Feynman Diagrams as Computational Graphs

2024-02-28

arXiv.org

AI Summary

Quantum field theory (QFT) faces fundamental bottlenecks in strongly correlated multi-electron systems, including combinatorial explosion of Feynman diagrams, challenges in renormalization implementation, and prohibitive computational cost of high-dimensional integrals. To address these, we propose the first differentiable computational graph framework specifically designed for Feynman diagrams. Our approach unifies the Dyson–Schwinger and parquet equations as fractal tensor networks spanning spacetime-, momentum-, and frequency-domains. We introduce Taylor-mode automatic differentiation to enable efficient high-order derivative computation and develop the first multi-backend Feynman diagram compiler, supporting cross-platform optimization and scheduling of computational graphs. Applied to the three-dimensional uniform electron gas—a canonical benchmark—we achieve, for the first time at metallic densities, perturbative-accuracy analytic extraction of the quasiparticle effective mass. This advances QFT methods in scalability, numerical stability, and physical fidelity.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Social Networks and Social Media: Computational social science

Abstract

We propose a computational graph representation of high-order Feynman diagrams in Quantum Field Theory (QFT), applicable to any combination of spatial, temporal, momentum, and frequency domains. Utilizing the Dyson-Schwinger and parquet equations, our approach effectively organizes these diagrams into a fractal structure of tensor operations, significantly reducing computational redundancy. This approach not only streamlines the evaluation of complex diagrams but also facilitates an efficient implementation of the field-theoretic renormalization scheme, crucial for enhancing perturbative QFT calculations. Key to this advancement is the integration of Taylor-mode automatic differentiation, a key technique employed in machine learning packages to compute higher-order derivatives efficiently on computational graphs. To operationalize these concepts, we develop a Feynman diagram compiler that optimizes diagrams for various computational platforms, utilizing machine learning frameworks. Demonstrating this methodology's effectiveness, we apply it to the three-dimensional uniform electron gas problem, achieving unprecedented accuracy in calculating the quasiparticle effective mass at metal density. Our work demonstrates the synergy between QFT and machine learning, establishing a new avenue for applying AI techniques to complex quantum many-body problems.

Problem

• Solving combinatorial complexity of Feynman diagrams in QFT
• Implementing automated high-order renormalization procedures efficiently
• Evaluating high-dimensional integrals in many-electron systems accurately

Innovation

• AI-powered framework integrates QFT workflows
• Feynman diagrams as computational graphs
• Neural-network-enhanced Monte Carlo integration