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Absence of Non-Trivial Supersymmetries and Grassmann Numbers in Physical State Spaces
This paper reviews the well-known fact that nilpotent Hermitian operators on
physical state spaces are zero, thereby indicating that the supersymmetries and
"Grassmann numbers" are also zero on these spaces. Next, a positive definite
inner product of a Grassmann algebra is demonstrated, constructed using a Hodge
dual operator which is similar to that of differential forms. From this
example, it is shown that the Hermitian conjugates of the basis do not
anticommute with the basis and, therefore, the property that "Grassmann
numbers" commute with "bosonic quantities" and anticommute with "fermionic
quantities", must be revised. Hence, the fundamental principles of
supersymmetry must be called into question.Comment: 10 page
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