The Gaussian Moments Conjecture is a pretty conjecture connected to probability and quantum field theory. Two days ago we suddenly learned it's false! - since it implies the Jacobian conjecture.
Let me explain this conjecture - it's very simple as these things go.
A lot of quantum field theory is about integrals like
E(P) = ∫ P(𝑥) exp(−x⋅x) dⁿx
where P is a polynomial in the variables x = (x₁,...,𝑥ₙ).
Feynman diagrams are a graphical trick for computing these integrals!
The Gaussian Moments Conjecture says that if a polynomial P in n variables has
E(Pᵏ) = 0
for all k = 1,2,3,... then also
E(PᵏQ) = 0
for all k and all polynomials Q.
Very pretty, and I could probably make up a nice physics interpretation.
But sorry, it's false! 😏
Warnings:
1) I don't know why the Gaussian Moments Conjecture implies the Jacobian Conjecture; this is claimed here:
• Harm Derksen, Arno van den Essen, Wenhua Zhao,The Gaussian Moments Conjecture and the Jacobian Conjecture, https://arxiv.org/abs/1506.05192
but they really just reduce the issue to work in an earlier paper by Zhao.
2) The expected value E(P) should really be normalized to make E(1) = 1. It's irrelevant here, but matters elsewhere. So, take my definition of E(P) and divide it by π^(n/2).
John Carlos Baez
npub1nf…3nqe4
2026-07-21 09:08:34 UTC
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2026-07-21 09:08:34 UTCEvent JSON
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"content": "The Gaussian Moments Conjecture is a pretty conjecture connected to probability and quantum field theory. Two days ago we suddenly learned it's false! - since it implies the Jacobian conjecture.\n\nLet me explain this conjecture - it's very simple as these things go.\n\nA lot of quantum field theory is about integrals like \n\nE(P) = ∫ P(𝑥) exp(−x⋅x) dⁿx \n\nwhere P is a polynomial in the variables x = (x₁,...,𝑥ₙ).\n\nFeynman diagrams are a graphical trick for computing these integrals! \n\nThe Gaussian Moments Conjecture says that if a polynomial P in n variables has\n\nE(Pᵏ) = 0 \n\nfor all k = 1,2,3,... then also\n\nE(PᵏQ) = 0 \n\nfor all k and all polynomials Q. \n\nVery pretty, and I could probably make up a nice physics interpretation.\n\nBut sorry, it's false! 😏 \n\nWarnings: \n\n1) I don't know why the Gaussian Moments Conjecture implies the Jacobian Conjecture; this is claimed here: \n\n• Harm Derksen, Arno van den Essen, Wenhua Zhao,The Gaussian Moments Conjecture and the Jacobian Conjecture, https://arxiv.org/abs/1506.05192\n\nbut they really just reduce the issue to work in an earlier paper by Zhao.\n\n2) The expected value E(P) should really be normalized to make E(1) = 1. It's irrelevant here, but matters elsewhere. So, take my definition of E(P) and divide it by π^(n/2).",
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