Experiments involving single or few elementary particles are completely described by Quantum Mechanics. Notwithstanding the success of that quantitative description, various aspects of observations, as nonlocality and the statistical randomness of results, remain as mysterious properties apart from the quantum theory, and they are attributed to the strangeness of the microscopic world. Here we restart from the fundamental relations of uncertainty to reformulate the probability law of Born including the temporal variable. Considering that both the spatial and the temporal variables play a symmetric role in the wave-function Ψ(x,t) , a temporal wavepacket is built and analysed. The probability density is written as p(x,t) = | Ψ(x,t) |^2, where the probabilistic interpretation for the temporal wavepacket is equivalent to Born's law for the spatial variable, x. For the convenience of the discussion of the role of the temporal variable, we write p(x_0,t) = | Ψ(x_0,t) |^2 for a free particle, expressing only the temporal wavepacket, then we discuss its spread. In the light of the evolution of this temporal wavepacket we analyse basic processes of matter-wave interaction, involving single and entangled entities. Nonlocality appears then as a consequence of the spread of the temporal wavepacket; and the position of each detected event in two-slits interferometry as due to the independent phases of the spatial and temporal wavepackets.
Experiments involving single or few elementary particles are completely described by Quantum Mechanics. Notwithstanding the success of that quantitative description, various aspects of observations, as nonlocality and the statistical randomness of results, remain as mysterious properties apart from the quantum theory, and they are attributed to the strangeness of the microscopic world. Here we restart from the fundamental relations of uncertainty to reformulate the probability law of Born including the temporal variable. Considering that both the spatial and the temporal variables play a symmetric role in the wave-function Ψ(x,t) , a temporal wavepacket is built and analysed. The probability density is written as p(x,t) = | Ψ(x,t) |^2, where the probabilistic interpretation for the temporal wavepacket is equivalent to Born's law for the spatial variable, x. For the convenience of the discussion of the role of the temporal variable, we write p(x_0,t) = | Ψ(x_0,t) |^2 for a free particle, expressing only the temporal wavepacket, then we discuss its spread. In the light of the evolution of this temporal wavepacket we analyse basic processes of matter-wave interaction, involving single and entangled entities. Nonlocality appears then as a consequence of the spread of the temporal wavepacket; and the position of each detected event in two-slits interferometry as due to the independent phases of the spatial and temporal wavepackets.
Abstract The so-called quantum measurement problems are solved from a new perspective. One of the main observations is that the basic entities of our world are particles, elementary or composite. It follows that each elementary process, hence each measurement process at its core, is a spacetime, pointlike, event. Another key idea is that, when a microsystem ψ gets into contact with the experimental device, factorization of ψ rapidly fails and entangled mixed states appear. The wave functions for the microsystem-apparatus coupled system for different measurement outcomes then lack overlapping spacetime support. It means that the aftermath of each measurement is a single term in the sum: a “wave-function collapse”. Our discussion leading to a diagonal density matrix, ρ = diag(|c 1|2,…,|cn |2,…) shows how the information encoded in the wave function <?CDATA $|\psi \rangle =\displaystyle {\sum }_{n}{c}_{n}|n\rangle $?> | ψ 〉 = ∑ n c n | n 〉 gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies Pn = |cn |2 for various experimental outcomes F = fn . Our discussion represents the first, significant steps towards filling in the logical gaps in the conventional interpretation based on Born’s rule, replacing it with a clearer understanding of quantum mechanics. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable.
The quantum measurement problems are discussed from a new perspective.
One of the main ideas of this work is that the basic entities of our world are various types of particles, elementary or composite. It follows that each elementary process, hence each measurement process at its core, is a spacetime, pointlike, event. Another key idea is that, when a microsystem ψ gets into contact with the experimental device, factorization of ψ rapidly fails and entangled mixed states appear. The wave functions for the microsystem-apparatus coupled systems for different measurement outcomes then lack overlapping spacetime support. It means that the aftermath of each measurement is a single term in the sum: a ``wave-function collapse.
Our discussion leading to a diagonal density matrix, ρ=diag(∣c1∣2,…,∣cn∣2,…) shows how the information encoded in the wave function ∣ψ⟩=∑ncn∣n⟩ gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies Pn=∣cn∣2 for various experimental results F=fn.
These results represent the first, significant steps towards filling in the logical gaps in the standard interpretation based on Born's rule, and replacing it with a more natural one. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable.
A few well-known puzzles such as the Schr\odinger cat conundrum and the EPR paradox are briefly reviewed: they can all be naturally explained away.
The quantum measurement problems are revisited from a new perspective. One of the main ideas of this work is that the basic entities of our world are various types of particles, elementary or composite. It follows that each elementary process, hence each measurement process at its core, is a space–time, pointlike, event. Another key idea is that, when a microsystem [Formula: see text] gets into contact with the experimental device, factorization of [Formula: see text] rapidly fails and entangled mixed states appear. The wave functions for the microsystem-apparatus coupled systems for different measurement outcomes then lack overlapping space–time support. It means that the aftermath of each measurement is a single term in the sum: a “wave function collapse.” Our discussion leading to a diagonal density matrix, [Formula: see text] shows how the information encoded in the wave function [Formula: see text] gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies [Formula: see text] for various experimental results [Formula: see text]. These results represent the first, significant steps towards filling in the logical gaps in the standard interpretation based on Born’s rule, and replacing it with a more natural one. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable. A few well-known puzzles such as the Schrödinger cat conundrum and the Einstein–Podolsky–Rosen (EPR) paradox are briefly reviewed: they can all be naturally explained away.
Born's statistical interpretation of quantum mechanics transformed the wave function from a dynamical object into a quantity whose squared modulus determines the probability distribution of measurement outcomes. Since the formulation of the Born rule in 1926, the relation between a continuously evolving quantum state and discrete detection events has remained one of the central conceptual questions of quantum theory. This paper proposes a physical reconsideration of that relation. Rather than treating the appearance of a localized detection event as an immediate consequence of wave-function collapse, we distinguish two physically different stages: continuous wave-packet propagation and discrete detector readout. In the first stage, a localized wave packet evolves through an interaction potential and undergoes phase reconstruction, deflection, focusing, or redistribution. In the second stage, the continuously distributed wave amplitude interacts locally with microscopic detector degrees of freedom, producing discrete detection events. The central hypothesis explored here is that the experimentally observed distribution of individual events may arise from a local detector-response functional acting on the incident wave amplitude. In an appropriate regime, a detector response of the form R(x)∝∣ψ(x)∣2could emerge from the microscopic interaction between the wave packet and the detector, rather than being imposed as a primitive probabilistic postulate. The paper does not claim a completed derivation of Born's rule. Instead, it identifies the physical assumptions that such a derivation would require, relates them to historical electron-scattering experiments, and formulates a concrete research program for deriving the observed quadratic detection law from wave amplitude, energy transfer, phase structure, and microscopic detector dynamics.
Waves and particles are alternative modes of being for the elements of atomic dimensions. This feature appears to be evident at the experimental level but remains a conundrum at the theoretical level. The squared wavefunction |Ψ|2 yields the probability of finding a particle at a particular position and time, namely, the wave is conceived as a derivation or a physical condition of the particle and not a condition of the quantum. What the wavefunction represents, whether it really exists, whether it is a purely formal representation and if the observer influences the wavefunction are the major arguments of quantum mechanics (QM). The double nature of electrons, photons, and so on has been clearly recognized, and the two natures should be established by precise formulations. This work adheres totally to this methodological criterion and puts forward two formal definitions. The measurement process and wave collapse problems are also discussed. Finally, we verify the theoretical conclusions using the double slit experiment and the photoelectric effect. Probability plays a crucial role in the quantum literature, and the definitions of the wave and the particle should be illustrated by exploiting the probability theory; however, the various probability theories are not univocal and propose irreconcilable views. Therefore, in a preliminary stage, we unify the ontological and epistemic interpretations of probability using four theorems that have been already published. In summary, this work is developed on the basis of two methodological criteria, which seek to improve the probability theory and fix the definitional formulas of QM.
The Schrödinger equation and the Born rule constitute two of the most successful structures in quantum mechanics. The former governs the continuous evolution of the wave function, while the latter connects its squared amplitude to the statistical distribution of experimental outcomes. Yet these two structures operate at different descriptive levels: the Schrödinger state is continuous, spatially extended, and phase-sensitive, whereas individual experiments terminate in localized physical events. This paper develops a theoretical framework for reconstructing the physical relation between these two levels. Rather than proceeding only in the conventional direction from Schrödinger dynamics to Born probabilities, we introduce an inverse explanatory strategy: from Born back to Schrödinger. Born statistics are treated as an empirical constraint from which the physical role of the wave function is reconsidered. We propose that, for the single-particle regime examined here, the wave function may be interpreted functionally as a dynamical propagation structure rather than as the localized particle itself. Its amplitude, phase, conserved current, gauge-covariant propagation momentum, and associated velocity field jointly define a structured family of possible motions. This interpretation leads to three separations: propagation from localization, propagation from physical population, and arrival from detector registration. A physical preparation generates a source measure (\mu_0^{\rm phys}); Schrödinger dynamics generates a propagation map (\Phi_\Psi); and microscopic detector dynamics is represented by a response map (\mathcal R_D). The Born problem can then be formulated as the closure condition RD[(ΦΨ)∗μ0phys]=?μBorn,D. This relation is not claimed here as a derived law, but as the central dynamical problem that a physical reconstruction of quantum probability must solve without inserting the Born measure at either boundary. The paper synthesizes six preceding studies on wave-packet localization, Born's scattering interpretation, the emergence of probability, the physical meaning of the wave function, source-to-detector transport, and unified propagation dynamics. It places the resulting framework in dialogue with Madelung and Hamilton–Jacobi formulations, Bohmian mechanics, decoherence, Quantum Darwinism, structural and symmetry-based derivations of the Born rule, Everettian approaches, objective-collapse theories, and detector models. The resulting proposal is best understood as a propagation-first theoretical reconstruction program: Schrödinger dynamics organizes possible propagation, while Born statistics constrain how that propagation becomes physically realized. Keywords: Schrödinger equation; Born rule; wave function; quantum foundations; propagation structure; quantum probability; quantum trajectories; source-to-detector transport; detector response; wave–particle duality; Bohmian mechanics; quantum measurement