Continuous monitoring and the introduction of a classical level in Quantum Theory

TL;DR

This paper proposes a modified formalism for continuous monitoring in quantum mechanics that introduces a classical level of observables, overcoming previous issues with dissipation and relativistic extension.

Contribution

It introduces a revised formalism for continuous quantum monitoring that maintains conservation laws and extends to relativistic models.

Findings

Successful modeling of harmonic oscillator and scalar field with coarse-grained observables.

Application to Spinor Electrodynamics with classical electric and magnetic fields.

Overcomes dissipation and relativistic extension issues of previous formalism.

Abstract

In ordinary Quantum Mechanics only ideally instantaneous observations of a quantity or a set of compatible quantities are usually considered. 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 Q. M. 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. The main aim was the introduction of a classical level in the context of Quantum Mechanics, treating formally a set of basic quantities {\it 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 and no matter related quantities are introduced as classical variables.