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Tropicalized quantum field theory and global tropical sampling

Michael Borinsky

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 $φ^4$ beta function at 50 loops with a proof-of-concept implementation.

Tropicalized quantum field theory and global tropical sampling

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 beta function at 50 loops with a proof-of-concept implementation.

Paper Structure

This paper contains 26 sections, 20 theorems, 93 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. Overview of the results
  4. Relation to previous works
  5. Plan of the paper
  6. A brief overview of tropicalized quantum field theory
  7. Preliminaries
  8. Perturbation theory
  9. Divergences of Feynman integrals and positivity assumptions
  10. Tropical algebra
  11. The Hepp bound
  12. Tropicalized quantum field theory
  13. The tropical effective action
  14. The tropical loop equation
  15. Perturbative tropicalized quantum field theory
  16. ...and 11 more sections

Key Result

Lemma 1

If $p$ is a homogeneous polynomial in $x_1,\ldots,x_n$ with only positive coefficients, then there are constants $C_1, C_2 > 0$ such that with $\mathbb P_{>0}^n = \{ [x_1:\ldots:x_n] \in \mathbb {RP}^{n-1} : x_i > 0 \text{ for } i =1,\ldots, n\}$.

Figures (4)

  • Figure 1: Factorization property of the Hepp bound
  • Figure 2: Five examples of beaded graphs and one example of a non-beaded graph.
  • Figure 3: Illustration of the tropical loop equation. Each orange blob stands for a 1PI graph. The right-hand graphs have one purple pointed edge. Cutting these edges yields beaded graphs.
  • Figure 4: Illustrations of the algorithms. The orange diagonal hatchings indicate 1PI graphs. The blue dotted pattern stands for beaded graphs.

Theorems & Definitions (41)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • Lemma 6
  • proof
  • ...and 31 more