  The familiar concepts of state vectors and operators in quantum mechanics
rely on associative products of observables. However, these notions do not
apply to some exotic systems such as magnetic monopoles, which have long been
known to lead to non-associative algebras. Their quantum physics has remained
obscure. This letter presents the first derivation of potentially testable
physical results in non-associative quantum mechanics, based on effective
potentials. They imply new effects which cannot be mimicked in usual quantum
mechanics with standard magnetic fields.
