Let $\Psi(t)$ be state of Dirac's electron in context of Dirac's equation and consider time-evolution operator $$\Psi(t) = U(t)\Psi(0)$$ is or is not $U$ an unitary (preserving length) operator? (note that we could not simply write $U(t)=e^{-itH}$ and say it is unitary. theorem is converse: if we know that operator in unitary we could represent it in this form. but arbitrary operators don't have this form.)
I mean use of non-free Dirac's equation and examine $\Psi(t)$ has constant length in time? And I really hope the answer be $U$ is "not" unitary. Because I almost sure that even non-second-quantized Dirac's equation must provide some evidence to probability of creation ar annihilation of Dirac's particle.
