Bounded Region Of Space Research Articles

On a partial solution of the diffusion equation

The process is considered of establishing the equilibrium spatial distribution of the concentration of particles in a one-dimensional bounded space region, subjected to a constant force normal to impermeable region boundaries. This process is described by the solution of the third boundary-value problem with homogeneous boundary conditions for the two-dimensional parabolic equation. It is shown that the found solution to the seemingly well-known problem of mathematical physics, but being of great importance in applications, cannot be obtained using theGreen’s function of this problem, known in the literature.

Multiple periodic solutions of Hamiltonian systems confined in a box

We consider a nonautonomous Hamiltonian system, $T$-periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many $T$-periodic solutions, by the use of a generalized version of the Poincaré-Birkhoff theorem.

Quantum mechanical reality according to Copenhagen 2.0

The long-standing conceptual controversies concerning the interpretation of nonrelativistic quantum mechanics are argued, on one hand, to be due to its incompleteness, as affirmed by Einstein. But on the other hand, it appears to be possible to complete it at least partially, as Bohr might have appreciated it, in the framework of its standard mathematical formalism with observables as appropriately defined self-adjoint operators. This completion of quantum mechanics is based on the requirement on laboratory physics to be effectively confined to a bounded space region and on the application of the von Neumann deficiency theorem to properly define a set of self-adjoint extensions of standard observables, e.g. the momenta and the Hamiltonian, in terms of certain isometries on the region boundary. This is formalized mathematically in the setting of a boundary ontology for the so-called Qbox in which the wave function acquires a supplementary dependence on a set of Additional Boundary Variables (ABV). It is argued that a certain geometric subset of the ABV parametrizing Quasi-Periodic Translational Isometries (QPTI) has a particular physical importance by allowing for the definition of an ontic wave function, which has the property of epitomizing the spatial wave function “collapse.” Concomitantly the standard wave function in an unbounded geometry is interpreted as an epistemic wave function, which together with the ontic QPTI wave function gives rise to the notion of two-wave duality, replacing the standard concept of wave-particle duality. More generally, this approach to quantum physics in a bounded geometry provides a novel analytical basis for a better understanding of several conceptual notions of quantum mechanics, including reality, nonlocality, entanglement and Heisenberg’s uncertainty relation. The scope of this analysis may be seen as a foundational update of the multiple versions 1.x of the Copenhagen interpretation of quantum mechanics, which is sufficiently incremental so as to be appropriately characterized as Copenhagen 2.0.

A family of inequivalent Weyl representations of canonical commutation relations with applications to quantum field theory

We consider a family of irreducible Weyl representations of canonical commutation relations with infinite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are established. As a simple application of one of these theorems, the well-known inequivalence of the time-zero field and conjugate momentum for different masses in a quantum scalar field theory is rederived with space dimension [Formula: see text] arbitrary. Also a generalization of representations of the time-zero field and conjugate momentum is presented. Comparison is made with a quantum scalar field in a bounded region in [Formula: see text]. It is shown that, in the case of a bounded space region with [Formula: see text], the representations for different masses turn out to be mutually equivalent.

Statistical acceleration of energetic particles and their diffusion in turbulent magnetic fields

Two processes of statistical cosmic ray (CR) acceleration and CR diffusion in turbulent magnetic fields are studied. It is assumed that the charged particles gain energy in a bounded space region and then they leave the acceleration region as a result of the spatial diffusion caused by particle scattering in the turbulent inhomogeneous medium. Both the second-order Fermi acceleration and the acceleration by the induced electric field (α-mechanism) are considered. Analytical solutions of the CR transport equation by the Fourier–Laplace method are obtained for various spatial diffusion coefficients. The equilibrium energetic spectra and the spatial distributions of CRs inside and outside the acceleration region are investigated.

Fermion localization and causality

The velocity of a fermion (neutrino) propagation from its source to its detector is investigated. It is shown that a simple approach to the problem in terms of the quantum packet spreading (Hegerfeldtet al.) is not applicable because the fermion cannot be localized in a bounded space region. Another approach is elaborated which uses relativistic quantum field description of the fermion source and detector. It is demonstrated that the velocity of the fermion propagation does not exceed the light velocityc within the precision of the source and detector localization.