Decidability of the Riemann Hypothesis

The Hamiltonian of a quantum mechanical system has an affiliated spectrum, and in order for this spectrum to be locally observable, the Hamiltonian should be Hermitian. Non-Hermitian Hamiltonians can be observed non-locally via parity, i.e. by taking the expectation value of the Wigner distribution evaluated at the orgin in phase space. Studies such as these quantum nonlocality analogies have led to the Bender-Brody-M\"uller (BBM) conjecture, which involves a non-Hermitian Hamiltonian eigenequation whose eigenvalues are the nontrivial zeros of the Riemann zeta function. Herein it is shown from symmetrization of the BBM Hamiltonian that the eigenvalues are not locally observable, i.e. \textit{the analytic continuation of the Riemann zeta function is not an analytically computable function at $\sigma=1/2$}. In the present case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a Hermitian Hamiltonian using a similarity transformation, and provide a trivial analytical expression for the eigenvalues of the results using Green's functions. A nontrivial expression for the eigensolution of the eigenequation is also obtained. A Gelfand triplet is then used to ensure that the eigensolution is well defined. The holomorphicity of the resulting eigenvalue spectrum is demonstrated, and it is shown that that the expectation value of the Hamiltonian operator is periodically zero such that the nontrivial zeros of the Riemann zeta function are not observable, i.e., the Riemann Hypothesis is not decidable. Moreover, a second quantization of the resulting Schr\"odinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigensolution it is shown that the real part of every nontrivial zero of the Riemann zeta function exists at $\sigma=1/2$, and a general solution is obtained by performing an invariant similarity transformation