The concept of phase space amplitudes for systems with continuous degrees of
freedom is generalized to finite-dimensional spin systems. Complex amplitudes
are obtained on both a sphere and a finite lattice, in each case enabling a
more fundamental description of pure spin states than that previously given by
Wigner functions. In each case the Wigner function can be expressed as the star
product of the amplitude and its conjugate, so providing a generalized Born
interpretation of amplitudes that emphasizes their more fundamental status. The
ordinary product of the amplitude and its conjugate produces a (generalized)
spin Husimi function. The case of spin-\half is treated in detail, and it is
shown that phase space amplitudes on the sphere transform correctly as spinors
under under rotations, despite their expression in terms of spherical
harmonics. Spin amplitudes on a lattice are also found to transform as spinors.
Applications are given to the phase space description of state superposition,
and to the evolution in phase space of the state of a spin-\half magnetic
dipole in a time-dependent magnetic field.Comment: 19 pages, added new results, fixed typo
