Freidel does it: BeeF, it's what's for dinner

[marcus]

Freidel does it: "BeeF, it's what's for dinner!"

This looks like the Quantum Gravity+Matter breakthrough I've been expecting for a few years. The paper's arxiv number is easy to remember----just think of July 14 as Bastille Day and write this year's Quatorze Juillet holiday as 0607014

http://arxiv.org/gr-qc/0607014

It is a paper by Laurent Freidel and two other gentlemen---a Pole and Russian, I believe---named Jerzy Kowalski-Glikman and Artem Starodubtsev.

Jerzy K-G is at Wroclaw University, Artem S is in Renate Loll's group at Utrecht, and Laurent Freidel, as I guess everybody knows, is jointly at Perimeter and the Lyon Ecole Normale.

The apparent breakthrough is not a sure thing. All I have to go on right now (besides some second-hand information) is a sense of confidence I get from reading the paper. It is an order of magnitude more on top of things than what I've met in the past.

Let's explicate the paper: this thread can be devoted to doing that---and also to examining the followups. there are two or more other Freidel et al papers expected to follow this one.

 

[marcus]

the paper is based on some unfamiliar things-----BeeF, the deSitter group SO(4,1) and it's algebra so(4,1)
We encountered these things in earlier papers (some by Smolin, Starodubtsev, Freidel, and extensively by John Baez in the case of BF theory) but for me the first prolonged exposure was from the January 2005 paper of Freidel and Starodubtsev.
http://arxiv.org/abs/hep-th/0501191

The story goes back to MacDowell and Mansouri (1977) who discovered that classical General Relativity can be written as a perturbed BF theory where the A and B differential forms are valued in so(4,1).

So we jump into this new Freidel paper at the bottom of page 3 where equation (2.4) gives a Gen Rel action S expressed as a perturbed BeeF action.
If you set alpha and beta equal zero, then the perturbing terms drop out and you get a pure BeeF action.

I guess one point to make is that it is known how to quantize a BF theory by the spinfoam method. So if they can get a comprehensive theory of gravity and matter in the form of a (perturbed) BF theory then
1. there will be things they can find out (like what is the FLAT limit, setting alpha equal zero, is it DSR? and
2. it is a known proceedure to quantize it.

I could use some help transcribing a few of the equations like (2.4) in Latex. have to go out, will do some more on this in an hour or so when I get back.

 

[Kea]

I could use some help transcribing a few of the equations like (2.4) in Latex...

$$S = \int ( B_{IJ} \wedge F^{IJ} - \frac{\alpha}{4} B_{IJ} \wedge B_{KL} \varepsilon^{IJKL} - \frac{\beta}{2} B^{IJ} \wedge B_{IJ} )$$

 

[marcus]

thanks Kea!
I just got back and was girding for Latex.

so this alpha is in a certain way a disguised version of the Newton G. when they want to have a FLAT or a "zero gravity" version of the theory they can turn gravity off by setting alpha = 0

and if they set both alpha and beta equal zero then they get
THE MOST VANILLA OF ALL THEORIES, the generic jack-of-all-trades action which John Baez teaches us to associate with spinfoam, namely

$$S = \int ( B_{IJ} \wedge F^{IJ} )$$

 

[marcus]

we should look at equation (2.12) and get some hint about the physical significance of alpha. It has the immirzi number in it, but that is just some order-one number like 0.2 or 0.3, so it is not what matters.

$$\alpha = \frac{G\Lambda}{3}\frac{1}{(1+\text{immirzi}^2)}$$

So for example if immirzi = 0.2, then 1+ immirzi² = 1.04
and the second fraction $$\frac{1}{(1+\text{immirzi}^2)}$$
is approximately equal to one and we don't worry about it.

The main term is Newton's G times the cosmological constant Lambda.

Alpha is going to be made zero either in the case we want to turn gravity off, or we want to have zero cosmological constant (no acceleration in the expansion of the universe.)

What is nice though. What I really like about this, is that this "most vanilla of all theories" this harmless nondescript BF is MANIFESTLY DIFFEOMORPHISM INVARIANT and manifestly does not require specifying any BACKGROUND GEOMETRY to start with.

Perturbation approaches (which are the standard proceedure for almost everything) almost always seem to have an explicit requirement of some definite (usually flat) background geometry. For example perturbative string theory. For example quantum field theories. They all make you start with flat Minkowski space, or the moral equivalent of that, and perturb around it. You get to just slightly wrinkle it.

But here we see them start with BF theory, which looks like a lot blanker slate, and perturb around THAT.
So it HAS the diffeoinvariant explict independence of background geometry that Quantum Gravitists customarily insist on.
============
there is still a long ways to go, but they sure are making a good start.

 

[marcus]

I will put a 4 in Kea's superscript. this is where they break SO(4,1) symmetry down to SO(3,1). It is part of the 1977 routine of MacDowell and Mansouri, who used SO(4,1) even though they always intended to do 4D Gen Rel.

$$S = \int ( B_{IJ} \wedge F^{IJ} - \frac{\alpha}{4} B_{IJ} \wedge B_{KL} \varepsilon^{IJKL4} - \frac{\beta}{2} B^{IJ} \wedge B_{IJ} )$$

the notation is explained in Freidel Starodubtsev January 2005 paper.
http://arxiv.org/abs/hep-th/0501191

===========================
another thing we need, for understanding the paper, and maybe someone will supply is
a handle on what could be the EIGENVALUES OF THE QUADRATIC AND QUARTIC CASIMIR OPERATORS of the so(4,1) algebra.

this is jumping to page 8. The two eigenvalues they call C₂ and C₄.

it will turn out that C₂, the eigenvalue of the quadratic Casimir operator, helps determine the MASS of the particle. And C₄, or rather the ratio C₄/C₂, determines the SPIN of the particle.

this is going to page 9, and looking at equations 3.13 and 3.15

the cosmological constant Lambda also enters into determining the mass.

 

[selfAdjoint]

thanks Kea!
I just got back and was girding for Latex.

so this alpha is in a certain way a disguised version of the Newton G. when they want to have a FLAT or a "zero gravity" version of the theory they can turn gravity off by setting alpha = 0

and if they set both alpha and beta equal zero then they get
THE MOST VANILLA OF ALL THEORIES, the generic jack-of-all-trades action which John Baez teaches us to associate with spinfoam, namely

$$S = \int ( B_{IJ} \wedge F^{IJ} )$$

Marcus you are aware of course, that you can have a BF theory basd on just about any Lie Group. If G is the group and $\mathfrac{g}$ is its Lie Algebra, then F is the curvature of G and B is a $\mathfrac{g}$ valued 2-form. the seemingly magical properties here are not due so much to the BF as to the de Sitter group, which gives the all-important cosmo constant.

 

[Kea]

The seemingly magical properties here are not due so much to the BF as to the de Sitter group, which gives the all-important cosmo constant.

Exactly. And the dependence of mass on $\Lambda$ follows from the magical parity cube. So we are beginning to look at a framework in which the gauge groups of the SM (and gravity) are derived.

 

[marcus]

the Casimirs C₂ and C₄ are important in this paper, so here is some review
Wiki on the Lie algebra symmetric bilinear form B(x,y)
http://en.wikipedia.org/wiki/Killing_form
you get it using Trace on the the compose of two ad(.) maps and B(. , .) behaves nicely with bracket.
=============
the Casimir operator or Casimir invariant is defined using this bilinear form
http://planetmath.org/encyclopedia/CasimirOperator.html
one can pick any basis x_i of the Lie algebra and then, using B(. , .), construct a DUAL basis x^j
with B(x_i, x^j) = 1 if i=j and zero otherwise.
Intuitively, this is THE NEXT BEST THING TO AN ORTHONORMAL basis, which you can't get because the symmetric bilinear form on the Lie algebra is not positive definite (not an "inner product"). So instead of making an orthornormal basis, you construct two baseez (two basisses)

And then one forms the Sum over i of these basis and dual basis thingees multiplied together formally in the "universal enveloping algebra" and summing
Sum_i x_i xⁱ
Multiplying things in the "universal enveloping algebra" is rather like tensoring----you do a formal multiplication and then factor out to validate the Lie bracket.

Wiki has an alternative definition, which may need clarification, but also an example and some discussion.
http://en.wikipedia.org/wiki/Casimir_operator
=================
Does anyone have a good online source about Casimirs?
=================
the rank of the Lie algebra is the number of independent Casimirs. Suppose we fix a representation. Then since a Casimir commutes with everything it is some number times the identity. So it is also a NUMBER namely its unique eigenvalue.
==================
Physicists are familiar with the Casimir of the rotations algebra so(3), as the squared angular momentum operator.
If we decide on some representation---corresponding to a particle---then this operator becomes a NUMBER, as indicated above, and that number is the spin of the particle.
==================
maybe this will provoke someone more expert to provide a GOOD tutorial about Casimirs and some GOOD online resources. But anyway I am doing a slapdash treatment because we need it right now.
===================
WHO CAN TELL US ABOUT THE TWO CASIMIRS OF so(4,1), the deSitter algebra?

apparently it is rank 2, and it has two Casimirs, which are the quadratic and the quartic. And if you pick a definite representation---as if one were specifying a particle---then one gets MASS AND SPIN numbers from the two Casimirs. Aiiyeee!
==================
maybe a nice online tutorial on so(4,1) and SO(4,1) will show up.
this smilie is intended to denote hopefulness

 

[marcus]

Marcus you are aware of course, that you can have a BF theory basd on just about any Lie Group...

Yes! I agree that BF can be done with any of the usual groups and what makes this special is SO(4,1). I would like to know more about the deSitter group SO(4,1) and its algebra so(4,1). Just about anything you or Kea can say would be a help---either to me or to other readers of the thread.

...we are beginning to look at a framework in which the gauge groups of the SM are derived.

that sounds quite promising Kea. Please elaborate, if you feel so inclined. It is a puzzle why the Standard Model gauge should be a mix of SU(3) x SU(2) x U(1).

 

[Careful]

Exactly. And the dependence of mass on $\Lambda$ follows from the magical parity cube. So we are beginning to look at a framework in which the gauge groups of the SM (and gravity) are derived.

Ah, the old idea to put the universe in a box (and put in a IR cutoff) - and then eventually take the infinite volume limit (Lambda --> 0). But the m^2 is merely just the p_ap^a no, the mass due the cosmological constant and spin being negligible. I don't know what you mean with the gauge groups of the standard model being derived here That the
SO(4,1) gives also spin (which contributes to the mass) and particle momentum hardly comes as a surprise: we knew that already for the Poincare group in 4 dimensions (which has exactly the same dimensionality) and this has been extensively studied in Einstein Cartan theory. Let's wait and see whether such split can cure the IR divergences in the quantum theory - there is no evidence presented for that at all !

 

[Kea]

Ah, the old idea to put the universe in a box...

Careful

You appear to be jumping to the most dreadful conclusions about what we are talking about. Please read a little more of the literature before making ill-informed remarks.

 

[Careful]

Careful

You appear to be jumping to the most dreadful conclusions about what we are talking about. Please read a little more of the literature before making ill-informed remarks.

Dear Kea,

If you add a cosmological constant, this serves as a volume regulator for the spacelike universe, and is crucial for renormalizability even for simple toy models such as 1+1 quantum gravity in the CDT approach (here is your IR cutoff ). Having finite volume on a compact spatial universe (de Sitter), is like putting in reflecting mirrors and it effectively boils down to ``putting the universe in a box'' - any child knows this. Frankly speaking, I was asking questions and not drawing any conclusions. You made some claim of seeing the standard model arise, so where is it ?

Now, in the thread about stroop theory, I was wondering why I should take this categorical relationship between nCob and Hilb seriously in any way. One of the reasons offered there was that making punctures destroys it all. Of course Hurkyl simply told that in TQFT this is forbidden, but quantum gravity is of course no TQFT. In going over to 3+1 gravity you will have to put on gravitational waves, that is you need quantized versions of the Riemann, Weyl, ricci scalar ... on the Hilbert space of (particle) states of this BF theory. The question is of course whether these operators will be bounded in this representation (and that the full states are normalizable !), and I see no good reason why this should be so. It is the same with the radiation problem in QFT, you start out with the coulomb states and then try to treat the radiation backreaction problem by adding photon per photon, it turns out that you need to put in a UV cutoff by hand to achieve this. Now people easily say, ohw but all operators in quantum gravity have bounded spectrum, and even the length, area and volume operators have discrete spectrum. But is this really so? None of these operators have any physical significance yet since the Hamiltonian constraint hasn't been implemented (and doing so might drastically change the spectral properties of the operators), and known work shows that the curvature operators on the FULL Hilbertspace of spin networks aren't bounded at all.

So, it might turn out that punctures need to happen... or that you still have to put in some regulator by hand - and why to take all this trouble then ?

I think these are all legitimate *questions* concerning those issues you believe in (I see this just as the next fad); it would be better to offer some comments/explanations than to guess about how well informed I am (on basis of a sentence where you seem to express your own ignorance).

Careful

 

[Kea]

Careful

Tell us something we don't know...and do some reading.

Kea (this is the FIRST time I have ever used grumpy)

 

[arivero]

the Casimirs C₂ and C₄ are important in this paper,

Amusingly, the Casimirs were also important in our scheme to formalise Hans' model of electroweak parameters,
http://arxiv.org/abs/hep-ph/0606171
Section 1 of this article was sent for peer review, and the referee did the following final coment:

"As a final comment, I note that there is nothing about Poincare invariance of quantum
field theory that uniquely chooses a gauge group and the corresponding gauge symmetry
breaking dynamics. I can write down an infinite number of models with different gauge
groups, different Higgs sectors and different spectra of gauge bosons and matter fields.
So without any further theoretical principle, the Casimir operators of the Poincare group
have nothing to say about the mass spectra of particles. The authors propose no additional
theoretical principle. Consequently, this paper offers nothing beyond a numerological co-
incidence."

I'd like to add that if we are into LQG or any other theory with a Planck Length, we are not interested anymore on plain Poincare but also in deformations of it. It has been an underground lore that a deformation of SU(2)xSU(2) could be the adequate substitute of Lorentz group.

 

[Careful]

Careful

Tell us something we don't know...and do some reading.

Kea (this is the FIRST time I have ever used grumpy)

Kea,

I am asking some legitimate questions about this approach (which you simply refuse to answer). As I remember some claims were made that category theory offers a deeper understanding between QM and GR, so it is my good right to ask where it is (since I don't see it). I do not claim to be a specialist in this kind of approach (far from), you have that position. Therefore it would be only normal that you answer my questions briefly, telling someone to read specific literature is not the way to sell your ideas (and it usually indicates that the ideas are pretty poor). If it is a good one, ``you should be able to explain it to the barmaid''. I am myself reading lots of different stuff (concerning local realism) and I really do not have the time to check all these latest fads into the details, that is why I ask (and if people ask me about the things I read, then I always offer the idea/explanation) since this one seems to excite people.

The reason why I am asking these similar questions over and over again when I hear these cries is because I believe a deeper failure in our theories to be responsible for the problems of QG while such lines of thought merely reflect a technical issue. Therefore, I would welcome any insight which shows me otherwise ...

Careful

 

[Careful]

**I'd like to add that if we are into LQG or any other theory with a Planck Length, we are not interested anymore on plain Poincare but also in deformations of it. It has been an underground lore that a deformation of SU(2)xSU(2) could be the adequate substitute of Lorentz group. **

Ok, never really looked into this but I have some (naive) questions :
(a) are these deformations, like doubly special relativity, merely not a fancy way to sneak in a(n ``observer independent'') cutoff ?
(b) the modified dispersion relations are ``non-quadratic'' so that the Minkowski metric becomes a cumbersome notion with respect to the inertial observers and only appears to be a good low energy approximation (in an observer independent way). So, how to ``geometrize'' such theory when there is no obvious geometric object present ?
Finsler geometry perhaps ?

Careful

 

[selfAdjoint]

Kea,

I am asking some legitimate questions about this approach (which you simply refuse to answer). As I remember some claims were made that category theory offers a deeper understanding between QM and GR, so it is my good right to ask where it is (since I don't see it). I do not claim to be a specialist in this kind of approach (far from), you have that position. Therefore it would be only normal that you answer my questions briefly, telling someone to read specific literature is not the way to sell your ideas (and it usually indicates that the ideas are pretty poor). If it is a good one, ``you should be able to explain it to the barmaid''. I am myself reading lots of different stuff (concerning local realism) and I really do not have the time to check all these latest fads into the details, that is why I ask (and if people ask me about the things I read, then I always offer the idea/explanation) since this one seems to excite people.

The reason why I am asking these similar questions over and over again when I hear these cries is because I believe a deeper failure in our theories to be responsible for the problems of QG while such lines of thought merely reflect a technical issue. Therefore, I would welcome any insight which shows me otherwise ...

Careful

The reason no-one is discussing your questions is because they are OFF-TOPIC. This thread is devoted to the RESULT, not to generalizing or big new fad theories or any such thing. As for Einstein-Cartan theory, did you or anyone else derive from it this RESULT connecting the de Sitter group to the Poincare group and GR, VIA WILSON LINES THAT ACT LIKE WORLD LINES OF PARTICLES? This is as far as I know new, and nothing in your posts on this thread suggests that you are prepared to refute it.
Just showing off your knowledge and being snarky isn't contributing.

 

[arivero]

Ok, never really looked into this but I have some (naive) questions :
(a) are these deformations, like doubly special relativity, merely not a fancy way to sneak in a(n ``observer independent'') cutoff ?

Perpahs, but it is not the purpose. The idea is to introduce representations of SU(2)_q with q a root of the unity and then relate them to particles and fields. It is lore, not published papers on it.

 

[Careful]

Perpahs, but it is not the purpose. The idea is to introduce representations of SU(2)_q with q a root of the unity and then relate them to particles and fields. It is lore, not published papers on it.

Ah doesn't matter (I am not the person to judge ideas to the phys rev qualifications it has). I also am not opposed to Lorentz symmetry being a low energy approximation (as is clear from many posts). I simply wonder whether there could not be an easier way to do all of this (without conflicting experiment), also one could wonder about the meaning of causality (and especially the high energy content of the ``no signalling faster than light theorem'' (I take c fixed here obviously)) in such context. Concerning (b), the Finsler suggestion might not be that bad, it reminds me at unification attempts between GR and EM which inevitably lead to violations of (usual) relativistic causality. Also, such type of geometry could lead to many appearantly non-local effects in QM.

Careful

 

[Careful]

The reason no-one is discussing your questions is because they are OFF-TOPIC. This thread is devoted to the RESULT, not to generalizing or big new fad theories or any such thing. As for Einstein-Cartan theory, did you or anyone else derive from it this RESULT connecting the de Sitter group to the Poincare group and GR, VIA WILSON LINES THAT ACT LIKE WORLD LINES OF PARTICLES? This is as far as I know new, and nothing in your posts on this thread suggests that you are prepared to refute it.
Just showing off your knowledge and being snarky isn't contributing.

How, how first of all, my questions aren't off topic ; the acceptance of future prospects here seems to be a function of how the impressions of me coincide with the tastes of others. But to come to the paper itself, first let me say it is a nice piece of work; but is it suprising ? Not really. To write gravity in the BF form was known for quite some time already; you start out from a broken SO(4,1) gauge theory, variate and throw out the topological invariants. Now, the A field (F(A) = dA + [A,A] ) is what gives the Einstein equations, pretty much like the ordinary gauge potential in Maxwell theory does that (with that difference that stuff is nonabelian here). So, how to introduce matter and couple it to gravity? Ah, we know already for long time that braking symmetries can give mass to the corresponding ``gauge particle'' (actually, it are the casimirs which label the gauge orbits and SO(4,1) just like the Poincare group (I guess) has two of them : mass and spin length). But we want point particles, so we want to break only this symmetry on a ``worldline'', moreover in analogy with Maxwell equations, one can reasonably expect the Einstein equations to emerge when the particle is minimally coupled to the full gauge field.
Hence, one adds int_t Tr KA^h(t) to the total action (where K is the symmetry destroyer), which gives the afore mentioned results, including other forces seems trivial (just put the interactions in the connection). Also, it seems pretty straightforward to generalize this scheme to string worldsheets and higher dimensional stuff. It would be a miracle however if this trick would renormalize gravity, but a nice feature of it is that a preferred (classical) vacuum is dynamically determined.

Careful

 

[selfAdjoint]

the Casimirs C₂ and C₄ are important in this paper, so here is some review
Wiki on the Lie algebra symmetric bilinear form B(x,y)
http://en.wikipedia.org/wiki/Killing_form
you get it using Trace on the the compose of two ad(.) maps and B(. , .) behaves nicely with bracket.
=============
the Casimir operator or Casimir invariant is defined using this bilinear form
http://planetmath.org/encyclopedia/CasimirOperator.html
one can pick any basis x_i of the Lie algebra and then, using B(. , .), construct a DUAL basis x^j
with B(x_i, x^j) = 1 if i=j and zero otherwise.
Intuitively, this is THE NEXT BEST THING TO AN ORTHONORMAL basis, which you can't get because the symmetric bilinear form on the Lie algebra is not positive definite (not an "inner product"). So instead of making an orthornormal basis, you construct two baseez (two basisses)

And then one forms the Sum over i of these basis and dual basis thingees multiplied together formally in the "universal enveloping algebra" and summing
Sum_i x_i xⁱ
Multiplying things in the "universal enveloping algebra" is rather like tensoring----you do a formal multiplication and then factor out to validate the Lie bracket.

Wiki has an alternative definition, which may need clarification, but also an example and some discussion.
http://en.wikipedia.org/wiki/Casimir_operator
=================
Does anyone have a good online source about Casimirs?
=================
the rank of the Lie algebra is the number of independent Casimirs. Suppose we fix a representation. Then since a Casimir commutes with everything it is some number times the identity. So it is also a NUMBER namely its unique eigenvalue.
==================
Physicists are familiar with the Casimir of the rotations algebra so(3), as the squared angular momentum operator.
If we decide on some representation---corresponding to a particle---then this operator becomes a NUMBER, as indicated above, and that number is the spin of the particle.
==================
maybe this will provoke someone more expert to provide a GOOD tutorial about Casimirs and some GOOD online resources. But anyway I am doing a slapdash treatment because we need it right now.
===================
WHO CAN TELL US ABOUT THE TWO CASIMIRS OF so(4,1), the deSitter algebra?

apparently it is rank 2, and it has two Casimirs, which are the quadratic and the quartic. And if you pick a definite representation---as if one were specifying a particle---then one gets MASS AND SPIN numbers from the two Casimirs. Aiiyeee!
==================
maybe a nice online tutorial on so(4,1) and SO(4,1) will show up.
this smilie is intended to denote hopefulness

Thank you for this mini-tutorial on Casimirs and the Killing Form. I am still trying to find something suitable on SO(1,4), or even SO(5), and I suppose you are too. It's not like SU(2) or SO(1,3) that everybody writes about!

BTW, just for the uninitiated, the names Lie and Killing have unfortunate homonyms in English. Those of us old enough to remember Trygve Lie when he was Secretary General of the UN know that name is pronounced "Lee". And Killing is rightly pronounced "Key Link" (a nice coincidence that!). Sophus Lie was the creater of Lie Groups, I believe he was an older cousin of Trygve. And Wilhelm Killing created most of the early theory of Lie Algebras (and was treated rather unfairly by Eli Cartan, who presented him to the world as a naive genius who couldn't do rigorous proofs, which he, Cartan had had to create. Later research has shown that although Killing sometimes made mistakes, many of the proofs Cartan said he'd come up with were Killing's.)

 

[marcus]

being of nonviolent disposition, I say to myself "Chilling form"----a cool bilinear form that let's the Leigh algebra chill out.

 

[selfAdjoint]

the Leigh algebra

So you could say we are on a Leigh Hunt?

"Jenny kissed me..."

 

[Mike2]

But to come to the paper itself, first let me say it is a nice piece of work; but is it suprising ? Not really. To write gravity in the BF form was known for quite some time already; you start out from a broken SO(4,1) gauge theory, variate and throw out the topological invariants. Now, the A field (F(A) = dA + [A,A] ) is what gives the Einstein equations, pretty much like the ordinary gauge potential in Maxwell theory does that (with that difference that stuff is nonabelian here). So, how to introduce matter and couple it to gravity? Ah, we know already for long time that braking symmetries can give mass to the corresponding ``gauge particle'' (actually, it are the casimirs which label the gauge orbits and SO(4,1) just like the Poincare group (I guess) has two of them : mass and spin length).

Do you know of a good book on this subject in modern form that seems complete? I'm aware of the references by JC Baez. But I'm wondering if there might be a more modern approach that should be taken. Thanks.

 

[marcus]

What a nice poem!

Jenny kiss'd me when we met,
Jumping from the chair she sat in;
Time, you thief, who love to get
Sweets into your list, put that in!
Say I'm weary, say I'm sad,
Say that health and wealth have miss'd me,
Say I'm growing old, but add,
Jenny kiss'd me.

===================
collecting some links. I will jot them down here. nothing especially good yet--just searching around:
http://math.ucr.edu/home/baez/week208.html
http://arxiv.org/abs/gr-qc/0511077
http://arxiv.org/abs/hep-th/0602002 (Krishnan and di Napoli).
http://etd.uwaterloo.ca/etd/astarodu2005.pdf (Starodubtsev's PhD thesis 2005)
http://arxiv.org/abs/gr-qc/0606122 ("de Sitter special relativity"---brief tutorial review---Jose Geraldo Pereira et al)
Wait! here is something that someone might find useful:
http://arxiv.org/abs/hep-th/0411154 (matrices, nuts-bolts, by Jerzy KowalskiGlikman, gets hands on the de Sitter group and algebra for a bit---"Quantum kappa-Poincaré algebra from de Sitter space of momenta")
Not sure to whom it might be useful, but if the Jerzy K-G paper is understandable it could the best I've found so far.

 

[selfAdjoint]

What a nice poem!

Jenny kiss'd me when we met,
Jumping from the chair she sat in;
Time, you thief, who love to get
Sweets into your list, put that in!
Say I'm weary, say I'm sad,
Say that health and wealth have miss'd me,
Say I'm growing old, but add,
Jenny kiss'd me.

It was sinful of me; I knew it was OT after I had been on careful's case about just that but (further OT warning!) I had just been thinking about my six month old grandaughter Elizabeth, and how her whole face lights up whenever she sees me, and she grins from ear to ear. And I thought I knew just how Leigh Hunt felt, and then you posted about Leigh Algebras and their Chilling Forms (why not Warming Forms?, I thought), and there I went again.

 

[selfAdjoint]

Do you know of a good book on this subject in modern form that seems complete? I'm aware of the references by JC Baez. But I'm wondering if there might be a more modern approach that should be taken. Thanks.

Freidel and company themselves cite

A.P. Balachandrian, G. Marmo, B.-S. Skagerstam, and A. Stern, Gauge Symmetries and Fibre Bundles. Application to Particle Dynamics, Lecture Notes in Physics 188, Springer 1983.

It's their reference 2. It's likely to be both very technical and out of print though. Springer books were known as "the yellow peril" in my grad school days, and they are not at all shy about pruning their backlists.

 

[marcus]

Jerzy K-G is actually quite helpful about SO(1,4) and so(1,4)

http://arxiv.org/abs/hep-th/0411154

He describes the "IWASAWA DECOMPOSITION" of the algebra which is a surprisingly nice decomposition into a direct sum of three things, one of which is simply so(1,3) the lorentz algebra

l + n + a = so(1,3) + n + a

he uses gothic letters but I just replaced the underlined lowercase L by what it really is, namely so(1,3)

He gets a decomposition of the GROUP too-----out of the Iwasawa decomposition of the algebra.

and he has a way of representing DESITTER SPACE as the quotient of two groups where you divide SO(1,4) by its subgroup SO(1,3)-----the coset space depicts de Sitter.

this is mostly all on page 3, with a little on page 4. Most efficient if anyone interested that doesn't know deSitter algebra already just looked at it, rather than me trying to paraphrase.
=====================

but I will try to paraphrase anyway just for practice. Mainly I have to say what are the algebras n and a.
Because the Iwasawa version of deSitter so(1,4) is equal to lorentz + n + a

and a is small potatos. It just has ONE GENERATOR
Actually it is late, past midnight. So I will not try to paraphrase at least right now. But Jerzy WRITES DOWN THE MATRICES for both n and a, the algebras, and also for N and A the groups. It is a patient and helpful paper.
good night Kea, time to turn in.
Anybody who wants, continue with the exposition. See y'all in the morning.

 

[Careful]

Do you know of a good book on this subject in modern form that seems complete? I'm aware of the references by JC Baez. But I'm wondering if there might be a more modern approach that should be taken. Thanks.

Hmm, usually we learn that by trying it out. But I guess you could find it out in the Nakahara book : geometry, topology and physics. Let me furthermore tell you that supermodern introductions mostly have severe limitations, the old books (which are much less formal) usually give a better intuition about the meaning of things.

Careful

 

[selfAdjoint]

Aiyeee! Nakahara of course! Now where did my copy get itself to?

Actually http://store.doverpublications.com/0486661814.html" has a useful section on representations of orthogonal groups that is accessible (pp. 391-399) and it's cheap at Dover.

 

[marcus]

a BTW comment:
Back in January 2005 this article (I think it was this one) was listed as "To appear", but I think the title changed.

In the first Freidel Starodubtsev article there were two "to appear" references

[6] L. Freidel, J. Kowalski-Glikman, A. Starodubtsev, “Background independent perturbation theory for gravity coupled to particles: classical analysis”. To appear.

[13] L. Freidel and A. Starodubtsev, “Perturbative gravity via spin foam” To appear.

I think what we are reading now is [6], retitled to be

"Particles as Wilson lines of gravitational field"

Anyway that is how I interpret it, because it is by the same three authors, and also it IS the classical analysis.

But I could be wrong, there could still be two further papers of Freidel with Starodubtsev (not just one, reference [13]) in the works.
===============
John Baez said Freidel has several in progress with Starodubtsev, and one with Baratin. I will look at the most recent Freidel Baratin http://arxiv.org/abs/gr-qc/0604016 and see if there is some mention.
Yes! their April 2006 paper was the 3D case of "gravity hidden in Feynman diagrams" and they say in the abstract that "We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results." The companion paper is their reference [1]

[1] A. Baratin, L. Freidel Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams, To appear.

 

[marcus]

Aiyeee! Nakahara of course! Now where did my copy get itself to?

Nakahara Geometry, Topology, and Physics IoP Bristol 2005
is reference [12] of Freidel, Kowalski-Glikman, Starodubtsev
gr-qc/0607014

Hope you found your copy!

 

[selfAdjoint]

Nakahara Geometry, Topology, and Physics IoP Bristol 2005
is reference [12] of Freidel, Kowalski-Glikman, Starodubtsev
gr-qc/0607014

Hope you found your copy!

Yes, 'twas in the living room next to my favorite chair. Studying it (section 5).

 

[john baez]

SO(4,1) is deSitter - a close relative of Poincare

Hi! I'm having too much fun in http://math.ucr.edu/home/baez/diary/july_2006.html#july5.06" to post much, but I'm happy to hear that Freidel is finally getting more of his work out.

Yes! I agree that BF can be done with any of the usual groups and what makes this special is SO(4,1). I would like to know more about the deSitter group SO(4,1) and its algebra so(4,1). Just about anything you or Kea can say would be a help---either to me or to other readers of the thread.

The main thing to realize is that SO(4,1) is the deSitter group. In other words, it's the symmety group of a 4d spacetime called deSitter spacetime - an exponentially expanding universe not unlike our own. The curvature of this spacetime is proportional to the cosmological constant, which is positive.

In the limit where the cosmological constant goes to zero, deSitter spacetime reduces to good old flat Minkowski spacetime - and the deSitter boils down - or contracts, in the usual jargon - to the Poincare group.

So, you should think of SO(4,1) as a close relative of the Poincare group. To ants like us who can't see far enough to notice that spacetime is curved, there's no way to tell if the symmetry group of the universe is the Poincare group or SO(4,1).

The group SO(p,n-p) has dimension given by the triangle number n(n-1)/2. Since 4+1 = 5, SO(4,1) has dimension 5(5-1)/2 = 10:

o
oo
ooo
oooo

The Poincare group also has 10 dimensions. That should be reassuring.

Another close relative of the Poincare group is SO(3,2), which also has dimension 10. This is the symmetry group of anti-deSitter spacetime, where the cosmological constant is negative.

You may find it odd to describe a group like SO(4,1) in a way that depends on a number - the cosmological constant - and see what happens as we send this number to zero. But, it's common in physics. One of the first guys to study this limiting process was Wigner. It may have been he who invented the term contraction for this process. For example, he noticed that the Poincare group contracts to the Galilei group as the speed of light goes to infinity. The Galilei group is the symmetry group of Newtonian physics, generated by translations, rotations, and Galilei boosts.