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arXiv:2609.29248

The Uncertainty Principle, Uncertainty Relations, and Underlying Trajectories: Feynman, Nelson, Bohm, and Persistent Kac-Dirac Dynamics

Partha Ghose

quant-phSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
v1 · Sep 24, 2026v2 · Oct 11, 2026v3 · Oct 15, 2026
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Abstract

A distinction should be made between quantum uncertainty relations and the broader uncertainty principle. The former are precise statistical consequences of quantum mechanics; the latter, when interpreted as excluding simultaneously definite conjugate variables and particle trajectories, is an additional ontological claim not implied by the uncertainty relations themselves. We examine this distinction using Feynman paths, Nelson's stochastic mechanics, Bohmian mechanics, and finite-speed persistent Kac dynamics. Feynman paths exhibit Brownian-like short-time scaling, while Nelson derives Schrödinger dynamics from Wiener diffusion supplemented by dynamical assumptions. Bohmian mechanics, by contrast, reproduces quantum predictions while retaining definite particle trajectories. Kac dynamics provides continuous, piecewise differentiable finite-speed trajectories whose diffusive limit approaches Wiener kinematics. Moreover, after an appropriate Wick rotation, its coupled two-sector transport equations can be mapped to the Dirac equation. These examples demonstrate that the experimentally established uncertainty relations are compatible with markedly different underlying path structures. They therefore do not, by themselves, rule out a microscopic trajectory description of quantum dynamics.

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A distinction should be made between quantum uncertainty relations and the broader uncertainty principle. The former are precise statistical consequences of quantum mechanics; the latter, when interpreted as excluding simultaneously definite conjugate variables and particle trajectories, is an additional ontological claim not implied by the uncertainty relations themselves.
We examine this distinction using Feynman paths, Nelson's stochastic mechanics, Bohmian mechanics, and finite-speed persistent Kac dynamics. Feynman paths exhibit Brownian-like short-time scaling, while Nelson derives Schrödinger dynamics from Wiener diffusion supplemented by dynamical assumptions. Bohmian mechanics, by contrast, reproduces quantum predictions while retaining definite particle trajectories. Kac dynamics provides continuous, piecewise differentiable finite-speed trajectories whose diffusive limit approaches Wiener kinematics. Moreover, after an appropriate Wick rotation, its coupled two-sector transport equations can be mapped to the Dirac equation.
These examples demonstrate that the experimentally established uncertainty relations are compatible with markedly different underlying path structures. They therefore do not, by themselves, rule out a microscopic trajectory description of quantum dynamics.
Maintaining the position that the wave function ψψψ provides a complete description of state, the traditional formalism of quantum mechanics is augmented by introducing continuous trajectories for particles which are sample paths of a stochastic process determined (including the underlying probability space) by ψψψ. In the resulting formalism, problems relating to measurements and objective reality are solved as in Bohmian mechanics (without sharing its weak points). The pitfalls of Nelson's stochastic mechanics are also avoided.
A history of the discovery of `new' quantum mechanics and the paradoxes of its probabilistic interpretation are briefly reviewed from the modern point of view of quantum probability and information. Modern quantum theory, which has been developed during the last 20 years for the treatment of quantum open systems including quantum noise, decoherence, quantum diffusions and spontaneous jumps occurring under continuous in time observation, is not yet a part of the standard curriculum of quantum physics. It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements, but they have all received an adequate mathematical treatment in quantum stochastics of open systems. Moreover, the only reasonable probabilistic interpretation of quantum mechanics put forward by Max Born was, in fact, in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schrödinger. They are reconsidered in this paper from the modern point of view of quantum stochastics and information. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or Bell's be ables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the actual histories with the potential future. This rule, together with the self-compatibility of the measurements ensuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, and also to the more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a dynamical solution, in the form of the quantum stochastic filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrödinger and Bohr. It has been recently proved that the quantum stochastic model for the continuous in time measurements is equivalent to a Dirac type boundary-value problem for the secondary quantized input `offer waves from future' in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the `trajectories of the output particles'. This supports the corpuscular-wave duality in the form of the thesis that everything in the future are quantized waves, while everything in the past are trajectories of the recorded particles.
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Analysis of the initial stages of the logical process followed by Louis de Broglie in establishing the electron phase wave equation in his 1924 thesis, which triggered the development of Wave Mechanics when Erwin Schrödinger formalized this concept with his vectorial wave equation. This development was soon followed by Quantum Mechanics, when Schrödinger proved that the Matrix Mechanics independently developed by Werner Heisenberg was equivalent to Wave Mechanics, with both theories leaving room for some degree of uncertainty as to the physical localization of the moving electron. This is what led Heisenberg to also formalize the Uncertainty Principle to take this situation into account. This principle was soon regarded as a fundamental axiomatic principle that seemed to make further exploration of the subatomic level of magnitude appear impossible to most researchers. We will analyze in this article the reason why the phase-wave velocity established by de Broglie generated this uncertainty in the localization of the moving electron in light of the current state of knowledge on the behavior of the electron in motion, in view of establishing the relevance of maintaining the Uncertainty Principle in the study of the subatomic level of magnitude.
Trajectory-based approaches to quantum mechanics include the de Broglie–Bohm interpretation and Nelson's stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schrödinger equation into a continuity equation and a modified Hamilton–Jacobi equation, fails for some quantum states. A very simple example is provided by a quantum particle in a box, described by a wavefunction that is initially uniform over the interior of the box. For this example, there is no corresponding continuity or modified Hamilton–Jacobi equation, and the space-time dependence of the wavefunction has a known fractal structure. Examples with finite average energies are also constructed.
100±Δt100\pmΔt100±Δt Years of Quantum Uncertainty: From Origins to Modern Insights
Heisenberg's uncertainty principle is a cornerstone of quantum mechanics, marking a decisive departure from classical physics. Conceived almost a century ago through a thought experiment showing that measuring an electron's position inevitably disturbs its momentum, it began as a deceptively simple idea that sparked countless studies and grew into the rich research field it is today. This review traces its development into a spectrum of mathematical formulations -- known as uncertainty relations -- and explores their interconnections and wide-ranging applications. We highlight its central role in quantum metrology, where it underpins strategies for extracting information from quantum systems with ever-increasing precision, and its links to multiparameter estimation and squeezed states. This review, dedicated to the centenary of the uncertainty principle, reflects on how it has deepened our understanding of quantum theory and driven practical advances, and looks ahead to a century poised for further surprising and transformative discoveries.