Matti Pitkanen (firstname.lastname@example.org)
Thu, 9 Sep 1999 08:22:50 +0300 (EET DST)
John Baez explains well my problem with QFT and symmetries!
In article <sNluueA1XM03Ewgw@upthorpe.demon.co.uk>,
Oz <Oz@upthorpe.demon.co.uk> wrote:
>In article <email@example.com>, John Baez
>>(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.)
[MP] I express point of view immediately. What breaks down, according
to my belief, is the approximate identity of psychological and geometric
time in time scale of order 10^4 Planck lenghts. Psychological time
is discrete (the center of mass temporal coordinate of cognitive
spacetime sheet increases the average amount by about 10^4 Planck times
in quantum jump). No revolutions in understanding of geometric time:
Riemann did something rather final!
>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.
[MP] I am really happy to see that things develop. Colleagues are slow
minded but it is pleasure to find that they are thinking hardly(;-). Even
string people are beginning to admit that there is something wrong and
this is great. Perhaps time is soon ripe for TGD(;-).
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.
[MP] Why not try something more simpler and less radical: already
Riemann tried this but too early when he proposed that 3-space
is curved surface in 4-space. Start from the
age old problem of General Relativity. How to define energy and momentum
when spacetime is not curved anymore and does not possess Poincare group
as its isometries? What about spacetime as surface in M^4_+xS?
You get Poincare! Plus isometries of S, color group perhaps! And You
get generalization of string model too! This should make bell ringing
in every head thinking about theoretical physics! But it does
not. I am frustrated(;-).
>Presumably a whole new category of things would have to replace the
Right! Or maybe even an n-category!
[MP] I looked the definition of category in separate
posting: objects and morphisms between
them. Is this all? I think it makes sense one speaks about
category of, say, Riemann spaces. Morphisms would be isometries.
Or groups, morphims would preserve group multiplication.
I am however sceptic about the idea that category theory could
describe physics. The space of 3-surfaces, infinite-dimensional
Riemann geometry, should be, and as I believe is, essentially unique.
Category of infinite-dimensional geometries (with some natural
restrictions) would contain only single member!
It could be interesting to find whether morphism idea could
somehow make sense in case of selves. Sensory experience
provides representation for other selves and a lot of else
as subself. Could the map of self in external world to subself
be regarded as a morphism in some sense? Sensory experience
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