Studies in noncommutative dynamics and multivariable operator theory

本文涉及两个松散连通区域：a)具有一定因果结构的Hilbert空间上所有算子的代数的单参数自同构群理论；b)作用于Hilbert空间上的n元算子的交换理论。多年来，我们一直试图了解非对易动力学的本质，最近我们的方法是基于E_0-半群的理论。最近在几个方面取得了令人振奋的进展，这些进展的全部影响仍在整理中。在多元算子理论中，我们引入了一个类似于黎曼流形的曲率积分的“曲率”不变量。我们已经通过将它与另一个不变量联系起来，证明了它在许多情况下是一个整数，即某个有限生成模的欧拉特征。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，本质上是可以与给定的n元算符相关联的“狄拉克算符”的索引。在量子理论中，可观测量由算符来表示。算符代数与数字代数有很大的不同，因为两个算符A和B相乘的结果取决于它们相乘的顺序：AB与BA不同。交换性定律的这种失败产生了深远的后果，最基本的后果就是测不准原理。量子理论中的时间流由某些变换来表示，每个变换都以微妙的方式移动算符，而量子理论的动力学就是对这些变换组的研究。这种“非对易动力学”与经典物理学中的对易动力学有很大不同。我们的方法是基于某种因果关系的概念，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--发现了概率论的新联系，在这种联系中，人们现在可以在非对易的时间流和某些随机过程之间来回传递，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，我们在(交换)算子集上的工作在算子集和基本几何概念(如曲率)之间建立了重要的联系。争论的焦点是运算符集的数值不变量。虽然这个不变量看起来是一个可以取任何值的实数，但它实际上是一个整数。什么整数？现在出现的答案似乎是，这个数字是某个狄拉克算符的数字索引。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.