Entwined Pairs and Schroedinger 's Equation

We show that a point particle moving in space-time on entwined-pair paths generates Schroedinger's equation in a static potential in the appropriate continuum linit. This provides a new realist context for the Schroedinger equation within the domain of classical stochastic processes. It also suggests that self-quantizing systems may provide considerable insight into conventional quantum mechanics.Comment: 16 pg. 1 fi

Entwined Paths, Difference Equations and the Dirac Equation

Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are `self-quantizing'. We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process in the same sense that the Diffusion and Telegraph equations are phenomenological descriptions of stochastic processes.Comment: 15 pages, 5 figures Replacement 11/02 contains minor editorial change

The Dirac Equation in Classical Statistical Mechanics

The Dirac equation, usually obtained by `quantizing' a classical stochastic model is here obtained directly within classical statistical mechanics. The special underlying space-time geometry of the random walk replaces the missing analytic continuation, making the model `self-quantizing'. This provides a new context for the Dirac equation, distinct from its usual context in relativistic quantum mechanics.Comment: Condensed version of a talk given at the MRST conference, 05/02, Waterloo, Ont. 7 page

Kinematics and uncertainty relations of a quantum test particle in a curved space-time

A possible model for quantum kinematics of a test particle in a curved space-time is proposed. Every reasonable neighbourhood V_e of a curved space-time can be equipped with a nonassociative binary operation called the geodesic multiplication of space-time points. In the case of the Minkowski space-time, left and right translations of the geodesic multiplication coincide and amount to a rigid shift of the space-time x->x+a. In a curved space-time infinitesimal geodesic right translations can be used to define the (geodesic) momentum operators. The commutation relations of position and momentum operators are taken as the quantum kinematic algebra. As an example, detailed calculations are performed for the space-time of a weak plane gravitational wave. The uncertainty relations following from the commutation rules are derived and their physical meaning is discussed.Comment: 6 pages, LaTeX, talk given in the session ``Quantum Fields in Curved Space'' at the VIII Marcel Grossmann Conference in Jerusalem, Israel, June 199

Monitoring Processes with Changing Variances

Statistical process control (SPC) has evolved beyond its classical applications in manufacturing to monitoring economic and social phenomena. This extension requires consideration of autocorrelated and possibly non-stationary time series. Less attention has been paid to the possibility that the variance of the process may also change over time. In this paper we use the innovations state space modeling framework to develop conditionally heteroscedastic models. We provide examples to show that the incorrect use of homoscedastic models may lead to erroneous decisions about the nature of the process. The framework is extended to include counts data, when we also introduce a new type of chart, the P-value chart, to accommodate the changes in distributional form from one period to the next.control charts, count data, GARCH, heteroscedasticity, innovations, state space, statistical process control

Is level of neighbourhood green space associated with physical activity in green space?

Background There is accumulating evidence that greater availability of green space in a neighbourhood is associated with health benefits for the local population. One mechanism proposed for this association is that green space provides a venue for, and therefore encourages, physical activity. It has also been suggested that socio-economic health inequalities may be narrower in greener areas because of the equalised opportunity for physical activity green spaces provide. However, research exploring associations between the availability of green space and physical activity has produced mixed results. Limits to the assessment of the type and amount of physical activity which occurs specifically in green space may account for these mixed findings. This observational study was therefore concerned with the extent to which green space is a venue for physical activity and whether this could account for narrower socio-economic health inequalities in greener neighbourhoods.<p></p> Method Secondary analysis of cross sectional data on 3679 adults (16+) living in urban areas across Scotland matched with a neighbourhood level measure of green space availability. Associations between green space availability and both total physical activity, and activity specifically within green space, were explored using logistic regression models. Interactions between socio-economic position and physical activity were assessed. All models adjusted for age, sex and household income.<p></p> Results The availability of green space in a neighbourhood was not associated with total physical activity or that specifically in green space. There was no evidence that income-related inequalities in physical activity within green space were narrower in greener areas of Scotland.<p></p> Conclusion Physical activity may not be the main mechanism explaining the association between green space and health in Scotland. The direct effect of perceiving a natural environment on physiological and psychological health may offer an alternative explanation.<p></p&gt

Probing the Improbable: Methodological Challenges for Risks with Low Probabilities and High Stakes

Some risks have extremely high stakes. For example, a worldwide pandemic or asteroid impact could potentially kill more than a billion people. Comfortingly, scientific calculations often put very low probabilities on the occurrence of such catastrophes. In this paper, we argue that there are important new methodological problems which arise when assessing global catastrophic risks and we focus on a problem regarding probability estimation. When an expert provides a calculation of the probability of an outcome, they are really providing the probability of the outcome occurring, given that their argument is watertight. However, their argument may fail for a number of reasons such as a flaw in the underlying theory, a flaw in the modeling of the problem, or a mistake in the calculations. If the probability estimate given by an argument is dwarfed by the chance that the argument itself is flawed, then the estimate is suspect. We develop this idea formally, explaining how it differs from the related distinctions of model and parameter uncertainty. Using the risk estimates from the Large Hadron Collider as a test case, we show how serious the problem can be when it comes to catastrophic risks and how best to address it