Zeta Functions in Quantum Mechanics: A New Mathematical Interpretation

This paper proposes a novel mathematical framework exploring quantum mechanics through the deep structure of the Riemann zeta function. We demonstrate that core quantum phenomena—wave-particle duality, superposition principle, measurement collapse, and quantum entanglement—can be understood through intriguing mathematical analogies with zeta function properties. By establishing correspondences between series divergence and quantum superposition, analytic continuation and measurement processes, functional equations and quantum entanglement, we provide fresh mathematical perspectives on quantum mechanics. Our key innovations include: (1) establishing inspiring analogy relations between wave functions and zeta functions; (2) proposing an analytic continuation theory of quantum measurement; (3) exploring mathematical analogies of EPR entanglement through functional equations; (4) characterizing quantum chaos statistics via zero distribution; (5) exploring AdS/CFT correspondence with zeta functions through holographic duality. This framework offers new mathematical insights into quantum mechanics and suggests potential quantum computational optimization directions.