# Exercise of Dirac Field Theory

## Homework Statement

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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?