[PD] tabread4~~

[Mathieu Bouchard]

On Fri, 23 Nov 2007, Uur Güney wrote:

> And she said that: "A simple harmonic oscillator makes a 1D motion (in 
> time). It goes back and forth. You can approximate a string as N 
> connected harmonic oscillator lying along a line. if N goes to infinity 
> we'll have a SHO at every point in space, which makes a 1D motion in 
> time. And this is a field, and hence it is a continuum." # This is in 
> accordance with your definition, an ideal string can have any shape, so 
> its possible shapes form "the set of all possible continous functions 
> over its length".

If she means Field as in Corps (fr) or Körper (de), then that's not 
necessarily a continuum. There are many finite fields, which are fields 
because they have regular +-*/, but still don't have fractions, because 
they work modulo-style. Infinite fields that contain all integers (Z) also 
contain all rationals (Q). Q is a field already.

You can extend Q quite a lot without ever getting to a continuum: add 
various square roots, cube roots, other roots, ... if you add all possible 
results of root operations, you get to the Algebraic Numbers, which are 
still not a continuum. You need to also add all limits of sequences before 
you get to a continuum. Depending on your mathematical religion, the 
continuum is either non-countable, or non-countability does not exist (i'm 
of the latter belief nowadays).

The idealness of a string depends on whether you base your ideas on 
classical physics or quantum physics. In the former, each harmonic has a 
"real" amplitude, whereas in the latter, you have a energy step 
proportional to the frequency and the amplitude is integer when expressed 
in units of the energy step. The latter theory is known to be more 
accurate, but when your string is not microscopic, you have no chance of 
noticing the difference, as steps are very small. Still, the total energy 
of a string can always be expressed as an integer multiple of the energy 
step of the fundamental frequency of the string.

Making an infinite number of integer dimensions may get you to 
non-countability of possible states (if you believe in it), but it still 
doesn't get you to a continuum.

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| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada