Spacetime Paths as a Whole

TL;DR

This paper introduces a novel perspective on spacetime and quantum paths by developing a 3+1D wave distribution framework that describes entire spacetime behavior without relying on an external time parameter, drawing parallels to holography.

Contribution

It formalizes a 3+1D spacetime wave distribution with no external time, offering a new way to understand quantum paths as whole spacetime entities rather than evolving over time.

Findings

Develops a 3+1D spacetime wave distribution framework.

Establishes a duality with 4-momentum representation encoding complete dynamics.

Draws parallels between spacetime behavior and hologram mathematics.

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.