On the relation of canonical and covariant formulations of Loop Quantum Gravity

Loop Quantum Gravity (LQG) is a background independent approach towards a quantum theory of gravity that splits into a canonical and a covariant branch the latter of which is also often called spin foam model. The spin foam model can only be derived formally from a constrained BF-theory that is discretized prior to quantization so that the resulting quantum theory is not continuous while canonical LQG rests on a true representation of the continuum canonical commutation relations at the kinematical level. The covariant approach therefore raises a lot of questions, for example, the faith of Dirac observables is unknown to a large extend and it is neither settled whether all constraints are implemented in the spin foam model nor how to take the continuum limit. In contrast, it is possible to even solve the quantum dynamics of deparametrized models coupled to a scalar field within the canonical approach. However, the full physical Hilbert space of canonical LQG can still not be determined satisfactorily even though a non-anomalous quantization of the Hamiltonian constraint is known. It is widely believed although not proven that spin foams could be a useful tool in order to solve this issue. A comparison of the two approaches could therefore give valuable insights for both branches of LQG. Semiclassical techniques provide an important tool to check the consistency of a model in the absence of experimental data. In canonical LQG such a limit is realized through so-called complexifier coherent states. These states could for example be used in the context of deparametrized models, e.g. dust models, where the Hamiltonian is not a constraint but defines a true evolution, in order to compare the semiclassical dynamics of the canonical theory with the prediction of a (to-be-defined) spin foam model coupled to dust. Since spin foams should already incorporate the dynamics it is crucial for such a comparison that the coherent states defined in the canonical theory stay (approximately) coherent throughout the evolution. In the first part of the thesis a stability criterion for finite systems is derived an analyzed. Although some important insights can be gained, the formalism developed turns out to be too restrictive to be applied to LQG. Despite some promising hints a further investigation of these issues would go beyond the scope of this thesis. Instead it is examined whether spin foam techniques can be directly applied to construct a projector onto the physical Hilbert space of the canonical theory. The fundaments for this ansatz where laid in a seminal work by Kaminski, Kisielowski and Lewandowski [1–3] who extended the definition of the (Euclidean) spin foam model to arbitrary boundary graphs. This finally allows to make contact to the canonical formulation whose Hilbert space contains all these graphs. The KKL-model used in the thrid part of the dissertation to build a spin foam operator Ẑ[κ] based on abstract 2-complexes κ that acts on the kinematical Hilbert space H0 of LQG by identifying the spin nets induced on the boundary graph of κ with states in H0. Following a common heuristic argument, a rigging map is then postulated by summing over all possible spin foams κ including a possible weight that is designed such that it does not violate a certain gluing property. In the analysis of the resulting object it is possible to identify a spin foam transfer matrix that allows to generate any finite foam as a finite power of the transfer matrix. That the would-be rigging map has the chance to define a projector is demonstrated in a simplified situation where the class of foams is restricted to a certain type with only one internal vertex and where the weight and Barbero-Immirzi parameter are set to one. The so-obtained sum indeed annihilates the Euclidean constraint whose matrix elements are calculated here as well. Despite these certainly encouraging results, the further analysis transpires that the full sum over all κ, as written, does not define a projector into the physical Hilbert space. This statement is independent of the concrete spin foam model and Hamiltonian constraint. However, the transfer matrix potentially contains the necessary ingredient in order to construct a proper rigging map in terms of a modified transfer matrix. There are several hints that the failure of the sum to define a rigging map is caused by the fact that the simplicity constraint used in current spin foam models does not only admit gravitational solutions but includes all Plebanski sectors. A similar effect also causes problems in the asymptotic expansion of the Euclidean 4-simplex amplitude but can be cured by an additional constraint as shown in [4–7]. These results are extended to the Lorentzian spin foam model in the last part.