Odlyzko showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. This gives some support to the Hilbert–Pólya conjecture, the "spectral viewpoint" of Riemann Hypothesis.
If we take this "spectral viewpoint" of Riemann Hypothesis, the operator, which supposes to generate the eigenvalues (or zeros of zeta function), will it correspond to an operator in quantum mechanics or in quantum field theory ?
My understanding is that, in quantum field theory, for a specific physical quantity, for example, energy, there will be collected of operators which are indexed by space and time (X, t), can we still find one "giant" matrix which can produce the all energy levels for the field ? or is that the matrix has to be indexed by (X, t) ?
If the matrix in quantum field has to be indexed by (X, t), does this imply that we shall NOT look for an operator in QFT to match RH ? We shall just try to search for an operator in Quantum mechanics.
The answer to this question seems to be yes. Check out this paper in PRL:
Brendy, Brody, Müller - Hamiltonian for the Zeros of the Riemann Zeta Function
Specifically, they construct an operator and show that the Riemann hypothesis is true if and only if this operator is Hermitian.