GrassmannCalculus | Wolfram Resource Object

GrassmannCalculus`

StandardOrderingFunction

StandardOrderingFunction[ A , B ]
	returns True if A and B are in the required standard order for GrassmannSimplify , and False otherwise.

Details and Options



Examples

(1)

Basic Examples

(1)

In[1]:=

<<GrassmannCalculus`

In[2]:=

★A;

★ℬ

5

;

The following all return

True

because the expressions are the same.

In[3]:=

{x,x},{

e

1

,

e

1

},{

e

1

⋀

e

2

,

e

1

⋀

e

2

},{x⋀y,x⋀y},{y⋀x,y⋀x},{

α

2

,

α

2

},{pQZ,pQZ}

StandardOrderingFunction

@@#&/@%

Out[3]=

{x,x},{

e

1

,

e

1

},{

e

1

⋀

e

2

,

e

1

⋀

e

2

},{x⋀y,x⋀y},{y⋀x,y⋀x},{

α

2

,

α

2

},{pQZ,pQZ}

Out[3]=

{True,True,True,True,True,True,True}

The following all return

True

because the first element precedes the second element in the standard Grassmann order. Basis symbols come before other types of factors.

In[4]:=

{

e

1

,

e

2

},{

e

1

,

e

1

⋀

e

2

},{

e

5

,x},{

e

5

,

α

2

},{

e

5

,pQZ}

StandardOrderingFunction

@@#&/@%

Out[4]=

{

e

1

,

e

2

},{

e

1

,

e

1

⋀

e

2

},{

e

5

,x},{

e

5

,

α

2

},{

e

5

,pQZ}

Out[4]=

{True,True,True,True,True}

Within basis symbols they are ordered by their position in the Basis.

In[5]:=

Table[{

e

3

,b},{b,Basis}]

StandardOrderingFunction

@@#&/@%

Out[5]=

{{

e

3

,

e

1

},{

e

3

,

e

2

},{

e

3

,

e

3

},{

e

3

,

e

4

},{

e

3

,

e

5

}}

Out[5]=

{False,False,True,True,True}

Here a Basis not in the standard Mathematica sort order is set. StandardOrderingFunction goes by the order in the Basis.

In[6]:=

Basis={pp,aa,qq,bb,hh};Table[{qq,b},{b,Basis}]

StandardOrderingFunction

@@#&/@%★A;

★ℬ

5

;

Out[6]=

{{qq,pp},{qq,aa},{qq,qq},{qq,bb},{qq,hh}}

Out[6]=

{False,False,True,True,True}

Vector symbols are ordered by their Mathematica sort order.

In[7]:=

Table{u,b},b,

★v



StandardOrderingFunction

@@#&/@%

Out[7]=

{{u,p},{u,q},{u,r},{u,s},{u,t},{u,u},{u,v},{u,w},{u,x},{u,y},{u,z}}

Out[7]=

{False,False,False,False,False,True,True,True,True,True,True}

The same for graded symbols.

In[8]:=

Table

β

2

,

b

2

,{b,{α,β,γ}}

StandardOrderingFunction

@@#&/@%

Out[8]=



β

2

,

α

2

,

β

2

,

β

2

,

β

2

,

γ

2



Out[8]=

{False,True,True}

In[9]:=

Table

β

3

,

β

b

,{b,0,5}

StandardOrderingFunction

@@#&/@%

Out[9]=



β

3

,

β

0

,

β

3

,

β

1

,

β

3

,

β

2

,

β

3

,

β

3

,

β

3

,

β

4

,

β

3

,

β

5



Out[9]=

{False,False,False,True,True,True}

In[10]:=

StandardOrderingFunction



α

3

,

β

2



Out[10]=

True

The following elements are in standard Grassmann order. Note that scalar symbols, vector symbols and arbitrary symbols are lumped together and other Grassmann expressions come last.

In[11]:=

testItems=

e

3

,a,pQZ,s,

α

2

,

e

1

⋀

e

3

Outer[{#1,#2}&,testItems,testItems]//MatrixFormMap

StandardOrderingFunction

@@#&,%,{2}//MatrixForm

Out[11]=



e

3

,a,pQZ,s,

α

2

,

e

1

⋀

e

3



Out[11]//MatrixForm=

e

3

e

3



e

3

a



e

3

pQZ



e

3

s



e

3

α

2

e

3

e

1

⋀

e

3



a

e

3





a





a

pQZ





a

s



a

α

2



a

e

1

⋀

e

3



pQZ

e

3



pQZ

a



pQZ



pQZ

s



pQZ

α

2

pQZ

e

1

⋀

e

3



s

e

3





s

a





s

pQZ





s



s

α

2



s

e

1

⋀

e

3



α

2

e

3

α

2

a

α

2

pQZ

α

2

s

α

2

α

2

α

2

e

1

⋀

e

3

e

1

⋀

e

3

e

3



e

1

⋀

e

3

a



e

1

⋀

e

3

pQZ



e

1

⋀

e

3

s



e

1

⋀

e

3

α

2

e

1

⋀

e

3

e

1

⋀

e

3

Out[11]//MatrixForm=

True	True	True	True	True	True
False	True	True	True	True	True
False	False	True	True	True	True
False	False	False	True	True	True
False	False	False	False	True	True
False	False	False	False	False	True

In[12]:=

Clear[testItems]

SeeAlso

SortToStandardOrdering

▪

▪

▪

▪

RelatedGuides

▪

Simplification

""