arXiv 0810.24042008-10-14gr-qc

随机算子的广义相对论

General Relativity on Random Operators

Michael Heller, Leszek Pysiak, Wieslaw Sasin

备注：: 17 LaTex pages

摘要：

我们提出了一个统一了广义相对论和量子力学数学结构的数学结构。它由紧支撑的复值函数${\mathcal A}$的非交换代数组成，卷积作为乘法，在广群$Γ$上，广群的基是时空$M$上的框架丛的总空间$E$。基于${\mathcal A}$的导子的微分几何适当地推广了时空的标准微分几何，并且代数${\mathcal A}$在希尔伯特空间的丛中表示时，定义了推广通常的量子力学的随机算子的冯诺依曼代数${\mathcal M}$。本文的主要结果是：存在一个空间${\mathcal M_0}$，它在${\mathcal M}$中稠密，与代数${\mathcal A}$同构.这种同构允许我们将所有微分几何结构，包括广义爱因斯坦方程，在${\mathcal A}$（及其衍生物）的帮助下转移到空间${\mathcal M_0}$。这样，我们得到了Hilbert空间丛上随机算子对广义相对论的推广。然而，这种推广不能扩展到整个${\mathcal M}$，这是主要的数学障碍，至少在这种方法中，完全统一引力理论与量子物理学。

英文摘要：

We present a mathematical structure which unifies mathematical structures of general relativity and quantum mechanics. It consists of the noncommutative algebra of compactly supported, complex valued functions ${\mathcal A}$, with convolution as multiplication, on a groupoid $Γ$ the base of which is the total space $E$ of the frame bundle over space-time $M$. A differential geometry based on derivations of ${\mathcal A}$ suitably generalizes the standard differential geometry of space-time, and the algebra ${\mathcal A}$, when represented in a bundle of Hilbert spaces, defines a von Neumann algebra ${\mathcal M}$ of random operators that generalizes the usual quantum mechanics. The main result of the present paper is that there exists a space ${\mathcal M_0}$, dense in ${\mathcal M}$, that is isomorphic with the algebra ${\mathcal A}$. This isomorphism allows us to transfer all differentially geometric constructions, generalized Einstein's equations including, made with the help of ${\mathcal A}$ (and its derivations) to the space ${\mathcal M_0}$. In this way, we obtain a generalization of general relativity in terms of random operators on a bundle of Hilbert spaces. However, this generalization cannot be extended to the whole of ${\mathcal M}$, and this is the main mathematical obstacle, at least in this approach, to fully unify theory of gravity with physics of quanta.

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摘要：

英文摘要：

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