Foldy-Wouthuysen Transformation, Scalar Potentials and Gravity

We show that care is required in formulating the nonrelativistic limit of generalized Dirac Hamiltonians which describe particles and antiparticles interacting with static electric and/or gravitational fields. The Dirac-Coulomb and the Dirac-Schwarzschild Hamiltonians, and the corrections to the Dirac equation in a non-inertial frame, according to general relativity, are used as example cases in order to investigate the unitarity of the standard and "chiral" approaches to the Foldy-Wouthuysen transformation, and spurious parity-breaking terms. Indeed, we find that parity-violating terms can be generated by unitary pseudo-scalar transformations ("chiral" Foldy-Wouthuysen transformations). Despite their interesting algebraic properties, we find that "chiral" Foldy-Wouthuysen transformations change fundamental symmetry properties of the Hamiltonian and do not conserve the physical interpretation of the operators. Supplementing the discussion, we calculate the leading terms in the Foldy-Wouthuysen transformation of the Dirac Hamiltonian with a scalar potential (of the (1/r)-form and of the confining radially symmetric linear form), and obtain compact expressions for the leading higher-order corrections to the Dirac Hamiltonian in a non-inertial rotating reference frame "Mashhoon term").Comment: 11 pages; RevTe

Dirac Hamiltonian and Reissner-Nordstrom Metric: Coulomb Interaction in Curved Space-Time

We investigate the spin-1/2 relativistic quantum dynamics in the curved space-time generated by a central massive charged object (black hole). This necessitates a study of the coupling of a Dirac particle to the Reissner-Nordstrom space-time geometry and the simultaneous covariant coupling to the central electrostatic field. The relativistic Dirac Hamiltonian for the Reissner-Nordstrom geometry is derived. A Foldy-Wouthuysen transformation reveals the presence of gravitational, and electro-gravitational spin-orbit coupling terms which generalize the Fokker precession terms found for the Dirac-Schwarzschild Hamiltonian, and other electro-gravitational correction terms to the potential proportional to alpha^n G, where alpha is the fine-structure constant, and G is the gravitational coupling constant. The particle-antiparticle symmetry found for the Dirac-Schwarzschild geometry (and for other geometries which do not include electromagnetic interactions) is shown to be explicitly broken due to the electrostatic coupling. The resulting spectrum of radially symmetric, electrostatically bound systems (with gravitational corrections) is evaluated for example cases.Comment: 11 page

Generalized Householder Transformations for the Complex Symmetric Eigenvalue Problem

We present an intuitive and scalable algorithm for the diagonalization of complex symmetric matrices, which arise from the projection of pseudo--Hermitian and complex scaled Hamiltonians onto a suitable basis set of "trial" states. The algorithm diagonalizes complex and symmetric (non--Hermitian) matrices and is easily implemented in modern computer languages. It is based on generalized Householder transformations and relies on iterative similarity transformations T -> T' = Q^T T Q, where Q is a complex and orthogonal, but not unitary, matrix, i.e, Q^T equals Q^(-1) but Q^+ is different from Q^(-1). We present numerical reference data to support the scalability of the algorithm. We construct the generalized Householder transformations from the notion that the conserved scalar product of eigenstates Psi_n and Psi_m of a pseudo-Hermitian quantum mechanical Hamiltonian can be reformulated in terms of the generalized indefinite inner product [integral of the product Psi_n(x,t) Psi_m(x,t) over dx], where the integrand is locally defined, and complex conjugation is avoided. A few example calculations are described which illustrate the physical origin of the ideas used in the construction of the algorithm.Comment: 14 pages; RevTeX; font mismatch in Eqs. (3) and (15) is eliminate

Gravitational Interactions and Fine-Structure Constant

Electromagnetic and gravitational central-field problems are studied with relativistic quantum mechanics on curved space-time backgrounds. Corrections to the transition current are identified. Analogies of the gravitational and electromagnetic spectra suggest the definition of a gravitational fine-structure constant. The electromagnetic and gravitational coupling constants enter the Einstein-Hilbert-Maxwell Lagrangian. We postulate that the variational principle holds with regard to a global dilation transformation of the space-time coordinates. The variation suggests is consistent with a functional relationship of the form alpha_QED being proportional to alpha_G^(1/2), where alpha_QED is the electrodynamic fine-structure constant, and alpha_G its gravitational analogue.Comment: 9 pages; LaTeX; Appeared in the Proceedings to the 17th Bled workshop "What Comes Beyond the Standard Models", Bled, Slovenia, July 2-28, 2014; Editors N. S. Mankoc Borstnik, H. B. Nielsen and D. Lukman; ISSN 1580-4992; book series: Bled Workshops in Physics vol. 15, no. 2, pp. 115-122 (2014

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

Nonrelativistic Limit of the Dirac-Schwarzschild Hamiltonian: Gravitational Zitterbewegung and Gravitational Spin-Orbit Coupling

We investigate the nonrelativistic limit of the gravitationally coupled Dirac equation via a Foldy-Wouthuysen transformation. The relativistic correction terms have immediate and obvious physical interpretations in terms of a gravitational Zitterbewegung and a gravitational spin-orbit coupling. We find no direct coupling of the spin vector to the gravitational force, which would otherwise violate parity. The particle-antiparticle symmetry described recently by one of us is verified on the level of the perturbative corrections accessed by the Foldy-Wouthuysen transformation. The gravitational corrections to the electromagnetic transition current are calculated

Dirac Equations with Confining Potentials

This paper is devoted to a study of relativistic eigenstates of Dirac particles which are simultaneously bound by a static Coulomb potential and added linear confining potentials. It has recently been shown that, despite the addition of radially symmetric, linear confining potentials, some specific bound-state energies surprisingly retain their exact Dirac--Coulomb values (in the sense of an "exact symmetry"). This observation raises pertinent questions as to the generality of the cancellation mechanism. A Foldy-Wouthuysen transformation is used to find the relevant nonrelativistic physical degrees of freedom, which include additional spin-orbit couplings induced by the linear confining potentials. The matrix elements of the effective operators obtained from the scalar, and time-like confining potentials mutually cancel for specific ratios of the prefactors of the effective operators, which must be tailored to the cancellation mechanism. The result of the Foldy-Wouthuysen transformation is used to explicitly show that the cancellation is accidental and restricted (for a given Hamiltonian) to only one reference state, rather than traceable to a more general relationship among the obtained effective low-energy operators. Furthermore, we show that the cancellation mechanism does not affect anti-particle (negative-energy) states.Comment: 11 pages; RevTe

Alkaline static feed electrolyzer based oxygen generation system

In preparation for the future deployment of the Space Station, an R and D program was established to demonstrate integrated operation of an alkaline Water Electrolysis System and a fuel cell as an energy storage device. The program's scope was revised when the Space Station Control Board changed the energy storage baseline for the Space Station. The new scope was aimed at the development of an alkaline Static Feed Electrolyzer for use in an Environmental Control/Life Support System as an oxygen generation system. As a result, the program was divided into two phases. The phase 1 effort was directed at the development of the Static Feed Electrolyzer for application in a Regenerative Fuel Cell System. During this phase, the program emphasized incorporation of the Regenerative Fuel Cell System design requirements into the Static Feed Electrolyzer electrochemical module design and the mechanical components design. The mechanical components included a Pressure Control Assembly, a Water Supply Assembly and a Thermal Control Assembly. These designs were completed through manufacturing drawing during Phase 1. The Phase 2 effort was directed at advancing the Alkaline Static Feed Electrolyzer database for an oxygen generation system. This development was aimed at extending the Static Feed Electrolyzer database in areas which may be encountered from initial fabrication through transportation, storage, launch and eventual Space Station startup. During this Phase, the Program emphasized three major areas: materials evaluation, electrochemical module scaling and performance repeatability and Static Feed Electrolyzer operational definition and characterization

Narrow Line Cooling and Momentum-Space Crystals

Narrow line laser cooling is advancing the frontier for experiments ranging from studies of fundamental atomic physics to high precision optical frequency standards. In this paper, we present an extensive description of the systems and techniques necessary to realize 689 nm 1S0 - 3P1 narrow line cooling of atomic 88Sr. Narrow line cooling and trapping dynamics are also studied in detail. By controlling the relative size of the power broadened transition linewidth and the single-photon recoil frequency shift, we show that it is possible to continuously bridge the gap between semiclassical and quantum mechanical cooling. Novel semiclassical cooling process, some of which are intimately linked to gravity, are also explored. Moreover, for laser frequencies tuned above the atomic resonance, we demonstrate momentum-space crystals containing up to 26 well defined lattice points. Gravitationally assisted cooling is also achieved with blue-detuned light. Theoretically, we find the blue detuned dynamics are universal to Doppler limited systems. This paper offers the most comprehensive study of narrow line laser cooling to date.Comment: 14 pages, 19 figure