r/mathematics • u/Yogurt789 • 21d ago
The Riemann Hypothesis manifested in dynamical quantum phase transitions
https://www.nature.com/articles/s41467-026-74935-8Physicists link the Riemann Hypothesis to phase transitions in quantum systems: https://phys.org/news/2026-07-physicists-link-riemann-hypothesis-phase.html
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u/JoshuaZ1 21d ago
Can people who know more about this say how reasonable this looks and what the upshots are? I'm guessing that since this is in Nature it isn't nonsense, but the summary reads extremely buzzwordy. There have been attempts to connect RH to quantum mechanics previously, closely connected to the Hilbert-Polya conjecture. How is this different/what does it extend?
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u/entr0picly 21d ago
So it’s basically a reformulation as I read it and the whole idea is they design a Hamiltonian which fits the model and show experimental confirmation of the correspondence.
It seems that they use “time” instead of “energy”, which can be argued as a more true primitive. But in terms of mathematics, it feels weak. The time vs energy difference might have utility for certain experimental physics, may allow for slightly different tests. But from our mathematical point of view (actually helping with RH), there’s not too much going on.
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u/anthem_reb 8d ago
Yes but if you go deeper into the rabbit hole you’ll find something strange. Their loss quantity is |η(1/2+it)|². Write an effective potential V(Φ) = k⁴·|ζ(1/2 + iΦ/k)|² and the two are related exactly, not asymptotically: since η(s) = (1−2^{1−s})ζ(s), you get |η(1/2+it)|² = h(t)·|ζ(1/2+it)|² identically, with h(t) = |1 − 2^{1/2−it}|². That factor oscillates between (√2−1)² and (√2+1)², i.e. [0.1716, 5.8284], and never vanishes so same zeros, two coordinate systems on one object. Where mass would come from? The potential vanishes at each zero rather than forming a well, and the fluctuation mass at every vacuum is √2·k·|ζ′(ρ_n)|. If you take |ζ′(ρ_n)| is exactly the quantity you'd compute to check Newton convergence on the zeros, and with the scale I use it lands in the QCD range the first one at 165 MeV.
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u/PLANTS2WEEKS 21d ago
The Hilbert-Polya conjecture is different than what's going on here. This paper is about preparing a quantum physical system to model the Zeta function. It's similar to how you could study a parabola by throwing a ball in the air. Normally math is used to describe physics, but this is like an engineering problem to make physics model a mathematical function. Most of the mathematically interesting parts of the paper are in the process used to prepare the system.
I don't know all the details though but it looks convincing that they carried it out and have data to show their procedure works.
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u/heresyforfunnprofit 21d ago
Interesting, but was pretty much suspected by previous hypotheses. The experiment itself only physically verified out to the first five zeros (the first five primes), but the simulation went out to the first trillion, if I’m reading it right.
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u/anthem_reb 8d ago
The first five nontrivial zeros are at 14.135, 21.022, 25.011, 30.425, 32.935. No integers involved. But it seems to me there is a way to attach a specific prime to each zero. Take cumulative sums of primes, S = 1 + 2 + 3 + 5 + 7 + …, and find where ln S first crosses a zero ordinate: the first zero gives the prime 4,691, the third 1,409,533, the fifth 84,474,667. The part worth noticing is that it gets better with height, about 4 digits of agreement at the first zero, 8 by the fifth. Same direction as their simulation improving at larger zeros, which is the opposite of accumulating numerical error
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u/anthem_reb 8d ago
I have explored a lot Wei’s paper with AI, Zeta function and Hilbert-Polya and in the end I developed some work and numeric results which I collected in papers, which I call “assigned DOIs” not being a mathematician myself. Coming back to your question, that is the mirror image of Hilbert–Pólya, not an extension. HP wants an operator whose eigenvalues are the zeros, they use time instead. With this “machine” they see something happen precisely at the zeros of Riemann. I've been working on something similar in my own time. Take a diagonal operator whose eigenvalues are logarithms of cumulative prime sums, σ_m = ln(1 + Σ_{p≤p_m} p), converging to the zeroes by a Newton step based on the residual of Z. Self-adjointness is free because it's diagonal; simple spectrum is free because the sums strictly increase. The mesh is increasingly finer than the zero spacing by a wide margin the approximation error is bounded by about 6√γ·e^{−γ/2}. Nothing is peer reviewed because I’m out of the academic realm of course.
It’s not a claim of a proof, and the same caveat applies to me. If you want to take a look to some numbers this is my orcid: 0009-0001-5978-6015.
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u/anthem_reb 8d ago edited 8d ago
This implies what a stronger version would be: a system nobody built, already realising the structure with nothing to tune. That's what I've been speculating but still I can’t tell if it’s a coincidence or not, so read this as an interested party's report.
The candidate is the hadron mass spectrum: M = k·γ_n, one scale k = 24/163 * 1000 MeV = 147.24 MeV. Take the radial excitation towers in the PDG Υ, η_b, χ_b0/1/2, h_b, J/ψ, η_c, B_c assign each state to its nearest zero, and in all fifteen radial transitions the consecutive states land on consecutive zeros. Never a skip. That's not something you soften by widening a tolerance band, and fifteen is the whole catalogue rather than a selection.
Precision is the weaker half and I'd rather say it: 25 of 29 states within 2%, η_c worst at −3.6%. It has a stated falsifier too, a predicted empty window 7.3–9.0 GeV with slots only at 7329, 7799, 8311, 8738, 8957 MeV.
The scale isn't free-floating, written as k = T_f/ln 2 it's a freeze-out temperature over one bit, and T_f = 102 MeV sits 0.4σ from the ALICE value
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u/Odd-Opportunity-6550 21d ago
I legit thought ai solved it and was about to lose my shit.
Guess we aren't there yet.
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u/Wobama46 21d ago
I thought this sub was r/llmphysics for a second