arXiv subjects

Maciej Kuna

Publications and source records attributed to Maciej Kuna.

At least 19 recordsLinked to original sources

arXiv2018

Simple fractal calculus from fractal arithmetic

Non-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a real-valued function with Sierpiński-set domain. The resulting formalism is as simple as the usual undergraduate calculus.

math.GN↗

arXiv2018

Systems, environments, and soliton rate equations (II): Toward realistic modeling

In order to solve a system of nonlinear rate equations one can try to use some soliton methods. The procedure involves three steps: (1) Find a `Lax representation' where all the kinetic variables are combined into a single matrix $ρ$, all the kinetic constants are encoded in a matrix $H$; (2) find a Darboux-Backund dressing transformation for the Lax representation $i\dot ρ=[H,f(ρ)]$, where $f$ models a time-dependent environment; (3) find a class of seed solutions $ρ=ρ[0]$ that lead, via a nontrivial chain of dressings $ρ[0]\to ρ[1]\to ρ[2]\to\dots$ to new solutions, difficult to find by other methods. The latter step is not a trivial one since a non-soliton method has to be employed to find an appropriate initial $ρ[0]$. Procedures that lead to a correct $ρ[0]$ have been discussed in the literature only for a limited class of $H$ and $f$. Here, we develop a formalism that works for practically any $H$, and any explicitly time-dependent $f$. As a result, we are able to find exact solutions to a system of equations describing an arbitrary number of species interacting through (auto)catalytic feedbacks, with general time dependent parameters characterizing the nonlinearity. Explicit examples involve up to 42 interacting species.

q-bio.PE↗

arXiv2016

Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus

Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the expected basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.

math-ph↗

arXiv2015

Crystallization of space: Space-time fractals from fractal arithmetic

Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.

gr-qc↗

arXiv2014

Quantum Structure in Competing Lizard Communities

Almost two decades of research on applications of the mathematical formalism of quantum theory as a modeling tool in domains different from the micro-world has given rise to many successful applications in situations related to human behavior and thought, more specifically in cognitive processes of decision-making and the ways concepts are combined into sentences. In this article, we extend this approach to animal behavior, showing that an analysis of an interactive situation involving a mating competition between certain lizard morphs allows to identify a quantum theoretic structure. More in particular, we show that when this lizard competition is analyzed structurally in the light of a compound entity consisting of subentities, the contextuality provided by the presence of an underlying rock-paper-scissors cyclic dynamics leads to a violation of Bell's inequality, which means it is of a non-classical type. We work out an explicit quantum-mechanical representation in Hilbert space for the lizard situation and show that it faithfully models a set of experimental data collected on three throat-colored morphs of a specific lizard species. Furthermore, we investigate the Hilbert space modeling, and show that the states describing the lizard competitions contain entanglement for each one of the considered confrontations of lizards with different competing strategies, which renders it no longer possible to interpret these states of the competing lizards as compositions of states of the individual lizards.

physics.bio-ph↗

arXiv2013

Systems, environments, and soliton rate equations: A non-Kolmogorovian framework for population dynamics

Soliton rate equations are based on non-Kolmogorovian models of probability and naturally include autocatalytic processes. The formalism is not widely known but has great unexplored potential for applications to systems interacting with environments. Beginning with links of contextuality to non-Kolmogorovity we introduce the general formalism of soliton rate equations and work out explicit examples of subsystems interacting with environments. Of particular interest is the case of a soliton autocatalytic rate equation coupled to a linear conservative environment, a formal way of expressing seasonal changes. Depending on strength of the system-environment coupling we observe phenomena analogous to hibernation or even complete blocking of decay of a population.

q-bio.PE↗

arXiv2008

Quantum chaotic systems with arbitrarily large Ehrenfest times

A class of time independent and physically meaningful Hamiltonians leads to evolution of observable quantities whose Ehrenfest times are arbitrarily large. This fact contradicts the popular claim that the true chaos is in quantum mechanics excluded by first principles.

quant-ph↗

arXiv2008

Solving the von Neumann equation with time-dependent Hamiltonian. Part I: Method

The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.

math-ph↗

arXiv2008

On the von Neumann equation with time-dependent Hamiltonian. Part II: Applications

This second part deals with applications of a general method to describe the quantum time evolution determined by a Schroedinger equation with time-dependent Hamiltonian. A new aspect of our approach is that we find all solutions starting from one special solution. The two main applications are reviewed, namely the Bloch equations and the harmonic oscillator with time-dependent frequency. Even in these well-known examples some new results are obtained.

math-ph↗

arXiv2006

Special solutions of nonlinear von Neumann equations

We consider solutions of the non-linear von Neumann equation involving Jacobi's elliptic functions sn, cn, and dn, and 3 linearly independent operators. In two cases one can construct a state-dependent Hamiltonian which is such that the corresponding non-linear von Neumann equation is solved by the given density operator. We prove that in a certain context these two cases are the only possibilities to obtain special solutions of this kind. Well-known solutions of the reduced Maxwell-Bloch equations produce examples of each of the two cases. Also known solutions of the non-linear von Neumann equation in dimension 3 are reproduced by the present approach.

math-ph↗

arXiv2004

Construction of exact solutions of Bloch-Maxwell equation based on Darboux transformation

A new strategy, using Darboux transformations, of finding self-switching solutions of $i\dotρ = [H, f(ρ)]$ is introduced. Unlike the previous ones, working for any f but for Hamiltonians whose spectrum contains at least three equally spaced eigenvalues, the strategy does not impose any restriction on the discrete part of the spectrum of H. The strategy is applied to the Bloch-Maxwell system.

quant-ph↗

arXiv2002

Photon fields in a fluctuating spacetime

We present a model of interacting quantum fields, formulated in a non-perturbative manner. One of the fields is treated semi-classically, the other is the photon field. The model has an interpretation of an electromagnetic field in a fluctuating spacetime. The model is equivalent with the quantization of electromagnetism proposed recently by Czachor. Interesting features are that standard photon theory is recovered as a limiting case, and that localized field operators for the electromagnetic field exist as unbounded operators in Hilbert space.

hep-th↗

arXiv2002

Covariance approach to the free photon field

We introduce photon theory following the same principles as for introduction of the quantum theory of a single particle, using a C*-algebraic approach based on covariance systems. The basic symmetries are additivity of the fields and additivity of test functions. We write down in explicit form a state of this covariance system. It turns out to reproduce the traditional Fock representation of the free photon field, with a Lorentz invariant vacuum. Properties of smeared-out photons are discussed.

quant-ph↗

arXiv2000

Model of a quantum particle in spacetime

Doplicher, Fredenhagen, and Roberts (1994, 1995) proposed a simple model of a particle in quantum spacetime. We give a new formulation of the model and propose some small changes and additions which improve the physical interpretation. In particular, we show that the internal degrees of freedom e and m of the particle represent external forces acting on the particle. To obtain this result we follow a constructive approach. The model is formulated as a covariance system. It has projective representations in which not only the spacetime coordinates but also the conjugated momenta are two-by-two noncommuting. These momenta are of the form P_mu-(b/c)A_mu, where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential A_mu coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Q_mu-(al^2/hbar c) Omega_mu where a is a generalized charge, l a fundamental length, and with vector potentials Omega_mu which are in some sense dual w.r.t. the A_mu.

hep-th↗

arXiv2000

Covariance systems

We introduce new definitions of states and of representations of covariance systems. The GNS-construction is generalized to this context. It associates a representation with each state of the covariance system. Next, states are extended to states of an appropriate covariance algebra. Two applications are given. We describe a nonrelativistic quantum particle, and we give a simple description of the quantum spacetime model introduced by Doplicher et al.

math-ph↗

arXiv1999

Nonlinear von Neumann-type equations

We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate the technique of binary Darboux integration of nonlinear density matrix equations and special attention is payed to the problem of how to find physically nontrivial `self-scattering' solutions.

quant-ph↗

arXiv1999

Nonlinear von Neumann-type equations: Darboux invariance and spectra

Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation given here in a new form: $ρ[1]=e^{P\ln(μ/ν)}ρe^{-P\ln(μ/ν)}$ where $P=P^2$ is explicitly constructed in terms of conjugated Lax pairs, and $μ$, $ν$ are complex. As a result spectra of $ρ$ and $ρ[1]$ are identical. Transformations allowing to shift and rescale spectrum of a solution are introduced, and a class of stationary seed solutions is discussed.

quant-ph↗

arXiv1998

Entanglement and pseudomixtures

In a recent paper Sanpera et al. have shown, that for the simplest binary composite systems any density matrix can be described in terms of only product vectors. The purpose of this note is to show that posibillity of decomposing any state as pseudomixtures does not depend on dimension of the subsystems.

quant-ph↗