We develop possible versions of supersymmetric single particle quantum
mechanics, with application to superstring-bit models in view. We focus
principally on space dimensions d=1,2,4,8, the transverse dimensionalities of
superstring in 3,4,6,10 space-time dimensions. These are the cases for which
``classical'' superstring makes sense, and also the values of d for which
Hooke's force law is compatible with the simplest superparticle dynamics. The
basic question we address is: When is it possible to replace such harmonic
force laws with more general ones, including forces which vanish at large
distances? This is an important question because forces between string-bits
that do not fall off with distance will almost certainly destroy cluster
decomposition. We show that the answer is affirmative for d=1,2, negative for
d=8, and so far inconclusive for d=4.Comment: 17 pages, Late
