Mario Paschke

Can (noncommutative) geometry accommodate leptoquarks

We investigate the geometric interpretation of the Standard Model based on noncommutative geometry. Neglecting the

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

S_0

On Spin Structures and Dirac Operators on the Noncommutative Torus

-reality symmetry one may introduce leptoquarks into the model. We give a detailed discussion of the consequences (both for the Connes-Lott and the spectral action) and compare the results with physical bounds. Our result is that in either case one contradicts the experimental results.

Quantum Gravity coupled to Matter via Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.The semi-classical analysis presented in this paper does away with most of the ambiguities found in the initial semi-finite spectral triple construction. The cubic lattices play the role of a coordinate system and a divergent sequence of free parameters found in the Dirac type operator is identified as a certain inverse infinitesimal volume element.

An essay on the spectral action and its relation to quantum gravity

We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that, similarly as in the classical case, the spectrum of the Dirac operator depends on the spin structure.

Multiple Noncommutative Tori and Hopf Algebras

We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac-type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of the Dirac-type operator on these states in a semi-classical approximation.

The Spectral Geometry of the Equatorial Podles Sphere

We give a brief, critical account of Connes’ spectral action principle, its physical motivation, interpretation and its possible relation to a quantum theory of the gravitational field coupled to matter. We then present some speculations concerning the quantization of the spectral action and the perspectives it might offer, most notably the speculation that the standard model, including the gauge groups and some of its free parameters, might be derived from first principles.

Local covariant quantum field theory over spectral geometries

Abstract We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite fibrations of the quantum double torus and generalize the construction for quantum multiple tori.