Asymptotic quantization: based on 1984 Naples lectures by Abhay Ashtekar

By Abhay Ashtekar

Chosen subject matters on quantum gravity are taken care of emphasizing extra geometrical me- tools and conceptual matters than practical research, perturbative expansions and computation of numbers, The lirst half bargains with Asymptotic Quantization sheding gentle at the foundation of the Bondi-Meizner-Sachs crew within the gravitational radiation concept. the second one half is dedicated to canonical quantization, which supplies the de- targeted quantum dynamics and enhances the in simple terms kinematic;)] asympotic descrip- description preset"!Led within the lirsf half.

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Assume that two pinball trajectories start out parallel, but separated by 1 ˚ Angstr¨ om, and the disks are of radius a = 1 cm and center-to-center separation R = 6 cm. Try to estimate in how many bounces the separation will grow to the size of system (assuming that the trajectories have been picked so they remain trapped for at least that long). Estimate the Who’s Pinball Wizard’s typical score (number of bounces) in game without cheating, by hook or crook (by the end of chapter 13 you should be in position to make very accurate estimates).

1 as a strange attractor. diagnosing and proving existence of a genuine, card carrying strange attractor is a tricky undertaking. Conversely, if we can enclose the non–wandering set Ω by a connected phase space volume M0 and then show that almost all points within M0 but not in Ω eventually exit M0 , we refer to the non–wandering set Ω as a repeller. An example of repeller is not hard to come by - the pinball game of sect. 3 is a simple chaotic repeller. It would seem that having said that the periodic points are too exceptional, and that almost all non-wandering points are aperiodic, we have given up the ancients’ fixation on periodic motions.

1 trace sum for a zero of the spectral determinant. The computation of the zeros of det (s − A) proceeds very much like the computations of sect. 3. 6 From chaos to statistical mechanics While the above replacement of dynamics of individual trajectories by evolution operators which propagate densities might feel like just another bit of mathematical voodoo, actually something very radical has taken place. Consider a chaotic flow, such as stirring of red and white paint by some deterministic machine.

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