Subject: Re: quantph/0202057
Date: Fri, 15 Feb 2002 13:13:05 +0200 From: "Dimiter G. Chakalov" <dchakalov@surfeu.at> To: "Zafiris, E" <e.zafiris@ic.ac.uk> CC: Chris Isham <c.isham@ic.ac.uk>, Ioannis Raptis <i.raptis@ic.ac.uk>, "John D. Fearns" <j.fearns@ic.ac.uk>, Joao Magueijo <j.magueijo@ic.ac.uk>, Ntina <k.savvidou@ic.ac.uk> Dear Elias, Thank you for your reply from Fri, 15 Feb 2002 08:11:04 0000. You wrote: > Regarding your question I would like to say the following:
It is not clear to me how do you model the infinitesimal, as in Thompson's lamp paradox (please see my preceding email printed below). It seems to me that all real numbers should literally emerge out of some seemingly simple algorithm, (t_2  t_0)  (t_1  t_0) = 2 , resembling renorm recipes. We have not two but three animals here, don't we? If so, where is that mysterious sliding cutoff, t_0 ? Please see also my email to Joao Magueijo, http://members.aon.at/chakalov/Magueijo.html Best regards, Dimi

Dear Dr. Zafiris, I am very much interested in your ideas [Ref. 1], but my math is too week to understand them. May I ask you for help in clarifying the issue of mapping infinitesimals to real numbers. We are inevitably forced to resort to real numbers, although we're fully aware, after Lucretius, Zeno, Leibnitz and Newton, that the infinitesimal is an entirely different entity. Consider, for example, the paradox of Thompson's lamp. Imagine a lamp that is 'on' for 1 min, then 'off' for 0.5 min, then 'on' for 0.25 min, etc. Do we have a limit? If you say 'yes', what is the state of the lamp at the instant of time at the "end" of this limit? We have not two but *three* puzzles here: the state of the lamp at the initial instant (t_1), the state of the same lamp after two min (t_2), and the nature of the cutoff t_0. The latter can approach The Beginning but can never reach it. Also, the "duration" of the two instants, t_1 and t_2, can only tend asymptotically toward the "singularity" (Planck length?). Also very intriguing to me is that the cutoff t_0, with which we measure any *finite* intervals, like (t_2  t_0)  (t_1  t_0) = 2 , can approach asymptotically t_1 but *not* t_2. Hence it seems that some timeasymmetry is prebuild in this story about the infinitesimal. Hence my questions: how do we move from the infinitesimal to real numbers, and back to the infinitesimal? Also, can you suggest some 'generalized projection postulate', which would include the case of measurements in classical physics as a limiting case of a broader situation including quantum observables? In both cases we have the infinitesimal "filtered" through Boolean reference frames, don't we? Do you believe one could unravel this "filter" with Synthetic Differential Geometry or topos theory? With kind regards, Dimi Chakalov

"Quantum theory stipulates that quantities admissible
as measured results must be real numbers. The resort to real numbers has
the advantage of making our empirical access secure, since real number
representability consists our form of observation.
"Hence it is reasonable to assert that an observable picks a specific Boolean algebra, which can be considered as a Boolean subalgebra of the Quantum lattice of events. "In essence an observable schematizes the Quantum event
structure by correlating its Boolean subalgebras picked by measurements
with the smallest Boolean algebra containing all the clopen sets of the
real line. In the light of this Boolean observables play the role of coordinatizing
objects in the attempt to probe the Quantum world. This is equivalent to
the statement that a Boolean algebra in the lattice of Quantum events picked
by an observable, serves as a reference frame, conceived in a precise topostheoretical
sense, relative to which the measurement result is being coordinatized,
suggesting a contextualistic perspective on the structure of Quantum events.
Philosophically speaking, we can assert that the quantum world is being
perceived through Boolean reference frames, regulated by our measurement
procedures, which interlock to form a coherent picture in a nontrivial
way.
"The passage from a system of prelocalizations to a system
of localizations for a quantum observable is achieved if certain compatibility
conditions are satisfied on the overlap of the modeling Boolean charts
covering the quantum observable under investigation."
