray}
&&\int_{0}^{\infty }drr^{\beta +2}G_{n\ell }\left( r,u\right) G_{n\ell
^{\prime }}\left( r,v\right)   \notag \\
&=&\int_{0}^{\infty }drr^{\beta +2}\frac{4}{n^{4}}\sqrt{\frac{\left( n-\ell
-1\right) !\left( n-\ell ^{\prime }-1\right) !}{\left( \left( n+\ell \right)
!\right) ^{3}\left( \left( n+\ell ^{\prime }\right) !\right) ^{3}}}%
e^{-2r/n}\left( \frac{2r}{n}\right) ^{\ell +\ell ^{\prime }}\times   \notag
\\
&&\times \frac{u^{2\ell +1}v^{2\ell ^{\prime }+1}}{\left( 1-u\right) ^{2\ell
+2}\left( 1-v\right) ^{2\ell ^{\prime }+2}}\exp \left\{ -\left( \frac{u}{1-u}%
+\frac{v}{1-v}\right) \frac{2r}{n}\right\}  \\
&=&\frac{4}{n^{4}}\sqrt{\frac{\left( n-\ell -1\right) !\left( n-\ell
^{\prime }-1\right) !}{\left( \left( n+\ell \right) !\right) ^{3}\left(
\left( n+\ell ^{\prime }\right) !\right) ^{3}}}\frac{u^{2\ell +1}v^{2\ell
^{\prime }+1}}{\left( 1-u\right) ^{2\ell +2}\left( 1-v\right) ^{2\ell
^{\prime }+2}}\times   \notag \\
&&\times \int_{0}^{\infty }dr\left( \frac{2r}{n}\right) ^{\beta +2+\ell
+\ell ^{\prime }}\exp \left\{ -\left( 1+\frac{u}{1-u}+\frac{v}{1-v}\right) 
\frac{2r}{n}\right\} .  \notag
\end{eqnarray}
It is well known that 
\begin{equation}
\int_{0}^{\infty }d\rho \rho ^{\beta }e^{-\rho }=\Gamma \left( \beta
+1\right)\, ,
\end{equation}
therefore 
\begin{eqnarray}
&&\int_{0}^{\infty }dr\left( \frac{2r}{n}\right) ^{\beta +2+\ell +\ell
^{\prime }}\exp \left\{ -\left( 1+\frac{u}{1-u}+\frac{v}{1-v}\right) \frac{2r%
}{n}\right\}   \notag \\
&=&\left( \frac{n}{2}\right) ^{\beta +3}\left( \frac{\left( 1-u\right)
\left( 1-v\right) }{1-uv}\right) ^{\beta +3+\ell +\ell ^{\prime }}\Gamma
\left( \beta +3+\ell +\ell ^{\prime }\right) 
\end{eqnarray}
and 
\begin{eqnarray}
&&\int_{0}^{\infty }drr^{\beta +2}G_{n\ell }\left( r,u\right) G_{n\ell
^{\prime }}\left( r,v\right)   \notag \\
&=&\frac{2^{-\beta -1}}{n^{-\beta +1}}\sqrt{\frac{\left( n-\ell -1\right)
!\left( n-\ell ^{\prime }-1\right) !}{\left( \left( n+\ell \right) !\right)
^{3}\left( \left( n+\ell ^{\prime }\right) !\right) ^{3}}}\times  \\
&&\times \frac{u^{2\ell +1}v^{2\ell ^{\prime }+1}\left( 1-u\right) ^{\beta
+1-\ell +\ell ^{\prime }}\left( 1-v\right) ^{\beta +1+\ell -\ell ^{\prime }}%
}{\left( 1-uv\right) ^{\beta +3+\ell +\ell ^{\prime }}}\Gamma \left( \beta
+3+\ell +\ell ^{\prime }\right) .  \notag
\end{eqnarray}
We expand this expression in a series, 
\begin{equation}
\int_{0}^{\infty }drr^{\beta +2}G_{n\ell }\left( r,u\right) G_{n\ell
^{\prime }}\left( r,v\right) =\sum_{\eta \eta ^{\prime }}C_{\eta \eta
^{\prime }}\left( n,\beta ,\ell ,\ell ^{\prime }\right) u^{\eta }u^{\eta
^{\prime }}.
\end{equation}
It is not difficult to show [19], that the coefficient $C_{n+\ell ,n+\ell
^{\prime }}$\ represents the integral\ 
\begin{equation}
C_{n+\ell ,n+\ell ^{\prime }}\left( n,\beta ,\ell ,\ell ^{\prime }\right)
=\int_{0}^{\infty }drr^{\beta +2}R_{n}^{\ell }(r)R_{n}^{\ell ^{\prime }}(r).
\end{equation}
Simple but tedious calculations show that this\ coefficient is zero for $%
\beta =-3$, $\ell =J-1$, $\ell^{\prime } =J+1$. Thus the kernel $\mathcal{K}_{12}$
does not contribute to the energy corrections to $O\left( \alpha ^{4}\right) 
$. The same result is obtained for the kernel $\mathcal{K}_{21}$.

%--------------------------  References -------------------------------

{\normalsize \vskip0.8truecm \centerline{\textbf{\large References}} }

{\normalsize \vskip 0.4truecm }

{\normalsize %\section{References}}

{\normalsize \enumerate}

1. J. W. Darewych, Annales Fond. L. de Broglie (Paris) \textbf{23}, 15
(1998).

2. J. W. Darewych, in \textit{Causality and Locality in Modern Physics}, G
Hunter et al. (eds.), Kluwer, Dordrecht, 1998, pp. 333-244.

3. J. W. Darewych, Can. J. Phys. \textbf{76}, 523 (1998).

4. M. Barham and J. W. Darewych, J. Phys. A \textbf{31}, 3481 (1998).

5. B. Ding and J. Darewych, J. Phys. G \textbf{26}, 907 (2000).

6. J. D. Jackson, \textit{Classical Electrodynamics} (John Wiley, New York,
1975).

7. A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles
(Dover, new York, 1980).

8. W. T. Grandy, Jr., Relativistic Quantum Mechanics of Leptons and Fields
(Kluwer, Dordrecht, 1991).

9. A. O. Barut, ''The Schr\"{o}dinger and Dirac Equation - Linear, Nonlinear
and Intergrodifferential'' in \textit{Geometrical and Algebraic Aspects of
Nonlinear Field Theory}. S. De Filippo, M. Marinaro, G. Marmo and G. Vilasi,
eds. Elsevier (North Holland), 1989, pp. 37-51.

10 J. W. Darewych and L. Di Leo, J. Phys. A: Math. Gen. \textbf{29}, 6817
(1996).

11. J. W. Darewych and M. Horbatsch, J. Phys. B: At. Mol. Opt. \textbf{22}%
,973 (1989); \textbf{2}3, 337 (1990).

12. W. Dykshoorn and R. Koniuk, Phys. Rev. A \textbf{41}, 64 (1990).

13. T. Zhang and R. Koniuk, Can. J. Phys. \textbf{70}, 683 (1992).

14. T. Zhang and R. Koniuk, Can. J. Phys. \textbf{70}, 670 (1992).

15. V. B. Berestetski, E. M. Lifshitz, and L. P. Pitaevski, \textit{Quantum
Electrodynamics }(Pergamon, New York, 1982).

16. G. Arfken and H. Weber \textit{Mathematical methods for physicists}
(2001), Academic Press.

17. M. A. Stroscio Positronium: A Review Of The Theory, Physics Reports C 22,
5, 215-277 (1975).

18. H. A. Bethe and E.E. Salpeter, \textit{Quantum Mechanics of One and
Two-Electron Atoms (Springer-Verlag/Academic, New York, 1957).}

19. M. Mizushima \textit{Quantum Mechanics of Atomic Spectra and Atomic
Structure }(W. A. Benjamin, Inc., New York, 1970).

\end{document}