Towards quantum noncommutative $\kappa$-deformed field theory

We introduce a new $\kappa$-star product describing the multiplication of quantized $\kappa$-deformed free fields. The $\kappa$ deformation of local free quantum fields originates from two sources: noncommutativity of space-time and the $\kappa$ deformation of field oscillators algebra; we relate these two deformations. We demonstrate that for a suitable choice of $\kappa$-deformed field oscillators algebra, the $\kappa$-deformed version of the microcausality condition is satisfied, and it leads to the deformation of the Pauli-Jordan commutation function defined by the $\kappa$-deformed mass shell. We show by constructing the $\kappa$-deformed Fock space that the use of the $\kappa$-deformed oscillator algebra permits one to preserve the bosonic statistics of $n$-particle states. The proposed star product is extended to the product of $n$ fields, which for $n=4$ defines the interaction vertex in perturbative description of the noncommutative quantum $\lambda {\phi }^4$ field theory. It appears that the classical four-momentum conservation law is satisfied at the interaction vertices.