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.
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.
Abstract We propose that the ontic understanding of quantum mechanics can be extended to a fully realistic theory that describes the evolution of the wavefunction at all times, including during a measurement. In such an approach the wave equation should reduce to the standard wave equation when there is no measurement, and describe state reduction when the system is measured. The general wave equation must be nonlinear and nonlocal, and we require it to be time-symmetric; consequently, this approach is not a new interpretation but a new theory. The wave equation is an integrodifferential equation (IDE). The time symmetry requirement leads to a retrocausal approach, in which the wave equation is solved subject to initial and final conditions to determine history at intermediate times. We propose that different outcomes from (apparently) identically prepared experiments may result from uncontrolled parameters; both the nonlocality and the retrocausality of the theory imply that Bell’s Theorem cannot rule out such “hidden variables.” Beginning with Hamilton’s principle, we demonstrate the construction of such a theory by replacing the action with a functional designed to give rise to a nonlinear, nonlocal IDE as the wave equation. This IDE reduces to the standard wave equation (a differential equation) in the absence of a measurement, but exhibits state reduction to a single eigenvalue when the system interacts with another system with the properties of a measurement apparatus. We demonstrate several desirable features of this theory; for other properties we indicate their plausibility and possible avenues to a proof.
In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component ŝz by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S + A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetization. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called collapse of the wavefunction or the reduction of the state appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.
We analyze in mathematical detail, within the framework of the QMUPL model of spontaneous wave function collapse, the von Neumann measurement scheme for the measurement of a 1/2 spin particle. We prove that, according to the equation of the model: i) throughout the whole measurement process, the pointer of the measuring device is always perfectly well localized in space; ii) the probabilities for the possible outcomes are distributed in agreement with the Born probability rule; iii) at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding to the measured eigenvalue. This analysis shows rigorously how dynamical reduction models provide a consistent solution to the measurement problem of quantum mechanics.
While Bell nonlocality can be explained through mechanisms such as retrocausality and superluminal influences, it is not clear why such mechanisms do not manifest at an everyday macroscopic level. In this paper, we show how measurement and nonlocality can be explained consistently with macroscopic realism and no-signaling, and causal relations for macroscopic quantities. Considering the measurement of a field amplitude x ̂ , we derive theorems that lead to an equivalence between a quantum phase-space probability distribution Q ( x , p , t ) and stochastic trajectories for real amplitudes x ( t ) and p ( t ) propagating backward and forward in time, respectively. We present forward-backward stochastic simulations that motivate a Q -based objective-field model of reality. Amplification plays a key role in the measurement. With amplification, similar to decoherence, contributions due to interference become unobservable, leading to “branches” in x that correspond to distinct eigenvalues λ . This elucidates how the system evolves from a superposition to an eigenstate, from which Born's rule follows. For superposition states, we find that the backward and forward trajectories are connected, leading to “hidden loops.” We deduce a hybrid causal structure involving causal deterministic relations for amplified variables, along with microscopic noise inputs and hidden loops for unobservable quantities. Causal consistency is confirmed, removing grandfather-type paradoxes. The simulations allow evaluation of a state inferred for the system, conditioned on a particular branch λ , from which we deduce a model for projection and the collapse of the wave function. The theory is extended to Einstein-Podolsky-Rosen correlations and Bell nonlocality. We demonstrate consistency with three weak local realistic premises: (1) the existence of real properties (defined for the system after operations that fix measurement settings); (2) a partial locality implying no-signaling; (3) “elements of reality” that apply to the predictions of a system by a meter, once meter settings are fixed. A Q -based hidden variable model and mechanism for Bell nonlocality is identified. Our work shows how forward-backward stochastic simulations lead to a hybrid causal structure, involving both deterministic causal relations and “hidden” stochastic loops, explaining measurement and entanglement, with paradoxes typically associated with retrocausality avoided.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.
Abstract We introduce the Quantum Consensus Principle (QCP), a first principles framework that provides a purely dynamical derivation of the quantum measurement process without modifying the Schrödinger equation. By treating the apparatus environment complex as an open quantum system, we show that measurement outcomes emerge from a thermodynamic selection process governed by large-deviation dynamics. Central to the theory is the derivation of a universal selection potential, Ф, which is uniquely determined by the BKM information geometry of the thermal environment and linked to microscopic Hamiltonian parameters via Green-Kubo relations. We rigorously demonstrate that the conditioned system state follows a Hellinger-contractive supermartingale, converging almost surely to unique pointer states. Unlike standard postulates, QCP derives the Born rule as a specific neutrality limit and predicts measurable, apparatus-dependent deviations for non-ideal observers. Furthermore, the theory establishes a characteristic non-monotonic scaling of collapse timescales, providing a clear path for experimental falsification. This work bridges the gap between quantum information theory and non-equilibrium thermodynamics, identifying measurement as a fundamental consensus phenomenon in macroscopi systems.