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The Dirac equation in Weyl representation

  1. Nov 20, 2017 #1
    1. The problem statement, all variables and given/known data
    Compute the antiparticle spinor solutions of the free Dirac equation whilst working in the Weyl representation.


    2. Relevant equations
    Dirac equation
    $$(\gamma^\mu P_\mu +m)v_{(p)}=0$$
    Dirac matrices in the Weyl representation
    $$

    \gamma^\mu=
    \begin{bmatrix}
    0 & \sigma^i \\
    -\sigma^i & 0
    \end{bmatrix}, \
    \gamma^0=
    \begin{bmatrix}
    0 & I \\
    -I & 0
    \end{bmatrix}
    $$

    3. The attempt at a solution
    I have worked through the algebra numerous times but I can't seem to get the correct energy momentum relation out, my workings are attached, apologies I'm not too strong with Latex yet. Any ideas on where I may be going wrong would be greatly appreciated :) DSC_0155.JPG
     
  2. jcsd
  3. Nov 20, 2017 #2

    TSny

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    Are you sure you have the correct form for ##\gamma^0## in the Weyl representation?

    Also, you haven't specified the sign convention that you are using for the Minkowski metric: (1, -1, -1, -1) or (-1, 1, 1, 1). The choice will determine which components of a 4-vector change sign when raising or lowering an index.
     
  4. Nov 20, 2017 #3
    Ah somehow I managed to shove a minus sign in the gamma zero matrix that shouldn't be there. Also, sorry, i'm using mostly minus :). I have tried again with the correct matrices but i'm having a similar issue...
    new.JPG
     
  5. Nov 20, 2017 #4

    TSny

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    How did you get E+m in the locations indicated below?
    upload_2017-11-20_11-17-9.png

    Another thing to consider. For most people, the vector ##\vec p## would denote the 3-vector with contravariant components ##(p^1, p^2, p^3)##. So, for example, ##\gamma^1 p_1 = \gamma^1 (-p^1) = -\gamma^1 p^1##.
     
  6. Nov 20, 2017 #5
    Hi,
    Its just the mass term from the Dirac equation which I have to add to each matrix element, unless I've done something daft. Ah so my momentum terms should have an overall change of sign?
     
  7. Nov 20, 2017 #6

    TSny

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    You don't want to add m to all the matrix elements. You should think of m in the Dirac equation as multiplied by the unit matrix.

    Yes.
     
  8. Nov 20, 2017 #7
    Thanks very much for your help :) Looks like I have the correct solution now.
     
  9. Nov 20, 2017 #8

    TSny

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    OK. Good work.
     
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