# Solvable Loop Quantum Cosmology: domain of the volume observable and semiclassical states

## TL;DR

This paper explicitly constructs the domain of the volume observable in solvable Loop Quantum Cosmology (sLQC) and examines the semiclassical properties of states, bridging different representations using the Wheeler-DeWitt approach.

## Contribution

It provides the explicit form of physical states in the volume domain within sLQC and clarifies the connection between different quantum representations.

## Key findings

- Explicit volume domain states in sLQC are derived.
- The connection between $v$-representation and sLQC representation is established.
- Semiclassical states in sLQC are analyzed for their properties.

## Abstract

The dynamics of a flat Friedmann-Lema\^{i}tre-Robertson-Walker model minimally coupled to a massless scalar field has been intensively studied in the context of Loop Quantum Cosmology. This model admits an appropriate solvable representation, named sLQC. The form of the domain of the volume, the main observable to track the quantum evolution, is not straightforward in this solvable representation, and its explicit construction has been overlooked so far. In this work we find the explicit form of physical states belonging to the domain of the volume in sLQC. Specifically, given a physical state in the $v$-representation where the volume acts diagonally, we derive its form in the representation employed in sLQC, making explicit the connection between both representations at the physical level. To this end, we resort to the Wheeler-De Witt (WDW) approach, which shares the physical Hilbert space with sLQC when cast in an analog solvable representation, while being analytically solvable as well in the $v$-representation. Then the domain of the volume for the WDW approach provides that for sLQC. Furthermore, we address the question of semiclassicality in sLQC.