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Awash on the Dirac sea

I've written two books about infinity, most recently the fun illustrated title Introducing Infinity, and it's a subject I enjoy writing and thinking about. But for physicists, infinity often means a problem. While we can conceive that the universe might be infinite, because we only ever deal with a part of it, when infinity rears its head in calculations, it usually means trouble. This most famously arises in quantum electrodynamics, the science of the interaction of light and matter on the quantum scale. The solution there has been renormalisation - in effect, putting in the real observed values of some quantities to make the infinities go away. And this works, but it's a bit uncomfortable. Elsewhere, such as at the moment of the big bang or in the heart of a black hole, the infinities are taken to mean that our current theories break down at that point and we need to find new ways to look at what's happening.

However, there is another class of infinite entities that is tolerated by some, because they produce useful results, but that others find a little uncomfortable. Two examples spring to mind, both from the quantum world - the Dirac sea and the many worlds interpretation of quantum physics. I'd like to take a look at the less frequently covered of these, that unusual infinite ocean.

Paul Dirac was a superb (if rather strange) British physicist who was one of the leading lights in quantum theory, though he tends to be less well known in the outside world than the likes of Heisenberg or Schrödinger. One of his crowning achievements was to extend Schrödinger's wave equation for some types of quantum particle, which describes the behaviour of those particles, so that it matches the real world. The original version was not relativistic - like Newton's laws it was an approximation that assumed particles moved fairly slowly. But an electron, for example, is often no slouch and it's Dirac's equation you need to keep track of it, not Schrödinger's.

However, something interesting emerged from Dirac's work. The equation has a kind of symmetry of solution that makes it equally possible to have positive and negative energy particles. Sometimes the negative parts of such equations have just been ignored. This happened most famously with 'advanced waves' - Maxwell's equations, describing electromagnetism and light, suggest there should be photons that travel backwards in time from destination to source as well as the usual forwards ones. These were simply ignored until Richard Feynman and John Wheeler realised they could be used to explain another oddity of physics. Dirac, though, did not simply cast his negative energy electrons away. But that led to a problem.

Light is typically produced when an electron drops an energy level. The electron loses energy and this is emitted as a photon of light. Eventually the electron gets to a 'ground state' below which it can't drop any more. But if negative energy levels were allowed, as Dirac's equation suggested, electrons should continue dropping in energy for ever, blasting out vast quantities of light. They don't. So Dirac came up with a the idea that the vacuum - empty space, if you like - contained an infinite sea of negative energy electrons, filling up all the negative energy levels, so your ordinary, everyday electron could never drop into negative energy.

This seems a very unlikely and highly wasteful proposition, requiring as it does this infinitely deep and wide sea of inaccessible particles. However it proved a very productive idea. The model predicts that there will sometimes be holes in the sea - gaps where a negative energy electron is missing. As it happens, a missing negative energy electron is identical to a present positive energy, positively charged equivalent of an electron. Dirac predicted these should exist, and a couple of years later, the positron was discovered. This hypothetical infinite negative energy sea enabled Dirac to predict there was antimatter.

Does the sea have a 'real' existence? That's a difficult one. Some physicists would say yes, while others would hedge their bets with philosophical waffle about the nature of reality. The fact remains that Dirac's infinite sea of negative energy particles has played a fundamental role in the development of physics.

That's what I call a big idea.


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