Notes on a Gaussian-Based Distribution Algebra for the Non-linear Wave Equation of the Shift Vector in Quantum Foam
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In these notes, a non-linear distributional renormalisation algebra is developed, tailored to the geometry of Gaussian Quantum Foam. The construction is based on sequences of smooth Gaussian functions restricted to spacelike hypersurfaces in a sequence of homotopic and globally hyperbolic spacetimes, converging in the sense of distributions to Quantum Foam.
A restricted subspace of Schwartz functions is defined, consisting of finite products of scaled Gaussians supported on the hypersurfaces. An associated distribution space is introduced as the space of distributional limits of such sequences. The resulting renormalisation algebra is closed under addition, multiplication, and arbitrary-order differentiation, with all non-linear operations defined at the level of smooth representatives prior to taking the limit.
This algebra is then applied to the non-linear scalar wave equation governing the shift vector field. In the distributional limit, the wave operator acting on the Gaussian sequence yields a linear combination of the Dirac measure and its second-order derivative, which together encode the singular curvature response of the collapsing Quantum Foam element.
The presence of the measure's second-order derivative signals a sharply localised curvature impulse, consistent with a quantum geometric source driving the displacement of the vacuum. Meanwhile, the measure term corresponds to a uniform shift across the hypersurfaces, reflecting residual translation in the emerging classical geometry.
In the classical limit, the non-linear wave equation reduces to the massless vacuum Klein–Gordon equation, thereby linking the quantum and classical regimes through a unified distributional formalism.
In this setting, the classical notion of singularity is replaced by a sharply localised but well-defined distributional structure. The singular support of the shift vector field defines the locus of curvature concentration, without requiring any geometric breakdown.
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- Preprint: 10.5281/zenodo.14911684 (DOI)
- Preprint: 10.5281/zenodo.14911730 (DOI)
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2025-05-28In the spring and early summer of 2025, I had already realised that Gaussians could help bypass the long–standing issue of products of distributions. Yet the field equation for Quantum Foam still eluded me. In the end, I overcame my ghosts—sitting down after work, having just dealt with securities amid the turmoil following the Rose Garden announcement—and defined a non-linear, renormalised distributional algebra. That was the turning point: I could finally construct the field equation of Quantum Foam. It was a real spark—the beginning of time (and the end of singularities), as Wheeler once put it, "for time is not a primordial and precise concept; it must be secondary, derivative, and approximate." My work aims at the beginning of time in a frozen state and with the bare mass. In this sense, the project is now ended and complete—though I know full well that such a statement is always false, and that with time there will be more. Nevertheless, the chapter on singularities—and the fact that none of Penrose's conditions for avoiding singularities is required once one works in distributional geometry—marks a natural end to my long quest. It has been a journey spanning more than thirty years: from my early work on unstable photon rings near rotating black holes, through the time-machine problem, and then a long absence from research until 2022. But even in those years, I was always thinking, always sketching in my notebooks, always searching for a resolution. The multiple versions of this work are intentional, allowing precise tracking of theoretical refinements and making it easier to compare, reuse, and reference key developments across the Quantum Foam project.