could anyone show me the first couple of steps in taking the trace of something like this, im not sure how to start.
'Tr[$\gamma($$\gamma k + $$\gamma p + $$\gamma q + m) $$\gamma ($$\gamma k + $$\gamma p+ $m)]'
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2$\begingroup$ A good first step would be to define what you have written down. Are you taking the trace of a matrix, the trace of an operator, ... ? $\endgroup$– NickCommented Oct 2, 2013 at 0:38
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$\begingroup$ taking the trace of matrices. $\endgroup$– alejandro123Commented Oct 2, 2013 at 0:44
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1$\begingroup$ what is the matrix $\gamma$? I have only seen expressions like these where $\gamma$ had a vector index. Typically then $k$ $p$ and $q$ also have vector indices which are contracted with the index from the $\gamma$. Also $k$ is being integrated over and we can get some mileage out of that fact. Is that the case here? $\endgroup$– Brian MothsCommented Oct 2, 2013 at 1:10
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$\begingroup$ the gamma matrices are the 4-tuple of dirac gamma matrices. the trace of the expression is supposed to be representative of the numerator of the propagator of two fermions in a Feynman diagram. and the gamma matrices followed by either a k, p or q are supposed to be equivalent to Feynman's slashe notation commonly used in fermion propagators. $\endgroup$– alejandro123Commented Oct 2, 2013 at 1:29
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1$\begingroup$ Would like to mention that once you understand how it works, it is straightforward to make a computer algebra system (CAS) like Mathematica or more hep-th specific Cadabra do these tedious & mechanical manipulations for you. Just look at some of the Cadabra samples to get a gist of it. Though like any CAS use case I don't recommend you blindly trust the system: make sure you understand how it's working first. $\endgroup$– MichaelCommented Oct 2, 2013 at 3:52
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1 Answer
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You should write the indices on the gamma matrices. So your expression is actually \begin{align} \text{tr}[\gamma^\mu(\gamma^\alpha k_\alpha + \gamma^\beta p_\beta + \gamma^\delta q_\delta + m) \gamma_\mu(\gamma^\rho k_\rho + \gamma^\sigma p_\sigma + m) ]. \end{align}
Then you use the trace technology to evaluate the traces. For example, the $m^2$ term has the coefficient 16 because $\text{tr}[\gamma^\mu \gamma_\mu m^2] = 16 m^2$.
The $k^2$ term has $\text{tr}[\gamma^\mu\gamma^\alpha \gamma_\mu \gamma^\rho]k_\alpha k_\rho = 4(g^{\mu \alpha}\delta_\mu^\rho - g^\mu{}_\mu g^{\alpha \rho} + g^{\mu \rho}\delta^\alpha{}_\mu) k_\alpha k_\rho = 4(k^2 - 4k^2 + k^2) = -8k^2$,
and so on.
