Grassmann Algebra
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Grassmann Algebra

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ExploringClifford<strong>Algebra</strong>.nb 24<br />

���Α<br />

�<br />

� Γ�� �Β<br />

� m p�<br />

k<br />

Α �<br />

m ���Γ<br />

� p<br />

�Α� m ���Γ<br />

� p<br />

�<br />

��� k�<br />

�<br />

� �� �<br />

k�<br />

���Α<br />

�<br />

�Γ�� � Β<br />

� m p�<br />

k<br />

Hence the right hand sides of the first equations may be written, in the orthogonal case, as:<br />

��1� mp � ����Α<br />

�<br />

�m<br />

�<br />

�<br />

����Α<br />

� Γ<br />

†����<br />

�<br />

�m<br />

p �<br />

�<br />

�<br />

���à p<br />

���à p<br />

�<br />

���� k�<br />

����<br />

���� Γ<br />

� p<br />

����Α<br />

� Γ<br />

†����<br />

�<br />

�m<br />

p �<br />

�<br />

�<br />

���à p<br />

��1� mp � ����<br />

�<br />

����Α<br />

�<br />

���� �<br />

� Β��� ���� Γ<br />

�m<br />

p�<br />

k�<br />

p<br />

Orthogonal intersection<br />

�<br />

� ��� �<br />

k�<br />

�<br />

� ��� �<br />

k�<br />

Αi ���� Βj � 0<br />

Αi ���� Βj � 0<br />

12.28<br />

12.29<br />

Consider the case of three simple elements Α, Β and Γ where Γ is totally orthogonal to both Α<br />

m k p<br />

p<br />

m<br />

and Β (and hence to Α � Β). A simple element Γ is totally orthogonal to an element Α if and only<br />

k<br />

m k<br />

p<br />

m<br />

if Α ���� Γi � 0 for all Γi belonging to Γ. Here, we do not assume that Α and Β are orthogonal.<br />

m p<br />

m k<br />

As before we consider the Clifford product of intersecting elements, but because of the<br />

orthogonality relationships, we can replace the generalized products on the right hand side by<br />

the right hand side of equation 10.39.<br />

���Γ<br />

� p<br />

�<br />

� Α�� �<br />

m�<br />

�<br />

�<br />

��à p<br />

Min�m,k��p<br />

�<br />

� �� � �<br />

k�<br />

Let Μ = Λ - p, then this equals:<br />

2001 4 26<br />

���Γ<br />

� p<br />

�<br />

� Α�� �<br />

m�<br />

�<br />

�<br />

��à p<br />

�<br />

� �� �<br />

k�<br />

�<br />

�<br />

Λ�0<br />

���1�<br />

�<br />

Min�m,k��p<br />

� ��1�<br />

��à p<br />

Λ�p<br />

Min�m,k�<br />

�<br />

���� ��� �<br />

p�<br />

Μ�0<br />

1<br />

��p�m�� � ����<br />

2<br />

Λ �Λ�1� ���Γ<br />

p<br />

�<br />

1<br />

��p�m�� � ����<br />

2<br />

Λ �Λ�1� �<br />

���à p<br />

���1�<br />

�<br />

�<br />

� Α������� �<br />

m�<br />

Λ �<br />

�<br />

��à p<br />

�<br />

���� Γ����Α � ����<br />

p�<br />

m<br />

�<br />

� �� k�<br />

Λ�p �Β k<br />

1<br />

��p���m�� � ���� ��p� ��p�1�<br />

2 ��Αm ����� �Β<br />

Μ k<br />

�<br />

�

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