Infinite matrices may violate the associative law

Abstract

The momentum operator for a particle in a box is represented by an infinite order Hermitian matrix $P$. Its square $P^2$ is well defined (and diagonal), but its cube $P^3$ is ill defined, because $P{P}^2\ne P^{2}P$. Truncating these matrices to a finite order restores the associative law, but leads to other curious results.

final version in J. Phys. A28 (1995) 1765-1770