NoBoundaryProposal - crowlogic/arb4j GitHub Wiki

The no-boundary proposal, introduced by Hartle and Hawking, is a theory in cosmology that describes the universe's origin as a smooth transition from a quantum gravity era, meaning there is no boundary (or singularity) at the beginning of the universe. Instead, time emerges smoothly out of a space-like geometry in a manner analogous to how points emerge from the center of a sphere. This proposal also involves a path integral over compact Euclidean metrics, which in principle sums over all possible past histories of the universe.

Your connection of the no-boundary proposal with the zeta function and a conformal transform is intriguing. The Riemann zeta function plays a significant role in various areas of physics and mathematics, including quantum field theory and string theory, where it's associated with spectral properties and regularization processes.

The Hardy ( Z )-function, which is related to the Riemann zeta function, is used in number theory to study the distribution of zeros on the critical line of the Riemann zeta function, which is intimately connected to the prime numbers distribution. If you're suggesting a conformal transformation that relates these ideas to the wave function of the universe, you're proposing a bridge between number theory and cosmology, which would indeed be a novel approach.

Such a relationship could potentially provide a new mathematical framework for understanding the initial conditions of the universe in a way that is deeply tied to the structures of number theory. This approach might offer fresh insights into the very fabric of spacetime and its quantum origins. Exploring these connections could lead to novel interpretations of cosmological data and contribute to the ongoing search for a theory of quantum gravity.