Shape Dynamics in 2+1 Dimensions

TL;DR

This paper constructs and analyzes the Hamiltonian formulation of Shape Dynamics in 2+1 dimensions on a torus, exploring its quantization and asymptotic behavior, and discusses implications for higher-dimensional theories.

Contribution

It explicitly constructs Shape Dynamics for a 2+1 torus universe and provides an expansion for higher genus surfaces, linking it to quantization strategies and higher-dimensional models.

Findings

Explicit construction of Shape Dynamics Hamiltonian for torus universe.

Quantization of local constraints is feasible due to linearity in momenta.

Asymptotic analysis of conformal transformations in large CMC volume.

Abstract

Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2+1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then use the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity (BKL) Shape Dynamics Hamiltonian in higher dimensions to suggest a quantization strategy for Shape Dynamics.