A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schrödinger equation and the measurement by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit necessary since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is necessary in order to eliminate the unknown initial and end time coordinate of the experiment in reference to the beginning of the universe.
The states (Schrödinger picture) and observables (Heisenberg picture) in the standard quantum theory evolve symmetrically in time, given by the unitary group with time extending over − ∞ < t < + ∞ . This time evolution is a mathematical consequence of the Hilbert space boundary condition for the dynamical differential equations. However, this unitary group evolution violates causality. Moreover, it does not solve an old puzzle of Wigner: How does one describe excited states of atoms which decay exponentially, and how is their lifetime τ related to the Lorentzian width Γ? These question can be answered if one replaces the Hilbert space boundary condition by new, Hardy space boundary conditions. These Hardy space boundary conditions allow for a distinction between states (prepared by a preparation apparatus) and observables (detected by a registration apparatus). The new Hardy space quantum theory is time asymmetric, i.e, the time evolution is given by the semigroup with t 0 ≤ t < + ∞ , which predicts a finite "beginning of time" t 0 , where t 0 is the ensemble of time at which each individual system has been prepared. The Hardy space axiom also leads to the new prediction: the width Γ and the lifetime τ are exactly related by τ = ℏ/Γ .
The Born rule provides the empirically successful statistical connection between the Schrödinger wave function and the outcomes of quantum experiments. Yet two logically distinct questions are often compressed into this rule: what the wave function describes during propagation, and why repeated localized particle events exhibit the statistics associated with its squared amplitude. In an earlier analysis, we proposed a deliberately restricted interpretation of the single-particle Schrödinger wave function as a complex-amplitude representation of the spatial--directional structure of quantum propagation. The present paper advances that proposal by one dynamical step. We begin not from an abstract probability distribution, but from the operational structure of electron-scattering experiments. Irrespective of the detailed electron-generation mechanism, a completed preparation procedure produces an actual electron flux crossing a source-exit surface. We therefore introduce a \textbf{preparation boundary}
Σ0, and define
N0 as the actual number of electrons crossing
Σ0 during a specified time interval. A second surface,
ΣD, represents an idealized detection boundary. For a Schrödinger state
Ψ(r,t), the exact continuity structure defines
ρΨ=∣Ψ∣2,jΨ=mℏIm(Ψ∗∇Ψ),and, wherever
ρΨ=0,
vΨ=ρΨjΨ. We investigate the trajectory hypothesis that a localized electron configuration crossing
Σ0 at
(X0,t0) evolves according to
dtdX=vΨ(X,t).This generates a boundary-to-boundary flow map
ΦΨ:(X0,t0)⟶(XD,tD). An ensemble of actual source crossings therefore defines a physical source measure
μ0, while the corresponding distribution of detector crossings is the push-forward
μD=(ΦΨ)∗μ0. The Born problem can then be reformulated sharply: under what physical conditions does the source-generated measure
μ0, transported by the Schrödinger flow, produce the standard Born distribution at
ΣD? The trajectory law and the equivariance of
∣Ψ∣2 are closely related to established de Broglie--Bohm theory, and the connection between quantum flux and asymptotic surface crossings is rigorously developed in the flux-across-surfaces literature. We do not claim these mathematical structures as new. The contribution of the present work is instead to reconstruct the quantum-statistical problem from the complete experimental chain
preparation→source crossing→Schro¨dinger flow→trajectory→detection crossing→statistics,and thereby isolate a distinct \textbf{source-measure problem}. If Born statistics are to be dynamically accounted for rather than imposed at the outset, the remaining question is not only how
∣Ψ∣2 is transported, but why the microscopic preparation process produces the particular source ensemble whose boundary transport yields Born statistics. The present paper does not claim to solve that final problem. It formulates it as a concrete boundary-transport problem that can be addressed analytically, numerically, and ultimately through explicit models of electron preparation.
On the level of quantum phenomena, there is no precise possibility of separating an examined physical system from a measuring instrument. This fact lends support to the assertion that the evolution of a quantum system ceases to be rigorously causal from the moment when the system is subjected to investigation. A quantum measurement is special in that it always causes a jump in the system under study to an eigenstate of the considered variable. Possible interactions between the measuring and measured objects will be neglected, so that a given quantum system, which is free from any disturbances, e.g. measurements or other perturbations, will evolve in a rigorously predictable manner. Globally speaking, a quantum-mechanical scheme of description of physical phenomena can be summarized by introducing the so-called Schrödinger picture. A physical field is obtained when a given physical feature is attributed to each point of a space in which the process under study is developing.
Textbook quantum mechanics treats time as a classical parameter and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}$, the term $\mathrm{\ensuremath{\Delta}}t$ must be interpreted as a time interval and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the ``time at which an event happens'' by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework we derive here two uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.
Abstract, This paper introduces the 0 → 1 hypothesis, a conceptual framework in which physical events and their relational ordering are treated as potentially more fundamental than macroscopic time. Rather than assuming time to be a universally flowing background parameter, the framework begins with distinguishable physical states, transitions between states, and causal or relational ordering among events. The notation 0 → 1 is used as a conceptual shorthand for physical becoming; it does not imply binary computation, creation from nothing, wave-function collapse, or a universal Planck-time clock. The central question is whether familiar temporal quantities can emerge from a more fundamental relational structure while remaining compatible with relativistic causal structure, quantum theory, and the thermodynamic arrow of time. A minimal mathematical scaffold is proposed in terms of an event set, a causal partial order, state transitions, and a coarse-grained duration functional. The framework is compared with relational time, the Page–Wootters mechanism, and causal-set approaches. At its present stage, the 0 → 1 hypothesis is a conceptual research framework rather than a complete physical theory. The paper therefore identifies the mathematical and empirical requirements that would be necessary for the hypothesis to become falsifiable and experimentally distinguishable from existing approaches.