Why do matter and anti-matter annihilate each other?

Title says it all. Thanks.

A general rule of particle and nuclear physics is that particles like to decay into things that have as little mass as possible (this is not always true, but good enough). You can imagine it, very roughly, like a ball on a hill. Mass is analogous to height - just like the ball rolls down the hill and speeds up, the particles want to reduce their mass and speed up.

When people say matter and antimatter annihilate, like in the PET scan thread, they mean the two matter and antimatter particles dissapear and produce photons, which are massless. Since massless is about as far down the hill as you can go, if you believe me about the rule above, then the real question should not be why does matter and antimatter annihilate, but why doesn’t everything annihilate all the time? If everything wants to go to the least massive particle possible, why don’t things just burst into photons all the time?

The answer is conservation laws. Consider, for example, an electron. Why doesn’t it just become a photon? Well, an electron has a negative charge. A photon has zero charge. Since charge is conserved, an electron then can’t just become a photon (or a pair of photons). Why doesn’t the electron just decay to something else? Well, if you think about it, an electron is the lightest charged particle, so it really has no where to go.

But bring in a positron. A positron has positive charge. In fact, ignore some more subtle things like spin and angular momentum and a positron has pretty much opposite values from an electron for every conserved quantity. As photons have zero for most conserved quantities, the positron and the electron cancel each other out, and now they’re able to produce photons. Since that’s what they wanted to do in the first place, and its no longer forbidden, it happens!

More generally, an antiparticle always has the opposite intrinsic quantum numbers (a fancy term for the conserved things I was talking about earlier) that its normal particle brother has, and so in general they will annihilate (i.e., positron-electron, proton-antiproton, etc.).

What an elegant explanation. I had no idea that I wanted to know that until I read it.

Perhaps it’s illuminating to look at how the idea of an ‘antiparticle’ got started. Sometime around 1930, Paul Dirac managed to devise an extension of quantum mechanics consistent with special relativity, encapsulated in what’s now known as the Dirac equation (an extension of the Schrödinger equation of non-relativistic quantum mechanics). That was all well and good, but the trouble was that the equation predicted infinitely many negative energy states that a particle – an electron, say – should be able to occupy; therefore, an electron ought to be able to shed energy via photons indefinitely (analogous to, for example, shell transitions in atoms), ‘plummeting’ ever deeper into the negative energy states. However, electrons aren’t observed to do that.

Dirac’s solution then was to appeal to the Pauli exclusion principle, which essentially says that no two electrons (or fermions in general) are allowed to occupy the same state, and postulate that all the negative energy states already are filled with electrons, so that the vacuum essentially corresponds to a ‘sea’ of filled negative states, in which only positive energy states are left for an electron to occupy.

However, what happens if one were to transfer some energy to one of those electrons in the sea, ‘elevating’ it to a positive energy state, via a photon, for instance? In the sea, that process would create a ‘hole’, a spot that’s missing an electron, and hence, a charge; thus, it would appear positively charged, and otherwise, act much like a particle that’s the opposite of an electron in some sense.

So, what you’d observe is that you can sometimes just shoot a photon at the vacuum, and have an electron and this curious ‘anti-electron’ pop out – this is, essentially, what we today call ‘pair production’ (though, for reasons of momentum conservation, you need another participant in this process in the real world, like for instance an atomic nucleus that takes a little ‘kick’).

But if that’s possible, there’s no reason why the electron, being in an excited state while having positive energy, shouldn’t ‘fall down’ back into the hole (or into another hole – a hole’s a hole, at least in the Dirac sea), shedding its excess energy in the form of photons – i.e., basically, an electron and an anti-electron, or hole, or positron, should be able to annihilate or recombine under the emission of energy quite naturally in this model.

However, while on the one hand, this is a quite successful model – it actually predicted the existence of the positron some years before it was first observed, plus it accounts for pair production and annihilation in a quite natural way --, on the other hand, it’s rather cumbersome and counter-intuitive: what does one make of an infinite sea of negative energy states occupied by electrons? What does that even mean? (Besides, as per Hilbert’s hotel, even a completely filled infinite sea of states should be able to accommodate yet more electrons – infinitely many, in fact.)

So today, the picture coming from quantum field theory is a different one, in which positrons are actually real, bona fide particles, and we don’t have to deal with electrons all the way down any more; but still, I think as an illustration it works quite well.