发表机构
Van Drie Research（范德里研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出最小合理关联原理：在给定约束下选取关联最小的分布或波函数，并证明香农熵、费希尔信息与动能均作为关联度量，建立了它们之间的恒等关系。
AI 中文摘要
本文表明，香农熵、费希尔信息和量子力学动能均可视为关联的度量。我们从一种基本的关联破坏映射出发：将联合概率密度 $\rho(x,y)$ 替换为其边缘分布的乘积 $\rho_x(x)\rho_y(y)$。在离散情形下，香农熵在该映射下非减。对于连续分布，这一基本映射导出一个性质良好、坐标不变的关联度量，类似于离散香农熵：\\[ I[\rho] = h[\rho_x] + h[\rho_y] - h[\rho]。 \\] 然而，在连续情形下，出现了另一种度量：费希尔信息。相对费希尔信息 $J(\rho\\|\rho_x\rho_y)$ 在基本映射下表现类似。对于归一化的实量子波函数，其减少量恰好与平均动能的减少成正比，\\[ \langle T\rangle_\psi - \langle T\rangle_\Phi = \frac{\hbar^2}{8m} J(\rho\\|\rho_x\rho_y), \qquad \Phi = \sqrt{\rho_x \rho_y}。 \\] 用杰恩斯（Jaynes）的语言来说，这一结果支持最小合理关联原理：在给定物理约束以及满足这些约束的一组可能分布或相关振幅的情况下，从该集合中选取关联最小的那些。这是两篇系列论文的第一部分。本文推导了上述熵、费希尔信息和动能恒等式，并陈述了它们所支持的原理。第二部分将处理第一部分提出但未解答的问题：多解、时间依赖性、自旋的作用、通向薛定谔方程本身的路径，以及拟议的实验检验。已使用人工智能。
英文摘要
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density $ρ(x,y)$ is replaced by $ρ_x(x)ρ_y(y)$, the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: \[ I[ρ] = h[ρ_x] + h[ρ_y] - h[ρ]. \] In the continuous case, however, another measure appears: Fisher information. The relative Fisher information $J(ρ\|ρ_xρ_y)$ behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, \[ \langle T\rangle_ψ- \langle T\rangle_Φ= \frac{\hbar^2}{8m} J(ρ\|ρ_xρ_y), \qquad Φ= \sqrt{ρ_x ρ_y}. \] In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation. This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.
Comments13 pages, 0 figures
