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Exercise of Dirac Field Theory

  1. Dec 30, 2014 #1
    1. The problem statement, all variables and given/known data

    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.

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

    3. 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
    Let's choose ##\gamma^0## in Dirac standard form:
    1 & 0 & 0 & 0\\
    0 & 1 & 0 & 0\\
    0 & 0 & -1 & 0\\
    0 & 0 & 0 & -1

    At this point i'm ok with all i have written. Now the solution says:
    So the equality becomes:

    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?
  2. jcsd
  3. Dec 31, 2014 #2


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    Education Advisor
    Gold Member

    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.
  4. Dec 31, 2014 #3
    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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