By combining relativity and quantum mechanics, we have uncovered two new phenomenon:
We could continue and develop a theory of spinless particles ( theory). This would enable us to calculate cross-sections for the scattering of spinless particles from spinless particles, and the decay of spinless particles to spinless particles. Also, nothing prevents us from trying to construct a theory for scalar (or pseudoscaler) bosons interacting with the electromagnetic field. The theory would be called ``scalar electrodynamics''. The importance of this theory is limited because there are no elementary charged scalar particles in nature. We will postpone the development of interactions until later.
The best candidates for the role of pseudoscalar mesons are the and K. They are unstable and decay by weak interactions. Since this lifetime is very long on a natural time scale, the pion can be considered stable to a good approximation. The more basic problem is that pions have an internal structure. It is well known that mesons are regarded as being composed of two quarks with spin 1/2. Scalar electrodynamics is completely inadequate for describing the coupling of mesons with each other, because the coupling is dominated by the strong interaction.
Many problems arise from attempting to apply a simple single-particle wave function picture to what is obviously a many-body situation. The correct way in which to handle all the subtlety of these problems is to use the formalism of quantum field theory. Nevertheless, the elementary wave function paradigm has allowed us to obtain an accurate sketch of the physics of spineless particles within the limitations of a one-particle theory.