Maciej Kuna

Complete positivity of nonlinear evolution: A case study

Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown in a concrete example.

General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach

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). In these cases we give explicit formulae for obtaining general solutions from special ones. 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. Progress can be made if one succeeds in finding a nontrivial special solution of the equations of motion. Then the general solution can be derived by means of the Lie theory. The problem of evaluating the evolution of the system is translated from a noncommutative integration in the sense of Dyson into an ordinary commutative integration. The two main applications of our method 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.

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 Bells 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.

Nonlinear von Neumann-type equations: Darboux invariance and spectra

Abstract 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.

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 arithmetic 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.

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

Abstract Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration, and complex structure. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the required 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.

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.

Model of a quantum particle in spacetime

In their paper Doplicher, Fredenhagen and Roberts (DFR) 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 Pµ-(b/c)Aµ, where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential Aµ coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Qµ-(al2/c)Ωµ, where a is a generalized charge and l a fundamental length, and with vector potentials Ωµ which are in some sense dual w.r.t. the Aµ.