论文标题
规范变异完成和4D高斯骨网重力
Canonical variational completion and 4D Gauss-Bonnet gravity
论文作者
论文摘要
最近，一项提案，以在\（d = 4 \）维度中从重新归一化的高斯 - 骨网术语中获得第二个导数顺序对重力场方程的有限贡献。它引发了讨论，使用的重新归化程序是否产生了明确的理论。主要的批评之一是基于以下事实：从差异性动作中，无法作为欧拉 - 拉格朗日方程获得所得的字段方程。在这项工作中，我们使用变体反向计算的技术指出，在任何维度上，根本无法从任何动作（是否不变形）中获得重新归一化的截短的高斯 - 邦网方程。然后，我们基于Vainberg -Tonti Lagrangian的概念采用规范变分的完成 - 该概念在于将规范定义的校正项添加到给定方程系统中，以使其从动作中衍生。为了将此技术应用于建议的$ 4 $ d重新归一化的高斯 - 骨网方程，我们将变异完成算法扩展到某些类别的PDE系统，通常为Vainberg-Tonti Lagrangian提供的常规积分。我们发现，在$ d> 4 $中，建议的字段方程可以在变化上完成，选择指标或逆向字段变量。两种方法都始终产生相同的拉格朗日，其变异导致了第四阶场方程。在$ d = 4 $中，在两种情况下，变体完成的理论的拉格朗日都在分歧。
Recently, a proposal to obtain a finite contribution of second derivative order to the gravitational field equations in \(D = 4\) dimensions from a renormalized Gauss-Bonnet term in the action has received a wave of attention. It triggered a discussion whether the employed renormalization procedure yields a well-defined theory. One of the main criticisms is based on the fact that the resulting field equations cannot be obtained as the Euler-Lagrange equations from a diffeomorphism invariant action. In this work, we use techniques from the inverse calculus of variations to point out that the renormalized truncated Gauss-Bonnet equations cannot be obtained from any action at all (either diffeomorphism invariant or not), in any dimension. Then, we employ canonical variational completion, based on the notion of Vainberg-Tonti Lagrangian - which consists in adding a canonically defined correction term to a given system of equations, so as to make them derivable from an action. To apply this technique to the suggested $4$D renormalized Gauss-Bonnet equations, we extend the variational completion algorithm to some classes of PDE systems for which the usual integral providing the Vainberg-Tonti Lagrangian diverges. We discover that in $D>4$ the suggested field equations can be variationally completed, choosing either the metric or its inverse as field variables; both approaches yield consistently the same Lagrangian, whose variation leads to fourth order field equations. In $D=4$, the Lagrangian of the variationally completed theory diverges in both cases.
