Three-space as a quantum hyper-layer in 1+3 dimensions

Abstract

We discuss a formalism where a universe is identified with the support of a wave function propagating through space-time. As opposed to classical cosmology, the resulting universe is not a spacelike section of some space-time, but a hyper-layer of a finite timelike width, a set which is not a three-dimensional submanifold of space-time. We test the formalism on the example of a universe that contains a single harmonic oscillator (a generalization of the curvature-dependent Cariñena-Rañada-Santander (CRS) model). As opposed to the original CRS formulation, here the curvature is not a parameter but a quantum observable, a function of the world-position operator. It is shown that asymptotically, for large values of the invariant evolution parameter $\tau$, one reconstructs the standard quantum results, with one modifiication: The effective (renormalized) mass of the oscillator decreases with $\tau$. The effect does not seem to be a peculiarity of harmonic oscillators, so one may speculate that masses of distant elementary quantum systems are greater from their values known from our quantum mechanical measurements.

A sequel to M. Czachor, "Cosmic-time quantum mechanics and the passage-of-time problem", Universe 9, 188 (2023). https://doi.org/10.3390/universe9040188

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