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

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 $x^0=ct$ 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 $\tau$, one reconstructs the standard quantum results, with one modification: 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 than the values known from our quantum mechanical measurements.