On the Correspondence Principle of Quantum Foam

This paper introduces a framework for modelling Quantum Foam as a sequence of homotopic, globally hyperbolic spacetimes converging to a limiting distributional structure at the Planck scale. The frameworkestablishes a Correspondence Principle for quantum gravity, illustrating how classical spacetime geometry emerges from quantum fluctuations. A sequence of normalized Gaussian functions extending the shift vector, which encodes localized perturbations and the universe’s contraction toward the Planck scale, serves as a sample model. The use of normalized Gaussian sequences ensures well-defined distributional products despite the theory’s inherent non-linearity. This model the analysis of inflationary dynamics under different lapse function scalings, providing insight into hypersurface evolution. The volume expansion rate for an Eulerian observer is used as a proxy for the Hubble parameter, characterising both localised and global inflation. This work bridges quantum and classical regimes, advancing the understanding of spacetime’s fundamental nature, evolution, and measurable implications, and provides a foundation for further exploration of quantum gravity and cosmology.