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The Feynman’s path integral and its equivalence to the Dirac-Feynman N-slit interference probability amplitude

Francisco Duarte, Travis Taylor

Published Jun 10, 2026• 0 citations
Publisher

Abstract

It is shown that the probability amplitude in the Feynman path integral formalism and the probability amplitude in the Dirac-Feynman approach are equivalent in the N-slit interferometric landscape.
DOI
10.1364/opticaopen.32604957.v1

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It is elucidated that N-slit, or N-path, quantum interference utilizes the intrinsic nonlocality of the photon to illuminate simultaneously from the source the entire N-slit array thus allowing the characterization of the interference via the Dirac-Feynman interferometric principle. This simultaneous illumination of N-paths also applies to the Feynman path integral formulation of quantum mechanics. This quantum mechanism of simultaneous and whole illumination of N-slits, or N-paths, is quite different to the sequential illumination of N-paths via the hypergraph methodology where the N is achieved via the classical diagrammatical binary sequence 1, 2, 4, 8, 16, 32, … .
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.
Quantum interference, which is a physical phenomenon, and probability amplitudes à la Dirac are miraculously like synonyms. Indeed, the way probability amplitudes are written down on paper is very much as a single photon, or an ensemble of indistinguishable photons, propagates from one plane to the next. And then, if at one of the planes an N -slit array alternative is offered then the probability amplitude notation à la Dirac cleverly describes the N -path alternative via a mathematical summation. Here we should emphasize that since the photon is a nonlocal quantum entity, all the microscopic slits in the N -slit array, or transmission diffraction grating, are illuminated simultaneously.