Odd coset quantum mechanics

TL;DR

This paper explores the quantum mechanics of Grassmann variables with a focus on symmetries, revealing a novel representation structure and interpreting certain wave-functions as BRST superfields.

Contribution

It introduces a quantization scheme for Grassmann-odd coset space models, highlighting the role of $SU(n|1)$ symmetry and the interpretation of wave-functions as BRST superfields for $n=2$.

Findings

States with non-zero norm form $SU(n|1)$ representations.

The model exhibits a non-linearly realized $SU(n|1)$ symmetry.

Wave-functions for $n=2$ can be viewed as BRST superfields.

Abstract

The standard quantum states of $n$ complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same `free-fermion' model has a non-linearly realized $SU(n|1)$ symmetry; it can be viewed as the mechanics of a `particle' on the Grassmann-odd coset space $SU(n|1)/U(n)$. We implement a quantization of this model for which the states with non-zero norm transform as a representation of $SU(n|1)$, the representation depending on the U(1) charge of the wave-function. For $n=2$ the wave-function can be interpreted as a BRST superfield.