J. Wess

University of Hamburg, Hamburg, Hamburg, Germany

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Publications (104)237.06 Total impact

  • International Journal of Modern Physics B 01/2012; 13(24n25). · 0.46 Impact Factor
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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.
    07/2009: pages 23-37;
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    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.
    07/2009: pages 3-21;
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    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.
    07/2009: pages 111-132;
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    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.
    07/2009: pages 167-190;
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    ABSTRACT: In this chapter we discuss another type of noncommutative space, the κ-deformed 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.
    07/2009: pages 73-85;
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    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.
    07/2009: pages 133-164;
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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.
    07/2009: pages 39-52;
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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.
    07/2009: pages 89-109;
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    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.
    07/2009: pages 191-197;
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    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.
    07/2009: pages 53-72;
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    ABSTRACT: In the first nine chapters, this book records approach to non-commutative differential geometry and its applications to physics developed over the past few years by the group led by Julius Wess. The final chapter summarises main themes of non-commutative geometry pursued by Julius Wess from 1989 until his death in 2007. The main theme of the book is physics and symmetries of spaces deformed by the twist or the *-product deformations (Moyal deformation of algebras of functions). The book is split into two parts. The first part is devoted to physical aspects of deformed field theory (chapters 1–5), the second part is devoted to foundations and applications of non-commutative geometries (chapters 6–9). The first three chapters, authored by Julius Wess, initially describe the *-product, differential calculus, differential operators and gauge transformations on deformed spaces. Subsequently gauge theory and the Einstein gravity are developed on deformed spaces. Chapters 4 and 5 are authored by Marija Dimitrijević and compare twisted gauge theory developed in Chapters 1–3 with the Seiberg-Witten approach to non-commutative field theory, and also describe gauge theory on the κ-deformed space. The second part starts with Chapter 6, written by Fedele Lizzi. This chapter is devoted to foundations of non-commutative geometry, and first describes the algebras of functions on classical spaces and then their deformation in terms of the Moyal *-product. Chapters 7 and 8 are written by Paolo Aschieri and start with an introduction to quantum groups (with particular attention paid to the deformations of Hopf algebras by twists) and culminate with the description of the twisted Poincaré algebra and the *-deformation of Einstein’s gravity. Chapter 9, authored by Petr Kulish, provides one with a brief introduction to the Quantum Inverse Scattering Method (of solving integrable systems in quantum mechanics, in particular spin chains), and then analyses twists of integrable spin chains. Some versions of some of the material presented in the book have been published elsewhere. The authors write: “Chapters 1–3 are, respectively, based on [J. Wess, Gen. Relativ. Gravitation 39, No. 8, 1121–1134 (2007; Zbl 1181.83144), J. Phys. Conf. Ser. 53, 752 (2006), and SIGMA, Symmetry Integrability Geom. Methods Appl. 2, Paper 089, 9 pages, electronic only (2006; Zbl 1188.83069)]. A preliminary version of Chaps. 4 and 5, originally coauthored with F. Meyer, appeared, respectively, in the proceedings of the 4th and 3rd Summer School in Modern Mathematical Physics, Belgrade and Zlatibor, published in SFIN, Institute of Physics, Zemun (Serbia). Chapter 7 is an extended version of contribution for the proceedings of the Second Modave Summer School in Mathematical Physics, International Solvay Institutes. Chapter 8 is an extended version of J. Phys. Conf. Ser. 53, 799 (2006).”
    Lecture Notes in Physics 01/2009;
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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 Wess-Zumino action is proposed and analysed in detail. Differently from most of the literature concerning this subject, we work in Minkowski space-time.
    Fortschritte der Physik 04/2008; 56(4-5):418-423. · 0.98 Impact Factor
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    J. Wess
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    ABSTRACT: We have asked how the Heisenberg relations of space and time change if we replace the Lorentz group by a q-deformed 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.
    01/2008: pages 227-234;
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    Julius Wess
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    ABSTRACT: A formalism is presented where gauge theories for nonabelian groups can be constructed on a noncommutative algebra.
    01/2008: pages 320-332;
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    Julius Wess
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    ABSTRACT: This lecture consists of two sections. In section 1 we consider the simplest version of a q-deformed 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 lattice-like structure.
    01/2008: pages 311-382;
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    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 Wess-Zumino action is proposed and analysed in detail. Differently from most of the literature concerning this subject, we work in Minkowski space-time. Comment: 23 pages, no figure, minor changed, refs. added
    Journal of High Energy Physics 10/2007; · 5.62 Impact Factor
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    Gaetano Fiore, Julius Wess
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    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 many-particle quantum mechanics and field quantization on a Moyal-Weyl 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 x-y is central and its Poincaré transformation properties remain undeformed. As a consequence, in quantum field theory (QFT) (even with space-time noncommutativity) one can reproduce notions (like spacelike separation, time- and normal-ordering, 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.
    Phys. Rev. D. 05/2007; 75(10).
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    Julius Wess
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    ABSTRACT: We consider a formalism by which gauge theories can be constructed on noncommutative space time structures. The coordinates are supposed to form an algebra, restricted by certain requirements that allow us to realise the algebra in terms of star products. In this formulation it is useful to define derivatives and to extend the algebra of coordinates by these derivatives. The elements of this extended algebra are deformed differential operators. We then show that there is a morphism between these deformed differential operators and the usual higher order differential operators acting on functions of commuting coordinates. In this way we obtain deformed gauge transformations and a deformed version of the algebra of diffeomorphisms. The deformation of these algebras can be clearly seen in the category of Hopf algebras. The comultiplication will be twisted. These twisted algebras can be realised on noncommutative spaces and allow the construction of deformed gauge theories and deformed gravity theory.
    General Relativity and Gravitation 01/2007; 39(8):1121-1134. · 1.90 Impact Factor
  • Julius Wess
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    ABSTRACT: Gauge theories are studied on a space of functions with the Moyal-Weyl 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.
    Journal of Physics Conference Series 11/2006; 53(1):752-763.

Publication Stats

8k Citations
237.06 Total Impact Points

Institutions

  • 1978–2008
    • University of Hamburg
      • I. Institut für Theoretische Physik
      Hamburg, Hamburg, Germany
  • 1991–2007
    • University Hospital München
      München, Bavaria, Germany
    • CSU Mentor
      Long Beach, California, United States
    • University of California, Berkeley
      • Department of Physics
      Berkeley, CA, United States
  • 2005
    • University of Belgrade
      • Faculty of Physics
      Beograd, Central Serbia, Serbia
  • 2001
    • Technische Universität München
      München, Bavaria, Germany
    • Humboldt-Universität zu Berlin
      • Department of Physics
      Berlin, Land Berlin, Germany
  • 1997–2001
    • Ludwig-Maximilians-University of Munich
      München, Bavaria, Germany
  • 1999–2000
    • Université Paris-Sud 11
      • Laboratoire de Physique Théorique d'Orsay
      Paris, Ile-de-France, France
  • 1994–2000
    • Max Planck Institute of Physics
      München, Bavaria, Germany
  • 1971–1991
    • CERN
      • Physics Department (PH)
      Genève, Geneva, Switzerland
  • 1980–1984
    • Karlsruhe Institute of Technology
      • Institute of Theoretical Physics
      Karlsruhe, Baden-Wuerttemberg, Germany
  • 1968
    • Harvard University
      Cambridge, Massachusetts, United States