Exercise of Dirac Field Theory

  • #1

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?
 

Answers and Replies

  • #2
DEvens
Education Advisor
Gold Member
1,203
457
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.
 
  • #3
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?
 

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