[time 441] Re: [time 427] Re: Conformal Invariance and related notions

Stephen P. King (stephenk1@home.com)
Tue, 13 Jul 1999 00:41:48 -0400

Dear Matti,

        I am finally able to respond. My computer is patched together enough to
write and send this...

Matti Pitkanen wrote:
> > 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?
> 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.

> > 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.
> You are right about the notion of critical temperature. I cannot
> say anything about Unruh effect because I do not know it well enough.

> 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.
> 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:


        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... :-)
> > [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.
> > 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. :-)
> 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
> > 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
> > 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"!
> > 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.
> > 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"!

Onward to the Unknown,

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