Free field states and conformal bases
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Stützpunkt <Mathematik>PunktMomentenproblemSortierte LogikÜberlagerung <Mathematik>Minkowski-Metriksinc-FunktionGruppenoperationMassestromAdditionQuantenchromodynamikMultiplikationsoperatorSymplektische GeometrieTrennschärfe <Statistik>KreisbewegungIsometrie <Mathematik>Nichtlinearer OperatorAggregatzustandDimensionsanalyseZeitdilatationPhysikalische TheorieTermHilbert-RaumKinematikDifferenteMatrizenrechnungt-TestKonforme FeldtheorieFreie GruppeKörper <Algebra>Hamilton-OperatorGruppentheorieFormation <Mathematik>KontinuumshypotheseInverser LimesTranslation <Mathematik>KoeffizientVollständigkeitProdukt <Mathematik>SymmetrieAlgebraische StrukturKonforme GruppeEnergiedichteKonforme AbbildungDirektes ProduktPhysikerVertauschungsrelationTheoremÜbergangSemidirektes ProduktSummierbarkeitRechter WinkelStellenringVorlesung/Konferenz
07:34
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11:26
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13:30
Finite-Elemente-MethodeTermRangstatistikReelle ZahlProdukt <Mathematik>MatrizenrechnungMengenlehreInnerer AutomorphismusNichtlineares GleichungssystemImpulsRadiusHelmholtz-ZerlegungMannigfaltigkeitDirektes ProduktArithmetisches MittelGruppendarstellungVertauschungsrelationFunktionalEinfache GruppeHarmonische AnalyseSchwerpunktsystemKomplexe EbeneVektorraumMinkowski-MetrikKreisbewegungRiemannsche ZahlenkugelRuhmasseGruppenoperationWeg <Topologie>Sortierte LogikSummierbarkeitHilbert-RaumDiagonale <Geometrie>DimensionsanalyseVektorpotenzialForcingZahlenwertJensen-MaßPhysikalische TheorieSymmetrieTensorproduktSpinorKörper <Algebra>EbeneGeometrieVarietät <Mathematik>Higgs-MechanismusPunktMultiplikationsoperatorDeltafunktionKomplex <Algebra>Konforme GruppePhasenumwandlungTranslation <Mathematik>Wurzel <Mathematik>Semidirektes ProduktStiefel-MannigfaltigkeitAggregatzustandStatistikSymmetrische MatrixVorlesung/KonferenzTafelbild
21:13
KugelflächenfunktionSchraubenlinieAggregatzustandImpulsPolynomringNichtlinearer OperatorFreiheitsgradStellenringTermArithmetisches MittelPolynomMultiplikationsoperatorAnalytische FortsetzungEntartung <Mathematik>GruppendarstellungAusdruck <Logik>Zusammenhängender GraphObjekt <Kategorie>Dimension 2Sortierte LogikOrdnung <Mathematik>SpieltheorieUntergruppeGruppenoperationKonforme AbbildungEndlichkeitEinfacher RingBasis <Mathematik>QuaderMomentenproblemFeldtheorieLeistung <Physik>Körper <Algebra>KugelFunktionalFreie GruppeKonforme GruppeSkalarfeldDimensionsanalyseMinkowski-MetrikKontraktion <Mathematik>KapillardruckQuadratische GleichungKombinatorikVektorraumDifferenteKonforme FeldtheorieBerechenbare FunktionAlgebraische StrukturFormation <Mathematik>Numerische MathematikStatistikHilbert-RaumSkalarproduktraumUnendlichkeitSimplexverfahrenPhysikalische TheorieTransformation <Mathematik>Endlich erzeugte GruppeTensorproduktAnnulatorMereologieHarmonischer OszillatorAuswahlaxiomVorlesung/Konferenz
28:38
PunktspektrumKomplex <Algebra>AggregatzustandEinsDreiSortierte LogikVektorraumNichtlinearer OperatorKonforme AbbildungNumerische MathematikSchwingungGruppendarstellungModulformTabelleImpulsEndlich erzeugte GruppeFundamentalsatz der Algebrap-BlockEinfacher RingAlgebraisches ModellTermDerivation <Algebra>TourenplanungObjekt <Kategorie>SchraubenlinieMengenlehreForcingFormation <Mathematik>DimensionsanalyseRuhmasseLeistung <Physik>Spezifisches VolumenOrdnung <Mathematik>Zentrische StreckungTeilbarkeitGruppenoperationTurm <Mathematik>PolynomKnotenmengeLineare DarstellungSkalarfeldKreisbewegungAusdruck <Logik>PhysikalismusStrömungsrichtungFeldtheorieTotal <Mathematik>Minkowski-MetrikWellenfunktionFamilie <Mathematik>GrenzwertberechnungMinimalgradMomentenproblemArithmetisches MittelKoeffizientSymmetrieRestklasseBootstrap-AggregationBerechenbare FunktionMultiplikationsoperatorPi <Zahl>Natürliche ZahlInnerer AutomorphismusKonforme GruppeIndexberechnungKörper <Algebra>AdditionVertauschungsrelationGammafunktionRadikal <Mathematik>MereologieZusammenhängender GraphUnendlichkeitPhysikalische TheorieUmwandlungsenthalpieDeterminanteWürfelProdukt <Mathematik>Freie GruppeDualitätstheorieOrdnungsreduktionTensorGravitationQuadratische GleichungGeschwindigkeitMathematikQuadratzahlDruckspannungEnergiedichteVorlesung/Konferenz
36:03
AggregatzustandUnendlichkeitSummierbarkeitPhysikalische TheorieDirektes ProduktDerivation <Algebra>MultiplikationsoperatorÜbergangEbene WellePolynomMinkowski-MetrikEinsProdukt <Mathematik>InvarianteEinfacher RingObjekt <Kategorie>IndexberechnungPolynomringGruppendarstellungQuaderWellenfunktionSortierte LogikKombinatorRestklasseKlasse <Mathematik>TermImpulsBasis <Mathematik>Endlich erzeugte GruppeHelmholtz-ZerlegungWellenlehreNichtlinearer OperatorVertauschungsrelationÄquivalenzklasseSchwingungTotal <Mathematik>Vorlesung/Konferenz
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Helmholtz-ZerlegungPunktSymmetrische MatrixEndlich erzeugte GruppeFreie GruppeMinkowski-MetrikAlgebraisches ModellAggregatzustandNumerische MathematikGammafunktionGruppendarstellungRechter WinkelSortierte LogikTourenplanungUnendlichkeitTermSkalarfeldArithmetisches MittelPhysikalismusDimensionsanalyseTensorFeldtheorieIndexberechnungAusdruck <Logik>Zusammenhängender GraphDruckspannungPartitionsfunktionErhaltungssatzSummierbarkeitKonforme GruppeDifferentePhysikalische TheorieRangstatistikProdukt <Mathematik>StellenringImpulsGruppenoperationKörper <Algebra>Objekt <Kategorie>Konforme AbbildungTeilbarkeitSchlussregelSymmetrieHarmonische AnalyseBerechenbare FunktionPolynomKontraktion <Mathematik>Algebraische StrukturKombinatorikSimplexverfahrenQuadratische GleichungLokales MinimumDualitätstheorieKoeffizientKombinatorKonforme FeldtheorieAnalytische FortsetzungTotal <Mathematik>AnnulatorRechenbuchSchraubenlinieTensorproduktGanze FunktionMaß <Mathematik>MultiplikationsoperatorNichtlinearer OperatorÜbergangBootstrap-AggregationLeistung <Physik>ModulformVektorraumMereologieNichtunterscheidbarkeitUmwandlungsenthalpieGebäude <Mathematik>PunktrechnungStrömungsrichtungSkalarproduktraumNatürliche ZahlInnerer AutomorphismusDerivation <Algebra>Gebundener ZustandKnotenmengeEinsp-BlockOrdnungsreduktionMathematikGüte der AnpassungSpieltheorieKinematikKomplex <Algebra>SchwingungEinfacher RingSpinorEntscheidungstheorieHilbert-RaumZentrische StreckungNichtlineares GleichungssystemIteriertes FunktionensystemFunktorFlächeninhaltVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:15
Thank you for having me here. It's a wonderful workshop. This is work that I've been doing in conjunction with Tom Milia.
00:23
It's in a writing stage right now. Tom's in the audience. And it's basically going to be an ingredients talk. Ingredients potentially hopefully of use for truncation. I'll say that I'm also, I have no plots in this talk,
00:41
but I'm currently working on some with a student, Jed Thompson, at Stanford. I won't say anything about this at the moment. So I think we've seen a lot of motivation for Hamiltonian truncation in this workshop.
01:03
And I think everyone knows the basic problem, but perhaps it's worthwhile just to say a few things again. So in general, we are sort of assuming that you have some initial theory, some UV point, and then you deform it and you have some flow to some IR point, some theory down here,
01:28
and it has an associated Hilbert space, H1 and H naught. And the sort of goal and hope is to be able to understand
01:41
what this looks like in terms of this or anywhere along this flow. So there's two questions here, one of in principle and one of in practice. So in principle, you might be asking you have some state
02:06
which belongs to the Hilbert space along this flow, and you want to expand it in terms of states inside the Hilbert space at H naught.
02:21
And you can ask in principle is this valid to do, which is a question about completeness. And in general, we will assume this to be true with an assumption on the fact that the deformation here to the Hamiltonian, here is some local interaction, then RG and things like the C theorem
02:42
will sort of guarantee that this can be done. There's another in principle question here. So I will be talking about free theories, and so I'll just be going through some essentially Fox space construction, but you need to make sure that you're then having the complete Hilbert space here,
03:00
and you might worry about things like super selection sectors and whatnot, but I'm not going to worry about that at the moment. So instantons, extended objects, whether or not these need to somehow be included up here or if they can be captured by the free field states themselves. That's a question I don't really know the answer to.
03:20
I believe it probably is true, but I'm not exactly sure. So the in practice question is we've removed this question mark, assume that we can do this, and it's how well does it work?
03:42
And that's really what this workshop is about. And this is not a new question, not new in the past few years. It's been around since the 80s, maybe even before that, and so you might ask what's new in 2018 for it?
04:02
Why are we here talking about it? And there's been proposals on different sorts of bases that you can use up here at H naught, which might be more efficient at capturing that data. So in the talks earlier today, you saw the use of conformal bases organized at the UVCFT,
04:20
and that's what I will also discuss. In earlier talks at this workshop, you've seen matrix product states, and that's very interesting. It's new to me, and both of these seem somehow more efficient at capturing the low energy dynamics. And for me, I have sort of a question about the matrix product states
04:43
of if they're somehow complementary to this picture of the continuum CFT states, but that's sort of an aside. So I will be taking the approach advocated in the previous talks that you kind of want to know the conformal data at the UV fixed point.
05:12
And to do that, that really means you need an explicit construction of the states in the Hilbert space.
05:28
And then there's scaling dimensions, and then from here you can also compute the overlaps and the three-point coefficients and so forth. So the organizing principle here, what we're really saying is that space-time symmetry is the key.
05:46
And this is the physicist's approach to everything since we learned what the word symmetry is. So this will really be an exercise in group theory, and in particular group theory of either the Poincare group or the conformal group.
06:06
And since we're starting at UV CFT, I'll be using this, but I like to phrase things a lot of times in terms of here. And just to remind everyone, this is the isometry group of Minkowski space, and it's just the Lorenz group with translations.
06:23
And the semi-direct product is here because translations don't commute with rotations or boosts. And the conformal group adds in another set. So these are your P mu's. Conformal group adds in a K mu and a dilatation operator, and this actually then closes into the nice group SO2 D.
06:50
So it has a little bit more rigid structure to it. Are you working at the level of the group or algebra? Yeah, that's a good question. So I'll discuss primarily algebraically, but since we're in the free-field limit,
07:02
this will be the double cover of the group is all perfectly well-defined. So a little just background of how I got into this. I come from an EFT background,
07:21
and there we like to write operators in the Lagrangian, and so you have some sort of kinetic term plus the sum on local operators, and you want some maybe operator basis for this. And these operators, you kind of want to get the minimum set.
07:43
It's a minimum set. Multiple operators could describe the same physics. So what we really mean is in EFTs, we're interested in computing the S matrix, which can be follows from Dyson's formula,
08:04
a time-ordered exponential of the Hamiltonian, and so that sort of dictates if you have a Hamiltonian, which follows from a Lagrangian built of scalar operators, it will guarantee the Lorentz invariance of this S matrix.
08:23
So throughout this talk, you'll see me end up focusing a few times on the scattering states in the theory, but they are just some sort of subsector of the whole Hilbert space. So what am I going to present to you?
08:43
I'm going to present to you an explicit construction of the free-field operator spectrum, or equivalently the Hilbert space, in four dimensions. I'm going to be using spinners in momentum space.
09:02
So it will have very much a flavor of the talk before us, which was formulated in momentum space. In fact, I was fortunate enough to have Zhu here come to Yale and give a talk on Hamiltonian truncation, which is what turned me on to this. And so Matt provided a very nice introduction
09:22
to what's sort of needed there on the ingredients. Why am I using spinners? Well, spinners trivialize the massless equation of motion, but it's going to also give me one other thing, which is it's going to allow me to get polarization.
09:44
In the talks earlier today, these basis functions were constructed purely out of the momentum, which is a limitation then to just scalar particles. And so with spinners, it's the only ingredient needed to capture polarizations in four and three dimensions,
10:00
so you can capture everything with it. But the basic idea is still the same. We're just going to get polynomials and spinners. But I think that there's going to be a very nice organization to this calculation, which perhaps sheds some light on the Hilbert space and how you might go about speeding up some of these computations that you need to do for truncation
10:21
when you're evaluating matrix elements. Do you mean that even if I'm interested in scalars, this method could still be... Yeah, absolutely. I mean, the momentum is just then lambda lambda, essentially.
10:42
It's not only going to get polarizations. I'm going to be able to get every massless particle primary. Operator is really the description of the result here. And there's going to be a geometric intuition of it, which I'll explain now, is that almost all this can be anticipated from phase space.
11:07
So I'm going to work in four dimensions for most of this talk and drop down to three and two later. So in four dimensions, any physical quantity that you have
11:20
involving n particles has in its phase space delta functions enforcing the masslessness and then a total momentum conserving delta function.
11:48
It enforces it to some total momentum of the state. Now, depending on what sort of coordinates you want to use, if you are working sort of an equal time commutation, you're using p0, p1, p2, p3,
12:03
and there should be theta functions here, and of course you have p0 squared, or p0 is equal to the square root of p squared. And so this defines some cone and light cone coordinates. You're going to have p plus is equal to p transverse squared over p minus,
12:23
so it's a parabola, some sort of geometry here. And then this aspect here defines your momentum components adding up to some total momentum and some linear set of equations, and so it defines some simplex. And to me, this is a rather intricate geometry to look at.
12:52
So if you move over to spinners, so for a massless particle in four dimensions,
13:01
and let p a a dot, well, this is general, so any vector you can put into a two by two matrix,
13:23
you have the light cone components on the diagonal and the transverse components on the off diagonal, and if it's massless, we all know that this means that this is going to be a rank one matrix, which means that I can write it in terms of the outer product of two spinners,
13:42
where this, for real momenta, is just the complex conjugate of this. So the value of this makes it manifest that this is a rank one matrix. So p squared equals zero is automatic,
14:01
and now this whole set over here reduces to a single equation, which, why don't we just write it out, it's the... So let me give the total momentum, go to its center of mass frame,
14:22
in which case it's on the diagonal, and this delta function forces you onto this sort of geometry. So just the lambda one one, some vector in the summing, and then lambda one dot lambda two star,
14:42
lambda two squared, and then the complex conjugate of this. So this is, to me now, geometrically a little bit more apparent, which is that you have some vector which sums to the mass,
15:04
another vector which sums to the mass, and then they are orthogonal. So you basically have two complex spheres,
15:23
which are orthogonal to each other. So this is defined as a manifold, and there's a very simple group theory definition of this manifold, which is that the total momentum now being equal to
15:45
the sum on lambda a i lambda tilde a dot i. This is invariant on a u n action, which rotates the lambdas,
16:01
and so you have some u n action, and these two higgs it down to a u n minus two. So we can go over to a real case if that's easier to think about. So you give a v to one radius that breaks o n down to o n minus one, then you have a second one which breaks o n minus one down to o n minus two.
16:22
So this manifold is described by u n over u n minus two, which is called the Stiefel manifold of two planes in n dimensions. It's over the complexes.
16:43
And the whole point of saying this here is that physical quantities only have support on this manifold. That's what the delta functions are telling you.
17:04
So there are harmonics on this, as in the mode decomposition on this manifold
17:21
tracking against the physical quantities in the Hilbert space states. I'll show this more explicitly, but this is the probably simple way of remembering what's going to appear.
17:44
Okay, so... You are saying that you should worry about the harmonic analysis of this. Exactly. Exactly.
18:05
So this is a free theory, which means I get to construct the Hilbert space as a Fock space,
18:20
meaning if I let H one denote the one-particle Hilbert space, the n-particle Hilbert space is just given by tensor products of the one-particle Hilbert space. And you might need to be careful about symmetrization or anti-symmetrization based on statistics.
18:44
So H one is going to be a very key ingredient, so let's spend a little bit of time with it. So what is the one-particle Hilbert space? It means that you're specifying the one-particle representations of either the conformal group or the Poincare group.
19:02
Turns out that those two are the same for massless particles in four dimensions, and let's have a little bit of a look at that. So again, I have a spinner, lambda a, and its conjugate, lambda tilde a dot. And I want to tell you why these are so nice.
19:21
So there's probably a variety of stories you've all read about this, and this is just another one of those, I suppose. So if I go to some, I'm going to frame this as I pick some momentum, and I kind of ask, what else can I have? And the value of spinners is that they allow you to get everything else. You make the full use of the spinner.
19:42
And so if I pick some sort of, I go to some frame for some massless particle, something like this. So, and this leads to lambda, the lambda which satisfies this is essentially square root of k.
20:05
And it's very easy to determine in terms of spinners what group is preserved by this, under this. So under for G and SL2C, the Lorentz group in four dimensions,
20:21
k transforms as G K G dagger, and little group by definition is what preserves this momentum, and it's very easy to see that these matrices are given by 1, 1, alpha, 0,
20:49
and then a potential phase thing here. So this is a complex number, and this here represents translations in two dimensions,
21:01
and this represents rotations. There's a semi-direct product between these because they don't commute, so the little group, the well-known statement, is that it's a Euclidean group in two dimensions. Now, in field theory, we want to say that you have a particle and it has a total momentum,
21:20
and then you're going to ask what other degrees of freedom can I give it for fixed momentum. So for fixed momentum, we, by choice, choose to only have a finite number of degrees of freedom in order to maintain a nice particle interpretation.
21:40
So that means that you're going to specify some representation of E2, induce it into the Poincare group or conformal group, but this, in general, has infinite dimensional representations due to the same reason why field theories have infinite dimensional representations. This is a continuous thing here.
22:01
And so what we want is something which only transforms under U1, and that's what's very nicely given by the spinners, which is that in this frame, the rest of the spinner, which sits here is the moduli space here. We call it lambda 1, the first component. Under an E2 transformation, lambda 1 goes to just e to the i theta lambda 1.
22:26
In particular, it only transforms under the U1, which is exactly the requirement that we want. So this is the reason why spinners are so nice, and it means that if you're going to then pick a representation of U1
22:43
that you want to do some helicity state. Of course, that's the interpretation of it, is that lambda 1 to the h interpolates that helicity state. So basically, polynomials in lambda are just going to allow you to get any sort of representation that you want.
23:03
And in particular, it's not just lambda 1 to the h which interpolates this. Let me now just go back and phrase it back in terms of this. It's that I can also add any power of momenta on top of this,
23:25
and this will also contain an object which transforms the same way when restricted down to the little group. So there's a notion that I'm using here, which is that we are inducing a representation from a subgroup
23:41
into a bigger group of some representation of h, and you can reciprocally view this as a sort of restriction from g to h as some representation of g. So if some representation of g, I'm going to use this again throughout the talk.
24:01
Some representation of g, if it restricts down to h and contains the representation you want here, then it necessarily is included when you induce that representation into g. What is g and h in your construction? Yeah, so g here is going to be a Poincare group,
24:22
and we'll see that the conformal group comes along for free. And then h is this e2 little group, and I specifically just want the representations of u1. So when I specify a u1 charge and go up, it means I'm going to get every single one of these terms.
24:54
That whole thing is actually very familiar. Perhaps its phrasing is not familiar to everyone.
25:05
So the sort of simplest thing that you can sort of think of for an example of this is to take a 2-sphere. So on a sphere and consider functions on the sphere,
25:21
which any scalar function on the sphere can be decomposed into spherical harmonics on the sphere. These are the modes. And so here what we've done is we've induced from SO2 to SO3
25:43
the singlet representation, singlet of SO2, meaning this was a scalar function, and then the modes would show up. So these are all the spin, integer spin modes, which when restricted down to SO2, they all have a singlet in them, like a vector, a 3-vector has a component,
26:02
which transforms into a 2-vector and a singlet. So now just in terms of polynomials, I can start looking at my helicity states. I'm confused.
26:23
These polynomials that you wrote, these polynomials in lambda, lambda a to the h times lambda lambda bar to the n. I guess they correspond to some local operators. Yeah, exactly. So if this is the momentum of the particle,
26:42
so this corresponds to adding d mu on top of, if we pick the helicity zero one and pick the first one here, it's going to correspond. I'll give a local operator interpretation in just a minute. Yeah, exactly.
27:03
So I can sort of draw this two-dimensional picture of, I have some sort of leading operator upon which I'm adding momenta on top.
27:25
Obviously, these are playing a distinguished role in this game. And so we see, in fact, that with just these polynomials, I can get every U1 helicity representation that I want.
27:46
And so, in fact, this itself is the regular representation of U1 showing up, which is consistent with this formula in its degenerate case of n equals one,
28:01
that U1 breaks down to nothing. And I can capture all these objects in terms of just, so every polynomial in lambda and lambda tilde plays a role here.
28:21
So this just corresponds to the basis of some polynomial ring. And for those of you who don't really know rings and so forth, it's not that fancy at the moment. Just view this as a box that I'm pulling polynomials out of.
28:41
Okay. So Slava just asked for the field interpretation of this. So what this corresponds to is what you would normally call the field strength,
29:05
a left-handed fermion, a scalar, a right-handed fermion, a right-handed field strength, gravitinos, gravitons, and all the way down. And what you can think of is in sort of some,
29:25
you're used to opening up a field theory textbook and seeing it written in terms of raising and lowering off modes. So this is going to be an integral over the momentum, v square lambda.
29:40
There's some volume factor that's important at the moment. And then the lambda lambda is exactly what you will get for the wave function p mu epsilon nu commutator, or anti-symmetric.
30:05
And this is e to the minus i lambda lambda tilde a dot a a dot. So if you think about Taylor expanding this operator
30:31
and evaluating at the origin, these are all just the modes of that. What about the stress energy? The stress energy tensor is a two-particle state,
30:41
which will be built out of this, and we'll talk about that. So does it mean that in field theory, I will not need more than the first five rows of this table because you will never have anything fundamental with more than two indices?
31:01
Well, if you want long-range interactions, yes.
31:23
So this furnishes a representation of the Lorentz group, obviously. It furnishes a representation of momentum as lambda lambda tilde. The Lorentz group in here is given by rotation generators,
31:44
which are traceless. These are the obvious derivatives associated with it. And then what comes along for free when you're looking at this is that obviously these have a well-defined total degree in the polynomial,
32:02
or any term in this thing here. So the total degree is just given by the number of overall lambdas plus the number of overall lambda tildes. And so I'm going to add in a factor of two just for sake at the moment.
32:26
And then I want to point out something here, which is that these are, one, all quadratic operators in terms of lambdas and derivatives. That's an important role here. And if you are thinking about this, these all act on lambdas
32:42
and return back lambdas, so they just act on that ring and return objects back in it. I could have equally formulated this in terms of a ring in terms of derivatives, which plays a role as well, so there should be another object here.
33:00
This is not surprising to anyone in this room, which is the special conformal generator. And this whole set here closes into the conformal algebra. That's why this factor is here. It tells you that the scalar has mass dimension one. That's all. Important dimensions.
33:22
So, yeah. Okay. Now... They act on the rows in the ring, right? They leave the rows in there. Pardon? They only act horizontally in your picture. That's right. They only act, so, yes.
33:41
They only act horizontally within that picture up there, which goes on to also prove to you that every massless particle in four dimensions is uniquely an irreducible conformal representation as well. So, if I want to...
34:00
The other nice thing that you notice about these basic terms here is that they are all holomorphic, so it's quite easy to see that they are annihilated by k, as in their primary operators. Not surprising to anyone. But I want a simple way of isolating these sort of primary operators.
34:21
And the way I've drawn this is that momentum keeps coming off to the side here. So, the space of primary operators, if I call that ring R1 and then on subscript P, can algebraically be captured by just modding out by momentum.
34:46
All right? So, this gives a mathematical definition to the physics question, besides momentum, what else can we observe?
35:06
What are the other states in the space? Where now we've translated in English this to modulo momentum. That's its proper definition. So, this here just means take polynomials, modulo momentum. And when you do this, this isolates out just these components,
35:23
the 1, lambda, lambda squared, et cetera. Okay? The last thing I want to point out before moving on is that in addition to these sets of generators here, there's another set of quadratic generators which commutes with all of them,
35:41
and that's something which labels the holicity. And so, the total holicity here is always determined by the number of lambdas minus the number of lambda tildes. All right? So, mathematically, lambda dot D minus lambda tilde dot D tilde.
36:03
This is a U1 generator, and the important thing here is that this object was invariant under this U1. So, it's not surprising then when I mod out by the invariant thing here, I'm foliating it into sort of the space of U1 representations.
36:22
All right. Now, I've hammered to death this one-particle ring because... Oh, yeah? Representation of these in terms of these lambdas and lambda tilde related to the oscillator representation?
36:41
Yeah, that's actually... I will get to that. That plays a very... Yeah. It's basically the same thing, right? It's basically the same thing? Yeah. Except maybe these... So, yeah, so what Bal is saying is that basically any polynomial ring is essentially a Fock space. I'm just considering objects if it's a ring, polynomials in X, and then X squared.
37:03
This is just like adding one extra energy level. It's our way of doing Fock spaces. And then, in particular, if you look at this and notice that lambda D is delta AB, it's very much a Fock space.
37:21
Commutes to it. And that plays a very important role in terms of the role of these as quadratic generators. And that's a story that I will get to maybe if I have time. But, yes. Okay. So, after beating to death one particle, I hope what I'm doing is sort of clear.
37:45
I'm trying to get away from... I want to go as algebraically as possible, just to get very simple algorithms that you can use to pull these out. And everything that I've said here will generalize nicely to n particles.
38:01
So, for n particles, what do you do? You have a spinner for each one or equivalently a momentum and a helicity state. And you could, of course, use as a basis for n particle states
38:24
just the sort of direct product basis, plain wave basis, and so on. But that's what's been used in the past. And that's not what this... We've seen that we can get sort of better convergence.
38:41
So, what I'm going to do is I'm going to couple these particles together and decompose them back into conformal representations. So, I couple them to some total momentum, p mu or p a a dot, and then ask the same question as I asked back there,
39:03
which is, besides momentum, what else can I see? And that gives you the answer of finding all conformal primaries. So, I will say what that looks like now is as follows.
39:26
Is that I'm considering polynomials in lambda and lambda tilde. I runs from 1 to n.
39:43
The claim is that this modding out by momentum will just leave you with the space of primaries, i.e. the polynomials will be annihilated by k.
40:00
So, to get a feel on that, how that whole decomposition looks like, what you're doing when you take any polynomial ring is its basis.
40:25
So, you're taking the objects here and just repeating them as polynomials, which means taking a symmetric product of them. No, that should be okay. That's okay.
40:41
And here I had two separate ones, which themselves have two indices. So, I will split this ring, this sum, and I'm not going to use direct sums all the time here because it's lazy for me. So, the two-particle question first,
41:06
because I'm confused by what does this mean operation. If I take two particles, yeah, understand what you mean. Construct like all derivatives, and these are the descendants. Now, we want to subtract them away. So, on a certain level, I see wave functions
41:21
where there is a derivative and there is something else. Now, I have to pick something else, which is a primary. I cannot just take any arbitrary combination, which is not a descendant, and declare that this is a primary. There's just going to be one. Very good, yeah. So, it's not specified by this algebraic. So, this specifies pick any representative in this equivalence class,
41:44
and that's your primary, but you don't have to specify any representative. There's just one particular representative. Yeah, that's right. So, I've been a little bit loose on my language. This is just a modulo, so it's a representative class, but there's a very natural candidate, which will arise,
42:01
and it will be specified in terms of UN. I want to say that story quickly, and then get to actually looking at a two-particle example or something like that so you can understand it as quickly as I can. So, anyways, the whole point of...
42:22
Okay, so I essentially need to look at the decomposition of these spaces, and if I say that lambda a i, let me represent it a little bit more abstractly. This is the space for the index a, which runs from one to two, and then w is a space for i, which runs from one to n.
42:43
And this is the sort of magic formula in this talk, which is that this decomposes into a sum on partitions of n, little n, of...
43:00
Now, these are Schur functors, which are symmetrizations of v left across the same symmetrization pattern applied to w. All right, this has a very simple physical... I mean, not physical, operational meaning. What I mean here is I've taken n copies of lambda.
43:24
That's what the symmetric product means. And if I apply a symmetrization procedure to the a indices, it's going to apply the same symmetrization procedure to the i indices. So that's why these are going to be in the same representation. And then the sum is restricted such that this is a partition.
43:45
It means that the sum is restricted by the sense that I can't anti-symmetrize. I can only anti-symmetrize so much. So since a runs from one to two, I can only do it like rank two anti-symmetric.
44:00
So this is restricted to the number of rows in row as a partition. And the sum is only two. So at the end of the day, I will have in this whole object things which look like something, a symmetrization pattern on v left for lambda ai,
44:24
which tells you the Lorentz spin on the a's. Some other symmetrization pattern on v right for lambda tilde. And then, so these are irreducible as Lorentz representations.
44:43
But here as the un objects, basically the i indices, it's not irreducible. And this is where pulling out the primary comes. And so what you essentially have is lambda a dot i j1.
45:08
So the only way to reduce this object in terms of un representations is, so this has some definite symmetry pattern, is you could start contracting the indices between here. And if I contract any indices, I'm going to pull out a factor of p.
45:24
So the claim is that the object which combines these two tableaux into un representation without contracting indices is primary. And with that, there's a very simple rule for it,
45:41
which is that if this represents the tableaux symmetrizing, I don't know, two states here. And actually, why don't we just do one because it will be very easy to see. So what I mean here is lambda a i lambda tilde a dot j
46:01
can be written as lambda lambda tilde minus 1 over n delta i j p a a dot and then plus the 1 over n delta i j p a a dot.
46:24
So this is equal to the adjoint representation plus a singlet. The singlet involved pulling out a factor of p, and this has no factor of p in it. It's the natural candidate for modding out by p.
46:42
This is primary, and it's very easy to see just by picking even some explicit weights inside of the representation. It will be holomorphic. There will be a component of it which you can just see is trivially annihilated by k.
47:04
So what's fascinating about this is that it means that they sort of, so you take kind of the object glued together in the maximal sort of way out of this, and that's primary. But that thing determines exactly the spin representation and vice versa.
47:23
And so they're controlling one another. So the conformal group, or we could say it as SL2C for this primary piece, and this U n together are sort of dual in the sense
47:47
that their decomposition in here determines one another. More precisely, you could look at this and say that the Casimirs of one group are determined by the Casimirs of the other group,
48:02
which you can do by explicit calculation with the formula generators here. So the upshot is that the representations as U n representations, actually, let me maybe not give all that detail because I'm going to end with that.
48:28
We've got to see an example or else this makes no sense. So in this two-particle ring, why don't we pick the stress tensor?
48:51
This was asked about before. So the stress tensor T mu nu is spinner indices.
49:02
It's going to be symmetric combinations of these two. So the tableau that I was ending up bringing in was one which symmetrized the left-handed indices, the A-indices, and then there's one which symmetrizes the right-handed indices.
49:21
The total continuous stress tensor of one theory. You just ask all the questions I want to get to. So the thing is that what this is at the level here is that I started with an object which contains every helicity representation. I took a tensor product of it. So it's just building the entire Hilbert space of an arbitrary.
49:42
You have to select your theory in here, and it's important to figure out what that means here. So you're going to need to be able to figure out how to interpret what comes out of here in terms of particles, and we're going to see something which is this explicitly contains only the vector fermion and scalar stress tensor,
50:05
which is not an accident. So as we have this U2, as U2 representations, I'm going to take this tensor product
50:22
and put them together in the one which forms a primary. And so this has some, this is,
50:40
I don't know how I'm going to explain this very shortly, is that if I take the dual of the, I'm just going to raise the indices on lambda tilde i, and that gives me the dual of it. And so now everything will have an upper index, and I can view this as this, and I want to take the one which causes no contractions. And so this thing is, the claim is primary.
51:05
It's relatively easy to check that it's annihilated by k. You just pick a single state in there. And so I have an overall object which has these indices on it, completely symmetric in i, j, k, l.
51:49
And these run over 1 and 2. So in particular, there's five states inside of this. If I pick i, j, k, l is equal to 1, 1, 1, 1, 1, 1, 1, 1, 2, etc.
52:09
So picking in, for example, all 1s on that, it's automatically symmetric. This thing is lambda 1, lambda 1, lambda tilde, upper 1,
52:20
which corresponds to lambda tilde lower 2. And so if you come back to this thing up here, it involves the first particle, the one with 1 on there, has two lambdas. So it's the left-handed field string. May I ask a question? Maybe it's a bigger question, but just to be sure I'm understanding.
52:44
So there are, say, scalar free theory. If I have a direct sum of two free scalar theory, and I want to compute primary by hand, then you realize that there are primaries. So momentum enters in primary, so it's not obvious that momentum will be condescended.
53:06
It's not true. You can have a combination. So once you want to compute a primary, a genetic primary, even in the direct sum of two free scalars, then you get, clearly you get three primaries, like phi 1 squared, phi 2 squared, and so on.
53:23
But then there are primaries, which are linear combinations, where derivatives do enter. Now some combinations will be annihilated by k, and the other will be the descendants. So the question is, can you see this primary here? I have the feeling that we will not be able to find this sort of...
53:41
A primary which is annihilated by... not annihilated by k? Yeah, they're not annihilated by k, no. In an un-trivial way. But here he has... Can you consider in your scheme theories where there are two types of scalar particles? Yeah. Yeah, but then I don't understand that.
54:05
So do you agree that... I mean, this is a very simple statement. So there are primaries that have momentum into... When you build a primary out of, you know, elementary fields, derivatives can enter. Let me tell you, as simple as that.
54:22
Yeah, yeah. Right? So do they appear here somehow? Yes, he's going to contract the lenses with the lambda tildes, and those are going to be the derivatives, right? When lambda and the tilde come together... Are you thinking that it even appears in this sort of sense, that here's an explicit form of momentum, a total derivative, inside this primary?
54:42
If you have a contraction, you will declare that it is the same. So I misunderstood it. So, well, I would say here that it's actually removing the total momentum component. So I think I need to see an explicit example of what you're talking about.
55:01
No, he just means, like, very simply, like, in the stress tensor, you're going to have d phi d phi. That's basically what he's saying. Yeah, oh, yeah, so what... Okay, so this is the stress tensor, and maybe we should say offline, because it will take a few things to say the indices out in here. But you actually get to, like, get away with not using that,
55:22
specifically because the derivative goes over to p mu, which goes over to lambda lambda tilde. And so, like, there's a way of this... If I wrote out the stress tensor here for the scale, this one turns out to be the scalar. It may not look immediately obvious to you that it's what you... And it's only...
55:41
The scalar of what? Because let's take a theory which contains... So Slava, this one here is... This one's very easy to look at. So I have two things here. It's f left with one particle, and f right for the second particle.
56:02
So your identity... Your identity contains two types of f's, or two types of scalars. Then I expect to see several fields. Yeah, so, good. Maybe I potentially oversold things, but that's a very simple thing to include by just adding another index to this and playing that game.
56:23
So I don't have a beautiful story for arbitrary... Alright, so there's some modification. Yeah, there's some modification to it. But what you do need to know is how you interpret what the state is made of, and that's all labeled by its UN representation. So that's just the number of basically lambdas minus lambda tildes for each i.
56:43
It tells you the sort of particle type which shows up in there. And if you play this game, here this would be a right-handed fermion for 2, left-handed derivative acting on... Sorry, the derivative acting left to right on 1.
57:03
Left-handed, this one is more or less, specifically because of what was just asked a second ago, but I'll write it in a more familiar way where everyone knows that this is sort of what this one's interpreting.
57:22
At first, it also looks like you might need to apply some sort of symmetrization procedure if these are supposed to be identical particles. So this itself is actually automatically symmetric under 1-2 exchange. All that happens when you add different types of flavors,
57:40
like U2 goes to UN, where N is the number of flavor of scalars you have, right? Yeah, this doesn't have to be U2 here, it could be UN where... Yeah, sure.
58:01
One of the things that's special about this explicit example, which you all know, is that the stress tensor is conserved. So instead of having 9 components in 4 dimensions, it has just 5 components from this conservation equation.
58:24
It's no accident that the U2 representation is dimension 5 here. And it's also no accident that the stress tensor, as determined in this sort of procedure, the one which has that symmetry, only contains vectors, fermions, and scalars,
58:41
as those are the only local stress tensors you can have for free theories in 4 dimensions. So I find that sort of remarkable that it's being controlled by this U2 as well. Okay, in the interest of time,
59:01
I need to say just that in D equals 3, it's basically the same story, just take lambda tilde and delete them in all my previous equations, call the UN, it's actually now an ON,
59:22
because you can write momentum PAB equals lambda A lambda B in 3 dimensions, the same sort of story here. But now if I have an I index on them, these are real, so it's ON. If you look at this whole story, all these modes which show up
59:42
in either the ON representation or UN representation, they all correspond to exactly the harmonic modes on UN minus UN minus 2 or ON minus 2. And this at the N equals 1...
01:00:00
level, just as before, we started with the U1 little group, this was a Z2 little group. So you're really only getting out scalars and fermions, which again, are the only free CFTs that you have in three dimensions, perfectly consistent.
01:00:20
And so the ON, again, like the UN is a direct generalization of taking N particles in the literal group and putting them into the full group that they sort of belong. In terms of computations of things, I've been a little sloppy throughout this by
01:00:46
just forcing it algebraically and calling these states, but this is a Hilbert space, this needs to be a unitary representation, there's an inner product which you need to use to compute anything that this audience wants to do. And so I won't say too much, but let's say in 3D, again, you had these sorts of things here,
01:01:07
dot dot dot, this disappears, but in particular, I want to focus on also this e to the i p dot x and then some polynomial in p, maybe p i, from our previous talk.
01:01:24
This just transforms into, maybe I have a problem, you know what I'm saying, okay. Transforms very nicely into, let's say in 3D, this sort of quadratic thing again,
01:01:46
and the exponential, which means you have Gaussian things which you're computing. And so everything turns out to be sort of with contractions again, and it's very familiar, and I think that there's sort of combinatoric structures there which become more obvious,
01:02:01
at least to me, the simplex was confusing. So to sort of summarize and say is that there's a very nice way, in my opinion,
01:02:30
of formulating how the primary fields in various space-time dimensions go. In 4D, we saw the role of the conformal group cross un in 3D.
01:02:46
It's the conformal group cross on, and then in 2D, I didn't say anything about this, but it's again the conformal group sl2r, and it's also on, and from the previous talk,
01:03:07
it just amounts to turning the momenta into spinners, which I just mean to say turn this into some quadratic thing where here, maybe I'll call it u. Just replace those variables, and you'll see this entire pattern show up in everything that you'd, in 2D.
01:03:24
So instead of working with p-, call that u2. When viewing this as harmonic oscillators, I didn't get into a story about that,
01:03:40
which I think is nice and complementary because there's actually different ways of computing in here that go between the spaces. That's worth exploring in terms of speeding up computations, but it also makes it clear what you would want to do if I want to do supersymmetry, which is just add in appropriate creation and annihilation operators,
01:04:05
which have anti-commuting statistics, just as we had Lambda d, which have commuting statistics, and you can probably pull out the representation story that way. It may or may not, hopefully it's obvious to everyone in this audience who's worked a little bit with CFTs,
01:04:22
but the OPE data is exactly what I'm computing here for the free field, which is that, so for free field theory, the OPE is literally just doing the tensor products of conformal representations and decomposing them. So you should be able to pull out the CFT three-point coefficients out of this procedure.
01:04:43
It'd be interesting to explore if that can be done in a sort of fast way, which has potential other uses, including maybe into the sort of input assumptions into the conformal bootstrap. So I think I'm over time. Thank you.
01:05:06
Thank you very much. So we are over time, but it's the last talk, so maybe we can take a few questions. So, yes. So, okay, it's the first exposure, so it's clear that something deep is going on, but just to understand,
01:05:24
suppose that I'm at equal four and I know that my unit theory just has only scalar fields, or, you know, or is equal to only scalar fields. And I don't want these vectors, fermions flying around. I just want, can I modify your procedure in a way that I only get operators made out
01:05:46
of scalar fields on the nodes? Or should I do this global computation and then at the end throw out the biggest part of the field because they contain some fermions which I did not have in the first place?
01:06:03
So I presented a very global computation. At the end of the day, you're just going to write down some tableau. And then what you want for your state, if you have a specific thing in mind, it's very easy. You just fill the tableau with the sort of correct helicity numbers.
01:06:22
You can view this as a computationally, you can do this automatically by looking for, using the UN generators in terms of, so the UN generators themselves are lambda DJ minus lambda sum on the A indices.
01:06:44
The Taurus generators of this, if I call this EIJ, determine the helicity of each individual particle so you can just look for the ones which EII minus the helicity you're interested in acting on just the block pulls out the sort of state that you want.
01:07:03
That's a very brute force way. I haven't given a lot of thought to implementing that in a fast way. And it's absolutely something that would need to be thought about. In D equals 2, can I understand the vertex operators in this framework?
01:07:21
That's a good question. I don't think so. It's very hard since this is all, again, locally based, but yeah. So they're not hiding in there somewhere? I haven't given it any thought. I thought about this when, an hour ago when this was being asked a lot, but I had to say
01:07:45
I didn't come up with anything clever. My guess would be no. Can I ask you a question of myself? I'm sorry, it wasn't, yeah, so I was a bit confused about this statement about the OBE coefficient. So normally we think that even free theories, these OBE coefficient encode something kinematical.
01:08:02
So normally they're very complicated products of gamma functions. And in general, they're not known for three arbitrary primaries and free theory, you don't have a simple way of computing them. So somehow all this complexity should be reproduced by your formulas, right? Yeah, that's, I would say, there isn't a ton of free lunch, necessarily.
01:08:28
But you have to go through the sort of usual exercise of how do you compute Clebsch-Gordan coefficients, which is you take the Casimir written in terms of the other one. Again, it's like something I've done maybe with a few of these, but haven't exploited.
01:08:42
So no, I think it's promising on some of it, but there is some algebra I'm hiding as well. That's right. And two, do you have any way to extract your sorrel primaries? No, not that I've thought of.
01:09:00
It'd be interesting, again, I think that there's lots of very easy routes to go on here. Yeah, basically I've just given you the generators of the global conformal group. So it's a building up on Zlava's question. So there has been, just to be concrete, in genetic dimension, if we say free scale,
01:09:24
you know, you have a unitary bounce and we give you a highest in currents, which are the only specific, you know, and these power operators, which people have been able to compute to the fact that they are conserved currents, the precise form in terms of the double trace Wi-Fi.
01:09:43
So that's how we can buy our database. Excellent question. So t mu nu is the first one in that entire tower, and then everything in this ring, except for the holomorphic polynomials. So this, if you compute d here, it's again just the number of lambdas,
01:10:05
lambda tildes, plus the overall constant is two. And so if it's not holomorphic, this goes over to J1 plus J2 plus two, which is the unitarity bound. So you have all these short multiplets, which are these currents.
01:10:23
And you're able to compute in terms of the phi, the elementary consequence, the specific form. I mean, you extract it from these. If I can use that also for, there's something there,
01:10:41
which is that you're essentially looking at these quadratic terms, which can turn the conserved currents. If this thing here has taught me anything, I wish I got to say it more, is the quadratic nature of these operators. And so I myself am curious as to if pi is the canonical conjugate variable to phi.
01:11:06
If these have any sort of meaning. And so there's an overall symplectic symmetry, which these are actually hiding in. And this is sort of an infinite dimensional version of it,
01:11:24
which if it's integrable, it would actually integrate to this sort of thing. So I've been a little bit curious if there's an interpretation of things here of that. And I should say that I think I forgot to mention, this is aspects of this story,
01:11:40
at least in terms of this duality go in the math literature by Howe duality, or reductive dual pairs. And this is called an oscillator representation or Siegel-Shale Weyl representation, who actually introduced it in field theory originally in this form,
01:12:01
but I don't know much about it. So yes, there's something I noticed that was very special about this two particle space. You do get all the conserved currents and get them explicitly. And one starts to wonder if you can actually just redo this whole procedure now in an infinite dimensional way and kind of look at it and get something from there.
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