Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 3 Jul 1999 16:32:35 +0300 (EET DST)
Here is continuation to the previous message.
On Fri, 2 Jul 1999, Stephen P. King wrote:
> Dear Matti,
>
> Matti Pitkanen wrote:
>
> On Fri, 2 Jul 1999, Stephen P. King wrote:
>
> > Dear Matti,
> >
> > I was wondering if you can get access to W. Schommers'
> book Symbols,
> > Pictures and Quantum Reality World Scientific, 1995? It has a
> discussion of
> > the issues that we are considering and could be very helpful.
> I am working
> > on some xeroxes for you on the statistics-geometry connection.
> Schommers
> > book links to this... I am about to do another write up on my
> ideas about
> > Weyl's gauge theory, I am very determined to communicate my
> idea and get
> > some feedback on it. It is making me crazy since I have not
> been able to
> > punch holes in this notion on my own!
>
> Unfortunately I cannot get the book of Schommers. By the way,
> its 1.45 at
> night and I we had a wonderful eveninng with friends and I have
> a little
> difficulty to type words correctly. Too many glasses of wine
> and
> wonderful coexistence of loving souls. It is wonderful to belong
> to human
> kind and have old friends.
>
> I know the feeling well! :-) I will see if I can be you a copy of
> Schommers' book... :-)
>
> I read the ps files about Weyl's gauge theory. I believe in
> conformal
> invariance, which is one manner to realize the idea of local
> scale
> invariance. In p-adic context conformal invariance generalizes
> to dimension four as infinite-dimensional symmetry. The magic
> conformal
> properties of the boundary of 4-dimensional light cone are
> crucial
> in the realization of conformal invariance in real context.
>
> This "infinite-dimensional symmetry" aspect of the p-adic context
> conformal invariance, is it related to the way that a given p-ary set is
> mapped to the Reals and back? There are many separate p-ary sets, there is
> only one set of Reals!
No. It is related to the existence of algebraic extensions of arbitrary
dimension. For real numbers you have just one: complex numbers and
functions representable as power series of z define conformal mappings
of complex plane (angles are preserved so that metric is mapped to
metric time conformal factor). For instance, for p-adics with p mod 4=3
4-dimensional algebraic extension Z= x+iy+ sqrt(p)(u+iv) is possible
since i=sqrt(-1) and sqrt(p) are not p-adic numbers. Hence also
analytic function f(Z) expressible as power series of Z define maps
of 4-dimensional space to itself. These are generalizations of conformal
mappings in the sense of being analytic and giving rise to the Super
Virasoro and Super Kac Moody algebras crucial for string models.
In four dimensions these maps are not however angle preserving: it is
analyticity which is generalize.
> I think that there is something very deep involved with the concept of
> conformal invariance that is being missed by mainstream physics. I am soo
> happy that you can see glimpses of this subtlety. ;-) There is something
> were strange going on with the quantum mechanical version of light-cone
> structures (LCS)! I say that every possible observation has a distinct LCS
> associated with it! Thus there is no Minknowsky spacetime for all! more...
Conformal invariance is not lost totally. Conformal field theories in two
dimensions has been entire industry and had physical basis. It was
found that two-dimensional critical systems are characterized by conformal
field theory allowing conformal symmetries as gauge symmetries.
This huge symmetry allowed to solve and classify these theories.
For instance, two point correlation function can be determined uniquely.
One motivation for the generalization of conformal symmetry in TGD is the
fact that TGD is quantum critical system: the Kahler couplings
strength has the mathematical role of temperature in the sense that
vacuum functional is exponent of Kaehler function inversely proportional
to alfa_K, just like partition function of thermodynamical system is
exp(-H/T). The reequirement that alfa_K is analogous to critical
temperature fixes the theory uniquely. It implies long range correlations
in all length scales, existence of macroscopic quantum systems, etc.
Interpretation of Kahler action as measure for cognitive resources leads
to the conclusion that quantum critical TGD describes the most interesting
and most complicated universe one can imagine in TGD framework.
So: conformal invariance in generalized form has deep connections to
physics and and consciousness theorizing!
>
> Kahler action is formally Weyl invariant but local scalings of
> induced
> metric cannot be realized as transformations of imbedding space.
> In this
> sense Weyl invariance is broken. CP_2 size is what breaks Weyl
> invariance.
> To sum up, I feel that Weyl invariance is broken invariance,
> otherwise we
> would not have elementary particle mass spectrum. The idea about
> scalar
> field developing expectation value determining the value of G is
> also nice
> but I believe that CP_2 'radius' determines G.
>
> I don't see the two notions as being mutually exclusive! What does
> "scalar field developing expectation value determining the value of G"
> imply?
G is dimensional quantity, length squared. The appearence G in action
breaks conformal invariance since conformal invariance implies scale
invariance as a special case. One can however solve the problem by
introducing scalar field phi and introduce term of form
Phi^2*Eintein-Hilbert action density (I think that it is
second power of Phi which makes action dimensionless). When Phi generates
vacuum expectation value, Einstein Hilber term results in action and one
obtains gravitation dynamically.
[SPK]
What does "CP2 'radius' determines G" imply? Could the radius of CP_2
> "evolve" dynamically just like how the scalar invarience is broken
> dynamically by the Higg's mechanism notion?
[MP]
Not in TGD framework. CP_2 radius sets the universal meter stick in TGD.
Everything can be expressed using it as a unit.
In string models imbedding space is taken to be dynamical, one speaks
of spontaneous compactification, etc.. I see this as the fatal flaw of
string models. In TGD M^4_+xCP_2 is fixed completely separately by
mathematical existence considerations. Configuration space geometry is the
unifying principle: its existence is extremely strong requirement.
> I think that p-adic physics
> fixes the problem of renormalization that supersymmetry has since it sets a
> quasi-absolute gauge scale on posets of quantum jumps.
[MP]
This might be the case but I am somehow convinced that making imbedding
space dynamics is completely unnecessary. In any case it would destroy
the whole TGD approach.
> But, I believe that
> this "quasi-absoluteness" is connected with the way that the p-adic metric
> is defined using the max function... The discreteness of the "elementary"
> particle mass spectrum, like the absoption and emision spectra, are NOT
> discrete in-themselves. It is the finiteness of the observation that makes
> it appear so. I think that Weyl was right but could not adaquately defend
> his intuition! :-(
[MP]
Some additional comments.
You are right about mass spectrum in the following sense. Super Virasoro
invariance implies universal mass squared spectrum of form
M^2 = M^2 n, n arbitrary integer, in principle also infinite as real
integer but finite as p-adic integer. The real counterpart of mass squared
spectrum is obtaine by mapping integers n to reals by canonical
identification. The image of n:s including also infinite n:s is the real
interval 0,p.
Elementary particles are not mass squared eigenstates. The masses of
physical particles are obtained by applying p-adic thermodynamics
assuming that besides massless ground state also thermally excited higher
masses (n>0) are possible. Also the predicted thermal mass squared
expectation spectrum is universal: only 1/integer valued temperature
appears as free parameter. Second parameter is the p-adic prime
characterizing elementary particle.
> We are assuming G is the curvature tensor, right?
Above I denoted gravitational constant by G.
>Or is it the
> non-integrable scale term? Or a combination of the two? It depends on the
> construction of the geometry of space-time. In Weyl's generalization of
> Riemannian geometry the curvature (gravity) is defined by the comparison of
> the angles of neighboring 4-vectors and the field defined by the comparison
> of the length scale of neighboring 4-vectors is identified by Weyl by the
> vector and scalar potential sources of the electromagnetic field. Weyl was
> right all along!
I think that Weyl's idea fails since the coupling of electromagnetic
potential is imaginary since gauge group is U(1), which is compact. For
scalings gauge group would be noncompact group R. This difference is
absolutely crucial in real context: for U(1) coupling to spinors is
imaginary, for R the coupling is real. In p-adic context situation is
unclear since all groups are compact in p-adic context as a consequence of
compact-open topology.
I will comment the rest later. I have pleasant social duties. Little
garden party and must leave.
........................
Well! I am now back from the garden party. We build 40 meters of
fence. Wine, bear, good food and wonderful experience of becoming one with
all these people talking and laughing around you. Mind in complete silence
and full of love.
> I have been proposing for a long time the idea that each observer is
> modelable as a partially ordered set of observations (p_o) taken "one at a
> time" (like your "quantum jump"!) (the inclusion relation is Binary for n =
> 2, fuzzy set inclusion for n > 2 n-ary relations ) from the Powerset P_o
> (P_o = N^p_i, p_i \elem. U; N is the number of mutually necessary qualia
> (wrong word!)) of all possible observations, all of which are aspects of U.
> I am conjecturing that the cardinality # of P_o is greater than that of the
> Reals! I propose that #U_o has an "undecidable by any finite enumerative
> procedure". I am not sure, but I believe that this # is only approximated
> asymptotically by the usual notion of a Limit. Thus, we can say that
> Eternity is the Time that it takes for Existence to exist. Umm, we
> generalize n to p for p-adic valued relations... I have to talk more about
> this with you. :-)
Sorry. I could not follow you idea. I got lost somewhere around
P_o=N^pi.
> That does this have to do with Weyl's theory? A lot! I am proposing that
> each poset p_o has its own basis of directions and gauge of length and there
> is not Absolute space-time, there are many! This idea is contrary to
> conventional notions that tacitly assume that there is an Absolute basis and
> gauge "imposed from Above"!
You might be right. In any case you must be able to produce breaking of
scale invariance since elemetary particle mass spectum is not continuous.
> I am identifying the scattering propagator of an
> LS with the evolution of the poset. It is a construction in the sense that
> either the poset (or LS), as a subset, nor its experiences, as an object,
> are "put together one piece at a time", but the "size" and "orientation" of
> the pieces is not a priori definite, they are defined "relationally"! :-)
> I will comment more on this in my reply to your last time list post. :-)
>
> But the best what can
> happen for two genuine thinkers is that they disagree as
> friends
> respecting each other's beliefs.
>
> It is good that we differ, otherwise we would be one and the same person
> and would miss out on the variety of experience that our differences offer.
> :-)
>
Best,
MP
>
>
>
>
>
>
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