In this paper, we examine measurement using the Q-based, objective-field model for quantum mechanics. Schrodinger considered a microscopic system prepared in a superposition of states which is then coupled to a macroscopic meter. We analyze the entangled meter and system, and measurements on it, by solving forward-backward stochastic differential equations for real amplitudes
x(t) and
p(t) that correspond to the phase-space variables of the Q function of the system at a time
t. We model the system and meter as single-mode fields, and measurement of
x^ by amplification of the amplitude
x(t). Our conclusion is that the outcome for the measurement is determined at (or by) the time
tm, when the coupling to the meter is complete, the meter states being macroscopically distinguishable. There is consistency with macroscopic realism. By evaluating the distribution of the amplitudes
x and
p postselected on a given outcome of the meter, we show how the
Q-based model represents a more complete description of quantum mechanics: The variances associated with amplitudes
x and
p are too narrow to comply with the uncertainty principle, ruling out that the distribution represents a quantum state. We conclude that the collapse of the wavefunction occurs as a two-stage process: First there is an amplification that creates branches of amplitudes
x(t) of the meter, associated with distinct eigenvalues. The outcome of measurement is determined by
x(t) once amplified, explaining Born's rule. Second, the distribution that determines the final collapse is the state inferred for the system conditioned on the outcome of the meter: information is lost about the meter, in particular, about the complementary variable
p.
We present and discuss some models for the quantum measuring process. Our models indicate that the coherent state approach to classical limits in quantum theories becomes a powerful method for a unified description of various measuring processes. Each detector in the models has a parameter N , the number of its dynamical variables; the whole process of an interaction between an object system and the detector is calculable for any finite number N . It is shown that the detector states after the interaction are characterized by a macroscopic parameter, which is closely connected with the “reliability” of the detector. The reliability plays an essential role in eliminating the quantum interference between two states corresponding to two different positions of the pointer. Finally, the quantum mechanics of the detector reduces to classical mechanics on suitable “phase space” as N becomes large, as is required.
An explicit model allowing a unified description of microscopic and macroscopic systems is exhibited. First, a modified quantum dynamics for the description of macroscopic objects is constructed and it is shown that it forbids the occurrence of linear superpositions of states localized in far-away spatial regions and induces an evolution agreeing with classical mechanics. This dynamics also allows a description of the evolution in terms of trajectories. To set up a unified description of all physical phenomena, a modification of the dynamics, with respect to the standard Hamiltonian one, is then postulated also for microscopic systems. It is shown that one can consistently deduce from it the previously considered dynamics for the center of mass of macroscopic systems. Choosing in an appropriate way the parameters of the so-obtained model one can show that both the standard quantum theory for microscopic objects and the classical behavior for macroscopic objects can all be derived in a consistent way. In the case of a macroscopic system one can obtain, by means of appropriate approximations, a description of the evolution in terms of a phase-space density distribution obeying a Fokker-Planck diffusion equation. The model also provides the basis for a conceptually appealing description of quantum measurement.