FeynCalc manual (development version)

DiracOrder

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.

See also

Overview, DiracSimplify, DiracTrick.

Examples

GA[\[Beta], \[Alpha]] 
 
DiracOrder[%]

γˉβ.γˉα\bar{\gamma }^{\beta }.\bar{\gamma }^{\alpha }

2gˉαβγˉα.γˉβ2 \bar{g}^{\alpha \beta }-\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }

DiracOrder also works with Dirac matrices in DD-dimensions.

GAD[\[Rho], \[Nu], \[Mu], \[Nu]] 
 
DiracOrder[%]

γρ.γν.γμ.γν\gamma ^{\rho }.\gamma ^{\nu }.\gamma ^{\mu }.\gamma ^{\nu }

(D2)γμ.γρ+2(2D)gμρ(D-2) \gamma ^{\mu }.\gamma ^{\rho }+2 (2-D) g^{\mu \rho }

By default γ5\gamma^5 is moved to the right.

GA[5, \[Mu], \[Nu]] 
 
DiracOrder[%]

γˉ5.γˉμ.γˉν\bar{\gamma }^5.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }

γˉμ.γˉν.γˉ5\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^5

GA[6, \[Mu], 7] 
 
DiracOrder[%]

γˉ6.γˉμ.γˉ7\bar{\gamma }^6.\bar{\gamma }^{\mu }.\bar{\gamma }^7

γˉμ.γˉ7\bar{\gamma }^{\mu }.\bar{\gamma }^7

orderlist comes into play when we need an ordering that is not lexicographic

GA[\[Alpha], \[Beta], \[Delta]] 
 
DiracOrder[%]

γˉα.γˉβ.γˉδ\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }

γˉα.γˉβ.γˉδ\bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\delta }

DiracOrder[GA[\[Alpha], \[Beta], \[Delta]], {\[Delta], \[Beta], \[Alpha]}]

γˉδ.γˉβ.γˉα+2γˉδgˉαβ2γˉβgˉαδ+2γˉαgˉβδ-\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]]]

γμ.γν+γν.γμ\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]

2gμν2 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]

γνgαμ+γμgανγαgμν\gamma ^{\nu } g^{\alpha \mu }+\gamma ^{\mu } g^{\alpha \nu }-\gamma ^{\alpha } g^{\mu \nu }