A Generalization of the Bargmann-Fock Representation to Supersymmetry by Holomorphic Differential Geometry

Abstract

In the Bargmann-Fock representation the coordinates $z^i$ act as bosonic creation operators while the partial derivatives ${\partial }_{z^j}$ act as annihilation operators on holomorphic $0$-forms as states of a $D$-dimensional bosonic oscillator. Considering also $p$-forms and further geometrical objects as the exterior derivative and Lie derivatives on a holomorphic ${\mathbf{C}}^D$, we end up with an analogous representation for the $D$-dimensional supersymmetric oscillator. In particular, the supersymmetry multiplet structure of the Hilbert space corresponds to the cohomology of the exterior derivative. In addition, a 1-complex parameter group emerges naturally and contains both time evolution and a homotopy related to cohomology. Emphasis is on calculus.

arXiv comment11 pages, LaTeX

References in corpus (1)

• Local Differential Geometry as a Representation of the SUSY Oscillator

Cited by in corpus (2)

• Generalization of Dirac conjugation in the superalgebraic theory of spinors
• Leibniz algebras of Heisenberg type