Helicity Op: Commuting Dirac Hamiltonian

[Safinaz]

Hi there,

The question about the helicity operator $h= S . \bf{p}$ ( where S is 2 by 2 matrix, with $\sigma^i$ on the diagonal ), that as mentioned in a reference as [arXiv:1006.1718], it commutes with the Dirac Hamiltonian $H = \gamma^0 ( \gamma^i p^i + m )$ equ. (3.3), due to Gamma matrices anticommutation relation, but this isn't clear for me ..

Bests,

 

[ChrisVer]

S is obviously not a 2x2 matrix if it has the sigma^i on the diagonal... it's a 4x4 (but you write it in block form)...

Let's see in the most "primitive" way whether the statement is correct:
S \cdot p = S_1 p_1 + S_2 p_2 + S_3 p_3

S \cdot p =\begin{pmatrix} p_3 & p_1 -ip_2& 0 & 0 \\ p_1+ip_2 & -p_3 & 0 & 0 \\ 0& 0 & p_3 & p_1-ip_2 \\ 0&0&p_1+ip_2& -p_3 \end{pmatrix}

On the other hand:

H=\begin{pmatrix} m & 0 & p_3 & p_1-ip_2 \\ 0& m & p_1+ip_2 & -p_3 \\ p_3 &p_1-ip_2&-m &0 \\ p_1+ip_2&-p_3&0& -m\end{pmatrix}

Then it's easy to take:
[H,h]=Hh -hH=? I did it roughly and I guess I was able to make it vanish.

I have to think a little for the "gamma matrix" reasoning...

 

[Safinaz]

I also checked out: Hh-hH and found it vanish ..

So thanks and in all casses may be Gamma matrices anticommution do something here..

 

[Hepth]

I'm pretty sure you can write $S$ in terms of gamma matrices. Depending on your basis (dirac or weyl) I think its something like $S^i = \gamma^0 \gamma_5 \gamma^i$, and then use commutation relations from there.

 

[ChrisVer]

yup I was looking for how to write S in term of the gamma matrices but I was unsuccessful in finding an expression.

 

[Hepth]

$S^i =\begin{pmatrix} \sigma^i & 0 \\ 0 & \sigma^i \end{pmatrix}$

Then you can in the dirac basis use:

$$\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}.$$ (from wikipedia)

And you just can use

$$ S^i =\begin{pmatrix} \sigma^i & 0 \\ 0 & \sigma^i \end{pmatrix} = \gamma^0 \gamma_5 \gamma^i $$

you can just do the matrix algebra by hand, i think it works. If you use weyl basis the gamma0 and gamma5 are switched.

 

[Safinaz]

So now we have the right formula ..