GrassmannParity - Maple Help

Physics[GrassmannParity] - compute the Grassmannian parity, as 0, 1 or undefined, according to whether an expression is commutative, anticommutative or noncommutative

 Calling Sequence GrassmannParity(expression)

Parameters

 expression - algebraic expression, or relation between them, or a set or list of them

Description

 • The GrassmannParity command computes the Grassmannian parity of expression, that is, 0, 1 or undefined, according to whether expression is commutative, anticommutative or noncommutative. In this sense, the parity here is equivalent to the type.

Examples

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

Set theta as an anticommutative prefix (see Setup)

 > $\mathrm{Setup}\left(\mathrm{anticommutativepre}=\mathrm{\theta }\right)$
 $\mathrm{* Partial match of \text{'}}{}\mathrm{anticommutativepre}{}\mathrm{\text{'} against keyword \text{'}}{}\mathrm{anticommutativeprefix}{}\text{'}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{\mathrm{\theta }}\right\}\right]$ (2)
 > $a\mathrm{\theta }\left[1\right]\mathrm{\theta }\left[2\right]+b$
 ${a}{}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{+}{b}$ (3)

The parity of (3) is 0 despite the presence of anticommutative variables: a product of two of them is overall commutative

 > $\mathrm{GrassmannParity}\left(\right)$
 ${0}$ (4)

A commutative function of commutative and anticommutative variables: its parity is zero

 > $F\left(x,y,\mathrm{\theta }\left[1\right],\mathrm{\theta }\left[2\right]\right)$
 ${F}{}\left({x}{,}{y}{,}{{\mathrm{\theta }}}_{{1}}{,}{{\mathrm{\theta }}}_{{2}}\right)$ (5)
 > $\mathrm{GrassmannParity}\left(\right)$
 ${0}$ (6)

A taylor expansion as well as an exact expansion for it respectively performed with Gtaylor and ToFieldComponents

 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right]\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){+}{{\mathrm{D}}}_{{3}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){}{{\mathrm{\theta }}}_{{1}}$ (7)
 > $\mathrm{ToFieldComponents}\left(\right)$
 ${\mathrm{_F1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{1}}{}{\mathrm{_Q1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}$ (8)

Note that the expansion performed with Gtaylor does not preserve the parity of (5) while the one performed with ToFieldComponents does:

 > $\mathrm{GrassmannParity}\left(\right)$
 ${\mathrm{undefined}}$ (9)
 > $\mathrm{GrassmannParity}\left(\right)$
 ${0}$ (10)

The coefficient of order zero of both expansions preserves the parity; the difference appears with respect to the coefficient of order 1

 > $\mathrm{Coefficients}\left(=,\mathrm{\theta }\left[1\right],0\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){=}{\mathrm{_F1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}\right)$ (11)
 > $\mathrm{map}\left(\mathrm{GrassmannParity},\right)$
 ${0}{=}{0}$ (12)
 > $\mathrm{Coefficients}\left(=,\mathrm{\theta }\left[1\right],1\right)$
 ${{\mathrm{D}}}_{{3}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){=}{-}{\mathrm{_Q1}}{}\left({x}{,}{y}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{{\mathrm{\theta }}}_{{2}}$ (13)
 > $\mathrm{map}\left(\mathrm{GrassmannParity},\right)$
 ${0}{=}{1}$ (14)
 > 

To understand this difference between the Taylor and the exact expansions performed with Gtaylor and ToFieldComponents see the expansion's definitions in the respective help pages

Compatibility

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