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Make the Dirac Equation Consistent with Relativity

  1. May 17, 2015 #1
    1. The problem statement, all variables and given/known data
    The free Dirac equation is given by ##(i\gamma ^\mu \partial _\mu -m)\psi = 0## where ##m## is the particle's mass and ##\gamma ^\mu## are the Dirac gamma matrices. Show that for the equation to be consistent with Relativity, the gamma matrices must satisfy ##[\gamma ^\mu ,\gamma ^\nu]=2g^{\mu \nu}##.

    2. Relevant equations
    Dirac equation
    Gamma matrices
    ##E^2=|\vec p|^2 + m^2## in natural units
    We use the ##+---## metric.

    3. The attempt at a solution
    I know that the Dirac Hamiltonian is ##\vec {\alpha} \cdot \vec p +\beta m##, so I have equated it to the energy-momentum relation as follows:

    ##E^2 = (\vec {\alpha} \cdot \vec p + \beta m)^2 = (\vec {\alpha} \cdot \vec p)^2 + 2\beta m\vec {\alpha} \cdot \vec p + \beta^2m^2 = |\vec p|^2 + m^2##

    It's clear that this leads to ##\beta^2=1##. I also know that we get ##\alpha^i\beta + \beta\alpha^i = 0##, although I'm less sure why. The bit that really puzzles me though is that we're supposed to get ##\alpha^i\alpha^j + \alpha^j\alpha^i = 2\delta^{ij}##, and I can't see how it follows from the equation above.

    Then even with those three pieces I'm not sure how to arrive at the required commutator. Any help is much appreciated.
     
  2. jcsd
  3. May 17, 2015 #2

    blue_leaf77

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    Science Advisor
    Homework Helper

    That should be an anticommutator.
    Expand ##(\alpha \cdot \mathbf{p})^2## and make it equal to ##|\mathbf{p}|^2##

    By the way you shouldn't write it like that since you know that ##\beta## does not commute with ##\alpha_i##.
     
  4. May 17, 2015 #3
    Yes you're right, my bad.

    That makes sense, I was just getting muddled with all the vector components. Thanks for your help.
     
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