Trace as a product of operators

[intervoxel]

I'm confused about index calculation in eq. 8.25, Mandl QFT textbook. Can anyone give me a detailed explanation showing the equality below?

$X=\frac{1}{2}A_{\delta \alpha}^+(\bf{p'})\Gamma_{\alpha \beta}(\bf{p'})A_{\beta \gamma}^+(\bf{p})\widetilde{\Gamma}_{\gamma\delta}$

$=\frac{1}{2}Tr[A^+(\bf{p'})\Gamma A^+(\bf{p})]$

Please be patient, I'm learning index notation from scratch.

Thanks.

 

[A. Neumaier]

I'm confused about index calculation in eq. 8.25, Mandl QFT textbook. Can anyone give me a detailed explanation showing the equality below?

$X=\frac{1}{2}A_{\delta \alpha}^+(\bf{p'})\Gamma_{\alpha \beta}(\bf{p'})A_{\beta \gamma}^+(\bf{p})\widetilde{\Gamma}_{\gamma\delta}$

$=\frac{1}{2}Tr[A^+(\bf{p'})\Gamma A^+(\bf{p})]$

Please be patient, I'm learning index notation from scratch.

Thanks.

A factor is missing in your second formula. You can get the corrected formula by application of $$Tr A = A_{jj}$$ and $$(AB)_{jk}=A_{jl}B_{lk}$$.

 

[intervoxel]

O.k. and

$Tr(XY^T)=\sum_{i,j} X_{i,j}Y_{i,j}$

Thank you.