Exploring the Role of SU(2) in Spin Networks

[tom.stoer]

In the context of LQG spin networks are derived based on Ashtekar's formulation implementing local SU(2) gauge symmetry in tangent space. Here SU(2) and 3+1 dim. spacetime are deeply related.

Forgetting about this derivation and starting with spin networks, talking about dimensions does no longer makes sense; a spin network or a graph w/o embedding in a manifold does not have a dimension. So all what remains from 3+1 dim. spacetime is the SU(2) spin network.

That means we can translate the question "why do we live in 3+1 dim. spacetime?" into "why does spacetime emerge from SU(2) spin networks?". So the basic question is "why SU(2)? Why not SU(N) or something else?".

The only reason for SU(2) I know is based on SO(3,1) ~ SU(2) * SU(2), but this argument is no longer valid as soon as we drop 3+1 dim. spacetime as our starting point.

Any ideas?

 

[marcus]

A smooth manifold has a lot of (a continuum of) other structural assumptions besides its dimensionality.

For example, embedded graphs can knot---get tangled.

You can throw away the continuum but keep ideas related to largescale perceived dimensionality---maybe SU(2) is such an idea.

If, as well, SU(2) were somehow inherent in another aspect of the situation, e.g. matter, I expect you would alert us to that.

 

[MTd2]

There is a list of similar threads on the bottom of the page when I accessed this thread and I found this:

https://www.physicsforums.com/showthread.php?t=164632

Is that useful? The thread was started by coin's 1s post.

 

[Fra]

So the basic question is "why SU(2)? Why not SU(N) or something else?".
...
Any ideas?

I want to first say I have no definite idea.

But fwiw, if I may add a general vision without necessarily having a specific idea that connects LQG directly?

Conceptually, as I currently interpret the LQG spin networks, it is a subset of the observer total microstructure system (what in TOTAL encodes the observers information, we can also loosely associate it with a subset of a hilbery space), and in particular the part that concerns certain relations about distance and inertia about neigbouring systems (what we usually call spacetime).

The kind of construction that I picture migth be able to answer the question Tom asks, is if you consider the constraints for a consistent interaction between several of these observers (but considering their FULL information sets). This consistency amounts to asking what relations exists between the observers if we assume that they are in equilibrium and thus are STABLE. So the quest is, what is the structure of the relations between then (ie. "spacetime" wether discrete or not! we have not made any assumptions yet) that an equilibirum condition requires.

So to answer the question, I would suspect thta we need to take matter into account. This is like a chicken and egg situation though, since all our abstractions of matter, depend on a RELATION to the external spacetime. IF spacetime as we know it breaks down, so does the structure of SM as spacetime is part of the inference machinery of SM. And as per my argument, the other way around.

This is why I think the picture is an evolution, where the answer Tom asks, and the question of what are the symmetries and structure of matter, evolve together and we must try to understand the evolution and seek for the conditions for an equilibrium condition.

This for me defines at least a direction where to make further investigations, which may or may not lead to a more definite idea at some point.

Edit: I'm not sure if any one-legged analogies are needed but I think the general idea is clear anyway. Otherwise it's not alien to nash equilibrium in economy, it's an equilibrium where are players have no benefit in changing their actions. So ath te equilibirum the strategys of all players(think observers;matter) are STABLE, and the relations between the players(~think spacetime) are also stable. The analogy sure isn't perfect but it's an alternative source of intuition for the idea.

/Fredrik

 

[marcus]

That means we can translate the question "why do we live in [macroscopic] 3+1 dim. spacetime?" into "why does spacetime emerge from SU(2) spin networks?".

I was stumped by your question.

I don't think we know the dimensionality of our space at very small scale, only that it appears 4D at ordinary medium to large, and to continue 4D in largescale limit. So I put "[macroscopic]" into your statement. That was just a quibble, though. I couldn't answer or even guess where an answer might come from.

CDT does not offer a clue either. One recognizes we live in 4D macro dimensionality---you use 4-simplices to construct CDT. That doesn't guarantee what the dimensionality will be at any given place or scale. It can still vary. And indeed it gets much less at small scale. But you put in a bias for macro 4D by your choice of building block.

In LQG you put in a bias for macro 4D by your choice of group. And as in CDT you seem able to get spontaneous reduction of dimensionality in LQG at small scale (I'm not as sure about that, but Steve Carlip's recent paper supports it.)

Maybe it is just how the QG game is played at this stage in history. The rules say you get to make one choice of something (a group, a simplex building block) that will in some way bias the outcome in favor of a preferred macro-scale dimensionality.
But your choice might not totally determine the dimensionality at all places and scales. It still might leave room for uncertainty, or for dimensionality at a given scale somewhere to be a quantum observable.

At least for now I don't feel I have much that's useful to say about this.

 

[tom.stoer]

https://www.physicsforums.com/showthread.php?t=164632

Is that useful?

I skimmed through the thread an some papers (some of them I studied some toime ago) but couldn't find something that relates to my question; of course I will check John Baez website as well.

I think I should explain what I have in mind: let's forget about the history of LQG. suppose somebody invents SU(2) spin networks, writes down an Hamiltonian or a path integral and shows how to derive a low energy effective action for gravity in 3+1 dim. spacetime. Presenting this idea he/she will be asked: "have you ever tried this with SU(N) instead of SU(2)?". The answer wil be something like "no, I know that SO(3,1) ~ SU(2)*SU(2) so it's natural to start with SU(2) in order to derive GR are as low energy limit". Of course this is OK but then one will check the details of his/her work. There it will be shown that the hard stuff is to make sense of something like dimension. "How do you derive that spacetime is 3+1 dim.?". The answer will be something like "this is really hard stuff; it's something like a spectral dimensions; the underlying colored graph does not have a dimension at all". OK, everybody understands, but then they will come back with "but how does this change if you use a different coloring? what about SU(N)? does N affect the spectral dimension?". OK, that no fair. Suppose you invent LQG from scratch and somebody asks you why not doing all this with SU(N). Have you ever compared explicit calculation in SU(N) with SU(2). It can become a nightmare! But of course that's not the right answer. Next question: "we understand that the basic reason to start with SU(2) is the symmetry structure of spacetime; but is there a second reason, unrelated to low-energy stuff which you want to derive? what would you do if nothing is known regarding 3+1 dim. spacetime? what would be your starting point?"

So my question is if there is a physical reason to select SU(2) - except for the fact that we know it must somehow be SU(2)?

 

[sheaf]

It may not really answer your question, but it's interesting to look at Roger Penrose's original motivation for spin networks

The most obvious physical concept that one has to start with, where quantum mechanics says something is discrete, and which is connected with the structure of space-time in a very intimate way, is in angular momentum. The idea here, then, is to start with the concept of angular momentum - here one has a discrete spectrum - and use the rules for combining angular momenta together and see if in some sense one can construct the concept of space from this.

(Quote is from Quantum Theory and Beyond, edited by Ted Bastin, Cambridge University Press 1971, pp. 151-180. )

His most primitive object with quantized angular momentum was a spin 1/2 particle, so he started from there and SU(2) is immediately relevant.

 

[Fra]

use the rules for combining angular momenta together and see if in some sense one can construct the concept of space from this.

The question is, if we seek a first principle motivation like I think Tom does, from where do you get the notions of angular momentum and it's specific combination rules if there is no space? As I've learned the spin ½ rules, the spacetime symmetries is essential for infering it. For example SR and spin ½ seem related, and I see no way to get there if there is no space at all to strat with. It's the consistency requirement of a spacetime-relativistic formulation of QM that pretty much gave us this.

This is the main reason I do not like the SPIN part of the entire spin netowork business. The units of spin is the same as units of action, I think a more spacetime netural picture, should be phrased more abstractly in terms of directed action networks;, where the SPIN network association only makes sense once some precursots of space is selected.

/Fredrik

 

[suprised]

The devilish landscape is lurking around every corner...hard to escape!

Of course other groups can have spin reps too.

 

[tom.stoer]

The devilish landscape is lurking around every corner...hard to escape!

You are absolutely right - unfortunately.

Of course other groups can have spin reps too.

Of course you are right again - unfortunately.


So the physical principle is that we observe 3+1 dim. spacetime and look for a structure from which it may emerge. SU(2) spin networks seem to be rather natural.

To be honest, I can think of no principle beyond that - except for the following crazy idea: many Lie groups have SU(2) or SO(3) as subgroups. So perhaps it's irrelevant from which group one starts, provided that there is a kind of symmetry breaking mechanism which reduces a given (arbitrary) symmetry down to SU(2) or SO(3).

 

[marcus]

...
If, as well, SU(2) were somehow inherent in another aspect of the situation, e.g. matter, I expect you would alert us to that.

Now you have alerted us, as might have been expected:

- except for the following crazy idea: many Lie groups have SU(2) or SO(3) as subgroups. So perhaps it's irrelevant from which group one starts, provided that there is a kind of symmetry breaking mechanism which reduces a given (arbitrary) symmetry down to SU(2) or SO(3).

Crazy idea = matter results from symmetry breaking in geometry, from some higher dimensionality down to four.

As a matter of honor we are obliged to briefly mention Zeitgeist crazy elusive ideas even though we refrain from pursuing what is (for the moment at least) too vague.

 

[suprised]

Such ideas, namely that gravity arises from SU(2) (in a sense the "simplest" object) and extra stuff arises from the embedding of SU(2) in larger groups, have been coming up from time to time in various contexts. For example W-gravity, which isn't at all motivated physically, but indeed one should always be aware that in a given construction several choices are in principle possible and one should always address the question: why this choice and not any other one. Typically one tries to answer this by saying choice XXX is the simplest, or an extremal "largest possible" one.

 

[sheaf]

Interesting discussion.

I wonder if you started with an G - spin network, where G is some bigger group having SU(2) as a subgroup, then performed the semiclassical coherent state approximation technique referred to in the New Look paper, what dimensionality of manifold you would end up with...

This of course is assuming you could define such a spin network consistently.

 

[Fra]

The discussion is converging! :)

Typically one tries to answer this by saying choice XXX is the simplest, or an extremal "largest possible" one.

Which I think brings the focus to the construction of a plausible measure of simplicity?

From the inference perspective, a natural construction of "simple" may be the "most probable" choice, in the sense of the choice that is in minimum disagreement from our expectations. Ie. the "simplest" or "minimalist" speculation. That would be a rational expectation.

But how can we get a physical handle on this? How do we even go about to count the set of possible choices?

why this choice and not any other one.

If we can get a more physical interpretation of these choices in some way, the situation becomes similar to biological evolution. Why does certain animals populate this planet and not others. This "landscape" is analogous to the problem we discuss here.

No sane educated evolutionary biologist would come up with the silly idea of constructing a fixed eternal mathematical measure of fitness. It is bound to fail; and it's just the wrong way to address the problem.

So I think the correct view even in physics, is to look at the evolutionary perspective, which in these terms would mean breaking as well as emergence of symmetries. So doesn't it seem very rational to focus in this? And try to get a physical abstraction around this?

/Fredrik

 

[suprised]

Yes of course this is nothing but the "landscape" problem in disguise. Typically the string theorists are accused for not being able to make predictions, while other approaches just make a choice beforehand and claim, their model would have "predictions" - be it Connes-, Lisi-, Randall-Sundrum models or whatever. Of course a string model, once a background choice is made, yields very concrete predictions as well, in fact infinitely many ones. But usually this is discounted because string theory does not provide a means to choose this background. But to be fair, one should also present a list of possible choices in any other approach. That includes e.g. also the choice of space-time dimension.

So basically there are only two possibilities: the "true" model is selected by some mathematical principle, being simplest or extremal according to some scheme, and physics somehow is sensitive to this principle, and selects this model by some unknown mechanism. This possibility has been long preferred but after decades of research, nothing concrete ever came out.
The other possibility is anthropic. .. there is nothing special about our world apart that from it is hospitable to us and that's why among many possible choices we see just what we can possibly see.

 

[marcus]

Interesting discussion.

I wonder if you started with an G - spin network, where G is some bigger group having SU(2) as a subgroup, then performed the semiclassical coherent state approximation technique referred to in the New Look paper, what dimensionality of manifold you would end up with...

This of course is assuming you could define such a spin network consistently.

It's interesting you mention the "new look" LQG paper (1004.1780) in this connection. In the new presentation of LQG some of the spin networks are labeled (not with SU(2) irreps but) with elements of the group SL(2,C).

Maybe there is some potential there, something to be gained by considering spin networks labeled by group elements--with the possibility of using a larger group containing SU(2).

The short condensed exposition is section E on page 4 of the April paper 1004.1780. But a longer more detailed discussion, as I recall, is found in the recent Bianchi Magliaro Perini papers on the holomorphic coherent states of LQG.

Now you have gotten me curious about equation (15) of the April paper which is a kind of "convolution integral" over the SU(2)^N group manifold. Since commenting might distract from the discussion here in this thread, I will take your idea over to the "Five QG principles" thread, which is specifically about things that come up in the April paper, and comment there. Hope this is OK. It will keep the side-discussion out of Tom's way.

 

[Fra]

Yes of course this is nothing but the "landscape" problem in disguise.

It seems several on here now can agree about this, in despite of having different perspectives :) Unusually constructive I'd say.

Let's see if we can dicsuss further on from this point and discuss possible "solutions" to this "landscape problem"; that seems to prevent our predictive power.

Typically the string theorists are accused for not being able to make predictions, while other approaches just make a choice beforehand and claim, their model would have "predictions"

I think this is a valid point, but I suggest that there is a detail in all this that may make a difference - not between ST and LQG, but to your classification of the options we have from this.

So basically there are only two possibilities: the "true" model is selected by some mathematical principle, being simplest or extremal according to some scheme, and physics somehow is sensitive to this principle, and selects this model by some unknown mechanism. This possibility has been long preferred but after decades of research, nothing concrete ever came out.
The other possibility is anthropic. .. there is nothing special about our world apart that from it is hospitable to us and that's why among many possible choices we see just what we can possibly see.

Maybe your simplifying but you describe here two extreme options, but for the discussion I think we can without taking any side can agree that we have here a something we may call the "landscape problem" in the general sense that is also a "landscape problem in disguise" as you said, even beyond the context of ST.

This "problem" can be phrased as a lack of uniqeness in the deductive system - THIS is what blurs the predictive power (or inference to connect to the). We rather end up with a set of POSSIBLE deductive inference.

So where does this leave us? We are in a situation where we need to not only make inferences, but we need to even INFER the inference system.

It may seem like a scientific method to simply enumerate the set of possible deductive systems, and try them one by one to be able to falsify them. But for large sets, or even uncountable sets, this just doesn't work well, even in theory.

So, is there a better way to think of this?

After all, the basic problem is still to make inferences about nature, right? IF that implies first inferrring the inference systems itself, so be it. It makes the problem more complex, but it doesn't change the basic quest. In despite of the apparent impossibility of making perfect deductions, this does NOT change the basic quest of making the BEST inference (of which deductions is a special case only), would you agree?

I'll stop there as a checkpoint and see how many of us that have our own pet ideas, that can still agree with me on this? I'm making it a point of beeing generic since it allows for program-independent agreement and I think it's good for discussions.

It was a progress that Tom and Surprised seems to reach some kind of agreement here on the problem of different possible deductive systems. It's tempting to try to push this further.

/Fredrik

 

[atyy]

If, as well, SU(2) were somehow inherent in another aspect of the situation, e.g. matter, I expect you would alert us to that.

Yes, something like that is my preferred line of thinking. The two lines of are contrasted by http://arxiv.org/abs/1004.0672

"Remarkably, summing over the gravitational degrees of freedom, the effective matter amplitude was seen to arise as the Feynman diagram of a non-commutative field theory [8]. To add to this position, it was shown that an explicit 2nd quantised theory of this gravity matter theory could be provided by group field theory, while later the non-commutative field theory was seen to arise as a phase around a classical solution of a related group field theory [9]. Of course, one may approach the subject with the view that one should discretise the field directly on the spin foam, since in the continuum theory, we expect that the field has a non-trivial energy-momentum tensor, and should affect the state sum globally. This method has yielded to a succinct initial quantisation for Yang-Mills and fermionic theories [4–6], but due to the non-topological nature of the resulting amplitudes, further calculations proved unwieldy. Now, it was not our intention that this work would or should settle this debate, but we find that this theory is more in line with the arguments of the former way."

 

[atyy]

Crazy idea = matter results from symmetry breaking in geometry, from some higher dimensionality down to four.

As a matter of honor we are obliged to briefly mention Zeitgeist crazy elusive ideas even though we refrain from pursuing what is (for the moment at least) too vague.

See, string theory works!

http://arxiv.org/abs/1004.0621
These variables encode extra dimensions.

 

[Fra]

So perhaps it's irrelevant from which group one starts, provided that there is a kind of symmetry breaking mechanism which reduces a given (arbitrary) symmetry down to SU(2) or SO(3).

I know Tom was asking a more specific question here, but if I may rephrase this into a more generic and less specific idea, that I think is not crazy, although controversial it's this:

Since it's clear here that the utility of the SU(2) here is that once it's chosen, the idea is that it will (together of course with other abstractions such as the network itself) loosely speaking "imply" or allow use to deduce certain specifics like dimensionality of spacetime.

So, what Tom said above can be rephrased so that "maybe it doesn't matter with what inference systems we start, provided there is a kind of process, which evolves it to the preferred system and thus helps us with the selection".

Perhaps Toms thinks in terms of deductive systems, but this is exactly how I think of infernence, and i think it's a good idea. It's also somehow also how I think anyone would picture the "ultimate" intelligence. Ie. ANY initial opinion or disturbed information should be able to by means of some process self-correct and adjust. This is the essence of systems that has learned how to learn. ie. something more like "true" intelligence rather than preprogrammed static search/fit/match/select algortihms.

So what think is that he process we would need to understand in Toms crazy idea, the symmetry breaking or arbitrary bigs ones into smaller ones, can be generalized to be the problem of how inference systems evolves. The SPECIFIC way how we exemplifies these systems is of course program dependent, LQG here their ideas, ST has theirs.

Also, I think that maybe reductionst ideas is not the only way: ie breaking big symmetries into smaller one by somehow tracing out degrees of freedom. This is somehow the easy way, not too unlike the decoherence iadeas to trace out the environment, But this is a strange idea since if we are forced to goto large and larger symmetrys to stay consistent then we must also ask whether the process whereby an inference system is increasing complexity is a physical process as well? and is this constrained by something? or does it make physical sense to without hesitation mathematicall just picture arbitrary amounts of information withing an infernece system?

My personal opinoin is that it does not. I think the complexity shouldl associate to mass or energy of the observer, and that if we take the reductionist view, it takes an observer of a certain complexity to ecode this. If this observer isn't around, we may need to resort ot a different idea as complementary. The reductionst idea does work to the exten that we are respecting the information bounds, but othe things must happen when we hit the limit.

I think there is a whole can of worms here, at the point where it seems we have a partial agreement that apparently any programs has "landscape-like" problems.

/Fredrik