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Journal of Physics Conference Series

Quantum extension for Newton’s law of motion

T. F. Kamalov

Published Jun 1, 2019• 5 citations
Publisher

Abstract

Abstract The main idea of I. Newton in Principia is a description of the laws of motion by a second-order differential equation. Classical Newtonian physics can describe stability trajectories in Inertial Reference Frames and some Non-Inertial ones. The variation of the action functional S of stability trajectories equals to zero. The observational error is including the influence of the random fields’ background to the particle. Can we use another description in the form of a high-order differential equation? The high-order derivatives can be used as additional variables accounting for the influence of random fields background. Trajectories due the influence of the random fields’ background can be called instability random trajectories. They can be described by high order derivatives. Then the stability classical trajectories to be complemented by additional instability random trajectories. Quantum objects are described by the trajectory with neighborhoods. Quantum Probability can describe quantum objects in random fields. The variation of the action functional S is defined by the Planck constant. For the common description of quantum theory and high-order theory, let us compare r-neighborhoods of quantum action functional with r-neighborhoods of the action functional of a high-order theory.

DOI
10.1088/1742-6596/1251/1/012022

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Abstract The main idea of I. Newton in Principia is a description of the laws of motion by a second-order differential equation. Classical Newtonian physics can describe stability trajectories in Inertial Reference Frames and some Non-Inertial ones. The variation of the action functional S of stability trajectories equals to zero. The observational error is including the influence of the random fields’ background to the particle. Can we use another description in the form of a high-order differential equation? The high-order derivatives can be used as additional variables accounting for the influence of random fields background. Trajectories due the influence of the random fields’ background can be called instability random trajectories. They can be described by high order derivatives. Then the stability classical trajectories to be complemented by additional instability random trajectories. Quantum objects are described by the trajectory with neighborhoods. Quantum Probability can describe quantum objects in random fields. The variation of the action functional S is defined by the Planck constant. For the common description of quantum theory and high-order theory, let us compare r-neighborhoods of quantum action functional with r-neighborhoods of the action functional of a high-order theory.
Abstract Refusal to use inertial reference frames in favor of non-inertial reference frames means significant changes in the axiomatics of both classical and quantum physics. Taking into account the influences of random small forces and fields makes the equations of classical physics, expanding, subject not only to deterministic laws, but also expands the equations of classical physics with additional variables that have an indeterministic, probabilistic behavior. Extending the scope of consideration of physical systems to non-inertial frames of reference leads to the already well-known new field of study known as quantum correction of Newton’s laws of motion. In this case, not only stable deterministic trajectories are considered, but also unstable, random ones, which usually constitute the measurement error in experiments in cases of applying classical physics and the probability distribution of quantum mechanics. Such a quantum correction can extend the action of Newton’s laws to the microworld by abandoning the basic idea of classical physics that physical systems should be described by differential equations of no higher than second order in favor of supplementing them with higher derivative variables that describe unstable random trajectories.
A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibrational) reference frames with the random initial conditions? One of the most general description (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher order temporal derivatives of the coordinates in the Lagrange function. Such formalism describes physical objects in the infinite dimensional space does not contradict quantum mechanics and infinite dimensional Hilbert space.
Quantum mechanics is traditionally formulated in terms of postulates inferred from experimentalobservations. Here we propose that the statistical regularities of quantum mechanics arise as macro-scopic manifestations of stochastic fluctuations of dynamical parameters in microscopic systems gen-erated by a non-inertial physical background. Although the specific nature of this background mayvary, the universality of quantum statistics is ensured by invariant properties of stochastic dynam-ics - analogous to the role of the equivalence principle in general relativity - and by the existence ofa fundamental action scale, Planck constant h. By extending classical mechanics to include higher-order derivatives induced by non-inertial fuctuations and imposing a stability condition on admissibletrajectories, the minimal action scale h naturally emerges. In this framework quantum mechanicsappears as an effective statistical description of systems embedded in a fluctuating non-inertial envi-ronment while preserving all standard predictions of the theory.
A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibration) reference frames with the random initial conditions? One of the most generalized ways of descriptions (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher temporal derivatives of the coordinates in the Lagrange function. Such formalism describing physical objects in the infinite dimensions space does not contradict to the quantum mechanics and infinite dimensions Hilbert space.
The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). A "functional" formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.