Anomalies in Neural Network Field Theory

TL;DR

This paper develops a formalism connecting neural network theory with quantum field theory, deriving equations to study anomalies and symmetries in both machine learning and physics contexts.

Contribution

It introduces a novel neural network field theory framework that derives Schwinger--Dyson equations and Ward identities to analyze anomalies and symmetries.

Findings

Derived Schwinger--Dyson equations and Ward identities in NN-FT.

Applied formalism to study $U(1)$ symmetry, scale, and Weyl anomalies.

Provided new insights into symmetries in quantum field theories from network parameter space.

Abstract

Neural network field theory (NN-FT) formulates field theory in terms of a network architecture and a density on its parameters. We derive Schwinger--Dyson equations and Ward identities in NN-FT and utilize them to study anomalies. The equations depend on a conserved parameter space current that characterizes symmetries and how they break. It is relevant even in non-local NN-FTs, but can recover local currents in the case of a local Lagrangian by an appropriate fiber-wise average. In machine learning, this formalism is applied to feedforward networks and the attention mechanism. In physics, we use this machinery to study $U(1)$ symmetry for a complex scalar, the scale anomaly in $4d$ massless $\phi^4$ theory, the Weyl anomaly for the bosonic string (including a new computation of the critical dimension), and examples involving discrete topological data, such as winding numbers and T-duality. Since the results are obtained in network parameter space rather than the standard field space, they represent a new way to understand symmetries in quantum field theories.