arXiv 2002.126912020-02-28math-phmath.MP

Feynman积分路径与Henstock积分：综述与待解决问题

The Feynman integral path a Henstock integral: a survey and open problems

Ricardo Gaitan, M. Guadalupe Morales

摘要：

费曼路径积分定义在所有可能路径的空间$\mathbb{R}^T$上，它是发展量子力学的有力工具。Feynman被积函数的绝对值是不可积的，因此Feynman不能使用Lebesgue积分理论。然而，它作为Henstock积分形式存在（不需要测度概念），并且是通常出现在路径积分中的普通积分的合适替代。费曼证明了他的理论与传统的量子力学公式的等价性，因为他的路径积分满足薛定谔方程。另一方面，费曼的路径积分与费曼图有关。为了在费曼图中应用这个积分，有必要交换积分$\int_{\mathbb{R}^T}$和级数。本文讨论了Henstock积分与控制收敛定理中积分与和互换的不可能性。即使它还没有被证明通过几个数学形式主义已被使用。

英文摘要：

The Feynman path integral is defined over the space $\mathbb{R}^T$ of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral $\int_{\mathbb{R}^T}$ and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.

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摘要：

英文摘要：

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