Grassmann Algebra
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Grassmann Algebra

Grassmann Algebra

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ExploringClifford<strong>Algebra</strong>.nb 20<br />

���Γ<br />

� p<br />

�<br />

� Α�� �<br />

m�<br />

�<br />

�<br />

��à p<br />

Min�m,k�<br />

�<br />

� �� � ��1�<br />

k�<br />

Ω � �<br />

Μ�0<br />

���1�<br />

1<br />

Μ��p�m�Μ� � ���� Μ �Μ�1� �<br />

2 ��Γ<br />

p<br />

��<br />

�<br />

�<br />

�<br />

�<br />

� Α<br />

m�<br />

������� Μ �Β k<br />

���<br />

���� Γ<br />

� p<br />

where �mp� 1 ��� 2 p �p � 1�. Hence we can finally cast the right hand side as a Clifford<br />

product by taking out the second Γ factor.<br />

p<br />

����Γ<br />

�p<br />

�<br />

� Α��� �<br />

m�<br />

�<br />

�<br />

���à p<br />

�<br />

� ��� � ��1�<br />

k�<br />

1<br />

mp���� p �p�1� �<br />

2 ���� �<br />

�<br />

�<br />

���à p<br />

�<br />

� Α��� �<br />

�Β��� ���� Γ<br />

m�<br />

k�<br />

p<br />

12.17<br />

In a similar fashion we can show that the right hand side can be written in the alternative form:<br />

����Γ<br />

�p<br />

�<br />

� Α��� �<br />

m�<br />

�<br />

�<br />

���à p<br />

By reversing the order of Α m and Γ p<br />

����Α<br />

�<br />

� ��� �<br />

�m<br />

p�<br />

�<br />

�<br />

���à p<br />

�<br />

� ��� � ��1�<br />

k�<br />

1<br />

��� p �p�1� �<br />

2 ����Α �<br />

�m<br />

�<br />

�<br />

���à p<br />

�<br />

� ��� k�<br />

����<br />

���� Γ<br />

� p<br />

in the first factor, formula 12.17 can be written as:<br />

�<br />

� ��� � ��1�<br />

k�<br />

1<br />

��� p �p�1� �<br />

2 ���� �<br />

�<br />

�<br />

���à p<br />

�<br />

� Α��� �<br />

�Β��� ���� Γ<br />

m�<br />

k�<br />

p<br />

12.18<br />

12.19<br />

† And by noting that the reverse, Γ of Γp is ��1�<br />

p<br />

1<br />

���� p �p�1�<br />

2 �Γp , we can absorb the sign into the<br />

† formulae by changing one of the Γ to Γ . For example:<br />

p p<br />

In sum: we can write:<br />

2001 4 26<br />

����Α<br />

� Γ<br />

†����<br />

�<br />

�m<br />

p �<br />

�<br />

�<br />

����Α<br />

� Γ<br />

†����<br />

�<br />

�m<br />

p �<br />

�<br />

�<br />

���à p<br />

����Α<br />

� Γ<br />

†����<br />

�<br />

�m<br />

p �<br />

�<br />

�<br />

���à p<br />

���à p<br />

�<br />

� ��� �<br />

k�<br />

����<br />

�<br />

�<br />

�<br />

���à p<br />

�<br />

� Α��� �<br />

�Β��� ���� Γ<br />

m�<br />

k�<br />

p<br />

�<br />

� ��� � ��1�<br />

k�<br />

mp � ����<br />

�<br />

����Α<br />

�<br />

� ��� �<br />

�Β��� ���� Γ<br />

�m<br />

p�<br />

k�<br />

p<br />

�<br />

� ��� � ��1�<br />

k�<br />

mp � ����Α<br />

�<br />

�m<br />

�<br />

�<br />

���à p<br />

�<br />

� ��� k�<br />

����<br />

���� Γ<br />

� p<br />

12.20<br />

12.21<br />

12.22

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