标题： 形状动力学中Weyl异常的第一个视角 显示英文标题

标题： A first look at Weyl anomalies in shape dynamics

摘要： 形状动力学的一个更流行的反对意见是怀疑在量子化过程中会出现空间Weyl对称性的反常。本文的目的在于建立研究可能产生的此类反常所需的工具。第一步是适应Barnich和Henneaux的哈密顿框架中的规范上同调公式，这可以将反常分解为空间部分和时间部分。反常的空间部分，即对称代数本身的反常（$[\Omega, \Omega]\propto \hbar$而不是消失），由与$\Omega$相关的哈密顿BRST微分的第二鬼场上同调的一个投影给出，模去空间导数。时间部分$[\Omega, H]\propto\hbar$由第一鬼场上同调的不同投影和来自泛函微分方程解的额外项组成。假设涉及的规范上同调群具有局域性，这一部分总是局域的。假设规范上同调群具有局域性，并利用Barnich和Henneaux的结果以及Boulanger对高阶鬼场上同调的分类和下降方程，我们在3+1维找到了反常的完整刻画。即使在形状动力学中，反常的空间部分和时间反常的第一部分在这些假设下总是局域的。从泛函微分方程解中出现的部分明确涉及形状动力学哈密顿量，因此可能是非局域的。如果限制时间反常的这部分也是局域的，则在3+1情况下不会出现\emph{局部的}类型的Weyl反常，无论是时间还是空间。 显示英文摘要

摘要： One of the more popular objections towards shape dynamics is the suspicion that anomalies in the spatial Weyl symmetry will arise upon quantization. The purpose of this short paper is to establish the tools required for an investigation of the sort of anomalies that can possibly arise. The first step is to adapt to our setting Barnich and Henneaux's formulation of gauge cohomology in the Hamiltonian setting, which serve to decompose the anomaly into a spatial component and time component. The spatial part of the anomaly, i.e. the anomaly in the symmetry algebra itself ($[\Omega, \Omega]\propto \hbar$ instead of vanishing) is given by a projection of the second ghost cohomology of the Hamiltonian BRST differential associated to $\Omega$, modulo spatial derivatives. The temporal part, $[\Omega, H]\propto\hbar$ is given by a different projection of the first ghost cohomology and an extra piece arising from a solution to a functional differential equation. Assuming locality of the gauge cohomology groups involved, this part is always local. Assuming locality for the gauge cohomology groups, using Barnich and Henneaux's results, the classification of Weyl cohomology for higher ghost numbers performed by Boulanger, and following the descent equations, we find a complete characterizations of anomalies in 3+1 dimensions. The spatial part of the anomaly and the first component of the temporal anomaly are always local given these assumptions even in shape dynamics. The part emerging from the solution of the functional differential equations explicitly involves the shape dynamics Hamiltonian, and thus might be non-local. If one restricts this extra piece of the temporal anomaly to be also local, then overall no \emph{local} Weyl anomalies, either temporal or spatial, emerge in the 3+1 case.