G. Menezes, N. F. Svaiter
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and P\'olya 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 assume that there is a class of hypothetical physical systems described by self-adjoint operators with countable infinite number of degrees of freedom with spectra given by the sequence of primes numbers. We prove a no-go theorem. We show that the generating functional of connected Schwinger functions of such theories cannot be constructed.
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http://arxiv.org/abs/1211.5198
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