John Carlos Baez on Nostr: Someone on Bluesky asked what's the relevance to quantum field theory. The first step ...

Someone on Bluesky asked what's the relevance to quantum field theory.

The first step is to find polynomials P with E[Pⁿ] = 0. After some thinking we notice P(x,y) = x + iy has this property, and indeed the Gaussian Moments Conjecture holds in this case, because for every polynomial Q(x,y) we have E[PⁿQ] = 0 except for finitely many n.

The polynomial z = x + iy is famous, of course, but in quantum physics multiplication by this is called the 'creation operator' in the Bargmann-Fock representation of the quantum harmonic oscillator, where quantum states are polynomials in z.

We can also do a multivariable version where zⱼ = xⱼ + i𝑦ⱼ for j = 1,...,n. This quickly leads to many things, including the so-called complex Ginibre model - a matrix model based on precisely this math:

https://arxiv.org/abs/1409.4494

But what's the physical meaning of "E[PⁿQ] = 0 except for finitely many n"? It means that for any operator Q built as a polynomial in annihilation and creation operators, its expected value must vanish on states with sufficiently high particle number!

Cool. But remember, all this involves cases where the Gaussian Moments Conjecture is *correct*. It's not affected by the new discovery that this conjecture has counterexamples. Someone should try to work out the consequences there. Since the counterexamples are a bit weirder, their impact on mathematical physics may be minor.

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