张量值和旋量值随机场及其在连续统物理学和宇宙学中的应用
Tensor- and spinor-valued random fields with applications to continuum physics and cosmology
摘要：
本文综述了两类随机函数谱理论的历史、现状和物理应用。一类是定义在欧几里得空间上的齐次各向同性随机场，其取值于实有限维线性空间。在连续介质物理的应用中，这样的场描述了均匀和各向同性连续介质在微结构附着于所有介质点的情况下的物理性质。场的值域是对称类的不动点集，其中两个紧李群通过正交表示作用。均匀介质的物质对称群在每一点上都是相同的，并且作用是平凡的，而物理对称群可能作用是不平凡的。在各向同性随机介质中，秩1（分别为秩2）场的相关张量在根据上述表示的物理对称性组的作用下变换（分别为它的张量平方），使场各向同性。另一类是紧李群陪集空间上齐次向量丛的迷向随机截面。在宇宙学的应用中，陪集空间模拟天球，而随机截面模拟宇宙背景。宇宙学原理保证了横截面是各向同性的。为了方便读者，从多重线性代数，表示论，微分几何的必要材料审查附录。
英文摘要：
In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp. rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp. its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.
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