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Optics Letters

Ray–wave duality in classical optics: crossing the Feynman bridge

James Babington

Published Nov 9, 2018• 10 citations
Publisher

Abstract

Classical ray and wave optics share a common structure with quantum mechanics. Much of this results from the representations given to abstract operators. It is possible to build a mathematical bridge that allows one to transform back and forth between the ray and wave picture. This is given by the Feynman path integral representation. The map between the ray and wave description is given in terms of a sum over histories integration. In particular, it allows for the computation of wave phenomena by the summation of multiple ray paths of a phase function connected to the classical optical path length.

DOI
10.1364/ol.43.005591

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Classical ray and wave optics share a common structure with quantum mechanics. Much of this results from the representations given to abstract operators. It is possible to build a mathematical bridge that allows one to transform back and forth between the ray and wave picture. This is given by the Feynman path integral representation. The map between the ray and wave description is given in terms of a sum over histories integration. In particular, it allows for the computation of wave phenomena by the summation of multiple ray paths of a phase function connected to the classical optical path length.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
The Feynman path integral plays a crucial role in quantum mechanics, offering significant insights into the interaction between classical action and propagators, and linking quantum electrodynamics (QED) with Feynman diagrams. However, the formulations of path integrals in classical quantum mechanics and QED are neither unified nor interconnected, suggesting the potential existence of an important bridging theory that could be key to solving existing puzzles in quantum mechanics. In this work, we delve into the theoretical consistency, completeness, and integration with established path integral theories, revealing this concealed path integral form. This newly uncovered form not only connects various path integral approaches but also demonstrates its potential in explaining quantum phenomena like the origin of spin and quantum nonlocal correlations. It transcends conventional quantum mechanics, proposing a more profound and fundamental physical principle.
Quantum mechanics presents two seemingly distinct theoretical pictures: the unitary evolution of state vectors in the Schrödinger picture, and the summation over all possible paths in Feynman's path integral formulation. This article systematically argues that the equivalence between these two pictures is a rigorous mathematical fact rather than a choice between competing physical mechanisms. We analyze the meaning of this equivalence at three levels—physical, mathematical, and philosophical—and explore its implications for the generation of entanglement, many-body systems, and quantum field theory. The complete mathematical proof of the equivalence, from unitary evolution to the path integral, is provided in the appendix, along with a detailed construction of quantum entanglement from the path integral framework. Building on this analysis, we distill a broader philosophical thesis: every physical theory, no matter how successful, is a representation of the natural world rather than the world itself. Physical theories are like maps to a territory—they can be extraordinarily precise, yet they are never the territory. The Schrödinger equation, the path integral, and Heisenberg's matrix mechanics are three different maps of the same "quantum territory." Asking whether the wave function "really exists" or whether a particle "really travels along all paths" commits a category mistake—conflating mathematical entities internal to a theory with ontological reality beyond it. The history of physics—from Newtonian gravity to general relativity, from thermodynamics to statistical mechanics, from field-based to action-at-a-distance formulations of electrodynamics—repeatedly teaches this lesson: ontological claims are relative to a given theoretical framework and carry no absolute significance beyond it. The principle of "representational freedom" is thus proposed: when multiple mathematically equivalent formulations of a physical theory yield strictly identical predictions for all observables, no experiment can adjudicate among them ontologically. Choosing one formulation as a calculational tool is a pragmatic decision, not an ontological verdict. The ultimate conclusion is that the path integral is simply the integral representation of the time-evolution operator in the position basis; neither picture enjoys ontological primacy. To claim that "the path sum really happens" or that "the state vector really evolves" is to cross the same boundary. True scientific wisdom lies in knowing where the boundary between representation and reality lies—and then, with ontological humility, continuing to explore the deeper structures that lie beyond it.