tag:blogger.com,1999:blog-86660912021-12-06T11:17:42.674+01:00The Reference FrameSupersymmetric world from a conservative viewpointLuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger1130125tag:blogger.com,1999:blog-8666091.post-70275720648282237612021-12-05T10:21:00.006+01:002021-12-05T19:30:44.120+01:00Predictive power of the world sheet conformal symmetryPhysics is a natural science but it is one that is really capable of explaining a lot (and very precisely) by assuming a very little, by assuming some important principles that actually imply powerful predictions that are often verified to be correct. The precise laws of physics are often derivable from some propositions that are almost comprehensible to linguists. The search for the deepest such principles has been vaguely underway since the conception of physics (in the strict sense) by Galileo, Newton, and pals but it was elevated to a new, industrial level when Einstein and pals switched physics to the era of modern physics.<br><br>A natural metaphysical hypothesis is that a single deep principle is sufficient to derive all correct predictions in physics, and therefore in natural sciences. There is no guarantee that such an assumption about a "theory of everything" and its underlying principle is a correct expectation. To say the least, it motivates some people. Other people (and sometimes the same ones) are satisfied with making some progress which, in the case of fundamental physics, always includes some deepening and refining of the primary assumptions, and some extending of their domain of validity towards more universal explanations than what the previous theories and principles could cover.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />There are lots of incredibly powerful conceptual physics ideas that imply a lot. For example, we have the conservation laws that allow us to make predictions about tons of seemingly diverse systems and exclude lots of possible scenarios that would otherwise be conceivable. Even more deeply, Noether's theorem shows the equivalence of such conservation laws and some symmetries of the laws of Nature. The existence of a conserved quantity that deserves to be called energy is equivalent to the symmetry of the laws of physics with respect to translations in time (i.e. the fact that the laws of physics don't change in time; and it's true with some possible clarifications but perhaps not others); the symmetry with respect to rotations is equivalent to the existence of a conserved pseudovector that we call the angular momentum, and so on.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />The principle of relativity, basically the Lorentz symmetry that allows us to use all inertial frames equally to formulate the laws of physics, has lots of implications through the reasoning of the special theory of relativity. The speed of light is universal, it is the maximum speed at which the information may propagate, the existence of the conserved energy implies the existence of momentum (or vice versa), the existence of the electric fields implies the existence of the magnetic fields (and vice versa), and many more things like that. The previous sentence is a bit simplified and an extensive discussion would be needed to clarify what the postulates of relativity do actually imply and what they only do if some extra assumptions are added. But morally speaking, the simple claims about the "power of relativity" are almost precisely right.<br><br>Quantum mechanics has transformed the very Ansatz of the laws of physics more profoundly than relativity did, and in some sense, it may be said to be a physics framework that is also derived from principles (the universal postulates of quantum mechanics). But from a different perspective, quantum mechanics is not a theory where lots of quantitative predictions may be immediately made from "verbal principles". For quite some time, the quantum mechanical theories that people used to describe the world were just "quantizations" of some corresponding classical theories. So if they couldn't have derived some dynamical details about the right theories within the classical framework, they couldn't have done it in the quantum mechanical framework, either.<br><br>At the end, the view that a quantum theory is the same as a classical theory with "hats" (a decoration or clothes that turn observables to operators; read this sentence both literally mathematically as well as poetically) is incorrect, as was increasingly clear in the last decades of the 20th century. Quantum mechanics really imposes qualitatively different rules and the "space of possible theories" only seems to coincide (well, be in one-to-one correspondence) with the "space of possible classical theories" if we only focus on quantum mechanical theories that have a classical limit, and on the regime where the limit is relevant. However, deeply in the quantum realm where the classical approximation becomes O(100%) wrong, the map to classical theories becomes wrong, misleading, impossible, and the whole classical thinking is an incorrect guide. We know that there are extra conditions (like the anomaly cancellation) that ban many quantum mechanical theories although these non-existent or inconsistent theories' classical counterparts seem to be OK; there are classical unknown monodromies; there are quantum mechanical theories whose observables just aren't continuous or they don't admit a classical limit for other reasons; and the "classical guide" to do quantum mechanics is always a sign of someone's immature approach to modern physics.<br><br>However, it is still true that even today, a large majority of the quantum mechanical theories that we use are "constructed" from pieces, much like "their classical counterparts", and they do have some classical counterparts, after all. The search for the simplest or textbook-like quantum mechanical theories is an engineering-like construction which requires one to invent some objects or pieces or variables; and some terms in the equations (or the Hamiltonian) etc. However, when we get to more specific quantum mechanical theories, they respect a general form, an Ansatz. And it is surely the case of gauge theories that describe all non-gravitational forces.<br><br>Within the space of gauge theories, there are again some principles that imply a lot but they are somewhat more engineering-style principles because they tell you what kind of variables and interactions the "good" theories are ultimately going to have. Gauge symmetries are a great example of a principle to build "nice" quantum mechanical theories. The gauge symmetry under \(SU(3)\times SU(2)\times U(1)_Y\) of the Standard Model implies "the behavior of all the non-gravitational forces". Electromagnetism, a \(U(1)\) residual of the breaking of the \(SU(2)\times U(1)\), leads to long-range Coulomb-like forces that drop like \(1/r^2\); the breaking itself needs something like the spin-zero Higgs boson; the broken part of the forces generates the massive W-bosons and Z-bosons. And then there is the whole QCD with the colorful \(SU(3)\) group that ends up being a short-range force as well because all finite-energy objects that are allowed to exist in isolation must be "color-neutral", and the forces between color-neutral objects are guaranteed to drop quickly with the distance (like the forces between protons and neutrons).<br><br>It is totally clear that gauge symmetries are very powerful and we need gauge theories for a reasonably effective description of the experiments in particle physics, especially (but not only) the high-energy ones such as the LHC collisions. On the other hand, the gauge theories are obviously not the final word (e.g. because they don't explain gravity or the spectrum of quarks and leptons) and there should be deeper principles that may be used as the starting point to derive an even wider and deeper set of correct statements about Nature, including the statement that "gauge symmetries in the spacetime are bound to arise and be useful to define the laws of physics". I am obviously talking about string/M-theory now which implies the existence of vacua whose dynamics may be approximated by "gauge theories coupled to Einstein's general relativity", the best classical theory of gravity, at long distances.<br><br>People may decide not to be interested in the deeper starting point which means to decide that they just want to remain shallow forever. But it is a fact that despite our ignorance about "which vacuum of string theory is really right", string theory is a deeper and more predictive system of laws that is not ruled out as of now and that seems to nontrivially imply lots of profound predictions about phenomena and patterns in fundamental physics. But is "string theory" a principle like the conservation laws of Noether's theorem? It sounds like an "object", a particular theory.<br><br>While string theory may be imagined as a system derived from "even deeper and more unifying principles" than e.g. gauge theories or Einstein's general relativity, string theory has been studied as a "totally constructive theory", almost by "engineers", for many decades. People who studied it between 1968 and the mid 1990s were facing some restrictions "what was possible and what dimension had to be picked" etc. but engineers are facing such restrictions, too. The derivation of all of string theory from some "deep principle" or "principles" (which would make it really analogous to the theories of relativity) remained a wishful thinking for quite some time. But in the 1990s, the situation changed and people started to have a clear idea how the "string theories" they had previously studied were connected or disconnected, and what the landscape of "all possible environments or 'theories' of this kind" looked like.<br><br>We might say that before the 1990s, people were playing on isolated "islands", like some ancient people with lame boats, but around the mid 1990s, the epoch of explorers arrived and people finally got credible plans to "map the whole world" of theories in this class (really "one theory" once they realized something about the "ocean floor" connecting the islands etc.). Finally, physicists could classify lots of things and see that the "accidentally constructed examples" are not just some random findings among googols of totally qualitatively different options. Instead, just like there are 5 or 6 continents, there are 5 or 6 maximally decompactified vacua of string theory in the maximal dimensions of the spacetime, ten or eleven.<br><br>There have been lots of progress that took us in unexpected directions etc. But at the end, I believe that the "massive industry of the mid 1980s", perturbative string theory, remains the proudest example of the "principles deeper than gauge symmetries that imply more". We know that what these deeper perturbative stringy principles imply only reliably applies to the case of the weak string coupling, \(g_s\ll 1\), which are regimes analogous to boats that aren't far from the shores. Less constructive or less universal descriptions have to be exploited if we want to sail very far from the shores of the islands or continents.<br><br>But the vicinity of the shores is an important region and in some counting, we know that it contains a very big part of the truly unequivalent insights or qualitatively different types of behavior. This is a view that people didn't have when they were playing with lame boats near the beaches. They saw a seemingly infinite ocean, to use the analogy further, and it filled them with humility. There must be so many things over there. But some down-to-Earth people didn't ever want to sail too far and in some sense, they were proven right once the world was mapped away from the shores. OK, if you are a Spanish queen, you can send a Columbus somewhere but what will he find? Some Americas which don't differ from Spain much, a few potatoes, and savages. Columbus found the old natives and could leave some whites there who transformed into new nutty savages 250 years after they established their most famous country, anyway.<br><br>I am not saying that the vast ocean or the Mariana Trench or the exotic life forms that live in the very deep ocean are uninteresting. But most of the stuff that is found there may be described using the concepts that are already useful near the shores, and in this explorers' sense, with down-to-Earth concepts. The ocean is vast but it is also a mostly boring place that (almost) only exists in order to consistently connect the landmasses and to make some intercontinental flights annoyingly long.<br><br>With this new perspective that looks at string/M-theory as a single connected theory that is partially known in its entirety (although we still don't have the completely precise definition that would cover all the places of the landscape, not even in principle), two seemingly contradictory changes took place. On one hand, our focus was redirected away from the shores to the ocean and the routes that exist there (which includes duality, the equivalences between previously "different" string theories or landmasses; India=America is an example of a wrong duality LOL but it is exactly the kind of dualities that the new tools allowed to establish). On the other hand, we could see that "the world is nice but it's the best thing to live at home". The European shores (and even landlocked countries like Czechia) basically contain everything that you need to know. The remaining places in the world are just combinations of three coordinates. Aside from the potatoes, some nutcases in the Silicon Valley, and McDonald's hamburgers that will only exist after a Czech American puts a small company on steroids, there is nothing interesting about the U.S. ;-)<br><br>Even the routes in the ocean (e.g. dualities) and other properties away from the shores have been derived from the concepts and laws that only claim to work very well near the shores, from perturbative string theory. Finally, what is really perturbative string based upon? What is the principle at least in this near-shore approximation?<br><br>I think that the "conformal symmetry on the world sheet" may be said to be the main technical principle that replaces the Yang-Mills theory and other "principles" of the quantum field theory in the spacetime. The conformal symmetry is a residual symmetry after some gauge-fixing, something that is left in a theory with a larger symmetry, the world sheet diffeomorphism (general coordinate transformations) times the Weyl symmetry (scaling of the world sheet metric tensor by a place-dependent scalar coefficient) after you make sure that the metric tensor looks locally Minkowskian (which is always possible in 2 dimensions of the world sheet).<br><br><b>Finally, some simple equations will start to appear. That is why you read it up to here.</b><br><br>Perturbative string theory applies to all environments in string/M-theory which contains light (low-tension) strings that are weakly interacting. In these regimes, the lightest massive objects may be described as slightly excited (near-ground-states of) vibrating strings in the spacetime. The qualitative behavior of such strings is pretty much universal, there is some quasi-exponential growth of the number of excited string states with the mass, and some general calculus how to compute the spectrum and interactions of all these strings, despite the fact that the strings are allowed in "many environments" that differ by some technical traits (the shape of the Calabi-Yau manifolds etc.).<br><br>A string is a 1-dimensional curve in the spacetime and its shape may be captured by the vector-valued function \(X^\mu(\sigma)\) where \(\sigma\) (read "sigma", Mr Biden and Fauci) is both a future variant of Covid-19 as well as a real coordinate parameterizing the one-dimensional string. Because the strings continue to exist in time, we need another time coordinate \(\tau\) (tau, also a virus) which may be identified with the spacetime time \(X^0\) in a certain choice of coordinates, but those coordinates are not really convenient at all. So we deal with two-dimensional world sheets (history of the shape of strings how they exist and evolve in the spacetime), \(X^\mu(\sigma,\tau)\).<br><br>These \(X\)'s are classical fields, soon to be turned into operators, and the whole relevant theory that we want to study requires operators in a spacetime-like space (the world sheet) labeled by the coordinates \(\sigma,\tau\). It is a two-dimensional quantum field theory. Even classically, the two-dimensional world sheet inherits some proper distances and therefore a metric tensor \(h_{\alpha\beta}(\sigma,\tau)\) from the embedding of the world sheet into the spacetime. It is useful to define this \(h_{\alpha\beta}\) on the world sheet but not directly calculate it from the spacetime metric. Instead, it is better to allow \(h_{\alpha\beta}\) to be a scalar-rescaled induced metric extracted from the spacetime. Once we allow this Weyl rescaling, the 3 components of the symmetric tensor, \(h_{\sigma\sigma},h_{\sigma\tau},h_{\tau\tau}\) may be locally set to the normal Minkowski form by 3-parameter (picked at each point) transformations, 2 for the general change of coordinates in 2D and one for the scaling of the metric tensor \(h_{\alpha\beta}\).<br><br>When it's done, we may see that even the most natural action describing the embedding of the world sheet, namely the proper area (which involves square roots and looks contrived), may be transformed to a nice, bilinear Klein-Gordon action on the world sheet, basically \[ \int \partial_\alpha X^\mu \partial^\alpha X_\mu. \] The scaling of the \(h\) tensor was a local symmetry on the world sheet, much like the Yang-Mills symmetry is a local symmetry in the spacetime (and the coordinate transformations of GR may be local symmetries both in the spacetime and the world sheet). That is why the residual symmetry coming out of this Weyl-diff symmetry must be required to annihilate physical states etc. It's really just the overall scaling of the world sheet (or the holomorphic functions of a complex variable which have just 1D-worth of parameters to be described, not 2D) and the invariance of the world sheet theories and of the physical states under the simple scaling of the world sheet metric is actually the single most powerful principle that implies all the big predictions, including <ul> <li>the existence of spin-2 gravitons in the spacetime that interact just like Einstein's GR dictates</li> <li>the existence of spin-1 gauge bosons in the spacetime that interacts just like Yang-Mills theories based on the spacetime gauge symmetries demand</li> <li>the existence of Dirac/Weyl/Majorana fermions in the spacetime that obey the expected kind of spacetime equations, plus the same for new scalars, Higgses/axions/inflatons, with their equations etc.</li> <li>lots of new insights that string theory discovered for the first time but that may indeed be verified to be important in the "effective quantum field theories in the spacetime" as well, including holography and other dualities</li></ul>How does the existence of a spin-two graviton follow from the "new powerful principle", the scale invariance of the world sheet coordinates? Well, the physical states of a vibrating string (energy or mass eigenstates) may be mapped to composite local operators on the world sheet. The map is really natural (and the only natural one) and it geometrically boils down to the fact that an infinite cyllinder (a history of a single closed string, one of a circular shape) is conformally equivalent to the whole complex plane.<br><br>In this state-operator correspondence which is a nice property of the 2D world sheet theories (depending on the conformal transformations of the complex plane and regions in it) and which is a major explanation "why the conformal symmetry ends up being so powerful", the overall momentum of the string \(p\) gets mapped to a prefactor in a seemingly artificial non-polynomial operator \(\exp(ip\cdot X(\sigma,\tau))\) in the world sheet-based quantum field theory. The Hilbert space of a string ends up being a Fock space (infinite-dimensional harmonic oscillator) and every excitation by \(\alpha^\mu_{-n}\) is mapped to an extra prefactor of \(\partial_z^n\) that you insert in front of the local operator in the 2D quantum field theory. (The corresponding \(\partial_{\bar z}^n\) is similarly representing the right-moving excitations \(\tilde \alpha_{-n}^\mu\), I don't want to go to similar technicalities such as the existence of the left-movers and right-movers in the Fock space.)<br><br>Now, on the world sheet, the fields \(X^\mu(\sigma,\tau)\) are dimensionless, like Klein-Gordon fields in 2D are (the Lagrangian density must be squared-mass which is just enough to cover the two derivative operators in the Klein-Gordon term; an alternative explanation is that \(X^\mu\) is an actual location in the spacetime so it mustn't scale in any way if you scale the auxiliary world sheet coordinates, so it must be dimensionless on the world sheet). Now, massless particles admit zero-momentum states where \(\exp(ip\cdot X)\) reduces to the identity operator. But the overall operator must still be a scale-invariant density (because its world sheet integral is the only natural operator representing the string state) so it must have the mass dimension two, too. It follows that the simplest local operators representing the simplest zero-momentum states of a string are\[ \partial_z X^\mu \partial_{\bar z} X^\nu. \] I simply needed two world sheet derivatives to get the mass dimension of two, so that the operator is a world sheet density. There was one \(z\) and one \(\bar z\)-derivative in order for the operator to be invariant under the rotations as well (those are also residual conformal symmetries, aside from the scaling, I forgot to mention that the multiplication by a complex number may both scale and rotate). We needed to insert two derivatives to obtain the right mass dimension on the world sheet, but because we needed to use the fields encoding the embedding of the world sheet into the spacetime, we produced two spacetime indices \(\mu,\nu\), and that is why we get a spin-two tensor-worth of spacetime states. Now, it is absolutely cool, and I have sketched it in an older blog post, that these massless string states integrate exactly as if they change the background metric in the spacetime. It is really possible to roughly see this fact from the very property that the operator above resembles the Klein-Gordon action in the world sheet. The Klein-Gordon action is actually also proportional to the spacetime metric and this "vertex operator for the graviton" clearly has the same form as the "variation of the world sheet action with respect to the change of the spacetime metric". So the presence of a coherent state of gravitons with this mode is exactly equivalent to an infinitesimal change of the theory resulting from the infinitesimal change of the spacetime metric.<br><br>Now, you can derive that quantum mechanically, the number of these bosons \(X\) must be right, the critical dimension. It's \(D=26\) spacetime dimensions for bosonic string theory; and \(D=10\) for the only (and truly) realistic improvement of that theory where you add fermions and the minimal supersymmetry relating the world sheet bosons and fermions. Because there are extra dimensions i.e. extra values for the indices \(\mu,\nu\) aside from the values \(0,1,2,3\) spanning the spacetime that we thought we knew, it means that aside from the spin-2 operators above, some components may also be \(\mu\nu=\mu 5\) or \(55\) so you may also produce massless states of the string that behave as spin-1 or spin-0 particles in the spacetime. With fermions, you may also get the spin 1/2 or 3/2 in a similar way. Elementary particles predicted from massless vibrating strings have the spins \(0,1/2,1,3/2,2\). The gravitino with \(j=3/2\) (the main new fermion predicted by supergravity theories) remains the so far only undetected particle that is being predicted by superstring theory.<br><br>Analogous arguments to those that imply that the spin-2 particles interact just like the spacetime gravitons are OK to prove that the spin-1 particles obey some Yang-Mills symmetries in the spacetime. You may derive the same "qualitative list of allowed fields, interactions, and symmetries" in the spacetime from a very different starting point than some "consistency conditions in the spacetime", from the scaling symmetry on the world sheet. It is surprising that a seemingly technical symmetry in a space that seems totally auxiliary from the viewpoint of a spacetime experimenter – a world sheet local symmetry – is at least equally powerful as the direct constraints "we need this and that to work in the spacetime, and that's why we need these fields in the spacetime" but it is so.<br><br>Aside from the right dimension of the vertex operators for the physical states (mass dimension of two for the densities), we may also derive lots of the characteristically new stringy phenomena like T-duality once we consider nontrivial topologies of the world sheet, starting with the torus. On the torus, another discrete residual symmetry of the diff-Weyl symmetry, the modular symmetry, is possible, and the theory must still be invariant under it which kills many wrong candidate theories with incorrectly allowed quasi-periodic boundary conditions on the closed string; and which is the first step to derive the T-dualities and other new important phenomena.<br><br>One usually needs to study things for some years "after she masters quantum field theory" to understand why the world sheet scaling-or-conformal symmetry is enough to predict and classify all these things and to indicate that the spacetime particle spectrum and interactions are indeed of the kind that we may expect for totally different reasons; i.e. why string theory implies the qualitatively right behavior near the shores. But the true power of string/M-theory ultimately seems to be the ability to connect all the islands, continents, and to allow us to sail far away from the shores. A principle that is equally potent as the "world sheet scaling symmetry" could underlie all the know truths about the behavior of the theory near the shores as well as in the deep ocean. Such an extension of the 2D world sheet and its scaling symmetry could be as straightforward as a space of values of infinite matrices and some unitary transformations of them; or something completely different, something that also appreciates that the topology change is equivalent to a quantum entanglement (ER=EPR) etc. To turn the voyages over the distant ocean completely controllable also means to understand maximum about the possible shapes of the islands and our "history of journeys" in between them. We may derive that we actually did come from Botswana and South Afica, perhaps just like our friend Omicron, and our ancestors simply had to first sail here or there, and that is why one compactification or another is the right one (or much more likely than others, to say the least). Again, one may ignore the deeper, more universal, and potentially more precise levels of explanations in physics but that means that he chooses shallowness and primitivism over wisdom, hard work, and curiosity. It is insane to represent this attitude as "good science".Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-79583195339143379182021-11-30T06:35:00.008+01:002021-11-30T06:45:24.109+01:00Metastrings and mysterious triality<b>Two new neat stringy papers</b><br><br>I want to mention two new hep-th preprints. First, Berglund, Hübsch, and Minić wrote <blockquote><b><a href="https://arxiv.org/abs/2111.14205">Mirror Symmetry, Born Geometry and String Theory</a></b></blockquote>where they sell their love for the "bosonic string as the parent of the superstring" and the "doubled degrees of freedom for T-duality" including the new term "metastring theory", not to mention the "Born geometry" (a structure on the doubled tori, with some symplectic structure and modular group), but the real new beef of the 5-page-short paper seems to be their ability to get a manifest mirror symmetry out of the doubled starting point. Some non-commutative generalizations of the Calabi-Yaus are automatically included.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />The last hep-th paper is <blockquote><b><a href="https://arxiv.org/abs/2111.14810">Mysterious Triality</a></b></blockquote>by Sati and Voronov. I've been in love with the <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20iqbal%20and%20a%20vafa%20and%20a%20neitzke">mysterious duality</a> by Vafa, Iqbal, and Neitzke, and debated the authors when they were making and releasing the paper. OK, the exceptional Lie groups' lattices which are relevant in the U-duality of M-theory on tori seem to be reproduced, along with other quantities, in the del Pezzo surfaces \({\mathbb B}_k\).<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />I've tried to complete a theory defined on the del Pezzos that could produce the string/M-theory spacetime as the "target space of the target space". Now these two authors have a third dual description i.e. the duality is extended to a triality. Their third descrpition involves <blockquote> the Sullivan minimal models of \({\mathcal L}_c^k S^4\), the iterated cyclip loop space of the four sphere. </blockquote>So start with the four-sphere \(S^4\) and find its loop space, i.e. the space of all maps from a circle \(S^1\) to the space. The space of all strings within \(S^4\), or the space of all strings in the space of all strings within \(S^4\), and so on. ;-) When you do it roughly 8 times, to get to \(E_8\), you will get a loopo-loopo-loopo-loopo-loopo-loopo-loopo-loopo-space of the four-sphere, and that will manifest the \(E_8\) symmetry and some other nice structures found in the U-duality of M-theory on tori, too.<br><br><a href="https://www.youtube.com/watch?v=K0XxozhMhYo"><img src="https://f4.bcbits.com/img/a1073619536_16.jpg" width=407></a><br><br><niciframe width="407" height="277" src="https://www.youtube.com/embed/K0XxozhMhYo" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></niciframe><nicbr><nicbr><em>Mysterious Duality Launch Video, press to play: I guess that even Vafa et al. fail to know it! ;-)</em><br><br>Needless to say, the idea of the "strings in the space of strings in the space of strings..." is even more aligned with my general program of "iteratively stringizing" the spaces, it is pretty much exactly what I always found relevant in these mysterious duality constructions in order to extend the pattern to a full-blown theory of everything in all forms. It is still mysterious for me why their sequence starts with a four-sphere.<br><br>Note that \(S^4\) is a real-four-dimensional compact manifold, and so is \({\mathbb C \mathbb P}^2\), the simplest del Pezzo surface (or Forefather Del Pezzo). But these two manifolds are distinct. In particular, the four-sphere isn't a complex manifold, doesn't admit complex coordinates. There are of course various ways to relate them or obtain them from one another but I have no idea why these seemingly non-unique constructions could be relevant for getting something as unique as M-theory, or even a semi-unique thing like M-theory on tori.<br><br>At any rate, my program would be that you define some appropriate enough string theory on \(S^4\), get some new target space out of it, define string theory in the same way on that, get something again, ... repeat it about 8 times, and then you get a complete description of M-theory on an eight-torus with the \(E_8\) U-duality group that is manifest.<br><br>You probably ask what the Sullivan model is. Roughly speaking, the Sullivan model is a rational homotopy model based on cohomology while the Quillen model is based on homology, so they are dual in some sense; see [Ta83][Maj00][FHT01]. I surely explained to you what a clipboard is now! ;-) Yes, the terminology is way too mathematical for me.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-43413408730158609322021-11-10T17:06:00.004+01:002021-11-10T17:10:48.781+01:00Unitarity conditions for the (de Sitter) wave function of the Universe?The Quanta Magazine's Charlie Wood wrote an interesting text <blockquote> <b><a href="https://www.quantamagazine.org/cosmologists-close-in-on-logical-laws-for-the-big-bang-20211110/">Laws of Logic Lead to New Restrictions on the Big Bang</a></b></blockquote>It starts with a nice self-similar <a href="https://d2r55xnwy6nx47.cloudfront.net/uploads/2021/11/Universe-expansion_1920_Lede.mp4">hypnotizing animation</a> by Dave Whyte (who is not racist, just racyst) whose value will exceed the value of the text according to many viewers (not readers).<br><br><img src="https://ars.els-cdn.com/content/image/1-s2.0-S1631070515001309-gr005.jpg"><br><br><em>Escher's picture of the hyperbolic space was used as a plan to spread the bats' and Fauci's Sino-American Covid-19 virus among an exponentially growing number of bats and people.</em><br><br>OK, Wood starts by popularizing some basics about the hyperbolic spaces, de Sitter space, and unitarity. Similar popular articles seem to be written for many different groups of readers. Each of them may find something appropriate, like cartoons that are sometimes more exciting for the adults than their kids. But I am afraid that the people who need to be explained what unitarity is won't really appreciate the new scientific results.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Great. I guess that most of the real "new science for experts" is about the unitarity conditions that may apply to the "wave function of the Universe". In "qualitatively static spacetimes", we often study the S-matrix, the unitary evolution from the infinite past to the infinite future.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Because the total probability of all outcomes is preserved for every initial state, we have\[ S\cdot S^\dagger = 1. \] The S-matrix is unitary, like all standard evolution or transformation matrices. This simple unitarity condition may be reinterpreted as a collection of infinitely many theorems for particular scattering amplitudes of many kinds. Various versions of the unitarity are still called <a href="https://en.wikipedia.org/wiki/Optical_theorem">the optical theorem</a> which was the first "very specific" identity (morally following from \(SS^\dagger =1 \)) which was derived by Werner Heisenberg.<br><br>That was originally "non-relativistic quantum mechanics" but you should look what <a href="https://www.google.cz/search?q=optical+theorem+feynman+diagrams&um=1&ie=UTF-8&hl=en&tbm=isch&source=og&sa=N&tab=wi&biw=1317&bih=708">the optical theorem looks like</a> in terms of Feynman diagrams. Like both sides of the \(SS^\dagger=1\) equation, the sides are "real" by construction, either the real part of something or the product of something and its complex conjugate.<br><br>Unitarity preserves the probability, the total probability of the final state is the total probability of the initial state, and the evolution has to respect it. But what if there is no variable initial state? A very complete theory of cosmology tells you what the right initial (and therefore final) state is, in terms of the wave function of the Universe, like the Hartle-Hawking wave function.<br><br>I think that Wood primarily promotes articles starting with the 2020 text by Goodhew, Jazayeri, and Pajer <blockquote><b><a href="https://arxiv.org/abs/2009.02898">The Cosmological Optical Theorem</a></b></blockquote>which has semi-respectable <a href="https://scholar.google.com/scholar?q=%22The+Cosmological+Optical+Theorem%22&hl=en&lr=&btnG=Search">two dozens</a> of followups by now (such as this <a href="https://arxiv.org/abs/2108.01695">Di Pietro et al. from August 2021</a>). For the diagramatic and \(\rm\LaTeX\) form of the optical theorems they propose for the wave function of the Universe, look at pages 15 and 19, among others (16 and 20 out of 58). Again, the equations are "real by construction" but we only qualitatively deal with "one moment", not a scattering process that relates two moments, "the past and the future". Four-point exchange diagrams are claimed to exhibit powerful constraints on the bispectrum and the trispectrum.<br><br>Without having any clear counterpart of \(SS^\dagger=1\), they claim to derive or "derive" all these constraints from some polology (poles in the complex plane and their residues) that are completely analogous to those in the flat space with the S-matrix. I don't understand whether this generalization of the flat-space polology depends on some extra unproven hypothesis or whether they claim their derivations to be "complete proofs". Pretty much equivalently, I would like to be clearly explained whether their "unitarity directly constraining de Sitter correlators" is something that remains important and precise near the very beginning of the Universe, or is just some "deformation of the flat space rules" that should only be relied upon when the space locally looks like the flat space qualitatively. In the latter case, I would say that the constraints don't really tell us much about the initial state of the newborn Universe.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-52669419530745591742021-10-18T07:19:00.009+02:002021-10-26T09:28:27.013+02:00\(J\bar T\) deformations: how a non-local theory stays UV-complete and conformally symmetric<blockquote> <b>Primordial gravitational waves:</b> <a href="https://arxiv.org/abs/2110.00483">A combo, when BICEP3 is added</a>, claims that there aren't any so far and \(r_{0.05}\lt 0.036\). <a href="https://www.google.com/search?aq=f&hl=en&gl=us&tbm=nws&btnmeta_news_search=1&q=bicep3#q=bicep3&hl=en&safe=off&gl=us&tbm=nws&source=lnt&tbs=sbd:1&sa=X&biw=925&bih=775&bav=on.2,or.r_gc.r_pw.r_cp.&cad=b">Google News</a>. </blockquote>Two or three decades ago, theoretical high energy physics was much more lively. An aspect of this liveliness was the constant stream of uprisings and minirevolutions. Sourballs and crackpots may have called them fads and they may have criticized the fact that many researchers were joining them because of FOMO. But it was an overwhelmingly good phenomenon. People were sufficiently excited and the system motivated them to work. So when someone found something new – about twistors, BMN string bits, and dozens of other topics (later – many people (including young ambitious people) worked on them. Researchers could have been sure that someone cared about their paper on the fashionable topic and it further stimulated their research.<br><br><iframe width="407" height="277" src="https://www.youtube.com/embed/YnopHCL1Jk8" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><br><br><em>It's not necessarily Monica's cup of tea but Ozone's Dragostea Din Tei is the greatest product of Romanian culture since Count Dracula. ;-) Leony was recently <a href="https://www.quora.com/Is-Dragostea-din-tei-a-good-song/answer/Lubo%C5%A1-Motl">inspired</a> (what is the polite word for plagiarism?) by that song, the new one is in English.</em><br><br>I am confident that the number of such signs of collective excitement and widespread activity has significantly dropped in a recent decade or so, and especially in the recent five years (some sourballs and crackpots probably celebrate). But there are still examples of mini-industries that look as healthy as we used to know them, their number is just smaller. On Saturday, it will have been four years since the launch of one such mini-industry. <blockquote><b><a href="https://arxiv.org/abs/1710.08415">An integrable Lorentz-breaking deformation of two-dimensional CFTs</a></b></blockquote>Monica Guica – whom I have known as a brilliant Harvard grad student (and who has been to Pilsen once) – has found a new class of UV-complete quantum field theories that are actually... not local. Well, this broader industry was founded by <a href="https://arxiv.org/abs/1608.05499">Smirnov and Zamolodčikov</a> (Sorbonne+Rutgers; the latter approved my spelling of his name when he was my instructor LOL) in August 2016, i.e. a year earlier. They added some composite operators \(T\bar T\) to the two-dimensional CFT (conformal field theory) action. The action has some extra term that is bilinear in the stress energy tensor.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />You might intuitively protest that it is a random contrived construction that messes the good properties of the CFT. But most of the magical traits of these deformations are all about the fact that the virtues are not really messed up at all. Instead, the \(T\bar T\) deformation of a CFT was found to be solvable. The spectrum is obtained from the original one by a universal formula, too.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />OK, in October 2017, Monica Guica (Saclay-Paris; sometimes add Nordita plus either Stockholm or Uppsalla University) presented a new deformation, the \(J\bar T\) deformation (don't get confused by the letters JT, the JT gravity means <a href="https://en.wikipedia.org/wiki/Jackiw%E2%80%93Teitelboim_gravity">Jackiw-Teitelboim gravity</a>, a different thing, which Wikipedia also tells you not to confuse with Liouville/CGHS gravity, not sure why you should). It is similar to \(T\bar T\) except that the first stress energy tensor is replaced with a \(U(1)\) current \(J\). The other factor \(\bar T\) is still the generator of translations in a null direction (or the antiholomorphic one, if you have a Euclidean world sheet). Back in 2017, she noticed that her deformation is somewhat less disruptive and preserves a finite subalgebra of the Virasoro-style algebra.<br><br>These "dipole" deformations may look <em>ad hoc</em> but what is nicely surprising is that despite the non-locality, the theories seem to be UV-complete QFTs. There are no inconsistency at any energy scale (energy in the 2D world sheet). This UV-completeness is surprising (I would say) because one wants to think that if the theory makes sense at arbitrarily short distances, it should admit some local lattice-like construction, and it's therefore local. But this reasoning is too sloppy and indeed, these deformations are examples of nonlocal but UV-complete quantum field theories.<br><br>Some 250+ followups of Smirnov-Zamolodčikov were written; and over 100+ of Monica's 2017 \(J\bar T\) paper. One of the important papers was the <a href="https://link.springer.com/article/10.1007/JHEP01%282019%29198">2019 article by Adam Bzowski and Monica</a> which asked what happens to the holographic bulk dual of the CFT (the AdS theory) if the CFT is deformed. The answer is that it is still gravity in an AdS space but it has some modified boundary conditions that mix the metric with the Chern-Simons gauge field.<br><br>I have always been uncertain about the basic status of these deformations. Are they a new, "more generic" class of QFTs that people should actually study? Or just classes of "fudged up" CFTs that actually lose some properties? Some trivial variations of the CFTs? Or even simple transformations of the CFTs to new variables? Even after the latest paper, I am uncertain but the latest paper surely changes a lot. Today, one second after the previous deadline (18:00:01 UTC), Monica released a new paper <blockquote> <b><a href="https://arxiv.org/abs/2110.07614">\(J\bar T\)-deformed CFTs as non-local CFTs</a></b>. </blockquote>She worked in the Hamiltonian formalism and the paper is full of Poisson brackets (and then commutators). But she adds some \(J\bar T\) deformations to the Hamiltonian and analyzes possible symmetries. Note that \(J\bar T\) consists of two factors, a holomorphic one and an antiholomorphic one. In the Minkowski signature, it means that the left-moving and right-moving sectors of the CFT are treated differently. One of them preserves a Kač-Moody algebra (she adds Witt- at the beginning). But the shocking thingg is that she can find the whole infinite-dimensional algebra on the other side as well. There are Kač-Moody algebras on both sides.<br><br>Despite the seemingly nonlocal deformation, the theory seems to have the same symmetry algebra as a rather normal CFT. But there is a catch. The Virasoro algebra has a zero mode and we automatically think that it is the same thing as the Hamiltonian, the generator of translations. However, in her deformation, the Kač-Moody zero mode generator is actually a quadratic function of the Hamiltonian!<br><br>Well, I think that this could have been found earlier and maybe by someone else than the mother of the \(J\bar T\) deformations. But she is just very bright and many people no longer feel the FOMO – or the incentives to work hard – so she had to find it herself. It's all very interesting because some of the old-fashioned assumptions we had about "every good UV-complete QFT" (and we surely tended to think that "most" good UV-complete QFTs are CFTs in the UV) are strictly speaking invalidated, but relatively mildly so.<br><br>A natural guess would be that all these theories are just some weird change of variables and they are otherwise totally equivalent to the undeformed CFT. This could lead to a rather trivial explanation of all these virtues. Well, it seems to me that it cannot be the case. The AdS dual of these deformed theories are just "slightly modified" but exactly because the modification looks so innocent (a linear mixing of bulk fields), we may see that the modification is non-empty. After all, it is just some deformation of the boundary conditions. When you change the allowed periodicity of world sheet scalars on a closed string, you are changing the radius of a circle in the target space which is a nonzero change. Well, you "unfreeze" some spacetime scalar field that you could have frozen or overlooked but you are surely doing something "new". In the same sense, the \(J\bar T\) deformation has to be "somewhat new". The deformation parameter \(\lambda\) is on par with the vev of something analogous to a field in the target spacetime or the dual gravitational bulk spacetime. Well, the parameter modifying the mixed boundary conditions has an easy interpretation but it isn't necessarily the only one.<br><br>I think that many HEP theorists have the same (almost?) black-and-white perspective on the field theories that are worth studying. There should be a unique list of "good properties" and only the theories that obey all of them are "truly interesting". Tens of thousands have been written that contributed something to the clarification of these precise rules but we're still uncertain about the "right virtues" and the "size of the space of the good theories", not only in string theory but even in field theory. If the theories with all the virtues don't have to be conformal, how many extra deformations and modifications preserve the kosher character of the QFTs? How big is the space that we actually remain ignorant of?<br><br>String theory really has an advantage because the string vacua are physically connected with each other. In the case of field theory, we might say that the questions in the previous paragraph are ill-defined because there may be many layers of "field theory" and the list of virtues doesn't have to be unique. In string theory, it is arguably unique and the list of vacua is well-defined. However, there may still be superselection sectors that are "just a step away from each other" from the perspective of one physical interpretation; but infinitely far from another vantage point.<br><br>These confusions will probably continue for some time but people should still try to settle similar more localized questions. Either the \(J\bar T\) deformations are a truly new original class of theories that open totally new possibilities, and in that case, the research of QFT should really be upgraded to that larger class; or all the new theories are really just "derivative objects" and cherries on a pie that only "quantatively and modestly change" something about the properties of the parent theories. People should better know which viewpoint is the more correct one.<br><br>Also, people should master the methods to rigorously prove whether such theories are unequivalent to the original ones by a redefinition of variables; and whether they are really nonlocal in all variables. We know other examples where the nonlocality is just an artifact of a choice of variables and using other variables, the locality is completely restored.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-57144288446245159142021-08-27T07:11:00.019+02:002021-08-27T10:07:24.584+02:00Axionic couplings guarantee stringy towers<a href="https://www.kudyznudy.cz/files/5a/5af3d5a8-6910-4cab-ae96-0e9e6fe93cdb.jpg?v=20210714125113" rel="nofollow"><img src="https://www.kudyznudy.cz/files/5a/5af3d5a8-6910-4cab-ae96-0e9e6fe93cdb.jpg?v=20210714125113" width=407></a><br><br><em>SOOS, a Western Bohemian national park (near Franzensbad and Sokolov) with fens and swamps.</em><br><br>Today, there is exactly one hep-th preprint on the arXiv that makes the authors excited and proud and this is why they submitted the preprint in the first second of the "arXiv day" (18:00:00 UTC) in order to guarantee the placing of the abstract at the top of the listing: <blockquote> <b><a href="https://arxiv.org/abs/2108.11383">The Weak Gravity Conjecture and Axion Strings</a></b></blockquote>Heidenreich, Reece, and Rudelius (UMass-Harvard-Berkeley) discuss a nice refinement of the swampland conjectures. Recall that "swampland" is a term introduced by Cumrun Vafa to point out that even though the string landscape (the set of mathematically consistent descriptions of quantum theories of gravity) may seem large by some measures, its size is actually tiny or negligible in comparison with the size of the "swampland" which is the set of effective (quantum) field theories dreaming to be coupled to the gravitational terms – but unable to be interacting with gravity consistently.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br /><a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20motl%2Cl%20and%20a%20vafa">The Weak Gravity Conjecture</a> is the most often studied example of the "swampland principles" that are the reasons why the landscape is such a small fraction of the swampland. It says, and the evidence has many components, that the gravitational force must be the weakest one at the fundamental level. More precisely, in some natural units, particle species must exist for which the electric-like forces trump the gravitational force. If these particles are imagined to be black holes, they are charged black holes surpassing the "extremality bound".<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />The consistent interactions with quantum gravity not only enforce the existence of a charged particle; it's been understood that there must be a whole tower. This tower of charged particles contributes a positive term to the beta-function which strengthens the electric force and the renormalization group rules guarantee that well beneath the Planck scale, the electric-like force becomes strongly coupled. For this reason, a trans-Planckian Landau pole which would be possible in a QED-like theory is really impossible even as a low-energy trajectory, thanks to the constraints that the consistency of quantum gravitational interactions imposes on everything.<br><br>The most relevant charged states that contribute to this strengthening of the electric-like interactions may either carry a low value of the spin; or a high spin. The former is a typical situation in Kaluza-Klein theories with a circle-like extra dimension; the latter scenario occurs when the excitations are carried by weakly-coupled strings, one-dimensional objects. These three authors argue that the latter, "stringy", possibility is unavoidable given an assumption: an axionic interaction term\[ {\mathcal L}_{\rm ax} = c\cdot \theta \cdot F \wedge F \] is present. Now I will leak something about the proof that is not included in the abstract. They had to find some reason (an exemption) why the Kaluza-Klein theories are "allowed" by the swampland principles to live without the strings. And the reason turned out to be the absence of the axionic coupling above. Why is this a sufficient excuse? It's because the left-movers and right-movers on the "preliminary" strings are allowed to be mixed when the coupling is absent. When the coupling is present, the mixing is impossible and the left-movers and right-movers must play their separate roles in an anomaly inflow argument which is the argument that traces the origin of the charged excitations.<br><br>When they consolidate and review their derived results, they find harmony with two other swampland conjectures, the Emergent String and Distant Axionic String Conjectures. On the other hand, they remain confused about the true "primordial" reasons that make it necessary for the axionic interaction to exist – in this sense, they have "only" rephrased one trait of some theories, the existence of strings at a certain scale, in terms of another "not really understood" assumption, the inevitability of the axionic coupling.<br><br>But the realization that these two assumptions are basically equivalent is clearly deep. The point is that in many situations, the consistency makes it unavoidable that string-like excitations exist; and they exist because the left-moving and right-moving excitations must stay separately light, instead of teaming up with each other via heavy mass terms that would mix the left with the right; and the segregation of the left- and right-movers is often made mandatory by the axionic coupling which would produce anomalies if the charged states only existed in a left-right-mixed setup.<br><br><b>Strings of a certain kind are unavoidable because the segregation of the left and the right is unavoidable; and the segregation is unavoidable because there would otherwise be anomalies (derived from the axionic coupling and an anomaly inflow argument).</b><br><br>Note that more generally, the arguments they extend say that with "seemingly purely field-theoretical" objects such as the axionic coupling, you are forced to immediately discover the existence of strings; or the existence of an extra dimension. If you have psychological or, more likely, psychiatric problems both with strings <em>and</em> extra dimensions, you should also better visit your psychiatrist before you are at risk that you will encounter an axionic coupling (or any other element from a long list of things that are dangerous for your psychiatric health).<br><br>The unavoidability of strings is understood a bit deeper than it was just very recently. Of course, such real scientific results don't prevent 100% scientifically illiterate inkspillers and hardcore imbeciles from writing childish articles saying "bye bye little SUSY and string theory" in <a href="https://www.economist.com/science-and-technology/physics-seeks-the-future/21803916">The Economist</a>. It is shocking that e.g. managers and financiers – who are pretty well-paid and used to be somewhat close to the intellectual elites – are being served this kind of absolute manure. They are encouraged to swallow the šit and happily smack their lips. You can't be shocked that when the people working with lots of money are being converted into giant vessels storing absolute šit, they make all the insane corporate decisions and they have effectively become little stinky tails of the unhinged SJW activists. Thank God, tomorrow The Economist will at least publish a <a href="https://www.economist.com/leaders/2021/08/28/fundamental-physics-is-humanitys-most-extraordinary-achievement">bullish article about fundamental physics</a>, too.<br><br>String theory is really a fact by now and the individuals whining when they hear about this simple fact are completely clueless when it comes to modern physics. Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-58795179573394671552021-08-23T07:04:00.012+02:002021-08-23T07:21:31.907+02:00Susskind: horizon complementarity, second law ban bigly ripped, cyclic, and bottled universes<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Leonard_Susskind_at_Stanford.jpg/407px-Leonard_Susskind_at_Stanford.jpg"><br><br>Bill has persuaded me to read the whole July 2021 hep-th paper <blockquote> <b><a href="https://arxiv.org/abs/2107.11688">Three Impossible Theories</a></b></blockquote>by Leonard Susskind. Let me say in advance that I saw the abstract last month, sort of agreed with it ("it is nothing new for me"), and I didn't read the body of the preprint which I did now. The body is visually funny because each page (and there are just 13 pages in the PDF file) contains a small number of letters, either because the "paper" is small or because the fonts are large.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />OK, Susskind starts by articulating the "central dogma" which says <blockquote> from the outside, a black hole may be described as a unitarily evolving quantum system operating in the Hilbert space of dimension \(\exp(A/4G)\) </blockquote>which I modified a little bit (an improvement, I think). This central dogma is especially supported by the gauge-gravity cases of the AdS/CFT correspondence, by Matrix theory, and perhaps by some other arguments, Now, something special which allows us to count the degrees of freedom through the area happens at the event horizon.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />An amusing observation that we considered potentially very deep from the 1990s (and I have discussed it on this website occasionally, for almost two decades) is that the cosmic horizons (the boundary of the faraway regions of a cosmological spacetime which can't affect you – an observer defined by a trajectory in the middle – for causal reasons; and which you can't affect, either) seem geometrically qualitatively equivalent to the black hole horizons. When it comes to the relationships between the regions, <blockquote> <b>the exterior</b> of the Universe behind the cosmic horizon of an observer (the faraway world) behaves just like <b>the interior</b> of a black hole. </blockquote>The bold face font was used to emphasize that the interior and the exterior are "exchanged" if you compare the single black hole scenario with the cosmological one. This qualitative "spherical inversion" has some consequences. In particular, the "cosmological interior", your causal patch, is bounded and lacks an asymptotic region. That is why the counterpart of the black hole, the "faraway world", is emitting thermal, Hawking-like radiation, just like a black hole does; but it is also absorbing the same amount of this (or analogous) radiation because all of this radiation (the former, emitted one) only moves through the patch for a finite amount of time. For this reason, the "faraway world" doesn't grow and doesn't shrink, unlike black holes which do shrink.<br><br>Great, the (less established, relatively to the black hole case) cosmological version of the "central dogma" says that the "faraway world" is similarly describable as a unitarily evolving Hilbert space of dimension \(\exp(A/4G)\). Taking this principle seriously, Susskind argues that three cosmological scenarios are impossible, namely: <ul> <li>big rip</li> <li>cyclic universe</li> <li>universe in a bottle</li></ul><b>Concerning the big rip,</b> it postulates a generalized cosmological constant with \(p\lt -1\cdot \rho\) i.e. \(w\lt -1\) (which is incidentally said to be suggested by some recent cosmological observations). In this scenario, the cosmological constant grows with time and so does the Hubble constant. The Hubble constant is inversely proportional to the radius of the cosmic horizon which means that the cosmic horizon shrinks! But if it shrinks, this area shrinks and so does the corresponding entropy. It is impossible for the entropy to shrink, by the second law of thermodynamics, so the big rip is impossible.<br><br>I like this argument (which has probably been around for decades as well, and I may have said it in the past as well) because it may potentially admit generalizations: Note that the big rip violates an energy condition and this violation is identified with a violation of the second law, through holography or complementarity. It is possible that <b>all truly valid energy conditions may be derived from the second law of thermodynamics</b> by a similar argument.<br><br><b>The second "impossible theory" is the cyclic universe.</b> I have been saying that the cyclic universes contradict the second law since the 1990s, in this case I am certain. In fact, I think it is also written in a popular book by Brian Greene. The oscillations produce some friction, heat, dissipation, so the cycles cannot be quite the same. They are either getting shorter or they are getting longer. If the duration of the cycle goes up (exponentially), then there was a beginning, after all; if they are getting shorter, there would be an end (because the geometric series are convergent). The former situation negates the key attractive feature, the infinite longevity in the past; the second leads to some arguably impossible version of compression at the end.<br><br>On top of the old argument, Susskind says that in the cyclic universe, a cosmological horizon (and therefore an area and an entropy) is oscillating as well, up and down and up and down. It is only possible for the entropy to go up, permanent oscillations like that violate the second law for 50% of the time and they resemble a perpetuum mobile device.<br><br><b>The third impossible theory is the universe in a bottle,</b> Farhi-Guth-Guven, the preparation of a new Universe in a laboratory. It's remotely possible within some vague frameworks that merge an informally quantized general relativity with the characteristic quantum effects such as the quantum tunneling (to justify the change of the spacetime topology, something that GR and especially its quantization seems to be tolerant about but you can't be quite sure whether a particular topology change has a nonzero probability in a given situation).<br><br>Here, the argumentation is a bit more complex but Susskind says that it was known to him or Guth or both in 2004. Again, the creation of the bottled universe may lead to a decrease of entropy but I think it is a "subjective entropy" of a cosmic horizon that is considered relevant by a feminist observer, Alice (at least I am pretty sure that Susskind wouldn't allow Alice to make it to his papers if she were not an obnoxious frigid feminist). She (her whole body? Is tunneling likely to preserve composite bodies, in the sense of conditional probabilities?) may tunnel from the bottle outside the bottle or vice versa, one of these processes would lead to the shrinking of the cosmic horizon around her, and it's therefore forbidden. I am not motivated enough to investigate which one.<br><br>This usage of the "cosmic central dogma" is even less reliable than the previous one because one is not talking about the area of a "fixed horizon" somewhere in the Universe but the area of a horizon attributed to someone "who can jump to faraway places just by being a victim of a tunneling event". It is less reliable but I personally think that even this usage of the principle could be justified.<br><br>But I feel uncomfortable about another issue, namely the low probability of the tunneling. He argues that the bottled universe must be banned because something contradicting the principles <b>could</b> happen, with a very low probability. Is it enough to rule out the bottled scenario? We could say that most of the principles of physics may be violated either with some insanely low probability or for a tiny amount of time. The tunneling effect itself may be described as a very short-lived violation of the energy conservation law. Like a leftist who wants to pay anything to prevent a scenario, even if it has a negligible probability (like a problem with the climate; or the extinction of a nation through Covid), Susskind thinks that the laws of Nature are "obliged" to do "prevention" and prevent even tiny probabilities of some events. I am not sure.<br><br>In other words, the reason why this argument could be wrong is that his calculation of the tunneling odds for Alice could be just an approximation and the precise one, in a full theory of quantum gravity, could yield precisely zero, and therefore no conflict with the principles. A tiny number and zero are very close which means that it is possible that his basically semiclassical calculation of the tunneling probability could be close to the exact result and there would be no material conflict with locality.<br><br>At any rate, these are three examples of the "principled thinking". Adopting some new principles in physics may lead to answers about seemingly very different questions. Theoretical physics has incorporated a massively profound and productive industry generating these clever thoughts at least since the golden years of Einstein and Susskind's playful preprint is a modern continuation of this approach.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-23602170028709007572021-07-23T13:07:00.004+02:002021-07-23T13:10:36.150+02:00Hatsuda, Siegel on their set of F-, M-, T-, S-theoriesMrs Mačiko Hatsuda (Tokyo+Tsukuba) and Mr Warren Siegel (Stony Brook) posted a fun new paper with their somewhat unusual cousins of M-theory: <blockquote><b><a href="https://arxiv.org/abs/2107.10568">Perturbative F-theory 10-brane and M-theory 5-brane</a></b></blockquote>Readers who expect colorful fonts (not only) on the title page won't be disappointed. This research is a part of the program to make dualities manifest by increasing the dimensionalities of spacetimes and branes' world volumes.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />The simplest T-duality of string theory on the torus \(T^k\) has the \(SO(k,k,\ZZ)\) T-duality group. Nonperturbatively, for M-theory on tori, this pseudoorthogonal symmetry is extended to the exceptional groups, roughly speaking to \(E_k(\ZZ)\). Can these symmetries be made manifest by writing the theory in a new way, with some bonus degrees of freedom (and dimensions)? Yes, you can. But you really need to add dimensions.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />In the T-duality case, you need a spacetime which has both the toroidally compactified dimensions \(X^\mu\); as well as their T-duals \(\tilde X^\mu\). However, these dimensions aren't quite independent from each other. In particular, string excitations have to obey the level-matching condition, \(L_0=\tilde L_0\). It means that only some excited string modes are allowed. But the condition isn't quite equivalent to \(N_L=N_R\), the equal number of left-moving and right-moving excitations. Instead, it is something like \(N_L = N_R + n\cdot w\) which depends both on the momentum \(n\) and the winding \(w\).<br><br>So the "fields" living on this doubled space are not quite arbitrary. They obey various constraints which generalize self-duality constraints. Things become even more mathematically advanced if the perturbative T-duality group is replaced by the full non-perturbative U-duality group where lots of things become exceptional in the group theory sense.<br><br>But this Lady and Gentleman are good mathematicians and they have lots of fun in generalizing the doubled spacetimes and world volumes to the exceptional case. Instead of simple "doubled dimensions", they have a master spacetime whose coordinates may transform as a spinor of a pseudoorthogonal group; and in this spinor-type spacetime, a vector-like brane is embedded. These objects are not just identical to the usual spacetimes because in usual spacetimes, we always consider the spacetime coordinates to be a "vector" of the most important spacetime-related group, the Lorentz group. Here they are spinors or other things.<br><br>Their research leads them to postulate a commutative diamond diagram with an F-theory at the top; M-theory on the left middle; T-theory on the right middle; and S-theory at the bottom. The letters M- and F- look like "the" usual Mother and Father theories and I think that it is not a deliberately misleading match in the notation. On the other hand, I think that their usage of the terms M-theory and F-theory is generalized relatively to what we normally consider M-theory and F-theory in 11 and 12 dimensions. But I just don't quite understand why the usual F-theory and M-theory are examples of their generalizations of the same name.<br><br>It's surely interesting to have constructions that make mathematically pretty properties such as U-duality groups manifest; but there are probably other things we have to pay. Perhaps more seriously, I am annoyed by the apparent conclusion that "their different theories" are not different theories at all; they are different formalisms to describe the same theory or theories. M-theory and F-theory in the normal sense are not just formalisms. They describe rather particular vacua (although you may adjust the axion-and-dilaton and in F-theory, aside from the shape of the 11 or 10 dimensions). And it is rather important that you may use <em>any of the equivalent formalisms</em> to deal with these theories. Changing the formalism to an equivalent one doesn't change the "theory".<br><br>For this reason, I think that they have redefined not only the terms M-theory and F-theory (let us generously ignore their really new masterpieces, the S-theory and T-theory); they have redefined the word "theory" itself. I would be afraid that extensive portions of work may end up being nothing else than contrived notation for simple things. Exceptional groups look cool and mysterious and there may be a contrived way to make the groups manifest; on the other hand, the overall mathematical difficulty of the theories can't be changed. In particular, you should better learn the exceptional groups at one moment or another! My suspicion is that these reshuffled definitions only change the pedagogical curriculum – i.e. the moment at which you learn the damn surprising group. They are not really made "more emergent" or "more explained" than they were before you started with the MSFT-theory formalism (which could be reasonably funded by Microsoft).<br><br>While it's interesting, we may have another general problem with the whole effort. The exceptional duality groups of the regular type are only relevant for the toroidal copmactifications. But there are many compactifications that are not toroidal. In fact, one might reasonably argue that the tori, while maximally supersymmetric, are "almost infinitely special". Do we learn something deep about the whole string theory – which surely has lots of non-toroidal vacua – if the generalized game with the formalism seems to depend on procedures that only work this simply for tori?Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-75806039193105066932021-07-21T06:23:00.020+02:002021-07-21T06:44:29.674+02:00Type IIB string theory with the cosmological constant \(10^{-144}\)<a href="https://cdn.zmescience.com/wp-content/uploads/2020/03/Untitled-design.jpg" rel="nofollow"><img src="https://iitk.ac.in/snt/blog/2013/07/24/dark-matter2.jpg" width=407></a><br><br>A Cornell-Northeastern-MIT quintuplet (Demirtas et al.) released two new hep-th papers (cross-listed in hep-ph) <blockquote> <b><a href="https://arxiv.org/abs/2107.09065">A Cosmological Constant That is Too Small</a></b> (short)<hr> <b><a href="https://arxiv.org/abs/2107.09064">Small Cosmological Constants in String Theory</a></b> (long) </blockquote>Demirtas, Kim, McAllister, Moritz, and Rios-Tascon claim to solve "one of the layers of the cosmological constant problem" as we have carefully distinguished it for over two decades.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />There are still some doubts whether the basic modern cosmological model is right – one in which the acceleration of the expansion is due to dark energy (70% of energy density in the Universe) which behaves like the cosmological constant or strictly <em>is</em> the cosmological constant. We will assume that the Universe has a positive cosmological constant, which has been the professionals' (I mean competent top-down theorists') preferred belief since the telescope observations of the late 1990s.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />In the 4D Planck units, the apprent cosmological constant is tiny, like \[ \Lambda \sim + 10^{-123} \] The inverse square root of this number approximately gives the linear size of the Universe which is \(10^{60}\) in the Planck units (that is a radius). The value of \(\Lambda\) above has some awkward, not so hidden, features. First, it is positive. Second, it is really tiny. Its being positive means (as we know from other sources, meaning particle physics) that our Universe cannot be unbroken supersymmetric. The tiny value indicates a naturalness-like problem: it seems extremely unlikely, like \(Prob\sim 10^{-123}\), that a positive dimensionless number ends up being as small as \(10^{-123}\).<br><br>In reality, the absence of very light superpartners means that supersymmetry is broken much more than needed to generate this tiny cosmological constant. The most straightforward order-of-magnitude estimates of the cosmological constant as a function of the observed SUSY breaking (meaning the observed superpartner masses) would produce \(\Lambda\) that is much closer to one than \(10^{-123}\) is, and consequently a much smaller Universe than \(10^{60}\) in the Planck units.<br><br>Various approximations (like non-gravitational or effective quantum field theory) and various "strategies to think and estimate" within them, like the naturalness argumentation, imply (to one level of belief or another) that it should be impossible to derive \(10^{-123}\) as the "only right" value of the cosmological constant. Many people have adopted the multiverse/landscape view that there are googols of vacua in string theory (the number is at least roughly the inverse cosmological constant), some of them end up having a tiny positive constant like ours by a coincidence, and those are overrepresented in the "research derived from observations" because life and observers are much more likely to arise in such universes (and they may be totally banned elsewhere). The more you emphasize that something like humans may be born in a world like ours and it should determine which of the vacua are likely as an arena for cosmology, the more "anthropic" you are. Clearly, many of the "hardcore anthropic" believers end up saying very wrong and stupid things, even basic things about the probability calculus.<br><br>But this whole scheme may be completely incorrect and the new papers we discuss returned to a previous stage of the research, namely to an intermediate question <blockquote>Can string theory refuse to respect the simplified probability estimates (mostly based on quantum field theory and naturalness) and rather naturally produce a tiny numerical value of the cosmological constant that looked unlikely? </blockquote>And their answer is Yes, it can. It is far from even a semi-realistic vacuum because the spectrum is unrealistic, some supersymmetry is unbroken, and correspondingly, the sign of the cosmological constant is negative, not positive. Many readers (and us) understand numbers so let me show you the value of the cosmological constant that they have calculated for a supersymmetric AdS background in type IIB string theory:\[ V_0 = -3M_{pl}^2 e^{\mathcal K} |W|^2 \approx -1.68\times 10^{-144} M_{pl}^4 \] The first equality is a standard supergravity formula for the vacuum energy. To get a small number, you might be tempted to use the exponential but you wouldn't get very far. Instead, they succumb to the fact that the exponential of the Kähler potential is "pretty much" comparable to one and they end up finding the configurations with a tiny superpotential \(W\) instead. <blockquote> <b>The trick behind the smallness is that the perturbative part of \(W\) cancels completely – which is much more possible than in field theory because the terms are determined by integers – and the small part comes from non-perturbative terms, here the world sheet instantons.</b></blockquote>To get the cosmological constant around \(10^{-144}\), they need world sheet instantons coming from a rather large (but not implausibly large) compactification manifold. Its volume in the string units turns out to be around 945. You are surely interested in the precise geometry. It's a threefold hypersurface \(X\) in a toric variety \(V\) constructed out of a reflexive polytope \(\Delta\) and you need a \(8\times 4\) matrix of vertices, you can determine the topological invariants, calculate \(g_s\sim 0.01,\) and you need to add fluxes and orientifold planes, to make it yummy. The work and expertise needed to settle this higher-dimensional geometry correctly clearly look much higher than the "creative" usage of this geometry for their clever phenomenological scenario – but the latter activity, while less intellectually demanding, may be more revolutionary and groundbreaking.<br><br>But this seems to be a controllable, supersymmetric, compactification with a cosmological constant whose absolute value is as "shockingly" tiny as the observed one, or a bit smaller. Clearly, they may find a whole class of such tiny-cosmological-constant vacua (but the number of elements in the class is much lower than the googols you need in the statistical, anthropic strategy to establish their existence).<br><br>The supersymmetric and \(AdS_4\) character of their vacuum may be considered a disadvantage because it's clearly not ready to be used directly for our Universe. From a conceptual perspective, when the goal is to make a solid step in the understanding of the cosmological constant problem, you may find the unbroken supersymmetry a great advantage because it makes the whole vacuum controllable and calculable. The results seem much more reliable than any results involving supersymmetry-breaking vacua.<br><br>So again, they negate or circumvent the "probabilistic estimates" of the typical magnitude of the cosmological constant.<br><br>They obtain a tiny (negative) vacuum energy because the superpotential is tiny. And the superpotential is tiny because the perturbative terms exactly cancel – which is possible because many quantities are discrete, determined by integers such as the topological invariants of the compactification manifold; and because the volume of the manifold is naturally "rather large" (of order one thousand), much like the largest Hodge numbers (which are several hundreds), and this "relatively large size" of the manifold, correlated with the complex topology ("several hundreds" is naturally the highest possible Hodge number \(h^{1,1}\) of Calabi-Yau threefolds that may mathematiucally exist) is enough to make the world sheet instantons "exponentially small" and that is what ultimately produces the tiny superpotential and therefore a tiny cosmological constant.<br><br>They cannot make a "directly analogous" construction of a SUSY-breaking de Sitter vacuum right now. No one can really reliably calculate de Sitter (or positive cosmological-constant) vacua in string theory at this moment, with no further refinements needed in this sentence, and claims about dS vacua almost always encode the authors' prejudices (some of which seem stupid). But these authors believe that a similar general Ansatz or strategy might lead to tiny positive cosmological constants and perhaps a realistic dS vacuum, too: perturbative terms exactly cancel which is not "unnatural in string theory" because it only amounts to some cancellation of (a priori) integers; and the observed tiny value therefore comes from non-perturbative (and therefore exponentially small) corrections.<br><br>Because SUSY seems broken intensely in our Universe, I think that their general strategy includes the implicit statement that string theory produces some discreteness-based cancellation of the vacuum energy whose strength is much stronger than the cancellations derived purely from supersymmetry. This ability would amount to another "string miracle" but one may argue that qualitatively similar string miracles are already known. The real punch line is a more general one: Top-down theorists as a community should avoid premature conclusions such as "the anthropic scenario is made unavoidable by the tiny value of the vacuum energy" because qualitatively different scenarios seem possible, at least if we look at slightly different classes of vacua than those that we need for the world around us.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-25292842047373574712021-07-11T10:27:00.006+02:002021-07-12T05:56:12.318+02:00Evolution, deterioration of the world's thinking about the deepest stringy ideas<blockquote> <b>A paper on Monday:</b> The first paper was posted in the initial second of the day (showing their pride) by a <a href="https://arxiv.org/abs/2107.04039">Munich triplet</a>. They develop a machine learning technique to guess correlations in the string landscape and look for viable vacua. With clever techniques, it could even be possible to find the right vacuum in a googol-like set. Anti-string simpletons don't get such things because their extremely modest brains can only understand brute force (which must be done by others because they don't possess it, either). </blockquote>David B. specifically asked me to rewatch this video<br><br><iframe width="407" height="277" src="https://www.youtube.com/embed/sMvbtgE-idQ" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><br><br>especially the part between 20:45 and 48:35 (in the first 20 minutes, both Green and Schwarz mostly review their important work around the 1970s). It is a discussion of selected string theory big shots at the end of "String 2021" which formally took place in Sao Paolo but which followed the format of an "online conference" of the Coronazi era. I don't think that Jair Bolsonaro would agree with this kind of a "conference in Brazil". Participating in the discussion were Gentlemen such as Seiberg, Witten, Green, Schwarz, Strominger, Vafa, Yin, Harlow, Berkovits, Ooguri, and a few more.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Witten started by a current version of his thoughts about the emergent spacetime, emergent quantum mechanics, and "what is string theory" – as he is ready to talk about these things in mid 2021. These comments rather similar to the comments he has said many times in the past and I have agreed with them to a similar extent. But I think that he hasn't really updated his conclusions about (and deep research into) all these deep topics in the light of the newest developments in string theory and adjacent fields – and what became important in them. It is my feeling that he hasn't really thought about these things much in recent years.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Concerning the updates, he said that holography and similar "spacetime developments" has superseded the idea of looking for the best formulation of a theory of everything in the form of an "extension of topological string theory". The latter may be too linked to perturbative string theory, we used to say, and that made it obsolete in the 1990s when strongly coupled string theory started to be understood. But I actually tend to think that this elimination was premature.<br><br>Witten seemed worried about a particular technicality, namely the "non-unitarity of world sheet CFTs" in the presence of a time coordinate, if I summarize it by a well-defined specific version of the trait of these theories. This (spacetime) time coordinate existing on the world sheet makes the dimensions "not bounded from below" which makes it possible to create near-zero-dimensional world sheet operators out of \(\exp(i\omega t)\) times an arbitrarily high excitation of the other degrees of freedom. That corresponds to the relevance of operators with arbitrarily many derivatives in the spacetime and that implies nonlocality and – as he slightly carefully says, but he does say it in between the lines – the death of a very general Ansatz for a final theory.<br><br>OK, I don't believe that any of these things are a "problem" in the sense of Witten's negatively sounding judgement. Indeed, string theory produces actions that look nonlocal in the spacetime and the existence of the time coordinate is a factor that makes it unavoidable. But as some basic string theory textbooks correctly pointed out, string theory still manages to remain free of ghosts – despite these nonlocal traits that would generically imply ghosts, an infinite increase of the degrees of freedom, and other pathologies in quantum field theories. I think that this is a purely positive news describing some special structure beneath string theory.<br><br>When you formulate string theory in the explicit Minkowskian spacetime and you think about the result in the language that tries to mimic quantum field theories, you will see "problems". But they're really problems of your way of thinking about the structure (and the QFT paradigm that you are implicitly using), not disadvantages of string theory itself. Instead, you could choose to think in terms of the Euclidean spacetime. And I think it's right to think about the continuation to the Euclidean spacetime where things simply look less pathological. And in the Euclideanized spacetime, many of the "pathologies" just cannot arise. The real, Minkowski spacetime dynamics is just the Wick continuation of "something more constrained, manifestly less pathological" that you can construct in the Euclidean spacetime. Maybe other signatures are relevant, too.<br><br>Daniel Harlow – a young man in front of fake mountains – pointed out that you can be professionally hired in "things like string theory" even though you don't really know string theory, supersymmetry, and other things. Well, he may have been talking about himself, too. I have been warning about these things since 1998 or so – it became increasingly "normal" to do undemanding things and the AdS/CFT correspondence was already misinterpreted as an invitation of "people who have no real clue about string theory" into "string theory".<br><br>While the AdS/CFT correspondence is very important and its main father is damn reasonable about all these matters, AdS/CFT just is very far from everything, as Cumrun Vafa was emphasizing. Back in the 1990s, I used to think that Vafa had to have rather different thoughts, more mathematical-physics-like thoughts, about all these matters than I had. After all, he was a big shot in topological string theory that looked (and, to some extent, still looks) rather unphysical to me. But I was completely wrong. My thinking about these matters may be more aligned with Vafa's than with the thinking by any other Gentleman in this conversation.<br><br>Aside from more amazing things, the AdS/CFT correspondence became just a recipe for people to do rather uninspiring copies of the same work, in some \(AdS_5/CFT_4\) map, and what they were actually thinking was always a quantum field theory, typically in \(D=4\) (and it was likely to be lower, not higher, if it were a different dimension!) whose final answers admit some interpretation organized as a calculation in \(AdS_5\). But as Vafa correctly emphasized, this is just a tiny portion of the miracle of string/M-theory – and even the whole AdS/CFT correspondence is a tiny fraction of the string dualities.<br><br>This superficial approach – in which people reduced their understanding of string theory and its amazing properties to some mundane, constantly repetitive ideas about AdS/CFT, especially those that are just small superconstructions added on top of 4D quantum field theories – got even worse in the recent decade when the "quantum information" began to be treated as a part of "our field". Quantum information is a legitimate set of ideas and laws but I think that in general, this field adds nothing to the fundamental physics so far which would go beyond the basic postulates of quantum mechanics.<br><br>"Quantum information" (including "quantum computers", "quantum error correction", and all stuff of this kind) is really just an engineering-like application of the basic postulates of quantum mechanics. It follows from them – plus the application of pure mathematics that is in principle perfectly understood. For this reason, you just can't learn "something new" about the quantum foundations (and even the foundations of quantum gravity) if you start to intensely work on "quantum information". In practice, "quantum information" has only provided us with some toy models that are neither realistic nor terrible deep.<br><br>When Cumrun correctly mentioned that the real depth of string theory is really being abandoned, Harlow responded by saying that there were some links of quantum information to AdS/CFT, the latter was a duality, and that was important. But that is a completely idiotic way of thinking, as Vafa politely pointed out, because string theory (and even string duality) is so much more than the AdS/CFT. In fact, even AdS/CFT is much more than the repetitive rituals that most people are doing 99% of their time when they are combining the methods and buzzwords of "AdS/CFT" and "quantum information". Many people are really not getting deeper under the surface; they are remaining on the surface and I would say that they are getting more superficial every day.<br><br>There is a sociological problem – coming from the terrifying ideological developments in the whole society – that is responsible for this evolution. I have been saying this for a decade or two as well – and now some key folks at Princeton and elsewhere told me that they agreed. The new generation that entered the field remains on the surface because it really lacks the desire to arrive with new, deep, stunning, revolutionary ideas that will show that everyone else was blind. Instead, the Millennials are a generation that prefers to hide in a herd of stupid sheep and remain at the surface that is increasingly superficial.<br><br>A real problem is that folks like Harlow openly talk about "being able to stay in the field while not being interesting in XYZ". This is not how my generation or the older generations thought about it at all. We (or at least the great among us) didn't join theoretical physics because we wanted to have comfortable chairs from which we could easily attack everyone who doesn't parrot insane SWJ-like lies which is what the members of the "Particles for Justice" are doing. We joined the field because we were and we still mostly are passionate about finding the deepest truths about the Universe. This is almost completely lost in the younger generation that sees "being in theoretical physics" as an entitlement, not a result of their deepest dreams that have nothing to do with careers.<br><br>So most of the stuff that is done in "quantum information within quantum gravity" is just the work of mediocre people who want to keep their entitlements but who don't really have any more profound ambitions. As the aforementioned anonymous Princeton big shot told me, their standards have simply dropped significantly. The toy models in the "quantum information" only display a very superficial resemblance to the theories describing Nature. That big shot correctly told me that in the early 1980s, Witten was ready to abandon string theory because it had some technical problems with getting chiral fermions and their interactions correctly.<br><br>Harlow says that many of the people – who may be speakers at the annual Strings conference and who may call themselves "string theorists" when they are asked – don't really know even the basics of string theory. And they can get away with it. Just like there is the "grade inflation" and the "inflation of degrees", there is "inflation in the usage of the term string theorist". Tons of people are using it who just shouldn't because they are not experts in the field at all. Harlow said that many of those don't understand supersymmetry, string theory etc. but it's worse. I think that many of them don't really understand things like chiral fermions, either. It's implicitly clear from the direction of the "quantum information in quantum gravity" papers and their progress, or the absence of this progress to be more precise. They just don't think it's important to get their models to a level that would be competitive with the previous candidates for a theory of everything – like the perturbative heterotic string theory, M-theory on \(G_2\) manifolds, braneworlds, and a few more. They are OK with writing a toy model having "something that superficially resembles a spacetime" and they want to be satisfied with that forever.<br><br>There are lots of questions and possible hypotheses that remain unanswered and that are tremendously exciting. But most of the people who should have a high probability to think about these matters don't think about it at all. They are increasingly repeating buzzwords – and repetitively applying well-known, not very important, methods in contexts that are increasingly similar to the previous ones – and they don't really do better because they are not the people who are passionate in getting the deepest ideas (or who are good at getting them, for that matter).<br><br>For example, it is totally possible (my ideas) that the correct interpretation of the "quantum information", starting with the ER/EPR correspondence, is that the topological string/field theory may be extended to contain the full dynamics of string/M-theory that we need to describe everything in the world. Why? Because ER/EPR may map a wormhole configuration into an entangled state of "states of matter" at two places and vice versa. In one of the directions, this map increases the fraction of the quantum information that is stored in the "information about the spacetime topology". It is totally possible that all microstates of matter may be parameterized in purely topological data in an appropriate space that isn't necessarily the same thing as the spacetime. There could be a basis of the black hole microstates (for a finite black hole mass, a finite-dimensional space) in which all the basis vectors describe objects that topologically differ from each other. The real task is either to prove that this picture is incorrect; or to prove that it is correct and then find particular realizations of this Ansatz.<br><br>None of these things (and hundreds of other important things) are really taking place, despite those 2500 people who were registered for the online Strings 2021, and it's mainly because an overwhelming majority of the 2500 people aren't really interested in deep questions, the most accurate statements and the most universally valid ideas, in the ideas that really encapsulate everything that has been learned. They are OK with "remaining a part of a comfortable enough herd of sheep" that uses fancy names for itself that it shouldn't be using.<br><br>I think that this general negative development must be seen by many others who no longer belong to the youngest generation (although I do remember some of them when they belonged there LOL). So Shiraz Minwalla was answering a question of Daniel Harlow's. Shiraz's answer was that some question about M2-branes and M5-branes was basically answered by the uniqueness of string theory. This important concept, the uniqueness of string theory, is the kind of thing that the likes of Harlow simply don't get. They are not even interested. You could have seen the intrinsically unscientific thinking of Harlow's at another place. He said that in AdS/CFT, there are qualitatively different CFTs for every AdS background of string theory or quantum gravity. Those "used to be" thought to be different solutions in the same theory and "this AdS/CFT work" therefore makes this picture of a unified theory with many solutions obsolete or incorrect or politically incorrect or whatever his adjective is exactly supposed to be – it is surely an adjective that means that "he won't do research on that".<br><br>Well, the problem with this thinking is that it is completely wrong and idiotic. Those vacua are still demonstrably related by dualities and they demonstrably are solutions to the same theory. The AdS/CFT description obscures this unity but it doesn't make the unity go away because there are other ways to see that the unity exists. But the likes of Harlow don't want to see any other ideas that give them "inconvenient" truths such as the truth that string theory unifies all the vacua. They don't really have the passion for the truth and they choose the "convenient lies" that allow them to call themselves string theorists, although they are not, and to terrrorize all sane people with their radical and criminally dishonest SJW organizations.<br><br>So sorry, Mr Harlow, but if you hide your head into the sand because you find the unity of the string theory vacua inconvenient for an easy job which is really why you arrived, that says something bad about you, not about string theory. If your whole herd adopts this strategy which really denies the important truth by spinning some superficial, distorted, and narrow-minded parts of the research, it only shows that the herd itself sucks. You may find it enough but it is enough for the actual science and actual scientists – and for the people who have some value. String theory does unify all the consistent quantum gravitational vacua and with a couple of extra conditions added (e.g. a 11D flat vacuum), it produces unique theories. That is the truth and you can't turn it into a non-truth just by joining a herd of dishonest sheep that decided to "research" only things that confirm your convenient lies.<br><br>I don't plan to proofread this text because the benefits would seem to exceed the costs.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-54954646424722422482021-06-23T08:04:00.003+02:002021-06-23T08:16:44.149+02:00Tetrahedron instantons, Magnificent Four, and the mother of all random partition modelsThe most interesting new hep-th arXiv paper today is the DESY-Rutgers-Beijing work <blockquote> <b><a href="https://arxiv.org/abs/2106.11611">Tetrahedron Instantons</a></b></blockquote>by Elli Pomoni, Yan, and Zhang. Instantons are localized configurations in the Euclidean spacetime (they are also localized in the Euclidean time, in this sense they live at an "instant", and that explains their names) and as local extrema of the path integral, they contribute to the transition amplitudes in Feynman's approach to quantum mechanics. The contribution is typically nonperturbative at small \(g\), scaling like \(\exp(-C/g^2)\) at least in the simplest cases.<br><br><a href="https://www.aliexpress.com/item/33002424584.html" rel="nofollow"><img src="https://ae01.alicdn.com/kf/HLB122RHRCzqK1RjSZPcq6zTepXam/Black-Obsidian-Tetrahedron-Pyramid-Stone-Stones-Crystals-Pyramide-Wicca-Cristal-Islande-Pierre-Naturelle-Cristaux-Healing-Home.jpg" width=407></a><br><br>In gauge theory, we have the simplest or "real" instantons in which the field strength is self-dual but arranged in a topologically nontrivial configuration that makes its magnitude decrease away from the center of the instanton. In string theory, there are other instantons that may be made of strings and branes, not just fields (gauge fields).<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />In particular, perturbative string theory has contributions from the world sheet instantons which are Euclidean 2-dimensional world sheets that are wrapped on some cycles of the spacetime geometry, ideally of the compactification manifold only. They are normally closed 2-surfaces but there exist very interesting open ones, too. In the realistic D-braneworld models of particle physics, world sheet instantons may have the shape of the triangle stretched between 3 intersections of D-brane stacks. The fact that the amplitude goes like \(\exp(-S)\) and the action includes a term scaling in the area of the triangle, \(TA\), is a reason why these D-braneworlds naturally produce exponentially small Yukawa couplings and therefore hierarchically "small and smaller" fermion masses, just like observed!<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />In this new paper, they study a type IIB string theory compactification with intersecting D7-brane stacks. And the stuff isn't made of the fundamental world sheet. Instead, it is the world sheet of a D1-brane. And the surface is close and topologically a sphere. But a very non-round sphere. In fact, it is a tetrahedron. "Where the D1-brane world sheet is wrapped" is one way to describe the instanton but a D1-brane (a lower-dimensional one) may also be produced from gauge field configurations within higher-dimensional D-branes (such as D7-branes) so they also describe this instanton gauge configuration as a more old-fashioned instanton in gauge theory (a configuration of classical fields).<br><br>They study the partition sum and find out that it "lies in between" (whatever this position exactly means) the Donaldson-Thomas invariants and the partition function of the <blockquote> <b><a href="https://arxiv.org/abs/1712.08128">Magnificent Four</a></b><br> <b><a href="https://arxiv.org/abs/1808.05206">Magnificent Four With Colors</a></b></blockquote>by Nikita Nekrasov (and Nicolo Piazzalunga, in the second case), from 2017. Nikita has calculated the index of a system of D0-branes moving inside the environment with D8-branes and anti-D8-branes. The relevant partition sum led him to four-dimensional generalizations of the Young diagrams and he has conjectured this to be the MOTHER OF ALL PARTITION FUNCTIONS. The present authors make a similar conclusion which is probably not too shocking because the D1-D7 environment seems to be a T-dual of the D0-D8 one.<br><br>It is really cool because I've been saying that the three- or four-dimensional Young diagrams are the MOTHER OF ALL PARTITION FUNCTIONS for more than two decades. You should be able to find relevant blog posts on this website as well but <a href="https://motls.blogspot.com/search?q=three+dimensional+Young&m=1&by-date=true">I can't do it right now</a>, it is easier for me to rediscover the things again. In the logic of the quantum foam by Vafa and pals, the three-dimensional Young diagrams' partition sums also rephrase the sum over some topologies of 3-complex-dimensional manifold represented by the toric geometry; they also naturally discretize some world sheet instantons. But Nekrasov needs to add a dimension.<br><br>But as I have realized for decades, there are other reasons why these could be the most general theory of deep physics or its elementary building blocks. In particular, via the Schur transformations, irreducible representations of a group are equally numerous as the conjugacy classes. Amplitudes in theories with symmetries may be organized in terms of the irreducible representations or the conjugacy classes. The three-dimensional extension of the two-dimensional Young diagrams may be analogous to the M-theoretical "3-index" extensions or the nearby "4-index ones" of the 2-index objects in the unitary-style gauge theories.<br><br>The degree of generality or complexity is likely to peak for three-dimensional Young diagrams, much like some topological problems in geometry peak around the dimensions 3 or 4 (like the difficulty of the Poincaré conjecture). The leadership of these 3- or 4-dimensional-like structures could be an equivalent observation as the supremacy of M-theory. Gauge theories with 2-index fields etc. may be obtained from the 3- or 4-index M-theory or F-theory but there aren't really strictly more amazing structures that could stand above M-theory, M-theory is the mother and this statement probably is rather close to Nekrasov's statement that his partition function is the mother of all partition functions.<br><br>I can't clarify the relationships between all these things beyond these vague suggestive remarks because there is not enough demand; and I don't understand all the relationships perfectly yet. But objects in this direction surely have a promising chance to be objects in a "most general formulation of a physics-like theory", something like a "matrix model" that has the ability to cover all allowed backgrounds or vacua, and you should believe that Nikita Nekrasov and (perhaps several) similar folks are damn mathematically deep and they don't study just some random nonsense (although the variables in his models ARE random nonsense LOL).Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-50468031848533075792021-06-09T07:17:00.011+02:002021-06-09T07:34:49.662+02:00M-theory on \(G_2\) manifolds, heterotic strings, oceans, climate, and carbon<a href="https://i.ytimg.com/vi/Gl3mQH_jLvQ/maxresdefault.jpg" rel="nofollow"><img src="https://i.ytimg.com/vi/Gl3mQH_jLvQ/maxresdefault.jpg" width=407></a><br><br><em>This unattractive Titanic-like building will become Czechia's tallest building (135 meters) and will be built just miles from the Prague Castle, in order to make us remember that the "architect" was impressed by the climate hysteria and willing to promote the lies about the climate.</em><br><br>Among the new <a href="https://arxiv.org/list/hep-th/new">hep-th papers that were released today</a>, five mention a "string" in the abstract. I think that the following paper is the most interesting one: <blockquote> <a href="https://arxiv.org/abs/2106.03886"><b>Non-Perturbative Heterotic Duals of M-Theory on \(G_2\) Orbifolds</b></a></blockquote>It's a nice paper that could have been written during the Duality Revolution in string theory of the mid 1990s but it waited up to 2021. It is a six-or-seven-dimensional or \(G_2\) counterpart of the statement that the \(T^4/ \ZZ_2\) orbifold is a special case of a \(K3\) manifold; and that the \(K3\) manifold may degenerate into two "half-\(K3\)" manifolds that are connected by a three-toroidal throat (times a line interval with irregular ends).<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />These \(G_2\) compactifications of M-theory which degenerate into \(T^7 / \ZZ_2^3\) orbifolds have their dual (equivalent) heterotic \(E_8\times E_8\) descriptions. But in order to distinguish between the precise shapes of the seven-dimensional \(G_2\) geometry, you need to imprint some extra data to the heterotic string, you need some extra data which are naturally "non-geometric" because one dimension is being subtracted. And the non-geometric data are some data about bundles describing instantons at the orbifold singularities.<br><br>There isn't sufficient demand for me to write more about technical physics questions although I have needed over 15 years to fully appreciate that I have always been throwing pearls before swine – swine like you who may prefer mediocre anti-physics activists' lies over world class physics.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />But what I found tragic is the analysis of the authors. They are <blockquote>Bobby Samir Acharya, Alex Kinsella, David R. Morrison </blockquote>Of course, I know Acharya and Morrison very well. They are brilliant senior physicists of higher-dimensional supersymmetric compactifications. Bobby (who has also worked as an experimental particle physicist for ATLAS) is tied with the Kings College in London and with Trieste; Dave is in Santa Barbara. It is the third name that is new and junior. Great, he is a PhD student in Santa Barbara, I thought, a sign that a younger generation is capable of doing serious exciting science.<br><br>My excitement didn't last long. I found this web page: <blockquote> <a href="https://kinsella.earth/">Kinsella.Earth</a></blockquote>Roughly at this time, Kinsella is getting his PhD under Morrison for "holonomy in geometry, analysis, and physics" and he will start a postdoc job in July 2021. So far so good. But keep on reading: <blockquote>...In July 2021, I will start as a postdoctoral investigator working with Amala Mahadevan at Woods Hole Oceanographic Institution.<br><br>My work in oceanography is motivated by the question of how ocean circulation and turbulence affect Earth’s climate. I’m interested in the ways that geophysical fluid dynamics in the ocean and atmosphere determine distributions of heat and carbon via transport phenomena and biogeochemistry. </blockquote>Holy cow, oceans, climate, and carbon? You can see that it's full of buzzwords that are convenient lies for the likes of Gr@tins Trautenberk. The new boss of Kinsela's, <a href="https://scholar.google.com/scholar?q=Amala+Mahadevan&hl=en&lr=&btnG=Search">Ms Amala Mahadevan</a>, has been at least a proper oceanographer and her papers were mostly about plankton. But how Kinsella describes his future "research", it is already a straight climate alarmist pseudoscience.<br><br>How may something as terrible as this thing happen? Was Kinsella unable to get a job in proper theoretical physics or geometry, something that he is clearly very good at? Was he blackmailed by the deformed job market in which governments – largely controlled by filthy and dishonest far left activists – favor pseudoscientific garbage over quality science? Or has Kinsella been truly brainwashed not to understand that what he is planning to do is a hardcore example of pseudointellectual prostitution? At any rate, I am utterly disgusted. This is an example showing how this whole generation has been turning and is still turning into a giant worthless cesspool.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-16408347148714696662021-05-20T06:28:00.030+02:002021-05-20T17:08:27.552+02:00The seemingly infinite volume of \({\mathrm{PSL}}(2,\RR)\) is \(-2 \pi^2\)<span class="isolimg"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/HyperboloidOfOneSheet.png/150px-HyperboloidOfOneSheet.png" width=144 align="left"></span><b>And you need just a few short steps to calculate the D-brane tensions from that fact</b><br><br>The today's most joyful hep-th abstract was written by two names that didn't click with me (<b>Eberhardt, Pal</b>; their publication records initiated around 2017 seem impressive) but their affiliation did, the IAS at Princeton (that's a place where Einstein has worked for 1/2 of his career). It's an appropriate place for the authors to work because I am really happy about the paper which is titled simply <blockquote> <b><a href="https://arxiv.org/abs/2105.08726">The Disk Partition Function in String Theory</a></b></blockquote>and it discusses a theme that I found cool since the high school: many seemingly infinite expressions must actually be assigned finite values. The most famous example is the <a href="https://motls.blogspot.com/2011/07/why-is-sum-of-integers-equal-to-112.html?m=1">sum of positive integers</a>\[ 1+2+3+4+5+ \dots = -\frac{1}{12}. \] This identity is important enough in string theory, including the (natural, bureaucracy-free) calculation of the critical spacetime dimension. I've played with some <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20motl%20and%20date%201995">generalizations</a> before I started to submit some serious papers. You may also check a <a href="https://motls.blogspot.com/2014/01/a-recursive-evaluation-of-zeta-of.html?m=1">recursive evaluation of zeta</a> of negative integers.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Other seemingly singular sums, products, and integrals have finite natural (regularized) values, too. For example, the <a href="https://www.youtube.com/watch?v=9OlrgzhjifM">product of all positive integers</a> is \(\sqrt{2\pi}\), yes, it is the same factor that you see in the Stirling approximation for the factorial (including the factorial of infinity) and you may even determine the <a href="https://www.google.com/search?q=regularized+product+primes&oq=regularized+product+primes&aqs=edge..69i57.3437j0j4&sourceid=chrome&ie=UTF-8" rel="nofollow">product of all primes</a> which is \(4\pi^2\). Because the number PI appears everywhere in this blog post, you should notice that we are currently using the newest <a href="https://www.mathjax.org/">MathJax 3.1.4</a> ;-). <a href="https://en.wikipedia.org/wiki/TeX#TeX82">The current version of \(\rm\TeX\)</a> is 3.141592653, I clearly share some of Donald Knuth's intellectual humor.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />This new paper discusses an identity with a very similar result (as these regularized sums and products),\[ {\rm Vol} ({\mathrm{PSL}}(2,\RR)) = -2 \pi^2. \] Cool. For the result to be this unambiguous, they had to pick a (usual) convention for the metric on the group manifold of this "projective special linear" group. The convention is basically equivalent to "the unit radius of all the spheres inside". The group is noncompact which means that its volume is infinite. However, this infinity is just an informal way of describing "that the group is noncompact" while the right value of the "volume" that you need to substitute to formulae is a finite number. Like in the case of the products of integers, it is a product of powers of rational numbers and \(\pi\); and like in the sum of integers, the right value turns out to be negative. Yes, the volume is \(-2\pi^2\).<br><br>Well, they dedicate the Appendix B to this constant. I can give you a motley, three-line derivation. The group is really the same as \({\mathrm{PSU}}(1,1)\). That differs from \({\mathrm{PSU}}(2)\) by a simple flip of the signature. But \({\mathrm{SU(2)}}\) is just a <a href="https://en.wikipedia.org/wiki/3-sphere#Elementary_properties">three-sphere whose surface is \(+2\pi^2\)</a>. I may get the volume of the noncompact group simply by Wick-rotating 2 of the 4 dimensions where \(SU(2)\) is embedded, and each Wick rotation produces a factor of \(i\). And \(i^2=-1\) gives me the minus sign. (Later on Thursday, the authors have written to me about some subtleties, e.g. that I should really divide the volume by 2 because of the P for projective, and I surely agree with that, and they seem to be infidels when it comes to the very possibility of fixing such a derivation.)<br><br><iframe width="407" height="277" src="https://www.youtube.com/embed/RijB8wnJCN0" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><br><br><em>Who you trying to get crazy with ése? Don't you know I'm loco? Insane in the membrane, insane in the brane.</em><br> <br>Note that in this post about <a href="https://motls.blogspot.com/2012/07/euler-characteristic.html?m=1">the Euler characteristic</a>, I argued that the Euler characteristic may be interpreted as a regularized number of points of a manifold (which is also seemingly infinite for almost all manifolds).<br><br>This volume enters calculations in perturbative string theory, too – and that is why they are doing it. Note that the oldest nontrivial result of string theory (in the Prague Spring year of 1968) was the <a href="https://en.wikipedia.org/wiki/Veneziano_amplitude">Veneziano amplitude</a>, the formula for the scattering amplitude of 4 open string tachyons in bosonic string theory. The result is \(B(u,v)\) where \(u,v\) are linear functions of the Mandelstam variables. It is the Euler Beta function, a ratio of the Gamma functions (generalized factorials).<br><br>You may obtain the Veneziano amplitude from a disk-shaped world sheet via a path integral. You need to insert those 4 vertex operators for the external four tachyons somewhere at the boundary of the disk. Three of them may be placed to arbitrary positions, by using the \({\mathrm{PSL}}(2,\RR)\) residual symmetry (which has 3 free real parameters). The location of the fourth vertex operator cannot be fixed and must be integrated over. So the <a href="https://en.wikipedia.org/wiki/Beta_function">Beta function</a> may be expressed as a one-dimensional (real) integral.<br><br>Cool. If you considered the scattering amplitude of 3 tachyons, there would be no residual integral – but the (spacetime) angles between the tachyons would have to be very specific for them to be on-shell and for the energy-momentum conservation to hold. But can't you go below 3 tachyons? Like 0 tachyons? If you do, the path integral has the unfixed residual symmetry group (more generally, a part of it which is 1,2,3-dimensional if you only place 2,1,0 tachyonic vertex operators on the disk, respectively). Because the configurations related by the \({\mathrm{PSL}}(2,\RR)\) are physically eequivalent (gauge symmetry), you need to avoid multiple counting and that is why you divide the (finite) integrand by the volume of the group. Because that denominator seems infinite, you naively get a zero.<br><br>But just like the sum of integers isn't really "infinite" in the deep physical sense, the volume of this group manifold isn't infinite, either. It is finite, it is \(-2\pi^2\) in the usual conventions, and you may substitute this correct value to the path integral and calculate the D-brane tensions from the simplest possible disk-shaped path integral in the appropriate string theory. They actually do so. The fact that the regularized volume of \({\mathrm{PSL}}(2,\RR)\) is negative is essential for the fact that the known D-brane tensions end up being positive. <br><br>They verify their result for the D-brane tension against <a href="https://www.kitp.ucsb.edu/joep/links/joes-big-book-string/errata">Joe's Big Book of String</a> which I have previously verified as well – well, I sent 128 valid corrections to Joe (almost beating the rest of the Earth combined), including numerous corrections of formulae dealing with D-brane tensions, indeed. ;-)<br><br>But yes, the simplest forms of the path integrals should be dealt in this way. The infinite values such as the volumes of similar noncompact groups are spurious and the right, finite volumes should be widely known (instead of computing the simple disk diagram without insertions, the usual method to determine the D-brane tensions involve much more complex and "reducible" diagrams). Note that \(-2\pi^2\) is a number of a similar form as the volumes of \(d\)-dimensional spheres. And indeed, the group manifold is basically a hyperboloid which is a Wick rotation of a sphere which is really why you had to get a similar form of the result. However, you need to be careful about the world sheets of the spherical topology. In superstring theory, you clearly get a vanishing result from those, after all (supersymmetry is a spacetime argument why it needs to be so).<br><br>So, an update: Yes, I think that the purely bosonic regulated volume of \(\mathrm{SL}(2,\CC)\) should better be related to the volume of \(\mathrm{SO}(3,1)\) which is a different signature of \({\mathrm{SO}}(4)\) which is \(\mathrm{SU}(2)\times {\mathrm{SU}}(2)\) which is two 3-spheres, so up to the powers of two (and, less likely, other rational factors), I do think that the volume of \(\mathrm{SL}(2,\CC)\) should be \(4\pi^4\) or so, as a product of two disks! I would be surprised if the regulated volumes were legitimate but this tempting configuration of "closed strings equal open squared" in the case of "sphere is a disk squared" were totally violated for volumes. (I won't write the group names with the mathrm command again because it is a pain in the aß.)Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-78030512320628320162021-05-06T09:11:00.014+02:002021-05-06T09:49:50.824+02:00Wormholes in general phase spaces<a href="https://www.thomann.de/blog/en/best-trumpet-beginners/"><img src="https://www.thomann.de/blog/wp-content/uploads/2018/01/header-einsteigertrompeten-thomann.jpg" width=407></a><br><br>Herman Verlinde (Princeton) has released two new papers about one topic <blockquote> <b><a href="https://arxiv.org/abs/2105.02129">Wormholes in Quantum Mechanics</a></b><hr> <b><a href="https://arxiv.org/abs/2105.02142">Deconstructing the Wormhole: Factorization, Entanglement and Decoherence</a></b></blockquote>He has studied general enough quantum mechanical systems – including simple enough ones such as a coupled harmonic oscillator – using the path integral. The path integral sums over all histories which are curves on the phase space (OK, he picks the phase space path integral; I don't exactly know why).<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />A funny fact is that in a local or quasi-local (field) theory, the action (which enters the exponent/phase of the path integral) may be integrated over some region of the spacetime or worldvolume only, \(\Sigma\) (capital Sigma). When the region \(\Sigma\) has several disconnected components, you may naively deduce that the action becomes the sum of the pieces from those components; and the path integral factorizes into a product over the components of \(\Sigma\).<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />However, as he points out, this is not generally the case because one may connect the components, too – by a wormhole with two or more entrances (these diagrams are called the "trumpets" i.e. the pro-Trump extraterrestrial aliens) – and it is these wormhole contributions that break the factorization rule. In general, he has to discuss nontrivial topologies that are allowed on the phase space. In the language of cohomology, the phase space always comes equipped with \(\Omega\), the capital Omega, a symplectic two-form that determines the Poisson bracket. This symplectic form has to be closed (yes, that is equivalent to the Jacobi identity for the implied Poisson bracket!) which means that it is locally exact but it doesn't have to be globally exact. <br><br>And the wormhole contributions exist exactly when this \(\Omega\) is not exact i.e. it's impossible to write it as \(\Omega=d\Lambda\) for some globally single-valued one-form \(\Lambda\), capital Lambda.<br><br>Intriguingly enough, he has linked the contribution to the partition sum from the "wormhole path integral" to an entropy. More precisely, it is the "path integral with \(n\) entrances" and it is equal to the \(n\)-th Renyi entropy of a thermal mixed state of the doubled system. In all the cases, the wormhole contribution may be seen to be a saddle term. In the second paper, these non-factorized terms are linked to decoherence with the environment.<br><br>It's all interesting and serious enough but maybe one should slow down. There are lots of fundamental questions here and answers should be proposed and defended by actual arguments. So first of all, are the other topologies always allowed and what does it mean? All interactions in perturbative string theory come from the freedom of the world sheets to have all allowed topologies. Once strings may split and join, all interactions are born. Is that true in an arbitrary quantum system that all topologies are allowed? May all these cases be considered "qualitatively the same"?<br><br>Second, we often say – and with good reasons – that the path integral isn't quite a fundamental description of a general quantum system. A path integral seems to depend on the existence of a classical limit. Theories such as the \((2,0)\) theory in six dimensions have no classical limit and no description in terms of a path integral, at least not one that would make the usual symmetries manifest. But we believe that there exists a full "set of operators" in an operator description of the \((2,0)\) theory. Is it true that the incorporation of the nontrivial wormhole topologies may always be represented as a correction that needs to be added <em>because</em> the classical-theory-based path integral would otherwise be inaccurate and wouldn't represent the precise "theory in the operator formalism"? And what principle would be exactly violated if we "demanded the nontrivial/wormhole topologies to be omitted"? I am asking because I feel that "the need to include all topologies in any chosen degrees of freedom" could very well be a badly understood law that may easily be strong enough to be the "master theory of everything". Such a law would implicitly require any quantum mechanical theory (of gravity) to have a classical limit/description with certain properties, too.<br><br>Third, a bit more technically. The wormhole term is also claimed to quantify decoherence. That's possible but decoherence is always just a "derived calculation", not a fundamental law. Decoherence only becomes possible when we – unavoidably arbitrarily – divide the system to the "degrees of freedom we trace" and "the environment". How is it possible that the wormhole contribution which seems fundamental even in the "complete theory" is said to be equivalent to a non-fundamental one that depends on the choice of "what is the environment"? Also, the path integrals tend to be used for mixed (and even thermal) states in this paper and similar papers. Do "pure state" counterparts always exist or does this reasoning push us to adopt the mixed states are more fundamental than pure states?Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-56414069796163109122021-05-04T08:10:00.010+02:002021-05-04T08:52:07.482+02:00Reorganizing a 4D S-matrix within a celestial CFTFor a couple of years, Andy Strominger and his so far lesser but not last collaborators have been immensely excited about the celestial sphere.<br><br><a href="https://planetary-science.org/astronomy/the-celestial-globe/"><img src="http://planetary-science.org/wp-content/uploads/2013/06/sphere.jpg" width=407></a><br><br>It's not hard to see why. As you can see, the celestial sphere (in the sky) is blue and has pretty stars in it.<br><br>Strominger et al. have studied the encoding of black hole information as seen on that sphere, scattering, hair, new conservation laws, and other related things. Note that if you imagine that we live in a 3+1-dimensional spacetime for a while, what we see at night (at one moment) is a two-sphere, \(S^2\), of possible directions. Each point on this \(S^2\) is actually a line in the 4D Minkowski space which goes to infinity. That's why there is no natural finite "length scale" defined on this \(S^2\). However, you may measure angles. After all, this \(S^2\) is conformally equivalent to a complex plane and the angle-preserving, conformal \(SL(2,\CC)\) transformations of this plane (or sphere) are equivalent to the \(SO(3,1)\) Lorentz group that transforms the Minkowski space.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />A big part of the insights in these papers are about the reorganization of the information – and dramatic reorganization, indeed. That's especially the case of a new paper <blockquote> <b><a href="https://arxiv.org/abs/2105.00331">New State-Operator Correspondence in Celestial Conformal Field Theory</a></b></blockquote>by an all-Harvard team consisting of Crawley, Miller, Narayanan, and – last but not least – Strominger (CMNS, not to be confused with Condensed Matter Nuclear Science which is a group of cold fusion crackpots).<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />OK, what did they do? The Minkowski spacetime phenomena must have some reinterpretation in terms of the celestial sphere and given the invariant structures, natural theories that live on the celestial sphere should respect the conformal symmetry – because the latter is isomorphic to the Lorentz symmetry in the 4D spacetime. So they decided that for a given 4D theory in the Minkowski spacetime (ideally just an off-shell theory given by the S-matrix; the reasons are analogous to the difficulties in getting bulk off-shell correlators in AdS/CFT), there should be a celestial conformal field theory (CCFT) on the celestial sphere.<br><br>I don't know whether they have a very good intuition what this CCFT is – it is surely better than mine by orders of magnitude. At any rate, it is a non-unitary CFT where the weights of operators may be negative. In fact, they are generically complex (and I hope that they realize that the OPEs therefore refuse to be single-valued and create a lot of subtleties). Like every CFT, however, it admits a state-operator correspondence and that is a focus of this new paper. (<a href="https://motls.blogspot.com/search?q=state+operator+correspondence&m=1&by-date=true">The state-operator correspondence</a> is mentioned in over a hundred of TRF blog posts; some of them are sketches of cutting-edge research. You won't find another personal website in the Solar System with this abundance of the state-operator correspondence and many other concepts LOL.)<br><br>If you want to calculate an \(N\)-particle scattering S-matrix element, you must find the corresponding states or operators (insertions) for the CCFT and, as they clarify, to calculate the inner product. Note that regular perturbative string theory uses a 2D world sheet CFT and external particles in the spacetime are represented by vertex operators that are pretty much independent of each other. The stringy perturbative calculation of the S-matrix amplitudes resembles a generalization of the "first quantized" calculation of Feynman diagrams: individual particles propagate as they do in 1-particle Hilbert spaces but they may split and join (in the Feynman diagram vertices). In normal perturbative string theory, the radial direction surrounding the operator insertions may be mapped to a "world sheet time" which is often correlated with some time in the spacetime but is a priori independent.<br><br>In their novel CCFT, things are fancier and less intuitive. The direction in which the operator insertions are evolved may be called a "Euclidean time" in the CCFT but it isn't close to the "spacetime time" at all. Instead, this "Euclidean time" in the CCFT is the <em>celestial latitude</em>. It means that you must slice the celestial sphere to circles of a fixed latitude (they are called "parallels" in the normal terrestrial geography). Note that Strominger often switches time to space and vice versa, for example, the holographic direction in his dS/CFT had to be timelike, not spacelike as it usually is. And there is one more major difficulty:<br><br>The individual particles in the in-and-out states of the 4D scattering are no longer independent insertions in the CCFT. Instead, if you insert a particle at some point of the celestial sphere, it creates a perturbation that propagates from North to South (or vice versa; and the opposite propagation is very unequivalent, like retarded and advanced propagators are) because, as we have just said, the natural "time" in the CCFT is always something like the celestial latitude. That's why the identification of the operator-or-state in the CCFT out of the external state is nontrivial because it isn't simply "factorized" to the external particles.<br><br>But when you do things right, the S-matrix element is given by a natural inner product in the CCFT. Well, it is the BPZ (Belavin-Poljakov-Zamolodčikov; the latter has agreed with me that this Czechoslovak-Yugoslav spelling of his name is cool; I learned the BPZ stuff mostly from Zamolodčikov) inner product. You must construct it out of a single circle in the CCFT; and acknowledge that circles are oriented and must run in the opposite direction for the BPZ product to be possible and nonzero. Note that the BPZ inner product is one that insists on naturally connecting 2 pieces of a world sheet along the boundary.<br><br>Such a bizarre reorganization should exist for a 4D S-matrix – arguably any 4D S-matrix, I guess. Out of a boring undergraduate system of Feynman diagrams linking particles in 4D, you should be able to get an equivalent but much prettier mathematical structure which builds everything out of expression that include the stringy, Veneziano-like \(\Gamma(h)\) objects, among other things, where \(h\) is the weight of an operator. Whether such a brutally different, creative reorganization is an important scientific breakthrough remains to be seen, as far as I can say. Whenever we get a qualitatively new description of physics and its degrees of freedom, some physical principles become opaque while others become transparent. Here they make the independence of separate particles in the 4D spacetime (basically the spacetime locality, even in the space of directions) opaque... while I don't exactly understand what has become more manifest – here I seem to assume that they would agree that the higher overall opacity would be a vice, not a virtue. ;-) My personal guess is that this celestial-sphere description must be ultimately equivalent to the twistor-based ones so e.g. the vanishing of the more-than-maximally-helicity-violating amplitudes and similar things should be manifest once you actually write down the rules to construct the CCFT.<br><br>Of course, I have deep sympathies for similar efforts – even though they may be called mere experiments so far. Note that I've described the basics of understanding Feynman diagrams as <a href="https://motls.blogspot.com/2015/08/feynman-sum-over-young-diagrams.html?m=1">sums over Young diagrams</a> in representation theory; and <a href="https://motls.blogspot.com/2019/01/quantum-gravity-from-self-collisions-of.html?m=1">locality on the configuration space</a> which is another reorganization from theirs (aside from a few related ideas). I believe that the configuration-space reorganization is the opposite one than theirs but I am not sure, maybe it is ultimately the same thing instead.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-13845584982630419662021-04-22T06:33:00.003+02:002021-04-22T19:47:05.110+02:00Swampland: light gravitinos bring their towers, slow down inflation<a href="https://www.youtube.com/watch?v=SrDxHGhxWkg" rel="nofollow"><img src="https://i.ytimg.com/vi/SrDxHGhxWkg/maxresdefault.jpg" width=407></a><br><br><em>A gravitino ball. Click the image for a video.</em><br><br>The swampland program deepens our understanding of quantum gravity by refining the conjectures stating that most of the seemingly consistent effective field theories have subtle bugs that prevent them from being completed to consistent theories of quantum gravity i.e. to string/M-theory vacua.<br><br>Gravitinos – the supersymmetric partners of gravitons – have entered several swampland hypotheses. For example, exactly one month ago, a <a href="https://motls.blogspot.com/2021/03/gravitino-must-never-be-motionless.html?m=1">paper said that gravitinos mustn't be motionless</a>. Now, in a fresh hep-th Madrid-French-German-Venezuelan paper <blockquote> <b><a href="https://arxiv.org/abs/2104.10181">A Gravitino Distance Conjecture</a></b></blockquote>Castellano, Font, Herraez and – last but surely not least – Ibáñez ;-) say that the gravitinos can't be too light, or at least they can't be adjusted to be light without additional consequences.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />In realistic \(\NNN=1\) string vacua, a light gravitino of mass \(m_{3/2}\) (here, 3/2 is the spin of the gravitino) imply that a whole Hagedorn-like tower of massive particles must exist as well which must also be light. The latter doesn't have to be as light as the gravitino itself but the mass scale of the tower obeys \[ M_{\rm tower} \approx m_{3/2}^\delta, \quad \frac 13 \leq \delta \leq 1. \] For an interesting example, if there is a collider-scale gravitino of mass close to \(1\TeV\), then there must be a tower of new particles that start at some intermediate \(10^5-10^{10}\TeV\).<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />On top of that, the Hubble constant determining the rate of expansion during the inflationary epoch must be rather low if the gravitino is light:\[ H \leq m_{3/2}^\delta M_{P}^{1-\delta} \] Again, it is some geometric average between the gravitino mass and the Planck mass. Isn't it exactly the same scale as \(M_{\rm tower}\)? That would mean that the inflationary epoch must be described along with the whole tower, wouldn't it?<br><br>At any rate, as expected from viable swampland papers, they accumulate evidence of various kinds in favor of their statement. An interesting fact is that their gravitino conjecture may be considered a strengthened (because it implies) the AdS distance conjecture.<br><br>I am sympathetic towards this proposal because gravitinos are "very quantum" from some perspectives. Well, one must be careful. Gravitons themselves are bosonic quanta of the metric field and they like to make Bose-Einstein condensates which simply change the background metric and that has a nice classical description – or a description in quantized effective field theories.<br><br>Due to supersymmetry, gravitinos must also exist and their properties must be pegged to those of the gravitons by supersymmetry. So they look rather classical, too. However, classical fermionic fields must be equal to zero – there are no nonzero anticommuting (fermionic) \(c\)-numbers. Nevertheless, gravitinos apparently can "do things", like produce condensates or perhaps bound states. These physical phenomena come with a mass scale that is also low if the gravitinos are light. If you think about a hydrogen-like bound state and its condensate, it is very obvious that you need the full quantum theory to study it, and these bound states may develop Bose-Einstein condensates as well. Any reduction to "the metric field only" must break down. You just need a tower.<br><br>Here, I only talked about the hydrogen-like bound states because they are very well-known ideas. I don't actually believe that it is the best way to think about the new unavoidable phenomena – but maybe even these things do exist. But there should be some new light portion of the spectrum and dynamics.<br><br>I don't quite get why the vacua with \(m_{3/2}\)=0 and unbroken supersymmetry don't immediately falsify the conjecture because the conjecture seems to predict a massless tower. Do I misunderstand something?Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-77909958343316020012021-04-15T20:23:00.004+02:002021-04-15T20:23:39.930+02:00Nima, singularities, and the fall of WilsonianismMmanuF at Twitter has kindly recommended me this recent, 2.5-hour-long video by Nima Arkani-Hamed:<br><br><iframe width="407" height="277" src="https://www.youtube.com/embed/5rEd_zGgboE?start=1199" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><br><br>Because he suspected that I may find it a bit long, being occupied with things like the insane anti-white and anti-Slavic racism thriving in the United Kingdom that has assaulted my homeland through soccer, MmanuF has also helpfully recommended a 10-minute-long excerpt to focus on, 0:20:00-0:30:00, and I did watch that.<br><br>Among other things, Nima has declared the end of Wilsonianism. The Wilson paradigm is just wrong, he stated at one point, and you can't organize physics according to the distance scales so that the theory at the distance scale \(L\) is always determined from the theory at a shorter length scale \(L(1-\epsilon)\). Instead, in a theory of quantum gravity (and he may apply it even to some other related theories), there are numerous UV-IR connections and communication in between the scales – aside from some role played by numerical coincidences.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Because Nima has been a hep-ph person and Wilsonianism defines it, while string-trained hep-th folks may be eager to throw away Wilsonianism, you might think that it would be bizarre if Nima abandoned Wilsonianism and I would defend it. But things have many extra layers of subtlety. In particular, my (former or forever) PhD adviser Tom Banks has been a great Ken Wilson worshiper. He admired that paradigm shift of the 1970s and I had to read some original papers about it. Note that Ken Wilson died in 2013.<br><br>Aside from that, Nima lives at Princeton where <a href="https://motls.blogspot.com/2020/06/madmen-at-princeton-remove-wilson-from.html?m=1">Wilson is being removed from all places</a> because "he was racist" according to the currently omnipotent unhinged racists there. In my city of Pilsen, we have bridges and many other things named after Wilson – who was a de facto co-father of Czechoslovakia. It was Woodrow Wilson, a Democratic president hated by most of the real Republicans, and a different Wilsonianism but who cares. ;-)<br><br>A substantial part of the aforementioned 10-minute segment was dedicated to a "warning" not to make a certain kind of an error. These insights may induce many reactions. Some people may fight them and claim that Nima must be wrong. Others may claim that he is saying obvious things that they have known for a long time. Others may feel illuminated.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />I tend to choose "Nima is saying things that I have known for decades" but there is a problem. Just a few days ago, <a href="https://motls.blogspot.com/2021/04/first-michio-kaku-is-releasing-his-new.html?m=1">I basically defended Wilsonianism</a> (although I haven't used these precise words) when I emphasized my view that string/M-theory may still be viewed as a framework to derive the (field and particle) spectrum of effective field theories and their parameters. OK, what is going on?<br><br>Nima said a nice and important thing that when we say that the Wilsonian effective theory fails at distances shorter than \(\ell\), we must interpret \(\ell\) as a proper Euclidean length... in the Euclidean spacetime. Why? Well, in our real world, we have both space and time and nearly light-like (null) intervals may be boosted to very short (timelike or spacelike) intervals. In the Minkowski spacetime, separations may look "very long" in the coordinate space but the proper distance or the proper time may be very short.<br><br>That is why the "balls" really look like "seemingly noncompact" solid hyperboloids in the Minkowski spacetime and the work with this topology is problematic. For this reason, the only "topological natural" discussion takes place in the Euclideanized spacetime where the short distances are confined in a ball and the proper distances inside the ball are shorter than \(\ell\) and equivalent to spacelike distances. Great. I agree with that.<br><br>However, as his discussion immediately realizes, we are not really behaving like that in most cases when we pretend to use the Wilsonian paradigm. In fact, by "long distance scales", we often mean the behavior of functions like Green's functions for a very long <em>real</em> time in the Minkowski space. But that long time is equivalent to a long <em>imaginary</em> spatial distance which isn't quite the same thing as the "small but real and positive" distances that define "short distances" in the Wilsonian paradigm. But in most cases, we assume that this "detail" (that we look at long times and not long spatial distances) doesn't matter because the relevant functions behave as Taylor series and those are qualitatively the same in all directions of the complex plane.<br><br>I have added this encapsulation of the discussion. I haven't heard it and it's possible that Nima has encapsulated the discussion unequivalently but I tend to believe that the disagreement wouldn't be substantial. Fine. At any rate, the Taylor expansion assumption may be incorrect and Nima states that it is incorrect in very typical Green's functions, indeed. Imagine that the experimenters carefully measure some Green-like function in the real Minkowski space, as a function of time, and they get\[ f(t) = \frac{2021}{t}. \] I added the numerator to prevent the future generations from repeating my blog posts and to encourage them to invent something new instead. That is great. It is the inverse function and you may calculate the counter-clockwise integral\[ \frac{1}{2\pi i} \oint f(t) dt = 2021. \] There is a nice singularity at \(t=0\), we may deduce, and the contour integral around that gives you \(2\pi i\) times the current year. The function is helpful because you may use it as a calendar but only up to the next New Year's Eve. Fine, it is getting time-consuming and I will switch to the units where \(1=43\cdot 47\) right now. :-) Do we know from the \(1/t\) asymptotic measurements that there is a singularity at \(t=0\)? No, we don't. The function could also be\[ f_2(t) = \frac{1}{t-1} = \frac{1}{t} \cdot \frac {1}{1-1/t} = \frac{1}{t} \zav{1 + \frac{1}{t}+\frac{1}{t^2}+\dots } \] The second factor comes from the geometric series. You may see that the original \(1/t\) is corrected by a power law expansion. This is quite a "textbook Wilsonian" expectation because such power law expansions in \(1/t\) are almost equivalent to the power law expansions in \(E\), i.e. the expansion according to the number of derivatives in the operators included in an effective theory. If your undergraduate Fourier knowledge is even deeper, you know that \(E\) doesn't quite get Fourier-translated to \(1/t\) but rather to \(\partial / \partial t\) but the effect is rather similar if these operators act on functions that admit a Taylor expansion and behave as a power law for \(|t|\to\infty\) in all directions of the complex plane \(t\in\CC\).<br><br>But again, as Nima says, it doesn't have to be the case. Consider the function\[ f_3(t) = \frac{1 - \exp(-kt)}{t} \] We have added an exponentially dropping correction, for \(t\to + \infty\), in the numerator. The effect may be unmeasurably small for a large \(t\) that the experimenters see. But we have changed the behavior in other parts of the complex plane. In particular, where are the singularities of \(f_3(t)\)? There are none. The value \(t=0\) used to be a singularity due to the denominator but that singularity and the \(t\) in the denominator is cancelled against \(1-\exp(-kt)\approx kt\) for \(t\to 0\) which appears in the expansion of the numerator. The exponential is smooth everywhere and doesn't introduce any new singularities. Well, more precisely, it doesn't introduce any singularities for a finite \(t\in\CC\). If you have listened to the right classification of singularities, you know that you mustn't hide the terrifying (or what is the right adjective LOL) singularity for \(|t|\to \infty\). OK, I think that it is called the essential singularity. Is it one?<br><br>While \(f_2(t)\) only moved the \(t=0\) singularity to \(t=1\), the new \(f_3(t)\) has erased it from everywhere. The flip side of this "beautification" is that \(f_3(t)\) behaves "terribly" far away from the real positive \(t\) axis. It oscillates on the imaginary \(t\) axis (which is the real Euclideanized time) and it exponentially grows for a negative \(t\).<br><br>Excellent. Functions may be messy and this may look artificial but as Arkani-Hamed assures you, this behavior has actually been seen everywhere in the AdS/CFT correspondence. The corrections similar to the exponential one are actually there, they are everywhere, and they are why the full theory of quantum gravity actually "erases" the singularities that seem unavoidable according to the Wilsonian or Taylor expansion.<br><br>Over 18 years ago, with Andy Neitzke, and Nima was watching us, we saw <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20neitzke%20and%20a%20motl%20and%20title%20quasinormal">some similar and perhaps even more subtle</a> kinds of behavior in the "asymptotic circle of the complex plane" spanned by \(r\), the tortoise coordinate, which isn't quite the same thing as the time \(t\) but it is geometrically equivalent, up to some Wick rotation. In our paper, we needed to compute some highly-damped quasinormal frequencies. To do so, we described some solutions as Bessel's function in the asymptotic region but these annoyingly advanced (but not too advanced) functions have some counterintuitive behavior such as the <a href="https://en.wikipedia.org/wiki/Stokes_phenomenon">Stokes phenomenon</a>. That is the trait which makes the asymptotic behavior look different in different directions of the complex plane. Also, the two-dimensional space of solutions to a second-order differential equation has exhibited a monodromy after you made a circle around the complex plane.<br><br>So a similar behavior which just prevents you from using a simple power law expansion for \(|r|\to\infty\) or \(|t|\to\infty\) is almost certainly omnipresent in a theory of quantum gravity – and our calculation with Andy was really some radiation from a black hole in the <em>classical</em> gravity although the purpose of this solution was to discuss radiation modes in a semiclassical approximation of quantum gravity and to say something about the full quantum gravity.<br><br>I agree with Nima that one has to be careful and the assumption about a universal \(1/t^M\) in all directions of the complex \(t\)-plane is broken at very many places. I only agree with him partially concerning another statement – but this partially problematic insight may be and probably should be illuminating for many – that this deviation from a simple, power-law-like behavior in all the directions of the complex plane is basically equivalent to the breakdown of the Wilson paradigm i.e. to some unavoidable deviations from the local effective quantum field theory.<br><br>Why do I say that this proposition is problematic? Well, it is only problematic if you actually did use functions like \(f_j(t)\) in all directions of \(t\in\CC\) in the complex plane. You could have taken a more experimental approach and talk about the behavior for \(t\) which is at (or very close to) the real positive \(t\) semi-axis. If that is how you approach it, then the power law expansions are just some pragmatic expansions of a function of the real variable and the behavior in the faraway regions of the complex plane (or even the possible non-existence of such analytic continuations) may be ignored.<br><br>Also, while the Wilsonian approach normally generates functions that are represented by power law expansion, we know that some other terms may hypothetically appear – such as the exponentials of field operators. Similar non-Taylor terms may be assumed to be non-perturbative corrections. Some of them may even be explicitly constructed in terms of instantons. You should be careful here, however. These two things are not equivalent, at least not self-evidently. The instantons scale like \(\exp(-C/g^2)\) as a function of a coupling constant while the non-Taylor corrections to the Green's functions require the decreasing exponentials in the field operators. They are <em>a priori</em> different things. But by this moment, you are probably open-minded enough to realize that both of these non-perturbative things, exponentials of inverse fields and exponentials of inverse coupling constants, may appear in the "full form" of Green's functions, along with some terms or the behavior that is hard to be expressed explicitly.<br><br>Whether these subtleties look like a minefield to you therefore primarily depends on the question whether your power law expansions are thought of as expansions of a function of a real variable or a function of a complex variable. Many functions may be extended to real or complex variables but the two methodologies are not quite the sime. In particular, you may have functions of a real variable that are zero to all orders but they are not zero. Well, it is true even in the complex plane but on the real axis, it looks "healthy" to connect such a function, like \(\exp(-1/g^2)\) for \(g\gt 0\), to the function \(0\) for \(g\lt 0\).<br><br>Many such subtleties, monodromies, divergent terms, perhaps singularities of slightly unexpected types may (and do) modify our expectations about Green's functions and reduce the direct relevance of the Wilson paradigm. I would still insist, as in my <a href="https://motls.blogspot.com/2021/04/first-michio-kaku-is-releasing-his-new.html?m=1">de Sitter post</a>, that an effective field theory may be considered a language into which an arbitrarily subtle or nonlocal theory of quantum gravity may be translated. The nonlocal interactions over roughly Planckian distances are unavoidable but the real question is whether they must be considered unavoidable even at very long distances. I still believe that in the Minkowski spacetime, the nonlocality should be just a Planckian subtlety.<br><br>However, if our world is close to a de Sitter space, I do find it possible that the expansion for \(|t|\to\infty\) may be confusing even at cosmological distances because the physically relevant region isn't \(|t|\to\infty\) but \(t\sim R_{dS}\) which is a finite cosmological infrared cutoff and the cosmological constant could be tiny due to some numerical coincidence or a "miracle". Subtleties similar to those above could be relevant for the cosmological constant problem and I guess that much of Nima's talk must be about their possible relevance for the hierarchy problem (why the Higgs is so much lighter than the Planck mass). I have been informed about a sketch of Nima's thinking and I've had some independent and overlapping thoughts. It may be very interesting to watch but 35 minutes from now, I must check that the best Czech soccer team won't be murdered by some visitors from Arsenal FC – let's hope that they won't be the same kind of unhinged fanatics and anti-Czech racist thugs as those in Rangers Glasgow. <br><br>No proofreading is planned, sorry.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-69184785792505389872021-04-14T07:30:00.014+02:002021-04-14T07:40:32.903+02:00Classifying viable supergravity theories in 8D, those with \({\mathfrak g}_2\) didn't make itThe Swampland program is a clever filter that eliminates a big portion of the effective field theories with gravity which are not good enough because they violate certain conditions that are necessarily to find a consistent realization of these theories as theories of quantum gravity, i.e. a realization within string/M-theory. In a new paper <blockquote> <b><a href="https://arxiv.org/abs/2104.05724">8d Supergravity, Reconstruction of Internal Geometry and the Swampland,</a></b></blockquote>Cumrun Vafa (a renowned professor) and Yuta Hamada (a Harvard postdoc) use the existing tools to say "you're fired" to many supergravity theories in 8 (large) spacetime dimensions. Note that 11 (spacetime) dimensions is the maximum dimension of a flat spacetime admitting a supersymmetric quantum theory of gravity, M-theory. 12 dimensions is still possible to some extent, like in F-theory, but two dimensions must be compactified on a torus (or perhaps something else?). Supersymmetric theories above 12 dimensions must be "even more subtle or less physical" if they exist and I want you to assume that they can't exist. In 10 large (spacetime) dimensions, there are 5 different consistent supersymmetric Minkowski-based theories of gravity, all of them are string theories (type I, IIA, IIB, heterotic \(E_8\times E_8\), heterotic \(SO(32)\)).<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />As you reduce the number of large spacetime dimensions, the diversity of the possible vacua increases (because so does the complexity of the possible geometries of the compactified dimensions, and decorations added inside these manifolds) but the laws of string/M-theory seem to guarantee that the number of stabilized vacua remains finite (or at least countable). What about the theories with 8 large (spacetime) dimensions?<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />You may obtain them by compactifying 3 (spatial) dimensions in M-theory or 2 (spatial) dimensions in 10-dimensional string theories. There are various shapes of the extra dimensions (aside from the torus, you already have a cylinder, Klein bottle, Möbius strip... to use, and K3 if you start from F-theory) and some extra choices may be made. However, the construction of the approximate effective field theory description proceeds very differently and seems to produce a greater number of choices.<br><br>They take various previously justified swampland criteria into account and remind you that the rank of the gauge group – the maximum number of \(A_\mu\) \(U(1)\) gauge potentials that you can embed into the field spectrum – must be 2 or 10 or 18 for the UV completion to exist. Those can be geometrically realized by type II on a torus or Klein bottle, Möbius strip, and a cylinder (with 0,1,2 boundaries), respectively; or as K3 with 2, 1, 0 frozen singularities, respectively. The rank changes by 8 (eight is the rank of \(E_8\) which lives on an M-theory boundary and it is no coincidence, either) for each boundary or extra singularity. As you may expect, all the consistent vacua have the rank of 2, 10, or 18.<br><br>The rank's being even may be justified by the global gravitational anomaly. Similar considerations about the anomalies in some spacetime or world volume directions produce additional constraints. The 1-brane unitarity bounds the rank from above. They review similar facts.<br><br>Their new constraints come from 3-branes. You know, a supergravity theory in 8 (spacetime) dimensions must have 3-branes. Why? Because the supersymmetry unavoidably adds a superpartner field to the graviton which may be written down using a 2-form gauge potential \(B_{\mu\nu}\). String theory (except for type I) is an example of that; this \(B\)-field is coupled to the fundamental strings (type IIB has D-strings as well, and a corresponding second \(B\)-field in the Ramond-Ramond sector). Well, this gauge potential is a 2-form and its field strength is a 3-form (higher by one). Because we are in 8 large (spacetime) dimensions, the (electro-magnetic) Hodge dual is a 5-form field strength (eight minus three) which is the "d" of a 4-form potential (five minus one). And 4-form potentials couple to 4-dimensional world-volumes, so the number of spatial dimension of the magnetic objects has to be 3. They are called (magnetic) 3-branes.<br><br>Now, such 3-branes must really be possible in a consistent theory approximated by an 8-dimensional supergravity theory. Why? Because this supergravity is just an extension of Einstein's general relativity, you may find classical black \(p\)-brane solutions for \(p=3\) i.e. black holes boringly extended in 3 extra spatial dimensions which are sourcing the magnetic field (five-form field strength). There must exist some effective world volume theory on these magnetic 3-branes and these two Harvard Gentlemen look at it carefully. Well, in particular, they look at the instanton 3-branes and via the swampland "cobordism" conjecture, they have to be connected (via the one-dimensional Coulomb branch on the brane, if you want to know).<br><br>These are highly constraining conditions and they end up with "dozens or hundreds" of possible constructions or vacua. Their gauge groups are diverse but some are visibly missing. In particular, one may construct 8-dimensional supergravity theories with the \({\mathfrak g}_2\) (gee-two) gauge algebra but it seems to be forbidden by the swampland argument: it belongs to the swampland.<br><br>This \({\mathfrak g}_2\) is a wonderful group, the smallest one among the five basic simple exceptional Lie groups. It is the automorphism group of the octonions. You may extend "2D" complex numbers to "4D" quaternions which are non-commutative and then "8D" octonions which are also non-associative. The octonions have 7 imaginary units and just like there is a \(\ZZ_2\) symmetry between \(+i\) and \(-i\) for complex numbers (and the \(SO(3)\) symmetry rotating the \(i,j,k\) quarternionic imaginary units), there is a symmetry, a subgroup of \(SO(7)\), between the 7 octonionic imaginary units that preserves (the addition and more nontrivially) the multiplicative table.<br><br>In grand unification (phenomenology in 4 dimensions), \({\mathfrak g}_2\) is no good as a gauge group because it has real representations only. In fact, the 7-dimensional representation encoding the "7 imaginary units of the octonions" (which is clearly a real rep) is a fundamental representation and all others may be built from its tensor powers. Well, in fact, the algebra \({\mathfrak e}_6\) yields the only exceptional group that has complex reps which is what turns it into an important grand unified group.<br><br>Here, the \({\mathfrak g}_2\) is forbidden because of the cobordism constraints – which are really still another generalization of the lore that "there are no global symmetries in a theory of quantum gravity". Many options remain viable, others are eliminated. Vafa and Yamada are really climbing down to lower number of large (spacetime) dimensions. If they climbed down from 11 or 12 to 4 which is the apparent number of spacetime dimensions around us, they would deal with a lot of options, lots of topologies, singularities in K3-like manifolds and their intersections and combinations, and at the end, they also want to break SUSY which makes things much less straightforward because one can no longer study "just the well-behaved vacua" where many basic things are determined by the BPS bounds – in the spacetime and the world volumes. But quite generally, one may reasonably believe that "do properly analyze, classify, and understand possible choices in D spacetime dimensions" is a viable way to find the right theory of everything (in phenomenology), crackpot movement's shouting to the contrary notwithstanding.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-22472177386789682422021-04-05T13:58:00.017+02:002021-04-06T10:15:46.133+02:00Susskind's de Sitter rant<blockquote> <b>LHC:</b> <a href="https://phys.org/news/2021-04-radar-stealthy-supersymmetry.html">Phys-Org about a 2.8-sigma CMS excess in the search for stealthy SUSY.</a></blockquote><iframe align="left" scrolling="no" frameborder="0" style="width:140px;height:245px;" marginheight="0" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ac&ref=tf_til&ad_type=product_link&tracking_id=lubosmotlsref-20&marketplace=amazon®ion=US&placement=0000000000&asins=0385542747&show_border=false&link_opens_in_new_window=false&price_color=BBBBBB&title_color=FFAA44&bg_color=002211" marginwidth="0"/></iframe>First, Michio Kaku is releasing his new popular book, <em>God Equation</em>, tomorrow. It could be a similar book about the theory of everything as some books that were published years ago but I guess that God may place a more central role here. <br><br>You may also read <a href="https://www.theguardian.com/science/2021/apr/03/string-theory-michio-kaku-aliens-god-equation-large-hadron-collider">an interview in the Grauniad</a> where he mentions string theory, colliders, and his view that it may be a stupid idea to obsessively contact the aliens. I can confirm the worries. Alza.cz, a Czech competitor of Amazon, allowed extraterrestrials to be hired. What happened? They have <a href="https://www.youtube.com/results?search_query=reklama+alza">completely dominated the commercials</a> from that moment on. Each ad has a stupid green extraterrestrial with a dwarf's voice. ;-)<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />But back to the main topic. David B. sent me this 6-year-old, 89-minute-long lecture titled "Old Man Leonard Susskind's Rant About de Sitter Space" which was introduced by Edward Witten.<br><br><iframe width="407" height="277" src="https://www.youtube.com/embed/4GKjr-y5MY0" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><br><br>The Covid-19 format sucks but many of you won't care, I guess. At the beginning, Susskind says that he more or less believes that the known wisdom about quantum gravity "softly predicts" (and incorrectly so) that we live in an anti de Sitter space or similar space (where the time is asymptotically bound to something); and the dynamics is supersymmetry, either with an unbroken supersymmetry or with an extremely weakly broken one.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Sorry, I don't agree it is the right summary of the knowledge. Anti de Sitter space and unbroken supersymmetry vacua play an important role in string theory and high-brow quantum gravity in general but that doesn't mean that they are predicted by the theory. Instead, the right interpretation is that these assumptions (AdS, SUSY) make the life of physicists who want to calculate something easier. But Nature doesn't give a damn whether something is easy to calculate.<br><br>In fact, I think it is right to say that relatively to an effective field theory in the swampland, our better sketch for quantum gravity – either explicit string/M-constructions or some more general wisdom respecting the paradigms and/or swampland criteria – only differs (1) at nearly Planckian distances, (2) by the rules that determine how larger massive objects are being built from the fundamental degrees of freedom and/or start to violate the regional independence or locality.<br><br>String/M-theory or quantum gravity do teach us new things about long distances, "infrared" non-locality around black holes, UV/IR-connections, and similar stuff but all these insights are really derived and from the possible dual descriptions, one may always find (at least) one that looks qualitatively just like the effective field theory. If you adopt this perspective of mine, the fundamental framework of quantum gravity doesn't – and cannot – predict something about the highly infrared distances, asymptotic regions of the spacetime, and maybe not even about the bubbling of the Universes from each other in the eternal inflation etc.<br><br>A heavily broken SUSY surely makes it hard to calculate something but you can't really prove (at least at this moment, as far as I know) that string/M-theory or quantum gravity ban a breaking of SUSY at energy scales that are much higher than the LHC scale. After all, there are still 15 orders of magnitude between that collider scale and the normal 4D Planck scale. The broken SUSY and/or the positive cosmological constant are still <em>emergent, long-distant effects</em>, even in string/M-theory or quantum gravity, and it is appropriate that it is difficult to calculate whether they occur. On the other hand, the apparent tiny cosmological constant is still a legitimate positive argument in favor of SUSY because SUSY makes the cancellations of the vacuum energy more likely or more plausible. The existing methods don't seem sufficient to get the tiny observed value but the qualitative statement is right, anyway. You make things <em>worse</em> by abolishing SUSY altogether.<br><br>Also, the degrees of freedom – what to calculate – are surely harder to be defined and calculated in de Sitter space (than in the AdS space). But it may even be impossible to define the precise ones. A fact is that even in the AdS setup which admits a precise, S-matrix-like set of observables, it is still possible to derive approximate predictions for a region which does <em>not</em> include the asymptotic region at infinity. Such a limited region is qualitatively the same as one in de Sitter space. That's why I think that it's obvious that if string/M-theory does allow the positive cosmological constant, it also makes it possible to calculate similar approximate quantities in a region of the de Sitter space. And everything else – predictions at a greater accuracy or observer-independence – may be physically prohibited or meaningless, due to a more massive version of the uncertainty principle.<br><br>I think that all existing versions of dS/CFT are wrong or at least physically useless and the whole logic that makes the people look for this kind of stuff is misguided for the reasons above. There is no reason why something like that should exist. On the other hand, one may derive effective field theories from the string vacua – and their parameters, like the gauge couplings and particle masses, are in principle calculable <em>precisely</em>. I think that this statement remains true and the possibility remains real even if the cosmological constant is positive and the spacetime ends up being de-Sitter-like. I think that nothing we have learned contradicts the possibility that we may derive totally precise parameters of an effective field theory expanded around a de Sitter space from string/M-theory. If you accept <em>this</em> way of linking string theory to detailed particle physics, the search for a "boundary of our de Sitter space" is perfectly useless and orthogonal to the things that physically matter.<br><br>Susskind also describes the de Sitter Penrose diagram, apparently identical to an enternal black hole, and discusses the entanglement between the "antipode" and the "pode" which Susskind believes to be the opposite of an "antipode". As a cultural European, I feel the moral duty to explain to the uncultivated American Mr Susskind that a "pode" isn't the opposite of an "antipode". An "antipode" is a person with the feet pointing in the opposite direction while a <a href="https://en.wikipedia.org/wiki/Antipodes#Etymology">"pode" is just a foot</a>, from the Greek πούς (poús). And a person with the misdirected feet isn't the opposite of the feet themselves. Thanks for your understanding. But I got distracted. ;-) I don't see much point in discussing the precise entanglement and correlations if one can't design an experimental protocol that will test this entanglement; and/or if one can't define the separate physics of the two Hilbert spaces. And it's obvious that we don't know how to make these exponentially precise measurements of the correlations in the dS space, isn't it? So why should we discuss it?<br><br>OK, I am still distracted. Juan Maldacena has been too shy for almost 3 decades but he should have taught Susskind and others that his name isn't "One" (at most "The One"). "One" may be OK in Mexican Spanish, e.g. for Susskind's Mexican janitor, but Maldacena is Argentinian and Juan is pronounced <a href="https://translate.google.com/?sl=auto&tl=en&text=Juan&op=translate" rel="nofollow">"Khoo-un"</a> Thank me very much, Juan, you are welcome. ;-)<br><br>As far as I can see, what is going on is that these de Sitter or dS/CFT rants have been trying to find "elegant, precise, or integrable constructions" analogous to the unbroken SUSY or AdS/CFT ones at least for 20 years – I would say it is at least 22 years now – but it seemed pretty self-evident to me (and it still does) that nothing like that exists. So why would you waste 20 years with such a thing? On the other hand, the fact that the ugly de Sitter space without a mixed-signature asymptotic region and without the niceties of unbroken SUSY prevents us from a nicely controllable, integrable construction does <em>not</em> mean that it isn't the right background recommended by string/M-theory or quantum gravity for our world.<br><br>Then there is the whole issue of the de Sitter temperature. Susskind says that it is the only temperature at which de Sitter may have the right symmetry. Well, let me tell you something. Quite generally, if the temperature is totally precise, it means that the system is described by the very precise density matrix \(\exp(-\beta H)\). <b>The tiny but nonzero temperature of the de Sitter space primarily means that you can't cool it to zero, without destroying the basic shape of the space, i.e. the very existence of pure states is banned in the de Sitter space!</b> There is almost certainly some (application of the) uncertainty principle that prevents you from building a complete set of commuting observables that would be consistent (all of them) even with the rough de Sitter geometry or topology. And that impossibility means that you can't prepare the system in a pure state and all the doable calculations are density-matrix-like. Any effort to discuss a precise construction of Hilbert spaces with pure states may very well be impossible and I think that it is. But this still doesn't mean that there is something wrong with de Sitter as a possible background of string/M-theory. It's just a background where you can't produce pure initial (or final) states. The preparation of the initial density matrix may very well depend on some priors, like in Bayesian inference, and this dependence on the priors may be as impossible to remove as the uncertainties for \(\Delta X,\Delta P\) in regular quantum mechanics. In de Sitter space, the thermal noise is unavoidable and the occupation numbers in the deeply infrared modes (whose wavelength is comparable to the de Sitter radius) are unavoidably uncertain! Also, you shouldn't compute scattering amplitudes but the inclusive cross sections (like in the usual treatment of infrared divergences, the de Sitter radius is the infrared cutoff)!<br><br>Susskind has also placed the holographic screen on the de Sitter horizon, surely not the first person who makes the guess. It may be right or wrong and neither possibility seems to have terribly interesting consequences. The horizon etc. is a fast scrambler but I couldn't see anything new in the comments using this sexy buzzword. OK, after some combinations of these ideas, Susskind talked about the Boltzmann fluctuations and combined them with the \(ij\)-bilocal degrees of freedom in Matrix theory. I have tried to generalize them in many contexts but here I didn't quite understand whether he claimed to "derive" the relevance of the Matrix-theory-like degrees of freedom from some properties of de Sitter, or he just combined two ideas that have nothing to do with each other, and if he did, I didn't understand what this combination was good for.<br><br>Some Susskind's Boltzmann fluctuation formula only worked for \(D=4\), as Juan pointed out, and Susskind didn't care, apparently suggesting that \(D=4\) is required for a consistent de Sitter. I don't believe it, it is an extraordinary claim. Instead, Juan's observation that the formula only had the right scaling for \(D=4\) meant that the justification of the formula was wrong.<br><br>We also saw Susskind's successful effort to simplify the everyone-to-everyone graph so that the system is still a fast scrambler and he decided that a graph that is as sparse as cabbage ("expander graphs") is still OK but parsley is no longer enough for fast scrambling. Excellent, probably a nontrivial mathematical result but I don't see how it could be interesting for a physicist. It seems to me that they have answered a similar question to the question "how many screws you may remove from a Michelson interferometer so that it still disproves the aether wind". No screws, one screw, 11 screws, a cabbage of screws? Unless you have another result that makes the answer important, I just don't care. It is not physically relevant.<br><br>Various comments and questions were contributed by Banks, Bousso, Maldacena, and Witten.<br><br>To summarize, I have the constant perception – and it gets manifested at so many, a priori independent, places – that Susskind conflates "what is true" (or compatible with string theory or quantum gravity) and "what is simple or admitting integrable structures". In particular, I find it almost self-evident that there is no AdS/CFT-like beautiful "integrable structure" in the de Sitter space and it's been foolish to look for one. De Sitter is warmer and messier than AdS, it has new kinds of unavoidable uncertainties and upper bounds on the precision of experiments. I would add that a complete set of commuting observables is impossible to be found in dS theories and consistent with the approximate dS locality, and that's why you can't work with pure states and you need to compute inclusive probabilities with the de Sitter radius as the infrared cutoff, just like in the calculations with infrared divergences.<br><br>The focus on the expander graphs has been a new example that struck me. Susskind and pals may be right that expander graphs produce "fast scramblers" much like some matrix models or Yang-Mills theories for black holes and their horizons. But clearly a complex or non-integrable enough systems are generically fast scramblers. It is not "unusual" to get a fast scrambler – a fast scrambler is some generic "non-integrable" dynamics which is the rule, not the exception. At most, one may get a psychological toy model for the fast-scrambling, complex behavior. But a "toy model for a complex behavior" sounds pretty stupid to me me because as long as the toy model is a good representation of what we're interested in, it will still be messy because by definition, we are interested in a messy thing! This worshiping of fast scramblers looks like Wolfram's worshiping of non-integrable systems of rules and I find it totally silly because non-integrable rules are the rule, not the exception.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-78169942330848927162021-04-04T08:15:00.010+02:002021-04-04T08:41:10.056+02:00An April Fools' Day paper on the swampland rules<a href="https://twitter.com/Satechi/status/1377633686066098182" rel="nofollow"><img src="https://b8a6h5u2.stackpathcdn.com/wp-content/uploads/2021/03/SATECHI_CyberMouse_9_REV.jpg" width=407></a><br><br>On April 1st, 2021 various individuals and companies came with their April Fools' Day pranks, see e.g. <a href="https://twitter.com/lumidek/status/1377656130491539458">this thread</a> where I found the Cybermouse to be particularly well-made. Some scientists have also posted preprints that weren't meant as a serious contribution to science. One of them was from <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20kinney%2Cw">William Kinney</a>, a cosmologist in Buffalo: <blockquote> <b><a href="https://arxiv.org/abs/2103.16583">The Swampland Conjecture Bound Conjecture</a></b></blockquote>For a few seconds, before I understood it was a prank, I thought that the double appearance of the word "conjecture" was a typo. But be sure it is not a typo. This paper is a "meta-research" that counts the number of papers with swampland conjectures, and tries to please you. Maybe the number of such papers won't be too huge, after all, just a cubed googol so that the papers will require a similar number of visible patches of the Universe to store all the physicists who write them etc. <br><br>The point of the paper is clearly to humorously express the author's opinion that the number of papers about the swampland is too high. The detailed jargon-filled, science-like-sounding propositions aren't hard to invent in real time, I would probably end up writing a similar April 1st paper if I wanted to express the same point (although I hope that mine would be funnier).<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />But is the serious point conveyed by the satirical paper correct? Is the number of the swampland papers too high? I don't think so. First, the "meme" and the very general foundations of this research were articulated by Cumrun Vafa – who had an office next to mine when he wrote it – in 2005: <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20vafa%20and%20title%20swampland%20%20and%20date%202005">The String Landscape And the Swampland</a>. The paper has 548 citations as of now.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Months later, I managed to submit the <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20motl%20and%20a%20vafa">first hep-th paper of 2006</a> where we accumulated the evidence that the gravitational interaction is the weakest one not just in this real world but in any world described by a consistent quantum mechanical theory including Einstein-like gravity. The Weak Gravity Conjecture (WGC) is at 795 citations now. It is higher than Vafa's original, "visionary" paper, perhaps because the visionary paper is a bit too vague and popular in character (a big wave of anti-string crackpots was just emerging and Vafa clearly stood on the same side as your humble correspondent; I think that the swampland paper was partly motivated by these interactions outside "quite professional science") and the WGC contains a bit more beef.<br><br>At any rate, we are talking about fewer than 1,000 papers that had a reason to refer to these two swampland papers. Only a smaller portion of them, a few hundreds, actually propose some entirely new swampland criteria or variations of some criteria in the literature. There are just a dozen of "truly different, new" swampland conditions as of now, I believe.<br><br>But let's ignore the degree to which the papers may be rightfully referred to as "papers about the swampland criteria". Even if we are generous, the total number doesn't surpass 1,000. How much has mankind paid for this research? The average swampland researcher (in the world) may be getting some $40,000 a year, he or she writes 4 papers a year. Research is done for 1/2 of his funding (he may also need to teach etc.) but this factor of 2 gets canceled by about 2 authors of a paper in average. So a swampland paper may cost some $10,000 to be made and there are a thousand of them.<br><br>So the swampland research has only cost something like $10 million, a ludicrously tiny amount of money. I think that a person who would propose that this is too much for this rather rare direction of research is a Luddite animal and enemy of the things that place the human race above other species. Make no mistake about it: If you would like to reduce this number further, you are a scumbag. Just to compare: tens of <em>billions</em> have been spent for the "research of climate change" that hasn't really taught us anything reliable yet new enough and that has only served to make some insane, far left projects in <em>politics</em> look more sciency. A trillion has been wasted for the misguided policies recommended along with the "climate research". And that's nothing compared to the totally pointless Covid lockdowns that have erased about $10 trillion from mankind's wealth (10% of the world GDP). The swampland has been 1,000 or 1,000,000 times cheaper than the "climate change research" or "lockdowns", respectively!<br><br>These costs are trivial which means that the only legitimate, similar question is one about the <em>relative attention</em> that is dedicated to the swampland (or the WGC) if this topic is compared to other things that these fundamental enough scientists can make. And it's simply true that there aren't too many things that are similarly, rightfully exciting as the swampland.<br><br>What I found amusing were the <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20kinney%2Cw">recent papers by this very Dr Kinney</a>. The most recent 3 have 0,0,1 citations, respectively. The fourth newest paper has 17 citations, quite a difference, and it is about the trans-Planckian censorship. If you look at it, you will realize that it is really a swampland paper itself! ;-) Kinney has actually written 3 serious papers with "swampland" in the very title and they have 12,53,133 citations as of now, respectively.<br><br>I guess that his own swampland papers end up being more successful than his other recent papers and he knows it. Is this success justified?<br><br>I think so. The real point is that the swampland is still a "new enough conjecture" and that is why it is so provocative for various people. But one paper co-authored by Kinney is titled "Primordial Non-Gaussianity" which I pick as my benchmark to be compared with the swampland.<br><br>Every professional (or graduate student) in the field of cosmology knows what it means. A fun fact that is easily overlooked is that the "primordial non-Gaussianity" is a conjecture, too, just like any swampland conjecture proposed in the literature. No statistically significant deviations from the Gaussianity have been observed but there are obvious reasons to think that the distribution shouldn't be <em>precisely</em> Gaussian. Various paradigms and/or detailed models have various reasons to predict a tiny non-Gaussianity or a substantial one (if they predict too much, they're excluded). The character of the argumentation in similar papers is really analogous to the argumentation in the swampland papers. And there are papers that actually talk both about the swampland and about non-Gaussianity, these concepts really belong to "almost the same subfield".<br><br>You may search for <a href="https://scholar.google.com/scholar?q=non-gaussianity+primordial&hl=en&lr=&btnG=Search">all papers that contain non-Gaussianity and primordial</a> and my Google Scholar shows over 9,000 such papers (some are important enough experimental searches for such non-Gaussianities). So the number of such papers almost certainly <em>is</em> much higher than the number of papers about the swampland. Kinney could write a satirical paper about the excessive attention dedicated to the primordial non-Gaussianities and it would be about 10 times more justified than the satirical paper about the swampland but he didn't. Are the new thousands of papers mentioning the primordial non-Gaussianity bringing us more value than the new swampland conjectures? I don't think so.<br><br>It is really great that new swampland criteria are being invented when people see some new patterns in the data and theories describing the Universe at the fundamental level. This approach, a form of "pattern recognition", has simply become a new methodology favored by many professionals. It is what makes this topic more creative and intellectually diverse than the discussion of the primordial non-Gaussianities – which are really a straightforward possible effect that may be either stronger or very weak.<br><br>To summarize, Kinney's satirical paper is somewhat funny but if or when there is a possibility that the reader extracts a message from the paper, one must say that this satirical paper is extremely demagogic, too. The swampland program is an example of a new enough, creative enough topic with quite some potential for progress and breakthroughs. To say the least, it makes many researchers genuinely excited – which isn't the case of the 9,071st paper describing the similar basic possibilities concerning the non-Gaussianities in a slightly different way. Why are the swampland conditions and WGC interesting? Because, as people see, they look analogous (especially) to the Heisenberg uncertainty principle which symbolized the most important revolution in the history of science – but unlike Heisenberg's principle, the swampland ideas aren't fully understood yet. When there is evidence for "something important" that we don't understand, real scientists (and curious people) focus their eyes, ears, and neurons.<br><br>More generally, ideas that came from string/M-theory or that were quickly tested within the string/M-theoretical framework simply <em>are</em> more intriguing according to the actual researchers and that is why the actual researchers must be allowed to focus on them. They know why they are excited if they are excited and a demagogic satirical paper simply isn't a legitimate counterargument to these reasons for excitement.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-58713479594075905292021-04-02T14:58:00.002+02:002021-04-02T20:01:54.318+02:00How holograms work<a href="https://en.wikipedia.org/wiki/Holography"><img src="https://upload.wikimedia.org/wikipedia/commons/5/5f/Holomouse2.jpg" width=407></a><br><br><em>You may really look at the mouse hologram from different directions.</em><br><br>The holographic principle states that a theory of quantum gravity is secretly hiding the information about (spatially) 3-dimensional objects in a plane ("photographic film" or "holographic screen") that is just (spatially) 2-dimensional i.e. whose dimension is lower by one. <br><br>In both cases, the information about the extra dimension, the "depth of a point", is encoded in the typical wavelength of an interference pattern that appears on the screen. In some generalized sense, <a href="https://motls.blogspot.com/2009/01/stereograms-and-dinograms.html?m=1">dinograms or stereograms</a> do exactly the same but the holograms can really store the information even about points that are overlapping each other, not just about the "closest layer".<br><br><a href="https://en.wikipedia.org/wiki/Holography#How_it_works"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Holograph-record.svg/1024px-Holograph-record.svg.png" width=407></a><br><br>Let's try to explain how the image of a 3D object such as <a href="https://en.wikipedia.org/wiki/Holography">this mouse</a> is encoded in a hologram a bit more quantitatively. We will need a coherent source of light which has the precise (angular) frequency \(\omega\). A laser. Dennis Gabor, a powerful Hungarian theorist, invented the holograms in December 1947 but the lasers only emerged as "real things" in 1960 and it took a few years for the holograms to be made, too. Gabor got the 1971 Physics Nobel Prize.<br><br><a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Fine, holograms have something to do with the wave properties of light. We require the light to have a fixed frequency i.e. fixed color which is why the basic hologram will be monochromatic. We may pretty much say that while the hologram "gains" one extra dimension of space, it "loses" another dimension, namely the information about the intensity of individual colors.<br><br>However, three holograms on top of each other, red-green-blue, are enough to emulate the color spectrum for the eyes that can only distinguish the three #RRGGBB intensities of colors, not the full function \(I(\omega)\). And three infinitely thin films are still close to one 2D image than to one 3D object.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Coherent light is an electromagnetic wave and each electromagnetic wave can be Fourier decomposed. It is a linear combination of plane waves\[ \psi_{\vec k}(x,y,z,t) = \exp[i(k_x x+k_y y + k_z z -\omega t)]. \] Because the waves are moving in the vacuum which will be enough for us (at almost all times), the phase velocity and the group velocity are \(c\) which means that \(kc = |\vec k| c = \omega\). The electromagnetic wave is a transverse wave: the direction of motion \(\vec k\) is orthogonal to the electric vector \(\vec E\) and the magnetic vector \(\vec B\) and the latter two are orthogonal to each other, too. In 3+1 dimensions (three spatial, one temporal one), there are two polarizations (e.g. linear \(x,y\) polarizations) of light. It means that the most general monochromatic wave is written as \[ (\vec E(\vec x,t),\vec B(\vec x,t) ) = \sum_{\lambda=1}^2 \int d^2 \Omega_{ \hat n} A_{\lambda,\hat n} (\vec E_{\lambda,\hat n},\vec B_{\lambda,\hat n}) \psi_{\lambda,\omega \hat n / c} \] The detailed distribution of light is encoded in the amplitudes \(A_{\lambda,\hat n}\) where \(\lambda=1,2\) labels which of the two polarizations we consider and \(\hat n\) is simply a unit vector that belongs to the sphere \(S^2\) and that's why we integrate over this variable or over \(d^2\Omega\). For each of the direction \(\hat n\) and the polarization \(\lambda=1,2\), there is some electric field, some magnetic field (which are orthogonal to \(\hat n\) and to one another) and \(\vec k = \omega \hat n / c\). The length of the vector is completely determined by the frequency (i.e. by the wavelength) so only the direction is variable!<br><br>Again, we reduced the dependence of \(\vec E,\vec B\) on four spacetime coordinates to the dependence of the complex amplitudes \(A_{\lambda,\hat n}\in\CC\) on two coordinates in \(\hat n\) – which may be parameterized as \(\theta,\phi\) – because the electromagnetic field we consider is time-independent (a static image); and because the frequency is fixed. For each of the two polarizations, there is some plane-wave solution to the wave equation \(\psi\) that we wrote before; and some choice of the electric and magnetic vectors.<br><br>The polarization of the light isn't essential. You may very well imagine that we are Fourier-decomposing just one particular nonzero component of the electromagnetic field, e.g. \(E_x\), and the remaining components of the electric and magnetic vectors are determined from it in some way. This comment is just to prepare you for a simplification that we will make. Instead of dealing with two polarizations which affect 6 components of the electric and magnetic field strength vectors, we will only deal with one polarization and a scalar field \(\Phi\) which you may imagine to be "the same thing" as \(E_x\). Excellent. So how will the hologram composed of the scalar field work?<br><br><a href="https://en.wikipedia.org/wiki/Holography#How_it_works"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Holograph-record.svg/1024px-Holograph-record.svg.png" width=407></a><br><br>Here I posted the Wikipedia image of the situation in which the hologram is being created. In the upper left corner, the coherent light is coming from a laser to the right where a beamsplitter splits it to beams going to the right or down. The beam going down is simply reflected from a mirror where it reflects and becomes the "reference beam". The other beam illuminates the mouse which partly reflects the light to the same photographic plate. That is where the object (mouse) beam and the reference beam interfere with each other and create an interference pattern. The intensity of the interference pattern at each point is recorded as the darkness of the photographic film.<br><br>Note that the object beam may be Fourier-decomposed according to the formula above and is encoded in the Fourier components \(A_{\hat n}\) where the direction \(\hat n\) is variable but "approximately" from the mouse to the photographic plate. However, this beam interferes with the reference beam whose \(\hat n\) is completely fixed (the beam is a plane wave so \(A\) is a multiple of the delta-function in the direction space).<br><br>It is not quite needed but let us assume that the overall intensity of the reference beam is much greater than the intensity of the object beam. At a given point \((x,y,z,t)\) of the photographic plate (the time doesn't matter), the squared intensity is\[ |\psi (x,y,z,t)|^2 = |\psi_{\rm ref} + \psi_{\rm obj}|^2 \approx |\psi_{\rm ref}|^2 + 2\,{\rm Re}\, \psi^*_{\rm ref} \psi_{\rm obj}. \] I neglected the absolute value of the object beam because it's much smaller than the last term that I kept which is the real part of the mixed term (if you don't neglect it, it has the capacity to at most create another annoying image in a different direction). Also, the absolute value of the reference beam is constant all over the photographic plate and won't affect the darkness variations. So it is only the mixed term that matters for the darkness and the interference pattern\[ 2\,{\rm Re}\, \psi^*_{\rm ref} \psi_{\rm obj}. \] Let us redefine the phase at each point so that the reference beam has a "positive real phase" at each point of the photographic plate. With this choice, the darkness is dictated by \[ 2 \,{\rm Re}\, \psi_{\rm obj}, \] the real part of the "object beam" wave evaluated on the photographic plate. That's nice. Recall that the object beam is Fourier-decomposed into the amplitudes \(A_{\hat n}\) where the direction \(\hat n\) is variable and approximately going "from the mouse to the film". What does the real part do to the complex coefficients \(A_{\hat n}\)? It's simple. The real part simply replaces a particular \(A_{\hat n}\) with \(A_{\hat n} + A_{-\hat n}\), with the Fourier coefficients in the original direction and in the exactly opposite direction. The addition of the "opposite \(\vec k\)" or "\(-\vec k\)" won't damage the hologram because when we're standing on one side of the hologram, we only see the waves with the right sign of \(k_z\) where \(z\) is the axis orhotgonal to the film, anyway. So the real part (which came from the squared absolute value) doesn't really matter at all.<br><br>Fine, we already have some interference pattern encoded in the intensity or darkness \(I(x,y)\) along the two directions of the photographic plate and we have good reasons to think that this interference pattern remembers the whole object beam as encoded in the complex amplitudes \(A_{\rm obj}(\hat n)\) where \(\hat n\) is roughly going from the mouse to the film. What happens if we illuminate this film with the reconstruction beam?<br><br><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Holography-reconstruct.svg/800px-Holography-reconstruct.svg.png" width=407><br><br>The reconstruction beam is coming from the same direction as the reference beam that was used to create the hologram. The intensity \(I(x,y)\) on a given place of the hologram is multiplying the incoming electric (and therefore also magnetic) fields. However, now the photographic plate is transparent and the laser light is allowed to continue to go to the viewed, in the downward direction. <br><br><b>The virtual image will be seen by the viewer if the electromagnetic light beneath the hologram is going to be the same as it would be in the first picture (when the hologram was being created) if the object beam had been just allowed to continue to go to the viewer.</b><br><br>And is that true? You bet. The funny thing is that the field \(\psi(x,y)\) defined as a function of the two dimensions of the hologram is basically enough to fully reconstruct the field \(\psi(x,y,z)\) at each point of the half-space (and each moment of time). Why is it so? It's because, as we have already emphasized, the Fourier coefficients \(A_{\hat n}\) only depend on the direction of the electromagnetic wave \(\hat n\). The plane waves calculated for different directions \(\hat n\) are linearly independent. And it is almost true even on the plane of the hologram, assuming the plane waves with a fixed \(|\vec k|\) only. The only exception is that the component of \(\vec k\) that is orthogonal to the hologram may flip the sign (which changes neither the plane wave's profile in the plane of the hologram, nor the length of the vector \(\vec k\)). That is why the coefficients \(A_{\rm obj, \hat n}\) can be extracted from the hologram \(I(x,y)\), up to an overall phase, and up to some mess that appears for uninteresting, opposite directions \(\hat n\) (that's the harmless effect of the "real part" business).<br><br>That's why the full electromagnetic wave coming from the hologram to the viewer will be the same – in the half-space and in the approximately correct direction – as the electromagnetic waves that would be coming from the mouse if the reference beam were totally omitted and if the photographic film were simply transparent. The frequency-\(\omega\) electromagnetic waves coming from the \(S^2\) sphere worth of directions are enough to remember the intensity of light at each point in the 3D space and the static hologram with the intesity \(I(x,y)\) is enough to encode all the information about the Fourier coefficients \(A_{\rm obj,\hat n}\) that are needed to reconstruct the whole electromagnetic field in the half-space.<br><br>There are just a few possible bugs that you need to investigate if you want to do high precision holographic engineering such as the 2 or 4 images coming from the various directions \(\vec k\) (4 images may exist because \(\vec k\) may have the overall sign flipped, and independently of that, the component transverse to the hologram's plane may be flipped); the effects of the terms quadratic in \(\psi\) i.e. the failure of the linearization; the effect of the two polarizations, and a few more details. But otherwise the hologram has to work.<br><br>The point is that the dependence of the "intensity of sources" on the third, holographically subtracted direction (the distance of a source from the hologram) may be eliminated because this depth is stored in the information about the "typical spacing of the interference pattern" instead. The very same encoding works in the holograms of quantum gravity, too.<br><br><em>I don't plan to proofread this text, sorry.</em>Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-18009951698227325292021-03-30T07:01:00.007+02:002021-03-30T11:27:04.130+02:00Pinball, heavy strings, black holes, saddles, and chaosThe new <a href="https://arxiv.org/list/hep-th/new">hep-th preprints</a> mention the word "string" 27 times today – in 4 abstracts of new hep-th papers, 2 new cross-listed ones, and 2 replacements. I think that the most interesting paper in this "string set" is <blockquote> <b><a href="https://arxiv.org/abs/2103.15301">Chaotic scattering of highly excited strings</a></b></blockquote>by (QCD co-father) David Gross and Vladimir Rosenhaus. It is about some high-energy scattering, in this case that of heavily excited strings, the the chaos that it exhibits. <a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Note that Gross (who was 80 last month, congratulations, David) has been investigating similar high-energy behavior of amplitudes for decades, see e.g. Gross-Mende. Here, they are calculating the tree-level perturbative string amplitudes for heavily excited string modes at high energies. Note that highly excited string states are "progenitors" of black hole microstates. And because the latter are very efficient in quickly and effectively mixing and scrambling the information, the excited string modes might be able to do it, too, and indeed, they do find out that the strings do behave in this chaotic way, too.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Their mental model of chaos is a <a href="https://cdn.htmlgames.com/Pinball/" rel="nofollow">pinball</a> with lots of collisions of balls. The elastic collisions finely depend on the precise angle and even the qualitatively character of the collisions has many possibilities how to evolve. I think that billiard is good enough for this concept – and I think that "success" in billiard becomes a matter of good luck when the number of ball collisions becomes large, like 5 or more.<br><br>In this paper, they calculate the Veneziano-like (or Virasoro-Shapiro-like) scattering amplitudes at a fixed angle, asymptotically high energies, and they realize that even for low-lying states, there are the so-called scattering equations that determine the positions of the saddles in the world sheet plane (or half-plane) as a function of the string momenta \(p^\mu\). The appearance of the saddle approximation is cool and sort of unavoidable. The number of such saddles tends to be \((n-3)!\) and the saddles provide us with a "dual" (and particle-like or pinball-like) reorganization of the whole calculation.<br><br>You know, the integrals that produce the Veneziano amplitude may look like some "totally chaotic functions with contributions from everywhere". But in the extreme limit of high energies, the integrand is highly non-uniform, being much larger at some positions of the vertex operators \(\{z_i\}\) than others. That's why the high-energy limit is also one where the integrals may be approximated by the saddle points. You just find out where the relevant saddle is located and what is the Gaussian-like approximation of the integrand around that point.<br><br>Some special properties occur when they consider particular highly excited strings instead of the low-lying string excitations and the mathematical structures they find generalize both the scattering equations and CHY (<a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find%20a%20cachazo%20and%20a%20he%20and%20a%20yuan">Cachazo-He-Yuan</a>, two influential papers from 2014), who used some KLT decompositions to find some link between the world sheet and spacetime expressions.<br><br>Some of the Gross-Rosenhaus formulae are compact and precise and they demonstrate the chaotic, pinball-like behavior of the high-mass, high-energy perturbative (tree-level) scattering amplitudes.<br><br>Unfortunately, the economy of the more nontrivial explanations doesn't add up so I reduce the old-fashioned texts about nontrivial topics by 80-90 percent and avoid proofreading.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-61828323725018921352021-03-22T06:30:00.005+01:002021-03-22T06:34:09.788+01:00Gravitino must never be motionless? A swampland hypothesisIn a new hep-th paper <blockquote> <b><a href="https://arxiv.org/abs/2103.10437">The Gravitino Swampland Conjecture</a></b></blockquote>Edward W. Kolb, Andrew J. Long, and Evan McDonough (Chicago/Houston) propose a new swampland principle – a new condition that must be obeyed for an effective field theory coupled to gravity to be capable of a consistent completion to a regime of quantum gravity and/or string theory. They postulate a simple rule: <blockquote> The gravitino speed must never drop to zero in any vacuum and any conditions. </blockquote>If true, such an assumption also has consequences.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />We have basically observed elementary particles with the spin 0 (Higgs), 1/2 (leptons and quarks), 1 (photon, gauge bosons), and 2 (graviton, although only in the classical limit of gravitational waves). From the interval 0 to 2 and the spacing of 1/2, the number 3/2 is missing. It is the spin of the gravitino, a superpartner of the graviton, and most people who do quantum gravity would agree with me that it is more likely than not that such a particle with a mass much lower than the Planck mass exists in Nature around us.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />These authors justify the claim that the gravitino speed mustn't ever vanish by a simple comment: if it vanished, it would mean that it would be easy to excite modes with an arbitrarily high momentum, and that is why the effective field theory would break down. Well, I think that the justifications of the swampland conditions should be more diverse and multi-dimensional. <br><br>Also, I think that this particular reasoning isn't quite a clear example of how the normal swampland restrictions work. The point of the swampland is that an effective field theory looks <em>totally OK</em> as an effective field theory – but there are still some hidden reasons (some potential problems with the completion that you may imagine to reside at the very high, Planckian scales) why it is wrong as a limit of a consistent theory of gravity. Here, they really find out that the effective field theory is self-defeating by itself. While it may be viewed as a "more direct problem" of the effective field theory, I think that it is actually a smaller problem in quantum gravity because there is nothing wrong about the situation when an effective field theory is an incomplete description. Maybe the very high-energy gravitino modes could be incorporated in some way, to get a better approximation of the full quantum gravity.<br><br>But if you believe that their principle is correct, it has consequences. You may write the speed as a sum of two ratios; each ratio has something like \(m^4\) terms in the numerator and the denominator, and all these terms are products of \(m^2\) (gravitino mass), \(\dot m\) (the time derivative), \(M_{\rm Pl}^2\), \(\rho\), and \(p\). In some condntions in unlucky enough theories, this may be adjusted to zero. The effective field theory would be killed by the newly proposed principle. Such dead theories would include theories with a very light (collider-scale) gravitino AND the B-mode tensor polarization modes from primordial gravitational waves. So these two things (light gravitino; tensor modes) couldn't exist simultaneously if these folks are right. This conclusion is extracted by efforts to quantize the gravitino at the end of the inflationary epoch.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-87940500564885780502021-02-05T07:05:00.004+01:002021-02-05T07:41:35.373+01:00JT supergravity and SYK described as type 0B strings<iframe align="left" scrolling="no" frameborder="0" style="width:140px;height:245px;" marginheight="0" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ac&ref=tf_til&ad_type=product_link&tracking_id=lubosmotlsref-20&marketplace=amazon®ion=US&placement=0000000000&asins=0262536080&show_border=false&link_opens_in_new_window=false&price_color=BBBBBB&title_color=FFAA44&bg_color=002211" marginwidth="0"/></iframe>One of the themes that string theorists and string-like theorists recently spent time with is the JT (<a href="https://www.sciencedirect.com/science/article/abs/pii/0550321385904481?via%3Dihub">Jackiw</a>-<a href="https://www.sciencedirect.com/science/article/abs/pii/0370269383900126?via%3Dihub">Teitelboim</a>) supergravity. It's some gravitational theory in 2 spacetime dimensions. Well, there aren't too many gravitons and other things propagating in such a low spacetime dimension. But you may calculate \(Z(\beta)\), a partition sum.<br><br>I consider this theme a "microrevolution" because it doesn't quite reach the threshold of excitement for a minirevolution. This JT supergravity, a 2D dilaton gravity, has been linked to the <a href="https://arxiv.org/abs/cond-mat/9212030">Sachdev-Ye</a>-Kitaev (SYK, random magnets) 1D quantum mechanical model which is pretty much a holographic description ("CFT") for this quantum gravitational theory.<br><br>OK, such "apparently non-stringy" quantum theories of gravity are always risky. The low spacetime dimension means that many of the methods and principles that operate in realistic quantum gravity in 4-11 spacetime dimensions may become invalid. Nevertheless, it is a highly justified position to assume that all truly consistent theories of quantum gravity should be a part of string/M-theory. Is that true for the JT supergravity?<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />Today, the answer "Yes" is advocated by Clifford Johnson, Felipe Rosso, and Andrew Svesko (USC/London) in the new paper <blockquote> <b><a href="https://arxiv.org/abs/2102.02227">A JT supergravity as a double-cut matrix model</a></b>. </blockquote>Clifford Johnson wrote a blog post about this manuscript: <a href="https://asymptotia.com/2021/02/04/full-circle-2/">Full Circle</a>. The title was chosen because he has <a href="https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=a%20C.V.Johnson.1">previously</a> discussed a connection with type 0A string theory; now it is about 0B and by listing all letters between A and B, he has probably covered a "full circle" LOL. Of course, Johnson was unusually attracted to 0A at least <a href="https://inspirehep.net/literature/633454">since 2003</a>.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />They basically adapted a 2003 paper by <a href="https://arxiv.org/abs/hep-th/0309168">Klebanov-Maldacena-Seiberg</a>. Those authors have connected some unitary and complex matrix model theories to type 0 string theories. Note that type 0 (0A or 0B) string theories are close siblings of type II (IIA or IIB) string theories except that you only impose the "left-right shared" GSO projection which isn't enough to remove the tachyon from the spectrum. Also, the absence of the chiral GSO projections in type 0 string theories means that the whole spectrum is bosonic; there are no fermions like in type II string theories. (In some brane-bound Hilbert spaces, you may have fermions only instead.)<br><br>On top of that, to describe these low-brow, low-dimensional matrix models, the type 0 tachyonic string theories have to be combined with the linear dilaton, the Liouville stuff etc. In the fresh paper, Johnson et al. have to consider a double-scaling limit that zooms in and replaces some moving packets by double cuts. This limiting treatment of the matrix models gives them the power to claim that they are able to calculate observables precisely, non-perturbatively. And the matrix model may be considered as a combination of (infinitely many) minimal type 0 string theories. Well, I don't understand why one is allowed to "combine" physical theories, what is the physical meaning of such a generic hybrid.<br><br>Well, some fishy steps and simplifications – tachyons to start with, linear dilatons, double limits, and combinations of several models – are being made to connect these simplified models with string theory. One may discuss the "degree and character of fishiness" of these extra features (or pathologies) but I still believe that the total fishiness is nonzero and these theories can't be considered full-blown siblings of the truly consistent quantum gravity setups resulting from string theory, those that are needed to describe the world around us, too.<br><br>It's clearly a paper with some merit (and Johnson had some reasons to feel that "this might be right") but I find this whole refocus of the high-energy theoretical physics community on the low-dimensional models (below 4D spacetimes) extremely frustrating and disappointing. This is the kind of stuff that I have associated with the phrase "mathematical physics" which is dull and to be contrasted with the hot and cool "theoretical physics". To do something that tells us important things about the life, universe, and everything, you simply need to find something about physics in 4 dimensions and higher (where the graviton physical polarizations exist), not in 3 dimensions and lower! I wouldn't have ever joined the "mathematical physics" community and I think that the growth of that, and the suppression of proper brilliant theoretical physics, is ultimately driven by the ideologically driven decay of the society.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-1566903553887918942020-12-29T16:16:00.008+01:002020-12-29T16:46:54.972+01:00Midsize miracles of perturbative string theoryA big part of the reason why all competent high-energy BSM+QG theoretical physicists are basically certain that string theory is the correct unifying theory of this Universe is that<br><br>* it makes the right qualitative predictions (without assuming something equivalent) of the qualitative traits of the approximate laws of physics that were determined empirically<br>* it solves problems with consistency and finiteness in a way that looks "infinitely unlikely" or "miraculous" in a generic similar yet non-stringy framework (and the alternative theories are the greatest examples)<br> * it interpolates between various kinds of effective theories in ways that would also look miraculous to someone not familiar with the "stringy exception"<br><br>Concerning the first point, we must realize that the progress in physics has been incremental and for quite some time, physicists were switching from a working hypothesis to a "theory that is one step deeper" than the previous one. String theory really does nothing else than that, too. You may ask whether the step is greater or smaller than previous steps but this question is operationally meaningless because there exists no way to compare the steps between completely different pairs of situations.<br> <br><a href="https://sketchfab.com/3d-models/calabi-yau-surface-3b66e71a7146433ca27f3affe50883cf"><img src="https://media.sketchfab.com/models/3b66e71a7146433ca27f3affe50883cf/thumbnails/7875a3c5ae504d6f9cdefa60b108cee2/1024x576.jpeg" width=407></a><br><br><em>Click the screenshot to 3D navigate the 6D Calabi-Yau with your mouse</em><br><br>Was the electroweak theory a bigger step than Maxwell's theory of electromagnetism? One cannot say, the conclusion is purely subjective or convention-dependent because the two advances are not commensurable.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />OK, string theory improves upon the previous framework which is a combination of <br><br>* a renormalizable quantum field theory, the Standard Model, loosely paired with<br>* the classical general theory of relativity due to Albert Einstein<br><br>In practice, both may be encoded in the action, \(S = \int d^4x \,{\mathcal L}\), where the Lagrangian density \({\mathcal L}\) is a sum of non-gravitational and gravitational pieces. Both may be interpreted as quantum theories and they should be because the world is quantum mechanical. In the case of the non-gravitational forces and matter, it's really essential (atoms exist etc.; fermionic fields are classically zero); in the case of the gravitational fields, the quantization leads to problems because Einstein's theory is not renormalizable.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />The gravitational action is something like the Einstein-Hilbert action\[ S_{EH} = \int d^4 x\,\frac{1}{16\pi G} R \sqrt{-g} \] which is the integral of the Ricci curvature scalar (multiplied by the invariant 4-volume measure), the only natural curvature invariant. On the other hand, the non-gravitational part, the Standard Model, is a sum of the kinetic terms for spin-1 gauge fields corresponding to the group \(SU(3) \times SU(2) \times U(1)\), fermions transforming as three copies (generations) of some reducible representation of that group, and scalar fields (the Higgses), along with the interactions (electromagnetic implied by the gauge invariance [for fermions and the Higgs] plus the similar Yukawa interaction; did I miss some? You bet, the Higgs quartic self-interaction).<br><br>A deeper theory should derive these traits without assuming them (or without assuming all of them). How do the proposed candidates succeeed in doing so? Take a theory by Lionel Volk, the magnetic raspberry (it's very important that it is not a strawberry). Does it predict the features of the Lagrangian above? Do we get the kinetic terms for the Yang-Mills fields out of the magnetic raspberries? No, we don't, of course (imagine that a one-hour-long monologue is inserted here, making fun of the fact that the raspberries don't even imply the right Dirac equation or anything else... nothing works). So Lionel's theory fails but it gives you magnetic raspberries. Too bad, if you do experiments with raspberries, you will find out that they are not really magnetic.<br><br>Out of the long list of Lionel's competitors, let's pick Stefan Tungsten. He has a Rule 30 theory. Define a binary coloring of the checkerboard based on some logical rule and you get something that Stefan finds surprising: the resulting coloring is neither "almost completely white", nor "almost completely black", nor "periodic", or otherwise simple. It is unsolvable. On top of that, after 3 bottles of wine, it looks like the skin of a tiger. Is that a theory of everything?<br><br>Well, no. He still didn't get the Yang-Mills action correctly. Or <em>any</em> physics, for that matter. The fact that a rule led to an unsolvable, "complex", system isn't because Rule 30 is interesting or worth attention. Instead, it is because Stefan's expectation was absolutely wrong and kind of shockingly dumb, too. Almost all equations or rules lead to "unsolvable" outcomes, solutions that cannot be expressed analytically or constructively (think about the three body problem in Newton's theory of gravity). So he got nothing surprising at all. And again, he got absolutely nothing out of the Maxwell-Dirac-Higgs-Yukawa-... action. Not even an epsilon about it.<br><br>There is a more elementary problem with the very basic "program" of all these armchair physicists. They seem impressed – and they want you to be impressed – with some solutions' appearances that resemble some objects in the real world like raspberries or the tiger's skin. Will a theory deeper than the Standard Model or Einstein's theory directly imply tigers and raspberries? Well, if that were the case, the Standard Model and Einstein's theory would be totally circumvented. These two (and hundreds of similar) geniuses' theory would construct the macroscopic object in the real world directly. It would mean that the theory is capable of making "several steps" at the same moment so that all the 20th century progress in physics could be ignored and replaced by something totally different. It is an extraordinary claim that requires extraordinary evidence. If you claim to have something deeper than the Standard Model or general relativity, you need evidence that is deeper, broader, and/or more precise than the evidence supporting those 20th century approximate theories. Of course, the geniuses don't have anything of the sort. They can't even get 1% of the correct predictions that the 20th century theories made correctly. They are self-evidently on the wrong track. Their whole thinking is an absolute non-starter.<br><br>One way to describe the failure is to say that they just don't understand reductionism. Before string theory, we didn't know the final theory but we still knew a lot about the structure of tigers and raspberries and the mechanisms that create them (evolved them and replicate them) and that keep them alive. And these mechanisms just don't follow from elementary magnetic raspberries; or elementary bits on a binary checkerboard. We know that these are wrong models of these objects, even as approximations. Natural processes may produce the structures on the tiger's skin that have similarities with Rule 30 of Stefan Tungsten. But we know that the mathematical ability of such hypothetical processes, even if they agreed (which they almost certainly don't), isn't a feature of the fundamental laws of physics. The processes that generate raspberries and the tiger's skin are self-evidently emergent i.e. not fundamental. You can't determine the fundamental laws of physics out of them directly!<br><br>There are lots of other armchair physicists who have their objects – playing the same role as LEGO pieces – and they are imagining that God constructed the world out of them. Again, the problem is that the only "correct" prediction of their paradigm is that "something is made of something" but this prediction was already incorporated as an assumption so it is not really a prediction at all! They only got what they assumed. And everything else is wrong. They don't get any electromagnetic fields, waves, Yang-Mills fields, Higgs fields, fermionic generations, electromagnetic, Yukawa, and Higgs quartic interactions.<br><br>There is a whole category of armchair physicists whose plan is similarly hopeless. Even though many of them are much older than 10, they still mentally play with LEGO and they imagine that everything is made of it. They don't ever get a single encouraging sign and they fail to notice. They haven't understood physics at all – or where it stands in the 21st century.<br><br>How does string theory succeed? String theory is a unique theory and since the 1990s, we have known tons of things about its nonperturbative behavior (that puts all the perturbative flavors into one theory that is even more connected and sexy than previously thought). But even if you focus your attention on perturbative string theory (where the goal is to calculate particles' scattering amplitudes as a Taylor expansion in the string coupling constant \(g_s\)), the success in reproducing the previous layer of the approximate theories is just amazing.<br><br>First, it's important to realize that perturbative string theory doesn't assume the existence of any "spacetime fields", especially not "fields with a nonzero spin". Nevertheless, their quanta are obtained – as strings in some energy eigenstates of the vibration – and the properties of these quanta exactly match the qualitative Ansatz we started with, namely the Higgs, Dirac-Weyl, Maxwell or Yang-Mills, and graviton fields with the desired interactions (and nothing else at low energies).<br><br>The fields' quanta are formally point-like, parameterized by their location (or momentum). A string becomes effective point-like in a long-distance approximation in which the dimension of the string seems negligible. It has some squared mass \(m^2\) that counts the amount of internal vibrations (which is quantized due to the usual magic of quantum mechanics, basically quantum harmonic oscillators). The string's (or particle's) spin is similarly nonzero (but a multiple of \(\hbar/2\)) and it is realized as some "orbital angular momentum of the string bits' relative motion".<br><br>These qualitative properties could be obtained for any "extended" object that replaces the point-like particles. But once you study the interactions between these particles (originated from strings), you find something way deeper, more specific, and more aligned with the world as we know it (the Lagrangian we started with). Unlike little green men, the strings directly imply what the interactions between all the particles are (at all energies, without adjustable parameters) and the resulting interactions precisely match the electromagnetic/gauge, Yukawa, and Higgs quartic interactions we started with. It's really a miracle of a sort because if a particle were a little green man, it could have a similarly quantized mass but the allowed interactions for the little green men would display (almost) no similarity with the Lagrangian of the Standard Model or Einstein's theory.<br><br>Fine. First, you get the qualitatively right spectrum of particles (which are fields quanta in quantum field theory; but they are derived "more directly" from strings, without fields, in string theory). In heterotic string theory using the RNS formalism, there is ground state tachyon \(\ket 0\) of a string that is ultimately filtered out; but the surviving excitations include \[ \alpha_{-1/2}^\mu \tilde \alpha_{-1/2}^\nu \ket 0 \] which has two indices and is capable of producing a spin-two graviton, too. By choosing \(\mu\) or \(\nu\) along some compactified dimensions, you may lower the spin to one or zero, too. And there are fermions in the sector with the periodic fermions, too. Let me not go into details. You may derive the correct spins between 0 and 2 at the massless level; all higher-spin particles are massive (which means very massive, comparable to the string scale which may be close to the Planck energy scale).<br><br>Fine, the single string's spectrum is a problem with infinitely many quantum harmonic oscillators and it's fun to play with it. But the true success and nontrivial checks validating string theory only begin once you start to consider the strings interactions. The strings (or particles that are made of strings) start to interact as soon as you allow the topology of the two-dimensional world sheet to be variable. Note that locally on the world sheet, the theory is already determined, so when you say "let the strings be free to have any topology of the world sheet", you don't really add any information about the interactions. The interactions are directly encoded in the rules defining the spectrum. It is not quite the case because one needs to determine the allowed boundary conditions and sometimes there are several options but it is still true that you are insanely far more constrained than if you were just defining a theory with "some objects" where you would have to invent "some interactions". The interactions of the strings are mostly determined automatically.<br><br>We have observed that spin-two massless particles are found among the excitations of the string. What are the interactions of these spin-two particles? Once you seriously learn the math from the first chapters of a string theory textbook, you will know that these interactions of spin-two particles are exactly what you would deduce for the interactions of gravitons, the quanta of the metric tensor field, that follow from the interaction terms involving the metric tensor and other fields. And those interactions of fields are determined by the equivalence principle. You get the same form of the interaction for the graviton-strings; you may derive the equivalence principle just from strings! This is looks like a miracle but once you know how to calculate with the stringy mathematics, it is a straightforward and rigorous proof.<br><br>Similarly, you may prove that even though the strings (e.g. propagating on a flat space time a torus) don't seem to have any non-Abelian isometries, some of the string states will behave as Yang-Mills fields and their interactions will be... exactly what you expect from the gauge symmetry! The proof is really completely analogous to the equivalence principle case in the previous paragraphs. It's not quite a new relationship between general relativity and Yang-Mills theory: already in the 1920s, the Kaluza-Klein theory knew how to get Yang-Mills theory from general relativity with extra dimensions. When some indices of the tensors go along the extra dimensions, you reduce the (four-dimensional) spin of the particles but there are still interesting patterns left. Gravitons may be reduced to gauge bosons; the equivalence principle is reduced to the gauge symmetry.<br><br>OK, you may classify all possible massless particles coming from string theory and their interactions. You get the usual spin 0 up to 2 particles (yes, including the spin-3/2 gravitino that is waiting to be discovered but we don't know whether it is light enough to make it in this century) with the right renormalizable, long-distance interactions: Yang-Mills self-interactions, gauge interactions with fermions, gauge interactions with scalars, Yukawa interactions, Higgs self-interactions. And some gravitino interactions that are almost completely determined by supersymmetry.<br><br>The gauge symmetry or the equivalence principle was assumed when we were building the Standard Model or general relativity. In string theory, these things may be derived from a completely different set of assumptions (namely that we have strings whose world sheet topology may be nontrivial). And it's equally important that we have a smaller number of assumptions to determine the theory. We only assume that there are the relativistic strings; and we may deduce the qualitative features of the Lagrangian of our (seemingly unrelated) approximate theory or theories.<br><br>Once you get to this point, you realize that the theory is really telling you much more than the approximate theories do. It's almost true that in the approximate theories, the spectrum and interactions were determined by direct measurements. They are not quite arbitrary and the equivalence principle and gauge symmetry are sort of needed for the consistency; but they may only apply once you start with fields with a spin and this assumption may look like a contrived one by itself. And all the pieces (fields of different spins) are independent building blocks that are coupled by interactions which are mostly independent, too. In string theory, all elementary particle species and their interactions come from the same string dynamics.<br><br>The stringy calculations are capable of calculating the interactions at arbitrarily high energies and the short-distance (ultraviolet, UV) divergences are completely avoided. It's another midsize miracle that you get. Again, you could imagine that particles are little green men and the extended nature of the little green men is enough to make the theory UV-finite. But it isn't really the case. The generic interactions will be UV-divergent just like the point-like particles' interactions. And even if you only allow some interactions that reflect the softness of the little green men (to get a soft behavior at high energies), you will hit virtually unavoidable problems. Such as those with causality.<br><br>If you integrate out all the excited modes of the little green men, you will unavoidably obtain a nonlocal theory for the particles as functions of their center-of-mass. That is true for string theory, too. But such a nonlocal theory will be almost always acausal. A particle at point P of the spacetime will interact with a particle at point Q which may be spacelike-separated as well as timelike-separated. That is a complete nightmare because a generic acausal and nonlocal interaction like that basically depends on arbitrarily high derivatives of the fields. So you need to specify a field and its infinitely many time derivatives. The initial data will depend on the reference frame etc.<br><br>You may see that string theory totally circumvents these deadly problems. And it's basically due to the local character of the string interactions – they are local on the world sheet. Locally, the world sheet always looks the same, as we said, so you can't really say where two strings join in the (Euclideanized) spacetime: there is no privileged point on the pants diagram. You may analyze possible "elementary objects" of higher dimensions than the 1D strings. You will find out that nothing works as a foundation for a perturbative string-like theory, basically because the worldvolume gets too high-dimensional and therefore suffering from the limited consistency of non-stringy yet gravitational quantum field theories in \(D\gt 2\).<br><br>There is a nice way to see why the UV divergences are absent in (closed) stringy one-loop diagrams. Topologically, the diagram is a torus (toroidal world sheet) and the shape of the torus is parameterized by \(\tau\). The imaginary part of \(\tau\) looks like a <a href="https://en.wikipedia.org/wiki/Schwinger_parametrization">Schwinger parameter</a> in one-loop quantum field theory diagrams. But such a Schwinger parameter \(u\) is integrated from \(0\) to \(\infty\). The UV-divergence arises from the \(u\to 0\) region. Nicely enough, the integral over the \(\tau\) plane only goes through the fundamental region\[ |\tau| \gt 1,\quad {\rm Re}(\tau) \leq \frac 12 \] which includes all the non-equivalent conformal shapes of the torus. The region with a small real part of \(\tau\) is completely avoided. It is due to the \(SL(2,\ZZ)\) symmetry of the torus, the "modular invariance". The modular invariance is responsible for removing the UV-divergences in a consistent way. The little green men (or any other objects you invent to be inside particles) fail to have this properties because the little green men don't have any modular invariance. If you imagine the world volume of the little green men in a periodic Euclideanized time, it's still a "circle times a little green men" and there is no other way to rewrite the Cartesian product because the two factors are so different. Because the string geometrically looks just like the circular periodic time coordinate, there is an extra symmetry and that symmetry is sufficient (and almost necessary) for the UV-finiteness. With some extra assumptions, you may say that this is a proof that you need strings in a similar UV-finite theory.<br><br>String theory does much more than to produce the qualitatively correct spectrum, the qualitatively correct interactions of particles, and UV-finite expressions (including loops of gravitons) at all energies. It really unifies all allowed "field theory Lagrangians" of this kind. In quantum field theory, a new Lagrangian means a new theory. The objects living in two QFTs must be said to be totally disconnected. People don't respecting the most fundamental laws of Nature are even more alienated aliens than aliens who believe in a different religion or irreligion. You won't ever meet them.<br><br>In string theory, you will find out that there is still a "landscape" of possible derived effective QFTs that comes from the non-uniqueness of the shape of extra dimensions (which is very constrained, however), from the extra branes and fluxes which may be added (which are also constrianed), and from some discrete-torsion-like discrete parameters that determine some allowed sets of boundary conditions and/or the phases they contribute to the Feynman path integral. This landscape seems rich enough to contain the Universe around us, including the more detailed properties of particles than the qualitative ones that were obtained correctly in the "previous sketch".<br><br>And what's amazing – another collection of midsize miracles – is that all these possibilities are ultimately connected into one theory. The connectedness of the possible "string theories" into something that is really just many vacuum states of one string theory becomes particularly striking once you appreciate the nonperturbative phenomena, topological transitions, and dualities. But even at the level of perturbative string theory, there exist many transformations of "one vacuum into another" that would look stunning or impossible in point-like particle-based quantum field theories.<br><br>Two of the five ten-dimensional flat-space supersymmetric string theories are the heterotic string theories. In the \(D=10\) spacetime (which is compactified in the environments like the Universe around us), there is a supergraviton multiplet and a super-Yang-Mills multiplet. The latter has a gauge group that must be either \(E_8\times E_8\) or \(SO(32)\). These are two possible gauge groups, two possible heterotic string theories. They cancel some anomalies which may already be calculated in quantum field theory; the anomaly cancellation is much more automatic and "derived from something more fundamental" in string theory.<br><br>Both of these gauge groups have the rank equal to 16; and, more interestingly, the dimension equal to 496. Note that\[ 248+248 = \frac{32 \times 31}{2 \times 1}. \] They actually share several other, harder to define, invariants. Now, at the level of quantum field theory, a quantum field theory with these two gauge groups would be "analogous to one another" but otherwise as disconnected as you can get. When you assume the gauge group to be \(E_8\times E_8\), it just can't be changed to \(SO(32)\) or vice versa because the gauge group is a part of the most unchangeable DNA of a field theory, model builders in QFT normally think.<br><br>But within string theory, you may see that this assumption is wrong. String theory is miraculously able to change one of the gauge groups to the other – and the intermediate steps are as consistent as the limiting ones. Just compactify one of the 9 spatial dimensions on a circle. Break the gauge groups to \(U(1)^{16}\) by the Wilson lines. The spectrum of possible Wilson lines and related \(B\)-fields may be adjusted to restore the symmetry either to \(E_8\times E_8\) or \(SO(32)\). So a \(U(1)^{16}\) theory at a point in between may be derived as a spontaneously broken theory with \(E_8\times E_8\) or from \(SO(32)\), via the Higgs mechanism, but the broken theory is exactly the same! At all energies!<br><br>String theory allows you to interpolate between the different options. Topology change is possible – flop transitions, conifolds. Even perturbatively, you may see T-duality that says that a toroidal compactification with the radius of a circle \(R\) is equivalent to another with a circle of radius \(\alpha' / R\), and so on. There is a large number of such amazing features of string theory that wouldn't hold if strings were replaced with the little green men (or raspberries or tigers or octopi or anything else, even membranes which are slightly more promising than the previous three). With some extra assumptions, you may really see that strings are the only fundamental objects "to be hidden inside elementary particles" that may generate these great properties. None of these proofs of the inevitability of string theory is completely rigorous and complete; but the number of highly suggestive patterns or nearly complete proofs is high and they seem very independent of each other.<br><br>The picture gets stronger within non-perturbative string/M-theory. Even the boundary conditions on the strings and other things we used to define "a perturbative string theory" become derived from a more elementary or more fundamental starting point. And the broader theory is more tightly unified and constrained than anything before. A person who looks for a deeper understanding of Nature within the wisdom outlined by string theory is really <em>centuries if not millenniums</em> ahead of a would-be competitor who wants to ignore string theory and hope that his tigers, octopi, or raspberries will miraculously do a better job than string theory. Be sure that they won't and they can't.<br><br>And that's the memo [Sorry for typos, I don't plan any proofreading].Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0tag:blogger.com,1999:blog-8666091.post-63232782854599269352020-12-13T08:55:00.022+01:002020-12-13T09:44:00.667+01:00The 2020 Nobel talks and the fate of singularitiesA few days ago, Roger Penrose and two less famous co-recipients gave the <a href="https://motls.blogspot.com/2020/10/penrose-genzelghez-share-2020-nobel.html?m=1">2020 Nobel Prize</a> lecture.<br><br><iframe width="407" height="277" src="https://www.youtube.com/embed/DWF1uNb9Q1Q" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><br><br>The Nobel Prize is a relatively famous award, arguably the most famous one in the world, and Penrose is one of the most famous living mathematical physicists. Still, the video above has 16,000 views now; <a href="https://www.youtube.com/watch?v=kJQP7kiw5Fk">this version of Despacito</a> has over 7 billion views. The ratio is about half a million which places the public interest in science into a perspective.<a name='more'></a><br /><br /><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block; text-align:center;" data-ad-layout="in-article" data-ad-format="fluid" data-ad-client="ca-pub-8768832575723394" data-ad-slot="4218709518"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br />And indeed, I haven't listened to the Nobel Prize lecture, either, arguably because I think that there would be nothing really new and interesting to learn (which may be a wrong expectation!). The Penrose-Hawking singularity work was done basically half a century ago and the fate of singularities according to the state-of-the-art theoretical physics has profoundly changed in several subsequent minirevolutions, most of which are unknown to Penrose himself.<br><br><script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-8768832575723394" data-ad-slot="0363397257"></ins><script> (adsbygoogle = window.adsbygoogle || []).push({}); </script><br /><br /><a href="https://en.wikipedia.org/wiki/Singularity">A singularity</a> is a point in space, either the real \(\RR^3\) space or the \(\RR^4\) spacetime or a moduli space or basically any other space, where some quantities are ill-defined, usually infinite in a limit. The previous sentence contains the word "infinite" and just like this word, the word "singularity" induces some religious feelings inside a layman (including a physicist-beginner).<br><br>Why? You know who is also infinite? God. He is great. The human stupidity and the Universe are infinite but I am not sure about the latter, Einstein properly quipped. We are so small relatively to the giant Infinity. We can't get there. We can't beat it. We must be humble. Infinity may be inconsistent in mathematics etc.<br><br>All these feelings are great and they represent the overlap of mathematics, science, theology, and philosophy. I haven't avoided them, either. At the same moment, mathematics and science ultimately becomes rational about everything including infinities and singularities. The laws of mathematics and physics and the tools to think about all the topics "beat" infinities and singularities – much like they tame God as long as He is meaningful or worth a discussion.<br><br>So you may discuss functions such as \(y=1/x\) which have a singularity, like one at \(x=0\) in this case. Singularities become a dull thing. They may also be classified. Something breaks down at these points but such a breakdown isn't the end of the world. It's just the end of quantitative discussions about the behavior at the given point – which is no catastrophe, partially because "almost all points" are away from this singularity (and the word "almost" may be given a rigorous meaning in measure theory). Similarly, manifolds may involve points where the geometry is not smooth. Those are singularities as well, some curvature invariants are infinite at these points, too.<br><br>One may ask whether singularities are physically possible. Again, rich people like these discussions because they make them feel metaphysically deep and theological. For example, Ray Kurzweil's ideas about the technological singularity represent a pile of clearly invalid gibberish that makes some billionaires feel like Jesus Christ, the carriers of some truly qualitative paradigm shift (that cannot arrive, partly because there always exist <a href="https://motls.blogspot.com/2020/04/the-doubling-time-always-gets-longer.html?m=1">new processes that slow down any exponentially growing process</a> in the real world).<br><br><a href="https://motls.blogspot.com/2020/09/most-laymen-completely-misunderstand.html?m=1">Most laymen still imagine the black hole to be an infinitely dense object</a>. The black hole is all about the singularity, they think. This view is completely wrong. The singularity is just a nearly untestable (and therefore uninteresting) episode at the "very end of the life" of the infalling observer. Instead, the truly shocking and defining traits of the black hole already start at the event horizon. The event horizon is what defines a black hole (the event horizon makes it a hole and it makes it black, too) – and the curvature and the matter density near the event horizon is perfectly finite. And the event horizon, the boundary between the interior and the exterior, plus the relationship between the interior and the exterior is what is responsible for the bulk of the information loss puzzle; the singularity is nearly irrelevant. <br><br>Nevertheless, Penrose and Hawking have used the singularity as an ally to prove that black holes unavoidably arise in rather generic astrophysical situations. The curvature invariants exceed any "classically reasonable" threshold which means that in a classical approximation which is still accurate enough, a singularity unavoidably arises. With some extra assumptions (that eliminate the naked singularities, in the sense of Penrose's Cosmic Censorship Conjecture), it is guaranteed that these unavoidable singularities are covered by the unavoidable event horizon, too.<br><br>Three days ago, Suvrat Raju, a brilliant physicist, my former TA, and a Marxist, posted a <a href="https://arxiv.org/abs/2012.05770">156-page-long review</a> of the black hole information puzzle. It's called a review but make no mistake about it, the title may be considered misleading because a "review" may be expected to contain "utterly uncontroversial knowledge". Lots of the stuff in that review is controversial enough (among researchers) although I agree with almost everything; many conclusions are extracted from very recent papers (the key ideas are partly conclusions, partly assumptions of the recent papers; the logic isn't "fully proven" yet). One of the conclusions that Suvrat considers settled is that "the black hole interior is completely dispensable", as I often emphasized. It's just a dead end street. Equivalently, a copy of all the physics is always encoded in the degrees of freedom that may be fully tied to the exterior.<br><br>Because the infinity is ill-defined (it can't appear on a voltmeter or any other "quantitative apparatus" used by an experimental physicist), amateur (and bad) physicists tend to jump to a premature conclusion. Infinities and singularities are totally evil, inconsistent, and they have to be banned. Everything must be smoothened out around the putative singularities. The failure to do so is politically incorrect.<br><br>Well, this is a wrong opinion based on a sloppy reasoning and exaggerations. Singularities appear in legitimate physics and they may be exact. For example, some heat capacity etc. may be infinite near a phase transition. This particular singularity may be blamed on an approximate, limiting theory – thermodynamics or statistical physics. If you send the number of molecules \(N\to\infty\), the functions describing the behavior of matter become really "pure" and they exhibit a singular behavior near the phase transitions. If you realize that you have a finite number \(N\) of atoms and if you study some quantities describing these atoms accurately, you will find out that the quantities become smooth in the very vicinity of the would-be singularity, too.<br><br>However, singularities and infinities appear even in quantum mechanics, a theory that properly explains why the atoms exist in the first place, a theory that is expected to make "everything smooth" and "ban singularities". Well, if you think about this "summary of quantum mechanics and its impact on physics", this expectation is very dumb for an obvious reason. Yes, it is true that quantum mechanics introduces atoms and it generally says that matter cannot be infinitely fine (and the electron can't sit infinitely close to a proton), and that is also why you cannot trust the formulae for the heat capacity etc. in an arbitrarily vicinity of the phase transitions. But does quantum mechanics make everything smooth? Not at all. The very term "quantum mechanics" was chosen because the energy (of an atom, for example) jumps discontinuously. And a discontinuous jump means an "infinite energy change per unit time", i.e. a singularity!<br><br>In fact, singularities are more omnipresent in quantum mechanics than they are in classical physics. You need to work with operators such as \(1/(H-E)\) where \(H\) is the Hamiltonian (operator). Those are important for the evolution and they display a pole when \(H=E\). Poles in the scattering amplitudes (written as functions of energy and/or momenta) are not inconsistencies. They are signs of the existence of stable states (initial, final, or intermediate). <br><br>The important lesson is that only the final predictions for "the numbers shown by an experimental apparatus" must be finite but any intermediate step in the calculation may involve infinite or singular quantities. It's just fine! So quantum field theory, a combination of the special theory of relativity and quantum mechanics, produces infinities in the loop diagrams. They may be removed by the renormalization, a "black magic" that nevertheless works just fine and allows far more precise predictions than a calculation where these loop diagrams would be completely banned (because their infinities may be said to be politically incorrect).<br><br>In the process of the renormalization, even terms contributing to the probabilities may be infinite. But as long as the "outcome" is well-defined and possible to be realized, its probability must be a finite number between zero and one. And a viable theory may include infinite terms contributing to the probability – which must perfectly cancel at the end, however. Again, the very notion that infinities may cancel may be called (and has been called) dangerous, a black magic, politically incorrect. But one may organize the calculations so that the calculation is damn real, controlled, and the result after the cancellation agrees with the experiments (almost) perfectly.<br><br>So results for the truly doable experiments must be finite; everything else that is used during the calculation is allowed to be infinite and any efforts to ban such infinite intermediate constructs or results are totally misguided symptoms of a knee-jerk reaction, the pathological precautionary principle, and sloppy reasoning. This is also true about the spacetime geometry in general. May the spacetime geometry include singularities or does it have to be smooth?<br><br>A certain widespread culture from popular books – which affected me as well when I was a kid – says that the spacetime must be smooth everywhere. Singularities must be banned and if they're present in any description by classical general relativity, it shows that the classical general relativity is incomplete. A more complete theory surely replaces the singular regions by similar but smooth ones. Well, not really. Sometimes, the singular classical geometry is indeed replaced by a smooth one (after corrections are incorporated). Sometimes, the corrected geometry only becomes smooth in some variables (string units vs Planck units). Sometimes, the corrected geometry remains singular and qualitatively equivalent to the classical one.<br><br>First of all, the singularities that arise in the Schwarzschild and similar black holes are the "final answer" in the sense that the life of an observer must really end there. There cannot be a consistent theory where the observer gets slightly deformed, the geometry behaves a bit differently than in the classical GR due to some corrections, and the observer survives and keeps on living "somewhere". Well, for curvature invariants smaller than the Planck scale, the corrections are still small and the classical GR may still be trusted which really means that the curvature invariants may get at least close to the Planck scale. But a Planck-like curvature is really so huge that not only carbon-based life is dead for a long time there; you may argue that the notion of a smooth spacetime has broken there, too. The corrections can't do much about it. On top of that, you may analyze the possible "loopholes" by which the spacetime could continue beyond the singularity and you will find out that none of them is really consistent with the basic rules such as causality. So yes, the black hole singularities are "real". And they don't mean any real inconsistency of the theory.<br><br>Don't get me wrong. I am not saying that the classical GR is exactly true. It surely isn't. I am just saying that the particular trait of classical GR, the fact that it may have singular points in the spacetime, isn't a trait that must be completely banned or rejected by a more complete theory. In fact, string theory is a perfectly consistent quantum theory of gravity. Nevertheless, it works just fine at spacetimes which aren't smooth everywhere. Also, Penrose has assumed that the classical GR must be complete enough so that naked singularities are banned at the level of classical GR, and that's why a deeper theory (which would be needed to predicted what a naked singularity emits) isn't too needed. This Cosmic Censorship Conjecture was pure faith. There have been some partial positive evidence in favor of weakened forms of the conjecture; any reasonably general form of the conjecture is believed to be wrong by now (counterexamples exist) although some interesting enough weakened versions are supported by new evidence, e.g. by the equivalence with the more tangible Weak Gravity Conjecture of ours.<br><br>The most innocent type of a singularity that is allowed in string theory is an orbifold singularity. An orbifold is a quotient of a space, like \(T^4/\ZZ_2\) where \(\ZZ_2\) flips the sign of the four coordinates of the torus. The 16 fixed points of this \(\ZZ_2\) map on the torus are singular; the space around them isn't equivalent to a piece of \(\RR^4\). Does string theory allow strings to propagate on the background of an orbifold spacetime geometry? Or does string theory confirm that even such singularities must be cancelled in the sense of the cancel culture warriors?<br><br>The answer is unambiguous. String theory perfectly allows orbifold backgrounds of this kind. In fact, the consistent treatment of orbifolds is one of the amazing features of string theory where the "stringiness" is most directly relevant. Aside from reducing the number of degrees of freedom by demanding that the fields are invariant under that \(\ZZ_2\) group that defines the orbifold, string theory does something else that the point-like particle theories don't. It also increases the number of degrees of freedom. By a factor that may be said to exactly cancel the reduction! String theory does so by adding the twisted sectors. New closed strings – whose points' position in the spacetime is periodic on the closed string, but only modulo the \(\ZZ_2\) transformation – have to be incorporated into the spectrum.<br><br>The stringy treatment of the orbifolds is a simple yet amazing piece of the string theory mathematics which already shows how simply clever string theory is. The particular \(T^4/\ZZ_2\) orbifold happens to be a singular point in the moduli space of the K3 surfaces. Most K3 surfaces are smooth but there are singular points in the space of shapes of these K3 surfaces, the "moduli space", where the K3 develops a singularity. \(T^4/\ZZ_2\) and \(T^4 / \ZZ_3\) are special points of K3 surfaces' moduli spaces. (Note that the singularities are present both in a K3 surface itself; as well as in the moduli space of possible shapes of the K3 surfaces).<br><br>Less trivial but equally consistent are orbifolds of curved manifolds; and conifolds. In fact, string theory vindicates the existence of any "spatial geometry with a singularity" that has some beauty and good reasons to expect that the spacetime where this geometry is stationary in time solves Einstein's or similar equations governing the evolution in time. Singularities that require time dependence are much less understood, partially because supersymmetry is almost unavoidably broken and many useful cancellations no longer hold. But I think that it is reasonable to say that string theory allows lots of time-dependent classically singular spacetimes, too.<br><br>All these constructs are physically consistent because the total probability amplitude for a process occurring within such a spacetime is ultimately finite. In the case of string theory, you don't even need the ultraviolet divergences as the intermediate results (terms whose infinite parts cancel). All terms are ultraviolet-finite. But they may still be terms describing the propagation of strings on a singular spacetime geometry. Well, the probability amplitudes for strings on top of an orbifold are just sums of e.g. \(2^2=4\) terms labeled by the boundary conditions in the two directions of the world sheet, \(\sigma\) and \(\tau\). All these four terms are completely analogous to terms on a non-singular smooth torus. The fact that the total spacetime geometry must be interpreted as a quotient (which therefore has an infinite curvature invariant at the fixed point) is not harmful at all. The terms adding up to the probability amplitudes are as finite and smooth as they were in the smooth space.<br><br>I must say that just like the religious feelings are suppressed by rationality in the case of "infinities and singularities in physics", the morally equivalent transformation has happened in mathematics, too. Are there infinite sets whose size is in between the continuum and the countable set (of integers)? This is the continuum hypothesis. The answer was impossible to reach for a "finite human being". But the 20th century mathematics has adopted a self-confident, less religious attitude. If some questions are really (and in principle) inaccessible to a human being, they are meaningless! This principle has the obvious importance in physics where every idea is ultimately a tool to explain the truly doable experiments. If ideas are in principle disconnected from observations, they're an uninteresting non-physics talk, not "big words that we must worship"! But it's important in mathematics, too. The question is whether you can prove the Continuum Hypothesis. And it was proven that you can neither prove it nor disprove it if you only have the other, more common axioms of set theory. Mathematicians say that "the Continuum Hypothesis is independent of the ZF or GB axioms [of modern set theory]". In this sense, many opinions about the "world of the infinite sets" are a matter of conventions. You can choose axiomatic systems that say "Yes" and systems that say "No" and those must be considered "equally good" because both may be consistent.<br><br>In the past, I've had goose bumps when I thought about lots of such things. Most of these goose bumps have gone away. Sometimes, I am a little bit sorry of this loss. The goose bumps were cool. It was great to be afraid of the infinity and the singularities. On the other hand, I only had the goose bumps because I really wanted to know the truth. It was largely unavoidable for me to learn about this truth; and understand (and perhaps, in some cases, help to discover) the answer to many such questions. The goose bumps have been replaced by the rational, cold understanding which may be boring. But in some cases, it is prettier than the unrefined fear and "infinity worshiping" that I started with. Various special, singular points may appear in the real space and moduli spaces. The symbol \(\infty\) may be a label placed on some special points of all these spaces (in most cases, more specific labels that say \(\infty\) but add some more detailed information about the singularity type are used instead). But the "big picture" of the theory including the "sometimes singular" spaces used by the theory is still mathematically and physically consistent. The mathematical consistency boils down to the absence of provable contradictions; the physical consistency involves the unique finite predictions for the experiments that are so "really" measurable that their results really have to be finite.<br><br>We may say that mathematicians and physicists have become the "masters of the infinities and singularities". In the past, the infinities and singularities were infinitely far, on the boundary of all the real or conceivable pictures, and therefore eternally inaccessible. (And the black hole singularity had to come and still has to come at the end of the observer's life.) However, by the late 20th century, mathematics and physics has become a full of diagrams of "theories and gadgets that really work fine" where the singularities and infinities may be spotted right in the middle of calculations and right at the center of many diagrams. We have tamed a lot of infinities and singularities and turned them into our employees, slaves, pets, and parrots in the cages, too.Luboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.com0