Electrodynamics in Dirac's Gauge: A Geometrical Equivalence

It is shown that, when classical non-relativistic electrodynamics is formulated in Dirac's gauge A A =const. and the vector potential A interpreted as a velocity field of the vacuum, the motion of a charged particle results from purely inertial effects. A metric is given for particles of a fixed charge to mass ratio.     

Proton dipolar spin-lattice relaxation in hexagonal ice

The ultraslow motion of defects in high purity hexagonal H₂O ice has been studied by proton dipolarT _1D measurements in the strong collision limit, using the Jeener technique. The obtained NMR correlation times agree rather well with both the Schottky H₂O diffusion timest _s=r ²/6D and the deuteron correlation times in D₂O ice, suggesting that Schottky rather than interstitial diffusion dominates spin-lattice relaxation in both H₂O and D₂O ice.On leave of absence from University of Ljubljana, Institute J. Stefan.     

Brane-World Models Emerging from Collisions of Plane Waves in 5D

We consider brane-world models embedded in a five-dimensional bulk spacetime with a large extra dimension and a cosmological constant. The cosmology in 5D possesses wave-like character in the sense that the metric coefficients in the bulk are assumed to have the form of plane waves propagating in the fifth dimension. We model the brane as the plane of collision of waves propagating in opposite directions along the extra dimension. This plane is a jump discontinuity which presents the usual Z ₂ symmetry of brane models. The model reproduces the generalized Friedmann equation for the evolution on the brane, regardless of the specific details in 5D. Model solutions with spacelike extra coordinate show the usual big-bang behavior, while those with timelike extra dimension present a big bounce. This bounce is an genuine effect of a timelike extra dimension. We argue that, based on our current knowledge, models having a large timelike extra dimension cannot be dismissed as mathematical curiosities in non-physical solutions. The size of the extra dimension is small today, but it is increasing if the universe is expanding with acceleration. Also, the expansion rate of the fifth dimension can be expressed in a simple way through the four-dimensional deceleration and Hubble parameters as – q H. These predictions could have important observational implications, notably for the time variation of rest mass, electric charge and the gravitational constant. They hold for the three (k = 0, + 1, – 1) models with arbitrary cosmological constant, and are independent of the signature of the extra dimension.     

Multicomponent field theories and classical rotators

It is shown that aD-component Euclidean quantum field, =(¹,...,^D), with ||⁴+|²| interaction, can be obtained as a limit of (ferromagnetic) classical rotator models; this extends a result of Simon and Griffiths from the caseD=1. For these Euclidean field models, it is then shown that a Lee-Yang theorem applies forD=2 or 3 and that Griffiths' second inequality is valid forD=2; a complete proof is included of a Lee-Yang theorem for plane rotator and classical Heisenberg models. As an application of Griffiths' second inequality forD=2, an interesting relation between the parallel and transverse two-point correlations is obtained.Research supported in part by the National Science Foundation under grant NSF MPS 74-04870.     

Multiparticle fractal aggregation

Kinetic fractal aggregation in a particle bath where a fractionf of the sites are initially occupied is studied withd=2 computer simulations. Independent particles diffusing to a fixed cluster produce an aggregate with fractal dimensionD 1.7 up to a correlation length(f). At larger lengthsD2.(f) asf 0. When the particles remain fixed but the cluster undergoes a rigid random walkD appears constant at larger scales but varies withf. D 1.95 at largef andD 1.7 asf 0. In both cases, the aggregate sizeN(t) grows with timet ^(f) . Aggregation on a surface by independently diffusing particles produces shapes reminiscent of electrochemical dendritic growth. The dependence of growth rate and geometry is studied as a function of particle concentration and sticking probability.     

Massless and massive quanta resulting from a mediumlike metric tensor

We formulate a simple model of the primordial scalar field theory in which the metric tensor is a generalization of the metric tensor from electrodynamics in a medium. The radiation signal corresponding to the scalar field propagates with a velocity that is generally less thanc. This signal can be associated simultaneously with imaginary and real effective (momentum-dependent) masses. The requirement that the imaginary effective mass vanishes, which we take to be the prerequisite for the vacuumlike signal propagation, leads to the spontaneous splitting of the metric tensor into two distinct metric tensors: one metric tensor gives rise to masslesslike radiation and the other to a massive particle.     

Induced geometry and confinement

A classical, Poincaré invariant dynamical system is developed which contains, besides the natural metric _v , an induced metricg _v that is generated by a real scalar dynamical field. It is shown that scalar fields whose dynamics are governed by the induced metric can be consistently introduced. Also, point particles which follow timelike quasi-geodesic trajectories can be introduced. The reaction forces acting ong _v due to the presence of these fields and particles are computed. A discussion of causality and geometrical confinement is given.     

Variable-G cosmology and creation

It has recently been asserted that a universe with a time-varying gravitational constantG necessarily implies creation if the rest mass of matter particles is constant. It is shown that this is not necessarily true. An example of a cosmological model with variableG and is presented, in which there is no creation and in which the rest mass of matter particles is constant.     

Self-diffusion in simple models: Systems with long-range jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self-diffusion coefficientD _N() in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set ofN neighboring sites. We obtain positive upper and lower bounds onF _N()=N{(1–)–[D_N()/D_N(0)]}/[(1–)]x for [0, 1]. Computer simulations for the square, triangular, and one-dimensional lattices suggest thatF _N becomes effectively independent ofN forN20.     

MPT versus: A manifestly covariant presentation of motion reversal and particle-antiparticle exchange

We show that particle-antiparticle exchange and covariant motion reversal are two physically different aspects of the same mathematical transformation, either in the prequantal relativistic equation of motion of a charged point particle, in the general scheme of second quantization, or in the spinning wave equations of Dirac and of Petiau-Duffin-Kemmer. While, classically, charge reversal and rest mass reversal are equivalent operations, in the wave mechanical case mass reversal must be supplemented by exchange of the two adjoint equations, implying .Denoting by M the rest mass reversal, P the parity reversal, T the Racah time reversal, and Z the exchange, the connection with the usual scheme of charge conjugation, parity reversal, and Wigner motion reversal, is with, of course,     

Self-gravitating three-dimensional solitons in nonlinear scale-invariant electrodynamics

A scale-invariant nonlinear modification of Maxwellian electrodynamics within general relativity is proposed. The starting point is the Mie model and its scale-invariant generalization in flat space-timeE ₄. We prove that all static, spherically symmetrical regular field configurations in this new theory, as well as those in the Mie model, possess negative energy. In search of solitonlike solutions with positive masses, we take into account their proper gravitational fields. We show first that in Riemannian space any gauge-invariant electrodynamic theory does not admit regular solutions. Supposing the gauge invariance to be broken inside the particle, we prove the existence of static particlelike solutions with spherical symmetry and positive energy in the scale-invariant electrodynamics described by a Lagrangian density of the form =-Y(I)R/(2)-W(I)F F _u/2+2X(I)R A A , withY, W, andX arbitrary functions of the invariantI=A A . The correspondence with the Maxwellian theory is required.     

БЕТАТРОННЫЕ КОЛЕБАН ИЯ В УСКОРИТЕЛЕ С ОБЩИМ ПО ЛЕМ II

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Betatron oscillations in an accelerator with a general field. II

In the second part of the theory of an accelerator with a generalized field the author solves the dynamics of particles in a stationary approximation (without acceleration). The trajectories of the particles in the neigbourhood of the equilibrium orbit are related to it as betatron oscillations. An analysis of betatron oscillations includes the region of stability determined by means of Routh-Hurwitz criterion, damping, resonance effects and field perturbations. A general matrix method is presented permiting the inclusion of all the focusing structures used in accelerators.

     

Inertial frames of reference: Mass coupling to space and time

We consider a new expression for the dependence of mass on velocity, more general than the corresponding law of the special theory of relativity (STR). The deviations from the STR become large with increasing rest mass. One should therefore measure the dependence of mass on velocity for objects with a large rest mass. The theory predicts that particles with real mass can travel with hyperlight velocities. The space-time picture discussed here is close to Mach's conception: It is assumed that the dynamical behavior of a particle in uniform translational motion is due to the action of all the other masses in the universe. Space-time is eliminated as an active cause and, in contrast to the STR, is not absolute within the theory discussed here. It turns out that effects based on the new transformation formulas (from the coordinates and time in a stationary frame to the coordinates and time in a moving frame) are identical to those expected from the Lorentz transformations. For example, it is known that rapidly moving mesons decay with a longer half-life than stationary mesons and the STR describes this effect quantitatively. However, there is no strong evidence for the validity of the STR because the theory given in this paper predicts the same result.     

Extra Force from an Extra Dimension: Comparison Between Brane Theory, Space-Time-Matter Theory, and Other Approaches

We investigate the question of how an observer in 4D perceives the five-dimensional geodesic motion. We consider the interpretation of null and non-null bulk geodesics in the context of brane theory, space-time-matter theory (STM) and other non-compact approaches. We develop a frame-invariant formalism that allows the computation of the rest mass and its variation as observed in 4D. We find the appropriate expression for the four-acceleration and thus obtain the extra force observed in 4D. Our formulae extend and generalize all previous results in the literature. An important result here is that the extra force in brane-world models with Z ₂-symmetry is continuous and well defined across the brane. This is because the momentum component along the extra dimension is discontinuous across the brane, which effectively compensates the discontinuity of the extrinsic curvature. We show that brane theory and STM produce identical interpretation of the bulk geodesic motion. This holds for null and non-null bulk geodesics. Thus, experiments with test particles are unable to distinguish whether our universe is described by the brane world scenario or by STM. However, they do discriminate between the brane/STM scenario and other non-compact approaches. Among them the canonical and embedding approaches, which we examine in detail here.     

Spin manifolds,killing spinors and universality of the Hijazi inequality

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (–) on the spinors (=(n/4(n–1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension 3, the Hijazi inequality [8] holds for every spinor field such that (P, P) = ²(, ) (=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of . Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the twistor operator.     

Conditional transformation of drift formula and potential theory for 1/2Δ+b(·)·∇

Using conditional Brownian motion and the transformation of drift formula (of Cameron-Martin, Girsarov, Maruyama) we give integral conditions on a vector fieldb which imply the harmonic measures and Green functions for 1/2 and 1/2+b(·)· on a bounded Lipschitz domainD are equivalent. By equivalent we mean there exist two-sided inequalities with constants depending only onb andD. This enables one to conclude the potential theory for 1/2+b(·)· onD and 1/2 onD are the same.     

Numerical Evidence for the Low-Mass Behavior of the One-Dimensional Rayleigh Gas with Local Interaction

Numerical investigations, supported by partial rigorous results suggest that the motion of a tagged particle of mass M on the line ¹ colliding with a free gas of particles of mass m in equilibrium is diffusive. It was conjectured that the diffusion constant D(M), for small mass M0, should approach D(m). (In dimension 1 this is a kind of continuity hypothesis.) Previous results of computer simulations are inconclusive. We report on some new computer results, which show clearly that there is no continuity, and the limit of D(M) as M0 is smaller than D(m). We compare with the corresponding results for a similar two-dimensional model, to which the continuity argument cannot be applied.     

Noncommutative Ricci Curvature and Dirac Operator on C q [SL2] at Roots of Unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C _q [ SL₂], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m] _q for m=0,1...,r–1 offset by the constant background curvature. We fully solve the Dirac equation for r=3.     

Extended class of metric tensors in relativity

An extended metric tensor that is a function of an internal vectory ^a(x) leads to a spin-1 massless field of gravitational origin. It is shown that this new field vanishes in the linear approximation for the extended metric.On leave of absence from the Universidade de Brasilia.     

A method for calculating the space-time metric inside a collapsing or expanding sphere

A class of solutions of Einstein's field equations is found in which the quantities can be expressed as power series in r. Using null coordinates it is shown that the spacetime metric inside a spherically symmetric distribution of matter can be calculated to any degree of accuracy by evaluating the terms in the power series step by step, provided we know the central density, _O,as a function of time, and the equation of state in the form p=f(), f() being a function analytic at _O.The method is found to be applicable to all cases met with in nature.Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.