Negative energy solutions - Dirac equation

[hendriko373]

The Dirac equation (http://en.wikipedia.org/wiki/Dirac_equation) permits negative energy states. This is a direct consequence of the fact that a dirac-solution is also a solution to the Klein-Gordon equation (http://en.wikipedia.org/wiki/Klein_gordon_equation), which itself is based upon the relativistic energy-momentum relation E^2 = p^2 + m^2 (natural units). And here comes my question then:

Why do we throw away the negative energy solutions in relativity but do we keep them when we combine it with quantum theory. Clearly this must have got something to do with the quantum part, but with what?

Thanks, Hendrik

[Avodyne]

The original thinking was something like this. In (for example) the hydrogen atom, the electron can emit a photon of definite energy, and drop to a lower energy level. So, what would stop an electron from emiting a high-energy photon and dropping to a negative energy level? And then keep going down and down?

Dirac's answer was the Dirac sea: all those energy levels are already filled, and so the electron cannot drop into them, by Pauil exclusion.

This doesn't work for bosons, though.

The modern viewpoint is that both the Dirac and Klein-Gordon equations apply to quantum fields rather than to probability amplitudes.

[samalkhaiat]

[QUOTE=hendriko373;1967209]
Why do we throw away the negative energy solutions in relativity but do we keep them when we combine it with quantum theory. Clearly this must have got something to do with the quantum part, but with what?


We cannot simply throw away the negative energy solutions as we are required by QM to work with a complete set of states, and this set inevitably includes the unwanted states.


regards

sam

[tiny-tim]

Hi Hendrik! :smile:
Why do we throw away the negative energy solutions in relativity but do we keep them when we combine it with quantum theory. Clearly this must have got something to do with the quantum part, but with what?
[We cannot simply throw away the negative energy solutions as we are required by QM to work with a complete set of states, and this set inevitably includes the unwanted states.

In non-quantum relativity, to describe a particle with a particular velocity (state), we just describe … well … that particle! … no other particles, and certainly no anti-particles.

But the whole basis of quantum theory is that a particle with a particular velocity (state) must be described by a field φ(x) which

i] is an integral (or "average") over the creation operators of particles with all possible velocities (ie, as samalkhaiat :smile: says, a "complete set of states"), and

ii] has φ(x) and φ(y) commuting (or anti-commuting) for any x and y whose separation is space-like.

Unfortunately, if that "complete set" means only particles, then (see, for example, p.202 of Weinberg's QTF, Vol I) condition ii] cannot work, and the only way to make it work is by using not only the creation operators of all possible particles, but also the annihilation operators of all possible anti-particles. :smile:

Why annihilation instead of creation? Mostly for "dimensional" reasons, but also because creating things with positive energy sort-of goes with destroying things with negative energy!