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arXiv:2009.03435

Time Evolution and Probability in Quantum Theory: The Central Role of Born's Rule

Stephen Bruce Sontz

Published Sep 7, 2020• 0 citations
arXivHTMLPDF

Abstract

In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution of the probabilities given by the Generalized Born's Rule. In other models, such as the Heisenberg model, the auxiliary equations are quite different, but the Generalized Born's Rule is the same formula (covariance) and gives the same results (invariance). Also some aspects of the Schrodinger model are not found in the isomorphic Heisenberg model, and they therefore do not have any physical significance. One example of this is the infamous collapse of the quantum state. Other quantum phenomena, such as entanglement, are easy to analyze in terms of the Generalized Born's Rule without any reference to the unnecessary concept of collapse. Finally, this leads to the possibility of quantum theory with other sorts of auxiliary equations instead of Schrodinger's equation, and examples of this are given. Throughout this treatise the leit motif is the central importance of quantum probability and most especially of the simplifying role of the time dependent Generalized Born's Rule in quantum theory.

DOI
10.48550/arxiv.2009.03435

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The Born rule, P(x)=∣ψ(x)∣2P(\mathbf{x})=|\psi(\mathbf{x})|^2P(x)=∣ψ(x)∣2 is one of the most successful relations in modern physics. Yet its physical meaning remains conceptually distinctive. Schrödinger's equation describes the continuous evolution of a complex wave amplitude, whereas an actual detector records discrete and localized events. The Born rule connects these two descriptions with remarkable precision, but the physical mechanism responsible for the quadratic relation between wave amplitude and event frequency is not contained explicitly in the Schrödinger equation itself. This paper revisits the historical and physical relationship between Schrödinger's wave mechanics and Born's statistical interpretation of scattering. The purpose is not to deny the empirical validity of the Born rule, but to identify the physical gap between continuous wave evolution and discrete detection statistics. We distinguish three levels of description: the preparation of localized quantum excitations, their continuous propagation and scattering, and the final microscopic detector event. This leads to a central question: \boxedtext{Why should the rate of discrete detection events be proportional to ∣ψ∣2|\psi|^2∣ψ∣2?} The paper argues that the answer cannot be obtained merely by defining ∣ψ∣2|\psi|^2∣ψ∣2 to be a probability density. Instead, a physical theory must explain how a continuously distributed complex amplitude produces a localized event rate. We investigate the possibility that the quadratic dependence emerges from a local interaction functional, energy-transfer mechanism, or bilinear response of the detector to the incident quantum amplitude. A particularly important distinction is made between the wave amplitude itself and the experimentally recorded event frequency. A single detection event is localized, whereas the Born distribution emerges only through repeated preparations and statistical accumulation. The paper therefore proposes that the Born rule may be viewed as the macroscopic statistical image of a microscopic event-generation process. No completed derivation of Born's rule is claimed here. Rather, the paper formulates the precise dynamical problem that such a derivation must solve and identifies several mathematical and experimental constraints that any successful mechanism must satisfy.
Schrodinger wave equation, Hilbert space formalism, born’s rule for probability and hermitian operator are mainly the core of quantum theory and quantum description is generally based on Hilbert space of states, observables are described by self adjoint operators on the space of states, physical symmetries are explained by unitary transformations, and time evolution is a one-parameter group of unitary transformations on the Hilbert space of states. In this paper, I have discussed why a generalized philosophy is required before the development of quantum probability to fill up the gap.
The main distinction between classical mechanics and quantum mechanics is the lack in the latter of a full mechanical determinism: different final states can arise from the same physical state, after the measurement. No hidden variable is supposed to exist, nothing can discriminate two apparently identical states even if they give a different result. In this paper we try to put the basis for a more fundamental theory that (approximately) coincides with quantum mechanics when comparing statistics, but it is more fundamental, since it mathematically describes measurement processes giving an explicit time evolution of the wave function during the collapse. The theory is deterministic even if the Heisenberg uncertainty principle is still valid. The theory distinguishes physical states that collapse and physical states that do not collapse. The theory can be made compatible with all experiments done in the past, but new phenomena such as violations of the Born law or the superposition principle could transpire. However, even if we have probably shown that it is possible to build ad hoc a theory that can describe both the wave function collapse and the Schrödinger linear evolution, a simple and unified construction is still missing.
Abstract It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated physical systems featuring events. A general statistical law replacing unitary Schrödinger evolution of states is then formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous “measurement problem.” Our general concepts and results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field. In the limit where the speed of light tends to infinity these models can be treated quite explicitly.
In order to make the quantum mechanics a closed theory one has to derive the Born rule from the first principles, like the Schroedinger equation (SEq), rather than postulate it. The Born rule was in certain sense derived in several articles, e.g. in [1] and [2]. In this work some arguments of previous authors are simplified and made more “physical”. It is shown how to derive the Born rule using the conservation of quantum state norm (Ψ|Ψ) that is the unitary evolution property determined by the SEq. It is this property that makes the probability equal to the square of the amplitude modulus. Possible modification of the Born rule is briefly discussed in the case when the evolution of the system is described by a nonlinear Schroedinger equation. We also present arguments in the spirit of the Many-World Interpretation to explain the origin of probabilistic behavior. Simply speaking, the randomness appears as a result of representing the wave function by using a detector of discrete nature that is found only in one state at a time, out of two or more possible states.
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. \boxed{\mathcal R_D\!\left[(\Phi_\Psi)_*\mu_0^{\rm phys}\right] \overset{?}{=} \mu_{\rm Born,D}.} 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