GrassmannCalculus`
ConvertComplements  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In a 3 dimensional space, the complement of a scalar is a 3-element, the complement of a 1-element is a 2-element, the complement of a 2-element is a 1-element, and the complement of a 3-element is a scalar.
In[2]:=
★A;
[%]
[%]
 | GrassmannBases |
 | GrassmannComplement |
| ConvertComplements |
Out[2]=
{{1},{,,},{⋀,⋀,⋀},{⋀⋀}}
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
Out[2]=
{},{,,},{⋀,⋀,⋀},{⋀⋀}
1
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
Out[2]=
{{⋀⋀},{⋀,-(⋀),⋀},{,-,},{1}}
e
1
e
2
e
3
e
2
e
3
e
1
e
3
e
1
e
2
e
3
e
2
e
1
Here we convert the complement of a general Grassmann number. First we compose it using .
ComposeGrassmannNumber
In[3]:=
★A;X=
[x]
 | ComposeGrassmannElement |
Out[3]=
x
0
e
1
x
1
e
2
x
2
e
3
x
3
x
4
e
1
e
2
x
5
e
1
e
3
x
6
e
2
e
3
x
7
e
1
e
2
e
3
To get the complement of this expression, you can enter , or simply enter by using the bar on the Basic Operations palette.
GrassmannComplement[X]
X
In[4]:=
X
Out[4]=
x
0
e
1
x
1
e
2
x
2
e
3
x
3
x
4
e
1
e
2
x
5
e
1
e
3
x
6
e
2
e
3
x
7
e
1
e
2
e
3
Applying to this expression expresses it directly as another Grassmann number.
ConvertComplements
In[5]:=
| ConvertComplements |
X
Out[5]=
e
3
x
4
e
2
x
5
e
1
x
6
x
7
x
3
e
1
e
2
x
2
e
1
e
3
x
1
e
2
e
3
x
0
e
1
e
2
e
3
In the case of a non-Euclidean metric, the complement of a basis element may be more complicated. Here we declare a 2-space with a general metric.
In[6]:=
| ★ℬ |
2
| DeclareMetric |
We can compose a Grassmann number as before:
In[7]:=
X=
[x]
| ComposeGrassmannElement |
Out[7]=
x
0
e
1
x
1
e
2
x
2
x
3
e
1
e
2
Applying to the complement of this expression now involves the metric elements.
ConvertComplements
In[8]:=
Xc=
[]
| ConvertComplements |
X
Out[8]=
★g
x
3
e
2
x
1
g
1,1
x
2
g
1,2
★g
e
1
x
1
g
1,2
x
2
g
2,2
★g
x
0
e
1
e
2
★g
The symbol ★g represents the determinant of the metric tensor. It is used as an alias for the determinant in order to visually simplify outputs of this sort. ★g can be evaluated by applying .
ToMetricElements
In[9]:=
| ToMetricElements |
Out[9]=
e
2
x
1
g
1,1
x
2
g
1,2
-+
2
g
1,2
g
1,1
g
2,2
e
1
x
1
g
1,2
x
2
g
2,2
-+
2
g
1,2
g
1,1
g
2,2
x
3
-+
+2
g
1,2
g
1,1
g
2,2
x
0
e
1
e
2
-+
2
g
1,2
g
1,1
g
2,2
ConvertComplements
In[10]:=
★A;
;X=
[x]
| ★ |
2
 | ComposeGrassmannElement |
Out[10]=
x
0
x
1
e
1
x
2
e
2
x
3
x
4
e
1
x
5
e
2
x
6
e
1
e
2
x
7
e
1
e
2
In[11]:=
| ConvertComplements |
X
Out[11]=
e
2
x
4
e
1
x
5
x
7
e
2
x
2
e
1
x
3
x
6
x
0
e
1
e
2
x
1
e
1
e
2
If we now change to a non-Euclidean metric and convert we get a result of the same form, but in which the scalars involve the metric tensor components.
X
In[12]:=
| DeclareMetric |
| ConvertComplements |
X
Out[12]=
★g
x
7
e
2
x
4
g
1,1
x
5
g
1,2
★g
e
1
x
4
g
1,2
x
5
g
2,2
★g
★g
x
6
e
2
x
2
g
1,1
x
3
g
1,2
★g
e
1
x
2
g
1,2
x
3
g
2,2
★g
x
0
e
1
e
2
★g
x
1
e
1
e
2
★g
In[13]:=
Clear[X,Xc]
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