%Title: Anomalous Magnetic Moment Interactions between Fermions 
%Author: H. Pilkuhn
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\begin{document}
\begin{center}\begin{Large}\begin{bf}Anomalous Magnetic Moment Interactions 
between Fermions \end{bf}\end{Large}\\
\vspace{1cm} \begin{Large}
Hartmut Pilkuhn\end{Large}\begin{large}\\
 Institut f\"ur Theoretische Teilchenphysik, Universit\"at,\\
D-76128 Karlsruhe, Germany\end{large}\\
(e-mail:hp@particle.uni-karlsruhe.de)
\end{center}
%\vspace {1cm}
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\begin{abstract}{The anomalous magnetic moments of both fermions are included
in the fully relativistic eight-component bound state equation. When these
moments are of order unity or larger, a discrepancy with the standard 
sixteen-component formalism appears in the hyperfine mixing of P and
D states for unequal masses at the leading order $Z^4\alpha^4$.}
\begin{center}{PACS numbers: 11.10St, 32.10Fn, 36.10Dr}\end{center}
\end{abstract}
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\begin{section}{Introduction}
Ever since the original
Dirac equation was applied to the hydrogen atom, it has been
improved by additional interactions. For the high-Z members of the hydrogen
iso-electric sequence, vacuum polarization, hyperfine interaction 
and radiative corrections are particularly important. 
Recoil corrections are important at low Z; the
present paper presents such corrections for the anomalous magnetic moment
interactions, including the magnetic hyperfine operator.

Recoil effects are of course largest for positronium (e$^-$e$^+$), but there
the traditional Dirac equation fails. 
To the order $\alpha^4$, one may instead use the nonrelativistic reduction 
of the 16-component Dirac-Breit equation. That equation was originally
formulated by Breit for the helium atom; its application to the isolated
two-fermion system is explained in \cite{1}. Today, the expression 
"Breit operators" is normally used for the spin-orbit, tensor etc operators
which appear in the nonrelativistic expansion; anomalous magnetic moments are
easily included. A major progress in recent years has been the
"nonrelativistic quantum electrodynamics" (NRQED), which is able to extend
the nonrelativistic reduction to the order $(Z\alpha)^6$.
The small anomalous magnetic moments of electrons and muons, $\kappa_e\approx
\kappa_\mu\approx\alpha/2\pi$, are automatically included in NRQED.
To the order $(Z\alpha)^6$, the NRQED method is presently more successful
than the fully relativistic Bethe-Salpeter equation. This has in fact been
interpreted to signal the end of relativistic quantum mechanics as a whole,
including the Dirac equation for hydrogen.

On the other hand, it has been shown that the Dirac-Breit equation can be
reduced to an 8-component equation which is still relativistic and in 
fact much simpler \cite{2}. The equation is Dirac-like, with seemingly four
components for the electron and two components for the nuclear spin. In
reality, however, it contains strictly relativistic two-body kinematics:
At large separation where the potential $V(r)$ vanishes,
 the following "asymptotic" equation describes the relative motion, with
$\bld p_1=-\bld p_2\equiv\bld p=-i\nabla$ (units $\hbar =c=1$), at total cms
energy $E$:
\beg{1}\bld p^2\psi_{as}=k^2\psi_{as},\quad k^2=[(E^2-m_1^2-m_2^2)^2
-4m_1^2m_2^2]/4E^2.\en
It applies for arbitrary particle spins. For
equal spins, $k^2$ is split symmetrically in $m_1$ and $m_2$,
\beg{2}k^2=\epsilon^2-\mu^2,\quad\epsilon=(E^2-m_1^2-m_2^2)/2E,\quad
\mu=m_1m_2/E,\en
where $\epsilon$ and $\mu$ are the reduced energy and reduced mass, 
respectively. For two spin-$\kalb$ particles, a factorized form of 
(\ref{1}) exists in which the potential $V(r)$ is simply added:
\beg{3}(\epsilon-V-\mu\beta-\gamma^5\bld\sigma_1\bld p-H_{hf,D}-H_{an}-\dots)
\psi_1=0.\en
Here $\bld\sigma_1$ is a set of $2\times2$ Pauli matrices; a second set 
appears in the "Dirac" hyperfine operator,
\beg{4}H_{hf,D}=i\bld\sigma^\times V\bld p/2E,\quad
\bld\sigma^\times=\bld\sigma_1\times\bld\sigma_2.\en
$\beta$ and $\gamma^5$ are Dirac matrices, they are related to the covariant $\gamma^\mu$ by
$\beta=\gamma^0,\;\gamma^5\bld\sigma_1=\bld\alpha=\gamma^0\bld\gamma$. 
Although they are usually written as $4\times4$-matrices, they are really
$2\times2$: $\beta,\;\gamma^5$ and $i\gamma^5\beta$ form a third $SU_2$
algebra. The $V$ in (\ref{3}) is the electric potential, viz. the
sum of Coulomb and vacuum polarization potentials. For point particles,
the Coulomb potential is $q_1q_2/r=-Z\alpha/r\equiv-\alpha_Z/r$, with
$\bld r=\bld r_1-\bld r_2$. $H_{an}$ is the desired contribution of the 
anomalous magnetic moments; the dots in (\ref{3}) indicate additional
operators, which lead to radiative corrections.

In the original derivation \cite{2} from the 16-component Dirac-Breit
equation, the radial variable $r_{\rm DB}$ which occurs both in the cms
kinetic energy operator and in the Coulomb potential, $V_{\rm
  DB}=-\alpha_Z/r_{\rm DB}$, appeared in the combination 
$2EV_{\rm DB}+V^2_{\rm DB}$, which was transformed into $2EV(r)$ by the
"quasidistance", $r=r^2_{\rm DB}(r_{\rm DB}+\alpha_Z/2E)^{-1}$. At the 
same time, the transformation eliminated the typical Breit retardation 
operators. The non-hermiticity of the resulting hyperfine operator (\ref{4})
was first regarded as a serious defect \cite{3}, until it was shown that
its anti-hermitian part made the equation applicable to positronium
\cite{4}. Finally, it was shown how to extract $V(r)$ and $H_{hf,D}$ directly
from the relevant T-matrix of perturbative QED, $S=1+iT$ \cite{5}, thereby
avoiding nonrelativistic approximations and distance transformations.
The resulting $\alpha_Z^6$-terms have been checked against the NRQED results
\cite{6}, but a number of radiative corrections are still missing, among
these the anomalous magnetic moments.

To understand that the number of components for the isolated two-fermion
system is eight, not sixteen, recall the
use of time translation invariance in the derivation of (\ref{1}):
$\psi_{as}$ satisfies two separate equations, $\bld
p^2\psi_{as}=(E_1^2-m_1^2)\psi_{as}$ and
$\bld p^2\psi_{as}=(E_2^2-m_2^2)\psi_{as}$, 
where the $E_i$ are eigenvalues of
the individual time shifters $i\partial/\partial t_i$. But as the interaction
commutes only with a common time shift $i\partial/\partial
t=i\partial/\partial t_1+i\partial/\partial t_2$, the two equations must be
combined into a single one that contains only $E_1+E_2=E$, which is an
eigenvalue of $i\partial/\partial t$. The form (\ref{1}) follows uniquely.
In analogy, after the two free single-particle Dirac equations have been
correctly combined into one 16-component equation which contains only
$E_1+E_2$,
\beg{6}(E_1+E_2-\gamma^5_1\bld p_1\bld\sigma_1-\gamma^5_2\bld p_2\bld\sigma_2
-m_1\beta_1-m_2\beta_2)\psi_{as}^{(16)}=0,\en
it remains to eliminate those components which arise from single-particle
parity transformations, $\psi_{as}^{P_i}(\bld r_i,\bld
r_j)=\beta_i\psi_{as}(-\bld r_i,\bld r_j)$. 
In other words, as the interaction commutes only with the total parity
transformation, the desired function
$\psi$ must only allow for that transformation,
\beg{7}\psi^P(\bld r_i,\bld r_j)=\beta\psi(-\bld r_i,-\bld r_j),\quad
\beta\equiv\beta_1\beta_2.\en

Although one may avoid $\psi^{(16)}$ altogether, we briefly indicate the
elimination method \cite{100}. $\psi^{(16)}$ is split into octets 
of eigenstates of the total chirality operator $\gamma^5_1\gamma^5_2$ which
commutes with $\beta$:
\beg{8}\psi^{(16)}=\left(\begin{array}{cc}\psi\\ \chi\end{array}\right),
\quad\gamma^5_1\gamma^5_2\psi=\psi,\quad\gamma^5_1\gamma^5_2\chi=-\chi.\en
The phases between the components of $\psi$ and $\chi$ are conveniently
chosen such that
\beg{9}\beta_2\psi=\chi,\quad\beta_2\chi=\psi.\en
Finally, $\psi$ and $\chi$ may be subdivided into large and small components
which are eigenstates of $\beta$. The asymptotic components of
$\psi$ and $\chi$ will be called $v$ and $w$, respectively. According to
(\ref{6}), they satisfy coupled equations, one of which is used for the
elimination of $w$:
\beg{10}w=m_+^{-1}[E-\gamma^5(\bld\sigma_1-\bld\sigma_2)\bld p]v,\quad
m_\pm=m_2\pm\beta m_1.\en
The distinction between $\gamma^5_1$ and $\gamma^5_2$ is now unnecessary, as
$\gamma^5_2v=\gamma^5_1v$. In the following, we generally set $\gamma^5_1\equiv\gamma^5$.
The only obstacle in the application of this method has been
the $m_+^{-1}$, which must be removed because it is singular for
$m_2=m_1$ and $\beta=-1$. This is achieved by the transformation $v=C_1v_1$, 
\beg{11}C_1=(m_+m_-)^{-1/2}(m_2+m_1\beta P_{\rm spin}),\quad
P_{\rm spin}=\kalb(1+\bld\sigma_1\bld\sigma_2),\en
where $P_{\rm spin}$ is the spin permutation operator. It also has the 
property of removing the $\bld\sigma_2$ of (\ref{6}), which
explains the absence of a term $\bld\sigma_2\bld p$ in (\ref{3}). A similar
transformation $C_2$ exists which removes instead the $\bld\sigma_2$ 
\cite{100}.

It is instructive to compare this formalism with the Bethe-Salpeter equation
and with NRQED, which both use the 16-component formalism. For this purpose,
we multiply (\ref{3}) by $2E$, obtaining the simpler "basic" form
\beg{12}[E^2-2EV-m_+^2-2E\gamma^5\bld\sigma_1\bld p-2i\gamma^5V
\bld\sigma^\times\bld p-2EH_{an}]\psi_1=0.\en
The reduction of the 16-component equation to 8 components produces the
operator $(E-V)^2$ instead of the
$E^2-2EV$ of (\ref{12}) (plus a more complicated
structure of operators linear in $V$). The $V^2$-contribution to the energy
levels is then cancelled by interactions derived from two-photon exchange
\cite{101}. The 16-component formalism is not incorrect, only less 
efficient.

The nonrelativistic reduction of (\ref{3}) produces the known tensor and
Fermi contact potentials between two Dirac particles, with 
$V'=dV/dr\approx \alpha_Z/r^2$,  
\beg{13}V_{t,D}=\sigma_tV'
/4m_1m_2r,\quad\sigma_t=3\sigma_{1r}\sigma_{2r}-\bld\sigma_1\bld\sigma_2,\en
\beg{14}V_{\rm con,D}=\textstyle{2\over3}\pi\alpha_Z\delta(r)\bld
\sigma_1\bld\sigma_2/m_1m_2,\en 
plus three spin-orbit potentials,
\beg{15}V_{s1l,D}=\bld\sigma_1\bld lV'/4\mu^2r,\quad
V_{s2l,D}=\bld\sigma_2\bld lV'/2m_1m_2r,\en
\beg{16}V_{s\times l,D}=i\bld\sigma^\times\bld lV'/4m_1m_2r.\en
$V_{s1l,D}$ is the spin-orbit potential of a Dirac particle of mass $\mu$,
$V_{s2l,D}$ is the "other-spin"-orbit potential emerging from $H_{hf,D}$, and
$V_{s\times l,D}$ comes from the antihermitian part of $H_{hf,D}$. 
The spin-orbit potentials differ
drastically from the two spin-orbit potentials of the 16-component reduction,
which were first calculated by Chraplyvy and by Barker and Glover \cite{7}:
\beg{17}V^{\rm CBG}_{s1l,D}=\bld\sigma_1\bld lV'(m_1^{-1}+2m_2^{-1})/4m_1r,
\quad V^{\rm CBG}_{s2l,D}=\bld\sigma_2\bld lV'(m_2^{-1}+2m_1^{-1})/4m_2r.\en
The results to order $\alpha_Z^4$ are nevertheless identical:
The sum of their expectation values (with
$\langle\bld\sigma_1\rangle=\langle\bld\sigma_2\rangle$ and
$\langle V_{s12l,D}\rangle=0$) are the same, and the inclusion of
$V_{s12l,D}$ provides the known mixing between singlet 
and triplet spin states in degenerate states. It is this latter mixing
which is modified by the inclusion of $\kappa_1$ and $\kappa_2$.
\end{section}

\begin{section}{The Matrices $T,\;M,\;M^C$}
The one-photon exchange of the QED T-matrix is,
with $\bld q=\bld k_1-\bld k_1'=\bld k_2'-\bld k_2$,
\beg{18}T^{(16\times16)}=4\pi\alpha_ZJ^\mu_{11'}g_{\mu\nu}J^\nu_{22'}/t,\quad
t=q^0{}^2-\bld q^2.\en
For $\bld q^2\ll m_1^2$, one may approximate the two fermion form factors by
$F_1=1,\;F_2=\kappa$,
\beg{19}J^\mu_{11'}=\overline u_1'(\gamma_1^\mu+\sigma_1^{\mu\nu}q_\nu
\kappa_1/2m_1)u_1.\en 
The free spinors $u_i$ and $u_i'{}^\dagger$ will now be omitted, and the 
remaining oprators will be distinguished by hats.
In the cms one has $q^0=0$, and with
$\sigma^{0j}=\gamma^5\sigma^j$,
$\sigma^{jk}=i\sigma_l\:(jkl\;{\rm cyclic})$,
\beg{20}\hat J^0_{11'}=1+\gamma^5_1\beta_1\bld{q\sigma}_1\kappa_1
/2m_1,\quad\hat{\bld J}_{11'}=
\gamma^5_1\bld\sigma_1+i\beta_1\bld q\times\bld\sigma_1\kappa_1/2m_1,\en
The corresponding $\hat J^\nu_{22'}$ has $\bld q_2=-\bld q$:
\beg{21}\hat J^0_{22'}=1-\gamma^5_2\beta_2\bld{q\sigma}_2\kappa_2/2m_2,
\quad\hat{\bld J}_{22'}=
\gamma^5_2\bld\sigma_2-i\beta_2\bld q\times\bld\sigma_2\kappa_2/2m_2.\en
Insertion of these expressions into $\hat T$ gives several terms, which may
be classified by their respective powers of anomalous magnetic moments:
\beg{22}\hat T=\hat T_{\rm D}+\kappa_1\hat T_{\kappa1}+\kappa_2\hat T_{\kappa2}+\kappa_1\kappa_2\hat T_{\kappa1,\kappa2}.\en
$\hat T_{\rm D}$ denotes the pure "Dirac" matrix element:
\beg{23}\hat T_{\rm D}=-4\pi\alpha_Z(1-\gamma^5_1\gamma^5_2\bld\sigma_1
\bld\sigma_2)/\bld q^2.\en
The remaining three matrices are 
\beg{24}\hat T_{\kappa1}=-4\pi\alpha_Z\bld q(\gamma^5_1\bld\sigma_1+i
\gamma^5_2\bld\sigma^\times)\beta_1/2m_1\bld q^2,\en
\beg{25}\hat T_{\kappa2}=-4\pi\alpha_Z\bld q(-\gamma^5_2\bld\sigma_2+i
\gamma^5_1\bld\sigma^\times)\beta_2/2m_2\bld q^2,\en
\beg{26}\hat T_{\kappa1,\kappa2}=-4\pi\alpha_Z[-\gamma^5_1\gamma^5_2
(\bld{q\sigma}_1)(\bld{q\sigma}_2)+(\bld q\times\bld\sigma_1)(\bld
q\times\bld\sigma_2)]\beta_1\beta_2/4m_1m_2\bld q^2.\en

Next, the 16 components of $u_1u_2$
are split into two octets, $v$ and $w$, which are eigenstates of
$\gamma^5_1\gamma^5_2$, with eigenvalues $+1$ and $-1$, respectively.
We rewrite (\ref{18}) as
\beg{27}T^{(16\times16)}=v'{}^\dagger\hat T_{vv}v+w'{}^\dagger\hat T_{ww}w
+v'{}^\dagger\hat T_{vw}w+w'{}^\dagger\hat T_{wv}v.\en
By using (\ref{10}) as well as the free-particle equations
(with $\bld k_1=-\bld k_2\equiv\bld k$),
\beg{28}w=m_+^{-1}(E-\gamma^5\Delta\bld\sigma\bld k)v,\quad v'{}^\dagger=
w'{}^\dagger(E-\gamma^5\bld{\sigma k}')m_+^{-1},\en
\beg{29}\Delta\bld\sigma=\bld\sigma_1-\bld\sigma_2,\quad\bld\sigma=
\bld\sigma_1+\bld\sigma_2,\quad\gamma^5=\gamma^5_1,\en
all four submatrices of (\ref{27}) may be put into the form of the last one,
\beg{30}T^{(16\times16)}=w'{}^\dagger m_+^{-1}Mv=
w_1'{}^\dagger m_+^{-1}M^Cv_1,\quad M^C\equiv C_1^{-1}MC_1.\en
(The form of $T$ between
Lorentz-irreducible spinors is necessarily asymmetric and non-hermitian,
anologous to the $u_l'{}^\dagger M^{(2\times2)}u_r$ between the right- and
lefthanded Weyl components of a single Dirac particle.)
The matrices $\hat T_{vw}$ and $\hat T_{wv}$ of $\hat T_{{\rm D}}$ and
$\hat T_{\kappa1,\kappa2}$ vanish; the other two give
\beg{31}M=m_+[(E-\gamma^5\bld{\sigma k}')\hat T_{vv}/m_+
+\hat T_{ww}/m_+(E-\gamma^5\Delta\bld{\sigma k})].\en
As $(1-\bld\sigma_1\bld\sigma_2)/4$ is the projector on the spin-singlet 
state, $\bld{\sigma k}'(1-\bld\sigma_1\bld\sigma_2)$ vanishes. The $m_+$
cancels, and
$-(1+\bld\sigma_1\bld\sigma_2)\Delta\bld\sigma=2i\bld\sigma^\times$ leads to
\beg{32}M_{D}=-4\pi\alpha_Z[2E+2i\gamma^5\bld{k\sigma}^\times]/\bld q^2=
M_{D}^C.\en
The Fourier transform of $M_D^C$ gives the Dirac
interaction of (\ref{6}); the Dirac potential in (\ref{3}) is the Fourier
transforms of $M_D^C/2E$.

As $\beta_1\beta_2\equiv\beta$ commutes with $\gamma^5_1\gamma^5_2$, it 
does not exchange $v$ with $w$:
\beg{33}\hat T_{\kappa1,\kappa2}{}_{ww}^{vv}=-4\pi\alpha_Z[\mp
(\bld{q\sigma}_1)(\bld{q\sigma}_2)+(\bld q\times\bld\sigma_1)(\bld
q\times\bld\sigma_2)]\beta/4m_1m_2\bld q^2.\en

Whereas $m_+$ has cancelled out in $M_D$, it does not cancel completely in 
$M_{\kappa1,\kappa2}$ calculated from (\ref{33}), because of
$m_+\gamma^5=\gamma^5m_-.$ $C_1$ has the following properties:
\beg{34}C_1^{-1}\gamma^5=\gamma^5C_1,\quad C_1\bld\sigma C_1=\bld\sigma
m_+/m_-,\en
\beg{35}C_1\Delta\bld\sigma C_1=\Delta\bld\sigma,\quad
C_1\bld\sigma^\times C_1=\bld\sigma^\times.\en
For (\ref{33}), the $C_1$-transformation commutes with $\hat T_{vv}$ and
$\hat T_{ww}$ which are symmetric under the exchange of 
$\bld\sigma_1$ and $\bld\sigma_2$:
\beg{36}C_1^{-1}(\sigma_{1i}\sigma_{2j}+\sigma_{1j}\sigma_{2i})C_1=
\sigma_{1i}\sigma_{2j}+\sigma_{1j}\sigma_{2i},\en
\beg{37}C_1^{-1}M_{\kappa1\kappa2}C_1=[(E-\gamma^5
\bld{\sigma k}')\hat T_{vv}+\hat T_{ww}(E-\gamma^5\Delta\bld{\sigma k})].\en

The leading terms $E(\bld{q\sigma}_1)(\bld{q\sigma}_2)$ in (\ref{33})
vanish. In the nonrelativistic limit, one may also set $\beta=1$:
\beg{38}(M_{\kappa1\kappa2}^C/2E)_{nr}=4\pi\alpha_Z[\bld\sigma_1\bld\sigma_2\bld q^2-(\bld\sigma_1\bld q)(\bld\sigma_2\bld q)]/4m_1m_2\bld q^2.\en

The other two matrices (\ref{24}) and (\ref{25}) contain a single factor
$\beta_i$, which implies $\hat T_{vv}=\hat T_{ww}=0$. We first treat
(\ref{25}), using $\beta_2w=v,\; v'{}^\dagger\beta_2=w'{}^\dagger$, 
\beg{39}T_{wv}=T_{vw}=-4\pi\alpha_Z\gamma^5\bld
q(\bld\sigma_2+i\bld\sigma^\times)/2m_2\bld q^2,\en
\beg{40}M_{\kappa2}=-4\pi\alpha_Z\gamma^5m_-\bld
q(\bld\sigma_2+i\bld\sigma^\times)/m_2\bld q^2.\en
The $C$-transformation replaces $m_-\bld\sigma_2$ by 
\beg{41}m_-\kalb(\bld\sigma m_+/m_--\Delta\bld\sigma)=\kalb(\bld\sigma m_
+-\Delta\bld\sigma m_-)=\bld\sigma_2m_2+\beta\bld\sigma_1m_1,\en
\beg{42}M_{\kappa2}^C=-4\pi\alpha_Z\gamma^5\bld
q[\bld\sigma_2+i\bld\sigma^\times+\beta(\bld\sigma_1-i\bld\sigma^\times)
m_1/m_2]/\bld q^2.\en
The calculation of $T_{\kappa1}$ and $M^C_{\kappa1}$ is formally different,
but the result follows from $M^C_{\kappa2}$ simply by exchange of $m_1$ and 
$m_2$. The $\beta_2$ is replaced by $\beta_1=\beta\beta_2$;
the $C_1$-transformation replaces $m_-\bld\sigma_1\beta$ by 
\beg{43}m_-\kalb(\bld\sigma
m_+/m_-+\Delta\bld\sigma)\beta=(\bld\sigma_1m_2+\beta\bld\sigma_2m_1)\beta=\bld\sigma_2m_1+\beta\bld\sigma_1m_2,\en
which follows from (\ref{41}) by the exchange of $m_1$ and $m_2$. As a result,\beg{44}M_{\kappa1}^C=-4\pi\alpha_Z\gamma^5\bld
q[\bld\sigma_2+i\bld\sigma^\times+\beta(\bld\sigma_1-i\bld\sigma^\times)
m_2/m_1]/\bld q^2.\en
\end{section}

\begin{section}{The Nonrelativistic Potentials and Hyperfine Mixing}
The interaction in the basic form (\ref{12}) is the Fourier transform of
the Born term,
\beg{45}I(\bld r)=2EH_I(\bld r)=(2\pi)^{-3}\smallint d^3qe^{-i\bld{qr}}M^C.
\en
$I_D=2EV+2i\gamma^5V\bld\sigma^\times\bld p$ results from $M_D^C$. Using 
\beg{46}(2\pi)^{-3}\smallint d^3qe^{-i\bld{qr}}\bld q(-4\pi\alpha_Z/q^2)
=i\nabla V=i\bld rV'/r,\en
\beg{47}I_{\kappa2}=V'/r\gamma^5\bld r[i\bld\sigma_2-\bld\sigma^\times
+\beta(i\bld\sigma_1+\bld\sigma^\times)m_1/m_2].\en
$I_{\kappa1}$ has $m_1/m_2$ replaced by $m_2/m_2$.

Next, we construct the nonrelativistic hyperfine operators, by eliminating 
the small components $\psi_f$ of the Dirac equation
(\ref{3}). We immediately restrict ourselves to the terms linear in $V'$,
\beg{48}V_{\kappa2}=\{\bld\sigma_1\bld p,(V'/r)\bld r(i\bld\sigma_2
-\bld\sigma^\times)\}/4\mu E+[\bld\sigma_1\bld p,V'/r\bld r(i\bld\sigma_1+
\bld\sigma^\times)m_1/m_2]/4\mu E.\en
The commutator arises from the extra factor $\beta$, because of 
$\beta\psi_g=\psi_g$, $\beta\psi_f=-\psi_f$.

The operator containing $\bld\sigma^\times\bld rV'/r$ in the anticommutator
is the hermitian part of the Dirac hyperfine operator, and we simply adopt 
the "Dirac" results (\ref{13})- (\ref{16}):
\beg{49}V_{s2l,\kappa2}=V_{s2l}=V'\bld\sigma_2\bld l/2E\mu,\quad
V_{t,\kappa2}=V_t=\kalb\sigma_tV'/2E\mu.\en
In the commutator, the part containing $i\bld\sigma_1$ is also hermitian,
giving
\beg{50}V_{s1l,\kappa2}=V'\bld\sigma_1\bld l/2m_2^2r.\en
For comparison, the CBG reduction finds only one potential \cite{8},
\beg{51}V_{\kappa2}^{\rm CBG}=\bld\sigma_2\bld lV'/2\mu m_2r.\en
Again, there is agreement in the expectation values,
\beg{52}\langle V_{s2l,\kappa2}+V_{s1l,\kappa2}\rangle=\langle
V_{s2l,\kappa2}^{\rm CBG}\rangle.\en

The part containing $\bld\sigma^\times$ is antihermitian and leads to
\beg{53}V_{s12l,\kappa2}=-iV'\bld\sigma^\times\bld l/4m_2^2r.\en
The operator $i\bld\sigma_2$ in the anticommutator is relatively complicated.
With $i\bld\sigma_1\bld p=\sigma_{1r}(\partial_r-\bld\sigma_1\bld
l/r)=(\partial_r+\bld\sigma_1\bld l/r)\sigma_{1r}$, 
one obtains, neglecting an operator $\sim \Delta V$,
\beg{54}V_{ah,\kappa2}=-[\sigma_{1r}\sigma_{2r},\bld\sigma_1\bld l]V'
/4E\mu r,\en where the index $_{ah}$ stands for "antihermitian".

As a result, the total operator linear in $\kappa_1$ and $\kappa_2$ is
\beg{55}V_{\kappa}=[(\bld\sigma_1\bld l-\kalb i\bld\sigma^\times\bld
l)\hat\kappa+(\bld\sigma_2\bld l+\kalb\sigma_t-\kalb
[\sigma_{1r}\sigma_{2r},\bld\sigma_1\bld l])\kappa]V'/2E\mu r,\en
\beg{56}\hat\kappa\equiv\kappa_1m_2/m_1+\kappa_2m_1/m_2,\quad
\kappa\equiv\kappa_1+\kappa_2.\en
\end{section}
\begin{section}{Results and Discussion}
Finally, we construct the matrix elements and energy shifts of the new
operators. Let $\bld f=\bld l+\bld\sigma_1/2+\bld\sigma_2/2$ denote the
total angular momentum operator, and $f(f+1)\equiv F^2$ an eigenvalue of
its square. In the following, we only consider states with $l=f>0$, where
spin-orbit and hyperfine operators mix already at the order $\alpha_Z^4$,
such that the antihermitian operators are essential. The triplet-singlet
spin basis will be used; the relevant spin states are denoted by
$\chi_{f,1}$ and $\chi_{f,0}$, respectively:
\beg{57}\chi=\left(\begin{array}{cc}\chi_{f,1}\\\chi_{f,0}\end{array}\right),
\quad\bld\sigma_1\bld l=\left(\begin{array}{cc}-1&F\\F&0\end{array}\right)
\quad
\bld\sigma_2\bld l=\left(\begin{array}{cc}-1&-F\\-F&0\end{array}\right).\en
In this two-component space, the tensor spin operator arises as follows:
\beg{58}\sigma_{1r}\sigma_{2r}=\left(\begin{array}{cc}1&0\\0&-1\end{array}
\right),\quad\bld\sigma_1\bld\sigma_2=
\left(\begin{array}{cc}1&0\\0&-3\end{array}\right),\quad\sigma_t=2
\left(\begin{array}{cc}1&0\\0&0\end{array}\right).\en
The following antihermitian matrices are also needed:
\beg{59}i\bld\sigma^\times\bld
l=2\left(\begin{array}{cc}0&-F\\F&0\end{array}\right)=-
[\sigma_{1r}\sigma_{2r},\bld\sigma_1\bld l].\en
Although all spin potentials except $V_{s1l,D}$ contain $2E\mu=2m_1m_2$ in 
their denominators, it is convenient to separate a factor $4\mu^2$, because
the expectation value of $V'/4\mu^2r$ is
simple for the hydrogenic Schr\"odinger equation for a particle of mass 
$\mu$:
\beg{60}\langle V'/4\mu^2r\rangle=\langle\alpha_Z/4\mu^2r^3\rangle=\alpha_Z^4
\mu/[4n^3(f+1/2)F^2].\en 
Collecting the matrices, one obtains
\beg{61}V_\kappa=\left(\begin{array}{cc}-\hat\kappa
& 2F(\hat\kappa-\kappa)\\
0&0\end{array}\right)\frac{V'}{2E\mu r},\quad\hat\kappa-\kappa=
(\kappa_1m_2-\kappa_2m_1)\frac{m_2-m_1}{m_1m_2}.\en

To this one has to add the tensor interaction
$\kappa_1\kappa_2I_{\kappa1,\kappa2}$ from (\ref{38}) and the Dirac spin
potentials (\ref{13})-(\ref{16}),
\beg{62}V_{s,tot}=\left(\begin{array}{cc}-1-2y&Fx\\F&0\end{array}\right)
\frac{V'}{4\mu^2r},
\quad y=(\hat\kappa-\kappa_1\kappa_2)\mu/E,\en
\beg{63}x=1-4(1-\hat\kappa+\kappa)\mu/E=(m_2-m_1)(m_2-m_1+4
\kappa_1m_2-4\kappa_2m_1)/E^2.\en
(In \cite{3}, both $\sigma_t$ and a minus sign were forgotten in eq (5.72).)
The eigenvalues of $V_{s,tot}$ have been called $-2a_l^{(\pm)}F^2$ \cite{4}
, with $F^2=f(f+1)=l(l+1)$. They are now found as
\beg{64}-2a_l^{(\pm)}F^2=-\kalb-y\pm\sqrt{(\kalb+y)^2+F^2x}.\en
They may be compared with those of the CBG spin matrix.
As the off-diagonal elements of a real $2\times2$ hermitian matrix are
equal, the corresponding $x^{\rm CBG}$ must be a square:
\beg{65}x^{\rm CBG}=(m_2-m_1+2\kappa_1m_2-2\kappa_2m_1)^2/E^2.\en
One thus finds 
\beg{66}x^{\rm CBG}=x+4(\kappa_1m_2-\kappa_2m_1)^2/E^2.\en
The difference vanishes both for positronium and for protonium. For muonium, $\kappa_1=\kappa_2=\alpha/2\pi$ makes
$x^{\rm CBG}-x=E^{-2}(m_1-m_2)^2\alpha^2/\pi^2$.
A contribution of this order could be cancelled by two-loop vertex graphs,
analogous to the $V^2$-cancellation observed in the Bethe-Salpeter equation.
 For hydrogen, the difference is
strongly suppressed relative to the main term $x\approx1$.
Antiprotonic atoms such as $\overline p\,^3$He
($g_1=5.58,\,\kappa_1=1.79\;g_2=-3.185,\,\kappa_2=-2.592$ \cite{10}) 
have $x^{\rm CBG}-x>1$. As
the baryon magnetic moments involve the strong interaction,
the results could in principle differ in different QED formalisms. The 
author is not aware of any relevant experiments.
There exist however atomic states for which the square in (\ref{66}) is
verified beyond doubt, namely the large-$l$ Rydberg states of helium
\cite{103}. The inner electron makes the core appear like a fermion of mass
$m_2=m(^4\!{\rm He})$ and $g$-factor $g_2=-2m_2/m_1\equiv-2(1+\kappa_2)$. The
resulting "hyperfine structure" is in fact much larger than the fine 
structure and is correctly reproduced by $x^{\rm CBG}$. Clearly, when the
square in (\ref{66}) is much larger than $x$, the 8-component method
breaks down, at least in its present form.

Even without the square, one may wonder how a perturbation theory with
non-hermitian operators works \cite{100}. In the case at hand, the
orthogonality relations are derived from an unperturbed operator which is
hermitian. In more general cases, there is in principle also the possibility
of expressing a non-hermitian operator $H$ in terms of an equivalent
hermitian operator $H_h$ by a similarity transformation,
$H=V_{sim}H_hV_{sim}^{-1}$. The $V_{sim}$ of the relativistic equation
(\ref{3}) has not yet been found; for the nonrelativistic reduction
(\ref{62}) it is a $2\times2$ matrix. The result is elegant for
$\kappa_1=\kappa_2=0$,
\beg{67}V_{sim}(\kappa_i=0)=\left(\begin{array}{cc}\delta\hat m^{1/2}&0\\
0&\!\delta\hat m^{-1/2}\end{array}\right),\quad V_{s,h}(\kappa_i=0)=
\left(\begin{array}{cc}-1&F\delta\hat m\\ \!F\delta\hat m&0\end{array}\right)
,\en  with $\delta\hat m=(m_2-m_1)/E$. A particular advantage of the 
hermitian form is presently not evident.
\end{section}
\bigskip                    
\begin{thebibliography}{99}
%\begin{references}                \baselineskip0pt
\bibitem{1}H. Bethe and E.E. Salpeter, "Quantum Mechanics of One- and
  Two-Eletron Atoms (Springer, Berlin 1975)
\bibitem{2}H. Pilkuhn, J. Phys. B 25 (1992), 289
\bibitem{3}M. Malvetti and H. Pilkuhn, Phys. Rep. C 248 (1994), 1
\bibitem{4}H. Pilkuhn, J.Phys.B 28 (1995), 421
\bibitem{5}R. H\"ackl et al, Phys. Rev. A 57 (1998), 3268
\bibitem{6}V. Hund and H.Pilkuhn, J. Phys. B 33 (2000), 1617

\bibitem{100}H. Pilkuhn, "Relativistic Quantum Mechanics" (Springer, Berlin
and Heidelberg 2002)
\bibitem{101}G.R.Sapirstein and D. Yennie, in "Quantum Electrodynamics"
(Kinoshita ed.,World Scientific, Singapore 1990)
%\bibitem{102}H. Pilkuhn, Phys. Letters B 512 (2001), 412
\bibitem{7}Z.V. Chraplyvy, Phys. Rev 91 (1953), 388; W.A. Barker and
F.N. Glover, Phys. Rev. 99 (1954), 317
\bibitem{8}S. Waldenstr\"om and H. Olsen, Nuovo Cim. A3 (1971), 491
\bibitem{10} S. Barmo et al, Z. Physik A 301 (1981), 283
\bibitem{103} E.A. Hessels et al, Phys. Rev. Letters 65 (1990), 2765
\end{thebibliography}
\end{document}





