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

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2014-08, Vol.47 (32), p.325204-41
• Main Author: Sierra, Germán
• Format: Article
• Language: English
• Subjects: Boundaries; Clocks; Delta function; Fermions; Mathematical analysis; Matrix theory; Orbits; Prime numbers; quantum chaos; quantum mechanics; Riemann zeros
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/47/32/325204

We construct a Hamiltonian H R whose discrete spectrum contains, in a certain limit, the Riemann zeros. H R is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1 + 1 dimensions, which 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 that are associated with the prime numbers p, whose periods, as measured by the observerʼs clock, are . Acting on the chiral components of the fermion , H R becomes the Berry-Keating Hamiltonian , where x is identified with the Rindler spatial coordinate and 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 ϑ, the spectrum is a continuum where the Riemann zeros are missing, as in the adelic Connes model. However, for some values of ϑ, related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues that are immersed in the continuum. We generalize this result to the zeros of Dirichlet L-functions, which are 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 the Random Matrix Theory.