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?)
 A: 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.
A: 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.
A: 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.
