Negative Solution for Dirac Equation The negative solution for the Dirac equation predicted the existence of positron. Can anyone show that basic solution for this?
 A: I'll give you a 'quick and dirty' answer, as I don't feel like I am capable of writing a comprehensive, full derivation. 
Of course, what one really needs to show is that the Dirac equation,
$$ (i\hbar \gamma^\mu\partial_\mu-mc)\psi=0$$
has solutions that come in pairs with energies $\pm E$. However, there is an easy way to 'understand' that this is to be expected, from the way that Dirac arrived at the equation. The motivation for looking for the Dirac equation was provided by the fact that the Schrödinger equation 
$$\left(i\hbar \frac{\partial}{\partial t}-H\right)\psi=0$$
is not invariant under general Lorentz transformations. Therefore, Dirac was looking for an equation that would be in accordance with special relativity.
Now, the Schrödinger equation can be interpreted as the operator equivalent of the usual Newtonian relation
$$E=\frac{p^2}{2m}+V $$
This, by the way, is another way to quickly see that it cannot be expected to apply in the more general setting of special relativity. If you're looking for a relativistic equation, then what better place to start than the famous 
$$E^2=m^2c^4+p^2c^2 $$
However, since we are considering quantum mechanics, we should be working with operators. The naive way to go is to simply substitute $p=i\hbar\nabla$ and $E=i\hbar \partial_t$. This would yield the Klein-Gordon equation, which is second order in both time and position. However, it turns out that this is not the right idea, for reasons that are outlined in e.g. Ryder's book on QFT and in the relevant wikipedia article
How to proceed? Dirac's idea was to try and find a differential equation that was first order in both time and space! This would roughly correspond to the operator-version of 
$$E=\pm c\sqrt{m^2c^2+p^2} $$
The precise story of how to convert this equation into the correct equation in terms of operators is not quite trivial (to say the least). However, assuming that this is the right way to go, we can already see that there will be a symmetry between positive and negative energy solutions! 
This is, to me, the easiest way to view this matter. I'm sure that someone more capable than me wil also be able to provide a more precise statement involving actually solving the Dirac equation.
