When Dirac found his equation for the electron $(-i\gamma^\mu\partial_\mu+m)\psi=0$ he famously discovered that it had negative energy solutions. In order to solve the problem of the stability of the ground state of the electron he invoked Pauli's exclusion principle and postulated that negative energy states well already filled by a "sea" of electrons. This allowed him to predict the positron, viewed as a hole in the sea.
This interpretation was ultimately discarded owing to its inapplicability to bosons and difficulties with explaining the invisibility of the infinite charge of the sea.
According to my understanding, the modern argument goes something like this. There is a discrete symmetry of the lagrangian called charge conjugation $\psi \rightarrow \psi^c$ which allows the negative energy solutions to be interpreted as positive energy solutions for a second mode of excitation of the electron field with opposite charge, called positrons. The decay of electrons to positrons is then suppressed by the $U(1)$ gauge symmetry of the lagrangian forcing conservation of electrical charge.
According to this interpretation What Dirac would have missed was the lagrangian formalism. Is this historically and physically correct?
Answer
Dirac's derivation of the existence of positrons that you described was a totally legitimate and solid argument and Dirac rightfully received a Nobel prize for this derivation.
As you correctly say, the same "sea" argument depending on Pauli's exclusion principle isn't really working for bosons. Modern QFT textbooks want to present fermions and bosons in a unified language which is why they mostly avoid the "Dirac sea" argument. But this fact doesn't make it invalid.
The infinite potential charge of the Dirac sea is unphysical. In reality, one should admit that he doesn't know what the charge of the "true vacuum" is. So there's an unknown additive shift in the quantity $Q$ and of course that the right additive choice is the choice that implies that the physical vacuum $|0\rangle$ (with the Dirac sea, i.e. with the negative-energy electron states fully occupied) carries $Q=0$. The right choice of the additive shift is a part of renormalization and the choice $Q=0$ is also one that respects the ${\mathbb Z}_2$ symmetry between electrons and positrons.
It is bizarre to say that Dirac missed the Lagrangian formalism. Dirac was the main founding father of quantum mechanics who emphasized the role of the Lagrangian in quantum mechanics. That's also why Dirac was the author of the first steps that ultimately led to Feynman's path integrals, the approach to quantum mechanics that makes the importance of the Lagrangian in quantum mechanics manifest.
It would be more accurate to say that Dirac didn't understand (and opposed) renormalization so he couldn't possibly formulate the right proof of the existence of the positrons etc. that would also correctly deal with the counterterms and similar things. Still, he had everything he needed to define a consistent theory at the level of precision that was available to him (ignoring renormalization of loop corrections): he just subtracted the right (infinite) additive constant from $Q$ by hand.
Your sentence
The decay of electrons to positrons is then supressed by the U(1) gauge symmetry of the lagrangian forcing conservation of electrical charge.
is strange. Since the beginning – in fact, since the 19th century – the U(1) gauge symmetry was a part of all formulations of electromagnetic theories. It has been a working part of Dirac's theory from the very beginning, too.
The additive shift in $Q$, $Q=Q_0+\dots$, doesn't change anything about the U(1) transformation rules for any fields because they're given by commutators of the fields with $Q$ and the commutator of a $c$-number such as $Q_0$ with anything vanishes: $Q_0$ is completely inconsequential for the U(1) transformation rules. All these facts were known to Dirac, too. The fact that the U(1) gauge symmetry was respected was the reason that there has never been anything such as a "decay of electrons to positrons" in Dirac's theory, not even in its earliest versions.
An electron can't decay to a positron because that would violate charge conservation while the charge has always been conserved. For historical reasons, one could mention that unlike Dirac, some other physicists were confused about these elementary facts such as the separation of 1-electron state and 1-positron states in different superselection sectors. In particular, Schrödinger proposed a completely wrong theory of "Zitterbewegung" (trembling motion) which was supposed to be a very fast vibration caused by the interference between the positive-energy and negative-energy solutions. However, there's never such interference in the reality because the actual states corresponding to these solutions carry different values of the electric charge. Their belonging to different superselection sectors is the reason why the interference between them can't ever be physically observed. The "Zitterbewegung" is completely unphysical.
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