560 research outputs found
Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
We employ new calculational technique and present complete list of classical
-matrices for complex homogeneous orthogonal Lie algebra
, the rotational symmetry of four-dimensional
complex space-time. Further applying reality conditions we obtain the classical
-matrices for all possible real forms of :
Euclidean , Lorentz , Kleinian
and quaternionic Lie algebras.
For we get known four classical Lorentz -matrices,
but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide
new results and mention some applications.Comment: 13 pages; typos corrected. v3 matches version published in PL
Quantum deformations of Euclidean, Lorentz, Kleinian and quaternionic symmetries in unified setting -- Addendum
In our previous paper we obtained a full classification of nonequivalent
quasitriangular quantum deformations for the complex Euclidean Lie
symmetry . The result was presented in the form of
a list consisting of three three-parameter, one two-parameter and one
one-parameter nonisomorphic classical -matrices which provide 'directions'
of the nonequivalent quantizations of . Applying
reality conditions to the complex -matrices we
obtained the nonisomorphic classical -matrices for all possible real forms
of : Euclidean , Lorentz
, Kleinian and quaternionic
Lie algebras. In the case of and
real symmetries these -matrices give the full
classifications of the inequivalent quasitriangular quantum deformations,
however for and the
classifications are not full. In this paper we complete these classifications
by adding three new three-parameter -real -matrices and
one new three-parameter -real -matrix. All
nonisomorphic classical -matrices for all real forms of
are presented in the explicite form what is
convenient for providing the quantizations. We will mention also some
applications of our results to the deformations of space-time symmetries and
string -models.Comment: 10 pages. We supplement results of our previous paper by adding new
and -matrices needed for the
complete classification of real classical -matrices for all four real
forms of $\mathfrak{o}(4;\mathbb{C})
Bicrossproduct construction versus Weyl-Heisenberg algebra
We are focused on detailed analysis of the Weyl-Heisenberg algebra in the
framework of bicrossproduct construction. We argue that however it is not
possible to introduce full bialgebra structure in this case, it is possible to
introduce non-counital bialgebra counterpart of this construction. Some remarks
concerning bicrossproduct basis for kappa-Poincare Hopf algebra are also
presented.Comment: 11 pages, contribution to the proceedings of the 7th International
Conference on Quantum Theory and Symmetries (QTS7), 7-13 August 2011, Prague,
Czech Republi
Once again about quantum deformations of D=4 Lorentz algebra: twistings of q-deformation
This paper together with the previous one (arXiv:hep-th/0604146) presents the
detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf
algebra in terms of complex and real generators. We describe here in detail two
quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of
the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation
an Abelian twist depending on Cartan generators of o(3,1) is used. The second
example of twisting provides a quantum deformation of Cremmer-Gervais type for
the Lorentz algebra. For completeness we describe also twisting of the Lorentz
algebra by standard Jordanian twist. By twist quantization techniques we obtain
for these deformations new explicit formulae for the deformed coproducts and
antipodes of the o(3,1)-generators.Comment: 17 page
Scalar field propagation in the phi^4 kappa-Minkowski model
In this article we use the noncommutative (NC) kappa-Minkowski phi^4 model
based on the kappa-deformed star product, ({*}_h). The action is modified by
expanding up to linear order in the kappa-deformation parameter a, producing an
effective model on commutative spacetime. For the computation of the tadpole
diagram contributions to the scalar field propagation/self-energy, we
anticipate that statistics on the kappa-Minkowski is specifically
kappa-deformed. Thus our prescription in fact represents hybrid approach
between standard quantum field theory (QFT) and NCQFT on the kappa-deformed
Minkowski spacetime, resulting in a kappa-effective model. The propagation is
analyzed in the framework of the two-point Green's function for low,
intermediate, and for the Planckian propagation energies, respectively.
Semiclassical/hybrid behavior of the first order quantum correction do show up
due to the kappa-deformed momentum conservation law. For low energies, the
dependence of the tadpole contribution on the deformation parameter a drops out
completely, while for Planckian energies, it tends to a fixed finite value. The
mass term of the scalar field is shifted and these shifts are very different at
different propagation energies. At the Planckian energies we obtain the
direction dependent kappa-modified dispersion relations. Thus our
kappa-effective model for the massive scalar field shows a birefringence
effect.Comment: 23 pages, 2 figures; To be published in JHEP. Minor typos corrected.
Shorter version of the paper arXiv:1107.236
Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit
We describe in detail two-parameter nonstandard quantum deformation of D=4
Lorentz algebra , linked with Jordanian deformation of
. Using twist quantization technique we obtain
the explicit formulae for the deformed coproducts and antipodes. Further
extending the considered deformation to the D=4 Poincar\'{e} algebra we obtain
a new Hopf-algebraic deformation of four-dimensional relativistic symmetries
with dimensionless deformation parameter. Finally, we interpret
as the D=3 de-Sitter algebra and calculate the contraction
limit ( -- de-Sitter radius) providing explicit Hopf algebra
structure for the quantum deformation of the D=3 Poincar\'{e} algebra (with
masslike deformation parameters), which is the two-parameter light-cone
-deformation of the D=3 Poincar\'{e} symmetry.Comment: 13 pages, no figure
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