The Dirac equation in Weyl representation

[Milsomonk]

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 :)

 

[TSny]

$$

\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.

 

[Milsomonk]

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...

 

[TSny]

How did you get E+m in the locations indicated below?

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, $\gamma^1 p_1 = \gamma^1 (-p^1) = -\gamma^1 p^1$.

 

[Milsomonk]

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?

 

[TSny]

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.

 

[Milsomonk]

Thanks very much for your help :) Looks like I have the correct solution now.

 

[TSny]

OK. Good work.