Showing papers by "Harald Grosse published in 2001"
TL;DR: In this article, the photon self-energy in quantum electrodynamics on noncommutative $\mathbb{R}^4$ is renormalizable to all orders (both in $\theta$ and $\hbar$) when using the Seiberg-Witten map.
Abstract: We show that the photon self-energy in quantum electrodynamics on noncommutative $\mathbb{R}^4$ is renormalizable to all orders (both in $\theta$ and $\hbar$) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the $\theta$-deformed classical action which are necessary to compensate the divergences coming from loop integrations.
127 citations
TL;DR: In this paper, it was shown that the photon self-energy in quantum electrodynamics on noncommutative 4 is renormalizable to all orders (both in θ and ) when using the Seiberg-Witten map.
Abstract: We show that the photon self-energy in quantum electrodynamics on noncommutative 4 is renormalizable to all orders (both in θ and ) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the θ-deformed classical action which are necessary to compensate the divergences coming from loop integrations.
125 citations
TL;DR: In this article, the authors reformulated the concept of connection on a Hopf-Galois extension B⊆P in order to apply it in computing the Chern-Connes pairing between the cyclic cohomology HC 2 n 1 (B) and K 1 n 0 (B).
Abstract: We reformulate the concept of connection on a Hopf–Galois extension B⊆P in order to apply it in computing the Chern–Connes pairing between the cyclic cohomology HC
2
n
(B) and K
0 (B). This reformulation allows us to show that a Hopf–Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz–Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern–Connes pairing for the super line bundles associated to a super Hopf fibration.
89 citations
TL;DR: In this article, the authors studied the q-deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity.
82 citations
TL;DR: In this paper, the feasibility of detecting noncommutative (NC) QED through neutral Higgs boson (H) pair production at linear colliders (LC) was studied.
Abstract: We study the feasibility of detecting noncommutative (NC) QED through neutral Higgs boson (H) pair production at linear colliders (LC). This is based on the assumption that H interacts directly with photon in NCQED as suggested by symmetry considerations and strongly hinted by our previous study on ${\ensuremath{\pi}}^{0}$-photon interactions. We find the following striking features as compared to the standard model (SM) result: (1) generally larger cross sections for an NC scale of order 1 TeV; (2) completely different dependence on initial beam polarizations; (3) distinct distributions in the polar and azimuthal angles; and (4) day-night asymmetry due to the Earth's rotation. These will help to separate NC signals from those in the SM or other new physics at LC. We emphasize the importance of treating properly the Lorentz noninvariance problem and show how the impact of the Earth's rotation can be used as an advantage for our purpose of searching for NC signals.
49 citations
TL;DR: In this paper, it was shown that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations.
Abstract: We show that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The noncommutative Yang-Mills action is invariant under combined conformal transformations of the Yang-Mills field and of the noncommutativity parameter \theta. The Seiberg-Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.
28 citations
TL;DR: In this article, a simple and reasonable generalization of the anomalous interaction between the neutral pion and two photons can induce the C-violating three photon decay in non-commutative quantum electrodynamics.
Abstract: We show that a simple and reasonable generalization of the anomalous interaction between the neutral pion and two photons can induce the C-violating three photon decay of the neutral pion in noncommutative quantum electrodynamics. We find that it is mandatory for consistency reasons to include simultaneously the normal neutral pion and photon interaction in which the neutral pion transforms under U(1) in a similar way as in the adjoint representation of a non-Abelian gauge theory. We demonstrate that the decay has a characteristic distribution although its rate still seems too small to be experimentally reachable in the near future. We also describe how to manipulate phase space integration correctly when Lorentz invariance is lost.
28 citations
TL;DR: In this paper, a simple and reasonable generalization of the anomalous interaction between the neutral pion and two photons can induce the C-violating three photon decay in non-commutative quantum electrodynamics.
23 citations
TL;DR: In this article, the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation based differential calculus were determined using quantized equivariant vector bundles over compact coadjoint orbits.
Abstract: Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation based differential calculus. The associated Chern numbers (topological charges) arise to be non-integer, in the commutative limit the well known integer Chern numbers of the complex line bundles over the 2-sphere are recovered.
21 citations
[...]
09 Jul 2001
TL;DR: In this article, the authors present a series of instanton-like solutions to a matrix model which satisfy a self-duality condition and possess an action whose value is, to within a fixed constant factor, an integer l 2.
Abstract: We present a series of instanton-like solutions to a matrix model which satisfy a self-duality condition and possess an action whose value is, to within a fixed constant factor, an integer l^2. For small values of the dimension n^2 of the matrix algebra the integer resembles the result of a quantization condition but as n -> \infty the ratio l/n can tend to an arbitrary real number between zero and one.
20 citations
TL;DR: In this paper, the authors introduce duality for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories.
Abstract: We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.
TL;DR: In this article, the integrability of two-dimensional spin systems is studied in terms of a simple trialgebraic symmetry, and the authors show that the same trialgebra can be realized on a kind of Fock space of q-oscillators.
Abstract: We suggest that trialgebraic symmetries might be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory.
TL;DR: In this paper, it was shown that the fermions form a spin-1/2 representation of the Lorentz algebra and the covariant splitting of the conformal transformations into a field-dependent part and a \theta-part implies the Seiberg-Witten differential equations for the Fermions.
Abstract: In this letter we apply the methods of our previous paper hep-th/0108045 to noncommutative fermions. We show that the fermions form a spin-1/2 representation of the Lorentz algebra. The covariant splitting of the conformal transformations into a field-dependent part and a \theta-part implies the Seiberg-Witten differential equations for the fermions.
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TL;DR: In this paper, the Chern characters of two projective modules over the fuzzy sphere were determined and the corresponding topological charges (Chern numbers) were calculated, compared to the commutative limit induced by the noncommutative structure of the three coordinates.
Abstract: We determine the Chern characters of two projective modules over the fuzzy sphere and calculate the corresponding topological charges (Chern numbers). These turn out to have corrections - compared to the commutative limit - induced by the noncommutative structure of the three coordinates.
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TL;DR: In this article, a non-commutative deformation of the commutative Hopf algebra of rooted trees is studied, which was shown to be related to the mathematical structure of renormalization in quantum field theories.
Abstract: We study a noncommutative deformation of the commutative Hopf algebra of rooted trees which was shown by Connes and Kreimer to be related to the mathematical structure of renormalization in quantum field theories The requirement of the existence of an antipode for the noncommutative deformation leads to a natural extension of the algebra Noncommutative deformations of the Connes-Kreimer algebra might be relevant for renormalization of field theories on noncommutative spaces and there are indications that in this case the extension of the algebra might be linked to a mixing of infrared and ultraviolet divergences We give also an argument that for a certain class of noncommutative quantum field theories renormalization should be linked to a noncommutative and noncocommutative self-dual Hopf algebra which can be seen as a noncommutative counterpart of the Grothendieck- Teichmueller group
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05 Mar 2001
TL;DR: In this paper, the Chern characters of two projective modules over the fuzzy sphere were determined and the corresponding topological charges (Chern numbers) were calculated, compared to the commutative limit induced by the noncommutative structure of the three coordinates.
Abstract: We determine the Chern characters of two projective modules over the fuzzy sphere and calculate the corresponding topological charges (Chern numbers). These turn out to have corrections - compared to the commutative limit - induced by the noncommutative structure of the three coordinates.
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TL;DR: In this article, the integrability of two-dimensional spin systems was studied in terms of a simple trialgebraic symmetry, and the authors showed that the same trialgebra can be realized on a kind of Fock space of q-oscillators.
Abstract: We suggest that trialgebraic symmetries migth be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory.
TL;DR: In this article, an auxiliary field theory for spinfoams has been proposed to generate spin foam models by using a group manifold and a topological quantum field theory with a 3-form field strength.
Abstract: We use the approach to generate spin foam models by an auxiliary field theory defined on a group manifold (as recently developed in quantum gravity and quantization of BF-theories) in the context of topological quantum field theories with a 3-form field strength. Topological field theories of this kind in seven dimensions are related to the superconformal field theories which live on the worldvolumes of fivebranes in M-theory. The approach through an auxiliary field theory for spinfoams gives a topology independent formulation of such theories.