Can anyone provide me with a rigorous mathematical definition of the fundamental particles (all fundamental bosons and fermions), reflecting the analogy of action of groups with interaction of particles, isospin and mediating forces? An answer would be preferable, but a link of an appropriate source would also suffice.
In your comment:
Physics theoretical models are not just mathematical models, where one starts from axioms and ends up with predictions of the theory. The models are chosen to describe known data, their predictions are checked continuously agains new data . Isospin, particle exchanges etc are all in the lagrangian and quantum field theory tools are used to calculate crossections etc to be checked by future data. Th SM developed over the years after observing the symmetries that existed in the experimental data , for example the eightfold way which led to the proposal of the quarks . The symmetries disclosed by the data led to the SM Lagrangian which was chosen to display these symmetries of SU(3)xSU(2)xU(1) .