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enables us to obtain the probability that an incoming state la> will evolve in time and be measured in the Ib> state. The out-states can represent the incoming states for a successive process. This is the case, for instance, when one prepares secondary beams. Therefore an isomorphism must exist among the in- and out-Fock space. Our goal is to relate the above transition amplitudes to the actual measurements. In other words, we want to establish the relativistic conventions of Fermi's golden rule and give the expression for cross sections.
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To take a simple case we first envisage the scattering of two distinct, spin less particles. The incoming state is written in terms of the incident wave packets in momentum space:
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With these wave packets are associated positive energy solutions f(x) of the Klein-Gordon equation with corresponding mass:
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and a flux given by
3 'f3 x f-* (x) oof(x) -f 2p d(2n)3 If(p) IZ p d <-+O
(5-3)
The transition probability to a final state If, out> is
HJ<-i =
I<f, out Ii, in>lz
(5-4)
In the absence of external sources, translation invariance implies that the matrix elements vanish if energy and momentum are not conserved. Furthermore, as discussed in Chap. 4, the assumption of an isomorphism between out- and instates implies the existence of a unitary operator S, commonly called the S matrix, such that
<f, out Ii, in>
<f, inl S Ii, in>
(5-5)
QUANTUM FIELD THEORY
Unitarity of S is necessary to conserve probabilities. Strictly speaking, S applies the out-states on in-states according to (5-5). We shall loosely say that we take its matrix elements in the in-space and unless necessary we will drop the index "in." It is then decomposed as
+ iT
(5-6)
where T contains the information on the interactions. If we idealize the final state as a plane wave state we can extract the delta function of energy momentum conservation according to
<II T Ipl, pz) = (2n)4c5 4(p f
PI -
pz)<II ff Ipl, pz)
(5-7)
and the reduced operator ff acts on the energy shell. When we substitute the decomposition (5-6) into the matrix element (5-5), the identity only contributes to forward scattering and represents a part of the incident wave packet unaffected by interactions. In most experiments we are only interested in the deflected part. This justifies keeping the sole contribution of ff in the transition probability
Vlj<-i
01 16
2.0(
dpl dpz dp'l dp'z
N(Pr)fz*(pz)Il(p'r)Iz(p'z)(2n)4c5 4(p1
+ pz -
P'l -
p'z)
(5-8)
2,d -,
x (2n)4c5 4(p f - PI -
Pz)<II ff Ipl, pz)*<II ff Ip'l, p'z)
If the final state is not a sharply defined eigenstate of momentum the above
formula has to be slightly generalized in an obvious way. In most cases, one prepares the initial particles with almost well-defined momenta with a negligible width on the scale of the variation of the matrix elements of ff. In short, J;(Pi) is peaked around a mean value Pi with a width LlPi so that
<II ff Ip'h p'z) ~ <II ff Ipl, pz) ~ <II ff Iph pz)
Using this approximation and the integral representation
(2n)4c5 4(pl
+ pz
- P'l -
p'z) =
d4 x e- ix '(P\+P2-Pl-P2)
we find
Vlj<-i =
d4 x I!l(X) IZI !z(x) IZ(2n)4c54(Pf - PI - pz) 1<11 ff Ipl, pz) IZ
(5-9)
which can be interpreted as a transition probability per unit time and unit volume:
(5-10)
Now ];(x) that
e-iiJ,'x Fi(X) with Fi(X) a slowly varying function of x in such a way
if*(x),%](X) ~ 2PJl I ](x) IZ
(5-11)
ELEMENTARY PROCESSES 201
To be concrete let us assume that particles of type 1 are the incident ones in the laboratory frame on particles of type 2 at rest. The number of particles in the target per unit volume is dnz/dV = 2p~ I]z(xW, p~ = mz. The incident flux is given by velocity Ip1l1p times density 2p 1 ]l(XW equal to 2Ip11111(xW. This gives the following interpretation of Eq. (5-10): dl1j<--;/dV dt = transition probability from state Ii> to state If> per unit time and unit volume = target density [2mzl ]z(x) x incident flux [21 P1 I I]1 (x) IZ] x cross section d(J:
(2n)4<5 4(Pf
P1 - pz)
1_ II <fl ff Iplo Pz> IZ 4mz P1
(5-12)
The cross section is the transition probability per scatterer in the target and per unit incident flux. The idealization of a final state with well-defined momentum is corrected by integrating over an energy and momentum resolution Ll. For instance, if we consider a final state of n distinct spinless particles (Fig. ;-1), the cross section is given by d(Jn<--2=4[( P1 PZ
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