tag:blogger.com,1999:blog-3832136017893749497.post7047093083993998302..comments2023-09-29T08:49:30.765-04:00Comments on Elliptic Composability: Florin Moldoveanuhttp://www.blogger.com/profile/01087655914212705768noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-3832136017893749497.post-79623454887523229502016-09-26T13:56:00.414-04:002016-09-26T13:56:00.414-04:00There is now an exposition at "PhysicsForums ...There is now an exposition at "<a href="https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/" rel="nofollow">PhysicsForums Insights</a>" of the close relation between spectral triples and string backgrounds encoded by 2d SCFTs. See here: <br /><a href="https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/" rel="nofollow">www.physicsforums.com/insights/spectral-standard-model-string-compactifications/</a> Urs Schreiberhttps://ncatlab.org/nlab/show/Urs+Schreibernoreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-46759416385763327692016-07-22T18:22:40.858-04:002016-07-22T18:22:40.858-04:00You are mentioning something with the worldline SU...You are mentioning something with the worldline SUSY that I have thought of. We can include this with a string world sheet in three dimensions. The geometry of the world lines is then dual to the string world sheets. The worldlines form knots and braids, which has a duality to string world sheets. The 2-space plus time (2, 1) metric means there is a Lorentz transformation in this space SO(2,1). This has π_1(SO(2,1)) = Z, which is different from the Z_2 = Z/2Z of higher dimensional Lorentz group. The spin (2,1) is then not a universal cover and PSO(2,1) and the exchange of two particle paths is obstructed. This obstruction leads to both bosonic and fermionic statistics. <br /><br />I like the ncatlab website BTWLawrence Crowellhttps://www.blogger.com/profile/12090839464038445335noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-54401860615324282082016-07-21T04:08:51.820-04:002016-07-21T04:08:51.820-04:00One should distinguish here between supersymmetry ...One should distinguish here between supersymmetry on target spacetime andon the worldvolume. Connes' original axtioms for spectral triples were formulated in terms of generalized Laplace operators, and axiomatized the structure of a bosonic quantum particle (its Hilbert space, Hamiltonian, and subalgebra of smooth poisition operators). Later he realized that the theory works much better if the Hilbert space is taken to be a Z/2Z-graded (a super-Hilbert space) and the generalized Laplace operator is assumed to be the square of a generalized Dirac operator. That gives the now standard axioms for spectral triples. In terms of thinking of them as encoding a quantum mechanical particle, the step means to pass from worldline translation symmetry (generated by the generalized Laplace operator regarded as a Hamiltonian) to worldline supersymmetry (generated by the generalized Dirac operator). In 1 dimension the supersymmetry algebra reduces to the relation H = D^2 between a Hamiltonian H and a corresponding Dirac operator D. In this sense spectral triples in the modern sense are an axiomatization of the ingredients of supersymmetric quantum mechanics.<br /><br />This is, incidentally, why modern spectral triples arise as the point particle limit of 2d superconformal field theories, instead of 2d bosonic conformal field theories (see <a href="http://fmoldove.blogspot.com/2016/07/what-is-noncommutative-geometry.html?showComment=1468934322483#c6802080472818966598" rel="nofollow">below</a>).<br /><br />Now, all this is still only worldline supersymmetric, not target space supersymmetric. Urshttps://ncatlab.org/nlab/show/Urs+Schreibernoreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-48075959214886220112016-07-20T21:15:26.010-04:002016-07-20T21:15:26.010-04:00I don't know what the relationship between NCG...I don't know what the relationship between NCG and supersymmetry would be. Superspace built from Grassmannian generators with<br /><br />{ξ_a, bar-ξ_b'} = -2iσ^μ_{ab'}∂_μ, p_μ = -i∂_μ<br /><br />has some appearances of being NCG. The primary difference is that this is anticommuting. One the other hand I see no reason why NCG can't be worked in a graded Lie algebraic context with generators g = h + k with rules<br /><br />[h, h] \subset h, [h, k] \subset k {k, k} \subset h.Lawrence Crowellhttps://www.blogger.com/profile/12090839464038445335noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-7855085653097008672016-07-20T06:54:57.591-04:002016-07-20T06:54:57.591-04:00Lifting a spectral triple to a 2d CFT means asking...Lifting a spectral triple to a 2d CFT means asking for much stronger constraints, even, since there is much more data in the 2d CFT than just its point particle limit reflected in the spectral triple. <br /><br />This is a version of the fact that there are much stronger constraints on a string background to be consistent (anomaly free) than on a random QFT. <br /><br />The article by Soibelman referenced above means to make use of this for saying something about the landscape of perturbative string vacua. A perturbative string vacuum is a 2d SCFT of central charge 15, which in addition satisfies modularity and sewing constraints. While the moduli space of all 2d SCFTs is hard to analyze, the shadow that it throws, via the point particle limit, in the space of spectral triples is more tractable. And Soibelman's article analyzes this shadow space.<br /><br />In this context one cannot help but notice the following coincidence:<br /><br />The spectral triples arising from 2d SCFTs of central charge 15 in string theory have, famously, KO-dimension 4+6 (mod 8). Now this is precisely the KO dimension that Connes claims in <br /><br />A. Connes, "Noncommutative Geometry and the standard model with neutrino mixing", JHEP0611:081 (<a href="http://arxiv.org/abs/hep-th/0608226" rel="nofollow">hep-th/0608226</a>)<br /><br />is necessary to get a viable standard model-like effective theory from a spectral triple.<br /><br />Maybe it's a coincidence. Or maybe it points to something deep.Urs Schreiberhttps://ncatlab.org/nlab/show/Urs+Schreibernoreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-2270451979893171942016-07-19T17:00:41.028-04:002016-07-19T17:00:41.028-04:00@Urs, one of the strengths of NCG is the contrains...@Urs, one of the strengths of NCG is the contrains it puts on model building (mostly the scalar sector but not only). Do the 2d super-conformal field theories having spectral triples as point particle limit provide the same type of constraints?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-68020804728189665982016-07-19T09:18:42.483-04:002016-07-19T09:18:42.483-04:00Connes' spectral triples arise as the point pa...Connes' spectral triples arise as the point particle limit of 2d (super-)conformal field theories. A commented list of references on this relation is here: <a href="https://ncatlab.org/nlab/show/2-spectral+triple#References" rel="nofollow">ncatlab.org/nlab/show/2-spectral%20triple#References</a><br /><br />Under this relation, the way spectral triples encode an effective target space geometry as seen by a quantum particle is analogous to how a 2d (S)CFT encodes an effective target space geomtry as seen by a quantum (super-)string. <br /><br />The first rigorous account of this is due to<br /><br />D. Roggenkamp, K. Wendland, "Limits and Degenerations of Unitary Conformal Field Theories" <a href="http://arxiv.org/abs/hep-th/0308143" rel="nofollow">arXiv:hep-th/0308143</a><br /><br />summarized in <br /><br />D. Roggenkamp, K. Wendland, "Decoding the geometry of conformal field theories" <a href="http://arxiv.org/abs/0803.0657" rel="nofollow">arXiv:0803.0657</a><br /><br />Yan Soibelman used this relation of 2d SCFT to Connes spectral triple in order to approach the analysis of aspects of the landscape of string vacua:<br /><br />Y. Soibelman, "Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry" (<a rel="nofollow">pdf</a>), in Sati et al. (eds.), "Mathematical Foundations of Quantum Field and Perturbative String Theory", Proceedings of Symposia in Pure Mathematics, AMS (2001)<br /><br />For a kind of reverse construction, in <br /><br />S. Carpi, R. Hillier, Y. Kawahigashi, R. Longo, "Spectral triples and the super-Virasoro algebra" (<a href="http://arxiv.org/abs/0811.4128" rel="nofollow">arXiv:0811.4128</a>)<br /><br />the authors realize 2d SCFTs essentially as local nets (in the sense of AQFT) of spectral triples.Urs Schreiberhttps://ncatlab.org/nlab/show/Urs+Schreibernoreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-56808801001203937612016-07-18T23:04:33.333-04:002016-07-18T23:04:33.333-04:00This comment has been removed by the author.Florin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-60333571160094049442016-07-18T22:53:27.896-04:002016-07-18T22:53:27.896-04:00This comment has been removed by the author.Florin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-89853821762529213232016-07-18T22:18:54.918-04:002016-07-18T22:18:54.918-04:00"I think QM and GR are the same thing, or tha..."I think QM and GR are the same thing, or that GR is a sort of classicality that emerges from QM entanglements with “large N.”"<br /><br />I tend to disagree. Let me explain. Cones showed that SM + GR is GR on a "noncommutative space" because it all follows from diffeomorphism invariance in the new space. Now enter exhibit A: supersymmetry.<br /><br />In SUSY you can have a formulation in the so-called "superspace" which at first sight looks a lot like Cones NCG space. However, this is not the case since SUSY's superspace does not come from a C* algebra and you don't get a spectral triple. So one has to be very careful with the details and make sure the gut feeling intuition is backed up by the math. <br /><br />Is GR an emergent QM phenomena? Indeed since everything is quantum at origin. Does it have to do with entanglement and large N? Of course. But how to go from one end to the other is a very tricky business and we need the meat of the proposal to pass judgement, otherwise it is only handwaving.Florin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-15988663134750200512016-07-17T17:52:27.997-04:002016-07-17T17:52:27.997-04:00The speed of light is the basis for projective geo...The speed of light is the basis for projective geometry of spacetime. A light cone as a projective variety is identical to a Riemann sphere. A point is then dual to a space. Similarly a null curve is a point. This constructs the twistor space. Quantum mechanics has a duality between conjugate variables, which in a discrete form is the reciprocal lattice map such as in solid state physics. Relativity → blow up of a point, QM → reciprocal lattice, point mapped to ∞. <br /><br />As I indicated the Tsirelson bound and the null interval are isomorphic to each other. In effect quantum entanglement builds up the symmetries of spacetime, such as defined by null Killing vectors. I think QM and GR are the same thing, or that GR is a sort of classicality that emerges from QM entanglements with “large N.”<br /><br />The SU(2,2):= (4,4) = (3,3)⊕(3,1)⊕(1,3)⊕(1,1), which for SU(2,2) ~ CL(4) → 1, 4, 6, 4, 1 in binomial distribution of 4, 3, 2, 1, 0 chains. The 6 = (3,3) corresponds to G_2(V^4), the (3,1) is CP^4 = CP^3∪C^3, the (1,3) is the dual CP^4 (twistor – ambitwistor) and the (1,1) are scalars. These are the “Higgs” scalars in a Hitchen bundle. Lawrence Crowellhttps://www.blogger.com/profile/12090839464038445335noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-6422235077913245382016-07-17T12:32:20.331-04:002016-07-17T12:32:20.331-04:00I would say that one encounters only discrete spec...I would say that one encounters only discrete spectra in quantum mechanics, but given a large number of discrete events, it is often convenient to use continuum math.<br /><br />Likewise, trajectories are always discrete particle actions and the notion of continuous trajectories in space and time is likewise a convenience of dealing with a very large number of events.<br /><br />Gravity then becomes a pairing of discrete yet complementary photon events whose now very, very large number represents the continuum of space and time trajectories very well.<br /><br />It seems to me that you already have the pieces of this puzzle and just need to assemble them correctly...steve agnewhttps://www.blogger.com/profile/00177693538649923112noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-59532553959851616482016-07-17T01:14:39.945-04:002016-07-17T01:14:39.945-04:00It's not a typo - you've written it twice....It's not a typo - you've written it twice. It's a proof that you're absolutely unfamiliar with basic literature and terminology in theoretical physics.<br /><br />When one does any physics of the sort, he is writing or reading the phrase Yang-Mills so frequently that it's simply impossible for him to misspell the names.<br /><br />I am just saying that your big-mouth pronouncements "On the other hand, writing off noncommutative geometry as a mathematical fantasy without physics merits is arrogant" are absolutely unbacked by any content.<br /><br />Concerning the last question:<br /><br />If you want to make new contributions in physics, does it make any sense to use the state-of-the-art mathematics from 100 years ago and ignore recent advances in math?<br /><br />Yes, of course, everyone who is working on cutting-edge physics is ignoring the recent developments in maths unless he's forced to incorporate them by the physical arguments. You recommend the exact opposite attitude: to force some content of recent math papers onto physics. Natural science doesn't work and cannot work in this way.Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-31051507839228923452016-07-16T23:29:11.115-04:002016-07-16T23:29:11.115-04:00"The Planck constant ħ is an intertwiner in q..."The Planck constant ħ is an intertwiner in quantum mechanics between uncertainty in momentum and position. It serves a role similar to the speed of light in spacetime that converts spatial distance to time."<br /><br />c and h have different origins. Planck's constant is categorical in nature and is responsible for a distinct way of composing two physical systems into a larger one. NCG is categorical in nature as well: it is category theory which provide the algebra-geometry duality.<br /><br />"Can we then do something similar with noncommutative geometry one does with spacetime"<br /><br />Maybe, I don't know. <br /><br />"In particular with twistor theory the light cone origin is a blow up of a point that is mapped to a Riemann sphere and a light ray to a point."<br /><br />Twistors are related to SU(2,2) which is isomprphic with the conformal group Spin(2,4). The same thing is the at the root of quantionic QM which is a constrained QM on SO(2,4) to obtain positivity (by BRST formalism). The resulting theory is SO(3,1)xSU(2)xU(1) or Connes' toy model for complex 2x2 matrices.<br /><br />"This gets to the question of locality and nonlocality."<br /><br />Unlikely, but it may depend on your definition of locality.<br /><br />"There is this correspondence between gravity in the bulk of an anti-de Sitter spacetime AdS_n and the conformal field theory on the boundary CFT_{n=1}. In addition there is a sort of duality between locality and nonlocality, where if gravity on the bulk is more local then the CFT on the boundary is more nonlocal, and visa versa. This is why a correspondence would potentially be interesting between the two geometries"<br /><br />This is highly speculative.<br /><br />"such as in your table." <br />Not my table. The credit goes to Connes.Florin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-71016217308057288902016-07-16T22:18:13.651-04:002016-07-16T22:18:13.651-04:00I have Connes' book on noncommutative geometry...I have Connes' book on noncommutative geometry. It has been a while since I studied it. This does lead me then to another question. If you have a noncommutative geometry it means in generality that <br /><br />[x_i, y_j] = iħθ_{ij}<br /><br />where the θ_{ij} is in the standard case a delta function. The Planck constant ħ is an intertwiner in quantum mechanics between uncertainty in momentum and position. It serves a role similar to the speed of light in spacetime that converts spatial distance to time. Can we then do something similar with noncommutative geometry one does with spacetime<br /><br />In particular with twistor theory the light cone origin is a blow up of a point that is mapped to a Riemann sphere and a light ray to a point. We in effect have correspondence maps between objects in spacetime and objects in projective geometry. In this way a sort of correspondence between standard QFT and NCG QFT might then exist.<br /><br />This gets to the question of locality and nonlocality. There is this correspondence between gravity in the bulk of an anti-de Sitter spacetime AdS_n and the conformal field theory on the boundary CFT_{n=1}. In addition there is a sort of duality between locality and nonlocality, where if gravity on the bulk is more local then the CFT on the boundary is more nonlocal, and visa versa. This is why a correspondence would potentially be interesting between the two geometries, such as in your table.Lawrence Crowellhttps://www.blogger.com/profile/12090839464038445335noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-59778140963987398142016-07-16T18:26:22.986-04:002016-07-16T18:26:22.986-04:00Lawrence,
NCG SM is simply a reformulation of usu...Lawrence,<br /><br />NCG SM is simply a reformulation of usual quantum field theory in the language of the spectral triple. The concept of locality is different in NCG because the very notion of distance and coordinates is generalized. For example, the gauge degrees of freedom are seen as "inner fluctuations of the metric". There is also a generalized concept of dimensionality called KO-dimension which tells apart different kinds of discrete sets of zero normal dimension.Florin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-7173762453847780912016-07-16T18:12:07.738-04:002016-07-16T18:12:07.738-04:00I have more of a question. What bearing does nonco...I have more of a question. What bearing does noncommutative geometry have with the locality or nonlocality of the field theory. Your last post was on a noncommutative geometry approach to the standard model, following Connes. I am not certain about the viability of this approach, but if one does field theory this way does the theory have the same locality (equal time commutators of fields etc) that one has with standard field theory?Lawrence Crowellhttps://www.blogger.com/profile/12090839464038445335noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-91380068005123858612016-07-16T17:26:35.157-04:002016-07-16T17:26:35.157-04:00Is this the best you come up with? Typos? Patheti...Is this the best you come up with? Typos? Pathetic.Florin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-36790403103968367842016-07-16T17:25:15.912-04:002016-07-16T17:25:15.912-04:00Oopsy... thanksOopsy... thanksFlorin Moldoveanuhttps://www.blogger.com/profile/01087655914212705768noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-51987076998996411252016-07-16T10:31:19.935-04:002016-07-16T10:31:19.935-04:00Dear Florin, as Kashyap observed, you misspelled (...Dear Florin, as Kashyap observed, you misspelled (twice, so no typo) Yang-Mills as Young-Mills. (Yes, C.N. Yang had the quote about the books.)<br /><br />You are trying to moralize and claim that physicists are wrong about some basic opinions on the Standard Model and Beyond the Standard Model physics. But you think that gauge theory was written down by Young and Mills.<br /><br />Try to be a bit sensible. How plausible is such an assertion that a person who can't distinguish Yang (Chinese) from Young (English) has something relevant to say to experts about cutting-edge particle physics?<br /><br />It's about as plausible as the assumption that the people who write about Einstien revolutionize relativity or those who refer to Hawkins revolutionize black holes. You seem to be like a classic crank. Have you ever calculated your Baez crackpot score?Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-3832136017893749497.post-63437081455628441622016-07-16T08:27:50.500-04:002016-07-16T08:27:50.500-04:00I think,it should be Yang-Mills, not Young-Mills i...I think,it should be Yang-Mills, not Young-Mills if you are talking about the famous field theorists.kashyap vasavadahttps://www.blogger.com/profile/10732897306667764590noreply@blogger.com