Grassmann Algebra

ExploringCliffordAlgebra.nb 44
algebra.
4) Verify that the resulting table is the same as the original table.
We perform these steps in sequence. Because of the size of the output, only the electronic
version will show the complete tables.
Step 1: Replace symbols for entities and operations
C3 � �C2 �� ReplaceNegativeUnit� �. �1 ��,e1 �Σ1 ,e2 �Σ2 ,e3 �Σ3� �.
Wedge � Dot; PaletteForm�C3 �
� Σ1 Σ2 Σ3 Σ1 .Σ2 Σ1 .Σ3 Σ2.
Σ1 � Σ1.Σ2 Σ1 .Σ3 Σ2 Σ3 Σ1 .Σ2
Σ2 �Σ1 .Σ2 � Σ2 .Σ3 �Σ1 �Σ1 .Σ2 .Σ3 Σ3
Σ3 �Σ1 .Σ3 �Σ2 .Σ3 � Σ1 .Σ2 .Σ3 �Σ1 �Σ
Σ1 .Σ2 �Σ2 Σ1 Σ1.Σ2 .Σ3 �� �Σ2.Σ3 Σ1.
Σ 1 .Σ 3 �Σ 3 �Σ 1.Σ 2 .Σ 3 Σ 1 Σ 2 .Σ 3 �� �Σ 1
Σ 2 .Σ 3 Σ 1 .Σ 2 .Σ 3 �Σ 3 Σ 2 �Σ 1.Σ 3 Σ 1 .Σ 2 �
Σ1 .Σ2.Σ3 Σ2 .Σ3 �Σ1 .Σ3 Σ1 .Σ2 �Σ3 Σ2 �Σ
Step 2: Substitute matrices and calculate
1 0
C4 � C3 �. �� ��
0 1 � , Σ1 � �
MatrixForm�C4�
0 1
1 0 � , Σ2
0 ��
� �
� 0 �, Σ3
1 0
� �
0 �1 ��;
� 1 0 0 1 0 �� 1 0 � 0 0 �1 0 �
����������������������������������������������������������������������������
� � � � � � � � � � � � �
0 1 1 0 � 0 0 �1 0 �� 1 0 � 0
�
�
0 1 1 0 � 0 0 �1 0 �� 1 0 � 0
� � � � � � � � � � � � �
1 0 0 1 0 �� 1 0 � 0 0 �1 0 � �
0 �� �� 0 1 0 0 � 0 �1 �� 0 1 0
� � � � � � � � � � � � �
� 0 0 � 0 1 � 0 �1 0 0 �� 0 �1 �
1 0 0 1 0 �� 1 0 � 0 0 �1 0 �
� � � � � � � � � � � � �
0 �1 �1 0 �� 0 0 1 0 � �1 0 �� 0 �
� 0 0 � 0 1 � 0 �1 0 0 �� 0 �1
� � � � � � � � � � � � �
0 �� �� 0 1 0 0 � 0 �1 �� 0 1 0 �
0 �1 �1 0 �� 0 0 1 0 � �1 0 �� 0
� � � � � � � � � � � � �
1 0 0 1 0 �� 1 0 � 0 0 �1 0 � �
0 � � 0 �1 0 0 �� 0 1 � 0 �1 0
� � � � � � � � � � � � �
� 0 0 � 0 1 � 0 �1 0 0 �� 0 �1
� 0 0 � 0 1 � 0 �1 0 0 �� 0 �1
� � � � � � � � � � � � �
0 � � 0 �1 0 0 �� 0 1 � 0 �1 0
Step 3:
Here we let the first row (column) of the product table correspond back to the basis elements of
the Grassmann representation, and make the substitution: throughout the whole table.
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