- #36
MTd2
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What is a semiclassical limit for you?
Why fitting cc could fit into an RG framework would be a fundamental question?
Why fitting cc could fit into an RG framework would be a fundamental question?
Kevin_Axion said:...
I think the real notion that must be addressed is the nature of space-time itself.
marcus said:...
The central quantity in the theory is the complex number ZC(h) and one can think of that number as saying
Zroadmap(boundary conditions)
I'm agnostic about what nature IS. I like the Niels Bohr quote that says physics is not about what nature is, but rather what we can say about it.Kevin_Axion said:So essentially quantum space-time is nodes connecting to create 4D tetrahedrons?
tom.stoer said:The problem with that approach was never the correct semiclassical limit (this is a minor issue) but the problem to write down a quantum theory w/o referring to classical expressions!
Kevin_Axion said:So essentially quantum space-time is nodes connecting to create 4D tetrahedrons?
So essentially quantum space-time is nodes connecting to create pentachorons?
marcus said:@Tom
post #35 gives an insightful and convincing perspective. Also it leaves open the question of what will be the definitive form(s) of the theory. Because you earlier pointed out that at a deeper level a theory can have several equivalent presentations.
I had a minor comment about that. For me, the best presentation of the current manifoldless version is not the absolute latest (December's 1012.4707) but rather October's 1010.1939. And I would say that the notation differs slightly between them, and also that (from the standpoint of a retired mathematician with bad eyesight) their notation is inadequate/imperfect.
If anyone wants to help me say this, look at 1010.1939 and you will see that there is no symbol for a point in the group manifold SU(2)L = GL = G x G x ... x G
Physicists think that they can write down xi and have this mean either xi or else the N-tuple (x1, x2,...,xN)
depending on context. This is all right to a certain extent but after a point it becomes confusing.
In many ways I think the presentation in 1010.1939 is the clearest, but it is still deficient.
Maybe I will expand on that a bit, if it will not distract from more meaningful discussion.
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BTW, in line with what Tom said in the previous post, there are obviously several different ways LQG can fail, not just one way. One failure mode is mathematical simplicity/complexity. To be successful a theory should (ideally) be mathematically simple.
As well as passing the empirical tests.
One point in favor of the 1010.1939 form is that it "looks like" QED and QCD, except that it is background independent and about geometry, instead of being about particles of matter living in fixed background. Somehow it manages to look like earlier field theories. The presentation on the first page uses "Feynman rules".
These Feynman rules focus on an amplitude ZC(h)
where C is a two-complex with L boundary or "surface" edges, and h is a generic element of SU(2) and h is (h1, h2,...,hL), namely a generic element of SU(2)L
The two-complex C is the "diagram". The boundary edges are the "input and output" of the diagram---think of the boundary as consisting of two separate (initial and final) components so that Z becomes a transition amplitude. Think of the L-tuple h as giving initial and final conditions. The notation h is my notational crutch which I use to keep order in my head. Rovelli, instead, makes free use of the subscript "l" which runs from 1 to L, and has no symbol for h.
The central quantity in the theory is the complex number ZC(h) and one can think of that number as saying
Zroadmap(boundary conditions)
My writing wasn't clear Kevin. The thing about only three meeting was just a detail I pointed out about the situation on the plane when you go from equilateral triangle tiling to the dual, which is hexagonal tiling. I wanted you to picture it concretely. That particular aspect does not generalize to other polygons or to other dimensions. I was hoping you would draw a picture of how there can be two tilings each dual to the other.Kevin_Axion said:... About connecting the points in the center of the triangles, so you always have an N-polygon with three N-polygons meeting at each vertex, what is the significance of that, will you have more meeting at each vertex with pentachorons (applying the same procedure) because there exist more edges?
marcus said:It would be a good brain-exercise, I think, to imagine how ordinary 3D space can be "tiled" or triangulated by regular tetrahedra. You can set down a layer of pyramids pointing up, but then how do you fill in? Let's say you have to use regular tets (analogous to equilateral triangles) for everything.
And when you have 3D space filled with tets, what is the dual to that triangulation? This gets us off topic. If you want to pursue it maybe start a thread about dual cell-complexes or something? I'm not an expert but there may be someone good on that.
marcus said:Oh good! You are on your own. I googled "dual cell complex" and found this:
http://www.aerostudents.com/files/constitutiveModelling/cellComplexes.pdf
Don't know how reliable or helpful it may be.
sheaf said:The dual skeleton is defined quite nicely on p31 in this paper http://arxiv.org/abs/1101.5061"
which you identified in the bibliography thread.
Helios said:Regular tetrahedra can not fill space...
Helios said:Regular tetrahedra can not fill space.
MTd2 said:But irregular tetrahedra can!
marcus said:@Tom
post #35 gives an insightful and convincing perspective. Also it leaves open the question of what will be the definitive form(s) of the theory. Because you earlier pointed out that at a deeper level a theory can have several equivalent presentations.
I had a minor comment about that. For me, the best presentation of the current manifoldless version is not the absolute latest (December's 1012.4707) but rather October's 1010.1939. And I would say that the notation differs slightly between them, and also that (from the standpoint of a retired mathematician with bad eyesight) their notation is inadequate/imperfect.
If anyone wants to help me say this, look at 1010.1939 and you will see that there is no symbol for a point in the group manifold SU(2)L = GL = G x G x ... x G
Physicists think that they can write down xi and have this mean either xi or else the N-tuple (x1, x2,...,xN)
depending on context. This is all right to a certain extent but after a point it becomes confusing.
In many ways I think the presentation in 1010.1939 is the clearest, but it is still deficient.
Maybe I will expand on that a bit, if it will not distract from more meaningful discussion.
============
BTW, in line with what Tom said in the previous post, there are obviously several different ways LQG can fail, not just one way. One failure mode is mathematical simplicity/complexity. To be successful a theory should (ideally) be mathematically simple.
As well as passing the empirical tests.
One point in favor of the 1010.1939 form is that it "looks like" QED and QCD, except that it is background independent and about geometry, instead of being about particles of matter living in fixed background. Somehow it manages to look like earlier field theories. The presentation on the first page uses "Feynman rules".
These Feynman rules focus on an amplitude ZC(h)
where C is a two-complex with L boundary or "surface" edges, and h is a generic element of SU(2) and h is (h1, h2,...,hL), namely a generic element of SU(2)L
The two-complex C is the "diagram". The boundary edges are the "input and output" of the diagram---think of the boundary as consisting of two separate (initial and final) components so that Z becomes a transition amplitude. Think of the L-tuple h as giving initial and final conditions. The notation h is my notational crutch which I use to keep order in my head. Rovelli, instead, makes free use of the subscript "l" which runs from 1 to L, and has no symbol for h.
The central quantity in the theory is the complex number ZC(h) and one can think of that number as saying
Zroadmap(boundary conditions)
tom.stoer said:The only (minor!) issue is the derivation of the semiclassical limit etc.
tom.stoer said:... I don't want to criticize anybody (Rovelli et al.) for not developping a theory for the cc. I simply want to say that this paper does not answer this fundamental question and does not explain how the cc could fit into an RG framework (as is expected for other couplings).
---------------------
We have to disguish two different approaches (I bet Rovelli sees this more clearly than I do).
- deriving LQG based on the EH or Holst action, Ashtekar variables, loops, ... extending it via q-deformation etc.
- defining LQG using simple algebraic rules, constructing its semiclassical limit and deriving further physical predictions
The first approach was developped for decades, but still fails to provide all required insights like (especially) H. The second approach is not bad as it must be clear that any quantization of a classical theory is intrinsically incomplete; it can never resolve quantization issues, operator ordering etc. Having this in mind it is not worse to "simply write down a quantum theory". The problem with that approach was never the correct semiclassical limit (this is a minor issue) but the problem to write down a quantum theory w/o referring to classical expressions!
Look at QCD (again :-) Nobody is able to "guess" the QCD Hamiltonian; every attempt to do this would break numerous symmetries. So one tries (tried) to "derive" it. Of course there are difficulties like infinities, but one has a rather good control regarding symmetries. Nobody is able to write down the QCD PI w/o referring to the classical action (of course its undefined, infinite, has ambiguities ..., but it does not fail from the very beginning). Btw.: this hasn't changed over decades, but nobody cares as the theory seems to make the correct predictions.
Now look at LQG. The time for derivations may be over. So instead of derived LQG (which by may argument explained above is not possible to 100%) one may simply postulate LQG. The funny thing is that in contradistinction to QCD we seem to be able to write down a class of fully consistent theories of quantum gravity w/o derivation, w/o referring to classical expressions, w/o breaking of certain symmetries etc. The only (minor!) issue is the derivation of the semiclassical limit etc.
From a formal perspective this is a huge step forward. If this formal approach is correct, my concerns regarding the cc are a minor issue only.
tom.stoer said:I think that the derivation of a certain limit is a minor issue compared to the problem that a construction of a consistent, anomaly-free theory (derived as quantization of a classical theory) is not available.
marcus said:...
One thing on the agenda, if we want to understand (4) is to see why the integrals are over the specified number of copies of the groups----why there are that many labels to integrate out, instead of some other number. So for example you see on the first integral the exponent 2(E-L) - V. We integrate over that many copies of the group. Let's see why it is that number. E and V are the numbers of edges and vertices in the foam C. So E-L is the number of internal edges.
marcus said:As I see it, the QG goal is to replace the live dynamic manifold geometry of GR with a quantum field you can put matter on. The title of Dan Oriti's QG anthology said "towards a new understanding of space time and matter" That is one way of saying what the QG researchers's goal is. A new understanding of space and time, and maybe laying out matter on a new representation of space and time will reveal a new way to understand matter (no longer fields on a fixed geometry).
Sources on the 2010 redefinition of LQG are
introductory overview: http://arxiv.org/abs/1012.4707
concise rigorous formulation: http://arxiv.org/abs/1010.1939
phenomenology (testability): http://arxiv.org/abs/1011.1811
adding matter: http://arxiv.org/abs/1012.4719
Among alternative QGs, the LQG stands out for several reasons---some I already indicated---which I think are signs that the 2010 reformulation will prove a good one:
- testable (phenomenologists like Aurelien Barrau and Wen Zhao seem to think it is falsifiable)
- analytical (you can state LQG in a few equations, or Feynman rules, you can calculate and prove symbolically, massive numerical simulations are possible but not required)
- similar to QED and lattice GCD (the cited papers show remarkable similarities---the two-complex works both as a Feynman diagram and as a lattice)
- looks increasingly like a reasonable way to set up a background independent quantum field theory.
- an explicitly Lorentz covariant version of LQG has been exhibited
- matter added
- a couple of different ways to include the cosmological constant
- indications that you recover the classic deSitter universe.
- sudden speed-up in the rate of progress, more researchers, more papers
These are just signs---the 2010 reformulation might be right---or to put it differently, there may be good reason for us to understand the theory, as presented in brief by the October paper http://arxiv.org/abs/1010.1939...