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Earlier derivations of simple Wick's theorems for operators of spin J and 1 (in units of fi) using the drone-fermion representation are applied to the Heisenberg model. The resulting diagrammatic-perturbational approach (Green's functions) is carried out in both the high-and low-temperature domains, where the expansion criteria of Stinchcombe et al. are closely followed. The present work effectively reexpresses the semi-invariant analysis of these authors in a much simpler manner, and many of their results are straightforwardly reproduced. The use of standard quantum-field-theory techniques enables renormalization to be undertaken in a simple, systematic manner. At low temperatures the present fermion analysis gives Dyson's T 4 contribution to the free energy from the first Born approximation to spin-wave scattering. Higher-order spin-wave contributions give a damping term, which, upon evaluation in the lowest approximation, is identical to that found by ter Haar and Tahir-Kheli. 9

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The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case,~$d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~$d=6$ by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~$q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

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We have extended our previous microscopic treatment of the dynamic transverse susceptibility for a random array of localized spins in a metal to include terms of second order in the exchange coupling constant /. Lattice relaxation of the localized and conduction electrons is included, as before, in such a way as to ensure relaxation to the instantaneous local field. The results, in the limit of equal conduction-electron and localizedspin g values and no lattice damping, reduce to the correct ("bottlenecked") limit. The linewidth for frequencies close to the localized-spin resonance frequency agrees with previous calculations. Bottlenecking of both the longitudinal (frequency-modulation) (T 2 ') and transverse (spin-flip) (7Y) parts of the localizedspin resonance linewidth is demonstrated for equal g values and no lattice relaxation. Similarly, the linewidth for frequencies close to the conduction-electron resonance frequency exhibits both 7Y-and 7Y-type terms, and again bottleneck effects are present. The results are compared with previous macroscopic treatments. It is demonstrated that it is unnecessary to introduce detailed balance conditions per se in the microscopic theory. The relation between the conduction-electron-hole relaxation width and the one-electron width calculated by Overhauser is examined in an appendix.

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