Absence of Non-Trivial Supersymmetries and Grassmann Numbers in Physical State Spaces

Abstract

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.