Emergence of Space in Quantum Shape Kinematics

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.