Supersymmetric Quantum Mechanics for String-Bits

TL;DR

This paper explores supersymmetric quantum mechanics models for superstring-bits across various space dimensions, analyzing the feasibility of replacing harmonic forces with more general forces without destroying cluster decomposition.

Contribution

It develops supersymmetric single particle quantum mechanics models tailored for superstring-bits in specific dimensions and investigates the conditions under which harmonic forces can be generalized.

Findings

Possible to replace harmonic forces in 1D and 2D cases

Impossible to replace harmonic forces in 8D case

Inconclusive results for 4D case

Abstract

We develop possible versions of supersymmetric single particle quantum mechanics, with application to superstring-bit models in view. We focus principally on space dimensions $d=1,2,4,8$, the transverse dimensionalities of superstring in $3,4,6,10$ space-time dimensions. These are the cases for which ``classical'' superstring makes sense, and also the values of $d$ for which Hooke's force law is compatible with the simplest superparticle dynamics. The basic question we address is: When is it possible to replace such harmonic force laws with more general ones, including forces which vanish at large distances? This is an important question because forces between string-bits that do not fall off with distance will almost certainly destroy cluster decomposition. We show that the answer is affirmative for $d=1,2$, negative for $d=8$, and so far inconclusive for $d=4$.