# Mass operator and dynamical implementation of mass superselection rule

May 07, 2012

We start reviewing Giulini's dynamical approach to Bargmann superselection
rule proposing some improvements. We discuss some general features of the
central extensions of the Galileian group used in Giulini's programme,
focussing on the interplay of classical and quantum picture, without making any
particular choice for the multipliers. Preserving other features of Giulini's
approach, we modify the mass operator of a Galilei invariant quantum system to
obtain a mass spectrum that is positive and discrete, giving rise to a standard
(non-continuous) superselection rule. The model is invariant under time
reversal but a further degree of freedom appears, interpreted as an internal
conserved charge. (However, adopting a POVM approach a positive mass operator
arises without assuming the existence of such a charge.) The effectiveness of
Bargmann rule is shown to be equivalent to an averaging procedure over the
unobservable degrees of freedom of the central extension of Galileo group.
Moreover, viewing the Galileian invariant quantum mechanics as a
non-relativistic limit, we prove that the above-mentioned averaging procedure
giving rise to Bargmann superselection rule is nothing but an effective
de-coherence phenomenon due to time evolution if assuming that real
measurements includes a temporal averaging procedure. It happens when the added
term $Mc^2$ is taken in the due account in the Hamiltonian operator since, in
the dynamical approach, the mass $M$ is an operator and cannot be trivially
neglected as in classical mechanics. The presented results are quite general
and rely upon the only hypothesis that the mass operator has point-wise
spectrum. These results explicitly show the interplay of the period of time of
the averaging procedure, the energy content of the considered states, and the
minimal difference of the mass operator eigenvalues.

Keywords:

Bargamann superselection rule, mass operator