Hilbert-P\'olya Conjecture, Zeta-Functions and Bosonic Quantum Field Theories

TL;DR

This paper investigates the possibility of linking the zeros of the Riemann zeta function and related number sequences to quantum field theories, concluding that certain spectral regularizations are impossible due to natural boundaries in associated series.

Contribution

It demonstrates that for specific number sequences related to zeta functions, the associated quantum field theories cannot be constructed because of natural boundaries preventing zeta regularization.

Findings

Functional integrals cannot be constructed for the considered systems.

Natural boundaries prevent the continuation of Dirichlet series.

Regularized determinants of Laplacians cannot be meromorphic.

Abstract

The original Hilbert and P\'olya conjecture is the assertion that the non-trivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However the suggestion of Hilbert and P\'olya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by Number Theory. In this paper we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularization, is that the various Dirichlet series attached to such sequences - such as those which are sums over "primes" of $(\mathrm{norm} \ P)^{-s}$ have a natural boundary, i.e., they cannot be continued beyond the line $\mathrm{Re}(s)=0$. The main argument is that once the regularized determinant of a Laplacian is meromorphic in $s$, it follows that the series considered above cannot be a regularized determinant. In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed.