Nilpotent Marsh and SUSY QM

TL;DR

This paper explores nilpotent modifications to classical trajectories in supersymmetric and nonsupersymmetric theories, linking nilpotence conditions to supersymmetric Lagrangians and spontaneous supersymmetry breaking.

Contribution

It introduces a novel analysis of nilpotent additions to trajectories, connecting nilpotence conditions to the structure of supersymmetric Lagrangians and topological charges.

Findings

Nilpotent conditions lead to Witten supersymmetric Lagrangian.

Spontaneous supersymmetry breaking relates to nilpotence of topological charge.

Half of Grassmann constants must vanish for balanced zero modes.

Abstract

We consider the nilpotent additions to classical trajectories in supersymmetric and nonsupersymmetric theories. The condition of anilpotence of action on some generalized solutions leads to the Witten supersymmetric Lagrangian. The condition of anilpotence of topological charge is the same as one of superpotential with spontaneous broken supersymmetry. We should vanish half of Grassmann constants of integration, because in this case only we obtain the same number of normalized bosonic and fermionic zero modes.