The quantum world does not like being pinned down to exact numbers. Physicists usually expand a complex interaction into a long series, like a sum of ever-smaller corrections. But in an expanding universe, where space itself stretches, this series behaves like a stubborn student: each successive term does not shrink but instead starts to grow without bound. The sum runs away, and no meaningful answer can be extracted.
The authors tackled this problem using the example of a massless field in de Sitter space — a simplified model of an eternally expanding universe. Instead of summing the series directly, they applied the method of autonomous equations. This approach resembles a pendulum that finds its own stable position no matter how it is pushed. The specially constructed equations naturally settle into finite, physically meaningful behavior, ignoring the mathematical tantrum of the original series.
The second approach proved even trickier. The scientists applied the same idea of autonomous equations, but not to the series itself; instead, they worked with its "Borel image"—the Borel–Leroy transform. This allowed them to perform what is known as Borel resummation: in essence, they took the divergent series, ran it through a mathematical filter, and obtained finite, stable predictions as output. The result matched much better what the stochastic picture gives — a cruder but more intuitive approach where quantum jumps are replaced by random kicks, like a dust particle in a beam of light.
In the end, both methods, the direct one and the one with Borel resummation, provided a reasonable description of how quantum correlations evolve over time. But the second approach came very close to the stochastic picture, as if it had brought the mathematics into sharp focus. This is not just playing with formulas: understanding such processes is needed to figure out how quantum fluctuations in the early universe could have given rise to the clumps of matter from which galaxies later grew.
as written by the authors arXiv:2605.09736
We investigate the divergent perturbative series of correlation functions for a massless, self-interacting scalar field in de Sitter space. First, we use our previously proposed method of autonomous equations to obtain finite time-dependent functions, and show that these functions approximate the time evolution of the correlation functions of the stochastic theory reasonably well. Second, we apply the technique of autonomous equations to the Borel-Le Roy transforms of correlation functions, and use solutions of these equations to perform Borel resummation. The results match the time evolution obtained in the stochastic picture substantially better. In addition, we propose an alternative method for extracting perturbative coefficients and provide a new derivation of our autonomous equation by truncating a system of Schwinger-Dyson-type differential equations.
Divergent series sometimes give the most accurate predictions in physics — if you "reassemble" them correctly. For instance, without summation methods for divergent series, we would not be able to calculate the properties of the electron with fantastic precision.