Exercise of Dirac Field Theory

[Luca_Mantani]

Homework Statement

[/B]
This is an excercise that was given by my professor in a previous test:
Consider the equation:
$$
\displaystyle{\not} p
=\gamma^\mu p_\mu= m$$
where the identity matrix has been omitted in the second member.
Find its most general solution.

Homework Equations

The equation is Lorentz invariant, so in another reference frame
$$
\displaystyle{\not} p'
=\gamma^\mu p'_\mu= m$$
holds true.

The Attempt at a Solution

I've got the solution but i can't understand it.
We choose a reference frame that is favorable, that is the one in which $\vec{p}=0$, so the equation become
$$\gamma^0p_0=m$$.
Let's choose $\gamma^0$ in Dirac standard form:
$$
\gamma^0=\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{pmatrix}
$$

At this point I'm ok with all i have written. Now the solution says:
So the equality becomes:
$$(p_0-m)^2=(p_0+m)^2=0$$

How did this happen? I can't understand it, i would have written the matrix equation and notice that for the equation to hold true i have $p_0=m$ and $p_0=-m$ simultaneously, so the equation is impossible.
What do you think?

 

[DEvens]

Don't forget the difference between p0 the Eigenvalue, and p0 the operator. You need to solve for both the Eigenvalue and the Eigenvector. The value p0=m corresponds to one vector, and the value p0=-m to another. The equation in your part 1 has to be an operator equation, and can't be true without acting on a wave function.

 

[Luca_Mantani]

Don't forget the difference between p0 the Eigenvalue, and p0 the operator. You need to solve for both the Eigenvalue and the Eigenvector. The value p0=m corresponds to one vector, and the value p0=-m to another. The equation in your part 1 has to be an operator equation, and can't be true without acting on a wave function.

Mmm, you mean that p0 is not the time component of the 4-momentum but it's the operator $-i\partial_0$ and both members of the equation are applied to a 4-spinor wave function?