Van der Waerden calculus with commuting spinor variables and the Hilbert-Krein structure of the superspace

TL;DR

This paper develops a van der Waerden calculus with commuting spinor variables to facilitate rigorous supersymmetric quantum field theory using operator-valued superdistributions.

Contribution

It introduces a novel van der Waerden calculus where Grassmann variables anticommute but fermionic components commute, aiding in the rigorous formulation of supersymmetric QFT.

Findings

Established a calculus suitable for supersymmetric test functions.

Provided a framework for operator-valued superdistributions.

Facilitated rigorous mathematical treatment of N=1 supersymmetry.

Abstract

Working with anticommuting Weyl(or Mayorana) spinors in the framework of the van der Waerden calculus is standard in supersymmetry. The natural frame for rigorous supersymmetric quantum field theory makes use of operator-valued superdistributions defined on supersymmetric test functions. In turn this makes necessary a van der Waerden calculus in which the Grassmann variables anticommute but the fermionic components are commutative instead of being anticommutative. We work out such a calculus in view of applications to the rigorous conceptual problems of the N=1 supersymmetric quantum field theory.