Grassmann Algebra

ExploringCliffordAlgebra.nb 31
����Α
�
� Ε � Θ � Γ��� �
�m
s z p�
�
�
���à p
�
� Θ � Β � ∆��� � �∆ � Ε � Ω � Θ� z k q�
q s r z
� ��1� sk � ����Γ
† �
���� Γp �����Θ
†
���� Θz ��
�p
� z
† †
�Ε ���� Εs ���∆ ���� ∆q ���Α � Β � Ω � Θ� s q m k r z
12.48
A mnemonic for making this transformation is then
1. Rearrange the factors in a Clifford product to get common factors adjacent to the Clifford
product symbol, taking care to include any change of sign due to the quasi-commutativity of the
exterior product.
2. Replace the common factors by their inner product, but with one copy being reversed.
3. If there are no common factors in a Clifford product, the Clifford product can be replaced by
the exterior product.
Remember that for these relations to hold all the elements must be totally orthogonal to each
other.
Note that if, in addition, the 1-element factors of any of these elements, Γ say, are orthogonal to
p
each other, then:
Γ ���� Γ � �Γ1 ���� Γ1 ���Γ2 ���� Γ2 �����Γp ���� Γp �
p p
Associativity of non-orthogonal elements
Consider the Clifford product �Α �Β��Γ where there are no restrictions on the factors Α, Β and
m k p
m k
Γ. It has been shown in Section 6.3 that an arbitrary simple m-element may be expressed in
p
terms of m orthogonal 1-element factors (the Gram-Schmidt orthogonalization process).
Suppose that Ν such orthogonal 1-elements e1 ,e2 , �, eΝ have been found in terms of which
Α, Β, and Γ
m k p
the ei as ej
m
can be expressed. Writing the m-elements, k-elements and p-elements formed from
, er , and es
k
p
Α m ��a j �ej
m
respectively, we can write:
Β k
��b r �er
k
Thus we can write the Clifford product as:
�Α �Β��Γ m k p
� ����
a
�
j �ej
m
��b r �er
k
Γ p
��c s �es
p
���
��c
�
s �es
p
����a j �b r �c s � ���ej
� m
� er
k
���
� es
� p
But we have already shown in the previous section that the Clifford product of orthogonal
elements is associative. That is:
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