ABSTRACT
This chapter analyses the implications of the unitarity of the https://www.w3.org/1998/Math/MathML" display="inline"> S -matrix, discussed in Chapter 7. From the requirement https://www.w3.org/1998/Math/MathML" display="inline"> S S † = 1 , with https://www.w3.org/1998/Math/MathML" display="inline"> S = 1 + i T , it follows that the operator T satisfies the equation https://www.w3.org/1998/Math/MathML" display="inline"> - i ( T - T † ) = T T † , which allows to establish a simple relation, known as optical theorem, between the total cross section of a scattering process and the imaginary part of the forward scattering amplitude. Using the representation of this result in terms of Feynman diagrams, the calculation of the discontinuity of the scattering amplitude can be obtained from the Cutkosky rule, which amounts to: (i) cutting the corresponding Feynman diagram, (ii) replacing the internal lines with https://www.w3.org/1998/Math/MathML" display="inline"> δ -functions representing on-shell particles with positive energy, and (iii) summing the contributions off all cuts. As an illustrative example, this method is first applied to the fermion-antifermion annihilation https://www.w3.org/1998/Math/MathML" display="inline"> u + u ¯ → d + d ¯ . The case of gauge theories, in which the Cutkosky rule does not guarantee unitarity, is also discussed in detail. For QED, it is shown that this problem, arising from the appearance of gauge-dependent terms in the propagator of the gauge fields, can be circumvented exploiting current conservation and the abelian—that is, commutative—nature of the theory. For non abelian theories, such as QCD, in which the charges do not commute with one another, Feynman demonstrated that unitarity can be satisfied by introducing, for each gluon loop, a loop of anticommuting scalar and massless fields, needed to cancel the contributions of the spurious polarisations introduced by the gauge. This argument provides a physical basis for the use of the gauge-fixing procedure of Faddeev-Popov, discussed in Chapter 14.
