Let's suppose we start with an energy eigenstate of the Hamiltonian of a relativistic system. This energy eigenstate could be in a degenerate subspace. Generically, evolution of the Schrodinger equation will cause the quantum state to evolve and become some kind of superposition of all the energy eigenstates within this subspace.

Then what can happen if you start with a quantum particle of mass $m$ with a given energy $E > 2mc^2$ (which is generically true for relativistic processes where energies are much larger than the mass), is that the degenerate subspace of energy eigestates will include states where both the particle and its antiparticle are present. If no conservation law or other obstruction prevents the original "particle-only" state from evolving to include a superposition of this new "particle+antiparticle" state, then the Schrodinger equation will generically cause the state to evolve to be a superposition with a non-zero coefficient with the "particle+antiparticle" state.

In more flowery language, "anything not forbidden is mandatory" in quantum mechanics. The quantum state will generically evolve into a superposition over every basis state it can consistent with the constraints of unitary Schrodinger evolution.