Physics: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 [math]\displaystyle{ N }[/math] protons and electrons. The reason is the angular momentum constraint in shape dynamics in the presence of spin-1/2 particles:[math]\displaystyle{ \vec J |\psi\rangle = \vec L |\psi\rangle + \vec S |\psi\rangle = 0 }[/math]Because of this constraint, the expectation value of [math]\displaystyle{ L^2 }[/math] and [math]\displaystyle{ S^2 }[/math] 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: [math]\displaystyle{ H|\psi\rangle = 0 }[/math].) However, there are states that yields a positive eigenvalue for [math]\displaystyle{ S^2 }[/math] and hence for [math]\displaystyle{ L^2 }[/math] while satisfying the total angular momentum constraint for spin-1/2 particles: [math]\displaystyle{ \vec J|\psi\rangle = 0 }[/math]. The article found out that the density of states in the total spin Hilbert space (which is of [math]\displaystyle{ 2^N }[/math] dimensional) that yields a zero expectation value for [math]\displaystyle{ S^2 }[/math] and [math]\displaystyle{ L^2 }[/math] as:[math]\displaystyle{ f_0^N = \frac{N!}{2^N (N/2+1)!(N/2)!} }[/math]when [math]\displaystyle{ N }[/math] is even, and zero when it is odd. The limit of this expression approaches zero as [math]\displaystyle{ N\to\infty }[/math]. Hence it is argued that the space is almost always emergent classically for kinematic states.