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L (21 + l).:rz Pz(cos 8)
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(5-168)
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1 8n 00 = ~ 1m .:r(cos 8 = 1) = ~ L (21 + 1) Im.:rz
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The last inequality turns into an equality below the inelastic threshold.
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The macroscopic requirement that effects become manifest after the action of causes not only requires that causes be identifiable without ambiguity but also implies the concept of thermodynamic irreversibility, a phenomenon outside the realm of microscopic physics. Fortunately, the finite velocity of signals allows a weaker formulation which does not imply the choice of a privileged direction for the time arrow. We adhere to the view that space-like separated regions do not influence each other. Stated differently, local observables pertaining to such regions commute. We have seen that this can be further specialized to the commutation (or anticommutation) of the fundamental fields. For simplicity let us only discuss here the case of Bose fields. Mathematically the corresponding properties of field commutators translate into analytic properties of their Fourier transforms. This
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observation is the basis of the original treatment by Kramers and Kronig of the diffractive index of light in a medium relating dispersion and absorption. Hence the name "dispersion relations" for the analytic representations of the scattering amplitudes. An example will illustrate these ideas. Consider the elastic scattering of particle A (mass ma) on a target particle B (mass mb). Both particles are assumed spinless to avoid cumbersome technical details without changing the essential conclusions, but may carry a charge. We therefore distinguish particle A from antiparticle A and use a complex field <p to describe both of them. If ql and q2 denote the initial and final momenta of particle A and Pi and P2 the corresponding ones of particle B, the connected part of the scattering amplitude may be expressed as
d4 x d4 y ei(q, y-qt x)(Oy
+ m~)(Ox + m~)<p2IT<pt(y)<p(x)lpl)
(5-169)
We have absorbed a factor Z-1/2 into the definition of the field. With ql and q2 inside the forward light cone, the time-ordered product in the right-hand side of Eq. (5-169) may be replaced by a retarded commutator
T<p t(y)<p(x) -+ O(yO - XO) [<p t(y), <p(x)]
without affecting the value of Sfi. This can be seen from the original derivation given in Sec. 5-1-3. Define the sourcej(x) of the field <p(x) through
+ m~)<p(x) =
j(x)
(5-170)
and assume for simplicity that <p andj commute at equal times. Taking translation invariance into account we may write
= (21lfc5 4 (P2 + q2 - Pi - qdiY"
(5-171)
Lorentz invariance implies that Y" depends only on the scalar products among the momenta, i.e., on two out of the three Mandelstam variables for on-shell particles. From locality, the retarded commutator <P21 O(ZO) [/(z/2), j( - z/2)] Ipl) vanishes unless z2 > 0, Zo > O. Inspection of Eq. (5-171) reveals that Y" is an analytic function of the four-vector q in the so-called forward tube defined by the condition that 1m q be a positive time-like vector. This follows from the assumption that the matrix elements of the fields are tempered (i.e., polynomially bounded) distributions. Indeed, if q = qR + iq[, the exponential in (5-171) provides a damping factor e - z q] when both z and q[ are positive time-like vectors. This example shows the direct relationship between the local properties of relativistic field theories and the analyticity of Green functions. Before we analyze the mathematical consequences of this result let us recall
ELEMENTARY PROCESSES
the property of cl19lsing ''yJllnll~t'.Y. Instead of the process A(qd + B(pd--+ A(q2) + B(P2) suppose that we were studying the reaction A(l]d + B(pd--+ A(1]2) + B(P2) involving the scattering of antiparticle A on the same target B. The corresponding amplitude ff is given by
I] = -!(1]1
+ 1]2)
(5-172)
Changing the integration variable z into - z, this can also be written as
This amplitude differs from Y given by Eq. (5-171) in two respects. The momentum q = -!(ql + q2) has been replaced by - ii = -!( - ii2 - iii) and the retarded commutator by the advanced one
We may study the quantities Y and ff for arbitrary values of their arguments, instead of dealing with the actual physical processes where q and I] lie in the forward light cone. Let us show that
Y(q) -
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