Entropy and Copula Theory in Quantum Mechanics

Abstract

In classical mechanics, we have individual particle and invariant density in the phase space. In quantum mechanics, any particle is sensitive in a different way from all other particles, for its position and also to the measure process. Thus, we substitute the classical probability in the phase space with the conditional probability in the network of communicating particles. Any probability and entropy are functions of the phase position conditioned by the position of the other particles. Therefore, for different measures we have different conditional entropies. The space of the entropies is a curved and possible torque multidimensional space where the derivative is the covariant derivative on a manifold of the entropic space. At the zero quantum field, the covariant derivative commutes and Fisher matrix is part of the kinetic terms in the Lagrangian where the derivative is the covariant derivative. With Lagrange minimum condition and the entropic space it is possible to show a connection between entropy space and Bohm potential in quantum mechanics. Entropy multidimensional space includes dependence and entanglement as geometric structure of the entropy. Now we can create a non-zero quantum field approach when the covariant derivative does not commute so we have curvature and torsion. The non-zero quantum field can be the Casimir field of forces. Therefore, Casimir force as gravity in the space-time is modelled by curvature and torsion of the entropic space. Useful connection between dependence and covariant derivatives are obtained by copula (dependence measure) and quantum mechanics.