Time evolution of semiclassical states in the one-vertex model of quantum-reduced loop gravity

Abstract

We compute numerically the time evolution of simple semiclassical states describing homogeneous and isotropic spatial geometries in quantum-reduced loop gravity under a deparametrized formulation of the dynamics, in which a reference matter field is taken as a relational time variable for the dynamics of quantum states of the gravitational field. The states which we consider are defined on the Hilbert space of a spin network graph formed by a single six-valent vertex. We find that the quantum dynamics is generally in close agreement with the semiclassical effective dynamics of a homogeneous and isotropic universe throughout the range of validity of the numerical approximation. In particular, an initial state describing a contracting geometry undergoes a dynamical "bounce", where the contraction is halted and turned into an expansion by the quantum dynamics.

Figures (14)

• Figure 1: Dynamics of a state corresponding to a classically static configuration in the Euclidean model. The parameters describing the initial state are given by $j_0 = 100$, $c_0 = 0$. The semiclassicality parameter is set to the value $t = 1/2j_0 = 1/200$. The expectation values of the quantum observables closely follow the effective semiclassical trajectory, and the occupation number $n_{\rm max}$ remains small throughout most of the time interval.
• Figure 2: Another example of a state in the Euclidean model corresponding to a classically static configuration. The initial state is again described by $j_0 = 100$, $c_0 = 0$ but the semiclassicality parameter has the value $t = 1/2\sqrt{j_0} = 1/20$. The state is initially sharply peaked on the volume operator but spreads rapidly. The occupation number $n_{\rm max}$ reaches a substantial value at a relatively early stage of the evolution, indicating the end of validity of the numerical approximation.
• Figure 3: Time evolution of a state describing an expanding universe in the Euclidean model. The initial state is defined by the parameters $j_0 = 50$, $c_0 = 0.4$. The quantum dynamics matches the semiclassical trajectory and the state remains well peaked throughout the time interval over which the computation can be expected to be accurate.
• Figure 4: Another example of a state describing an expanding universe in the Euclidean model. Here the parameters of the initial state are $j_0 = 100$, $c_0 = 0.1$. The qualitative features of the dynamics are essentially identical to the example shown in Fig. \ref{['fig:H_E-expanding-1']}.
• Figure 5: An example of an initially contracting state undergoing a dynamical "bounce" in the Euclidean model. The initial state is described by $j_0 = 150$, $c_0 = -1.2$. The state remains sharply peaked and the expectation values follow the semiclassical trajectory through the bounce, beginning to deviate only around the time the occupation number $n_{\rm max}$ has grown to a significant value.