Spontaneous symmetry breaking in Loop Quantum Gravity

In this paper we investigate the question how spontaneous symmetry breaking works in the framework of Loop Quantum Gravity and we compare it to the results obtained in the case of the Proca field, where we were able to quantise the theory in Loop Quantum Gravity without introducing a Higgs field. We obtained that the Hamiltonian of the two systems are very similar, the only difference is an extra scalar field in the case of spontaneous symmetry breaking. This field can be identified as the field that carries the mass of the vector field. In the quantum regime this becomes a well defined operator, which turns out to be a self adjoint operator with continuous spectrum. To calculate the spectrum we used a new representation in the case of the scalar fields, which in addition enabled us to rewrite the constraint equations to a finite system of linear partial differential equations. This made it possible to solve part of the constraints explicitly.Comment: 24 pages, two appendix. v2 modified abstract, amended each section, 28 pages, two appendi

The Proca-field in Loop Quantum Gravity

In this paper we investigate the Proca-field in the framework of Loop Quantum Gravity. It turns out that the methods developed there can be applied to the symplectically embedded Proca-field, giving a rigorous, consistent, non-perturbative quantization of the theory. This can be achieved by introducing a scalar field, which has completely different properties than the one used in spontaneous symmetry breaking. The analysis of the kernel of the Hamiltonian suggests that the mass term in the quantum theory has a different role than in the classical theory.Comment: 15 pages. v2: 19 pages, amended sections 2 and 6, references added v3: 20 pages, amended section 6 and minor correction

Properties of the Volume Operator in Loop Quantum Gravity I: Results

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a `volume gap' (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5--7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum on 4-valent vertices are included, for which the presence of a volume gap is proved. This paper presents our main results; details are provided by a companion paper arXiv:0706.0382v1.Comment: 36 pages, 7 figures, LaTeX. See also companion paper arXiv:0706.0382v1. Version as published in CQG in 2008. See arXiv:1003.2348 for important remarks regarding the sigma configurations. Subsequent computations have revealed some minor errors, which do not change the qualitative results but modify some of the numbers presented her