Q-based, objective-field model for wave-function collapse: Analyzing measurement on a macroscopic superposition state

The measurement problem remains unaddressed in modern physics, with an array of proposed solutions but as of yet no agreed resolution. 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 $\hat{x}$ by amplification of the amplitude $x(t)$. Our conclusion is that the outcome for the measurement is determined at (or by) the time $t_{m}$, 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 $p$.