Studies in noncommutative dynamics and multivariable operator theory

摘要：本文研究了Hilbert空间上具有一定因果结构的所有算子的代数自同构的单参数群理论和Hilbert空间上的算子的交换n元组理论。多年来，我们一直试图理解非交换动力学的本质，最近我们的方法是基于e_0半群理论。最近在几个方面取得了令人兴奋的进展，其全部含义仍在整理中。在多变量算子理论中，我们引入了一个“曲率”不变量，它有点类似于黎曼流形的曲率积分。我们已经证明，在许多情况下，它是一个整数，通过将它与另一个不变量，某有限生成模的欧拉特征联系起来。然而，关键公式只在某些情况下有效，我们现在正在将曲率不变量与更微妙的整数不变量联系起来，本质上是“狄拉克算子”的索引，它可以与给定的n元算子相关联。在量子论中，可观测量由算符表示。算子的代数与数字的代数有很大的不同，因为两个算子A和B相乘的结果取决于它们相乘的顺序：AB和BA不一样。交换律的失败产生了深远的影响，其中最基本的就是测不准原理。量子理论中的时间流是由特定的变换来表示的，每个变换都以微妙的方式移动算子，量子理论的动力学就是对这些变换群的研究。这种“非对易动力学”与经典物理学的对易动力学非常不同。我们的方法是基于一定的因果关系概念，这涉及到自同态半群的技术思想。最近的进展非常令人鼓舞——随着概率论的新联系的发现，人们现在可以在非交换时间流和某些随机过程之间来回穿梭，这些随机过程可以被认为是“非白”噪声，因为它们接近白噪声，但不完全是白噪声。在一个不同但相关的方向上，我们关于（交换）算子集的工作在算子集和基本几何思想（如曲率）之间建立了重要的联系。问题是操作符集合的数值不变量。虽然这个不变量看起来是一个实数，可以取任何值，但实际上它是一个整数。整什么?现在出现的答案是，这个数是某个狄拉克算子的数值指标。它似乎与数学的其他部分有重要的联系，包括黎曼几何和代数几何。

AbstractAversonThis work relates to two loosely connected areas, a) the theory of one-parameter groups of automorphisms of the algebra of all operators on a Hilbert space which posses a certain causal structure, and b) the theory of commuting n-tuples of operators acting on a Hilbert space. We have been attempting to understand the nature of noncommutative dynamics for many years, and our approach recently has been based on the theory of E_0-semigroups. There has been exciting recent progress on several fronts, the full implications of which are still being sorted out. In multivariable operator theory, we have introduced a "curvature" invariant which is somewhat analogous to the integral of curvature of a Riemannian manifold. We have shown that it is an integer in many cases by relating it to another invariant, the Euler characteristic of a certain finitely generated module. However, the key formula is valid only in certain cases, and we are now in the process of relating the curvature invariant to a more subtle integer invariant, essentially the index of a "Dirac operator" that can be associated with the given n-tuple of operators.In quantum theory the observable quantities are represented by operators. The algebra of operators differs sharply from the algebra of numbers because the result of multiplying two operators A and B depends on the order in which they are multiplied: AB is not the same as BA. This failure of the commutativity law has profound consequences, the most basic one being the uncertainty principle. The flow of time in quantum theory is represented by certain transformations, each of which moves operators in subtle ways, and the dynamics of quantum theory is the study of such groups of transformations. This "noncommutative dynamics" is very different from the commutative dynamics of classical physics. Our approach is based on a certain notion of causality, which involves the technical idea of semigroups of endomorphisms. Recent progress has been very encouraging - with the discovery of new connections with probability theory, in which one may now pass back and forth between noncommutative flows of time and certain random processes which can be thought of as "off-white" noise, in that they are close to white noise, but not exactly white noise. In a different but related direction, our work on sets of (commuting) operators establishes significant connections between sets of operators and fundamental geometric ideas such as curvature. At issue is a numerical invariant for sets of operators. While this invariant appears to be a real number capable of taking on any value, it is in fact an integer. What integer? The answer that appears to be emerging now is that this number is the numerical index of a certain Dirac operator. There appear to be significant connections with other parts of mathematics, including Riemannian geometry and algebraic geometry.