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WITHDRAWN: Further refinement to the mathematical foundation of quantum mechanics

Kong Lingfang

Published Feb 27, 2023• 0 citations
Publisher

Abstract

Abstract The theoretical framework of quantum mechanics consists of five assumptions, and academic circle generally believed that the correctness of the assumptions can only be proved by experiments and cannot be proved theoretically. In this paper, according to the stochastic process problem in probability theory, we used Fourier transform, delta function and other methods to study stochastic motion, derived the wave function, superposition state and other concepts, thus giving the theoretical proof process of five assumptions. This paper is based on the basic concepts of quantum mechanics, the derivation process further improves the mathematical basis of quantum mechanics, but the derivation conclusion can be applied to both quantum mechanics and classical mechanics.

DOI
10.21203/rs.3.rs-2597673/v1

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Abstract The theoretical framework of quantum mechanics consists of five assumptions, and academic circle generally believed that the correctness of the assumptions can only be proved by experiments and cannot be proved theoretically. In this paper, according to the stochastic process problem in probability theory, we used Fourier transform, delta function and other methods to study stochastic motion, derived the wave function, superposition state and other concepts, thus giving the theoretical proof process of five assumptions. This paper is based on the basic concepts of quantum mechanics, the derivation process further improves the mathematical basis of quantum mechanics, but the derivation conclusion can be applied to both quantum mechanics and classical mechanics.
Abstract For a long time, the academic community believes that quantum mechanics and classical mechanics are two greatly different mechanical systems, and the algorithms of quantum mechanics cannot be used to solve classical mechanics problems [1] . However, starting from classical mechanics, this paper proves through rigorous mathematical methods that classical mechanics and quantum mechanics have exactly the same mathematical algorithms. For example, concepts such as wave function, probability amplitude and superposition state can also describe the random motion states of macroscopic objects. This paper mainly adopts mathematical calculation, supplemented by computer simulation experiments, and verifies the correctness of the mathematical calculation process from both theoretical and experimental perspectives. No assumptions are needed in this paper, and all concepts are derived and proved through strict mathematical deduction. These derivation steps show that concepts such as probability amplitude and wave function are universal concepts applicable to random motion processes, which can be used for both macroscopic objects and microscopic particles, and are not unique calculation methods of quantum mechanics. PACS: 03.65.Ta, 03.50.Kk
Quantum mechanics contains certain novel mathematical concepts. Among these are complex numbers, Hilbert spaces with their unitary and self-adjoint operators, states represented by complex vectors, superpositions of states, collapse of wave functions, Born's rule for probabilities and others. If we accept that quantum mechanics is probabilistic, then these concepts can be derived and they become secondary. In this work, we begin with what we call a conditional stochastic process, which is based on real numbers and probabilities. As we shall see, such processes are defined by three simple axioms. We then use conditional stochastic processes to derive quantum mechanics by employing a correspondence called a dictionary. We also show that the converse holds. That is, beginning with a quantum system, we employ the dictionary to derive a conditional stochastic process.
Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.
This book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness . Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or stochastic systems. The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below. Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization. Part two is on Stochastic processes . Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes.
The time-dependent Schrddinger equation has been derived from three assumptions within the domain o f classical and stochastic mechanics. The continuity equation is not used in deriving the basic equations o f the stochastic theory as in the literature. They are obtained by representing Newton's second law in a time-inversion consistent equation. Integrating the latter, we obtain the stoehastie Hamilton-Jacobi equation. The Schr6dinger equation is a result o f a transformation o f the Hamihon-Jacobi equation and linearization by assigning the arbitrary constant h : 2mD. An experiment is proposed to determine h and to test a hypothesis o f the theory directly. A mathematical apparatus is formulated from the Jacobian formalism to derive physical parameters from ¢(x, t) and to obtain operators for the boundary eases o f the theory. The operator formalisms are compared by means o f a well-known solution in the quantum theory.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.