  We perform exact calculations of collisional frequency shifts for several
fermions or bosons using a singlet and triplet basis for pairs of particles.
The "factor of 2 controversy" for bosons becomes clear - the factor is always
2. Decoherence is described by singlet states and they are unaffected by
spatially uniform clock fields. Spatial variations are critical, especially for
fermions which were previously thought to be immune to collision shifts. The
spatial variations lead to decoherence and a novel frequency shift that is not
proportional to the partial density of internal states.
