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The Quantum Extension: C: P(G) → C - Summing Complex Amplitudes over Consistent Paths Gives the Feynman Path Integral for a Free Particle in 4D Minkowski Spacetime

Eliam Raell

Published Mar 14, 2026• 0 citations

Abstract

This paper completes the Boolean-to-quantum transition. The consistency predicate C, previously taking values in {0,1}, is extended to take values in C. Each consistent path from x to y is assigned amplitude exp(i * tau(path)), where tau is the sum of Minkowski proper times along the path. The propagator K(x,y) is the sum of these amplitudes over all consistent paths.Four experiments confirm that this extension gives quantum mechanics. Phases span the full circle: paths interfere, cancelling and reinforcing. The geodesic (longest chain) maximises the action tau(path), and near-geodesic paths reinforce while off-geodesic paths cancel the stationary phase approximation. The propagator |K|^2 grows with proper time with correlation r=0.84, p=0.036. All twelve pairs analysed show constructive interference with |K|/sqrt(n) greater than 1 in every case.In the continuum limit, K(x,y) converges to the free particle Feynman propagator exp(i * m * tau_proper(x,y)). The action S = proper time emerges from J(2,C) geometry. No physics is assumed. Quantum mechanics emerges from the algebraic structure of distinction.

DOI: 10.5281/zenodo.19012761

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This paper completes the Boolean-to-quantum transition. The consistency predicate C, previously taking values in {0,1}, is extended to take values in C. Each consistent path from x to y is assigned amplitude exp(i * tau(path)), where tau is the sum of Minkowski proper times along the path. The propagator K(x,y) is the sum of these amplitudes over all consistent paths.Four experiments confirm that this extension gives quantum mechanics. Phases span the full circle: paths interfere, cancelling and reinforcing. The geodesic (longest chain) maximises the action tau(path), and near-geodesic paths reinforce while off-geodesic paths cancel the stationary phase approximation. The propagator |K|^2 grows with proper time with correlation r=0.84, p=0.036. All twelve pairs analysed show constructive interference with |K|/sqrt(n) greater than 1 in every case.In the continuum limit, K(x,y) converges to the free particle Feynman propagator exp(i * m * tau_proper(x,y)). The action S = proper time emerges from J(2,C) geometry. No physics is assumed. Quantum mechanics emerges from the algebraic structure of distinction.

The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory.

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Abstract Feynman's path integral formulation describes quantum propagation as a sum over all possible paths, weighted by a phase factor e^{iS/ℏ}. The classical path emerges via constructive interference of paths with stationary action. We reinterpret this mechanism within the Generalized Mass as Twisted Time (GMaTT) framework. The phase factor is identified as the accumulated phase difference Δφ of a G₂ root vector rotating around the √α axis. The condition for constructive interference is shown to be equivalent to the TUT (TwistUntwist Threshold) coherence condition |δφ| < √α. We clarify that the TUT window is not a dynamic laboratory filter but a static geometric boundary condition of the phase space. Classical paths are not "chosen" by the particle; they are the trajectories that maintain phase coherence within the TUT threshold during propagation. This provides a geometric origin for the path integral and the principle of least action, grounding Feynman's sum in the pregeometric phase geometry of GMaTT.

Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of a timeless path integral is rigorously derived. The transition amplitude is reformulated as the sum, or functional integral, over all possible paths in the constraint surface specified by the (relativistic) Hamiltonian constraint, and each path contributes with a phase identical to the classical action divided by ℏ. The timeless path integral manifests the timeless feature as it is completely independent of the parametrization for paths. For the special case that the Hamiltonian constraint is a quadratic polynomial in momenta, the transition amplitude admits the timeless Feynman's path integral over the (relativistic) configuration space. Meanwhile, the difference between relativistic quantum mechanics and conventional nonrelativistic (with time) quantum mechanics is elaborated on in light of the timeless path integral.

We show that the quantum-mechanical probability distribution involving complex probability amplitudes can be derived from three natural conditions imposed on a relativistically invariant probability function describing the motion of a particle that can take multiple paths simultaneously. The conditions are: (i) pairwise Kolmogorov additivity, (ii) time symmetry, and (iii) Bayes' rule. The resulting solution, parameterized by a single constant, is the squared modulus of a sum of complex exponentials of the relativistic action, thereby recovering the Feynman path-integral formulation of quantum mechanics.