# PT Symmetry and the Riemann Hypothesis

Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The flurry is over the paper Hamiltonian for the zeros of the Riemann zeta function by Bender, Brody, and Müller in Physical Review Letters. The new idea is to look for a Hamiltonian which is not Hermitian, but rather PT-symmetric.

PT symmetry was introduced to Quantum Theory by Bender; there’s a nice survey by him Introduction to PT symmetric Quantum Theory on the arXiv, and a short expository paper PT symmetry in quantum physics: from a mathematical curiuosity to optical experiments in Europhysics News. Here’s an extended quote from the survey:

The central idea of $$\mathcal{PT}$$-symmetric quantum theory is to replace the condition that the Hamiltonian of a quantum theory be Hermitian with the weaker condition that it possess space-time reflection symmetry ($$\mathcal{PT}$$ symmetry).

$$\ldots$$

Two important discrete symmetry operators are parity (space reflection), which is represented by the symbol $$\mathcal{P}$$, and time reversal, which is represented by the symbol $$\mathcal{T}$$. The operators $$\mathcal{P}$$ and $$\mathcal{T}$$ are defined by their effects on the dynamical variables $$\hat x$$ (the position operator) and $$\hat p$$ (the momentum operator). The operator $$\mathcal{P}$$ is linear and has the effect of changing the sign of the momentum operator $$\hat p$$ and the position operator $$\hat x$$: $${\hat p}\to-{\hat p}$$ and $${\hat x}\to-{\hat x}$$. The operator $$\mathcal{T}$$ is antilinear and has the effect $${\hat p}\to-{\hat p}$$, $${\hat x}\to{\hat x}$$, and $$i\to-i$$.

$$\ldots$$

We describe here an alternative way to construct complex Hamiltonians that still guarantees the reality of the eigenvalues and the unitarity of time evolution and which also includes real, symmetric Hamiltonians as a special case. We will maintain the symmetry of the Hamiltonians in coordinate space, but we will allow the matrix elements to become complex in such a way that the condition of space-time reflection symmetry ($$\mathcal{PT}$$ symmetry) is preserved. The new kinds of Hamiltonians discussed in this paper are symmetric and have the property that they commute with the $$\mathcal{PT}$$ operator: $$[H,\mathcal{PT}]=0$$. In analogy with the property of Hermiticity $$H=H^\dagger$$, we will express the property that a Hamiltonian is $$\mathcal{PT}$$ symmetric by using the notation $$H=H^ \mathcal{PT}$$. We emphasize that our new kinds of complex Hamiltonians are symmetric in coordinate space but are not Hermitian in the Dirac sense. To reiterate, acceptable complex Hamiltonians may be either Hermitian $$H=H^\dagger$$ or $$\mathcal{PT}$$-symmetric $$H=H^\mathcal{PT}$$, but not both. Real symmetric Hamiltonians may be both Hermitian and $$\mathcal{PT}$$-symmetric.

Although Bender originated the idea of PT symmetry himself, the new paper is not the first to look at PT symmetry in the context of the Hilbert- Polya conjecture. The paper A pseudo-unitary ensemble of random matrices, PT symmetry and the Riemann Hypothesis by Ahmed and Jain dates back to 2006 (In their terminology, PT symmetric matrices are pseudo-unitary.) In that paper, and also in the survey Gaussian-random ensembles of pseudo-Hermitian matrices, he even investigates the nearest neighbor spacing for ensembles of such matrices.

With long introduction out of the way, I’ll try to formulate an appropriate MO question. The nearest neighbor spacings computed by Ahmed are similar to, but not the same as, those of the Gaussian Unitary Ensemble. Give the extraordinarily good fit to the data of the Riemann zeros computed by Odyzko and others, why would one still believe this is a viable approach?

Update: Since posting I've been pointed to this blog post (with comments) in Not Even Wrong, and this MSE question (with answers and comments). These and the comments below don't answer my specific question about nearest neighbor spacings, but do make other points which counteract the enthusiasm of the popular media.

• a rather severe criticism by Jean Bellisard of this approach has just been posted on arXiv. – Carlo Beenakker Apr 11 '17 at 19:55
• The critique by Bellisard (noted by @CarloBeenakker) appears to me to be very accurate. – paul garrett Apr 11 '17 at 20:43
• About the spacing, I do not pretend to have any sort of definitive answer, but we do know that certain behaviors don't kick in until $\log\log T$ is large, which is probably forever beyond numerical experimentation. That is, there is some precedent for meaningful phenomena occurring beyond our sight. It might be silly to "bet" in that direction, but it'd be possible, etc. – paul garrett Apr 11 '17 at 22:22
• A response to Jean Bellisard's criticism was recently posted on the arXiv: arxiv.org/abs/1705.06767 – Pace Nielsen May 22 '17 at 15:54
• Any takes on the response to Bellisard's criticism @PaceNielsen mentions? – Rudi_Birnbaum Oct 5 '17 at 19:41