A clarification of LQG by Lubos Motl

[Tyger]

Just plug "ball lightning"

into MSN or Google, you'll get a lot of stuff.

 

[marcus]

Originally posted by Tyger
into MSN or Google, you'll get a lot of stuff.

Thanks Tyger, right now would you believe it
I am trying to review some basic facts about classical
Lie groups and Lie algebras. Otherwise would find out about ball lightning on Google.

I would guess you know some things about O(3)
the real orthogonal group, det = +/- 1 and so on.
Transpose equals inverse etc. etc. At least its real so
you don't have to take complex conjugate all the time
Are you comfortable talking about SO(3) and the algebra so(3)?

And, if comfortable, are you at all interested?
I might start a thread about SO(3) and Loop Quantum Gravity
to go thru the easy facts with others interested in LQG.

 

[jeff]

Originally posted by Sauron
Your argument about particle labels don´t totally convince me. In fact it doesn´t convince me at all. You first begin with a 4 manifold provided with an vielbein and an spin connecton. So your departure point allows you to define spin 1/2 particles without any problem. that´s the usuall way it is done in particle physics(with the "curved" dirac matrices and all that).

Ok, later you do the ADM slicing and you pass form lagrngian to hamiltonian and get your constraints and all that. But think these. You initially have L=Lgravity + Lmatter. When you go to the hamiltonian the Lmatter would be transformed In Hmatter. That doesn´t introduce any difficoultie (constraint) for spiin 1/2 particles.

In fact even in mikownsky space i don´t understand why you claim you need to get the SU(2) covering of the SO(3) subgroup of SO(3,1) to get spin 1/2. I studied that topic in the Ramond´s book on field theory (and also i ht kaku book of field theory) and don´t remember to have seen that statement.

So you disagree, both with the claim that the lowest spin particles allowed in SO(3) and SU(2) LQG are spin-1 and spin-1/2 respectively, and with my attempt to account for this. Correct?

 

[Sauron]

>>So you disagree, both with the claim that the lowest spin >>particles allowed in SO(3) and SU(2) LQG are spin-1 and spin-1/2 >>respectively

Now you say that the lowest allowed particles are spin -1 and spin-1/2. But if spin-1/2 are fermions and your first claim was that fermions were not allowed. Now instead your problem is with the spin-0, i.e. thte hiporthethical and still unobserved Highs bosson (that tachyonic particle in the unbroken symmetry phase...).


I guess that you may explain your change of opinion before i give any answer to these aspect.

 

[jeff]

Originally posted by Sauron
Now you say that the lowest allowed particles are spin -1 and spin-1/2. But if spin-1/2 are fermions and your first claim was that fermions were not allowed. Now instead your problem is with the spin-0, i.e. thte hiporthethical and still unobserved Highs bosson (that tachyonic particle in the unbroken symmetry phase...). I guess that you may explain your change of opinion before i give any answer to these aspect.

Of course you're right about the implication of my statement for spin-0 particles, but that's a problem for the LQG people, not me. Besides, spin-1/2 is more important than spin-0.

Now, I know you're saying that you think I've changed my previously stated position. But I'm having a bit of trouble decoding precisely why. Anyway, I haven't, as the following shows.

Originally posted by marcus
Motl is kind of outrageous in places----like at the end in the Unsolved Questions section he raises the question "could it be that LQG does not allow for the existence of fermions?"

Who's right lubos motl at harvard or marcus? It's clear what marcus thought. Similarly, because of this it was clear to me that marcus didn't understand what was going on. He needed to learn about spin, so I gave a brief overview of this subject in very general terms unrelated to LQG in any specific way. Here's what I said (you can check that I haven't changed a single word)

'Okay guys, briefly - as all children learn in kindergarten (maybe you guys skipped?) - particles are defined in terms of their mass and spin which label - and this reflects their deeper significance - the representations of the inhomogeneous lorentz group under which their states must transform to respect invariances related to the geometry of spacetime that give rise by noether's theorem to conservation of mass-energy-momentum. In particular, spin specifies the representation of the rotational subgroup SO(3) of the homogeneous lorentz group SO(3,1). For even and odd spin particles - i.e. bosons and fermions - the reps are tensorial and spinorial respectively. For instance, in the case of spin-1 particles - the photon is an example - the rep is SO(3) itself. On the other hand, states of the spin-1/2 particles out of which all matter in our universe is composed transform under the double cover SU(2) of SO(3)."

The point is that I've said nothing about LQG in this: This is a lesson on spin, not LQG. Now, what marcus needed to understand to see that lubos motl was in fact not being "outrageous" was why SO(3) LQG doesn't couple to spin-1/2 particles - i.e. to matter, clearly a bad thing - while SU(2) does, the rest is irrelevant. Thus to make my point, it was sufficient to simply give marcus the examples in the last line of my explanation, and with those examples given, I could end with the following conclusion (again you can check that I haven't changed a word)

"We'll marcus, you now have the answer to the question "why SU(2)?" I challenged you to answer a while back, which is that the LQG gauge group G is some representation of the rotational subgroup SO(3) of the lorentz group, with the observed existence of matter requiring G=SU(2), the double cover of SO(3), with G=SO(3) being by itself insufficient to incorporate this basic empirical fact."

Everything I wrote is correct and designed simply to convince marcus of the special import of lubos's result with respect to LQG's consistency with the existence of matter. Pointing out that once G - which can simply be taken as some compact group - is chosen, the complete spectrum (open questions about the precise manner and consistency of couplings to gravity - if they can ever be sensibly defined at all - aside) is given by it's irreducible representations was irrelevant, since it was only the lowest spin reps that were at issue.

But your not marcus. I responded directly to your post in which you voiced general disagreement with the above remarks, but I wasn't completely sure how to respond because you said you hadn't learned about what I'd discussed above, so I simply asked the general question I did to set the compass, and it in no way reflects a change of opinion on my part.

Still, based on this post, I don't know whether you agree with marcus. If you don't agree than I'll simply tell you to keep reading until you come to it. The SO(3) problem is well-known, I didn't make it up.

If you do agree but still aren't convinced that my way of looking at it is correct, then let me know and we'll figure out the right argument if mine turns out to be wrong (though I'm fairly sure this is not the case). In any event, I think it's important to understand the stuff I was saying about spin.

I'm aware that in GR, bosons can live on the spacetime manifold from the start, while fermions require a vierbein. But in LQG there is no fixed background geometry and the gauge group would determine the spectrum (assuming that this idea can ever work).

One last thing. When you post a response, be sure to quote the most recent version of my posts because I'm building an arsenal of explanations so I try to give answers that are precise and detailed, and that requires an editing process.

 

[thermonuclear]

I’m wondering that there is no reply to this question. Is this a trivial, uninteresting point for all who have participated in this thread? For me it is not, but may be this is just due to my ignorance.

I agree that the restriction to SO(3) excludes the fermions from the theory, since spin-1/2 transforms under SU(2) and not under SO(3). Why does L. Motl claim that SO(3) is more “natural” than SU(2) at this place? According L. Motl, SU(2) was chosen by Ashtekar and Baez hoping to overcome the problem of the fermions but not logically derived from the theory, which describes only vacuum solutions.

It would be great if some of you could explain the derivation of this gauge group.

The Lorentz group is SO(3,1) and restricting to a three dimensional vector space (by use of the ADM formalism) it reduces to SO(3). But what is the reason to take only Lorentz symmetries and not to use a representation of the Poincaré group --SO(3,1) and the translation group R^4-- as a gauge group? Or, let’s say better, since we are dealing with a complex manifold in the Ashtekar formulation, we may use SL(2,C) and the translation group R^4 (covering the Poincaré group) as the gauge group...?

If there is indeed no interest may be someone could give me some references anyway.

Regards.

 

[Sauron]

Well, it is not that the thrad would have lost interest for me.

I am just triying to find where the exposition i had about the lorent´z groups representations is.

Also jeff has pointed an aspect i hadn´t realized. I know what a vielbein (n-bein in general) is, but it usually is exposed related to a metric. If we have no metric, as it hapens inLQG i must revisate the detaills.


The complementary thread that marcus created is being very informative about the detais of spin networks, and it consumes time to read it. Most of the literature i had readed in LQG beguined with Wilson loops adn was quickly into topological BF theories, and the detaills of spin networks were not addressed.

Last, but not least, i am studiying some aspects of statitisthical mechanics (not related to LQG, it is just that my basic in that area is very weak and i am triying to remediate that, just for fun)

So let´s say i am being a reader most of time. I am basically new to LQG and i still need to meditate a lot about it.


Jeff, i understand that people eidt it´s messages, but it makes a bit confusing to follow the thread, why not just to indicate your previous mistakes in a new post?

And i still see a self-contradiction in the posts. You say LQG allows particles of spin not minor than 1/2, but that means fermions are allowed, and in other places ou say LQG doesn´t couple to fermionic matter. And your argument says that the absence of SU(2) is forbiding spin 1/2, so ,why are them allowed and only spin 0 particles are not allowed?.


To end these point a clarification. I say i dón´t like too much string theory, but for sure string physicans are great physicians. In fact previous to LQG i defended string physics, that´s why i studied it. But now i see LQG like a most natural creature. I think LQG is the theory to study now, and only when it is proved it failss(if it fails) we would return to strings. Let's say that in my opinion if LQG woule have existed in the 80´s (almost) nobody would have studied strings.

 

[jeff]

Originally posted by Sauron
Jeff, i understand that people eidt it´s messages, but it makes a bit confusing to follow the thread, why not just to indicate your previous mistakes in a new post?

Okay, sorry about that.

Originally posted by Sauron And i still see a self-contradiction in the posts. You say LQG allows particles of spin not minor than 1/2, but that means fermions are allowed, and in other places ou say LQG doesn´t couple to fermionic matter. And your argument says that the absence of SU(2) is forbiding spin 1/2, so ,why are them allowed and only spin 0 particles are not allowed?.

The point was simply that lubos's result strengthened a previous result suggesting that the lowest spin particles that can be coupled to gravity in LQG are spin-1 so that the gauge group should be taken as SO(3) rather than SU(2). Since all matter is composed of spin-1/2 particles this would make LQG inconsistent with the existence of matter. If you continue to feel that I'm contradicting myself, quote the contradictory passages and I'll comment on them.