SCALAR FIELD THEORY ON κ-MINKOWSKI SPACE–TIME AND TRANSLATION AND LORENTZ INVARIANCE

We investigate the properties of κ-Minkowski space–time by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg–Weyl algebra. The deformed algebra consists of κ-Poincaré algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an advantage of this approach to consistently construct a star product, which has a property that under integration sign, it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal but not for κ-Minkowski space–time. This star product also has generalized trace and cyclic properties, and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and requiring it to be Hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachyonic modes and basically of the very same form. The issue of Lorentz invariance of the theory is also discussed.