The Dirac equation in Weyl representation

1. Nov 20, 2017

Milsomonk

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 :)

2. Nov 20, 2017

TSny

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. Nov 20, 2017

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...

4. Nov 20, 2017

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

5. Nov 20, 2017

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?

6. Nov 20, 2017

TSny

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.

7. Nov 20, 2017

Milsomonk

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

8. Nov 20, 2017

TSny

OK. Good work.