Grassmann Algebra

ExploringCliffordAlgebra.nb 17
The D form
X � ToInteriorProductsD�Α �Β� 4 k
Β k
� Α4 � Α3 � Α2 � Α1 �Α1 ��Β� Α4 � Α3 � Α2 � �
k
Α2 ��Β� Α4 � Α3 � Α1 � �Α3 ��Β� Α4 � Α2 � Α1� �
k
k
Α4 ��Β� Α3 � Α2 � Α1 � �Α1 � Α2 ��Β� Α4 � Α3� �Α1 � Α3 ��Β� Α4 � Α2 � �
k
k
k
Α1 � Α4 ��Β� Α3 � Α2 � �Α2 � Α3 ��Β� Α4 � Α1� �Α2 � Α4 ��Β� Α3 � Α1 � �
k
k
k
Α3 � Α4 ��Β� Α2 � Α1 � �Α1 � Α2 � Α3 ��Β� Α4� �Α1 � Α2 � Α4 ��Β� Α3 � �
k
k
k
Α1 � Α3 � Α4 ��Β� Α2 � �Α2 � Α3 � Α4 ��Β� Α1� �Α1 � Α2 � Α3 � Α4 � Β
k
k
k
The C form
Note that since the exterior product has higher precedence in Mathematica than the interior
product, Α1 � Β ���� Α2 is equivalent to �Α1 � Β� ���� Α2 , and no parentheses are necessary in the terms
k
k
of the C form.
ToInteriorProductsC�Α �Β� 4 k
Β k
� Α4 � Α3 � Α2 � Α1 �Α1 � Β � Α4 � Α3 � Α2 �Α2 � Β � Α4 � Α3 � Α1 �
k
k
Α3 � Β � Α4 � Α2 � Α1 �Α4 � Β � Α3 � Α2 � Α1 �Α1 � Α2 � Β � Α4 � Α3 �
k
k
k
Α1 � Α3 � Β � Α4 � Α2 �Α1 � Α4 � Β � Α3 � Α2 �Α2 � Α3 � Β � Α4 � Α1 �
k
k
k
Α2 � Α4 � Β � Α3 � Α1 �Α3 � Α4 � Β � Α2 � Α1 �Α1 � Α2 � Α3 � Β � Α4 �
k
k
k
Α1 � Α2 � Α4 � Β � Α3 �Α1 � Α3 � Α4 � Β � Α2 �Α2 � Α3 � Α4 � Β � Α1 �Α1 � Α2 � Α3 � Α4 � Β
k
k
k
k
Either form is easy to write down directly by observing simple mnemonic rules.
� A mnemonic way to write down a general Clifford product
We can begin developing the D form mnemonically by listing the Αi . This list has the same
Λ
form as the basis of the Grassmann algebra of a 4-space.
� 1. Calculate the Α i
Λ
These are all the essentially different combinations of factors of all grades. The list has the same
form as the basis of the Grassmann algebra whose n-element is Α m .
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