Operations involving momentum variables in non-Hamiltonian evolution equation

Non-Hamiltonian evolution equations have been recently considered for the description of various physical processes. Among these types of equations the class which has been more extensively studied is the one usually referred to as quantum-dynamical semi-group equations (QDS). In particular an equation of the QDS type has been considered as the basis for a model, called quantum mechanics with spontaneous localization (QMSL), which has been shown to exhibit some very interesting features allowing us to overcome most of the conceptual difficulties of standard quantum theory. QMSL assumes a modification of the pure Schroedinger evolution by assuming the occurrence, at random times, of stochastic processes for the wave function corresponding formally to approximate position measurements. In this paper the consequences of modifying and/or enlarging the class of the considered stochastic processes, by considering the spontaneous occurrence of approximate momentum and of simultaneous position and momentum measurements, are investigated. It is shown that the considered changes in the elementary processes have unacceptable consequences. In particular they either lead to drastic modification in the dynamics of microsystems or are completely useless from the point of view of the conceptual advantages that one was trying to get from QMSL. The present work supports therefore the idea that QMSL, as originally formulated, can be taken as the basic scheme for the generalizations which are still necessary in order to make it appropriate for the description of systems of identical particles and to meet relativistic requirements