Dirac Gamma Matricies and Angular Momentum Commutation Relations

[variety]

Homework Statement

This isn't really the problem, but figuring this out will probably help me with the rest of the problem. I want to know what $$[\gamma^0, L_x]$$ is.

Homework Equations

I know the commutation (or rather anticommutation) relations between the gamma matricies, and I know the relations between angular momentum, momentum, and position. Also $$\gamma^0=diag(1, 1, -1, -1)$$.

The Attempt at a Solution

My guess is that $$[\gamma^0, L_x]=0$$ because that would make my original problem work out. But I don't know how to justify those because $$\gamma^0$$ doesn't ordinarily commute with any matrix.

 

[Jhero]

Thank you for your question. The commutation (or anticommutation) relations between the gamma matrices are given by:

{γ^μ, γ^ν} = 2η^μν

where η^μν is the Minkowski metric. In your case, we are interested in the commutator between γ^0 and L_x, which can be written as:

[γ^0, L_x] = γ^0L_x - L_xγ^0

To simplify this, we can use the fact that γ^0 is diagonal and has the form diag(1, 1, -1, -1). Therefore, we can write:

γ^0L_x = diag(L_x, L_x, -L_x, -L_x)

and

L_xγ^0 = diag(L_x, -L_x, L_x, -L_x)

Substituting these into the commutator, we get:

[γ^0, L_x] = (diag(L_x, L_x, -L_x, -L_x) - diag(L_x, -L_x, L_x, -L_x))

= diag(0, 2L_x, -2L_x, 0)

Therefore, we can see that the commutator is not equal to zero, but instead has non-zero components along the y and z directions. This means that [\gamma^0, L_x] is not equal to zero and cannot be used to simplify your original problem. I hope this helps you in your further calculations.