The Dirac equation in Weyl representation

  • Thread starter Milsomonk
  • Start date
  • #1
91
11

Homework Statement


Compute the antiparticle spinor solutions of the free Dirac equation whilst working in the Weyl representation.


Homework 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}
$$

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
 

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Answers and Replies

  • #2
TSny
Homework Helper
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$$

\gamma^\mu=
\begin{bmatrix}
0 & \sigma^i \\
-\sigma^i & 0
\end{bmatrix}, \
\gamma^0=
\begin{bmatrix}
0 & I \\
-I & 0
\end{bmatrix}
$$
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.
 
  • #3
91
11
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
 

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  • #4
TSny
Homework Helper
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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##.
 

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  • #5
91
11
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?
 
  • #6
TSny
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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.
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.

Ah so my momentum terms should have an overall change of sign?
Yes.
 
  • #7
91
11
Thanks very much for your help :) Looks like I have the correct solution now.
 
  • #8
TSny
Homework Helper
Gold Member
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OK. Good work.
 

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