[time 131] Reply to Kitada

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Apr 1999 12:55:37 +0300 (EET DST)

Dear Matti,

Thanks for an interesting post. Let me make a comment and some

>On Thu, 1 Apr 1999, Ben Goertzel wrote:
>Actually the recent picture about quantu jump provides generalization for
>von Neumann's intuitions about brain as ultimate reducer.
>In TGD framework cognitive spacetime sheets, which are nearly vacuum and
>have finite time duration. [Energy and other conserved quantities flow
>from material spacetime sheets to cognitive sheets when they are formed
>and back to material spacetime sheets when cognitive spacetime sheets

The difference here between your view and mine seems to be in that you
think one sheet disappears when another sheet comes in, while I think
both sheets (local systems and the outer classical world, in my case)
exist at least on the unobservable level. But this seems just a difference
in expressions:

[MP] ...or that there clock here and another clock (spacetime sheet)
somewhere else whereas you associate clock (R^6 fiber) to every point of

In my
formulation, all possible local systems "exist," and the classical world
appears according to the requests of observers, at which instant a
set of local systems are selected. This corresponds to your "...
flow from material sheets [disappearance of material sheets] to cognitive
sheets [apperance of cognitive sheets]" (and vice versa).

In the sense that observer (or its brain) is the ultimate reducer, our
views are on the same ground (althought there seems a difference in
geometries as Stephen pointed out in [time 102]).

> The entanglement of cognitive spacetime sheets, 'Mind' with
>spacetime sheets carrying matter, 'Matter' is reduced in allowed quantum
>I think that any periodic phenomenon provides a clock: the basic
>problem is to find someone to perceive the reading of the clock(:-).
>In quantum jumps between quantum histories picture the nondeterminism of
>Kahler action comes at rescue and makes possible conscious experiences
>with time localized contents.

Does "the nondeterminism of Kahler action" continue only for a _finite_
in an exact sense? My question is if it ends precisely in a finite time
or in some approximate sense of "finiteness."

One can distinguish between two kinds of nondeterminisms: nondeterminism
corresponds to multifurcation (typically bi-) and the N (2) branches last
infinitely long or at least very long time. One could call this kind of
nondeterminism 'volitional': our volitional acts correspond to quantum
reducing quantum entanglement between many particle states and this kind
of branches. One can also consider
'cognitive nondeterminism' in which different branches of spacetime
surface differ in finite spacetime region only. Cognitive spacetime
sheets with finite duration should represent cognitive nondeterminism.
You are probably correct in that one probably has to resort to approximate
sense of finiteness: the gluing of cognitive spacetime sheet to material
spacetime sheet has probably effect which decreases, say exponentially,
rather than having literally finite duration. Essential point is however
that there is not effect on 3-surface
X^3 to which the spacetime surfaces X^4_i(X^3) are associated.
Multifurcation type nondeterminism satisfies this criterion.

>The oscillations of Josephson junctions formed by wormhole super
>conductors indeed generate clocks if one believes that EEG is a clock.
>Amusingly, simplest EEG clock corresponds to sequence of solitons of Sine
>Gordon, which is mathematically nothing but gravitational pendulum
>rotating. Also EEG oscillations equivalent with oscillating
>gravitational pendulum are possible. In latter case EEG is equivalent
>with the clock in the wall!
>p-Adicity leads in a natural manner to lattice like structure. You can
>form from real axis 1-dimensional lattice by cutting, say decimal
>expansion, from n:th decimal. In p-adic context cutting of pinary
>expansion of pinary number so that O(p^n) part of p-adic number is put
>zero is analogous procedure but defines equivalence relation in p-adic
>context. Hence one can define entire hierarchy of discrete coset spaces
>R_p/E_n by this equivalence relation (denoted by E_n).
>This hierarchy of lattices defines extremely rapidly converging
>approximation procedure for physically interesting primes p (p=2^127-1
>electron!). Various physical fields become in this approximation fields
>What is especially nice is that p-adic counterpart of, say, Poincare
>respects these lattice structures. I told about how p-adic Poincare
>group leaves finite p-adic spacetime cube invariant in some earlier
>posting few weeks ago. One can quite well say that p-adics are Taylor
>for lattice approximation.
>Personally I however believe that geometry is continuous at basic level.

>The basic reason for this is that infinite-dimensional geometry is highly
>unique: in TGD case the sole requirements that Riemann connection exists
>mathematically + some other general requirements fix the entire
>geometry and also imbedding space itself essentially uniquely. In TGD
>framework this means unique physics also since physics is just
>infinite-dimensional spinor geometry. The inability of physicists to
>find divegence free QFT:s reflects also this high uniquess of
>infinite-dimensional mathematics.

Could you give any calculation procedures of the quantities physicists
want to calculate on the basis of your infinite-dimensional mathematics?

Only in principle. In practice the general framework combined with
general wisdom about particle physics leads to rather detailed models and
predictions for new physics. Everything is 'TGD inspired' in practice.

a) What I have done is the construction of configuration space geometry
and spinor structure: it seems that construction works for M^4_+xCP_2 and
is essentially unique and there are detailed formulas for the matrix
elements of metric (which cannot
be totally wrong!). What gives hope of practical realization is that
configuration space is union of symmetric spaces G/H (all points (now
3-surfaces) of symmetric space are metrically equivalent: 'physics
 depends on point of configuration space only through zero mode

c) The construction of quantum states reduces to the construction of Super
Virasoro, Kac Moody and super canonical representations. Ground states of
these representations give the physics: the rest is physics at CP2 length
scale. Super Virasoro algebra and Virasoro generator L_0 in the role of
Hamiltonian. New element is super canonical algebra (CP_2 canonical
transformations localized with respect to lightcone boundary).
Super Virasoro structure has been tested physically: the whole game with
p-adics game with the calculation of elementary particle masses using
p-adic thermodynamics
with energy replaced by L_0. Predictions are excellent: in particular
the mysterious mass scales (typically 10^19 Planck masses) reduce to
number theory. p-Adic thermodynamics which is already infinite-dimensional
mathematics, has led to very detailed predictions: just now I am working
with TGD predictions for CP breaking and new physics implied by higher
gluon generations. A potential killer prediction is that Higgs does not
exist: if it found next year (this has been the belief for 10 years at
least!), TGD is in difficulties.

d) Calculation of S-matrix element involves configuration space
In principle configuration space integration reduces to perturbation
theory: this can be done perturbatively and I have identified mechanisms
taking care of the cancellation of known divergences (or those which I
have been able to identify). Propagator is just contravariant Kahler
metric and is calculable
thanks to union of symmetric spaces property. p-Adicization is necessary
in order to perform functional integral over zero modes characterizing
shape and size
and induced Kahler field associated with spacetime surface.
 p-Adicization of configuration space geometry must be
carried out in order to calculate S-matrix: configuration space of
3-surfaces decomposes into sectors D_p such that in sector R_p defines the
effective topology.
Next project is to look whether p-adicization means simply the replacement
of real formulas with p-adic counterparts and understanding of
new effects related to number theoretical existence.

e) p-Adic integration is now understood (so I believe!): there is explicit
discrete sum formula for the p-adic integral, which defines perturbation
theory in powers of p: it converges extremely rapidly for physically
interesting primes. But a lot of physical intuition is required before
anything practical can be done.
At the limit of infinite primes the two lowest terms give the finite part
of S-matrix: remaining terms give infinitesimals and it seems that recent
day experimental physics is not yet able to deal with them(;-).

f) It would require huge amount of work and mathematical genius to
derive practical Feynmann rules for S-matrix: analytic Schwinger type
mind would be required to achieve this. I think that I will spend rest
of my life just refining the basic conceptual apparatus(;-).

>p-Adic approach fits very naturally with bits and bytes philosophy.
>For instance, even infinite-dimensional configuration space integral
>reduces to a discrete sum.

Could you explain the machinery that reduces the infinite-dimensional
configuration space integral to a discrete sum? If so, it seems to me that
you can define e.g. the (problematic) Feyman's path integral rigorously by
your formulation. Is my expectation true?

I can send some pages of tex file as a separate posting: it would take too
much space here.

I do not use Feynmann's path integral in the basic
formulation of the theory (sum over all possible spacetime surfaces): I
have vacuum functional exp(K), where K is Kahler function: exponent is
*real* rather than imaginary. exp(K) is uniquely determined by the
requirement that Gaussian determinant associated with functional integal
cancels metric determinant: covariant Kahler metric partial_K
partial_(barL) determines the kinetic part of 'action' in perturbation
theory and propagator is just the contravariant Kahler metric.
Propagator has thus purely geometric interpretation.

Kahler function is NOT a local functional of 3-surface (although Kahler
action is local functional of 4-surface). This means that there are no
local interaction vertices so that the standard divergences coming from
these should be absent.

The expansion of sqrt(G) gives Ricci scalar in perturbation theory:
for symmetric space Ricci tensor is proportional to metric and Ricci
scalar divergences in real context unless configuration space satisfies
vacuum Einstein equations.

The remaining problem is integration over zero modes and at this stage
one must introduce p-adics: actually entire CH geometry must be p-adicized
but it seems that the mechanisms listed above work still since p-adic
definite integral can be defined also analytically, not only numerically.

>My own view is that
>objective realities=quantum histories are continuous object but that our
>consciousness is able to work with bits and bytes
>only. TGD however leads naturally infinite primes and p-adic number
>associated with infinite primes (which are actually very much like
>also infinite hierarchy of consciousnesses is predicted. Perhaps these
>Godlike consciousness above us are not limited to play with bits and
bytes only(;-).

I suspect that each of your infinite hierarchy of consciousness is still
finite. If this is the case, each God on each hierarchy looks like being
able to play just with "bits and bytes."

Certainly able but also willing?(;-) I tend to believe that infinite
primes and rationals and reals transcend the bit and byte level. The
concept of infinite primes forces to introduce also infinitesimals. It
might be that this generalization of the concept of number is necessary to
describe things like emotions physically: ordinary real numbers are
perhas not sophisticated enough a concept for this purpose.

At the spacetime level infinite primes would correspond to hierarchy of
spacetime sheets with *literally infinite* size bounded however above
(by infinite bound).

Also would you explain the meaning of the second sentence of what you
wrote in
[time 43]?:

>d) Note that M^4 metric is of standard form dt^2 -dx^2-... in p-adic
>context. Now however there seems to be no sharp difference between
>Euclidian and Minkowskian signature of metric.

My point was that for p-adic numbers are not well ordered so that it is
not possible to define distinction between negative and positive numbers.
For p mod 4=3 one -1 =(p-1)/1+p+p^2....) does not possess squared root
so that one can introduce i by performing algebraic extension. Hence
one could consider the possibility of defining p-adics whose square root
is proportional to i as negative numbers and this could define the
difference between Minkowski and Euclidian signature.

More concretely: both S(3,1) and SO(4) are compact in p-adic context.
To get better grasp on situation, one can consider p-adic counterpart of
2-dimensional Lorentz group: the matrices describing Lorentz boosts are
of form A11= cosh(eta) = A22; A12= A21= sinh(eta).
These matrices can be obtained as exponentiation of
Lie-algebra element if eta is of order O(p): that is its p-adic norm
is 1/p or smaller. This group is *compact* whereas in real case it is
noncompact [p-adic topology is compact-open: all open sets are compact].
There is also discrete group of 'big boosts' not obtainable as
exponentiation: cosh(eta) and sinh(eta) are both rationals m/p and n/p
with the property (m/p)^2 -(n/p)^2=1: non-Euclidian
analog of Pythagorean triangle is in question.

[For rotation group SO(2) rotation angles correspond to Pythagorean
triangles having integer sides. By the way, Pythagoras seemingly lived
in p-adic world. Unfortunately, no one had told him about algebraic
extensions of p-adics: the famoous discovery of sqrt(2) meant death
sentence for one his pupils. I am not so phanatic(;-).]


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