Regularization of 2d supersymmetric Yang-Mills theory via non commutative geometry

The non commutative geometry is a possible framework to regularize Quantum Field Theory in a nonperturbative way. This idea is an extension of the lattice approximation by non commutativity that allows to preserve symmetries. The supersymmetric version is also studied and more precisely in the case of the Schwinger model on supersphere [14]. This paper is a generalization of this latter work to more general gauge groups

Operator identities in q-deformed Clifford analysis

In this paper, we define a q-deformation of the Dirac operator as a generalization of the one dimensional q-derivative. This is done in the abstract setting of radial algebra. This leads to a q-Dirac operator in Clifford analysis. The q-integration on R(m), for which the q-Dirac operator satisfies Stokes' formula, is defined. The orthogonal q-Clifford-Hermite polynomials for this integration are briefly studied

A New Noncommutative Product on the Fuzzy Two-Sphere Corresponding to the Unitary Representation of SU(2) and the Seiberg-Witten Map

We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a unitary representation. This is done by constructing a star product, $\star_{\lambda}$, for an arbitrary representation of SU(2) which depends on a continuous parameter $\lambda$ and searching for the values of $\lambda$ which give unitary representations. We will find two series of values: $\lambda = \lambda^{(A)}_j=1/(2j)$ and $\lambda=\lambda^{(B)}_j =-1/(2j+2)$, where j is the spin of the representation of SU(2). At $\lambda = \lambda^{(A)}_j$ the new star product $\star_{\lambda}$ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order $\ell \leq 2j$ and then $\star_{\lambda}$ reduces to the star product $\star$ obtained by Preusnajder. The star product at $\lambda=\lambda^{(B)}_j$, to be denoted by $\bullet$, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order $\ell \leq 2j$. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg-Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg-Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point $\lambda=0$.Comment: 15 pages, typos corrected, references added, a note adde

Arthroscopy or ultrasound in undergraduate anatomy education: a randomized cross-over controlled trial

Background: The exponential growth of image-based diagnostic and minimally invasive interventions requires a detailed three-dimensional anatomical knowledge and increases the demand towards the undergraduate anatomical curriculum. This randomized controlled trial investigates whether musculoskeletal ultrasound (MSUS) or arthroscopic methods can increase the anatomical knowledge uptake. Methods: Second-year medical students were randomly allocated to three groups. In addition to the compulsory dissection course, the ultrasound group (MSUS) was taught by eight, didactically and professionally trained, experienced student-teachers and the arthroscopy group (ASK) was taught by eight experienced physicians. The control group (CON) acquired the anatomical knowledge only via the dissection course. Exposure (MSUS and ASK) took place in two separate lessons (75 minutes each, shoulder and knee joint) and introduced standard scan planes using a 10-MHz ultrasound system as well as arthroscopy tutorials at a simulator combined with video tutorials. The theoretical anatomic learning outcomes were tested using a multiple-choice questionnaire (MCQ), and after cross-over an objective structured clinical examination (OSCE). Differences in student's perceptions were evaluated using Likert scale-based items. Results: The ASK-group (n = 70, age 23.4 (20--36) yrs.) performed moderately better in the anatomical MC exam in comparison to the MSUS-group (n = 84, age 24.2 (20--53) yrs.) and the CON-group (n = 88, 22.8 (20--33) yrs.; p = 0.019). After an additional arthroscopy teaching 1 % of students failed the MC exam, in contrast to 10 % in the MSUS- or CON-group, respectively. The benefit of the ASK module was limited to the shoulder area (p < 0.001). The final examination (OSCE) showed no significant differences between any of the groups with good overall performances. In the evaluation, the students certified the arthroscopic tutorial a greater advantage concerning anatomical skills with higher spatial imagination in comparison to the ultrasound tutorial (p = 0.002; p < 0.001). Conclusions: The additional implementation of arthroscopy tutorials to the dissection course during the undergraduate anatomy training is profitable and attractive to students with respect to complex joint anatomy. Simultaneous teaching of basic-skills in musculoskeletal ultrasound should be performed by medical experts, but seems to be inferior to the arthroscopic 2D-3D-transformation, and is regarded by students as more difficult to learn. Although arthroscopy and ultrasound teaching do not have a major effect on learning joint anatomy, they have the potency to raise the interest in surgery

Introduction to the AfricaGEO 2018 Conference Special Issue of the South African Journal of Geomatics

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Examples of q-regularization

An Introduction to Hopf algebras as a tool for the regularization of relavent quantities in quantum field theory is given. We deform algebraic spaces by introducing q as a regulator of a non-commutative and non-cocommutative Hopf algebra. Relevant quantities are finite provided q\neq 1 and diverge in the limit q\rightarrow 1. We discuss q-regularization on different q-deformed spaces for \lambda\phi^4 theory as example to illustrate the idea.Comment: 17 pages, LaTex, to be published in IJTP 1995.1

Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator

It is shown that the local axial anomaly in $2-$dimensions emerges naturally if one postulates an underlying noncommutative fuzzy structure of spacetime . In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown to contain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$ axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant expansion of the quark propagator in the form $\frac{1}{{\cal D}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where $a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ is the covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effective Dirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$ but differes from it on the UV modes. Most remarkably is the fact that both operators share the same limit and thus the above covariant expansion is not available in the continuum theory . The first bit in this expansion $\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuum limit, its contribution to the anomaly is exactly the canonical theta term. The contribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other hand equal to the toplogical Chern-Simons action which in two dimensions vanishes identically .Comment: 26 pages, latex fil

Nonpointlike Particles in Harmonic Oscillators

Quantum mechanics ordinarily describes particles as being pointlike, in the sense that the uncertainty $\Delta x$ can, in principle, be made arbitrarily small. It has been shown that suitable correction terms to the canonical commutation relations induce a finite lower bound to spatial localisation. Here, we perturbatively calculate the corrections to the energy levels of an in this sense nonpointlike particle in isotropic harmonic oscillators. Apart from a special case the degeneracy of the energy levels is removed.Comment: LaTeX, 9 pages, 1 figure included via epsf optio

Gauge Theory on Fuzzy S^2 x S^2 and Regularization on Noncommutative R^4

We define U(n) gauge theory on fuzzy S^2_N x S^2_N as a multi-matrix model, which reduces to ordinary Yang-Mills theory on S^2 x S^2 in the commutative limit N -> infinity. The model can be used as a regularization of gauge theory on noncommutative R^4_\theta in a particular scaling limit, which is studied in detail. We also find topologically non-trivial U(1) solutions, which reduce to the known "fluxon" solutions in the limit of R^4_\theta, reproducing their full moduli space. Other solutions which can be interpreted as 2-dimensional branes are also found. The quantization of the model is defined non-perturbatively in terms of a path integral which is finite. A gauge-fixed BRST-invariant action is given as well. Fermions in the fundamental representation of the gauge group are included using a formulation based on SO(6), by defining a fuzzy Dirac operator which reduces to the standard Dirac operator on S^2 x S^2 in the commutative limit. The chirality operator and Weyl spinors are also introduced.Comment: 39 pages. V2-4: References added, typos fixe

The Fuzzy Ginsparg-Wilson Algebra: A Solution of the Fermion Doubling Problem

The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly.Comment: references added, typos corrected, section 4.2 simplified. Report.no: SU-4252-769, DFUP-02-1