Calculus of Variations and Geometric Measure Theory

S. Albeverio - N. Cangiotti - S. Mazzucchi

A rigorous mathematical construction of Feynman path integrals for the Schrödinger equation with magnetic field

created by cangiotti on 05 Dec 2023
modified on 10 May 2024


Published Paper

Inserted: 5 dec 2023
Last Updated: 10 may 2024

Journal: Communications in Mathematical Physics
Volume: 377
Pages: 1461-1503
Year: 2020
Doi: 10.1007/s00220-020-03744-x

ArXiv: 1907.11928 PDF


A Feynman path integral formula for the Schr\"odinger 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 Schr\"odinger and heat equation with magnetic field.