TL;DR
This paper examines two different scalar products in solvable loop quantum cosmology, highlighting their differences at small volumes and implications for quantum state coarse graining.
Contribution
It clarifies the distinction between two scalar product proposals in sLQC and emphasizes their importance at small volumes for quantum state analysis.
Findings
Two scalar products differ at small volumes
The difference impacts coarse graining of quantum states
Large volume limit shows agreement between scalar products
Abstract
Solvable loop quantum cosmology provides a simple model of spatially flat, homogeneous, and isotropic quantum cosmology where the initial singularity is resolved. A close inspection of the literature reveals that there exist two different proposals for a scalar product that are usually assumed to be identical, but agree only in the large volume limit. The small volume limit, and thus the reported difference, turns out to be important for questions such as coarse graining fundamental quantum states in full theory embeddings of these cosmological models.
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