For two free leptons, the uncoupled equations are first taken as Pauli
equations (2.58), including external vector potentials. Using again the com-
pact notation
π
i
=
π
i
σ
i
, they may be written in forms which also comprise
KG particles,
p
02
i
ψ
as
=
K
i
ψ
as
,
K
i
=
m
2
i
+
π
2
i
.
(4.388)
The first step is to combine the two equations into a single one which
contains only a common time shift in the form
p
0
1
+
p
0
2
≡
p
0
. With
p
02
1
−
p
02
2
=
p
0
(
p
0
1
−
p
0
2
), the difference of the equations gives
(
p
0
1
−
p
0
2
)
p
0
ψ
as
= (
K
1
−
K
2
)
ψ
as
.
(4.389)
4.11 Dirac Structures of Binary Bound States
197
An eigenvalue
K
0
of
p
0
will be assumed. As
p
0
1
−
p
0
2
commutes with
K
1
and
K
2
, a second application of
p
0
1
−
p
0
2
gives
(
p
0
1
−
p
0
2
)
2
ψ
as
= (
K
0
)
−
2
(
K
1
−
K
2
)
2
ψ
as
.
(4.390)
Next, consider the sum of the two equations (4.388),
(
p
02
1
+
p
02
2
−
K
1
−
K
2
)
ψ
as
= 0
.
(4.391)
With
p
0
1
=
1
2
K
0
+
1
2
(
p
0
1
−
p
0
2
) etc, this leads to
[
K
02
/
2 + (
K
1
−
K
2
)
2
/
2
K
02
−
K
1
−
K
2
]
ψ
as
= 0
.
(4.392)
In terms of the triangle function
λ
(4.76), (
K
0
)
−
2
λ
(
K
02
, K
1
, K
2
)
ψ
as
= 0.
In the “constraint Hamiltonian mechanics” which dates back to Dirac, the
coupling between the two particles is introduced already in equations (4.388),
but practical success has been limited (Crater and Van Alstine 1994).
For particles of equal spins,
λ
may be decomposed symmetrically in the
indices 1 and 2:
λ
= (
K
02
−
K
1
−
K
2
)
2
−
4
K
1
K
2
.
(4.393)
With
m
2
2
−
m
2
1
=
m
+
m
−
,
λ
= (
K
02
−
m
2
1
−
m
2
2
)
2
−
2
π
2
1
(
K
02
+
m
+
m
−
)
−
2
π
2
2
(
K
02
−
m
+
m
−
)
−
4
m
2
1
m
2
2
+ (
π
2
1
−
π
2
2
)
2
.
(4.394)
For
A
= 0, the transformation (4.291), (4.307) of variables,
p
1
=
p
lab
+
K
E
1
/E,
p
2
=
−
p
lab
+
K
E
2
/E,
E
≡
(
K
02
−
K
2
)
1
/
2
,
(4.395)
yields together with
p
lab
,x
=
p
x
, p
lab
,y
=
p
y
, p
lab
,z
=
γp
z
,
λ
=
γ
2
[(
E
2
−
m
2
1
−
m
2
2
)
2
−
4
m
2
1
m
2
2
−
4
E
2
(
p
2
x
+
p
2
y
+
p
2
lab
,z
/γ
2
)]
.
(4.396)
The factor
γ
2
in front of the square bracket corresponds to the separation of
one factor
γ
from (4.308). The square bracket is factorized by means of the
Dirac matrices
β, γ
5
=
β
x
and
iγ
5
β
=
β
y
as follows:
λ/γ
2
= (
f
0
+
f
z
β
+
f
x
β
x
+
f
y
β
y
)(
f
0
−
f
z
β
−
f
x
β
x
−
f
y
β
y
)
,
(4.397)
f
0
=
E
2
−
m
2
1
−
m
2
2
,
f
z
= 2
m
1
m
2
,
f
x
= 2
E
pσ
1
,
f
y
= 0
.
(4.398)
The second factor of (4.398) is precisely that of the leptonium equation
(4.237). More general forms include a rotation by an arbitrary angle
ω
D
about
the
β
-axis in Dirac space (2.103),
f
x
= 2
E
pσ
1
cos
ω
D
, f
y
= 2
E
pσ
1
sin
ω
D
, or
a rotation (4.270) in the Pauli product space. For constant
f
0
, one may also
take
f
y
= 2
E
pσ
2
sin
ω
D
, as
σ
1
i
σ
1
j
and (
σ
1
i
cos
ω
D
+
iσ
2
i
sin
ω
D
)(
σ
1
j
cos
ω
D
−
iσ
2
j
sin
ω
D
) are identical in their symmetric tensor components. (In the quark
198
4 Scattering and Bound States
model,
f
y
is not excluded a priori even when
f
0
is
r
-dependent. For small
ω
D
, it would mainly affect the hyperfine structure of quarkonium.)
The inclusion of
A
is trivial in the 16-component equation (3.186) or
(3.190), whereas (3.218) is still unclear. The first-order relativistic Zeeman
shift
δE
of the ground state has been calculated by Faustov (1970) and by
Grotch and Hegstrom (1971). Besides the factor (1
−
α
2
Z
/
3) from
H
Zee
(2.285)
in the Pauli reduction of the GY-equation (4.378), these authors find essen-
