Feynman Tree Theorem and the Gelfand--Yaglom Formula

Ipak Fadakar; Guilherme L. Pimentel; Behrang Tafreshi

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arXiv · 2607.24923

Feynman Tree Theorem and the Gelfand--Yaglom Formula

Ipak Fadakar·Guilherme L. Pimentel·Behrang Tafreshi

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Abstract

The Gelfand--Yaglom formula equates the one-loop determinant of a Schr\"odinger operator with Dirichlet boundary conditions to the solution of an initial value problem. Following a suggestion by Polyakov, we provide a new diagrammatic proof of this formula as a direct application of the Feynman Tree Theorem. By cutting the one-loop determinant, we re-express it as a sum of tree diagrams. These trees explicitly encode the solution to the Gelfand--Yaglom initial value problem. We extend this diagrammatic framework to general boundary conditions and comment on a potential generalization to quantum field theory.

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Ipak Fadakar, Guilherme L. Pimentel, Behrang Tafreshi. 2026-07-27. Feynman Tree Theorem and the Gelfand--Yaglom Formula. https://arxiv.org/abs/2607.24923

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