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## 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?