A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a non-perturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space. Bilocal fields arising in the harmonic decomposition of the operator product expansion prove to be a powerful instrument in exploring the field content. In particular, in the theory of a field of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scalar field contribution to the 6-point function of the twist 2 bilocal field is analyzed with the aim to separate the free field part from the nontrivial part.