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We have appended subscripts to the product of y matrices in order to distinguish the electron from the positron variables. The structure Yllly~/k2, with k = (p - p'), arises from the use of the Feynman gauge. We can, however, split this interaction into an instantaneous and a retarded part in the total rest frame, P = (E, 0), according to (10-87) The second term on the right-hand side contains both retardation and magnetic interaction. It will be treated perturbatively (we wish we could do better); so will be the annihilation contribution. The zeroth-order approximation for K is obtained by separating in VB a Coulomb part Vo from a remaining part noted Vb, according to (10-87). We write therefore in the instantaneous approximation
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(10-88)
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This suggests the derivation of an equation for
<p(p)
dpo K(pO, p)
(10-89)
by dividing both sides of (10-88) by the wave operators and integrating over pO. Thus
<p(p) = -i ~
4n 3
(fl2 + p + mh (fl2 - P+ mh (P12 + p)2 - m2 + is (P12 - p)2 - m2 + is
(p _ p,2) Yl Y2 <p(p )
d3p'
,
(10-90)
The is prescription in the denominators follows from the derivation of the equation. The pO integral converges and may be computed by closing the contour in the upper half plane encountering poles at Po = =+= EI2 + w - is where w = ~p2 + m 2. If we use the Dirac notations 13 = yO and a: = 131' and the definitions
H(p)
= a: p + 13m
A (p) =
H(p)
(10-91)
Af(p)Ai(-p)
the result of the integration over pO takes the form
A ++
<p(p) = 2" w - EI2
+ w + EI2
A- -
2n 2
d 3 p' <p(p') (p _ p,)2
(10-92)
Note the equivalence between the two formulations of Feynman's propagator
s= P2 Since
+m 2 . + 18
[A +(p)
Po W
. + 18 + Po + W
A -(p)
H(p)A (p)
= wA (p)
Eq. (10-92) can also be written as an effective one-body equation
[Hl(P) + H2(-p)-E]<p(p)= (A++ - A--)2:2
f(p~~')2 <p(p')
(10-93)
Denoting <p = A <p, Eq. (10-93) is equivalent to a set of coupled equations d3 , ++ ++ ()( P ++, --, (2w - E)<p (p) = A 2---Z ( ')2 [<p (p) + <p (p )] , n p- p
(2w+E)<p--(p)=A-- 2:2
f f(P_:y
[<p++(p')
+ <p--(p')]
(10-94)
QUANTUM FlliLD THEORY
Neither (10-88) nor (10-92) is sufficient for our purposes since it omits the retarded part Vb and the annihilation kernel crucial for the hyperfine splitting. When both of these will be included, we obtain the correct version of an equation originally discussed by Breit for such problems. To discuss this effect, we will satisfy ourselves here with the form of perturbation theory originally derived by Sal peter. It is designed to circumvent the difficulty arising from the presence of the energy E, quadratically in the differential operator of the equation (and parametrically in the general kernel) and is suited to an instantaneous unperturbed interaction. Returning to Eq. (10-88) we multiply both sides by y y~ and obtain
[~ + Po - Hl(P)J[~ - Po -
-P)JK ;n
vqJ(p)
(10-95)
vqJ(p)
2: f
d3p' (p _ py qJ(p')
Assuming qJ to be known and using the same it; prescription to invert the operator D acting on K we have
D- 1 -vqJ 2in
h [E12 + Po +
(10-96)
(w - it;)] [E12
- Po + (w
- it;)]
Integration over Po would lead back to Eq. (10-92). For shortness let us define in such a way that
(H - E)1]
(10-97)
where we have set
H(p)
From Eq. (10-93) it follows that
Hl(P)
+ H2 ( -p)
-)1]
qJ = (A + + - A -
(10-98)
and K is given by
K = (H - E) 2in D- 1]
(10-99)
For a fixed 1] corresponding to a given solution of the equation we define the amplitude Q through an expression similar to (10-99) but with the energy E replaced by E'. Recall that D also depends on E so that
Q = (H - E') ~ D- 1 (E')1] 2m
(10-100)
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
Consider the combination
pO [ D(E') - 2m fd vJQ
(H - E')
~ 11 - ~ fd pO v(H 2m 2m
~ D- 1(E')11 2m
On the left-hand side when E' reduces to E and Q to K we find zero. On the right-hand side the only dependence on the variable pO is in the propagator D- 1 (E'). The integral is the same as above, leading to
pO [ D(E') - 21n fd vJQ = (H - E') 21n 11- 21n v(A + +
A - -)11
(10-101)
= - ~ (E' - E)11
The last equality follows from Eqs. (10-97) and (10-98). Similarly, if
Q = I1t(H - E') -1 D- 1 (E') . 2m
(10-102)
the hermiticity of H and v leads to
QD(E') -
~ fd pO Qv = 21n
bV = y y~(J'b
~ I1t(E' 21n
(10-103)
We are now in a position to discuss the effect of a perturbation
+ Va)
(10-104)
to the instantaneous Coulomb interaction. The new wave function K' and energy
E' will satisfy
(10-105) We multiply this equation to the left by Q, defined as above relative to E', and integrate over the four-momentum p. From Eq. (10-103) we obtain
(E' - E)
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