经典引力场的观测量与上同调类
Observables and cohomology classes for classical gravitational field
摘要：
由于我们对时空几何的整体结构和整体时空同构群在量子引力中的作用的认识有限，引力和几何领域中的一些最突出的问题仍然没有得到解决。通过考察经典引力理论中广义协方差和可观测量的某些方面，可以对这些重要问题有一些了解。本文将定义为引力场的经典几何观测量集，即定义在洛伦兹结构上的Diff（M）-规范不变上同调类。它们代表了物理引力现象的全局特征，与时空的拓扑结构相联系，并可用于构造新的拉格朗日量。从可观测量中找到一组完整数据的问题可能与以下事实有关：目前，四维及四维以上的流形不能有效地分类。人们可以将这个结果解释为一个指针，指向可能存在具有不同拓扑结构的时空（即，不同的全局特征），其具有不可区分的局部时空几何。
英文摘要：
Some of the most outstanding questions in the field of gravitation and geometry remain unsolved as a result of our limited understanding of the global structure of the spacetime geometry and the role played by global spacetime diffeomorphism group in quantum gravity. Some insight into these important questions may be gained by looking at certain aspects of general covariance and observables in classical gravitational theory. In this paper I shall define as set of classical geometric observables of the gravitational field by which I mean Diff(M)-gauge invariant cohomology classes defined on a Lorentzian structure. They represent global characteristics of the physical gravitational phenomena, are linked to the topology of the spacetime, and can be used in constructing new Lagrangians. The problem of finding a complete set of data out of observables is related perhaps to the fact that at present moment, manifolds in dimension 4 and above cannot be effectively classified. One could interpret this result as a pointer to the possibility that there might be spacetimes with different topologies (i.e., different global characteristics) which have indistinguishable local spacetime geometry.
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