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{e-iWp(xo-xo)+ip {x -
x'f_
eiwp{xo-xo)+ip.{x-x')}
We wish to give a physical interpretation of this formula. To do so, we may imagine the system enclosed in a very large cubic box of size L. The momentum integral is replaced by a Riemann sum, and each component of the momentum is of the form 2nniL with n an arbitrary integer, in such a way that [1/(2nn J d3 p ~ LnIL3 .
QUANTUM FIELD THEORY
We then have
Gret(x - x') -4 W(xo - xo)[!: o/+,p(x)o/+,p(x') -!: o/_,p(x)o/'I<.,p(x')] (1-171)
The momentum p assumes only discrete (but very dense) values and the functions
0/ + (x) =
1 L3 / 2 J2w p
e iwpxo +ip' x
(1-172)
stand for periodic solutions of the homogeneous Klein-Gordon equation. The plus or minus sign corresponds to the sign of the frequency. The relative minus sign in (1-171) [together with the factor i in front to build a real quantity: o/+,p(x) = 0/_, _p(x)] comes from the fact that the solutions of the equations are to be normalized according to a nondefinite "norm" [see formula (2-5)]. We shall see that while 0/+ has a positive norm, 0/- has a negative one. In any case the expression (1-171) leads to an interesting interpretation of propagation according to Gret(x - x'). Positive and negative frequencies are respectively propagated forward in time by the first and second term of (1-171). For Gadv(x - x') we have a similar expression with x and x' interchanged. The difference
G(-)(x) = Gret(x) - Gadv(x)
(2n)3
_i_ fd4p
e-ip'x
)(5(~2 _ m2 )
(1-173)
= -i-
(2n
3 d p e-'wpxo+'P'x _ e'wpxo+'P'X) . . . __ ( .
is an odd function vanishing outside the light cone such that
(1-174)
It clearly satisfies the homogeneous Klein-Gordon equation. When m 2
0, it
reduces to
(1-175)
= lim _1 [ 2
1 1 ] ~~o 4n i (x - i1'/)2 (x + i1'/)2
where 1'/ is an infinitesimal positive time-like vector. Finally, the half sum
G(+)(x) 1 [G () L: ret X
+ Gadv ()] - - (2n)4 PP X -
fd pe
- ip x
1 p2 _ m2
(1-176)
is an even solution of (1-165). In (1-176) the principal part symbol applies to the Po integral, and we recall that
CLASSICAL THEORY
---- =
Po - wp
ie
1 Po - wp
inJ(po - w p )
J(p2 _ m2) = J(po - w p)
+ J(po + w p)
(1-177)
The above discussion takes place in a purely classical context. We shall, however, encounter in the quantized version another even solution to the very same equation, first introduced by Stueckelberg and Feynman. One reason to explain why it does not naturally appear in classical physics is that it is a complex distribution defined through
GF(X)=-(21)4jd4pe-iP'X 2 1 n p -m 2
It therefore satisfies
+,Ie
(1-178)
If we discretize as above we find the equivalent form:
GF(x - x') ~ i[8(xo - xo) L CP+ ,p(x)cp'!t- ,p(x') + 8(xo - XO)L cp- ,p(x)cp!.,p(x')]
(1-179) While the previous Green functions were zero outside the light cone, this is not the case for GF , which has an exponential tail for negative X2. Let Xo = and set r = 1 I. Then x
GF(O, r) = (2n)2 r
P _ r p2 _ m2 e p
i e'"
(nmr)1/2
(2n r2
(1-180)
From (1-179) we read off that, according to GF , positive (negative) frequencies propagate forward (backward) in time. This distinction according to frequency explains why GF is intrinsically complex. Note also that GF(P) is a meromorphic function of the complex variable p2 in contradistinction to the previous Green functions.
These expressions can be generalized to more general cases such as the electromagnetic one. Corresponding to the fact that Maxwell's equation alone
a~(a
(1-181) we find in (1-182)
is not sufficient to determine the potential A" in terms of the conserved current momentum space that the matrix
is singular. Its determinant vanishes identically. The operator L is in fact proportional to a projector since (1-183) To avoid this difficulty, we can either add a small mass term or proceed as in Sec. 1-2 and introduce
QUANTUM FIELD THEORY
an extra term (A./2)(8 A)' in the lagrangian. The modified equations of motion are
(1-184)
and the propagator becomes
(1-185)
The apparent singularity in the numerator plays no role when is convoluted with a conserved current. For A. = 1 we recover the Feynman propagator g"'GF(x). We shall have more to say about this in Chap. 3.
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