**Stephen P. King** (*stephenk1@home.com*)

*Tue, 13 Jul 1999 00:41:48 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 442] Re: [time 427] Re: Conformal Invariance and related notions"**Previous message:**Matti Pitkanen: "[time 440] Re: [time 439] Re: [time 437] Dissipation"**In reply to:**Stephen P. King: "[time 439] Re: [time 437] Dissipation"**Next in thread:**Matti Pitkanen: "[time 442] Re: [time 427] Re: Conformal Invariance and related notions"

Dear Matti,

I am finally able to respond. My computer is patched together enough to

write and send this...

Matti Pitkanen wrote:

[SPK]

*> > These "algebraic extensions of arbitrary dimension", is the
*

*> > dimensionality that of R^n? Is there a relation to the spaces of linear
*

*> > functionals, e.g. tangent subspaces, I am thinking of these algebraic
*

*> > identities as being identifiable with some type of vector notion?
*

[MP]

*> They are linear spaces, just like R^n. Isomorphic as linear spaces to
*

*> R_p^n just like C is isomorphic with R^2. The key idea is that n:th order
*

*> polynomial has algebraic numbers as its roots in real domain.
*

*> These roots do not exist as p-adic numbers in general. One can however
*

*> introduce extension of p-adics consisting of numbers
*

*> x+theta_1y+ thetaz+.... so that one can say that roots exist in the
*

*> extended number field.
*

*>
*

*> Also rationals allow algebraic extensions in the same manner:
*

*> for instance, the numbers of form r+sqrt(2)s+ sqrt(3)t + sqrt(6)v,
*

*> r,s,t,v rational, is 4-dimensional algegbraic extension of rationals.
*

*> Products, sums ratios below to the algebraic extnsion as one easily finds.
*

snip

[SPK]

snip

*> > The assumption of Einstein et
*

*> > al, there there exist only a single space-time for all observers in U, is
*

*> > admitted to be a very problematic notion by even Chris Isham and company! It
*

*> > directly contradicts the fundamental properties of QM! The problem of time
*

*> > and the inner product of the Hilbert space of "Universal" wavefunctions is a
*

*> > corollary effect of this problem!
*

*>
*

*> Yes. I agree here.
*

*>
*

snip

*>
*

*> You are right about the notion of critical temperature. I cannot
*

*> say anything about Unruh effect because I do not know it well enough.
*

See:http://members.home.net/stephenk1/Outlaw/Unruh.html

*> Kahler function is of form
*

*>
*

*> K= (1/16*pi*alpha_K) *INT J^munuJ_munu d^4x
*

*>
*

*> The integral is essentially Maxwell action for spacetime surface.
*

*> Coefficient involves alpha_K= e_K^2/4*pi, which is completely analogous
*

*> to fine structure constant, e_K being unit of 'Kahler electric charge'.
*

*> This is standard variational principles. Any introduction to quantum field
*

*> theories or book about classical mechanics contains short summary of
*

*> variational principles or action principles as they are also called.
*

*> Action is what economists would call cost function. The solutions of field
*

*> equations typically extremize action so that action is stationary with
*

*> respect to small variations. Kahler function is not only extremum
*

*> of Kahler action but actually absolute minimum: thus interpretion as 'cost
*

*> function' makese sense.
*

*>
*

*> exp(-H/T)/Z, Z normalization factor appears in classical thermodynamics
*

*> and is essentially Boltzmann weight, the probability of configuration
*

*> with given value of classical energy. Hamiltonian as a function
*

*> of physical configuration gives the energy of that configuration.
*

*> In classical mechanics one would typically have H=T+V, T and V denoting
*

*> kinetic and potential energies of system consisting of point particles.
*

*> T is temperature. In Maxwell ED H would be some of magnetic and electric
*

*> field energies.
*

*>
*

*> When system is critical, partitition function
*

*>
*

*> Z= INT(configurations)exp(-H/T),
*

*>
*

*> where INT denotes integral over all configurations, diverges.
*

*> Some book about statistical mechanics would help.
*

*>
*

*> I hope I good remember some references. In any case: Books
*

*> on classical mechanics and QFT contain typically the essentials about
*

*> variational/action principles. Books on statistical mechanics containg
*

*> the essentials about partition functions and how they are used to code
*

*> everything about thermodynamical system to partition function.
*

*>
*

snip

snip

*> No. Gravitation breaks scale invariance. G emerges when one
*

*> derives simplest action principle giving rise to Einstein equations, which
*

*> themselves follow from very simple tensorial considerations. The reason
*

*> is that curvature scalar
*

*>
*

*> INT R d^4x ,
*

*>
*

*> which is the simplest action involving metric,
*

*> has dimension length squared and must be multiplied by constant G with
*

*> dimension 1/length squared to get dimensionless quantity (I am assuming
*

*> hbar=c=1).
*

*>
*

*> I think that theoreticians have quite a lot of imagination but the simple
*

*> fact is that experimental physics demonstrates unquivocably the breaking
*

*> of scale invariance! In fact, the notion of Higgs relies on breaking of
*

*> scale invariance by Higgs vacuum expectation: Yang Mills action is scale
*

*> invariant as is also Maxwell action. The approximate scale and conformal
*

*> invariance at high energy limit of, say QCD, provides very strong
*

*> tool to understand the dynamics of quarks and is routinely used.
*

*>
*

*> Note that in standard model one could imagine the possibility that
*

*> Higgs expectation depends on spacetime point so that elementary particle
*

*> mass scale would be different on different parts of the world. There is
*

*> however no experimental support for this.
*

Ah, but I am arguing that since our observations are restricted by

logical chaining, we can not directly observe such. I will try to

explain my self mor ein the future... :-)

*> > [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.
*

*> >
*

*> > Umm, I see no Fundamental meter stick, I see an undecidable infinity of
*

*> > them. Could we discuss the meaning of "CP_2"?
*

*> >
*

*> > 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.
*

*> >
*

*> > It's not string theory's only flaw, as I explained above! But, to me,
*

*> > there is really little difference between "spontaneous compactification" and
*

*> > "spontaneous symmetry breaking"! The former is just a special case of the
*

*> > latter.
*

*>
*

*> Spontaneous compactification involves also the assumption that topology of
*

*> 10-dimensional Minkowski space somehow spontaneously compactifies in
*

*> 10-4 =6 dimensions. Infinite R^6 would become Calabi-Yau with finite size.
*

*> This is something which I cannot eat!
*

Umm, it might not taste so bad! :-) We do need to talk about this more!

*> From one of the earlier postings
*

*> of yours, I learned that string model people are finally beginning to
*

*> realize that they must return to the roots and consider the basic
*

*> philosophical questions and that the notion of spontaneous
*

*> compactification is one of these questions. I learned that they even had a
*

*> meeting in which they pondered what to do next: quite a symptomatic
*

*> situation! Only two years ago there there was media campaing about second
*

*> string revolution!
*

Have you been reading about M-Theory?

*> I understand very little of the concepts involved in "Configuration
*

*> > space geometry" of M^4+xCP_2. :-( M^4 is a Minkowski spacetime manifold and
*

*> > CP_2 is a complex projective surface, right? I say that there as at least
*

*> > #Reals of locally indistinguishable M^4 and CP_2;s! Are you familiar with
*

*> > the Poincare conjecture in topology concerning 3-dimensional manifolds?
*

*> >
*

*> Your are right about identification of M^4 and CP_2.
*

*> The point is that M^4 is completely fixed by the requirement of
*

*> Poincare invariance of metric. CP_2 is also fized by the requirement that
*

*> color symmetries SU3 acts as its isometries.
*

*>
*

*> Does Poincare conjecture say that homology
*

*> of 3-sphere fixes the topology of 3-sphere uniquely?
*

Here are some links about the Poincare Conjecture:

http://www.math.unl.edu/~mbritten/ldt/poincare.html

http://www.maths.warwick.ac.uk/~cpr/ftp/algorithms.ps

http://www-sal.cs.uiuc.edu/~edels/P-27.ps

I am thinking that there are an undesidable infinity of 3-dimensional

manifolds that differ in some way. I think that what we call "the

Universe experiencing itself" is the "exploration" of each 3-manifold to

find a way to smothly map it to all others. We can think of an act of

observation as an action of the Universe to compare one 3-manifold to

another. I have not proof of this idea other than an intuition... :-)

snip

*> > [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.
*

[SPK]

*> > I avoid this problem by making space-time (your M^4) a construction
*

*> > generated by the interactions of quantum mechanical Local Systems, as per
*

*> > Hitoshi's model... I, unfortunately do not understand TGD well enough to be
*

*> > sure that it is not adversely affected. But, if TGD is anything like
*

*> > Wheeler's spacetime foam ideas, I think that it is actually well modeled in
*

*> > the LS theory in my thinking. :-)
*

[MP]

*> In GRT nontrivial topology of spacetime emerges in Planck length scale.
*

*> In TGD nontrivial topology is present in all length scales (by the way
*

*> this means scale invariance!: Kahler action is
*

*> Maxwell action whose scale invariance is broken only by CP_2 size!)
*

Umm, but I still do not understand how this "size" is derived. :-(

*> > [MP]
*

*> > Some additional comments.
*

*> > You are right about mass spectrum in the following sense. Super
*

*> > Virasoro
*

*> > invariance implies universal mass squared spectrum of form
*

*> >
*

*> > Could you explain "Super Virasoro invariance"? What is being considered
*

*> > as "rigid" under the transformation involved?
*

*> >
*

*>
*

*> Super Virasoro is same as Super conformal. Virasoro probably invented the
*

*> conformal algebra in context of hadronic string models 25 years ago or so.
*

*> Conformal transformations preserve angles between vectors of complex
*

*> plane. This symmetry is extended to super conformal/Virasoro symmetry.
*

*> Besides ordinary conformal transformations also super conformal
*

*> transformations which transform bosons into fermions and vice versa and
*

*> which are 'square roots' of conformal transformations.
*

Is it true that supersymmetry transformations of a particle result in

displacement in space-time?

*> The notion of symmetry is actually generalized. This means that
*

*> the Lie algebra of infitesimal conformal transformations
*

*> is extended by super conformal generators, which anticommute to
*

*> conformal generators.
*

*>
*

*> I recommend some book on conformal field theories or on string models.
*

I have a very hard time with the math! I am a philosopher not a

mathematician... But, I will try harder... :-)

*> > 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.
*

*> >
*

*> > But note that there are as many primes as there are Real numbers! (I
*

*> > don't know if this is a proven mathematical fact!)
*

*>
*

*> Probably you mean that the number p-adics is same as reals?
*

*> The number of integers allowing infinite integers is same as reals.
*

*> This is obvious from the pinary expansion:
*

*>
*

*> x= SUM x_np^n interpreted as p-adic number
*

*> can be mapped to a finite real number by canonical identification
*

*> inverting p^n to p^(-n) in the sum formula. The arrays giving the pinary
*

*> digits of p-adic number and its real image are same.
*

*>
*

*> I would say that the number of finite primes is that of integers: is this
*

*> what you mean? If one allows infinite primes as I do, then the
*

*> number of primes is very probably larger than the number of reals.
*

Yes, my first thought was mistaken! Umm, these infinite primes, are

they like the cardinals in the set of Surreal numbers that Conway wrote

about?

snip

[MP]

*> > 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.
*

*> >
*

*> > Well, I don't understand that! :-( I forget what compactness is... I
*

*> > assume that R is the Reals?
*

*>
*

*> Yes.
*

Umm, I am learning more about this in a paper by Yau..

*> > The "known" properties of U(1) worry me. :-( The thinking involving
*

*> > groups still contains the vestiges of classical assumptions! Weyl himself
*

*> > discusses how this is wrong in his Space-Time-Matter book! The properties of
*

*> > observables or entities, particle or otherwise, are not "a priori", they are
*

*> > given only in relation to the interactions involved. Mach Principle has this
*

*> > notion at its root! The reductionistic attitude of material monism is the
*

*> > problem!
*

*>
*

*> My answer is that consistency implies existence. Infinite-dimensional
*

*> physics is unique. QFT theorists have spent for more than fifty years
*

*> without being able to find physical QFT free of divergencies.
*

*> The construction of string models also demonstrated this: string theory
*

*> was almost unique!
*

*>
*

*> In TGD same occurs.
*

*>
*

*> Finite-dimensional groups provide excellent example for my phisophy.
*

*> Finite-dimensional groups are classified and listed. Cartan was one of the
*

*> persons involved. If one is able to identify the correct axioms
*

*> for physical theory one can also give list of physical theories. Even
*

*> better, this list could contain only single item! I believe that the
*

*> axioms making possible to achieve this are contained in TGD approach(;-).
*

*>
*

*> Conformal quantum field theories are also a good example: they can be more
*

*> or less listed.
*

Umm, "listed"; what do you mean? The finiteness of these groups is, to

me, only an indication of the finiteness of a given observation. It does

not imply that the set (or powerset) of possible observations is finite

or even enumerable. There is a bubtle point here that I need to explain

better, but it requires that we can communicate about "computational"

issues... :-)

*> > Can we not have a complex valued coupling such that one can only observe
*

*> > the square resultant?
*

*>
*

*> I think that unitary would be problem. Certainly the dropping of i
*

*> from covariant derivative partial_i +iA_i would make this operator
*

*> nonhermitian. But I am not sure whether I am talking about right thing.
*

*> What is clear is that this does not work for electromagnetism: fine
*

*> structure constant would become negative.
*

Unitarity is suspect in my thinking! We assume that all possible

observable states are "available", like the faces on a dice cube. The

actuality of a given entity is a finite sample of the totality, which is

infinite. Unitarity is an idealization used to "patch over" the holes

that this causes. I think that we should discuss unitarity more in

detail! I may be very wrong...

*> > My friend Paul and I have been discussing the notion
*

*> > that we only observe 1/2 of the EMF group, this derives from Dirac or
*

*> > Pauli's ideas of how magnetic monopoles and electric charges are
*

*> > transformations of each other that involve a conjugate to M^4 (where the
*

*> > time coordinate is considered as imaginary and the spatial dimensions as
*

*> > real), M^4* (having 3 real dimensions of "time" and one imaginary
*

*> > dimension of space). I think that these are labeled as M^3,i1 and M^1,i3.
*

*> > The spacetime inside a black hole has this property, we believe, as spatial
*

*> > motions are constrained toward the singularity and the time-like "motions"
*

*> > are not.
*

*> > Paul has been working on this for a while but has not given me any paper
*

*> > to publish for him. :=( He is very shy but brilliant.
*

*> >
*

*> > snip
*

[MP]

*> > Sorry. I could not follow you idea. I got lost somewhere around
*

*> > P_o=N^pi.
*

*> >
*

*> > The Powerset P_o is the set of all subsets of the Universe U, U is
*

*> > included. (which generates a Russellerian paradox for those that only see
*

*> > the world as binary!) Thus P_o equals N to the power of p_i where p_i are
*

*> > the individual subsets of U. We use N instead of 2, since it is assumed that
*

*> > binary relations are merely a special case of interactions in general, and
*

*> > qualia are defined only by interactions, we say that free particles have no
*

*> > qualities! Interactions, I believe, are modelable by powerset inclusion. I
*

*> > will try explain this more in detail in the future.
*

*> > Did you understand the proposal that the cardinality of U, #U, is
*

*> > greater than the Reals or the algebraic functionals, or any other a priori
*

*> > enumerational scheme?
*

*> >
*

*> I think I understood the latter. Power set idea resembles construction of
*

*> infinite primes, which reduces repeated second quantization. Very roughly,
*

*> infinite primes at given level of infinity correspond to states of super
*

*> symmetric quantum field theory. The state basis constructed at given
*

*> level of infinity correspond to power set for the state basis constructed
*

*> at previous level. One forms power set and power set of this and so on...
*

*> Ad infinitum. One just quantizes again and again. First quantization,
*

*> second quantization, third quantization,....such that many particle
*

*> states of given quantization become single particle states of
*

*> next quantization.
*

This is very interesting. Finkelstein has talked about levels of

quantization... Look at how Pratt uses the powerset idea.

*> > > 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"!
*

[MP]

*> > 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.
*

[SPK]

*> > Interactions are always relative to some finite basis, thus a discrete
*

*> > (?) scaling invariance group for each poset, but these are not "static". The
*

*> > construction of generative aspect of observation implies an action of
*

*> > asymptotic approximation, like the notion of a limit. We say with the
*

*> > mystics that we seek after the Grail of Perfection forever. It is the Quest
*

*> > that defines us!
*

*> > Anyhow, the discrete nature of spectra, attributed to the finiteness of
*

*> > the Planck constant, is not, I am claiming Universal! I say that we just
*

*> > happen to have the common experience of a finite space-time with a
*

*> > particular value of minimum action. It should never be assumed that this is
*

*> > EVERYTHING! That line of thinking is the first mistake made by people about
*

*> > our world! Just because an individual can not experience or communicate
*

*> > about something does not mean that such do not "exist". Existence is
*

*> > independent of observation. Actuality, now that is a different story
*

*> >altogether. :-)
*

*>You are of course right. The spectra seem to be same in
*

*>the known world and theory must explain this. Certainly there is much more
*

*>involved: for instance, TGD predicts huge number of exotic particles not
*

*>yet observed and entire hierarchy of p-adic mass scales.
*

If my suspicion is correct, the existence of these particles is

necessitated by the fact that the Universe is infinite. Once we realize

that a given observation is always finite, we see that the Obler's

paradox is a "red herring"!

(http://madsci.wustl.edu/posts/archives/dec96/844241598.Ph.r.html)

Onward to the Unknown,

Stephen

**Next message:**Matti Pitkanen: "[time 442] Re: [time 427] Re: Conformal Invariance and related notions"**Previous message:**Matti Pitkanen: "[time 440] Re: [time 439] Re: [time 437] Dissipation"**In reply to:**Stephen P. King: "[time 439] Re: [time 437] Dissipation"**Next in thread:**Matti Pitkanen: "[time 442] Re: [time 427] Re: Conformal Invariance and related notions"

*
This archive was generated by hypermail 2.0b3
on Sun Oct 17 1999 - 22:36:55 JST
*