I'm no nuclear physicist, but here's what I've gathered from various confusingly technical websites:

Assume that all matter is energy. Everything we deal with on a daily basis has a positive energy. Protons, elections, neutrons, photos, radio waves, all of it.

Dirac's equations (based on Einstein's, i believe) predicted that there should also be negative energy. Today we know this energy as antimatter. But how can you have negative energy? Less than none?

Dirac explained this by saying that all measurements are taken relative to empty space, in which all of the negative energy levels have been filled.

Imagine a graph with positive energy levels at the top and negative energy at the bottom. All energy levels below zero are filled in. This is the "Sea of Dirac" or "Dirac Sea".

According to Dirac's theory (as I understand it), antimatter would not be matter with negative mass, but an absence of "negative mass". Antiparticles are the holes in the negative energy sea.
The Dirac equation is an extension of the Klein-Gordon equation, which is in turn an extension of the Schrodinger equation.

The Schrodinger equation is not relativistically invariant. The Klein-Gordon equations is but admits negative energy soloutions to the euqation. (Schrodinger wrote down the Klein-Gordon equation before Klein and Gordon but rejected it due to it's negative energy soloutions.) The K-G equation also admitted negative proabaility soloutions.

I remember once I was in Copenhagen, that Bohr
asked me what I was working on and I told him
I was trying to get a satisfactory relativistic
theory of the electron, and Bohr said
'But Klein and Gordon have already done that!'
That answer first rather disturbed me. Bohr
seemed quite satisfied by Klein's soloution, but 
I was not because of the negative proabailities that
it led to. I just kept on with it, worrying
about getting a theory which would have only
positive proabailities.
(from The Quantum Theory of Fields by Stephen Weinberg)

The DE solved this but the negative energies remained. Dirac suggested that in the energy level diagram of the universe all of the lowest energy states (negative ones) were occupied, and because they were all occupied no transitions were observed. This set of infinte particles (all occupied) is the Dirac sea. He went on to say that every now and again a particle would get photoionised and a positve energy particle could make the downward transition realseing a photon. This was the first predicion for anti-matter. A few years later the positron was observed in particle accelerator expermints and Dirac won a Nobel prize.

Nowadays we do not follow this interpretaion. It would indicate a lot more mass in the universe for one thing. (Anti-matter does not have negative mass !). Instead Feynman proposed that anti-matter is matter travelling backwards in time. This is a nice interpretation.

There are plenty of systems with negative energy. Any gravitationally bound system has a net negative energy, other wise it would blow itself apart.

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