Question about the solutions of the dirac equation

[SheikYerbouti]

I am working through Greiner's text on relativistic quantum mechanics and I am confused about what appear to be two somewhat contradictory ways of presenting the solutions of the Dirac equation. In chapter 2, he just treats the equation as a system of coupled differential equations and solves the thing by inspections. He ends up with the term $\frac{i}{\hbar}(\vec{p}\cdot \vec{x} - \lambda E t)$ in the exponent, where E is the magnitude of the energy and $\lambda$ determines the sign of the energy. The energy with the sign in front of it also appears in the normalization constant and in the vector. In chapter 6, he finds the solutions at rest and uses a Lorentz transformation to generate an arbitrary solution. However, he does so by assuming that the phase is given by $\frac{i}{\hbar}\epsilon_r p^{\mu}x_{\mu}$, where $\epsilon_r = \pm 1$ depending on which type of solution we are using. However, he defines the time component of the four-momentum to always be positive. Additionally, when he finds the general form of a Lorentz boost $S[\Lambda] = cosh(\frac{\chi}{2})I + (\hat{\chi}\cdot \vec{\alpha})sinh(\frac{\chi}{2})$ where $\chi$ is the boost parameter, he states that this always leads to strictly positive time components of the four-momentum (other notes that I have found do so as well). It appears as if these two ways of presenting the solutions produce inconsistent results with the relative signing of terms in the phase factor. In the other notes I have found, the latter way of finding the solutions/ the same form was presented in the context of the Feynman-Stueckelberg interpretation, but I am having some difficulty with this since I can't really reason this out since I don't understand the form of the solutions used to reason it out. I would greatly appreciate it if someone could elaborate on this and clear things up for me. If my problem isn't clear, just let me know and I will do what I can to elaborate.

 

[SheikYerbouti]

It doesn't make too much of a difference as far as my issue is concerned, but for the sake of accuracy, the second expression for the phase should be $-\frac{i}{\hbar}\epsilon_r p^{\mu} x_{\mu}$. I see that this is in agreement for the positive energy solutions, but my problem still exists for the negative energy solutions.