Dynamics in canonical models of loop quantum gravity

Loop quantum gravity is a tentative but relatively well-developed attempt at a manifestly background independent theory of quantum gravity. The kinematical structure of loop quantum gravity is well understood, with the elementary kinematical states of the theory having a physically appealing interpretation as states describing quantized, discrete spatial geometries. However, formulating the dynamics of the theory in a fully satisfactory way has proven to be surprisingly challenging, and to a significant extent, the issue of dynamics still remains an open problem in loop quantum gravity. In this thesis we consider the problem of dynamics in canonical loop quantum gravity, primarily in the context of deparametrized models, in which a scalar field is taken as a physical time variable for the dynamics of the gravitational field. The dynamics of the quantum states of the gravitational field is then generated directly by a physical Hamiltonian operator, instead of being implicitly defined through the kernel of a Hamiltonian constraint. The deparametrized approach therefore bypasses many of the considerable technical difficulties one encounters when working with the constraint operator. We introduce a new construction of a Hamiltonian operator for loop quantum gravity, which has both mathematical and practical advantages in comparison to earlier proposals. Most importantly, our Hamiltonian can be constructed as a symmetric operator, and is therefore a mathematically consistent candidate for a generator of unitary physical time evolution in deparametrized models. Furthermore, by employing the recently introduced curvature operator in our construction, we obtain a significant practical simplification in the structure of the Hamiltonian operator. After having completed the definition of several variations of the new Hamiltonian, we develop methods for evaluating the dynamics generated by a given physical Hamiltonian in an approximate form, even if an exact solution to the eigenvalue problem of the Hamiltonian cannot be achieved. In particular, we show that for large values of the Barbero–Immirzi parameter, the Euclidean part of the Hamiltonian can be considered as a perturbation over the comparatively simple curvature operator, and hence standard perturbation theory can be used to approximate the spectrum of the complete Hamiltonian in terms of the known eigenvalues and eigenstates of the curvature operator. We illustrate the use of our approximation methods with simple numerical calculations. We also introduce a new representation for intertwiners in loop quantum gravity. By considering intertwiners projected onto coherent states of angular momentum, we obtain a description in which intertwiners are represented as polynomials of certain complex variables, and operators in loop quantum gravity are expressed as differential operators acting on these variables. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors parametrizing the angular momentum coherent states, as opposed to complicated, non-intuitive expressions involving combinations of the Wigner nj-symbols of SU(2) recoupling theory. The new representation could therefore turn out to be a useful tool for analyzing the dynamics in loop quantum gravity. In addition to reviewing the results of the author’s scientific work, this thesis also gives a thorough introduction to the basic framework of canonical loop quantum gravity, as well as a self-contained presentation of the graphical formalism for SU(2) recoupling theory, which is the invaluable tool for performing practical calculations in loop quantum gravity in the spin network basis. The author therefore hopes that parts of this thesis could serve as a comprehensible source of information for anyone interested in learning the elements of loop quantum gravity.