Quantum Proof of Riemann Hypothesis

Enlightened by the G-dynamics being identified as a part of quantum covariant Hamiltonian system (\textbf{QCHS}) for constructing the geometric wave equation based on the quantum covariant Poisson bracket (\textbf{QCPB}) theory, we find a new quantum physical system associated with classical Hamiltonian operator ${\hat{H}}^{\left( cl \right)} $ in Schr\"{o}dinger equation $\sqrt{-1}\hbar \frac{\partial}{\partial t} \varphi ={\hat{H}}^{\left( cl \right)} \varphi$ behind the Riemann non-trivial zeroes, such an unbounded linear self-adjoint operator ${{\hat{w}}^{\left( cl \right)}}={{b}_{c}}\hat{Q} $ to solve the Hilbert-P\'{o}lya conjecture (\textbf{HPC}) really exists that is given by one of geometric wave equation, that's exactly what Hilbert-P\'{o}lya and Berry-Keating still expect to find such peculiar quantum system in proving the Riemann hypothesis over the long time, and with identity ${{\hat{w}}^{\left( cl \right)}}{{u}^{-1/2}}=\hat{Q}{{u}^{-1/2}}=0$ holds, then Riemann hypothesis (\textbf{RH}) is then proven. Meanwhile, we discuss the curvature operator $\hat{Q}$ in different dimensional case, especially, Laplacian-Bertrami operator is considered for curvature operator. The relation between the Schr\"{o}dinger equation and geometric wave equation associated with geometric Hamiltonian operator is given.For the Berry-Keating conjecture (\textbf{BKC}), we need to find this unknown physical system, so that we comprehensively consider the Riemann manifold, Einstein metric and Ricci flow to ensure what the imaginary part really stands for, then it connects the vacuum field equation in two dimensional case, the scalar curvature forms a discrete set in 2D Riemann manifolds.本研究受启发于将G动力学（G-dynamics）识别为量子协变哈密顿系统（quantum covariant Hamiltonian system, QCHS）的组成部分，该系统依托量子协变泊松括号（quantum covariant Poisson bracket, QCPB）理论构建几何波动方程。我们发现了一类与黎曼非平凡零点背后薛定谔方程$sqrt{-1}hbar frac{partial}{partial t} varphi ={hat{H}}^{left( cl ight)} varphi$中的经典哈密顿算子$hat{H}^{(cl)}$相关的新型量子物理系统；而用于求解希尔伯特-波利亚猜想（Hilbert-Pólya conjecture, HPC）的无界线性自伴算子$hat{w}^{(cl)} = b_c hat{Q}$确实存在，且可通过某几何波动方程推导得到——这正是希尔伯特-波利亚与贝里-凯特长期以来在证明黎曼假设的过程中一直期望探寻的这类特殊量子系统。结合恒等式$hat{w}^{(cl)} u^{-1/2} = hat{Q} u^{-1/2} = 0$，即可完成黎曼假设（Riemann hypothesis, RH）的证明。同时，我们探讨了不同维度下的曲率算子$hat{Q}$，尤其针对曲率算子研究了拉普拉斯-贝尔特拉米算子（Laplacian-Bertrami operator）。本文还给出了与几何哈密顿算子相关的薛定谔方程与几何波动方程之间的内在关联。针对贝里-凯特猜想（Berry-Keating conjecture, BKC），我们需要找到这一未知物理系统，因此综合考量黎曼流形、爱因斯坦度量与里奇流，以明确虚部的实际物理内涵；该系统与二维情形下的真空场方程相联系，且在二维黎曼流形中，标量曲率构成离散集合。