Commutators with the Dirac Equation

[Greener1387]

Homework Statement

(Introduction to Elementary Particles, David Griffiths. Ch 7 Problem 7.8 (c))
Find the commutator of H with the spin angular momentum, S=\frac{\hbar}{2}\vec{\Sigma}. In other words find [H,S]

Homework Equations

For the Dirac equation, the Hamiltonian H=c\gamma^0(\vec{\gamma} \cdotp+mc) where p=\frac{\hbar}{i}\vec{\nabla}


Where <br /> <br /> \vec{\gamma} = \left(<br /> \begin{array}{cc}<br /> 0 &amp; \vec{\sigma}\\<br /> -\vec{\sigma} &amp; 0<br /> \end{array}<br /> \right)<br /> <br /> where \vec{\sigma} are the pauli spin matrices

And <br /> \vec{\Sigma} = \left(<br /> \begin{array}{cc}<br /> \vec{\sigma} &amp; 0\\<br /> 0 &amp; \vec{\sigma}<br /> \end{array}<br /> \right)<br /> <br />

The Attempt at a Solution

In a previous part of the problem we determined that the commutator for orbit angular momentum L, [H,L]=-i\hbarc\gamma^0 (\gammaxp). The point of the problem is to show that Total Angular Momentum J=L+S commutes with the Hamiltonian, so we know that [H,S] should be -[H,L].

My initial attempt at a solution was to explicitly write out all the matrices and decompose p into it's cartesian coordinates and then hopefully recognize a cross product after factoring out an i. The problem I ran into was that there were a few terms on the backwards diagonal that did not involve i and did not cancel out. If nothing else works I'll triple check that, but I think what is needed is to make the quantized p substitution. But when I do that, I'm at a loss for how to handle the expression for the commutator with that definition of p. It also seems a bit strange to me to have a cross product where the first term in the cross product has three components of 4x4 matrices and the second term is a three component vector.

Can anybody verify which method I should follow or where I made any bad assumptions?

Any bold letters are three component vectors, not four

 

[sgd37]

all you have to do is compute

[\gamma^0 \gamma^i , \Sigma^j ] = \begin{pmatrix} 0 &amp; [\sigma^i ,\sigma^j ] \\ [\sigma^i ,\sigma^j ] &amp; 0 \end{pmatrix} = 2i \epsilon_{ijk} \begin{pmatrix} 0 &amp; \sigma^k \\ \sigma^k &amp; 0 \end{pmatrix} = 2i \epsilon_{ijk} \gamma^0 \gamma^k

i'm sure you can manage the rest like putting in factors and writing it in vector notation

 

[Greener1387]

Thanks, sgd37, writing it in \epsilon_{ijk} notation definitely cleared things up for me.