$\kappa$-Poincar\'e-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory

We discuss the obstruction to the construction of a multiparticle field theory on a $\kappa$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $\kappa$-Poincar\'e group. This necessitates a braided tensor product. We study the representations of this product, and prove that $\kappa$-Poincar\'e-invariant N-point functions belong to an Abelian subalgebra, and are therefore commutative. We use this construction to define the 2-point Whightman and Pauli--Jordan functions, which turn out to be identical to the undeformed ones. We finally outline how to construct a free scalar $\kappa$-Poincar\'e-invariant quantum field theory, and identify some open problems.