论文标题
从一般拉格朗日密度得出的高旋转量规理论的曲率张量
Curvature tensors of higher-spin gauge theories derived from general Lagrangian densities
论文作者
论文摘要
一段时间以来，高自旋理论的曲率张量已知。过去，它们是使用Riemann张量的对称属性的概括（每个旋转$ n $ curl curl curl curl curl curl curl curl curl）。因此，它们有时被称为广义的“ Riemann”张量。在本文中，提出了一种从第一原理中得出这些曲率张量的方法。该推导是在没有任何先验知识的情况下完成的，即riemann张量的存在或高旋转量规理论的曲率张量。为了执行此推导，最近开发的程序，用于从$ n $衍生剂的二次组合和$ m $ tensor潜力的二次组合中得出确切的量规lagrangian密度，适用于$ n = m = n $ case the Spin-$ n $ n $ gemuge转换。此过程在$ n = M = 1 $ case中唯一地产生了Lagrangian的经典电动力学，而Lagrangian则在$ n = M = 2 $案例中用于更高的衍生性重力（`riemann''和``riemann''和`ricci'平方）。在这里通过直接计算的$ n = m = 3 $情况证明了这一过程的唯一解决方案是Spin-3曲率张量及其收缩。对于$ n = M = 4 $ case，Spin-4曲率张量也是唯一得出的。换句话说，在这里证明，对于以$ n $衍生剂和$ m $张量的电势构建的标量最通用的线性组合，最高$ n = m = 4 $，存在着一种独特的解决方案，可以作为合同的$ n $ curvature curvature tensors的线性方程系统。讨论了有关较高旋转的解决方案的猜想-N $ $ n = m = n $。
Curvature tensors of higher-spin gauge theories have been known for some time. In the past, they were postulated using a generalization of the symmetry properties of the Riemann tensor (curl on each index of a totally symmetric rank-$n$ field for each spin-$n$). For this reason they are sometimes referred to as the generalized 'Riemann' tensors. In this article, a method for deriving these curvature tensors from first principles is presented; the derivation is completed without any a priori knowledge of the existence of the Riemann tensors or the curvature tensors of higher-spin gauge theories. To perform this derivation, a recently developed procedure for deriving exactly gauge invariant Lagrangian densities from quadratic combinations of $N$ order of derivatives and $M$ rank of tensor potential is applied to the $N = M = n$ case under the spin-$n$ gauge transformations. This procedure uniquely yields the Lagrangian for classical electrodynamics in the $N = M = 1$ case and the Lagrangian for higher derivative gravity (`Riemann' and `Ricci' squared terms) in the $N = M = 2$ case. It is proven here by direct calculation for the $N = M = 3$ case that the unique solution to this procedure is the spin-3 curvature tensor and its contractions. The spin-4 curvature tensor is also uniquely derived for the $N = M = 4$ case. In other words, it is proven here that, for the most general linear combination of scalars built from $N$ derivatives and $M$ rank of tensor potential, up to $N=M=4$, there exists a unique solution to the resulting system of linear equations as the contracted spin-$n$ curvature tensors. Conjectures regarding the solutions to the higher spin-$n$ $N = M = n$ are discussed.
