Solving the Equation for Trace: Gamma Matrices Explained

[Fr34k]

Homework Statement

Solve the equation. What is it's trace?

Homework Equations

[STRIKE]k[/STRIKE] γ_μ γ₅ [STRIKE]o[/STRIKE] γ^$\nu$ γ₅

The Attempt at a Solution

I don't think this is reduced enough.
γ^μk_μγ₅γ^$\nu$o_$\nu$γ^$\nu$γ₅

trace: just got rid of gamma5 with anticommutation.
-Tr[γ^μk_μγ^$\nu$o_$\nu$γ^$\nu$]

 

[gabbagabbahey]

Homework Statement

Solve the equation. What is it's trace?

Homework Equations

[STRIKE]k[/STRIKE] γ_μ γ₅ [STRIKE]o[/STRIKE] γ^$\nu$ γ₅

You haven't posted an equation, you've posted an expression. You can't solve an expression.

The Attempt at a Solution

I don't think this is reduced enough.
γ^μk_μγ₅γ^$\nu$o_$\nu$γ^$\nu$γ₅

I see the same index, $\nu$, 3 times in this term, so it is not valid according to the rules of the Einstein summation convention.

 

[Fr34k]

My bad, you re right it is an expression. I should have said evaluate the expression or simplify or something similar.

As for index $\nu$. Doesn't [STRIKE]o[/STRIKE] =γ^$\nu$o?
Would really appreciate some help here.

 

[gabbagabbahey]

My bad, you re right it is an expression. I should have said evaluate the expression or simplify or something similar.

So the problem is to simplify the expression [strike]k[/strike] γ_μ γ₅ [strike]o[/strike] γ_ν γ₅, and then find the trace? If so, are you given a particular space-time and metric (or Lagrangian)? Are $k_{\nu}$ and $o_{\nu}$ arbitrary vectors, or do they have some meaning here?

As for index $\nu$. Doesn't [STRIKE]o[/STRIKE] =γ^$\nu$o?
Would really appreciate some help here.

Do you mean ${\not}{o}=\gamma^{\nu}o_{\nu}$, where $o_{\nu}$ is some covariant vector in your spacetime? If so, then realize that there is an implied summation over the index $\nu$ in the term $\gamma^{\nu}o_{\nu}$, according to the Einstein summation convention. This makes the $\nu$ in this term a so-called "dummy" index which can be replaced with any other Greek index.

When you have something like ${\not}{k}{\not}{o}$ and you want to write it in terms of the Dirac matrices and the covariant vectors, according to the Einstein summation convention, you should use a different index for each implied sum, so that it is clear which terms belong to which summation. For example, ${\not}{k}{\not}{o}=\gamma^{\nu}k_{\nu}\gamma^{\nu}o_{\nu}$ is meaningless and incorrect, but ${\not}{k}{\not}{o}=\gamma^{\mu}k_{\mu}\gamma^{\nu}o_{\nu}$ is correct and consistent with the Einstein summation convention. Likewise, you wouldn't write $\gamma_{\nu}{\not}{k}=\gamma_{\nu} \gamma^{\nu} k_{\nu}$, but rather you would use a dummy index that is not already used in the term like $\mu$, and write $\gamma_{\nu}{\not}{k}=\gamma_{\nu} \gamma^{\mu} k_{\mu}$

Now, that said, if this sort of index notation is not immediately clear to you, you should almost certainly brush up on your mathematics before trying to study spinors and Dirac matrices.

 

[Fr34k]

k_γ and o_ν are just arbitrary vectors.

I think I kinda get this notation now.

As for mathematics behind I couldn't agree more. The problem is we (the students) first saw this kind of notation when actually studying scattering of Dirac particles and didn't really get much (or any) of the math behind it.

But thank you for clearing things out a bit.