Q-deformed Minkowski Space Research Articles

ANALYSIS ON q-DEFORMED QUANTUM SPACES

A q-deformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. The subject is presented in a rather complete and self-contained way. All relevant notions are introduced and explained in detail. The different possibilities to realize the objects of q-deformed analysis are discussed and their elementary properties are studied. In this manner attention is focused on star products, q-deformed tensor products, q-deformed translations, q-deformed partial derivatives, dual pairings, q-deformed exponentials, and q-deformed integration. The main concern of this work is to show that these objects fit together in a consistent framework, which is suitable to formulate physical theories on quantum spaces.

Superanalysis on quantum spaces

Attention is focused on antisymmetrised versions of quantum spaces that are of particular importance in physics, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each of these quantum spaces we provide q-analogs for elements of superanalysis, i.e. Grassmann integrals, Grassmann exponentials, Grassmann translations and braided products with supernumbers.

Grassmann variables on quantum spaces

Attention is focused on antisymmetrized versions of quantum spaces that are of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each case standard techniques for dealing with q-deformed Grassmann variables are developed. Formulae for multiplying supernumbers are given. The actions of symmetry generators and fermionic derivatives upon antisymmetrized quantum spaces are calculated. The complete Hopf structure for all types of quantum space generators is written down. From the formulae for the coproduct a realization of the L-matrices in terms of symmetry generators can be read off. The L-matrices together with the action of symmetry generators determine how quantum spaces of different type have to be fused together.

Q-Exponentials on quantum spaces

We present explicit formulae for q-exponentials on quantum spaces which could be of particular importance in physics, i.e. the q-deformed Minkowski-space and the q-deformed Euclidean space with two, three or four dimensions. Furthermore, these formulae can be viewed as 2-, 3- or 4-dimensional analogues of the well-known q-exponential function.

Q-Integration on quantum spaces

In this article we present explicit formulae for q-integration on quantum spaces which could be of particular importance in physics, i.e., q-deformed Minkowski space and q-deformed Euclidean space in three or four dimensions. Furthermore, our formulae can be regarded as a generalization of Jackson’s q-integral to three or four dimensions and provide a new possibility for an integration over the whole space being invariant under translations and rotations.

Operator representations on quantum spaces

In this article we present explicit formulae for q-differentiation on quantum spaces which could be of particular importance in physics, i.e., q-deformed Minkowski space and q-deformed Euclidean space in three or four dimensions. The calculations are based on the covariant differential calculus of these quantum spaces. Furthermore, our formulae can be regarded as a generalization of Jackson's q-derivative to three and four dimensions.

*-products on quantum spaces

In this paper we present explicit formulas for the *-product on quantum spaces which are of particular importance in physics, i.e., the q-deformed Minkowski space and the q-deformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation parameter q. In addition, we worked out an expansion in powers of $h={\rm ln} q$ up to second order, for all considered cases.

Q-deformed Lorentz-algebra in Minkowski phase space

In the present paper we show that the Lorentz algebra \(\cal L\) as defined in [5] is isomorphic to an algebra \(\hat{\cal U}\) closely related to a q-deformed \(SU_q(2) \otimes SU_q(2)\) algebra. On this algebra \(\hat{\cal U}\) we define a Hopf algebra structure and show its action on q-spinor modules. This algebra is related to the q-deformed Minkowski space algebra by a non invertible factorisation.

Q-Deformed Minkowski Space based on a q-Lorentz Algebra

The Hilbert space representations of a non-commutative q-deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone.

A QUANTUM LORENTZ GROUP

We examine the properties of the quantum Lorentz group SO q(3, 1) using the [Formula: see text] matrix given in Ref. 14. We show that this matrix together with the q-deformed metric C provide a representation of a BWM algebra. Using the projection operators which decompose the [Formula: see text] matrix into irreducible components, we give the general definition of the corresponding quantum space, i.e. the q-deformed Minkowski space and the q-deformed Clifford algebra. We also construct the q analog of Dirac matrices and show that they form a matrix representation of the q-deformed Clifford algebra.