**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Wed, 14 Jul 1999 16:48:02 +0900*

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Dear All,

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Best wishes,

Hitoshi

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From: Matti Pitkanen <matpitka@pcu.helsinki.fi>

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To: "Stephen P. King" <stephenk1@home.com>

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Subject: Re: [time 442] Re: [time 427] Re: Conformal Invariance and

related

notions

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On Wed, 14 Jul 1999, Stephen P. King wrote:

*> Dear Matti,
*

*>
*

*> Matti Pitkanen wrote:
*

*> >
*

*> > On Tue, 13 Jul 1999, Stephen P. King wrote:
*

*> >
*

*> > > 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.

*>
*

*> Question: How could we think of these algebraic extensions as
*

*> 4-dimensional spaces? Do these act like co-ordinates with which to
*

*> locate objects in them or do they describe the behaviours of the
*

objects

*> or both or other? I am not understanding their value with regard
*

to the

*> construction of models of space-time. The answer to the question
*

that I

*> have about how it is that events are partitioned into light cone
*

*> structures is eluding me. :-(
*

All dimensions for algebraic extensions are possible. 4-dimensional

are special in the sense that the existence of the square root

for *'real'* p-adic number leads to four dimensional extension:

Z= x+iy+ sqrt(p)(u+iv) for p mod 4=3 and p>2.

For p=2 this algebraic extension is 8-dimensional.

This extension is requirement by basic formulas of QM:

for instance, Glebsc-Gordan coefficients contain typically square

roots.

*>
*

*> snip
*

*> [MP]
*

*> > > > 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.
*

*>
*

*> How is this extremum computed by Nature? Against what standard
*

can be

*> measure its value? To say that a value is an absolute implies that
*

there

*> is no other possibility and this caries a very high ontological
*

price!

This question makes sense in Universe as Computer philosophy

in style of artificial intelligence. If this action is computed

in Nature, certainly approximately, it is computed by the Big

Physicists

near the top of the hierarchy of conscious intelligences predicted

by

TGD based model of self and binding (see separate posting about

this).

The measurement standard for Kahler action is provided by so

called

CP_2 extremals for which action is negative and quantized: I do not

remember the precise value just now.

[CP_2 extremals are what black holes are for GRT and provide a model

of

elementary particle. Amusingly, the semiclassical quantization of

CP_2

extremals leads to Super Virasoro algebra of string models!]

Absolute minima can be degenerate and they are: one can glue

cognitive

spacetime sheets to given absolute minimum to get new absolute

minima

with same action. This degeneracy is absolute crucial for the

TGD based model of cognition. Degeneracy forces the generalization

of concept of 3-surface: by allowing also 3-surfaces containing

several disjoint 3-surfaces with time like separations one can

get rid of degeneracy and achieve determinism in generalized sense.

Amusingly, absolute minimum property implies duality in the

following

sense. Very rouhgly:

a) One can fix configuration space geometry from

symmetric principles almost uniquely. Matrix elements of metric

can be expressed in terms of *magnetic fluxes* for Kaehler field.

b) Other construction starts from absolute minima of Kaehler

action and and now appropriate quantities appearing in metric

are derivable from the action and are very much like *electric

fluxes*.

Magnetic=electric duality guarantees that the two constructions

yield the

same metric. This is the TGD counterpart of duality in string models

and even more the counterpart of duality in Euclidian YM theories

where absolute minima of Euclidian action correspond to selfdual

field configurations.

*> > > > 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.
*

*>
*

*> This is good for a mathematician, but not for a philosopher. What
*

does

*> it mean experienciably? What does the Maxwell action mean? You say
*

that

*> the action is stationary and extremum and I ask: according to what
*

*> standard? Perhaps I am appealing to visual and mechanistic lines
*

of

*> thinking, but, still, how is it that these extrema are actualized?
*

Extremum principles state that the change of action vanishes under

small variations to lowest order: variational derivatives of the

action vanish. This allows saddle points, minima, maxima.

For absolute minimum principle this is not enough: one must find

local minima and compare them. Of course, there is no mechanical

procedure for achieving this.

In TGD framework the duality about which I told above might solve

the

problem to extend that one could calculate the absolute minimum of

Kahler action associated given 3-surface X^3. Construction of

4-surfaces

X ^4(X^3) is probably totally outside of intelligences in this

mundane level we are living(;-). The picture is actually very much

analogous to the electric-magnetic duality of Euclidian Yang-Mills

theories which solves the absolute minimization of Yang Mills action

in

terms of instantons.

Standard physicist does not even try to say anything about the

meaning

of Maxwell action or any other action but is satisfied with the fact

that these action principles follow from rather general simplicity

and symmetry arguments.

In TGD framework Kahler action is desperately needed to assign

unique spacetime surface to given 3-surface: otherwise one could

not realize General Coordinate Invariance in 4-dimensional form

in the space of 3-surfaces.

Frieden's approach, modified suitably in TGD framework, provides

interpretation for the exponent of the negative of the

absolute minimum of Kahler action. The hypothesis

is that this number represents approximately the number

of degenerate absolute minima of Kaehler action associated with

given

3-surface. This number is measure for the cognitive resources

of 3-surface in TGD inspired theory of consciousness.

This hypothesis leads to the understanding of quantum criticality

but is still an unproven hypothesis.

*> Does the Universe compute them or are they somehow "out there"
*

like

*> entries in a book to be looked up when needed? The difficulty
*

involved

*> in finding the n-body Lagrangian is my case in point! How does the
*

*> n-body system "know" what configuration to take? Integration is
*

*> impossible, as Prigogine points out many times!
*

I do not see why universe should be able to calculate Kahler action.

One could however think the possibility that the p-adic evolution

of the universe (with infinite p-adic primes forced by

infinite size of universe) would make ultimately make possible

arbitrary precise modelling of universe and this would involve

actual calculation of Kaehler action.

In fact, one could quite well construct Monte Carlo problam trying

to find

absolute minima of Kaehler action already now! I would not however

try

this: something like 10^8 lattice points in in 8-dimensional space:

ten

points for one coordinate! Does not look very promising!

*>
*

*> > > > 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.

*>
*

*> I have read many of these books and they, at best, beg the
*

question!

*> But, I will read them again! The problem I have is with the
*

*> idealizations and hand waving assumptions that are used in
*

statistics.

*> The fundamental assumptions are really what I am interested in
*

*> discussing! :-)
*

*>
*

In quantum theory key object is vacuum functional: in case of

TGD vacuum functional in the space of 3-surfaces.

In TGD exponent of Kahler function (absolute minimum

of Kaehler action) is unique vacuum functional. The reason

is that only for this vacuum functional integration

over configuration space is well defined. Gaussian determinant

coming from perturbative integration around maximum of Kaehler

function

cancels the poorly defined metric determinant. There

are also other divergence cancellation mechanisms involved.

*> > > 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).
*

*>
*

*> Weyl's action is very simple also and it makes a lot of sense
*

since it

*> makes a connection between the curvature of the subuniverse of a
*

*> particle and its size, if I remember correctly. This is why I am
*

so

*> enthusiastic about his ideas.
*

There are action principles which are quadratic in curvature tensor:

these actions are automatically dimensionless. I think however that

curvature scalar term must be generated somehow to yield

gravitation.

*>
*

*> > > > 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.

*>
*

*> The Higgs mechanism is an abstract model constructed to try to
*

explain

*> the observation of finite mass in particles. I am asking if the
*

breaking

*> of scale invariance is related to how observation takes place. I
*

agree

*> with you that scale invariance is broken, I am merely trying to
*

discuss

*> the notion of scale invariant geometry at the philosophical level,
*

and

*> deal with applications later.
*

Breaking of scale invariance, and more generally breaking of

symmetries,

is very probably related to how observations are made:

moments of consciousness=quantum jumps!

TGD does not apply Higgs mechanism to elementary particle mass

calculations but quite recently I found that the mere

hypothesis that quantum jump can be regarded as a quantum

measurement

implies that in each quantum jump *localization in zero modes*.

This follows from the general structure of configuration space

spinor fields.

This in turn implies TGD counterpart of Higgs mechanism and symmetry

breaking caused by quantum jumps: system approaches quantum jump by

quantum jump to the minimum of effective potential and if there are

several minima, selects some of them so that symmetry breaking

typically results. Observation implies symmetry breaking.

*>
*

*> > > > > [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"?
*

*> >
*

*> > CP_2 radius is the metric stick. One can assign to it arbitrary
*

*> > value of length: but this does not affect physic since there is
*

no other

*> > fundamental length scale to compare. One can quite well put
*

value of CP2

*> > radius equal to one or denote it just by R. All other
*

dimensional

*> > units (every dimension is expressible as some power of length
*

for

*> > hbar=c=1) is expressible using some power of CP_2 size.
*

*> > Elementary particle masses are expressible in terms of inverse
*

*> > of CP_2 radius, etc...
*

*>
*

*> Umm, if the CP_2 radius is the metric stick, is it considered to
*

be

*> "separate" from the objects that it is used to measure?
*

I am not quite sure what you mean. Every physical prediction for any

dynamical quantity is in principle expressible in terms of CP_2

radius.

*> I would think
*

*> so. This idea, as I interpret what you write, is what I am trying
*

to

*> discuss with regards to Weyl's notion. I am thinking that each
*

poset of

*> observations that make up an observer, or more generally, any
*

system

*> that can be considered to be able to make irreversible
*

measurements.

*> Here, Hitoshi's Local Systems are as good example. Each LS has
*

its own

*> clocking mechanism that gives it its own measure of time. Time is
*

not

*> considered to be something external to the LS. This idea of a
*

clock

*> associated with each LS can be generalized to the notion of a unit
*

of

*> length associated with each LS, with the relationship between the
*

"time"

*> and "interval" of each LS being something like the \gamma of
*

relativity.

*> Since each LS has its own standards with which to measure other
*

LSs, we

*> have a system that looks at a simple level like a sophistry! But,
*

it

*> escapes that criticism by if we consider the alternatives. The
*

notions

*> introduced by Newton et al that there exist a priori absolute
*

standards

*> of time and length is, to quote Dirac (?), "not even wrong!"
*

Weyl's

*> discussion of this in his Space-Time-Matter book is very
*

illuminating!

Absolute standard if or course nonsensical unless it is based on

geometry

itself. The length of CP_2 geodesic is however based on geometry and

provides an absolute standard.

This is very important point actually. One of my criticisms against

string model is the introduction of gravitational constant G

as a dimensional constant multiplying action which is area of

string world sheet. This is indeed introducing a priori absolute

standard in completely ad hoc manner: G is not related to any

geometrical structure. By the way, curvature scalar

would yield dimensionless action for strings but this action is

trivial: theory would reduce to topological field theory.

The next dimension is d=4 (TGD) and Maxwell action is the simplest

dimensionless action in this case.

*> But, I need to discuss how the LS model can generate the
*

illusion that

*> we "all live in one and the same space-time". The idea that I have
*

is

*> that the sets of possible measurements that are associated with
*

each LS

*> allow for the possibility of "overlap" and "underlap" among the
*

LSs.

*> This notion is described metaphorically by saying that "those
*

aspects

*> that we can agree upon as being "real" are really only those
*

aspects

*> that are common to the LSs that 'are' us". This idea that we
*

construct a

*> common reality by interacting with each other was critiqued by
*

Robert

*> Fung, but his argument is incorrect. The fact is that we can only
*

*> communicate meaningfully about events that do not entail logical
*

*> conflicts with each other. An example of this is to consider why
*

we do

*> not experience closed time-like loops.
*

*>
*

Yes. I understand your basic philosophy. I can only provide my own

versions of the problem. I see the solution of

'overlapping' in the possibility of quantum entanglement between

selves.

The members in the infinite hierarchy of more and more

intelligent selfs can get quantum entanglement mutually:

entanglement

by enlightment. This explains why animals like us can have sciences

and

moral philosophy(;-). Summation of experiences of subselves

within self to abstraction experienced by self is what makes

conceptual

thinking and categorization possible.

You could translate to LS by replacing self with LS.

Self as subsystem able to remain unentangled indeed provides

fundamental

definition for 'observer'.

*> [MP]
*

*> > > > 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!
*

*> [SPK]
*

*> > > Umm, it might not taste so bad! :-) We do need to talk
*

about this more!

*> > >
*

*> >
*

*> > To me it tastes really bad! For instance, quantum field theory
*

limit

*> > is nonrenormalizable because imbedding space is dynamical.
*

*>
*

*> Please, the problem is caused by the continued insistence that
*

the

*> field quanta are embedded in a unique infinitely integrable
*

space-time.

*> By using the notion that each quantum particle (LS) has its own
*

*> space-time associated we can easily avoid the problem.
*

I understand the point. In TGD manysheetedness corresponds to this

idea.

But nonrenormalizability is related to spacetime dimension only and

this is why *any* theory in which *dynamical* spacetime with

dimension

higher than four, appears, is bound to be nonrenormalizable.

*> The prediction of
*

*> a cosmological constant that is 10^123 times that observed is a
*

strong

*> indication that something is very wrong with assuming a single
*

unique

*> space-time for all.
*

Yes. I completely agree. I have probably told about how manysheeted

spacetime explains the observations suggesting small cosmological

constant. The cosmological constant problem is problem of General

Relativity: this is the point. It is not accident that non-dynamical

imbedding space of TGD makes impossible generation of cosmological

constant for spacetime surfaces.

*> The idea that what an infinite R^6 for one
*

*> observation is a finite Calabi-Yau (manifold) for another is
*

really not

*> so far fetched once we over come our prejudice that our
*

measurement is

*> absolute. The point is that absolute measures or standards are
*

*> idealization at best and we should consider them as harmful to a
*

*> physical theory. The only standards that should be postulated are
*

those

*> associated with a finite set of measurements that could be made
*

*> therewith. This is the notion that Mach advanced and one that
*

Smolin and

*> Schommers put to good use.
*

This philosophy relies on subjective existence as the only 'real'

existence. I am Platonist and believe that subjective existence,

in particular ours(!), gives only shadows about that-which-is.

*> About string theory: The insistence that the string's 10 or
*

whatever

*> dimensional space-time collapses somehow to 3+1 space-time is an
*

*> exercise in futility since it is assumed that such is absolute and
*

*> unique. Icould be done, but for only a single string! We need to
*

*> relativize everything! This is why I really like the work that
*

David

*> Finkelstein is doing! Hitoshi's LS can be infinite on the
*

"inside", I

*> think, and still look like infinitesimal point particles to
*

another

*> LS...
*

Unless consistency implies existence philosphy is at work and

excludes

all these possibilities: all field theories in d>4 spacetimes

are nonrenormalizable... I see internal consistency as fundamental

principle.

*>
*

*> > > > 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?
*

*> >
*

*> > Not much. I have heard a couple of seminars and I was surprised
*

that

*> > they are just playing with formulas: great principles are
*

lacking.

*> > For instance, the concept of p-brane looks for me something what
*

*> > theoretician can produce at the moment of extreme despair when
*

*> > nothing works nicely(;-).
*

*>
*

*> They are mainly worried about figuring out ways to extend their
*

grants!

*> :-(
*

*>
*

*> > > > 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.
*

*>
*

*> Well, why do we "require" Poincare invariance of "the" metric?
*

This is

*> a perpetuation of the error!
*

As a physicists I would answer: because physics is Poincare

invariant.

As a mathematician I would answer:

a) Because the boundary of the

4-dimensional future lightcone (moment of big bang) has miraculous

mathematical properties. The conformal invariance of 3-manifolds

generalizes and this is what makes possible Super Virasoro

invariance and

related symmetries and the generalization of string models

by replacing strings with 4-surfaces.

b) Configuration space Riemann connection exists only because the

metric is extremely symmetric: all points of configuration space

for given values of zero modes are metrically equivalent: physics

does not depend on configuration space point. This is achieved

only by choosing imbedding space to have similar property. M^4_+ and

CP_2 are indeed symmetric spaces.

By General Coordinate Invariance construction of configuration

space geometry reduces to deltaM^4_+ x CP_2, 3-surfaces on the

boundary

of imbedding space and generalized conformal invariance

can be indeed realized.

String modelists discovered somehing

analogous ten years later: Maldacena's conjecture states that the

construction of string models or whatever-they-call-it reduces

to the boundary of 10- or whatever-dimensional space.

*> The Poincare invariance only applies
*

*> individually to the poset of observations of an LS, not to all
*

*> observations in general! This again is a logical derivation from
*

the

*> incorrect notion that all observers (posets) exist in one single
*

*> space-time. WE DO NOT! We just have subsets or partitions of our
*

posets

*> in common, they overlap, and we only can communicate to each other
*

about

*> them.
*

You are correct. LS identified as spacetime sheets do not allow

Poincare symmetries as isometries. It is IMBEDDING SPACE which

allows Poincare invariance as symmetries. It is just this

what makes it possible to identify elementary particles

as CP_2 extremals which have SU(3) rather than Poincare group

as isometries and whose metric has Euclidian signature.

In standard General Relativity this kind of identification would not

be possible.

*> Remember logical entailment is part and parcel with causality!
*

Those

*> events that are causally ordered in a given LS's Minkowski
*

structure are

*> defined relative to the partition that is logically consistent.
*

Pratt

*> argues that Logic "goes backwards" and physical effects go forward
*

in

*> time, this is the mechanism that "choices" what is observed! There
*

are

*> limits to free will. We are free to chose from the "menu" that
*

Nature

*> presents us but we are not free to write it directly. But we can
*

*> influence what is writen by our choices since we can modify Nature
*

to a

*> finite degree! :-)
*

As I notices LS= spacetime sheet in imbedding space identification

allows even Euclidian metric for LS's. About the menu

represented by eigenstates of density matrix I agree.

*>
*

*> > > > 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... :-)

*> > >
*

*> > Perhaps I should add 'conscious comparison' to the list of
*

*> > thinkabouts of TGD inspired theoryofcs.
*

*> Your paper about this is very interesting! :-)
*

*>
*

*> > > 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!)

*> [SPK]
*

*> > > Umm, but I still do not understand how this "size" is
*

derived. :-(

*> [MP]
*

*> > CP_2 size is not derived, it is fundamental unit. Elementary
*

particle

*> > size is derived and also Planck length. The prediction is that
*

*> > Planck length is about 10^(-4) CP2 sizes.
*

*>
*

*> But, I am asking: "How many such "fundamental units" are possible
*

to

*> exist. Here I am talking about ontology, not experienciability!
*

This is

*> equivalent to asking if there exist a universe with a slightly
*

different

*> "Planck length". I fail to see how this prediction could be
*

"wrong". :-(

One could of course formally consider superposition of parallel

universes

with varying CP_2 radius: as such this kind of generalization would

be

interesting. Rather, one should make imbedding space metric

dynamical. In

this case however QFT limit would be nonrenormalizable.

Neither me nor string modellists knows of any manner of combining

dynamical imbedding space with dynamical spacetime surface.

But who knows..

*>
*

*> > Of course, in reality I have deduced the values of CP_2 size in
*

terms

*> > of Planck length from elementary particle mass calculations.
*

*> > The assumption that electron corresponds to Mersenne prime
*

*> > M_127=2^127-1 plus p-adic thermodynamics for electron mass
*

squared

*> > fixes CP2 size (mass unit is essentially 1/CP_2 size).
*

*>
*

*> My point exactly! We need to be able to predict the masses from
*

basic

*> Principles or, at least show why they are observed to have such
*

observed

*> values based on observations of unrelated quatities. The relation
*

of the

*> electron mass to the Mersenne prime is what I like to see! :-) I
*

guess

*> that I am very Eddingtonian. :-)
*

*>
*

Mersenne primes seem indeed to correspond to important

physical mass/length scales. M_89 corresponds to intermediate

gauge bosons and M_107 to hadrons. After M_127 there is huge

gap: the next Mersenne prime corresponds to completely

superastronomical

length scale.

*> > This size leads to sensical value for the tension of cosmic
*

strings

*> > allowing to construct model of galaxies and dark matter based
*

*> > on cosmic strings. If CP_2 size where of order Planck length,
*

cosmic

*> > strings would be by factor 10^8 too heavy.
*

*>
*

*> We really have not need to model "dark matter"! Such is a fantasy
*

*> created by people who do not wish to consider that most matter in
*

the

*> visible universe is electrically charged (plasma) and thus do not
*

wish

*> to be bothered with electrical and magnetic terms in their
*

cosmological

*> toy theories! Eric Lerner's discussion of how plasma physics
*

explains

*> the behaviour of cosmological objects ranging from solar systems
*

to

*> quasars to galactic clusters is very illustrative. The main
*

problem of

*> distribution of angular momentum in a galaxy is easily solved. He
*

shown

*> computer graphical solutions using their equations and they are
*

*> stunningly similar to real pictures of galaxies, and their
*

mathematical

*> model did not include gravity! See The Big Bang Never Happend...
*

*>
*

Interesting claim. Almost all imaginable and even unimaginable

explanations for dark matter have been proposed. I do not whether

Lerner's

theory explains also other problems of cosmology such as quasars

and

gamma ray busters.

*> > 10^(-4) Planck mass has been realized to
*

*> > be a fundamental mass scale also in string models. They try to
*

produce

*> > it by tricks in eleventh dimension (size of circle in that
*

*> > dimension would be of order CP_2 size). One cannot get rid of
*

Planck

*> > length in string models since string tension determines directly
*

*> > gravitational constant
*

*>
*

*> How is it that a geometical "object" can have "tension"? I know
*

that

*> this is a very silly question, but really! We have gone a long way
*

from

*> models of guitar strings to models of abstractions that can't even
*

be

*> observed in principle! Is physics of metaphysics? What keeps it
*

from

*> collapsing? Zero point energy?
*

Cosmic strings are spacetimes of form X^2xS^2: X^2 is

orbit of string in M^4_+ (minimal surface) and S^2 is

homologically nontrivial geodesic sphere of CP_2. Cosmic

strings are one of the simplest extremals (not absolute minima)

of Kahler action.

String tension is simply energy per unit length:

E=T L, where L is length of string and T is string tension.

For TGD string string tension is essentially

T=about 1/R^2, R geodesic length of CP_2. At length

of R there is 10^(-4) masses of blackhole with radius R.

Strings are extremelymassive objects.

Strings have positive Kaehler action. Absolute minimization

of Kaehler action requires however action to be negative. Free

cosmic

strings must be unstanble and

appear only as intermediate states: very early TGD:eish cosmology

can be visualized as soup of cosmic strings decaying

to elementary particles within 10^4 Planck times.

Cosmic strings could however topologically condense on spacetime

surface and generate Kahler electric field whose action cancels

the magnetic action: this makes the string stable. The hypothesis

is that quasars and galaxies correspond to this kind of objects.

They gather around themselves ordinary matter and also decay to

ordinary matter.

This model explains quite many basic numbers. The sizes of recent

galaxies are predicted correctly at order of magnitude level.

The rate of increase for galactic nuclei is predicted correctly.

Energy production rate by quasars and gamma ray bursters is

predicted

correctly. The prediction is that the energy from cosmic strings

is emitted as two jets: this has been discovered experimentally

quite recently. The rotation velocities of distant starts around

galaxies

are predicted correctly.

I would say that cosmic strings are for the modelling of galaxies

what Schwarschild metric is for the modelling of stars.

*>
*

*> > > > > [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?
*

*> >
*

*> > For Super Poincare The anticommutator of two infinitesimal
*

*> > supersymmetries is infinitesimal translation.
*

*>
*

*> Could you elaborate? Does this imply that the movement of an
*

object in

*> space is generated by the "anticommutator's" chance in state? What
*

*> "causes" this?
*

Super symmetry is purely *algebraic symmetry* generalizing

the theory of representations of Lie-algebras.

There are books about super geometry but I do not believe

that this the correct approach (I might be wrong). One introduces

coordinates (x,theta) where x is ordinary commuting coordinate and

theta

anticommuting spinorial coordinate. Super symmetries are

translations

in theta. Infinitesimally *very roughly* something like follows:

theta --> theta + epsilon

x--> x+ ibar(theta)*gamma^k*epsilon

One can indeed realize super symmetries as translations for spinors

in certain dimensions in which spinors can be taken to be real

(Majorana property which implies D=3,4,6,10, and is not possible

for D=8 encountered in TGD).

In TGD super symmetry is geometrized in different manner.

Lie-algebra

generators of isometries and complexified gamma matrices defining

configuration space spinor structure combine to form a

super algebra. This is all that is needed for Super Virasoro

representations: no need for super-space. Majorana property is not

possible in D=8 and this implies generalization of Super Virasoro

algebra.

Super generators are not Hermitian and carry fermion number.

*> It looks like an infinite regress of causes! We
*

*> philosophers are quite familiar with this! :-) "Its turtles all
*

the way

*> down!" I like the idea, but how is it that I feel like I can
*

decide

*> whether or not the "infinitesimal supersymmetries" exist such that
*

I can

*> move my finger. Umm, my wording is wrong! :-(
*

*>
*

*> > For Super Virasoro the anticommutator of constant
*

supersymmetries

*> > is Virasoro generator L_0 which acts as complex scaling.
*

*> >
*

*> > {Super,Super}= Lie, [Lie,Super]= Super,[Lie,Lie] =Lie
*

*> > is the general structure of Super algebra
*

*>
*

*> I, unfortunately, do not follow the braket notation... :-(
*

*>
*

[A,B]=AB-BA, {A,B}= AB+BA.

*> snip
*

*> > > 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?
*

*> > >
*

*> >
*

*> > They are *not cardinals* but integers. The great idea is extend
*

*> > the concept of *divisibility* to apply in infinite context.
*

*> > The divisors of infinite and infinite+1 are different!
*

*> > This is really something genuinely new and motivated by
*

*> > both p-adico-physical and consciousness-theoretic
*

considerations!

*>
*

*> Have you read about non-standard numbers and/or surreal numbers?
*

*>
*

I tried to read a popular book about surreals and did not

have the needed patience. It would be much easier to learn them

by reading some less popular. I have some faint ideas about

nonstandard numbers: it would be nice if someone would see the

trouble of writing from these objects to theoretical physicists.

*> snip
*

*> [MP]
*

*>
*

*> > Compact group U(1) is expressible as phase phase factors
*

*> > U=exp(i*phi). This representation is unitary since 1+1 matrix
*

*> > in question satisfies UdaggerU=1.
*

*>
*

*> And "dagger" is the transposition operation? e.g. a matrix times
*

its

*> transpose is equal to unity? Man, do I easily forget such things.
*

:-(

Transposition plus conjugation.

*>
*

*> > Noncompact R is expressible as
*

*> > exponentials U=exp(x) and UdaggerU =U^2=exp(2x) is not equal to
*

*> > 1x1 unit matrix. Therefore the norm of states is not preserved
*

*> > under the action of U: U is not Hilbert space isometry.
*

Conservation

*> > of probability however requires unitarity in QM.
*

*>
*

*> Umm, I would like to discuss this further. The notion of
*

"conservation

*> of probability" seems to tacitly assume that *all* of the
*

possibilities

*> are "available" in any given observation. The notion of
*

*> "superselection", as I understand it, has been a way to limit the
*

"size"

*> of the ensemble in order to make sence of all the linear
*

combinations

*> that the ensemble of Quantum Cats contains.
*

Super selection is related to spin and statistics: bosons and

fermions cannot appear in superposition: this has

good mathematical justification. State would not suffer mere

phase multiplication under rotation of 2*pi.

Also superpositions of quarks and leptons are assumed to not occur.

Now

color rotations not changing physics but changing the phase

of quark exclude the superpositions.

*> Umm, I am not getting my point across. Physicists have no problem
*

*> identifying time with R^1 as a parameter of the changes in what
*

they

*> think is the Universe and they go on to construct a rigid
*

4-dimensional

*> cube model of space-time, and then wonder why their model does not
*

allow

*> for something as obvious as consciousness.
*

*> When Weyl described his idea of generalizing Riemannina geometry:
*

*>
*

*> "The metric (ds^2 = \SUM g_ik dx_i dx_k (g_ik = g_ki) )
*

*>
*

*> ik
*

*> to be compared, not only at the same point, but at any two
*

arbitrary

*> separated poijnts. A true infinitesimal geometry should ...
*

recognize

*> only a principle for transfering the magnitude of a vector to an
*

*> infinitesimally close point and then, on transfer to an arbitrary
*

*> distant point, the integrability of the magnitude of a vector is
*

no more

*> to be expected than the integrability of its direction." (pg. 25
*

of The

*> Dawning of Gauge Theory by Lochlainn O'Raifeartaigh, Princeton U.
*

*> Press), he was, to me, uncovering the bias inherent in classical
*

*> thinking.
*

*> The obvious problem is that: "...the lengths of measuring rods
*

and time

*> measurements of clocks would be rescaled by the non-integrable
*

factor

*> e^e/\gamma *INT dx_nu A^nu and would therefore depend on their
*

history.

*> This is in clear contradiction with the fact that the atomic
*

spectra

*> (known very accurately at the time) depend only on the nature of
*

the

*> atoms and not their histories."
*

*> This argumant stand only if the assuption the "histories" are not
*

*> subject to quantum superposition. The use of the "kick the stone"
*

*> argument that "the fact that the atomic spectra..." and its appeal
*

to

*> experience is good for the naive realist, but not for me!
*

One must take cautiously all arguments. In fact, scale

invariance is excellent approximation also in string models

and TGD. Conformal invariance at light cone boundary generalizes

the scale invariance but not in the same sense as Weyl invariance

does. Our ideas are not so far: only our views about their

realization

are different.

*> What I am arguing is that an act of observation is an act of
*

selection

*> from a set that is infinite. This makes the mystery that Penrose
*

*> discussed even more important. Penrose points out in The Emperor's
*

New

*> Mind that the current QFT says that the observed universe is one
*

in

*> 10^123 possible. I am saying that the observed universe is one in
*

*> infinity!
*

I agree here completely: C implies E must be however taken into

account...

So how is it that we can communicate consistently with each

*> other at all? Perhaps, it is because logical entailment plays a
*

role,

*> and that the "histories" of particles are important since the
*

history of

*> interactions that lead up to a particular measurement involves the
*

*> notion of logical entailment or implication. I see this idea in
*

your

*> analysis of entanglement!
*

Yes. I believe in enlightement by entanglement and on summation of

selves. Amen.

*> The use of unitary operators to model the evolution of a quantum
*

*> mechanical system is an idealization, since it is explicitly
*

stated that

*> such systems can not be interacted with. If I can't interact with
*

a

*> system, I can make not statements whatsoever about its properties.
*

The

*> use of closed systems in classical physics and thermodynamics has
*

the

*> same flaw!
*

*>
*

You are right of course. On the other hand, the notion of self

as subsystem able to remain unentangled gives precise content for

observer

and shows the limits of this approach. Observer is now a particular

kind of state rather than something external. In principle we

should model entire universe but it is quite difficult to model God.

Even

hydrogen atom is quite a difficult thing: one usually assumes

it to be not only unentangled but also stationary: in state

of pure awareness using the terminology of consciousenss

theory(;-).

*> [MP]
*

*> > Of course one can find for R also unitary representations but
*

*> > for the representation appearing in gauge coupling the
*

representation

*> > would be nonunitary.
*

*>
*

*> We deal with this! I am wanting to work out a mathematical model
*

of how

*> logical entailment restricts the systems that can interact by
*

*> segregating their space-times acording to which can agree with
*

each

*> other. But I need help!
*

Fields provide a nonunitary representation for scalings since they

typically have definite scaling dimensions. Vector potential

has dimension -1, for instance. This means that the transformation

is of form A--> A/lambda, under scaling of coordinates

by lambda. Hence is difficult

to see how one could make scaling local gauge invariance

using standard covariant derivative construction.

One manner to deal with situation would be to consider the space

of fields. In this space one could realize covariant derivative

as hermitian operator. This would however lead from spacetime to

the space of field configurations.

*> Such "agreements" are the mutual information that is involved in
*

an act

*> of observation. Basically, we can not observe events that
*

contradict

*> what we can locally "prove". Thus the "fact" that "...the atomic
*

spectra

*> (known very accurately at the time) depend only on the nature of
*

the

*> atoms and not their histories." only reenforces my point. Since
*

*> interactions with entities that have histories that are
*

inconsistent

*> with our own causes all sorts of paradoxes, we can turn this
*

around and

*> think of it as a universal principle that restricts observations.
*

*> In this way, we can easily deal with difficulties like tachyons,
*

time

*> operators conjugate to Hamiltonian operators, Closed Time-like
*

Loops,

*> and I propose, explain why the sky is dark and cold at night when
*

the

*> Universe is really infinite.
*

*>
*

You could of course be right, but as I said, the localization

of scaling invariance might not be the correct realization:

rather, the generalization to conformal invariance might be the

correct

approach: it indeed works in case of TGD.

*> snip
*

*> [SPK]
*

*> > > > > 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!
*

*>
*

*> I go further can say that logical consistency constructs local
*

reality!

*> With the caveat, of course, that consistency is only definable up
*

to an

*> epsilonic! (I think that is how it is said.) Anyway, the
*

relationship

*> between thermodynamic entropy necessarily generated by an act of
*

*> observation that reduces the information entropy or extremizes the
*

*> negentropy, detailed in the quote from Pierce, shows us that given
*

a

*> finite system LS_i with a finite amont of available free energy,
*

LS_i

*> can only have a finite number of observations available. And thus
*

we can

*> get a space-time that looks like it has a spontaneous breaking of
*

its

*> inherent scale invariance.
*

*>
*

I cannot show that your argument is wrong. If we rely on mere

sensory

observations everything is finite. But was is logical thinking and

construction of theories and application of mathematical consistency

requirement: isn't it just observations at metalevel:

extremely powerful sensory perception which makes it possible to

choose

between theories?

*> > > > 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 subtle point here that I need
*

to explain

*> > > better, but it requires that we can communicate about
*

"computational"

*> > > issues... :-)
*

*> > >
*

*> > The theories are classified and even solved to certain degree in
*

the

*> > sense that there are recipes for correlation functions. There
*

are

*> > beatiful connections with theory of Lie groups but all this goes
*

badly

*> > over my head.
*

*>
*

*> Me too! :-( But, we do not have an understanding of what such
*

theories

*> predict in terms of what would it feel like if... This, again,
*

involves

*> computational issues. The subsets of the Universe, LSs or p-adic
*

*> subuniverses, must expend "free energy" in order to compute what
*

will

*> happend next. Subsets that have no "next" are static by definition
*

and

*> have no time or scale associated... No computation <=> no time <=>
*

no

*> observations <=> no consciousness.
*

*>
*

This free energy point of view is interesting. The ability

of self being able to remain p-adically un-entangled: what this

ability means physically. Certainly it requires macroscopic

quantum phases. I do not know whether there is connection with

thermodynamics.

Certainly selves dissipate. Strong NMP predicts that subsystems

of unentangled subsystem perform quantum jumps and this implies

that ageing is price paid for having self is dissipation. Could the

need of free energy be related to this? There should be feed

of quantum entanglement entropy to the interior of self. Something

like

this?

*> > There is misunderstanding here: I meant finite-*dimensional*
*

Lie-groups,

*> > not finite!
*

*>
*

*> Ok...
*

*>
*

*> > > > > 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.
*

*>
*

*> It would be unobservable! We can not apply the "consistency
*

implies

*> existence" only when it supports our pet theories! Tachyons are
*

*> consistent and even predicted by SR, for instance!
*

No, no! CE is much more that this. More than fifty years of futile

attempts by particle theorists to find divergence free QFT

demonstrate

this better than any rational argument.

I am very doubtful about consitency

of tachyonic world (ground state in TGD is tachyon as also in string

models but physical states have mass squared >=0).

*>
*

*> > > 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...
*

*> > >
*

*> >
*

*> > I see unitarity as a generalization of probability conservation
*

*> > to quantum theory, as one of these CE things(;-).
*

*>
*

*> CE? Oh, Consistency implies Existence...
*

*>
*

Yes.

*> 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.
*

*> > >
*

*> >
*

*> > There is analogy with Finkelstein's idea of repeated
*

*> > quantization. I see this connection as magic relationship
*

between

*> > different disciplines: infinite primes could be regarded as
*

mathematical

*> > decadence but magically, it has direct connection with basic
*

theories of

*> > physics.
*

*>
*

*> I would like to better understand this concept! :-)
*

There is chapter on 'TGD inspired theory of consciousness' at

my homepage about this.

*>
*

*> snip
*

*> [SPK]
*

*> > > 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)

*> [MP]
*

*> > The existence of masless exotics is related to the ground states
*

*> > of Super Virasoros. There are finite number of ground states. Of
*

*> > cousre, besides this every Super Virasoro representation
*

contains

*> > infinite number of very massive states with natural mass unit
*

given

*> > by 10^(-4) Planck massess: this follows from the extension of
*

*> > point like particle to 3-surface bringing in infinite number of
*

*> > 'vibrational' degrees of freedom. These particles are not seen
*

*> > in recent day accelerations but the mixing of massless states
*

*> > with them gives rise to the tiny masses (in scale of Planck
*

mass) of the

*> > observed particles.
*

*>
*

*> Umm, I don't understand the details of how this works, but it
*

sounds

*> interesting. :-) The concept of a "ground state", this is an
*

extremal or

*> minimun relating to vacua?
*

*>
*

Ground state refers to Hilbert space ground state: the calculation

of

particle masses relies on Super Virasoro invariance and p-adic

thermodynamics alone. Only the symmetries are needed. Reduction of

everything at the level of

configuration space would be quite a challenge.

Best,

MP

*> Onward to the Unknown,
*

*>
*

*> Stephen
*

*>
*

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