
#109
Aug2005, 06:04 PM

P: 1,308

So the question becomes, how did particle and/or the metric come into existence to begin with? How was the initial total symmetry broken? Did the metric have to start out with zero distance between particles? I'm sure without a metric to start with, we have to rely on topological characteristics to answer how a metric came to be. 



#110
Aug2005, 06:08 PM

Emeritus
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PF Gold
P: 16,101

At least once you have a C1 manifold, there's a unique way to turn it into a C∞ manifold.




#111
Aug2005, 06:18 PM

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PF Gold
P: 16,101

What you're touching upon is the problem of measurement. In fact, the metric isn't even fundamental  General Relativity can be reformulated without any reference to a metric. (At least if I understand correctly) 



#112
Aug2005, 06:53 PM

P: 1,308

As I recall, it requires matter to produce curved space in Einstein's eq. Perhaps you are refering to massive particles only? I'm trying to imagine what measure one would use when there are no objects to measure with respect to, or no center, or no edge. It would seem one measure would be just as effective an any other. 



#113
Aug2005, 07:08 PM

Astronomy
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PF Gold
P: 22,809

I think one can argue that neither is more fundamental they are just different ways. Maybe other people have differing views on this. Thiemann's postdoc Bianca Dittrich (one of the strongest LQG researchers now) just posted a paper in which she chose to work with the metric instead of the connection formulation (Ashtekar style). Several others have made this choice also in some if not all of their recent papers (Reuter, Husain, Winkler, Modesto). So the metric continues in use in quantum gravity and there seems no clear choice for the moment. Dittrich's paper was http://www.arxiv.org/abs/grqc/0507106 Partial and Complete Observables for Canonical General Relativity B. Dittrich 33 pages "In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the spacetime picture, i.e. quantities invariant under spacetime diffeomorphisms, and Dirac observables in the canonical picture." 



#114
Aug2005, 07:46 PM

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However, if you build up geometry from the tangent spaces of the points of the manifold, then you can arrange for the symmetry breaking to occur in spacetime itself. This is a modification of the ideas of David Hestenes with the Geometric Algebra. The GA takes the tangent vectors of the manifold and uses them as the generators of a Clifford algebra. The signature of the Clifford algebra is typically taken to be (+++) or (+); this is a feature that doesn't show up in the manifold but has to be added. Anyway, if you begin with the GA, you end up with same symmetry that the usual version of spacetime possesses, but it is possible to generalize the relationship between the tangent vectors and the Clifford algebra in a manner that reproduces the symmetry breaking that distinguishes between the symmetry of spacetime and the symmetry of the observed vacuum. Carl 



#115
Aug2005, 08:18 PM

Astronomy
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PF Gold
P: 22,809

what I want to focus attention on here is WHAT CAN YOU DO WITH NO PRIOR METRIC? so there is no bilinear form on the tangent space at a point. (when you talk about "signature" you are assuming some bilinear form on the tangent space, I want to stop well before that point and look around) the B.I. viewpoint is all you have is the manifolda continuum without prior assumed geometryand then the gravitational field arises dynamically AS the geometry. So we are going in opposite directions here: you are looking for more prior structure (which could be mathematically very nifty, like Clifford algebras) and I want to illustrate (in case anyone is interested in Background Independence) what it looks like with LESS prior structure. The various nonstring QG approaches tend to be built on a manifold WITHOUT metric, or to have even less structure. For example in Loll CDT Triangulations YOU DON'T EVEN ASSUME THAT THE CONTINUUM IS A MANIFOLD. You just approximate it, in a certain sense, by manifolds. And of course there is no prior metric and no Clifford algebra or any of that stuff. Background Independent means "no frills" you try to assume as little as possible to get started with and the surprise is when something we associate with familiar macroscopic space EMERGES. Like 4D dimensionality, as reported here: http://arxiv.org/hepth/0404156 this is one of the articles I gave links to some 8 or 9 posts back. Maybe I should bring up that list of links. 



#116
Aug2005, 08:28 PM

Astronomy
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PF Gold
P: 22,809

It is really remarkable, Carl. They dont even put in that space is supposed to be 4D and it COMES OUT THAT WAY at macroscopic scale, although at very short range the spectral dimension measured by diffusion processes comes out less. Carl I think you have read some CDTweren't we discussing that in the "Introduction" thread? But in case anyone else is reading along with us I will bring up that list of CDT links from a few posts back




#117
Aug2105, 02:35 AM

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PF Gold
P: 9,185

Ouch. I am amazed that background independence is somehow irrelevant. It seems a difficult and awkward position from which to propose a 'theory of everything'.




#118
Aug2105, 12:36 PM

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P: 1,204

However, it is also possible to treat time as an independent variable. That is, one can treat time as separate from the geometry of space. If you do this, then the signature becomes (++++), and you don't need to specify a bilinear form. Instead, one defines the tangent vectors as velocity vectors. In other words, the metric is a result of the continua having a characteristic velocity. This is a classical way of treating space and time, that is separately. Having read the links you've provided, I must say that I am singularly unimpressed with their lack of assumptions about the physical world. I saw no "emergence of a 4D World". Instead they begin with 4D simplices and end up with a 4D world. This is no more surprising to me than starting with little cubes and ending up with big cubes. Please correct me here. I see this as just a gravity from QM paper, not something that separates metric from manifold. Carl 



#119
Aug2105, 02:04 PM

Astronomy
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PF Gold
P: 22,809

It may be that you have not read the first page of the article, Carl. this is page 2 (the abstract occupies page 1). Here is a quote from page 2: quote from "Emergence of a 4D world Note that the dynamical nature of “dimensionality” implies that the Hausdorff dimension of the quantum geometry is not a priori determined by the dimensionality at the cutoff scale a, which is simply the fixed dimensionality d of the building blocks of the regularized version of the theory. An example in point are the attempts to define theories of quantum geometry via “Euclidean Dynamical Triangulations”, muchstudied during the 1980s and ‘90s. In these models, if the dimension d is larger than 2, and if all geometries contribute to the path integral with equal weight, a geometry with no linear extension and d_{Hausdorff}= infinity is created with probability one. If instead – as is natural for a gravityinspired theory – the Boltzmann weight of each geometry is taken to be the exponential of (minus) the Euclidean EinsteinHilbert action, one finds for small values of the bare gravitational coupling constant a firstorder phase transition to a phase of the opposite extreme, namely, one in which the quantum geometry satisfies d_{Hausdorff}= 2. This is indicative of a different type of degeneracy, where typical (i.e. probability one) configurations are socalled branched polymers or trees (see [11, 12, 13, 14, 15, 16, 17] for details of the phase structure and geometric properties of the fourdimensional Euclidean theory). end quote The Dynamical Triangulations literature all through the 1990s is a history of frustration where they would put together, say, 4simplices and the result would be something of small dimensionality like 2 or the dimensionality would go off to infinity. the 2004 result reported in "Emergence..." was highly nontrivial, as they say, and as they explain by reference to the earlier work. this behavior has been discussed in quite a few papersnot just in 4D case but also in 3D For instance look around page 7 of Loll's introductory paper "A discrete history..." hepth/0212340 which was written for grad students entering the field. She describes the 3D case, which is easier to picture. in the 3D case, one randomly assembles 3simplices (tetrahedrons), but for a decade or so the result was always something highly branched out or highly compacted either 2 dimensional or very high, essentially infinite, dimensional. Loll provides some pictures, which I can't. 



#120
Aug2105, 02:36 PM

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But it's not starting from a point of "NO PRIOR METRIC". Instead they're talking about starting without a coordinate system. For example, from your very useful link:
In fact, each of the simplices that these guys are adding up does possess a metric structure. That's what gives the squared lengths of their edges. For that matter, if one knows the squared lengths of the edges, it's easy enough to define a coordinate system and metric for the simplice (which is assumed to be flat in the above link). This concept of getting space back from just the edge lengths of simplices smells to me of pure mathematics. It's just not amazing to me except that so many people would work so hard on it. It's like a chapter from Bourbaki. What's more, it appears to provide no explanation for any physical phenomena such as masses or coupling constants or anything else not already covered by the standard model. Carl Also see: It seems to me that the whole difficulty in this endeavor comes from the requirement that the result be Lorentz symmetric. But there is also an apparent assumption of the existence of a global time: By the way, Hestenes believes that there is a method of putting gravitation onto a flat copy of his space time algebra (STA). Thus the underlying manifold would be flat. The method was found by Lasenby, Doran and Gull. If this is the case, wouldn't it make the whole problem of having to sum over bizarre geometries trivial? Here's a link to his article, please comment (as I know little about gravitation): http://modelingnts.la.asu.edu/pdf/NEW_GRAVITY.pdf Carl 



#121
Aug2105, 04:00 PM

P: 1,545

The tune up in my thinking was where I’d though of Special and General Relativity both as being Classical 4D ideas. My problem was thinking Classical as 4 D. But as said here: This classical way was fine for SR with the SR equations being more precise solutions to the ones Newton provided. But the classical was unable to depict how gravity worked. So we have the first really significant application of Riemannian geometry (from mid 1800’s I think) in order to build General Relativity. As 4D thinking to create “Warped spacetime” was needed. Thus I shouldn’t think of Time by itself as being a dimension independent of three spatial ones where all four would have a metric. But instead : In the QM arena : On the issue of “perturbative” (String & M Theory) and “non perturbative” (CDT, Triangulations) Background Independence are both of these significantly different that the BI of Gen Rel? Is QM by definition Background Independent? with perturbative just one way of recognizing that aspect of QM. Or is there even such a thing a Background Dependent QM theory? Thanks for the links, and comments from all. RB 



#122
Aug2205, 07:39 AM

P: 416





#123
Aug2205, 10:25 AM

Astronomy
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PF Gold
P: 22,809

Juan, you are doing the right thing to refocus the discussion on the main question. I will try to prevent any misunderstanding by also giving CONTEXT of quotation. It is a very good idea. Here is my post #96
loops is more indep, assuming a manifold WITHOUT prior metric, and has had something like 50% increase in research output, people entering the field CDT triangles is a small field, but it is the MOST indep, and its spacetime is not even a differentiable manifold (although piecewise flat manifs are used in an approximation method). So this is radical, taking independence to a new level, and this approach has made the most pronounced progress, with percentagewise more people entering the field Juan, you can challenge any of this because this is largely my personal perspective. Progress is hard to measure objectively and one must use individual judgement. But I am giving you this comparison chart so you will better understand my point of view. You have argued that it would NOT help string become more predictive (that is: falsifiable) to develop a version that DOES NOT ASSUME A PRIOR METRIC. This would be the first kind of independence to ask fora version that you can CALCULATE from without depending on a prior metric on the manifold. My guess is that, on the contrary, it WOULD help theorists arrive at a falsifiable theory, if they would focus effort on making it nonperturbative. Nonperturbative theories are harder to construct, and the difficulty narrows down the range of options. By denying themselves the convenience of a priorchosen metric, the researchers might very well arrive at a theory that could be falsified through inconsistency or by experiment. This is how scientific theories are supposed to be and would constitute a kind of longdelayed success. And so i see it as a hopeful possibilitybut I certainly confess that it is not a certainty! 



#124
Aug2305, 06:31 AM

P: 416

In short, string theory fails because is NOT a theory about our universe. This validation of the theory is rather broad and is not based in specific issues like BI. Our universe is TODAY 4D and non supersymmetric, therefore we may develop a quantum theory for 4D and nonsupersimmetriy. Perhaps tomorrow new experiments discover hidden dimensions or super partners of currently known particles, but FIRST one may develop a theory for the uinverse that we know TODAY. The problem of 40 year of impressive failure of ST is in the violation of scientific method. String theorists followed an initial "beatiful" idea and develop a theory for 26, 10 or 11D with supersimmetry and other stuff according to mathematical incosistency of the beatiful initial idea. How there is no posibility for developing a consistent theory for 4D without supersymmetry, there is possibility for computing nothing of this world from ST. Precisely this is the history of the field on last 30 years. Nothing computed or when computed with wrong behavior (nuclear force), wrong models (spliting of metric violating GR) or discrepancies of 50 orders of magnitude between theory and data. And all of that even ignoring recent advanced stuff that is developed in other fields of theoretical science and ignored by super masterminds string theorists (of course some are respectful and hones but others are not). Stuff known in chemistry during 30 years (see Nobel lecture by Prigogine) is being introduced these days by string theorists in a new revolution. That is, that was known 20 or 30 years ago in other fields is the last fad for ignorant (but very arrogant) people like Witten, Vafa, Greene, Schwartz, Motl, etc. already explained that even with 2 or 3 new revolutions, string theory continue to be a joke (irrelevant) for people working in serious stuff. Finalize saying that the idea of nondifferentiable spacetimes is also one of my ideas, but string and M theorists (yes those that claim for the Final theory the theory most sophisticated of the world, etc, etc.) continue working with "old" differentiable manifolds (e.g. famous CY of string theory or the new G2 of Mtheory). The arrogance of many string theorists permit to me writte this hard words (that i newer wrote for other honest researchers, including trinagulation ones) I would say that there is posibility for reduction of dimensionality on my work and contacted with the author of paper you cite time ago. We discussed the rumour that a decrease on Newton force has been measured. If finally true this is another hard knock for ST which always has claimed that Newton force may be stronger on small distances. F = (1/r^(2+D)) with D additional dimensions. Curiously doing D = 2, that is, reduction of dimensionality, one obtains less force (if confirmed experimentally) and absence of divergences for r = 0. This argument is not riguroius but offer an idea of the surprising things that one can learn from alternative points of view. 



#125
Aug2305, 08:50 AM

P: 416

According to
http://www.physicsforums.com/showthread.php?t=85971 there is not violations of Newtonian force known. 



#126
Aug2305, 08:58 AM

Astronomy
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PF Gold
P: 22,809

But I couldnt think of anything more to say right now. Yes, about the short range newton law measurments. I know. Ohwilleke noted this in a thread in this section also. If you want to post on Ohwilleke's thread you might get some discussion. I dont connect this immediately to string theory because I dont find string theory very interesting and in the long run it might not be all that important. but verifying newton law of gravity at short range does seem interesting. maybe Ohwilleke or somebody will expand on this subject GRAVITY NORMAL AT SMALL SCALES http://www.physicsforums.com/showthread.php?t=85989 check it out. Haelfix and Chronos have already replied on that thread.  by the way, something different. Do you know the story (arivero told me) of the two men discussing whether a whiteish block of material soap or cheese one says it is soap, the other says it is cheese, and to prove it he cuts a sample and starts chewing it uphe will show it is cheese by eating some. After a while he begins to foam and bubble at the mouth, and he stops chewing and says: "Sabe a jabon, pero es queso." We might translate this as IT TASTES LIKE EPICYCLES, BUT IT'S REALLY A THEORY OF EVERYTHING. 


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