Stephen P. King (firstname.lastname@example.org)
Wed, 08 Sep 1999 22:55:05 -0400
John Baez explains well my problem with QFT and symmetries!
attached mail follows:
>>(Personally I suspect that the whole idea of spacetime as
>>a manifold breaks down at this point, but we really know
>>rather little about these things - though we calculate
>>endlessly and publish lots of papers.)
>Wouldn't this be a stressful break to how we view spacetime and
>possibly (certainly?) make GR just an approximation?
About the only thing that everyone working on quantum gravity
agrees upon is that general relativity is just an approximation.
It must be, because it doesn't take quantum mechanics into account,
and the world is quantum-mechanical.
So the big question is: how radically must we break from the picture
of spacetime provided by general relativity?
It makes sense to try the most conservative things first, then
if those don't work, more radical things, and so on. People have
been working on this for about 50 or 60 years, so by now they are
getting desperate and trying some fairly radical things. In the
conferences on quantum gravity that I went to earlier this spring,
I noticed a surprising unanimity of opinion about one thing. People
from string theory, loop quantum gravity, noncommutative geometry
and so on disagreed about almost everything, but they almost all
seemed to agree that we need to move away from the picture of
spacetime as a manifold.
But you're right, this is very stressful. This is especially true
because general relativity and quantum field theory - our two best
theories of physics - both assume that spacetime IS a manifold.
People have been assuming something like this at least since Descartes,
so most of our mathematical tools are suited to dealing with situations
where spacetime is a manifold. If we want to switch to something new,
it's not easy or quick. It's very hard to build up the necessary new
tools to replace all the old ones.
>Has anyone any
>sensible idea as to what structure might replace manifolds in this
>situation let alone how to manipulate objects in it?
Various people have different ideas: spin networks, spin foams, the
Regge calculus, matrix models, dynamical triangulations, noncommutative
geometry, and so on. I talk about them a lot here on sci.physics.research,
because this is my main interest: figuring out what spacetime is really
like. As you probably know, I'm a fan of using spin networks to describe
space and spin foams to describe spacetime. Thus it's my job to cook up
lots of nice tools to work with these objects.
>Presumably a whole new category of things would have to replace the
Right! Or maybe even an n-category!
>>By the way, note that I say "Lorentzian" rather than "pseudo"
>>or "quasi" or "kindasorta" Lorentzian. Here's the deal: folks
>>use "Riemannian" when the metric looks like +++++++ (all
>>directions are spacelike), they use "Lorentzian" when it looks
>>like -++++++ (all but one is spacelike, one is timelike), and
>>"semiRiemannian" or "pseudoRiemannian" for the general case,
>>like ---++++ (any number of spacelike or timelike directions).
>Hmmmmm. This looks a very tempting area for study so I guess lots of
>work has been done on it.
You bet. I'll be visiting you soon, I hope, and I expect you to
have read Barrett O'Neill's book "Semi-Riemannian Geometry, with
Applications to Relativity" by the time I get there.
>Would the er, um (dang, I've forgotten the
>guys KK) 5-D spacetime that gets EM & GR to fall out be Lorentzian
The Kaluza-Klein model features a 5-dimensional Lorentzian spacetime -
1 dimension of time, 4 of space. More precisely, this spacetime is
R^4 x S^1 - the usual Minkowsk spacetime you know and love, good old
R^4, times a wee little circle, or S^1. See? You're beginning to
understand this math notation! (Please, tell me you are....)
The 10-dimensional spacetime of string theory is also Lorentzian -
it's R^4 x M, where M is some 6-dimensional Riemannian manifold
(they're, ahem, still not sure which one).
Only a few very bold people have considered "2-time" theories where
spacetime is semi-Riemannian but not Lorentzian. We talked about
these a while ago here on sci.physics.research.
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