Why spin foams have int/semi-int values?

[MTd2]

Why spin foams have int/half-int values?

These values are due quantization of space time with 4 or more dimensions. But there is a catch here. There is no space time or space, just nodes and links at fundamental level, so why constrain so much those values?

 

[Naty1]

Roger Penrose discusses his selection in THE ROAD TO REALITY page 946m section 32.6.

It is apparently the result of input to the model, not an output from the model. I can't be positive that his rationale is identical to that in spin foam models in use currently.

This was based on his belief that

...spacetime structure should be based at root on discreteness rather than continuity...

His rationale was that total spin of a system j = 1/2n, taken in units of h-bar :

...had seemed to me that total spin as measured by the natural number n, was was an ideal quantity to fix attention upon if one were interested in building up from scratch some discretye combinational structure that leads to a notion of actual physical space.

 

[MTd2]

So, that was due his original guess. Why didn't anyone question that?

 

[tom.stoer]

I don't see the problem.

The idea is to describe "something" by using SU(2) spins. Of course they are integer and half-integer. This is motivated by the fact that SO(3,1) ~ SU(2)*SU(2). Of course one can start with a different idea which would require a different motivation.

It's questioned a lot: string theory says that one must use strings. ST is motivated by the fact that it always contains a spin-2 mode identified with the graviton.

 

[f-h]

SO(3,1) ~ SU(2)*SU(2).

Well it is based on representation theory and no it's not always discrete.

SO(4) ~ SU(2)*SU(2) Thus a theory based on SO(4) representations will naturally have half integer labels. SO(3,1) has an SU(2) subgroup, but the original BC model didn't use that but "simple" representations those are labeled by an element $p \in R$, thus this has continuous labels. The EPRL Lorentzian models fixes p in terms of the representation of the SU(2) subgroup of SO(3,1) and thus reduces to a discrete model again.

One way to see why these representations arise is that we want to impose conditions on a connection taking values in SO(4) or SO(3,1) and our starting point is decomposing the delta function on the groups into a sum over characters, that is, a sum over irreps.

 

[Naty1]

MTD: Penrose's model did NOT eventually pan out... I don't know why nor exactly how current spin foam models are different.

But the motivation implied by Tom in posts above, I think, is that particles including force particles have been found to have certain limited spin options...in the Standard Model; I guess people suspect that's also a characteristic of space and time...but whether space and time are really discrete is still an open question.

More recently, black hole horizons have also suggested a discrete nature to space...with hidden information being displayed in a bit per Planck sized area.

you can search here under "emergent spacetime" for a number of other perspectives. also "emergent gravity" for some ideas of Verlinde, Jacobsen, and others.

Here is one paper from Rovelli, last month:
http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1780v2.pdf

I came across this quote within Rovelli's paper regarding spin foams :
Such geometrical pictures are helps for the intuition, but there is no microscopic geometry at the Planck scale and these pictures should not be taken too literally in my opinion….These geometrical pictures can play a very useful role in various situations, but what the theory is about is expectation values of physical observables, not mental pictures of the geometry of individual states.

and this current discussion thread:
https://www.physicsforums.com/showthread.php?p=2793581#post2793581

 

[MTd2]

@Naty: I did not say it pan out. I want to know why an unfixed value was spin was not attempted. It could be even more interesting than it is now.

@tom.stoer and f-h: SU(2) being a complex group and a double cover of SO(3), makes me thing of a theory in 3d with a complex evolution parametrized by 4 variables, presumably reflecting a 4 space embedding of that space. So, I picture SU(2) describing a 3 space like slice of the universe with parameters being space time, or Hamiltonian time.

Right?

 

[MTd2]

There are nice things one can do with SU(2) is that the roots of E8 form a finite subgroup of SU(2), by using the affine extension of 2 copies of the 600 cell roots, which is also a subgroup of SU(2) But I want to know something important here:

Does SO(2,1) embed in SU(2)? Tell that, I am telling you people something really cool.

 

[f-h]

SU(2) is a "real" group. Its parameters are three real numbers. e.g. a unit vector and an angle of rotation. Just as SO(3), except that the angle runs not from 0 to 2pi but from 0 to 4pi.

SU(2) doesn't give you the evolution in the fourth dimension it gives you the rotations in 3 dimensions, nothing more.

The complexification of SU(2) is SL(2,C) which is the Lorentz group SO(3,1), and this 6 dimensional group allows you to boost out of the 3 dimensional hyper plane. This is why the dynamics of the new models are built by injecting SU(2) representations into SL(2,C) representations.

SO(2,1) is, if i remember correctly, isomoprhic to SL(2,R) which sits in SL(2,C) but not in SU(2).

 

[MTd2]

su(2) can be an algebra of the quartenions and given that this group is unitary, we can think of SU(2) as an S3 embedded in R4. Since the center of this sphere is arbitrary, we can walk with the coordinate of its center along R4. Thus 3 coordinates for time and one to time, although this is a kind of parametric time.

The fact that it is so related to SO(3) can be seen that its antipodal points can define a unit circle around the equator of th S3, which is an S2. So SU(2) can be seen to cover 2 SO(3) groups.

 

[f-h]

su(2) can be an algebra of the quartenions and given that this group is unitary, we can think of SU(2) as an S3 embedded in R4. Since the center of this sphere is arbitrary, we can walk with the coordinate of its center along R4. Thus 3 coordinates for time and one to time, although this is a kind of parametric time.

The fact that it is so related to SO(3) can be seen that its antipodal points can define a unit circle around the equator of th S3, which is an S2. So SU(2) can be seen to cover 2 SO(3) groups.

So? The first thing you describe seems to be SU(2) \times R^4.

The double cover can be seen in many ways, I like the geometric picture of SO(3) as a ball of radius pi with antipodal on the surface of the ball identified.

 

[MTd2]

So? The first thing you describe seems to be SU(2) \times R^4.

So that is what a I wanted to mean here, as one of the possible motivations to use SU(2) in spin foams.

@tom.stoer and f-h: SU(2) being a complex group and a double cover of SO(3), makes me thing of a theory in 3d with a complex evolution parametrized by 4 variables, presumably reflecting a 4 space embedding of that space. So, I picture SU(2) describing a 3 space like slice of the universe with parameters being space time, or Hamiltonian time.

 

[MTd2]

Well, I'd like to know here if it is possible to use anyons to generate spin, getting a vertex from a given polygon of the spin network and defining an SO(2,1) group there. The inductive "magneitic" flux would come from the opposite face.

http://en.wikipedia.org/wiki/Anyon

 

[f-h]

So that is what a I wanted to mean here, as one of the possible motivations to use SU(2) in spin foams.

This isn't really how we use it though. The isomorphism SU(2) - S^3 doesn't really play a strong geometric role in the models considered.

Remember that the Ponzano Regge model based on SU(2) describes flat 3-space.

 

[MTd2]

Alright. I just thought it would be reasonable given that a spacelike slice of the universe is an S3, at least in FRW. Well, at least S3 is extensively used here, section 6:

http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.3319v2.pdf

 

[f-h]

Sure, it's mathematically important, but not in the physics/geometric interpretation of the state sum.

More important is the Hopf Bundle:

http://en.wikipedia.org/wiki/Hopf_fibration

Roughly speaking you have S^3 as a bundle with fiber S^1 over S^2. Since the phase S^1 doesn't matter so much quantum mechanically you are left with the base space S^2 which naturally lives in R^3.

This is how we get 3 dimensional geometry out of SU(2) (or SU(2) out of the quantisation of 2-spheres).

 

[MTd2]

Regge Ponzano model is a simplicial model of Eintein's field equations, so I jumped into a wrong conclusion thinking it was a backward reasoning from a 3-sphere slice of FRW. So, it seems the usual justification for spin statistics in 4 dimensions.

But I am even more confused than in the beginning. I asked for spin networks, not Regge Calculus! Conceptualky each node can see just other nodes, it does not know any kind of global information, that is, at best part of the bundle enough to a coherent linking. So, shouldn't spin quantization as spin and half spin arise just as an aproximation due to interactions?

 

[f-h]

The SU(2) is the local group. Each node of the network has its own SU(2) invariance. each link its own SU(2) matrix in the spin-j representation.

Spins are the representations of SU(2), they are quantized because SU(2) is a compact group. If you build models with other local groups you get other results. There's nothing sacred about SU(2) unless you want to connect to the Ashtekar-Barbero formulation of GR which is a SU(2) gauge theory.

The Ponzano Regge model is build from spin networks as are all spin foam models.

Things like spin statistics are in a completely different game and rely on poincare invariance in Lorentzian 4d space time.

 

[MTd2]

The SU(2) is the local group. Each node of the network has its own SU(2) invariance. each link its own SU(2) matrix in the spin-j representation.

Yes, I know that, but it feels like cheating. It seems like rushing to get the final result by trying to restrict too much from the most fundamental entities of the theory. It looks like a got a bunch of intermadiate, simple, structures that could be much better used. For example, take a face on a spin foam. It generates a 2+1 world volume, which is the dimension of topological quantum computations, but the spin has continuous values. Maybe one is missing a way to make particles out of this. So, shouldn't the SU(2) be given to the whole tetrahedron?