On Quantum Mechanics in Friedmann-Robertson-Walker Universe

TL;DR

This paper investigates how quantum mechanics and canonical quantization behave in different Friedmann-Robertson-Walker universes, showing that only flat cosmologies preserve standard quantum commutation relations.

Contribution

It demonstrates that in flat FRW space-times, quantum mechanics aligns with canonical quantization, while in curved cases, it results in deformed commutation relations.

Findings

Canonical quantization matches quantum field theory in flat FRW space-times.

Deformation of CCR occurs in non-flat FRW universes.

Spatially flat cosmology is uniquely compatible with standard quantum principles.

Abstract

It is shown that only in the space-times admitting a 1+3-foliation by flat Cauchy hypesurfaces (i.e., in the Bianchi I type space-times the isotropic version of which the spatially flat Friedmann-Robertson-Walker space-times are) the canonical quantization of geodesic motion and quantum-mechanics obtained as an asymptotics of the quantum theory of scalar field lead to the same canonical commutation relations (CCR). Otherwise, the field-theoretical approach leads to a deformation of CCR (particularly, operators of coordinates do not commute), and the Principle of Correspondence is broken in a sense. Thus, the spatially flat cosmology is distinguished intrinsically in the quantum theory.