Stephen P. King (stephenk1@home.com)
Thu, 09 Sep 1999 11:17:03 -0400
Dear Matti,
Matti Pitkanen wrote:
[SPK]
> 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 <7qpnl4$78f@charity.ucr.edu>, John Baez
> ><baez@galaxy.ucr.edu> writes
>
> >>(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!
[SPK]
I see that you could what I has thinking right away! :-) But, you are a
minority in not seeing a problem with the concept of "geometric time"!
One problem I have is how do you model the communication between two
observers, do you propose wormholes connecting their "cognitive
space-time sheets". I am having trouble translating your fixed geometry
ideas over to my "every thing is process" way of thinking...
> >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(;-).
> ********
I hope so! :-)
> 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(;-).
I really would like to see this mental picture you have. I am just not
bisimulating your thinking at all, I see too many contradictions, but
that is, more than likely, due to my way of thinking... :-)
> >Presumably a whole new category of things would have to replace the
> >manifold approach.
>
> 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!
[SPK]
The primitive quantities in Category theory are objects and morphisms,
yes. This gives us an easy way of, literally, graphing out the
situations, since we can consider objects as nodes in a graph and
morphisms as edges... :-) I am VERY interested in what the "natural
restrictions" would be such that InfDimGeo (the category of infinite
dimensional geometries) would be a monoid! See Baez definition of such
in: http://math.ucr.edu/home/baez/week74.html
The way that Pratt discusses how CABAs "collapse the whole algebra into
a singlet" when a "new equation" is added, seems to me to relate to what
you are saying! The difficulty that I see is that the "classification of
3-surfaces" is NP-Complete computationally! We can not just assume
non-constructive arguments! I am thinking that each particular
"experience" is a particular "classification" (a morphism from a subset
of MEM to a subset of InfDimGeo, see below...) of a 3-surface. We can
just assume that the 3-surfaces are "out there" already sorted for us.
This thought is equivalent to the idea that there exist a single
absolute space-time and all events are like bubbles frozen in the 4-cube
and the subjective flow of time is an illusion! (See [time 623])
> 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
> as morphism?
Yes! the map of "self in external world to subself" is, I believe, best
described as a morphism! Pattern recognition, of sense data, would be
defined as a functor between the Category MEM (of memories of PAST) and
some subcategory of InfDimGeo, maybe! :-)
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