 Rewrite functions of anticommutative variables in terms of functions of commutative variables - Maple Programming Help

Physics[ToFieldComponents] - Rewrite functions of anticommutative variables in terms of functions of commutative variables

Physics[ToSuperfields] - rewrite expressions with field components in terms of the corresponding superfields

 Calling Sequence ToFieldComponents(expression, F, ...) ToSuperfields(expression)

Parameters

 expression - algebraic expression, or relation between them, or a set, list or rtable of them F - optional, a set of functions restricting the action of ToFieldComponents; if not given, all functions in expression are considered anticommutativeparameter = ... - optional, default to _lambda, the right-hand-side can be any anticommutative symbol anticommutativefunction = ... - optional, default to _lambda, the right-hand-side can be any anticommutative symbol query = ... - optional, to query about the expansions performed by ToFieldComponents during the current Maple session reset = ... - optional, to reset the tracking of expansions performed useonlycommutativefunctions = ... - optional, can be true (default) or false, to use or not anticommutative functions of commutative variables in the result returned

Description

 • 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

$F\left(x,{\mathrm{\theta }}\right)=\mathrm{_F1}\left(x\right)-{\mathrm{\theta }}{\mathrm{_Q1}}\left(x\right)$

 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

$F\left(x,{\mathrm{\theta }}\right)=\mathrm{_F2}\left(x\right)+\mathrm{_F3}\left(x\right){\mathrm{_λ1}}{\mathrm{\theta }}$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

Set first $\mathrm{\theta }$ and $Q$ as prefixes for variables of type/anticommutative (see Setup)

 > $\mathrm{Setup}\left(\mathrm{anticommutativepre}=\left\{Q,\mathrm{\theta }\right\}\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'} against keyword \text{'}}\mathrm{anticommutativeprefix}\text{'}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{Q}{,}{\mathrm{\theta }}\right\}\right]$ (2)

A commutative function (its name $F$ is of type commutative) of commutative $x,y,z$ and anticommutative variables ${\mathrm{\theta }}_{1},{\mathrm{\theta }}_{2},{\mathrm{\theta }}_{3}$

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

The expansion of (3) is a polynomial in ${\mathrm{\theta }}_{1},{\mathrm{\theta }}_{2}$ and ${\mathrm{\theta }}_{3}$ with terms of degree 0 and 1 with respect to each of ${\mathrm{\theta }}_{1},{\mathrm{\theta }}_{2}$ and ${\mathrm{\theta }}_{3}$. Recall that any product of these variables is also of degree 0 or 1 with respect to each of them, so the expansion contains all the monomials that can be constructed with products of ${\mathrm{\theta }}_{1},{\mathrm{\theta }}_{2}$ and ${\mathrm{\theta }}_{3}$

 > $\mathrm{ToFieldComponents}\left(\right)$
 ${\mathrm{_F1}}{}\left({x}{,}{y}{,}{z}\right){-}{{\mathrm{\theta }}}_{{1}}{}{\mathrm{_Q1}}{}\left({x}{,}{y}{,}{z}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}{,}{z}\right){-}{{\mathrm{\theta }}}_{{3}}{}{\mathrm{_Q3}}{}\left({x}{,}{y}{,}{z}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}{,}{z}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{+}{\mathrm{_F3}}{}\left({x}{,}{y}{,}{z}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{3}}{+}{\mathrm{_F4}}{}\left({x}{,}{y}{,}{z}\right){}{{\mathrm{\theta }}}_{{2}}{}{{\mathrm{\theta }}}_{{3}}{-}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{}{{\mathrm{\theta }}}_{{3}}{}{\mathrm{_Q4}}{}\left({x}{,}{y}{,}{z}\right)$ (4)

This expansion preserves the parity of (3)

 > $\mathrm{GrassmannParity}\left(=\right)$
 ${0}{=}{0}$ (5)

The expansion of (3) using only commutative functions

 > $\mathrm{ToFieldComponents}\left(,\mathrm{useonly}\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{useonly}\mathrm{\text{'} against keyword \text{'}}\mathrm{useonlycommutativefunctions}\text{'}$
 ${\mathrm{_F5}}{}\left({x}{,}{y}{,}{z}\right){+}{\mathrm{_F6}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{_λ1}}{}{{\mathrm{\theta }}}_{{1}}{+}{\mathrm{_F7}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{_λ2}}{}{{\mathrm{\theta }}}_{{2}}{+}{\mathrm{_F8}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{_λ3}}{}{{\mathrm{\theta }}}_{{3}}{+}{\mathrm{_F9}}{}\left({x}{,}{y}{,}{z}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{+}{\mathrm{_F10}}{}\left({x}{,}{y}{,}{z}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{3}}{+}{\mathrm{_F11}}{}\left({x}{,}{y}{,}{z}\right){}{{\mathrm{\theta }}}_{{2}}{}{{\mathrm{\theta }}}_{{3}}{+}{\mathrm{_F12}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{_λ4}}{}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{}{{\mathrm{\theta }}}_{{3}}$ (6)

Compare the expansion (6) returned by ToFieldComponents with a multivariable taylor expansion of (3) (see Gtaylor)

 > $\mathrm{Gtaylor}\left(,\left[\mathrm{\theta }\left[1\right],\mathrm{\theta }\left[2\right],\mathrm{\theta }\left[3\right]\right]\right)$
 ${F}{}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){+}{{\mathrm{D}}}_{{4}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{1}}{+}{{\mathrm{D}}}_{{5}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{2}}{+}{{\mathrm{D}}}_{{4}{,}{5}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{+}{{\mathrm{D}}}_{{6}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{3}}{+}{{\mathrm{D}}}_{{4}{,}{6}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{3}}{+}{{\mathrm{D}}}_{{5}{,}{6}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{2}}{}{{\mathrm{\theta }}}_{{3}}{+}{{\mathrm{D}}}_{{4}{,}{5}{,}{6}}{}\left({F}\right){}\left({x}{,}{y}{,}{z}{,}{0}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{}{{\mathrm{\theta }}}_{{3}}$ (7)

The reverse transformation, expressing (6) in terms of superfields

 > $\mathrm{ToSuperfields}\left(\right)$
 ${F}{}\left({x}{,}{y}{,}{z}{,}{{\mathrm{\theta }}}_{{1}}{,}{{\mathrm{\theta }}}_{{2}}{,}{{\mathrm{\theta }}}_{{3}}\right)$ (8)
 > 

Compatibility

 • The Physics[ToFieldComponents] command was introduced in Maple 16.