Quantum Field Theories and Prime Numbers Spectrum

2012/11/22 by G. Menezes, Menezes, G., N. F. Svaiter +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #hep-th #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1211.5198

10 pages; revised version

arxiv created 2013/07/11 · arxiv updated 2013/07/12

Abstract

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line \Re(s)=1/2. Hilbert and Pólya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors formulated the question: is there a quantum mechanical system related to the sequence of prime numbers? In this Letter we show that such a sequence is not zeta regularizable. Therefore, there are no physical systems described by self-adjoint operators with countably infinite number of degrees of freedom with spectra given by the sequence of primes numbers.

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