[Paper Review] Breaking the uniqueness of the Shape Dynamics Hamiltonian

Henrique Gomes|arXiv (Cornell University)|Jan 19, 2012

Protein Structure and Dynamics5 references3 citations

TL;DR

This paper demonstrates that Shape Dynamics' global Hamiltonian is not unique when coupled to certain matter fields or cosmological constants, as the underlying differential operator develops a non-trivial kernel. Instead of a single Hamiltonian, the theory admits a finite set of weakly commuting global Hamiltonians, preserving local conformal invariance while extending the framework to non-vacuum cases via the implicit function theorem applied to Lichnerowicz-type equations.

ABSTRACT

In earlier works on Shape Dynamics (SD), a linear method of solving a particular set of Lichnerowicz-type equations through the implicit function theorem was developed in order to implicitly construct SD's global Hamiltonian and eliminate second class constraints. This method was later used for extending Shape Dynamics (SD) to the non-vacuum case, showing how other fields are coupled to the theory. In that study it was found that unlike the vacuum case the use of such methods yielded puzzling bounds on the density of some types of fields. Here we show that the original SD cannot be extended beyond such bounds, but that a slight modification of the original can withstand any type of coupling. When the bound is broken, the theory does not come equipped with a single Hamiltonian as in vacuum SD, but with a finite set of weakly commuting Hamiltonians, which we describe.

Motivation & Objective

• To resolve the breakdown of the original Shape Dynamics construction when extended to non-vacuum systems with matter fields or cosmological constants.
• To identify the mathematical obstruction—non-positive linear terms in the differential operator—that invalidates the original linear method for Hamiltonian construction.
• To show that despite the loss of uniqueness, a coherent Shape Dynamics theory still exists by allowing multiple compatible global Hamiltonians.
• To clarify the role of the Stuckelberg field and the kernel of the matter-modified differential operator in determining the number of independent time-evolution generators.
• To establish a robust framework for non-vacuum Shape Dynamics using the implicit function theorem, ensuring local solvability of constraint equations.

Proposed method

• Apply the implicit function theorem to solve Lichnerowicz-York-type equations arising from second-class constraints in the extended phase space.
• Use canonical transformations and the Stuckelberg mechanism to extend the ADM phase space with auxiliary scalar fields and their conjugate momenta.
• Decompose the scalar constraint into parts with invertible Poisson brackets to isolate the second-class constraints for elimination.
• Analyze the kernel of the matter-modified differential operator $\Delta_{\text{matter}}$ to determine the number of independent solutions for the lapse function.
• Construct a finite set of global Hamiltonians corresponding to the dimension of the kernel of $\Delta_{\text{matter}}$, ensuring weak commutativity.
• Use ellipticity of the differential operator to guarantee local solvability and well-posedness of the constraint system in the non-vacuum case.

Experimental results

Research questions

• RQ1 What happens to the uniqueness of the Shape Dynamics Hamiltonian when matter fields or a cosmological constant are introduced?
• RQ2 How does the failure of positivity in the linear term of the differential operator affect the solvability of the constraint system?
• RQ3 Can a consistent Shape Dynamics theory still be constructed in the non-vacuum case if the original Hamiltonian is no longer unique?
• RQ4 What is the physical interpretation of multiple commuting global Hamiltonians in the context of time evolution in Shape Dynamics?
• RQ5 How does the kernel of the operator $\Delta_{\text{matter}}$ determine the number of independent time-evolution generators in the non-vacuum theory?

Key findings

• The original Shape Dynamics construction fails to produce a unique global Hamiltonian when the linear term in the differential operator becomes non-positive due to matter coupling.
• The number of weakly commuting global Hamiltonians in the non-vacuum case is equal to the dimension $n$ of the kernel of the operator $\Delta_{\text{matter}}$, which governs the lapse function solutions.
• Despite the loss of uniqueness, the theory remains consistent and dynamically equivalent to ADM gravity, with local conformal invariance preserved.
• The implicit function theorem provides a linear criterion for local solvability of the Lichnerowicz-type equations, replacing the need for non-linear methods like Leray-Schauder theory.
• The Stuckelberg field $\phi$ and the lapse solutions $N_i$ are no longer in one-to-one correspondence; instead, the kernel structure determines the number of independent reductions.
• Perturbative analysis around ADM solutions still yields $\phi_o = 0$ for physical solutions, enabling systematic construction of non-trivial Shape Dynamics solutions.