Analysis of observables in FLRW spacetime

We consider a positively curved Friedmann–Lemaitre–Robertson–Walker spacetime as a background in which a nonrelativistic quantum particle propagates according to the Schroedinger equation. The probability fluid for the corresponding quantum states is taken as a model for the cosmological fluid filling this Friedman–Lemaitre–Robertson–Walker Universe. The Hamiltonian operator governing this fictitious particle is proportional to the Laplacian operator derived from the Friedman–Lemaitre–Robertson–Walker metric, while the mass of the particle equals the overall matter of the Universe (baryonic and dark). A complete, orthonormal set of quantum eigenstates of the Hamiltonian is obtained. Restricting to radially symmetric states, the latter are then used to compute matrix elements and expectation values of two observables for which quantum operators are identified, namely, the cosmological constant and the gravitational Boltzmann entropy. This value of the entropy is regarded as corresponding to a positively curved Friedman–Lemaitre–Robertson–Walker geometry when the cosmological fluid filling the Universe occupies a given quantum state. Our results comply closely with the upper bound on the entropy set through the holographic principle stated independently by 't Hooft and Susskind in the 1990's.