# [time 445] Fw: BOUNCE time: Message too long (>40000)

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

Dear All,

Since Matti's recent post seems over 40KB, the majordomo program
bounced it. I changed the default limit of the size to 100KB. I hope
this forwarding will be accepted by the program!

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

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
> > > > 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

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
> > > > > > "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

> 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
> > >
> >
> > 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,

> 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
> > > > meeting in which they pondered what to do next: quite a
symptomatic
> > > > situation! Only two years ago there there was media
> > > > string revolution!
> > >
> >
> > 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
> 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
represented by eigenstates of density matrix I agree.

>
> > > > Does Poincare conjecture say that homology
> > > > of 3-sphere fixes the topology of 3-sphere uniquely?
> > >
> > >
> > > 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.

>

> > > 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]
> > > > > 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
> > >
> >
> > 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!
>
>
I tried to read a popular book about surreals and did not
have the needed patience. It would be much easier to learn them
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
> 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
>
> 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

>
> 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"!
> > >
> [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
>

This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:36:56 JST