In an old paper of our group in Milano a formalism was introduced for the continuous monitoring of a system during a certain interval of time in the framework of a somewhat generalized approach to quantum mechanics. The outcome was a distribution of probability on the space of all the possible continuous histories of a set of quantities to be considered as a kind of coarse grained approximation to some ordinary quantum observables commuting or not. In fact the main aim (...) was the introduction of a classical level in the context of QM, treating formally a set of basic quantities, to be considered as beables in the sense of Bell, as continuously taken under observation. However the effect of such assumption was a permanent modification of the Liouville-von Neumann equation for the statistical operator by the introduction of a dissipative term which is in conflict with basic conservation rules in all reasonable models we had considered. Difficulties were even encountered for a relativistic extension of the formalism. In this paper I propose a modified version of the original formalism which seems to overcome both difficulties. First I study the simple models of an harmonic oscillator and a free scalar field in which a coarse grain position and a coarse grained field respectively are treated as beables. Then I consider the more realistic case of spinor electrodynamics in which only certain coarse grained electric and magnetic fields are introduced as classical variables and no matter related quantities. (shrink)

A formalism developed in previous papers for the description of continual observations of some quantities in the framework of quantum mechanics is reobtained and generalized, starting from a more axiomatic point of view. The statistics of the observations of continuous state trajectories is treated from the beginning as a generalized stochastic process in the sense of Gel'fand. An effect-valued measure and an operation-valued measure on the σ-algebra generated by the cylinder sets in the space of trajectories are introduced. The properties (...) of the characteristic functional for the “operation-valued stochastic process” are discussed and, through a suitableansatz, a significant class of such processes is explicitly constructed, which contains the examples of the preceding papers as particular cases. (shrink)