Feynman Tree Theorem and the Gelfand--Yaglom Formula

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.