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arxiv: 1907.11928 · v1 · pith:OOEBSDJAnew · submitted 2019-07-27 · 🧮 math-ph · math.FA· math.MP

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

Sergio Albeverio , Nicol\`o Cangiotti , Sonia Mazzucchi This is my paper

Pith reviewed 2026-05-24 14:32 UTC · model grok-4.3

classification 🧮 math-ph math.FAmath.MP
keywords Feynman path integralsSchrödinger equationmagnetic fieldStratonovich integraloscillatory integralscountertermheat equation
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The pith

Requiring the Feynman path integral for the magnetic Schrödinger equation to be independent of approximation forces a counterterm into the action.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides a rigorous construction of the Feynman path integral for the Schrödinger equation with magnetic field in terms of infinite dimensional oscillatory integrals. It demonstrates using a linear vector potential that independence from the approximation procedure requires adding a counterterm to the classical action functional. This explains the natural appearance of the Stratonovich integral in the path integral expressions for both the Schrödinger and heat equations with magnetic field. A reader cares because it clarifies how to properly define these integrals mathematically, avoiding ambiguities from different discretization choices.

Core claim

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.

What carries the argument

Infinite dimensional oscillatory integrals, with independence from the approximation procedure used to force addition of a counterterm to the action.

Load-bearing premise

That realizing the path integral as an infinite dimensional oscillatory integral and demanding independence from the approximation procedure correctly identifies the proper mathematical definition.

What would settle it

An explicit computation for the linear vector potential showing that the oscillatory integral without the counterterm yields different values depending on the approximation procedure chosen.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to rigorously realize a Feynman path integral formula for the Schrödinger equation with magnetic field via infinite-dimensional oscillatory integrals. Using the concrete example of a linear vector potential, the authors show that requiring the integral to be independent of the approximation procedure forces the addition of a specific counterterm to the classical action functional; this counterterm accounts for the appearance of the Stratonovich integral in the path-integral expressions for both the Schrödinger and heat equations with magnetic field.

Significance. If the detailed construction and proofs hold, the work supplies a mathematically rigorous foundation for path integrals in the presence of magnetic fields and derives the Stratonovich prescription from the single, natural requirement of approximation independence. This is a substantive contribution to mathematical physics; the oscillatory-integral approach and the explicit linear-potential example are strengths that give the argument a parameter-free character within the treated case.

minor comments (2)
  1. [Abstract] Abstract: the statement that the construction applies to 'the Schrödinger equation with magnetic field' would be clearer if the precise regularity assumptions on the vector potential (beyond the linear example) were indicated.
  2. The manuscript would benefit from an explicit statement, early in the text, of the precise function space on which the infinite-dimensional oscillatory integral is defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so we have no points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; construction rests on external oscillatory integral framework

full rationale

The derivation realizes the Feynman path integral via infinite-dimensional oscillatory integrals (an established external mathematical tool) and applies an approximation-independence consistency condition to the linear vector potential example to identify the required counterterm. This selects a specific form (Stratonovich) without any equation reducing to its own input by construction, without fitted parameters renamed as predictions, and without load-bearing self-citation chains. The central claim is self-contained against the cited external theory and does not rely on renaming known results or smuggling ansatzes via self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; main assumption is that infinite-dimensional oscillatory integrals provide a valid rigorous realization of the path integral.

axioms (1)
  • domain assumption Feynman path integrals for Schrödinger equation with magnetic field can be realized via infinite dimensional oscillatory integrals.
    Central to the construction stated in the abstract.

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discussion (0)

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