  Quaternionic quantum Hamiltonians describing nonrelativistic spin particles
require the ambient physical space to have five dimensions. The quantum
dynamics of a spin-1/2 particle system characterised by a generic such
Hamiltonian is worked out in detail. It is shown that there exists, within the
structure of quaternionic quantum mechanics, a canonical reduction to three
spatial dimensions upon which standard quantum theory is retrieved. In this
dimensional reduction, three of the five dynamical variables are shown to
oscillate around a cylinder, thus behaving in a quasi one-dimensional manner at
large distances. An analogous mechanism is shown to exist in the case of
octavic Hamiltonians, where the ambient physical space has nine dimensions.
Possible experimental tests in search for the signature of extra dimensions at
low energies are briefly discussed.
