Emergence of Space in Quantum Shape Kinematics

TL;DR

This paper investigates how space emerges from quantum shape kinematics in a universe with multiple particles, showing that space almost always appears for large systems due to non-zero angular momentum expectations.

Contribution

It demonstrates the existence of quantum shape space for N≥4 particles and links the emergence of space to non-zero angular momentum in quantum states.

Findings

Quantum shape space exists for N≥4 particles.

Space emergence is linked to non-zero $\

For odd N, space always emerges; for large even N, space almost always emerges.

Abstract

A model universe is analyzed with $N$ protons and electrons where there are electromagnetic and spin interactions in the Hamiltonian is investigated in the context of quantum shape kinematics. We have found that quantum shape space exists for $N\geq 4$ particles and has $2N-7$ functional degrees of freedom in the case of spin-1/2 particles. The emergence of space is associated with non-vanishing expectation value $\langle L^2 \rangle$. We have shown that for odd $N$ space always emerges, and for large even $N$ space almost always emerges because $\langle L^2 \rangle \neq 0$ for almost all states. In the limit $N\to\infty$ the density of states that yields $\langle L^2 \rangle=0$ vanishes. Therefore we conclude that the space is almost always emergent in quantum shape kinematics.