The metron model: Towards a unified deterministic theory of fields and particles

A summary is given of the principal concepts of a unified deterministic theory of fields and particles that have been developed in more detail in a pre- vious comprehensive four-part paper (Hasselmann, 1996a,b, 1997a,b). The model is based on the Einstein vacuum equations, Ricci tensor RLM = 0, in a higher-dimensional space. A space of at least eight dimensions is re- quired to incorporate all other forces as well as gravity in Einstein’s gen- eral relativistic formalism. It is hypothesized that the equations support soliton-type solutions (”metrons”) that are localized in physical space and are periodic in extra (”harmonic”) space and time. The solitons represent waves propagating in harmonic space that are locally trapped in physical space within a wave guide produced by a distortion of the background met- ric. The metric distortion, in turn, is generated by nonlinear interactions (radiation stresses) of the wave field. (The mutual interaction mechanism has been demonstrated for a simplified Lagrangian in Part 1 of the previous paper). In addition to electromagnetic and gravitational fields, the metron solutions carry periodic far fields that satisfy de Broglie’s dispersion relation. These give rise to wave-like interference phenomena when particles interact with other matter, thereby resolving the wave-particle duality paradox. The metron solutions and all particle interactions on the microphysical scale (with the exception of the kaon system) satisfy strict time-reversal symmetry, an arrow of time arising only at the macrophysical level through the introduc- tion of time-asymmetrical statistical assumptions. Thus Bell’s theorem on the non-existence of deterministic (hidden variable) theories, which depends crucially on an arrow-of-time, is not applicable. Similarly, the periodic de Broglie far fields of the particles do not lead to unstable radiative damping, the time-asymmetrical outgoing radiation condition being replaced by the time-symmetrical condition of zero net radiation. Assuming suitable polarization properties of the metron solutions, it can be shown that the coupled field equations of the Maxwell-Dirac-Einstein sys- tem as well as the Lagrangian of the Standard Model can be derived to low- est interaction order from the Einstein vacuum equations. Moreover, since Einstein’s vacuum equations contain no physical constants (apart from the introduction of units, namely the velocity of light and a similar scale for the harmonic dimensions, in the definition of the flat background metric), all physical properties of the elementary particles (mass, charge, spin) and all universal physical constants (Planck’s constant, the gravitational constant, and the coupling constants of the electroweak and strong forces) must fol- low from the properties of the metron solutions. A preliminary inspection of the structure of the solutions suggests that the extremely small ratio of gravitational to electromagnetic forces can be explained as a higher-order nonlinearity of the gravitational forces within the interior metron core. The gauge symmetries of the Standard Model follow from geometrical symme- tries of the metron solutions. Similarly, the parity violation of the weak interactions is attributed to a reflexion asymmetry of the metron solutions (in analogy to molecules with left- and right-rotational symmetry), rather than to a property of the basic Lagrangian. The metron model also yields further interaction fields not contained in the Standard Model, suggesting that the Standard Model represents only a first-order description of elemen- tary particle interactions. While the Einstein vacuum equations reproduce the basic structure of the fields and lowest-order interactions of quantum field theory, the particle content of the metron model has no correspondence in quantum field theory. This leads to an interesting interpretation of atomic spectra in the metron model. The basic atomic eigenmodes of quantum electrodynamics appear in the metron model as the scattered fields generated by the interaction of the orbiting electron with the atomic nucleus. For certain orbits, the eigenmodes are in resonance with the orbiting electron. In this case, the eigenmode and orbiting electron represent a stable self-supporting configuration. For circular orbits, the resonance condition is identical to the integer-action condition of the Bohr orbital model. Thus the metron interpretation of atomic spectra yields an interesting amalgam of quantum electrodynamics and the original Bohr model. However, it remains to be investigated whether higher-order computations of the metron model are able to reproduce atomic spectra to the same high degree of agreement with experiment as QED. On a more fundamental level, the basic questions of the existence, structure, stability and discreteness of the postulated metron solutions still need to be addressed. However, it is encouraging that, already on the present exploratory level, the basic properties of elementary particles and fields, including the origins of particle properties and the physical constants, can be explained within a unified classical picture based on a straightforward Kaluza-Klein extension to a higher dimensional space of the simplest vacuum form of Einstein’s gravitational equations

Intertemporal accounting of climate change - Harmonizing economic efficiency and climate stewardship

Continuing a discussion on the intertemporal accounting of climate-change damages initiated by Nordhaus, Heal and Brown in response to the recent demonstration of Hasselmann et al. that standard exponential discounting applied uniformly to all goods and services invariably leads to a `climate catastrophe' in cost-benefit analyses, it is argued that (1) there exists no economically satisfactory alternative to cost-benefit analysis for the determination of optimal climate protection strategies, and (2) it is essential to allow for the different long-term evolution of climate damage costs relative to mitigation costs in determining the optimal cost-benefit solution. A climate catastrophe can be avoided only if it is assumed that climate damage costs increase significantly in the long term relative to mitigation costs. Cost-benefit analysis is regarded here in the generalized sense of optimizing a social welfare function that incorporates all relevant `quality-of-life' factors, including not only consumption and the value of the environment, but also the ethical values of equitable intertemporal and intrasocietal distribution. Thus, economic efficiency and climate stewardship are not regarded as conflicting goals, but as synonyms for a single encompassing economic optimization exercise. The same reasoning applies generally to the problem of sustainable development. To quantify the concept of sustainable development in cost-benefit analyses, the projected time evolution of the future values of natural resources and the environment (judged by the present generation, acting as representative agents of future generations) must be related to the time-evolution of all other relevant quality-of-life factors. Different ethical interpretations of the concept of sustainable development can be readily operationalized by incorporation in a generalized cost-benefit analysis in which the evolution paths of all relevant material and ethical values are explicitly specified

A similarity relation for the non-linear energy-transfer in a finite-depth gravity-wave spectrum

The energy transfer in a finite-depth gravity-wave spectrum is investigated in the approximation of a narrow spectrum. It is shown that for ocean depths larger than approximately one tenth of the wavelength (kh [ges ] 0·7) the finite-depth case can be reduced to Longuet-Higgins’ (1976) result for an infinitely deep ocean by a similarity transformation involving changes in scale of the angular spreading function and the transfer rate. For shallower water (kh < 0·7) Longuet-Higgins’ expansion technique is no longer applicable without modification, as the nonlinear coupling coefficient develops a discontinuity at the origin of the expansion. In the range kh [ges ] 0·7 both the magnitude and the two-dimensional frequency-directional distribution of the energy transfer are found not to differ significantly (to within variations by a factor of 2) from the case of an infinitely deep ocean. The transformation rules relating the infinite-depth and finite-depth cases may provide a useful guide for constructing parametrizations of the nonlinear transfer for finite-depth wave prediction models

Techniques of linear prediction, with application to oceanic and atmospheric fields in the tropical Pacific

The problem of constructing optimal linear prediction models by multivariance regression methods is reviewed. It is well known that as the number of predictors in a model is increased, the skill of the prediction grows, but the statistical significance generally decreases. For predictions using a large number of candidate predictors, strategies are therefore needed to determine optimal prediction models which properly balance the competing requirements of skill and significance. The popular methods of coefficient screening or stepwise regression represent a posteriori predictor selection methods and therefore cannot be used to recover statistically significant models by truncation if the complete model, including all predictors, is statistically insignificant. Higher significance can be achieved only by a priori reduction of the predictor set. To determine the maximum number of predictors which may be meaningfully incorporated in a model, a model hierarchy can be used in which a series of best fit prediction models is constructed for a (prior defined) nested sequence of predictor sets, the sequence being terminated when the significance level either falls below a prescribed limit or reaches a maximum value. The method requires a reliable assessment of model significance. This is characterized by a quadratic statistic which is defined independently of the model skill or artificial skill. As an example, the method is applied to the prediction of sea surface temperature anomalies at Christmas Island (representative of sea surface temperatures in the central equatorial Pacific) and variations of the central and east Pacific Hadley circulation (characterized by the second empirical orthogonal function (EOF) of the meridional component of the trade wind anomaly field) using a general multiple‐time‐lag prediction matrix. The ordering of the predictors is based on an EOF sequence, defined formally as orthogonal variables in the composite space of all (normalized) predictors, irrespective of their different physical dimensions, time lag, and geographic position. The choice of a large set of 20 predictors at 12 time lags yields significant predictability only for forecast periods of 3 to 5 months. However, a prior reduction of the predictor set to 4 predictors at 10 time lags leads to 95% significant predictions with skill values of the order of 0.4 to 0.7 up to 6 or 8 months. For infinitely long time series the construction of optimal prediction models reduces essentially to the problem of linear system identification. However, the model hierarchies normally considered for the simulation of general linear systems differ in structure from the model hierarchies which appear to be most suitable for constructing pure prediction models. Thus the truncation imposed by statistical significance requirements can result in rather different models for the two cases. The relation between optimal prediction models and linear dynamical models is illustrated by the prediction of east‐west sea level changes in the equatorial Pacific from wind field anomalies. It is shown that the optimal empirical prediction is statistically consistent in this case with both the first‐order relaxation and damped oscillator models recently proposed by McWilliams and Gent (but with somewhat different model parameters than suggested by the authors). Thus the data do not allow a distinction between the two physical models; the simplest acceptable model is the first‐order damped response. Finally, the problem of estimating forecast skill is discussed. It is usually stated that the forecast skill is smaller than the true skill, which in turn is smaller than the hindcast skill, by an amount which in both cases is approximately equal to the artificial skill. However, this result applies to the mean skills averaged over the ensemble of all possible hindcast data sets, given the true model. Under the more appropriate side condition of a given hindcast data set and an unknown true model, the estimation of the forecast skill represents a problem of statistical inference and is dependent on the assumed prior probability distribution of true models. The Bayesian hypothesis of a uniform prior distribution yields an average forecast skill equal to the hindcast skill, but other (equally acceptable) assumptions yield lower forecast skills more compatible with the usual hindcast‐averaged expressio

On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion

A new, closed nonlinear integral transformation relation is derived describing the mapping of a two-dimensional ocean wave spectrum into a synthetic aperture radar (SAR) image spectrum. The general integral relation is expanded in a power series with respect to orders of nonlinearity and velocity bunching. The individual terms of the series can be readily computed using fast Fourier transforms. The convergence of the series is rapid. The series expansion is also useful in identifying the different contributions to the net imaging process, consisting of the real aperture radar (RAR) cross-section modulation, the nonlinear motion (velocity bunching) effects, and their various interaction products. The lowest term of the expansion with respect to nonlinearity order yields a simple quasi-linear approximate mapping relation consisting of the standard linear SAR modulation expression multiplied by an additional nonlinear Gaussian azimuthal cutoff factor. The cutoff scale is given by the rms azimuthal (velocity bunching) displacement. The same cutoff factor applies to all terms of the power series expansion. The nonlinear mapping relation is inverted using a standard first-guess wave spectrum as regularization term. This is needed to overcome the basic 180° mapping ambiguity and the loss of information beyond the azimuthal cutoff. The inversion is solved numerically using an iteration technique based on the successive application of the explicit solution for the quasi-linear mapping approximation, with interposed corrections invoking the full nonlinear mapping expression. A straightforward application of this technique, however, generally yields unrealistic discontinuities of the best fit wave spectrum in the transition region separating the low azimuthal wave number domain, in which useful SAR information is available and the wave spectrum is modified, from the high azimuthal wave number region beyond the azimuthal cutoff, where the first-guess wave spectrum is retained. This difficulty is overcome by applying a two-step inversion procedure. In the first step the energy level of the wave spectrum is adjusted, and the wave number plane rotated and rescaled, without altering the shape of the spectrum. Using the resulting globally fitted spectrum as the new first-guess input spectrum, the original inversion method is then applied without further constraints in a second step to obtain a final fine-scale optimized spectrum. The forward mapping relation and inversion algorithms are illustrated for three Seasat cases representing different wave conditions corresponding to weakly, moderately, and strongly nonlinear imaging conditions

Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part I: A new method for efficient computations of the exact nonlinear transfer integral

A more efficient method of computing the nonlinear transfer in a surface wave spectrum is developed which is symmetrical with respect to all wavenumbers of the resonant interaction quadruplets. This enables a large number of computations to be carried out, as required for investigations of the spectral energy balance or the development of parameterizations. New results are presented for finite-depth surface waves. By filtering out regions in interaction phase space, the assumptions involved in the narrow-peak and local-interaction approx-imations are investigated. Both approximations are found to be useful but are generally not sufficiently accurate to replace exact computations or provide adequate parameterizations for wave models

On the existence of a fully developed wind-sea spectrum

We consider the energy transfer equation for well-developed ocean waves under the influence of wind, and study the conditions for the existence of an equilibrium solution in which wind input, wave-wave interaction and dissipation balance each other. For the wind input we take the parameterization proposed by Snyder and others, which was based on their measurements in the Bight of Abaco and which agrees with Miles's theory. The wave-wave interaction is computed with an algorithm given recently by S. Hasselmann and others. The dissipation is less well-known, but we will make the general assumption that it is quasi-linear in the wave spectrum with a factor coefficient depending only on frequency and integral spectral parameters. In the first part of this paper we investigate whether the assumption that the equilibrium spectrum exits and is given by the Pierson-Moskowitz spectrum with a standard type of angular distribution leads to a reasonable dissipation function. We find that this is not the case. Even if one balances the total rate of change for each frequency (which is possible), a strong angular imbalance remains. Thus the assumed source terms are not consistent with this type of asymptotic spectrum. In the second part of the paper we choose a different approach. We assume that the dissipation is given and perform numerical experiments simulating fetch-limited growth, to see under which conditions a stationary solution can be reached. For the dissipation we take K. Haseelmann's form with two unknown parameters. From our analysis it follows that for a certain range of values of these parameters, a quasi-equilibrium solution results. We estimate the relation between dissipation parameters and asymptotic growth rates. For equilibrium spectra, the input, dissipation and nonlinear-transfer source functions are all significant in the energy-containing range of the spectrum. The energy balance proposed by Zakharov and Filonenko in 1966 and Kitaigorodskii in 1983, in which dissipation is assumed to be significant only at high frequencies, yields a spectrum that grows too rapidly and does not approach equilibrium. One of our equilibrium solutions has a one-dimensional spectrum that lies close to the Pierson-Moskowitz spectrum. However, the angular distribution differs in some important features from standard spreading functions. The energy balance of this equilibrium spectrum is analysed in detail

Computations of the response of a wave spectrum to a sudden change in wind direction

The response of a wind-sea spectrum to a step function change in wind direction is investigated theoretically for a sequence of direction changes ranging from 30° to 180°, in increments of 30°. Two spectral energy balance models are used: the model EXACT-NL, in which the nonlinear transfer is represented exactly, and the model 3G-WAM, in which the nonlinear transfer is approximated by the discrete interaction parameterization. In both modes the input and dissipation source functions are taken from the energy balance proposed by Komen et al. The operational model 3G-WAM reproduces fairly closely the EXACT-NL results. For wind direction changes less than 60°, the wind-sea direction adjusts smoothly. The high-frequency components relax more rapidly to the new wind direction than the low-frequency components. The computed relaxation rates are generally consistent with the analysis of measured directional spectra by D.E. Hasselman et al. and Allender et al. However, the relaxation rate is found to be a function of wind speed as well as frequency. For wind direction changes greater than 60°, a second, independent wind-sea spectrum is generated in the new wind direction, while the old wind-sea gradually decays as swell