Grassmann Algebra

ExpTheGeneralizedProduct.nb 29
�Α����� �Β������ �Γ
m 1 k 1 p
� ��1�Ξ � � ���Β
�k
����� 1 �à p
���������
�Α � ��1�
� 1 m Ψ � �
�
���Γ����� �Α
p 1 m
���������
�Β � 0
� 1 k
10.28
The signs ��1� Ξ and ��1� Ψ are at this point unknown, but if the conjecture is true we expect
them to be simple functions of m, k, p and their products. We conjecture therefore that in any
particular case they will be either +1 or –1.
� Exploring the conjecture
The first step therefore is to test some cases to see if the formula fails for some combination of
signs. We do that by setting up a function that calculates the formula for each of the possible
sign combinations, returning True if any one of them satisfies the formula, and False
otherwise. Then we create a table of cases. Below we let the grades m, k, and p range from 0 to
3 (that is, two instances of odd parity and two instances of even parity each, in all
combinations). Although we have written the function as printing the intermediate results as
they are calculated, we do not show this output as the results are summarized in the final list.
TestPostulate��m_, k_, p_�� :�
Module��X, Y, Z, Α, Β, Γ�, X� ToScalarProducts��Α � Β� � Γ�; m 1 k 1 p
Y � ToScalarProducts� ����Β
�
� Γ��� � Α�; � k 1 p�
1 m
Z � ToScalarProducts� ����Γ
�
� Α��� � Β�; � p 1 m � 1 k
TrueQ�X � Y � Z �� 0 �� X � Y � Z �� 0 �� X � Y � Z �� 0 �� X � Y � Z �� 0��
Table�T � TestPostulate��m, k, p��; Print���m, k, p�, T��;
T, �m, 0, 3�, �k, 0, 3�, �p, 0, 3�� �� Flatten
�True, True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True�
It can be seen that the formula is valid for some values of Ξ and Ψ in the first 64 cases. This
gives us some confidence that our efforts may not be wasted if we wished to prove the formula
and/or find the values of Ξ and Ψ an terms of m, k, and p.
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