Ever since I learned about the Dirac equation as an undergraduate student, it bothered me that it was introduced - both in my course and many others - on the back of concluding that the time component of the conserved current density $j^0$ for the Klein-Gordon equation was not positive definite, and so could not be a probability density in the way that might be hoped for a relativistic upgrade of the Schrödinger equation, only for this issue to be immediately thrown away altogether by then jumping to quantum field theory.

There are plenty of other posts that touch on the existence of this positive-definite Dirac $j^0$, such as here, here and here.

Having read the introductions of Weinberg's books on quantum mechanics and QFT Volume 1, I now realise that this strange feature of my course was a product of history. Dirac originally looked for a relativistic wave equation which had a positive-definite $j^0$. At the time, Dirac concluded that the lack of a positive-definite $j^0$ for the K-G equation really meant that relativistic particles could not be described by it.

Other issues that still plagued this new equation, such as dealing with the negative energy states, eventually led to the invention of QFT. There's also this post which points out that the interpretation of the Dirac spinors as quantum states is inherently problematic.

With all of this in mind, is it therefore just a coincidence that Dirac's original reasoning led to the discovery of an equation describing relativistic particles? In other words, with the benefit of hindsight, was Dirac's original requirement that $j^0$ be positive-definite justified or important?

Although I believe one could construct a reason for why it does mathematically, from the answer under this post and the relationship between the positive-definiteness of $j^0$ and the equation being first-order in time rather than second-order (EDIT: also see the answer below from J.G.), this doesn't seem to actually have any bearing at all on the second quantisation procedure that emerged soon after historically, or indeed on QFT in general.


1 Answer 1


Since the Dirac equation improves our perspective on $E<0$ solutions (thereby predicting spin-$1/2$ electrons and positrons), a less historical motivation would make it look like antimatter and spin are put in by fiat, as the latter is in pre-relativistic QM. But this can leave a student with your impression:

is it therefore just a coincidence that the Dirac equation has a positive-definite $j^0$ with respect to its discovery as an equation describing relativistic particles?

The free-particle TDSE $i\partial_t\psi=m\psi$ (where we interpret mass as energy to anticipate relativity) should have a Lorentz-invariant generalization of the form $i\gamma^\mu\partial_\mu\psi=m\psi$, from which we can prove $j^\mu:=\psi^\dagger\gamma^0\gamma^\mu\psi$ is a conserved $4$-current, with $j^0=\psi^\dagger\psi\ge0$. You're essentially asking whether this theorem is "coincidental". I'd say no, because we're generalizing a Galilean-invariant equation that was already known to imply a $\rho\ge0$ continuity equation, rather than starting from the relativity-inspired KGE. In other words, we don't have to pass through a $\partial_t^2$ experiment that would doom us to $E^2=\cdots\implies j^0\propto E=\pm\sqrt{\cdots}$ from the start.

  • $\begingroup$ I'll update my post if I can work out how to improve its wording. Your answer is useful for seeing more plainly why it isn't a coincidence that it was possible for Dirac to find an equation yielding a positive-definite $j^0$, but in hindsight, is Dirac's original reasoning justifiable, i.e. is it actually important that $j^0$ is positive-definite at all, or was it a "coincidence" that this unjustifiable reasoning allowed him to find his equation? $\endgroup$ Dec 23, 2021 at 12:25
  • $\begingroup$ @turbodiesel4598 Where physicists really started is the KGE, but it couldn't deal with probability as the TDSE had. I'd say Dirac was right to expect that should be fixed, as he didn't know about antimatter yet. In some courses, we start from "force $E>0$" rather than "force $j^0>0$". This makes his work easier to mandate, as we want ground states, and can discuss his sea proposal. $\endgroup$
    – J.G.
    Dec 23, 2021 at 12:40
  • $\begingroup$ Maybe a hypothetical scenario would allow me to express my confusion a little better. Contrary to the argument you've given in your answer, let's imagine the Dirac equation did not have a positive-definite $j^0$. Would Dirac conclude that relativistic quantum mechanics was doomed? If it was an important requirement to him, maybe he would, and yet we know from the KG equation's validity for describing scalars that it isn't actually a make-or-break property. $\endgroup$ Jan 13, 2022 at 16:10
  • $\begingroup$ @turbodiesel4598 Your question amounts to, "how would Dirac have reacted to his equation having the same pros & cons as the KGE has?" Well, scalar equations with negative-norm solutions lead to all sorts of very interesting QFT complications, so with any luck Dirac would react with the same problem-solving that went into addressing those in real life. (Birrell & Davies is a good reference for this.) $\endgroup$
    – J.G.
    Jan 13, 2022 at 16:54
  • $\begingroup$ Thanks for all the replies, think I have a better understanding of the matter now :) $\endgroup$ Jan 13, 2022 at 17:27

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