2007/12/05 by Germán Sierra, German Sierra
Engineering · Mathematics · Physics and Astronomy · #Quantum #Quantum and Classical Electrodynamics #Quantum field theory #Riemann hypothesis #Riemann surface #Spectral Theory in Mathematical Physics #Sports Dynamics and Biomechanics #cond-mat.other #hep-th #math-ph #math.MP #math.NT #quant-ph
paper · pdf · doi:10.1088/1367-2630/10/3/033016
published as NewJ.Phys.10:033016,2008 · 42 pages, 12 figures
arxiv created 2007/12/05 · openalex publication_date 2008/03/11 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In 1999, Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper, we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wavefunctions associated with the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wavefunctions associated with the Berry–Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wavefunctions, giving rise to the Riemann zeros, are found using the Riemann–Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.