Classification of Finite Spectral Triples

It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings.Comment: Latex, 29 pages with 2 figures, reference adde

On the tensor convolution and the quantum separability problem

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product is separable or entangled. We show that the tensor convolution defined for certain mappings on an almost arbitrary locally compact abelian group, give rise to formulation of an equivalent problem to the separability one.Comment: 13 pages, two sections adde

K\"all\'en-Lehmann representation of noncommutative quantum electrodynamics

Noncommutative (NC) quantum field theory is the subject of many analyses on formal and general aspects looking for deviations and, therefore, potential noncommutative spacetime effects. Within of this large class, we may now pay some attention to the quantization of NC field theory on lower dimensions and look closely at the issue of dynamical mass generation to the gauge field. This work encompasses the quantization of the two-dimensional massive quantum electrodynamics and three-dimensional topologically massive quantum electrodynamics. We begin by addressing the problem on a general dimensionality making use of the perturbative Seiberg-Witten map to, thus, construct a general action, to only then specify the problem to two and three dimensions. The quantization takes place through the K\"all\'en-Lehmann spectral representation and Yang-Feldman-K\"all\'en formulation, where we calculate the respective spectral density function to the gauge field. Furthermore, regarding the photon two-point function, we discuss how its infrared behavior is related to the term generated by quantum corrections in two dimensions, and, moreover, in three dimensions, we study the issue of nontrivial {\theta}-dependent corrections to the dynamical mass generation