Tropicalized quantum field theory and global tropical sampling

TL;DR

This paper introduces a tropicalized approach to scalar quantum field theory, providing an exactly solvable model and an efficient polynomial-time sampling algorithm for moduli space points, with applications to high-loop computations.

Contribution

It presents a novel tropicalization of quantum field theory, deriving an exact recursive solution and an efficient sampling algorithm for moduli spaces, advancing computational methods in the field.

Findings

Tropicalized scalar QFT is exactly solvable via a recursion relation.

The sampling algorithm operates in polynomial time and memory.

Successfully evaluated the 50-loop primitive contribution to the $4$ beta function.

Abstract

We explain how to tropicalize scalar quantum field theory and show that tropicalized massive scalar quantum field theory is exactly solvable. This exact solution manifests as a non-linear recursion equation fulfilled by the expansion coefficients of the quantum effective action. Geometrically, this recursion computes specific 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, we construct an algorithm that samples points from the moduli space of graphs approximately proportional to their perturbative contribution. Remarkably, this algorithm requires only polynomial time and memory, suggesting that perturbative quantum field theory computations lie in the polynomial-time complexity class, while all known algorithms for evaluating individual Feynman integrals are exponential in time and memory. To demonstrate the capabilities of the algorithm, we evaluate the primitive contribution to the $\phi^4$ beta function at 50 loops with a proof-of-concept implementation.