Time evolution in deparametrized models of loop quantum gravity

An important aspect in understanding the dynamics in the context of deparametrized models of loop quantum gravity (LQG) is to obtain a sufficient control on the quantum evolution generated by a given Hamiltonian operator. More specifically, we need to be able to compute the evolution of relevant physical states and observables with a relatively good precision. In this article, we introduce an approximation method to deal with the physical Hamiltonian operators in deparametrized LQG models, and we apply it to models in which a free Klein-Gordon scalar field or a nonrotational dust field is taken as the physical time variable. This method is based on using standard time-independent perturbation theory of quantum mechanics to define a perturbative expansion of the Hamiltonian operator, the small perturbation parameter being determined by the Barbero-Immirzi parameter $\beta$. This method allows us to define an approximate spectral decomposition of the Hamiltonian operators and hence to compute the evolution over a certain time interval. As a specific example, we analyze the evolution of expectation values of the volume and curvature operators starting with certain physical initial states, using both the perturbative method and a straightforward expansion of the expectation value in powers of the time variable. This work represents a first step toward achieving the goal of understanding and controlling the new dynamics developed in Alesci et al. [Phys. Rev. D 91, 124067 (2015)] and Assanioussi et al. [Phys. Rev. D 92, 044042 (2015)].

Figure 1

Evolution of the expectation value $⟨V⟩$ of the volume operator in the scalar field model with an initial eigenvector with eigenvalue $v=0.5730$.

Figure 2

Evolution of the expectation value $⟨R⟩$ of the curvature operator in the scalar field model with an initial eigenvector with eigenvalue $v=0.8725$.

Figure 3

Evolution of the expectation value $⟨V⟩$ of the volume operator in the scalar field model with an initial eigenvector with eigenvalue $v=8.3177$.

Figure 4

Evolution of the expectation value $⟨R⟩$ of the curvature operator in the scalar field model with an initial eigenvector with eigenvalue $v=5.1078$.

Figure 5

Evolution of the expectation value $⟨V⟩$ of the volume operator in the dust field model with an initial eigenvector with eigenvalue $v=0.8725$.

Figure 6

Evolution of the expectation value $⟨R⟩$ of the curvature operator in the dust field model with an initial eigenvector with eigenvalue $v=0.5730$.

Figure 7

Evolution of the expectation value $⟨V⟩$ of the volume operator in the dust field model with an initial eigenvector with eigenvalue $v=5.1078$.

Figure 8

Evolution of the expectation value $⟨R⟩$ of the curvature operator in the dust field model with an initial eigenvector with eigenvalue $v=8.3177$.

Figure 9

Evolution of the expectation value $⟨V⟩$ of the volume operator with an initial eigenvector with eigenvalue $v=0.5730$.

Figure 10

Evolution of the expectation value $⟨V⟩$ of the volume operator with an initial eigenvector with eigenvalue $v=0.5730$.

Figure 11

Evolution of the expectation value $⟨R⟩$ of the curvature operator with an initial eigenvector with eigenvalue $v=0.8725$.

Figure 12

Evolution of the expectation value $⟨R⟩$ of the curvature operator with an initial eigenvector with eigenvalue $v=0.8725$.

Figure 13

Evolution of the expectation value $⟨V⟩$ of the volume operator with an initial eigenvector with eigenvalue $v=32.1396$.

Figure 14

Evolution of the expectation value $⟨V⟩$ of the volume operator with an initial eigenvector with eigenvalue $v=32.1396$.

Figure 15

Evolution of the expectation value $⟨R⟩$ of the curvature operator with an initial eigenvector with eigenvalue $v=20.1761$.

Figure 16

Evolution of the expectation value $⟨R⟩$ of the curvature operator with an initial eigenvector with eigenvalue $v=20.1761$.

Figure 17

Comparative plot of the evolution of the volume expectation value $⟨V⟩$ between the time expansion and the $\beta$ expansion, with an initial eigenvector with eigenvalue $v=0.5730$ and $\beta =50$.

Figure 18

Comparative plot of the evolution of the volume expectation value $⟨V⟩$ between the time expansion and the $\beta$ expansion, with an initial eigenvector with eigenvalue $v=5.1078$ and $\beta =100$.