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This paper is devoted to detailed investigations of free scalar field theoryon $\kappa$-Minkowski space. After reviewing necessary mathematical tools wediscuss in depth the Lagrangian and solutions of field equations. We analyzethe spacetime symmetries of the model and construct the conserved chargesassociated with translational and Lorentz symmetry. We show that the version ofthe theory usually studied breaks Lorentz invariance in a subtle way: There isan additional trans-Planckian mode present, and an associated conserved charge(the number of such modes) is not a Lorentz scalar.

Field theory on $\kappa$--Minkowski space revisited: Noether charges and breaking of Lorentz symmetry