# Dirac equation, why not unitary, why not single-particle formalism?

I am reading the first chapter of Akhiezer, Berestetskii QED (1981). They state that Dirac was wrong to assume that the evolution of the wave function is described by

$\psi(t) = e^{-iHt} \psi(t_0)$

They say that in relativity it is also not legit to try to write a single particle formalism; that we only can describe an ensemble of the particles. Why is that?

Actually, why do we assume time reversal invariance in QM at the first place?

• sorry but you seem to be asking three different and independent questions, right? In that case, consider breaking your post up into three different posts. For example, the first question is essentially answered in physics.stackexchange.com/questions/257787; in short: no, Dirac was not wrong, and your first equation is correct. Jan 19, 2017 at 19:20
• @ AccidentalFourierTransform Yes, but I think they should be deeply interconnected on the level of QED axiomatics. Ex., time reversal is the reason for demanding unitarity in QM; QM axiomatics was adopted by Dirac when he was working out his spin-1/2 theory; and his logic was faulty because his theory appeared to be relativistic, where for some reason single particle approach is not applicable. Jan 19, 2017 at 19:26
• @ AccidentalFourierTransform Nobody says that he was wrong in his final results. I am talking about the axiomatics which he used initially and which apparently was wrong. To begin with, he was solving negative energy problem and didn't even have a clue about spin-1/2... Jan 19, 2017 at 19:27
• 1) No, time reversal has nothing to do with unitarity. There are plenty of healthy (read, unitary) quantum theories that break the time-reversal symmetry. 2) What I meant is that "They state that Dirac was not wrong to assume that the evolution of the wave function is described by..." Jan 19, 2017 at 19:28
• 1) see physics.stackexchange.com/questions/65793 and physics.stackexchange.com/questions/191034 2) without the context, I don't know. For example, in QED there are no wave-functions, so maybe it has to do with that. Or maybe because the Hamiltonian is time-dependent, and so the exponential has to be time-ordered. There are many possibilities. Jan 19, 2017 at 19:37

"They say that in relativity it is also not legit to try to write a single particle formalism; that we only can describe an ensemble of the particles. Why is that?"

Because even if initially you have only one particle, extra particles can be created later (say, in external fields).

• Aha, you mean because of the vacuum oscillations? Right, because he was probably still working in QM formalism, not QFT... Thank you! Jan 19, 2017 at 19:34
• The quote is a well-known piece of lore that's entirely untrue. Clearly, in the absence of interactions one can describe single particles just fine. It's less obvious that one can also incorporate interactions in a 1-particle formalism, but that's also possible; see e.g. arxiv.org/abs/hep-ph/0011331 for a review.
– user21299
Jan 19, 2017 at 21:39
• @alexarvanitakis: The Dirac equation, when considered as a single particle equation, typically includes external fields, and in this case pairs can be created, as the Klein paradox shows. Therefore, I stand by my answer. Jan 19, 2017 at 22:58
• @alexarvanitakis what about p.13 of Peskin Shroeder? Wrong? Or maybe I just didn't get your comment... Jan 26, 2017 at 16:16