Variational Neural Network Approach to QFT in the Field Basis

TL;DR

This paper introduces a neural network variational method for solving quantum field theories in momentum space, validated against the exactly solvable Klein-Gordon model, and sets the stage for future complex models.

Contribution

It systematically benchmarks neural network variational methods in momentum space for the Klein-Gordon model, providing a foundation for extending to interacting theories.

Findings

Accurately reproduces the ground-state energy and correlators of the Klein-Gordon model.

Demonstrates the effectiveness of momentum space for neural network quantum field theory benchmarks.

Provides a framework for future neural network applications to interacting quantum field theories.

Abstract

We present a variational neural network approach for solving quantum field theories in the field basis, focusing on the free Klein-Gordon model formulated in momentum space. While recent studies have explored neural-network-based variational methods for scalar field theory in position space, a systematic benchmark of the analytically solvable Klein-Gordon ground state -- particularly in the momentum-space field basis -- has been lacking. In this work, we represent the ground-state wavefunctional as a neural network defined on a discretized set of field configurations and train it by minimizing the Hamiltonian expectation value. This framework enables direct comparison to exact analytic results for a range of key observables, including the ground-state energy, two-point correlators, expectation value of the field, and the structure of the learned wavefunctional itself. Our results provide quantitative diagnostics of accuracy and demonstrate the suitability of momentum space for benchmarking neural network approaches, while establishing a foundation for future extensions to interacting models and position-space formulations.