A Feynman path integral formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich integral in the path integral formula for both the Schrödinger and heat equation with magnetic field.

中文翻译：

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带磁场的薛定谔方程的费曼路径积分的严格数学构造

具有磁场的薛定谔方程的费曼路径积分公式是根据无限维振荡积分在数学上严格实现的。我们表明（通过线性向量势的例子）逼近过程中积分独立性的要求迫使引入反项添加到经典动作泛函中。这为薛定谔方程和磁场热方程的路径积分公式中出现 Stratonovich 积分提供了自然的解释。