Paper detail

The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime

We construct a Hamiltonian H whose discrete spectrum contains, in a certain limit, the Riemann zeros. H is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1+1 dimensions, that has a boundary given by the world line of a uniformly accelerated observer. The action contains a sum of delta function potentials that can be viewed as partially reflecting moving mirrors. An appropriate choice of the accelerations of the mirrors, provide primitive periodic orbits associated to the prime numbers $p$, whose periods, measured by the observer's clock, are log p. Acting on the chiral components of the fermion, H becomes the Berry-Keating Hamiltonian (x p + p x)/2, where x is identified with the Rindler spatial coordinate and p with the conjugate momentum. The delta function potentials give the matching conditions of the fermion wave functions on both sides of the mirrors. There is also a phase shift for the reflection of the fermions at the boundary where the observer sits. The eigenvalue problem is solved by transfer matrix methods in the limit where the reflection amplitudes become infinitesimally small. We find that for generic values of the phase shift the spectrum is a continuum, where the Riemann zeros are missing, as in the adelic Connes model. However, for some values of phase shift, related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues immersed in the continuum. We generalize this result to the zeros of Dirichlet L-functions, associated to primitive characters, that are encoded in the reflection coefficients of the mirrors. Finally, we show that the Hamiltonian associated to the Riemann zeros belongs to class AIII, or chiral GUE, of Random Matrix Theory.