It happens when matter meets anti-matter. That there is such a thing called anti-matter never ceases to amaze me. Paul Dirac, when he arrived at the famous equation for electron-, realized that the equation predicted two particles: one was the electron and the other a as yet unknown particle which he called the positron. Great physicists are guided towards Truth by the subtle hand of Beauty. Dirac's equation was exquisite. Hence, it must be true, he declared in 1928. The positron was discovered in 1932. (Dirac was awarded the Nobel Prize for Physics the following year.)
The positron is electron's anti-particle. When an electron meets a positron, all matter is converted into energy. Quite appropriately, we call this annihilation. Compared to an atomic explosion where only a very small amount of matter is converted to energy (according to Einstein's equation), annihilation produces a lot of heat and light. There have been various speculative stories with potential annihilation of our world. They have not come to pass because the particles involved are very small that an annihilation event may only produce as much energy as a lighted matchstick. This has not been a problem in particle accelarators where annihilations happen routinely. In fact, we use positrons to image our body in PET scans.
The real puzzle, however, is: according to well known theories, during the creation of the universe matter and antimatter we created in equal quantities. If we are in the matter region, then there is possibly the antimatter region, with perhaps anti-me and anti-you wondering the same as we do. From far, antimatter's behavior is exactly the same as matter. The day of recknoning for material beings may come only when they meet the antimaterials.
There are theories that predict that there is a subtle difference between matter and antimatter which may have led to a matter-antimatter annihilating in early universe leaving a small residual matter particles that make up our present universe.[link] I am not a physicist, but I think this asymmetry is ugly, and would like it to be untrue. But, Natural laws are not beholden to human wishes, so I'll keep my fingers crossed while wishing for beauty, symmetry and eternal happiness.
I'll leave you with Dirac's words, said in his 1933 Nobel Prize Lecture:
I think it is probable that negative protons can exist, since as far as the theory is yet definite, there is a complete and perfect symmetry between positive and negative electric charge, and if this symmetry is really fundamental in nature, it must be possible to reverse the charge on any kind of particle. The negative protons would of course be much harder to produce experimentally, since a much larger energy would be required, corresponding to the larger mass. If we accept the view of complete symmetry between positive and negative electric charge so far as concerns the fundamental laws of Nature, we must regard it rather as an accident that the Earth (and presumably the whole solar system), contains a preponderance of negative electrons and positive protons. It is quite possible that for some of the stars it is the other way about, these stars being built up mainly of positrons and negative protons. In fact, there may be half the stars of each kind. The two kinds of stars would both show exactly the same spectra, and there would be no way of distinguishing them by present astronomical methods.
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Interestingly, in a moment of weakness, Dirac first thought his positive particle must be the proton. Robert Oppenheimer in a 1930 paper concluded that it could not be the proton. And yet Oppenheimer's skepticism about the Dirac equation stopped him from deducing the existence of the positron. This was one small tragedy among many in his life.
Emilio Segre, Fermi's friend and protege won the Nobel Prize with Owen Chamberlain for discovering the anti proton. A professor of mine studied with Chamberlain and thought he had inherited all of Fermi's rigor and simplicity, no doubt some of it from Segre.
Your closing Dirac quote seems to suggest to us physics lay people that Dirac thought that the anti-matter he had stumbled upon existed not in an alternate universe/place, but right here in our own universe - just in differential quantities in various places. The analogy that comes to mind is the amount of water on a tank with a sloping bottom. There is a fixed amount of water, but as you move down slope there appears to be more because depth increases.
Am I even in the ballpark?