From: Richard McGough (richard@biblewheel.com)
Date: Thu Jul 03 2003 - 17:30:51 EDT
In response to David Bowman's post:
http://www.calvin.edu/archive/asa/200307/0052.html
I said:
>Consider a single electron. Consider just its spin. Its state is a
>linear superposition of the two eigenstates:
>
>|s> = a|u> + b|d> with aa* + bb* = 1
>
>There is a continuous infinity of possible states for the spin of this
>one electron. ...
David repsonded to my argument:
>Of course all this hinges on just what one wants to consider as distinct states. Does one want to consider as distinct the eigenstates of a complete set of mutually commuting observables as is done, for instance, in statistical mechanics? Or does one want to count as distinct the set of all *amplitudes* for the various unresolved superpositions for the various values of the observables? The first of these counts the dimensionality of the Hilbert space, and is the set of all possible distinct outcomes of a fully interrogative mutually consistent set of observations. But the second counts all of the rays through the origin (i.e. 1-d projections) of that Hilbert space, and is the set of all possible distinct amplitudes for all manner of measurements that are not to be done.
Point 1) "Of course all this hinges on just what one wants to consider as distinct states."
Answer: True. But this is has nothing to do with what Tegmark wrote in his sciam article, where he simply counted the binary "there is/is not a proton at point (x,y,z)." This is a totally inadequate model of all possible physical configurations in a real hubble volume. No one has yet refered to anything Tegmark has written that addresses the issue of continuous eigenvalues. Does anyone know if Tegmark has ever addressed this question?
Also, we don't want to confuse the assumptions made in statistical mechanics that are used to calculate energy distributions of equilibrium states with Tegmark's calculations which are supposed to count all possible physical configurations, including those that are so far from equilibrium as to support life, like our real universe.
Point 2) "The first of these counts the dimensionality of the Hilbert space ..."
Answer: Correct. And since the dimensionality of the Hilbert space for even a single free proton is infinite, Tegmark's calculation fails.
I didn't notice anything else in the post that might impact the validity of my criticism of Tegmark. Please let me know if I have missed something.
I calmly look forward to your congenial response,
-- Richard Amiel McGough Discover the sevenfold symmetric perfection of the Holy Bible at http://www.BibleWheel.com --
This archive was generated by hypermail 2.1.4 : Thu Jul 03 2003 - 17:37:07 EDT