Calculating the S-matrix with Casimir's Trick: A QED/QFT Approach

[sean_mp]

Homework Statement

There's a similar discussion on the forum at https://www.physicsforums.com/showthread.php?t=162812, but I'm not sure it's correct, and it doesn't entirely apply to my problem.

From QED/QFT, we use Casimir's trick when calculating the S-matrix to find
[tex] \sum_{s=1}^2u^{(s)}_{ \alpha}( \mathbf{p}) \bar u^{(s)}_{ \beta}( \mathbf{p}) = \bigg( \frac{-i \gamma \cdot p+m}{2m} \bigg)_{ \alpha \beta} [/tex]
[tex]\begin{multline} \frac{1}{2} \sum_s \sum_{s'} \lvert \bar u' \gamma_4 u \lvert^2= \frac{1}{2} \sum_s \sum_{s'} \bar u^{(s)}_{ \delta}( \mathbf{p}) ( \gamma_4)_{ \delta \gamma} u^{(s')}_{ \gamma}( \mathbf{p}') \bar u^{(s')}_{ \beta}( \mathbf{p}') ( \gamma_4)_{ \beta \alpha} u^{(s)}_{ \alpha}( \mathbf{p})
\\~ = \frac{1}{2}( \gamma_4)_{ \delta \gamma} \bigg( \frac{-i \gamma \cdot p+m}{2m} \bigg)_{ \gamma \beta}( \gamma_4)_{ \beta \alpha} \bigg( \frac{-i \gamma \cdot p+m}{2m} \bigg)_{ \alpha \delta}= \frac{1}{2} \text{Tr} \bigg[ \gamma_4 \bigg( \frac{-i \gamma \cdot p' +m}{2m} \bigg) \gamma_4 \bigg( \frac{-i \gamma \cdot p -m}{2m} \bigg) \bigg]\end{multline}[/tex]

I'm using old notation, so in most modern texts you would probably do something more aligned with Griffiths, http://scienceworld.wolfram.com/physics/CasimirTrick.html.

My problem involves a slight variation, and I'm not sure how to deal with it. Instead of
[tex] \lvert \bar u' \gamma_4 u \lvert , [/tex]
I've reached a point where I'm stuck with
[tex] \lvert \bar u' u \lvert [/tex]

The Attempt at a Solution

Well, I still have
[tex] \sum_{s=1}^2u^{(s)}_{ \alpha}( \mathbf{p}) \bar u^{(s)}_{ \beta}( \mathbf{p}) = \bigg( \frac{-i \gamma \cdot p+m}{2m} \bigg)_{ \alpha \beta} [/tex]
But my sticking point is
[tex] \frac{1}{2} \sum_s \sum_{s'} \lvert \bar u' u \lvert^2= \frac{1}{2} \sum_s \sum_{s'} \bar u^{(s)}_{ \delta}( \mathbf{p}) u^{(s')}_{ \gamma}( \mathbf{p}') \bar u^{(s')}_{ \beta}( \mathbf{p}') u^{(s)}_{ \alpha}( \mathbf{p})= \frac{1}{2} \bigg( \frac{-i \gamma \cdot p+m}{2m} \bigg)_{ \gamma \beta} \bigg( \frac{-i \gamma \cdot p+m}{2m} \bigg)_{ \alpha \delta}[/tex]

The sad thing is that I think my situation is actually easier to deal with. Any insight would be appreciated.

 

[mark726]

I can understand your confusion with the notation and the slight variation in your problem. However, I believe I can provide some insight to help you solve it.

Firstly, it is important to note that the expression in question is actually the same as before, just with a different notation. The notation \lvert \bar u' u \lvert is equivalent to \lvert \bar u' \gamma_4 u \lvert, as the gamma matrices are just a representation of the spinor space. Therefore, you can use the same approach as before and apply the Casimir's trick to simplify the expression.

Secondly, the Casimir's trick is a useful tool for simplifying expressions involving spinors, but it is not the only method. If you are still stuck, I suggest looking into other techniques such as using the Dirac equation or the Fierz identities. These may provide a different perspective and help you understand the problem better.

I hope this helps and good luck with your calculations. Don't hesitate to reach out for further assistance if needed.