Lorentz transform on the Dirac equation

[Bill Foster]

Homework Statement

Show that a Lorentz transformation preserves the sign of the energy of a solution to the Dirac equation.

The Attempt at a Solution

I'm not sure how to approach this.

So I apply the Lorentz transform to the Dirac equation, and work through it to obtain the energy solutions?

Or do I apply the Lorentz transform to the energy solutions?

 

[arkajad]

You choose a positive energy solution of the equation, apply Lorentz transformation to the solution (The Lorentz transformation can be even infinitesimal, because you can argue from there) and check that the transformed function is again a solution with positive energy.

 

[Bill Foster]

You choose a positive energy solution of the equation, apply Lorentz transformation to the solution (The Lorentz transformation can be even infinitesimal, because you can argue from there) and check that the transformed function is again a solution with positive energy.

Since the energy solution contains a square root of the momentum (and the Lorentz transform acts on the momentum), I need to expand the function first before applying the Lorentz transform?

 

[arkajad]

Well, I am assuming that you went through the exercise of checking in all details that when you apply Lorentz transformation to a solution, you will get another solution. So, you have the formula of the transformed solution. Do you?

 

[Bill Foster]

The energy solutions to the Dirac equation are

$$E=\pm\sqrt{p^2+m^2}$$

So I apply a Lorentz transform to that. But first, I need to expand it, right?

$$\sqrt{p^2+m^2}=m\left(1+\frac{1}{2}\left(\frac{p}{m}\right)^2-\frac{1}{8}\left(\frac{p}{m}\right)^4+\frac{1}{16}\left(\frac{p}{m}\right)^6...\right)$$

Or...do I just consider the momentum as a 4-vector: 4-momentum...

$$\left[E, \vec{p}\right]$$

and apply the Lorentz transform to that?