The Principle of Minimum Justified Correlation

Abstract

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.