Publications (110)262.19 Total impact

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ABSTRACT: In this chapter we discuss another type of noncommutative space, the κdeformed space. It is an example of Lie algebra type of deformation of the usual commutative space. In the first part derivatives and the symmetry of this space are discussed. We start with the abstract algebra of operators and using the *\star product approach represent everything on the space of commuting coordinates. In the second part we describe how to construct noncommutative gauge theory on this space using the Seiberg–Witten approach. 
Chapter: Noncommutative Spaces
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ABSTRACT: In this chapter we present some of the basic concepts needed to describe noncommutative spaces and their topological and geometrical features. We therefore complement the previous chapters where noncommutative spaces have been described by the commutation relations of their coordinates. The full algebraic description of ordinary (commutative) spaces requires the completion of the algebra of coordinates into a C*C^\star algebra, this encodes the Hausdorff topology of the space. The smooth manifold structure is next encoded in a subalgebra (of “smooth” functions). Relaxing the requirement of commutativity of the algebra opens the way to the definition of noncommutative spaces, which in some cases can be a deformation of an ordinary space. A powerful method to study these noncommutative algebras is to represent them as operators on a Hilbert space. We discuss the noncommutative space generated by two noncommuting variables with a constant commutator. This is the space of the noncommutative field theories described in this book, as well as the elementary phase space of quantum mechanics. The Weyl map from operators to functions is introduced in order to produce a *\star product description of this noncommutative space.  [Show abstract] [Hide abstract]
ABSTRACT: In this chapter, led by examples, we introduce the notions of Hopf algebra and quantum group. We study their geometry and in particular their Lie algebra (of left invariant vector fields). The examples of the quantum sl(2)sl(2) Lie algebra and of the quantum (twisted) Poincaré Lie algebra isoq(3,1)iso_\theta(3,1) are presented. 
Chapter: Einstein Gravity on Deformed Spaces
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ABSTRACT: A differential calculus, differential geometry, and the Einstein gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the deformed gravity is based on a deformation of the Hopf algebra.  [Show abstract] [Hide abstract]
ABSTRACT: Deformed gauge transformations on deformed coordinate spaces are considered for any Lie algebra. The representation theory of this gauge group forces us to work in a deformed Lie algebra as well. This deformation rests on a twisted Hopf algebra, thus we can represent a twisted Hopf algebra on deformed spaces. That leads to the construction of Lagrangian invariant under a twisted Lie algebra.  [Show abstract] [Hide abstract]
ABSTRACT: Spacetime geometry is twisted (deformed) into noncommutative spacetime geometry, where functions and tensors are now star multiplied. Consistently, spacetime diffeomorphisms are twisted into noncommutative diffeomorphisms. Their deformed Lie algebra structure and that of infinitesimal Poincaré transformations is defined and explicitly constructed. We can then define covariant derivatives (that implement the principle of general covariance on noncommutative spacetime) and torsion and curvature tensors. With these geometric tools we formulate a noncommutative theory of gravity.  [Show abstract] [Hide abstract]
ABSTRACT: Julius Wess first work on noncommutative geometry dates June 1989. Since then he gradually became more and more interested and involved in this research field. We would like to describe briefly his interests, motivations, and main contributions, which could be divided into four periods. Therefore, we shall trace a short account of his last 18 years of scientific activity and hence of an approach to the subject that has become a reference point for the scientific community.  [Show abstract] [Hide abstract]
ABSTRACT: Twist deformations of spacetime lead to deformed field theories with twisted symmetries. Twisted symmetries are quantum group symmetries. Most integrable spin systems have dynamical symmetries related to appropriate quantum groups. We discuss the changes of the properties of these systems under twist transformations of quantum groups. A main example is the isotropic Heisenberg spin chain and the jordanian twist of the universal enveloping algebra of sl(2)sl(2) . It is shown that the spectrum of the XXX label XXX model spin chain is preserved under the twist deformation while the structure of the eigenstates depends on the choice of boundary conditions. Another example is provided by abelian twists, these give physical deformations of closed spin chains corresponding to higher rank Lie algebras, e.g., gl(n)gl(n) . The energy spectrum of these integrable models is changed and correspondingly their eigenvectors.  [Show abstract] [Hide abstract]
ABSTRACT: In this chapter we discuss two possible ways of introducing gauge theories on noncommutative spaces. In the first approach the algebra of gauge transformations is unchanged, but the Leibniz rule is changed (compared with gauge theories on commutative space). Consistency of the equations of motion requires enveloping algebravalued gauge fields, which leads to new degrees of freedom. In the second approach we have to go to the enveloping algebra again if we want noncommutative gauge transformations to close in the algebra. However, no new degrees of freedom appear here because of the Seiberg–Witten map. This map enables one to express noncommutative gauge parameters and fields in terms of the corresponding commutative variables. 
Chapter: Deformed Gauge Theories
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ABSTRACT: Gauge theories are studied on a space of functions with the Moyal product. The development of these ideas follows the differential geometry of the usual gauge theories, but several changes are forced upon us. The Leibniz rule has to be changed such that the theory is now based on a twisted Hopf algebra. Nevertheless, this twisted symmetry structure leads to conservation laws. The symmetry has to be extended from Lie algebra valued to enveloping algebra valued and new vector potentials have to be introduced. As usual, field equations are subjected to consistency conditions that restrict the possible models. Some examples are studied. 
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ABSTRACT: We discuss a deformation of the Hopf algebra of supersymmetry (SUSY) transformations based on a special choice of a twist. As usual, algebra itself remains unchanged, but the comultiplication changes. This leads to a deformed Leibniz rule for SUSY transformations. Superfields are multiplied by using a ☆product which is noncommutative, hermitian and finite when expanded in power series of the deformation parameter. One possible deformation of the WessZumino action is proposed and analysed in detail. Differently from most of the literature concerning this subject, we work in Minkowski spacetime.  [Show abstract] [Hide abstract]
ABSTRACT: We have asked how the Heisenberg relations of space and time change if we replace the Lorentz group by a qdeformed Lorentz group (Lorek et al. 1997). By the Heisenberg relations we mean: $ {*{20}c} {X^a X^b = X^b X^a , } \\ {P^a P^b = P^b P^a , } \\ {X^a P^b = P^b X^a + i\eta ^{ab} .} \\ $ \begin{array}{*{20}c} {X^a X^b = X^b X^a , } \\ {P^a P^b = P^b P^a , } \\ {X^a P^b = P^b X^a + i\eta ^{ab} .} \\ \end{array} ((1)) The indices a; b run from 0 to 3, 0 being the time component, ηab is the Lorentz metric. This relation is covariant under the Lorentz group, X a and P a are four vectors, that is representations or equivalently modules of the Lorentz group. The relations are compatible with an involution $ \overline {X^a } = X^a , \overline {P^b } = P^b . $ \overline {X^a } = X^a , \overline {P^b } = P^b . ((2)) Dividing the free algebra generated by X a, P b by the ideal generated by the relations (1) we obtain an algebra.  [Show abstract] [Hide abstract]
ABSTRACT: A formalism is presented where gauge theories for nonabelian groups can be constructed on a noncommutative algebra. 
Chapter: q Deformed Heisenberg Algebras
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ABSTRACT: This lecture consists of two sections. In section 1 we consider the simplest version of a qdeformed Heisenberg algebra as an example of a noncommutative structure. We first derive a calculus entirely based on the algebra and then formulate laws of physics based on this calculus. Then we realize that an interpretation of these laws is only possible if we study representations of the algebra and adopt the quantum mechanical scheme. It turns out that observables like position or momentum have discrete eigenvalues and thus space gets a latticelike structure.  [Show abstract] [Hide abstract]
ABSTRACT: We discuss a deformation of the Hopf algebra of supersymmetry (SUSY) transformations based on a special choice of twist. As usual, algebra itself remains unchanged, but the comultiplication changes. This leads to the deformed Leibniz rule for SUSY transformations. Superfields are elements of the algebra of functions of the usual supercoordinates. Elements of this algebra are multiplied by using a $\star$product which is noncommutative, hermitian and finite when expanded in power series of the deformation parameter. Chiral fields are no longer a subalgebra of the algebra of superfields. One possible deformation of the WessZumino action is proposed and analysed in detail. Differently from most of the literature concerning this subject, we work in Minkowski spacetime. Comment: 23 pages, no figure, minor changed, refs. added  [Show abstract] [Hide abstract]
ABSTRACT: We explore some general consequences of a proper, full enforcement of the “twisted Poincaré” covariance of Chaichian et al., Wess, Koch et al., and Oeckl upon manyparticle quantum mechanics and field quantization on a MoyalWeyl noncommutative space(time). This entails the associated braided tensor product with an involutive braiding (or ⋆tensor product in the parlance of Aschieri et al.) prescription for any coordinate pair of x, y generating two different copies of the space(time); the associated nontrivial commutation relations between them imply that xy is central and its Poincaré transformation properties remain undeformed. As a consequence, in quantum field theory (QFT) (even with spacetime noncommutativity) one can reproduce notions (like spacelike separation, time and normalordering, Wightman or Green’s functions, etc.), impose constraints (Wightman axioms), and construct free or interacting theories which essentially coincide with the undeformed ones, since the only observable quantities involve coordinate differences. In other words, one may thus well realize quantum mechanics (QM) and QFT’s where the effect of space(time) noncommutativity amounts to a practically unobservable common noncommutative translation of all reference frames. 
Article: Deformed Gravity
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ABSTRACT: A deformation of the EinsteinRimann gravity theory is presented in this talk. Deformation is based on the deformation of the product of functions, the star product. A short introduction to this deformation algorithm is first presented. It is shown how gauge groups and the algebra of diffeomorphisms can be deformed in the general setting of twisted star product. In this lecture only the MoyalWeyl star product is treated explicitely. Finally the deformed theory of gravity is constructed as a theory covariant under deformed diffeomorphisms.  [Show abstract] [Hide abstract]
ABSTRACT: A differential calculus, differential geometry and the EinsteinRiemann Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the deformed gravity is based on a deformation of the Hopf algebra. See also P. Aschieri et al., Classical Quantum Gravity 22, No. 17, 3511–3532 (2005; Zbl 1129.83011), ibid. 23, No. 6, 1883–1911 (2006; Zbl 1091.83022).
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Institutions

19912009

University Hospital München
München, Bavaria, Germany 
Max Planck Institute for Astrophysics
Arching, Bavaria, Germany


20062008

University of Hamburg
 I. Institut für Theoretische Physik
Hamburg, Hamburg, Germany


19972008

LudwigMaximiliansUniversity of Munich
München, Bavaria, Germany


19912006

Max Planck Institute for Physics
München, Bavaria, Germany


2005

University of Belgrade
 Faculty of Physics
Beograd, Central Serbia, Serbia


2004

Technische Universität München
München, Bavaria, Germany


2001

HumboldtUniversität zu Berlin
 Department of Physics
Berlín, Berlin, Germany


19711984

CERN
 Physics Department (PH)
Genève, Geneva, Switzerland


1979

University of Geneva
Genève, Geneva, Switzerland


19681969

CUNY Graduate Center
New York, New York, United States
