Mathematical proof for antimatter? I've heard (BBC Documentary Atom-The Illusion of Reality) that the Dirac Equation implies the existence of antimatter. Can someone tell me how that mathematical proof is done. Just for knowledge. Thanks in advance.
 A: Just to flesh out Frederic Thomas's point a bit:
The relativistic energy-momentum relation $E^2=p^2+m^2$ (I'm working with $c=1$) has solutions with $E<0$, which would prevent atoms having a ground state. And worse still, if you try to extend-to-relativity the non-relativistic quantum-mechanical continuity equation that makes probability non-negative and sum to $1$ in the obvious way, you find the negative-energy solutions also give negative probabilities. What the...
This led Paul Dirac to seek a linear relation, say $E=\alpha\cdot p+\beta m$, consistent with the above equation. This turns out to require $\alpha,\,\beta$ to be at least $4\times 4$ matrices. If we identify $E\psi =i\hbar\dot{\psi},\,p\psi=-i\hbar\nabla\psi$, we'll need $\psi$ to be a column vector with $4$ components too.
There was some good news and bad news when Dirac was done. The bad news was his new equation still had negative-energy solutions! The good news was he could get around this by interpreting the four parts of $\psi$ in the right way. As a theory of the spin-$1/2$ electron with spin degeneracy $2S+1=2$, it explained why electrons had spin states (which until then was a mysterious brute fact) but also required a further factor of $2$ to give $\psi$ four parts. That factor of $2$ comes from matter vs antimatter, and he realised the "negative probabilities" just meant antiparticles were outnumbering particles in those solutions. In other words, the quantity we'd misinterpreted as probability density is really total particle number density, where antimatter scores negatively.
in one fell swoop Dirac had made relativistic quantum mechanics work, shown it gives you spin for free, and made a novel prediction that there would exist something that looks a bit like an electron, with the usual mass but the wrong sign to its charge. It didn't take long to find this positron, and it also didn't take long to build on Dirac's work to develop a fully-fledged theory of how in general relativity gives you antimatter and spin. The details vary by particle species, but the electron example gives you the gist of it.
A: The Drake equation is a way of estimating the possible number of extraterrestrial civilizations in our galaxy, not something to imply the existence of antimatter. See also What is Drake equation?
A: Among the solutions of the Dirac-equation negative energy solutions appear. The (first part) existence of these solutions  can be indeed mathematically proven. However, it was up to physicists, on first place of course Dirac himself and later Feynman and Stueckelberg to interpret these solutions as related with anti-particles. This second part, however, is by far not evident as the existence of negative energy solutions on first sight gets in conflict with another principle in physics that the energy of particles should be bounded from below.  Fortunately the anti-particles could be already found 4 years after the establishment of the Dirac-equation. If the latter had not been the case, the given interpretation of the negative energy solutions would have certainly been not accepted in the physics community. Therefore, I would not consider the interpretation of these solutions as mathematical proof. So in conclusion, anti-matter cannot be mathematically proven. And I would say,  only a few things in physics can be actually mathematically proven, it is not up to physics to do so. Physics is at the basis an empirical science.  
