  The position operator (defined within Schroedinger representation as usual)
becomes meaningless when the usual Born-von Karman periodic boundary conditions
are adopted: this fact is at the root of the polarization problem. I show how
to define the position expectation value by means of rather peculiar many-body
(multiplicative) operator acting on the wavefunction of the extended system.
This definition can be regarded as the generalization of a precursor work,
apparently unrelated to the polarization problem. For uncorrelated electrons,
the present finding coincides with the so-called "single-point Berry phase"
formula, which can hardly be regarded as the approximation of a continuum
integral, and is computationally very useful for disordered systems.
Simulations which are based on this concept are being performed by several
groups.
