  We construct the algebra of operators acting on the Hilbert spaces of Quantum
Mechanics for systems of $N$ identical particles from the field operators
acting in the Fock space of Quantum Field Theory by providing the explicit
relation between the position and momentum operators acting in the former
spaces and the field operators acting on the latter. This is done in the
context of the non-interacting Klein-Gordon field. It may not be possible to
extend the procedure to interacting field theories since it relies crucially on
particle number conservation. We find it nevertheless important that such an
explicit relation can be found at least for free fields. It also comes out that
whatever statistics the field operators obey (either commuting or
anticommuting), the position and momentum operators obey commutation relations.
The construction of position operators raises the issue of localizability of
particles in Relativistic Quantum Mechanics, as the position operator for a
single particle turns out to be the Newton-Wigner position operator. We make
some clarifications on the interpretation of Newton-Wigner localized states and
we consider the transformation properties of position operators under Lorentz
transformations, showing that they do not transform as tensors, rather in a
manner that preserves the canonical commutation relations.
