GrassmannCalculus`
Condense  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
★A;
Here is a simple expression which we expand using . Condense reverse-expands the result.
GrassmannExpand
In[3]:=
(x+y+z)⋀(u+v+w)
[%]
[%]
 | ★ℰ |
| Condense |
Out[3]=
(x+y+z)⋀(u+v+w)
Out[3]=
x⋀u+x⋀v+x⋀w+y⋀u+y⋀v+y⋀w+z⋀u+z⋀v+z⋀w
Out[3]=
(x+y+z)⋀(u+v+w)
Of course, if one or more terms are missing, the condensing will not be as complete. Here we remove the last product z⋀w to get a modified expression.
In[4]:=
x⋀u+x⋀v+x⋀w+y⋀u+y⋀v+y⋀w+z⋀u+z⋀v
[%]
| Condense |
Out[4]=
x⋀u+x⋀v+x⋀w+y⋀u+y⋀v+y⋀w+z⋀u+z⋀v
Out[4]=
(x+y)⋀(u+v+w)+z⋀(u+v)
Here is the Clifford product of two Grassmann numbers in a 2-space. We expand the product and then Condense to recover the original expression
In[5]:=
★A;
;Z=
⋄
[%]
[%]
 | ★ℬ |
2
 | ★ |
a
| ★ |
b
| ★ℰ |
| Condense |
Out[5]=
(+++⋀)⋄(+++⋀)
a
0
a
1
e
1
a
2
e
2
a
3
e
1
e
2
b
0
b
1
e
1
b
2
e
2
b
3
e
1
e
2
Out[5]=
a
0
b
0
a
0
b
1
e
1
a
0
b
2
e
2
a
0
b
3
e
1
e
2
a
1
e
1
b
0
a
1
e
1
b
1
e
1
a
1
e
1
b
2
e
2
a
1
e
1
b
3
e
1
e
2
a
2
e
2
b
0
a
2
e
2
b
1
e
1
a
2
e
2
b
2
e
2
a
2
e
2
b
3
e
1
e
2
a
3
e
1
e
2
b
0
a
3
e
1
e
2
b
1
e
1
a
3
e
1
e
2
b
2
e
2
a
3
e
1
e
2
b
3
e
1
e
2
Out[5]=
(+++⋀)⋄(+++⋀)
a
0
a
1
e
1
a
2
e
2
a
3
e
1
e
2
b
0
b
1
e
1
b
2
e
2
b
3
e
1
e
2
You can also condense more general expressions.
In[6]:=
A+2
[(x+y+z)⋀(u+v+w)]+3
⋄
⊖
[(x+y+z)⋀(u+v+w)]
[%]
 | ★ℰ |
| ★ℰ |
 | ★ |
a
| ★ |
b
| ★ℰ |
| Condense |
Out[6]=
A+3(⋄+⋄()+⋄()+⋄(⋀)+()⋄+()⋄()+()⋄()+()⋄(⋀)+()⋄+()⋄()+()⋄()+()⋄(⋀)+(⋀)⋄+(⋀)⋄()+(⋀)⋄()+(⋀)⋄(⋀))⊖(x⋀u+x⋀v+x⋀w+y⋀u+y⋀v+y⋀w+z⋀u+z⋀v+z⋀w)+2(x⋀u+x⋀v+x⋀w+y⋀u+y⋀v+y⋀w+z⋀u+z⋀v+z⋀w)
a
0
b
0
a
0
b
1
e
1
a
0
b
2
e
2
a
0
b
3
e
1
e
2
a
1
e
1
b
0
a
1
e
1
b
1
e
1
a
1
e
1
b
2
e
2
a
1
e
1
b
3
e
1
e
2
a
2
e
2
b
0
a
2
e
2
b
1
e
1
a
2
e
2
b
2
e
2
a
2
e
2
b
3
e
1
e
2
a
3
e
1
e
2
b
0
a
3
e
1
e
2
b
1
e
1
a
3
e
1
e
2
b
2
e
2
a
3
e
1
e
2
b
3
e
1
e
2
Out[6]=
A+3(+++⋀)⋄(+++⋀)⊖(x+y+z)⋀(u+v+w)+2(x+y+z)⋀(u+v+w)
a
0
a
1
e
1
a
2
e
2
a
3
e
1
e
2
b
0
b
1
e
1
b
2
e
2
b
3
e
1
e
2
 |
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