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
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