Matter and anti-matter collision energy problem From Beyond Einstein, by Michio Kaku and Jennifer Thompson, Chapter 13, Antimatter :

Dirac, also focused on the fact that Einstein's equation $E=mc^2$ wasn't totally true. (Einstein was aware that the true equation was $E=\pm mc^2$ but as his theory was based on forces, he ignored the minus sign.) As Dirac was creating a new kind of equation (known as Dirac's equation) for the electron, he shouldn't ignore the minus sign. This sign was intriguing people, because this, seems like he predicted a whole new type of matter.

So if I understand correctly, matter (as always known) follow the equation $E=mc^2$ whereas anti-matter follows the equation $E=-mc^2$. But a few paragraph later, it says :

When matter and anti-matter collide, they annihilate each other and an enormous amount of energy comes out.

When I thought a little bit, if a particle carrying $mc^2$ amount of energy, and it's anti-patricle carrying $-mc^2$ amount of energy then the total energy carried by both particles should be $E_{total}=E_{particle}+E_{anti-particle}=mc^2-mc^2=0$. I think that everyone can see here the problem : there are not any energy in total with both particles. So where this huge energy comes from when the particle and anti-particle collide? Or is it me misinterpreting what I've read?
 A: The negative energy states are called "the Dirac sea "

The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.



Dirac sea for a massive particle.  yellow•  particles,  blue •  antiparticles

So  in this model  a negative energy hole  appears in our positive energy world as an antiparticle , with positive mass /energy and opposite quantum numbers: the opposite to the electron is the  positron. 
It is a very successful model of the half spin  particles we have observed.
A: Ben Franklin assigned what were to be electrons a negative charge.  The math then says electrons and current flow in opposite directions.  That is consistent, but one would be a fool to believe something material counterflows a lightning strike.
A gram of matter or antimatter (constituting same stuff, of course, say of AlF3 that is monoisotopic in both elements) have identical mass equivalents, mc^2.  Photons are their own antiparticles.  Lepton-antileptons (lazily - no acceleration) annihilate into photons - easy to balance the books by measurement, including kinetic energy and binding energy (that is negative) such as electron-positron direct merge vs. positronium collapse (and then singlet vs. triplet positronium and excited states of each).
HOWEVER, hadrons (stuff with quarks) lazily annihilate to give about half their mass equivalent as energetic neutrinos. 2mc^2 than appears to be about mc^2.  USS Enterprise NCC-1701 only got about half its dealer sticker mileage.  Nothing has changed in the 23rd century.
Finally, if you are at an accelerator smashing stuff-antistuff, the kinetic energy adds to the stew.  You cannot get more mass-energy out than you put in, but you can put in 14 Tev.  14 TeV is (14,900 protons)(c^2).  All sorts of things happen and output.
