Wednesday, October 10, 2018

Das, Kalauni claim to have a 3-page SUSY proof of Riemann Hypothesis

Cafeinst has pointed out that one week ago, Ashok Das (Rochester) and Pushpa Kalauni (Oklahoma) have published a 3-page preprint
A simple derivation of the Riemann hypothesis from supersymmetry
on the math.GM arXiv. So I immediately opened it. And yes, you can immediately see that the authors are less than 89 years old.

I have personally tried a hundred of strategies to prove the Riemann Hypothesis so I could immediately recognize their operators \(O,O^\dagger\) from equations (10) and (11): I have played with the very same operators for quite some time, too. You may imagine that there's an operator whose spectrum is the set of values of \(\lambda\), the imaginary part of \(s\) such that \(\zeta(s)=0\) nontrivially.

The corresponding eigenstates may be considered "delta functions" located at \(\lambda\). However, if you perform some kind of a generalized Fourier transform, the corresponding eigenstates could go like \(x^{i\lambda}\) and be defined on \(x\in \RR^+\). Exactly when you add the factor of \(x^{-1/2}\), these wave functions become somewhat naturally normalizable and orthogonal in a Dirac sense. There's no mystery about the normalizability – they are just standard plane waves as functions of the variable \(q=\ln x\) and the extra \(1/x\) in the norm comes from the Jacobian.

Well, a problem is that the plane waves are normalizable to the Dirac delta-function if you allow any continuous \(\lambda\in\RR\) while we only want \(\lambda\) to match the spectrum of the zeta function zeroes. So when \(\lambda\) is restricted to the zeta function zeroes, they span a smaller Hilbert space than the Hilbert space of all functions of \(x\in\RR^+\). We should have an independent description of the Hilbert space of the "restricted" functions of the positive \(x\). What are they? They don't really discuss this problem at all.

Except for these lethal problems, we could say "so far so good."

I couldn't have ever completed a proof of the Riemann Hypothesis with these intriguing toys because I couldn't have eliminated the possibility that there are roots of the zeta function that produce some non-normalizable formal solutions or non-normalizable "eigenstates" by this procedure. In recent years, I realized that this is a problem that I have with all strategies based on the Hilbert-Pólya program: the program doesn't seem to exclude the existence of "quasinormal modes" that may still correspond to a pole in some Green's functions but that don't enhance the Hilbert space because the corresponding solutions simply aren't normalizable.

Fine. So Das and Kalauni claim to have something extra that uses SUSY. Well, I have obviously tried to use SUSY as well but in ways that weren't equivalent to their claim.

How do they use SUSY? They write \[

H=\{Q,Q^\dagger\}={\rm diag}(A^\dagger A, A\cdot A^\dagger)

\] which may be obtained from the \(2\times 2\) matrix \(Q= ((0,0),(A,0))\) and its Hermitian conjugate. Their incorporation of the zeta function is that \(A\), the matrix element of the supercharge \(Q\), produces \(\zeta(1/2+i\lambda)\) when acting on the desired \(x^{i\lambda}\)-like wave function. The Hamiltonian itself, when acting on these power law wave functions, is supposed to produce a factor \(|\zeta(s)|^2\).

If the zeta function of \(s\) in the critical strip vanishes, then their Hamiltonian has to annihilate the power law test function, and this test function must therefore be "BPS". And they claim that for it to be "BPS", the test function has to be normalizable, and therefore the real part of \(s\) must sit at the critical line.

So far, again, I don't understand how they avoid my general problem with any Hilbert-Pólya proof: Cannot there be non-normalizable, formally "BPS" states as well? So sadly, at least so far, I cannot confirm that they have solved the big open mathematical problem. The authors promise to release a more complete "calculation". I would be willing to bet that it won't become a proof – they won't fix the problems. They seem to be exactly as naive as I was about an hour after my enthusiastic playing with the \(x^{i\lambda}\) test functions. ;-)

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