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We may change k into - k in the second term of the right-hand side, the coefficient of (y. k vanishes, and we obtain
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Kret(xz, Xl) may also be expressed in terms of the scalar retarded Green function [Eq. (1-169)] as Kret(xz -
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xd = -i(iilz + m)Gret(xz - xd
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G ret :
Equation (2-110) also follows from the identity (1-165) satisfied by (Oz From
+ mZ)Gret(XZ - xd = (j4(XZ
- Xl)
it follows that
K ()
ret X =
(2n)4
fd4k
-ik.x
~+m + ie)Z _ k Z _
where the ko integration has to be performed first, along the dashed contour of Fig. 2-3.
Tq.e hole theory suggests the introduction of a different Green function, the Feynman propagator already discussed in the last section of Chap. 1. It will
THE DIRAC EQUATION 91
Figure 2-3 Contour of integration for the definition of Green functions. The broken line corresponds to the retarded propagator, the solid line to the Feynman one.
appear in a natural way in the quantized field theory. Nevertheless, let us sketch the ideas that led Feynman and Stueckelberg to its construction. A Green function may be considered as describing three successive steps:
1. Appearance of an electron at (tb Xl) 2. Propagation of the electron from (t1, xd to (t2' 3. Disappearance of the electron at (t2, X2)
As long as the electron has a positive energy, this process is physically acceptable for t2 > t 1. On the other hand, if we deal with a negative energy electron, we would like to interpret its vanishing as the appearance of a positron, and vice versa. The second step should be then considered as the propagation of the positron from (t2' X2) to (tb Xl), which makes sense only for t2 < t1' Therefore in the hole theory, we would like to construct a Green function which propagates the positive energy solutions only for t 2 > t1. the negative energy ones (more precisely the positrons) only for t 1 > t2:
SF(X2, Xl)
J(2:)33~E
+ e(t1
[e(t2 t2)b(~
t1)a(~ + m)
- m)
e-ik'(x,-xl)
eik.(x,-x 1 )]
The constants a and b are determined by imposing that
(i 2 - m)SF(X2, Xl) = c5 4 (X2
(2-111)
[Notice the change of normalization with respect to (2-110)]. It follows from a straightforward calculation that
(i 2 - m)SF(X2, Xl)
= ic5(t2 - td
-b(~
J2~~~)3 yO[a(~ +
m) e ik '(X2- X l)
- m) e- ik '(X2- X d]
QUANTUM FIELD THEORY
= ib(tz - tl)
3 f 2E(2n)3 eik (x,-x,j d k
yO[a(EyO - yo k
+ m) -
b(EyO
+ yo k -
and the condition (2-111) is fulfilled provided a = - b = 1/i. The result is thus
S F(X2 Xl) =
1 d k i f 2E(2n)3
[&(tz - tl)(~
. + m) e- ,k (x,
-Xl)
(2-112)
Kret(xz - Xl) and iSF(xz - Xl) may differ only by a solution of the homogeneous Dirac equation. This is indeed what is found by a direct calculation: 3 d k "" K ret (Xz - Xl ) - I'S F (Xz - Xl ) 2E(2n)3 (IJ - m) e ik (X'-Xl)
A covariant expression is obtained by means of the integral representation
&(t) = lim
--+0+
dw eirot -.- - - - . 21n w - lG
The limit e -+ 0 + will be understood in what follows. The quantities we are dealing with are distributions acting on smooth test functions. After insertion of this expression into (2-112), we get
SF(X)=-
d3k . . - - - 1 [ (~+m)e-'k'x foo -dw . e,wt (2n)4 2E -00 W lG dw . . m) e''k x foo _ _ e-'w
+ mZ)1/2] and in the
In the first integral we set pO = E - w, P = k[E = kO = (kZ second one pO = W - E, P = - k :
S F(X) =
f - - -2E(2n)4
d4p e- ip . x [EYO E
+ yop-m + pO - ie
-----'----;c-=---
Eyo- yop+m] E - pO - ie
For a vanishing positive e, we may write
+ pO _
ie)(E - pO - ie) = - (P5 - pZ - m 2
+ ie)
(2-113)
Finally,
d4p S()-f -ip.x p+m FX- (2)4 e n p 2 -m 2 + ' lG
The ie term gives the prescription for the momentum integration. The integration over pO is performed first, along the solid contour shown on Fig. 2-3, and then we integrate over p. If we consider m as complex me = m - ie, <;-+0+, we have
THE DIRAC EQUATION
p+ m p + me 1 Z + is = pZ - m; = p - me = pZ _ m
S()-
IP - m + is
Therefore, we may write the Fourier transform of SF(X) as
F P - pZ _ mZ + is -
+ is
(2-114)
Notice the relationship between SF(X) and the Feynman propagator of the scalar field (1-178) : (2-115) To summarize, the role of the Feynman propagator is to propagate the positive frequencies toward positive times and the negative ones backward in time. Let tjJ(+)(tl,XO) and tjJ(-)(tz, x) be the positive and negative frequency components of a solution, given for tl and tz respectively, tl < tz and all x. The propagator SF allows us to find tjJ(t, x) at intermediate times t:
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