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arXiv:0912.2474

Quantum theory is classical mechanics with non-local existence

John J. Hegseth

Published Dec 13, 2009• 0 citations
arXivHTMLPDF

Abstract

I propose a new and direct connection between classical mechanics and quantum mechanics where I derive the quantum mechanical propagator from a variational principle. This variational principle is Hamilton's modified principle generalized to allow many paths due to the non-local existence of particles in phase space. This principle allows a physical system to evolve non-locally in phase space while still allowing a representation that uses many classical paths. Whereas a point in phase space represents a classical system's state, I represent the state of a non-local system by a mixed trajectory. This formulation naturally leads to the transactional interpretation for resolving the paradoxes of the measurement problem. This principle also suggests a more flexible framework for formulating theories based on invariant actions and provides a single conceptual framework for discussing many areas of science.

DOI
10.48550/arxiv.0912.2474

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I propose a new and direct connection between classical mechanics and quantum mechanics where I derive the quantum mechanical propagator from a variational principle. This variational principle is Hamilton's modified principle generalized to allow many paths due to the non-local existence of particles in phase space. This principle allows a physical system to evolve non-locally in phase space while still allowing a representation that uses many classical paths. Whereas a point in phase space represents a classical system's state, I represent the state of a non-local system by a mixed trajectory. This formulation naturally leads to the transactional interpretation for resolving the paradoxes of the measurement problem. This principle also suggests a more flexible framework for formulating theories based on invariant actions and provides a single conceptual framework for discussing many areas of science.
Many quantization schemes rely on analogs of classical mechanics where the connections with classical mechanics are indirect. In this work I propose a new and direct connection between classical mechanics and quantum mechanics where the quantum mechanical propagator is derived from a variational principle. I identify this variational principle as a generalized form of Hamilton's principle. This proposed variational principle is unusual because the physical system is allowed to have imperfect information, i.e., there is incomplete knowledge of the physical state. Two distribution functionals over possible generalized momentum paths a[p(t)] and generalized coordinates paths b[q(t)] are defined. A generalized action is defined that corresponds to a contraction of a[p(t)], b[q(t)], and a matrix of the action evaluated at all possible p and q paths. Hamilton's principle is the extremization of the generalized action over all possible distributions. The normalization of the two distributions allows their values to be negative and they are shown to be the real and imaginary parts of the complex amplitude. The amplitude in the Feynman path integral is shown to be an optimal vector that extremizes the generalized action. This formulation is also shown to be directly applicable to statistical mechanics and I show how irreversible behavior and the micro-canonical ensemble follows immediately.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
This paper explores the profound connection between the Eikonal principle, primarily known in optics for describing wavefront propagation, and the Hamilton-Jacobi formulation of classical mechanics. We explicitly demonstrate how the Hamilton-Jacobi equation, a powerful partial differential equation that unifies various aspects of classical dynamics, naturally gives rise to an Eikonal-like behavior when considering the surfaces of constant action. By drawing an explicit analogy between the optical path length and Hamilton's principal function, we show that the trajectories of particles in classical mechanics can be interpreted as "rays" that are orthogonal to these propagating "wavefronts" of constant action in configuration space. This framework provides an elegant geometric interpretation of classical mechanics, where the evolution of a system is described by the propagation of a scalar field. The paper elucidates how the principle of least action, a cornerstone of classical mechanics, finds its optical counterpart in Fermat's principle of least time, thereby achieving a deeper, unified understanding of these seemingly disparate physical phenomena. This unification not only offers a powerful method for solving complex mechanical problems but also provides conceptual insights bridging classical wave theory and particle dynamics, laying foundational groundwork for quantum mechanics.
Abstract Application of the Ramsey Infinite Theorem to the fundamental variational principles of physics is addressed. The Hamilton Least Action Principle states that, for a true/actual trajectory of a system, Hamilton’s Action is stationary for the paths, which evolve from the preset initial space-time point to the preset final space-time point. The Hamilton Principle distinguishes between the actual and trial/test trajectories of the system in the configurational space. This enables the transformation of the infinite set of points of the configurational space (available for the system) into the bi-colored, infinite, complete, graph, when the points of the configurational space are seen as the vertices, actual paths connecting the vertices/ points of the configurational space are colored with red; whereas, the trial links/paths are colored with green. Following the Ramsey Infinite Theorem, there exists the infinite, monochromatic sequence of the pathways/clique, which is completely made up of actual or virtual paths, linking the interim states of the system. The same procedure is applicable to the Maupertuis’s principle (classical and quantum), Hilbert-Einstein relativistic variational principle and reciprocal variational principles. Exemplifications of the Infinite Ramsey Theorem are addressed.