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M. Czachor

Publications and source records attributed to M. Czachor.

6 recordsLinked to original sources
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↗
arXiv2001

Schroedinger-picture correlation functions for nonlinear evolutions

The well known interpretational difficulties with nonlinear Schrödinger and von Neumann equations can be reduced to the problem of computing multiple-time correlation functions in the absence of Heisenberg picture. Having no Heisenberg picture one often resorts to Zeno-type reasoning which explicitly involves the projection postulate as a means of computing conditional and joint probabilities. Although the method works well in linear quantum mechanics, it completely fails for nonlinear evolutions. We propose an alternative way of performing the same task in linear quantum mechanics and show that the method smoothly extends to the nonlinear domain. The trick is to use appropriate time-dependent Hamiltonians which involve "switching-off functions". We apply the technique to the EPR problem in nonlinear quantum mechanics and show that paradoxes of Gisin and Polchinski disappear.

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↗
arXiv1995

Roentgen term makes dipole approximation more divergent

The Roentgen correction to the dipole interaction term leads to an additional divergency which can be eliminated for infinitely heavy atoms. For M < infinity a probability of emission in a given direction is represented by a divergent integral.

quant-ph↗
arXiv1995

What happens to spin during the SO(3)->SE(2) contraction? (On spin and extended structures in quantum mechanics)

It is shown that the spin operator can be described by an algebra which is in between so(3) and e(2). Relativistic version of the singlet state for two Dirac electrons is discussed. It is shown that a measure of massless particle's extension can be naturally constructed and that this measure corresponds at the classical level to the radius of the Robinson congruence.

quant-ph↗