# Riemann zeta and quantum physics?

Sometimes I read about connections between "advanced math" and quantum physics, but I am skeptical of these claims. I can believe or understand the connections to calculus, vector calculus, differential equations, or linear algebra, but when I read about connections with prime numbers and the Riemann zeta function, I get very skeptical and confused. As a "toy model" or a math exercise for physics students, okay, but is it seriously used in physics?

I'm not an expert at math nor quantum physics but I will let the links speak for themselves:

And others. What are they talking about? Is this real? What is meant with energy levels? What does the real and imaginary part in the domain and range of zeta mean physically? Is the position of the zeros of the zeta function related to the radii of the electrons around a nucleus?

I have never seen this in school!

• en.wikipedia.org/wiki/Primon_gas is an example. It is not a real thing, it is an imaginary quantum system with properties from the zeta function. Mar 1, 2017 at 0:07
• This recent article may be relevant: arxiv.org/abs/1608.03679# ("A Hamiltonian operator...is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function") Mar 1, 2017 at 10:25
• I think the riemann zeta can use the gamma function in its definition which is basically the factorial. But that is not a direct use of riemann zeta so maybe just a fun fact :-)
– Emil
Mar 1, 2017 at 15:27
• Primon gas is artificial and theoretical. Not a natural real thing. So it seems like a toy model or exercise. Also it is not clear - even dubious how studying Primon would help understand RH. Perhaps less clear but i think the same applies to hamiltonians and other proclaimed applications/connections. Nothing personal, i just adress this.
– mick
Mar 1, 2017 at 17:30
• Related: physics.stackexchange.com/q/122905/2451 , physics.stackexchange.com/q/190712/2451 , and links therein. Mar 1, 2017 at 17:36

The precise theorem is that suppose you take the zeroes of the Riemann zeta function as $1/2+i \gamma_n$. Then you look at the normalized differences between neighboring zeroes. If you pretend these were drawn from a random variable and calculate the correlations, you get the same result as if you just took a random Hermitian matrix like the kind that is used to model heavy nuclei. The statement is not about them being equivalent, but that there is a universal property in the sense that they are asking two questions that have the same answers. If you ask a different question about the Riemann zeta function, there is no reason for it to still keep this connection.

You can look up this as it was known as Montgomery's pair correlation conjecture.

First let me say that this article was published in 2009 and obviously the buzz doesn't seem to have concretized, so I'll take it with caution.

I haven't read the article yet (maybe later, it is quite a long one...).

Although this is not the first time we see such convergence of phenomena. For instance when Conway showed some resemblance between nuclear disintegration process and look-and-say sequence, it felt really surprising at the time. But apparently it stayed at the curiosity stage and deeper examination showed discrepancies with the current knowledge.

I'm guessing it is possibly the same with these prime numbers vs levels of energy, but why not, lets be open minded, someday it could result in a fascinating theory.

Anyway, you mentioned the relation of quantum physics and the zeta function, there is actually a well known one which is the Casimir effect. When calculating the force between two plates, we have to sum the energy modes of the vacuum for all frequencies, something like $\sum\frac{n\pi}{\delta}$ appears, although this sum is infinite.

Yet in fact the waves are vanishing when their wavelength do not agree anymore with the (very small) distance $\delta$ separating the plates. The calculation uses the wave functions that act as what we call smoothing functions in mathematics and we find the famous $\sum n=\frac{-1}{12}=\zeta(-1)$ as a proportionnality factor for the amplitude of the force. So at least in this case, zeta and quantum physics were related.

• Well barely. You used a summability method. The complex Numbers , primes or zeta zero's are not related here ?
– mick
Mar 1, 2017 at 0:30
• But +1 for the physics anyway.
– mick
Mar 1, 2017 at 0:31
• Who's to stop you from using a different regularization? Regularization is a choice. Your final answer shouldn't depend on this choice. Mar 1, 2017 at 2:18
• I suspect that Conway's use of the elements to label the "atomic" look and say sequences was a linguistic/numerical joke, not a serious connection to physics. That the distribution of zeroes of the the zeta function may match the one that comes from some Hamiltonians may be of physical interest. See en.wikipedia.org/wiki/Look-and-say_sequence#Cosmological_Decay Mar 1, 2017 at 14:20

Best to read where the author is coming from, and where he wants to go to.

In the blog you quoted, the author says "The landscape created by the Riemann hypothesis it turns out, directly matches the quantum energy levels of atoms. This is amazing! It’s like in the Matrix where neo sees the code that builds his world."

But when you read his source, extract below, you get this:

Mathematicians have long suspected that there might be a way to convert the Riemann hypothesis into an equation similar to those used in quantum physics. The zeros of the zeta function could then be calculated the same way physicists calculate the possible energy levels for an electron in an atom, for example.

Following ideas by Keating and Michael Berry of the University of Bristol and also by Alain Connes of the Collège de France in Paris, Sierra and Townsend have now made that connection a bit more concrete. They have suggested that an electron constrained to move in two dimensions, and subjected to electric and magnetic fields, might have energy levels that precisely match the zeros of the zeta function. Demonstrating the existence of such a system, even on paper, would confirm the Riemann hypothesis.

If you subject an electron to electronic and magnetic fields, and constrain it to 2 dimensions, well you can get it to fit pretty much any pattern you want.........

• So ... The blog is nonsense ? And so is Every claim relating Riemann zeta to physics ?
– mick
Mar 1, 2017 at 0:13
• Honestly, I like his blog and his fascination with physics, and also I don't know enough to give you any worthwhile opinion of the complete link between the zeta to physics. But I own enough popular science books to know you can tell in the first 10 minutes of reading if it's data or drama. So get another source, or another answer from this site...Without Riemann, building directly on the work of Gauss, we would not have General Relativity, but that was because his work was directly connected with describing space, so you have to do the research the blogger doesn't.
– user146020
Mar 1, 2017 at 0:28

As Countto10 pointed out, the claim in the blog is at best hyperbole and at worst nonsense. This is not to say that there are no connections/applications of Riemann Zeta to physics. There is a good paper that goes over some of the ways in which this function is used in physics: Physics of the Riemann Hypothesis. Another connection is Montgomery's Conjecture.

Many times these claims of connections are tenuous at best, but there is undoubtedly deep connections between higher math and physics - definitely more than just vector calc, diff eq's, and linear algebra.

Incidentally, I found this post that has a few more examples, including the two above.