Crucial to all the preceding chapters is the assumption that we are able to produce beams and targets of polarized particles and that we are able to analyse the state of polarization of these particles.

In the production of targets and beams we are dealing with stable particles (or at least particles stable on the time scale involved) and the physics involved is basically a mixture of classical and quantum mechanics.

There has been extraordinary progress in the design and construction of polarized proton sources at Argonne and Brookhaven and in the development of highly polarized, radiation-resistant targets of various materials by workers at CERN, Fermilab, HERA, Basel, Virginia, SLAC and Ann Arbor.

Great advances have been made in the resolution of problems involved in the acceleration of polarized protons by groups at Bloomington and at Brookhaven. The electron beams at LEP and at HERA have been successfully polarized and a superb polarized electron source is in use at SLAC.

Also quite remarkable has been the building of secondary and tertiary beams of polarized hyperons at Fermilab. Who would have believed it possible that one can measure the magnetic moment of the Ω–?!

Firstly we shall provide a brief discussion of the physical principles of polarized proton sources and targets and of the problems involved in accelerating beams of polarized protons without loss of polarization.

We also discuss a relatively new development, the attempt to polarize protons and antiprotons via the Stern–Gerlach effect.

Elastic scattering is in some sense the most fundamental type of reaction, but it is also the most difficult to understand theoretically. There is a huge amount of spin-dependent data at low to medium energies, but little understanding of the mechanisms at work. In several instances, however, spin-dependent data have played a crucial rôle in nailing the coffin of a current theoretical picture. Somehow, simple-minded ideas, which succeed in explaining gross features of cross-sections, angular distributions etc., run aground when faced with the more probing questions involved in spin-dependent reactions. Because of the lack of clear-cut theoretical ideas and because of the difficulty of the experiments there has generally been a lack of experimental effort in this field since the mid-1980s, but this situation is about to change with the commissioning of the RHIC collider at Brookhaven. There, besides a major programme of heavy-ion physics, it will be possible to study pp collisions, with both beams polarized and up to an energy of 250 GeV per beam. Consequently we shall concentrate in this chapter on nucleon–nucleon scattering.

Broadly speaking there are two kinematic regions of interest, small to medium values of momentum transfer and large momentum transfer. The first is, strictly speaking, in the domain of non-perturbative QCD, so there are no precise theoretical predictions, though there are very interesting suggestive hints. In the second region perturbative QCD ought to be applicable and, indeed, very powerful theoretical results have been derived.

In the previous chapters we have dealt with the production of the polarized states that serve as initial states in reactions. Here we turn to the measurement of the state of polarization of an ensemble of particles, i.e. to polarimetry.

In the analysis of the state of polarization we may be dealing with stable or unstable particles. If the particles are stable it may be possible to rely on well-understood reactions, such as those of QED, to achieve the polarization analysis, via, e.g. Coulomb interference or scattering off a laser beam. Or, if this is impracticable, it is sometimes possible to use a double-scattering technique even if the reaction mechanism is unknown. The only assumption needed for this is time-reversal invariance. If the particles are unstable their decay angular distribution gives information on their state of polarization prior to decay. This is not surprising if the decay is electromagnetic, so that the decay amplitudes are precisely known. What is remarkable, however, is that even when the decay mechanism is not known certain decays are ‘magic’ and still provide information on the polarization state of the decaying particle. Examples are p → ππ, ω → γπ, D* → γD,Ψ → pπ, α2 → pπ etc.

For electron beams, where we can rely on QED, it has been possible to construct very accurate and rapidly acting polarimeters.

One of the most interesting challenges at the moment is to construct efficient high energy proton polarimeters for use at RHIC, UNK and possibly at Fermilab.

This is not a book on field theory, so we do not wish to get involved in a comprehensive discussion of field equations. But the transformation laws for particle states examined in Section 2.4 shed an interesting light upon the problem of constructing fields for arbitrary spin-particles and upon the wave equations they satisfy.

In particular, concerning the Dirac equation, many readers will have followed the beautiful derivation by Dirac of his famous equation for spin-1/2 particles (See Dirac, 1947). Here we shall look at the Dirac equation from a different point of view which provides an alternative insight into the origin and meaning of the equation.

Relativistic quantum fields

The essence of the physical states that were discussed in Chapter 1 is that for a particle at rest they transform irreducibly under rotations. It would be possible to deal with quantum field operators that also had this property (Weinberg, 1964a), i.e. spin-s fields, which have only 2s + 1 components. This, as we shall see, is not very convenient for constructing Lagrangians and building-in symmetry properties so that, for example, we normally use a four-component field for spin-1/2 Dirac particles and a 4-vector Aμ to describe spin-1 mesons or photons etc.