κ-Minkowski spacetime and the star product realizations

Abstract

We investigate a Lie algebra-type κ-deformed Minkowski spacetime with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of κ-Minkowski space. The coproduct and the star product corresponding to each of them are found. An explicit connection between realizations and orderings is established and the relation between the coproduct and the star product, provided through an exponential map, is proved. Utilizing the properties of the natural realization, we construct a scalar field theory on κ-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space. This result is universal and does not depend on the realizations, i.e. orderings used.

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

“…In [14,17] a universal formula was given for the representation of Lie algebra generators as formal power series of the corresponding structure constants with the coefficients in Bernoulli numbers. In [18] the above procedure was applied to obtain the closed formula of the polydifferential representation for the κ-Minkowski Lie algebra on R d . Here we will follow the same approach.…”

“…the function X (t) is given by 18) and the expression X i l −θ 2 M /2 is understood as a function of the matrix M in the sense of power series,…”

On κ-deformation and triangular quasibialgebra structure

“…This paper is a continuation of earlier work [6,7], following [8], whose aim is systematically to construct κ-deformed quantum field theory from the following particular perspective. (Other approaches can be found in [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].) We recall the viewpoint on quantum field theory taken by Weinberg in [25], namely that quantum field theory takes the form it does because this is essentially the only way to construct a quantum mechanical theory of point particles with Poincaré symmetry -given only a very limited number of additional physical principles, like cluster decomposition.…”

Covariant particle exchange for κ-deformed theories in dimensions

“…Given the existence of this deformation, one would like to know how much of relativistic physics carries over to the κ-deformed case; in particular, there has been much interest in understanding κ-deformed quantum field theory [6,7,8,9,10]. Beyond the intrinsic mathematical appeal 3 , a physical motivation for this interest is that κ-Poincaré is known to be a symmetry of the kinematics (i.e.…”