V. the Semiclassical Foldy–wouthuysen Transformation and the Derivation of the Bloch Equation for Spin–1/2 Polarised Beams Using Wigner Functions

A complete picture of spin polarisation in accelerators and storage rings, either with or without synchrotron radiation, can only be obtained on the basis of evolution equations for combined spin–orbit distributions. See Article I. Moreover, if we are concerned with the effects of radiation, its simulation by classical white noise and damping does not suffice for all situations. For example we cannot obtain the Sokolov– Ternov effect by that means. In fact to include all the subtleties of radiation, a quantum mechanical approach is needed and then obtaining the ‘complete picture’ implies that we must begin by finding the equation of motion for the spin–orbit density operator in the presence of radiation. To ensure some level of transparency and trackability one begins by ignoring direct and indirect inter–particle effects so that at the classical level the beam would be described by a single-particle density depending on the six orbital phase space variables, the spin variables and on time as in statistical mechanics in ‘μ-space’. In the single-particle approximation, only positive energy two-component spinorbit wave functions are needed. The appropriate quantum Hamiltonian is provided by a Foldy − Wouthuysen (FW) transform [1] of the Dirac Hamiltonian and in order to get explicit results for time-dependent electromagnetic fields one has to use perturbation theory. Since we are interested in high energy behaviour in storage rings we do semiclassical perturbation theory, where the expansion parameter is Planck’s constant, not 1/m etc. Before launching into the full blown calculation of the effects of radiation one should first obtain the transformed Hamiltonian for motion due to the Lorentz forces in the fields of the storage ring and then the corresponding equations of motion for the spin and orbital parts of the density operator. The evolution equations for the resulting classical distributions should then be derived. These are the tasks of this paper. Radiation will be considered elsewhere.