# Holographic / entropic approaches - why SU(2)?

1. Feb 8, 2010

### tom.stoer

Let's forget about the history of LQG, that means let's forget about that it has been derived from GR in an appropriate formulation (Ashtekar variabes), let's forget about the constraints that had to be implemented; then we can simply use SU(2) spin networks as the starting point for a theory of quantum gravity.

The new discussion regarding entropic and holographic approaches (Verlinde, Smolin) are pointing into the same direction. One does not need the GR roots, the theory could be viable w/o its own history. Instead one has to turn this around and derive 4-dim spacetime, GR, dynamics etc.

Then there is one central question: why SU(2)

SU(2) is the unique remainig structure; it knows something about the 4-dim. spacetime as it is essentially one-half of SO(3,1) ~ SU(2)*SU(2). But if we forget about the history we must forget about SO(3,1), too.

So I can rephrase my question: why not SU(N), or SO(N), or some exceptional group?

Last edited: Feb 8, 2010
2. Feb 8, 2010

### arivero

Last edited by a moderator: Apr 24, 2017
3. Feb 8, 2010

### tom.stoer

The main point is that we cannot answer the question "why SU(2)?" w/o having the result already in mind.

4. Feb 8, 2010

### Fra

I sure don't have any answer but I do expect a satisfactory argument would go along the lines that the symmetries we see in nature are the simplest possible distinguishable symmetry, BUT where simplicity has a specific meaning of minimally speculative, and where the set of possible distinguishable symmetries are CONSTRAINED by the complexit of the observer. And if we start from the low complexity limit, then the complexity is so low that there choices are finite. Then question is then how these preferencs SCALE with complexity, up to the point where continuum like structures start to appear. The continuum does not exist at the low complexity limit IMO.

This is why I think that models like ST, are IGNORING part of the construction when STARTING with continuum structures. I think this may even be the seed to the landscape problems - or in particular the lack of selection principle.

I think this is an interesting question and I think there ought to be a good answer to this out there.

/Fredrik

5. Feb 8, 2010

### arivero

Hmm are we sure that this SU(2) is defined as coming from SO(3,1)?

The fundamental object flowing across spin networks is $$\hbar$$. Perhaps SU(2) is imposed because of it.

6. Feb 8, 2010

### arivero

In the original paper (which, incidentally, is titled "Angular momentum: an approach to combinatorial space-time") Penrose insinuates that SU(2) would be fixed by some kind of self-consistency: there should be rules to build a space from a group, and a group from a space, and you should get the original space (or the original group) after going across all the process and back. He seems to insinuate that only the peculiar properties of SU(2) allow for the game he is playing.

7. Feb 8, 2010

### Fra

Thanks for reminding of that paper, I've probably seen that paper before but I'll read it again.

What you describe in principle makes sense, as it describes an evolving almost circular process, where the so called "self-consistency" condition sound like it really is a simply some assumption of the existence of a steady state? (equilibrium) in this game, and this equiblirium state would then be characterised by this symmetry?

This could be an alternative view of things, that would be more to my taste than the way rovelli describes it.

Wether his reasoning is right is anothre story, but the general reasoning seems rational enough to me,. I'll get back when I read that

/Fredrik

8. Feb 8, 2010

### tom.stoer

I know that Penrose introduced spin networks quite early, but I have never seen this paper before; is there an online version available somewhere?

I would appreciate hints regarding self consistency for the "space-time symmetry structure".

@arivero: yes, I would say we can be sure; where else should this symmetry come from? We start with SO(3,1), reformulate it according to Ashtekar's approach and end up with SU(2) connections. The local gauge invariance formulated in the tangent space is just this SU(2) symmetry. Constructiing the Hilbert space uses SU(2) holonomies and cylindrical functions which eventually result in SU(2) spin networks. As far as I can see there is no choice.

9. Feb 8, 2010

### arivero

There is a hidden choice: that we have chosen to preserve angular momentum. This is a logical choice in the origin of mechanics; it is needed to formulate an action (which has units of angular momentum), or to formulate a contact transformation (which is the origin of Feynman method, following a idea of Dirac), or to grant Kepler's second law. Probably it is also the cause for the use of Riemaniann Curvature.

Now, it is true that we can preserve angular momentum in an arbitrary n-dimensional space time, so we are not limited to SO(3,1). On other hand, now I am not immediately sure about if it is possible, for any dimensionality, to represent and angular momentum J as a product of linear momentum p cross position x.

10. Feb 8, 2010

### Fra

http://math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf

/Fredrik

11. Feb 8, 2010

### Fra

Given what I know of penrose general reasoning since before, I'm quite sure I would interpret his own reasoning at right angle, but still share the main conclusion. But that has happened before regarding his gravitationally induced wave collapse ideas. He tries to use gravity to construct a collective collapse; I think the better way is to use the subjective collapses to EXPLAIN the emergence of gravity.

My hunch is that I am similarly inverting his "consistency requirement" regarding the symmetry. I'll try to catually read the paper before adding more ramblings.

/Fredrik

12. Feb 8, 2010

### Prathyush

Hi,
Can some one explain the question to me. I am familiar with the recent interest in holography and connection to gravity . I will be very helpful if some one can point me to resources for further reading on what spin networks are and their importance in quantum gravity and holography.

13. Feb 8, 2010

### tom.stoer

This is not a hidden choice; if you chose n-dim. Minkowski space you have SO(n-1, 1) as symmetry group. The generators Ta with (Ta)00 = 1 generate angular momentum.

It is rather simple; in 3 dim. angular momentum is defined as

Ja = εabc xb pc; a=1..3

Now you can identify εabc with the bc-component of the generator εa; that means you write it as

Ja = (εa)bc xb pc

Going from SO(3,1) to SO(n-1,1) means that you replace (εa)bc with (ta)bc where ta is the generator a of the symmetry group of n-dim. spacetime.

Be careful: in 4 dim. spacetime you have SO(3,1) ~ SU(2)*SU(2); this is a special situation and does not hold in arbitrary dimensions. In addition the dimension of the rotation group is no longer identical to the dimension of space. In n-dim spacetime i.e. in n-1 dim. space you have

dim SO(n) = dim SO(n-1,1) = n(n-1)/2

The "spacelike" subgroup SO(n-1) generates angular momentum.

Summary: chosing n-dim. Minkowski space automatically fixes the structure group. So my question "why SU(2)" is related to the question "why 4-dim. spacetime". Now if we forget Minkowski spacetime we need longer longer answer the question "why 4-dim. spacetime", but instead we have to answer the question "why SU(2)". Perhaps the latter question is easier ...

14. Feb 8, 2010

### arivero

Yes it seems that for 3 dimensions the interesting object is Ja while for generic dimensions the object to look is (xb pc-xc pb).

Were we to need really a vector J, is should justify three dimensions, should it?

I think that a way to understand why the dimension of the rotation group must be greater than the dimension of space is that the rotation group actually needs to label "areas" where the cosines and sines will act.

15. Feb 8, 2010

### arivero

You can guess the answer. Choose between:

- It is because of the coincidences between Lie algebras in low dimensions.

- It is because of R C H O.

Usually it is one or another, at the end :-D

16. Feb 8, 2010

### tom.stoer

Yes, for generic dimensions the object is slightly more complicated. Three dimensions are singled ouf if you insist on the pseudo-vector structure. You can trace it back similar to the tensor structure for (2,0) tensors like F for the electromagnetic field; only in four dimensions the dual *F has the same number of indices: let F be 2-form in D space; then its dual *F is an D-2 form, which means that 2 = D-2 is solved by D=4.

The dimension of the rotation group is fixed by the requirement that is generated by real, hermitean matrices. Geometrically the number of Euler angles grows faster than the number of dimensions.

Last edited: Feb 8, 2010
17. Feb 8, 2010

### atyy

18. Feb 9, 2010

### Fra

I just got half way through the paper last night, but I think Penrose isn't radical enough. I also recognise this so I read this sometime in that past.

These traits of his, are the ones I fully share in my own thinking:

+) "get rid of the continuum and build up physical theory from discreteness"
+) "build up space-time and QM simultaneously from combinatorical principles"
+) "continous concepcts emerget in a limit when we take more complex systems"
+) His search for so called "pure probabilities" which are rational numbers, rather than real numbers.

This SOUNDS ambitious, but then comes these things

-) "build up space-time and QM simultaneously from combinatorical principles - but not (at least in the first instance) to try and change physical theory"

This means he does not quite aim to exaplain things properly from first principles, he is - just like rovelli - just doing a REformulation and an alternative REconstruction, to arrive at an already set result.

-) "the most obvious physical concept that one has to start with, where QM says something is discrete and which is conncted to the structure of spacetime in a very intimate way is angular momentum. The idea here is to start with the concept of angular momentum - here one has a discrete spectrum - and use the rules for combining angular momentum tp see of one cn construct space from this"

This is not the most obvious choice of starting point to me. It also seems that far too much baggge is taken in, motivated by what we are looking for.

I am trying to connect to my own reasoning and find IMO a more sensible interpretation of Penrose networks, as more general action networks or state history networks, but I need to keep thinking of this. It's at least sure that while I share a number of his inspirational starting points, his ideas are not "first principle" enough for me.

I do not think we should just REconstruct, using what we know as constraints, I think we should REconstruct, without cheating, and if the idea is right the right physics should come out by itself, if not, the idea is wrong. Otherwise, what have we proven? Nothing as far as I can see.

/Fredrik

19. Feb 9, 2010

### tom.stoer

I started to read the paper. Because now we know that there is a derivation of spin networks in LQG, it is more interesting to learn, why Penrose - w/o knowing this derivation - was interested in these mathematical objects.

In addition I would like to see if there is some principle of uniqueness forcing him to study SU(2). His spinor program was somehow specific for 4-dim spacetime, even though I was never able to find out why not something similar should by possible in other dimensions (there seems to be no no-go theorem).

Regarding R, C, H, O: could be that it's related to these structures, but it is not clear what the argument for uniqueness should be. Simplicity singles out R, largest commutative divison algebra singles out C, largest divison algebra singles out O, ...

John Baez was very interested in these topics for years, but I could not find a single hint for C, he seems to be more interested in O. The unique structure from Lie algebras is ceratinly E(8) as it is defined entirely in terms of e(8). But there's a long way down from E(8) to SU(2).

20. Feb 9, 2010

### Fra

I'm not sure how you reason, but to make any progress and be able to find something that is more properly an explanation, rather just a constructed explanation, I think we need to step back alot.

To me, starting from repetitive occurences of distinguishable events, one of the most fundamental symmetry is permutation symmetry (operating on historical sequences). And this symmetry probably somehow breaks into sub symmetries as the complexity increases, because the permutation symmetry are likely to be be broken in the sense that the breakaing becomes distinguishable only at a certain complexity level, since the breaking is yet another measures, requiring represenation cmoplexity.

Each time I've went this road, it seems to me the simplest possible continuum-like structure, appearing after permutation symmetry is broken is like a "string", but one with unknown embedding, and the string index is defined as the limit of the rational probability values, with in the limit is the continuum ]0,1]. Next one would ponder what happens when this grows more complexity, new dimensions are bound to grow.

I have not yet come as far as to infer symmetries from this, since I was stalled by trying to understand the process by which complexity is in fact increasing (generation of complexity).

Penrose also ponders the "large N" limit of what he called "directions". He also agreed that to define aconnection these must further be connected to a common network, which ha labeled K. This is interesting except I don't quite see it the same way.

But the exact "scaling" of the N->inf is important, and I think the reason for this scaling is too. How interacting networks compete for complexity.

I have to admit that I do nto yet understand penrose endavours in any other way that he is actually using a designed construction to arrive at something he wants, rather than using real first principles?

Does anyone agree or is it just me?

/Fredrik

21. Feb 9, 2010

### arivero

Dunno. This comparison between spin networks and Wilson lines seems to miss the point of angular momentum, which was the main motivation of Penrose, as it has been told. A spin network link is not (only) a kind of particular "gauge current line".

22. Feb 9, 2010

### tom.stoer

Angular momentum emerges from spin networks b/c one starts with SU(2) or SO(3) which means one starts with angular momentum ...

Does group field theory tell us something regarding spin networks based on different groups?

23. Feb 9, 2010

### atyy

GFT links to spin foams. I don't know if spin foams in general link to spin networks.

Another place that spin networks are popping up is condensed matter! http://arxiv.org/abs/0907.2994

24. Feb 9, 2010

### tom.stoer

I do not care if spin networks or spin foams; what I would like to understand is whether different groups can be used and what the results are. I googled "SU(N) spin networks" and "SU(N) spin foams" and found nothing!

25. Feb 9, 2010

### Physics Monkey

I can tell you one situation where the answer is reasonably well understood. I have in mind the situation of topological quantum field theory in 2+1 dimensions. This is clearly not quantum gravity in 3+1, but at least quantum gravity in 2+1 seems quite like one of these topological theories. No background metric is required, for example.

In this setting there is no sense in which SU(2) is unique. One may in general have various kinds of so called Chern-Simons theories which can be reformulated in a spin network language. One of the simplest is the "quantum group" called SU(2) level k (a truncated version of SU(2) ) that can be used to define a regularized version of Regge quantum gravity. But you can have many different groups. Of course, demanding that some kind of semiclassical limit gives 2+1 GR may fix the group, but I think the group is not fixed by very general considerations like background independence, etc.