# [time 444] Re: [time 442] Re: [time 427] Re: Conformal Invariance and related notions

Stephen P. King (stephenk1@home.com)
Wed, 14 Jul 1999 21:44:55 -0400

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. :-(

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!

> > > 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?
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 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! :-)

> > 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

> > > 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.

> > > > [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

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 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!
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.

[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]
> >
>
> 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. 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. 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.
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...

> > > 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!
> >
>
> 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! 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.
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! :-)

> > > 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". :-(

> 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. :-)

> 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...

> 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?

> > > > [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? 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... :-(

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!

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. :-(

> 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.
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!
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! 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!
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!

[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!
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.

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.

> > > 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

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.

> 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!

> > 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...

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! :-)

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?

Onward to the Unknown,

Stephen

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