Noncommutative QFT and Renormalization

Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to $\phi^3$ models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a $\theta$-deformed space and derive noncommutative gauge field actions.Comment: To appear in the proceedings of the Workshop "Noncommutative Geometry in Field and String Theory", Corfu, 2005 (Greece

Novel Symmetry of Non-Einsteinian Gravity in Two Dimensions

The integrability of $R^2$-gravity with torsion in two dimensions is traced to an ultralocal dynamical symmetry of constraints and momenta in Hamiltonian phase space. It may be interpreted as a quadratically deformed $iso(2,1)$-algebra with the deformation consisting of the Casimir operators of the undeformed algebra. The locally conserved quantity encountered in the explicit solution is identified as an element of the centre of this algebra. Specific contractions of the algebra are related to specific limits of the explicit solutions of this model.Comment: 17 pages, TUW-92-04 (LaTeX

Witten index, axial anomaly, and Krein's spectral shift function in supersymmetric quantum mechanics

A new method is presented to study supersymmetric quantum mechanics. Using relative scattering techniques, basic relations are derived between Krein’s spectral shift function, the Witten index, and the anomaly. The topological invariance of the spectral shift function is discussed. The power of this method is illustrated by treating various models and calculating explicitly the spectral shift function, the Witten index, and the anomaly. In particular, a complete treatment of the two‐dimensional magnetic field problem is given, without assuming that the magnetic flux is quantized

Geometry of the Grosse-Wulkenhaar Model

We define a two-dimensional noncommutative space as a limit of finite-matrix spaces which have space-time dimension three. We show that on such space the Grosse-Wulkenhaar (renormalizable) action has natural interpretation as the action for the scalar field coupled to the curvature. We also discuss a natural generalization to four dimensions.Comment: 16 pages, version accepted in JHE

Renormalisation of \phi^4-theory on noncommutative R^2 in the matrix base

As a first application of our renormalisation group approach to non-local matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean two-dimensional noncommutative \phi^4-theory. It is widely believed that this model is renormalisable in momentum space arguing that there would be logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can indeed be computed to any loop order, the logarithmic UV/IR-divergence appears in the renormalised two-point function -- a hint that the renormalisation is not completed. In particular, it is impossible to define the squared mass as the value of the two-point function at vanishing momentum. In contrast, in our matrix approach the renormalised N-point functions are bounded everywhere and nevertheless rely on adjusting the mass only. We achieve this by introducing into the cut-off model a translation-invariance breaking regulator which is scaled to zero with the removal of the cut-off. The naive treatment without regulator would not lead to a renormalised theory.Comment: 26 pages, 44 figures, LaTe

Spectral noncommutative geometry and quantization: a simple example

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D).Comment: 7 pages, no figure

Mass-symmetry breaking in three-body ions

The ground-state energy of three-body ions $(M^+,M^+,m^-)$ evolves when the like-charge constituents are given different masses. The comparison of $(m_1^+,m_2^+,m^-)$ with the average of $(m_1^+,m_1^+,m^-)$ and $(m_2^+,m_2^+,m^-)$ reveals a competition between the symmetric term and the antisymmetric one. The former dominates in the Born--Oppenheimer regime such as the (p,t,e) case, while the latter wins for H$^-$-like systems with two negative light particles surrounding a heavy nucleus. A comparison is also made with the case of baryons in simple quark models with flavour independence.Comment: 4 pages, 3 figure

Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II

In the first part, we have constructed several families of interacting wedge-local nets of von Neumann algebras. In particular, there has been discovered a family of models based on the endomorphisms of the U(1)-current algebra of Longo-Witten. In this second part, we further investigate endomorphisms and interacting models. The key ingredient is the free massless fermionic net, which contains the U(1)-current net as the fixed point subnet with respect to the U(1) gauge action. Through the restriction to the subnet, we construct a new family of Longo-Witten endomorphisms on the U(1)-current net and accordingly interacting wedge-local nets in two-dimensional spacetime. The U(1)-current net admits the structure of particle numbers and the S-matrices of the models constructed here do mix the spaces with different particle numbers of the bosonic Fock space.Comment: 33 pages, 1 tikz figure. The final version is available under Open Access. CC-B

Perturbative dynamics of fuzzy spheres at large N

We clarify some peculiar aspects of the perturbative expansion around a classical fuzzy-sphere solution in matrix models with a cubic term. While the effective action in the large-N limit is saturated at the one-loop level, we find that the ``one-loop dominance'' does not hold for generic observables due to one-particle reducible diagrams. However, we may exploit the one-loop dominance for the effective action and obtain various observables to all orders from one-loop calculation by simply shifting the center of expansion to the ``quantum solution'', which extremizes the effective action. We confirm the validity of this method by comparison with the direct two-loop calculation and with Monte Carlo results in the 3d Yang-Mills-Chern-Simons matrix model. From the all order result we find that the perturbative expansion has a finite radius of convergence.Comment: 21 pages, 9 figures, (v2) all order analyses added, (v3) some typos correcte

Prediction of regulatory targets of alternative isoforms of the epidermal growth factor receptor in a glioblastoma cell line.

BackgroundThe epidermal growth factor receptor (EGFR) is a major regulator of proliferation in tumor cells. Elevated expression levels of EGFR are associated with prognosis and clinical outcomes of patients in a variety of tumor types. There are at least four splice variants of the mRNA encoding four protein isoforms of EGFR in humans, named I through IV. EGFR isoform I is the full-length protein, whereas isoforms II-IV are shorter protein isoforms. Nevertheless, all EGFR isoforms bind the epidermal growth factor (EGF). Although EGFR is an essential target of long-established and successful tumor therapeutics, the exact function and biomarker potential of alternative EGFR isoforms II-IV are unclear, motivating more in-depth analyses. Hence, we analyzed transcriptome data from glioblastoma cell line SF767 to predict target genes regulated by EGFR isoforms II-IV, but not by EGFR isoform I nor other receptors such as HER2, HER3, or HER4.ResultsWe analyzed the differential expression of potential target genes in a glioblastoma cell line in two nested RNAi experimental conditions and one negative control, contrasting expression with EGF stimulation against expression without EGF stimulation. In one RNAi experiment, we selectively knocked down EGFR splice variant I, while in the other we knocked down all four EGFR splice variants, so the associated effects of EGFR II-IV knock-down can only be inferred indirectly. For this type of nested experimental design, we developed a two-step bioinformatics approach based on the Bayesian Information Criterion for predicting putative target genes of EGFR isoforms II-IV. Finally, we experimentally validated a set of six putative target genes, and we found that qPCR validations confirmed the predictions in all cases.ConclusionsBy performing RNAi experiments for three poorly investigated EGFR isoforms, we were able to successfully predict 1140 putative target genes specifically regulated by EGFR isoforms II-IV using the developed Bayesian Gene Selection Criterion (BGSC) approach. This approach is easily utilizable for the analysis of data of other nested experimental designs, and we provide an implementation in R that is easily adaptable to similar data or experimental designs together with all raw datasets used in this study in the BGSC repository, https://github.com/GrosseLab/BGSC