Linear Transformations in Hilbert Space: III. Operational Methods and Group Theory.
Linear Transformations in Hilbert Space: III. Operational Methods and Group Theory.
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DOI：
10.1073/pnas.16.2.172
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发表时间：
1930-02
影响因子：
11.1
通讯作者：
M. Stone
中科院分区：
文献类型：
--
作者：
M. Stone
如果 (X) 12dx 存在：微分算子定义了一个自伴随变换，相关的运算微积分是赫维赛德微积分。在这种情况下，我们的定义与维纳给出的定义一致。 2 运算微积分的第一个重要应用对自伴随变换的酉等价理论进行了精彩的讨论，该理论由 Hellinger3 和 Hahn 以基本相同但不太明显的形式阐述了有界变换。 4 我们不会在这里给出细节。第二个应用是迄今为止尚未发表结果的领域，涉及量子力学中具有根本重要性的群论问题。这些问题之一是研究直线到自身的平移组的统一表示。该问题的解决方案体现在以下语句中：
I lf (X) 12dx exists: the differential operatordefines a self-adjointtrans-formation, and the related operational calculus is the Heaviside calculus. In this instance, our definitions coincide with those givenby Wiener. 2 A first important application of the operationalcalculus yields a beautiful discussion of the theory of the unitary equivalence of self-adjoint transformations, a theory elaborated for bounded transformations in an essentially identical but less perspicuous form by Hellinger3 and Hahn. 4 We shall not give details here.A second application, to a field in which few results have been pub-lished hitherto, concerns group-theoretic questions of fundamental importance in the quantum mechanics. Among these questions is the study of the unitary representations of the group of translations of a straight line into itself. The solution of this problem is embodied in the following statement: