# Three-Space as a Quantum Hyperlayer in 1+3 Dimensions: A Case Study in Quantum Space and Time

## TL;DR

This paper explores a quantum model of the universe as a hyperlayer in space-time, where curvature is a quantum observable and mass changes over time.

## Contribution

The paper introduces a new quantum formalism where curvature is a quantum observable and the universe is a hyperlayer with time-like width.

## Key findings

- The universe is a hyperlayer with a finite timelike width, not a three-dimensional submanifold of space-time.
- Curvature is a quantum observable dependent on the world-position operator, not a fixed parameter.
- The effective mass of a quantum oscillator decreases over time, suggesting distant systems may have higher masses.

## Abstract

We discuss a formalism where a universe is identified with the support of a wave function propagating through space–time. The dynamics is of a squeezing type, with shrinking in time and expanding in space. As opposed to classical cosmology, the resulting universe is not a spacelike section of some space–time but a hyperlayer of a finite timelike width, a set which is not a three-dimensional submanifold of space–time. The universe is in superposition of different localizations in both space and time so that x0=ct has the same formal status of a position operator as the remaining three coordinates. 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 τ, one reconstructs the standard quantum results, with one modification: The effective (renormalized) mass of the oscillator decreases with τ. 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 than the values known from our quantum mechanical measurements.