🚀 go-pugleaf

In article <1a3j6vINNsh9@agate.berkeley.edu> ted@physics.Berkeley.EDU (Emory F. Bunn) writes:
>In article <17584@ucdavis.ucdavis.edu> carlip@landau.ucdavis.edu (Steve Carlip) writes:
>>In article <19tid9INNo1h@agate.berkeley.edu> ted@physics.Berkeley.EDU (Emory F. Bunn) writes:
>>>[much deleted...]
>[more deleted...]
>>> What bothers me about
>>>this is that there is no natural measure one can impose on the set
>>>of possible initial conditions for Universes in order to make that
>>>statement meaningful.
>>>[...]
>[more deleted...]
>>
>>If I understand you correctly, you are claiming that there is no
>>possible measure on the space of three-geometries (or maybe
>>three-geometries plus initial configurations of matter fields).
>>This seems unlikely to me; do you have a proof or a reference?
>>Would you argue that a path integral is impossible in principle
>>in general relativity, for instance?

>I was making the assertion you attribute to me, and I made it with
>more confidence than I probably should have.  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?

There is certainly nothing definitive known about this question.
It's best to begin by pondering the situation with field theories
that are a lot better understood than quantum gravity.
Honest-to-goodness measures on some space of Euclidean field 
configurations have only been constructed in 2 or 3 spacetime
dimensions.  I hear that Balaban, a student of Jaffe, claims to have
done it for 4d Yang-Mills theory, but that the proof was so long
he isn't going to bother publishing it (in a curious twist on 
Fermat's last theorem).  

Of course, people can calculate perturbatively in ordinary QFT
quite well as if these measures really existed.

On the other hand, every set has lots of measures on it, and being
"big" is no obstacle.  What we want in physics is measures satisfying
certain properties, and one can only discuss whether they exist or
not after we have some idea of what properties we want.  That's
what happened in ordinary QFT: the Osterwalder-Schroeder axioms
state what your measure should satisfy, and people work on
constructing measures that satisfy those axioms.  In quantum gravity
I doubt it is even known what the "axioms" should be.

By the way, don't you want a measure on 4-geometries, rather than
3-geometries, if you're doing path integration... and you live in
3+1 dimensions (unlike Steve :-)).