Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 18 Apr 1999 18:08:08 +0300 (EET DST)
On Sun, 18 Apr 1999, Ben Goertzel wrote:
>
> Well, I tried to read some of Matti's book while in Brazil, but I found it
> tough going.
> It would take a long time to really understand that stuff because the
> mathematics
> is so unfamiliar. And I'm afraid I don't have the motivation to push
> through it right now ;(
> It does look interesting, but there are so many interesting things in the
> world...
>
The mathematics and manner of talking might be easier to follow with
string model background, the symmetry philosophy of particle physicists,
and intuitive grasp about submanifold geometry.
> In reading the first 30 pages, however, I noted that Matti, like Hitoshi,
> posits that the universe is
> of the form
>
> G x H
>
> where G is GR spacetime, and H is some kind of quantum space
>
> Differences are:
>
> -- Matti restricts the 4-D spacetime continuum G to a subset of 8-dim space
> (a restriction that would not affect any of Hitoshi's conclusions)
Yes this this true. As a matter fact,
I am not actually assuming GxH but rather X^4 subset H as you already
noticed, that is classical spacetime is 4-surface in 8-dimensional H: both
spaces are completely classical. 3-surface is basic dynamical unit: also
classical spinor fields of H induced to spacetime surface are dynamical.
Quantum theory is formulated in the space CH of all possible 3-surfaces
of H since 3-surface is basic dynamical object. This space is direct
counterpart for Wheeler's superspace. 3-metrics are replaced by
3-surfaces with metric and other geometric structures induced from H. CH
is also classical space: in fact everything except quantum jump
is classical in quantum TGD. Oscillator operator algebras have
interpretation in terms of classical Kahler geometry in infinite
dimensional context.
By the way, the induction of bundles structures is basic procedure,
which can be found from the first pages of any book about fiber bundles:
imbedding map defines a new bundles for which base space is imbedded
manifold. For some mysterious reason string model people have not realized
the possibility that induction might be useful for string theory. In TGD
it is fundamental concept.
>
> -- Hitoshi makes H a standard Hilbert space, whereas Matti makes H a more
> complicated structure. However, Matti attempts to account for strong and
> weak forces,
> and in order for Hitoshi to account for these he might need to build out H
> into a more
> intricate structure.
Hilbert space emerges in TGD as the space of all possible *classical
spinor fields* defined in the infinite-dimensional space of 3-surfaces:
what is done is to generalize the concept of spinor field to infinite
dimensional context. The surprise is that spinor fields in infinite
dimensions are physically equivalent with second quantized spinor fields
in finite dimensions! Gamma matrices in infinite dimensional Kahler
manifold form an algebra equivalent with fermionic oscillator operator
algebra and fermionic anticommutations become geometrized.
I think that one of the basic difference between Hitoshi's and my approach
is that I follow Kaluza-Klein type philosophy: the symmetries of H dictate
particle spectrum and types of interactions and indeed fix the
choice of H uniquely.
>
>
> So on a very cursory level, it does seem to me that there's a possibility
> that the two theories
> are getting at the same thing. But this conclusion ~may~ be due to my not
> having looked deeply
> enough into Matti's theory to really understand it.
I think that our pictures are so different that there is no equivalence.
The concept of local system understood at sufficiently general level is
however common and is realized in TGD topologically as many-sheeted
spacetime time.
>
> Comments?
>
> Ben
>
Matti
>
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