Distributions in Hilbert space and canonical systems of operators
Distributions in Hilbert space and canonical systems of operators
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希尔伯特空间中的分布和算子规范系统
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发表时间：
1958
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通讯作者：
I. Segal
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作者：
I. Segal
虽然无限维线性空间上的概率分布理论的代数特征与我们熟悉的有限维空间上的概率分布理论的代数特征大致相似，但它们之间存在一些重要的差异，量子场论的某些数学困难就来源于这些差异。在本论文中，我们将这样一个方面的分布，其绝对连续性和变换性质，并开始应用的结果的唯一性和分类问题所产生的量子统计理论。从纯数学的观点来看，与有限（-维）情形的对比表现在无限维希尔伯特空间上存在不同类的准不变分布的连续体（即关于所有平移绝对连续的分布），而不是有限情形下的唯一类。这里使用的分布概念是一种重新表述，这是为了适应重要的具体情况以及理论目的所必需的，但它要求（在无限情况下）放弃将分布作为空间子集上的可数可加概率测度的想法。空间的线性和拓扑结构大致弥补了可数可加性的损失，特别是在希尔伯特空间的情况下，因此出现了一个有效的理论，在其初步的形式方面平行于通常的理论。在量子场论中，满足（部分形式）关系的自伴算子P1，P2，· · ·和Qx，Q2，· · ·起着重要作用
Although the algebraic features of the theory of probability distributions on an infinite-dimensional linear space generally resemble those of the familiar theory on finite-dimensional spaces, there are some important differences, in which certain of the mathematical difficulties of quantum field theory originate. In the present paper we treat one such aspect of distributions, their absolute continuity and transformation properties, and initiate the application of the results to uniqueness and classification problems arising in the theory of quantum statistics. From a purely mathematical viewpoint, the contrast with the finite (-dimensional) situation is shown by the existence of a continuum of distinct classes of quasi-invariant distributions on an infinite-dimensional Hilbert space (i.e. distributions absolutely continuous with respect to all their translates), as opposed to the unique class in the finite case. The notion of distribution that is used here is a reformulation, which is required in order to accommodate significant concrete cases as well as for theoretical purposes, but which requires (in the infinite case) the abandonment of the idea of a distribution as a countably additive probability measure on subsets of the space. The linear and topological structure of the space roughly compensates for the loss of countable additivity, particularly in the case of Hilbert space, so that there emerges an effective theory parallel in its preliminary formal aspects to the usual one. In quantum field theory an important part is played by self-adjoint operators Pi, P2, • • • and Qx, Q2, • • satisfying the (partially formal) relations