a new paper by Girelli and Livine
the time seems propitious for taking postive Λ into account
which seems to mean using the q-deformed SL(2,C) and its representations on the foams and networks
Karim Noui and Philippe Roche have done this for Barrett-Crane foams, producing a q-BC Lorentzian spin foam model
Livine's thesis provides a bridge between covariant loop gravity and the ordinary BC spin foam, so I would imagine the next thing will be to explore the bridge between q-deformed covariant loop gravity and the q-BC foam model.
the physical consequences of positive Λ in q-BC foam, as explained in this short paper by Girelli/Livine, seem extremely interesting
Livine is talking about the "time normal" c which is defined at each vertex of the graph and takes values in Minkowski space on an "upper hyperboloid" H+.
The time normal is quantized and H+ becomes a stack of fuzzy spheres. This is from Noui and Roche.
Imagined in 3D the hyperboloid is a surface of revolution around the time axis and consists of a pile of rings of expanding radius so in 4D it seems reasonable it would be a stack of 2-spheres.
We know this vector c from Livine's thesis*. One of the main ideas is that the quantum state cylinder function is defined not only on a bunch of group elements one for each edge but also on a bunch of x-vectors one for each vertex. f(g1, g2,...,gE, x1, x2,...xV). We've seen this before in both the new (covariant) canonical quantization and in BC spin foams.
And it is this c that, by getting quantized, means that the speed and momentum are quantized. This is not because space has some discrete or latticey structure---says Livine--but for a subtler reason and is more related to the discreteness of time in this model. (the model may indeed have a continuous area operator spectrum---space here is less obviously quantized than it may be in some other models)
but also from papers such as Alexandrov/Livine
and Livine's http://arxiv.org/gr-qc/0207084
at present the cosmological constant l (in planck units) is
Girelli/Livine equation (3) says that the way speed gets quantized there is a series of possible values of the rapidity η = v/(1 - beta2)1/2
ηn = (1/2) arc cosh (cosh(nl)/cosh(l)) = approx. nl/2
It is the rapidity that one would multiply by a particle's rest mass to get momentum. Approx. is good for classical or everyday situations---n large but nl<<1. Girelli/Livine propose a conservation of momentum law which conserves the sum of the n's. It comes to the same thing unless the cosmological constant is large or unless the speed is small compared with the cosmological constant.
With our present small l, no imaginable instrument could detect the quantization (they give a couple of thought experiments about that) BUT suppose that sometime in the very early expansion of the U, suppose l was large, whoah, then the quantization of speed might have had some noticeable effect. And some remnant might conceivably be observable in the cosmic background.
comment on style
notice this from page 3 second paragraph of Girelli/Livine
"We can turn now to the physical consequences of the quantization of speed. Ordinarily the effects of quantum gravity are expected to occur at very high energy, for example when dealing with Gamma Ray Bursts . In fact until recently quantum gravity was not expected to be testable due to this high energy problem. If one tries to measure the quantized speed, then, strangely the effect we want to measure is at very low energy! The effect that we are suggesting has therefore the originality to be present at very low energy, contrary to the usual predictions..."
the relation of theory to nature is often amusing and in earlier days some good physicists have been humorists. My impulse is to use italics or bold and write "the effect that we are suggesting has therefore the originality to be present"----this is a good 4-page paper
Yes, I agree. I don't know if some their effects are deliberate or the products of uncertain English (I once wrote a letter in my College French on behalf of a fellow Air Force officer to his French girl friend. He later told me her family found my letter hilarious).
I do think that if other thinkers work on their suggestions, we might see some valid interpretations of WMAP that support LQG. Or not, of course, as the empirical cookie crumbles.
this has got to be one of the coolest 4-page papers ever.
I've been looking a Girelli/Livine equation (3) and with a cosmological constant like ours (namely 1.3E-123) there is no way of distinguishing cosh(lambda) from one. And so the quantized boost parameter etaN is just (1/2)N lambda. The approximation is extremely close so that no instrument could ever tell the difference.
It reminds me of the formulas in SR for low beta----never know youre not in the world of euclid/newton. sly of Ma Nature (if she is doing that)
So all the observers in the world are moving relative to each other by speeds which come in tiny steps on order of 10-123.
the speed knob has clicks (how to say this, Aiiiieee, Selah, Ah so, no exclamation is quite appropriate)
This seems very strange to me. In order to quantize speed, you would have to quantize the speed of an allowable reference frame. That just doesn't seem right to me.
Consider a particle moving at on quanta of speed. Now adjust the reference frame by half of a speed quanta and the particle is moving at a non-integral quanta.
Consider in one reference frame 2 particles move away from their common origin at one quanta of speed, with only a slight divergence in their directions. Now consider them in the reference frame of one of the particles. The other particle appears to be moving away with just a fraction of the minimum quanta of speed.
In these cases, would an observer see erratic motion? IE, the observed particle sits still, then moves, sits still, then moves. Would it be a regular oscillation of movement, or a probability density funtion of velocities each of which is an integral number of speed quanta?
I haven't read the paper, but ... in a theory in which speed is quantized, is it physically possible for two frames to differ by half of a speed quantum? (By "speed quantum" I guess I mean the smallest nonzero speed in the spectrum.)
you've got it Njorl
the speeds of allowable observers relative to each other are quantized in their proposal. they explicitly say this in the paper as I recall
so the example you mentioned of adjusting a frame to move by "half" a quantum does not come up
AmbiT, I do hope you have a look at this.
I'd much appreciate your comments or clarifactions.
The paper says: "The speed (with respect to a given observer) is quantized." Notice that we no longer have the Poincare group, or even its covering SL(2,C), but rather the q-deformed version of SL(2,C). Wouldn't this imply that the usual Lorenz boosts would be replaced by q-deformed, i.e. quantized versions?
When Cosmological constant was large.
Another interesting affect of Livine's new paper is the effects quantizised speed has when Cosm. const. is big (as it were in the early phases of big bang). His paper proclaims that speed then only could have a few certain values, something that has a great impact on the structure of the universe.
It would be great if someone could reply to this.
Re: When Cosmological constant was large.
A hearty welcome to you Martin! I too am curious about this and hope that someone will reply. Sometimes David Louapre (who knows Livine's work) has posted here and he may reply.
I think maybe here is one approach to thinking about this: first, to understand the present situation with very small lambda (order E-123 in planck units).
this means the deformation parameter q is extremely near to one
q = exp(- lambda) = exp(- 1.3E-123)
only just barely less than one
so to deform the Lorentz group is to do almost nothing at all,
qualitatively everything remains the same.
(almost the same: there is a cosmological horizon beyond which we will never see but it is far away and its area is very large, like E123)
With that in mind, one goes through the development in Livine's
Lambda = 0 (ordinary SL(2,C)) case in
"Projected Spin Networks..." gr-qc/0207084
or in "Thesis" gr-qc/0309028
so one knows what he means by "projected spin network" and then
one looks at the key paragraph on page one:
"In the q-deformed model, using the representation theory of Uq(SL(2,C)), it turns out that the areas become bounded...Moreover the time normal becomes quantized: the hyperboloid H+ becomes a pile of fuzzy spheres with quantized radii..."
[[[Soon thereafter he remarks that classically a vector on H+ defines "a boost or equivalently a boosted observer." in continued response to Njorl, and agreement with selfAdjoint's comment]]]
NOW LOOK AT page 17, equation (58) of Karim Noui and Philippe Roche
"Cosmological deformation..." http://arxiv.org/gr-qc/0211109.
Here we have the formula which Livine uses next. It is the Girelli/Livine equation (1) on their page 2.
I believe that this may be what Livine means when he says "it turns out that....the time normal becomes quantized..."
Now in equation (1) there is the dependence on Lambda that we were both wondering about----which has such interesting consequences for speeds in the very early universe when Lambda was large.
(and when, reciprocally, the area of the cosmological horizon was small)
I realize I have raised questions rather than answered them but we can still hope that David Louapre will reply to your post or one of the other knowledgeable people--and in your post you did not specifically ask for answers but only simply for a reply
well in case you are still around Martin,
I just checked in a mainstream cosmology reference
and the present distance to the cosmological event horizon
is some 60 billion lightyears.
That is how far away objects are whose light could ever reach us if we could wait forever.
It is a finite distance because of accelerating expansion so it is a new length in cosmology. Before there was the radius of the observable universe---what is observable at present. This is like that but bigger: the radius of what is in principle observable if you wait forever. More precisely, with the accepted value of Lambda, it is 62 billion LY, but I say 60E9 LY
And a LY is 5.8E50 planck, so the cosmo horizon distance is 3.6E61 planck.
This is why the AREA of the cosmo horizon is essentially E123
the reciprocal of Lambda the dark energy density expressed in planck.
This recalls what Girelli/Livine say towards the end of page one.
they say area is bounded by pi over Lambda
A < p/L
part of the interest to me of Girelli/Livine is the connection between a local thing, the quantization of speeds right around me, and a giant enormity: the radius and area of the cosmological event horizon
the cosmological event horizon is somewhat like a black hole event horizon because no information from beyond there can ever get to us even if we wait forever---like we are in a bubble enclosed in an everted BH horizon
Now you asked about the early universe, when the cosmological event horizon had a very small radius and a small area and so the reciprocal of that area, which is Lambda, was very large
and Girelli/Livine are saying that this means the SPEED STEPS in that early universe were therefore big
only a few steps and you are already up to the speed of light!
therefore all things forced to adopt from a small menu of speeds
during that tiny very early instant of time when Lambda was large
it is worth thinking about and discussing
If you are still around and want to
Yes, Marcus, it was this kind of discussion I wanted. Thank you.
However it would also be interesting if someone who knows Livine's work could answer (I remember you mentioned someone).
If the theory is accepted and furthermore taken in account in today's other big bang theories, I believe it'll have a great impact.
But this is all I can say, I'm not a physician (yet - I'm only 17 so I've just started on Quantum phys.) and I do not really have the knowledge required to understand this paper in its wholeness (yet).
But thank you for your replies and comments, Marcus. <S>
It is a young person's field, quantum gravity. there is a bunch of young people (under 30, mostly postdocs or not even with PhD yet) who are doing much of the creative work. One of the world centers is near Potsdam (the Albert Einstein Institute at Goelm, part of the MPI for Gravitation Physics at Potsdam). Martin Bojowald is at AEI-Goelm.
If you are in Munich, there is Marcus Gaul at Munich and several others. Other centers for quantum gravity research are at Marseille and Lyon in France.
A recent short paper by Martin Bojowald is "Quantum Gravity and the Big Bang". It is online at http://arxiv.org.
Bojowald's email address is given in the paper
In fact, as you say, recent work on quantum gravity has exciting consequences for understanding the early universe----inflation, the "big bang singularity", the domination of matter over antimatter.
the person I mentioned earlier is from France and is named David Louapre. I think he is busy writing a paper just now and cannot answer questions.
Why dont you have a look at Bojowald's "Quantum Gravity and the Big Bang" Any search engine, e.g. Google, would give a link to the paper at arxiv.org (which is where almost all physics papers seem to appear online)
Separate names with a comma.