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What is the recent development of Loop Quantum Gravity 
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#55
Feb812, 09:40 PM

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We could bring this thread up to date on recent developments.
Today I noted something really strange in Ashtekar Pawlowski paper on Lqc with Λ>0. It makes me think of the Cai Easson picture where inflation is driven by a brief epoch of huge cosmological constant, so you don't need a "graviton", and a "curvaton" field supplies the fluctuations in the CMB. What Ashtekar Pawlowski get is a PLANCK SCALE LIMIT ON THE SIZE OF LAMBDA. So if it is running up as you go back in time (as the energy scale k is increasing) there is a limit to how big it can get. I'll get the link http://arxiv.org/abs/1112.0360 Positive cosmological constant in loop quantum cosmology Tomasz Pawlowski, Abhay Ashtekar (Submitted on 1 Dec 2011) The k=0 Friedmann Lemaitre Robertson Walker model with a positive cosmological constant and a massless scalar field is analyzed in detail. If one uses the scalar field as relational time, new features arise already in the Hamiltonian framework of classical general relativity: In a finite interval of relational time, the universe expands out to infinite proper time and zero matter density. In the deparameterized quantum theory, the true Hamiltonian now fails to be essentially selfadjoint both in the Wheeler DeWitt (WDW) approach and in LQC. Irrespective of the choice of the selfadjoint extension, the big bang singularity persists in the WDW theory while it is resolved and replaced by a big bounce in loop quantum cosmology (LQC). Furthermore, the quantum evolution is... 36 pages Look on page 20, right after equation (4.9): ==quote Ashtekar Pawlowsk page 20i== However, because this Θ_{Λ} is negative, the physical Hilbert space is now zero dimensional! (For proofs, see [18].) Thus, in striking contrast to the WDW theory, in LQC a nontrivial quantum theory exists only when the cosmological constant Λ is less than a critical value, Λ_{c}. Although this result is not phenomenologically relevant because Λ_{c} is of Planck scale, it is of considerable conceptual interest. In the rest of this section, then, we will with work Λ < Λ_{c}. ==endquote== To connect that to the Cai Easson picture: Think of k as momentum or wavenumber or as inverse length. Then k^{2} is inverse area. The cosmological constant Λ is also curvature quantity, an inverse area. So the dimensionless couping number which presumably runs to safety is λ = Λ/k^{2}. This is what goes to a finite limit as k→∞. The only way this can happen is if the dimensionful cosmo constant Λ becomes huge as k increases. But Ashtekar and Pawlowski find that it can only get so large. Nice to have a mathematical handlea grip on the cosmological constant. In the Loop bounce the Hubble expansion rate parameter, an inverse time, reaches Planck scale in a natural period of inflation that does not require assuming an "inflaton" field. this is even without a positive cosmo constant. The Hubble parameter reaches approximately Planck frequency, as I recall. 


#56
Feb812, 10:18 PM

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I think the Ashtekar Pawlowski and the Cai Easson papers offer a remote chance of linking LQC and AS cosmologies, with a running cosmological constant driving inflation.
Here is some bibliography on the Cai Easson paper, to have for convenient reference if anyone is interested: http://arxiv.org/abs/1202.1285 Higgs Boson in RG running Inflationary Cosmology YiFu Cai, Damien A. Easson (Submitted on 6 Feb 2012) An intriguing hypothesis is that gravity may be nonperturbatively renormalizable via the notion of asymptotic safety. We show that the Higgs sector of the SM minimally coupled to asymptotically safe gravity can generate the observed near scaleinvariant spectrum of the Cosmic Microwave Background through the curvaton mechanism. The resulting primordial power spectrum places an upper bound on the Higgs mass, which for canonical values of the curvaton parameters, is compatible with the recently released Large Hadron Collider data. 5 pages ==Cai Easson page 1== ...In this paper, we propose that the Higgs boson may play an important role in the early inflationary universe if the gravitational theory is asymptotically safe. In the frame of AS gravity, the gravitational constant G and cos mological constant Λ are running along with the energy scale, and thus vary throughout the cosmological evolution. It has been argued that if there are no intermediate energy scales between the SM and AS scales, the mass of the Higgs boson is predicted to be m_{H} = 126 GeV with only several GeV uncertainty [14]. We find a suitable inflationary solution can be obtained in a cosmological system which contains a Higgs boson and AS gravity, along the lines of [15]. In this model, there are effectively two scalar degrees of freedom, one being the adiabatic mode and the other being an isocurvature mode. We find the corresponding perturbation theory leads to both the primordial power spectrum for the curvature perturbation and the entropy perturbation. When the cutoff scale runs lower than a critical value, inflation abruptly ends and the Higgs field can give rise to a reheating phase. During this phase, the fluctuations seeded by the Higgs field can be converted into the curvature perturbation through the curvaton mechanism [16, 17]. We derive a relation between the spectral index of the primordial power spectrum and the Higgs mass. We confront this relation with the latest cosmological observations and collider experiment data, and find they are consistent under a group of canonical values of curvaton parameters. ==endquote== Cai Easson references: [14] M. Shaposhnikov and C. Wetterich, Phys. Lett. B 683, 196 (2010) http://arxiv.org/abs/0912.0208 [15] Y. F. Cai and D. A. Easson, Phys. Rev. D 84, 103502 (2011) http://arxiv.org/pdf/1107.5815.pdf (warning: involves JordanBransDicke variant of GR.) [16]D. H. Lyth and D. Wands, Phys. Lett. B 524, 5 (2002) http://arxiv.org/abs/hepph/0110002 Generating the curvature perturbation without an inflaton David H. Lyth, David Wands (Submitted on 28 Sep 2001) We present a mechanism for the origin of the largescale curvature perturbation in our Universe by the late decay of a massive scalar field, the curvaton. The curvaton is light during a period of cosmological inflation, when it acquires a perturbation with an almost scaleinvariant spectrum. This corresponds initially to an isocurvature density perturbation, which generates the curvature perturbation after inflation when the curvaton density becomes a significant fraction of the total. The isocurvature density perturbation disappears if the curvaton completely decays into thermalised radiation... 8 pages. 


#57
Feb812, 11:18 PM

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I'll try to assemble a select bunch of links I think relevant to current directions in LQG research
Google "alesci rovelli hamiltonian arxiv" and get http://arxiv.org/abs/1005.0817 [second hit] Google "ashtekar introduction 2012" and get http://arxiv.org/pdf/1201.4598.pdf [review] Google "rovelli zakopane" and get http://arxiv.org/abs/1102.3660 [tutorial, research problems] Google "pawlowski positive cosmological arxiv" and get http://arxiv.org/abs/1112.0360 [loop with lambda > 0] Google "freidel geiller ziprick" and get http://arxiv.org/abs/1110.4833 [loop classical gravity] Google "jonathan ziprick pirsa" and get http://pirsa.org/12020096 [loop classical gravity video] Google "freidel speziale BF" and get http://arxiv.org/abs/1201.4247 [ways to get GR from BF] Google "hossenfelder emission spectra" and get http://arxiv.org/abs/1202.0412 [curious dark matter conjecture, idea for QG test] Google "wise symmetry gravity" and get http://arxiv.org/abs/1112.2390 [different approach to hamiltonian] 


#58
Feb912, 02:04 PM

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Latest on the classical/semiclassical limit that I'm aware of. There may be more recent.
http://arxiv.org/abs/1108.2258 Emergence of gravity from spinfoams Elena Magliaro, Claudio Perini (Submitted on 10 Aug 2011) We find a nontrivial regime of spinfoam quantum gravity that reproduces classical Einstein equations. This is the double scaling limit of small Immirzi parameter (gamma), large spins (j) with physical area (gamma times j) constant. In addition to quantum corrections in the Planck constant, we find new corrections in the Immirzi parameter due to the quantum discreteness of spacetime. The result is a strong evidence that the spinfoam covariant quantization of general relativity possesses the correct classical limit. 9 pages, Europhysics Letters 95:30007,2011 http://arxiv.org/abs/1110.5899 EinsteinRegge equations in spinfoams Claudio Perini (Submitted on 26 Oct 2011) We consider spinfoam quantum gravity on a spacetime decomposition with many 4simplices, in the double scaling limit in which the Immirzi parameter γ is sent to zero (flipped limit) and the physical area in Planck units (γ times the spin quantum number j) is kept constant. We show that the quantum amplitude takes the form of a Reggelike path integral and enforces Einstein equations in the semiclassical regime. In addition to quantum corrections which vanish when the Planck constant goes to zero, we find new corrections due to the discreteness of geometric spectra which is controlled by the Immirzi parameter. 4 pages, based on a talk given at Loops '11 in Madrid, to appear in Journal of Physics: Conference Series (JPCS) http://arxiv.org/abs/1109.6538 Lorentzian spinfoam propagator Eugenio Bianchi, You Ding (Submitted on 29 Sep 2011) The twopoint correlation function is calculated in the Lorentzian EPRL spinfoam model, and shown to match with the one in Regge calculus in a proper limit: large boundary spins, and small BarberoImmirzi parameter, keeping the size of the quantum geometry finite and fixed. Compared to the Euclidean case, the definition of a Lorentzian boundary state involves a new feature: the notion of past and futurepointing intertwiners. The semiclassical correlation function is obtained for a timeoriented semiclassical boundary state. 13 pages http://arxiv.org/abs/1105.0216 Regge gravity from spinfoams Elena Magliaro, Claudio Perini (Submitted on 1 May 2011) We consider spinfoam quantum gravity in the double scaling limit γ → 0, j → ∞, with γj=const., where γ is the Immirzi parameter, j is the spin and γj gives the physical area in Planck units. We show how in this regime the partition function for a 2complex takes the form of a path integral over continuous Regge metrics and enforces Einstein equations in the semiclassical regime. The Immirzi parameter must be considered as dynamical in the sense that it runs towards zero when the small wavelengths are integrated out. In addition to quantum corrections which vanish for h → 0, we find new corrections due to the discreteness of geometric spectra which is controlled by γ. 11 pages =============== Incidental information I just noticed an interesting lineup of speakers at Princeton Institute for Advanced Study this spring. Princeton has a regular High Energy Theory Seminar, which is sometimes held in the IAS Bloomberg Lecture Hall and sometimes in a seminar room at the PCTS (Princeton Center for Theoretical Science) http://www.princeton.edu/physics/events/ High Energy Theory Seminar  IAS  Andrew Strominger, Harvard University Apr 9, 2012 · 2:30 p.m.– 3:30 p.m. · Bloomberg Lecture Hall High Energy Theory Seminar  Erik Verlinde, University of Amsterdam  TBA Apr 16, 2012 · 2:30 p.m.– 3:30 p.m. · PCTS Seminar Room High Energy Theory Seminar  IAS  Carlo Rovelli, AixMarseille University, France  Loop quantum Gravity: Recent Results and Open Problems Apr 23, 2012 · 2:30 p.m.– 3:30 p.m. · Bloomberg Lecture Hall Description: The loop approach to quantum gravity has developed considerably during the last few years, especially in its covariant ('spinfoam') version. I present the current definition of the theory and the results that have been proven. I discuss what I think is still missing towards of the goal of defining a consistent tentative quantum field theory genuinely background independent and having general relativity as classical limit. I checked out the IAS calendar and they have a neat thing planned for the 23rd April. From 12:30 to 1:30 they have a LUNCH DISCUSSION on Early Universe Cosmology. Then an hour for leisurely reflection followed by Rovelli's talk at 2:30. http://www.ias.edu/calendar/201204...ndar%2F201204 Nice menu planning...timing. 


#59
Feb912, 05:36 PM

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Continuing to bring this thread on current LQG developments up to date, as I noted elsewhere there was (IMHO) an important January paper by Bee, Leonardo, and Isabeau which had this key conclusion paragraph.
Google "hossenfelder emission spectra" and get http://arxiv.org/abs/1202.0412 [curious dark matter conjecture, idea for QG test] ==QUOTE 1202.0412== 4 Conclusion We have derived here an approximate analytic expression for the emission spectrum of selfdual black holes in the mass and temperature limits valid for primordial black holes evaporating today. The idea that primordial black holes are dark matter candidates is appealing since it is very minimalistic and conservative, requiring no additional, so far unobserved, matter. This idea has therefore received a lot of attention in the literature. However, the final stages of the black hole evaporation seem to be amiss in observation, and so there is a need to explain why primordial black holes were not formed at initial masses that we would see evaporating today. The selfdual black holes we have studied here offer a natural explanation since they evaporate very slowly. The analysis we have presented here allows to calculate the particle flux from such dark matter constituted of selfdual black holes, and therefore is instrumental to test the viability of this hypothesis of dark matter constituted of selfdual black holes against data. ==endquote== In short, the main (perhaps only) problem with tiny primordial BH as DM is that by conventional Hawking model temperature rises as the thing evaporates, going as mass inverse, so tiny BH are hot and evaporate too fast. But rightly or not in the interesting mass range Modesto's Loop BH temperature goes down with mass. So in the very long run, the tiny BH could conceivably even come into equilibrium with the CMB or at any rate last a long time. This gives a conservative (and testable!) way to account for Dark Matter. One does not need a new particle. Testable because if the observed clouds of DM do indeed consist of these tiny Loopish BH then the clouds should have a characteristic radiation spectrum I think it is not only this interesting and testable idea which is important, but also the *minimalistic conservative* THEME which is underscored in the paper. Figuring out how to explain stuff without imagining exotic unobserved particles/fields. That's a desideratum to keep in mind when looking over current research. We may start seeing more of it. To me it's strongly represented in the Cai Easson paper which seems aimed at explaining inflation without the need for an inflaton field! I talked about that in post #56: What I want to do is try to use this *minimalistic conservative* THEME that comes out explicitly in Bee and Leonardo and Isabeau's paper, to try to organize how I view current research. What other recent papers bear out this trend? If it is a trend. How does research by other people (like Freidel, Bianchi, Dittrich...) fit into this picture, if it does? 


#60
Feb1012, 02:38 PM

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I looked just now at the PERIMETER INSTITUTE colloquium schedule for Winter term 2012. Their winter term is JanApril. Spring in Canada does not start until May.
I'll get the link http://www.perimeterinstitute.ca/en/...rs/Colloquium/
http://www.perimeterinstitute.ca/en/...antum_Gravity/ They don't have anything scheduled yet for April. A couple of posts back I noted that Princeton IAS is having Rovelli give a colloquium talk on Loop gravity in April, paired with a discussion of Early Universe Cosmology earlier in the day. http://physicsforums.com/showthread....81#post3756881 


#61
Feb1012, 06:47 PM

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Some interesting recent developments in LQG.
http://arxiv.org/abs/1201.2187 A spinfoam vertex amplitude with the correct semiclassical limit Jonathan Engle (Submitted on 10 Jan 2012) Spinfoam models are hoped to provide a dynamics for loop quantum gravity. All 4d spinfoam models of gravity start from the Plebanski formulation, in which gravity is recovered from a topological field theory, BF theory, by the imposition of constraints, which, however, select not only the gravitational sector, but also unphysical sectors. We show that this is the root cause for terms beyond the required Feynmanprescribed exponential of i times the action in the semiclassical limit of the EPRL spinfoam vertex. By quantizing a condition isolating the gravitational sector, we modify the EPRL vertex, yielding what we call the proper EPRL vertex amplitude. This provides at last a vertex amplitude for loop quantum gravity with the correct semiclassical limit. 4 pages see also Alesci Rovelli's proposal for new Hamiltonian: Google "alesci rovelli hamiltonian arxiv" and get http://arxiv.org/abs/1005.0817 and the Freidel Geiller Ziprick paper: Google "freidel geiller ziprick" and get http://arxiv.org/abs/1110.4833 More discussion here: http://physicsforums.com/showthread....88#post3637688 http://physicsforums.com/showthread....30#post3643430 http://physicsforums.com/showthread....56#post3624456 


#62
Feb1112, 03:35 AM

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In order not to confuse the reader:
A "proper vertex amplitude" to recover the correct semicalssical limit is a necessary but not a sufficient condition for the model to be "correct". Of course one must recover GR as low energy theory, but in the deep QG regime there may very well be a whole bunch of inequivalent theories with the same semiclassical limit. This is the main reason why some people insist on 1) a completion of the canonical formulation (constructing a "correct" H) plus 2) a consistent quantization in terms of spin foams plus 3) a proof of equivalence of (1) and (2) All three pathways are being investigated, but up to now neither completion of 1) or 2) nor convergence in the sense of 3) can be claimed. @marcus: did you spent some time in looking into the 2011 Thiemann papers? There are some interesting aspects like going beyond dim=4 and incorporating SUSY. 


#63
Feb1412, 03:00 PM

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Another recent development. The cosmological constant Λ put into spinfoam cosmology and one gets a nontrivial solution to the Einstein equation out: de Sitter space.
It's kind of beautiful. The Friedmann equation with cosmo constant is derived from the Zakopane spinfoam amplitude (with Λ inserted). And this turns out to be compatible with the treatment where Lambda is a quantum group deformation parameter. Several things brought together in one paper. Google "bianchi krajewski spinfoam cosmology" and get http://arxiv.org/abs/1101.4049 Cosmological constant in spinfoam cosmology Eugenio Bianchi, Thomas Krajewski, Carlo Rovelli, Francesca Vidotto (Submitted on 20 Jan 2011) We consider a simple modification of the amplitude defining the dynamics of loop quantum gravity, corresponding to the introduction of the cosmological constant, and possibly related to the SL(2,C)q extension of the theory recently considered by FairbairnMeusburger and Han. We show that in the context of spinfoam cosmology, this modification yields the de Sitter cosmological solution. 4 pages, 2 figures for the treatment where Λ appears in quantum group, related to the qdeformation, see papers by Muxin Han and by Fairbairn Meusberger. It's fascinating that in that treatment one would expect that Λ running to large values (as in asym safe gravity) with high energy density corresponds to a decline in angular resolutionangles get fuzzy things are either in the same direction or they are not, lacking fine angular distinctions. It's intriguing. But one does not have to deal with the quantum group idea of how Λ arises. One can simply insert it in the Zakopane spinfoam amplitudeor in the Friedmann equationand treat it as a constant the way cosmologists customarily do. this paper was a "sleeper". I'm not sure we recognized its importance back in the first quarter of 2011. 


#64
Feb1512, 01:51 PM

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Tom, raised an interesting point in the ShapoWetter thread, which this paper could serve as partially answering.
You can see from 1101.4049 equation (2) that in LQG the cosmological constant can indeed be treated as a "standard running coupling", as it is in the Asymptotic Safe approach. And it can also be treated as a qdeformation of SL(2,C) as per Han, Meusberger, Fairbairn and others. The paper tentatively suggests the two ways of including Λ are "possibly related"! 


#65
Feb1612, 06:26 PM

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Since it may be possible to MERGE Asym Safe QG with Loop QG (and get something that works better than AS currently does at the cosmo singularity) I want to pay close attention to recent AS talks and papers. Here is the abstract for Frank Saueressig's 15 February video lecture:
Google "saueressig pirsa fractal" and get http://pirsa.org/12020088/ Fractal Spacetimes Under the Microscope: a RG View on Monte Carlo Data Frank Saueressig The emergence of fractal features in the microscopic structure of spacetime is a common theme in many approaches to quantum gravity. In particular the spectral dimension, which measures the return probability of a fictitious diffusion process on spacetime, provides a valuable probe which is easily accessible both in the continuum functional renormalization group and discrete Monte Carlo simulations of the gravitational action. In this talk, I will give a detailed exposition of the fractal properties associated with the effective spacetimes of asymptotically safe Quantum Einstein Gravity (QEG). Comparing these continuum results to threedimensional Monte Carlo simulations, we demonstrate that the resulting spectral dimensions are in very good agreement. This comparison also provides a natural explanation for the apparent conflicts between the short distance behavior of the spectral dimension reported from Causal Dynamical Triangulations (CDT), Euclidean Dynamical Triangulations (EDT), and Asymptotic Safety. Date: 15/02/2012  4:00 pm and also recall the Bianchi et al paper from 2 posts back: Google "bianchi krajewski spinfoam cosmology" and get http://arxiv.org/abs/1101.4049 Cosmological constant in spinfoam cosmology Eugenio Bianchi, Thomas Krajewski, Carlo Rovelli, Francesca Vidotto (Submitted on 20 Jan 2011) We consider a simple modification of the amplitude defining the dynamics of loop quantum gravity, corresponding to the introduction of the cosmological constant, and possibly related to the SL(2,C)q extension of the theory recently considered by FairbairnMeusburger and Han. We show that in the context of spinfoam cosmology, this modification yields the de Sitter cosmological solution. 4 pages, 2 figures They derive the Friedman equation for deSitter space starting from the Zakopane dynamics equation with a λ term inserted for cosmo constant. 


#66
Feb1812, 01:24 AM

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Several interesting parallels between AsymSafe QG and Loop are appearing. One is an explanation of dark matter as clouds of small black holes. We see this from Modesto in the Loop case and from Easson in the Safe QG case. Modesto has been working on this for several yearsI'll get a recent paper of his with Hossenfelder and you can check the references.
Google "modesto emission spectra" and get http://arxiv.org/abs/1202.0412 Emission spectra of selfdual black holes Sabine Hossenfelder, Leonardo Modesto, Isabeau PrémontSchwarz (Submitted on 2 Feb 2012) We calculate the particle spectra of evaporating selfdual black holes that are potential dark matter candidates. We first estimate the relevant mass and temperature range and find that the masses are below the Planck mass, and the temperature of the black holes is small compared to their mass. In this limit, we then derive the numberdensity of the primary emission particles, and, by studying the waveequation of a scalar field in the background metric of the black hole, show that we can use the low energy approximation for the greybody factors. We finally arrive at the expression for the spectrum of secondary particle emission from a dark matter halo constituted of selfdual black holes. 15 pages, 6 figures, Small conventional BH don't last long since they get hotter as they lose mass and evaportion speeds up. By contrast, small Loop BH last a very long time since they get colder as they lose mass. Curiously enough Easson has come up with a similar conclusion in the Safe QG case. Google "easson safe black hole" and get http://arxiv.org/abs/1007.1317 Black holes in an asymptotically safe gravity theory with higher derivatives YiFu Cai, Damien A. Easson (Submitted on 8 Jul 2010) We present a class of spherically symmetric vacuum solutions to an asymptotically safe theory of gravity containing highderivative terms. We find quantum corrected Schwarzschild(anti)de Sitter solutions with running gravitational coupling parameters. The evolution of the couplings is determined by their corresponding renormalization group flow equations. These black holes exhibit properties of a classical Schwarzschild solution at large length scales. At the center, the metric factor remains smooth but the curvature singularity, while softened by the quantum corrections, persists. The solutions have an outer event horizon and an inner Cauchy horizon which equate when the physical mass decreases to a critical value. Superextremal solutions with masses below the critical value correspond to naked singularities. The Hawking temperature of the black hole vanishes when the physical mass reaches the critical value. Hence, the black holes in the asymptotically safe gravitational theory never completely evaporate. For appropriate values of the parameters such stable black hole remnants make excellent dark matter candidates. 22 pages, 3 figures; version to appear in JCAP ==links to some recent papers== New Hamiltonian: Google "arxiv alesci rovelli hamiltonian" and get http://arxiv.org/abs/1005.0817 Intro, Survey, Tutorial, Open Problems for Research: Google "ashtekar introduction 2012" and get http://arxiv.org/abs/1201.4598 Google "rovelli zakopane" and get http://arxiv.org/abs/1102.3660 Cosmological Constant: Google "bianchi cosmic constant spinfoam" and get http://arxiv.org/abs/1101.4049 Google "pawlowski cosmic constant" and get http://arxiv.org/abs/1112.0360 Loop Classical Gravitythe right version of GR to quantize: Google "freidel geiller ziprick" and get http://arxiv.org/abs/1110.4833 Google "jonathan ziprick pirsa" and get http://pirsa.org/12020096 Small black holes and dark matter: Google "modesto emission spectra" and get http://arxiv.org/abs/1202.0412 Google "easson safe black hole" and get http://arxiv.org/abs/1007.1317 Miscellaneous: Google "freidel speziale BF" and get http://arxiv.org/abs/1201.4247 [ways to get GR from BF] Google "wise symmetry gravity" and get http://arxiv.org/abs/1112.2390 [different approach to hamiltonian] 


#67
Feb2012, 07:51 AM

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Thanks marcus for the recent overview and thanks tom for the very interesting discussion.
Like Tom, I also have some trouble with LQG. Especially I think some of the problems came form the lack of understanding the 'localglobal' problems (or sometimes mix them). But no one considers the relation (and the special features) between 3 and 4 dimensions. After the breakthrough of Perelman, we know that the relation between homogenous gemetries in 3 dimensions and 3manifold topology is very close. A general 3manifold consist of a mix of 8 possible gemetries (related to the Bianchi model I to IX). This close relation is the reason that the EinsteinHilbert action in 3 dimensions is a topological invaraint (ChernSimons invaraint). The situation changes dramatically in 4 dimensions. The relation between geometry and topology is lost but there is a new one between the smoothness structure (the maximal smooth atlas) and the topology. This new relation is essential for the understanding of the dynamics: the sum over 3geometries will automatically lead to the inclusion of pathes like: spherical 3geometry > hyperbolic 3geometry > spherical 3geometry The corresponding 4manifold can have the topology [math]S^3\times R[/math] but an exotic smoothness structure (I considered this case in http://arxiv.org/abs/1201.3787 ) This fact has also an impact on spin foam models. Usually one try to relate thise models to a triangulation of the 4manifold. But smoothness structures and piecewiselinear structures (as a kind of triangulations) are equivalent. Therefore oen has something like an exotic triangulation. To address these questions, one has to consider the class of topspin models (Marcolli, Dustopn et. al.) using branched coverings. In this approach one sees the problem: every 3manifold can be obtained by a 3fold covering of the 3sphere branched along a 1dimensional complex, a knot or link, and every 4manifold can be obtained by a 4fold covering of the 4sphere branched along a 2dimensional complex, a surface, but the surface contains 2 singularities: the cusp and the fold. The appearance of cusp singularities was already discussed in the spin foam literature as conical singularities. I agree with marcus and tom, that the Alesci/Rovelli hamiltonian is a real breakthrough, it considers a more global change of spacetime. So again thanks for the overview, I will go more deep into these new papers. Torsten 


#68
Feb2012, 04:47 PM

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Hi Torsten!
You could give some general words on what you expect first and foremost from a QG theory, to provide a context. For me the basic requirement is a clear testable one that reproduces classical geometry (where applicable) and resolves the cosmo singularity. That's what I want from QG first and foremost, and then if there are several QG theories successful in this basic way then I will ask which one follows Dirac quantization plan most transparently, which one has both hamiltonian and path integral versions most clearly equivalent and so on. Because if you have more than one theory that works, these niceties can be useful in selecting from among them. But right now i do not see a multitude of QG theories that meet the basic requirements. As for Loop, I see steady progress, a growing understanding of how to set up the classical phase space, quantize it, and get a hamiltonian version, mounting evidence that classical GR is recovered, that the cosmological singularity is resolved, and that it is testable. Numerous papers on all these fronts. So I *expect* a hamiltonian version to be constructed that will be equivalent to the Zako spinfoam version or whatever it has evolved into by that time. The present formulation is remarkably clear and simple so it is hard to imagine how it could change, but it could of course. But my basic desiderata are not that (unless there are several equally good theories to choose from). My requirements, as I said, are a clearly formulated testable theory which reproduces classical GR where valid and can model the start of expansionforming the basis for cosmology. I'm curious about what you would say instead of this. You are actively engaged in your own QG program. You must have some basic goals, primary objectives. You may have summed up your philosophy in one or more of your papers and can just give a page/paragraph reference or paste something in here. Or maybe it is something you can say informally in just a few words. 


#69
Feb2112, 01:05 AM

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Here I have one central question: what is the fundamental structure of (L)QG: 1) PL or smooth manifolds with diffeomorphisms factored away  resuting in triangulations? 2) generic spin networks? Not all generic spin networks are dual to some triangulation (of a manifold), and therefore there are spin networks for which no triangulation of a manifold does exist (at least the dimension of the manifold can be rather large). My impression is that the SF models rely in some sense on some fundamental structures of the underlying 4manifold, whereas the generic spin networks do have no such limitations. It's interesting that spin networks arise from manifolds with rather severe restrictions (3space foliations of globally hyperbolic 4manifolds, local diffeomorphisms, i.e. no singularites) but that once the construction is completed they seem to be agnostic regarding these restrictions. So spin networks are a much richer structure than triangulations. 


#70
Feb2112, 11:02 AM

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Interesting comments, Tom, I hope Torsten will discuss some of your questions. About your central question you know there are different formulations, and some do use 3D and 4D manifolds. "Do we have to?" It seems not since not every formulation of the theory does. The version I am most familiar with does not have these structures embedded. It uses both spin networks and spinfoams but they are not immersed in any continuum.
You are totally correct that "not all generic spin networks" are dual to triangulations! For one thing a spin network is not restricted to having just 4valent nodes (which would correspond to tetrahedra in the dual). It's normal to have nodes with valence > 4 corresponding (fuzzily, indefinitely) to manysided polyhedral chunks of space. 


#71
Feb2112, 01:51 PM

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marcus, there may very well be nvalent nodes which do not correspond to triangulations but which may describe Voronoicelllike structures; but I think that not even this structure need always be sufficient. I am afraid that an arbitrary graph need not comply with any celllike structure embedded in lowdimensional manifolds.



#72
Feb2112, 02:26 PM

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PF Gold
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I was responding to your talking about triangulations. The overwhelming majority of graphs, of any given size, are NOT dual to a triangulation. So I wanted to agree with emphasis! I think you can probably extend that to a division of a 3D manifold into 3D cells which are NOT simplices. Is this the kind of thing you mean? Most graphs would not be dual to that sort of structure either. Or so I believe (haven't thought about it.) I was puzzled by your saying you are afraid such and such might not be so. Don't see why it matters. 


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