[time 427] Re: Conformal Invariance and related notions

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Jul 1999 08:58:58 +0300 (EET DST)

On Sat, 3 Jul 1999, Stephen P. King wrote:

> Dear Matti.
>
> Matti Pitkanen wrote:
>
> Here is continuation to the previous message.
>
> snip
> >       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.
>
>     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.

>
> >     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.
>
>
>     Umm, I don't understand what you are meaning here. :-(

There are books about conformal field theories. For instance Goddard and
Olive have written a classica. I hope I good remember bibliodata for it.

>
>     Most modern models of physics are constructed on a null surface, e.g. on
> a light-cone hypersurface S_null, in order to "borrow" the assumed
> invariances of it. F.

This is true in TGD. Entire construction reduces to lightcone boundary
xCP_2. Ligtcone boundary is indeed null surface. An most importantly: it
possesses conformal invariance *in real sense* , which is usually regarded
to be a property of *2-dimensional* Riemann surfaces. This does not
however mean that one would lose psychological time: the nondetermism
of Kaehler action saves the day and brings in psychological time.

Maldacena's conjecture, which came 10 years after construction of
configurations space geometry in TGD(!), states that also string models
reduces to a theory defined on the boundary
of 10-dimensional or 11-dimensional space.

> David Peat points this out as a fundamental flaw of
> all superstring theories! They assume a fixed background in which to embed
> the strings... The problem is that it is assumed that there exists only a
> single such null hypersurface!
>That is simply not true! There are an
> undecidable infinity of such since for each observation there is associated
> a space-time framing and such  includes an S_null, and there "exist" an
> undecidable infinity of possible observations.

This is a good counter argument to string models. Second counter argument
is the presence of two gravitions. String dynamics yields graviton and
the dynamics of imbedding space yields graviton. Third counterargument is
non-renormalizability of the QFT limit resulting dimension D>4 of
dynamical imbedding space. In TGD QFT limit is defined on spacetime
surface and D=4 gives hopes renormalizability and p-adicity implies
finiteness!

> 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.

>
> 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
> 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!
>
>    You lost me again. :-(  What are "alpha_K and "exp(-H/T)" and "vacuum
> functionals"? Is this critical temperature notion similar to the critical
> temperatures involved in the magnetization of a piece of iron or
> condensation of liquid water from vapor? What about the Unruh effect
> relating temperature to acceleration to gravitational field curvature?
>

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.

> >       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
> >       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 Hilbert term results in
> action and one
> obtains gravitation dynamically.
>
>     Is not the introduction of G the "cause" of the supposed symmetry
> breaking in the model since it is a "hand entered" finite physical constant?
> It is not derived from purely geometrical considerations! The rest of the
> meaning of this statement eludes me. :-( I have serious philosophical issues
> with theoreticians that think that the Universe bows to the limitations of
> their imaginations!

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.

>
> [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!

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!

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?
>
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?

> > 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.
>
>     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!)

>
> > But, I believe that
> > this "quasi-absoluteness" is connected with the way that the
> > 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
> > his intuition! :-(
>
> [MP]
> 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.

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.

> 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.

>But, nevertheless, the
> relation between p-adic valued functions and real valued functions is
> identical to the relationship between potential and actual infinities! The
> former are, piece-wise, a priori elements of U, the latter, like limits and
> asymptotic approximations, are identifiable with the "observations" of the
> subsets of U or their "quantum histories".; they do not have a priori
> ontological status. The p-adics are "experiences" they have to be
> constructed!
>
> 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.
>
>     Ok.
>
> >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.
>
>     Well, I don't understand that! :-( I forget what compactness is... I
> assume that R is the Reals?

Yes.
>
>     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.

>
>     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.

>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
>
> >     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
> > this with you. :-)
>
> 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.

> >     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.

>     Is the world in a state of expansion or is our common Knowledge
> expanding, or are these just complementary aspects? :-)
>
> > 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"! :-)
>
> Onward to the Unknown,
>
> Stephen
>
Best,
MP

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