In analogy to non-relativistic quantum field theory where the Galilei invariant plane t = const. is used for a description of bound (and scattering) states we take the hyperboloid x 2 = const. which is invariant under homogeneous Lorentz transformations. We give the appropriate complete set of basis functions for the expansion of any scalar and spin 1 2 field on the hyperboloid. The free Klein Gordon and Dirac fields are discussed in detail. In this formulation a multi-particle system can be described by a Lorentz covariant wave function.