Abstract
The mathematical similarities between non-relativistic wavefunction propagation in quantum mechanics and image propagation in scalar diffraction theory are used to develop a novel understanding of time and paths through spacetime as a whole. It is well known that Feynman’s original derivation of the path integral formulation of non-relativistic quantum mechanics uses time-slicing to calculate amplitudes as sums over all possible paths through space, but along a definite curve through time. Here, a 3+1D spacetime wave distribution and its 4-momentum dual are formally developed which have no external time parameter and therefore cannot change or evolve in the usual sense. Time is thus seen “from the outside”. A given 3+1D momentum representation of a system encodes complete dynamical information, describing the system’s spacetime behavior as a whole. A comparison is made to the mathematics of holograms, and properties of motion for simple systems are derived.
Highlights
In this proposal, the well-developed connection between image propagation in scalar diffraction theory (SDT) and non-relativistic quantum wavefunction propagation (QWP) will be used to develop a 3+1D formulation of QWP and interpret the result. (Throughout this paper, 3+1D or “four-dimensional” refers to the usual three spatial dimensions and one temporal dimension of spacetime, or the three wavenumber dimensions and one angular frequency dimension (ω) of momentum space.)In Feynman’s path integral formulation [1], a spacetime path is defined as “a sequence of configurations for successive times”
The continuity of time has led to complications in physics, such as the higher order corrections to solutions of the time-dependent Schrödinger equation, and path integrals in quantum field theory
Events are contextual in time, relying on both past and future all at once. Such “timeless”, non-local, or as-a-whole descriptions exist in Lagrangian dynamics, quantum field theory, and Fourier optics
Summary
The well-developed connection between image propagation in scalar diffraction theory (SDT) and non-relativistic quantum wavefunction propagation (QWP) will be used to develop a 3+1D formulation of QWP and interpret the result. (Throughout this paper, 3+1D or “four-dimensional” refers to the usual three spatial dimensions and one temporal dimension of spacetime, or the three wavenumber dimensions (kx, ky, kz) and one angular frequency dimension (ω) of momentum space.). The equation of motion of a system will be encoded as a whole into the phase profile of the 3+1D wave distribution in the frequency domain This leads to a novel description of a spacetime “path as a whole”. The path predicts the spacetime coordinates at which one could find the system, but until an interaction occurs, all spacetime coordinates along the path are equivalent This suggested approach addresses the previously stated shortcomings by relating the advancement of time with discrete instances of convolution. This isometry motivates an “image formation” model of quantum measurement based on recursive 3+1D Fourier transforms, in which measurable time advances in discrete steps of arbitrary duration as a result of interactions. A description of dynamics is arrived at without continual time-slicing, in favor of a novel view of “time” which advances discretely between interactions and spacetime paths which must be described “as a whole”
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