Fermions and Supersymmetry in Neural Network Field Theories

The Big Picture

Bosons (photons, Higgs bosons) are sociable: they pile into the same quantum state without complaint. Fermions (electrons, quarks, neutrinos) are antisocial loners. No two fermions can occupy the same quantum state at once, a rule called the Pauli exclusion principle. This single distinction drives the structure of chemistry, nuclear physics, and the universe itself.

For years, physicists have known that wide neural networks share deep mathematical similarities with quantum field theories. But the connection only described bosons. Fermions, with their rule that swapping any two of them flips the sign of the whole expression, seemed out of reach.

A new paper by Samuel Frank, James Halverson, Anindita Maiti, and Fabian Ruehle fills that gap. They introduce fully fermionic neural network field theories, then go further: they show how to build supersymmetry, a proposed symmetry of nature pairing every boson with a fermion partner, directly into neural network architecture.

Key Insight: By replacing ordinary numbers in neural networks with Grassmann variables (the anticommuting “numbers” that describe fermions), the authors prove that neural networks can realize fermionic quantum field theories, including interactions and supersymmetry, purely from their statistical structure.

How It Works

The starting point is a result from the 1990s: in the infinite-width limit, a randomly initialized neural network becomes a Gaussian process, a probability distribution over functions determined entirely by its mean and correlations. This is equivalent to a free field theory, the quantum field theory of non-interacting particles. Finite-width corrections introduce interactions. Physicists call this the NN-FT correspondence.

Gaussian processes describe bosons, though. Fermions require Grassmann variables, mathematical objects that anticommute: if η and ξ are Grassmann numbers, then ηξ = −ξη. Swapping the order flips the sign, and that single property encodes everything strange about fermionic statistics.

The central mathematical result is a Grassmann Central Limit Theorem. In ordinary probability theory, summing many independent random variables produces a Gaussian distribution. Frank et al. prove the analogous statement for Grassmann-valued random variables: a scaled sum of N independent, identically distributed Grassmann variables converges to a Gaussian Grassmann distribution as N → ∞. Free fermionic field theories then follow as infinite-width limits of Grassmann-valued neural networks.

The paper constructs Grassmann neural network field theories in two ways:

• Grassmann weights: Output layer weights are drawn from Grassmann-valued distributions; hidden layer computations remain ordinary
• Grassmann post-activations: Intermediate network features are Grassmann-valued, flowing into the outputs

Both give free fermionic theories at infinite width. Finite-width corrections generate interactions through the same mechanism that produces interacting bosonic theories in the standard NN-FT framework.

What does this buy you physically? At infinite width, the authors explicitly realize the free Dirac spinor, the field theory describing free electrons and quarks. At finite width, they find a four-fermion interaction, a direct contact interaction that shows up across models of nuclear and particle physics.

They also introduce Yukawa couplings, the interactions between fermionic and bosonic fields that give particles mass through the Higgs mechanism. These arise by correlating the weight distributions of the fermionic and bosonic sectors: a simple architectural choice with real physical content.

The most ambitious part of the paper tackles supersymmetry. Implementing SUSY requires superspace, a geometric arena with both ordinary and Grassmann-valued coordinates where bosons and fermions coexist in a single framework. The authors realize superspace via super-affine transformations on the network’s inputs: coordinate changes that mix ordinary inputs with Grassmann-valued inputs according to superspace geometry.

Networks built this way automatically satisfy the Ward identities of supersymmetry (the consistency conditions confirming a theory genuinely has the symmetry) without any fine-tuning. This single architectural choice generates a broad class of supersymmetric quantum mechanics and field theory models, including models with supersymmetry breaking, where the symmetry is present in the theory’s structure but absent from its ground state.

Why It Matters

For theorists, this is a new constructive tool. The parameter-space formulation lets you compute expectation values by sampling neural network weights rather than evaluating path integrals. Monte Carlo sampling of randomly initialized Grassmann networks could give access to fermionic correlators in interacting theories that are otherwise hard to compute.

On the machine learning side, neural network architecture encodes more physics than previously recognized. Yukawa couplings pop out of weight correlations. Supersymmetry falls out of super-affine input transformations. Physical symmetries can be baked in at the level of initialization and architecture, before any training happens. Do supersymmetric inductive biases help learning in specific domains? Do fermionic weight statistics offer new regularization tricks? These are now testable questions.

The work fits into a broader program treating neural networks as a laboratory for field theory, alongside lattice methods and perturbation theory. It may prove especially useful for probing strongly coupled, finite-N regimes with specific symmetry constraints.

Bottom Line: Frank, Halverson, Maiti, and Ruehle have extended the neural network–field theory correspondence to fermions and supersymmetry, proving that Grassmann-valued neural networks realize free and interacting fermionic theories and that supersymmetric architectures naturally encode SUSY Ward identities.