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The probabilities of emission are expressed in terms of these transverse components. Indeed, for Po we find
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1<00utIOin>12 exp { -
I< Oin ISIOin>12
dk[1 J1(kW
+ 1J2(kWJ }
(4-21)
To compute the probability Pn corresponding to the emission of n photons when neither the momenta nor the polarizations are observed, we recall that an n photon state is expressed by W'!)(k 1)'" aW.n)(kn) 10> 1 k1A1' k2A2,"" knAn> = a with the normalization
<k1 Ab"" knAnlk'l Xb
.. ,
k~A~>
LD A1AP' D (2n)3 2k D3(k 1 AnAP
k p J"'(2n)3 2k~D3(kn - kl'J
where the sum runs over permutations. Owing to Bose statistics, the projector over the n photon states reads
Pn = f 1 n.
f- -
dk1.. dkn L 1k1 A1,'''' knAn> <k1 A1,"" knAnl '<,=1,2
(4-22)
INTERACTION WITH AN EXTERNAL FIELD
We have therefore to consider the matrix element
<k1At, ... ,knAn,outIOin) = <k1A1, ... ,knAn,inISIOin)
ex p { -i f dkClJ1(kW
+ IJ2(k)J2]}
x <k1A1, ... ,knAn,inlexp[ -i fd4XAl:)(x)oj(X)J!oin)
(4-23) Now
f d4x Al:)(x) j(x)
f dk f dk
;.to a(;.lt(k)e(;')(k)
;'~'2 d;')t(k)J;.(k)
J(k)
Current conservation has again limited the sum to the transverse degrees of freedom; if we include the "longitudinal" and "scalar" photons in the projector Pn , their contribution cancels automatically. In Eq. (4-23), the term with n creation operators gives the only contribution:
(~:t <k 1At, ... ,knAnl f dq1"' dqn d"I)t(Q1)J"I(qd .. d" )t(qn)J".(qn)IO)
= (-
it J;'I(k 1) J;..(k n)
(4-24)
The factor lin! has disappeared, since there were n! equal terms. The probability Pn reads
Pn= <OinIPnIOin)
{f dq [I J1(qW
+1 J2(qWJ
exp {- f dk [I J1(kW
+ P2(kWJ}
(4-25)
Defining (4-26) we obtain the Poisson distribution
e- n -
_ fin
(4-27)
The distribution is normalized:
LPn = 1
QUANTUM FIELD THEORY
and the average number of emitted photons is ii:
The Poisson distribution (4-27) reflects the statistical independence of the emission of successive photons, which may also be seen on the factorized structure of the matrix element (4-24). On the other hand, let us examine the nature of the final state. We start at time - 00 from the vacuum state 10 in) which is an eigenvector of the annihilation part AfJ+)(x):
AfJ+)(x)
10 in) = 0
(4-28)
We may replace the time evolution of the state by that of the operator, and thus consider
Here A~l+) is the positive frequency part of the classical field involved in Eq.
(4-11):
_1_' A Il\+)(x ) - (2n fd3k-1- e -ik,xJIl(k)1 kO=lkl c 2ko
(4-29)
Hence we have
<0 inl A~ut(x) 10 in) = A~I(X) =
f d y G(-)(x 4
y)jll(y)
(4-30)
Equation (4-28) means that the final state is a coherent state (compare Chap. 3). This is not contradictory with the Poisson distribution of the emission. Indeed, there is a deep connection between this property of the state to be an eigenstate of the operator A(+) and the statistical independence of successive emissions. In a loose sense, neglecting quantum fluctuations, the field in the final state is Acl(X). This is what is expressed by Eq. (4-30).
4 :1-2 Emitted Energy and the Infrared Catastrophe
To obtain an average value of the energy emitted in the process we consider
g = <0 inl H(A out ) 10 in)
<0 inl S-l H(Ain)S 10 in) <0 inl S-l f
dk kO
0 A~.2 a!~)t(k)a!~)(k)S 1 in)
(4-31)
and observe that the contributions of unphysical degrees of freedom cancel in the sum as expected. From Eq. (4-11), it follows that
A = 1,2
INTERACTION WITH AN EXTERNAL FIELD
and thus
g = f
ik kO[IJl(kW + IJz(kWJ
~ f (~;~3 [IJl(kW + IJz(kWJ
(4-32)
This result agrees with the classical theory! Indeed, the radiated electromagnetic field FIlV corresponding to the potential A~l is
Fmx) = Oil A~l - OV A~l = f
(~:~3 e(kO)c5(P) e- ik . X[kll J"(k) -
kV JIl(k)]
The energy density is
e(x)
[Fg1"(xW
1 [F~r(XW} ';"~P';3
and its integral, averaged over a long period of time, reads
~l= fd 3Xe(x) = fdk2~oLtllkOr(k)-k"JO(k)IZ
+ 1 ,;,,~p';31 k"JP(k) - kPr(kW ]
Using the decomposition (4-19) on the light cone, the longitudinal component of j does not contribute, and we get
~l =
f dk kO [lJl(kW
+ IJz(kWJ
(4-33)
in agreement with (4-32). The interpretation of this result and of the one provided by Eq. (4-26) is that the number of photons emitted in a phase space element dk is (4-34) and their energy is (4-35) In particular, if a finite amount of energy is emitted at low frequency, the number of soft photons dn = djjhkO tends to blow up. Typically, it may happen for certain currents j that Sdg is finite, whereas SdjjhkO diverges. This is the famous infrared catastrophe, already encountered in Chap. 1. We saw there that when a charge is suddenly accelerated from momentum Pi to momentum Ph it creates a current
jll(k)
ie (PIll _ Pill) PI k Pi k
As a consequence, it radiates a flow of photons, of finite total energy iff '" S dko. Unfortunately, the total number of particles diverges as n'" SdkOjhko. Therefore
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