John H. Van Drie | Searcharxiv

arXiv2000

A generalization of Jaynes' principle: an information-theoretic interpretation of the minimum principles of quantum mechanics and gravitation

By considering the "kinetic-energy" term of the minimum principle for the Schrödinger equation as a measure of information, that minimum principle is viewed as a statistical estimation procedure, analogous to the manner in which Jaynes ({\it Phys. Rev.},{\bf 106}, 620, 1957) interpreted statistical mechanics. It is shown that the entropy formula of Boltzmann and Jaynes obey a property in common with the quantum-mechanical kinetic energy, in which both quantities are interpreted as measures of correlation. It is shown that this property is shared by the key terms in the minimum principles of relativistic quantum mechanics and General Relativity. It is shown how this principle may be extended to non-Riemannian nonEuclidean spaces, which leads to novel field equations for the torsion.

math-ph↗

arXiv2000

The Boltzmann/Shannon entropy as a measure of correlation

IIt is demonstrated that the entropy of statistical mechanics and of information theory, $S({\bf p}) = -\sum p_i \log p_i $ may be viewed as a measure of correlation. Given a probability distribution on two discrete variables, $p_{ij}$, we define the correlation-destroying transformation $C: p_{ij} \to π_{ij}$, which creates a new distribution on those same variables in which no correlation exists between the variables, i.e. $π_{ij} = P_i Q_j$. It is then shown that the entropy obeys the relation $S({\bf p}) \leq S({\bf π}) = S({\bf P}) + S({\bf Q})$, i.e. the entropy is non-decreasing under these correlation-destroying transformations.

math-ph↗