Stochastic Analysis and Quantum Field Theory

我们已经研究了随机微积分和量子动力学中的几个当前主题。为了解决这一问题，我们解释了我们所研究的一个重要课题：白色噪声分析可以看作是由无限维旋转群0* 产生的调和分析，由于引入到广义函数空间上的白色噪声测度在旋转群的作用下是不变的，很自然地，我们会想到一个无限维的类似物，它类似于有限维旋转群作用于其上的有限维矩阵上的调和分析。根据0*.1子群的选择，我们的调和分析有几个步骤。有限维旋转群的归纳极限，记为G*。研究了Fock空间<symmetry>H_n，Laplace-Beltrami算子DELTA*，G* 在H_n上的第一类不可约酉表示等.存在一个子群Y，称为Levy群，它包含无穷多个 ...更多信息 e维旋转，远离有限维旋转，由基本函数空间的坐标排列定义。为了进行与1类似的演算，与Levy群相关，我们需要向前走一步。也就是说，自然要引入一个广义白色噪声泛函空间，然后我们讨论了有效作用在广义白色噪声泛函空间上的Levy拉普拉斯算子.然后，我们得到一个非常有趣的子群W，它包含了所有来自参数空间的自同构的变换。W在性质上与G* 和Levy群完全不同。为了引入一些解析结构，我们考虑W的连续单参数子群（称为须子）。然后我们取一个随机变量族{X（C）}，其中参数C穿过解析的、封闭的、简单流形的某个子类，并且其中C被W的子群的作用变形。一个普遍的理论尚未建立，但我们可以发现许多有趣的问题，在分析以及一些应用。少

We have investigated several current topics in stochastic calculus and quantum dynamics. To fix the idea, we explain one of the most important topics of what we have studied.White noise analysis may be viewed as the harmonic analysis arising from the infinite dimensional rotation group 0*.Since the white noise measure, which is introduced on the space of the generalized functions, is invariant under the action of the rotation group, it is natural to think of an infinite dimensional analogue of the harmonic analysis on a finite dimensional sphery on which the finite dimensional rotation group acts.There are several steps for our harmonic analysis depending on the choice of subgroup of 0*.1. Inductive limit of finite dimensional rotation groups, denoted by G*. The Fock space <symmetry>H_n, the Laplace-Beltrami operator DELTA*, class-one irreducible unitary representations of G* on H_n, and so forth, are well investigated.2. There is a subgroup Y, called Levy group, which contains infinit … More e dimensional rotations, far from the finite dimensional ones, defined by permutations of coordinates of the basic function space. To carry on a similar calculus as in 1, associated with the Levy group, we need to go one step ahead. Namely, a space of generalized white noise functionals should naturally be introduced, then we are led to discuss the Levy's Laplacian operator that acts effectively on the space of the generalized white noise functionals.3. We then come to a very interesting subgroup W consisting of all the transformations that come from diffeomorphism of the parameter space. The W is quite different, in character, from G* and the Levy group. To introduce some analytic structure, We consider continuous one-parameter subgroups (called whiskers) of W. Then we take a family of random variables {X(C)}, where the parameter C runs through a certain subclass of analytic, closed, simple manifolds, and where C is deformed by actions of the subgroup of W. A general theory is not yet established, however we can find many interesting questions in analysis as well as several applications. Less