  We consider the projected SO(5) bosonic model introduced in order to connect
the SO(5) theory of high-T$_c$ superconductivity with the physics of the
Mott-insulating gap, and derive the corresponding effective functional
describing low-energy degrees of freedom. At the
antiferromagnetic-superconducting transition, SO(5) symmetry-breaking effects
due to the gap are purely quantum mechanical and become irrelevant in the
neighborhood of a possible finite-temperature multicritical point separating
the normal from the antiferromagnetic and the superconducting phases. A
difference in the magnon and hole-pair mobility always takes the system away
from the SO(5)-symmetric fixed point towards a region of instability, and the
phase transition between the normal and the two ordered phases becomes first
order before merging into the antiferromagnetic-superconducting line. Quantum
fluctuations at intermediate temperatures, while introducing symmetry-breaking
terms in the case of equal mobilities, tend to cancel the symmetry-breaking
effects in the case of different mobilities.
