SortProducts receives an expression involving products and sorts the operands of these products according to the ordering indicated as the second argument, a list containing some or all of the operands of the product. The sorting of operands performed automatically takes into account any algebra rules set using Setup.

By default, the sorting between two operands A and B is performed when the value of their Commutator or AntiCommutator is known to the system (can be resolved). Note that if a Commutator can be resolved, so is the AntiCommutator and viceversa. To have the sorting performed even when the Commutator cannot be resolved, use the options usecommutator or useanticommutator, in which case the output will respectively contain an unresolved (unevaluated) Commutator or AntiCommutator. The options usecommutator and useanticommutator cannot be used simultaneously.

When the options usecommutator or useanticommutator are used, commutators or anticommutators introduced by SortProducts will be expressed using the active Commutator and AntiCommutator commands unless the operands of these commutators or anticommutators involve products, in which case they will be represented using the inert %Commutator or %AntiCommutator commands. This has the purpose of allowing for further manipulations of the output without having these commutators introduced by SortProducts automatically expanded by the corresponding active command (for details on this automatic expansion, see Commutator).

To override that default behavior, you can pass the optional argument useinert, or useinert = true. In that case, all the Commutators and AntiCommutators introduced by SortProducts will be expressed using the inert commands %Commutator or %AntiCommutator, regardless of whether these commutators can be resolved.

Equivalently, passing useinert = false all the Commutators and AntiCommutators introduced by SortProducts will be expressed in active form, using the Commutator or AntiCommutator commands, regardless of whether their operands involve products, in which case their expansion will be automatically performed as explained in the help page for Commutator.

SortProducts sorts the operands of the products 'in place', optionally positioning, when possible, the sorted operands to the left or to the right of the unsorted operands, when respectively passing the optional arguments totheleft or totheright. The option totheright cannot be used simultaneously with either of usecommutator and useanticommutator; so when using this option the sorting of say A and B is performed only if their Commutator can be resolved.

SortProducts is relevant mainly to sort the operands in noncommutative products expressed using the Physics:-`*` product operator, although it can also sort the operands in commutative products expressed with Maple's standard product operator, *. Note that the sorting of operands in commutative products within a sum may be reverted by the Maple kernel.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$P≔abcd$

$P≔abcd$

$\mathrm{SortProducts}\left(P\,\left[c\,b\right]\right)$

$acbd$

$\mathrm{SortProducts}\left(P\,\left[c\,b\right]\,\mathrm{totheleft}\right)$

$cbad$

$\mathrm{SortProducts}\left(P\,\left[c\,b\right]\,\mathrm{totheright}\right)$

$adcb$

$\mathrm{Setup}\left(\mathrm{noncommutativeprefix}=Z\,\mathrm{\%Commutator}\left(\mathrm{Z1}\,\mathrm{Z2}\right)=0\,\mathrm{\%Commutator}\left(\mathrm{Z1}\,\mathrm{Z3}\right)=0\,\mathrm{\%Commutator}\left(\mathrm{Z2}\,\mathrm{Z3}\right)=0\,\mathrm{\%Commutator}\left(\mathrm{Z3}\,\mathrm{Z4}\right)=0\,\mathrm{\%Commutator}\left(\mathrm{Z2}\,\mathrm{Z4}\right)=0\,\mathrm{\%Commutator}\left(\mathrm{Z1}\,\mathrm{Z4}\right)=0\right)$

$\left[\mathrm{algebrarules}=\left\{{\left[\mathrm{Z1},\mathrm{Z2}\right]}_{-}=0\,{\left[\mathrm{Z1},\mathrm{Z3}\right]}_{-}=0\,{\left[\mathrm{Z1},\mathrm{Z4}\right]}_{-}=0\,{\left[\mathrm{Z2},\mathrm{Z3}\right]}_{-}=0\,{\left[\mathrm{Z2},\mathrm{Z4}\right]}_{-}=0\,{\left[\mathrm{Z3},\mathrm{Z4}\right]}_{-}=0\right\}\,\mathrm{noncommutativeprefix}=\left\{Z\right\}\right]$

$P≔\mathrm{Z1}\mathrm{Z2}\mathrm{Z3}\mathrm{Z4}$

$P≔\mathrm{Z1}\mathrm{Z2}\mathrm{Z3}\mathrm{Z4}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z3}\,\mathrm{Z2}\right]\right)$

$\mathrm{Z1}\mathrm{Z3}\mathrm{Z2}\mathrm{Z4}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z3}\,\mathrm{Z2}\right]\,\mathrm{totheright}\right)$

$\mathrm{Z1}\mathrm{Z4}\mathrm{Z3}\mathrm{Z2}$

$\mathrm{SortProducts}\left(\mathrm{Z1}\mathrm{Z5}\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\right)$

$\mathrm{Z1}\mathrm{Z5}$

$\mathrm{SortProducts}\left(\mathrm{Z1}\mathrm{Z5}\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\,\mathrm{usecommutator}\right)$

$\mathrm{Z5}\mathrm{Z1}+{\left[\mathrm{Z1},\mathrm{Z5}\right]}_{-}$

$\mathrm{expand}\left(\right)$

$\mathrm{Z1}\mathrm{Z5}$

$\mathrm{SortProducts}\left(\mathrm{Z1}\mathrm{Z5}\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\,\mathrm{useanticommutator}\right)$

$-\mathrm{Z5}\mathrm{Z1}+{\left[\mathrm{Z1},\mathrm{Z5}\right]}_{+}$

$\mathrm{expand}\left(\right)$

$\mathrm{Z1}\mathrm{Z5}$

$P≔P\mathrm{Z5}$

$P≔\mathrm{Z1}\mathrm{Z2}\mathrm{Z3}\mathrm{Z4}\mathrm{Z5}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\right)$

$\mathrm{Z1}\mathrm{Z2}\mathrm{Z3}\mathrm{Z4}\mathrm{Z5}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\,\mathrm{usecommutator}\right)$

$\mathrm{Z5}\mathrm{Z1}\mathrm{Z2}\mathrm{Z3}\mathrm{Z4}-{\left[\mathrm{Z5},\mathrm{Z1}\mathrm{Z2}\mathrm{Z3}\mathrm{Z4}\right]}_{-}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z3}\right]\,\mathrm{usecommutator}\right)$

$\mathrm{Z1}\mathrm{Z2}\mathrm{Z5}\mathrm{Z3}\mathrm{Z4}-\mathrm{Z1}\mathrm{Z2}{\left[\mathrm{Z5},\mathrm{Z3}\mathrm{Z4}\right]}_{-}$

$\mathrm{value}\left(\right)$

$\mathrm{Z1}\mathrm{Z2}\mathrm{Z5}\mathrm{Z3}\mathrm{Z4}-\mathrm{Z1}\mathrm{Z2}\left(\mathrm{Z3}{\left[\mathrm{Z5},\mathrm{Z4}\right]}_{-}+{\left[\mathrm{Z5},\mathrm{Z3}\right]}_{-}\mathrm{Z4}\right)$

$\mathrm{ee}≔\mathrm{Z5}+{\left(\mathrm{Z1}\mathrm{Z5}\right)}^2$

$\mathrm{ee}≔\mathrm{Z5}+{\left(\mathrm{Z1}\mathrm{Z5}\right)}^2$

$\mathrm{SortProducts}\left(\mathrm{ee}\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\,\mathrm{usecommutator}\right)$

$\mathrm{Z5}+{\left(\mathrm{Z5}\mathrm{Z1}+{\left[\mathrm{Z1},\mathrm{Z5}\right]}_{-}\right)}^2$

$\mathrm{Setup}\left(\mathrm{\%Commutator}\left(\mathrm{Z1}\,\mathrm{Z5}\right)=\mathrm{Z6}\right)$

$\left[\mathrm{algebrarules}=\left\{{\left[\mathrm{Z1},\mathrm{Z2}\right]}_{-}=0\,{\left[\mathrm{Z1},\mathrm{Z3}\right]}_{-}=0\,{\left[\mathrm{Z1},\mathrm{Z4}\right]}_{-}=0\,{\left[\mathrm{Z1},\mathrm{Z5}\right]}_{-}=\mathrm{Z6}\,{\left[\mathrm{Z2},\mathrm{Z3}\right]}_{-}=0\,{\left[\mathrm{Z2},\mathrm{Z4}\right]}_{-}=0\,{\left[\mathrm{Z3},\mathrm{Z4}\right]}_{-}=0\right\}\right]$

$P≔\mathrm{Z1}\mathrm{Z5}$

$P≔\mathrm{Z1}\mathrm{Z5}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\right)$

$\mathrm{Z5}\mathrm{Z1}+\mathrm{Z6}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\,\mathrm{useinert}\right)$

$\mathrm{Z5}\mathrm{Z1}+{\left[\mathrm{Z1},\mathrm{Z5}\right]}_{-}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z1}\right]\,\mathrm{useanticommutator}\,\mathrm{useinert}\right)$

$-\mathrm{Z5}\mathrm{Z1}+{\left[\mathrm{Z1},\mathrm{Z5}\right]}_{+}$

$P≔\mathrm{Z2}\mathrm{Z3}\mathrm{Z5}$

$P≔\mathrm{Z2}\mathrm{Z3}\mathrm{Z5}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z2}\right]\,\mathrm{usecommutator}\right)$

$\mathrm{Z5}\mathrm{Z2}\mathrm{Z3}-{\left[\mathrm{Z5},\mathrm{Z2}\mathrm{Z3}\right]}_{-}$

$\mathrm{SortProducts}\left(P\,\left[\mathrm{Z5}\,\mathrm{Z2}\right]\,\mathrm{usecommutator}\,\mathrm{useinert}=\mathrm{false}\right)$

$\mathrm{Z5}\mathrm{Z2}\mathrm{Z3}-\mathrm{Z2}{\left[\mathrm{Z5},\mathrm{Z3}\right]}_{-}-{\left[\mathrm{Z5},\mathrm{Z2}\right]}_{-}\mathrm{Z3}$