The Factor command generalizes the standard factor command in that it can factorize expressions involving noncommutative variables. Factor complements the set of Physics commands for handling expressions with noncommutative operands, `*`, `.`, `^`, diff, Expand, Normal, Simplify, Gtaylor, and Coefficients.

The approach used is similar to the one used in the other commands of this kind, (see for instance the experimental Physics command, PerformOnAnticommutativeSystem), that is, to transform the problem into one that can be treated with the commands that work only with commutative variables, and from there extract the result for expressions involving noncommutative variables. This approach has limitations (see the Examples) but the class of problems that can be handled is well defined and the cases covered are relevant.

Note: For the conventions adopted to represent noncommutative and anticommutative objects, see Setup and the types anticommutative and noncommutative.

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

$\mathrm{Setup}\left(\mathrm{quantumoperators}=\left\{a\,b\,c\,d\,e\right\}\,\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{quantumoperators}=\left\{a\,b\,c\,d\,e\right\}\right]$

${\mathrm{\alpha }}^2{a}^2+\mathrm{\alpha }\mathrm{sqrt}\left(2\right)ab+4\mathrm{sqrt}\left(2\right)\mathrm{\lambda }{b}^{2}c+4\mathrm{\lambda }\mathrm{\alpha }bca+4\mathrm{\lambda }\mathrm{sqrt}\left(2\right)bcb+16{\mathrm{\lambda }}^2{\left(bc\right)}^2+4\mathrm{\alpha }\mathrm{\lambda }abc+\mathrm{sqrt}\left(2\right)\mathrm{\alpha }ba+2b^2$

${\mathrm{\alpha }}^2{a}^2+\sqrt{2}\mathrm{\alpha }ab+4\sqrt{2}\mathrm{\lambda }{b}^{2}c+4\mathrm{\lambda }\mathrm{\alpha }bca+4\sqrt{2}\mathrm{\lambda }bcb+16{\mathrm{\lambda }}^2{\left(bc\right)}^2+4\mathrm{\alpha }\mathrm{\lambda }abc+\sqrt{2}\mathrm{\alpha }ba+2b^2$

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

${\left(4\mathrm{\lambda }bc+\mathrm{\alpha }a+\sqrt{2}b\right)}^2$

$\mathrm{PDEtools}:-\mathrm{declare}\left(\left(a\,b\,c\,g\right)\left(x\,y\right)\right)$

$a\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}a$

$b\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}b$

$c\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}c$

$g\left(x\,y\right)\mathrm{will\; now\; be\; displayed\; as}g$

$\mathrm{Intc}\left({\left(4\mathrm{Dagger}\left(b\left(x\,y\right)\right)c\left(x\,y\right)\mathrm{\lambda }+\mathrm{\alpha }f\left(t\right)a\left(x\,y\right)\mathrm{Dagger}\left(a\left(x\,y\right)\right)+\mathrm{sqrt}\left(2\right)g\left(x\,y\right)b\left(x\,y\right)\right)}^2\,x\,y\right)$

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left(4\mathrm{\lambda }{b}^{†}c+\mathrm{\alpha }f\left(t\right)a{a}^{†}+\sqrt{2}gb\right)}^2\ⅆx\right)\ⅆy$

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

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(16{\mathrm{\lambda }}^2{b}^{†}c{b}^{†}c+4\mathrm{\lambda }\mathrm{\alpha }f\left(t\right)b^{†}ca{a}^{†}+4\sqrt{2}\mathrm{\lambda }g{b}^{†}cb+4\mathrm{\alpha }f\left(t\right)\mathrm{\lambda }a{a}^{†}{b}^{†}c+{\mathrm{\alpha }}^2{f\left(t\right)}^{2}a{a}^{†}a{a}^{†}+\sqrt{2}\mathrm{\alpha }f\left(t\right)ga{a}^{†}b+4\sqrt{2}g\mathrm{\lambda }b{b}^{†}c+\sqrt{2}g\mathrm{\alpha }f\left(t\right)ba{a}^{†}+2g^2{b}^2\right)\ⅆx\right)\ⅆy$

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

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left(4\mathrm{\lambda }{b}^{†}c+\mathrm{\alpha }f\left(t\right)a{a}^{†}+\sqrt{2}gb\right)}^2\ⅆx\right)\ⅆy$

$\mathrm{Commutator}\left(a\,b\right)c$

${\left[a,b\right]}_{-}c$

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

$abc-bac$

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

$\left(ab-ba\right)c$

$\left(ab-ba\right)\left(a+\mathrm{\beta }b+c^2\right)$

$\left(ab-ba\right)\left(\mathrm{\beta }b+c^2+a\right)$

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

$aba+\mathrm{\beta }a{b}^2+ab{c}^2-b{a}^2-\mathrm{\beta }bab-ba{c}^2$

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

$\left(ab-ba\right)\left(\mathrm{\beta }b+c^2+a\right)$

${\left(\mathrm{Commutator}\left(a\,b\right)+c\right)}^2$

${\left({\left[a,b\right]}_{-}+c\right)}^2$

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

$abab-a{b}^{2}a+abc-b{a}^{2}b+baba-bac+cab-cba+c^2$