🚀 go-pugleaf

In article <1a3j6vINNsh9@agate.berkeley.edu> ted@physics.Berkeley.EDU (Emory F. Bunn) writes:
>[...]
>I believe, based on my
>admittedly limited understanding of measure theory, that the space
>of 3-geometries is simply "too big" to admit a well-behaved measure
>that would allow one to talk about probabilites.  (I have formed
>some vague arguments in my mind in support of this position,
>but they're far from being rigorous enough to convince anyone else.)
>
>I hope that I am wrong about this, as I think it would be very nice 
>to have such a measure.  Is there anyone out there who can be more 
>definitive about this than I can?
>
Well, there are certainly plenty of "physicists' measures" on the
space of three-geometries.  This space admits a natural metric, the
DeWitt metric, and (as is traditional in physics) one can ignore
the fact that the space is infinite-dimensional and use this metric
to define a measure.  Once you have one such measure, you have a
lot, since you can multiply by any reasonable positive function on
the space of geometries.  

(For anyone who wants to be picky, it is true that the DeWitt metric
is really a metric on the space of _metrics_ rather than the space
of geometries.  To get the metric on the space of geometries, you
have to "gauge-fix" and factor out the coordinate transformations.
But this is fairly standard, and in fact, unlike the rest of the
construction, can be done in a reasonably rigorous fashion, thanks
to the work of David Ebin, a mathematician who has studied the space
of geometries fairly carefully.)

Of course, the fact that the space of geometries is infinite-dimensional
does cause problems in defining these measures rigorously.  There are
a number of possible approaches.  You can approximate the smooth
geometries by flat simplicial complexes (the three-dimensional version
of the approximation of a hemisphere by a geodesic dome), for instance,
or you can define a measure by requiring that Gaussian integrals be
properly normalized.  For more formally inclined physicists like John
Baez, these procedures are not completely satisfying, but for most of
us they'll do for the moment.

Having said this, though, I completely agree with you that we have no
idea of the _correct_ measure on the space of three-geometries.  One
of the main goals of quantum cosmology is precisely to find such a 
measure.  A lot of the fine-tuning and naturalness arguments you
criticize can be translated into statements about this measure; for
instance, we would like to have it sharply peaked around geometries
for which the cosmological constant is very small (in Planck units).
There are a number of proposals for how to do this --- for instance,
Hawking's "no boundary" wave function would define a measure on
three-geometries --- but we're a long way from a believable model.

Steve Carlip
carlip@dirac.ucdavis.edu