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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 Fx,θ be a commutative function F of x (commutative) and θ (anticommutative); the expansion returned by ToFieldComponents is according to

Fx,θ=_F1xθ_Q1x

  

This expansion involves only functions of commutative variables _F1x and _Q1x, where the function _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 (_Q1) of commutative variables (x), introduced by ToFieldComponents, you can optionally indicate the prefix to be used (instead of _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

Fx,θ=_F2x+_F3x_λ1θ

  

Note the introduction of anticommutative parameters prefixed by _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

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Set first θ and Q as prefixes for variables of type/anticommutative (see Setup)

Setupanticommutativepre=θ,Q

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

_______________________________________________________

anticommutativeprefix=Q,θ

(2)

A commutative function (its name F is of type commutative) of commutative x,y,z and anticommutative variables θ1,θ2,θ3

Fx,y,z,θ1,θ2,θ3

Fx,y,z,θ1,θ2,θ3

(3)

The expansion of (3) is a polynomial in θ1,θ2 and θ3 with terms of degree 0 and 1 with respect to each of θ1,θ2 and θ3. 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 θ1,θ2 and θ3

ToFieldComponents

_F1x,y,zθ1_Q1x,y,zθ2_Q2x,y,zθ3_Q3x,y,z+_F2x,y,zθ1θ2+_F3x,y,zθ1θ3+_F4x,y,zθ2θ3θ1θ2θ3_Q4x,y,z

(4)

This expansion preserves the parity of (3)

GrassmannParity=

0=0

(5)

The expansion of (3) using only commutative functions

ToFieldComponents,useonly

* Partial match of 'useonly' against keyword 'useonlycommutativefunctions'

_F5x,y,z+_F6x,y,z_λ1θ1+_F7x,y,z_λ2θ2+_F8x,y,z_λ3θ3+_F9x,y,zθ1θ2+_F10x,y,zθ1θ3+_F11x,y,zθ2θ3+_F12x,y,z_λ4θ1θ2θ3

(6)

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

Gtaylor,θ1,θ2,θ3

Fx,y,z,0,0,0+D4Fx,y,z,0,0,0θ1+D5Fx,y,z,0,0,0θ2+D4,5Fx,y,z,0,0,0θ1θ2+D6Fx,y,z,0,0,0θ3+D4,6Fx,y,z,0,0,0θ1θ3+D5,6Fx,y,z,0,0,0θ2θ3+D4,5,6Fx,y,z,0,0,0θ1θ2θ3

(7)

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

ToSuperfields

Fx,y,z,θ1,θ2,θ3

(8)

See Also

anticommutative, Coefficients, commutative, GrassmannParity, Gtaylor, PerformOnAnticommutativeSystem, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, taylor

Compatibility

• 

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

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For more information on Maple 16 changes, see Updates in Maple 16.