DiracOrder[exp] orders the Dirac matrices in
exp lexicographically.
DiracOrder[exp, orderlist] orders the Dirac matrices in
exp according to orderlist.
DiracOrder is also an option of DiracSimplify
and some other functions dealing with Dirac algebra. If set to
True, the function DiracOrder will be applied
to the intermediate result to reorder the Dirac matrices
lexicographically.
Overview, DiracSimplify, DiracTrick.
GA[\[Beta], \[Alpha]]
DiracOrder[%]\bar{\gamma }^{\beta }.\bar{\gamma }^{\alpha }
2 \bar{g}^{\alpha \beta }-\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }
DiracOrder also works with Dirac matrices in D-dimensions.
GAD[\[Rho], \[Nu], \[Mu], \[Nu]]
DiracOrder[%]\gamma ^{\rho }.\gamma ^{\nu }.\gamma ^{\mu }.\gamma ^{\nu }
(D-2) \gamma ^{\mu }.\gamma ^{\rho }+2 (2-D) g^{\mu \rho }
By default \gamma^5 is moved to the right.
GA[5, \[Mu], \[Nu]]
DiracOrder[%]\bar{\gamma }^5.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }
\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^5
GA[6, \[Mu], 7]
DiracOrder[%]\bar{\gamma }^6.\bar{\gamma }^{\mu }.\bar{\gamma }^7
\bar{\gamma }^{\mu }.\bar{\gamma }^7
orderlist comes into play when we need an ordering that
is not lexicographic
GA[\[Alpha], \[Beta], \[Delta]]
DiracOrder[%]\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }
\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }
DiracOrder[GA[\[Alpha], \[Beta], \[Delta]], {\[Delta], \[Beta], \[Alpha]}]-\bar{\gamma }^{\delta }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\alpha }+2 \bar{\gamma }^{\delta } \bar{g}^{\alpha \beta }-2 \bar{\gamma }^{\beta } \bar{g}^{\alpha \delta }+2 \bar{\gamma }^{\alpha } \bar{g}^{\beta \delta }
Reordering of Dirac matrices in long chains is expensive, so that
DiracSimplify does not do it by default.
DiracSimplify[GAD[\[Mu], \[Nu]] + GAD[\[Nu], \[Mu]]]\gamma ^{\mu }.\gamma ^{\nu }+\gamma ^{\nu }.\gamma ^{\mu }
However, if you know that it can lead to simpler expressions, you can
activate the reordering via the option DiracOrder.
DiracSimplify[GAD[\[Mu], \[Nu]] + GAD[\[Nu], \[Mu]], DiracOrder -> True]2 g^{\mu \nu }
Reproduce Eq. 18.128 from An Introduction to Quantum Field Theory by M. Peskin and D. Schroeder.
DiracSimplify[1/2 (GAD[\[Mu], \[Alpha], \[Nu]] + GAD[\[Nu], \[Alpha], \[Mu]]), DiracOrder -> True]\gamma ^{\nu } g^{\alpha \mu }+\gamma ^{\mu } g^{\alpha \nu }-\gamma ^{\alpha } g^{\mu \nu }