Abstract

According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrödinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by Feynman (1948). In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.

中文翻译：

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具有非二次哈密顿函数的无穷维哈密顿系统的Schrödinger量化

摘要

根据安德烈·韦尔（Andre Weil）的一个定理，在任何无限维局部凸空间上都没有标准的Lebesgue测度。因此，通常使用非平移不变的sigma加法来定义无限维哈密顿系统的Schrödinger量化。在本文中，使用了一种完全不同的方法：我们使用广义的Lebesgue测度，该测度是平移不变的。Feynman（1948）在第一篇论文中以隐式形式使用了这种方法。在这种情况下，与有限维情况一样，正式定义了符号为经典哈密顿函数的伪微分算子。特别是，他们使用单一傅立叶变换，将函数（在有限维空间上）映射为函数。