The continuum limit of the non-commutative lambda phi^4 model

We present a numerical study of the \lambda \phi^{4} model in three Euclidean dimensions, where the two spatial coordinates are non-commutative (NC). We first show the explicit phase diagram of this model on a lattice. The ordered regime splits into a phase of uniform order and a ``striped phase''. Then we discuss the dispersion relation, which allows us to introduce a dimensionful lattice spacing. Thus we can study a double scaling limit to zero lattice spacing and infinite volume, which keeps the non-commutativity parameter constant. The dispersion relation in the disordered phase stabilizes in this limit, which represents a non-perturbative renormalization. From its shape we infer that the striped phase persists in the continuum, and we observe UV/IR mixing as a non-perturbative effect.Comment: 3 pages, 3 figures, talk presented by W.B. at the 11th Regional Conference on Mathematical Physics, Tehran, May 3-6, 200

Developing an understanding of the nature of science : a case study exploring the potential of implicit approaches in open learning situations using chemistry teaching as an example

Schülern gezielt Wege naturwissenschaftlicher Erkenntnisgewinnung bewusst zu machen, ist spätestens seit der Veröffentlichung der Bildungsstandards wieder stärker in den Blick geraten. Leider wird naturwissenschaftlich orientiertes Arbeiten aber nicht selten auf die systematische Durchführung einer Forschungshandlung reduziert, bei der naturwissenschaftliche Prozesse wie \u27Beobachten\u27 oder \u27Schlussfolgerungen ziehen\u27 rezeptartig angewandt werden müssen. Ein sinnvoller Einsatz naturwissenschaftlicher Prozesse erfordert jedoch neben konkreten Inhalten sowie einer Kenntnis der Prozesse selbst auch immer eine Methodologie, also eine Reflexion über naturwissenschaftliche Methoden. Dieses Meta-Wissen umfasst weder in Lernstandserhebungen prüfbare Kenntnisse noch das \u27korrekte\u27 Anwenden naturwissenschaftlicher Prozesse, sondern Wertvorstellungen und erkenntnistheoretische Grundannahmen, auf denen naturwissenschaftliches Wissen und naturwissenschaftliche Tätigkeiten basieren. Ohne ein solches Meta-Wissen ist weder die Entwicklung eines naturwissenschaftlichen Problembewusstseins oder gar einer naturwissenschaftlichen Beurteilungskompetenz, noch eine Anerkennung der Leistungen der Naturwissenschaften möglich. Während sich für dieses Wissen über die \u27Geartetheit\u27 der Naturwissenschaften im deutschen Sprachraum die Wendung \u27Natur der Naturwissenschaften\u27 durchzusetzen scheint, spricht man im internationalen Diskurs von nature of science (NOS). Bei der derzeitigen Fokussierung auf messbaren, unterrichtlichen Output werden Möglichkeiten eines Wissenserwerbs über NOS nur randständig beachtet. In der vorliegenden Fallstudie werden zwei Wege zum Erwerb von Wissen über nature of science in praxisgerechten, geöffneten Lehr-Lern-Arrangements vergleichend untersucht: Einerseits ein rein impliziter Weg, bei dem SchülerInnen in der Rolle naturwissenschaftlicher Forscher agieren (open inquiry), indem sie die Wirkungsweise einer Babywindel naturwissenschaftlich untersuchen. Andererseits ein implizit-vorstrukturierter Weg, der in Form eines Gruppenpuzzles die historische Entwicklung unterschiedlicher Säure-Base-Konzeptionen ex post als Sprungbrett für eine Reflexion über Naturwissenschaften nutzt. Das Potenzial beider Wege wird anhand einer explorativen Fallstudie mit insgesamt 60 SchülerInnen der Jahrgangsstufe 11 ausgelotet. Als Erhebungsinstrument kommt eine adaptierte Fassung des Fragebogens "VNOS-C" (Lederman et al. 2002) zum Einsatz. Forschungsleitende Fragestellungen sind dabei: - Welche Vorstellungen über NOS bringen die beforschten Schüler mit in den Unterricht? - Welches Potenzial weist ein bezogen auf NOS implizit-vorstrukturiertes Lernarrangement beim Erwerb von Wissen über nature of science auf, bei dem Schüler eine eigenständige Rekonstruktion historischer und explizit dargelegter Forschungsprozesse leisten müssen? - Welche Effekte zeigt ein lerntheoretisch attraktiver Weg eines impliziten Erfahrungslernens auf Schülervorstellungen zum Bereich NOS, bei dem Schüler in einer Forscherrolle einem relativ offenen naturwissenschaftlichen Problem (open inquiry) nachgehen müssen?Due to the current focus on the measurable output of teaching lessons, opportunities to learn about the nature of science (NOS) are often sidelined. In the present case study, two ways of learning about NOS are compared and examined in practice-oriented, open learning situations. One follows a purely implicit approach (open inquiry) with 11th grade students acting in the role of scientists investigating the functionality of babies\u27 diapers; the other uses an "implicit pre-structured" approach in a jigsaw classroom by reconstructing the historical development of different acid-base definitions as a springboard for students\u27 reflections about science processes and the nature of science afterwards. The potential of both teaching methods was explored in a case study with 60 students using an adapted version of the questionnaire VNOS-C (Lederman et al. 2002). The study explored in detail the following questions: ? What preconceptions about the NOS did the students bring to the classroom? ? What is the potential of "implicit pre-structured" learning situations in which students reconstruct historical research processes, in helping them to understand the nature of science? ? What effects does an open inquiry have on the students’ ideas of NOS

Scalar fields in a non-commutative space

We discuss the lambda phi**4 model in 2- and 3-dimensional non-commutative spaces. The mapping onto a Hermitian matrix model enables its non-perturbative investigation by Monte Carlo simulations. The numerical results reveal a phase where stripe patterns dominate. In d=3 we show that in this phase the dispersion relation is deformed in the IR regime, in agreement with the property of UV/IR mixing. This "striped phase" also occurs in d=2. For both dimensions we provide evidence that it persists in the simultaneous limit to the continuum and to infinite volume ("Double Scaling Limit"). This implies the spontaneous breaking of translation symmetry.Comment: 15 pages, LaTex, 13 figures, 1 table, talk presented at the XIV Mexican Workshop on Particles and Fields, Oaxaca, November 201

Optimally Stabilized PET Image Denoising Using Trilateral Filtering

Low-resolution and signal-dependent noise distribution in positron emission tomography (PET) images makes denoising process an inevitable step prior to qualitative and quantitative image analysis tasks. Conventional PET denoising methods either over-smooth small-sized structures due to resolution limitation or make incorrect assumptions about the noise characteristics. Therefore, clinically important quantitative information may be corrupted. To address these challenges, we introduced a novel approach to remove signal-dependent noise in the PET images where the noise distribution was considered as Poisson-Gaussian mixed. Meanwhile, the generalized Anscombe's transformation (GAT) was used to stabilize varying nature of the PET noise. Other than noise stabilization, it is also desirable for the noise removal filter to preserve the boundaries of the structures while smoothing the noisy regions. Indeed, it is important to avoid significant loss of quantitative information such as standard uptake value (SUV)-based metrics as well as metabolic lesion volume. To satisfy all these properties, we extended bilateral filtering method into trilateral filtering through multiscaling and optimal Gaussianization process. The proposed method was tested on more than 50 PET-CT images from various patients having different cancers and achieved the superior performance compared to the widely used denoising techniques in the literature.Comment: 8 pages, 3 figures; to appear in the Lecture Notes in Computer Science (MICCAI 2014

Numerical results on the Non-commutative \lambda \phi^4 Model

The UV/IR mixing in the \lambda \phi^4 model on a non-commutative (NC) space leads to new predictions in perturbation theory, including Hartree-Fock type approximations. Among them there is a changed phase diagram and an unusual behavior of the correlation functions. In particular this mixing leads to a deformation of the dispersion relation. We present numerical results for these effects in d=3 with two NC coordinates.Comment: 3 pages, 6 figures, Talk presented at Lattice2003(theory

Non--Commutative Field Theories beyond Perturbation Theory

We investigate two models in non-commutative (NC) field theory by means of Monte Carlo simulations. Even if we start from the Euclidean lattice formulation, such simulations are only feasible after mapping the systems onto dimensionally reduced matrix models. Using this technique, we measure Wilson loops in 2d NC gauge theory of rank 1. It turns out that they are non-perturbatively renormalizable, and the phase follows an Aharonov-Bohm effect if we identify \theta = 1/B. Next we study the 3d \lambda \phi^{4} model with two NC coordinates, where we present new results for the correlators and the dispersion relation. We further reveal the explicit phase diagram. The ordered regime splits into a uniform and a striped phase, as it was qualitatively conjectured before. We also confirm the recent observation by Ambjorn and Catterall that such stripes occur even in d=2, although they imply the spontaneous breaking of translation symmetry. However, in d=3 and d=2 we observe only patterns of two stripes to be stable in the range of parameters investigated.Comment: 8 pages, 8 figures, talk presented at 35th Ahrenshoop Symposiu

Field Theory Simulations on a Fuzzy Sphere - an Alternative to the Lattice

We explore a new way to simulate quantum field theory, without introducing a spatial lattice. As a pilot study we apply this method to the 3d \lambda \phi^4 model. The regularisation consists of a fuzzy sphere with radius R for the two spatial directions, plus a discrete Euclidean time. The fuzzy sphere approximates the algebra of functions of the sphere with a matrix algebra, and the scalar field is represented by a Hermitian N x N matrix at each time site. We evaluate the phase diagram, where we find a disordered phase and an ordered regime, which splits into phases of uniform and non-uniform order. We discuss the behaviour of the model in different limits of large N and R, which lead to a commutative or to a non-commutative \lambda \phi^4 model in flat space.Comment: 6 pages, 3 figres, talk presented at LATTICE0