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.
[\[Beta], \[Alpha]]
GA
[%] 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.
[\[Rho], \[Nu], \[Mu], \[Nu]]
GAD
[%] 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.
[5, \[Mu], \[Nu]]
GA
[%] DiracOrder
\bar{\gamma }^5.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }
\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^5
[6, \[Mu], 7]
GA
[%] 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
[\[Alpha], \[Beta], \[Delta]]
GA
[%] DiracOrder
\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }
\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }
[GA[\[Alpha], \[Beta], \[Delta]], {\[Delta], \[Beta], \[Alpha]}] DiracOrder
-\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.
[GAD[\[Mu], \[Nu]] + GAD[\[Nu], \[Mu]]] DiracSimplify
\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
.
[GAD[\[Mu], \[Nu]] + GAD[\[Nu], \[Mu]], DiracOrder -> True] DiracSimplify
2 g^{\mu \nu }
Reproduce Eq. 18.128 from An Introduction to Quantum Field Theory by M. Peskin and D. Schroeder.
[1/2 (GAD[\[Mu], \[Alpha], \[Nu]] + GAD[\[Nu], \[Alpha], \[Mu]]), DiracOrder -> True] DiracSimplify
\gamma ^{\nu } g^{\alpha \mu }+\gamma ^{\mu } g^{\alpha \nu }-\gamma ^{\alpha } g^{\mu \nu }