Tropicalized quantum field theory

Quantum field theory (QFT) is one of the most accurate methods for making phenomenological predictions in physics, but it has a significant drawback: obtaining concrete predictions from it is computationally very demanding. The standard perturbative approach expands an interacting QFT around a free QFT, using Feynman diagrams. However, the number of these diagrams grows superexponentially, making the approach quickly infeasible. I will talk about arXiv:2508.14263, which introduces an intermediate layer between free and interacting field theories: a tropicalized QFT. Often, this tropicalized QFT can be solved exactly. The exact solution manifests as a non-linear recursion equation fulfilled by the expansion coefficients of the quantum effective action. Geometrically, this recursion computes volumes of moduli spaces of metric graphs and is thereby analogous to Mirzakhani’s volume recursions on the moduli space of curves. Building on this exact solution, an algorithm can be constructed that samples points from the moduli space of graphs approximately proportional to their perturbative contribution. Via a standard Monte Carlo approach we can evaluate the original QFT using this algorithm. Remarkably, this algorithm requires only polynomial time and memory, suggesting that perturbative quantum field theory computations actually lie in the polynomial-time complexity class, while all known algorithms for evaluating individual Feynman integrals are at least exponential in time and memory. The (potential) capabilities of this approach are remarkable: For instance, we can compute perturbative expansions of massive scalar D=3 phi^3 and D=4 phi^4 quantum field theories up to loop orders between 20 and 50 using a basic proof-of-concept implementation. These perturbative orders are completely inaccessible using a naive approach.