# Why didn't the Klein-Gordon equation suggest antimatter like the Dirac equation did?

I have heard the story that the Dirac equation suggested the existence of antimatter due to the existence of negative energy solutions. The Klein-Gordon equation also has negative energy solutions. Therefore does it not also suggest the existence of antimatter?

(I know historically it was Dirac who proposed antimatter using his equation, but would the same argument work for the Klein-Gordon equation?)

In the beginnings of quantum theory, people were looking at the K-G and the Dirac equation as equations for wave functions (or at least something similar that would give them a probability density like the non-relativistic wavefunctions did) - the notion of a "quantum field" did not yet exist.

As an equation for such (generalized) wavefunctions, the K-G equation is rather obvious "nonsense" - not only does it have "negative energy solutions", but its solutions also produce negative probability densities (see e.g. this answer by gented). So negative energy solutions to the K-G equations weren't really hinting at antiparticles, since everyone knew the solutions to the K-G equation didn't produce meaningful quantum states anyway.

In contrast, the Dirac equation as a first-order equation gives solutions with positive probability densities, so its solutions can be interpreted as defining quantum states, and so its negative energy solutions "suggest" antiparticles.

• While this is largely just a question of definitions, I would disagree that the K-G equation implies negative probability densities. In nonrelativistic QM, the probability density is $\rho := |\psi(x)|^2$, the probability (3-)current is ${\bf j} := Im[\psi^* \nabla \psi]/m$, and they satisfy the continuity equation $\nabla \cdot {\bf j} + \partial \rho/\partial t = 0$. In relativistic QM, these expressions aren't covariant, so you have two choices for the probability density: you can keep it as $|\phi(x)|^2$, which preserves positive definiteness and the Born statistics but isn't Lorentz ... Mar 27 at 22:52
• ... covariant, or you can redefine it to be $Im[\phi^* \dot{\phi}]/m$, which transforms with the spatial probability current as a conserved four-vector, but is neither positive-definite nor respects the usual Born statistics. IMO, the former option is much more natural, not the latter option as you have assumed. From this perspective, the probability with the naive interpretation of a solution of the KG equation as a wavefunction isn't that it leads to negative probability densities, but that it isn't Lorentz-covariant. Mar 27 at 22:55
• This answer may well reflect why at the time the Klein-Gordon equation was considered wrong. This opinion is widespread even today. However, the KG equation is far from wrong. It is the wave mechanics equivalent of $E^2=m^2c^4+p^2c^2$ and free matter should therefore obey it. The free Dirac solutions also obey KG. For the hydrogen atom the KG and Dirac energies have the same form: only where the Dirac energy has $j={\cal l} \pm s$, the KG energy has $\cal l$. Compare Itzykson&Zuber, p72, Eqs 2.86 and 2.87. Also the Schrödinger equation is the nonrelativistic limit of the KG equation. Mar 29 at 13:49
• Continued. When the proper spin term is added to the KG equation it gives the same answers a the Dirac equation. In view of the high score on this answer it must be foreseen that the underappreciation of the KG equation will continue. Mar 29 at 13:55
• @ColinMacLaurin The squared Dirac equation has the form of the KG equation extended with the relativistic extension of the Pauli spin term. See Itzykson&Zuber. Mar 30 at 22:19

The reason why Dirac suggested that the negative energy solutions are related to antimatter is a little more complicated. By proposing that the negative energy states are all filled with electrons, it solved the problem of not having an electron in a cascade effect in which it continues to fall down energy levels to -infinity. Electrons obey the Pauli exclusion principle since they are fermions, so since the negative energy states are all filled, positive energy electron cannot occupy it, unless there was a “hole” in the negative energy states, in which the hole is interpreted as an anti-electron (not the electron itself). The Klein Gordon equation describes a spin-0 boson, so it is not subject to the Pauli exclusion principle. Ergo, proposing that the negative energy solutions were all occupied did not solve the problem of spin-0 particles falling down energy levels.

• I'm not sure this is correct - you need the spin-statistics theorem for the claim that spin-0 objects are necessarily bosons, and that's a feature of relativistic QFT. When we're just looking at the equations as equations for potential wavefunctions, why would you assume that the solutions to the KG equation cannot be subject to the exclusion principle? Mar 27 at 2:02
• @ACuriousMind This objection does not hold. Pauli formulated the exclusion principle as early as in 1925, when QFT did not yet exist. en.wikipedia.org/wiki/Pauli_exclusion_principle Mar 29 at 10:35
• @my2cts I'm not saying you can't have Pauli exclusion without relativistic QFT, I'm saying you can't determine that it should be spin-1/2 particles that show exclusion - without relativistic QFT, there is nothing that tells you half-spin particles should be fermions showing exclusion and integer-spin particles should be bosons, you could just as well have integer-spin fermions. Mar 29 at 11:20
• @ACuriousMind It was clear even in 1925 that Pauli exclusion applies to electrons. Mar 29 at 13:58

The initial rejection(s) of the Klein-Gordon equation were not entirely because of the problems it created theoretically—with negative energies and negative probability densities. There was a practical, numerical problem also. At the time that quantum mechanics was being created in the 1920s, the fine structure of the hydrogen spectrum had already been measured. The spectroscopic measurements were not especially precise, but they were accurate enough that it could seen that they did not agree with the energy eigenvalues found by treating the electron as a Klein-Gordon particle. Order of magnitude estimates suggested (correctly) that the fine structure was due to relativistic corrections, and so the fine structure ought to be adequately explained by a relativistic equation like the Klein-Gordon equation. So it was inferred that electrons were not Klein-Gordon paticles.

This was in stark contrast to what happened with the Dirac theory that came along a few years later. The Dirac equation has interpretational problems, although they are not as severe as those for the Klein-Gordon equation. However, it was also critically important that using the Dirac equation to describe the electron gave correct predictions for the magnitude (and spin structure) of the hydrogenic fine structure.