The Principle of Minimum Justified Correlation

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 $\rho (x,y)$ is replaced by ${\rho }_x(x){\rho }_y(y)$, 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: $$I[\rho ]=h[{\rho }_x]+h[{\rho }_y]-h[\rho ]\mathrm{.}$$ In the continuous case, however, another measure appears: Fisher information. The relative Fisher information $J(\rho \mathrm{\parallel }{\rho }_x{\rho }_y)$ behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, $$⟨T{⟩}_{\psi }-⟨T{⟩}_{\mathrm{\Phi }}=\frac{{\mathrm{\hslash }}^2}{8m}J(\rho \mathrm{\parallel }{\rho }_x{\rho }_y),\mathrm{\Phi }=\sqrt{{\rho }_x{\rho }_y}\mathrm{.}$$ 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.