GrassmannCalculus`
CliffordProduct (⋄)  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the book default and turn on Precedence showing.
In[2]:=
★A;
 | ★P |
These inputs are equivalent:
In[3]:=
{CliffordProduct[x,y],Diamond[x,y],x⋄y}
Out[3]=
{x⋄y,x⋄y,x⋄y}
The Clifford product is listable.
In[4]:=
u∘{w,x}{u,v}⋄{w,x}⋄{y,z}
Out[4]=
{u∘w,u∘x}
Out[4]=
{u⋄w⋄y,v⋄x⋄z}
The Clifford product transformed to conventional notation.
In[5]:=
p⋄q%//
| ToInnerProducts |
Out[5]=
p⋄q
Out[5]=
p⊖q+p⋀q
Notice that the precedence of Clifford products is higher than exterior products so, unless , the two following expressions give different results.
p⊖x=0
In[6]:=
p⋀q⋄x%//
%//
| ToInnerProducts |
| ★ |
Out[6]=
p⋀(q⋄x)
Out[6]=
p⋀(q⊖x+q⋀x)
Out[6]=
p(q⊖x)+p⋀q⋀x
In[7]:=
(p⋀q)⋄x%//
%//
| ToInnerProducts |
| ★ |
Out[7]=
(p⋀q)⋄x
Out[7]=
-q(p⊖x)+p(q⊖x)+p⋀q⋀x
Out[7]=
-q(p⊖x)+p(q⊖x)+p⋀q⋀x
A Clifford product in terms of basis vectors. Here we do a step by step reduction.
In[8]:=
(⋀)⋄(⋀)%//
%//
%//
%//
e
1
e
2
e
2
e
3
| ConvertCliffordToGeneralized |
| ConvertGeneralizedToInterior |
| ToScalarProducts |
| ToMetricElements |
Out[8]=
(⋀)⋄(⋀)
e
1
e
2
e
2
e
3
Out[8]=
(⋀)(⋀)-(⋀)(⋀)-(⋀)(⋀)
e
1
e
2
△
0
e
2
e
3
e
1
e
2
△
1
e
2
e
3
e
1
e
2
△
2
e
2
e
3
Out[8]=
-((⋀)⊖(⋀))-((⋀)⊖)⋀+((⋀)⊖)⋀+⋀⋀⋀
e
1
e
2
e
2
e
3
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
1
e
2
e
2
e
3
Out[8]=
(⊖)(⊖)-(⊖)(⊖)-(⊖)⋀+(⊖)⋀-(⊖)⋀
e
1
e
3
e
2
e
2
e
1
e
2
e
2
e
3
e
2
e
3
e
1
e
2
e
2
e
2
e
1
e
3
e
1
e
2
e
2
e
3
Out[8]=
e
1
e
3
This could be processed in a single step.
In[9]:=
(⋀)⋄(⋀)%//
e
1
e
2
e
2
e
3
| ToMetricElements |
Out[9]=
(⋀)⋄(⋀)
e
1
e
2
e
2
e
3
Out[9]=
e
1
e
3
The following expands a Clifford product of graded symbols.
In[10]:=
a⋄b%//
%//
%//
%//
α
3
β
2
| SimplifyCliffordProducts |
| ConvertCliffordToGeneralized |
| ConvertGeneralizedToInterior |
| ToScalarProducts |
Out[10]=
(a)⋄(b)
α
3
β
2
Out[10]=
ab⋄
α
3
β
2
Out[10]=
ab+-
α
3
△
0
β
2
α
3
△
1
β
2
α
3
△
2
β
2
Out[10]=
ab-⊖+(⊖)⋀-(⊖)⋀+⋀
α
3
β
2
α
3
β
2
β
3
α
3
β
3
β
2
α
3
β
2
Out[10]=
-ab(-(⊖)(⊖)+(⊖)(⊖))+ab(-(⊖)(⊖)+(⊖)(⊖))-ab(-(⊖)(⊖)+(⊖)(⊖))-ab(⊖)⋀⋀+ab(⊖)⋀⋀+ab(⊖)⋀⋀-ab(⊖)⋀⋀-ab(⊖)⋀⋀+ab(⊖)⋀⋀
β
2
α
6
β
3
α
5
β
2
α
5
β
3
α
6
α
4
β
2
α
6
β
3
α
4
β
2
α
4
β
3
α
6
α
5
β
2
α
5
β
3
α
4
β
2
α
4
β
3
α
5
α
6
α
6
β
3
α
4
α
5
β
2
α
6
β
2
α
4
α
5
β
3
α
5
β
3
α
4
α
6
β
2
α
5
β
2
α
4
α
6
β
3
α
4
β
3
α
5
α
6
β
2
α
4
β
2
α
5
α
6
β
3
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