The ToFieldComponents command computes an exact expansion of expression, containing functions of anticommutative variables, by rewriting them in terms of functions of commutative variables. In some frameworks, functions of anticommutative variables are also called superfields, and the functions of commutative variables entering the output of ToFieldComponents are called the field components. Note: mathematical functions such as exp, sin are not expanded - for them you can use Gtaylor.

The ToSuperfields command reverses in a given expression the expansions performed by ToFieldComponents.

Each expansion performed with ToFieldComponents is a polynomial in the function's anticommutative variables, with arbitrary functions of commutative variables as coefficients. This polynomial has terms of degree 0 and 1 with respect to each of its anticommutative variables. In the simplest case, for instance, let $F\left(x,\mathrm{\theta }\right)$ be a commutative function $F$ of $x$ (commutative) and $\mathrm{\theta }$ (anticommutative); the expansion returned by ToFieldComponents is according to

This expansion involves only functions of commutative variables $\mathrm{\_F1}\left(x\right)$ and $\mathrm{\_Q1}\left(x\right)$, where the function $\mathrm{\_Q1}$ itself is anticommutative (not its argument $x$), and preserves the grassmannian parity of the expression: if the left-hand-side is (anti)commutative, so is the sum of terms on the right-hand-side.

Regarding the anticommutative functions ($\mathrm{\_Q1}$) of commutative variables ($x$), introduced by ToFieldComponents, you can optionally indicate the prefix to be used (instead of $\mathrm{\_Q}$) for the function's name by passing it on the right-hand-side of the optional argument anticommutativefunction = ....

To avoid introducing anticommutative functions of commutative variables in the returned result and perform the expansion using only commutative functions of commutative variables, pass the optional argument useonlycommutativefunctions; for the example presented above, for instance, the result would be according to

Note the introduction of anticommutative parameters prefixed by $\mathrm{\_\λ}$, necessary to preserve the parity of the right-hand-side the same as that of the left-hand-side. You can optionally indicate the anticommutative parameter to be used as a prefix by passing it in the right-hand-side of the optional argument anticommutativeparameter = ....

To restrict the expansion of the functions found in expression to only a subset of them, pass this set as second argument to ToFieldComponents.

ToFieldComponents keeps track of the expansions performed so that ToSuperfields can revert them. To query about the expansions tracked pass the optional argument query. To reset the tracking of expansions (equivalent to forget the ones performed) use the optional argument reset.

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

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\left\{Q\,\mathrm{\theta }\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\,\mathrm{\theta }\right\}\right]$

$F\left(x\,y\,z\,\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\,\mathrm{\theta }\left[3\right]\right)$

$F\left(x\,y\,z\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\,{\mathrm{\theta }}_3\right)$

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

$\mathrm{\_F1}\left(x\,y\,z\right)-{\mathrm{\theta }}_1\mathrm{\_Q1}\left(x\,y\,z\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\,z\right)-{\mathrm{\theta }}_3\mathrm{\_Q3}\left(x\,y\,z\right)+\mathrm{\_F2}\left(x\,y\,z\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\mathrm{\_F3}\left(x\,y\,z\right){\mathrm{\theta }}_1{\mathrm{\theta }}_3+\mathrm{\_F4}\left(x\,y\,z\right){\mathrm{\theta }}_2{\mathrm{\theta }}_3-{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\theta }}_3\mathrm{\_Q4}\left(x\,y\,z\right)$

$\mathrm{GrassmannParity}\left(=\right)$

$0=0$

$\mathrm{ToFieldComponents}\left(\,\mathrm{useonly}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{useonly}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{useonlycommutativefunctions}\text{'}$

$\mathrm{\_F5}\left(x\,y\,z\right)+\mathrm{\_F6}\left(x\,y\,z\right)\mathrm{\_\λ1}{\mathrm{\theta }}_1+\mathrm{\_F7}\left(x\,y\,z\right)\mathrm{\_\λ2}{\mathrm{\theta }}_2+\mathrm{\_F8}\left(x\,y\,z\right)\mathrm{\_\λ3}{\mathrm{\theta }}_3+\mathrm{\_F9}\left(x\,y\,z\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\mathrm{\_F10}\left(x\,y\,z\right){\mathrm{\theta }}_1{\mathrm{\theta }}_3+\mathrm{\_F11}\left(x\,y\,z\right){\mathrm{\theta }}_2{\mathrm{\theta }}_3+\mathrm{\_F12}\left(x\,y\,z\right)\mathrm{\_\λ4}{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\theta }}_3$

$\mathrm{Gtaylor}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\,\mathrm{\theta }\left[3\right]\right]\right)$

$F\left(x\,y\,z\,0\,0\,0\right)+{\mathrm{D}}_4\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_1+{\mathrm{D}}_5\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_2+{\mathrm{D}}_{4,5}\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+{\mathrm{D}}_6\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_3+{\mathrm{D}}_{4,6}\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_1{\mathrm{\theta }}_3+{\mathrm{D}}_{5,6}\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_2{\mathrm{\theta }}_3+{\mathrm{D}}_{4,5,6}\left(F\right)\left(x\,y\,z\,0\,0\,0\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\theta }}_3$

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

$F\left(x\,y\,z\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\,{\mathrm{\theta }}_3\right)$