Local Differential Geometry as a Representation of the SUSY Oscillator

TL;DR

This paper extends the Bargmann-Fock representation to supersymmetric systems using differential geometry, showing that key SUSY oscillator features can be captured by geometric notions on real Euclidean space.

Contribution

It introduces a geometric framework for SUSY oscillators that simplifies understanding their structure by relying on differential geometry on real spaces.

Findings

Essential SUSY oscillator structures are captured by differential geometry.

Euclidean evolution corresponds to dilation groups.

Scalar product is the only additional structure needed.

Abstract

This work proposes a natural extension of the Bargmann-Fock representation to a SUSY system. The main objective is to show that all essential structures of the n-dimensional SUSY oscillator are supplied by basic differential geometrical notions on an analytical R^n, except for the scalar product which is the only additional ingredient. The restriction to real numbers implies only a minor loss of structure but makes the essential features clearer. In particular, euclidean evolution is enforced naturally by identification with the 1-parametric group of dilations.