Jonathan Gratus

Quantum mechanics on graphs

We analyse the problem of one-dimensional quantum mechanics on arbitrary graphs as idealized models for quantum systems on spaces with non-trivial topologies. In particular we argue that such models can be made to accommodate the physical characteristics of wavefunctions on a network of wires and offer several derivations of a particular junction condition. Throughout we adopt a continuity condition for the wavefunction at each primitive node in the network. Results are applied to the problem of the energy spectrum of a system containing one and infinitely many junctions.

Asymptotic analysis of ultra-relativistic charge

This article offers a new approach for analysing the dynamic behaviour of distributions of charged particles in an electromagnetic field. After discussing the limitations inherent in the Lorentz–Dirac equation for a single point particle a simple model is proposed for a charged continuum interacting self-consistently with the Maxwell field in vacuo. The model is developed using intrinsic tensor field theory and exploits to the full the symmetry and light-cone structure of Minkowski spacetime. This permits the construction of a regular stress-energy tensor whose vanishing divergence determines a system of non-linear partial differential equations for the velocity and self-fields of accelerated charge. Within this covariant framework a particular perturbation scheme is motivated by an exact class of solutions to this system describing the evolution of a charged fluid under the combined effects of both self and external electromagnetic fields. The scheme yields an asymptotic approximation in terms of inhomogeneous linear equations for the self-consistent Maxwell field, charge current and time-like velocity field of the charged fluid and is defined as an ultra-relativistic configuration. To facilitate comparisons with existing accounts of beam dynamics an appendix translates the tensor formulation of the perturbation scheme into the language involving electric and magnetic fields observed in a laboratory (inertial) frame.

New Perspectives On the Relevance of Gravitation for the Covariant Description of Electromagnetically Polarizable Media

By recognizing that stress–energy–momentum tensors are fundamentally related to gravitation in spacetime it is argued that the classical electromagnetic properties of a simple polarizable medium may be parameterized in terms of a constitutive tensor whose properties can in principle be determined by experiments in non-inertial (accelerating) frames and in the presence of weak but variable gravitational fields. After establishing some geometric notation, discussion is given to basic concepts of stress, energy and momentum in the vacuum where the useful notion of a drive form is introduced in order to associate the conservation of currents involving the flux of energy, momentum and angular momentum with spacetime isometries. The definition of the stress–energy–momentum tensor is discussed with particular reference to its symmetry based on its role as a source of relativistic gravitation. General constitutive properties of material continua are formulated in terms of spacetime tensors including those that describe magneto-electric phenomena in moving media. This leads to a formulation of a self-adjoint constitutive tensor describing, in general, inhomogeneous, anisotropic, magneto-electric bulk matter in arbitrary motion. The question of an invariant characterization of intrinsically magneto-electric media is explored. An action principle is established to generate the phenomenological Maxwell system and the use of variational derivatives to calculate stress–energy–momentum tensors is discussed in some detail. The relation of this result to tensors proposed by Abraham and others is discussed in the concluding section where the relevance of the whole approach to experiments on matter in non-inertial environments with variable gravitational and electromagnetic fields is stressed.

Covariant constitutive relations, landau damping and non-stationary inhomogeneous plasmas

The distributional form of the Maxwell-Vlasov equations are formulated. Submanifold distributions are analysed and the general submanifold distributional solutions to the Vlasov equations are given. The properties required so that these solutions can be a distributional source to Maxwell’s equations are analysed and it is shown that a sufficient condition is that spacetime be globally hyperbolic. The cold fluid, multicurrent and water bag models of charge are shown to be particular cases of the distributional Maxwell-Vlasov system. PACS numbers: 52.65.Ff, 03.50.De, 41.75.Ht, 02.30.Cj AMS classification scheme numbers: 46F66, 53Z05, 78A25 Submitted to: J. Phys. A: Math. Gen.Models of covariant linear electromagnetic constitutive relations are formulated that have wide applicability to the computation of susceptibility tensors for dispersive and inhomogeneous media. A perturbative framework is used to derive a linear constitutive relation for a globally neutral plasma enabling one to describe in this context a generalized Landau damping mechanism for non-stationary inhomogeneous plasma states.

A kinetic model of radiating electrons

A kinetic theory is developed to describe radiating electrons whose motion is governed by the Lorentz-Dirac equation. This gives rise to a generalized Vlasov equation coupled to an equation for the evolution of the physical submanifold of phase space. The pathological solutions of the 1-particle theory may be removed by expanding the latter equation in powers of τ ≔ q 2/6πm. The radiation-induced change in entropy is explored and its physical origin is discussed. As a simple demonstration of the theory, the radiative damping rate of longitudinal plasma waves is calculated.

An improved method of gravicraft propulsion

Abstract There is an interaction between gravitational tidal forces and the instantaneous moment of inertia of any massive extended body. A mechanism that changes the altitude of a spacecraft orbiting a central mass by periodically changing its instantaneous moment of inertia tensor is discussed. This can be achieved most simply by connecting two bodies with a tether of adjustable length and does not rely on any chemical propellant. By using a series of reasonable approximations, a formula for the rate of change of height is given in terms of accessible data. This formula predicts that by suitably varying the length of a 50 km connecting tether a connected system can rise at about 300 m an hour in low earth orbit.

A natural basis of states for the noncommutative sphere and its Moyal bracket

An infinite-dimensional algebra which is a nondecomposable reducible representation of su(2) is given. This algebra is defined with respect to two real parameters. If one of these parameters is zero, the algebra is the commutative algebra of functions on the sphere, otherwise it is a noncommutative analog. This is an extension of the algebra normally referred to as the (Berezin) quantum sphere or “fuzzy” sphere. A natural indefinite “inner” product and a basis of the algebra orthogonal with respect to it are given. The basis elements are homogeneous polynomials, eigenvectors of a Laplacian, and related to the Hahn polynomials. It is shown that these elements tend to the spherical harmonics for the sphere. A Moyal bracket is constructed and shown to be the standard Moyal bracket for the sphere.

Spatially dispersive inhomogeneous electromagnetic media with periodic structure

Spatially dispersive (also known as non-local) electromagnetic media are considered where the parameters defining the permittivity relation vary periodically. Maxwells equations give rise to a difference equation corresponding to the Floquet modes. A complete set of approximate solutions is calculated which are valid when the inhomogeneity is small. This is applied to inhomogeneous wire media. A new feature arises when considering spatially dispersive media, that is the existence of coupled modes.

Topology change and the propagation of massless fields

The massless wave equation on a class of two‐dimensional manifolds consisting of an arbitrary number of topological cylinders connected to one or more topological spheres are analyzed herein. Such manifolds are endowed with a degenerate (nonglobally hyperbolic) metric. Attention is drawn to the topological constraints on solutions describing monochromatic modes on both compact and noncompact manifolds. Energy and momentum currents are constructed and a new global sum rule discussed. The results offer a rigorous background for the formulation of a field theory of topologically induced particle production.

On Spacetime Transformation Optics: Temporal and Spatial Dispersion

The electromagnetic implementation of cloaking, the hiding of objects from sight by diverting and reassembling illuminating electromagnetic fields has now been with us ten years, while the notion of hiding events is now five. Both schemes as initially presented neglected the inevitable dispersion that arises when a designed medium replaces vacuum under transformation. Here we define a transformation design protocol that incorporates both spacetime transformations and dispersive material responses in a natural and rigorous way. We show how this methodology is applied to an event cloak designed to appear as a homogeneous and isotropic but dispersive medium. The consequences for spacetime transformation design in dispersive materials are discussed, and some parameter and bandwidth constraints identified.