Building Quantum Field Theories Out of Neurons

Abstract

An approach to field theory is studied in which fields are comprised of $N$ constituent random neurons. Gaussian theories arise in the infinite-$N$ limit when neurons are independently distributed, via the Central Limit Theorem, while interactions arise due to finite-$N$ effects or non-independently distributed neurons. Euclidean-invariant ensembles of neurons are engineered, with tunable two-point function, yielding families of Euclidean-invariant field theories. Some Gaussian, Euclidean invariant theories are reflection positive, which allows for analytic continuation to a Lorentz-invariant quantum field theory. Examples are presented that yield dual theories at infinite-$N$, but have different symmetries at finite-$N$. Landscapes of classical field configurations are determined by local maxima of parameter distributions. Predictions arise from mixed field-neuron correlators. Near-Gaussianity is exhibited at large-$N$, potentially explaining a feature of field theories in Nature.