tag:blogger.com,1999:blog-23819000801065224412024-02-20T20:02:22.949+01:00relatively quantumA blog around my research in quantum gravityNorberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.comBlogger36125tag:blogger.com,1999:blog-2381900080106522441.post-45598437435442476602019-10-08T14:48:00.001+02:002019-10-09T08:33:25.589+02:00Thermal Phase Transition in the Bosonic BFSS Matrix ModelI recently got involved with numerically simulating the BFSS and BMN matrix models as a means to study quantum gravity via holographic dualities. These models are quantum mechanical models with hermitian N x N matrices as degrees of freedom, i.e. 0+1-dimensional quantum field theories.<br />
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In its field theory description, the BFSS model arises from dimensionally reducing maximally supersymmetric SU(N) Yang-Mills theory in 9+1 dimensions down to 0+1 dimensions. From string theory, it can be obtained as a description of N D0-branes on flat space, which can form a bound state with the properties of certain black hole solutions in type IIa supergravity in the large N and strong coupling limit.<br />
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The BMN model is a massive deformation of the BFSS model that features a more complicated vacuum structure on the field theory side, but better stability of the numerical simulations. On the gravity side, the massive deformation corresponds to changing the flat background into a certain plane wave.<br />
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Despite being quantum mechanical models, both the BFSS and BMN models are quite hard to simulate in interesting regimes, as they feature fermions and require the matrix size N to be large enough to accurately probe the large N limit which connects to the description via classical gravity.<br />
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An interesting starting point is to consider only the bosonic part of the BFSS model, which arises as the high-temperature limit of 1+1-dimensional supersymmetric Yang-Mills theory. This theory again has a gravity description and is of particular interest to study the transition from black strings to localised black holes, see e.g. <a href="https://arxiv.org/abs/1806.11174" target="_blank">this recent paper</a>. Using this truncation, numerical simulations are much faster and access to N=64 was feasible using the ATHENE cluster in Regensburg.<br />
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In our <a href="https://arxiv.org/abs/1909.04592" target="_blank">first joint paper</a>, we investigated the thermal confinement / deconfinement phase transition in the bosonic BFSS model. Conflicting results on the order of the transition were reported in the literature and we set out to clarify the situation with a careful study at sufficiently large N.<br />
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It turned out that N=64 was required to obtain a clear signal and previous investigation reported contradictory results as a consequence of accessing only N=32 due to limited computational resources.<br />
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Our simulations revealed that the deconfinement transition is of first order. The order parameter of the transition is the (absolute value of the) Polyakov loop P, the trace of the closed loop parallel transport of the gauge field in the time direction (periodic boundary conditions in the time-like direction result in a finite temperature). The first oder nature of the transition can be seen in the following plot.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/--vgWiLN_BuQ/XZyDf-_NgII/AAAAAAAABPk/WZ2mdQEuyd00vG5FhdKx_zAPSt7AdPwsgCLcBGAsYHQ/s1600/plot_jointbins_Polyak_GBN64S24M0D9.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="1600" height="400" src="https://1.bp.blogspot.com/--vgWiLN_BuQ/XZyDf-_NgII/AAAAAAAABPk/WZ2mdQEuyd00vG5FhdKx_zAPSt7AdPwsgCLcBGAsYHQ/s400/plot_jointbins_Polyak_GBN64S24M0D9.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Histogram of the order parameter |P|. At the transition temperature T=0.885, two peaks are visible. In the large N, these peaks become sharply localised at |P|=0 (confined) and |P|=0.5 (dedonfined). </td></tr>
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The first order nature of the transition becomes apparent due to the system spending most time in either the small |P| or |P|=0.5 phase at the critical temperature (green). In the Monte Carlo history, repeated tunnelling between those phases is visible. For slightly lower or higher temperature, the system remains in either phase.<br />
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Future work will address similar issues in the full supersymmetric models. In an upcoming post, I will also discuss interesting results on "partial deconfinement" that can be found in the above paper.<br />
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<br />Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com1tag:blogger.com,1999:blog-2381900080106522441.post-75564459260055260012019-10-01T11:34:00.001+02:002019-10-01T11:34:57.612+02:008th Tux Workshop on Quantum Gravity<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-fDlggGZm5Lg/XZMdR1j9uNI/AAAAAAAABOY/TiGjAKvt8jEzz42zX07ClZy30yCH2KIpQCLcBGAsYHQ/s1600/IMG_0718.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-fDlggGZm5Lg/XZMdR1j9uNI/AAAAAAAABOY/TiGjAKvt8jEzz42zX07ClZy30yCH2KIpQCLcBGAsYHQ/s400/IMG_0718.JPG" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">View from the top of the Eggalm-Bahn. </td></tr>
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As in previous years, I am co-organising a winter workshop on quantum gravity in Tux, Austria. The dates are February 10-14, 2020, and registration is open now. Further details can be found on our website:<br />
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<a href="https://www.gravity.physik.fau.de/events/tux8/" target="_blank">https://www.gravity.physik.fau.de/events/tux8/</a>Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com1tag:blogger.com,1999:blog-2381900080106522441.post-28336760447384634562019-09-16T14:38:00.000+02:002019-09-16T14:38:00.454+02:00Coarse Graining with SU(1,1)This is the second post in a series about SU(1,1) techniques for coarse graining loop quantum cosmology. While the <a href="https://relatively-quantum.bodendorfer.eu/2019/05/quantum-cosmology-with-su11.html" target="_blank">previous post</a> was about the general idea of group quantisation techniques, this one will focus on coarse graining.<br />
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Coarse graining implements the idea that coarse physical systems, such as the universe at large, are built up from smaller systems. The physics at large scales is determined by "integrating out" physics at small scales, i.e the description at large scales is reduced to a few parameters that take into account how the details of the system behave on average.<br />
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Obtaining such an effective description at a coarse scale may be very difficult, in particular via an analytic computation. Therefore, explicit examples, even in simplified models, are of great interest.<br />
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<a href="https://arxiv.org/abs/1811.02792" target="_blank">In a recent paper </a>with a former Bachelor student, I was able to provide such an example using systems quantised with group quantisation techniques. The result is quite general and and requires only that one quantises a classical Poisson algebra isomorphic to su(1,1) in which the 3 generators scale like extensive quantities with the system size and interactions between neighbouring subsystems can be neglected. This is for example the case in the application to quantum cosmology discussed before. <br />
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It turns out that the representation label $j$ of certain SU(1,1) representations behaves as the scale of the system if one uses certain coherent states. $N$ systems with representation label (scale) $j_0$ then jointly behave like a single system with representation label $j = N j_0$.<br />
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The agreement of the $N$ fine and the single coarse systems is correct for all moments of the 3 generators and the eigenvalues with associated probabilities. The derivation necessary to arrive at this result is the core part of the cited paper. The coarse graining procedure is even dynamically stable if dynamics is generated by one of the group generators. Again, this can be achieved in the quantum cosmology application.<br />
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In a future post, another paper with a former Master student will be discussed which lifts these results directly onto the Hilbert space of loop quantum cosmology. This application is interesting because it gives an example of a renormalisation group flow of Hamiltonian operators from small to large spins.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-9351986939265540682019-09-13T13:21:00.000+02:002019-09-13T13:21:06.099+02:00Public talk (in German)<div class="separator" style="clear: both; text-align: left;">
Last year, I gave a public talk (in German) about quantum gravity in Regensburg. It is available on youtube:</div>
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<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/k6y3fNGh3uI/0.jpg" src="https://www.youtube.com/embed/k6y3fNGh3uI?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
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It was done as an invited presentation to a talk series called <a href="http://was-ist-wirklich.de/" target="_blank">Was-Ist-Wirklich</a> (What is real), featuring various topics in science.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-72627258699965865652019-05-10T18:16:00.000+02:002019-05-10T18:17:18.985+02:00Quantum cosmology with SU(1,1)This is the first post in an upcoming series covering some recent papers that deal with quantum cosmology models that are strongly relying on an SU(1,1), or equivalently SL(2,R), structure. In brief, one identifies classical phase space functions whose Poisson algebra is isomorphic to the Lie algebra su(1,1) and then quantises the cosmological model by promoting those functions to the generators of su(1,1) in some representation. The main advantage of this quantisation method is that the representation theory of the group under consideration is well known, so that the crucial "find a representation" step in constructing the quantum theory is essentially trivial.<br />
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The exercise of finding such phase space functions can for example be successfully completed for classical spatially flat homogeneous and isotropic cosmology. There, the phase space is described by the canonical pair $\{v,b\}=1$, where $v$ is the spatial volume and $b$ the mean curvature. Instead of writing the details for the case of classical cosmology, we will in the following give a flavour of a similar analysis in the context of a modification of classical cosmology inspired by effective loop quantum cosmology. For a brief recap, see <a href="https://relatively-quantum.bodendorfer.eu/2016/03/how-does-loop-quantum-cosmology-work.html" target="_blank">this post</a>. In short, the transition from classical cosmology to the effective loop quantum cosmology model is performed by substituting $b$ by functions such as $\sin(b)$, which are capturing some key quantum effects like a limiting curvature at the Planck scale.<br />
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The first main work along these lines was <a href="https://arxiv.org/abs/1204.0539" target="_blank">this paper</a>. It turns out that the identification $j_z=v$, $k_x = v \cos(b)$, $k_y = v \sin(b)$ reproduces the su(1,1) algebra with the standard identifications of the generators. Quantisation can then proceed as said before by promoting them to the su(1,1) generators in a certain representation space.<br />
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This is only one identification of su(1,1) generators from the cosmology phase space and others are possible. Several examples are given in this <a href="https://arxiv.org/abs/1705.03772" target="_blank">follow-up work</a>. The details of the identification are relevant for the correct choice of the representation, as they determine the classical value of the Casimir operators as either positive, zero, or negative. Corresponding SU(1,1) representation are available in all cases, but differ in their properties. <br />
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There are several avenues that can be followed from here:<br />
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<li>On the SU(1,1) representation spaces, one can define standard Perelomov coherent states. They have the nice property that one can transfer the action of su(1,1) generators on them directly to a spinor label in the defining representation, thus allowing an explicit computation of the quantum dynamics via exponentiating su(1,1) generators. This also allows to easily compare evolution in different representations. </li>
<li>Expectation values of the su(1,1) generators feature a splitting of the dependence on the spinor label and representation label (for the discrete representation with $j=1/2, 3/2, \ldots$) such that the extensive state of the system (its size) decouples from its intensive properties. This suggests to study coarse graining based on such states. </li>
<li>An alternative way of quantising is to not start with classical phase space functions forming the su(1,1) algebra, but to start with a given quantum cosmological system and to find compound operators built from the elementary operators, e.g. $\hat v$ and $\hat b$, that have the su(1,1) algebra. This may reduce quantisation ambiguities if interesting operators, such as the Hamiltonian, are linear combinations of the generators.</li>
<li>It would be interesting to check whether there is a geometric interpretation of the action of the conformal group SL(2,R) on spacetime. </li>
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Some results have already appeared in the literature and we will comment on them in future posts.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-72302840372077359172019-05-03T11:10:00.000+02:002019-05-03T11:14:30.943+02:00Singularity resolution in LQG inspired black holes It is expected that quantum gravity will somehow resolve the singularities that are generically present in classical gravitational theories. For example, this may be the Big Bang singularity that one encounters when applying Einstein's theory of General Relativity all the way to the beginning of the universe. An example of how this singularity is resolved in the context of loop quantum cosmology was discussed in <a href="https://relatively-quantum.bodendorfer.eu/2016/03/how-does-loop-quantum-cosmology-work.html" target="_blank">this post</a>.<br />
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Another place where singularities prominently occur is inside black holes. Matter that falls through the horizon of a black hole will eventually hit this singularity and the theory describing its propagation breaks down in this instant. On the other hand, a compete theory of quantum gravity should provide a well-defined description of such a process.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-F1v5SmF1IlQ/XMwCHN6Al-I/AAAAAAAABLc/UeFRSqZegDkkEFfG0VYtk9YAaRLl2VurACLcBGAs/s1600/penrose_diag_bh.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="609" data-original-width="1001" height="241" src="https://1.bp.blogspot.com/-F1v5SmF1IlQ/XMwCHN6Al-I/AAAAAAAABLc/UeFRSqZegDkkEFfG0VYtk9YAaRLl2VurACLcBGAs/s400/penrose_diag_bh.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Penrose diagram of a maximally extended Schwarzschild spacetime. Physical motion bounded by the speed of light is possible only within the future light-cones drawn in orange. Due to the structure of the spacetime, any observer crossing the black hole horizon will eventually hit the black hole singularity. </td></tr>
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While the general problem of dynamical collapse, leading classically to the formation of singularities, is a very hard problem to solve in quantum gravity, some insights can be gained by considering the simpler situation of eternal black holes, in particular the Schwarzschild solution. It turns out that the interior of a Schwarzschild black hole can be described by a cosmological spacetime so that the black hole singularity is mapped to a cosmological singularity of the Big Crunch (future Big Bang) type. This in particular allows to apply techniques from quantum cosmology to such spacetimes.<br />
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Within loop quantum gravity, this line of thought has a long history with many publications subsequently improving the involved effective models. Most work is at the level of effective classical equations that capture quantum effects as $\hbar$-corrections. The main problem that researchers are currently faced with is to make a good educated guess about the effective theory that captures the main relevant quantum effects.<br />
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Key insights for how to construct physically viable theories capturing the main aspects of full loop quantum gravity were previously obtained within loop quantum cosmology, leading to the so-called notion of $\bar{\mu}$-schemes. They should ideally be understood and derived as coarse grained versions of the fundamental quantum dynamics. So far however, existing derivations mainly involve heuristic steps that may or may not reflect the physics of full loop quantum gravity. In any case, the effective models obtained along these lines are interesting in their own right and can simply be considered as modified gravity theories with interesting phenomenology.<br />
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In a <a href="https://arxiv.org/abs/1902.04542" target="_blank">recent paper</a> together with my PhD students Fabio Mele and Johannes Münch, we proposed a new effective model that is inspired by two main ideas:<br />
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<li><i>Physical viability of the dynamics</i>: most previous works were facing severe problems in that quantum effects were large in low curvature regions were this wasn't expected. This was due to an unsuitable choice of quantisation scheme, ultimately due to a choice of variables that weren't tailored to the situation. We proposed a new set of variables that directly link quantum effects to high curvature regions when applied in a loop quantum gravity type quantisation. </li>
<li><i>Existence of a corresponding quantum theory:</i> one is ultimately interested in a full theory embedding of the quantum theory corresponding to the effective model used. For this, the effective Hamiltonian should exist as a well-defined operator. We showed that this is the case in our model due to being able to work in a simple so-called $\mu_0$ scheme due to our adapted choice of variables, reminiscent of the $b,v$-variables in loop quantum cosmology.</li>
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The physics captured in the model is as follows: an observer falling into the black hole crosses the horizon like in classical general relativity as quantum effects are small in this low-curvature region. The singularity that would eventually be hit in the classical theory is replaced by a transition surface to a new spacetime region where the observer will emerge from a white hole. This journey can be repeated an infinite number of times, each time to a new spacetime region.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-vZ7C7_jVys4/XMwCtkd-UJI/AAAAAAAABLk/5PB5tWU0phA66FCeoMAwDcYEZK_4ikG5QCLcBGAs/s1600/penrose_diag_qbh.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1092" data-original-width="967" height="400" src="https://1.bp.blogspot.com/-vZ7C7_jVys4/XMwCtkd-UJI/AAAAAAAABLk/5PB5tWU0phA66FCeoMAwDcYEZK_4ikG5QCLcBGAs/s400/penrose_diag_qbh.jpg" width="353" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Quantum corrected maximally extended Schwarzschild spacetime according to our (and other) papers. The diagram can be extended an infinite number of times into the future and past. In the text, we refer to the lower region as the black hole region and the upper region as the white hole region, as this would be nomenclature used by the indicated observer. The masses of the spacetimes oscillate between black and white hole masses. The dotted line is a regular surface of high but finite curvature denoted as the transition surface. </td></tr>
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This picture is not new to our article and was presented before. The main new input was to achieve both goals 1 and 2 above simultaneously via an adapted choice of variables. Moreover, our paper identifies a new Dirac observable in systems of this type that can be interpreted as the white hole mass (in addition to the black hole mass) and exists non-trivially only in the quantum theory. This allows to (in principle) already measure the white hole mass on the black hole side, although its signature is very faint. This new observable and suitable analogues thereof in other effective theories should be relevant to future investigations on the topic.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-88559188154824672672018-05-18T15:04:00.000+02:002018-09-12T10:59:00.751+02:00PhD Position availableA PhD Position in my group is available starting this fall. The full advertisement is below:<br />
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EDIT: The position has been filled.<br />
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The Junior Research Group “Quantum Gravity Techniques for Real World Applications of the Gauge / Gravity Duality” funded by the Elite Network of Bavaria is planning to hire a PhD student starting in September 2018 or later. The group is located at the University of Regensburg within a large and very active quantum field theory group and headed by Dr. Norbert Bodendorfer. It employs up to three PhD students (of which two positions are currently filled) and hosts a varying number of Bachelor and Master students. <br />
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The salary is 2/3 of an TVL-E13 position, resulting in approximately 1500 EUR monthly after taxes and benefits (including health insurance and retirement funds). The position will be funded for at least 3 years. A moderate amount of teaching in accordance with the university’s regulations will be required. The position comes with an annual travel budget of 2000 EUR. Applicants should hold a MSc degree in physics or equivalent before signing the contract.<br />
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The research focus of the group will be to apply loop quantum gravity techniques to the gauge / gravity correspondence with the eventual goal to better understand finite N gauge theories via quantum gravity. Individual research projects may include lattice gauge theory and will in this case be executed in collaboration with experts from Regensburg.<br />
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PhD students in the Junior Research Group have access to the activities of the Elite Graduate Programme “Physics Advanced” jointly run by the Universities of Erlangen-Nuremberg and Regensburg (workshops, summer schools, …) as well as to those of the Elite Network of Bavaria (soft-skill seminars, …). Likewise, activities of the graduate school “Particle Physics and High Performance Computing” in Regensburg will be open to them.<br />
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In order to apply, please send an email to norbert.bodendorfer@physik.uni-r.de including a CV, transcripts, and a concise letter of motivation stating <br />
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– your research interests<br />
– your motivation to join the group<br />
– whatever you think is relevant.<br />
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Letters of recommendation are optional but welcome and should be sent by the author to the same email address.<br />
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Applications will be reviewed on a rolling basis. The final selection will not be made prior to August 1, 2018, so that all applications received by this date will receive full consideration. Later application will be considered until the position is filled.<br />
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EDIT: The position has been filled.<br />
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<br />Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-61442237783987948172018-04-12T09:47:00.000+02:002018-04-12T09:47:30.827+02:00Quantum Gravity meets Lattice QFTTogether with some colleagues, I am organizing a workshop on the intersection of quantum gravity and lattice QFT at ECT* in Trento, Italy, September 3-7.<br />
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<a href="http://indico.ectstar.eu/event/21/" target="_blank">Click here for the conference website. </a><br />
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<br />The conference abstract goes as follows:<br /><br />AdS/CFT has been one of the most fruitful approaches to analyse the qualitative aspects of the dynamics of strongly interacting QFTs, most prominently QCD. As an approach to understanding the early stage of high energy heavy ion collisions, but also proton-proton collisions at LHC, it is, in fact, one of very few systematic approaches. However, it is not clear how reliable the description is quantitatively, because QCD is not a N=4, supersymmetric, conformal, SU(N) gauge theory with infinite N and the QCD coupling constant is of limited size. Individual contributions exist on both sides of the duality calculating the size of the relevant corrections (like the perturbative calculation of quantum corrections on the gravity side for finite N and finite coupling strength, the lattice simulation of SU(N) gauge theories with N>3, the calculation of perturbative corrections from non conformality on the QFT side, lattice simulation with partial supersymmetry …) but no systematic effort. In addition, more general scenarios for gauge/gravity dualities have been studied, extending beyond the realms of AdS, CFT, and string theory. The probability is high that quantitative contact can only be made on the basis of non-perturbative calculations on both sides, which is a very tall order. On the QFT side, lattice QFT is the best established tool to do so, while on the quantum gravity side resummed string theory is the main approach. In addition, there is an increased recent interest within loop quantum gravity in holographic computations.<br /><br />The aim of the workshop is to bring some of the internationally leading experts in these fields together, formulate a more systematic strategy, and realize a few projects in the direction of a quantitative application of quantum gravity techniques to QCD in subsequent months.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com5tag:blogger.com,1999:blog-2381900080106522441.post-42972313112781247482017-07-24T10:59:00.000+02:002017-08-25T13:29:44.451+02:00New group + PhD positions<div class="separator" style="clear: both; text-align: center;">
<a href="https://idw-online.de/de/newsimage?id=108185&size=screen" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="618" data-original-width="800" height="247" src="https://idw-online.de/de/newsimage?id=108185&size=screen" width="320" /></a></div>
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The <a href="https://www.elitenetzwerk.bayern.de/elite-network-home/elite-network-home/?L=2" target="_blank">Elite Network of Bavaria</a> recently awarded me with a <a href="https://www.elitenetzwerk.bayern.de/elitenetzwerk-home/aktuelles/artikel/einrichtung-neuer-elitestudiengaenge-und-internationaler-nachwuchsforschergruppen-226/" target="_blank">grant</a> to start my own research group at the University of Regensburg starting this fall. I am currently looking for PhD students to work with me on projects at the intersection of loop quantum gravity and string theory, more precisely in applying ideas about quantum gravity corrected geometries in the context of the gauge / gravity duality. The announcement for the positions can be found here.<br />
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EDIT: All currently open positions have been filled.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com4tag:blogger.com,1999:blog-2381900080106522441.post-83001114152525095922017-04-03T19:00:00.000+02:002017-04-03T19:00:30.628+02:00Chamseddine and Mukhanov reinvent loop quantum cosmologyIn a <a href="https://arxiv.org/abs/1612.05860" target="_blank">recent paper</a>, Chamseddine and Mukhanov have proposed a modification of general relativity that features a limiting curvature. The model is in the framework of their earlier work on "<a href="https://arxiv.org/abs/1403.3961" target="_blank">mimetic gravity</a>" and coupled general relativity to a scalar field $\phi$ that is constrained as<br />
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g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi = 1<br />
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in the mostly minus signature convention. In addition to this constraint, the action contains the term $f(\Box \phi)$. In the homogeneous and isotropic sector, $\Box \phi$ is proportional to the Hubble rate $\dot a / a$.<br />
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The main input that is needed to fully define this model is the function $f$, which can a priori be chosen arbitrarily up to a consistency requirement with general relativity at low curvatures. In their paper, Chamseddine and Mukhanov make the proposal<br />
\[<br />
f(\chi) = - \chi_m^2 \, g\left( \sqrt{\frac{2}{3}} \frac{\chi}{\chi_m} \right), ~~~ g(y) = -1 -\frac{y^2}{2} + y \arcsin y + \sqrt{1-y^2}<br />
\]<br />
which leads in the homogeneous and isotropic sector to a very simple correction to the Friedmann equation of the form<br />
\[<br />
\frac{\dot a}{a} \propto \rho \left(1- \frac{\rho}{\rho_\text{crit}}\right)<br />
\] <br />
where $\rho$ is the matter energy density and $\rho_\text{crit}$ a free parameter. This equation leads to a bouncing universe at the critical matter energy density $\rho_\text{crit}$.<br />
<br />
In the homogeneous but non-isotropic sector, one can again solve the equations of motion to find singularity resolution, however with somewhat more complicated details. An analogous modification for Schwarzschild black holes is discussed in a <a href="https://arxiv.org/abs/1612.05861" target="_blank">second paper</a>.<br />
<br />
It is interesting to note that the corrected Friedmann equation above is already known from loop quantum gravity, more precisely from the effective equations of loop quantum cosmology. This already suggests that at least in the homogeneous and isotropic sector, the two theories should agree.<br />
<br />
In fact, the choice for $f$ can be read off from the effective action of loop quantum cosmology, as computed first in <a href="https://arxiv.org/abs/0811.4023" target="_blank">this paper</a>, equation 7. A more detailed investigation in based on the Hamiltonian formulation of the model proposed by Chamseddine and Mukhanov confirms this. The details have been written up in <a href="https://arxiv.org/abs/1703.10670" target="_blank">this paper</a>, see also <a href="https://arxiv.org/abs/1703.10812" target="_blank">this paper</a> for an intendedly simultaneous publication of related results.<br />
<br />
In addition, a quantum mechanical argument is given there why this choice of $f$ is the simplest one consistent with quantum mechanics: Mukhanov and Chamseddine motivate the structure of their model with non-commutative geometry, in particular the quantisation of three-volume. In a canonical quantisation setting, this leads one directly to the simplest version of loop quantum cosmology. The choice for $f$ then follows by a Legendre-transform. <br />
<br />
If one goes beyond the homogeneous and isotropic setting, the effective loop quantum cosmology deviates from the model of Chamseddine and Mukhanov in the details. Still, the qualitative behavior is the same for homogeneous but non-isotropic metrics. Beyond this, a comparison is hard to make because the effective LQG dynamics are in general not known. This on the other hand suggests to simply use the new model as a toy model for loop quantum gravity, i.e. as a toy model effective action that is completely known. The main application of this would be to obtain an understanding for the effective geometries that can emerge from the quantum corrections build into loop quantum gravity beyond the cosmological setting.<br />
<br />Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-42846460046936375422016-12-23T11:04:00.000+01:002016-12-23T11:04:12.696+01:00Holographic signatures of resolved cosmological singularitiesAfter some longer silence partly due to moving to a new location (LMU Munich) and teaching my first regular lecture (Theoretical mechanics for lyceum teachers and computational science at Regensburg University), I hope to write more regularly again in the future.<br />
<br />
As a start, a new paper on using loop quantum gravity in the context of AdS/CFT has finally appeared <a href="https://arxiv.org/abs/1612.06679" target="_blank">here</a>. Together with Andreas Schäfer and John Schliemann from Regensburg University, we asked the question of what happens in the dual CFT if you assume that the singularity on the gravity side is resolved in a manner inspired by results from loop quantum gravity.<br />
<br />
Building (specifically) on recent work by <a href="https://arxiv.org/abs/1503.08838" target="_blank">Engelhardt, Hertog, and Horowitz</a> (as well as many others before them) using classical gravity, we found that a finite distance pole in the two-point-correlator of the dual CFT gets resolved if you resolve the singularity in the gravity theory. Several caveats apply to this computation, which are detailed in the papers. We view this result therefore as a proof of principle that such computations are possible, as opposed to some definite statement of how exactly they should be done.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com1tag:blogger.com,1999:blog-2381900080106522441.post-64269873564641612112016-09-21T14:43:00.000+02:002016-09-21T14:43:57.993+02:00Coarse graining and state refinements<div class="separator" style="clear: both; text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/UfN-KoNCWag/0.jpg" src="https://www.youtube.com/embed/UfN-KoNCWag?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
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<br />Coarse graining has become an increasingly important topic in loop quantum gravity with several researchers working on it. As usual in physics, one is interested in integrating out microscopic degrees of freedom and doing computations on an effective coarse level. How exactly the states, observables, and dynamics of the theory should change under such a renormalisation group flow is only poorly understood at the moment.<br /><br />Recently, I have written <a href="http://arxiv.org/abs/1607.06227">a paper</a> about this topic in a simplified context which is tailored to reproduce loop quantum cosmology from loop quantum gravity. Here, one can explicitly coarse grain the relevant observables, the scalar field momentum and the volume of the spatial slice, and check that their dynamics remains unchanged under such a coarse graining.<br /><br />The reason that this works is rooted in an exact solution to loop quantum cosmology which can be imported in this full theory setting. In particular, the form of the dynamics of this solution is independent of the volume of the universe. It then follows that if we concentrate the volume of some set of vertices at a single one, the evolution of this coarse grained volume will agree with the evolution of the sum of the individual volumes.<br /><br />The content of the paper is sketched in the brief talk linked above.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-68360686157361893392016-07-27T09:35:00.000+02:002016-07-27T09:35:34.748+02:00Heuristics for Lorentz violations in LQG<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-jyXTnlxxpNg/V5DjCijsm-I/AAAAAAAAAfY/thhn_qN3tk8aBGvs_q3bsmh18WGQYGvWgCLcB/s1600/Lorentz%2Bviolation%2Bdifference%2Bin%2Bspeed%2Bof%2Blight%2Bsmall.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="182" src="https://1.bp.blogspot.com/-jyXTnlxxpNg/V5DjCijsm-I/AAAAAAAAAfY/thhn_qN3tk8aBGvs_q3bsmh18WGQYGvWgCLcB/s400/Lorentz%2Bviolation%2Bdifference%2Bin%2Bspeed%2Bof%2Blight%2Bsmall.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Slight energy dependences in the speed of light can accumulate over time,<br />
allowing for possible detection or exclusion of such effects. Highly energetic<br />
photons emitted e.g. by a supernova are in particularly useful for such studies.</td></tr>
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Predictions for possible Lorentz violations are a key area of interesting phenomenology in quantum gravity. Within loop quantum gravity, it has not been possible so far to reliably extract predictions for Lorentz violations. Existing claims were based on simplified toy models inspired by LQG, but not implied. This situation hasn’t changed so far, but there is an interesting development which hints at which order we might expect such Lorentz violations to be found.<br />
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In recent years, investigations in the anomaly freedom of effective constraints for <a href="http://arxiv.org/abs/1111.7192" target="_blank">cosmological perturbations</a> and <a href="http://arxiv.org/abs/1507.00329" target="_blank">spherical symmetry</a> have brought forward evidence for a<a href="http://arxiv.org/abs/1503.09154" target="_blank"> smooth signature change </a>at densities close to the Planck density. This effect shows up in a deformation of the hypersurface deformation algebra, where the commutator<br />
$$ \{ H[M], H[N] \} = H_a[\beta ~ q^{ab}(M \partial_b N - N \partial_b M)]$$<br />
between two Hamiltonian constraints is modified by a function $\beta$. In the simplest case of holonomy corrections, we have $\beta = 1 - 2 \rho / \rho_\text{crit}$, where $\rho$ is the background energy density and $\rho_\text{crit} \sim 1 / G^2 \hbar$ is the critical energy density at which the quantum bounce substituting the big bang occurs. Since the hypersurface deformation algebra of Euclidean gravity has the opposite sign in the previous Poisson bracket, we can interpret this as a smooth signature change from Lorentzian to Euclidean. <br />
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As a physical effect, one finds that also the propagation equation for matter fields gets modified as<br />
$$ \frac {\partial^2 \phi^2}{\partial t^2} - \frac{\beta}{a^2} \Delta \phi = S[\phi] $$<br />
where $S[\phi]$ contains source and lower derivative terms. It follows that $\beta \neq 1$ implies a change in the speed of light. <br />
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So far, $\beta$ only depends on the background energy density, meaning that no particle-energy dependent speed of light can be deduced. However, one should also take into account backreaction of a particle on the background. This has not been done rigorously so far, but we can try a heuristic estimate of the effect to be expected. <br />
<br />
Consider a photon. Its energy density is roughly given by </div>
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$$\rho_\text{Photon} \sim E_\text{Photon} / \lambda_\text{Photon}^3 = \hbar \omega^4 / (2 \pi)^3 \text{.}$$ </div>
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The difference in the speed of two photons thus depends on the difference of their energy densities that we input in $\beta$. We estimate<br />
$$ 1-\beta \sim \frac{\rho_\text{Photon}}{\rho_\text{crit}} \sim \hbar^2 G^2 \omega^4 \sim \frac{E_\text{Photon}^4}{E_\text{Planck}^4} \text{.} $$<br />
We conclude that based on our argument, we would expect an energy dependence of the speed of light which deviates from the standard speed of light by a term of order $4$ in $\frac{E_\text{Photon}}{E_\text{Planck}}$. Given <a href="http://arxiv.org/abs/1304.5795" target="_blank">current experimental data</a>, there seems to be no contradiction with experiment. <br />
<br />
It has to be stressed that this argument was very qualitative and has several weak points. While signature change has been observed by several authors in different contexts, it is not established yet in full loop quantum gravity. Also, our treatment of backreaction was very naive and a more detailed study should be undertaken. Still, it seems that the possible deduction of Lorentz violations from the deformed algebra approach is an interesting subject of study, and that the appearance of the effect only at order 4 goes against common expectations. <br />
<br />
This argument has been written up <a href="http://arxiv.org/abs/1607.05129" target="_blank">here</a> in other words. I am currently aware <a href="http://arxiv.org/abs/1112.1899" target="_blank">of</a> <a href="http://arxiv.org/abs/1304.2208v1" target="_blank">some</a> <a href="http://arxiv.org/abs/1605.05979" target="_blank">other</a> <a href="http://arxiv.org/abs/1605.00497" target="_blank">works</a> trying to find modified dispersion relations from the deformed algebra approach, but I could not find the argument made here in there. In any case, I would be very happy about comments on this topic.</div>
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Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com1tag:blogger.com,1999:blog-2381900080106522441.post-53761672313978566482016-07-19T18:44:00.000+02:002016-07-19T18:44:13.133+02:00Elements of loop quantum gravity<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-jYcLFGVV1O0/V45Xn33zEzI/AAAAAAAAAfA/hTql4u7dY3AobzWIQGocrcbPeH_IMkMRgCLcB/s1600/Introduction_to_loop_quantum_gravity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Elements of loop quantum gravity lecture outline" border="0" height="185" src="https://2.bp.blogspot.com/-jYcLFGVV1O0/V45Xn33zEzI/AAAAAAAAAfA/hTql4u7dY3AobzWIQGocrcbPeH_IMkMRgCLcB/s320/Introduction_to_loop_quantum_gravity.jpg" title="Elements of loop quantum gravity lecture outline" width="320" /></a></div>
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After some longer silence on this blog, I am happy to announce that the <a href="http://arxiv.org/abs/1607.05129">introductory lectures</a> from which excerpts have appeared here before are finally online. <br /><br />The lectures start with a general introduction to quantum gravity, including a theoretical motivation, possible experimental tests, and the previously posted <a href="http://relatively-quantum.bodendorfer.eu/2016/04/approaches-to-quantum-gravity.html">list on approaches</a> to the subject. There is also an improved (as compared to <a href="http://relatively-quantum.bodendorfer.eu/2016/04/updates-on-some-criticisms-of-loop.html">here</a>) estimate on the local Lorentz invariance violation based on anomaly freedom of effective constraints. I am planning to write about it in more detail in the future. <br /><br />Next, an introduction to loop quantum cosmology is given, a draft of which has appeared <a href="http://relatively-quantum.bodendorfer.eu/2016/03/how-does-loop-quantum-cosmology-work.html">here</a> before. The new version features some improvements in the presentation and some simplifications in the derivation.<br /><br />The remaining part of the lectures introduces full loop quantum gravity with minimal technical details. The derivation of geometric operators is sketched and different approaches to the dynamics are discussed. Promising lines of current research are mentioned and evaluated. Exercises are included at the end of each section. <br /><br />The present lecture notes are somewhat complementary to several other sets of lecture notes existing in the literature in that they refrain from technical details and give a broad overview of the subject, including motivations and current trends. If someone spots mistakes or has suggestions for a better presentation, I would be happy to hear about it. Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com2tag:blogger.com,1999:blog-2381900080106522441.post-70994479344554368052016-06-08T13:55:00.000+02:002016-06-09T08:47:27.850+02:00Strings meet Loops via AdS/CFT (Helsinki Workshop on Quantum Gravity 2016)<div class="separator" style="clear: both; text-align: center;">
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<br />
This rerecorded talk was originally given at the <a href="http://research.hip.fi/hwp/qg_helsinki/" target="_blank">Helsinki Workshop on Quantum Gravity</a> on June 2, 2016. I was invited by the organisers to talk about a <a href="http://arxiv.org/abs/1509.02036" target="_blank">recent paper</a>, which was intended as an invitation for people to become interested in the subject, as opposed to giving concrete and detailed results. <br />
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In particular, I am very much looking forward to discussions about this topic and criticism of the ideas, in particular from experts in string theory. In the long run, much can be gained in my point of view from intensifying the exchange between researchers in loop quantum gravity and string theory.<br />
<br />
In this context, it is certainly worth pointing out a recent <a href="https://www.quantamagazine.org/20160112-string-theory-meets-loop-quantum-gravity/" target="_blank">article</a> by Sabine Hossenfelder for Quanta Magazine, as well as a <a href="http://backreaction.blogspot.com/2015/10/when-string-theorists-are-out-of-luck.html" target="_blank">blogpost</a> of hers on the paper I wrote.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com2tag:blogger.com,1999:blog-2381900080106522441.post-17471160787918181912016-06-01T13:27:00.000+02:002016-06-01T13:27:00.190+02:00Graph superspositions and improved regularisations<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-lMSEGtlN90A/V07CNpWl2JI/AAAAAAAAAdg/LEHSIZfyzagO8Q0DFjlqz_tJNIo41VslgCLcB/s1600/Picture_Summation_Of_Graphs_LQG_small.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Sum of graphs in loop quantum gravity" border="0" height="151" src="https://1.bp.blogspot.com/-lMSEGtlN90A/V07CNpWl2JI/AAAAAAAAAdg/LEHSIZfyzagO8Q0DFjlqz_tJNIo41VslgCLcB/s400/Picture_Summation_Of_Graphs_LQG_small.jpg" title="Sum of graphs in loop quantum gravity" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">In loop quantum gravity, the elements of a certain basis in the Hilbert space<br />can be (roughly) interpreted as lattices. Generic quantum states can be constructed <br />as superpositions thereof and a priori have different properties which cannot<br />be realised at the level of individual lattices. </td></tr>
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Let us pick up the topic of extracting effective cosmological dynamics from loop quantum gravity again. Today’s post is about a <a href="https://arxiv.org/abs/1604.02375">recent paper</a> by Emanuele Alesci and Francesco Cianfrani within their framework of “Quantum reduced loop gravity”. This version of loop quantum gravity is arrived at by a gauge fixing to the diagonal metric gauge at the quantum level (as opposed to the classical level as e.g. <a href="https://arxiv.org/abs/1410.5608">here</a>), along with a truncation of the Hamiltonian constraint consistent with a spatially homogeneous setting (in particular, the non-trivial shift resulting from the gauge fixing is dropped). The simplified quantum dynamics resulting form this gauge fixing have allowed to <a href="https://arxiv.org/abs/1402.3155">compute the expectation value</a> of the Hamiltonian constraint in suitable coherent states, leading to the effective Hamiltonian that one finds in loop quantum cosmology. <br /><br />An open issue in this context has been to properly derive the so called <a href="https://arxiv.org/abs/gr-qc/0607039">“improved “ dynamics</a> of loop quantum cosmology, which are consistent with observation and do not feature some unphysical properties of the original formulation. This is somewhat tricky if one uses standard connection variables along with a gauge group like SU(2) or U(1) for the following reason:<br /><a name='more'></a><br />In loop quantum gravity, only holonomies of the connection are well defined operators. In the process of regularising the Hamiltonian, the connection, or rather, the field strength thereof, has to be approximated by holonomies. In practise, this roughly means that one performs the substitution $A \rightarrow \sin (A)$, which is clearly good only as long as A is small. If one now integrates the connection A along a “long” path, one finds that in the context of cosmology the resulting object scales as “matter energy density” times “distance”. In quantum gravity, we expect deviations from known physics as soon as we reach the Planck scale, i.e. when the matter energy density is at the order of the Planck density. Therefore, in order to be consistent with standard physics at large distances, one needs to modify the polymerisation $\int A \rightarrow \sin (\int A)$ to $\int A \rightarrow \sin (\bar \mu \int A) / \bar \mu$, where $\bar \mu$ scales as the inverse distance. This prescription was originally put forward <a href="https://arxiv.org/abs/gr-qc/0607039">using a coarse graining argument</a> and rederived <a href="https://arxiv.org/abs/1410.5608">here</a> and <a href="https://arxiv.org/abs/1512.00713">here</a> using the above argument. <br /><br />The problem now is that the inverse distance scaling necessitates to use non-integer representation labels in the quantum theory, which is not possible with SU(2) or U(1). A way out is to construct the theory from beginning using the gauge group $\mathbb R_{\text{Bohr}}$ after a classical gauge fixing, which was done <a href="https://arxiv.org/abs/1410.5608">here</a>. However, this leaves open how the improved LQC dynamics can arise also from standard loop quantum gravity. <br /><br />In their paper, Emanuele and Francesco now show that at least at the level of the expectation value of the Hamiltonian constraint, one can also get the improved dynamics without resorting to $\mathbb R_{\text{Bohr}}$. Due to the above argument, this cannot be achieved on a fixed graph. Instead, they propose to use a superposition of graphs of all possible sizes, weighted by a natural statistical factor. After a suitable saddle point approximation in the resulting state, they show that the expectation value of the Hamiltonian is given by that of loop quantum cosmology with the improved dynamics. Also, next-to-leading order corrections are computed, with the result that they don’t affect the LQC dynamics qualitatively $\omega < 1$ (see <a href="https://en.wikipedia.org/wiki/Equation_of_state_(cosmology)">here</a> for what $\omega$ is in cosmology), as well as for $\omega = 1$ imposing the universe at the bounce point is sufficiently large w.r.t. the Planck scale.<br /><br />This is, to the best of my knowledge, the first example where graph superpositions were used in loop quantum gravity to derive a result which one cannot simply get by working on a fixed graph. The relevance of such graph superpositions for the theory, in particular for the classical limit and the issue of Lorentz invariance, has been discussed in an <a href="http://relatively-quantum.bodendorfer.eu/2016/04/is-loop-quantum-gravity-based-on.html">earlier post</a>. The possibility to consider quantum superpositions of graphs is the key feature of loop quantum gravity that distinguishes it from approaches based on fixed lattices. <br /><br />To sum up, the paper contains a very interesting new result based on a feature of loop quantum gravity (graph superpositions) that has been mostly ignored so far. The insights gained here can be potentially very important in studies of the classical limit of the theory.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-42860242642176678622016-05-04T14:39:00.000+02:002016-05-04T14:39:41.166+02:00Progress on low spins<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-33XcSQ5Ci8Y/VynVNT9-nSI/AAAAAAAAAcM/3facAA19EWITEfUeo9owTqKefUmNjXdQwCLcB/s1600/Low_spins_Transplanckian_and_continuum_small.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Transplanckian large spins decay into small spins under the dynamics" border="0" height="305" src="https://3.bp.blogspot.com/-33XcSQ5Ci8Y/VynVNT9-nSI/AAAAAAAAAcM/3facAA19EWITEfUeo9owTqKefUmNjXdQwCLcB/s400/Low_spins_Transplanckian_and_continuum_small.jpg" title="Transplanckian large spins decay into small spins under the dynamics" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fundamental (<a href="http://arxiv.org/abs/1303.4752" target="_blank">trans-Planckian</a>) large spins are expected to decay into<br />many small spins under the dynamics. An <a href="http://arxiv.org/abs/1604.06023" target="_blank">explicit calculation</a> showing<br /> this has been given now within group field theory. At the same time,<br />one can coarse grain the many small spins into few large spins to have<br />an effective continuum description. The precise relation between those<br />two, a priori distinct large spin regimes is so far unclear. </td></tr>
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The fundamentally important issue of low and high spins in the dynamics of LQG has been discussed in <a href="http://relatively-quantum.bodendorfer.eu/2016/04/updates-on-some-criticisms-of-loop.html" target="_blank">previous</a> <a href="http://relatively-quantum.bodendorfer.eu/2016/02/what-does-it-mean-to-derive-lqc-from-lqg.html" target="_blank">posts</a>. In short, large geometries can be described using either many low or few large quantum numbers (SU(2) spins), but the respective dynamics needs to be interpreted with care. In particular, using large spins to describe continuum geometries requires to understand the renormalisation group flow of the theory. Most work in LQG has so far been in the context of large spins (without considering renormalisation), where calculations simplify drastically due to the availability certain asymptotic formulae for the SU(2) recouping coefficients with nice geometric interpretations. However, there seems to be some progress now on the low spin front. </div>
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Last month, <a href="http://arxiv.org/abs/1604.06023" target="_blank">Steffen Gielen showed in a paper</a> that a low spin regime dynamically emerges within the group field theory framework. <a href="http://arxiv.org/abs/1408.7112" target="_blank">Group field theory</a> (GFT) is a quantum field theory defined on a group manifold and constitutes a quantum field theory <i>of</i> spacetime (as opposed to a quantum field theory <i>on</i> spacetime). Its basic excitations above the “no space” Fock vacuum are the quanta of geometry that one also considers in loop quantum gravity. In fact, spin foam amplitudes can be obtained from the perturbative expansion of group field theories, linking it to the path integral formulation of loop quantum gravity. </div>
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More precisely, Steffen showed that the group field theory dynamics, in a suitable approximation and for the simplest choice of kinetic term, exponentially suppress all but the lowest spins. The work has been in the context of <a href="http://relatively-quantum.bodendorfer.eu/2016/02/what-does-it-mean-to-derive-lqc-from-lqg.html" target="_blank">deriving loop quantum cosmology from loop quantum gravity</a> and thus naturally relies on some assumptions. A weak coupling limit is used, where the interaction term in the group field theory Lagrangian is neglected. The kinetic term is taken to be a a constant plus the Laplacian on the group (which is well motivated by GFT renormalisation) as well as for a coupled massless scalar field. The dynamics are computed in a hydrodynamic limit using the Gross-Pitaevskii equation. The wave function is taken to be the simplest possible condensate state and the tetrahedra are restricted to be equilateral. Then, for a certain choice of coupling parameters in the group field theory action, the above result follows. For other parameters, no semiclassical geometry seems to emerge. In addition, one has to exclude vanishing spins $j=0$ by hand (which is dynamically stable), as these play a little different role in GFT than in standard LQG. </div>
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The calculation is then extended to a more general class of models previously considered <a href="https://arxiv.org/abs/1602.05881" target="_blank">here</a> and <a href="https://arxiv.org/abs/1602.08271" target="_blank">here</a> by Daniele Oriti, Lorenzo Sindoni, and Edward Wilson-Ewing. By adjusting additional parameters, the dominating spin can be set to some value $j_0$. The case $j_0 = 1/2$ obtained already in the simplest case is however favoured by naturalness and also enters the (heuristic) <a href="http://arxiv.org/abs/gr-qc/0607039" target="_blank">derivation</a> of the Hamiltonian constraint in loop quantum cosmology. In any case, these models then agree with the Friedmann dynamics for large volumes of the universe. The effective dynamics of loop quantum cosmology is obtained up to a correction term which is suppressed by the volume of the universe in Planck units. In particular, higher spins are excited in the Planck regime, which is not taken into account in LQC and thus constitutes a genuine full theory correction to the mini-superspace quantisation in LQC. </div>
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Even despite these assumptions, the result is very interesting and seems for the first time to demonstrate that a low spin regime dynamically emerges from loop quantum gravity, or, more precisely, the group field theory approach to it. This result then underlines the importance of studying the effective dynamics of large spin descriptions of continuum geometries via a renormalisation group analysis starting at the lowest possible spins. Otherwise, a <a href="http://arxiv.org/abs/1303.4752" target="_blank">transplanckian interpretation</a> of the large spin regime seems more appropriate. </div>
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Previous results where low spins were dominant were computations of black hole entropy. However, these computations are by construction insensitive to the quantum dynamics. </div>
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Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-64683596010217432172016-04-25T16:02:00.000+02:002016-04-25T16:05:45.329+02:00Is loop quantum gravity based on discretisations?<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-H4nsvYEuJDs/Vx4eU1We-zI/AAAAAAAAAbw/ruEkYnXPmvgoxoK5dsuP8ndXxZdruC7pgCLcB/s1600/sum_of_two_spin_networks.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Sum of two spin networks for continuum limit of loop quantum gravity" border="0" height="152" src="https://4.bp.blogspot.com/-H4nsvYEuJDs/Vx4eU1We-zI/AAAAAAAAAbw/ruEkYnXPmvgoxoK5dsuP8ndXxZdruC7pgCLcB/s400/sum_of_two_spin_networks.jpg" title="Sum of two spin networks for continuum limit of loop quantum gravity" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Two spin networks, which can be interpreted as Hilbert space elements <br />
describing truncations (or discretisations) of general relativity, <br />
are summed. What is the physics of such states? </td></tr>
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This post is related to last week’s about <a href="http://relatively-quantum.bodendorfer.eu/2016/04/updates-on-some-criticisms-of-loop.html">criticism of loop quantum gravity</a> and a <a href="http://backreaction.blogspot.com/2016/04/dear-dr-b-why-is-lorentz-invariance-in.html?showComment=1461159421001#c3341633063568452673">comment</a> I found in Sabine Hossenfelder’s blog. Saying that loop quantum gravity is based on discretisations quickly leads one to doubt that Lorentz invariance may be a property of LQG, as happened in the comment. So it seems worth to clear up this issue and precisely say in what context discretisations appear, in what context they don’t, and what this means for the physics that are described by LQG.<br />
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<a name='more'></a><br />
The canonical formalism of LQG is most suited to discuss this question, since one has a direct relation to general relativity as the theory which is quantised to obtain LQG. In particular, one really quantises continuum general relativity and the Hilbert space that one obtains thus has to be interpreted as the continuum Hilbert space. However, the elementary excitations on this Hilbert space, <a href="https://arxiv.org/abs/1310.7786">comparable to particles on a Fock space</a>, are discrete quanta of geometry. They are the quantum operators corresponding to certain holonomies and fluxes that one can compute in the classical theory. A finite number of such holonomies however only captures a limited amount of the continuum geometry that is roughly equivalent to a discretisation of GR. Therefore, the (simplest) quantum states that one usually discusses in LQG, which also form a basis of the Hilbert space, have a certain interpretation as discrete geometries. This however doesn’t mean that there cannot exist any continuum states in this Hilbert space, since we can of course take arbitrary linear combination of basis elements, and thus, morally speaking, arbitrary superpositions of lattices. Gaining control over such states and understanding their physics is currently the main open problem in LQG. A computation in a <a href="https://arxiv.org/abs/gr-qc/0405085">simplified toy model</a> explicitly showed that such quantum geometries can be consistent with local Lorentz invariance. <br />
<br />
In conclusion, the Hilbert space of LQG contains elements which can be interpreted as discrete quantum geometries. In different approaches to LQG, these states are given a more or less fundamental status which might suggest that one is really dealing with a discretisation. However, one always needs to keep in mind that one can arbitrarily superpose such states and that the state of quantum geometry that correctly describes our universe at all scales should be searched for in a suitable continuum limit.<br />
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(As a short remark, this does not mean that states containing only a few quanta of geometry cannot be useful when one is considering certain truncations of the theory, such as cosmology. However, one then has to interpret those states in a coarse grained setting and properly understand the renormalisation group flow which relates the dynamics of continuum states to those of coarse grained few quanta states.) </div>
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Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com34tag:blogger.com,1999:blog-2381900080106522441.post-34595221896958763382016-04-21T10:55:00.000+02:002016-04-21T10:55:34.232+02:00Updates on some criticisms of loop quantum gravity<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-T3Ywl-OIYLk/VxiMyww-glI/AAAAAAAAAbY/l_-olg6K5xsUmvQ43oJQVv7J_BVN1mzWwCLcB/s1600/constructive_criticism.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Criticism of loop quantum gravity: Lorentz violations, general relativity limit, ambiguities" border="0" height="288" src="https://3.bp.blogspot.com/-T3Ywl-OIYLk/VxiMyww-glI/AAAAAAAAAbY/l_-olg6K5xsUmvQ43oJQVv7J_BVN1mzWwCLcB/s400/constructive_criticism.png" title="Criticism of loop quantum gravity: Lorentz violations, general relativity limit, ambiguities" width="400" /></a></td></tr>
<tr><td class="tr-caption"><span style="background-color: white; color: #222222; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 12px;">Image from </span><a href="http://www.mysafetysign.com/constructive-criticism-in-progress-workplace-bullying-sign/sku-s-9268">MySafetySign.com</a></td></tr>
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In this post, we will gather some criticism which has been expressed towards loop quantum gravity and comment on the current status of the respective issues. The points raised here are the ones most serious in my own opinion, and different lists and assessments could be expressed by other researchers. <div>
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<h3>
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<li><b>Obtaining general relativity in the appropriate limit</b></li>
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</h3>
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The kinematics and dynamics of loop quantum gravity are defined in a ultra high energy quantum gravity regime, where the usual notion of continuum spacetime or the idea of fields propagating on a background do not make sense any more. It is thus of utmost importance to understand how general relativity and quantum field theory on curved spacetime emerges in a suitable limit, and what the quantum corrections are. There does not seem to complete agreement on how such a limit should be constructed, and depending on the route chosen, one finds different statements about the status of this endeavour in the literature. The level of complexity of this task can be compared having a theory of atoms, and the aim to compute the properties of solids. <br /><br />In order to understand the current status, let us remark that there are two limits which can be taken in order to obtain a spacetime of large scale in loop quantum gravity: large quantum numbers (spins), or many quantum numbers, i.e. very fine spin networks. Much is known about the limit of large spins, where the number of quanta is fixed. Here, one finds strong evidence that the theory reproduces general relativity on large scales both in <a href="http://arxiv.org/abs/gr-qc/0607100" target="_blank">the</a> <a href="http://arxiv.org/abs/gr-qc/0607101" target="_blank">canonical</a> and in <a href="http://arxiv.org/abs/0907.2440" target="_blank">the</a> <a href="http://arxiv.org/abs/1109.0499" target="_blank">covariant</a> approach. The resulting semiclassical picture however corresponds to Regge-gravity on a given lattice which is specified by the underlying graph on which the (coherent) quantum state labelled by the large spins lives. This limit is usually referred to as the “semiclassical” limit in the literature and should not be confused with the following: <br /><br />On the other hand, one can leave the quantum numbers arbitrary, in particular maximally small, and only increase the number of quanta. This corresponds to a continuum limit and it should be a priori preferred over the large spin limit in my opinion. In particular, the large spin and continuum limit do not need to commute and may in principle lead to different physics, even on macroscopic scales. The problem with this approach is that we know only very little about the dynamics in this sector of the theory, apart from several concrete proposals for its implementation. The dynamics on large scales is then expected to emerge via a coarse graining procedure. More discussion on this point of view can be found <a href="http://arxiv.org/abs/1409.1450" target="_blank">here</a>. See also this <a href="http://arxiv.org/abs/1604.06023v1" target="_blank">brand new paper</a> for a dynamical emergence of such a sector consistent with FRW cosmology. <br /><br />As an additional subtlety, we can consider arbitrary superpositions of spin networks, in other words we can have quantum superpositions of “lattices”. The impact on the dynamics of this feature is so far unclear and may strongly depend on the regularisation details of the Hamiltonian, e.g. whether it is graph preserving or not, and thus superselecting. <div>
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<div>
Concerning a limit to obtain quantum field theory on curved spacetimes, we point out these <a href="http://arxiv.org/abs/gr-qc/0207030" target="_blank">pioneering</a> <a href="http://arxiv.org/abs/1504.02171" target="_blank">works</a>.<br /><br />To conclude, the situation of whether general relativity emerges in the continuum limit is so far unclear, whereas there is strong evidence for a Regge-truncation thereof emerging in the large spin limit. Whether one is satisfied with one or the other limit, or a combination of both, also depends on the following problem. </div>
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<br /><br /><h3>
<ul>
<li><b>Local Lorentz invariance</b></li>
</ul>
</h3>
</div>
<div>
<br /> It is sometimes suggested that loop quantum gravity is not locally Lorentz invariant in the sense that modified dispersion relations might arise which could be in conflict with observation. Unfortunately, our current understanding of loop quantum gravity does not allow us to answer whether there are Lorentz violations, and how severe they might be. In order to make a meaningful statement, one would essentially have to identify a quantum state corresponding to Minkowski space, which should be thought of as a (possibly infinite) superposition of lattices, put matter fields thereon, and track their dynamics, including back-reaction, on a coarse grained scale where geometry can already be considered smooth. This is a formidable task, and currently out of reach not because of lacking proposals for how to define the problem, but mainly due to computational complexity. <br /><br />In order to judge certain statements that one might find in the literature or on the internet, one should keep the following in mind to avoid confusion: <br /></div>
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<div>
<b>Discrete eigenvalues of geometric operators don't imply Lorentz violations</b></div>
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<b><br /></b> One might naively think that discrete eigenvalues of geometric operators violate special relativity: if an observer at rest measures a certain discrete eigenvalue of, say, an area, what does another observer measure who is not at rest? The short answer is that he might in principle measure any value for the area, as long as it is in the (discrete) spectrum of the area operator. However, the expectation value can still transform properly according to special relativity. A well-known example of this is the theory of angular momentum: while the eigenvalues of one of the components of the angular momentum are always (half)-integers, expectation values transform properly according to the continuous rotation symmetry. This point has been made for example <a href="http://arxiv.org/abs/gr-qc/0205108" target="_blank">here</a>, with further discussion <a href="http://arxiv.org/abs/gr-qc/0405085" target="_blank">here</a>. Similar conclusions are also drawn in <a href="http://journals.aps.org/pr/abstract/10.1103/PhysRev.71.38" target="_blank">other</a> <a href="http://arxiv.org/abs/gr-qc/0311055" target="_blank">contexts</a>. <br /></div>
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<div>
<b>Internal gauge groups do not determine isometries of the spacetime </b></div>
<div>
<b><br /></b>Different formulations of loop quantum gravity, canonically or covariant, use different <i>internal</i> gauge groups. The analogue of these groups in QCD is thus SU(3), and not the Lorentz group, which is a global symmetry of Minkowski space. </div>
<div>
While the covariant path integral formulations of the Lorentzian theory use either <a href="http://arxiv.org/abs/1012.1739v3" target="_blank">SL(2, C)</a> or <a href="http://arxiv.org/abs/0711.0146" target="_blank">SU(2)</a> in a gauge-fixed version, the Lorentzian canonical theory in 3+1 dimensions can be formulated using either <a href="http://arxiv.org/abs/gr-qc/9410014" target="_blank">SU(2)</a> [45], SO(1,3) (see <a href="http://arxiv.org/abs/gr-qc/0201087" target="_blank">here</a>, <a href="http://arxiv.org/abs/0811.1916" target="_blank">here</a>, <a href="http://arxiv.org/abs/1103.4057" target="_blank">here</a>, and <a href="http://arxiv.org/abs/1105.3703" target="_blank">here</a>), or <a href="http://arxiv.org/abs/1105.3703" target="_blank">SO(4)</a>. This is because one is coding the spatial metric and its momentum in a connection, whereas the signature of spacetime in the canonical formalism is determined by the Hamiltonian constraint, more precisely a relative sign between two terms. In fact, the structure of spacetime, coded in the hypersurface deformation algebra, is already set at the level of metric variables, and completely insensitive of the additional gauge redundancy that one introduces by passing to connection variables. Also, it does not matter for this whether the connection that one uses can be interpreted as the pullback of some manifestly covariant spacetime connection. While it is a possibility that only a certain choice of variables or internal gauge group leads to a consistent quantum theory in agreement with current bounds on Lorentz violations, such a conclusion cannot be drawn given our current understanding of the theory. </div>
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<br /></div>
<div>
<b>At which order do we expect possible Lorentz violations? </b></div>
<div>
<br />In order to parametrise the violation of Lorentz invariance in a model-independent way, one usually constrains the free parameters $c_n$ in a modified dispersion relation </div>
<div>
<div>
$$ E^2 = m^2 + p^2 + \sum_{n\geq 3} c_n \frac{p^n}{E_{\text{pl}}^n} $$</div>
</div>
<div>
where $E_{\text{pl}} = \sqrt{\hbar c / G}$ is the Planck mass. It is worthwhile to formulate a general expectation at what order one expects the first quantum corrections to appear. The two reasonable alternatives here are n = 3 or n = 4, with the difference being a first quantum correction in $\sqrt{\hbar}$ or in $\hbar$. While an appropriate calculation as outlined above will finally have to decide this question, we can make two observations. First, in the case of loop quantum cosmology, where we know how to compute the quantum corrections to the classical theory, we obtain a correction of the order $\hbar$, i.e. the effective Friedmann equation </div>
<div>
$$ H^2 \sim \rho \left(1-\frac{\rho}{\rho_{\text{crit}}} \right), ~~~ \rho_{\text{crit}} \sim \frac{1}{G^2 \hbar} $$</div>
<div>
Second, the concept of area has a much more fundamental status in loop quantum gravity than that of length, and one would thus expect quantum corrections as an expansion in $l_{\text{pl}}^2 = \hbar G / c^3$, rather than $l_{\text{pl}}$. Both of these observations hint that one would expect that a first non-vanishing correction in the dispersion relation should appear at order n=4. </div>
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<a href="http://arxiv.org/abs/1304.5795" target="_blank"> Experimentally</a>, it turns out that there are very strong constraints on the n = 3 terms, whereas constraints on n = 4 are much weaker, yet still restrictive, and currently still plagued by some astrophysical uncertainties. Thus, even if one expects Lorentz- violating effects to arise from loop quantum gravity at order n = 4, this is not necessarily in conflict with current experiments. Still, the stringent bounds put by experiment might rule out loop quantum gravity once we will be able to properly derive Lorentz violations from it. </div>
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<br /><br /><h3>
<ul>
<li><b>Testability and ambiguities</b></li>
</ul>
</h3>
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<br /> A general problem for theories of quantum gravity is to come up with testable predictions which are independent of free parameters in the theory that can be tuned in order to hide any observable effect below the measurement uncertainty. Furthermore, one would like to have a uniquely defined fundamental theory whose dynamics depend only on a finite number of free parameters. <br /><br />In loop quantum cosmology, much progress towards predictions for observable effects <a href="http://arxiv.org/abs/1302.0254" target="_blank">have</a> <a href="http://arxiv.org/abs/1504.07559" target="_blank">been</a> <a href="http://arxiv.org/abs/1510.08766" target="_blank">made</a>, however the choice of parameters in the models, such as the e-foldings during inflation, can so far hide any observable effects. Also, there are <a href="http://arxiv.org/abs/1602.04452" target="_blank">different approaches</a> to the dynamics of cosmological perturbations. <br /><br /><div>
Within full loop quantum gravity (and also loop quantum cosmology), the Barbero-Immirzi parameter $\beta$, a free real parameter entering, e.g., the spectra of geometric operators, constitutes a famous ambiguity. It enters the classical theory in a canonical transformation which is not implementable as a unitary transformation or even an algebra automorphism at the quantum level and thus constitutes a quantisation ambiguity. Since this parameter can enter physical observables, one would like to fix it by an experiment or derive its (only consistent) value by theoretical means. An example for how to fix it by theoretical means is to consider black hole entropy computations and match them to the expected Bekenstein-Hawking entropy. Within the <a href="http://arxiv.org/abs/gr-qc/0005126" target="_blank">original approach</a> to black hole entropy from loop quantum gravity, this gives a certain value for $\beta$, however this computation neglects a possible running of the gravitational constant, i.e. it identifies the high and low energy Newton constants. More recently, it has been observed that the Bekenstein-Hawking entropy can also be reproduced by an <a href="http://arxiv.org/abs/1212.4060" target="_blank">analytic continuation</a> of $\beta$ to the complex self-dual values $\pm i$, which interestingly also <a href="http://arxiv.org/abs/1303.4752" target="_blank">agrees</a> with a computation of the effective action. In fact, the classical theory is easiest when expressed in self-dual variables, so that it might turn out that the value $\beta = \pm i$ could be favoured also in the quantum theory. The current problem is however that the quantum theory is ill-defined for complex $\beta$ and the above mentioned results were obtained via analytic continuation from real beta.</div>
<br />In addition to quantisation ambiguities resulting from the choice of variables, as above (see also <a href="http://relatively-quantum.bodendorfer.eu/2016/03/yang-mills-analogues-of-ambiguities-in.html" target="_blank">this</a> and <a href="http://relatively-quantum.bodendorfer.eu/2016/03/are-spectra-of-lqg-geometric-operators.html" target="_blank">this post</a>), there are quantisation ambiguities in the regularisation of the Hamiltonian constraint, and thus the dynamics of the theory. These go somewhat beyond factor ordering, as the techniques used in regularising the Hamiltonian also involve classical Poisson bracket identities which are used to construct otherwise ill-defined operators. The requirement of anomaly freedom of the quantum constraint algebra already removed many of those ambiguities in <a href="http://arxiv.org/abs/gr-qc/9606089" target="_blank">Thiemann’s original construction</a>. However, the precise notion of anomaly freedom has been <a href="arxiv:hep-th/0501114" target="_blank">criticised</a> on the ground that it corresponds to a certain “on-shell” notion, which is however the one that makes sense in the context of Thiemann's construction. More recent work based on a slightly changed quantisation seems to make significant progress towards the goal of implementing a satisfactory “off-shell” version of the quantum constraint algebra, at least <a href="http://arxiv.org/abs/1105.0636" target="_blank">in</a> <a href="http://arxiv.org/abs/1210.6869" target="_blank">simplified</a> <a href="http://arxiv.org/abs/1204.0211" target="_blank">toy</a> <a href="http://arxiv.org/abs/1401.0931" target="_blank">models</a>. The possible regularisations then seem to be very constrained, although no unique prescription has emerged so far. <br /><br />To conclude, it is so far not possible to extract definite predictions from full loop quantum gravity due to the existence of ambiguities in its construction, whose influence on the dynamics is not well understood. However, there is promising recent work on the removal of such ambiguities, which could eventually allow for clear predictions which would allow to falsify the theory. <div class="page" title="Page 11">
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Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com12tag:blogger.com,1999:blog-2381900080106522441.post-72367036659464730892016-04-12T15:46:00.000+02:002016-04-13T11:11:34.905+02:00Approaches to quantum gravity<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-CrpDfBxLSFU/Vwz76MUnwKI/AAAAAAAAAbA/KF2fMBLpRMEU5_fuIUCj5FvQtKHthVyZgCLcB/s1600/approaches_to_quantum_gravity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="245" src="https://4.bp.blogspot.com/-CrpDfBxLSFU/Vwz76MUnwKI/AAAAAAAAAbA/KF2fMBLpRMEU5_fuIUCj5FvQtKHthVyZgCLcB/s400/approaches_to_quantum_gravity.jpg" width="400" /></a></div>
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Again as a part of a lecture series in preparation, I compiled a list of the currently largest existing research programmes aimed at finding a quantum theory of gravity (while unfortunately omitting <a href="https://en.wikipedia.org/wiki/Quantum_gravity" target="_blank">some smaller</a>, yet very interesting, approaches). A much more extensive account is given in <a href="http://www.cambridge.org/us/academic/subjects/physics/cosmology-relativity-and-gravitation/approaches-quantum-gravity-toward-new-understanding-space-time-and-matter">here</a>. For an historical overview, I recommend <a href="http://arxiv.org/abs/gr-qc/0006061">this paper</a>.<br />
<a name='more'></a><br />
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<b>Semiclassical gravity </b><br />
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Semiclassical gravity is a first step towards quantum gravity, where matter fields are treated using full quantum field theory, while the geometry remains classical. However, semiclassical gravity goes beyond quantum field theory on curved spacetimes: the energy-momentum tensor determining the spacetime geometry via Einstein’s equations is taken to be the expectation value of the QFT energy-momentum tensor. The state in which this expectation value is evaluated in turn depends on the geometry, and one has to find a self-consistent solution. Many of the <a href="http://arxiv.org/abs/gr-qc/9602052" target="_blank">original problems</a> of semi-classical gravity have been addressed recently and the theory can be applied <a href="http://arxiv.org/abs/0801.2850" target="_blank">in practise</a>. <br />
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<b>Ordinary quantum field theory </b><br />
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The most straight forward approach to quantising gravity itself is to use ordinary perturbative quantum field theory to quantise the deviation of the metric from a given background. While it turned out that general relativity is non-renormalisable in the standard picture, it is possible to use <a href="http://arxiv.org/abs/1209.3511" target="_blank">effective field theory techniques</a> in order to have a well-defined notion of perturbative quantum gravity up to some energy scale lower than the Planck scale. Ordinary effective field theory thus can describe a theory of quantum gravity at low energies, whereas it does not aim to understand quantum gravity in extreme situations, such as cosmological or black hole singularities. <br />
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<b>Supergravity</b> <br />
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Supergravity has been invented with the hope of providing a unified theory of matter and geometry which is better behaved in the UV than Einstein gravity. As opposed to standard supersymmetric quantum field theories, supergravity exhibits a local supersymmetry relating matter and gravitational degrees of freedom. In the symmetry algebra, this fact is reflected by the generator of local supersymmetries squaring to spacetime-dependent translations, i.e. general coordinate transformations. <br />
<br />
While the local supersymmetry generally improved the UV behaviour of the theories, it turned out that also supergravity theories were <a href="http://arxiv.org/abs/hep-th/9905017" target="_blank">non-renormalisable</a>. The only possible exception seems to be N = 8 supergravity in four dimensions, which is known to be <a href="http://arxiv.org/abs/0905.2326" target="_blank">finite at four loops</a>, but it is unclear what happens beyond. Nowadays, supergravity is mostly considered within string theory, where 10-dimensional supergravity appears as a low energy limit. Due to its finiteness properties, string theory can thus be considered as a UV-completion for supergravity. Moreover, 11-dimensional supergravity is considered as the low-energy limit of M-theory, which is conjectured to have the 5 different string theories as specific limits. <br />
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<b>Asymptotic safety </b><br />
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<div>
The underlying idea of <a href="http://arxiv.org/abs/1202.2274" target="_blank">asymptotic safety</a> is that while general relativity is perturbatively non- renormalizable, its renormalisation group flow might possess a non-trivial fixed point where the couplings are finite. In order to investigate this possibility, the renormalisation group equations need to be solved. For this, the “theory space”, i.e. the space of all action functionals respecting the symmetries of the theory, has to be suitably truncated in practise. Up to now, much evidence has been gathered that general relativity is asymptotically safe, <a href="http://arxiv.org/abs/1311.2898" target="_blank">including matter couplings</a>, however always in certain truncations, so that the general viability of the asymptotic safety scenario has not been rigorously established so far. Also, one mostly works in the Euclidean. At microscopic scales, one finds a fractal-like effective spacetime and a reduction of the (spectral) dimension from 4 to 2 (or 3/2, which is <a href="http://arxiv.org/abs/1104.5505" target="_blank">favoured by holographic arguments</a>, depending on the calculation). Moreover, a <a href="http://arxiv.org/abs/0912.0208" target="_blank">derivation of the Higgs mass</a> has been given in the asymptotic safety scenario, correctly predicting it before its actual measurement. <br />
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<b>Canonical quantisation: Wheeler-de Witt </b><br />
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The oldest approach to full non-perturbative quantum gravity is the Wheeler-de Witt theory, i.e. the canonical quantisation of the Arnowitt-Deser-Misner formulation of general relativity. In this approach, also known as quantum geometrodynamics, one uses the spatial metric and its conjugate momentum as canonical variables. The main problems of the Wheeler-de Witt approach are of mathematical nature: the Hamiltonian constraint operator is extremely difficult to define due to its non-linearity and a Hilbert space to support is is not known. It is therefore strongly desirable to find new canonical variables for general relativity in which the quantisation is more tractable. While the so called “problem of time”, i.e. the absence of a physical background notion of time in general relativity, is present both in the quantum and the classical theory, possible ways to deal with it are <a href="http://arxiv.org/abs/gr-qc/9210011" target="_blank">known</a> and <a href="http://arxiv.org/abs/1206.3807" target="_blank">continuously developed</a>. <br />
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<b>Euclidean quantum gravity </b><br />
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In Euclidean quantum gravity, a Wick rotation to Euclidean space is performed, in which the gravity path integral is formulated as a path integral over all metrics. Most notably, this approach allows to extract thermodynamic properties of black holes. In practise, the path integral is often approximated by the exponential of the classical on-shell action. Its main problematic aspect is that the Wick rotation to Euclidean space is well defined only for a certain limited class of spacetimes, and in particular dynamical phenomena are hard to track. An overview can be found <a href="http://jdsweb.jinr.ru/record/51903/files/Quantum%20Gravitation.pdf" target="_blank">here</a>.<br />
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<b>Causal dynamical triangulations </b><br />
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Causal dynamical triangulations <a href="http://arxiv.org/abs/1302.2173" target="_blank">(CDT)</a> is a non-perturbative approach to rigorously define a path integral for general relativity based on a triangulation. It grew out of the Euclidean dynamical triangulations programme, which encountered several difficulties in the 90’s, by adding a causality constraint on the triangulations. The path integral is then evaluated using Monte Carlo techniques. The phase diagram of CDT in four dimensions exhibits three phases, one of which is interpreted as a continuum four-dimensional universe. Moreover, the transition between this phase and one other phase is of second order, hinting that one might be able to extract a genuine continuum limit. More recently, also Euclidean dynamical triangulations has been reconsidered and evidence for a good semiclassical limit has been <a href="http://arxiv.org/abs/1604.02745" target="_blank">reported</a>. <br />
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<b>String theory </b><br />
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<a href="https://en.wikipedia.org/wiki/String_theory" target="_blank">String theory</a> was initially conceived as a theory of the strong interactions, where the particle concept is replaced by one-dimensional strings propagating in some background spacetime. It was soon realised that the particle spectrum of string theory includes a massless spin 2 excitation, which is identified as the graviton. Moreover, internal consistency requirements demand (in lowest order) the Einstein equations to be satisfied for the background spacetimes. String theory thus is automatically also a candidate for a quantum theory of gravity. The main difference of string theory with the other approaches listed here is thus that the quantisation of gravity is achieved via unification of gravity with the other forces of nature, as opposed to considering the problem of quantum gravity separately. <br />
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The main problem of string theory is that it seems to predict the wrong spacetime dimension: 26 for bosonic strings, 10 for supersymmetric strings, and 11 in the case of M-theory. In order to be compatible with the observed 3 + 1 dimensions at the currently accessible energies, one therefore needs to compactify some of the extra dimensions. In this process, a large amount of arbitrariness is introduced and it has remained an open problem to extract predictions from string theory which are independent of the details of the compactification. Also, our knowledge about full non-perturbative string theory is limited, with the main exceptions of D-branes and using AdS/CFT as a definition of string theory. <br />
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<b>Gauge / gravity </b></div>
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The gauge / gravity correspondence, also known as AdS/CFT, has <a href="http://arxiv.org/abs/hep-th/9711200" target="_blank">grown out of string theory</a>, but was later recognised to be applicable more widely. Its statement is a (in most cases conjectured) duality between a quantum gravity theory on some class of spacetimes, and a gauge theory living on the boundary of the respective spacetime. Once a complete dictionary between gravity and field theory computations is known, one can in principle use the gauge / gravity correspondence as a definition of quantum gravity on that class of spacetimes. <br />
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The main problem of gauge / gravity as a tool to understand quantum gravity is the lack of a complete dictionary between the two theories, in particular for local bulk observables. Also, it is usually very hard to find gauge theory duals of realistic gravity theories already at the classical gravity level (i.e. in an appropriate field theory limit), and many known examples are very special supersymmetric theories. <br />
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<b>Loop quantum gravity </b></div>
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<b><br /></b>
Loop quantum gravity originally started as a canonical quantisation of general relativity, in the spirit of the Wheeler-de Witt approach, however based on connection variables parametrising the phase space of general relativity, e.g. the Ashtekar-Barbero variables based on the gauge group SU(2). The main advantage of these variables is that one can rigorously define a Hilbert space (containing Wilson-"loops" of the connection, hence the name) and quantise the Hamiltonian constraint. The application of the main technical and conceptual ideas of loop quantum gravity to quantum cosmology resulted in the subfield of <a href="http://arxiv.org/abs/1108.0893" target="_blank">loop quantum cosmology</a>, which offers a quantum gravity resolution of the big bang singularity and successfully makes <a href="http://arxiv.org/abs/1309.6896" target="_blank">contact to observation</a>. <br />
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The main problem of loop quantum gravity is to obtain general relativity in a suitably defined classical limit. In other words, the fundamental quantum geometry present in loop quantum gravity has to be coarse grained in order to yield a smooth classical spacetime, while the behaviour of matter fields coupled to the theory should be dictated by standard quantum field theory on curved spacetimes in this limit. The situation is thus roughly the opposite of that in string theory. Also, it has not been possible so far to fully constrain the regularisation ambiguities that one encounters in quantising the Hamiltonian constraint. In order to cope with these issues, a path integral approach, known as spin foams, has been developed, as well as the group field theory approach, which is well suited for dealing with the question of renormalisation. <br />
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Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com5tag:blogger.com,1999:blog-2381900080106522441.post-45848212475438231892016-03-25T12:31:00.000+01:002016-04-08T05:46:26.211+02:00How does loop quantum cosmology work?<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-STbA_hY4x7o/VvTxfXe6vcI/AAAAAAAAAac/BLor-drPFe8ty5htNQARsBtadLU-nniJg/s1600/loop_quantum_cosmology_big_bounce_big_bang.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Loop quantum cosmology big bounce vs Wheeler-de Witt big bang" border="0" height="268" src="https://1.bp.blogspot.com/-STbA_hY4x7o/VvTxfXe6vcI/AAAAAAAAAac/BLor-drPFe8ty5htNQARsBtadLU-nniJg/s400/loop_quantum_cosmology_big_bounce_big_bang.jpg" title="Loop quantum cosmology big bounce vs Wheeler-de Witt big bang" width="400" /></a></td></tr>
<tr><td class="tr-caption">The evolution of the expectation value of the volume v of the universe is plotted. <br />
The blue and orange lines are the expanding and contracting branches in the <br />
Wheeler-de Witt theory. The green curve follows from loop quantum cosmology <br />
and exhibits a quantum bounce close to Planck density. </td></tr>
</tbody></table>
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These days I have been working on lecture notes for an introductory lecture series on loop quantum gravity. An <a href="http://arxiv.org/abs/gr-qc/0702030" target="_blank">introductory article</a> by Abhay Ashtekar introduced this subject via a discussion on loop quantum cosmology (LQC), where already many of the essential features of loop quantum gravity are present and can be studied in a simplified setting. I think that this is a very useful pedagogical approach and I also wanted to incorporate it into my lectures. I just finished my first draft of a lecture about this subject, focussing on a specific exactly soluble LQC model and its comparison to a similar quantisation using the Wheeler-de Witt framework. Mostly I follow the <a href="http://arxiv.org/abs/0710.3565" target="_blank">original paper,</a> with some additional comments, slight rearrangements, and omission of more advanced material that is not necessary in an introductory course in my point of view. The current draft is available <a href="https://drive.google.com/file/d/0B30fkrPRtkVRanJCa1ZESjkwVzQ/view?usp=sharing" target="_blank">here</a>, and comments are always welcome. <br />
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The short version goes as follows:<br />
<a name='more'></a><br />
In both cases, Wheeler-de Witt and loop quantum cosmology, one quantises general relativity minimally coupled to a massless scalar field in a homogeneous and isotropic setting. At the classical level, one constructs a Dirac observable corresponding to the volume of the universe at a specific point in time specified by the value that the scalar field takes. As is well known, such models lead to either big bang or big crunch singularities. <br />
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One can now quantise using Wheeler-de Witt techniques (both the spatial volume and its conjugate momentum are well defined operators) and show that one can reduce the problem of finding the physical Hilbert space to that of solving a 1+1-dimensional Klein-Gordon equation due to a suitable choice of variables. On the physical Hilbert space, one can represent a Dirac observable corresponding to the volume at some specific value of the internal scalar field time $\phi$. One sees that the expectation value of these observables follows the classical trajectories, i.e. $V_0 e^{\pm \phi}$. A more <a href="http://arxiv.org/abs/1006.3837" target="_blank">in depth analysis</a> based on consistent histories (not detailed in the lecture) also shows that superpositions of contracting and expanding branches do not lead to singularity resolution (see however <a href="http://arxiv.org/abs/1501.04181" target="_blank">this paper</a> based on the Bohmian approach to quantum theory). <br />
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In loop quantum cosmology on the other hand one finds some small, but important differences in the computation. Due to the loop quantum gravity inspired Hilbert space, the spatial volume is a well defined operator, whereas its conjugate momentum $b$ exists only in exponentiated form $e^{i n b}$, $n \in \mathbb Z$. This reflects the underlying discrete quantum geometry. Therefore, one needs to regularise $b$ in terms of such ``U(1)-holonomies'', with the simplest choice being $\sin(b)$. Since we are working in Planck units, $b$, corresponding on-shell to the matter energy density, is small and $\sin(b)$ is an excellent approximation to $b$ if we are far away from the Planck density. At Planck density however, a departure from classical physics can be expected. Indeed, passage to the Klein-Gordon equation now requires a slightly different choice of variables, leading to a different form of the operator corresponding to the spatial volume on the physical Hilbert space. It turns out that the expectation value of this operator behaves as $V_{\text{min}} \cosh(\phi)$. The big bang / big crunch singularities are thus substituted by a quantum bounce.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com1tag:blogger.com,1999:blog-2381900080106522441.post-53117595487110455862016-03-15T15:58:00.000+01:002016-03-15T15:58:14.421+01:00Yang-Mills analogues of the ambiguities in defining LQG geometric operators<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-00MEm1PoEj8/VughofKSixI/AAAAAAAAAaM/4LhwPcGtjPcPmNwU9rVAp_oXwLJ4jP5IQ/s1600/ambiguity_geometric_operator_loop_quantum_gravity_yang_mills.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Ambiguity in the geometric operators in loop quantum gravity from changing connection variables" border="0" height="270" src="https://2.bp.blogspot.com/-00MEm1PoEj8/VughofKSixI/AAAAAAAAAaM/4LhwPcGtjPcPmNwU9rVAp_oXwLJ4jP5IQ/s400/ambiguity_geometric_operator_loop_quantum_gravity_yang_mills.jpg" title="Ambiguity in the geometric operators in loop quantum gravity from changing connection variables" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Changing the connection variables underlying loop quantum gravity<br />also changes the geometric operators. They measure geometry with respect<br />to the metric encoded in the Yang-Mills electric field $E^a_i$. </td></tr>
</tbody></table>
<b>tl;dr</b>: In some cases there are such analogues, but they are rather awkward. <br /><br />A comment to a <a href="http://relatively-quantum.bodendorfer.eu/2016/03/are-spectra-of-lqg-geometric-operators.html">recent post</a> on the current status of the issue of the spectra of geometric operators in loop quantum gravity raised the question of whether such ambiguities can also be found in Yang-Mills theory. The question if of course very interesting, however I am not aware of any reference commenting on it. So let me try.<br /><a name='more'></a><br />To begin with, let us recall what is the issue in the context of loop quantum gravity. There, since one is in the context of general relativity, observables (= gauge invariant quantities) have to be constructed. One way to do this is to construct an observable algebra at the classical level and then to quantise it. Such a choice of observable algebra then effectively leads to a specific choice of quantisation variables, which as a result may change the properties of geometric operators, e.g. their spectra. <br /><br />An instructive example is provided by considering general relativity coupled to a scalar field. Here, the canonical variables are the spatial metric $q_{ab}$, its conjugate momentum $P^{ab}$, the scalar field $\phi$, as well as its momentum $\pi$. One can now construct connection variables in the gravitational sector as follows. We first extend our theory to incorporate a local SO(3) gauge invariance by going over to the variables $e_{ai}$ and $K_{ai}$, where $e_{ai}$ is simply a tetrad for the metric, $q_{ab} = e_{ai} e_{b}^i$, and $K_{ai} e_{b}^i = K_{ab}$ defines the extrinsic curvature (entering $P^{ab}$). Instead of $e_{ai}$, we will actually use the densitised co-tetrad $E^{a}_j$, which is conjugate to $K_a^i$. In order to erase the additional degrees of freedom, we have to introduce the Gauss law $K_{a[i} E^{b}_{j]} = 0$. <br /><br />The key step in order to arrive at the connection variables underlying loop quantum gravity is now the following. From the tetrad, we construct the spin connection $\Gamma_a^i$, which annihilates the tetrad in a suitably defined covariant derivative. We can now construct the connection $A_{a}^i = \Gamma_a^i + \beta K_{a}^i$, which turns out to have canonical brackets with $E^{a}_i / \beta$. Here, $\beta$ is a priori a non-zero real number that we are free to choose. We can now proceed to the quantum theory, where the configuration space is the space of all holonomies obtained from $A_a^i$. <br /><br />If we now consider the area operator, then its eigenvalues will scale with $\beta$. This follows since fluxes in the quantum theory are constructed from $E^{a}_i / \beta$, while the physical geometry is measured only by $E^{a}_i$. However, it is the (squared) fluxes rescaled with $\beta$ which act via multiplication with $\sqrt{j(j+1)}$ on spin network edges labelled by the spin $j$. Therefore, the real number $\beta$ (known as the Barbero-Immirzi parameter) already introduces an ambiguity. <br /><br />Let’s come back now to the scalar field. Instead of just taking $\beta$ to be a real number, we could also take it to be a function of the scalar field. In order to ensure canonicity of the variables, some changes in the scalar field momentum $\pi$ are then necessary, however this introduces no problems. Then, the area operator is changed in such a way that it does not measure area any more, but $\beta(\phi) \cdot \text{area}$. An operator measuring area could then be recovered by inserting an appropriate operator for the scalar field. <br /><br />While this last construction may seem a bit awkward, it can in fact be well motivated: The choice $\beta = \phi^2$ follows if one wants an observable algebra which is defined in relation to a constant mean curvature gauge fixing in the context of a conformally coupled scalar field, as discussed <a href="http://arxiv.org/abs/1203.6525">here</a> in detail. <br /><br />Let us now come back to the issue of a similar phenomenon in Yang-Mills theory. As we have seen above, the appearance of such ambiguities in LQG is related to the fact that the connection used there is assembled from an affine piece, the spin connection, and a tensorial piece, the extrinsic curvature. The tensorial piece can be modified by an arbitrary scalar function. <br /><br />Without such a split however, a similar ambiguity does not occur. Consider for example the Gauss law $D_a E^a_i = 0$ of Yang-Mills theory. It acts as $A_a^i \rightarrow A_a^i - D_a \lambda^i$. If we would simply rescale $A_a^i$ by a constant, we would immediately find that the transformation properties of $A_a^i$ under the Gauss law have been spoiled, since the affine piece would now be rescaled. Instead, if we rescale the metric as above and then construct the affine piece of the connection from it, the constant would drop and the affine part of the connection would remain intact. In the case of a non-trivial scalar function, the spin connection would change, since it would then be the spin connection with respect to the rescaled metric. But then the new Gauss law would again be intact. <br /><br />So can we make such splits in general? To the best of my knowledge not. In the case of SU(2) Yang-Mills theory in 3 dimensions, we could just repeat the construction outlined above. We could solve the Gauss law and introduce a metric and its conjugate momentum as SU(2) - invariant degrees of freedom. For this, we would need to demand that $E^{ai} E^{b}_i$ is a positive definite 3-metric, something that one naturally demands in GR, but no necessarily in Yang-Mills theory. Then however, the above construction can be repeated and a new connection, based on the rescaled “metric” and its momentum could be performed. <br /><br />The more radical changes in the geometric operators which resulted from reducing the components of the metric via gauge fixing don’t seem to have an analogue in Yang-Mills theory, as there simply is nothing left to gauge fix once one would be at the “metric” level.<br /><br />What about other gauge groups? It <a href="http://arxiv.org/abs/1105.3703">seems</a> that similar constructions are not possible in general, at least when reducing to a metric and its conjugate. The only other working example in 3 dimensions that I know is to use the groups SO(4) or SO(1,3). Then however, the corresponding Yang-Mills electric fields $E^{a}_{IJ}$ with $I,J = 0,1,2,3$ have to be subject to an additional constraint, known as the simplicity constraint $E^{a}_{[IJ} E^{b}_{KL]} = 0$. This is natural in the context of general relativity, but again probably not in the case of Yang-Mills theory. In $D+1$ dimensions, it turns out that SO(D+1) and SO(1,D) work, again subject to the simplicity constraints. <br /><br />To conclude, the reason for the ambiguity in the geometric operators in loop quantum gravity is the freedom in choosing a metric on which the connection variables are based. An example for a good reason to change this metric is the construction of observables in order to access the physical Hilbert space. In Yang-Mills theory on the other hand, one always starts with connection variables, and it is highly unnatural to temporarily pass to similar metric variables in order to construct a different connection. Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com3tag:blogger.com,1999:blog-2381900080106522441.post-65918650949928302962016-03-11T17:07:00.000+01:002016-03-11T17:07:02.095+01:00Some comments on the properties of observable algebras<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-z1M6FCmwvws/VuGyXk2o64I/AAAAAAAAAZ4/noxfviwNxmM/s1600/Geodesic_rerouting_in_radial_gauge.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="248" src="https://3.bp.blogspot.com/-z1M6FCmwvws/VuGyXk2o64I/AAAAAAAAAZ4/noxfviwNxmM/s400/Geodesic_rerouting_in_radial_gauge.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">An example of non-locality in an observable algebra. Observables are <br />
defined as fields at the endpoints of geodesics. Changing the metric<br />
along the geodesic via $P^{rA}$, the field conjugate to some of the relevant<br />
components of the spatial metric, leads to an apparent non-locality due to<br />
a rerouting of the geodesic.</td></tr>
</tbody></table>
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A <a href="http://relq.blogspot.com/2016/03/the-algebra-of-observables-in-gauian.html" target="_blank">recent post</a> contained a talk of mine of a specific construction of observables in general relativity, where a physical coordinate system was specified by the endpoints of certain geodesics. The so constructed coordinates are simply Gaußian normal coordinates and the usual relational construction of observables as fields at a physical point could proceed in the standard manner.<br />
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The motivation for the underlying paper came from AdS/CFT, where people were interested in constructing <a href="http://arxiv.org/abs/1311.3020" target="_blank">CFT operators corresponding to scalar fields in the bulk</a>, localised at the endpoint of a geodesic. While the scalar fields were argued to commute in this construction, a <a href="http://arxiv.org/abs/1507.07921" target="_blank">contradictory statement</a> was recently made in a perturbative calculation. To settle this question, we upgraded a <a href="http://arxiv.org/abs/1506.09164" target="_blank">calculation referring to spatial geodesics</a>, based on <a href="http://arxiv.org/abs/1403.8062" target="_blank">this seminal work</a>, to the <a href="http://arxiv.org/abs/1510.04154" target="_blank">case of spacetime geodesics</a> considered in the AdS/CFT case. <br />
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The calculation which we performed illustrates nicely some points about the properties of observables which are often not spelled out, however seem of relevance for researchers interested in quantum gravity and should not be confused. In this post, we will gloss over global problems in defining observables, see e.g. <a href="http://arxiv.org/abs/1508.01947" target="_blank">this paper</a> and references therein for a recent interesting discussion. See also <a href="http://relq.blogspot.com/2016/01/do-we-really-not-have-observables-for-gr.html" target="_blank">this post</a> for an earlier discussion of observables and the control we have on them. <br />
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<b>1. Structure of the (sub-)algebra of observables </b><br />
When we quantise, we always pick some preferred subalgebra of phase space functions that we want to quantise. It is well known that we cannot quantise all functions at the same time, as formalised in the Groenewold-van Hove theorem. Therefore, also in canonical quantum gravity such a choice has to be made, independently of whether we first quantise or first solve the constraints. We are interested here in the latter case, i.e. in the quantisation of a complete (= point separating on the reduced phase space) set of Dirac observables, or equivalently a complete set of phase space functions after gauge fixing the Hamiltonian and spatial diffeomorphism constraints, although similar statements should in principle also hold in the former case. <br />
The precise nature of this choice will determine the properties of our observable algebra. In particular, examples can be given where <br />
<ol>
<li>the resulting algebra has a perfectly local structure (e.g. when employing dust to specify a reference frame),</li>
<li>the resulting algebra is local, but some phase space functions not contained in the complete sub-algebra have non-local commutation relations</li>
<li>the resulting algebra is non-local</li>
</ol>
Roughly, the following happens. If we localise some field with respect to some locally defined structure, such as the values that four scalar fields take at a point, then the algebra remains local, where local means that Poisson brackets will be proportional to a delta-distribution in the coordinates defined by the scalar field coordinates. Another such example is to use <a href="http://arxiv.org/abs/gr-qc/9409001" target="_blank">dust</a>. We can however also localise some field with respect to a more global structure, such as the <a href="http://arxiv.org/abs/1403.8062" target="_blank">endpoint of a spatial geodesic</a> originating from some observer or the boundary of the spatial slice. Then, phase space functions with a non-vanishing Poisson bracket with components of the metric specifying this geodesic will have non-local (in the coordinates specified by the geodesics) Poisson brackets with fields localised at the endpoints of geodesics, just because they change the geodesic which defines the point where the field is evaluated. However, it is still possible to pick a complete set of such observables which have local Poisson brackets among themselves if we pay attention not to include fields not commuting with the components of the metric determining the path of the geodesics. This is in fact what is happening <a href="http://arxiv.org/abs/1403.8062" target="_blank">here</a>. For the <a href="http://arxiv.org/abs/1510.04154" target="_blank">last case</a>, we simply locate our fields with respect to some extended structure that is specified using non-commuting phase space functions, e.g. a spacetime geodesic. This doesn’t mean that locality breaks down, it simply means that we are considering observables which are smeared over regions of spacetime, and two such regions associated to different localisation points might overlap.<br />
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<b>2. Locality of the physical Hamiltonian </b><br />
Another question is whether the physical Hamiltonian that one derives with respect to a certain choice of time will be non-local. Again, in the case of dust as a reference field, it is local. However, already in the second case above one finds that is non-local, simply because it contains the phase space functions which have non-local Poisson brackets. Since a generic Hamiltonian will contain all fields of the theory, it seems that one has to avoid relational observables with respect to non-local structures if one wants to obtain a local Hamiltonian. However, it should be stressed again that one still might have a complete local observable algebra, even though the Hamiltonian is non-local. A detailed example is given <a href="http://arxiv.org/abs/1506.09164" target="_blank">here</a>.<br />
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<b>3. Independence of spacetime boundaries</b><br />
All arguments made above are independent of the structure of the spacetime at infinity. In particular, the math involved in constructing observables and computing their algebra, see e.g. <a href="http://www.sciencedirect.com/science/article/pii/S0003491684711146" target="_blank">here</a>, <a href="http://arxiv.org/abs/gr-qc/0411013" target="_blank">here</a>, and <a href="http://arxiv.org/abs/1203.6526" target="_blank">here</a>, does not refer to infinity. Also, the fact that the on-shell Hamiltonian constraint is a boundary term is not of relevance here, as it generates evolution via its Hamiltonian vector field, which is non-vanishing on-shell. Once a clock field is chosen on the other hand, one obtains a physical Hamiltonian with respect to that clock which is non-vanishing on-shell. Therefore, as long as one can locally define a clock, one can also have evolution. </div>
Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com1tag:blogger.com,1999:blog-2381900080106522441.post-31252653706884394912016-03-08T15:45:00.000+01:002016-03-08T15:46:31.921+01:00The algebra of observables in Gaußian normal space-(time) coordinates<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/FYQARS2iGn8/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/FYQARS2iGn8?feature=player_embedded" width="320"></iframe></div>
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This rerecorded talk was originally given at the spring meeting of the German Physical Society on February 29, 2016. It is based on <a href="http://arxiv.org/abs/1510.04154">this paper</a>, where the algebra of a set of gravity observables constructed using a coordinate system specified by geodesics is computed. My interest in this question, next to general considerations in quantum gravity, comes from the AdS/CFT correspondence, where this construction has been used in the construction of the bulk / boundary dictionary. I plan to comment a little more on the content of the paper in the near future, highlighting some in my point of view important conceptual conclusions on the structure of general relativistic observable algebras.Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com0tag:blogger.com,1999:blog-2381900080106522441.post-75646678539759192882016-03-05T12:01:00.000+01:002016-03-07T09:12:32.466+01:00Are the spectra of the LQG geometric operators really discrete?<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-l5q7OdUPTWs/Vtq1rpfL4oI/AAAAAAAAAYg/p7Mb-cCklmU/s1600/Discrete-geometric-operators-loop-quantum-gravity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Are the spectra of geometric operators in loop quantum gravity really discrete?" border="0" height="208" src="https://4.bp.blogspot.com/-l5q7OdUPTWs/Vtq1rpfL4oI/AAAAAAAAAYg/p7Mb-cCklmU/s400/Discrete-geometric-operators-loop-quantum-gravity.jpg" title="Are the spectra of geometric operators in loop quantum gravity really discrete?" width="400" /></a></div>
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<b>tl;dr</b>: Sometimes yes, sometimes no, depending on how one passes to the physical Hilbert space. <br />
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The title for this post is shamelessly borrowed from <a href="http://arxiv.org/abs/0708.1721" target="_blank">this paper</a>, where Bianca Dittrich and Thomas Thiemann have discussed this question some time ago, concluding that the answer could be negative at the level of the physical Hilbert space, i.e. after solving the Hamiltonian constraint. Carlo Rovelli then wrote a <a href="http://arxiv.org/abs/0708.2481" target="_blank">rebuttal</a>, arguing that discreteness should remain also at the physical level. Since those papers, some time has passed and new light has been shed on this issue from various angles, to be reviewed in this post.<br />
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<a name='more'></a><br />
To begin with, let us briefly recall the area operator in loop quantum gravity. It is constructed roughly as $\text{A}(S) = \int_S \sqrt{ E^i E_i }$ and acts diagonally on spin networks, with an eigenvalue proportional to $l_p^2 \sum_i \sqrt{j_i(j_i+1)}$, where $j_i$ is the spin label of the edges intersecting the surface $S$. The operator thus has a discrete spectrum and a spectral gap. <br />
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This operator is a priori only defined on the kinematical Hilbert space, where the spatial diffeomorphism constraint and the Hamiltonian constraint still have to be implemented. In particular, it is not invariant w.r.t. their gauge flow. The question is thus whether solving these constraints in some suitable way might change the spectrum of the area operator.<br />
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Constraints can be solved classically (reduced phase space quantisation) as well as at the quantum level (Dirac quantisation). At the quantum level, little is known about the issue of physical spectra, since we do not have a sufficient understanding of the solution space of the Hamiltonian constraint operator. As for the diffeomorphism constraint, we know how to solve it (roughly by going over to diffeomorphism averages of spin networks), but we then have the problem that the area operator does not preserve the diffeomorphism-invariant states. However, we can simply go over to the volume operator, which when evaluated on the whole spatial slice (e.g. for compact slices) does preserve these states. Then, the spectral properties of the volume operator simply pass to the diffeomorphism-invariant Hilbert space, indicating that the physical spectra might coincide with the kinematical ones. However, a <a href="http://arxiv.org/abs/1306.3246" target="_blank">study in 2+1 dimensions</a> has shown that that the spectra can become continuous at the level of the physical Hilbert space.<br />
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As soon as we allow ourselves to solve the constraints classically, many things can happened. This is connected with a freedom in the choice of quantisation variables, or more precisely, the choice of Dirac observables and gauge fixings. As a seminal example, Kristina Giesel and Thomas Thiemann have considered <a href="http://arxiv.org/abs/0711.0119" target="_blank">LQG coupled to Brown-Kuchar dust</a>, where the dust fields provide us with a physical coordinate system and the Ashtekar-Barbero variables at given values of the physical dust coordinates are Dirac observables. Since the dust fields don’t interfere with the gravitational sector at the level of the symplectic structure, the LQG geometric operators simply pass to the level of the physical Hilbert space, thus giving and example of where their physical spectra indeed are discrete and furthermore coincide with the kinematical ones. <br />
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However, different choices of gauge fixings or Dirac observables are possible. <a href="http://arxiv.org/abs/1203.6525" target="_blank">Another example</a> has been discussed in the context of a constant mean curvature clock for GR conformally coupled to a scalar field. Here, due to the non-minimal coupling, matter and geometry mix in the symplectic structure and the Dirac observables commuting with the constant mean curvature gauge condition correspond to a conformally rescaled metric $\tilde q_{ab} = \phi^2 q_{ab}$. From this metric, one can again pass to connection variables and construct a loop quantisation, resulting in a “twiddled” area operator which has the same spectral properties as the standard area operator, however a different interpretation. Since it is constructed from the twiddled metric, it measures a twiddled area, which is related to the usual area by a multiplication of $\phi^2$. Yet another example of canonical variables leading to a changed are spectrum due to non-minimal coupling has been given <a href="http://arxiv.org/abs/gr-qc/0305082" target="_blank">here</a>.<br />
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So are the physical (untwiddled) areas discrete in this case? This obviously depends on the quantisation of $\phi$, since we have to extract the geometric area from the twiddled one. For scalar fields, different representations are available, e.g. following <a href="http://arxiv.org/abs/gr-qc/9705021" target="_blank">Thiemann’s original proposal</a>, <a href="http://arxiv.org/abs/gr-qc/0211012" target="_blank">its upgrade </a>to the Bohr compactification of the real line as a gauge group, or the representation discussed <a href="http://arxiv.org/abs/1507.01149" target="_blank">here</a>. As an example, in the last case $\hat \phi(x)$ acts by multiplication of an arbitrary (sufficiently smooth) function $\phi(x)$, and thus the eigenvalues of the induced untwiddled area operator can take any value. However, this simple argument only makes sense if we forget about our intended gauge fixing and simply use the variables suggested by it. Otherwise, we should solve the Hamiltonian constraint for $\phi$, resulting in a much more complicated and generally non-local expression. <br />
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While an area operator could be reconstructed from the scalar field and the twiddled area operator, this might be too complicated for other choices of variables. An example is provided by Lovelock gravity, a higher-derivative generalisation of general relativity with the same phase space. <a href="http://arxiv.org/abs/1304.3025" target="_blank">In the LQG context</a>, one finds that the appropriate area operator analogue now measures Wald entropy (on non-rotating and non-distorted isolated horizons) for a suitable choice of variables suggested by the canonical analysis of the Lovelock action. The area of the surface can not simply be recovered from this expression, and the fate its spectrum remains unclear. <br />
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While in the above examples the basic structure of the connection variables remained the same, e.g. having gauge group SU(2), this can also be tampered with. For example, choosing the <a href="http://arxiv.org/abs/1410.5608" target="_blank">diagonal metric gauge</a>, one obtains Abelian connection variables with the Bohr compactification of the real line as a gauge group. Then, all real numbers are eigenvalues of the area operator, as opposed to the SU(2) case. Somewhat differently, one can choose the <a href="http://arxiv.org/abs/1410.5609" target="_blank">radial gauge</a> and corresponding connection variables, where some of the area operates (the radial-angular ones) remain the same as in standard LQG, while others (the angular-angular) become volume operators of 2+1 gravity, which then have different spectral properties. <br />
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In conclusion, we saw that the spectrum of geometric operators may be strongly influenced by the choice of variables we make, which is in turn influenced by the choice of gauge fixings / Dirac observables, i.e. ultimately the passage to the physical sector of the theory. This shows that the final spectra of geometric operators do depend on this choice and may or may not have discrete eigenvalues. One may conjecture that similar effects might also appear when solving the constraints at the quantum level, but this remains to be shown in concrete examples. </div>
Norberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.com2