On Integral Relations for Invariants Constructed from Three Riemann Tensors and their Applications in Quantum Gravity
On Integral Relations for Invariants Constructed from Three Riemann Tensors and their Applications in Quantum Gravity
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黎曼三张量构造的不变量的积分关系及其在量子引力中的应用
DOI：
10.1063/1.523128
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发表时间：
1977
影响因子：
1.3
通讯作者：
C. C. Wu
中科院分区：
文献类型：
--
作者：
P. Nieuwenhuizen;C. C. Wu
纯引力的最低阶量子修正是有限的，因为两个黎曼张量的乘积之间存在积分关系（高斯-博内定理）。本文确定了四维和六维时空中三个黎曼张量积之间的几个代数关系和积分关系。在这两种情况下，当Rμν=0时，只剩下一个不变量，即F(−g)1/2（RαβμνRμνρσRρσ αβ）。明确地表明，即使当Rμν=0时，这个不变量也不会消失。因此，纯引力的双环量子修正只有在这个不变量的系数由于奇迹般的抵消而消失时才会是有限的。
The lowest order quantum corrections to pure gravitation are finite because there exists an integral relation between products of two Riemann tensors (the Gauss–Bonnet theorem). In this article several algebraic and integral relations are determined between products of three Riemann tensors in four‐ and six‐dimensional spacetime. In both cases, one is left with only one invariant when Rμν=0, viz., F (−g)1/2(RαβμνRμνρσRρσ αβ).It is explicitly shown that this invariant does not vanish, even when Rμν=0. Consequently, the two‐loop quantum corrections to pure gravitation will only be finite if, due to miraculous cancellation, the coefficient of this invariant vanishes.