Also a physicist. It's not quite right to call basis vectors particles, even in a free field theory. The particles correspond better to the creation / ladder

Also a physicist. It's not quite right to call basis vectors particles, even in a free field theory. The particles correspond better to the creation / ladder operators that take you from one Hamiltonian eigenstate to another. In a perturbative interacting case, people still think of particles as these same ladder operators, but they don't connect eigenstates so simply (the interactions generically mix all the states with the same quantum numbers). In a strongly interacting case the story is even more subtle, because composite operators may be closer to the ladder operators between the asymptotic states, even though they're built of other... particles? Language isn't great in this instance.