Quantum shape dynamics

Quantum Shape Dynamics is an emerging research endeavour that aims to provide a quantum understanding of quantum mechanics of shapes, quantum fields on shape dynamics background, and quantization of shape dynamics.

Emergence of Space in the Kinematic Structure

In an article recently revised, a quantum gravity phenomenon known as the emergence of space is observed for a universe consisting of $N$ protons and electrons. The reason is the angular momentum constraint in shape dynamics in the presence of spin-1/2 particles:$${\vec {J}}|\psi \rangle ={\vec {L}}|\psi \rangle +{\vec {S}}|\psi \rangle =0$$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 $S^{2}$ and hence for $L^{2}$ while satisfying the total angular momentum constraint for spin-1/2 particles: ${\vec {J}}|\psi \rangle =0$. The article found out that the density of states in the total spin Hilbert space (which is of $2^{N}$ dimensional) that yields a zero expectation value for $S^{2}$ and $L^{2}$ as:$$f_{0}^{N}={\frac {N!}{2^{N}(N/2+1)!(N/2)!}}$$when $N$ is even, and zero when it is odd. The limit of this expression approaches zero as $N\to \infty$. Hence it is argued that the space is almost always emergent classically for kinematic states.