# A rigorous mathematical construction of Feynman path integrals for the Schr\"odinger equation with magnetic field

## TL;DR

This paper rigorously constructs Feynman path integrals for the Schrödinger equation with magnetic fields using infinite dimensional oscillatory integrals, revealing the necessity of counterterms and explaining the emergence of Stratonovich integrals.

## Contribution

It provides a rigorous mathematical framework for Feynman path integrals with magnetic fields, including the role of counterterms and the connection to Stratonovich integrals.

## Key findings

- Counterterms are necessary for independence of approximation procedures.
- The path integral naturally involves Stratonovich integrals.
- The construction applies to Schrödinger and heat equations with magnetic fields.

## Abstract

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.