
#37
Jan2611, 12:25 PM

Astronomy
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PF Gold
P: 22,808

@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} = G^{L} = G x G x ... x G Physicists think that they can write down x_{i} and have this mean either x_{i} or else the Ntuple (x_{1}, x_{2},...,x_{N}) 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 Z_{C}(h) where C is a twocomplex with L boundary or "surface" edges, and h is a generic element of SU(2) and h is (h_{1}, h_{2},...,h_{L}), namely a generic element of SU(2)^{L} The twocomplex C is the "diagram". The boundary edges are the "input and output" of the diagramthink of the boundary as consisting of two separate (initial and final) components so that Z becomes a transition amplitude. Think of the Ltuple 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 Z_{C}(h) and one can think of that number as saying Z_{roadmap}(boundary conditions) 



#38
Jan2611, 01:02 PM

P: 920

The thing I like about LQG is that although the ideas may be incorrect or the redefinition for that matter, they are making progress and aren't afraid to delve into these unique concepts. I've never seen so many original papers come out in a year in one specific research program!
All I see now from String Theory research programs is [tex]AdS_5 \times S^5[/tex] and holographic superconductors, they haven't really ventured into other ideas. Is [tex]AdS/CFT[/tex] even a physical theory at this point, is it possible in our universe? I don't know, but many interesting things are going on in LQG and it's relatives such as CDT that appear much more interesting then the plateau that ST is facing, what the "heck" is a holographic superconductor anyways? I think the real notion that must be addressed is the nature of spacetime itself. I feel that all of our ideas in Physics rely on a specific spacetime backgrounds and therefore having a quantum description of spacetime at a fundamental level is a more clear approach  which LQG does. Does ST address this idea, is [tex]AdS/CFT[/tex] a valid idea? Anyways enough with the merits of ST, what is LQG lacking? 



#39
Jan2611, 01:47 PM

Astronomy
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PF Gold
P: 22,808

The issue is whether or not is is time to replace the manifold with something lighter, more finite, more minimal, more "informatic" or informationtheoretical. If the historical moment is ripe to do this, then Rovelli and associates are making a significant attempt which may show the way. If the historical moment is not ripe to replace the manifold (as model of spacetime) then they will be heading off into the jungle to be tormented by savages, mosquitoes and malaria. At the present time the proposed minimalist/informatic structure to replace manifold is a 2complex. Or, ironically, one can also work with a kind of "dual" which is a fullblown 4D differential manifold which has a 2complex of "defect" removed from it and is perfectly flat everywhere else. A twocomplex is basically just like a graph (of nodes and links) except it has one higher dimensionality (vertices, edges, faces). A twocomplex is mathematically sufficient to carry a sketch of the geometric information (the curvatures, angles, areas between eventmarked regions,...) contained in a 4D manifold where this departs from flatness. A twocomplex provides a kind of finite combinatorial shorthand way of writing down the geometry of a 4D continuum. So we will watch and see how this goes. Is it time to advance from the 1850 spacetime manifold beachhead, or not yet time to do that? 



#40
Jan2611, 01:52 PM

P: 920

So essentially quantum spacetime is nodes connecting to create 4D tetrahedrons?




#41
Jan2611, 02:37 PM

Astronomy
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PF Gold
P: 22,808

Also another favorite is the Rovelli quote that QG is not about what spacetime is but about how it responds to measurement. (there was a panel discussion and he was trying to say that arguments about whether it is really made of chainlinkfence, or tinkertoy, or legoblocks, rubberbands, or tetrahedra, or the 4D analog of tets, called 4simplices, or general Nface polyhedra....are not good arguments. How one sets up is really just a statement about how one intends to calculate. One calculates the correlations between measurements/events. The panel discussion was with Ashtekar and Freidel, at PennState in 2009, as I recall. I can get the link if anyone is interested. It told me that QG is about geometric information, i.e. observables. not about "ontology". So I liked that and based my agnosticism on it.) BTW I think human understanding grows gradually, almost imperceptibly, like a vine up a wall. Nothing works if it is too big a step, or jump. Therefore, for me, there is no final solution, there are only the small steps that the human mind can take now. The marvel of LQG, for me, is that it actually seems as if it might be possible to take this step now, and begin to model spacetime with something besides a manifold, and yet still do calculations (not merely roll the Monte Carlo simulation dice of CDT and Causets.) But actually, Kevin, YES! Loosely speaking, the way almost everyone does speak, and with the weight on "essentially" as I think you meant it, in this approach spacetime essentially is something like what you said! 



#42
Jan2611, 03:06 PM

P: 362

Moreover, I think the real culprit is differential equations, they are inherently a guess work, the technique is always to "add terms" to get it to fit experiment, not to mention its limited relating points to the neighbors and the notorious boundary condition requirement. It has served us well for a long time,but No fundamental theory should be like that. As for LQG, the original idea was just the only option to make GR look like the quantum and to "see what happens", only for rovelli to conclude that spacetime and matter should be related. But how, LQG is giving hints which has not been capitalized on. I still think spacetime is ""unphysical ""and must be derived from matter and not the other way around. 



#43
Jan2611, 04:16 PM

Astronomy
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PF Gold
P: 22,808

Tedrahedron means "four sides" and a tetrahedron does have four (triangular sides). At tet is also a "3simplex" because it is the simplex that lives in 3D. Just like a triangle is a 2simplex. The official name for a 4simplex is "pentachoron" choron means 3D room in Greek. the boundary of a pentachoron consists of five 3D "rooms"five tetrahedrons. To put what you said more precisely The blue pattern will look like a honeycomb hexagon tiling of the plane. The blue pattern is dual to the black triangulation. Each blue node is connected to three others. Then imagine it in 3D where you start by triangulating regular 3D space with tetrahedra. Then you think of putting a blue dot at the center of each tet, and connect it with a blue line to each of the 4 neighbor blue dots in the 4 adjacent tets. In some versions of LQG, the spin networksthe graphs that describe 3D spatial geometry are restricted to be dual to triangulations. And in 4D where there are foams (analogous to graphs), only foams which are dual to triangulations are allowed. These ideassimplexes, triangulations that chop up space or spacetime into simplexes, duals, etc.become very familiar and nonpuzzling. One gets used to them. So that would be an additional wrinkle to the general idea you expressed. Finally, it gets simpler aqain. You throw away the idea of triangulation and just keep the idea of a graph (for 3D) and a foam thought of either as 4D geometry, or as the evolution of 3D geometry. And you let the graphs and foams be completely general, so no more headaches about the corresponding dual triangulation or even if there is one. You just have general graphs and twocomplexes, which carry information about observables (area, volume, angle,...) =============================== Kevin, one could say that all this stuff about tetrahedrons and pentachorons and dual triangulations is just heuristic detail that helps people get to where they are going, and at some point becomes extra baggageunnecessary complicationand gets thrown out. You can for instance look at 1010.1939. In fact it might do you good. You see a complete presentation of the theory in very few pages and no mention of tetrahedrons Nor is there any mention of differentiable manifolds. So there is nothing to chop up! There are only the geometric relations between events/measurements. That is all we ever have, in geometry. Einstein pointed it out already in 1916 "the principle of general covariance deprives space and time of the last shred of objective reality". Space has no physical existence, there are only relations among events. We get to use all the lego blocks we want and yet there are no legoblocks. Something like that... 



#44
Jan2611, 06:25 PM

Astronomy
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PF Gold
P: 22,808

At any rate, let's get back to the main topic. There is this new formulation, best presented in http://arxiv.org/abs/1010.1939 or so I think, and we have to ask is it simple enough and also wonder if it will be empirically confirmed. It gives Feynman rules for geometry leading to a way of calculating a transition amplitude a certain complex number, which I wrote
Z_{roadmap}(boundary conditions) the amplitude (like a probability) of going from initial to final boundary geometry following the Feynman diagram roadmap of a certain twocomplex C. A twocomplex is a finite list of abstract vertices, edges, faces: vertices where the edges arrive and depart and faces bordered by edges (the list says which connect with which). Initial and final geometry details come as boundary edge labels which are elements of a group G = SU(2). There is some finite number L of boundary edges, so the list of L group elements labeling the edges can be written h = (h_{1}, h_{2},...,h_{L}). So, in symbols, the complex number is Z_{C}(h). The theory specifies a formula for computing this, which is given by equation (4) on page 1 of http://arxiv.org/abs/1010.1939 , the paper I mentioned. Here is an earlier post that explains some of this: 



#45
Jan2611, 08:58 PM

Astronomy
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PF Gold
P: 22,808

The way the equation (4) works is you let boundary information ( h ) percolate into the foam from its outside surface, and you integrate up all the other labels that the twocomplex C might have compatible with what is fixed on the surface.
The foam is like an informationsponge, with a certain welldefined boundary surface (actually a 3D hypersurface geometry, think initial + final) and you paint the outside of the sponge with some informationpaint h and the paint seeps and soaks into the inside, and constrains what colors can be there to some extent. Then you integrate out, over all what can be inside, compatible with the boundary. So in the end the Z amplitude depends only on the choice of the unlabeled roadmap C, a pure unlabeled diagram, plus the L group element labels on the boundary graph. If the grouplabeled boundary graph happens to have two connected components you can call one "initial geometry" and one "final geometry" and then Z is a "transition amplitude" from initial to final, along the twocomplex roadmap C. BTW Etera Livine just came out with a 90page survey and tutorial paper on spinfoam. It is his habilitation, so he can be research director at Lyon, a job he has already be performing from the looks of it. Great! Etera has posted here at PF Beyond sometimes. His name means Ezra in the localtradition language where he was raised. A good bible name. For some reason I like this. I guess I like the name Ezra. Anyway he is a firstrate spinfoam expert and we can probably find this paper helpful. http://arxiv.org/abs/1101.5061 A Short and Subjective Introduction to the Spinfoam Framework for Quantum Gravity Etera R. Livine 90 pages (Submitted on 26 Jan 2011) "This is my Thèse d'Habilitation (HDR) on the topic of spinfoam models for quantum gravity, which I presented in l'Ecole Normale Supérieure de Lyon on december 16 2010. The spinfoam framework is a proposal for a regularized path integral for quantum gravity, inspired from Topological Quantum Field Theory (TQFT) and statesum models. It can also be seen as defining transition amplitudes for the quantum states of geometry for Loop Quantum Gravity (LQG)." It may interest you to go to page 61 where begins Etera's Chapter 4 What's Next for Spinfoams? 



#46
Jan2611, 09:27 PM

P: 920

Awesome, thanks for the detailed explanation marcus! I'm in grade 11 so the maths only makes partial sense to me but the words will be good enough for now. About connecting the points in the center of the triangles, so you always have an Npolygon with three Npolygons 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?




#47
Jan2611, 10:08 PM

Astronomy
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PF Gold
P: 22,808

It would be a good brainexercise, 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 cellcomplexes or something? I'm not an expert but there may be someone good on that. 



#48
Jan2611, 10:12 PM

P: 920

The Wiki article is good: "The 5cell can also be considered a tetrahedral pyramid, constructed as a tetrahedron base in a 3space hyperplane, and an apex point above the hyperplane. The four sides of the pyramid are made of tetrahedron cells."  Wikipedia: 5cell, http://en.wikipedia.org/wiki/Pentach...ernative_names
Anyways, I digress. I'm sure this is slightly offtopic. 



#49
Jan2611, 10:14 PM

Astronomy
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PF Gold
P: 22,808

Oh good! You are on your own. I googled "dual cell complex" and found this:
http://www.aerostudents.com/files/co...lComplexes.pdf Don't know how reliable or helpful it may be. 



#50
Jan2611, 10:24 PM

P: 920

I understand some vector calculus and that appears to be what the math being used is. Thanks I'm sure that will be useful!




#51
Jan2711, 12:46 AM

P: 144

Regular tetrahedra can not fill space. Tetrahedra combined with octahedra can fill space. See isotropic vector matrix or octettruss. ...and I think the dual is packed rhombic dodecahedra 



#52
Jan2711, 02:05 AM

P: 203

which you identified in the bibliography thread. 



#53
Jan2711, 03:04 AM

Astronomy
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PF Gold
P: 22,808

I will quote a sample passage from page 31: ==quote Livine 1101.5061 == Starting with the simpler case of a threedimensional spacetime, a spacetime triangulation consist in tetrahedra glued together along their triangles. The dual 2skeleton is defined as follows. The spinfoam vertices σ are dual to each tetrahedron. Those vertices are all 4valent with the four attached edges being dual to the four triangles of the tetrahedron. Each edge e then relates two spinfoam vertices, representing the triangle which glues the two corresponding tetrahedra. Finally, the spinfoam faces f are reconstructed as dual to the triangulation’s edges. Indeed, considering an edge of the triangulation, we go all around the edge and look at the closed sequences of spinfoam vertices and edges which represent respectively all the tetrahedra and triangles that share that given edge. This line bounds the spinfoam face, or plaquette, dual to that edge. Finally, each spinfoam edge e has three plaquettes around it, representing the three triangulations edges of its dual triangle. To summarize the situation: 3d triangulation ↔ spinfoam 2complex ___________________________________ tetrahedron T ↔ 4valent vertex σ triangle t ↔ edge e edge ↔ plaquette f The setting is very similar for the fourdimensional case. The triangulated spacetime is made from 4simplices glued together at tetrahedra. Each 4simplex is a combinatorial structure made of 5 boundary tetrahedra, glued to each other through 10 triangles. Once again, we define the spinfoam 2complex as the dual 2skeleton: .... ==endquote== 



#54
Jan2711, 04:21 AM

Astronomy
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PF Gold
P: 22,808

Suppose I allow two kinds of tet. Can it be done? Please tell us if you know. [This may not be absolutely on topic, because all we need to accomplish what Etera is talking about is some sort of tetrahedral triangulation of space, which I'm pretty sure exists (if we relax the regularity condition slightly). But it's not a bad exercise for the imagination to think about it. Helios might be a good teacher here.] 


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