Seminar “Factorization of Numbers, Schrödinger Cats and the Riemann Hypothesis”
Professor Wolfgang P. Schleich, Chair of Theoretical Physics, Institut für Quantenphysik, Universität Ulm, Germany
Professor Wolfgang P. Schleich is 2018/2019 Sackler Lecturer in the Mortimer and Raymond Sackler Institute of Advanced Studies.
Abstract
In this talk we connect the three different topics of factorization of numbers, Schrödinger cats and the Riemann hypothesis. The bridge between these areas is the concept of a Gauss sum.
Gauss sums manifest themselves in various phenomena such as the Talbot effect, wave packet dynamics or quantum carpets. Moreover, Gauss sums can be used to efficiently factor numbers. The talk summarizes these activities [1] and discusses a new approach [2] based on a potential with a logarithmic energy spectrum.
Moreover, we propose an elementary quantum system which provides us with the Riemann zeta function. We show [3] that its zeroes are a consequence of the interference of two quantum systems with opposite phases. However, the preparation of such a superposition state (Schrödinger cat) is impossible unless one takes advantage of entangled quantum systems. In this sense analytic continuation familiar from complex analysis finds entanglement as its analogue in quantum mechanics.
We conclude by introducing a geometrical approach [4] towards the Riemann hypothesis based on the lines of constant phase.
[1] S. Wölk, W. Merkel, W.P. Schleich, I.Sh. Averbukh, B. Girard, and G.Paulus, Factorization of numbers with Gauss sums: I. Mathematical background, New J. Phys. 13, 103007 (2011)
[2] F. Gleisberg, F. Di Pumpo, G. Wolff, and W.P. Schleich, Prime factorization of arbitrary integers with a logarithmic energy spectrum, J. Phys. B: At. Mol. Opt. Phys. 51, 035009 (2018)
[3] C. Feiler and W.P. Schleich, Entanglement and analytical continuation: an intimate relation told by the Riemann zeta function, New J. Phys. 15, 063009 (2013)
[4] W.P. Schleich, I. Bezdekova, M.B. Kim, P.C. Abbott, H. Maier, H.L. Montgomery, and J.W. Neuberger, Equivalent formulations of the Riemann hypothesis based on lines of constant phase, Phys. Scr. 93, 065201 (2018)
