Quantum shape dynamics

In an article[2] recently revised, a quantum gravity phenomenon known as the emergence of space is observed for a universe consisting of ${\displaystyle N}$ protons and electrons. The reason is the angular momentum constraint in shape dynamics in the presence of spin-1/2 particles:

$${\displaystyle {\vec {J}}|\psi \rangle ={\vec {L}}|\psi \rangle +{\vec {S}}|\psi \rangle =0}$$

Because of this constraint, the expectation value of ${\textstyle L^{2}}$ and ${\textstyle S^{2}}$ is the same. Classical shape dynamics requires the expectation value of the former operator to vanish. It is related to non-existence of an absolute space. (En passé, non-existence of absolute time is related to the Hamiltonian constraint: ${\textstyle H|\psi \rangle =0}$.) However, there are states that yields a positive eigenvalue for ${\displaystyle S^{2}}$ and hence for ${\displaystyle L^{2}}$ while satisfying the total angular momentum constraint for spin-1/2 particles: ${\displaystyle {\vec {J}}|\psi \rangle =0}$. The article[2] found out that the density of states in the total spin Hilbert space (which is of ${\displaystyle 2^{N}}$ dimensional) that yields a zero expectation value for ${\displaystyle S^{2}}$ and ${\displaystyle L^{2}}$ as:

$${\displaystyle f_{0}^{N}={\frac {N!}{2^{N}(N/2+1)!(N/2)!}}}$$

when ${\displaystyle N}$ is even, and zero when it is odd. The limit of this expression approaches zero as ${\displaystyle N\to \infty }$. Hence it is argued that the space is almost always emergent classically for kinematic states.