Why is spacetime four-dimensional

[arivero]

The sphere compactifications are not chiral, as Witten showed in the 80s. It's irrelevant that a SM-type group shows up there, since modifying these models to introduce chiral matter will change the gauge group as well.

Have you read the proof by Witten? Actually it is not easy to find, as it is published in a non periodic proceedings volume, "Shelter Island II". It is not so pessimistic as the people who quotes it.

To me, the point is that the SM-type spaces have some interesting structure that could allow to orbifold into chiral matter and still keep the gauge group there. Note the branched covering of S4 by CP2, it is a very peculiar discrete relationship (and again not widely know), note also that all lens spaces can be used as fiber instead of S3, and finally note that we are playing near of the world of exotic 7-spheres. Theres is a lot of interesting math here, ad it could provide some scape. It was mainly a historical/social problem, the availability in 1984 of the heterotic string and their huge symmetry groups make that any person with reponsabilities of publication were directed to work with E8xE8 etcetera instead of keeping playing with these structures.

 

[czes]

I agree with that. This is the missing link.

This is why I think the idea would be if one can: starting from discrete measure theoretic plausible abstractions (the physical inferential connection); show that any sufficiently complex (large) system are more likely to infer 4D spacetime (in a continuum approximation) than anyone one.

The argument could be that given observers with unknown microstructure, only given that they are "communicating" and thus develops relations... it would be the most probable outcome that the microstructures selected by evolution is 4D (based on counting possbilities within the scheme).

/Fredrik

It is very important how to join a mathematics with the physics.
May be if we encode a constant Planck time dilation for each quantum interaction it creates the space-time of the General Relativity.
I am sorry that I am coming with the Planck time again and again. I made some calculations which shows the gravitational time dilation close to a massive object and other . Nobody proved it is wrong, till now.
Am I mistaken with that Plankian time dilation ?

 

[suprised]

This is why I think the idea would be if one can: starting from discrete measure theoretic plausible abstractions (the physical inferential connection); show that any sufficiently complex (large) system are more likely to infer 4D spacetime (in a continuum approximation) than anyone one.

The argument could be that given observers with unknown microstructure, only given that they are "communicating" and thus develops relations... it would be the most probable outcome that the microstructures selected by evolution is 4D (based on counting possbilities within the scheme).

I got lost...why is 4D here preferred as compared to, say, 5D?

 

[jal]

Fra always says ... “from a given observers "inside view"”

Fra, take what you you say to the level of the universe of what a QUARKION would say.

1. the universe is confined to 10^-15m
2. A dimension is as big as the universe, (10^-15m)
3. The degrees of freedom, (numbers of space dimensions) are NOW 3.
4. In the beginning, we were 12 quarkions in the universe. Now we are 3.
5. In the beginning, It appeared that our degrees of freedom were limited to 2 and that we were organized so that we could only move from a cubic to a hex. pattern.
6. In the beginning, Everything was perfect.
7. In the beginning, it appeared that we lived within another, (4th), degree of freedom/motion/dimension).

I await to hear what else you think the QUARKIONS WOULD SAY about their universe.
---
See the following blog entry to get an explanation that is a little more technical.

http://blogs.uslhc.us/known-knowns-of-the-standard-model
“Known knowns” of the Standard Model
Posted by Flip Tanedo on 08 Dec 2010

The top two particles are the up and down quarks. These are the guys which make up the proton (uud) and neutron (udd). As indicated in the chart, both the up and down quarks come in three “colors.” These aren’t literally colors of the electromagnetic spectrum, but a handy mnemonic for different copies of the quarks.
Below the up and down we have the electron and the electron-neutrino (νe), these are collectively known as leptons.* The electron is the usual particle whose “cloud” surrounds an atom and whose interactions is largely responsible for most of chemistry. The electron-neutrino is the electron’s ghostly cousin; it only interacts very weakly and is nearly massless.
As we said, this first column (u, d, e, and νe) is enough to explain just about all atomic phenomena. It’s something of a surprise, then, that we have two more columns of particles that have nearly identical properties as their horizontal neighbors. The only difference is that as you move to the right on the chart above, the particles become heavier. Thus the charm quark (c) is a copy of the up quark that turns out to be 500 times heavier. The top quark (t) is heavier still; weighing in at over 172 GeV, it is the heaviest known elementary particle. The siblings of the down quark are the strange (s) and bottom (b) quarks; these have historically played a key role in flavor physics, a field which will soon benefit from the LHCb experiment. Each of these quarks all come in three colors, for a total of 2 types x 3 colors x 3 columns = 18 fundamental quarks. Finally, the electrons and neutrinos come with copies named muon (μ) and tau (τ). It’s worth remarking that we don’t yet know if the muon and tau neutrinos are heavier than the electron-neutrino. (Neutrino physics has become one of Fermilab’s major research areas.)

So those are all of the particles.

 

[PAllen]

Well, having a mathematical structure which is unique in 4d does not explain anything why a physical world should be 4dim, without a concrete physical mechanism that makes actual use of that structure. There are plenty of distinguished mathematical structures in almost any dimension, so such an argument doesn't explain anything, at most it hints at a direction to explore.

Let me remind about octonions, E8, Leech lattice, K3...they are all distinguished in some way. See also the recent paper http://arxiv.org/abs/1102.3274 , which remains utterly incomprehensable to me.

Neither Freund-Rubin explains 4d, there is no energetic reason why such a compactification would be preferred over other ones, or over no compactification at all.

I completely agree with this spirit, which is why I phrased it as a personal suspicion without (at the present time) any physical motivation.

 

[friend]

I got lost...why is 4D here preferred as compared to, say, 5D?

You're asking one of the most fundamental questions possible, not just who are the actors (particles) and why do they act as they do (move). But you're asking where does the stage come from (spacetime).

There is not going to be any physical answer to this question because you are basically asking what is the beginning of physics, what's the stage on which physics plays out. It will have to come from completely abstract general principles. You're (and by that I mean, we all) are going to have to go back to purely mathematical concepts as to what a "dimension" is to begin with and then develop some consistent model from there.

Dimensions are parameters we assign to points. And it seems arbitrary as to how many parameters are needed to label many points. For you could label a point, say, (0,0,0,0,0,1,3) with only two parameters really needed. Since the first five parameters are all constant at zero, this is really a two dimensional subspace of a larger 7 dimensional space. So in order to result in 4 spacetime dimension, out of the infinite number of possible dimensions that can be used, there must be some way to assign a number to the dimensionality and some calculation that takes all possible dimensions and results in the 4 spacetime dimension we see. Perhaps there is a way of putting all the possible dimensions in a quantum mechanical superposition and calculate an expectation value of 4 spacetime dimensions.

 

[tom.stoer]

Hmm? Tell Milnor. http://en.wikipedia.org/wiki/Exotic_sphere "a differentiable manifold that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere." To me it sounds as Topology.

Of course it's topologically, but there are many examples for homeomorphic but not diffeomorphic manifolds in higher dimensions. R⁴ is exceptional as there are uncountably many such non-diffeomorphic R⁴'s. This is not possiblein any other dimension!

 

[PAllen]

This is really grasping at straws beyond my level of expertise, but may a bridge between differential structure properties of 4-D and approaches that don't want to use smooth manifolds can be found in Jenny Harrison's work on chainlets as a more fundamental basis for analysis (just search for Jenny Harrison in the math section of arxiv).

 

[marcus]

http://arxiv.org/abs/math-ph/0601015
I remember her, happened to sit next to her at a lecture by Penrose in 2006.
Here is what is I guess a moderately recent, fairly representative paper about doing calculus with chainlets.

It could be interesting. I looked at the paper but did not check to see if it had been much cited. I'm not up to speed on this discussion, so cannot guess how it fits into the Big Question ("Why is it 4D?")

 

[arivero]

Of course it's topologically, but there are many examples for homeomorphic but not diffeomorphic manifolds in higher dimensions.

Yep, but higher than 7 dimensions always, isn't it? Or is it only for spheres? And actually, I am not sure if both facts are completely unrelated.

 

[czes]

Marcus wrote a link to Harrison paper:
"The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of ``pointed chains, culminating in the chainlet complex."
http://arxiv.org/abs/math-ph/0601015

What if dimensions are nothing more than the relation between the information. If there is something like a reduction of the dimensions (AdS/CFT) may be dimensions are not fundamental at all.
There is almost empty vacuum in our environment. Vacuum is a relation between virtual particles and atiparticles. We have to define us in relation to the Vacuum.
Due to reduction of the dimensions (holography) the Quantum Events are the relation between information on a screen of the Events Horizon.
Sometimes it is a relation between virtual quarks and antiquarks in the gluons space in the Vacuum and we need more relation then.
If each Quantum Relation encodes a constant Planck time dilation we get the space-time of the General Relativity then.
If there are more quantum relations we put it together in the approximate 4-th dimension.

Is a dimension something more than a relation ?

 

[Fra]

I got lost...why is 4D here preferred as compared to, say, 5D?

Assuming it was clear that I didn't suggest we have the complete argument on the table. Several thing are unclear to me. But I merely lined out roughly how the form of such an argument could be, inspired by counting distinguishable continuum constructions in different dimensions. I think the general form of the argument is what one could expect.

Ie. supposed we start out with an abstraction for the representation and decision/action problem of a given observer. Then this observer IS it's on measure, and it's contraint, of it's environement (ie. all other observers).

Since you like strings, to TRY to give you an incomplete handle on what I mean: this is related to something I say long time ago that my best projection of string theory is to view the string (or rather the microstructure that corresponds to the string states, including it's motion) as a measure-complex on the environment. The difference is that the STRING in string theory is a continuum construction, I picture a discrete starting whose continuum limit MAYBE is soemthing like a string or brane. Now, this measure-complex IMPLICTLY encodes as frozen information the structure also of external space in a holographic sense (at least in equibrium-but this is subject oevolution)

Then the mechanism to allow the embedded space of this measure-complex to EVOLVE and be selected, requires to understand how these measure complexes (measure each other/interact). In string language this would translate to that the background spacetime really is emergent between interacting strings. The only problem of course is that you can't describe the string without first having what is supposed to emerge. This problem isn't solved yet in string theory as far as I know.

The remeby IMO is to consider an evolutionary model, where the "landscape" from the perspective of the measure complex itself is much smaller. The massive landscape is only apparent from the laboratory frame.

If we can understand, HOW say this measure complex (thing discrete string if you like) can INFER from the inside the structure of the environment, and hences at some poitn reach stable relations with other "string/measure"complexes in the environemtn... these external relations is spacetime... (there can still be internal spces, like you have Calabi–Yau manifolds in ST that replaces the points in spacetime, this exists also on my picture but it's different).

The trick would be to caputer the correct physical INSIDE counting, and look at the large complexity limit (ie. very large continuum limit, or low energy limit as you would but it) then it could be that a pure statistcal case can be made that it's simply overwhealminglyt likely that a "RANDOMLY" interacting RANDOM structures at some point would decompose spontaneously into 4D+xD structures, where xD. Wether this is true I don't know, but it could be.

If this is true, in my view 4D is not fundamental, and it would just correspond to a continuum limit of some more fundamental interaction models.

Meaning that the string itself isn't a starting poitn because it's already a continuum.

The 4D structure would then maybe correspond to say a nash equilibrium in a silly senset. Each measure complex views the environemtn as a "black box" and the holographic duality corresponds to when all players have not benefiy from changing their assumptions or strategy. Maybe 4D is selected here in the sense that picking a manifold at random (given that the samples are genrated as per the correct construction, and not just some ad hoc "all mathematicall possible manifolds"; obviously the consistency constraint is that these structures need to be able to interact in a stable holographic scenario.) is most likely to be 4D.

To think that there are a hard logical necessity between 4D does not seem reasonable to me. I don't think that's ever going to be. Definig the form of argument you expect, makes it easier to find it I think.

Edit: I also don't think in terms of DEGREES of freedom, I prefer to just think of complexions. A continuum of complexions is only a limit (a non-physical one IMO). This is why the emergent 4D is just a form of sprinkling the complexions AS IF the would happen to find a 4D manifold. But this is IMO just an approximation.

/Fredrik

 

[tom.stoer]

Solving an action principle with a langrangian has been accepted traditionally as a good alternative to a energetic reason?

Of course.

I still haven't found a paper which explains Freud-Rubin in detail.

Is this a strict result, e.g. a "classical solution" which is stable under quantum corrections? Are there non-perturbative effects which would allow for tunneling and may destroy 4+7? And the most important question: are there other classical solutions - and if yes, what favours 4+7 instead of anything else? I am not asking if 4 is a possibility, but if there is a good reason why it must be 4.

I agree that my idea from the first post (counting differentiable structures w/o explaining what exp(-S) means) is just mathematics ("statistices") . Would you agree that contructing S and giving the summation a precise meaning would be a physical argument?

 

[suprised]

And the most important question: are there other classical solutions - and if yes, what favours 4+7 instead of anything else? I am not asking if 4 is a possibility, but if there is a good reason why it must be 4.

That's the hitch. Nothing forbids eg. d=10, that is, no compactifiction. Or simple torus compactification with maximal susy to any d up to 9.

In all those sugra compactifications like Freund-Rubin one always assumes some background, or some class of background, and goes from there (ie, check whether it's stable etc). AFAIK a convincing physical reason why d=4 would be favored is nowhere seen.

Of course there are many features unique to d=4, eg the log running of gauge couplings etc, but as said, it is unclear whether and how such features play any role concretely.

 

[tom.stoer]

That's the hitch. ... In all those sugra compactifications ... one always assumes some background, or some class of background ... AFAIK a convincing physical reason why d=4 would be favored is nowhere seen.

But this is exactly what I was asking for.

I am interested in a dynamical physical principle (of course with a sound basis in mathematics) that is able to explain why we live in dim=4. Afaik there is no such principle, neither in string / M-theory (you just confirmed that) nor in the context of LQG (the spin foam is basically dimension-agnostic, but the very construction introduces a "hidden" relationship to dim=4, namely Spin(4)).

My idea was to relax the assumptions to start with some specific dimension (dim=4 in LQG, dim=10/11 in string/M-theory) and allow for any dimension. Then we must look for a principle that selects dim=4. One idea was to "count" diffeomorphic structures. This could single out dim=4 rather easily, but of course it misses a physical concept, e.g. explicit construction of exp(-S).

 

[suprised]

Yes this is of course exactly the question that is pressing everyone, but AFAIK no one knows how to translate this (or another mathematical) proporty to a physical selection or extremality principle.

There has been circumstantial evidence here and there over the years, see eg: http://www-spires.dur.ac.uk/cgi-bin/spiface/hep/www?eprint=hep-th/0511140
but nothing came out really concrete.

Many people tend today to believe in some kind of anthropic or evolutive cosmological principle but that's of course a matter of heavy disputes.

 

[Fra]

Ie. supposed we start out with an abstraction for the representation and decision/action problem of a given observer. Then this observer IS it's on measure, and it's contraint, of it's environement (ie. all other observers).

To keep to conceptually relating to strings, string theory can with some imagination be seen as an attempt at exactly this.

Ie. the string action beeing somehow a fundamental action, from which a lot then more or less follows together with generic lessions from QFT.

In this, sense, it's not a bad attempt at all. This is also I think almost the essence of what some string researchers think with string theory beeing theory of theories. That's an impressing ambition, and the logic isn't alien to me.

But, my main problem with ST, is that string theory is NOT supposed to be an inferential theory in the proper sense (like I try to suggest; becuse it's my wild imagination that sees a remote connection here, I know well that string theorist does not make this conenction). For example the fundamental string action is pretty much a classical starting point, building purely from the mental picture of a litteral excited string. The ACTION of the string has no proper inferential interpretation or meaning.

But a pretty much similar theorizing such as in string theory, BUT if based on a proper inferential starting point where the fundamental action is a pure probaiblistic or information divergence view with a representation that fits with histories of events, would MAYBE be able to overcome many of the issues that ST has. The landscape problem beeing one of them.

This is why I've rambled several times that max ent principles and action principles can be udnerstood as purely inferential. Thus the fundamental action should be understood as purely inferential. No association to "classical strings" or anything else that is just confusing should be necessary.

Rather a finite string can be associated maybe even with the [0,1] interval, of a probability measure, when this measure no longer can accommodate the environment, conservation laws require that the measure itself maps out more complexions and dimensions. In this way the original string can be understood as living in a higher dimensional space. The an action can be defined by pure combinatorics.

This would do away with th baggge of ST starting points such as background space where QFT applies, and the background "string action" (which is really just taken from classical mechanics mentality).

If what I suggest is right, maybe one can udnerstand why string research might have stumbled upon some interesting ideas, even though the deepest understanding is still lacking.

/Fredrik

 

[Fra]

This could single out dim=4 rather easily, but of course it misses a physical concept, e.g. explicit construction of exp(-S).

It seems we all agree here where the issue is.

The physical basis of exp(-S) is the essence of seeking the physical basis for inference. This is yet a deep argument for acknowledging the inferential nature of theory and physical law in any research program.

The selection of the MEASURE and understanding the relativity of measures is at the heart of all this. And these things are also at the CORE of the inferential perspective.

/Fredrik

 

[friend]

One idea was to "count" diffeomorphic structures. This could single out dim=4 rather easily, but of course it misses a physical concept, e.g. explicit construction of exp(-S).

"count diffeomorphic structures"? Remind me again what so special about 4D? What does this have to do with diffeomophism invariance, or what? Thanks.

 

[tom.stoer]

"count diffeomorphic structures"? Remind me again what so special about 4D? What does this have to do with diffeomophism invariance, or what? Thanks.

Look at the topological manifold R³. Try to construct a differential structures on top of it. It works - and you'll get exactly one such structure; nothing else but the standard differential structure we are used to. This applies to many other manifolds as well: one topological manifold - one differential structure. It applies especially to all Rⁿ except n=4.

Now take the famous S⁷. You get 28 different topological structures, i.e. exotic spheres which are differentiable manifold that are homeomorphic but not diffeomorphic to the standard S⁷. Again this applies to many other manifolds as well: one topological manifold - N different differential structures (with N>1).

Now look at the topological manifold R⁴ (and afaik other non-compact 4-manifolds). There is not one differential structure, not N differential structure, but a continuum of differential structures. That means that dim=4 is unique in the following way: only in dim=4 one can have uncountably many manifolds that are all homeomorphic but not diffeomorphic to each other.

My idea is to "count" all differentiable manifolds, or to use something like a set of all differentiable manifolds. By the above reasoning it follows that manifolds with dim != 4 are a null-set in this set of all manifolds.

 

[Fra]

but of course it misses a physical concept, e.g. explicit construction of exp(-S).

Now I think I realize what you meant something else here.

The way I picture the counting, does not misses this weight. I rather think that if the counting procedure is taken seriously, these factors will pop out. I think if you look at the physics of counting, and in particular when the counting events com from non-commuting sets, the counting will in addition to the classical "probability weight", contain a transformation factors that corresponds to the connection-weight so to speak, betwene the non-commuting eventspaces. This conenction weight would measure the information loss during "transporting" between eventspaces. Just like one need to parallelltransport vectors in curved space into the same tangentspace in order to be able to comapre them. The same applies to the evidence orevents. A transport is need, before they can be comapred and this will introduce some further factors.

So if we take the counting more serious than just a CLASSICAL counting, giving rise to a classical probability, then a full expectation combining counting from non-commutative evidence, will introduce nonclassical terms in Z.

This is not in doubt in me, what I find unclear is the details, and wether the program will succeed. But I don't see such counting as beeing "simple" and missing those action terms. It would rather probably explain these terms, including quantum logic.

The idea beeing something like

/Fredrik

 

[friend]

Now look at the topological manifold R⁴ (and afaik other non-compact 4-manifolds). There is not one differential structure, not N differential structure, but a continuum of differential structures. That means that dim=4 is unique in the following way: only in dim=4 one can have uncountably many manifolds that are all homeomorphic but not diffeomorphic to each other.

One idea was to "count" diffeomorphic structures. This could single out dim=4 rather easily, but of course it misses a physical concept, e.g. explicit construction of exp(-S).

Let's try this: In the Feynman Path Integral, each path is continuous but not necessarily differentiable. In other words, paths can take sharp turns where no tangent exists at the turning point. So one path would not be diffeomorphic to another, but it would be homeomorphic. And you would need an infinite number of these non-diffeomorphic paths to construct the path integral. That only exists in R⁴.

Or, perhaps the whole path integral might be calcuated in one diffeomorphic manifold. And since the path integral is valid everywhere, you might need an entirely different manifold not diffeomorphic to the first to calculate the path integral somewhere else. Clearly then, you'd need an infinite number of non-diffeomorphic structures to insure that you could calculate the path integral everywhere so that the laws of physics would be the same everywhere. How does this sound?

 

[tom.stoer]

Let's try this: In the Feynman Path Integral, each path is continuous but not necessarily differentiable. In other words, paths can take sharp turns where no tangent exists at the turning point. So one path would not be diffeomorphic to another, but it would be homeomorphic. And you would need an infinite number of these non-diffeomorphic paths to construct the path integral. That only exists in R⁴.

No; what you are discribing is possible in any dimension. But I am not talking about a path in spacetime, but about spacetime itself.

Or, perhaps the whole path integral might be calcuated in one diffeomorphic manifold. And since the path integral is valid everywhere, you might need an entirely different manifold not diffeomorphic to the first to calculate the path integral somewhere else. Clearly then, you'd need an infinite number of non-diffeomorphic structures to insure that you could calculate the path integral everywhere so that the laws of physics would be the same everywhere. How does this sound?

I think that's not really what I am talking about.

I'll try to give you a simple example.

In bosonic string theory you try to define something like that:

$$\int dg\,e^{iS}$$

Here g is the Riemann metric on the two-dim. worldsheet of the string (forget about the 10-dim. target space; it's not relevant here). Then you recognize that you have different manifolds, in two dimensions simply identified via their genus; so you write the integral as

$$\sum_\text{genus}\int dg\,e^{iS}$$

where now the integral is over all metrics for fixed genus. But of course two different metrics g and g' with same genus are homeomorphic to each other and therefore should be identified physically. So formally one writes

$$\sum_\text{genus}\int \frac{dg}{\text{Vol}(\text{Diff})}dg\,e^{iS}$$

But here something interesting has been hidden: in two dim. two homeomorphic manifolds are also diffeomorphic and vice versa. This is no longer the case in higher dimensions. The first example are the famous exotic 7-spheres. They are all homeomorphic to the standard S⁷, but there is no smooth map between them, they are pair-wise non-diffeomorphic. Of course on each such S⁷ there are diffeomorphisms, but not between them.

My idea was to make use of this concept and treat non-diffeomorphic manifolds as physically different. So for the 7-spheres I would have to calculate the integral on each S⁷ and I would have to sum over all 28 7-spheres. In 4-dim spacetime the same will happen: I have numerous different manifolds. Usually we say that one of them is the R⁴. So when e.g. Hawking tries to write down a path integral over Riemann metrics he counts every manifolds (Minkowski, deSitter, ...) exactly once. But what I am saying is that even for the stadard R⁴ (required in the euclidean version) he has just one R⁴ topologically, but uncountably many different R⁴ which are homeomorphic but not diffeomorphic to each other. Therefore there should be a sum (or better: an integral) over all different R⁴'s.

Now the funny thing is that this is unique to dim=4. There are examples for higher dimensional spaces which are homeomorphic but not diffeomorphic (the 7-spheres have been discovered first), but usually you only get a finite number of non-diffeomorphic manifolds. Only in dim=4 you get uncountably many.

 

[Fra]

Therefore there should be a sum (or better: an integral) over all different R⁴'s.

We've learned to understand the feynmann path integral as that the action needs to account for all distinguishable possibilities (because somehow nature does). So we just count them, like we would count outcomes in probability theory.

But there are two things in this picture, which isn't understood well and that I think need to be understood to implement your idea too.

1. The quantum logic way of counting is different. Why? And how can be understand this?

2. When do we know that all physical possibilities are counted, but not overcounted? We need to understand the counting process within the right context.

The first issue is I think related to the decision problem where we have several sets of non-commuting information (that simply can't be added). It could be that BOTH sets contain information or evidence that supports a certain event, and then we need to ADD the "counts" from both sets... somehow, this is where quantum logic (and other generalisations) enter. This would amount to the classical expressions for probabilities from "classical counting" having forms such as (probability of possibility i)

$$P(i) = w(i)e^{-S(i)}$$

Where S is a kind of information divergence, w is just the factor from statistical uncertaint, going to 1 in the infinity limit.

would by necessaity incomplex more complex computations where w and S are generalized (just like it is in PI vs classical statistics).

The NEXT problem(2) is that of normalization and making sure we count all options, but to not over count. IMHO, the key not here is to understand that any counting must be specificed with respect to a physical COUNTER, and record. This is the equivalent to counting, so the subjective bayesian view to rpobability. Call this context observer O.

This the expression further changes as

$$P(i|O) = w(i|O)e^{-S(i|O)}$$

the complex formalism of QM is still real in the dend. I mean all expectations values are real. The to complex math is only in the comptuation.

Now, if the non-commuting sets are related by a Fourier transform, then obviously these transforms will enter the expressions. Any other relation and these will also reflect the comptation.

In particular will the context, put a bount on the number of possible distinguishable states if you think that the COUNTER and record can only distinguish, resp, encode a certain amount of information. This is what I think is ithe physical argument behind why it does not make sense to think we have to sum all mathematically possibilites.

Past attempts suchs as hawkings euclidian summation etc really does not even seem to ask this question: ie. the fact that the context of the counter is important, and has physical significance and that there is a good amoutn of relativity in the counting.

That two observers disagrees on how to count evidence is expected, it's not an inconsistency per see. I think is the reason for interactions in the first place.

As long as one is clear what is mean here, and doesn't think it means that two scientists will disagree upong PI calculations -they shouldn't. IT's just that if we play with the idea that a quark was about the perform the PI calculation, I am pretty sure it would be different, and this would explain the behaviour of the quark. The action ofthe quark reflects, it's expectations as defined by "renormalizing" this PI to quark level.

So I really think that we need to understand the physics of this counting itself.

/Fredrik

 

[tom.stoer]

I agree to most of the problems you are describing (and of course Hawking doesn't talk about these mathematical subtleties at all).

Yes, the biggest problem is how to define the counting including the weights. It is clear that we should count different topologies, but that we mustn't count physical identical entities twice. So the question is: what are physical identical entities? Usually one says that the same manifold equipped with different coordinates must be counted only once (if we would talk about world lines: each world line with different parameterizations is counted only once). But that means that we need diffeomorphisms between these different coordinates such that we are allowed to identify the two manifolds. As far as I can see it's exactly this step that fails when introducing homeomorphic but not diffeomorphic manifolds: the construction of a complete set of diffeomrphisms between the two atlases is no longer possible - therefore we should count them twice.

But there are additional problems: is it reasonable to start with manifolds at all? Wouldn't it be better to start with discrete structures from which manifolds can be recovered in a certain limit? If we try to do that, how can one save my argument, i.e. is there any discrete structure which is agnostic regarding dimensions in the very beginning (graphs are in some sense) but from which manifolds do arise, and which is somehow peaked around dim=4? I don't think that graphys will do the job as I don't see why dim=4 shall be favoured. What about causal sets, for example?

The problem is that all approaches I have seen so far seem to select dim=4 based on input + a dynamical approach (causal sets are constructed from dim=4 space and they recover dim=4 in some limkit defined by dynamics). My approach would be different in that sense that dim=4 is no input, dim=4 is not favoured by dynamics but by counting w/o dynamics. So the dynamics (that is still missing) should not be constrcuted such that dim=4 is selected (this is already done by counting) but that this selections not spoiled (i.e. that dim=4 is not supressed too much by exp(-S)).

So instead of having a dim-fixed starting point + dim=4 selecting dynamics it's the other way round: one as a dim-free setup + non-dyn. selection principle + dim-agnostic dynamics.

The major weakness is that I need manifolds. So any other (discrete) structure that could do the same job would be welcome.

 

[friend]

So instead of having a dim-fixed starting point + dim=4 selecting dynamics it's the other way round: one as a dim-free setup + non-dyn. selection principle + dim-agnostic dynamics.

Couldn't 4D be selected because first principles require an infinite number of homeomorphic but non-diffeomorphic structures. So I was looking for where such structures might be used in a physical context and thought about how Feynman paths might be homeomorphic but not diffeomorphic to each other, and you'd need an infinite number of them. Although, you'd probably have to do a path integral of 4D space (paths) that are homeomorphic but not diffeomorphic to each other. So if one could justify the use of Feynman type path integrals, then 4D might become logically necessary, right?

 

[Fra]

So the question is: what are physical identical entities

We agree on the question.

This is also a different but deeper perspective to the old question of what the important observables; I mean do we quantize observer invariants, or do we form new invariants from quantized variants?

Because "quantization" is not just a mechanical procedure although one somtimes get that impression. It's is just "taking the inference perspective seriously". The choice reflects how seriously we take the inferencial status of physics. They way QFT "implements this" mathematically can IMHO be understood as necessarily a special case.

Namely: Who is counting? An inside observer, or an external observer? That's the first question.

I'd suggest that current QFT, makes sense in this perspective if the counter is an external observer. And here external is relatively speaking, not external to the universe of crouse. Just external to the interaction domain, which is the case in particle experiments. The external observer is the labframe. In this sense current understanding is purely descriptive, it is not really the basis for decision making.

But this is not the general case, therefore the exact mathematical abstraction of QFT, breaks down for a "general inside counter", and an inside counter is not merely doing descriptive science, it bets it's life on it's counting, since the action of this inside observer is dependent on predicting the unknown environment.

To imagine inside counters, also in a deep sense touches upon RG. Since it is like scaling the counting context. So that you count naked events or events from the much more complex screene/antiscreened original system. Again current RG, describes this scaling descriptively relative to a bigger context. Ie. from assumptions of some naked action and a environment with screening antiscreening effects this is predictable; and this can be described and tested against experiment. Again this theory or theory scaling is not a proper inside view in RG.

So the same idealisation exists there. RG and counting, are integral parts, and both these things will need reconstruction in such a counting scheme you seek ( and I see it too, so it think we share the quest here).

So I think it's not possible to resolve this, by keep taking the same of PI formalism for granted and ONLY focus on various spacetime topologies and diffomorphisms... I agree that needs to be done, but I feel quite confident in my hunch that clarifying this, in the sense you suggest... is probably possible, but it will require a deepening of many things.. including foundations of QM and RG.

But if we can agree on a common question here, that's still quite nice. If I understand surprised right he seems to more or less share the same quest, except the question may be formulated different from within ST?

More later...

/Fredrik

 

[Fra]

Since I sometimes think of evolution, one should maybe clarify the difference to "dynamical" evolution.

So instead of having a dim-fixed starting point + dim=4 selecting dynamics it's the other way round: one as a dim-free setup + non-dyn. selection principle + dim-agnostic dynamics.

If I understand you right, you by "dynamical selection principle" mean a deterministic law (although it can of course stil lbe probabilistic; just like QM) that rules the dynamics of the system, and this then selects the 4D structure.

Then I fully agree that such an "dynamical selection" does in fact no explain anyitnh, it's just a recoding of the same problem, but where the "why 4D" then transforms into "why this particular dynamical law(that "happens" to select 4D;)"

I do however think of the mechanism of evolution, that does select 4D. But not one which is ruled by deterministic evoluton laws, but more a darwinian evolution.

Of course the details of this must be clarified. I see this as work in progress. But this can explain things like; we do NOT count all "past possibilities" in the action integral, we only count the FUTURE possibilities. Because for a real bounded observer, I think that part of the history must necessarily be forgotten.

So evolution of law can still be seens as a random walk, ande here the number of possibilities and favouring of 4D may still have a place like you suggest. But this I see not as a "dynamical evolution" but rather as an selective and adaptive evolution.

I figure you will think that this is starting to just get more foggy and foggy, but I think there are some expoits here that to my knowledge has never been explored.

Namely to reconstruct the counting, in depth, and consider "artifically" probably evolutionary steps and come up with arguments for why nature looks like it does, that are more like rational inferences, rather than logical necessities.

I really do not have much time at all myself, although I try to make progress with the little tiny time I hade. I do enjoy and hope to see some of the promising professionals that are working in a promising direction make some progress here.

/Fredrik

 

[Fra]

So any other (discrete) structure that could do the same job would be welcome.

All I can say at the moment, is that I have some fairly specificf ideas here, but they are very immature. But I think this way is the right now.

My exploit is to start my reconstruction in the low complexity end of the RG. And consider how the evolving interactions develop relations (seed to spacetime) and how the set of possibilities increase as complexity does. The point is that at the low complexity limit, you can pretty much manually look a the possibilities. I think this would correspond to a level beyond the continuum beyond "strings" or other continuum measures. Something like at causet level... but still for some reason causet papers tend to get a different turn that I want to see. But the basic abstraction of ordered sets (corresponding to events) and historeise or chains of events corresponding to observers are I think plausible to me.

The continuum structures you think about, should emergen in some large complexity limit, and I am not crazy enough to think that a physical theory need to model every information bit in the universe... rather at some point we wil lconenct to ordinary continuum models, but very enriched with the new strong guidance we apparently need.

/Fredrik

 

[arivero]

That's the hitch. Nothing forbids eg. d=10, that is, no compactifiction. Or simple torus compactification with maximal susy to any d up to 9.

In all those sugra compactifications like Freund-Rubin one always assumes some background, or some class of background,

Just to be sure, have you read the paper of F-R and do you remember that it assumes some background, or are you guessing? My recollection was that it was a dynamical argument, from a lagrangian and an action.

Also, I remember there was papers such as "10 into 4 doesn't go", showing that the F-R arguments were very particular of 11=7+4.

I think that in this kind of threads we are dangereously near of the mechanisms of consensus science: someone guess some content, it coincides with another guess, and nobody checks. I can try to xerox some papers for interested people, but if you guys don't have access even to commonplace journals that are available in any university campus, I am not sure if it is worthwhile.

 

[fzero]

Just to be sure, have you read the paper of F-R and do you remember that it assumes some background, or are you guessing? My recollection was that it was a dynamical argument, from a lagrangian and an action.

The FR paper is available at KEK http://ccdb4fs.kek.jp/cgi-bin/img_index?198010222

There's no dynamical argument at all. The whole point of FR solutions is that they are maximally supersymmetric, however that means that they are at the same energy as the uncompactified theory. So there is no dynamical argument selecting FR without additional physics that we do not as yet know about.

Also, I remember there was papers such as "10 into 4 doesn't go", showing that the F-R arguments were very particular of 11=7+4.

Again, FR solutions, in their original sense, were maximally supersymmetric solutions. There are many more options available if you only want to preserve one supersymmetry in 4d. That these were not known in 1980 does not mean that we should ignore them.

 

[arivero]

The FR paper is available at KEK http://ccdb4fs.kek.jp/cgi-bin/img_index?198010222

There's no dynamical argument at all. The whole point of FR solutions is that they are maximally supersymmetric, however that means that they are at the same energy as the uncompactified theory. So there is no dynamical argument selecting FR without additional physics that we do not as yet know about.

Thanks, my recollection was different! My reading was that maximal supersymmetry limits the choosing to the 3-index antisymmetric tensor, and that then Einstein-Hilbert equations imply that any separation, if it exists, must me 4+7.

EDIT: In fact, my re-reading of the paper doesn't contradict my previous recollection, first they proof that the existence of a s-indexed antysym tensor implies that compactifications must be of the form (s+1), (D-s-1). They use Einstein-Hilbert equations, not susy, to prove this argument. Then D=11 Sugra in maximal susy has a s=3 tensor, ann they get the announced result. But the compactification argument does not use susy at all, it seems to me.

 

[fzero]

Thanks, my recollection was different! My reading was that maximal supersymmetry limits the choosing to the 3-index antisymmetric tensor, and that then Einstein-Hilbert equations imply that any separation, if it exists, must me 4+7.

EDIT: In fact, my re-reading of the paper doesn't contradict my previous recollection, first they proof that the existence of a s-indexed antysym tensor implies that compactifications must be of the form (s+1), (D-s-1).

They make the assumption that the (s+1)-form must be proportional to the volume form of the compact manifold. It is a worthwhile class of solutions to study, but it is by far not the only class. In fact, one reason not to do so is that the VEV of the kinetic term for the form becomes the negative cosmological constant of the AdS part of the solution. While there are models like Bousso-Polchinski, where the fluxes partially cancel the naive $$10^{120}~\text{eV}$$ scale CC, they are all incredibly fine-tuned. Other examples of moduli stabilization rely on much more modest amounts of flux.

They use Einstein-Hilbert equations, not susy, to prove this argument. Then D=11 Sugra in maximal susy has a s=3 tensor, ann they get the announced result. But the compactification argument does not use susy at all, it seems to me.

True, there are various internal manifolds that one can consider. The round spheres are maximally supersymmetric. This, together with hints at gauge groups from deformed spheres was what made these models interesting.

Incidentally, it is important to check the stability of these solutions in the absence of supersymmetry. I don't remember any relevant references, but I think most non-SUSY solutions would be unstable to decay to flat space.

 

[arivero]

They make the assumption that the (s+1)-form must be proportional to the volume form of the compact manifold.

Ah, so proportionality of the s-form + application of Einstein-Hilbert action imply (s+1), and then susy implies s=3. And it uses an action principle (Einsten-Hilbert).

Of course it is not the right solution. If it were, we should not be here discussing about how to find solutions.

I think that the question of stability was studied too in the eighties, for spheres and deformed spheres, with both good and bad results, depending of parameters. In any case, as the problem of fermions show, spheres are not the complete solution neither, just interesting models that seem to be close to the real thing. Probably the deformed 7 spheres and the spaces with standard model isometries are connected from the fact that CP2 is a branched covering of the 4-sphere, a very singular situation.

The point of 11 SUGRA=7+4 being near of the real thing is that it was a serious justification to study M-theory. In fact it is better justification that to study it "because it is cool", or "because I am going to get more citations". Blame the split between hep-ph and hep-th.

 

[Fra]

Fra always says ... “from a given observers "inside view"”

Fra, take what you you say to the level of the universe of what a QUARKION would say.


I await to hear what else you think the QUARKIONS WOULD SAY about their universe.

Jal, you're right that asking what a "quark would see" does fit into my intrinsic inference quest :)

Though it's too early for me to speculate in this. The main reason is that before quarks enter the picture I expect the formation of continuum like structure comes first. Now, even if someone would argue that it's 4D rather than 2D, 2D is neverthelss a countinuum.

So to attach my envisions construction into the standard big bang timeline, the starting points is somehow the Planck epoch. As early as this, is where the "discrete picture applies". When we get to the quark formation we first need to understand how the complexions separated out from gravity and how the continnuum approximation is formed.

/Fredrik