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Physical Review D

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Integral Hilbert spaces and the dynamics of loop quantum cosmos

Janek Kozicki ^*

Tomasz Pawłowski ^†

Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland and Advanced Materials Center, Gdańsk University of Technology, 80-233 Gdańsk, Poland

University of Wrocław, Faculty of Physics and Astronomy, Institute for Theoretical Physics, Plac Maxa Borna 9, 50-204 Wrocław, Poland

^*Contact author: jkozicki@pg.edu.pl
^†Contact author: tomasz.pawlowski@uwr.edu.pl

Phys. Rev. D 114, 106005 – Published 4 November, 2026

DOI: https://doi.org/10.1103/37d1-65t2

Abstract

Polymer quantization program applied in loop quantum gravity/cosmology leads to nonseparable Hilbert spaces. Commonly, one sidesteps this problem by singling out and working with a separable superselection sector. This is, however, often no longer accessible in more involved models beyond isotropic ones. In an alternative approach, one builds a separable Hilbert space as an integral over all available sectors. Here, we test the dynamics following from the latter on the example of a flat isotropic universe admitting a negative cosmological constant and a massless scalar field. There, numerical evolution of (initially) semiclassical states shows that there is no relevant difference in their long-term semiclassicality properties, in comparison to those in a single-sector approach. Further, the older (problem-specific) numerical methods are compared against an application of more common and more efficient standard tools (eigen library).

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