M. Kuna | Searcharxiv

arXiv2003

Quantum morphogenesis: A variation on Thom's catastrophe theory

Non-commutative propositions are characteristic of both quantum and non-quantum (sociological, biological, psychological) situations. In a Hilbert space model states, understood as correlations between all the possible propositions, are represented by density matrices. If systems in question interact via feedback with environment their dynamics is nonlinear. Nonlinear evolutions of density matrices lead to phenomena of morphogenesis which may occur in non-commutative systems. Several explicit exactly solvable models are presented, including `birth and death of an organism' and `development of complementary properties'.

quant-ph↗

arXiv2000

Darboux-integration of idρ/dt=[H,f(ρ)]

A Darboux-type method of solving the nonlinear von Neumann equation $i\dot ρ=[H,f(ρ)]$, with functions $f(ρ)$ commuting with $ρ$, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities $f(ρ)$ are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian $H$ of a one-dimensional harmonic oscillator and $f(ρ)=ρ^q-2ρ^{q-1}$ with arbitary real $q$. It is shown that self-scattering solutions possess the same asymptotics for all $q$ and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.

quant-ph↗