compute the Grassmannian parity, as 0, 1 or undefined, according to whether an expression is commutative, anticommutative or noncommutative - Maple Help

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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

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Set theta as an anticommutative prefix (see Setup)

Setupanticommutativepre=θ

* Partial match of 'anticommutativepre' against keyword 'anticommutativeprefix'

anticommutativeprefix=_λ,θ

(2)

aθ1θ2+b

aθ1θ2+b

(3)

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

GrassmannParity

0

(4)

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

Fx,y,θ1,θ2

Fx,y,θ1,θ2

(5)

GrassmannParity

0

(6)

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

Gtaylor,θ1

Fx,y,0,θ2+D3Fx,y,0,θ2θ1

(7)

ToFieldComponents

_F1x,yθ1_Q1x,yθ2_Q2x,y+_F2x,yθ1θ2

(8)

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

GrassmannParity

undefined

(9)

GrassmannParity

0

(10)

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

Coefficients=,θ1,0

Fx,y,0,θ2=_F1x,yθ2_Q2x,y

(11)

mapGrassmannParity,

0=0

(12)

Coefficients=,θ1,1

D3Fx,y,0,θ2=_Q1x,y+_F2x,yθ2

(13)

mapGrassmannParity,

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

See Also

anticommutative, Coefficients, commutative, Physics, Physics conventions, Physics examples, relation, series, Setup, ToFieldComponents


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