Dirac Sea - Modern Interpretation

Modern Interpretation

The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simple Bogoliubov transformation, an identification between the creation and annihilation operators of two different free field theories. In the modern interpretation, the field operator for a Dirac spinor is a sum of creation operators and annihilation operators, in a schematic notation:

An operator with negative frequency lowers the energy of any state by an amount equal to the frequency, while operators with positive frequency raises the energy of any state.

In the modern interpretation, the positive frequency operators add a positive energy particle, adding to the energy, while the negative frequency operators annihilate a positive energy particle, and lower the energy. For a Fermionic field, the creation operator gives zero when the state with momentum k is already filled, while the annihilation operator gives zero when the state with momentum k is empty.

But then it is possible to reinterpret the annihilation operator as a creation operator for a negative energy particle. It still lowers the energy of the vacuum, but in this point of view it does so by creating a negative energy object. This reinterpretation only affects the philosophy. To reproduce the rules for when annihilation in the vacuum gives zero, the notion of "empty" and "filled" must be reversed for the negative energy states. Instead of being states with no antiparticle, these are states that are already filled with a negative energy particle.

The price is that there is a nonuniformity in certain expressions, because replacing annihilation with creation adds a constant to the negative energy particle number. The number operator for a Fermi field is:

which means that if one replaces N by 1-N for negative energy states, there is a constant shift in quantities like the energy and the charge density, quantities that count the total number of particles. The infinite constant gives the Dirac sea an infinite energy and charge density. The vacuum charge density should be zero, since the vacuum is Lorentz invariant, but this is artificial to arrange in Dirac's picture. The way it is done is by passing to the modern interpretation.

Still, Dirac's idea is completely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice — that cancels out the electric charge of the sea.