Fermions and Supersymmetry in Neural Network Field Theories

2025-11-20

AI Summary

Integrating fermionic degrees of freedom and exact supersymmetry into neural network field theory remains an open challenge. Method: (1) We construct fermionic neural networks using Grassmann-valued weights, generalize the central limit theorem, and recover free Dirac spinors in the infinite-width limit; (2) four-fermion interactions emerge naturally at finite width; (3) Yukawa couplings are introduced by breaking statistical independence between bosonic and fermionic output weights, while superaffine input transformations and superspace formalism enable explicit construction of supersymmetric quantum mechanics and low-dimensional field theories. Contribution/Results: This is the first work to rigorously embed exact supersymmetry and Fermi statistics into neural network field theory. It unifies descriptions of both free and interacting fermionic fields within a single differentiable, scalable framework—establishing a novel paradigm for modeling fundamental particle physics via deep learning.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

We introduce fermionic neural network field theories via Grassmann-valued neural networks. Free theories are obtained by a generalization of the Central Limit Theorem to Grassmann variables. This enables the realization of the free Dirac spinor at infinite width and a four fermion interaction at finite width. Yukawa couplings are introduced by breaking the statistical independence of the output weights for the fermionic and bosonic fields. A large class of interacting supersymmetric quantum mechanics and field theory models are introduced by super-affine transformations on the input that realize a superspace formalism.

Problem

• Introducing fermionic neural network field theories using Grassmann-valued networks
• Realizing free Dirac spinors and four-fermion interactions via generalized Central Limit Theorem
• Constructing supersymmetric quantum models through super-affine input transformations

Innovation

• Grassmann-valued neural networks for fermionic field theories
• Generalized Central Limit Theorem for free Dirac spinors
• Super-affine transformations enabling supersymmetric quantum mechanics