David Gross: Millennium Prize Problem: Yang Mills Theory

okay so welcome to the grand finale the final lecture in the series on the millennium prize problems and we are very grateful for kitv4 hosting this event at KITV our speaker today really needs no introduction greatly honored and delighted to have Nobel laureate former director KTP professor David gross who's going to tell us about his perspective on the millennium problem Yaya's theory well when I was asked to do this it was what a year ago or something it was one of these invitations so far in the future you'd say sure and then a few days ago I started to think what the hell am I going to tell these mathematicians about quantum field theory and so I wrote a talk and then I tore it up another one last night was and since it's a math talk I'll use a blackboard and since it's at the KTP if there any questions stop me and ask them so this is I guess you've already solved the other six problems and and today's subject is probably the hardest or at least that's what edward witten tells me of the seven Millennium Prizes now Arthur Jaffe was an old friend and colleague of mine when he started putting together the clay foundation before he was he he had a brilliant idea of how to get a lot of publicity as you know for the clay Institute I think it's called by formulating prizes that would be announced at the beginning of the millennium then he would you know the clay foundation you get a lot of publicity and aroused a lot of interest in mathematics and maybe even in these problems but not have to pay any money and so far they've been very successful they've got millions of dollars worth of publicity and only had to pay out one price that somebody who maybe they even knew wouldn't accept it so it's cost them nothing but it clearly has aroused a lot of interest and to win the prizes remarkably I mean some of us physicists were surprised that two of the seven prizes were in physics one of them being as you know like navier-stokes equations seven I gather seven four different formulations of that prize and that I certainly hasn't been solved yet and then a prize for the proof of the existence and a mass cap in a non abelian yang-mills theory or which is no surprise given that Arthur Jaffe was one of the the presidents of clay and stood at the time and Edward Witten on their board this you know is a prize that a problem that is even hard to formulate I mean you could state what they said which I think I don't even have it's roughly to prove the existence of a knows theory in I've been engaged theory and four dimensions for an arbitrary age group and and to prove the existence of a mass gap but even to explain what that prize is takes more than a one lecture and probably a full course and I'm going to give a very limited to really explain even what the problem is can't really be adequately done in in an hour or a semester but I'll try to give some perspective on it but I would like to say a few words just from the point of view of a physicist who was educated in a period where mathematics were a lot of these ideas that form the basis for this prize were being developed but the attitude towards mathematics was quite different one of my mentors Murph Goldberger at Princeton wrote in his famous book he wrote on dispersion relations which were an alternative and even more opaque approach to problems now adjust by quantum field theory he wrote that mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes which by and large is the Attic to what or at least used to be there to the most physicists and it did allow us to get awfully far on rather flimsy foundations foundations are still pretty flimsy and that's since the essence of this problem the problem is asking for lets you actually have the formulation upon someplace you've included it in the abstract but roughly speaking in words it's the existence in the sense of quantum field theory of Mills theory in four dimensions or dimensions with a gauge group G an arbitrary semi simple in each group g plus the finite mass cap now ditch theories are examples of quantum field theories the ones we use to describe nature of aya the standard model and four dimensions have never been proven to exist in fact I'm not even sure at nontrivial except for very special cases of enormous symmetry quantum field theories have been of any kind have been proven to exist so this is not like a problem that you know a more complicated problem of where there are examples of what an existence proof would he would consist of and the attempts in the past I must say have coincided with another quotation of my mentor Goldberger where he discussed proofs of dispersion theory which he said are like man's teats neither useful nor rental and by and large that has always been my attitude towards proof of the existence of the quantum field theory if you actually look at the literature of what has been proven for rather trivial quantum field theories we certainly don't have the richness of this that allows for the second step they are ugly and useless and well so many of my colleagues I must admit don't aren't that excited about this clay prize yeah no now because you're not you're not really sure what it is you're talking about besides which existence is a little well if you don't worry about all those details details which shall we regard it's sort of essential for describing the real world and our then and it's easy so it's trivial yeah I mean you know the mathematics is neither beautiful nor useful I mean none of the techniques I mean a lot of people have done an incredible amount of work on trying to prove the existence and I'm not gonna say too much about this since I find it so urgent myself but but I will say a bit and it just the the reason this prize has formulated is I think of great interest after putting it down I'll say why it is of interest is twofold one of these these theories that we're talking about are you know the theory the standard model of the theory of elementary particles well tested and it's in some sense the best physical most complete ambitious successful theory we've ever had we'd like to know that it's based on solid foundations that's one but we sort of know it is I mean physicists do have ways of discovering truth which isn't they you know as I said in my first quote which isn't guided solely by mathematical rigor if first part of this problem it's hard to know how this would fail since it's hard to know what it would mean to prove the non-existence of quantum field theory you could not succeed in proving it as people have for many years yes and that indeed is one of the issues and and I'll say a word about that but not in great detail because this involves much too much background but it would be possible to prove the existence of of say the continuum limit of a well-defined way of defining yang-mills theory and and and to show that it not have a finite mass scale so it would be possible to come up with a negative result here for the combined problem and that of course would shock us all since we know it's false but our evidence is based on a way of thinking that many mathematicians find difficult to absorb and when I started to think about how to give a lecture like this I went back and looked at the attempt that had Witten and others at the Institute did an experiment ten years ago in Princeton to try to educate to try to teach a bunch of mathematicians quantum field theory and string theory eighteen years ago actually just before I left Princeton I even lectured in that only normalization group and and the students like David cosgrove who took my class rewrote the lecture notes and I couldn't read them understand after it's impossible more or less anyway it was not greatly successful because you know we think in a way that's based on you know sort of walking on a rope over a precipice without looking down and I get pretty far that way because of the successful history of physics the fact that we have nature to guide us so we sort of know that these things are correct for sure based on lots of internal consistency analog examples in other simpler cases in physics and an experiment okay that's why it still might be interesting so if one were to find a counter of profile at mass gap here it would be a total shock and you know I'll take strong bets against it a million dollars I but but more interesting I think is the feeling that if the mathematics could be developed to answer such questions that given the beauty of the theory as physicists cieth those methods very powerful enough to answer these questions but have to be both beautiful and useful but it's clear were for nobody has really a good ideas as far as I couldn't see or the mathematical tools and so the prize is might be useful in and in pushing mathematicians to develop those tools and they probably will be very different than than anything that's been tried so far so so I have to tell you a bit about gauge theories and the 1 gauge theory I'm sure you know about is electromagnetism which was the first field theory in which there were dynamical objects that were functions if you want of space-time and could be measured could influence other object and Maxwell's theory of electricity magnetism is based on what we call a gauge theory for historical reasons non-abelian an abelian gauge theory based on the gauge group u1 it's dynamical field is a connection on a u1 bundle over Minkowski space so space and time three dimensional four dimensional space with Lorentzian signature and the physical observables are the field strengths so this is a one form and the field strength which is just da is to form whose components are the familiar electric fields and magnetic fields so Maxwell unified electricity magnetism into this theory whose equations of motion are simple by definition second-order PDEs which follow from an action principle from which is just the integral over space of the of this curvature of this connection squared or now Maxwell's equations were the first field theory invented and that concept of having dynamics described by a function of space and time like a magnetic or electric field has dominated physics ever since it was also the first theory his equations of motion was action is invariant under Lorentz transformations so3 one the eyes AMA tree of this flat this Minkowski space with flat metric it these linear equations have wave solutions which are Maxwell discovered are just light the waves of ripples and this field strength governed by Maxwell's equations so in the absence of sources charges Maxwell's equations in the vacuum are very simple easily soluble described not linear waves that pass through each other don't interact traveling at the speed of light and all the theories that will discuss hearing those theories are generalizations of that so this is a dynamical variable this connection field strength the in this action there is a constant here which plays no role classically since it doesn't affect the equations of motion but is known as the electric charge and if we introduce objects that transform nontrivial under this age group carry electric charge they will be related to the electric charge of that field but we're just going to discuss back in solution and classically this charge is irrelevant now at the classical level the theory is very simple linear equations very symmetric it actually has a bigger symmetry group than Lorentz transformations because of the lack of any scale explicit length scale here so since we're going to be talking about masses and couplings we say a word about units physics you always need to specify the units of the physical objects you're talking about so objects that can be measured you measure things you measure you compare the some meter stick you measure length you measure time some counts of some clock beats and you measure energy or mass so MKS metre kilogram second CGS that are you need in physics always to specify the units we do so of course with answer amorphic units but in fundamental physics we tend to use units in which at least two of the basic dimensional constants of nature appear one is the velocity of light and the other is Planck's back which has units of mass times length squared divided by time and we can define units in which these are 1 in regard these as units of time and of get rid of two of these fundamental units express everything all velocities are fractions of the velocity of light which is 1 in these units all actions mass times velocity times lens are also Planck expressed in units of Planck's constant which plays a fundamental role in quantum mechanics sometimes we complete this with another unit but we won't hear the natural unit be Newton's constant which defines the unit of mass not going to be discussing gravity at all so having using units like this it means that everything is expressible as a length okay time is a length divided by the velocity of light so now it's easy to see that the correct dimension for the gauge field is one over length and for the electric field or the magnetic field which has an extra derivative one over length squared and so you see F F has dimensions of one over length and fourth volume integral has dimensions of length to the fourth and therefore this parameter here of charge is dimensions in these units the way we normally say it is that the electric charge squared over H bar see Planck's constant divided by velocity of light is dimensionless so e squared has dimensions of h-bar see but setting them to one and is a number as usually as defined physically 101 over hundred thirty seven point zero something zero three five so it's a small number and it's dimensionless and so you see in this theory there are no parameters which have dimensions of length under a scale of all lengths the volume increases like the scale factor to the fourth power both time scales like length so on and so this scale invariant under rescaling of all length all times all energies and E is dimensionless it's just a pure number in these units so this theory is scale and very in fact it's more it's conformally invariant full symmetry of this action is so4 comma 2 which is arrange transformations plus scaling of position plus special conformal transformations inversions so that's for electrodynamics in the vacuum which is very trivial it's even trivial to quantize this theory this is essentially a theory of not interacting infinite the new mobile number of non interacting harmonic oscillators and and before a transforming and can be easily couldn't easily construct a Hilbert space that describes the quantum mechanics of long en M in the vacuum now and this was the first quantum field theory that was attacked after the birth of quantum mechanics electrodynamics existed very successful classically but it obviously had to be quantized and was already in a sense quantized by by Einstein applying Planck's ideas to radiation so coupled to electrons which were also described by fields I'm not going to describe matter here so I'm not going to go into fermionic fields and describe electrons and the rock equation but coupled to matter electrodynamics was no longer so linear because of the coupling the quantum mechanical effects pictorially we say that two light rays described here by these wavy lines or quanta particle description of these field can interact what by creating pairs so-called virtual pairs in the vacuum of charged particles like electrons which then Reece quatre and these effects were a challenge for actually something like 30 years to really control and calculate for a lot of technical reasons but in the case of electrodynamics that was worked off quite successful and has led to the most extraordinary tests of fundamental physics that have ever been calculated for example the electron which has spin 1/2 so it's a as a trinsic rotation also because of its charge as an intrinsic magnetic moment which is measured in some dimensional units by what's called G and which for a relativistic electron not interacting with the electromagnetic field is one and there was a deviation the measured deviations of this from one that led to the necessity developed QED as a quantum theory and it is now measured to 12 digit point so this is the actual measured value of the trinsic magnetic moment of the electron 1 point zero zero one one five nine six five two one eight zero eight five and the last two digits are uncertain plus or minus seventy six this is the experimental value it's quite unbelievable that experimentalist can perform a measurement of this little magnet to one part in a trillion not only that I don't really need to write the theoretical value which again is an calculational tour de force because it agrees up to this point with a measured value in fact you can't really test at this level of accuracy experiment in theory because we don't know the fine-structure constant the parameter that enters into this theory to sufficient precision in fact you can turn it around use the experiment to get one of the best determinations of the fine-structure constant now this is the case even though nobody's ever proven the existence of QED as a quantum version of electromagnetism couples to electrons we don't believe it actually exists and to this and finally this is the result of an expansion in powers of this fine structure constant with coefficients call them GN which we believe although again nobody's proved go like n factorial for large n and therefore this series that were summing to get this number as zero radius of convergence so here is a good example of why mathematicians and physicists find it hard to communicate we're talking QED which I haven't really described because I haven't told you how to couple these gauge fields E and B to matter doesn't really exist we believe for good physical reasons the same reasons we believe that yang-mills theory non-abelian meeting this theory does exist furthermore even if it existed we don't really know how to calculate well the only handle we really have is this perturbative expansion in powers of this very small number luckily small and we have good reasons to believe physical reasons as well as analogies to other quantum field theories with the same structure which definitely have this divergent behavior that these perturbative expansions we use to one part in a trillion of 0 radius of convergence what we don't know that we don't know that either but anyway I'm not going to complicate whether it's n or n factorial n over 2 factorial so so this is you know in a sense the most spectacular example of the ability of physics to predict quantitatively make predictions that can be tested and agree it's a test of the basic but it's an example where we don't actually believe the theory makes any sense without further input which I haven't told you about or or or the method of calculation is under control now the yeah well let me let me let me just yes well you know the answer is so the reason so what this the way we look at this nowadays is that when we say here the existence of you know theory what is being asked in the clay prize is something that most quantum field theories most visitors would say today is much too much to ask what we really are interested in is not quantum field theory what effective field theory which means that you see quantum field theory assumes the existence of fields functions of points in space-time and here in the formulation of the field now this actually depends on space-time so does this here we're taking products of fields at the same point now it turns out that such products are extremely singular it was one of the technical difficulties of the theory but the fact that we have products of fields at the same point mean that we're really taking these points seriously we're really saying that we believe that physics is formulated in a continuum where we can bring objects fields together arbitrarily closely now we have no evidence for that because at best we can look a few centimeters with our eyes microscopes accelerators we can go down the distances of 10 to the minus 18 centimeters but there's a long way between 10 to the minus 18 and 0 especially on a logarithmic scale and it is a bit of hubris to say that I should imagine that this described by a quantum field theory which is assuming that it makes sense to talk about these physical objects down to arbitrarily small distances sorry well we actually don't in fact but anyway so the modern point of view would be to say let's be more humble and you know we and put in some cutoff not at zero but at the smallest distance that we've probed we've seen nothing destroy the point like structure of space-time but there might be and then that removes a lot of problems and allows us to define a theory without cut off destroying if you want the continuum at very short distances for example replacing the continuum by a set of discrete points as we will shortly do with in order to the finding Mills theory with the lattice what we've learned is that if the cut off say is at 10 to the minus 33 centimeters to choose an arbitrary distance it's not going to affect physics much up here at these much larger distances and the non-existence of QED that we'll get to later occurs at distances which turned out to be let's say in distance scale cut off of e to the - 1 over alpha times some standard distance scale say if you want the radius of the electron distance scale defined by the size of the electron it's 10 to the minus 11 centimeter so this is you know e to the minus 137 extraordinarily short distance much much smaller than anything that occurs here this is where the problems we run into will kill us we believe from showing that you that this QED this simple abelian gauge theory exists as a well-defined theory and furthermore that's fine so we don't have to worry about the theory existing a much less ambitious theory one that is cut off at some credibly short scale doesn't regiments we do it much larger scale does exist and we can argue that it's asymptotic expansion it is perfectly valid for measurements done at large scales and that this albeit although not convergent is asymptotic so this is an asymptotic expansion in which if these coefficients a and grow like n factorial until we get to the 2n of order 1 over alpha this should give better and better results like asymptotic expansions tend to do and that the errors the inherent errors we make will be of order e to the well alpha to the to the order one over alpha so until we get to this is to get this result you have to work to fourth or fourth or fifth order fourth order I think but for loops and until you get to order you know maybe a hundred loops are you going to have a problem nobody's ever going to do that so you can use a synthetic expansions where you don't have any rigorous control over the errors and you can't make the precision arbitrarily good and furthermore the sicknesses that prevent this theory from really existing occur only at some incredibly I cut off and we know for many reasons that there will be other physical effects other phenomena coming in which will change things at these high energies now yang-mills theory which I'm coming to now we're going to demand much more from one we actually for this for much the same reasons we believe that electromagnetism has a complete quantum field theory taken to arbitrarily short distances doesn't exist we believe and that's the first part of the proof on abelian gauge theories even just vacuum non abelian gauge theories which are highly non-trivial do exist and you can really go to arbitrarily short distances it's a complete theory you don't have to regard it just as a large distance approximation to something you don't know about so you cut it off and second of all it has this remarkable factor fact of producing a scale which this theory in the vacuum doesn't have dynamically quantum mechanically so we're going to be discussing the non abelian version of electromagnetism it's just like electromagnetism evily has no scale as a dimensionless coupling and yet will produce a scale a mass cap from quantum effects now what does it really mean to construct or define or prove the existence of a quantum field theory so nobody really has ever done it except in a very trivial example or in examples where the symmetries are so strong that certain parts of the quantum field theory can be explicitly constructed using the large symmetry the inter kabila T it's difficult to really explain this in great detail but there are certain features that are necessary any quantum theory one needs a overt space states in hilbert space one wants the hilbert space to have certain positive metric positive norm states because our interpretation of quantum mechanics requires you know identifies probability with the norm of states and probability should be positive it's sort of trivial it's it's not an enjoyable and that's right so it's a theory right indeed so there have been theories in less than four dimensions like life or theory which from the point from our point of view have don't have these dimensionless couplings they have dimensional couplings like masses or or nonlinear couplings with dimensions of mass and those are simple enough at short distances that one can actually prove the existence of the theory now as I said those don't teach you anything interesting about the structure of quantum field theories we need to describe nature but but they you know there are very difficult pieces of work and you know they took a lot of effort it doesn't appear however that that effort allows you to address these more difficult methods more difficult theories and I'll say what's been done for four-dimensional theories but not much so anyway we also most important we want relativistic invariance so we have a symmetry much like classical enm Lorentz invariance in fact punk array and variance which is translations of space and time plus the Ren's transformations which is so we're not dealing with fermions just so3 and in quantum mechanics such symmetries of our theory are represented by operators so we want unitary Opera unitary operators that under which states and the operators the fields that transform covariantly and and we want a vacuum a state a that is invariant under Lorentz transformations and translations ground state of the system should be symmetric these are properties that I'm naive ly true and true using our limited tools to solve for quantum field theories but should be true of what we're looking for we also want that the generators of these symmetry transformations which are for translations the momenta energy and momenta are have definite spectral properties so particular the energy which is a generator the hermitian operator and this Hilbert space should have should have a spectrum which is bounded from below the vacuum is invariance of the vacuum zero energy and zero momentum its translationally in time under time and spatial translations invariant and in fact representation theory of the Lorentz group tells us that states that transformed irreducibly under the Lorentz group are characterized by momenta and other discrete labels and the Kasmir the energy squared minus the momentum squared is what we call the mass the particle now if we look at electromagnetism this theory here we have a unbroken conformal Verity of scale invariance under scale transformations length scale up energy scale down so in the case of enm the spectrum of the energy is continuous but in that's not what we observe if we ignore the existence of light rays photons we see particles of definite mass particles the mass is just the energy of a particles momentum is zero okay and the spectrum of the energy or a theory of which we're going to be discussing particular the spectrum of the mass operator as a discrete state the vacuum with zero mass zero energy zero momentum and then the lowest particle mass a discrete state with some mg/g standing for gap and then for sure pairs of these particles with twice the mass and some extra kinetic energy a continuum of multi particle States so in a quantum field theory where we have massive particles and no massless particles like light rays there is a gap between the vacuum the ground state and the first state that can be created with a minimal energy equal to the mass of that particle and then multi particle States perhaps other isolated discrete particle states as well and it's this gap that is the issue the really interesting dynamical issue in the second part of this problem how to produce a yang-mills theories which classically will have roughly the same structure is that electromagnetism and classically have scale invariance and managed to produce a mass gap in the spectrum of the energy operator but anyway in order to exotic why don't feel thinner you double spaced you'd like to have states that are representations of the transform coherently under great transformations I'd like to have some kind of field operators that depend on space and time that transform coherently under Lorentz transformations you'd like to be able to define matrix elements of product of these operators and and so on but that is a lot of problems sorry well the fact that you're here you have a so you're made out of let me come a moment to theory well let's take it let's take one of these theories which is representative of the class of theories that falls into the hyper QCD is an example of it sorry goes to zero as what in what goes to zero sorry the gap is the so in any quantum field theory where you have you know this set up there's a gap I mean there is a well-defined lowest energy state and the spectrum of P naught above orthogonal to the vacuum and that might be zero it is in quantum electrodynamics quantum electrodynamics one has the vacuum and then a single light ray can have an energy equal you know to its frequency and you can have as low frequency as you want arbitrarily long wavelength so the one photon state has any energy starting to zero all the way up to infinity two photon state has any energy starting with zero you just ever continue which is what you'd expect because after all there's no scale in the theory there are no periodic boundary conditions and oh you mean sorry if you put the theory in a box well we're not in a box for an hour three one now when yang-mills theory so yang-mills theory briefly speaking is exactly the same action except we now have a this is a connection now on some some bundle some gauge group and su3 is the relevant gauge group for the real world for QCD but this problem is stated for any gauge group but this is now I'm not an abelian gauge group and the field strength is nonlinear and the gauge field and that transforms covariantly under each transformation and the action involves the trace a quadratic form in the lie algebra of the same beast but that is now as quadratic terms and cubic terms and quartic terms in the field strength in the equations of motion which are now d f or d f+ are nonlinear but again naively the this coupling is dimensionless and the theory classically is invariant not just under want great transformations but under conformal transformations and scale invariance and so classically although it's very hard to you know the equations of angles equations classically are almost as complicated design science equations although somewhat simpler still it's clear that the spectrum is continuous under scale transformations the energy scales and the theory is invariant now when yang-mills theories were first proposed in the 1950s as a generalization of electromagnetism because people had discovered approximate symmetries non abelian symmetries of nuclear particles one of the reasons why it took was so difficult to imagine applying these the real world was the fact that there weren't any particles with a continuous spectrum like this except for the photon so the only long-range arbitrarily light particles that exists to get the origin of the long-range force are believed to be photons light rays or gravitons which were ignoring the quanta of einstein's theory if you look at this classical theory it would would be a continuous spectrum there would be some kind of nonlinear waves which would be like electromagnetism except that there they do interact two waves cannot be superimposed to get another solution of in those equations it's a horribly nonlinear theory but just because the symmetry is includes scaling of the energies clearly there's a continuous spectrum and quantum mechanics that gives rise to the long-range interactions and the forces that one were trying and was trying to describe namely the strong and weak so-called nuclear forces that act within the nucleus act within the nucleus their short range and the if they're going to be described by quantum fields and it wasn't obvious by the way back then then the quanta of those fields the spectrum of the Hamiltonian of energy would have to have a gap the gap in fact the value of this gap this mass is related through the range of the force one over the range of the force mediated by such by particles of this mass so this was the problem we're trying to apply gauge theories to the real world now it was solved in two different ways in the case of the weak interaction by the so-called Higgs mechanism that you all heard about it was final form has been verified what in the case of the weak interaction you add new degrees of freedom another field the Higgs field which you arrange so that classically it's minimal energy state breaks the underlying symmetry here and in that vacuum state it turns out these broken scale and very to introduce the scale for sure but you also create a gap classically very simply it's a classical mechanism doesn't require a quantum mechanic and in fact most of us I must say 40 years ago were somewhat suspicious that such a cheap solution could explain the weak nuclear force but it turned out to be the cheapest classical solution survives quantum mechanically it explains the weak nuclear force that's the so-called Higgs mechanism but what you've done is you know you've added something else but new parameters you've broken this scale conformal symmetry by hand and you generated a mass cap and then you can treat the theory much like electromagnetism calculate perturbatively with the Virgin expansions correlative corrections and they work does the theory exist well probably not it's mixed with electromagnetism it's really a su 2 cross u 1 gauge Theory who is diagonal u 1 is preserved and that's electromagnetism and as such together with this Higgs field probably doesn't strictly exist either but nobody cares because we all know at some cutoff someplace the physics is going to change and we can use that approximate Theory way below the color but then there's the strong interactions and that is a purely quiet explanation here of why the forces short-range and why the basic particles the basic quanta of this field and the quarks which I I haven't won't discuss but these light rays of issue three gauge group are not observable which is another property of the theory which I want this gas called confinement that is a consequence of rather surprising and beautiful quantum structure of UCD and it's also the reason we believe that this among all these other examples is a theory which actually does exist without any cutoff down to arbitrarily short distances again somewhat of academic because there is no physics that will come in from here from these other forces and ultimately from gravity so who needs an exact complete mathematically rigorous theory anyway it's nice to have one what mathematicians which is why they should prove that it exists so let's discuss QCD a bit so QCD is classically is described by this connection on an su 3 bundle with a non abelian field strength and classically nonlinear equations now classically a Mills Theory has not has been studied a bit but it's complicated its nonlinear and it's also unobservable you know classical electromagnetism is something we use all the time in the real world and that's actually because there is no ASCAP so it's easy to excite photons lots of photons and quantum the quantum mechanics of quanta of light are well described by classical fields when the number of photons is very big and you can easily construct states with lots in fact an infinite number of photons without requiring an infinite amount of energy but that's not true once there's a gap so then you need a minimum amount of energy to construct one quanta and and a very large frequency remember the energy is like a frequency and if the minimal energy is about a minimal energy of what would be the lightest state of a in QCD vacuum QCD that minimum frequency would be something like 10 to the minus 24 Hertz now it's pretty hard to shake something and create a classical chromodynamic field I mean we don't do it unimaginable that there's any regime a microscopic phenomenon that is described by classical QCD just because of this gap okay so anyway let me not try to describe anymore what what the proving the existence of a theory isn't let me write down the answer that we the answer to the theory as we use it one of the reasons we believe in the positive answer to this problem is that this theory works in two ways one the predictions of this theory which give can be calculated using this kind of expansion as before where these coefficients grow like in factorial but these are functions of diable physical quantities like the momenta and energy of beams and these big colliders and it turns out that albeit an asymptotic expansion whose coefficients grow like sorry whose coefficients grow like in factorial the couplings here depend on the scale of the phenomena being measured and for large enough momentum energy z' these couplings can become in principle arbitrarily small so one has an asymptotic expansion with a parameter that can be physically made as small as you want in principle by going to buy enough energies and so although this so as long as you take P bigger and bigger these coefficients alpha will get smaller in principle you have predictions that get better and better as you make do experiments at higher and higher energies so that's one reason and you know there are zillions and zillions of predictions they are used all the time in calculating processes for the LHC they're the background the uninteresting things but the ability to calculate a crown and for those calculations to work give extraordinary tests of the theories so that's one reason we believe that this theory exists and its predictions are valid and that these perturbative expansions make sense now by the way one way that hasn't been used so far of constructing the theory right in the end turn out to be a proof of both the existence and its properties is to make mathematical sense out of these diversion expansions much like you can make mathematical sense out of other asymptotic series in for mathematical objects if you can control the various singularities you get when you construct Borel transforms or other things there is in fact a revived interest in making physical sense out of such expansions and that might be actually a way of a different way than people typically used to define the theory and prove its existence but the way people used to prove our physics point of view the existence of a mascot and to calculate the mascot which is the same as calculating the spectrum of the theories and masses of elementary particles is to define the physical observables in terms of a truncated cut off discretized version of the theory and then take a limit so this relies on path integrals which physicists used all the time to derive beautiful mathematical results that have really no mathematical basis at all except as limits of finite dimensional integrals so in the case of gauge Theory one needs to replace this action which is usually a path integral and this is going to happen so much so in quantum mechanics briefly speaking the basic entities which are products of fields or products of fields that's based on points can be calculated by integrals so called path integrals integrals over the basic field variables oven phase which is just the action of those field variables and then the operators express this functionals of those field variables such path integrals can only be really defined for quantum mechanical systems of the finite number of degrees of freedom and in the infinite dimensional case in order to get a wealth of the beginnings of a well-defined expression one first rotates rotates time to imaginary time or our three one two are four three places mikowski space by euclidean space and then there is a well defined procedure for having calculated these so-called correlation functions in the clinton space of going backwards to the so called wipin functions or the objects that one can use to construct a Hilbert space and the physical observables of a quantum field theory reason one does that is that one then has a real measure here and one can think of evaluating this integral no Merrick Lee but one needs to first replace this product over functions by a ordinary integral of products over variables at each space time point on some lattice that approximates four-dimensional space and to do that one has to introduce some kind of discretization of space-time with a lattice separation and then the action will depend on the fields and on this lattice spacing now in the case of an abelian gauge theories or gauge theories in general the variables are these connections a connection really tells you how to transport it feels a gives you a lot of me from one point to another from the transformation of the group at one point to that of another and so in the lattice definition of gauge theories the basic variables live on these links and our unitary operators so one has some kind of lattice in four dimension the field strength the curvature tells you what happens if you do parallel transport around a little loop so the field strength is defined on a black hat a little square and will be the product of these unitary operators in the group around this little plug it and what replaces this action is just so this is a unitary matrix that describes what happens when you go around a little bucket and this is summed over all buckets so for each plug that you have so in a sense you of a link is the parallel transport along the link of the gauge connection and you P if you work this out is roughly something like the exponential of I times F DX mu DX nu where this bucket is in the you new direction my sum over all ply cats take the trace trace of F a0 the first term that survives here is the sum roughly over all pockets of a to the fourth F to the four F squared plus terms of water a to the sixth and this is just if a kit becomes arbitrarily small the same as this integral so this is a discretization of the action and in terms of these placate variables which are products of linked variables and one integrates over all links on the lattice with our measure this is bounded and for finite finite lattice a finite number of links this certainly is a well-defined in a row the idea is that as this lattice expands to fill all of space and become finer and finer this should up to terms which are irrelevant vanished like powers of a-squared compared to this term this is like the integral over space of trace F where if these turn these higher-order terms have extra powers of a in them and I easily go to zero but this will define a continuum theory now I've gone through this much unless you've seen this before you'll never understand but essentially you replace space-time by a lattice of points in space-time you might put in periodic boundary conditions helps numerically so you do this on a torus and then you want to use this as a well-defined measure to calculate correlation functions of operators and the only problem is that you have to carefully choose this coupling that appears here as the analog of the fine-structure constant to depend on the lattice spacing as well now the remarkable property of an abelian gauge series is how you have to choose the coupling dependence on a on the cutoff as you take the cutoff zero you want to take this whole thing in the limit where this lattice of points fills up the space and a goes to zero the famous the Virgin says ultraviolent divergences interesting dynamical behavior of quantum field theories has to do with the behavior of a has you let the continuum over let the lattice spacing or the cutoff go to zero and here is where the big difference between a theory like QED and theory like UCD happens and so in a in QCD there is a definite prescription that you must thank as a goes to zero in order for this limit to exist and that requires that the coupling defined on this lattice of scale a vanishes like 1 over log a times some parameter of dimensions actually may define it as 1 over lambda as a goes to zero this blows up this vanish is like what over log and lambda is an arbitrary parameter of dimensions 1 over length or mass that you introduce to define this limit what we know is that if you evaluate correlations function using this measure by perturbation theory by expanding around the Gaussian fixed point the point of which this is quadratic then you can perturbatively calculate all of these sums and that is known as perturbation theory and in that turbit of limit you can prove that if you let G squared vanish coupling that at out-of-scale a vanish logarithmically in this fashion then term by term in that expansion all physical observables are finite furthermore even though those perturbative expansions are asymptotic you can always go to small fa where this behavior makes G as small as you want so that means you you can you would think you could trust an asymptotic expansion and if you're a coupling isn't small enough you just make the lattice a little bit smaller and then it gets better and better so the strong reasons to believe that that that this so-called asymptotic freedom the vanishing of a coupling which governs the scale of interactions at distance a vanishes as a goes to zero that that is what's necessary to make this limit exist that innocence is what needs the first thing that needs to be proved that the limit exists the limit exists for using this measure of course inserting in here some appropriate product of observables formulated in terms of these lattice approximations to the field strength now if you could prove that the limit exists then you could probably also prove all the properties of the correlation functions of expectation you know the correlation functions of various operators that you need to satisfy the constructive axioms of quantum filter the existence of unique vacuum represent you'd have to prove that the theory possesses the symmetry the Lorentz invariants of the original theory which you've broken by putting the theory on the last we just all you have our discrete subgroup of the Lorentz group notation group and translation currents that to we believe is true because the terms that violate Lorentz invariants are these terms which have extra powers of the lattice spacing and order by order and Britta Bayesian theory you do this by saddle-point order by order in the saddle point approximation they do indeed vanish like powers of a so order by order in this asymptotic expansion it gets better and better as you go to spoil and support or lattices everything works but that's not a proof that the limit exists for physics or approval but it's not a mathematical proof and now there has been some progress by some by Balaban and some other constructive field theorists were who got pretty far I think for non abelian gauge theory in a finite box so in a finite box you keep the size of the system of the torus if you want fixed as you take the lattice spacing to zero and from a physical point of view that makes a job a lot easier because one you in a finite box you don't have this problem of a continuum you have discrete states and you never get through a region you can make the box small enough so if the asymptotic expansion is better and better and you can put in downs and control try to control a limit but even there nobody really isn't a proof of the existence even in a box that I know of and the papers are really impossible to read and by and large workers stopped them that effort as far as I know in the last decade now that's just the first part proving the limit exists the next part is actually proving that there's a mass gap and there what you have to do is do what people do anyway which is calculate the mass spectrum of this theory and they do that and one of the reasons we believe in the theory beyond just the perturbative asymptotic expansion is that by doing the numerical integration Monte Carlo integration of these integrals and then trying to take the limit for bigger and bigger lattice they can calculate the masses and the fine that as you take this limit chi-squared vanishes like 1 over a you've introduced some parameter with dimensions of mass 1 over length and that the masses you calculate and there's a specific procedure for doing that our finite and our numbers depending on the state as the pure number in units of this arbitrary scale that you have introduced to define this limit now you say what is the scale I don't know let's put an equal to 1 with nothing compared with because this theory has no scale so when you have a theory that you've introduced the scale in this way to define the theory then all masses are numbers times this scale that's what we call dimensional transportation we've traded this dimensionless number for a dimension full number and it makes no meaning to say what the value of this is it simply is what it is take one of these masses one of these particles say the one in the real world we might take the proton and call it mass one and then all other masses particle 1 particle to the ratios will just be pure numbers and this is what people in calculating compare with experiment and by now after 30 years or more of doing these calculations massive effort we have calculated the spectrum of all the low-lying hadrons particles made out of mostly quarks which I haven't looked yeah but if in pure QCD which we can't compare with experiment it would be the masses of particles made out of these interacting quanta of the gauge field to one and they agree with experiment about one percent accuracy their various ways of estimating theoretical errors as well but there's nothing rigorous about this one nobody's ever shown the limit exists and to nobody has a rigorous way of controlling these errors however we have enough you know every state that exists in nature appears in the spectrum there are no states with zero mass so there's a gap indeed no maracle e and that's why we believe the answer should be yes this limit exists and yes if one calculates the spectrum of the energy there's a gap and an interesting spectrum so unfortunately I didn't have time to discuss this remarkable phenomena which is the basis of this the fact if you ask what about QED quantum electrodynamics you discover order by order and perturbation theory you have to arrange for this coupling as a function of lattice spacing to get bigger and bigger and bigger which one means you don't trust what you're doing because perturbation expansion begins to diverge and to every indication is that before you reach the continuum limit before you get to zero this parameter blows up and you lose all sense of what you're doing it doesn't mean that this theory QED doesn't describe the real world we know it does but something's missing need other ingredients you might need one or two other ingredients it might be part of string theory it might be part of a bigger unified theory it might be involve an infinite number para we have no idea all by itself it just means that it's not what we would call a UV complete theory that's extendable to arbitrary short distances in fact in four dimensions we believe that the only theories that have a chance of existing in this sense are these very special theories and those are all contained in the definition of the clay problem so these aren't just any alcoholic field theories they one describe the real world - the only areas that manage to do all these miracles and therefore they deserve a proof everything we know is true actually is thank you [Applause] so the mathematicians of any question I'm sure you started off by talking about you won and how mathematical point of view it's a sick theory in some ways yes but then of course you wrote down G over 2 to 13 or 14 decimal places right would you say this has to do with the connection between math and physics I guess would you say that G over 2 is a is it a number in the sense of mathematics because it's not defined at the 137th place or something right couldn't could you say that so there is a measurement where is the ratio of masses of particles in well we discussed here I discussed here by the way vacuum QCD if you are QCD with no matter no quarks just the nonlinear interacting fields which themselves are charged they're very complicated and and in the real world there are also quarks and that introduces other parameters unfortunately but if we didn't have those quarks we just had this theory there are these numbers which are not going to be simple rational numbers or anything know mathematically nice but they were they are numbers infinite set of numbers QCD this QCD without matter you see started out with one dimensionless parameter look like a freedom number you could put in it ends up with a scale a scale which means nothing because you're a nine compare with so all dimensionless measure durable quantities in qcdr calculus so mathematically are kind of interesting all these numbers are pure numbers and they depend on the gauge group G and nothing else so in the gauge if the gauge group is issue and they have a interesting dependence on n for example one which we exploit because life does get simpler and is big but they're just numbers with a characteristic of the gauge group if one had some really powerful mathematical way of of describing this dynamics they might turn out to be very interesting mathematically certainly from a physics point of view there are very few theories we have in physics that have such a rich and powerful structure where there an infinite set of quantities that are just numbers very determined I have a naive question the existence of mass calf does it have anything to do with the namibian nature of the gauge group yeah definitely remember there's only one abelian theory and it doesn't have a mascot for sure so well this has to do there are ways of the abelian theory is trivial there's two trivial to have a gap in the vacuum it's just free waves with continuous spectrum non-abelian theory is an interactive theory and there are lots of ways of understanding the where the gap comes from the dynamics that creates these massive bound States but it's not like there many of the abelian theories and all non-abelian theories should have this the same behavior right well you know it also so its ambitious in the sense that it doesn't you know it requires the real world as opposed to less than four dimensions which makes life easier or extra symmetries like supersymmetry that make life easier and it also but it takes the easy way out doesn't ask for this to be true an arbitrary manifold but we have very good reasons to believe that if quantum field theory exists on on flat space with a flat metric that small deformations these small deformations of the background geometry are interchanging because all the problems occur you know in the vicinity at very short distances so as long as you deform the geometry in a way that's smooth that again is a folk theorem but one that are surely true so this would be the starting point now the applications are quantum field theory which I've been of great interest to mathematicians and I suppose they would like to have some firm foundation for some of those applications often have you know don't work in flat space they work with some curved manifold but as long as the curvature is you know smooth then locally the theory is flat it won't change so I I would imagine that once if you had methods powerful enough to prove the existence for flat space you could prove it on on an arbitrary smooth well but you know like in this definition you don't really use what you use is that your break the Lorentz in symmetry and defining the object you recover it in the limit yeah it's hard it's much harder especially if if you have some you know the global structure of space-time is different than [Applause]

Channel: Kavli Institute for Theoretical Physics

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Length: 107min 7sec (6427 seconds)

Published: Tue Jan 02 2018