From: David Bowman (David_Bowman@georgetowncollege.edu)
Date: Wed Jul 02 2003 - 22:56:17 EDT
Regarding Richard McGough's objection to the Tegmark article in SciAm:
>What has happened to sciam?
>
>The article under discussion was featured on the cover of the May 2003
>Scientific American, where it was flatly asserted, with no question or
>qualification, that parallel universes really exist. This is ludicrous
>in the extreme. At best Tegmark has some evidence for his hypothesis.
>He certainly has nothing like scientific proof. This makes the cover
>of sciam look rather like the National Enquirer.
>
>How Tegmark ever got his article past peer review is beyond me. When
>I first looked at his basic claim, I knew it was false. ...
> ...
>It appears Tegmark only counted the eigenstates. He ignored the fact
>that each superposition of these eigenstates corresponds to a different
>physical configuation of the system.
>
>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. ...
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.
It is a fact that the states counted by the discipline of statistical
mechanics is the first enumeration. Only such an enumeration is capable
of resolving such classical paradoxes as the UV catastrophe of black
body radiation and other problems with infinite entropy for finite
systems at finite temperatures, violations of the 3rd law of thermo,
etc. The second type enumeration vastly overcounts the physical states
because any two states that are not orthogonal have a finite probability
amplitude of actually having the same physical values of a complete
set of commuting observables upon a fully interrogative ideal
measurement.
However, the problem that I noticed was somewhat different. The
relevant quote from Glenn's post is:
>"The article notes that the region of the universe which we can
>observe, is cT big, where c is the speed of light and T is the age of
>the universe.
This is not correct. It would only be correct if the universe was not
expanding while the light we see from distant objects was on its way to
us. If D represents the current proper distance to the particle horizon
for our universe, then it is straightforward to show (in a homogeneously
expanding universe) that the rate at which D is increasing with time is:
dD/dt = c + H_0*D where H_0 is the current value of the Hubble
parameter. Since D is necessarily positive and since H_0 is positive in
an expanding universe we see that dD/dt > c in any homogeneously
expanding universe (but if the universe was contracting and had a
negative value of H_0 then dD/dt < c, and dD/dt = c for a static
universe).
If we use the recent results from the WMAP project we find that the
value of H_0 is about 71 km/s/Mpc = 1/(13.77 Gly) and D is about
47.5 Gly. This results in the particle horizon's proper distance
receding from us currently at about 4.45*c. The most distant actual
stuff we can see is the CMB. The current location of the surface of
last scattering of the CMB is about 46 Gly away and it is receding from
us at about 4.34*c.
>This is approximately 10^26 meters away in all directions.
The 47.5 Gly for the theoretical horizon particle distance translates to
4.49 x 10^26 m. The 46 Gly distance of the CMB's surface of last
scattering translates to 4.35 x 10^26 m.
>Now, each second this region grows by c meters
As mentioned above the radius of this region grows by over 4*c meters
each second (assuming the WMAP parameters are correct and the
universe is sufficiently modelable by a flat homogeneous FRW universe).
>bringing a part of the universe which was
>previously unobservable within view."
True.
David Bowman
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