Quantum Mechanics and Special Relativity merge in the basic Principle of
Quantum Field Theory, the Principle of *Locality*, but it is not known
whether this is a limit of an equally sharp principle taking its place in
General Relativistic Theories.

A pale shadow of a Principle where Quantum Mechanics and Classical General Relativity merge, is the principle of gravitational stability of Spacetime against localization of events. It leads to a quantum structure of Spacetime at the Planck scale, most easily implemented by the "basic model".

We will review the Quantum Geometry of Quantum Spacetime, the problems in formulating the interaction of fields over Quantum Spacetime, the partial fulfillment of the hope that Quantum Spacetime "regularizes" Quantum Field Theory, and some possible cosmological consequences related to the open problems.

In the theory of noncommutative spaces (in the sense of spectral triples) the notion of dimension spectrum plays an important role. For a classical manifold it reduces to the standard definition of the dimension, but in general it may be an arbitrary discrete subset of the complex plane. In my talk I will present the results concerning the dimension spectrum of the Podles sphere. Moreover, I will show how the introduction of an auxiliary twist affects the dimension spectrum.

From the point of view of physics, the knowledge of the dimension spectrum and associated spectral functions is crucial to define the spectral action. Hence, it is the first step towards gauge theories on the quantum sphere.

We discuss some relevant spectral and ergodic properties of quantum disordered systems including quantum (Edward Anderson) spin glasses, and models based on the CAR algebra (describing Fermi Particles). That allows us to study some relevant properties of the convex set of temperature states of the model, and the structure and the type of the von Neumann algebras generated by such KMS states.

Loop quantum gravity has been established as a major attempt to quantize general relativity. Although its dynamical part is still widely unknown territory, some remarkable achievements have been obtained: construction of geometric operators for area and volume, derivation of black hole entropy, or to transfer the ideas to cosmology. In our talk, we will review the mathematical foundations of loop quantum gravity. We start with Ashtekar's variables that turn gravity into an SU(2) gauge field theory with constraints. Then, generalized connections, the Ashtekar-Lewandowski measure and geometric operators will be described. If time permits, we will discuss the relation between loop quantum gravity and loop quantum cosmology.

Certain quantum fields over deformed space-time lead to a bounded renormalization group flow.

Using a Ward identity, it was possible to prove the vanishing of the beta function for the coupling constant to all orders in perturba- tion theory. With the help of the Schwinger-Dyson equations we obtain non-linear integral equations for the renormalized two and four point functions alone. A nonperturbative construction of this model is almost done.

Next we discuss a model with two deformed quantized space direc- tions, which we have proven to be renormalizable too. We perform the one-loop renormalization explicitly. The Euclidean model is connected to the Minkowski one via an analytic continuation. At a special value of the parameters a nontrivial fixed point of the renormalization group occurs again.

A recently derived theorem allowing the analytic continuation from a Euclidean model to its Minkowski version is mentioned.

We review precanonical quantization and our recent result on the explicit construction of the Schroedinger wave functional in terms of the continuous product of precanonical wave functions (to appear in arxiv) which allows us to derive the functional Schroedinger equation from the precanonical partial differential covariant analogue of the Schroedinger equation. Then we show how this precanonical quantization approach can be applied to general relativity in vielbein - spin-connection formulation and potentially sheds new light on the "issue of time", "emergent space-time" and "Planck-scale physics".

We show how the bosonic spectral action emerges from the fermionic action by the renormalization flow in the presence of a dilaton and the Weyl anomaly. The induced action comes out to be basically the Chamseddine-Connes spectral action introduced in the context of noncommutative geometry. The entire spectral action describes gauge and Higgs fields coupled with gravity. We then consider the effective potential and show, that it has the desired features of a broken and an unbroken phase, with the roll down.

In this talk I will present a generalization of Rieffel's deformation procedure by actions of \(\mathbb R^n\) which can be used to construct spacetimes which are noncommutative only locally, i.e. in a fixed region. As will be explained, models of locally noncommutative spacetimes require the study of unbounded actions on quite generic target spaces, and the necessary framework for the corresponding symbol spaces and their oscillatory integrals will be discussed. First steps towards quantum field theory models on such spacetimes are also indicated. (joint work with Stefan Waldmann)

I will consider the regularization of a gauge quantum field theory based on a Poincare' invariant deformation of the ordinary point-wise product of fields. I show that it yields, through a limiting procedure of the cutoff functions, to a regularized theory, preserving all symmetries at every stage. The new gauge symmetry yields a new Hopf algebra with deformed costructures, which is inequivalent to the standard one.

We demonstrate that the covariance of the algebra of quantum NC fields under quantum- deformed Poincare symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way", in particular braided commutator of deformed field oscillators as well as the braid between the field oscillators and noncommutative Fourier exponentials. As example of braided quantum-deformed NC QFT we describe the NC scalar free fields on noncom- mutative canonical (Moyal-Weyl) space-time with braided c-number field commutator which implies braided locality. Finally we shall comment on kappa-deformed braided free quantum fields

These results were obtained with Mariusz Woronowicz"

We shall address the metric aspect of quantum space, confronting two notions: the quantum length as the spectrum of an operator \(L\) (as in the DFR model) on the one side, Connes' distance formula in noncommutative geometry on the other side. Although these two notions do coincide in the commutative case, there is an obvious discrepancy in the quantum case: the non-zero minimum of the spectrum of \(L\) is interpreted as the emergence of a minimal length, whereas Connes' distance on the state space of the algebra of the quantum spacetime can be as small as desired. We show how to solve this discrepancy by using a natural tool in noncommutative geometry, consisting in doubling the spectral triple. Thanks to some Pythagoras equalities that are proved for the occasion, we show that the DFR quantum length and Connes' distance in a double quantum space coincide exactly on the set of states of optimal localization, and asymptotically (i.e. at high energy) on a larger class of states, including the eigenstates of the quantum harmonic oscillator. On the latest, we interpret the difference between the DFR length and Connes' distance as two distinct ways of integrating the same quantum line element along two different kinds of quantum geodesic.

The aim of the present paper is to present the construction of a general family of \(C^*\)-algebras that includes, as a special case, the "quantum space-time algebra" first introduced by Doplicher, Fredenhagen and Roberts. To this end, we first review, within the \(C^*\)-algebra context, the Weyl-Moyal quantization procedure on a fixed Poisson vector space (a vector space equipped with a given bivector, which may be degenerate). We then show how to extend this construction to a Poisson vector bundle over a general manifold \(M\), giving rise to a \(C^*\)-algebra which is also a module over \(C_0(M)\). Apart from including the original DFR-model, this method yields a "fiberwise quantization" of general Poisson manifolds.

*A first version of this paper cam be found in arXiv:math.OA/1201.1583

I would like to talk about a noncommutative counterpart of Witten's Chern-Simons Theory which I developed in my Ph. D. thesis. As the main result of this thesis I define a proper noncommutative generalization of the Chern-Simons action. "Proper" in this context means an action which is gauge invariant and concurs with the classical action if we return from noncommutative geometry to ordinary differential geometry. In contrast to the classical case, in noncommutative geometry the Chern-Simons action contains a linear term which shifts the vacuum state. Equipped with this denition of a noncommutative Chern-Simons action I developed the notion of a noncommutative Feynman Path Integral. I computed explicetly the Chern-Simons action and the first coefficient of the Taylor expansion of this Feynman Path Integral for the noncommutative 3-torus. Compared with the classical 3-torus everything remains nearly unchanged for the noncommutative counterpart. For this reason I computed explicetly the Chern-Simons action for the Quantum Sphere \(SU_0(2)\), where noncommutative effects really emerge. For the \(SU_0(2)\) the linear part of the Chern-Simons action, which is always zero in the commutative case of ordinary differential geometry, does not vanish. This causes a shift of the vacuum state which makes an explicit computation of the Feynman Path Integral impossible. But despite this fact I was able to prove some interesting properties of the Path Integral in the \(SU_0(2)\) setting.

Basing on recent ideas I shall present some new possibilities of introducing the notion of Noncommutative Riemnnian Geometry for noncommutative spaces. First, which comes from studies of connections on circle bundles makes possible to introduce the generalized metric tensor - for example, for noncommutative tori. The second possibility explores first steps towards producing a version of "twisted" Riemannian geometry, which links geometric construction to (possibly) twisted derivations.

I present a first numerical investigation of a non-commutative gauge theory defined via the spectral action for Moyal space with harmonic propagation. This action is approximated by finite matrices. Using Monte Carlo simulation we study various quantities such as the energy density, the specific heat density and some order parameters, varying the matrix size and the independent parameters of the model. We find a peak structure in the specific heat which might indicate possible phase transitions. However, there are mathematical arguments which show that the limit of infinite matrices is very different from the original spectral model.

The mechanism for emergent gravity on brane solutions in Yang-Mills matrix models is clarified. Newtonian gravity and Ricci-flat 4-dimensional vacuum geometry can arise at least in the linearized regime from the basic matrix model action, without invoking an Einstein-Hilbert- type term. This requires the presence of compactified extra dimensions with extrinsic curvature \(\mathscr M^4 \times \mathscr K \subset\mathbb R^D\) and split noncommutativity, such that the Poisson tensor \(\theta^{ab}\) links the compact with the noncompact directions. The dominant degrees of freedom for gravity are encoded in the moduli of the compactification, which are transmitted to the noncompact directions via the Poisson tensor. The Ricci tensor is sourced by the energy-momentum tensor in a modified way, depending on the specific compactification. The effective Newton constant is determined by the scales of noncommutativity and the compactification, and deviations from general relativity depend on the compactification. This gravity theory is well suited for quantization and argued to be perturbatively finite for the IKKT model, possibly avoiding the cosmological constant problem.

I will present recent results of joint work with Frank Pfäffle in which we derive a formula for the Spectral Action for non-symmetric Dirac operators on 4-dimensional manifolds. Evaluating the heat-trace on left-handed or right-handed spinors we recover the Holst action which is a starting point for Loop Quantum Gravity. The Chamseddine-Connes Dirac operator then couples gravity with matter fields and leads to a geometric interpretation of the Barbero-Immirzi parameter.

We present some recent results on the renormalizability of the spectral action for the Yang-Mills system on a flat 4-dimensional background manifold, focusing on its asymptotic expansion. Interpreting the latter as a higher-derivative gauge theory, a power-counting argument shows that it is superrenormalizable. From BRST-invariance of the one-loop effective action, we conclude that it is actually renormalizable as a gauge theory. If time permits, we will discuss the relation with noncommutative geometry by presenting a more general result on so-called almost commutative manifolds.

We study infrared divergences due to ultraviolet-infrared mixing in quantum field theory on Moyal space with Lorentzian signature in the Yang-Feldman formalism. Concretely, we are considering the \(\phi^4\) and the \(\phi^3\) model in arbitrary even dimension. It turns out that the situation is worse than in the Euclidean setting, in the sense that we find infrared divergences in graphs that are finite there. We briefly discuss the problems one faces when trying to adapt the nonlocal counterterms that render the Euclidean model renormalizable.

Gherardo Piacitelli©2011 (Credits)