打破形状动力学哈密顿量的唯一性
Breaking the uniqueness of the Shape Dynamics Hamiltonian
摘要：
在形状动力学（SD）的早期工作中，开发了一种通过隐函数定理求解一组特定的Lichnerowicz型方程的线性方法，以隐式地构造SD的全局哈密顿量并消除第二类约束。这种方法后来被用于扩展形状动力学（SD）到非真空的情况下，显示其他领域是如何耦合到理论。在那项研究中，人们发现，与真空情况不同的是，使用这种方法对某些类型的场的密度得出了令人困惑的界限。在这里，我们表明，原来的SD不能扩展超出这样的界限，但原来的轻微修改可以承受任何类型的耦合。当边界被打破时，理论不像真空SD那样配备单个哈密顿量，而是配备有限的弱对易哈密顿量集，我们描述了这一点。
英文摘要：
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.
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