13 research outputs found
Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
Families of vector-like deformed relativistic quantum phase spaces and
corresponding realizations are analyzed. Method for general construction of
star product is presented. Corresponding twist, expressed in terms of phase
space coordinates, in Hopf algebroid sense is presented. General linear
realizations are considered and corresponding twists, in terms of momenta and
Poincar\'e-Weyl generators or generators, are constructed
and R-matrix is discussed. Classification of linear realizations leading to
vector-like deformed phase spaces is given. There are 3 types of spaces:
commutative spaces, -Minkowski spaces and -Snyder
spaces. Corresponding star products are associative and commutative (but
non-local), associative and non-commutative and non-associative
and non-commutative, respectively. Twisted symmetry algebras are considered.
Transposed twists and left-right dual algebras are presented. Finally, some
physical applications are discussed.Comment: 20 pages, version accepted for publication in EPJ
Twist for Snyder space
We construct the twist operator for the Snyder space. Our starting point is a
non-associative star product related to a Hermitian realisation of the
noncommutative coordinates originally introduced by Snyder. The corresponding
coproduct of momenta is non-coassociative. The twist is constructed using a
general definition of the star product in terms of a bi-differential operator
in the Hopf algebroid approach. The result is given by a closed analytical
expression. We prove that this twist reproduces the correct coproducts of the
momenta and the Lorentz generators. The twisted Poincar\'{e} symmetry is
described by a non-associative Hopf algebra, while the twisted Lorentz symmetry
is described by the undeformed Hopf algebra. This new twist might be important
in the construction of different types of field theories on Snyder space.Comment: 15 pages, references added, matches published versio
Toward the classification of differential calculi on κ-Minkowski space and related field theories
Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincar\'e Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications
Remarks on simple interpolation between Jordanian twists
In this paper, we propose a simple generalization of the locally r-symmetric
Jordanian twist, resulting in the one-parameter family of Jordanian twists. All
the proposed twists differ by the coboundary twists and produce the same
Jordanian deformation of the corresponding Lie algebra. They all provide the
-Minkowski spacetime commutation relations. Constructions from
noncommutative coordinates to the star product and coproduct, and from the star
product to the coproduct and the twist are presented. The corresponding twist
in the Hopf algebroid approach is given. Our results are presented symbolically
by a diagram relating all of the possible constructions.Comment: 12 page
Noncommutative spaces and Poincaré symmetry
We present a framework which unifies a large class of noncommutative spacetimes that can be described in terms of a deformed Heisenberg algebra. The commutation relations between spacetime coordinates are up to linear order in the coordinates, with structure constants depending on the momenta plus terms depending only on the momenta. The possible implementations of the action of Lorentz transformations on these deformed phase spaces are considered, together with the consistency requirements they introduce. It is found that Lorentz transformations in general act nontrivially on tensor products of momenta. In particular the Lorentz group element which acts on the left and on the right of a composition of two momenta is different, and depends on the momenta involved in the process. We conclude with two representative examples, which illustrate the mentioned effect