On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.Comment: 45 pages, latex, no figure

On the construction of group invariant non expansive operators

In questa tesi vengono illustrati metodi di topologia computazionale negli ambiti di topological data analysis e shape comparison. Nello specifico, dato uno spazio topologico X e un sottogruppo G di Homeo(X), viene studiato un set di dati Φ composto da funzioni definite su X, continue e limitate, a valori reali. Per farlo si utilizzano degli operatori G-invarianti non espansivi (GINO), che si sono dimostrati efficaci per approssimare la pseudo-distanza naturale. In particolare in questo lavoro vengono studiati metodi per la costruzione di tali operatori, sfruttando le proprietà algebriche delle variabili del problema. È importante osservare che il gruppo G viene sempre considerato come variabile, in quanto un cambio dell’osservatore può generalmente coincidere con un cambio dell’invarianza a cui si è interessati

The complex crown for homogeneous harmonic spaces

A theory of holomorphic extension of eigenfunctions on homogeneous harmonic spaces is developed.Comment: 17 pages, 1 figure

Vacuum Polarisation and the Black Hole Singularity

In order to investigate the effects of vacuum polarisation on mass inflation singularities, we study a simple toy model of a charged black hole with cross flowing radial null dust which is homogeneous in the black hole interior. In the region $r^2 \ll e^2$ we find an approximate analytic solution to the classical field equations. The renormalized stress-energy tensor is evaluated on this background and we find the vacuum polarisation backreaction corrections to the mass function $m(r)$. Asymptotic analysis of the semiclassical mass function shows that the mass inflation singularity is much stronger in the presence of vacuum polarisation than in the classical case.Comment: 12 pages, RevTe

Quantum Field Theory of Open Spin Networks and New Spin Foam Models

We describe how a spin-foam state sum model can be reformulated as a quantum field theory of spin networks, such that the Feynman diagrams of that field theory are the spin-foam amplitudes. In the case of open spin networks, we obtain a new type of state-sum models, which we call the matter spin foam models. In this type of state-sum models, one labels both the faces and the edges of the dual two-complex for a manifold triangulation with the simple objects from a tensor category. In the case of Lie groups, such a model corresponds to a quantization of a theory whose fields are the principal bundle connection and the sections of the associated vector bundles. We briefly discuss the relevance of the matter spin foam models for quantum gravity and for topological quantum field theories.Comment: 13 pages, based on the talk given at the X-th Oporto Meeting on Geometry, Physics and Topology, Porto, September 20-24, 200

The Heat Kernel on AdSAdS

We explicitly evaluate the heat kernel for the Laplacian of arbitrary spin tensor fields on the thermal quotient of (Euclidean) $AdS_N$ for $N\geq 3$ using the group theoretic techniques employed for $AdS_3$ in arXiv:0911.5085. Our approach is general and can be used, in principle, for other quotients as well as other symmetric spaces.Comment: Added references, added appendix on heat kernel in even dimensio

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