arXiv 1206.02392012-06math-phmath.APmath.MP

de Sitter时空中Klein-Gordon方程的惠更斯原理

Huygens' Principle for the Klein-Gordon equation in the de Sitter spacetime

Karen Yagdjian

发表信息：: J. Math. Phys. 54, 091503 (2013)
DOI：: 10.1063/1.4821115

摘要：

本文证明了de Sitter时空中的Klein-Gordon方程只有在标量场的物理质量$m$和空间变量的维数$n\geq 2$由方程$m^2=(n^2-1)/4 $联系时才服从惠更斯原理。定义了不完全惠更斯原理，即限制在第二初始数据为零的惠更斯原理，并揭示了de Sitter时空中的无质量标量场只对$n=1,3$维服从不完全惠更斯原理而不服从惠更斯原理.因此，在德西特时空中，存在两个不同的标量场（实际上，m=0和$m^2=(n^2-1)/4 $），它们服从不完全惠更斯原理，等价于条件$n=3$（实际上，物理世界的空间维度）。对于$n=3$，这两个质量值是量子场论中所谓的Higuchi界的端点。物理质量的值$m^2=(n^2-1)/4 $也使我们能够获得大时间解的完全渐近展开。关键词：惠更斯原理;克莱因-戈登方程;德西特时空; Higuchi边界

英文摘要：

In this article we prove that the Klein-Gordon equation in the de Sitter spacetime obeys the Huygens' principle only if the physical mass $m$ of the scalar field and the dimension $n\geq 2$ of the spatial variable are tied by the equation $m^2=(n^2-1)/4 $. Moreover, we define the incomplete Huygens' principle, which is the Huygens' principle restricted to the vanishing second initial datum, and then reveal that the massless scalar field in the de Sitter spacetime obeys the incomplete Huygens' principle and does not obey the Huygens' principle, for the dimensions $n=1,3$, only. Thus, in the de Sitter spacetime the existence of two different scalar fields (in fact, with m=0 and $m^2=(n^2-1)/4 $), which obey incomplete Huygens' principle, is equivalent to the condition $n=3$ (in fact, the spatial dimension of the physical world). For $n=3$ these two values of the mass are the endpoints of the so-called in quantum field theory the Higuchi bound. The value $m^2=(n^2-1)/4 $ of the physical mass allows us also to obtain complete asymptotic expansion of the solution for the large time. Keywords: Huygens' Principle; Klein-Gordon Equation; de Sitter spacetime; Higuchi Bound

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