tag:blogger.com,1999:blog-58266329603566940902018-06-18T02:07:53.839-05:00International Loop Quantum Gravity Seminar<a href="http://ilqgse.blogspot.com">En Español</a>
<p>
The International Loop Quantum Gravity Seminar is held every two weeks via teleconference among the main research groups in loop quantum gravity. Slides are distributed in advance and audio posted after the seminar at the Seminar's website.
This blog presents summaries for the general public of the content of the seminars.</p>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.comBlogger43125tag:blogger.com,1999:blog-5826632960356694090.post-51043436815786659042018-05-01T11:54:00.000-05:002018-05-01T11:54:44.931-05:00Cosmological perturbations in terms of observables and physical clocksTuesday, Apr 17th<br /><b></b><br /><b>Kristina Giesel, FAU Erlangen-Nürnberg</b><br /><b>Title: Gauge invariant observables for cosmological perturbations </b><br /><a href="http://relativity.phys.lsu.edu/ilqgs/giesel041718.pdf">PDF</a> of the talk (8M)<br /><a href="http://relativity.phys.lsu.edu/ilqgs/giesel041718.mp4">Audio+Slides</a> of the talk (15M)<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Zv8zh7SNvqI/Wtzfgidil0I/AAAAAAAAKLw/Hweh9nuAoZwwxM1yNfaSG2QF3mCY2l71wCEwYBhgL/s1600/giesel-a016249e0816edcd70c9ee3a55b0ccb85ee20ac838fb1980b6a652d657de1840.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="220" data-original-width="220" src="https://3.bp.blogspot.com/-Zv8zh7SNvqI/Wtzfgidil0I/AAAAAAAAKLw/Hweh9nuAoZwwxM1yNfaSG2QF3mCY2l71wCEwYBhgL/s1600/giesel-a016249e0816edcd70c9ee3a55b0ccb85ee20ac838fb1980b6a652d657de1840.jpg" /></a></div>By Jorge Pullin, LSU<br /><br />When one sets up to quantize general relativity something unusual happens. When one constructs a key quantity for describing the evolution called the Hamiltonian, it turns out it vanishes. What the framework is telling us is that since in general relativity one can choose arbitrary coordinates, the coordinate t that one normally associated with time is arbitrary. That means that the evolution described in terms of it is arbitrary.<br /><br /><br /><br />Of course this does not mean that the evolution predicted by general relativity is arbitrary. It is just that one is choosing to describe it in terms of a coordinate that is arbitrary. So how can one get to the invariant part of the evolution? Basically one needs to construct a clock out of physical quantities. Then one asks how other variables evolve in terms of the variable of the clock. The relational information between such variables is coordinate independent and therefore characterizes the evolution in an invariant way.<br /><br />Cosmological perturbation theory is an approximation in which one assumes that the universe at large scales is homogeneous and isotropic plus small perturbations. One can then expand the Einstein equations keeping only the lower order terms in the small perturbations. That makes the equations much more manageable. Up to now most studies of cosmological perturbations were done in coordinate dependent fashion, in particular the evolution was described in terms of a coordinate t. This talk discusses how to formulate cosmological perturbation theory in terms of physical clocks and physically observable quantities. Several choices of clocks are discussed.Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-26217553631515450372018-04-22T11:58:00.003-05:002018-05-01T12:22:43.117-05:00Quantum gravity inside and outside black holesTuesday, Apr 3rd<br /><b></b><br /><b>Hal Haggard, Bard College</b><br /><b>Title: Quantum Gravity Inside and Outside Black Holes </b><br /><a href="http://relativity.phys.lsu.edu/ilqgs/haggard040318.pdf">PDF</a> of the talk (5M)<br /><a href="http://relativity.phys.lsu.edu/ilqgs/haggard040318.mp4">Audio+Slides</a> of the talk (19M)<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-jBxhPI0_-Kg/Wty9LR7Ha_I/AAAAAAAAKLg/x2jkKO65hyAaZNnicC70uhPJKZUlGaQgwCLcBGAs/s1600/Hal-Haggard.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="415" data-original-width="285" height="320" src="https://1.bp.blogspot.com/-jBxhPI0_-Kg/Wty9LR7Ha_I/AAAAAAAAKLg/x2jkKO65hyAaZNnicC70uhPJKZUlGaQgwCLcBGAs/s320/Hal-Haggard.jpg" width="218" /></a></div>By Jorge Pullin, Louisiana State University<br /><br />The talk consisted of two distinct parts. The second part discussed black holes exploding into white holes. We have <a href="http://ilqgs.blogspot.com.uy/2018/01/black-holes-exploding-into-white-hole.html">covered the topic in this blog before</a>, and the new results were a bit technical for a new update, mainly a better handle on the time the process takes, so we will not discuss them here.<br /><br />The first part concerned itself with how the interior of a black hole would look like in a quantum theory. Black holes are regions of space-time from which nothing can escape and are bounded by a spherical surface called the horizon. Anything that ventures beyond the horizon can never escape the black hole. Black holes develop when stars exhaust their nuclear fuel and start to contract under the attraction of gravity. Eventually gravity becomes too intense for anything to escape and a horizon forms.<br /><br />The interior of the horizon however, is drastically different if a black hole has rotation or not. If the black hole does not rotate, anything that falls into the black hole is crushed in a central singularity where, presumably, all the mass of the initial star concentrated. If the black hole has rotation however, the structure is more complicated and infalling matter can avoid hitting the singularity and move into further regions of space-time inside the black hole.<br /><br />This raises the question: what happens with all this in a quantum theory of gravity. Presumably a state representing a non-rotating black hole will consist of a superposition of black holes with rotation, peaked around zero rotation, but with contributions from black holes with small amounts of rotation. How does the interior of a non-rotating quantum black hole look when it is formed through a superposition of rotating black holes? This is an interesting question since the interior of rotating black holes are so different from their non-rotating relatives.<br /><br />The talk concludes that the resulting interior actually does resemble that of a non-rotating black hole. The key observation is that one cannot trust the classical theory all the way to the singularity and that leads to the superposition having large curvatures where one would have expected the singularity of the non-rotating black hole to be.Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-19318046612949908252018-03-25T14:36:00.002-05:002018-03-25T14:36:24.723-05:00Cosmological non Gaussianity from loop quantum cosmologyTuesday, Mar 6th<br /><b></b><br /><b>Ivan Agullo, LSU</b><br /><b>Title: Non-Gaussianity from LQC </b><br /><a href="http://relativity.phys.lsu.edu/ilqgs/agullo030618.pdf">PDF</a> of the talk (22M)<br /><a href="http://relativity.phys.lsu.edu/ilqgs/agullo030618.mp4">Audio+Slides</a> [.mp4 19MB]<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-0FXS3_LYtNs/WraJQDu_lCI/AAAAAAAAKFE/ajDrx0OXn2EXJ9ZZ_bPKim_Aj1h4ZfE7wCLcBGAs/s1600/agullo.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="266" data-original-width="200" src="https://2.bp.blogspot.com/-0FXS3_LYtNs/WraJQDu_lCI/AAAAAAAAKFE/ajDrx0OXn2EXJ9ZZ_bPKim_Aj1h4ZfE7wCLcBGAs/s1600/agullo.jpg" /></a></div>By Jorge Pullin, LSU<br /><br />The standard picture of cosmology is that the universe started in the "big bang" and then underwent a period of rapid expansion, called inflation. During those initial moments, densities are very high and matter is fused into a primordial "soup" that is opaque, light cannot travel through it. As the universe expands and cools, eventually electrons and protons form atoms and the universe becomes transparent to light. The afterglow of that initial phase can then travel freely through the universe and eventually reaches us. Due to the expansion of the universe that light "cools" (its frequency is lowered). In the 1960's to Bell Telephone Co. engineers were working on a microwave antenna and discovered a noise they could not get rid of. That noise was the afterglow of the Big Bang, that by then had cooled off into microwaves. That afterglow has been measured with increasing precision using satellites. It is remarkably homogeneous, if one looks into two different directions of the universe, the difference in temperature (frequency) of the microwave radiation is equal to one part in 100,000. The diagram below has those temperature differences magnified 100,000 times to make them visible, different colors correspond to different temperatures. The whole celestial sphere is mapped into the oval.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-D1zCK6PoI8w/WraKjOYU7TI/AAAAAAAAKFQ/twrfTfIl_dwIbZfyxnSswEpCdP-hDNaOwCLcBGAs/s1600/Ilc_9yr_moll4096%2B%25281%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="165" data-original-width="330" height="160" src="https://1.bp.blogspot.com/-D1zCK6PoI8w/WraKjOYU7TI/AAAAAAAAKFQ/twrfTfIl_dwIbZfyxnSswEpCdP-hDNaOwCLcBGAs/s320/Ilc_9yr_moll4096%2B%25281%2529.png" width="320" /></a></div>At first, it appears that the distribution of temperature is sort of random. But it is not, it has a lot of structure. To characterize the structure, one picks a direction and then moves away from it a certain angle and draws a circle of all directions forming the same angle with the original direction one picked. One then averages the temperature along the circle. Then one averages the result for all possible initial choices of direction. If the distribution were truly random, if one plotted the average computed as a function of the angle, one would get a constant, no angle would be preferred over others. But what one gets is shown in the following diagram,<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-YJ9rhuKLmw8/WraLUs3eHxI/AAAAAAAAKFY/OiwA8405xHYWGEBu5w9UchKB7fzreRyfwCLcBGAs/s1600/450px-PowerSpectrumExt.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="368" data-original-width="450" height="261" src="https://2.bp.blogspot.com/-YJ9rhuKLmw8/WraLUs3eHxI/AAAAAAAAKFY/OiwA8405xHYWGEBu5w9UchKB7fzreRyfwCLcBGAs/s320/450px-PowerSpectrumExt.svg.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">In the vertical are the averages, in the horizontal, the angles. The dots are experimental measurements. The continuous curve is what one gets if one evolves a quantum field through the inflationary period, starting from the most "quiescent" quantum state possible at the beginning, called "the vacuum state". The incredibly good agreement between theory and experiment is a great triumph of the inflationary model. The quantity plotted above is technically known as the "two point correlation". Loop quantum cosmology slightly changes the predictions of standard inflation, mostly for very large angles. There, the experimental measurements have a lot of uncertainty and are not able to tell us if loop quantum cosmology or traditional inflation give a better result. Perhaps in a few years better measurements will allow us to distinguish between them. If loop quantum cosmology is favored it would be a tremendously important experimental confirmation. But we are not there yet.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">One can generalize the construction we made with two directions and an angle between them to three directions and three angles between them, and so on for higher number of directions. These would be known technically as the three point correlation, four point correlation, etc. If the distribution of temperatures were given by a probabilistic distribution known as a Gaussian, all the higher order correlations are determined by the two point correlation, there is no additional information in them. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">In this talk a study of the three point correlations for loop quantum cosmology was presented. It was shown that non-Gaussianities appear. That is, the three point correlation is not entirely determined by the two point one. Satellites are able to measure non-Gaussianities. In the talk it was shown that depending on the values chosen for the quantum fields at the beginning of the universe, the non-Gaussianities predicted by loop quantum gravity can be made compatible with experiment. This is not strictly speaking an experimental confirmation since one had a parameter one could adjust. But the good news is that the values needed to fit the data appear very natural. Again, future measurement should place tighter bounds on all this.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Image credits: Cosmic microwave background <a href="https://en.wikipedia.org/wiki/Cosmic_microwave_background">Wikipedia page.</a></div><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-58037684182797996382018-03-25T14:05:00.000-05:002018-03-25T14:05:02.435-05:00Quantum spacetimes on a quantum computerTuesday, Mar 20th<br /><b></b><br /><b>Keren Li, Tsinghua University</b><br /><b>Title: Quantum spacetime on a quantum simulator </b><br /><a href="http://relativity.phys.lsu.edu/ilqgs/li032018.pdf">PDF</a> of the talk (3M)<br /><a href="http://relativity.phys.lsu.edu/ilqgs/li032018.mp4">Audio+Slides</a> [.mp4 11MB]<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-K8tq55jyXPo/WrfwN4Zf_hI/AAAAAAAAKGo/YxWw-JxzE4QEIIK_WTohjdYPuw5UgC8ygCLcBGAs/s1600/keren.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1067" data-original-width="1600" height="213" src="https://1.bp.blogspot.com/-K8tq55jyXPo/WrfwN4Zf_hI/AAAAAAAAKGo/YxWw-JxzE4QEIIK_WTohjdYPuw5UgC8ygCLcBGAs/s320/keren.jpeg" width="320" /></a></div>By Jorge Pullin, LSU<br /><br /><br />In loop quantum gravity the quantum states are labeled by objects known as "spin networks". These are graphs in space with intersections. If one evolves a spin network in time one gets a "spin foam". If one had a static situation, the various spatial slices of a spin foam would be the same, as shown in the figure,<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-uyda9VyR7As/WraQsC4D77I/AAAAAAAAKFo/kJAZ-Z9z_FIv8v9K7SibyDCut6mgUVW1gCLcBGAs/s1600/Screen%2BShot%2B2018-03-24%2Bat%2B12.53.08%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="289" data-original-width="300" src="https://4.bp.blogspot.com/-uyda9VyR7As/WraQsC4D77I/AAAAAAAAKFo/kJAZ-Z9z_FIv8v9K7SibyDCut6mgUVW1gCLcBGAs/s1600/Screen%2BShot%2B2018-03-24%2Bat%2B12.53.08%2BPM.png" /></a></div>If one were in a dynamical situation, new vertices are created,<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Rv5KxBUwXQ0/WraQ2Yj8NQI/AAAAAAAAKFs/DCgoduAsHbkLLF_xzMy8zmuCklmPdS1pwCLcBGAs/s1600/Screen%2BShot%2B2018-03-24%2Bat%2B12.54.18%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="280" data-original-width="301" src="https://1.bp.blogspot.com/-Rv5KxBUwXQ0/WraQ2Yj8NQI/AAAAAAAAKFs/DCgoduAsHbkLLF_xzMy8zmuCklmPdS1pwCLcBGAs/s1600/Screen%2BShot%2B2018-03-24%2Bat%2B12.54.18%2BPM.png" /></a></div>To compute the probability of transitioning from a spin network to another is what calculations in spin foams are about. The details of these computations resemble computations people do in quantum mechanics of systems with spins. This allows to make a parallel between these computations and the ones that are involved in setting up a quantum computer, specifically the qubits that are constructed using nuclear magnetic resonance systems (NMR). In this talk it was described how the evolution of a very simple spin foam known as the tetrahedron can be simulated on an NMR quantum computer of four qubits and how the experimental measurements reproduce very well theoretical calculations of spin foam models.<br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-32369955036710580762018-02-06T15:20:00.000-06:002018-02-06T15:20:37.115-06:00Using symmetries to determine the dynamics<span style="background-color: white;">Tuesday, Feb 6th</span><br /><b></b><br /><b>Ilya Vilensky, Florida Atlantic University</b><br /><b>Title: The unique form of dynamics in LQC </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/vilensky020618.pdf">PDF</a><span style="background-color: white;"> of the talk (0.5M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/vilensky020618.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 11MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-70KfZBUF37I/WnoZHLTZzTI/AAAAAAAAJ8c/L_-wVJaUi-kibyRLIrmpYFs3_GzrjNk5QCLcBGAs/s1600/photo18.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1217" height="320" src="https://4.bp.blogspot.com/-70KfZBUF37I/WnoZHLTZzTI/AAAAAAAAJ8c/L_-wVJaUi-kibyRLIrmpYFs3_GzrjNk5QCLcBGAs/s320/photo18.jpeg" width="243" /></a></div><span style="background-color: white;"><br /></span><br /><br /><br />By Jorge Pullin, LSU<br /><br />Loop quantum cosmology is the application of ideas of loop quantum gravity to the context of cosmology, where one freezes most degrees of freedom and studies just a few large scale ones, like the volume of the universe or its anisotropy. Loop quantum cosmology is not "derived" from loop quantum gravity, in the sense of choosing in the full theory quantum states that are very symmetric with only a few degrees of freedom and study their evolution. That is at the moment, too complicated. In loop quantum cosmology one first freezes the degrees of freedom one wishes to ignore and then proceeds to quantize the remaining ones. It is not clear that this coincides with "quantizing and then freezing". It is therefore important to run cross checks to make sure that at least within the approximation considered, things are consistent.<br /><br />In spite of the enormous simplification one obtains when one first freezes most degrees of freedom and then quantizes, there are still quite a few ambiguities in the quantization process. This talk showed in the example of anisotropic universes, how imposing the residual symmetries and left after freezing most degrees of freedom, and demanding that the correct classical limit follow, allows to cut down on the number of ambiguities present. This increases the confidence in results previously obtained in loop quantum cosmology, some of which may have observable implications for the anisotropies of the cosmic microwave background radiation.Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-60086072692864652422018-01-29T12:38:00.000-06:002018-01-29T12:38:10.410-06:00New dynamics for quantum gravity<span style="background-color: white;">Tuesday, Jan 23rd</span><br /><b></b><br /><b>Cong Zhang, Univ. Warsaw/Beijing</b><br /><b>Title: Some analytical results about the Hamiltonian operator in LQG </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/zhang012318.pdf">PDF</a><span style="background-color: white;"> of the talk (1.7M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/zhang012318.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 10MB]</span><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MmshFvtLeKY/Wm9jhb20VAI/AAAAAAAAJ4s/RBKHF_PFqhE8Kju5ViViZMs2ylnLvisggCLcBGAs/s1600/image1.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://1.bp.blogspot.com/-MmshFvtLeKY/Wm9jhb20VAI/AAAAAAAAJ4s/RBKHF_PFqhE8Kju5ViViZMs2ylnLvisggCLcBGAs/s320/image1.jpeg" width="240" /></a></div><br /><br />by Jorge Pullin, LSU<br /><br />One of the central elements when building quantum theories using the approach known as "canonical" is to define a quantity known as the Hamiltonian. This quantity is responsible for the time evolution of the system under study. In general relativity, when one tries to construct such quantity one notices it vanishes. This is because in general relativity one can choose any arbitrary time variable and therefore there is not a uniquely selected evolution. One needs to make a choice. One such choice is to use matter to play the role of a clock. That leads to one having a non-vanishing Hamiltonian. In this work a detailed construction for the quantum operator associated with such Hamiltonian in loop quantum gravity was presented. The implementation presented differs from others done in the past. Among the attractive elements is that it can be shown in certain circumstances that the operator has the desirable mathematical property known as "self-adjointness". This property ensures that physical quantities in the theory are represented by real (as opposed to complex) numbers.<br /><br />A discussion was also presented of how the operator acts on certain states that behave semi-classically known as "coherent states", in particular in the context of cosmological models. It was observed that it leads to an expanding universe.Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-35550724827347434762018-01-15T16:00:00.000-06:002018-01-15T16:00:03.668-06:00Construction of Feynman diagrams for group field theory<span style="background-color: white;">Tuesday, Dec 5th</span><br /><b></b><br /><b>Marco Finocchiaro, Albert Einstein Institute</b><br /><b>Title: Recursive graphical construction of GFT Feynman diagrams </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/finocchiaro120517.pdf">PDF</a><span style="background-color: white;"> of the talk (1M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/finocchiaro120517.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 24MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-eEwO3ce1Hh4/Wl0gt3tfLsI/AAAAAAAAJ2M/ot7suhY8Tz0DmoJWyqNWHy4gtuYYdLm0wCLcBGAs/s1600/Screen%2BShot%2B2018-01-15%2Bat%2B3.44.01%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1309" data-original-width="1111" height="320" src="https://1.bp.blogspot.com/-eEwO3ce1Hh4/Wl0gt3tfLsI/AAAAAAAAJ2M/ot7suhY8Tz0DmoJWyqNWHy4gtuYYdLm0wCLcBGAs/s320/Screen%2BShot%2B2018-01-15%2Bat%2B3.44.01%2BPM.png" width="271" /></a></div><span style="background-color: white;"><br /></span><span style="background-color: white;">By Jorge Pullin, LSU.</span><br />A common technique for computing probability amplitudes in quantum field theory consists in expanding such objects as power series in term of the coupling constant of the theory. Each term in the expansion, usually involving complicated expressions, can be represented in a pictorial way by using diagrams. This graphical technique (known as "Feynman diagrams method") allows to write down and organize the terms in the perturbative series in a much easier way. <br /><br />Group field theories (GFTs) are ordinary quantum field theories on group manifolds. Their Feynman amplitudes (i.e. amplitudes associated to Feynman graphs) correspond by construction to Quantum Gravity Spinfoam amplitudes. There exists an analogue situation in 1+1 dimensional theories known as matrix models, which are quantum field theories whose Feynman diagrams are related to the path integrals for gravity in 1+1 dimensions. From this point of view group field theories can be seen as a four dimensional generalization of matrix models. <br /><br />The seminar, articulated in three parts, dealt with several aspects concerning the construction of GFT's Feynman diagrams and the evaluation of the corresponding amplitudes. In the first part a general introduction to group field theory was provided, stressing the importance of studying the divergences appearing in the amplitudes' computations. Indeed they can be used as tools to constraint and test the type of theories that can be built. In the second part the main methods to extract the amplitudes' divergences were briefly reviewed. Moreover a new GFT/Spinfoam model for Euclidean quantum gravity was presented. The last part was devoted to the seminar's main topic, namely the generation of Feynman graphs in group field theory. Beyond the leading order in the power series expansion this is often a difficult task. It was shown how to construct GFT's Feynman diagrams using recursive graphical relations that are suitable for implementations in computers. Future works will deal with making the computations parallelizable.<br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-36060725028730805982018-01-15T15:41:00.000-06:002018-01-15T15:41:19.039-06:00Entanglement in loop quantum gravity<span style="background-color: white;">Tuesday, Nov 7th</span><br /><b></b><br /><b>Eugenio Bianchi, PennState</b><br /><b>Title: Entanglement in loop quantum gravity </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/bianchi110717.pdf">PDF</a><span style="background-color: white;"> of the talk (9M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/bianchi110717.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 19MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-oUgukYNpXts/WlZAg-ngaVI/AAAAAAAAJ08/oGG5Y9hOZi0LbZaQuHci8MzgwuZShMT2gCLcBGAs/s1600/Eugenio_Bianchi3.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="661" data-original-width="992" height="213" src="https://4.bp.blogspot.com/-oUgukYNpXts/WlZAg-ngaVI/AAAAAAAAJ08/oGG5Y9hOZi0LbZaQuHci8MzgwuZShMT2gCLcBGAs/s320/Eugenio_Bianchi3.jpg" width="320" /></a></div><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><span style="background-color: white;">By Jorge Pullin, LSU</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">Entanglement is one of the most fascinating new concepts introduced in quantum mechanics. When quantum systems interact, the resulting systems properties cannot be described by considering the properties of the individual systems. One needs to consider global properties of the set of systems as a whole. Not only one cannot reconstruct the properties of the whole from the properties of the constituent parts. It turns out that the properties of the constituent parts cannot be determined if one does not know the properties of the whole. Entanglement entropy is a quantity that measures "how much entanglement" there is in a set of quantum systems. This seminar dealt with the application of this concept to the quantum states of loop quantum gravity. Here one tries to understand how different regions of space become entangled with each other in a quantum geometry and how the entanglement entropy measures such entanglement. </span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">This is not a mere theoretical development. Quantum theory plays an important role in cosmology. We now know that the fluctuations we see in the cosmic microwave background radiation are the product of the evolution of the vacuum state of the inflaton field during inflation. If one assumes that before inflation the field was in a vacuum state and evolves it, the state develops non-trivial correlations that are precisely the ones observed in the cosmic background radiation fluctuations. </span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-efnKnkBtits/WlZDmryLgII/AAAAAAAAJ1I/FSJQR82LRX8osEqeksads3CcmgdLk99BwCLcBGAs/s1600/Ilc_9yr_moll4096.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="256" data-original-width="512" height="160" src="https://4.bp.blogspot.com/-efnKnkBtits/WlZDmryLgII/AAAAAAAAJ1I/FSJQR82LRX8osEqeksads3CcmgdLk99BwCLcBGAs/s320/Ilc_9yr_moll4096.png" width="320" /></a></div><div style="text-align: center;"><span style="background-color: white;">The cosmic microwave background fluctuations. Credit: NASA/WMAP team.</span></div><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><span style="background-color: white;">The vacuum state of a quantum field is a highly entangled state. Therefore the correlations one observes in the cosmic microwave background are directly related to entanglement. This seminar raises the mesmerizing possibility that the particular type of entanglement that occurs in the states of loop quantum gravity could leave an observable imprint in the cosmic microwave background radiation. This occurs through their evolution from the big bounce that loop quantum cosmology replaces the big bang with up to the beginning of inflation influencing the type of vacuum the inflaton starts in. </span>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-78772014554800120332018-01-10T11:59:00.000-06:002018-01-10T11:59:37.207-06:00Black holes exploding into white hole fireworks<span style="background-color: white;">Tuesday, Oct 24th</span><br /><b></b><br /><b>Marios Christodoulou, Aix Marseille U/SUSTec Shenzen</b><br /><b>Title: Geometry transition in covariant LQG: black to white </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/christodoulou102417.pdf">PDF</a><span style="background-color: white;"> of the talk (3M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/christodoulou102417.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 11MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-AFw_EUGRdgg/WlZES_T69cI/AAAAAAAAJ1Q/rypcATz_iDkrBjJtmXURu-DMRM0x068kwCLcBGAs/s1600/photo_bilbao.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="180" src="https://4.bp.blogspot.com/-AFw_EUGRdgg/WlZES_T69cI/AAAAAAAAJ1Q/rypcATz_iDkrBjJtmXURu-DMRM0x068kwCLcBGAs/s320/photo_bilbao.jpeg" width="320" /></a></div><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><span style="background-color: white;">By Jorge Pullin, LSU</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">Black holes are regions of space-time where gravity is so intense that nothing, including light, can escape, hence they are black. They are expected to form as stars exhaust their nuclear fuel and start to contract due to gravitational attraction. Eventually they become so dense that a black hole forms. According to classical general relativity, the star matter continues to contract inside the black hole until the density diverges. That is what is known as a "singularity". Obviously nothing can diverge in nature so it is believed that the singularities are an indication that one has pushed general relativity beyond its domain of validity. One expects that at high densities quantum effects should arise and a theory of quantum gravity is needed. There has been some progress in spherically symmetric loop quantum gravity that indicates that the singularity is replaced by a highly quantum region that eventually leads to another classical region of space-time beyond it. </span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">At the same time Hawking showed in the 70's that if one puts quantum fields to live on the classical background of a black hole, radiation is emitted as if the black hole behaved as a black body with a temperature inversely proportional to the black hole's mass. There is no contradiction with the black hole radiating because the radiation is produced by the quantum field outside the black hole. If the black hole radiates, then it should lose energy. Hawking's calculation cannot study this, because it assumes the quantum field lives in a fixed black hole background. It is expected that more precise calculations including the back-reaction of the field on the background should make the black hole shrink as it emits radiation. As the temperature increases as the black hole loses mass (it is inversely proportional to the mass) the black hole heats up and radiates more. Eventually it should evaporate completely. No detailed analysis of such evaporation is available at present. Such evaporation raises many questions, in particular what happened to the singularity inside the black hole (or the highly quantum region that apparently replaces it). What happened to all the information of the matter that formed the black hole? Is it lost?</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">The work described in this seminar posits that the highly quantum region inside the black hole transitions into the future into a "white hole" (the time reverse of a black hole). A great explosion in which all the information that entered the black hole exits. This scenario is known as "fireworks". An important question is: does the explosion happen fast enough for it to make the loss of information through Hawking radiation irrelevant? In this seminar spin foams are used to try to address the question. The calculation at hand is to compute the probability of transition from a black hole to a white hole. There are many assumptions needed to make such calculation, so the results are at the moment tentative. However, the main conclusion seems to be that the explosion takes as long as the process of Hawking evaporation to take place. This may rule out the "fireworks" as candidates for fast radio bursts that have been observed by astronomers, but may keep in play other astrophysical predictions associated with them.</span>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-43354621585021449542018-01-10T10:00:00.000-06:002018-01-10T10:00:09.443-06:00Cosmological dynamics from full loop quantum gravity<span style="background-color: white;">Tuesday, Sept 26th</span><br /><b>Andrea Dapor and Klaus Liegener, FAU Erlangen</b><br /><b>Title: Cosmological Effective Hamiltonian from full Loop Quantum Gravity </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/dapor092617.pdf">PDF</a><span style="background-color: white;"> of the talk (2.2M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/dapor092617.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 13MB]</span><br /><span style="background-color: white;"><br /></span><br /><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-If-X3veskOQ/WlEeHv4kgNI/AAAAAAAAJ0M/ahG0IPCBoiQ9GynX-Jko_zoBR21z0TQNQCLcBGAs/s1600/Screen%2BShot%2B2018-01-06%2Bat%2B1.05.47%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="393" data-original-width="754" height="166" src="https://3.bp.blogspot.com/-If-X3veskOQ/WlEeHv4kgNI/AAAAAAAAJ0M/ahG0IPCBoiQ9GynX-Jko_zoBR21z0TQNQCLcBGAs/s320/Screen%2BShot%2B2018-01-06%2Bat%2B1.05.47%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white;">By Jorge Pullin, LSU</span></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white;">Due to the complexity of theories like general relativity, a common line of attack to understand the theory is to consider situations with high symmetry. In them, one freezes almost all degrees of freedom but a few and studies them. Examples are the studies of homogeneous cosmologies, where the only degrees of freedom left are the volume of the universe and perhaps variables characterizing its anisotropy. In some cases people choose to freeze most of the degrees of freedom and then quantize the remaining ones. This is called "minisuperspace" quantization. In the case of cosmologies that means that one is left with just a handful of degrees of freedom, turning a field theory with infinitely many degrees of freedom like general relativity into a "mechanical system" in the sense of having only a finite number of degrees of freedom. The resulting quantization is therefore much simplified and a lot of progress can be made. The field of study of these quantizations is known as "loop quantum cosmology". The hope is that the resulting theories resemble what happens when one follows the evolution of a highly symmetric state in the full theory. This is, however, not guaranteed. There are known examples where "reducing then quantizing" does not yield the same result as "quantizing then reducing".</span></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white;">The seminar dealt with an attempt to "quantize then reduce" loop quantum gravity and see if the results of loop quantum cosmology follow for such an approach. This requires choosing quantum states in the full theory whose probabilities are "peaked" around homogeneous geometries and that evolve maintaining the homogeneity. States that are peaked around certain classical solutions and follow their evolution in quantum theory are known as "coherent states". In this talk such states for loop quantum gravity based on cubic lattices were constructed and their evolution was studied. It was noted that the resulting evolution does not coincide with the one usually chosen in loop quantum cosmology. When one quantizes theories there are ambiguities in how one proceeds and choices need to be made in how one write certain classical equations as quantum operators. It turns out that one of the choices usually made in loop quantum cosmology does not match with what one gets in the "quantize then reduce" approach. This suggests novel dynamics to study in the context of loop quantum cosmology that may affect the emerging picture of how our universe's Big Bang got replaced by a Big Bounce. In the traditional loop quantum cosmology approach the bounce is preceded by a large classical universe like ours. In the new dynamics suggested in this talk the bounce is preceded by a large but very quantum universe with a large Planck-scale cosmological constant. In the distant past our universe asymptotes to a very symmetrical universe known as De Sitter space. Further studies need to be done to check the consistency of the approach. </span></div><div class="separator" style="clear: both; text-align: left;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="background-color: white;"><br /></span>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-45919510721758186682018-01-10T09:36:00.000-06:002018-01-10T09:36:18.617-06:00Intrinsic time geometrodynamics<span style="background-color: white;">Tuesday, Sept 12th</span><br /><b>Eyo Eyo Ita, University of South Africa</b><br /><b><b>Title: Intrinsic time quantum geometrodynamics: emergence of General Relativity and cosmic time</b> </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/ita091217.pdf">PDF</a><span style="background-color: white;"> of the talk (1.5M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/ita091217.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 8MB]</span><br /><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-zALrAFrbQBI/WlEXQWIz1NI/AAAAAAAAJz0/5mH66gzHhqcZFHH84UtZYhrH0-EWK6nfgCLcBGAs/s1600/Eyo3small.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="154" data-original-width="150" src="https://2.bp.blogspot.com/-zALrAFrbQBI/WlEXQWIz1NI/AAAAAAAAJz0/5mH66gzHhqcZFHH84UtZYhrH0-EWK6nfgCLcBGAs/s1600/Eyo3small.jpg" /></a></div>By Jorge Pullin, LSU<br /><br />Usual Newtonian mechanics describes the motions of systems with respect to an absolute time variable usually called t. Already special relativity introduces the idea that time is not absolute and that it ticks at different rates for different observers. General relativity goes beyond that: one can pick any variable to play the role of time. The result of that is that if one tries to understand the dynamics of the theory as an "evolution in time" one runs into difficulties. This is important because many of our ideas of how to quantize theories are implemented dynamically. One needs what is known as a "Hamiltonian formulation" of the theory in order to implement what is known as "canonical quantization". In the Hamiltonian formulation there is a quantity known as the Hamiltonian that is responsible for time evolution. If one attempts to construct a Hamiltonian formulation for general relativity one discovers that the Hamiltonian vanishes. This reflects the fact that if one is allowed to pick any time variable one essentially can get any evolution one wants. This was the source of quite a bit of confusion and explains why a suitable Hamiltonian formulation took almost 50 years to emerge, being general relativity from 1915 and the Hamiltonian formulation only finally understood in the early 60's. Today we know that if one wants to have a defined Hamiltonian and evolution one needs to choose a time variable. The intrinsic geometrodynamics essentially chooses the volume of space as time variable. This seminar discussed the details and its implications for quantization in particular in the so-called "path integral quantization". Among the results a natural vacuum for the theory is found that involves the well known mathematical invariant related to the Chern-Simons form, suggesting perhaps that general relativity could be turned into a renormalizable quantum field theory.Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-33541736008860561832018-01-06T16:08:00.002-06:002018-01-06T16:08:33.360-06:00Gravitational path integral and group theory<b>Pietro Dona, Penn State</b><br /><b>Title: SU(2) graph invariants, Regge actions and polytopes </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/dona101017.pdf">PDF</a><span style="background-color: white;"> of the talk (10M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/dona101017.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 16MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-MwOcrfuqvmY/WlE1cUcYUlI/AAAAAAAAJ0c/HQVDpntkEHUuVN0yGlRGZgmYyyhYuHddQCLcBGAs/s1600/Screen%2BShot%2B2018-01-06%2Bat%2B2.45.16%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="416" data-original-width="183" height="320" src="https://1.bp.blogspot.com/-MwOcrfuqvmY/WlE1cUcYUlI/AAAAAAAAJ0c/HQVDpntkEHUuVN0yGlRGZgmYyyhYuHddQCLcBGAs/s320/Screen%2BShot%2B2018-01-06%2Bat%2B2.45.16%2BPM.png" width="140" /></a></div><span style="background-color: white;"><br /></span><span style="background-color: white;">By Jorge Pullin, LSU</span><br /><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">In the loop quantum gravity approach to quantum gravity the quantum states are given by spin networks, graphs with intersections and "colors" associated with each link. The colors are a shorthand to characterize that each link in the graph is associated with a mathematical quantity known as an element of a group. A group is a type of mathematical set with a composition law that is associative, has a neutral element and has an opposite element. For instance, real numbers form a group under addition. Matrices of numbers also form groups under multiplication. When links of a spin network meet at an intersection, the respective group elements associated with them get multiplied into a mathematical entity known as "intertwiner". Such intertwiners are constructed with what are known as invariant tensors in the group. </span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">One of the approaches to quantizations of field theories is the path integral approach. In it, one assigns probabilities to each physical trajectory and sums over all possible trajectories. When applied in the context of loop quantum gravity one gets trajectories in time of spin networks, which give rise to what are known as "spin foams". The probability of a given trajectory is quantified in terms of a number related to how the spin networks branch out into the future known as a "vertex". There are several proposals for such vertices to represent the dynamics of general relativity, at present it is not clear which one of the proposed ones represents nature more accurately. One of the most studied ones is the EPRL (Engle-Pereira-Rovelli-Livine) vertex. Other vertices that have simpler nature have also been proposed. This seminar deals with the evaluation of these vertices. This requires calculations in group theory. These calculations may have broader applicability than in just quantum gravity as these types of mathematical entities appear in many physical domains. Numerical calculations of the vertices have been carried out and asymptotic analyses performed for some of the more simplified vertices. The objective is to later extend the results to the EPRL vertex.</span>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-46665885834707400542017-08-20T12:45:00.000-05:002017-08-20T12:45:15.988-05:00Simplicial group field theory<span style="background-color: white;">Tuesday, May 2nd</span><br /><b>Marco Finocchiaro, Albert Einstein Institute</b><br /><b>Title: Simplicial Group Field Theory models for Euclidean quantum gravity: recent developments </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/finocchiaro050217.pdf">PDF</a><span style="background-color: white;"> of the talk (2M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/finocchiaro050217.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 15MB]</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">by Jorge Pullin, Louisiana State University</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-40_svw_O03Y/WZnGxtv8vnI/AAAAAAAAJkE/0HpAT5Q4QeYoGlYbVKNwTIAkCbEdvOFTACLcBGAs/s1600/PictureI.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1358" height="320" src="https://2.bp.blogspot.com/-40_svw_O03Y/WZnGxtv8vnI/AAAAAAAAJkE/0HpAT5Q4QeYoGlYbVKNwTIAkCbEdvOFTACLcBGAs/s320/PictureI.jpeg" width="271" /></a></div><span style="background-color: white;"><br /></span>The approach to quantum gravity known as “spin foams” is based on the quantization technique known as the path integral. In this technique probabilities are assigned for a given slice of space to transition to a future slice in a space-time. Since in loop quantum gravity spatial slices are associated to spin networks, as these evolve in time transitioning to slices of the future one gets the “spin foams”. Group field theory is a technique in which an ordinary (but non-local) quantum field theory is constructed in such a way that its Feynman diagrams yield the probabilities of the spin foam approach. There is an analogue en 1+1 dimensions known as “matrix models” that were widely studied in the 1990’s. Group field theories can be viewed as their generalization to four dimensions.<br /><br /> Formulating spin foams in terms of group field theories has several advantages. Results do not depend on the triangulations picked, as one expects it should be but is not obvious in terms of spin foams. One can import techniques from field theories, in particular to introduce notions of renormalizability and a continuum limit.<br /><br /> In this talk a particular group field theory model is presented and discussed in some detail. In particular a numerical analysis of the resulting probabilities was made. And results were compared to a popular model of spin foams, the EPRL model. Certain insights on the possible choices in the construction of the model and how it could influence the ultraviolet behavior and possible singularities present were discussed. Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-43263310373110407692017-08-08T15:43:00.000-05:002017-08-08T15:43:21.067-05:00Loop quantum gravity with homogeneously curved vacuum<span style="background-color: white;">Tuesday, Apr 18th</span><br /><b>Bianca Dittrich, Perimeter Institute</b><br /><b>Title: (3+1) LQG with homogeneously curved vacuum </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/dittrich041817.pdf">PDF</a><span style="background-color: white;"> of the talk (8M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/dittrich041817.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 17MB]</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">by Jorge Pullin, Louisiana State</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-gIUvp3AW7Ws/UnK032ccxEI/AAAAAAAAEUA/OYMCf2I87xM/s1600/bianca.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-gIUvp3AW7Ws/UnK032ccxEI/AAAAAAAAEUA/OYMCf2I87xM/s1600/bianca.jpg" /></a></div><span style="background-color: white;"><br /></span>The way geometries are studied mathematically one starts with a set of points that has a notion of proximity. One can say when points are close to each other. This is not the same as being able to measure distances in the set. That requires the introduction of an additional mathematical structure, a metric. The sets of points with notion of proximity are known as “manifolds”. General relativity is formulated on a manifold and is a theory about a metric to be imposed on that manifold. Ordinary quantum field theories, like quantum electrodynamics, require the introduction of a metric before they can be formulated, so they are of different nature than general relativity. Theories that do not require a metric in order to be formulated are known as “background independent”. Interestingly, although general relativity is a theory about a metric, it can be formulated without any prior metric. There exist quantum field theories that can be formulated without a metric. They are known as topological field theories and they typically, contrary to ordinary field theories, have only a finite number of degrees of freedom. This means that they are much easier to treat and to quantize. <br /><br />An example of a topological field theory is general relativity in three space-time dimensions. In one dimension less than four, the Einstein equations just say that the metric is flat, except at a finite number of points. So space-time is flat everywhere with curvature concentrated at just a few points. An example of a space that is flat everywhere except at a point is a cone. The only place that is curved is the tip. One has to remember that the notion of curvature we are talking here is one that can be measured from inside the space-time (typically by going around a circle and seeing if a vector carried around returns parallel to itself). If you do that in a cone on any circle that does not thread the tip, the vector comes back parallel to itself. So space-times in three dimensional general relativity are said to have “conical singularities” at the points where the curvature is non-zero. As other topological field theories, general relativity in three space-time dimensions has a finite number of degrees of freedom. This explains why Witten was able to complete its quantization in the mid 1980’s whereas the quantization of four dimensional general relativity is still a big outstanding problem today. <br /><br /><br />In this talk a generalization of three dimensional general relativity to four dimensions was presented. The resulting theory in four space-time dimensions has curvature concentrated at edges (strings) –as opposed to points as we had in the three dimensional case- and elsewhere the metric is flat. This makes them much easier to quantize than general relativity. Among the results was the construction of four dimensional quantum geometries similar to those in a previous model by Crane and Yetter. Also a role for quantum groups, that had been conjectured to arise when one considers a cosmological constant was found providing more evidence to this assertion. The space of quantum states (Hilbert space) was rigorously constructed and leads to insights about how the continuum limit of the theory could emerge. The hope is that one could build on these theories to construct new representations for loop quantum gravity in four space-time dimensions and hopefully to implement on them the (quantum) dynamics of general relativity. <br /><br /><br />Also a notion of duality emerges. In this context, duality means a certain relationship between the metric and the curvature of the space-time at a classical level. Here it can be implemented at a quantum level and quantum space of states associated with the metric (areas) and curvatures can be introduced and are dual to each other. Similar spaces had been proposed for general relativity, but here there is much more mathematical control over them, so this provides a controlled arena to test ideas that are being put forward in the context of loop quantum gravity in four space-time dimensions. <br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-67588452880396895032017-04-28T12:51:00.000-05:002017-04-28T12:51:22.954-05:00Transition times through the black hole bounce<span style="background-color: white;">Tuesday, Apr 4th</span><br /><b>Parampreet Singh, LSU</b><br /><b>Title: Transition times through the black hole bounce </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/singh040417.pdf">PDF</a><span style="background-color: white;"> of the talk (2M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/singh040417.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 18MB]</span><br /><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-RoMaXdHqZYQ/WQNwkMR7ROI/AAAAAAAAJWs/BE_i6AfsUxw9vhWUqe0Rs3HIGQQmr5G0gCLcB/s1600/param2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-RoMaXdHqZYQ/WQNwkMR7ROI/AAAAAAAAJWs/BE_i6AfsUxw9vhWUqe0Rs3HIGQQmr5G0gCLcB/s320/param2.jpg" width="304" /></a></div><span style="background-color: white;">by Gaurav Khanna, University of Massachu</span><span style="background-color: white;">setts Dartmouth</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span> Loop quantum cosmology (LQC) is an application of loop quantum gravity theory in the context of spacetimes with a high degree of symmetry (e.g. homogeneity, isotropy). One of the main successes of LQC is the resolution of "singularities" that generically appear in the classical theory. An example of this is the "big bang" singularity that causes a complete breakdown of general relativity (GR) in the very early universe. Models studied within the framework of LQC replace this "big bang" with a "big bounce" and do not suffer a singular breakdown like in the classical theory. <br /><br /><br />It is, therefore, natural to consider applying similar techniques to study black holes; after all, these solutions of GR are also plagued with a central singularity. In addition, it is plausible that a LQC model may shed some light on long-standing issues in black hole physics, i.e., information loss, Hawking evaporation, firewalls, etc. <br /><br /><br />Now, if one restricts the model only to the Schwarzschild black hole interior region, the spacetime can actually be considered as a homogeneous, anisotropic cosmology (the Kantowski-Sachs spacetime). This allows techniques of LQC to be readily applied to the black hole case. In fact, a good deal of study has been performed in this direction by Ashtekar, Bojowald, Modesto and many others for over a decade. While these models are able to resolve the central black hole singularity and include important improvements over previous versions, they still have a number of issues. <br /><br /><br />Recently, Singh and Corichi (2016), proposed a new LQC model for the black hole interior that attempts to address these issues. In this talk, Singh describes some of the resulting phenomenology that emerges from that improved model.<br /><br />The main emphasis of this talk is on the following questions:<br /> <br /><br />(1) Is the "bounce" in the context of a black hole LQC model, i.e., transition from a black hole to a white hole, symmetric? Isotropic and homogeneous models in LQC have generally exhibited symmetric bounces. But, that is not expected to hold in the context of more general models.<br />(2) Does quantum gravity play a role only once during the bounce process?<br />(3) What quantitative statements can be made about the time-scales of this process; and what are the full implications of those details?<br /> (4) Do all black holes, independent of size, exhibit very similar characteristics? <br /><br /><br />Based on detailed numerical calculations that Singh reviews in his presentation, he uncovers the following features from this model: <br /><br /><br />(1) The bounce is indeed not symmetric; for example, the sizes of the parent black hole and the offspring white hole are widely different. Other details on this asymmetry appear below.<br />(2) Two distinct quantum regimes appear in this model, with very different associated time-scales.<br />(3) In terms of the proper time of an observer, the time spent in the quantum white hole geometry is much larger than in the quantum black hole. And, in particular, the time for the observer to reach the white hole horizon is exceedingly large. This also implies that the formation of the white hole interior geometry happens a lot quicker than the formation of its horizon.<br />(4) The relation of the bounce time with the black hole mass, does depend on whether the black hole is large or small. <br /><br /><br />On the potential implications of such details on some of the important open questions in black hole physics, Singh speculates: <br /><br /><br />(1) For large black holes, the time to develop a white hole (horizon) is much larger than the Hawking evaporation time. This may suggest that for an external observer, a black hole would disappear long before the white hole appears!<br />(2) For small black holes, the time to form a white hole is smaller than Hawking time, i.e., small black holes explode before they can evaporate! <br /><br /><br />These could have some interesting implications for the various proposed black hole evaporation paradigms. Given the concreteness of the results Singh presents, they are also likely to be relevant to the many previous phenomenological studies on black hole to white hole transitions including Planck stars. <br /><br /><br />The two main limitations of Singh's results are: (1) the current model ignores the black hole exterior entirely; and (2) the conclusions rely on effective dynamics, and not the full quantum evolution. These may be addressed in future work. <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-32067506403398625262017-03-28T16:40:00.001-05:002017-03-28T16:40:17.981-05:00Holographic signatures of resolved cosmological singularities<span style="background-color: white;">Tuesday, March 21st</span><br /><b>Norbert Bodendorfer, LMU Munich</b><br /><b>Title: Holographic signatures of resolved cosmological singularities </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/bodendorfer032117.pdf">PDF</a><span style="background-color: white;"> of the talk (2M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/bodendorfer032117.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 10MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-vP0SUwQJ2V4/WNrQ1FdcHwI/AAAAAAAAJNU/mGU6cKh9WPgQBJnhswsI_QWgkzDVb1duQCLcB/s1600/Norbert_Bodendorfer.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-vP0SUwQJ2V4/WNrQ1FdcHwI/AAAAAAAAJNU/mGU6cKh9WPgQBJnhswsI_QWgkzDVb1duQCLcB/s1600/Norbert_Bodendorfer.png" /></a></div><span style="background-color: white;">By Jorge Pullin, Louisiana State University</span><br /><span style="background-color: white;"><br /></span>One of the most important results in string theory is the so called “Maldacena conjecture” or “AdS/CFT correspondence” proposed by Juan Maldacena. This conjecture states that given a space-time with cosmological constant (known as anti De Sitter space-time or AdS) the behavior of gravity in it is equivalent to the behavior of a field theory living on the boundary of the space-time. These field theories are of a special type known as “conformal field theories”. Hence the AdS/CFT name. Conformal field theories are considerably better understood than quantum gravity so to make the latter equivalent to them opens several new possibilities. The discussion of AdS/CFT has mostly taken place in the context of string theory which has general relativity as a classical limit. This opens the question of what kind of imprint the singularities that are known to exist in general relativity leave in the conformal field theory. <br /><br />On the other hand, loop quantum gravity is known for eliminating the singularities that arise in general relativity. They get replaced by regions of high curvature and fluctuations of it that are not well described by a semiclassical geometry. However, nothing is singular, physical variables may take large –but finite-values. If AdS/CFT were to hold in the context of loop quantum gravity the question arises of what imprint would the elimination of the singularity leave on the conformal field theory. The seminar dealt with this point by considering certain functions known as correlation functions in the conformal field theory that characterize its behavior. In particular how the singularities of general relativity get encoded in these correlation functions and how their elimination in loop quantum gravity changes them. The work is at the moment only a model in five dimensions of a particular space-time known as the Kasner space-time. <br /><br />Future work will consist in expanding the results to other space-times. Of particular interest would be the extension to black hole spacetimes, which loop quantum gravity also rids of singularities. As is well known, black hole space-times have the problem of the “information paradox” stemming from the fact that black holes evaporate through the radiation that Hawking predicted leaving in their wake only thermal radiation no matter what process led to the formation of the black hole. It is expected that when the evaporation is viewed in terms of the conformal field theory, this loss of information about what formed the black hole will be better understood.<br /><br />In addition to the specific results, the fact that this work suggests points of contact between loop quantum gravity and string theory makes it uniquely exciting since both fields have developed separately over the years and could potentially benefit from cross pollination of ideas. Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-29109902011762004102017-02-22T12:09:00.000-06:002017-02-22T12:09:14.265-06:00Gravity as the dimensional reduction of a theory of forms in six or seven dimensions<span style="background-color: white;">Tuesday, February 21st</span><br /><b>Kirill Krasnov, University of Nottingham</b><br /><b>Title: 3D/4D gravity as the dimensional reduction of a theory of differential forms in 6D/7D </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/krasnov022117.pdf">PDF</a><span style="background-color: white;"> of the talk (5M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/krasnov022117.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 16MB]</span><br /><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-vkrWIIbzM64/WK3OaVBRo8I/AAAAAAAAJI4/DcIpF_d2slI4TroeawZdApAXRECwfn3PQCLcB/s1600/re.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-vkrWIIbzM64/WK3OaVBRo8I/AAAAAAAAJI4/DcIpF_d2slI4TroeawZdApAXRECwfn3PQCLcB/s320/re.png" width="320" /></a></div><span style="background-color: white;">by Jorge Pullin, Louisiana State University</span><br /><span style="background-color: white;"><br /></span><br /><div class="MsoNormal">Ordinary field theories, like Maxwell’s electromagnetism, are physical systems with infinitely many degrees of freedom. Essentially the values of the fields at all the points of space are the degrees of freedom. There exist a class of field theories that are formulated as ordinary ones in terms of fields that take different values at different points in space,<span style="mso-spacerun: yes;"> </span>but that whose equations of motion imply that the number of degrees of freedom are finite. This makes some of them particularly easy to quantize. A good example of this is general relativity in two space and one time dimensions (known as 2+1 dimensions). Unlike general relativity in four-dimensional space-time, it only has a finite number of degrees of freedom that depend on the topology of the space-time considered. This type of behavior tends to be generic for these types of theories and as a consequence they are labeled Topological Field Theories (TFT). These types of theories have encountered application in mathematics to explore geometry and topology issues, like the construction of knot invariants, using quantum field theory techniques. These theories have the property of not requiring any background geometric structure to define them unlike, for instance, Maxwell theory, that requires a given metric of space-time in order to formulate it.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Remarkably, it was shown some time ago by Plebanski, in 1977 and later further studied by Capovilla-Dell-Jacobson and Mason in 1991 that certain four dimensional TFTs, if supplemented by additional constraints among their variables, were equivalent to general relativity. The additional constraints had the counterintuitive effect of adding degrees of freedom to the theory because they modify the fields in terms of which the theory is formulated. Formulating general relativity in this fashion leads to new perspectives on the theory. In particular it suggests certain generalizations of general relativity, which the talk refers to as deformations of GR. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The talk considered a series of field theories in six and seven dimensions. The theories do not require background structures for their definition but unlike the topological theories we mentioned before, they do have infinitely many degrees of freedom. Then the dimensional reduction to four dimensional of these theories was considered. Dimensional reduction is a procedure in which one “takes a lower dimensional slice” of a higher dimensional theory, usually by imposing some symmetry (for instance assuming that the fields do not depend on certain coordinates). One of the first such proposals was considered in 1919 by Kaluza and further considered later by Klein so it is known as Kaluza-Klein theory. They considered general relativity in five dimensions and by assuming the metric does not depend on the fifth coordinate, were able to show that the theory behaved like four-dimensional general relativity coupled to Maxwell’s electromagnetism and a scalar field. In the talk it was shown that the seven dimensional theory considered, when reduced to four dimensions, was equivalent to general relativity coupled to a scalar field. 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mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} </style><br />--> <!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin;} </style><![endif]--> <!--StartFragment--> <!--EndFragment--><div class="MsoNormal"><span style="mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">At the moment is not clear whether these theories can be considered as describing nature, because it is not clear whether the additional scalar field that is predicted is compatible with the known constraints on scalar-tensor theories. However, these theories are useful in illuminating the structures and dynamics of general relativity and connections to other theories.</span><o:p></o:p></div><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-82186023070346698362017-02-07T15:02:00.002-06:002017-02-07T15:02:26.175-06:00Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy <span style="background-color: white;">Tuesday, February 7th</span><br /><b>Muxin Han, Florida Atlantic University</b><br /><b>Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/han020717.pdf">PDF</a><span style="background-color: white;"> of the talk (2M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/han020717.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 18MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-im79Xdc4yWk/WJowVPRr0DI/AAAAAAAAJGg/GzA_11oJtJUuZbEWvCPOQ6yDLDlyjYytQCLcB/s1600/re.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="213" src="https://4.bp.blogspot.com/-im79Xdc4yWk/WJowVPRr0DI/AAAAAAAAJGg/GzA_11oJtJUuZbEWvCPOQ6yDLDlyjYytQCLcB/s320/re.JPG" width="320" /></a></div><span style="background-color: white;">by Jorge Pullin, Louisiana State University</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span>The cosmological constant is an extra term that was introduced into the equations of General Relativity by Einstein himself. At the time he was trying to show that if one applied the equations to the universe as a whole, they had static solutions. People did not know in those days that the universe expanded. Some say that Einstein called the introduction of this extra term his “biggest blunder” since it prevented him from predicting the expansion of the universe which was observed experimentally by Hubble a few years later. In spite of its origin, the term is allowed in the equations and the space-times that arise when one includes the term are known as de Sitter space-times in honor of the Dutch physicist who first found some of these solutions. Depending on the sign of the cosmological constant chosen, one could have de Sitter or anti-de Sitter (AdS) space-times. <br /><br /><br />It was observed in the context of string theory that if one considered quantum gravity in anti-de Sitter space-times, the theory was equivalent to a certain class of field theories known as conformal field theories (CFT) living on the boundary of the space-time. The result is not a theorem but a conjecture, known as AdS/CFT or Maldacena conjecture. It has been verified in a variety of examples. It is a remarkable result. Gravity and conformal field theories are very different in many aspects and the fact that they could be mapped to each other opens many possibilities for new insights. For instance, an important open problem in gravity is the evaporation of black holes. Although nothing can escape a black hole classically, Hawking showed that if quantum effects are taken into account, black holes radiate particles like a black body at a given temperature. The particles take away energy and the black hole shrinks, eventually evaporating completely. This raises the question of what happened to matter that went into the black hole. Quantum mechanics has a property named unitarity that states that ordinary matter cannot turn into incoherent radiation, so this raises the question of how it could happen in an evaporating black hole. In the AdS/CFT picture, since the evaporating black hole would be mapped to a conformal field theory that is unitary, that would provide a way to study quantum mechanically how matter turns into incoherent radiation. <br /><br /><br />Several authors have connected the AdS/CFT conjecture to a mathematical construction known as tensor networks that is commonly used in quantum information theory. Tensor networks have several points in common with the spin networks that are the quantum states of gravity in loop quantum gravity. This talk spells out in detail how to make a correspondence between the states of loop quantum gravity and the tensor networks, basically corresponding to a coarse graining or averaging at certain scales of the states of quantum gravity. This opens the possibility of connecting results from AdS/CFT with results in loop quantum gravity. In particular the so-called Ryu-Takahashi formula for the entropy of a region can be arrived from in the context of loop quantum gravity. <br /><br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-73560028788857376502017-01-25T14:33:00.000-06:002017-01-25T14:33:07.313-06:00Symmetries and representations in Group Field Theory<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-4l7mxEBa--A/WIkBF9i_ABI/AAAAAAAAJDU/AoCkF_vVDbkWQHhTXyIEz6mgvsiWkAURQCLcB/s1600/kegeles.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-4l7mxEBa--A/WIkBF9i_ABI/AAAAAAAAJDU/AoCkF_vVDbkWQHhTXyIEz6mgvsiWkAURQCLcB/s320/kegeles.png" width="239" /></a></div><span style="background-color: white;">Tuesday, January 24th</span><br /><b>Alexander Kegeles, Albert Einstein Institute</b><br /><b>Title: Field theoretical aspects of GFT: symmetries and representations</b><span style="background-color: white;"> </span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/kegeles012417.pdf">PDF</a><span style="background-color: white;"> of the talk (1M)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/kegeles012417.mp4">Audio+Slides</a><span style="background-color: white;"> [.mp4 11MB]</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><span style="background-color: white;">by Jorge Pullin, Louisiana State University</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span>In loop quantum gravity the quantum states are labeled by loops, more precisely by graphs formed by lines that intersect at vertices and that are “colored”, meaning each line is associated with an integer. They are known as "spin networks". As the states evolve in time these graphs "sweep" surfaces in four dimensional space-time constituting what is known as a “spin foam”. This is a representation of a quantum space-time in loop quantum gravity. The spin foams connect an initial spin network with a final one and the formalism gives a probability for such “transition” from a given spatial geometry to a future spatial geometry to occur. The picture that emerges has some parallel with ordinary particle physics in which particles transition from initial to final states, but also some differences. <br /><br />However, it was found that one could construct ordinary quantum field theories such that the transition probabilities of them coincided with those stemming from spin foams connecting initial to final spatial geometries in loop quantum gravity. This talk concerns itself with such quantum field theories, known generically as Group Field Theories (GFTs). The talk covered two main aspects of them: symmetries and representations. <br /><br />Symmetries are important in that they may provide mathematical tools to solve the equations of the theory and identify conserved quantities in it. There is a lot of experience with symmetries in local field theories, but GFT’s are non-local, which adds challenges. Ordinary quantum field theories are formulated starting by a quantity known as the action, which is an integral on a domain. A symmetry is defined as a map of the points of such domain and of the fields that leaves the integral invariant. In GFTs the action is a sum of integrals on different domains. A symmetry is defined as a collections of maps acting on the domains and fields that leave invariant each integral in the sum. An important theorem of great generality stretching from classical mechanics to quantum field theory is Noether’s theorem, that connects symmetries with conserved quantities. The above notion of symmetry for GFTs allows to introduce a Noether’s theorem for them. The theorem could find applicability in a variety of situations, in particular certain relations that were noted between GFTs and recoupling theory and better understand various models based on GFTs. <br /><br />In a quantum theory like GFTs the quantum states structure themselves into a mathematical set known as Hilbert space. The observable quantities of the theory are represented as operators acting on such space. Hilbert spaces are generically infinite dimensional and this introduces a series of technicalities both in their own definition and in the definition of observables for quantum theories. In particular one can find different families of inequivalent operators related to the same physical observables. This is what is known as different representations of the algebra of observables. Algebra in this context means that one can compose observables to form either new observables or linear combinations of known observables. An important type of representation in quantum field theory is known as Fock representation. It is the representation on which ordinary particles are based. Another type of representations is the condensate representation which, instead of particles, describes their collective (excitations) behaviour and is very convenient for systems with large (infinite) number of particles. A discussion of Fock and condensate like representations in the context of GFTs was presented and the issue of when representations are equivalent or not was also addressed. <br /><br />Future work looks at generalizing the notion of symmetries presented to find further non-standard symmetries of GFTs. Also investigating “anomalies”. This is when one has a symmetry in the classical theory that may not survive upon quantization. The notion of symmetry can also be used to define an idea of “ground state” or fundamental state of the theory. In ordinary quantum field theory in flat space-time this is done by seeking the state with lower energy. In the context of GFTs one will invoke more complicated notions of symmetries to define the ground state. Several other results of ordinary field theories, like the spin statistics theorem, may be generalizable to the GFT context using the ideas presented in this talk. <br /><br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-87896684992091873782016-03-11T10:00:00.000-06:002016-03-11T10:00:07.821-06:00Symmetry reductions in loop quantum gravity<span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-cvWAc6lmksE/VuLq7IIBukI/AAAAAAAAIYU/02Z6ppHbxkonAr5s6EiE5QYElVsYjQxDQ/s1600/Screen%2BShot%2B2016-03-11%2Bat%2B9.57.10%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="https://2.bp.blogspot.com/-cvWAc6lmksE/VuLq7IIBukI/AAAAAAAAIYU/02Z6ppHbxkonAr5s6EiE5QYElVsYjQxDQ/s320/Screen%2BShot%2B2016-03-11%2Bat%2B9.57.10%2BAM.png" width="320" /></a></div><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><span style="background-color: white;"><br /></span><span style="background-color: white;">Tuesday, Dec. 8th</span><br /><b>Norbert Bodendorfer, Univ. Warsaw </b><br /><b>Title: Quantum symmetry reductions based on classical gauge fixings </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/bodendorfer120815.pdf">PDF</a><span style="background-color: white;"> of the talk (1.4MB)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/bodendorfer120815.wav">Audio</a><span style="background-color: white;"> [.wav 35MB]</span><br /><a href="https://www.youtube.com/watch?v=6pDvzlCYJx4">YouTube.</a><br /><br /><span style="background-color: white;">Tuesday, Nov. 10th</span><br /><b>Jedrzej Swiezewski, Univ. Warsaw </b><br /><b>Title: Developments on the radial gauge </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/swiezewski111015.pdf">PDF</a><span style="background-color: white;"> of the talk (4MB)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/swiezewski111015.mp3">Audio</a><span style="background-color: white;"> [.mp3 40MB]</span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">by Steffen Gielen, Imperial College</span><br /><span style="background-color: white;"><br /></span><br /><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><span style="font-family: "liberationserif"; font-size: 12.000000pt;">A few months ago, physicists around the world celebrated the centenary of the field equations of general relativity, presented by Einstein to the Prussian Academy of Sciences in November 1915. Arriving at the correct equations was the culmination of an incredible intellectual effort by Einstein, driven largely by mathematical requirements that the new theory of gravitation (superseding Newton's theory of gravitation, which proved ultimately incomplete) should satisfy. In particular, Einstein realized that its field equations should be generally covariant – they should take the same general form in any coordinate system that one chooses to use for the calculation, say whether one uses Cartesian, cylindrical, or spherical coordinates. This property sets the equations of general relativity apart from Newton's laws of motion, where changing coordinate system can lead to the appearance of additional “forces” such as centripetal or Coriolis forces. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> <span style="font-family: "liberationserif"; font-size: 12.000000pt;">Many conferences were held honoring the anniversary of Einstein's achievement. What was discussed at those conferences was partially the historical context, the beauty of the form of the equations, or the precise mathematical and conceptual significance of general covariance. However, the most important legacy of general relativity and the main inspiration for modern research have been the new physical phenomena that appear in general relativity but not in Newtonian gravity: </span><span style="font-family: "liberationserif"; font-size: 12.000000pt; font-style: italic;">black holes </span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">are regions of spacetime where gravity becomes so strong that not even light can escape; the strong gravitational field outside a black hole leads to a </span><span style="font-family: "liberationserif"; font-size: 12.000000pt; font-style: italic;">time dilation </span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">so strong that an hour nearby a black hole can correspond to years on Earth, as used recently in the film </span><span style="font-family: "liberationserif"; font-size: 12.000000pt; font-style: italic;">Interstellar; </span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">and we now believe that the universe as a whole </span><span style="font-family: "liberationserif"; font-size: 12.000000pt; font-style: italic;">is expanding</span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">, and has been since the Big Bang which is thought of as the beginning of space and time. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> <span style="font-family: "liberationserif"; font-size: 12.000000pt;">In order to understand these dramatic consequences of the Einstein equations, physicists had to find solutions to these equations. This is rather challenging in general: the Einstein equations are complicated differential equations for ten functions, depending on one time and three space dimensions, that encode the gravitational field of spacetime. Furthermore, the conceptually appealing property of general covariance means that apparently different solutions of the equations can be simply the same physical configuration looked at in different coordinates. Indeed, both issues – finding solutions to the equations at all and understanding their meaning – were challenges in the early days of the theory, when physicists tried to make sense of Einstein's equations. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> <span style="font-family: "liberationserif"; font-size: 12.000000pt;">Despite this formidable challenge, the Prussian lieutenant of the artillery Karl Schwarzschild, while serving on the Eastern front in World War I, was able to derive an exact solution of Einstein's equations in vacuum within weeks of their publication, much to the surprise of Einstein himself. This solution, now called the </span><span style="font-family: "liberationserif"; font-size: 12.000000pt; font-style: italic;">Schwarzschild solution</span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">, describes a black hole, and is one of the most important solutions of general relativity. What Schwarzschild did in order to solve the equations was to assume a </span><span style="font-family: "liberationserif"; font-size: 12.000000pt; font-style: italic;">symmetry </span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">of the solution: he assumed that the configuration of the gravitational field should be spherically symmetric. In spherical coordinates, where each point in space is specified by one radial and two angular coordinates, it should be independent of any change in the angular directions. This means that one describes space as a collection of regular, concentric spheres. What Schwarzschild found was that the spheres did not have to be glued together to simply give normal flat space, but one could form a curved geometry out of them, with curvature increasing as one heads towards the centre (eventually forming a black hole), while still solving Einstein's equations. To be able to do the calculation, Schwarzschild had to choose a particularly suitable coordinate system, hence exploiting the property of general covariance in his favor. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> <span style="font-family: "liberationserif"; font-size: 12.000000pt;">This strategy of finding solutions is typical for practitioners of general relativity: cosmological solutions could similarly be found by assuming that the universe looks exactly the same at each point and in each direction in space (in mathematical terms, it is homogeneous and isotropic), and only changes in time. This reduces the problem of solving Einstein equations to a much simpler task, and explicit solutions could be written down, again in a suitable coordinate system. These simplest solutions already exhibit the main features of our universe (overall expansion and an initial </span><span style="font-family: liberationserif; font-size: 12pt;">Big Bang singularity) and are fairly realistic – indeed our Universe seems to display only small variations between different large-scale regions, and at the very largest scales is, within an approximation, well described by a geometry that simply looks the same everywhere in space.</span></div></div></div><div class="page" title="Page 2"><div class="layoutArea"><div class="column"><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">Loop quantum gravity is an approach at a quantization of general relativity, aiming to extend general relativity by making it compatible with quantum mechanics. What distinguishes it from other approaches is that the main property of general relativity, general covariance, is taken as a central guiding principle towards the construction of a quantum theory. In some respects, the status of loop quantum gravity can be compared to the early days of general relativity: while it is now known that a quantum theory compatible with general covariance can be constructed, and its mathematical structure is well understood, one now needs to understand the new physical phenomena implied by the quantization, beyond general relativity. Just as in the time after November 1915, today's physicists should find explicit solutions to the equations of loop quantum gravity that can be used to study the physical implications of the (relatively) new framework. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> <span style="font-family: "liberationserif"; font-size: 12.000000pt;">One of the main successes of loop quantum gravity has been its application to cosmology. Homogeneous solutions of the Einstein equations that approximately describe our universe have been shown to receive modifications once loop quantum gravity techniques are used, leading to a resolution of the Big Bang singularity by a Big Bounce, and potentially observable quantum effects. However, the resulting models of the universe are not solutions of the full theory of loop quantum gravity: rather, they arise from quantization of a reduced set of solutions of classical general relativity with loop quantum gravity techniques. There is no reason, in general, to expect that these are exact solutions of loop quantum gravity. Quantum mechanics is funny: quantization can lead to many inequivalent theories, depending on how one decides to do it. By assuming that the universe is homogeneous from the outset, one obtains a quantum theory of only a finite, rather than an infinite number of “degrees of freedom”. It is well known that quantum theories can behave differently depending on whether they have a finite or infinite number of degrees of freedom. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span><span style="font-family: "liberationserif"; font-size: 12.000000pt;">In their ILQGS seminars, Jedrzej and Norbert presented work towards resolving this tension. Namely, they presented an approach in which, similar to how Schwarzschild and contemporaries proceeded 100 years ago, one identifies a suitable coordinate system in which the spacetime metric, representing the gravitational field, is represented. In a quantum theory where general covariance is implemented fundamentally, this means one has to perform a “gauge-fixing”; the freedom of changing the coordinate system must be “fixed” consistently in the quantum theory. Gauge-fixings mean that one works with fewer variables, and has to worry less about different but physically equivalent solutions that are only related by changes in the coordinate system. Achieving them is often quite hard technically. Together with collaborators in Warsaw, Jedrzej and Norbert have made progress on this issue in recent years. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> <span style="font-family: "liberationserif"; font-size: 12.000000pt;">The second step, after a convenient coordinate system (think of spherical coordinates for treating the Schwarzschild black hole) has been chosen, is to do a “symmetry reduction” in the full quantum theory: rather than on the most general quantum universes, one now focusses on those that have a certain symmetry property. Norbert showed a detailed strategy for how to do this. One identifies an equation satisfied by all classical solutions with the desired symmetry, such as isotropy (i.e. looking the same in all directions). The quantum version of this equation is then imposed in loop quantum gravity, leading to a full quantum definition of symmetries like “isotropy” or “spherical symmetry” in loop quantum gravity. The obvious applications of the mechanism, which are being explored at the moment, are identifying cosmological and black hole solutions in loop quantum gravity, studying their dynamics, and verifying whether the resulting effects are in accord with what has been found in the simpler finite-dimensional quantum models described above. In particular, one would like to know whether singularities inside black holes and at the Big Bang, where Einstein's theory simply breaks down, can be resolved by quantum mechanics, as is hoped. </span><br /><span style="font-family: "liberationserif"; font-size: 12.000000pt;"><br /></span> </div></div></div><br /><div class="page" title="Page 3"><div class="layoutArea"><div class="column"><span style="font-family: "liberationserif"; font-size: 12.000000pt;">Jedrzej also showed how the methods developed in different “gauge-fixings” for classical general relativity could be used to resolve a disputed issue in the context of the AdS/CFT correspondence in string theory, where one faces a similar problem of fixing the huge freedom under changes in the coordinate system in order to identify the invariant physical properties of spacetime. In particular, a certain choice of gauge-fixing has been discussed in AdS/CFT, which leads to unfamiliar consequences such as non-locality in the gauge-fixed version of the theory. The tools developed by Jedrzej and collaborators could be used to clarify precisely how this non-locality occurs. They hence provide a somewhat unusual example of the application of methods developed for loop quantum gravity in a string theory-motivated context, clearly a positive example that can inspire more work on closer connections between methods used in these different communities. </span></div></div></div>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-83862033447230556392015-05-25T15:43:00.000-05:002015-05-29T15:12:10.508-05:00Separability and quantum mechanics<br /><a href="https://www.blogger.com/blogger.g?blogID=5826632960356694090" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><span style="background-color: white;">Tuesday, Apr 21st</span><br /><b>Fernando Barbero, CSIC, Madrid </b><br /><b>Title: Separability and quantum mechanics </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/barbero042115.pdf">PDF</a><span style="background-color: white;"> of the talk (758k)</span><br /><span style="background-color: white;"><a href="http://relativity.phys.lsu.edu/ilqgs/barbero042115.wav">Audio</a> [.wav 20MB]</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-NnrZKO65IAs/VWC9yP-725I/AAAAAAAAHlM/CZ6hGcWR-Wk/s1600/barbero.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-NnrZKO65IAs/VWC9yP-725I/AAAAAAAAHlM/CZ6hGcWR-Wk/s1600/barbero.jpg" /></a></div><br /><span style="background-color: white;">by Juan Margalef-Bentabol, UC3M-CSIC, Madrid</span> <br /><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> <br /><h2>Classical vs Quantum: Two views of the world</h2><a href="https://www.blogger.com/blogger.g?blogID=5826632960356694090" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a>In classical mechanics it is relatively straightforward to get information from a system. For instance, if we have a bunch of particles moving around, we can ask ourselves: where is its center of mass? What is the average speed of the particles? What is the distance between two of them? In order to ask and answer such questions in a precise mathematical way, we need to know all the positions and velocities of the system at every moment; in the usual jargon, we need to know the dynamics over the state space (also called <a href="https://en.wikipedia.org/wiki/Configuration_space">configuration space</a> for positions and velocities, or <a href="https://en.wikipedia.org/wiki/Phase_space#Conjugate_momenta">phase space</a> when we consider positions and momenta). For example, the appropriate way to ask for the center of mass, is given by the function that for a specific state of the system, gives the weighted mean of the positions of all the particles. Also, the total momentum of the system is given by the function consisting of the sum of the momenta of the individual particles. Such functions are called <strong>observables</strong> of the theory, therefore an observable is defined as a function that takes all the positions and momenta, and returns a real number. Among all the observables there are some ones that can be considered as <strong>fundamental</strong>. A familiar example is provided by the generalized position and momenta denoted as <script type="math/tex;">q^i</script> and <script type="math/tex;">p_i</script>.<br /><br />In a quantum setting answering, and even asking, such questions is however much trickier. It can be properly <a href="https://en.wikipedia.org/wiki/Matrix_mechanics">justified</a> that the needed classical ingredients have to be significantly changed:<br /><ol><li>The state space is now much more complicated, instead of positions and velocities/momenta we need a (usually infinite dimensional) complex vector space <script type="math/tex;">\mathcal{H}</script> with an inner product that is <a href="https://en.wikipedia.org/wiki/Complete_metric_space">complete</a>. Such vector space is called a <a href="https://en.wikipedia.org/wiki/Hilbert_space">Hilbert space</a> and the vectors of <script type="math/tex;">\mathcal{H}</script> are called states (up to a complex multiplication).</li><li>The observables are functions <script type="math/tex;">A</script> from <script type="math/tex;">\mathcal{H}</script> to itself that "behave well" with respect to the inner product (these are called <a href="https://en.wikipedia.org/wiki/Self-adjoint_operator">self-adjoint</a> operators). Notice in particular that the outputs of the quantum observables are complex vectors and not numbers anymore!</li><li>In a physical experiment we do obtain real numbers, so somehow we need to retrieve them from the observable <script type="math/tex;">A</script> associated with the experiment. The way to do this is by looking at the <strong>spectrum</strong> of <script type="math/tex;">A</script>, which consists of a set of real numbers called <a href="https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvalues</a> associated with some vectors called <a href="https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvectors</a> (actually the number that we obtain is a probability amplitude whose absolute value squared is the probability of obtaining as an output a specific eigenvector).</li></ol>The questions that arise naturally are: how do we choose the Hilbert space? how do we introduce <strong>fundamental</strong> observables analogous to the ones of classical mechanics? In order to answer these questions we need to take a small detour and talk a little bit about the algebra of observables.<br /><h2>Algebra of Observables</h2>Given two classical observables, we can construct another one by means of different methods. Some important ones are:<br /><ul><li> By adding them (they are real functions) <script type="math/tex;">h_1=f+g</script></li><li> By multiplying them <script type="math/tex;">h_2=f\cdot{}g</script></li><li> By a more sophisticated procedure called the <a href="https://en.wikipedia.org/wiki/Poisson_bracket">Poisson bracket</a> <script type="math/tex;">h_3=\{f,g\}</script></li></ul>The last one turns out to be fundamental in classical mechanics and plays an important role within the <a href="https://en.wikipedia.org/wiki/Hamiltonian_mechanics">Hamiltonian form</a> of the dynamics of the system. A basic fact is that the set of observables endowed with the Poisson bracket forms a <a href="https://en.wikipedia.org/wiki/Lie_algebra">Lie algebra</a> (a vector space with a rule to obtain an element out of two other ones satisfying some natural properties). The fundamental observables behave really well with respect to the Poisson bracket, namely they satisfy simple <a href="https://en.wikipedia.org/wiki/Canonical_commutation_relation">commutation relations</a> <script type="math/tex;">\{q^i,p_j\}=\delta^i_j</script> i.e. if we consider the <script type="math/tex;">i</script>-<script type="math/tex;">th</script> position observable and "Poisson-multiply" it by the <script type="math/tex;">j</script>-<script type="math/tex;">th</script> momentum observable, we obtain the constant function <script type="math/tex;">1</script> if <script type="math/tex;">i=j</script>, or the constant function <script type="math/tex;">0</script> if <script type="math/tex;">i\neq j</script>.<br /><br />One of the best approaches to construct a quantum theory associated with a classical one, is to reproduce at the quantum level some features of its classical formulation. One way to do this is to define a Lie algebra for the <strong>quantum</strong> observables such that some of such observables mimic the behavior of the Poisson bracket of some <strong>classical</strong> fundamental observables. This procedure (modulo some technicalities) is known as finding a <a href="https://en.wikipedia.org/wiki/Representation_theory#Unitary_representations">representation</a> of this algebra. In order to do this, one has to choose:<br /><ol><li>A Hilbert space <script type="math/tex;">\mathcal{H}</script>.</li><li>Some fundamental observables that reproduce the canonical commutation relations when we consider the <a href="https://en.wikipedia.org/wiki/Commutator#Ring_theory">commutator</a> of operators.</li></ol>In standard Quantum Mechanics the fundamental observables are positions and momenta. It may seem that there is a great ambiguity in this procedure, however there is a central theorem due to <a href="https://en.wikipedia.org/wiki/Stone-von_Neumann_theorem">Stone and von Neumann</a> that states that, under some reasonable hypothesis, all the representations are essentially the same.<br /><h2> Separability </h2>One of the hypotheses of the Stone-von Neumann theorem is that the Hilbert space <script type="math/tex;">\mathcal{H}</script> must be <strong>separable</strong>. This means that it is possible to find a <strong>countable</strong> set of orthonormal vectors in <script type="math/tex;">\mathcal{H}</script> (called <a href="https://en.wikipedia.org/wiki/Orthonormal_basis">Hilbert basis</a>) such that any state -vector- of <script type="math/tex;">\mathcal{H}</script> can be written as an appropriate countable sum of them. A separable Hilbert space, despite being infinite dimensional, is not "too big", in the sense that there are Hilbert spaces with uncountable bases that are genuinely larger. The separability assumption seems natural for standard quantum mechanics, but in the case of quantum field theory -with infinitely many degrees of freedom- one might expect to need much larger Hilbert spaces i.e. non separable ones. Somewhat surprisingly, most of the quantum field theories can be handled with our beloved and "simple" separable Hilbert spaces with the remarkable exception of LQG (and its derivative LQC) where non separability plays a significant role. Henceforth it seems interesting to understand what happens when one considers non separable Hilbert spaces [3] in the realm of the quantum world. A natural and obvious way to acquire the necessary intuition is by first considering quantum mechanics on a non-separable Hilbert space.<br /><h2> The Polymeric Harmonic Oscillator </h2>The authors of [2,3] discuss two inequivalent (among the infinitely many) representations of the algebra of fundamental observables which share a non familiar feature, namely, in one of them (called the position representation) the position observable is well defined but the momentum observable <strong>does not even exist</strong>; in the momentum representation the roles of positions and momenta are exchanged. Notice that in this setting, some familiar features of quantum mechanics are lost for good. For instance, the position-momentum Heisenberg uncertainty formula makes no sense at all as both position and momentum observables need to be defined.<br /><br />To improve the understanding of such systems and gain some insight for the application to LQG and LQC, the authors of [1] (re)study the <script type="math/tex;">1</script>-dimensional Harmonic Oscillator (PHO) in a non separable Hilbert space (known in this context as a polymeric Hilbert space). As the space is non separable, any Hilbert basis should be uncountable. This leads to some unexpected behaviors that can be used to obtain exotic representations of the algebra of fundamental observables.<br /><br />The motivation to study the PHO is kind of the same as always: the HO, in addition to being an excellent toy model, is a good approximation to any 1-dimensional mechanical system close to its equilibrium points. Furthermore, free quantum field theories can be thought of as ensembles of infinitely many independent HO's. There are however many ways to generalize the HO to a non separable Hilbert space and also many equivalent ways to realize a concrete representation, for instance by using Hilbert spaces based on:<br /><ul><li> the <a href="https://en.wikipedia.org/wiki/Bohr_compactification">Bohr compactification</a> of the real line.</li><li> the <a href="https://en.wikipedia.org/wiki/Almost_periodic_function#Besicovitch_almost_periodic_functions">Besicovitch almost periodic functions</a>.</li><li> constructions that generalize the usual separable Hilbert spaces based on sequences (<script type="math/tex;">\ell^2(\mathbb{R})</script> spaces).</li></ul>The eigenvalue equations in these different spaces take different forms: in some of them they are <a href="https://en.wikipedia.org/wiki/Recurrence_relation#Relationship_to_difference_equations_narrowly_defined">difference equations</a>, whereas in others they have the form of the standard Schrödinger equation with a periodic potential. It is important to notice nonetheless that writing Hamiltonian observables in this framework turn out to be really difficult, as only one of the position or momentum observables can be strictly represented. This means that for the other one it is necessary to rely on some kind of approximation (that can be obtained by introducing an arbitrary scale) and choosing a periodic potential with minima corresponding to the one of the quadratic operator. The huge uncertainty in this procedure has been highlighted by Corichi, Zapata, Vukašinac and collaborators. The standard choice leads to an equation known as the <a href="https://en.wikipedia.org/wiki/Mathieu_function#Mathieu_equation">Mathieu equation</a> but other simple choices have been explored, as the one shown in the figure.<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><figure><center></center><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-il2_uOj7reM/VWC_dQBWgJI/AAAAAAAAHlY/WpTwbNcNhTY/s1600/figurablog.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="288" src="http://4.bp.blogspot.com/-il2_uOj7reM/VWC_dQBWgJI/AAAAAAAAHlY/WpTwbNcNhTY/s320/figurablog.jpg" width="320" /></a></div><figcaption><strong>Energy eigenvalues (bands) of a polymerized harmonic oscillator.</strong> The horizontal axis shows the position (or the momentum depending on the chosen representation), the vertical axis is the energy and the red line represents the particular periodic extension of the potential used to approximate the usual quadratic potential of the HO. The other lines plotted in this graph correspond to auxiliary functions that can be used to locate the edges of the bands that define the point spectrum in the present example.</figcaption></figure> <br />As we have already mentioned, the orthonormal bases in non separable Hilbert spaces are uncountable. A consequence of this is the fact that the orthonormal basis provided by the eigenstates of the Hamiltonian must be uncountable, i.e. the Hamiltonian must have an uncountable infinity worth of eigenvalues (counted with multiplicity). A somewhat unexpected result that can be proved by invoking classical theorems on functional analysis in non-separable Hilbert spaces is the fact that these eigenvalues are gathered in bands. It is important to point out here that only the lowest-lying part of the spectrum is expected to mimic reasonably well the one corresponding to the standard HO, however it is important to keep also in mind the huge difference that persists: even the narrowest bands contain a continuum of eigenvalues.<br /><h2> Some physical consequences </h2>The fact that the spectrum of the polymerized harmonic oscillator displays this band structure is relevant for some applications of polymerized quantum mechanics. Two main issues were mentioned in the talk. On one hand the statistical mechanics of polymerized systems must be handled with due care. Owing to the features of the spectrum, the counting of energy eigenstates necessary to compute the entropy in the microcanonical ensemble is ill defined. A similar problem crops up when computing the partition function of the canonical ensemble. These problems can probably be circumvented by using an appropriate regularization and also by relying on some superselection rules that eliminate all but a countable subset of energy eigenstates of the system.<br /><br />A setting where something similar can be done is in the polymer quantization of the scalar field (already considered by Husain, Pawłowski and collaborators). As this system can be thought of as an infinite ensemble of harmonic oscillators, the specific features of their (polymer) quantization will play a significant role. A way to avoid some difficulties here also relies on the elimination of unwanted energy eigenvalues by imposing superselection rules as long as they can be physically justified.<br /><h2>Bibliography</h2>[1] J.F. Barbero G., J. Prieto and E.J.S. Villaseñor, <i>Band structure in the polymer quantization of the harmonic oscillator</i>, Class. Quantum Grav. <strong>30</strong> (2013) 165011.<br />[2] W. Chojnacki, <i>Spectral analysis of Schrodinger operators in non-separable Hilbert spaces</i>, Rend. Circ. Mat. Palermo (2), <strong>Suppl. 17</strong> (1987) 13551.<br />[3] H. Halvorson, <i>Complementarity of representations in quantum mechanics</i>, Stud. Hist. Phil. Mod. Phys. <strong>35</strong> (2004) 45-56.Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-12674970544961376372015-05-05T12:19:00.000-05:002015-05-05T12:19:01.547-05:00Cosmology with group field theory condensates<span style="background-color: white;">Tuesday, Feb 24th</span><br /><b>Steffen Gielen, Imperial College </b><br /><b>Title: Cosmology with group field theory condensates </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/gielen022415.pdf">PDF</a><span style="background-color: white;"> of the talk (136K)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/gielen022415.wav">Audio</a><span style="background-color: white;"> [.wav 39MB]</span><br /><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-38P0Kz1NgRA/VTuqFHfCzyI/AAAAAAAAHbg/Ll1nIzrzcSw/s1600/re.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-38P0Kz1NgRA/VTuqFHfCzyI/AAAAAAAAHbg/Ll1nIzrzcSw/s1600/re.jpg" /></a></div><span style="background-color: white;">by Mercedes Martín-Benito, Rabdoud University</span><br /><span style="background-color: white;"><br /></span><br /><div class="p1">One of the most important open questions in physics is how gravity (or in other words, the geometry of spacetime) behaves when the energy densities are huge, of the order of the Planck density. Our most reliable theory of gravity, general relativity, fails to describe the gravitational phenomena in high energy density regimes, as it generically leads to singularities. These regimes are achieved for example at the origin of the universe or in the interior of black holes, and therefore we do not have yet a consistent explanation for these phenomena. We expect quantum gravity effects to be important in such situations, but general relativity, being a theory that treats the geometry of the spacetime as classical, do not take those quantum gravity effects into account. Thus, in order to describe black holes or the very early universe in a physically meaningful way it seems unavoidable to quantize gravity.</div><div class="p2"><br /></div><div class="p1">The quantization of gravity not only requires attaining a mathematically well-described theory with predictive power, but also the comparison of the predictions with observations to check that they agree. The regimes where quantum gravity plays a fundamental role, such as black holes or the early universe, might seem very far from our observational or experimental reach. Nevertheless, thanks to the big progress that precision cosmology has undergone in the last decades, in the near future we may be able to get observational data about the very initial instants of the universe that could be sensitive to quantum gravity effects. We need to get prepared for that, putting our quantum gravity theories at work in order to extract cosmological predictions from them.</div><div class="p2"><br /></div><div class="p1">This is the main goal of Steffen's analysis. He bases his research in the approach to quantum gravity known as Group Field Theory (GFT). GFT defines a path integral for gravity, namely, it replaces the classical notion of unique solution for the geometry of the spacetime with a sum over an infinity of possibilities to compute a quantum amplitude. The formalism that it uses is pretty much like the usual quantum field theory formalism employed in particle physics. There, given a process involving particles, the different possible interactions contributing to that process are described by so-called Feynman diagrams, that are later summed up in a consistent way to finally lead to the transition amplitude of the process that we are trying to describe. GFT follows that strategy. The corresponding Feynman diagrams are spinfoams, and represent the different dynamical processes that contribute to a particular spacetime configuration. GFT is thus linked to Loop Quantum Gravity (GFT), since spinfoams are one main proposal for defining the dynamics of LQG. The GFT Feynman expansion extends and completes this definition of the LQG dynamics by trying to determine how these diagrams must be summed up in a controlled way to obtain the corresponding quantum amplitude. </div><div class="p2"><br /></div><div class="p1">GFT is a fundamentally discrete theory, with a large number of microscopical degrees of freedom. These degrees of freedom might organize themselves, following somehow a collective behavior, to lead to different phases of the theory. The hope is to find a phase that in the continuum limit agrees with having a smooth spacetime as described by the classical theory of general relativity. In this way, we would make the link between the underlying quantum theory and the classical one that explains very well the gravitational phenomena in regimes where quantum gravity effects are negligible. To understand this, let us make the analogy with a more familiar theory: Hydrodynamics. </div><div class="p2"><br /></div><div class="p1">We know that the fundamental microscopical constituents of a fluid are molecules. The dynamics of this micro-constituents is intrinsically quantum, however these degrees of freedom display a collective behavior that leads to macroscopic properties of the fluid, such as its density, its velocity, etc. In order to study these properties it is enough to apply the classical theory of hydrodynamics. However we know that it is not the fundamental theory describing the fluid, but an effective description coming from an underlying quantum theory (condense matter theory) that explains how the atoms form the molecules, and how these interact among themselves giving rise to the fluid. </div><div class="p2"><br /></div><div class="p2">The continuum spacetime that we are used to might emerge, in a similar way to the example of the fluid, from the collective behavior of many many quantum building blocks, or atoms of spacetime. This is, in plane words, the point of view employed in the GFT approach to quantum gravity.</div><div class="p2"><br /></div><div class="p1">While GFT is still under construction, it is mature enough to try to extract physics from it. With this aim, Steffen and his collaborators, are working in obtaining effective dynamics for cosmology starting from the general framework of GFT. The simplest solutions of Einstein equations are those with spatial homogeneity. These turn out to describe cosmological solutions, which approximate rather well at large scales the dynamics of our universe. Then, in order to get effective cosmological equations from their GFT, they postulate very particular quantum states that, involving all the degrees of freedom of the GFT, are states with collective properties that can give rise to a homogeneous and continuum effective description. The similarities between GFT and condense matter physics allows Steffen and collaborators to exploit the techniques developed in condense matter. In particular, based on the experience on Bose-Einstein condensates, the states that they postulate can be seen as condensates. </div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><div class="p2"><br /></div><br /><div class="p1">The collective behavior that the degrees of freedom display leads, in fact, to a homogeneous description in the macroscopic limit. The effective equations that they obtain agree in the classical limit with cosmological equations, but remarkably retaining the main effects coming from the underlying quantum theory. More specifically, these effective equations know about the fundamental discreteness, as they explicitly get corrections (non-present in the standard classical equations) that depend on the number of quanta (spacetime “atoms”) in the condensate. These results form the basis of a general programme for extracting effective cosmological dynamics directly from a microscopic non-perturbative theory of quantum gravity. </div>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-14067246795120705562014-11-24T13:16:00.000-06:002014-11-24T13:16:27.198-06:00Quantum theory from information inference principles<div class="" style="font-size: 18px; margin: 0px;"><span style="background-color: white; font-size: small;">Tuesday, Nov 11th</span><br /><b style="font-size: medium;">Philipp Hoehn, Perimeter Institute </b><br /><b style="font-size: medium;">Title: Quantum theory from information inference principles </b><span style="background-color: white; font-size: small;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/hoehn111114.pdf" style="font-size: medium;">PDF</a><span style="background-color: white; font-size: small;"> of the talk (800k)</span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/hoehn111114.wav" style="font-size: medium;">Audio</a><span style="background-color: white; font-size: small;"> [.wav 40MB]</span></div><div class="" style="font-size: 18px; margin: 0px;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Zvaow2vNRzE/VHNb_tIGxfI/AAAAAAAAHIs/ZMEsYHatPiQ/s1600/hoehn_philipp.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-Zvaow2vNRzE/VHNb_tIGxfI/AAAAAAAAHIs/ZMEsYHatPiQ/s1600/hoehn_philipp.jpg" /></a></div><div class="" style="font-size: 18px; margin: 0px;"><br /></div><div class="" style="font-size: 18px; margin: 0px;">by Matteo Smerlak, Perimeter Institute</div><div class="" style="font-size: 18px; margin: 0px;"><br /></div><div class="" style="font-size: 18px; margin: 0px;"><br /></div><div class="" style="font-size: 18px; margin: 0px;"><br /></div><div class="" style="font-size: 18px; margin: 0px;">When a new theory enters the scene of physics, a succession of events normally takes place: at first, nobody cares; then a minority starts playing with the maths while the majority insists that the theory is obviously wrong; farther down the road, we find the majority using the maths on a daily basis and all arguing that the theory is so beautiful, it can only be right; along the way, thanks to many years of practice, a new kind of intuition grows out of the formalism, and our entire picture of reality changes accordingly. This is the process of science.</div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="font-size: 18px; margin: 0px;">For some reason, though, the eventual shift from formalism to intuition never happened for quantum mechanics (QM). Ninety years after its discovery, specialists still call QM “weird”, teachers still quote Feynman claiming that “nobody really understands QM”, and philosophers still discuss whether QM requires us to be “antirealist”, “neo-Kantian”, “Bayesian”… you name it. Niels Bohr wanted new theories to be “crazy enough”, but it seems this one is just <i class="">too </i>crazy. And yet it works!</div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="font-size: 18px; margin: 0px;">In the face of this puzzle, a school of thought initiated by Birkhoff and von Neumann in the thirties has declared it its mission to <i class="">reconstruct </i>QM. The idea is simple: if you don’t get how the machine works, then roll up your sleeves, take the machine apart, and build it again—from scratch. Indeed this is how Einstein delt with the symmetry group of Maxwell’s equations (and its mysterious action on lengths and durations): he found intuitive two physical principles—the relativity principles—and <i class="">derived </i>the Lorentz group (the set of symmetries of Maxwell's equations) from them. Thus special relativity was “really understood”.<br /><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-YXQ6OR_0Ihc/VHNap9NnaPI/AAAAAAAAHH8/h7LCoo73diU/s1600/Pasted%2BGraphic.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-YXQ6OR_0Ihc/VHNap9NnaPI/AAAAAAAAHH8/h7LCoo73diU/s1600/Pasted%2BGraphic.tiff" height="163" width="320" /></a></div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="margin: 0px;"><br /></div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="font-size: 18px; margin: 0px;">Much recent work towards a reconstruction of QM has taken place within a framework called “generalized probability theories” (GPT). This approach elaborates on basic notions such as <i class="">preparations, transformations</i> and <i class="">measurements</i>. The main achievement of GPT has been to locate QM within a more general landscape of possible modifications of classical probability theory. It has showed for instance that QM is not the most non-local theory consistent with what is known as no-signaling property: stronger correlations than quantum entanglement are in principle possible, though they are not realized in nature. To understand what <i class="">is</i>, we must know what else<i class=""> could have been</i>—thus speak GPT proponents. </div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="font-size: 18px; margin: 0px;">Philipp uses a different language for his reconstruction of QM: instead of <i class="">measurements </i>and <i class="">states, </i>he talks about <i class="">questions </i>and <i class="">answers. </i>The semantic shift is not innocent: while a “measurement” uncovers the intrinsic state of a system, a “question” only brings information to whoever asks it—that is, a question relates to <i class="">two </i>entities (the system <i class="">and</i> the observer/interrogator) rather than just one (the system). Because there isn’t anybody out there to ask questions about <i class="">everything</i>, there is no such thing as the “state of the universe”, Philipp says!<br /><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-MAppSI7xsFQ/VHNaxYCea4I/AAAAAAAAHIE/mC8IFn0LPCU/s1600/world.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-MAppSI7xsFQ/VHNaxYCea4I/AAAAAAAAHIE/mC8IFn0LPCU/s1600/world.tiff" height="208" width="320" /></a></div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br /></div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="font-size: 18px; margin: 0px;">This so-called “relational” questions/answers approach to QM was advocated twenty years ago by Rovelli, who emphasized its similarity with the structure of gravitation (time is relative, remember?). He also proposed two basic informational principles: one states that the total information that an observer O can gather about a system S is limited; the second specifies that, even when O has obtained the maximum amount of information about S, she can still learn something about S by asking other, “complementary” questions. Thence non commuting operators! Similar ideas where discussed independently by Zeilinger and Brukner—and Philipp embraces them wholeheartedly.</div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div><div class="" style="font-size: 18px; margin: 0px;">But he also takes a big step further. Adding four more postulates to Rovelli’s (which he calls completeness, preservation, time evolution and locality), Philipp shows how to reconstruct the set Σ of all possible states of S relative to O (together with its isometry group, representing possible time evolutions). For a quantum system allowing only one independent question—a qubit—Σ is a three-dimensional ball, the Bloch sphere. (Note that a 3-ball is a much bigger space than a 1-ball, the state space of a classical bit—enter quantum computing…) For systems with more independent questions, i.e. N qubits, Σ is the mathematical structure known as the convex cone over some complex projective space—not quite what is known as a Calabi-Yau manifold, but still a challenge for the mind to picture.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-oiNjs4H60WQ/VHNbWwfe6TI/AAAAAAAAHIc/A7CpjMYR0TE/s1600/Screen%2BShot%2B2014-11-24%2Bat%2B10.22.29%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-oiNjs4H60WQ/VHNbWwfe6TI/AAAAAAAAHIc/A7CpjMYR0TE/s1600/Screen%2BShot%2B2014-11-24%2Bat%2B10.22.29%2BAM.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;">N=2 turns out to be the most difficult case: once this one is solved—Philipp says this took him a full year, with inputs from his collaborator Chris Wever—, higher N’s follow rather straightforwardly. This is a reflection of a crucial aspect of QM: quantum systems are “monogamous”, meaning that they can establish strong correlations (aka “entanglement”) with just one partner at a time. Philipp’s questions/answers formulation provides a new and detailed understanding of this peculiar correlation structure, which he represents as a spherical tiling. “QM is beautiful!”, says Philipp.<br /><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-EAE5kTcSdxw/VHNbcmlaA3I/AAAAAAAAHIk/i3eurB82TZw/s1600/pentagons.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-EAE5kTcSdxw/VHNbcmlaA3I/AAAAAAAAHIk/i3eurB82TZw/s1600/pentagons.tiff" height="194" width="320" /></a></div><div class="" style="font-size: 18px; margin: 0px;">One limitation of Philipp’s current approach—also pointed out by the audience—is the restriction to binary (or yes/no) questions. A spin-1 particle, for instance, falls outside this framework, for it can give <i class="">three</i> different answers to the question “what is your spin in the z direction?”, namely “up”, “down” or “zero”. Can Philipp deal with such ternary question, and reconstruct the 8 dimensional state space of a quantum “trit”? We wish him to find the answer within… less than a year! </div><div class="" style="font-size: 18px; margin: 0px; min-height: 22px;"><br class="" /></div>Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0tag:blogger.com,1999:blog-5826632960356694090.post-52116688555892828342014-04-28T12:52:00.000-05:002014-04-28T12:52:39.873-05:00Holographic special relativity: observer dependent geometry<h3><b style="font-size: medium;">Derek Wise, FAU Erlangen</b><br style="font-size: medium; font-weight: normal;" /><b style="font-size: medium;">Title: Holographic special relativity: observer space from conformal geometry </b><br style="font-size: medium; font-weight: normal;" /><a href="http://relativity.phys.lsu.edu/ilqgs/wise101513.pdf" style="font-size: medium; font-weight: normal;">PDF</a><span style="background-color: white; font-size: small; font-weight: normal;"> of the talk (600k) </span><a href="http://relativity.phys.lsu.edu/ilqgs/wise101513.wav" style="font-size: medium; font-weight: normal;">Audio</a><span style="background-color: white; font-size: small; font-weight: normal;"> [.wav 38MB] </span><a href="http://relativity.phys.lsu.edu/ilqgs/wise101513.aif" style="font-size: medium; font-weight: normal;">Audio</a><span style="background-color: white; font-size: small; font-weight: normal;"> [.aif 4MB] </span></h3><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-LJdSR1gXqQk/UH8mtAKDvtI/AAAAAAAADmA/Qdbh6bqWnoo/s1600/wise.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-LJdSR1gXqQk/UH8mtAKDvtI/AAAAAAAADmA/Qdbh6bqWnoo/s1600/wise.jpg" height="320" width="185" /></a></div><h3>by Sean Gryb, Radboud University</h3><h3><br /></h3><h3>Introduction</h3><br />In Roman mythology, <a href="http://en.wikipedia.org/wiki/Janus">Janus</a> was the god of gateways, transitions, and time, whose two distinct faces are depicted peering in opposite directions, as if bridging two different regions (or epochs) of the Universe. The term “Janus-faced” has come to mean a person or thing that simultaneously embodies two polarized features, and the Janus head has come to represent the embodiment of these two distinct features into one.<br /><br />In this talk (based off the paper <a href="http://arxiv.org/abs/1305.3258" quot="" relativity="" special="" target=""_blank"" title=""Holographic">[1]</a>), <a href="http://www.math.ucdavis.edu/~derek/index.html" quot="" s="" target=""_blank"" title=""Derek" webpage="" wise="">Derek Wise</a> explores the possibility that spacetime itself might be Janus-faced. He explores an intriguing relationship between the structure of expanding spacetime and the scale-invariant description of a sphere. What he finds is a mathematical relationship providing a bridge between these two Janus faces that distinctly represent events in the Universe. This bridge is remarkably similar to the picture of reality proposed by the <a href="http://en.wikipedia.org/wiki/Holographic_principle" principle:="" quot="" target=""_blank"" title=""Holographic" wikipedia=""><em>holographic principle</em></a> and, in particular, the <a correspondence:="" href="http://en.wikipedia.org/wiki/AdS/CFT_correspondence" quot="" target=""_blank"" title=""AdS/CFT" wikipedia=""><em>AdS/CFT correspondence</em></a> where, on one side, there is the usual spacetime description of events and, on the other, there is a way to imprint these events onto the 3-dimensional boundary of this spacetime. <br /><br />Aside from providing an alternative to spacetime, Derek's picture may even help illuminate the deeper structures behind a recent formulation of g<a href="http://en.wikipedia.org/wiki/General_relativity" quot="" relativity:="" target=""_blank"" title=""General" wikipedia="">eneral relativity (GR)</a> called <a dynamics:="" href="https://www.blogger.com/2http://en.wikipedia.org/wiki/Shape_dynamics" quot="" target=""_blank"" title=""Shape" wikipedia=""><em>Shape Dynamics</em></a>, which I will come to at the end of this post. But to begin, I will try to explain Derek's result by first giving a description of the spacetime aspect of the Janus face and then describe how a link can be established to a completely distinct face, which, as we will see, is a description of events in the Universe that is completely free of any notion of scale. The key points of the discussion are summarized beautifully in the depiction of Janus given below, by <a href="http://www.bumblenut.com/" quot="" target=""_blank"" title=""Marc's" webpage="">Marc Ngui</a>, who has provided all the images for this post. The diagram shows how, as I will describe later, events seen by observers in spacetime can be described by information on the boundary. I encourage the reader to revisit this image as its main elements are progressively explained throughout the text. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-09paOzh0T3w/Uz22kMnKtfI/AAAAAAAAE0k/ojWJCpv6NYY/s1600/janus_final.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-09paOzh0T3w/Uz22kMnKtfI/AAAAAAAAE0k/ojWJCpv6NYY/s1600/janus_final.jpg" height="320" width="266" /></a></div><h3>Relativity, Observers, and Spacetime</h3>In 1908, <a href="http://en.wikipedia.org/wiki/Hermann_Minkowski" minkowski:="" quot="" target=""_blank"" title=""Hermann" wikipedia="">Hermann Minkowski</a> made a great discovery: Einstein's new theory of <a href="http://en.wikipedia.org/wiki/Special_relativity" quot="" relativity:="" target=""_blank"" title=""Special" wikipedia="">Special Relativity</a> could be cast into a beautiful framework, one that Minkowski recognized as a kind of union of space and time. In his own words: “space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”<a 1952.="" 75="" a="" and="" ch.="" collection="" dover="" general="" he="" href="http://books.google.nl/books/about/The_Principle_of_Relativity.html?id=S1dmLWLhdqAC&redir_esc=y" memoirs="" minkowski="" new="" of="" on="" original="" pp.="" principle="" quot="" relativity:="" relativity="" space="" special="" target=""_blank"" the="" theory="" time="" title=""H." york:="">[2]</a> To understand what Minkowski meant, let's go back to 1904-1905 in order to retrace the discoveries that spawned Minkowski's revolution. <br /><br />Relativity concerns the way in which different observers organize information about ‘when’ and ‘where’ events take place. Einstein realized that this system of organization should have two properties: i) it should work the same way for each observer, and ii) it should involve a set of rules that allows different observers to consistently compare information about the same events. This means that different observers don't necessary need to agree on <em>when</em> and <em>where</em> a particular event took place, but they do need to agree on how to compare the information gathered by different observers. Einstein expressed this requirement in his <em>principle of relativity</em>, to which he gave primary importance within physical theories. The key point is that relativity is fundamentally a statement about <em>observers</em> and how they collect and compare information about events. Minkowski's conception of spacetime comes afterwards, and it comes about through the specific mathematical properties of the rules used to collect and compare the relevant information. <br /><br />To try to understand how spacetime works, we will use a slightly more modern version of spacetime than the one used by Minkowski — one with all the same essential properties as the original, but which can accommodate the observed accelerated expansion of space. This kind of spacetime was first studied by Willem de Sitter, and is named <a href="hhttp://en.wikipedia.org/wiki/De_Sitter_space" quot="" sitter="" spacetime:="" target=""_blank"" title=""de" wikipedia="">de Sitter (dS) spacetime</a> after him. It has the basic shape depicted by the blue grid in the Janus image above. Because this space is curved, it is most convenient to describe it by putting it into a larger dimensional flat space (just like the 2D surface of a sphere depicted in a 3D space). This means that we can label events in this spacetime by 5 numbers: 4 space components, labeled <em>(x, y, z, w)</em> and one time component, <em>t</em>, that obey the relation<br /><br /><center><em>x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> - t<sup>2</sup> = ℓ<sup>2</sup></em>. (1)</center><br />This restriction (which serves as the definition of this spacetime) means that the 4 space components are not all independent. Indeed, the single constraint above removes one independent component, leaving the 3 space directions we know and love. The parameter <em>ℓ</em> is related to the cosmological constant and dictates how fast space is expanding. Adjusting its value changes the shape of the spacetime as illustrated in the figure below. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-f092bTS3TNo/Uz229kr0PjI/AAAAAAAAE0s/M-7h5UcEvTA/s1600/dS_limits_final.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-f092bTS3TNo/Uz229kr0PjI/AAAAAAAAE0s/M-7h5UcEvTA/s1600/dS_limits_final.jpg" height="192" width="320" /></a></div><br />The middle spacetime in blue depicts a typical dS spacetime. Increasing the parameter ℓ decreases the rate of expansion so that, if <em>ℓ → ∞</em>, the spacetime barely expands at all and looks more like the purple cylinder on the right. The opposite extreme, when <em>ℓ → 0</em>, is the yellow <em>light cone</em>, which is named that way because the space is expanding at its maximum rate: the speed of light. This extreme limit will be very important for our considerations later. <br /><br />Although this model of spacetime is dramatically simplified, it remarkably describes, to a good approximation, two important phases of our real Universe: i) the period of exponential expansion (or <a href="http://en.wikipedia.org/wiki/Inflation_(cosmology)" quot="" target=""_blank"" title=""Inflation:" wikipedia=""><em>inflation</em></a>), which we believe took place in the early history of our Universe, and ii) the present and foreseeable future. Different observers compare the labels they attribute to each event by performing transformations that leave the form of (1), and thus the shape of dS spacetime, unchanged. Because of this property, these transformations constitute <em>symmetries</em> of dS spacetime. Since the transformations that real observers must use to compare information about real events just happen to correspond to spacetime symmetries, it is no wonder that the notion of spacetime has had such a profound influence on physicists’ view of reality. However, we will shortly see that these rules can be recast into a completely different form, which tells a different story of what is happening. <br /><!-- SPACETIME'S JANUS FACE --> <br /><h3>Spacetime's Janus Face</h3>We will now see how the symmetries of observers in dS spacetime can be rewritten in terms of symmetries that preserve angles, but not necessary distances, in space. In particular, all information about scale is removed. In mathematics, these are called <a href="http://en.wikipedia.org/wiki/Conformal_symmetry" quot="" symmetry:="" target=""_blank"" title=""Conformal" wikipedia=""><em>conformal symmetries</em></a>. This means that different observers have a choice when analyzing information that they collect about events: either they can imagine that these events have taken place in dS spacetime, and are consequently related by the dS symmetries; or they can imagine that these events are representing information that can be expressed in terms of angles (and not lengths), and are consequently related by conformal symmetries. <br /><br />To understand how this can be so, consider the very distant future and the very distant past: where dS spacetime and the light cone nearly meet. This extreme region is called the <em>conformal sphere</em> because it is a sphere and also because it is where the dS symmetries correspond to conformal symmetries. <br /><br />In fact, any cross-section of the light cone formed by cutting it with a spatial plane (as illustrated in the diagram below) is a different representative of the conformal sphere since these different cross-sections will disagree on distances but will agree on angles. Although the intersection looks like a circle (represented in dark green), it is actually a 3-dimensional sphere because we have cut out 2 of the spatial dimensions (which we can't draw on a 2 dimensional page). <br /><br />To see how events on this 3d sphere can be represented in a scale-invariant way on a 3d plane, we can use a handy technique called a <a href="http://en.wikipedia.org/wiki/Stereographic_projection" projection:="" quot="" target=""_blank"" title=""Stereographic" wikipedia="">stereographic projection</a>. The stereographic projection is often used for map drawing where the round earth has to be drawn onto a flat map. One of its key properties, namely that it preserves angles, means that maps drawn in this way are useful for navigating since an angle on the map corresponds to the same angle on the Earth. It is precisely this property that will make the stereographic projection useful for us here. <br /><br />To perform a stereographic projection, imagine picking a point on a sphere, which we can interpret as the location of a particular observer on the sphere (represented by an eye in the diagram below), and call this the South Pole. Now imagine putting a light on the North Pole and letting it shine through the space that the sphere has been drawn in. Suppose our sphere is filled with points. Then, the shadow of these points will form an image on the plane tangentially to the sphere on the South Pole. The picture below illustrates what is going on. Points on the sphere are represented by stars and the yellow rays indicate how their image is formed on the plane. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-LGWn9Y21ysQ/Uz23QnJn6zI/AAAAAAAAE00/Gjr04rppXKM/s1600/stereo_project_final.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-LGWn9Y21ysQ/Uz23QnJn6zI/AAAAAAAAE00/Gjr04rppXKM/s1600/stereo_project_final.jpg" height="300" width="320" /></a></div><br />It is now a relatively straightforward mathematical exercise to show that the symmetries of the light cone represent transformations on the plane that may change the size of the image, but will preserve the angles between the points. Thus, the symmetries of the cone can be understood in terms of the conformal symmetries of this plane. <br /><br />If we now move our cross-section ever further into the future or the past, then the dS spacetime begins to resemble more and more the light cone. Thus, if we can represent arbitrary events in dS spacetime by information imprinted on two cross-sections in the infinite future and infinite past, then these events can be represented in terms of the images they induce onto our projected planes, and we have obtained our objective. <br /><br />There is a simple way that this can be done. Imagine taking, as shown in the figure below, an arbitrary event in dS spacetime and drawing all the events in the distant past that could affect things that happen at this point (this region is a finite portion of the spherical cross-sections because no disturbance can travel faster than the speed of light). The result is a 2 dimensional spherical region, called the <em>particle horizon</em> indicated by the red regions in the diagram below, which grows steadily over time. You can think of this region as the proportion of dS spacetime that is visible at any particular place. In fact, you can use the relative size of this region as an indication of the time at which that event occurs. Because this is a notion of time that exists solely in terms of quantities defined in the distant past, it will transform under conformal symmetries. To give an idea of what this looks like, the motion of an observer from some point in the distant past to a new point in the distant future is represented by a series of concentric spheres, starting at the initial point and then spreading out to eventually cover the whole sphere. The diagram below shows how this works. The different regions <em>(a,b,c,d)</em> represent progressively growing regions corresponding to progressively later times. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-opqBQCLupWM/Uz23dxNfW1I/AAAAAAAAE08/claQbsA0rMY/s1600/past_horizon_final.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-opqBQCLupWM/Uz23dxNfW1I/AAAAAAAAE08/claQbsA0rMY/s1600/past_horizon_final.jpg" height="230" width="320" /></a></div><br />In this way, you can map information about events in dS spacetime to information on the conformal sphere. In other words, the picture of reality that one gets from Einstein's theory of Special Relativity is a story that can be told in two very different ways. In the first way, there are events which trace out histories in spacetime. ‘Where’ and ‘when’ a particular event takes place depends on who you are, and the information about these events can be transformed from one observer to another via the global symmetries of spacetime. In the new picture, it is the information about angles that is important. ‘Where’ and ‘how big’ things are depends on your point of view and the information about particular events can be transformed from one observer to another using conformal transformations. <br /><!-- From Special to General Relativity --> <br /><h3>From Special to General Relativity</h3>We have just described how to relate two very different views of how observers can collect information about the world. Until now, we have only been considering <a href="http://en.wikipedia.org/wiki/Homogeneous_space" quot="" space:="" target=""_blank"" title=""Homogeneous" wikipedia=""><em>homogeneous spaces</em></a>: i.e., those that look the same everywhere. The class of observers we were able to consider was resultantly restrictive. It was Einstein's great insight to recognize that the same mathematical machinery needed to describe events seen by <em>arbitrary</em> observers could also be used to study the properties of gravity. The machinery in question is a generalization of Minkowski's geometry, named after Riemann. <br /><br />In order to describe <a geometry:="" href="http://www.blogger.com/2http://en.wikipedia.org/wiki/Riemannian_geometry" quot="" target=""_blank"" title=""Riemannian" wikipedia="">Riemannian geometry</a>, it is easiest to first describe a generalization of it (which we will need later anyway), and then show how Riemannian geometry is just a special case. The generalization in question is called <a geometry:="" href="http://en.wikipedia.org/wiki/Cartan_geometry" quot="" target=""_blank"" title=""Cartan" wikipedia=""><em>Cartan geometry</em></a>, after the great mathematician <a cartan:="" href="http://en.wikipedia.org/wiki/%C3%89lie_Cartan" quot="" target=""_blank"" title=""Élie" wikipedia="">Élie Cartan</a>. Cartan had the idea of building general curved geometries by modelling them off homogeneous spaces. The more general spaces are constructed by moving these homogeneous spaces around in specific ways. The geometry itself is defined by the set of rules one needs to use to compare vectors after moving the homogeneous spaces. These rules split into two different kinds: those that change the point of contact between the homogeneous space and the general curved space and those that don't. These different moves are illustrated for the case where the homogeneous space is a 2D sphere in the diagram below. <br /> The moves that don't change the point of contact (in the case above, this corresponds to spinning about the point of contact without rolling) constitute the local symmetries of the geometry and could, for example, correspond to what different local observers would see (in this case, spinning observers versus stationary ones) when looking at objects in the geometry. Einstein exploited this kind of structure to implement his general principle of relativity described earlier. The moves that change the point of contact (in the case above, this means rolling the ball around without slipping) give you information about the curved geometry of the general space. Einstein used a special case of Cartan geometry, which is just Riemannian geometry, where the homogenous space is Minkowski space. He then exploited the analogue of the structure just described to explain an old phenomenon in a completely new way: gravity. In the process, he produced one of our most radical yet successful theories of physics: General Relativity. The figure below shows how the different kinds of geometry we've discussed are related. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-8yVTcUA0Qf0/Uz23yX3IopI/AAAAAAAAE1M/Q3S5dr71Fow/s1600/marc_geo_diag_final.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-8yVTcUA0Qf0/Uz23yX3IopI/AAAAAAAAE1M/Q3S5dr71Fow/s1600/marc_geo_diag_final.jpg" height="264" width="320" /></a></div><br />Now, consider what happens when we substitute, as we did in the last section, Minkowski's flat spacetime for de Sitter's curved, but still homogenous, spacetime. We can still describe gravity, but in a way that naturally includes a cosmological constant. However, the conformal sphere is also a homogeneous space. Moreover, as we described earlier, the symmetries of this homogeneous space can be related to the dS symmetries. This suggests that it might be possible to describe gravity in terms of a Cartan geometry modelled off the conformal sphere. <br /><h3><br /></h3><h3>From the Conformal Sphere to Shape Dynamics?</h3>Cartan geometries modelled on the conformal sphere are called <em>conformal geometries</em> because the local symmetries of these geometries preserve angles, and not scale. Although we have laid out a procedure relating the model space of conformal geometries to the model space of spacetimes with a cosmological constant, it is quite another thing to rewrite gravity in terms of conformal geometry. This is, in part, because the laws governing spacetime geometry are complicated and, in part, because our prescription for relating the model spaces is also not straightforward, since it relates local quantities in spacetime to non-local quantities in the infinite future and past. Nevertheless, this exciting possibility provides an interesting future line of research. Furthermore, there are other hints that such a description might be possible. <br /><br />Using very different methods, it is possible to show that General Relativity is actually <em>dual</em> to a theory of evolving conformal geometry <a 3d="" a="" as="" conformally="" gravity="" href="http://iopscience.iop.org/0264-9381/28/4/045005/" invariant="" quot="" target=""_blank"" theory="" title=""Einstein">[3]</a>. However, the kind of conformal geometry used in this derivation has not yet been written in terms of Cartan geometry (which makes use of slightly different structures). This new way of describing gravity, called <a dynamics:="" href="http://en.wikipedia.org/wiki/Shape_dynamics" quot="" target=""_blank"" title=""Shape" wikipedia=""><em>Shape Dynamics</em></a>, is perhaps making use of the interesting relationship between spacetime symmetries and conformal symmetries described here. Understanding exactly the nature of the conformal geometry in Shape Dynamics and its relation to spacetime could prove valuable in being able to understand this new way of describing gravity. Perhaps it could even be a window into understanding how the quantum theory of gravity should work? <br /><ul><li>[1] D. K. Wise, <em>Holographic Special Relativity, </em><a href="http://arxiv.org/abs/1305.3258" target=""_blank"">arXiv:1305.3258 [hep-th]</a>.</li><li>[2] H. Minkowski, <em>The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity</em>, ch. Space and Time, pp. 75–91. New York: Dover, 1952.</li><li>[3] H. Gomes, S. Gryb, and T. Koslowski, <em>Einstein gravity as a 3D conformally invariant theory</em>, <a href="http://iopscience.iop.org/0264-9381/28/4/045005/" target=""_blank"">Class. Quant. Grav. 28 (2011) 045005</a>, <a href="http://arxiv.org/abs/1010.2481" target=""_blank"">arXiv:1010.2481 [gr-qc]</a>.</li></ul><br />-->Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com1tag:blogger.com,1999:blog-5826632960356694090.post-86730738444006412052014-04-01T13:30:00.000-05:002014-04-01T13:30:59.082-05:00Spectral dimension of quantum geometries<b>Johannes Thürigen, Albert Einstein Institute </b><br /><b>Title: Spectral dimension of quantum geometries </b><span style="background-color: white;"></span><br /><a href="http://relativity.phys.lsu.edu/ilqgs/thueringen112613.pdf">PDF</a><span style="background-color: white;"> of the talk (1MB) </span><a href="http://relativity.phys.lsu.edu/ilqgs/thueringen112613.wav">Audio</a><span style="background-color: white;"> [.wav 39MB] </span><a href="http://relativity.phys.lsu.edu/ilqgs/thueringen112613.aif">Audio</a><span style="background-color: white;"> [.aif 4.1MB] </span><br /><span style="background-color: white;"><br /></span><span style="background-color: white;">By Francesco Caravelli, University College London</span><br /><span style="background-color: white;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-LyrMqwCQGZM/UziOA6rSqJI/AAAAAAAAEz8/kNA_TfXm1LQ/s1600/thur.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-LyrMqwCQGZM/UziOA6rSqJI/AAAAAAAAEz8/kNA_TfXm1LQ/s1600/thur.jpg" height="320" width="213" /></a></div>One of the fundamental goals of quantum gravity is understanding the structure of space-time at very short distances, together with predicting physical and observable effects of having a quantum geometry. This is not easy. Since the introduction of fractal dimension in Quantum Gravity, and its importance emphasized in the work done in Causal Dynamical Triangulations (Loll et al. 2005) and Asymptotic Safety (Lauscher et al. 2005), it has become more and more clear that space-time, at the quantum level, might have a radical transformation: the number of effective dimensions might change with the energy of the process involved. Various approaches to Quantum Gravity have collected evidences of a dimensional flow at high energies, which was popularized by Carlip as Spontaneous Dimensional Reduction (Carlip 2009, 2013). (The use of the term reduction is indeed a hint that a dimensional reduction is observed, but the evidences are far from conclusive. We find dimensional flow more appropriate.) <br /><br />Before commenting on the results obtained by the authors of the paper discussed in the seminar<br />(Calcagni, Oriti, Thuerigen 2013), let us first step back for a second and spend some time introducing the concept of fractal dimension, which is relevant to this discussion.<br /><br />The concept of non integer dimension was introduced by the mathematician Benoit Mandelbrot half a century ago. What is this fuss about fractals and complexity? What is the relation with spacetimes, quantum space-times? <br /><br />Everything start from an apparently simple question asked by Mandelbrot: What is the length of the coast of England (or more precisely, Cornwall)? As it turned out, the length of the cost of England, depended on the lens used to magnify the map of coast, and depending on the magnifying power, the length changed with a well defined rule, known as scaling, which we will explain shortly. <br /><br />There are several definitions of fractal dimension, but let us try to keep things as easy as possible, and see why a granular space-time might indeed imply a different dimensions at different scales (i.e., our magnifying power). The easy case is the one of a regular square lattice, which for the sake of clarity we consider infinite in any direction. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-7A37ZJl57Xw/Uzcl8P-kFWI/AAAAAAAAEzc/CaJS3t394g0/s1600/manny.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-7A37ZJl57Xw/Uzcl8P-kFWI/AAAAAAAAEzc/CaJS3t394g0/s1600/manny.jpg" height="270" width="320" /></a></div><br /> Source: Manny Lorenzo<br /><br /> The dimension of the lattice might look two dimensional, as the lattice is planar: it can be embedded into a two dimensional surface (this is what is called embedding dimension). 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Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles></xml><![endif]--> <!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin;} </style><![endif]--> <!--StartFragment--><!--EndFragment-->.<br /><br />If d is not too big, the value of gamma changes if the underlying structure is not a continuum, or is granular, and gamma can take non-integer values. This can be interpreted in various ways. For the case of fractals, this implies that the real dimension of fractals is not integer. Analogously to the case of the number of points within a certain distance d, it is possible to define a diffusion operation which will do the work of counting for us. However, the counting process depends on the operator which defines the diffusion process: how a swarm of particles move on the underlying discrete space. This is a crucial point of the procedure. <br /><br />In the continuum, the technology is developed to the point that it can to show that such an operator can be defined precisely**. The problem then is that the scaling not precise: for too long times, the scaling relation is not exact (as curvature effect might contribute). Thus, the time given to the particle to diffuse has to be appropriately tuned. This is what the authors define in Section 2 of the paper discussed in the talk and is a standard procedure in the context of the spectral dimension. Of course, what discussed insofar is valid for classical diffusion, but the operator can be defined for quantum diffusion as well, which is, put in simple terms, described by a Schroedinger unitary evolution like in ordinary quantum mechanics.<br /><br />It is important to understand that the combinatorial description of a manifold (how these are represented in the discrete setting), rather than the actual geometry, plays a very relevant role. If you calculate the fractal dimension of these lattices, although at large scale they give the right fractal dimension, on small scale they do not. This shows that in fact discreteness does have an effect on the spectral dimension, and that results do indeed depend on the number of dimensions. But more importantly the authors observe that the spectral dimension, even in the classical case, depends on the precise structure of the underlying pseudo-manifold, i.e. how the manifold is discretized. If you combine this with the fact that insofar the fractal dimension is the global observable saying in how many dimensions you live in (concept very important for other high energy approaches), the interest might be quite well justified. <br /><br />The case of a quantum geometry, considered using Loop Quantum Gravity (LQG), is then put forward at the end. The definition is different from the one given previously (Modesto 2009, assuming that the scaling is given by the area operator of LQG), and it leads to different results. <br /><br />Without going into the details (described anyway quite clearly in the paper), probably it is noteworthy to anticipate the results and explain the difficulties involved in the calculation. The first complication comes from the calculation itself: it is in fact very hard to calculate the fractal dimension in the full quantum case. However, in the semiclassical approximation (when the geometry is in part quantum and in part classical), the main "quantum" part can be neglected. The next issue is that, in order to claim the emergence of a clear topological dimension, the fractal dimension has to be constant for a wide range of distances of several orders of magnitude. It is important to say that, if you use the fractal dimension as your definition of dimension, it is not possible to assign a given dimensionality unless the number of discrete points under consideration is large enough. This is a feature of the fractal dimension which is very important for Loop Quantum Gravity in many respects, as there as been for long time a discussion on what is the right description of classical and quantum spacetime. Still, this approach gives the possibility of a bottom-up definition of dimension (in the top-down, there would not be any dimensional flow). <br /><br />As a closing remark, it is fair to say that this paper goes one step further into defining a notion of fractal dimension in Loop Quantum Gravity. The previous attempt was made by Modesto and collaborators using a rough approximation to the Laplacian. That approximation exhibited a dimensional flow towards an ultraviolet 2-dimensional space, which seems to be not present using a more elaborated Laplacian.<br /><br /> *For a square lattice, if d is big enough, \gamma is equal to two: this is the Haussdorf dimension of the lattice, and indeed this dimension can be defined through the following equation: gamma=\partial log(N)/ \partial d<br /><br />** Using the technical terminology, this is the Seeley-De Witt expansion of the heat kernel on curved manifolds. This is usually called spectral dimension. The first term of the expansion depends explicitly on the spectral dimension, while in the terms at higher orders there are also contributions from the curvature. <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Jorge Pullinhttp://www.blogger.com/profile/07465581283254332265noreply@blogger.com0