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arXiv:2601.02767

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

Channa Hatharasinghe, Ashleigh Willis, Run Yan Teh, P. D. Drummond, M. D. Reid

quant-phSubmitted Jan 6, 2026• Announced Jan 7, 2026• 0 citations• Revised Sep 7, 2026 (v2)
v1 · Jan 6, 2026v2 · Sep 7, 2026
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Abstract

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 $p$.

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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)x(t)x(t) and p(t)p(t)p(t) that correspond to the phase-space variables of the Q function of the system at a time ttt. We model the system and meter as single-mode fields, and measurement of x^\hat{x}x^ by amplification of the amplitude x(t)x(t)x(t). Our conclusion is that the outcome for the measurement is determined at (or by) the time tmt_{m}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 xxx and ppp postselected on a given outcome of the meter, we show how the QQQ-based model represents a more complete description of quantum mechanics: The variances associated with amplitudes xxx and ppp 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)x(t)x(t) of the meter, associated with distinct eigenvalues. The outcome of measurement is determined by x(t)x(t)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 ppp.
This thesis investigates a novel proposal for quantum reality and measurement, based on phase-space representations and forward-backward stochastic differential equations, that is consistent with macroscopic realism: a Q-based objective-field model for quantum mechanics. We study the model through macroscopic quantum paradoxes, quantum measurement, and continuous-variable Bell tests. We investigate techniques for enhancing the coherence of two-mode cat states and analyze macroscopic versions of Wigner’s friend and the three-box paradox. The measurement problem is examined through the dynamics of the Q function, providing an account of wave-function collapse. This work contributes to understanding quantum reality and measurement and provides a foundation for developing quantum technologies.
The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities (or eigen states).This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation.However, when a measurement is carried out on the system to determine the value of that property (say position), the system instantaneously transforms to one of the eigen states and thus we get only a single value as an outcome of the measurement.Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation.In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real-world and will identify the root cause behind the stochastic nature of events.Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement.In the process of development of model, we will explain Schrodinger cat paradox and will show how Born's rule for probability becomes a natural consequence of the measurement process.
Abstract: The quantum measurement problem has been the deepest conceptual difficulty since the birth of quantum mechanics. Standard quantum mechanics accepts wavefunction collapse as an independent postulate but never provides a physical explanation. The decoherence approach attributes collapse to the interaction between the system and its environment, but faces two fundamental limitations: the “infinite regress” problem and the “insufficiency of decoherence” problem. Within the framework of the Self-Construction Theory (SCT), this paper provides a complete resolution of the measurement problem. The core thesis is that quantum measurement is not a fundamental process independent of the laws of physical evolution, but rather a local manifestation of constructive renormalization—the scale evolution of the spacetime complex field KKK from microscopic degrees of freedom to macroscopic effective descriptions. The origins of wavefunction collapse, decoherence, and the Born rule receive a unified first‑principles interpretation in this framework. SCT’s spontaneous decoherence mechanism is the key to the resolution. Constructive renormalization inherently involves information loss—the integrated‑out microscopic degrees of freedom no longer exist in the macroscopic effective description. During a measurement, the measuring apparatus possesses a huge number (∼1023\sim10^{23}∼1023) of microscopic degrees of freedom. When the apparatus interacts with the measured system, constructive renormalization triggers information compression of these degrees of freedom—the apparatus can no longer distinguish which branch of the superposition it is in, and quantum coherence is thus spontaneously lost. No “external environment” is needed—the constructive renormalization dynamics of spacetime itself is the ultimate source of decoherence, thereby completely solving the “infinite regress” problem. Among the probability distribution given by the Born rule, the weights of non‑maximal probability branches are further suppressed by information compression under the huge number of microscopic degrees of freedom, so that macroscopic observers can perceive only a definite outcome in the effective description. The information loss caused by constructive renormalization is ontological—it does not rely on any approximation or truncation, thus fundamentally solving the “insufficiency of decoherence” problem. The Born rule—that measurement outcomes occur with probabilities given by the squared amplitudes—is directly derived from the constructive integral structure in SCT. In the constructive renormalization limit, interference terms between different configurations are naturally suppressed by off‑diagonal phase randomization, and the system automatically switches to a classical probabilistic description. In the diagonal density matrix representation, the squared amplitude is the only natural weight that is simultaneously real and non‑negative. Hence the Born rule is not an additional postulate of quantum mechanics but a necessary and unique manifestation of the relative frequencies of the constructive integral in the constructive renormalization limit. This derivation is rigorous within the mathematical framework of constructive renormalization—it does not rely on any approximations or truncations. SCT’s quantum measurement theory is deeply unified with the black hole information paradox. The “collapse” in quantum measurement and the “information loss” in black hole evaporation are different manifestations of the same core mechanism—the redistribution of information due to constructive renormalization—in two extreme physical situations. In quantum measurement, information is redistributed into the degrees of freedom of the macroscopic apparatus; in black hole evaporation, information is redistributed into the time‑point configurations on the horizon. This unification is the strongest proof of the universality of constructive renormalization as an information‑processing mechanism. Keywords: quantum measurement; spontaneous decoherence; constructive renormalization; Born rule; wavefunction collapse; information redistribution; Self-Construction Theory
A model of quantum measurement, illustrated using the spin--boson model, is formulated in terms of a cascading pair of quantum phase transitions. The first produces the desired superposition of macroscopic responses to the microscopic state under investigation, while the second provides a mechanism for subsequent "wavefunction collapse," suppressing superpositions of distinct macroscopic states, producing instead a density matrix that implements the expected classical observation outcome via the Born probability rule. Motivated by numerous carefully designed measurements that may occur during the course of a quantum computation, effects of entanglement are investigated when the measurement is performed on only a subset of the microscopic degrees of freedom.
We describe a quantum mechanical measurement as a variational principle including interaction between the system under measurement and the measurement apparatus. Augmenting the action with a nonlocal term (a double integration over the duration of the interaction) results in a theory capable of describing both the measurement process (agreement between system state and pointer state) and the collapse of both systems into a single eigenstate (or superposition of degenerate eigenstates) of the relevant operator. In the absence of the interaction, a superposition of states is stable, and the theory agrees with the predictions of standard quantum theory. Because the theory is nonlocal, the resulting wave equation is an integrodifferential equation (IDE). We demonstrate these ideas using a simple Lagrangian for both systems, as proof of principle. The variational principle is time-symmetric and retrocausal, so the solution for the measurement process is determined by boundary conditions at both initial and final times; the initial condition is determined by the experimental preparation and the final condition is the natural boundary condition of variational calculus. We hypothesize that one or more hidden variables (not ruled out by Bell's Theorem, due both to the retrocausality and the nonlocality of the theory) influence the outcome of the measurement, and that distributions of the hidden variables that arise plausibly in a typical ensemble of experimental realizations give rise to outcome frequencies consistent with Born's rule. We outline steps in a theoretical validation of the hypothesis. We discuss the role of both initial and final conditions to determine a solution at intermediate times, the mechanism by which a system responds to measurement, time symmetry of the new theory, causality concerns, and issues surrounding solution of the IDE.
Quantum mechanics famously posits two distinct evolution rules for the state of a system: a deterministic, unitary time evolution under the Schr¨odinger equation, and a probabilistic “collapse” of the wavefunction upon measurement as dictated by the Born rule. The apparent inconsistency of these two dynamical laws – one continuous and one discontinuous – constitutes the quantum measurement problem. This foundational problem, which underlies the emergence of definite outcomes from quantum superpositions, has persisted since the theory’s inception. In this review, we present a comprehensive and rigorous examination of the measurement problem and survey the leading approaches that seek to resolve or circumvent it. We begin by formulating the measurement problem and its conceptual challenges. We then discuss in detail the major interpretations of quantum mechanics and theoretical frameworks addressing the problem: the Copenhagen interpretation, Everett’s many-worlds interpretation, de Broglie–Bohm pilot-wave theory, objective collapse models, the role of decoherence and environment-induced superselection, and epistemic approaches such as QBism. For each interpretation, we describe the core principles and mathematical formalism, assessing how (and whether) it attempts to solve the measurement problem. We also review modern developments and experiments relevant to quantum measurements, including tests of Bell inequalities, observations of decoherence and macroscopic superpositions, weak measurements and quantum eraser experiments, and thought experiments ‘a la Wigner’s friend. We highlight how these results inform the ongoing debate. Finally, we discuss open problems and challenges that remain in fully resolving the measurement problem.