On the Heisenberg commutation relation. I
On the Heisenberg commutation relation. I
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关于海森堡交换关系。
DOI：
10.1016/0022-1236(83)90058-7
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发表时间：
1983
影响因子：
1.7
通讯作者：
K. Schmüdgen
中科院分区：
文献类型：
--
作者：
K. Schmüdgen
在Hjlbert空间L&R中，P=-i-1，Q= x。根据Stone-von Neumann唯一性定理，满足（2）的每对自伴算子P，Q酉等价于薛定谔对的直和。一个无穷小的变体的无尾定理是由于Rellich和Dixlane。关于这一行的细节和更多结果，我们参考Putnam的书[111]。从李群表示论的观点来看，（1）和（2）之间的联系非常清楚。方程（2）定义了Heisenberg群的酉表示，方程（1）是相应的李代数表示的公式。因此，（2）总是导致（1）在网格域，而（1）并不意味着（2）一般。（1）的物理意义在于量子力学的测不准原理。在作者对量子理论的理解中，
P=-i-&, Q= x in the Hjlbert space L&R,). By the Stone-von Neumann uniqueness theorem, each pair of self-adjoint operators P, Q satisfying (2) is unitarily equivalent to a direct sum of Schrodinger couples. An infinitesimal variant of the uniqueless theorem is due to Rellich and Dixmier. For details and more results alo lg this line we refer to Putnam’s book [111. From the viewpoint of representation theory of Lie groups, the connection between (1) and (2) is very clear. Equation (2) defines a unitary representation of tie Heisenberg group and (1) is the formula for the associated Lie algebra re) resentation. Therefore,(2) always leads to (1) on the Girding domain, while (1) does not imply (2) in general. The physical importance of (1) lies in the uncertainty principle of quantum mechanics. In the author’s understanding of quantum theory, the