Building Quantum Field Theories Out of Neurons

The Big Picture

What if the universe is, in some deep sense, a neural network? That sounds like science fiction, but physicist James Halverson at Northeastern University has taken the question seriously and built a mathematical framework that puts quantum field theory and machine learning on the same footing. Not metaphorically. Literally.

Quantum field theory (QFT) is the language of modern physics. It describes electrons, photons, quarks, and the Higgs boson, every fundamental particle and force we’ve ever measured. Traditionally, physicists write down a Lagrangian, a recipe encoding how particles move and interact, and derive all physics from there. Choosing the right recipe feels like an art. Halverson asks: what if you could skip the recipe entirely and build a quantum field out of neurons instead?

The result is a construction called neural network quantum field theory (NN-QFT). In the usual approach, quantum randomness comes from summing over all possible field configurations. In Halverson’s version, randomness comes from the statistical properties of the neurons themselves. The construction can satisfy all the mathematical requirements of a QFT that respects the symmetries of special relativity.

Key Insight: By building fields out of random neurons, the Central Limit Theorem does the work that Lagrangians usually do. Free theories emerge automatically at large N, while interactions arise from finite-N corrections and neuron correlations.

How It Works

Start with a scalar field φ, which assigns a real number to every point in space. In Halverson’s construction, this field is a sum of N random neurons:

φ(x) = a₁h₁(x) + a₂h₂(x) + … + aₙhₙ(x)

Each neuron hᵢ is a randomly initialized function. The coefficients aᵢ are drawn from a distribution with variance σ²/N. No Lagrangian appears anywhere.

What happens next depends on two limits. When N → ∞ and neurons are independently distributed, the Central Limit Theorem kicks in. Just as averaging millions of coin flips produces a bell curve, summing millions of independent random neurons produces a Gaussian theory: the field-theory equivalent of a free particle. This mirrors a well-known result in machine learning called the Neural Network Gaussian Process (NNGP) correspondence, where infinitely wide neural networks behave like Gaussian processes.

Interactions enter in two ways:

• Finite-N corrections: at large but finite N, the CLT isn’t exact, and the 4-point connected correlation function picks up contributions of order 1/N
• Independence breaking: if neurons are correlated with each other, interactions appear even at infinite N through a term called I⁽⁴⁾_IB

The CLT assumes independence. Break that assumption, and you get non-trivial physics.

Going from a Euclidean field theory to a genuine QFT requires satisfying the Osterwalder-Schrader (OS) axioms. These are mathematical conditions guaranteeing the theory can be analytically continued into real spacetime, with a well-defined Hilbert space and no ghost states that would break unitarity. The hardest condition to satisfy is reflection positivity: correlation functions must behave correctly under time reflection. For Gaussian NN-QFTs, Halverson shows this reduces to a constraint on the power spectrum G⁽²⁾(p), and he constructs explicit neuron architectures that satisfy it. These are the first concrete NN-QFTs.

The paper also turns up a result about duality. Different neuron architectures, with different internal symmetries and parameter distributions, can produce identical theories at N → ∞ but diverge at finite N. Think of it as the neural analog of known dualities in physics: two models that look identical from far away reveal their differences under a microscope.

Without an action to minimize, classical field configurations take on a new meaning. Classical solutions correspond to local maxima of the parameter distribution P(θ). The landscape of classical physics becomes the landscape of likely neuron parameters.

Predictions in NN-QFT come from mixed field-neuron correlators, quantities that involve both the output field φ and the constituent neurons hᵢ. Ordinary QFT has no “constituent” degrees of freedom to probe. Here, the internal structure of the field is directly accessible, connecting the macroscopic field to its microscopic building blocks.

Why It Matters

For physics, NN-QFT offers a non-Lagrangian route into quantum field theory. That matters in regimes where writing down an action is hard: strongly coupled theories, exotic geometries, situations where the standard toolkit breaks down.

The near-Gaussianity of real-world quantum fields has always seemed like a coincidence. Most QFTs in nature are weakly coupled and nearly free. In the NN-QFT framework, there’s a natural explanation: large-N suppression of non-Gaussian terms is baked into the architecture, just as it is in real neural networks.

For machine learning theory, the payoff runs deeper than analogy. There’s a growing research program that imports QFT tools (Feynman diagrams, renormalization group flows) to understand how networks learn. Halverson’s paper inverts that program: instead of applying QFT to neural networks, it uses neural networks to build QFTs. The two directions together point toward a tighter mathematical relationship between learning systems and physical theories than anyone had previously shown.

Bottom Line: Neural networks can construct quantum field theories from scratch, no Lagrangian required. Gaussian theories emerge naturally at large N, and interactions arise from finite-N corrections. The framework offers both a new approach to QFT and a principled explanation for why real-world fields are nearly free.