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If I have:

$Tr(\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\gamma^{\beta}\gamma^{\rho}\gamma^{\gamma}\gamma^{\sigma}\gamma^{\delta})$

and I want to get it re-ordered like

$Tr(\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\delta}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}) + something, $

what should I do? Is there any "fast" trick to compute that?

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  • 1
    $\begingroup$ That might be more appropriate to the Mathematics . $\endgroup$ – stafusa Aug 30 '17 at 16:35
  • $\begingroup$ Tricks aside, you do know how to utilize the Clifford algebra to move such matrices around, right? $\endgroup$ – Cosmas Zachos Aug 30 '17 at 16:43
  • $\begingroup$ Yes, I do. I tried to use the anticommutation rule, but too many terms appear. I thought there was a simpler way. $\endgroup$ – LorisAm Aug 30 '17 at 16:47
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This is hard work by hand as you have to repeatedly apply the commutators, multiplying the terms. Nowadays everybody uses a computer algebra system for this chore. The most used is FeynCalc. It took me 5s by typing the following Mathematica code

GA[:mu:, :alpha:, :nu:, :beta:, :rho:, :gamma:, :sigma:, :delta:]
DiracOrder[%]

to get the following result (it works as simply because you wanted to re-order the indices alphabetically, otherwise it would have taken a few more steps, by first replacing the indices by a,b,c,... for example, then reorder, and then replace back).

$$\begin{align} \bar{\gamma }^{\mu }&.\bar{\gamma }^{\alpha }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\sigma }.\bar{\gamma }^{\delta }\\ &= \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }+2 \bar{g}^{\alpha \mu } \bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }\\ &-2 \bar{g}^{\beta \mu } \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }+2 \bar{g}^{\beta \nu } \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } -4 \bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \nu } \bar{g}^{\alpha \mu }\\ &+4 \bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \nu } \bar{g}^{\alpha \mu }-4 \bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \nu } \bar{g}^{\alpha \mu }-8 \bar{\gamma }^{\delta }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \nu } \bar{g}^{\gamma \rho } \bar{g}^{\alpha \mu }\\ &+8 \bar{\gamma }^{\beta }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \nu } \bar{g}^{\gamma \rho } \bar{g}^{\alpha \mu }-4 \bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \rho } \bar{g}^{\alpha \mu }+8 \bar{\gamma }^{\gamma }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \nu } \bar{g}^{\delta \rho } \bar{g}^{\alpha \mu }\\ &-8 \bar{\gamma }^{\beta }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \nu } \bar{g}^{\delta \rho } \bar{g}^{\alpha \mu }+4 \bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \rho } \bar{g}^{\alpha \mu }-8 \bar{\gamma }^{\gamma }.\bar{\gamma }^{\rho } \bar{g}^{\beta \nu } \bar{g}^{\delta \sigma } \bar{g}^{\alpha \mu }\\ &+8 \bar{\gamma }^{\beta }.\bar{\gamma }^{\rho } \bar{g}^{\gamma \nu } \bar{g}^{\delta \sigma } \bar{g}^{\alpha \mu }+16 \bar{g}^{\beta \nu } \bar{g}^{\gamma \rho } \bar{g}^{\delta \sigma } \bar{g}^{\alpha \mu }-8 \bar{\gamma }^{\beta }.\bar{\gamma }^{\nu } \bar{g}^{\gamma \rho } \bar{g}^{\delta \sigma } \bar{g}^{\alpha \mu }\\ &-4 \bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho } \bar{g}^{\delta \sigma } \bar{g}^{\alpha \mu }+2 \bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\alpha \mu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \nu } \bar{g}^{\beta \mu }\\ &-8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \rho } \bar{g}^{\delta \nu } \bar{g}^{\beta \mu }+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \nu } \bar{g}^{\beta \mu }+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \rho } \bar{g}^{\beta \mu }\\ &+8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \nu } \bar{g}^{\delta \rho } \bar{g}^{\beta \mu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \rho } \bar{g}^{\beta \mu }-8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\rho } \bar{g}^{\gamma \nu } \bar{g}^{\delta \sigma } \bar{g}^{\beta \mu }\\ &+8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\nu } \bar{g}^{\gamma \rho } \bar{g}^{\delta \sigma } \bar{g}^{\beta \mu }+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho } \bar{g}^{\delta \sigma } \bar{g}^{\beta \mu }-2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \mu }\\ &+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \nu } \bar{g}^{\gamma \mu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \nu } \bar{g}^{\gamma \mu }+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \rho } \bar{g}^{\gamma \mu }\\ &+8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\rho } \bar{g}^{\beta \nu } \bar{g}^{\delta \sigma } \bar{g}^{\gamma \mu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho } \bar{g}^{\delta \sigma } \bar{g}^{\gamma \mu }+2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \mu }\\ &+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \nu } \bar{g}^{\delta \mu }+8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \nu } \bar{g}^{\gamma \rho } \bar{g}^{\delta \mu }-2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \mu }\\ &-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \mu } \bar{g}^{\beta \nu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \rho } \bar{g}^{\beta \nu }-8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \mu } \bar{g}^{\delta \rho } \bar{g}^{\beta \nu }\\ &+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \rho } \bar{g}^{\beta \nu }-8 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\mu } \bar{g}^{\gamma \rho } \bar{g}^{\delta \sigma } \bar{g}^{\beta \nu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho } \bar{g}^{\delta \sigma } \bar{g}^{\beta \nu }\\ &+2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\beta \nu }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \rho } \bar{g}^{\gamma \nu }+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho } \bar{g}^{\delta \sigma } \bar{g}^{\gamma \nu }\\ &-2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \nu }+4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \rho } \bar{g}^{\delta \nu }+2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \nu }\\ &-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \mu } \bar{g}^{\gamma \rho }-4 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu } \bar{g}^{\delta \sigma } \bar{g}^{\gamma \rho }+2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\gamma \rho }\\ &-2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\sigma } \bar{g}^{\delta \rho }+2 \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho } \bar{g}^{\delta \sigma }+\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma } \end{align}$$

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  • $\begingroup$ Sorry, I had read your question too quickly: I edited my answer to answer your re-ordering. $\endgroup$ – user154997 Aug 30 '17 at 17:37

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