Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 2 Sep 1999 10:22:55 +0300 (EET DST)
On Wed, 1 Sep 1999, Stephen P. King wrote:
> Dear Matti,
>
> Cutting to the basics...
>
> Matti Pitkanen wrote:
>
> > [MP] In TGD approach UPsi_i represents the state of entire Universe
> > after dispersion from single sector D_p to all sectors of configuration
> > space: Djinn has just left the bottle. In practical modelling
> > one of course cuts the universe to single self and forgets the rest.
>
> Is your Universe the totality of Existence?
My universe, quantum history, GCI configuration space spinor field.
Totally of objective existence could be defined as the set of all
solutions of field equations. Subjective existence would correspond to
hopping around this totality of objective existence.
Totality of subjective existence decomposes into selves having subjective
memories. At the top is entire universe with rather long short term
memory: it remembers something about the infinite
number of all quantum jumps occurred hitherto.
>
> snip
> [SPK]
> > > Ok, but how would the existence of many observers affect this argument?
> >
> > [MP]
> > In TGD each self is observer and each UPsi_i decomposes
> > into infinitely many selves. Each self behaves like its own
> > sub-Universe. Therefore existence of many observers does not
> > affect the argument in no ways. Actually, it took quite
> > a lot of effort to find the proper form of strong NMP localizing
> > appropriately and only the concept of self solved the
> > problem finally.
>
> How do your selves talk to each other?
>
[MP]
They communicate only if they have provided they have achieved that level
of intelligence when conscious intentional communication is possible.
I explained this in more detail in posting of yesterday.
The point is that my approach does not require any communication,
nor its strong from 'bisimulation'.
> [MP]
> > The need to transform observers to each other seems to be
> > the special feature of LS theory reflecting the attempt
> > to geometrize mind, OK?. In TGD there is no need for
> > mapping selves to each other. In fact the idea of transforming DNA self to
> > human self would be non-sensical: Human self can containg somehing like
> > 10^17 of DNA selves!
> >
> > Presumably LS approach assumes kind of hierarchy of
> > intelligences such that observers with different IQ cannot trasnformed
> > to each other smoothly, correct?
>
> It is not forced into such an anthropocentric box. It merely allows it.
>
> [SPK]
> > > I see the
> > > q-jump as a tournament between the possibilities! It is interesting to
> > > note that the number n of pair's of "matches" needed in the tournament
> > > is \upperbound Log_2 n + \Remainder_2 n. This comes from the question
> > > "how many steps does it take for a given number of ladies to gossip such
> > > that each ends up knowing what others know and they can only communicate
> > > in pairs (no "conference calls")".
> >
> > [MP] I am afraid that the modelling of quantum jump as tournament or
> > process could lead to problems. I do not believe that this kind of
> > modelling is consistent with what we know about quantum jump. How to cope
> > with Bell's inequalities is the problem. Everything else
> > in quantum mechanics, even quantum entanglmeent
> > at the level of configuration space, can be regarded as
> > classical, but not quantum jump.
>
> Do the possible quantum histories within the Psi_i compete?
[MP]
There must be some typo: Psi_i IS the quantum history. You probably refer
to selves or subsystems...
One can say that subsystems within self (UPsi_i has decomposition into
selves) compete for having moment of consciousness making them
unentangled subsystem and candidate for
new subself (if they can keep themselves p-adically unentangled during
next Psi_i-->UPsi_i-->Psif:s they get self, that is they wake up).
This is nothing but competition for self consciousness. Systems
intelligent enough realize this and do their best to guarantee that they
wake-up next morning. This is fight for survival. Perceptive rivalry is
known to occur inside brain.
>From information processing point of view, my quantum computing Universe
is such that the information gain described by entanglement entropy
reduction is maximal. Somehow maximal amount of conscious information
is abstracted from each quantum computation. In fact, localization in zero
modes maximimes position information in zero modes. So that two different
kinds of information gain are maximized.
>
> > I understand the need to model the quantum jump when one
> > believes in dualism: accepting the quantum jump as it is and adding
> > to the dualism gives tripartism.
>
> Ok...
>
> snip
> [MP]
> > > > Quantum measurement replaces clocking and gaugeing in TGD framework.
> [SPK]
> > > So would you agree that they are equivalent?
> [MP]
> > No. Quantum measurement is much more general and primitive concept and
> > clocking and gauging are higher level operations involving
> > the concept of self (observer!) in an essential manner.
>
> How?
[MP]
This is clear for me from the fact that our close relatives apes do not
perform this operations! Note that I define these operations as what
they are: operations performed by selves. Entire dynamical self cascades
are generated within these selves during these operations. At geometrical
level I am satisfied with mere identification of distances and angles,
etc... When lecturing about basic calculus in University, I had a habit
of characterizing Riemann geometry as a model of length and angle
measurement: I was wrong!
> > [MP] Note how economical the scenario is: self/observer is
> > just heap of observations!
>
> What is an observation?
>
[MP]
Quantum jump=quantum measurement for density matrix is building brick
of observation, 'elementary particle' of subjective world. Selves
represent heaps of observations occurred after wake-up, 'composite
particles' of subjective world in terms of ORP ('Ontogeny Recapitulates
Phylogeny').
> [SPK]
> > > I do not agree! Your definition of a "self" is far to anthropocentric!
> > > Self-awareness emerges when we have "quantum measurement of quantum
> > > measurements", but clocking and gauging are primitive acts! They are the
> > > essence of quantum measurement! We see this when we consider what a
> > > measurement *is*.
> > >
> > [MP] You agree that clocking and gauging require observer. But
> > observer is self. The idea about self-awareness as
> > quantum measurement of quantum measurement does not make sense to
> > me. Reflective level of consciousness could be identified
> > either the fermionic contribution to conscious experience
> > or as pure self awareness in contrast to the contribution
> > of the experiences of subselves of self to experience:
> > my thoughts are my subselves.
>
> I am saying that "clocking and gauging" are what an observer does. The
> observer is defined in terms of what they observe. Otherwise we are
> saying the same thing!
>
[MP]
Actually we say same thing: observer=self is heap of quantum measurements.
Observer is what observer observes. We identify basic observations
differently.
> > The definition of self could not be more general! It sounds almost
> > trivial in its generality: self as sub-system behaving effectively as
> > p-adic sub-universe. Not a single word about brain or biosystems! Self as
> > subsystem allowing description using pure quantum state.
>
> Why should we? I am by nature very wary of Theories of Everything! DO
> think that one can "expalin everything" with in a single perspective is
> "not even wrong"! It is like saying that a map of the entire infinite
> totality of existence Universe is constructible in finite time!
My background is theoretical physics. Every great theory of physics is
huge abstraction and free of references to any particular system.
Explaining is quite different thing. Models are constructed to be
consistent with this Grand Scenario and if
unresolvable anomalies arise, one starts to rethink about basics of
the Grand Scenario. Therefore single tiny anomaly can change the entire
world picture.
They have tried also different approaches. Bootstrap philosophy was tried
in hadronic physics but it did not work: quark model and QCD had so
dramatic success that bootstrap models died within few years.
>
> > Why I like Buddhist ideas is not detailed assumptions but similar
> > generality: the Westner models for brain and consciousness
> > are extremely culture biased: few centuries ago brain was
> > clockwork, then it was hydrodynamical system, then it became
> > computer, some regard it as a hologram,...
>
> Sure, these are example of how the popular technologies of the time are
> used as paradigms to make sense of experiences.
Note that all these analogies are *mechanical*: brain is something
which we already understand. Not a single attempt to
base definition on, say concept of self(;-)!
> snip
> [SPK]
> > > The key question is" Is this "reality" unique and absolute?! How could
> > > this be knowable?! We must consider the "other minds" problem!
> > > http://members.home.net/stephenk1/Outlaw/othermind.html
> [MP]
> > One of the first victories of TGD approach was solution to the other minds
> > problem, which I understand as problem about why we do not
> > have direct access to other minds. Actually we have access
> > to our subselves: we experiences their lifehistories as
> > our thoughts, mental images, sensory experiences.
> > We do not have direct accces to the mind of selves at the same
> > level as us or to subselves of other selves not to mind
> > of higher level selevs.
>
> I fail to understand your solution! It looks like you just say that TGD
> does and do not provide a reasoning chain that others can follow. DO you
> really wish for us to agree with you or just believe you? "Trust but
> verify!"
>
The solution is nothing but few *plausible* hypothesis about how
experiences of selves integrate. Time integration gives rise to subjective
memory and selves. The integration of experiences of subselves inside
self gives rise to abstraction. These are just natural hypothesis about
architecture of experiences. Since they say something about subjective
experience these hypothesis do not and cannot follow from any field
equations.
I would of course be happy if someone would take me seriously. But after
the experience of twenty years I tend to believe that having graduate and
phD students is the only solution in case that one is not just
satisfied with personal understanding and wants academic status. To
have graduate student is to make trade: graduate student *believes* and
has hopes of becoming *phD* in future. This is the origin of schools of
thought in science. Of course, I have nothing to offer in this kind of
trade!
> > In fact, split brain patients are good test for any theory of
> > consciousness. I have not yet checked thoroughly whether the little
> > I know about this topic is certainly consistent with the concept
> > of self.
>
> Do we really need to combine physics and phychology? You are trying to
> put alot of people of of work. It is a property of Nature to specialize!
TGD is unified theory. Unified theories are necessarily theories of
consciousness. In TGD framework this became clear when I realized that
quantum jump between quantum histories concept forced by GCI forces to
consider the problem of psychological time. There was no other option
than simply construct theory of consciousness. You see how great
ideas like GCI force to answer fundamental questions.
I regard the specialization as the basic problem in sciences. Biology
and neurosciences are excellent examples this. We have mountains of data
and practically no understanding.
I know extremely well that those who want to ridiculize TGD find
consciousness to be excellent tool. This sounds solen but intellectual
honesty means for me much more than personal career.
Already now consciousness theory has shown its power in construction of
basic quantum TGD. It might have taken much longer time to discover the
necessity of localizationin zero modes unless I had to understand strong
NMP. I would be still worrying about pinary cutoff unless I had not
found that it is precisely the key element in understanding information
content of conscious experience and making the notion of self realizable.
No hope of understanding psychological time and its arrow if I had not
spent 10 years in building of consciousness theory.
> snip
> [SPK]
> > > Yes, thus I am thinking that interactions among LSs is best modeled by
> > > bisimulations! The "faithfulness" issue is VERY important! The idea of
> > > mutation is very much implicit! This relates directly to the idea of
> > > "forgetfulness" in Chu spaces! In the involution transform Mind_i ->
> > > Body -> Mind_f (and its dual Body_i -> Mind -> Body_f), if the initial
> > > and final states (events) are not exactly the same we get
> > > mutation/motion! (In the gene configuration space for biological
> > > organisms, or in the configuration space for LSs!)
> >
> > [MP] I already wondered whether you have hierarchy of LSs such
> > that bisimulation is possible only for LSs at same level of
> > hierarchy. Do LSs experience the experiences of their subLSs
> > somehow?
>
> It seems that LSs can only bisimulate other LSs that have equal or less
> complexity! So it seems that LSs experience the experiences of their
> subLSs somehow, but this is a very delicate question! For example, how
> do we experience the "experiences" of our body's cells? Pain is one
> thing that comes to mind...
[MP] Yes.
> [MP]
> > I am not at all sure that brain only generates the idea of spacetime.
> > I cannot yet exclude the possibility that also my body contributes
> > to my consciousness and primary sensory experiences occur at
> > the level of sensory organs. This hypothesis explains elegantly
> > Libet's experiments showing that subjective experience of skin stimulation
> > occurs about 1/2 seconds before the neural activity in sensorimotor
> > cortex. [Also the location of experience at level of brain nuclei could
> > do this.]
>
> Oh, I did not mean to imply that the brain is the only thing involved
> in generation the idea of space-times! It is obvious, if we are going to
> use this metaphorical line of thinking, that the interaction and
> interdependence of the brain and body are involved! But that is really
> not my point. I am saying that any observation is framed in spatial and
> temporat terms. It was this notion that led Kant to his ideas about
> space and time.
[MP] This is true. I would speak of zero modes: localization in zero modes
characterizing classical spacetime macroscopically determines the content
of sensory experience and perhaps even contents of thought.
>
> > Cognitive processing certainly occurs dominantly in brain but again
> > TGD suggests that also organs and even skin possess some genuine
> > intelligence: there are experimental claims about this. Hearthmath
> > homepage contains something about intelligence of heart.
> > Also eyes seem to perform cognitive processing as I learned
> > some months ago (article in Nature or Science?)
> >
> > For instance, proprioception might be occur in
> > body. Brain would only cognitively process this
> > experience. The situations in which patient loses his body
> > experience could be explained simply as dissociation of
> > bodily and brainy self to separate parallel selves.
> > This would mean that we basically directly experience
> > the 3-space, brain only analyzes this experience.
>
> Yes. But is this really a justification for infering general physics
> from particular properties?
[MP] I do not quite understand your point. The speculations about
the body consciousness are just natural hypothesis consistent with
TGD based concept of time and deserve serious consideration since
they solve Libet's and Radin's causal anomalies. They might be wrong:
perhaps it is after all only brain which is able to have self. TGD
certainly allows also this.
> [MP]
> > Physical realism boils down to experimental tests. The
> > Bell inequalities state this kind of test. If the world
> > is classical in the sense that quantum jump is modellable
> > by a process, this can be seen experimentally. Aspen
> > experiments demonstrated that Bell inequalities hold true
> > and therefore modellability hypothesis is excluded.
>
> Please be more specific!
There are articles related to Bell inequalities in quant-ph. For
instance. http://xxx.lanl.gov/abs/quant-ph/9608047.
The *violation* of Bell inequalities is basic signature of quantum
mechanical reality: not that they are satisfied as I erraneously
stated above. More precisely, local objectivism is violated.
In the most general situation BEs are formulated for a collection of N
objects (particles) characterized by three binary variables a,b,c (3
components of spin). a=+/-1, etc...
a) In classical world picture there is *joint probability distribution*
P(a,b,c) for these variables. For this collection of N objects one can
define the numbers objects such that two of these variables have fixed
values. For instance,
n(a=1,b=-1)= N*SUM(c=-1,1) P(a=1,b=-1,c).
b) By direct and simple calculation one finds that
n(a=1,b=-1) <= n(a=1,c=-1) + n(b=-1,c=1).
This just substituting the expressions for n(..) in terms of
joint probability distribution. These inequalities hold true in
*classical world*.
c) Quantum world is not classical. a, b and c can represent mutually
noncommuting observables. This is the case when they correspond to
components of angular momentum or spin. If a and b and c represent
mutually noncommuting observables, joint probability distribution
does not simply exist and one cannot derive Bell's inequalities.
This looks nice but the problem is that it is difficult to make sense
what the numbers n(a=1,b=-1) mean in case of quantum world
since a and b are mutually noncommuting observables!
****
This forces a more refined approach involving measurement
of spin for two particles in spin singlet state: spin components
in z directiona are opposite and state is antisymmetric under exchange
of particles: state is proportional to |+1,-1>-|-1,+1>.
a) One considers 3 measurements. A corresponds to measurement of
J_z of particle. B corresponds to measurement of component of J
for first particle in direction making angle theta with z-axis.
A' corresponds to measurement of J_z for particle 2 and C
corresponds to the measurement of component of J making angle phi
with z-axis.
b) Now if the measurements performed for particle 1 and particle 2
are *independent as local realism asserts*, then measurements A and B
do not affect the outcomes of the measurements A' and C.
This assumptions does *not* hold true in quantum world.
The outcomes of measurements satisfy
ab+ac-bc<=1
This follows from the fact that measurement results have values +/-1.
In quantum world this statement does not make sense since
one cannot have simultaneous eigenstates of Jx,Jy,Jz.
But one could claim that quantum measurement is just a black box
description and actually quantum measurement of Ji simultaneously makes
sense. In any case, the outcomes are +/-1 experimentally so that in this
case the statement would make sense.
c) Rest is algebra: one takes cyclic permutations of the first inequality
and takes statistical averages to get
<ab>+<ac>-<bc><=1
<ab>-<ac>+<bc><=1
-<ab>+<ac>+<bc><=1
Taking second and third inequalities one obtains
|<ab>-<ac>|+<bc> <=1.
This is Bell inequality. Note that everything is now *well defined*
quantum mechanically! Unlike in the first form of inequality.
Expectation values <ab>, etc are quantum mechanical correlations
for components of spin in certain directions and directly calculable.
This inequality however involves the assumption about local realism
and might be violated.
d) The *failure* of Bell's inequalities provides a test for whether
local realism makes sense or whether quantum reality is more real.
Aspen experiments demonstrate that these inequalities are violated
and this is really tight constraint on theories trying to
reduces quantum jump/quantum measurement to some
dynamical process.
> >
> > [MP]
> > To the possible peril or glory of the theory predicting that this
> > symmetry structure is very special physically and mathematically.
> > Here I cannot never sigh of relief(;-).
>
> Do you understand my consern? I do not understand your justification!
> How can I evaluate your beautiful model if I can not have hope of
> understanding it? Please!
I can only explain and hope that you understand my explanations.
Configuration space geometry requires a lot of mathematical background
and one must learn it by doing. We must realize that we have very
different backgrounds.
I am not ashamed that I find technical string model papers completely
incomprehensible: this is not fault of string model people.
To understand them would require complete devotion of years and would
require strong belief in string model ideas. Even experts are said to
not understand what is going on: they just look where Witten's finger
points to. I can only read popular expositions about what is happening
in string models.
>
> > >
> > > Ah, "microlocality"! Indeed! And what is that? Do you mean
> > > "microcausality"? See:
> > > http://mist.npl.washington.edu/npl/int_rep/ti_over/node7.html
> > > http://physics.indiana.edu/~kostelec/faq.html
> > > http://xxx.lanl.gov/abs/quant-ph/9805008
> [MP]
> > Microcausality and microlocality are essentially the same thing.
>
> Ok, well then how does "locality is lifted to the level of
> configuration space of 3-surfaces" avoid the non-locality problem. It
> looks to me that it makes it worse! But, umm, since you are mapping
> whole 3-surfaces in q-jumps, I think that I can see a glimmer of hope!
> :-) I am just not sure how you avoid the problem of defining the norm
> uniquely, maybe the p-adics does this for you, but I do not understand
> the mechanism. :-(
>
Locality at the level of configuration space is in practices very much
like microcausality at the level of spacetime: configuration space spinor
is essentially Fock state constructed using fermionic oscillator
operators of second quantized free fermionic QFT in *spacetime*: this
gives the connection to QFT and standard physics.
In standard classical field theory you have
fields in spacetime. Say spinor field Psi(x) as in old fashioned Dirac
theory of electron. They describe probability amplitudes for point like
object to exist at x. In TGD I replaced point like object with
3-surface and have configuration space spinor field Psi(X^3).
In standard QFT one must second quantize Psi(x): this
Dirac and his cotemporaries learned soon. All worked nicely
except that perturbative QED contained horrible divergences.
In TGD this second quantization is *not needed* since configuration space
spinors are essentially second quantized free spacetime spinors.
Classical Psi(X^3) is enough. Universe is formally classical
single "fermion" state in configuration space. Universe as particle:
this is one of nice features of TGD. TGD is like old fashioned
Dirac theory in infinite-dimensional context but avoiding its problems
(negative energies, negative probabilities).
> snip
> > [MP] I am not expecting that you believe blindly what I say.
> > If you want to learn basic maths of TGD it is quite possible.
> >
> > The locality in configuration space means
> > only that physical states are described by
> > *classical* configuration space spinor fields which are
> > *functionals of 3-surface*.
>
> Are their mathematical objects that are dual to these functionals?
I would say no. One can of course construc dual as dual space of Hilbert
space but I think that this does not bring in anything interesting.
>
> > If I were to second quantize
> > these fields, physical states would become nonlocal with respect
> > configuration space. I would construct manyfermion states
> > with fermions located at different points of configuration
> > space (3-surfaces), etc..
> >
> > Since QFT states correspond to states of field theory in fixed spacetime,
> > the values of configuration space spinor field for given 3-surface
> > are counterparts of states of quantum field theory.
> >
> > Qmind files contain more detailed description of this.
>
> Could I post a text file of your Qmind posts on my web site so that all
> could look at them easily?
Of course.
> [SPK]
> > > "geometries do not precede observation"!!!!!!!!! Geometries are *how*
> > > events are related to each other within the contents of an observation.
> > > It is a very serious mistake to assume that the geometries are
> > > pre-selected prior to the act of observation. You are clinging to the
> > > vanishing sugar cube of "naive realism" that is being eroded by the
> > > waves of experimental verification of such things as the EPR effect, the
> > > delayed choice, the quantum erasure, etc.
> > > WHY?
> [MP]
> > Well you state your belief very strongly. Explanatory power
> > and internal consistency of the
> > theory is the only criterion for our beliefs.
> >
> > And as I have explained many times, observation
> > UPsi_i-->Psi_f selects very few classically equivalent
> > spacetime surface from the huge superposition represented
> > by UPsi_i. But somehow you fail to see this.
>
> Yes, because you are not explaining it to me!
>
I think that the problem is that you do not have the general
QFT intuition. It is not about detailed formulas but just some basic
working with QM and QFT: the formal generalization to new
situations is easy for mathematician if he has the basic QM
intuition in spine.
The lack of computationalistic intuition makes it
difficult for me to understand details of Pratt's work and your
motivations, for instance.
> > TGD of course reproduces EPR effect, delayed choice, etc..
> > I would not spend an hour with TGD if it would not.
> > The analysis of these experiments involves in essential
> > manner the approximation about pre-existing spacetime
> > but this is not essential. What matters are the
> > basic rules of quantum mechanics: quantum jump as something
> > irreducible, Born rule, the basic rules for constructing
> > many particle states by tensor product construction.
>
> The more time you spend working on TGD, the more you will resist giving
> it up! This is an example of why people get stubborn as they get older!
> The effort and resorces used must be justified to oneself...
Why should I give it up? I have been done huge amount of work
in constructing TGD and showing that it really exists and is not only a
dream and a lot of wishful thinking. I am happy to spend the rest of my
life polishing it and even making real progress at deeper level. There's a
lot to do and I can do only small part of it with my miserable analytical
skills.
>
> > The question is about what *new* effects TGD predicts.
> > Libet's experiments, which I discussed in earlier posting
> > and Radin-Bierman experiment are examples of these new effects
> > and find nice explanation in terms of new time concept.
>
> Ok...
>
> >
> > [MP] I agree completely. In infinite-dimensional context however the
> > set of all possible geometries reduces to a set containing single
> > element(;-)!
>
> Could you please at least give me some details or explicit references
> so that I can understand your reasoning!
>
The first chapters of 'TGD' at my homepage contain the exposition
of configuration space geometry.
http://www.physics.helsinki.fi/~matpitka/tgd.html
contains the table of contents and by clicking chapters
like. Chapters 'Kahler function' and 'Construction of configuration
space geometry from symmetry principles' are the key chapters.
There are detailed identifications for zero mode and fiber coordinates
of configuration space here. There is also chapter 'Configuration space
spinors structure'.
> snip
> [MP]
> > > > Of course not. Each self has its own *representation* for Psi_i.
> [SPK]
> > > Yes, and thus each self has its own geometry (defined by its own
> > > triangle inequality)! (I see this by considering a "self" and its
> > > "world" as a pair of points in the space of possibilities. This follows
> > > from my assumption that the self-world relation is symmetric, e.g. when
> > > I observe my world, my world observes me back.)
> [MP]
> > In TGD self has not geometry: only the information contents of conscious
> > of self can be *approximately* located to cognitive spacetime sheet.
>
> I agree, it is what is observed that is framed geometrically!
>
> [MP]
> > > > I take as quantum measurement the basic model of quantum measurement:
> > > > measuring is to have moment of consciousness: make world classical
> > > > by localization in zero modes, etc..
> [SPK]
> > > Sure! I just fail to understand what "zero modes" are... :-(
> >
> > [MP] Zero modes are somewhat technical concept, explained
> > in qm postings. Configuration space has fiber space structure.
> > Fiber corresponds to the quantum fluctuating degrees of freedom
> > in which metric is nontrivial. Propagator is actually contravariant
> > metric, which shows how beautifully everything geometrizes.
> > One functionally integrates
> > over them just as in QFT.
>
> Thank you! This is the detail that we need to understand you!
>
> > In zero modes metric of configuration space is trivial: line element just
> > vanishes. There are now quantum fluctuations in these degrees of
> > freedom.
> >
> > Zero modes characterize size, shape and classical Kahler fields
> > of spacetime surface. They characterize what we are used to call classical
> > physics. Quantum jump indeed involves localization in zero modes
> > so that every final state of quantum jump is classical in well defined
> > sense.
> >
> > Zero modes are purely TGD:eish feature and derive from the
> > fact that particles are not more point like. They are absent
> > in QFT. In string models they emerge as so called moduli characterizing
> > the conformal equivalence classes of metrics for string world
> > sheets.
> >
> > Zero modes make it possible to unify classical physics with
> > the physics of quantum field theories. Fiber<--> quantum field theories.
> > Base (zero modes)<-->classical physics.
>
> I will think about this! :-)
>
[MP9
The chapter 'Construction of configuration space geometry from symmetry
principles' contains also detailed formulas for coordinates
characterizing zero modes.
Best,
MP
This archive was generated by hypermail 2.0b3 on Sat Oct 16 1999 - 00:36:39 JST