The Universe from a Single Particle III

TL;DR

This paper extends previous work on symmetry-breaking metrics in Lie algebras to the fermionic case, finding metrics that exhibit properties relevant to emergent spatial structure and black hole scrambling, with preliminary results on $ ext{su}(4)$ and $ ext{su}(8)$.

Contribution

It introduces fermionic versions of symmetry-breaking metrics, called kam metrics, and demonstrates their approximate adherence to the Brown-Susskind penalty schedule in specific cases.

Findings

Kam metrics approximate the Brown-Susskind schedule on $ ext{su}(4)$ and $ ext{su}(8)$.

The toy model exhibits localized degrees of freedom and preference for low-body interactions.

Further constraints on interaction neighborhoods require advanced analytic techniques.

Abstract

In parts I [6] and II [7] we studied how metrics $g_{ij}$ on $\mathfrak{su}(n)$ may spontaneously break symmetry and $\textit{crystallize}$ into a form which is kaq, $\textit{knows about qubits}$. We did this for $n=2^N$ and then away from powers of 2. Here we address the Fermionic version and find kam metrics, these $\textit{know about Majoranas}$. That is, there is a basis of principal axes $\{H_k\}$ of which is of homogeneous Majorana degree. In part I, we searched unsuccessfully for functional minima representing crystallized metrics exhibiting the Brown-Susskind penalty schedule, motivated by their study of black hole scrambling time. Here, by segueing to the Fermionic setting we find, to good approximation, kam metrics adhering to this schedule on both $\mathfrak{su}(4)$ and $\mathfrak{su}(8)$. Thus, with this preliminary finding, our toy model exhibits two of the three features required for the spontaneous emergence of spatial structure: 1. localized degrees of freedom and, 2. a preference for low body-number (or low Majorana number) interactions. The final feature, 3. constraints on who may interact with whom, i.e. a neighborhood structure, must await an effective analytic technique, being entirely beyond what we can approach with classical numerics.