Time evolution in deparametrized models of loop quantum gravity
AbstractAn 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 β. 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)PRVDAQ1550-799810.1103/PhysRevD.91.124067] and Assanioussi et al. [Phys. Rev. D 92, 044042 (2015)PRVDAQ1550-799810.1103/PhysRevD.92.044042].