Fermions and Supersymmetry in Neural Network Field Theories

TL;DR

This work extends the neural network–field theory framework to fermions by introducing Grassmann-valued networks and establishing a Grassmann Central Limit Theorem that yields a free Grassmann Gaussian process in the infinite-width limit while preserving meaningful finite-width interactions. It constructs explicit realizations of the Euclidean free Dirac spinor and Yukawa-type couplings within Grassmann NN-FTs, and calculates leading finite-N four-fermion interactions, highlighting their controllable $1/N$ scaling. The authors then develop supersymmetric NN-FTs in both quantum mechanics and four-dimensional field theory by promoting inputs to superspace, showing SUSY correlators remain invariant under SUSY transformations and that a broad class of interacting SUSY theories can be realized, including chiral variants and SUSY breaking scenarios. Overall, the paper provides a formal bridge between fermionic quantum field theories and neural network field theories, enabling fermionic and SUSY models to be studied and simulated within NN-FT frameworks with controlled large-N limits and finite-N corrections.

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.