Characterization of canonical Bose–Fermi systems by ‘‘anti-Hermitian’’ symplectic forms

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Abstract
A complexification of graded manifold theory is given, following Kostant’s procedure, but with a reality concept defined by ‘‘classical’’ correspondent to Hermitian conjugation in quantum mechanics. Presented herein are definitions of graded manifolds with ‘‘Hermite’’ coordinates and of ‘‘Hermiticity’’ on graded differential forms and graded vector fields, all in the coordinate independent way, and characterization of ‘‘classical’’ Bose–Fermi systems by graded symplectic forms ω which are, here, ‘‘anti‐Hermitian’’ nonsingular closed 2‐forms of z2 grading 0. Also given are Frobenius’ theorem on the graded manifold with ‘‘Hermiticity,’’ and Darboux’s theorem, ω=∑kdpkΛdqk+i∑j +i∑j(εj/2)dsjΛdsj, where all coordinates are ‘‘Hermite,’’ pk† =pk, qk†=qk, sj† =sj. Naive quantization procedures fit in with these systems.