I was skimming through this recent PRL paper [preprint here], which purports to show that the non-trivial zeroes of the Riemann zeta function correspond to the energy levels of a quantum mechanical system. My initial reaction was that it’s a cute result, but not one that’s going to help prove/disprove the Riemann Hypothesis. The final paragraph of the paper caught my eye, though:
Apart from providing a new tool in the spectral approach to the Riemann hypothesis, there is also the possibility that it will allow a laboratory construction of a system for which the physics is described by the ‘‘Riemann Hamiltonian’’, the existence of which would prove the Riemann hypothesis.
The mathematical existence (whatever that means) of that Hamiltonian will prove the Riemann hypothesis (RH from now on). But I wonder how physically constructing a system with that Hamiltonian will have any relevance to RH. Firstly, the model is an approximation anyway (to get the correct energy levels, you have to consider the ‘semiclassical limit’), so if we think that quantum mechanics is the ‘true’ theory, then in the real world, that system will not have energy levels giving us the nontrivial solutions of the Riemann zeta function. But more importantly, it seems that the existence of a mathematical operator should be independent of whether there is a physical system whose Hamiltonian is that operator.
So why would one think that the existence of such a physical system would be relevant to RH? Here’s one possible line of thought:
P1. The existence of a self-adjoint operator H having such-and-such mathematical properties implies RH.
P2. If a physical system whose Hamiltonian is a self-adjoint operator H having such-and-such mathematical properties could exist (in our world),* then there exists a self-adjoint operator H having such-and-such mathematical properties.
C1. If a physical system whose Hamiltonian is a self-adjoint operator H having such-and-such mathematical properties could exist (in our world), then RH is true.
I phrased P2 in terms of the possible existence instead of the actual existence of a physical system having that kind of Hamiltonian, since I am assuming that it cannot be that RH is false before physicists construct a certain physical system and then true thereafter — that the truth value of RH is independent of what happens in the physical world, and insofar as the construction of the physical system in question is relevant to the mathematical problem, it can only help us discover if RH is true.
Even if we accept the above argument, it’s not clear how the construction (in the laboratory) of the physical system in question is relevant to RH. One might think that the construction of such a system would show that yes, that system could exist. That would provide us with the antecedent of P2 and hence (if we accept the argument leading to C1) prove RH. But there is an ambiguity in our imagining the construction of ‘such a system’. The authors derive the Hamiltonian they need from a system consisting of a charged particle on the xy plane in a constant uniform perpendicular magnetic field and an electric potential . Suppose we do construct a system with a charged particle in a plane and the requisite electromagnetic field. How are we to determine if its Hamiltonian really is the one derived by the authors? We could simply say that we know it is so theoretically, but in that case, we need not have physically constructed the system. If we want to say more than that, we can try to find experimental evidence for its Hamiltonian being what we suspect it is. We can measure and count the system’s energy levels. We might then ‘prove’ RH as follows: those energy levels are best explained by the fact that in the semiclassical limit the system has a Hamiltonian with such-and-such properties, so we infer that the system has a Hamiltonian with such-and-such properties, so a Hamiltonian with such-and-such properties exists, and hence RH is correct. That is, the only motivation for physically constructing such a system would be to check that it in fact has the energy levels we expect it to have, otherwise we could just sit pretty on the existing theoretical derivations. But since the inference outlined above is essentially an inference to the best explanation, it won’t be a proof of RH — if someone disproved it by mathematical deduction, then we would discount the ‘experimental’ proof and conclude that the explanation for the energy levels we observed is not the existence of a certain type of self-adjoint operator.
So much for physically constructing the system. What about the theoretical derivation itself? Would the mere theoretical possibility of the system suffice to prove RH? Here’s one way it might. If we think the authors’ derivation of the Hamiltonian for the system in question implies that a system with such a Hamiltonian could exist in our world, and we accept the argument from P1 and P2 to C1, then yes, the theoretical possibility of the system would prove the Riemann Hypothesis. But I think it’s very difficult to think that the theoretical derivation alone implies that the system could exist. As a matter of scientific practice, it isn’t the case that all physical systems that are consistent with fundamental laws are considered systems that could exist — many physicists would reject systems that have backwards causation, for example. This is to say that there are factors other than consistency with fundamental theories that we use to determine if a system ‘could’ exist. So if RH was disproved by mathematical deduction, I suspect many would take that as a reason to view the derivation of the Hamiltonian in question as suspect — perhaps to the point of wanting to revise the theories used in the derivation. That is, we would reject quantum mechanics if it implies a mathematical impossibility, but we would not reject mathematics if quantum mechanics says something inconsistent with it.
All this without considering whether the Hamiltonian of the system should be considered as an instance of a mathematical operator that exists, or whether the Hamiltonian itself even exists, etc.
*The ‘in our world’ provision is to say that the system could exist in a world with the laws of nature as this world.
Germán Sierra, Paul K. Townsend (2008). Landau Levels and Riemann Zeros Physical Review Letters, 101 (11) DOI: 10.1103/PhysRevLett.101.110201