Where does quantum field theory come from?
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Where does quantum field theory come from?

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12

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2013

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English

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Abstract 
This will be an interim report on a longrunning project to construct a mechanism that produces spacetime quantum field theory; to indentify possible exotic, noncanonical low energy phenomena in SU(2) and SU(3) gauge theories produced by this mechanism; and to calculate signals of these phenomena to see if they can be used to check whether the proposed mechanism operates in the real world. The last effort is still ongoing.

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03:01
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20:32
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30:55
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58:48
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1:00:56
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00:01
and I was a time the and in a I'm going to be talking about the project to which has been going on for a ridiculously long time I should mention it makes use of discoveries by many of the people some from more in the room and at appropriate times all tried Hanshin moment but if I forget that I think it will be obvious OK so the the sketching out a proposal a proposed mechanism that by producers quantum field theory it's based on a real quantum field theory a measure on spacetime Shields to which is equivalent to Hambletonian alone and at the same time produces the the client string background and I am reasonably confident that that's the correct idea of quantum string background as things stand at a purely formal question whether this is how actual quantum field theory is produced by which I mean the 1 that describes the real world that I don't know of but other than that I would like to have know so at the moment I'm trying to derived From this proposal they predictions of 4 unexpected lowemission phenomena all what actually the scene we're not seen experiments the and the more explicitly that the possibility that this mechanism produces some what will the the bizarre lowenergy degrees of freedom whenever there is as you to gage invariant Bloomberg and these will be associated with well known nontrivial homotopy groups of the space this you to gage fields on you couldn't spacetime but still a on this is
03:04
I should put in this can be added this is extremely speculative fundamental physics it was a long shot but I like that where it comes from when the British were shooting at the French Open of alright that goes back as I said a very long time the beginnings of so it goes back to World twodimensional of generalize nonlinear model where alone during twodimensional quantum chill theory that the still lies in some manifold and in the action yeah food given by way of what mathematicians call the energy function at all the parameters in the action or comprised in a money metric on the target manifold and this is real normalized so they're infinitely many coupling constants which you can think of as the Taylor series of the metric at some point action not in there then you get the metric its 1st derivative 2nd ad infinitum infinitely many couplings but they are already normalizable they the renormalization can be done covariant live with respect to the target manifold and renormalization group equation has an expansion in sort of the inverse size of Cordon manifold which starts with the Ricci so this work was inspired by such a player costs the discovery that the twodimensional nonlinear sigma model that there Israeli normalizable were the target manifold there is to see With all 3 symmetry the round to secure so the only coupling constant is the inverse size of the 2 sphere and Sasha discovered that that wasn't hasn't publicly frequently comes so this can be thought of as a generalization of that so what the renormalization was that I was a graduate student at the time and can Wilson's you of the renormalization group who once How I Learned renormalization and 1 thinks of the renormalization group as a machine that acts on quantum field theories in this case 2 dimensional quantum field theories and drives them 2 a drives them to something new at larger and larger distances so here the the twodimensional renormalization group so and then it's doing this at very short distances it drives this metric on the manifold hordes of 6 points which would be the solution of reach intensity equals 0 so it is in the twodimensional renormalization group isn't a mechanism that producers what looks like a classical field theory not by mechanism I'm not talking about a mechanism that acts in real time it's an abstract mechanisms that so actually OK so at the time that was extremely excited and so I think only of it seemed to me that it it could be a clue to where the laws of physics come it looked like that Einstein's equations of not exactly but it looked like object so the questions that that that arose work 1st of all it is important integration it's a field equation but not a physical 1 so the question was of his there's something related that would produce From a realistic feel theory by which I mean it should come from an action principle and it should have realistic collection of feels like just a metric on space time the forgot to mention that for this picture 1 takes the target for them to be spaced so it should have gage fields firmly on fields scalar field then there is the question I was a bit sloppy when I said that the renormalization group drives you to a fixed point will it only does that it that 6 point is an attractive if that fits 1 has unstable directions then you have to do some training to get driven there and at the time but I didn't see that you can eliminate of all but a finite number of directions but I couldn't see how to get rid of the weight of the possibility of a finite number of unstable directions so how do you get stability in this sort of thing and then the big question and this is what this producers as a classical field theory where is quantum field theory come from what makes quantum field so within a few years the 1st 2 were injured when this twodimensional field theory was interpreted as the string background so the idea is that the world surface of the string propagating in space is described by a twodimensional field theory and of the spacetime has a curved metric that twodimensional field theory will be this twodimensional nonlinear model and the condition that the beta function vanished that you bear a fixed point is a consistency conditions to get the virus sorrow algebra and there by a Unitarian S matrix from this the twodimensional construction of the string as metrics for cases so that I I possibly mention of how the names involved in that picture of string theory but I would like to mention my late colleague Claude Loveless who amid the really remarkable discovery that string theory of To be consistent requires the spacetime dimensions to be a certain number 26 for the books on so that was I think the 1st example of the properties of space time being determined by consistency conditions of OK now it was so it was realized that this condition baby equals 0 but but I would do the work of physical of field equation coming from action principle if you added In addition to the target metric additional couplings In the twodimensional theory this was frightened said who introduced but I think it's much called dilettante handcuffed and then if 1 goes to the thermal comic strip when you add pharaonic twodimensional degrees of freedom to get twodimensional supersymmetry you get more we normalizable couplings in this twodimensional 30 besides the metric and the deal
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upon you can Our cook things up so you get spacetime gage fields family and fields Steelers and of course 2 of the main people responsible for the fund managers in the the 100 John In the family on extreme the twodimensional version of stability turned out to be basically given by the the GSO projection of which gets rid of tacky correspond to relevant directions is under the renormalization group so you're getting rid of the the main source of real directions and then on other complications which I did .period took don't do reach a cancer can have marginal directions of the can have a manifold of solutions reach equals 0 the higherorder terms in the function curvature squared and so on can produce nontrivial flow on that manifold of of of sort of solutions of reach equals 0 and that can produce instabilities so there we but there was a remarkable discovery by but they are on average Goldman Friedman that thought at higher orders the if the tournament theory has added additional supersymmetry alone this higher terms canceled when they discovered that the SEC to term canceled and then later was sent to all orders OK so now we have additional questions doesn't seem right for the during backgrounds should be given by a classical field you would think that this string backroom but that those the environment in which strings scatter should be a quantum states all of them 1 of the the laboratory where you're doing the same experiment for a hypothetical strings shot so the question is not all of the additional question is what it what would what is this what is the quantum string back and then is there a mechanism that produces a quantum field theory and space time and in such a rate that back at the end of the classical background the biggest that the renormalization group for the twoday renormalization group flow is producing a solution that equals 0 at producing a string Bakr the question is is there a generalization that would produce for quantum streamed you're going to creations of all of us all these things in the all In the beginning of the year In space but the question is the only way to people look at this maybe this is the case of Europe from the installation of all is the only to that's sort of the motivation for this questions 1 would like to go beyond that 0 OK I have I don't know this is probably personal I feel that the urgent need 4 but up a mechanism that produces quantum field theory that there are too many selfconsistent quantum field theories if is what were to march in the position of sort of groping in this huge space a possible quantum field theories the world is described by 1 so a mechanism that produces quantum field theory seems like it might give some explanatory power beyond their binges just quantum field theory so after quite a while I came up with a proposal all so abstract a little bit right when the idea for the twodimensional coupling constants so that when the or the more loads of whatever spacetime fields the described the twodimensional filter my mojo Mina about momentum modes in this time and then let alone and what what's it called can become talent in the text of the manifold of spacetime shields so the land or clean on this matter equivalently we can think of it as since this is the most general quantum field theory in the summer neighborhood of a certain class of quantum field theories we can think of them as the manifold of Trudy quantum field theory so to be slightly more concrete the show and allied couples to some true do field which abstractly alright far but think of it as say vertex operators in but a guy I'm always taking space time to be cleared him too and leaving with quotation for some future date so this 1 and I have spelling dimensions the 1st turn in the beta function which are of the form spacetime momentum squared minus slightly negative where amusing dimensional units for space time distance and these are all going to be very small it in interventionalist units grown on only interested in very long distance physics and dimensional scientists so In the language of the renormalization group views coupling constants are slightly irrelevant they have slightly negative dimension so under the flow you're driven but they're driven to 0 under the normalization group that's the idea of stability no let them become sources In 2 dimensions so let the coupling constants vary over the surface and then 2nd fluctuating make them into twodimensional fields with natural action so this is a the sort of men of nonlinear model where the target space is that curling em the space of spacetime fields land eyes coordinated it said the action is given by the natural metric on the space of spacetime fields roughly the at the Yale 2 metric modular vision variants which is In the limit where where the target manifold is big is the natural metric on the space of 2 D quantum field theories as this the sessions on logic craft 4 and then but little GE would be the the coupling constant in spacetime string company so here on the right we have the coupling of the sources through the local twodimensional fields and a in middle
20:34
there we have the exponential of the actions of this medal nonlinear model OK so In this functional integral over 1 of our sources I want to do the twodimensional renormalization and Wilsonian cents and want to integrate out fluctuations of the land II at small threedimensional distances less than capital below a lender inverse so when we integrate out fluctuations we're going to be producing insertions of the 5 In the surface smeared by by some functions of 0 the following local at that hour are up over distances less than this distance from Denver OK and this action was designed so that these insertions of the file I would be this as the insertions produced by tightening handles attached to the surface what would be produced by string theory plant and corrections but only the quantum corrections that are local intruder mentions where the ends of the handle or close together at this twodimensional distance scale so this when I see the land the model acts within the UN the twodimensional renormalization group in the land the model the to produced on effective these insertions of the firehouse produces only about a minute twodimensional the authority but with but a cutoff this this scale bigger landed in and this is what I claim is the quantum string back this twodimensional feel cut off at this scale inverse so what does the land model is doing at smaller distance scales it is calculating the effects that of froth of the froth of small handles would have produced not we will only understand it even in principle had to calculate the effects of those small handles put readable but this twodimensional field theory at a sensible twodimensional feel Theriot said it it's not perturbed so this is perturbed bit of construction of the quantum corrections to the world's surface at short twodimensional distances In the crucial design principle here that this limitation through handles that the act locally on the surface is twodimensional locality so that at twodimensional distances larger than this cut off we have a local twodimensional field theory so string theory calculations of the string S matrix 0 0 will work and they depend on integrating correlation functions that are not on singular thought except when vertex operators chloride so here this this effective through the field theory will will have the property of the distances larger then this cut off OK now at the same time on Malaysia on the land is being produced so before which allowed the land eyes to fluctuate locally you would have some value for in some quantum field theory at very short distance and it would evolve under the renormalization group along a trajectory so parents In in curly we evolve to different points in curly and the bigger function is that the renormalization group is a flow In the space of quantum field but now the land a fluctuating so when you integrate out the short distance fluctuations you might start with a point some specific treaty ,comma feel 3rd but you immediately get a measure on introduce quantum field theory as youth as as you evolve and the 2 renormalization group of the land the model drives that measured to limiting equilibrium measure that's the generalization of driving to a fixed point so the picture as well 2 studies what's going on take some observable adds in this theory that a function on this on the spacetime fields function on the 2 d coupling constants functional news Curly and on and on the change of style well the coupling constants will evolve in the better function but there are also be Norris and you you you can see this as expanding landed in in the the soft modes plus the fluctuations and you'll get a twopoint function of the fluctuations that producers to derivatives of the and shortdistance correlation that will go as the inverse of the coupling in them action so there then it if you just look at the evolution of the dual it evolves as they are a random walk in the space of quantum field theory is driven by data so Europe which was the picture I I tried to give up In the previous life so you're being driven to the fixed .period but there's no lies so you windup that the asymptotic equilibrium is a measure concentrated near the fixedpoint bought with some of the some distribution now it's a bit of function was 0 it would just be random walk over random walk on the manifold city had been driven to the metric volume element on the space of quantum field the on the space of spacetime fields if the beta function as the gradient of some function asks on spacetime field than the equilibrium measure will be the cover the vote netted volume element .period you the minus apps so 1 that "quotation mark filtering in spacetime as equilibrium measure under the
29:18
flow another way to to picture this production of a measure is in the radio conversation so then you have loops in the target manifold or evolving under dilation twodimensional dilation the nonzero modes I strongly suppressed so the ground state wave function will be concentrated on the client's near the constant loop so you know that so the ground state wave function under the radio conversation will its evolution will be just the sensualist Acosta colonization of of the of the spacetime .period feel theory it's just not what would be driven under the gradient of the spacetime action With and then the evolution of the I'm sorry that the distribution around the constant mold would be governed by the target metric you have some soon but way function around the the constant modes 0 0 plus corrections and that contains all the twodimensional physics all right
30:57
now what's the role of this cut off all of them in the effective twodimensional surface this OK so we don't think there was always just uses coordinated and takes absolute 0 minus 0 minors into such such qualities I where I was think in terms of morphine physical twodimensional so I'm Rajinder or twodimensional quantum field theory with which has couplings land it renormalised at some twodimensional distance Nunavut and or cut off his the next much smaller then the scale at which the 2 dimensional quantum field theory Israel minimalist so this number big landed 20 new inverse is very large OK so we know where we have a twodimensional theory renormalised :colon macroscopic scale distance film universe and we're during these fluctuations in the coupling constant at a very tiny twodimensional scale the Glendening OK so when you have an irrelevant operator but only relevant coupling constant the gift facts where we want to use them at scale new inverse analysts well are suppressed by a power of the differences scale with the with the explanation is the dimension of the coupling constant so there is the dimension of the coupling constant bears the racial scales and I write it in this form you don't mind elsewhere times the spacetime momentum swing where elsewhere the logarithm of the ratio of twodimensional said so but what we see is the hope that the copilot but the William modes of the spacetime fields and hiring tho all are cut off so we can simply they are they are really irrelevant the ones it's monumental are almost marginal but the ones with large momentum in space time or are genuinely irrelevant and we can disregard them they have no effect the corresponding institutions had no effect on the the macroscopic twodimensional physics that we're going to make use of its chief restraint as major calculation so this space distance he is also an effective ultraviolet cut off in space which renders the target manifold effect all of this 1 the model effectively finite dimensional so it really is a sensible twodimensional so we have a correspondence between the twodimensional distance scale that which which divides where the land the model fluctuations take place from the effective the world surface that corresponds to a space distance but now OK usually as a senator beginning we just had no direct weekend new equals 1 and then this Land inverse would be a dimension was number which would cut off the integral over marginalized in string theory calculations but I prefer this more physical feeling interpretation now when we're doing string theory calculations right all and and have a twodimensional in the effective World Service we have this becomes but the ultraviolet twodimensional trouble and when we integrate all over the marginalized for a troupe in the world surface we lined up with the propagator at that is not 1 over Peace Square .period for each mode but has a contribution a cut off from ultraviolet cut off into dimensions so you notice that for for large piece where this is a lot the number in parenthesis is very large so of a large piece where we just the 1 repeat square but for small piece squared the poll it is eliminated so we have in the string theory in this effect the world surface we have a cut off in the space infrared so this effective string as matrix is only describe scattering at spacetime distances up to the and the quantum field theory that the lender model has only land allies wave modes of the spacetime fields at distance is larger than so the the the the the quantum field theory that produced describes physics at distances larger than old and the string scattering matrix is at distance is smaller than in space and all that choice of the Glenda was arbitrary so we can vary L as well as long as it's large and dimension units and this construction of what I suggest that the fact that both at the start the the World surface and the special employment field there are constructed by this twodimensional model guarantees that these will be consistent so if you choose L large and described some physics by a string as major or to its smaller and described the same physics by quantum field theory they should agree so I think this is the only real estate version of the string background but where you have a quantummechanical description of the apparatus in which scattering pieces experiments take place of that's what we ought to have before going to do a physical strength alone and on a philosophical level it seems like a practical theory the the religious S matrix theory but where everything is mess and that but I think that the idealization that doesn't have anything to do with how we really have don't visit we described all that we say "quotation mark mechanics in particularly describe certain by quantum mechanics but on the other hand the notion that we have climbed fuel theory all the way down indefinitely In Distance Batson seems also unnecessary idealization up and finally in this To presentation of the proposal at point guard and intriguing properties the the twodimensional of physics this tridimensional renormalization group in the landmark that's operating from small twodimensional distances to larger Bextra Wilson taught us to think of renormalization but I guess it goes back to the beginnings of a normalization the picture but given this correspondence between twodimensional distance in spacetime distanced the
39:24
quantum field theory and spacetime is being built from the largest distances the downward which is very unlike picture of effective spacetime .period feel I find that intriguing of thank it also might have end up having some explanatory power broken so at this point given a proposal the proposal offers endless number of formal internal questions I know described strings Scott those things like the reduction some analog of the reduction formula described string scattering as well but in in in a state of the quantum field theory and that's just 1 of them the problem but I am on 1 of this is writers along so I was on but I would like it to say something that can be checked by against experiment and no 1 could fantasize that this machine you know you crank it in the end and and pay careful attention to what it does and you will end up with the standard model but given the the history of that fantasy over the last couple decades I think it's not I'm not going to happen in my lifetime N and I really want to find this out in my life 0 and I don't see much possibility of verifying strengthen experiment with so I think I took another tack I said OK let's suppose that it does produce but the only way it could be right is if it does somehow that under some circumstances produced the standard model so let's say it does produce something roughly like the Standard Model something that could include the standard model could be that it produces something else something exotic that you would not expect from the canonical condensation of the standard model OK now this slender model is a true dimensional field theory it is against Acosta colonization of the classical field theory action it's a twodimensional feel so could there be no and prepare a bid of Trudy effects that would show up in this measure it produces on spacetime fields on OK so what remind you that all concerned that given this assumption that we've been we've produced a spacetime field theory that that includes the standard model of curly and is manifold of of whatever the spacetime fields or the gage fields of the standard model of spacetime metric family on fields some Scala's whatever N the space where we're going to do experiments which bar for less than universal yeah I don't know I just mean that the victims all connections in MSU to bundle of our foreign right but that's what metrics to show it a quantum field theory cut off quantum field theory of the metric is gonna look like classical field to us so that's not a particularly interesting you can even match accounted for the yeah but it's also a somewhat and its place in the well not I'm in fourdimensional few of so that any and everything I'm assuming it's happening that we coupling so so not perturb of effects would be sent a classical effects and in twodimensional feel 3rd Senate classical effects the ones that I know about are given by winding modes which come from nontrivial loops in the target manifold a twodimensional instant which comes from CPI pie 2 of the manifold so I guess this is the 1st to be more mentions of instant but 2 of the inventors of bad 2 of the discovers that Sasha and Sasha political often below the ownership of some of so the mathematicians have long known that I had that that they're all art nontrivial homotopy groups in the space of gage fields on our for all and when I say the space of gage field I mean of course the physical space gage field gage feels gage transformation so Taiwan for S U 2 gage fields there's 1 nontrivial blue in the space of gage fields that 1st came into physics I believe with witnessed the discussion of global anomalies in this year to and power 1 of the space of S U 3 days all the other player once vanish so the only interesting 1 this 1 winding road that might be interesting and that stress due to gage fields which of course we should have applied to lose are always nonzero there is 1 nontrivial to see friends you to gage fields and those of whole integer worth of such to spheres for as you 3 so winding awards In a twodimensional filtering given local fields so those you would think would correspond to new degrees of freedom In the spacetime theory so the question is are there lower these these lowenergy degrees of freedom Of course it would be not canonical coz they have nothing to do with the the spacetime Classical Action and and then this 2 serious this would give 20 instant ponds N 1 might expect they would give noncanonical couplings in attitude in the field but I find that the 1st question but more interested so I haven't done much With this OK but I I wanted to know what those loops look like right because if you're doing a Quantum in the UN nonlinear model with the spaces target manifold vote vote the winding road would be given by the middle of the shortest loop you can and if that Lopez finite length it's going to be a highenergy but massive excitation and we can forget about it the only possibility is is that minimal if there's a loop of of 0 length that's topologically nontrivial so I I I got I I investigated this in a very roundabout way and it turns out after I replied reported my results to the mathematicians that thing they basically knew about it but I had to find out for myself and I didn't really look around like that that's it's a somewhat long story what will what I found was that the nontrivial who in the space of this to gage fields is a loop of of gage fields which consist of an innocent time and airplanes and parts of these a fourdimensional incident wants the original incident time alone and the loan
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all of and then to fresh due to gage fields you need to win sometimes and to sometimes glued together and then you modify the glowing parameters and you find a nontrivial to severe and for pipe to of S U 3 it's again and some time and patience and compare but the interesting thing is that I the minimal long has 0 length and the minimal Troussier has 0 area so there's a chance of lowenergy physics as I said I find the check the possibility of noncanonical degrees of freedom given that mysteries we haven't In physics that that's tantalizing surpassed the 1 I've been thinking about OK so under which tend to have a reliable the continuance of the trial OK so here's the construction of this nontrivial you take an instant time on our 4 and then you blew in a very small airplanes so they plastered it is the instant plans and it has sides role plus and it has a center X plus 2 x classes in our and this and that's and there is a possibility of a all of the rotation in internal rotation in to but you fix that by a by global gage and then you can entertain kind of very small size remind centered at another point X minus and you rotate back by an element of this you to acting in joint and then you blew it To the incident power well the possible ways you can rotate that airplanes and Parliament are given by S U 2 acting in the adjoint so the parameter space is as you to modest the center plus minus 1 2 SO 3 and that has a nontrivial opener time Warner vessel 3 is too and that little in this family of gage fields is a representative of the nontrivial in the space gage field government will ensure that very explicitly volatility in the 2nd half of the year was not yet you you you you know you can explicitly show that jet around the loot you go around the loop and you get a gage transformations that is explicitly the transmission that cannot be contracted to the identity of the physical fitness of his days of the year .period as companies in the event of the year from constructing a little in that substation in verifying that that Lopez nontrivial indispensable auditory utilities will be forwarded to the views of the Missionaries of the bid comes only 2 days it was made by 1 of the things the good glowing in if to group but this is this has topological number 0 this is in them the for it so I want to but what we are doing we need to have the matter we want find out what's the length of this nontrivial who so again this is From the mathematicians we so we parameterize that plain simple that we're doing 1 not looking for leave the location fixed we parameterize the size and and the internal rotation it is necessary to buy an element in in situ so we take a reference point in C 2 replied the S U 2 rotation to it and we get element Institute the standard parameterization unless you to buy sea to OK and then we multiplied but it's also other Tusa Thresa in this incident and then we we take the radius To be remind so we get an element in the minor since too which parameterize is the size and yes you to rotation OK and then we calculate the metric take small perturbations the deal took might by gage transformations to fail to win product and the metric is smooth in this parameterization at the origin cat in its instead that the actual metric has a the instant and modular space has nice properties but here we we we we only talking about near the origin where the incident on the contents and 20 couples from the incident on at when real minus is finite you have to glue them together not in the space of self dual you don't have been what nice mathematical problem but anyway the metric remember has a oneover over coupling constant square and then it's just nuclear geometric in this Spanish city to end the nontrivial loop it backs look in Shimane switched book alone in the 3 severe right over the is only find up to plus or minus 1 because GM minuses only defined the proposed so we're really looking at the to Margita OK so we're really looking at another trip the nontrivial loop from the North Pole of S 3 to the South Pole of S 3 for every any given size Romanesque if we take the minus 2 0 the length of that that all art that that that longitudinal like goes to 0 so the minimal nontrivial because his 0 length OK so In the land model the vote the winding mold for such a little is just the twist field for this wonderful OK and we did this for fixed values of EC of the location of the incident on Monday and paints parts and 4 fixed sides parameter for the incident on so their collective quarter and now With again given this assumption that we somehow got out of a sort of field theory that includes the standard model will have their eye on field and those coupled to the issue to gage field and they'll have 0 modes localized in the incident on an Indian planes Parliament and it's it's it's very pretty you can actually show hello as you move in this nontrivial the 0 more Indian planes and that the idea of changes and not too explicit realization of the this issue to global OK but when we do on on a folding with projectile although to date twodimensional degrees of freedom that under the Zito so we project out the 0 modes in the plants and Poland this tiny and we're land will leave this 0 modes in the sometime they summer OK so it looks like there are possibilities from foreign interest in quantum numbers in the subject but at this
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point I want to emphasize that it's a bizarre looking object it's by local it depends on 2 points in space time X plus and minus and then this additional parameter worldclass there's an analogous object the CPA transformed with which you have empire in time with tiny sometime added anyway it's looks very bizarre on right and I this but it but in this all folding right we will eliminate the the the degrees of freedom the odd under the Zito idiots like that or to make sense of of classic of a field theories which have global issue to anomalous because this gets rid of the global issue to anomaly but we leave that OK Will you have and what we need to understand the flow in this neighborhood and 1 actually calculated the Yang Mills action of which the flows the gradient American integral to the details but you find that there is 1 unstable direction so there's a trajectory right you have a fixed point II normalizing those actions of the instant time has action 1 so we have an instant prominent and found that it's too plus something proportional to the size of we became 1 of at least 1 of which is small end of the most of the form that is determined by by the conformal invariance in or for bronze but if you study the behavior Nearer to You have near the 6 . is when when when the the little and pension fund has 0 saw better fix .period pedestal a genuinely self dual fuel is studied the behavior knew that you find 1 unstable direction so there's a flow that leaves from near that this 6 .period this nontrivial fixed point and flows down to the verge of flat "quotation mark connection and an interesting things happen when both In sometimes become small but the flow become sort of like the asymptotic free you don't have actual instability but just marginal stability so at this
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point of my thinking is still quite model the question is how of various degrees of freedom will contribute to the spacetime physics if they do so my room my best understanding of how to proceed is to introduce more coupling constants that couple through this winding road and then study that dynamics mean that no field 1 can fantasize all sorts of things a condensation England results but in order to study the dynamics of the scuttling 1 has to know the beta function of these new degrees of freedom so has stopped understand how products of the twist field an ordinary twodimensional fields evolve under twodimensional stale and in particular 0 length for a for a Theory of Justice guarantees that the energy will be 0 classically but we know From ordinary or befalls that the twist field the scaling dimension of the twist field its quantum corrections but immediately that at 1st but 1 low and the question is they would rule on the side idea of lowenergy excitation so that has to be checked and then this it seems to me the basic object that is this unstable trajectory from the instant patience and Parliament Taylor down to the flat connection but I have yet to work that out the question is the fact
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that the model sections of the model on other parts of the
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region without a whole is normal or I mean you can think of of wealthy you can think of it that way I think it is yeah but but but but alone it seems to me to analyze the structure it's important to understand that the the coupling slammed which in this picture on the field and space time that they have a nontrivial Renong was different dimension that they have dimensions that come from quantum corrections in the original nonlinear model and 1 1 that structure of the land of models and is based on on properties that quantum properties of that twodimensional of the original twodimensional feel so I don't think it's and it's all that useful to study there are 2 quantities and simultaneously the picture presented his 1st few quantized the original model and then you add fluctuations of its quantum coupling constants I mean it might be that there is some fruitful way to do it all at once but I I and hasn't seen like that of the of the last movie theaters the 1st was the reason he was known in the that that interesting yeah I I know I have no idea how to but that the interest so you get an effective theory just yet that might be that might Of course this is maybe thinking to the previous question with so and if this phase of all the people of the theories and use it to the end of the year is also held recently that underlies the point of entries in the top it's a it's in is that it's not all altered Yukon filters at a special class all I I forgot 1 thing I really want to set back when I in when I started talking about string theory that that this is where the American National Can I intersected we wish we were both interested back in the mid80's in finishing the covariance condensation of the firm and extra using such a player costs super diffuse morphism Trudy ghost fields yes I definitely wanted to mention that at my motivation was this need a covariant twodimensional filters but I'm so it's a special class of 2 decline field theories the original tradition of theirs and the lender model is not a globally defined to do feel theory it's only operating at short threedimensional distances so I don't think that the bill the a picture of him as you is the 1st said increasing the use of the substance of the meeting of the you I want you to know that you have with he said he was well I mean you hear OK to have this parameter space which is the size of the tiny and plants and and then 3 sphere of orientations unless you too 1 moment logic too but the thing of it is a threeship care and the nontrivial goes from the north to to the South Pole of that tradition OK To take the size to and 9 0 you we really well what would you can think of this as describing you have all the other modes of the gage field all the other ways that that the irrelevant directions you can go away from this family and what this is describing is a cycle In some it so there are not any nontrivial will any looked at the that surrounds this cycle will be nontrivial but for the twodimensional physics I'm interested in what the 1 that tracks down to 0 in the 5th and of likelihood the result of this have analyzing give it the other way around need those exotic to Israel's because they're in the land of model I mean who could go you could go and study them but I wouldn't have the motivation