Noncommutative Model Research Articles

Quantum Field Theory on a Discrete Space and Noncommutative Geometry

We analyse in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feyman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies.

The Low Energy Limit of the Noncommutative Wess-Zumino Model

The non-commutative Wess-Zumino model is used as a prototype for studying the low energy behaviour of a renormalizable non-commutative field theory. We start by deriving the potential mediating the fermion-fermion and boson-boson interactions in the non-relativistic regime. The quantum counterparts of these potentials are afflicted by irdering ambiguities but we show that there exists an ordering prescription which makes them hermitean. For space/space noncommutativity it turns out that Majorana fermions may be pictured as rods oriented perpendicularly to the direction of motion showing a lack of localituy, while bosons remain insensitive to the effects of noncommutativity. For time/space noncommutativity bosopns and fermions can be regarded as rods oriented along the direction of motion. For both cases of noncommutativity the scattering state described scattered waves, with at least one wave having negative time delay signalizing the underlying nonlocality. The superfield formulation of the model is used to compute the corresponding effective action in the one- and two-loop approximations. In the case of time/space noncommutativity, unitarity is violated in the relativistic regime. However, this does not preclude the existence of the unitary low energy limit.

Low-energy dynamics of noncommutative CP1 solitons in 2+1 dimensions

We investigate the low-energy dynamics of the BPS solitons of the noncommutative CP^1 model in 2+1 dimensions using the moduli space metric of the BPS solitons. We show that the dynamics of a single soliton coincides with that in the commutative model. We find that the singularity in the two-soliton moduli space, which exists in the commutative CP^1 model, disappears in the noncommutative model.We also show that the two-soliton metric has the smooth commutative limit.

THE NONCOMMUTATIVE SUPERSYMMETRIC NONLINEAR SIGMA MODEL

We show that the noncommutativity of space–time destroys the renormalizability of the 1/N expansion of the O(N) Gross–Neveu model. A similar statement holds for the noncommutative nonlinear sigma model. However, we show that, up to the subleading order in 1/N expansion, the noncommutative supersymmetric O(N) nonlinear sigma model becomes renormalizable in D=3. We also show that dynamical mass generation is restored and there is no catastrophic UV/IR mixing. Unlike the commutative case, we find that the Lagrange multiplier fields, which enforce the supersymmetric constraints, are also renormalized. For D=2 the divergence of the four-point function of the basic scalar field, which in D=3 is absent, cannot be eliminated by means of a counterterm having the structure of a Moyal product.

Noncommutativity in string theory and D-branes

By considering the B-field dynamical and studying its interaction with Ramond-Ramond (RR) background we observe the breaking of the B-field gauge symmetry in the effective action. This effect takes place due to non-perturbative coupling of the B-field to membrane topological charge. As a result, the B-field is renormalized in the RR backgrounds, making it impossible to obtain consistent non-commutative models with constant B-field. We argue that the gauge invariance is restored by introducing appropriate external D-brane configuration.

Isotropic representation of the noncommutative 2D harmonic oscillator

We show that a 2D noncommutative harmonic oscillator has an isotropic representation in terms of commutative coordinates. The noncommutativity in the new mode induces energy level splitting and is equivalent to an external magnetic field effect. The equivalence of the spectra of the isotropic and anisotropic representation is traced back to the existence of the SU(2) invariance of the noncommutative model.

Operator regularization and noncommutative Chern–Simons theory

We examine noncommutative Chern–Simons theory using operator regularization. Both the ζ function and the η function are needed to determine one-loop effects. The contributions to these functions coming from the two-point function is evaluated. The U(N) noncommutative model smoothly reduces to the SU(N) commutative model as the noncommutative parameter θμν vanishes.PACS Nos.: 11.10.–z

ON LOW-ENERGY EFFECTIVE ACTION OF NONCOMMUTATIVE HYPERMULTIPLET MODEL

We consider the noncommutative hypermultiplet model within harmonic superspace approach. The one-loop four-point contributions to the effective action of self-interacting q-hypermultiplet are computed. This model has two coupling constants instead of a single one in commutative case. It is shown that both coupling constants are generated by one-loop quantum corrections in the model of q-hypermultiplet interacting with vector multiplet. The holomorphic effective action of q-hypermultiplet in external gauge superfield is calculated. For the fundamental representation there is no uv/ir-mixing and the holomorphic potential is a ⋆-product generalization of a standard commutative one. For the adjoint representation of U(N) gauge group the leading contributions to the holomorphic effective action are given by the terms respecting for the uv/ir-mixing which are related to U(1) phase of U(N) group.

Fuzzy [formula omitted

Regularization of quantum field theories (QFT's) can be achieved by quantizing the underlying manifold (spacetime or spatial slice) thereby replacing it by a non-commutative matrix model or a ``fuzzy manifold''. Such discretization by quantization is remarkably successful in preserving symmetries and topological features, and altogether overcoming the fermion-doubling problem. In this paper, we report on our work on the ``fuzzification'' of the four-dimensional CP2 and its QFT's. CP2 is not spin, but spin${}_c$. Its Dirac operator has many unique features. They are explained and their fuzzy versions are described.

The three-dimensional noncommutative nonlinear sigma model in superspace

We study the superspace formulation of the noncommutative nonlinear supersymmetric O(N) invariant sigma-model in 2+1 dimensions. We prove that the model is renormalizable to all orders of 1/N and explicitly verify that the model is asymptotically free.

Implications of ocular kinematics for the internal updating of visual space.

Recent studies have suggested that during saccades cortical and subcortical representations of visual targets are represented and remapped in retinal coordinates. If this is correct, then the remapping processes must incorporate the noncommutativity of rotations. For example, our three-dimensional (3-D) simulations of the commutative vector-subtraction model of retinocentric remapping predicted centripetal errors in saccade trajectories between "remembered" eccentric targets, whereas our noncommutative model predicted accurate saccades. We tested between these two models in five head-fixed human subjects. Typically, a central fixation light appeared and two peripheral targets were flashed. With all targets extinguished, subjects were required to saccade to the remembered location of one of the peripheral targets and saccade between their remembered locations. Subjects showed minor misestimations of the spatial locations of targets, but failed to show the cumulative pattern of errors predicted by the commutative model. This experiment indicates that if targets are remapped in a retinal frame, then the remapping process also takes the noncommutativity of 3-D eye rotations into account. Unlike other noncommutative aspects of eye rotations that may have mechanical explanations, the noncommutative aspects of this process must be entirely internal.

Superfield effective action in the noncommutative Wess–Zumino model

We introduce the concept of superfield effective action in noncommutative N=1 supersymmetric field theories containing chiral superfields. One and two loops low-energy contributions to the effective action are found for the noncommutative Wess-Zumino model. The one loop Kahlerian effective potential coincides with its commutative counterpart. We show that the two loops nonplanar contributions to the Kahlerian effective potential are leading in the case of small noncommutativity. The structure of the leading chiral corrections to the effective action and the behaviour of the chiral effective potential in the limit of large noncommutativity are also investigated.

Critical behavior of a class of noncommutative aperiodic models on hierarchical lattices

We propose two schemes to construct aperiodic spin models on hierarchical lattices. As in previous works, we choose nearest-neighbor exchange interactions according to a substitution rule. In contrast to these works, however, we are able to deal with Thue–Morse substitutions, a → ab , b → ba , where the geometrical fluctuations are due to the noncommutative nature of the letter sequences. For the Ising case, we show some examples where the critical behavior belongs to the same universality class of the underlying uniform system.

The substructure of space-time and some related issues

Abstract We re-examine space-time in the light of El Naschie's Cantorian set and other discrete space-time models, including the author's own non-commutative model and comment upon issues of metric, dimensionality and other ramifications.

Noncommutative linear sigma models

We examine noncommutative linear sigma models with U(N) global symmetry groups at the one-loop quantum level, and contrast the results with our previous study of the noncommutative O(N) linear sigma models where we have shown that Nambu-Goldstone symmetry realization is inconsistent with continuum renormalization. Specifically we find no violation of Goldstone's theorem at one-loop for the U(N) models with the quartic term ordering consistent with possible noncommutative gauging of the model. The difference is due to terms involving noncommutative commutator interactions, which vanish in the commutative limit. We also examine the U(2), and O(4) linear sigma models with matter in the adjoint representation, and find that the former is consistent with Goldstone's theorem at one-loop if we include only trace invariants consistent with possible noncommutative gauging of the model, while the latter exhibits violations of Goldstone's theorem of the kind seen in the fundamental of O(N) for N>2.

Non-commutative Gross-Neveu model at large N

The non-commutative O(N) Gross-Neveu model is solved in the large N limit in two and three space-time dimensions. The commutative version of the two dimensional model is a renormalizable quantum field theory, both in a coupling constant expansion and an expansion in 1/N. The non-commutative version has a renormalizable coupling constant expansion where ultraviolet divergences can be removed by adjusting counterterms to each order. On the other hand, in a previous work, we showed that the non-commutative theory is not renormalizable in the large N expansion. This is argued to be due to a combined effect of asymptotic freedom and the ultraviolet/infrared mixing that occurs in a non-commutative field theory. In the present paper we will elaborate on this result and extend it to study the large N limit of the three dimensional Gross-Neveu model. We shall see that the large N limit of the three dimensional theory is also trivial when the ultraviolet cutoff is removed.

Large-N matrix models and applications

Abstract String theory is coming to the stage of constructive definition and numerical simulations. In this talk, various candidates for non-pertubative definition of superstring are discussed, and some of the interesting numerical results are introduced. The equivalence between non-commutative field theories and matrix models is also discussed and applied to a few problems.

AXIAL GAUGED NONCOMMUTATIVE U(2)/U(1) WESS–ZUMINO–WITTEN MODEL

We construct various kinds of gauged noncommutative WZW models. In particular, axial gauged noncommutative U (2)/ U (1) WZW model is studied and by integrating out the gauge fields, we obtain a noncommutative nonlinear σ-model.

Gauged noncommutative Wess–Zumino–Witten models

We investigate the Kac-Moody algebra of noncommutative Wess-Zumino-Witten model and find its structure to be the same as the commutative case. Various kinds of gauged noncommutative WZW models are constructed. In particular, noncommutative $U(2)/U(1)$ WZW model is studied and by integrating out the gauge fields, we obtain a noncommutative non-linear $\sigma$-model.

The superfield formalism applied to the noncommutative Wess-Zumino model

We introduce the notion of superoperators on noncommutative R^4 and re-investigate in the framework of superfields the noncommutative Wess-Zumino model as a quantum field theory. In a highly efficient manner we are able to confirm the result that this model is renormalizable to all orders.