Why is spacetime four-dimensional

[tom.stoer]

Are there any indications in different theories or approaches to QG explaining what could possibly single out 4-dim. spacetime?

Is there an idea why string theory favours compactification of 6 dimensions?
Is there an idea why purely algebraic spin networks (w/o any dependence on triangulations of PL manifolds) should results in 4 dim.?
Has anybody thought about spin networks based on Spin(n) or SU(n) and determined a "dimension" in the low-energy limit?
Does CDT really predict 4-dim. spacetime - or does it "only" reproduce 4 dimensions based on 4-dim. triangulations?

Assume for a moment that even in QG theories we can still use differential manifolds. What about the following idea: Assume we have something like

$$Z \sim \sum_\text{dim}\,\sum_\text{top}\,\sum_\text{diff}\,e^{-S}$$

I would like to "sum" or "integrate" over all dimensions, over all topologies (non- homeomorphic manifolds) per dimension, and over all differential structures (non-diffeomorphic manifolds). Then (regardless what S could be!) by simply "counting" manifolds the non-compact 4-dim. manifolds are singled out (continuum of non-diffeomorphic differentiable structures of R⁴; Clifford Taubes).

 

[ptalar]

Is there an idea why string theory favours compactification of 6 dimensions?

The 6 or 7 compacted dimensions are what is required to make string theory work, in theory, in our Universe, as we understand physics today in the world we observe. But I do recall reading somewhere on this forum that there is speculation that there may well be a Darwinian mechanism that decides why certain dimensions are compacted and others are not. And the mix of dimensions may change from Universe to Universe if a multi-verse exists.

 

[sheaf]

Maybe we should be looking for smaller dimensionality (this paper has been brought up previously in this forum I think):

Several lines of evidence hint that quantum gravity at very small distances may be effectively two-dimensional. I summarize the evidence for such ``spontaneous dimensional reduction,'' and suggest an additional argument coming from the strong-coupling limit of the Wheeler-DeWitt equation. If this description proves to be correct, it suggests a fascinating relationship between small-scale quantum spacetime and the behavior of cosmologies near an asymptotically silent singularity.

http://arxiv.org/abs/1009.1136"

 

[tom.stoer]

I knew this paper already. The methods described are used to derive 4-dim. as well (at least in CDT these two results are related). But somehow 4 is used as an input as well; CDT, Regge, LQG, Horava-Lifgarbagez are not agnostic regarding dimensions in the very beginning.

 

[arivero]

Is there an idea why string theory favours compactification of 6 dimensions?

Because the M theory sugra limit has a 3-indexed antisymmetric tensor.

 

[Fra]

I definitely expect an "explanation" of the 4D choice, in the sense you seek. My idea still in progress should finally end up with something like that the choice of STRUCTURE of the set of distinguishable events that constitute topology, dimensionality somhow has an explanation in terms of something like "most efficient" representation and the this structure on the relations between matter is thus somehow the "most stable" one. So while it is not the only LOGICally possible one, it may be understood as the most rational one.

$$Z \sim \sum_\text{dim}\,\sum_\text{top}\,\sum_\text{diff}\,e^{-S}$$

I think the spirit of your summation here is not too unlike set theoretic approaches.

$$Z \sim \sum_\text{dist complexions}\ e^{-S}$$

Thus, I'd propose that the question should not be how to make sense out of extending the state space in the equipartition to insanity, but rather how to understand, how a controlled set of complexions with NO further structure besides possibly a partial order or so, beeing the from a given observers "inside view", spontaneously forms the structure that by
"conincidence" sprinkles just as a dim x top x diff or whatever. The classification seems about as arbitrary as decomposing a set into subsets and then construct a direct sum or something like that. I mean a distinguishable possibility (state or path) counts as one regardless, right?

I mean does it seem like a very plausible abstraction to start out with an instrinsic view where we simply count the distinguishable states. Wether they classify as different topologies or dimensions, SEEMS to me to almost be close to just a classification label that does not make a difference to the counting (from a first principle starting point). The points where it does make a difference is where the sprinkle patterns are stable enough to appear frozen, and thus suddently we forget to COUNT (or factor it out of the Z) as constants - but which really aren't constants.

The thinking is I suppose along the lines you think of. I really do think that it's a good way to approach this, and eventually there will be an answer.

/Fredrik

 

[tom.stoer]

Because the M theory sugra limit has a 3-indexed antisymmetric tensor.

Is there some deep reason behind that statement? What is the difference to any other dimension or any other tensor in any other theory? If this SUGRA limit favours dim=4, what favours exactly this SUGRA limit?

 

[Orbb]

This is not quite a deep thing, and definitely not singling out 4D over anything else, but what's peculiar about 4D is that the hodge dual of any 2-form is again a 2-form. In GR, this is what makes possible the switch from the palatini action to the holst action, which is equivalent on a classical level. There is an analogous modification to the ordinary yang mills action in 4D. The specific form of the additional term is unique to 4D.
This was quite imprecise as I don't have much time, if you're confused about what I mean, I am appy to expand.

Edit: as for the SUGRA limit, I think this is just the unique low energy limit of 11D M theory. It is unique because the representation theory of SO(10,1) (or more precisely its little group SO(8) ) and SUSY constraints uniquely fix the massless field content in 11D. I don't know about the relation of the 3-form to 4D though.

 

[oldman]

Folk here are looking for a reason consistent with the maelstrom of mathematical models that is current theoretical physics --- I can't help in that context

Perhaps if the answer to the old philosophical chestnut "why is there something rather than nothing" is sought in topology a clue might emerge.

We've known for a long time that for minimalist knots (in a one-space-dimensional line) to exist, two extra space dimensions are needed --- one for the line to curve in, and another for the line to cross itself in. This makes three space dimensions in all. More than two extra dimensions allow knots to come untied.

So knots (which are "something"?) need at least three (but no more than three) space dimensions in which to exist (in a sea of unknots, which is "nothing"?). As for a fourth dimension, time --- well, I can't help there, either!

Could topology be important in the microcosm?

 

[tom.stoer]

This is not quite a deep thing, and definitely not singling out 4D over anything else, but what's peculiar about 4D is that the hodge dual of any 2-form is again a 2-form. In GR, this is what makes possible ...

There are a lot of special features of 4-dim. manifolds (you mention some of them; Penrose tried to use some others in his Twistor approach) but up to now I haven't seen how to use them in a dynamical setup, i.e. where these special features by themselves DO something - instead of being used CONSTRUCT something. This is what my last idea should indicate: treat all manifolds on the same footing and find a "dynamical" reason that singles out D=4.

 

[tom.stoer]

Could topology be important in the microcosm?

This is an interesting question. Many believe that at the microscopic level spacetime becomes discrete, so there may be no topology, just algebra. Nevertheless I try to start with a topological idea ...

 

[Fra]

Ane may ask which algebra? do we pick one at random? infer from experiment? infer from interactions? IE. even infer from te structure of measurements and ordering of events?

An idea is to infer this from more basic distinguishable partially ordered set relations. Any measurement process and observations naturally imply an order.

For example (this is one of the few papers I'm aware of that's at lesat in the ballpark)

Information Physics: The New Frontier, Kevin Knuth
"At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the process of inference, which make it clear that these are inferential theories. That is, rather than being a description of the behavior of the universe, these theories describe how observers can make optimal predictions about the universe. In such a picture, information plays a critical role. What is more is that little clues, such as the fact that black holes have entropy, continue to suggest that information is fundamental to physics in general.
In the last decade, our fundamental understanding of probability theory has led to a Bayesian revolution. In addition, we have come to recognize that the foundations go far deeper and that Cox's approach of generalizing a Boolean algebra to a probability calculus is the first specific example of the more fundamental idea of assigning valuations to partially-ordered sets. By considering this as a natural way to introduce quantification to the more fundamental notion of ordering, one obtains an entirely new way of deriving physical laws. I will introduce this new way of thinking by demonstrating how one can quantify partially-ordered sets and, in the process, derive physical laws. The implication is that physical law does not reflect the order in the universe, instead it is derived from the order imposed by our description of the universe. Information physics, which is based on understanding the ways in which we both quantify and process information about the world around us, is a fundamentally new approach to science."

"First, I will rely on ordering relations to obtain algebraic operations that have specific symmetry properties. I will then use these symmetries to place strong constraints on any quantified description. The resulting constraints correspond to the physical laws."

"We can choose to view the join and meet as algebraic operations that take any two lattice elements to a unique third lattice element. From this perspective, the lattice is an algebra."

"We will see that this is equivalent to extending an algebra to a calculus by defining functions that take lattice elements to real numbers."

"The logic of the process of measuring served to generate the algebra, which implicitly defines a poset of measurement sequences."
-- http://arxiv.org/abs/1009.5161

I think that's a really good paper, although there is quite a leap from there to selecting some 4D structures. But I think there is some good analysis in that paper, even though as always not compelte or perfect (I do not like his mapping to real numbers, but I imagine the construction be be done starting in a similar faishon, but instead use other counting system); which means it won't be continuum calculus though. but I think this may be the step needed to also unify and therefore select dimensionality, so that you can measure sets of different dimensionality against each other in a rational way.

/Fredrik

 

[czes]

This is an interesting question. Many believe that at the microscopic level spacetime becomes discrete, so there may be no topology, just algebra. Nevertheless I try to start with a topological idea ...

According to developed recently holographic principle at the microscopic level may be is just an alone algebra.
What if the dimenssions are the relations only ?
The vacuum is relation between virtual particle and antiparticle. To see the empty vacuum we have to relate it with a third information. Therefore a perfect empty vacuum is 3-dimensional. Any distortion in the vacuum (massive particle) needs more relations and it creates Minkowski's spacetime. A complex structure of the proton needs even more relation to describe it as a particle.

The space around us is almost empty so it is enough 3 relations as in empty vacuum. Sometimes we use a fourth dimension as an approximate sum of the more relations (4, 10, 150, number of particles in the Universe as in Bohm)

 

[Fra]

The interesting part is, how to understand the choice of algerbra.

To in some ad hoc postulate an algebraic structures then one might as well postulate the 4D directly, since I don't see one more or less plausible.

The set theoretic approch, tries to argue from starting with a partially ordered set, then argue that certain operations on this set, then can defined the algebratic operations, thus extending the set to an algebra. But the interesting part is to try to understand that starting points.

Why is it plausible to assume that posets are good abstractions? I don't think was outlined in that paper but IMO the measurement perspective, naturally leads to ordering structures of the set of events.

Then algebraic operations can then be interpreted as construced from logical operations on these sets. And the sets themselves can be interpretes as constituting the information state (historical events). And the logical operations would correspond to physical internal processes.

This allows for an IMO pretty deep stance, and if things can be constructed from this it would be very nice as it would rest on plausible ground. Formally these would also be "postulates" but IMHO extremely plausible ones and as close to minimalist thinking one can imagine.

This may suggest that at some level of development, there should be some reason for a particular algebra, or group can be understood in terms of basic postulates of structures of measurement records and processing of data; unifying at deep level representation with processing and evolution.

Instead of thinkg of just abstract "mathematical" representation and operations, there seems to be an infinity of them. So the exploit I think underlying the paper above is to let measurements, and the presumed plausible properties of representation and processing of INFORMATION guide is in this process.

Edit: This is why I do not think we will ever find an answer in terms of "logical necessity". The answer will be of the form, that it would be the most rational constructing principle; all in the original spirit of "inferential theories". The status of these theories are not logical necessity, beucase the whole counter argument is that there are generally an infinity of "logically consistent" such views and NO SELECTION principle. (Just look at ST). This is one of the basic motivators for inferential theories. It seems to be even the heart of a good scientific method (IMHO at least); all scientific theories are in fact inferential, wether we think so or not. Sometimes we forget and thing they correspond to some eternal reality. This is a completely irrational viewpoint IMHO.

/Fredrik

 

[Fra]

If were looking for precursors of lorentz symmetry etc, Knuth and Bahreyni has this paper as well (I know this was discussed be fore but I figure it would fit in here).

A Derivation of Special Relativity from Causal Sets, Knuth K.H., Bahreyni N. 2010

"This leads to the concept of a partially-ordered set of events, which is called a causal set. Quantification proceeds by selecting two chains of coordinated events, each of which represents an observer, and assigning a valuation to each chain...

...Pairs can be decomposed into a sum of symmetric and antisymmetric pairs, which correspond to time-like and space-like coordinates. From this pair, we derive a scalar measure and show that this is the Minkowski metric..."

-- http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.4172v1.pdf

Of course the do not say anything about dimenstionality! But I think these types of analysis, may be building blocks in the full picture.

/Fredrik

 

[czes]

If were looking for precursors of lorentz symmetry etc, Knuth and Bahreyni has this paper as well (I know this was discussed be fore but I figure it would fit in here).

A Derivation of Special Relativity from Causal Sets, Knuth K.H., Bahreyni N. 2010

"This leads to the concept of a partially-ordered set of events, which is called a causal set. Quantification proceeds by selecting two chains of coordinated events, each of which represents an observer, and assigning a valuation to each chain...

...Pairs can be decomposed into a sum of symmetric and antisymmetric pairs, which correspond to time-like and space-like coordinates. From this pair, we derive a scalar measure and show that this is the Minkowski metric..."

-- http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.4172v1.pdf

Of course the do not say anything about dimenstionality! But I think these types of analysis, may be building blocks in the full picture.

/Fredrik

I agree with you, Fredrik. The idea that Quantum Events are fundamental is very helpful.
I would like to add that if each Quantum Event encodes the constant Planck time dilation we may very easy derive the space-time of General Relativity.
The sum of Quantum Events (Knuth) creates the General Realtivity (Einstein) then.

 

[arivero]

Because the M theory sugra limit has a 3-indexed antisymmetric tensor.

Is there some deep reason behind that statement? What is the difference to any other dimension or any other tensor in any other theory? If this SUGRA limit favours dim=4, what favours exactly this SUGRA limit?

Yes, there is a deep reason.

What is nowadays called "The M theory limit" was called time ago N=1 D=11 SUGRA, and it was the maximal supergravity theory, ie the biggest susy theory with spin less or equal than 2. In this theory, there is a single fermion with 128 components, but the graviton only has 44. So one must provide another 84 bosonic components. The answer is this 3-indexed antisymmetric tensor, with in dimension 11 has exactly 84 components. The 3-index has two roles: it is the source of a 2-brane "C-field" in D=11, and it favours a compactification of the metric into two separate, 4+7, sectors. Google for Freund-Robin.

 

[tom.stoer]

... and it favours a compactification of the metric into two separate, 4+7, sectors

What is the dynamical reason for favouring 4+7 and not something else?

 

[arivero]

What is the dynamical reason for favouring 4+7 and not something else?

Freund-Rubin mechanism.

 

[arivero]

http://www.google.es/search?q=Freund-Rubin+mechanism

 

[fzero]

There's no particular reason that Freund-Rubin singles out 4+7 over 7+4.

 

[Fra]

A question.

I have no detailed knowledge and motivation of how/why this supposed M-theory limit or SUGRA 11d theory is constructed but just to see if I get the overall argument.

I don't know what "status" the Freund-Rubin mechanism has, but let's SUPPOSE it's a theorem that can be proved (I don't konw, I tried to google but the orignal paper seems to be ones you have to pay for).

The as far as I can see it seems the inference goes likes this.

IF we have a theory X (m-theory limit, sugra 11d or what it may be called) that contains a 3 rank antisym tensotr field, THEN a dynamical stabilisation into 3+1; and 11-(3+1) ie 4+7.

Is this correct?

Then assuming the mechanism is a theorem and true, then the weight of the argument rests upon what reason we have to believe in theory X; including it's 3 tank antisym. tensor field in the first place?

Isn't rather this fact (assuming it us, I can not judge) rather than "explaning 4D", merely indirectly support the theory X in the first place? Then the question is what is the biggest assumption in the first place. The complex theory X, or spacetime beeing 4D?

Am I missing something?

It's an interesting question, so hopefully someone can explain briefly the overall inference. (details such as what steps qualify as proofs, or merely conjectures is not the primarily interesting thing)

/Fredrik

 

[PAllen]

I have no physical motivation, but a strong personal suspicion 4-d reality is tied the the fact that 4-d is the only case where the number of differential structures is not finite.

 

[Fra]

I have no physical motivation, but a strong personal suspicion 4-d reality is tied the the fact that 4-d is the only case where the number of differential structures is not finite.

Is there something somewhere about this to read? papers?

I sounds like a interesting thougt. It sounds like you suggest that it may be the continuum itself (beeing infinite) simply has no other place to "hide"? I like the sound if it but I'm curious as to exactly how you "count" these structures (this would be the key), as to arrive that 4D is the most probabl state.

Edit: it sounds like your idea must include a whole framework for constructing reprenstation systems in a way that they can be counted and thus assigned probabilities. This is quite in line with my thinking...so I'm curious to see if it's possible something existing that may be of interest for me.

/Fredrik

 

[PAllen]

Is there something somewhere about this to read? papers?

I sounds like a interesting thougt. It sounds like you suggest that it may be the continuum itself (beeing infinite) simply has no other place to "hide"? I like the sound if it but I'm curious as to exactly how you "count" these structures (this would be the key), as to arrive that 4D is the most probabl state.

Edit: it sounds like your idea must include a whole framework for constructing reprenstation systems in a way that they can be counted and thus assigned probabilities. This is quite in line with my thinking...so I'm curious to see if it's possible something existing that may be of interest for me.

/Fredrik

Here is one recent paper:

http://arxiv.org/abs/1005.3346

However the result goes back a few decades. 4-d unique in this way among all dimensionalities.
See the wikipedia article on differential structures, but I encountered this fact first when reading about proofs of the Poincare conjecture for dimensions greater than 3 (at the level of the 'sophisticated layperson', which is the strongest claim I can make for myself).

 

[arivero]

However the result goes back a few decades. 4-d unique in this way among all dimensionalities.
See the wikipedia article on differential structures

Yep, 4 is very unique, but 7 is also unique in a close sense, think Milnor spheres. While these results are important, I don't know of any proposal to relate them to the compactification mechanism, ie to the 4+7 split we were discussing. IMO, these results could contribute to control the isomorphisms of the compact manifold, note that the 7 sphere is a fiber of the 3-sphere over the 4-sphere, than the 4-sphere has a branched covering by CP2, and that the family of fiber bundles of 3 spheres over CP2 have as group of isometries the infamous U(1)xSU(2)xSU(3) (or SU(2)xSU(2)xSU(3) in some extreme cases).

 

[fzero]

Yep, 4 is very unique, but 7 is also unique in a close sense, think Milnor spheres. While these results are important, I don't know of any proposal to relate them to the compactification mechanism, ie to the 4+7 split we were discussing. IMO, these results could contribute to control the isomorphisms of the compact manifold, note that the 7 sphere is a fiber of the 3-sphere over the 4-sphere, than the 4-sphere has a branched covering by CP2, and that the family of fiber bundles of 3 spheres over CP2 have as group of isometries the infamous U(1)xSU(2)xSU(3) (or SU(2)xSU(2)xSU(3) in some extreme cases).

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.

 

[Fra]

Here is one recent paper:

http://arxiv.org/abs/1005.3346

However the result goes back a few decades. 4-d unique in this way among all dimensionalities.
See the wikipedia article on differential structures, but I encountered this fact first when reading about proofs of the Poincare conjecture for dimensions greater than 3 (at the level of the 'sophisticated layperson', which is the strongest claim I can make for myself).

Thanks for the hints PAllen! I'l look that up.

/Fredrik

 

[tom.stoer]

I have no physical motivation, but a strong personal suspicion 4-d reality is tied the the fact that 4-d is the only case where the number of differential structures is not finite.

This is exactly what I wanted to express in the starting post. The problem seems to be that this argument relies on differentiable manifolds - which contradicts the expectation that the fundamental structure of spacetime may not be smooth but discrete.

Yep, 4 is very unique, but 7 is also unique in a close sense, think Milnor spheres. While these results are important, I don't know of any proposal to relate them to the compactification mechanism, ie to the 4+7 split we were discussing.

Topologically (purely mathematically) 7 and S⁷ is not very special. It becomes special only if you use it in the 11-dim. SUGRA context and if you discuss 11=4+7.

 

[Fra]

The problem seems to be that this argument relies on differentiable manifolds - which contradicts the expectation that the fundamental structure of spacetime may not be smooth but discrete.

I started to skim the references and my first impression is along with Toms, BUT, OTOH it may not be as much of a problem as there is nothing whatsoever I know of that suggests (on the contrary) that 4D is correct at microstructure level... thus a reasoning somewhat in the line with this:

"Given a random selection among discrete structures, in the large complexity limit (continuum limit) then the probability of ending up with 4D at large scle is high (or Dominating))"

Though, I assume the exact premises in that theorem must be analysed to see that it doesn't somehow break down by considering a particular limiting process instead of starting out with an actual continuum to start with. Because to merge these ideas, the EXACT 4D limit is then strictly speaking not physical, but then close enough to 4 would be good enough to be consistent with what we know.

For me that is the form of an argument that I would think is good. However, that means that some OTHER guidiing principle is neede to understand the microstructures of things (including spacetime), and that at THIS level dimensionality has nothing to do with anytihng.

So I think this general reasoning is good. Except I think that just like Tom says, there is a huge gap to be analysed here... and I have to spend more time to understand how they are counted and wether the counting has physical justification (which is far from obvious to me at least).

This is why I think the way is to; replace the continuum stuff with discrete constructions; associate this to a real measurement case and a real observer; and they see if the scaling of the construction as the observer complexity -> infinity, still makes the theorem converges nicely to the actual limit os that the 4D conclusion holds also for sufficiently large complexity and not just the acual limit (probalby it does).

Then the idea I envisoin is that an intrisic observer dependent probability concept is part of the construction, so that one can justify what it means with "probability of a theory"... this is why I insiste to get the discrete measurement/obserer picture entangled with this.

If one just starts, without observer, and just flatly talks about continuum structtures in the pure mathematical sense, I think the connection to physics is somewhat lost. But it seems as per above that this may be curable, at least it does not seem obviously hopeless.

/Fredrik

 

[suprised]

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.

 

[Fra]

One immediate fear that comes up is, wether the counting atually can be constructed in the limiting process, or wether it's merely the limits that are counted, and then if the limits are non-physical, it seems the theorem needs first to be generalized to say "discrete differentials".

By conincidence these things appear natural in the ponderings I'm working on. I'm starting with sets of discrete sets (to be thought of as ordinary microstructures corresponding to mesaurement events; encoded by internal processing (this is the cause of non-commutative structures)) and each such microstructure naturally by means of combinatorical reasoning (~relative entropy) defines differentials living on a kind of tangent plane to to each point(microstate) on the microstructure.''. The dynamics in each picture is given by randomwalks.

One problem is to predict what large scale structure that appears in the space part of communicating structures... it would be head on if an argument could be made along the lines that as the complexity increases; the probability for 3D space (4D spacetime) is dominating. Thats exactly the kind of argument thta would be beautiful; provided that a physical justification for the coutning (and thus probability) is supplemented.

/Fredrik

 

[Fra]

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.

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

 

[arivero]

Topologically (purely mathematically) 7 and S⁷ is not very special. .

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.

 

[arivero]

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.

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