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.
The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.
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, … .
We derive the intensity and fringe visibility of an N-slit quantum interference system directly from the Feynman path integral by partitioning the amplitude into slit-conditioned path classes and incorporating internal quantum states to account for distinguishability. Applying Born's rule and tracing over unobserved degrees of freedom yields the standard coherence-theoretic intensity expression, establishing an explicit equivalence between the path-integral and Dirac formulations of multi-slit interference. We show that a naive uniform-coherence model leads to unphysical visibility values for N>2, motivating a normalization-consistent decomposition into indistinguishable and distinguishable photon populations. The resulting visibility relation depends explicitly on the relative populations of coherent and incoherent photons and reproduces the standard two-slit limit while exhibiting the correct fully coherent and fully distinguishable limits. These results connect path-integral interference directly to experimentally measurable coherence properties and provide a normalization-consistent description of multi-slit visibility.
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, … .