[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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. DOI:10.1002/prop.200710514 · 2.44 Impact Factor
[Show abstract][Hide abstract] 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.
[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 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.
[Show abstract][Hide abstract] ABSTRACT: A formalism is presented where gauge theories for nonabelian groups can be constructed on a noncommutative algebra.
[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 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; 0712. DOI:10.1088/1126-6708/2007/12/059 · 6.11 Impact Factor
[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 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.
Physical Review D 05/2007; 75(10). DOI:10.1103/PhysRevD.75.105022 · 4.64 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A deformation of the Einstein-Riemann 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 Moyal-Weyl star product is treated explicitly. Finally the deformed theory of gravity is constructed as a theory covariant under deformed diffeomorphisms.
[Show abstract][Hide abstract] 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. DOI:10.1088/1742-6596/53/1/049