Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 13 May 1999 09:21:05 +0300 (EET DST)
Asnwer to a couple of questions in previous very long email.
[MP]
> Quantum theory for dynamical imbedding space would also lead to
> horribly nonrenormalizable theory: for instance, low energy limit of
> string models is nonrenormalizable theory since it is defined in
> 10-dimensional spacetime. Also physics would come out wrong: for
> instance, TWO gravitons would be predicted since the
> dynamical metric of the imbedding space would also give rise to graviton
> besides the graviton predicted by quantized dynamics of 3-surfaces.
[Stephen]
Question: In the TGD model, how would we make
observational measurements of gravity waves and/or gravitons?
[Matti]
Low energy limit of TGD does not differ much from that of standard
model. Gravitons are predicted and same experimental techniques
as used to test GRT predictions should work. Take this with a grain
of salt, I am not experimentalist.
[MK]
> > Comment: Already at this stage one notice precise analogy with
> > quantum measurement theory. SO(2) belongs to and U(1)xU(1) is
> > the group spanned by maximal commuting set of observables associated
> > with isometries of H!
[SPK]
> How many "mutually commutative" sets of observables could exist
> if we allow for each to be "almost" convex?
>
> [MP]
> You are probably speaking of G= SO(3,1)xSU(3). Observables correspond
> group theoretically to Lie-algebra of G. Mutually commuting observables
> correspond to Cartan subalgebra generating maximal abelian subgroup of
G.
> For G under consideration Cartan subgroup is H= SO(1,1)xSO(2)
xU(1)xU(1).
> Lorenz boosts in given direction and rotations around that direction
> plus rotations generated by color isospin and hypercharge. Any
> Cartan subalgebra gives a set of mutually commuting observables.
> The space of them is the coset space G/H, H Cartan subgroup.
[Stephen]
Ok, but what is the cardinality?
[Matti]
I would speak of dimension as manifold (or orbifold):
for Lorentz group it is 6-2=4. For
SU3 it is 8-2=6.
> BTW, for SU(3) this is SU(3)/U(1)xU(1), which a mathematician Barbara
> Shipman found to be related in mysterious manner to the mathematical
model
> of the dance of honeybee and guessed that quarks must somehow be
involved!
> TGD inspired explanation of the appearence of this space is on my
homepage
> and relies crucially on the fact that TGD predicts classical color
> fields in all scales (although gluons are confined) and to to the fact
> that macrosopic 3-surfaces have besides ordinary rotational degrees
> of freedom also color rotational degrees of freedom: color rotating
> 3-surfaces has color charges just like rotating body has angular
momentum.
[Stephen]
I need to buy your book... ;)
[Matti]
No need to buy, you can print the interesting chapters from my
homepage. This requires some patience, however!
You wondered how the mapping of the the phase of complex number
number to its p-adic counterpart is discontinuous so that
the rays of complex plane are permuted randomly. The point is
that the modulus of complex number is mapped continuously
using canonical identification whereas
rational phases are only reinterpreted as p-adic numbers
and this identification is discontinuous.
Rational phases are characterized by two two integers r and s having no
common divisors. Small change of the
phase factor corresponds in real context to small change of r or s
or both. In p-adic topology the change, of say r by one unit, is not
however small in general. p-Adically small changes of phase correspond to
replacements r--> r+ r_1p^N, s--> s+ s_1p^M, N and M larger integers
and these changes in turn correspond to large changes of real
phase in general. Hence the radial rays in complex plane are permuted
'randomly'.
Only for phase angles very near to 0 or pi situation is different as is
seen by studying
sin(phi) =2rs/(r^2+s^2)
for r= r_1p^K and s= O(p^0). Both real sine and p-adic sine are
near to zero. Hence one can say that angles near to zero and pi
are exceptional: in these regions identification seems
to be continuous. Interestingly, these angles are physically
very interesting since scattering in long range fields occurs mostly
to angles near 0 or pi and this leads to infrared divergences.
Still one comment: p-adic phase does not in general exists as p-adic
number. Only the p-adic phase factor exists! Exception is formed
by phases for which sin(phi) is of order O(p^k):
k>0 p-adically that is sin(phi) has p-adic norm smaller than 1.
This is just the exceptional case corresponding to region near phi=0
or pi.
MP
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