Kostant's quantisation program & twistor space

0:00 The first part of the theory of quantization. The theory breaks up quite nicely into two parts. And the first part is what Crosstonton calls pre-contradiction. I should perhaps say that Suryo had very similar ideas going about the same thing at the time. I think for a mathematician, Crosstonton is a little bit easier to follow. Now both Crosstonton and Suryo noticed that some of the sorts of things that people were doing in quantizing classical systems were very similar to the sorts of things which they were doing in group representations. And also, in Surya's case, you notice the similarities of things which are going on in statistical mechanics. And they were able to exploit these similarities. In some cases, the theory of quantization and classical systems was more widely developed. In other cases, the theory of building and proof representation was more widely developed. And by exploiting these, they managed to build up a perhaps fuller theory of each. I suppose the one which is more related about the theory of the group representation and indeed it's the case of assignment that really the only systems, classical systems, which can be quantized with any degree of ease and certainty are those where there is some kind of group working around the back of it. And sort of systems just forming representations of the connotation of it, starting off with the quantum operators which, satisfied by that, can be regarded as the problem.

2:30 There are many representations of a group, a Lie group, whose Lie algebra has three members such that P and Q both commute with A, so that case of commutation relations seems sufficient to the general idea of any group representation. And similarly other classical problems like Harlan's oscillator are transitive to other groups, such as floating around and one can replace the problem by a group theory problem, and again the cases are wider than that. To some extent, the problem of computation is very closely linked to theory. If one really wants to go through and exploit this, one of the things which one wants to do is to make it a little clearer how groups act in classical situations. See, one of the things which is very often being said is that group theory only became important when one could draw a group symmetry. But I think in fact when one starts to examine this more carefully... When they discover that that statement isn't really true, at least not terminally, then in fact the existence of new symmetries puts a much stronger restriction on classical mechanics than people think. One of the reasons is that it's not sufficient that in classical mechanics we just have the symmetry group acting arbitrarily on the phase space of the system, but it has to act in a way which is compatible.

5:00 With the mechanical structure, it has to act as a group of complex functions, and moreover we really wanted to act a little bit better than that, we wanted to act in such a way that each of the one parameter subgroups of the group of symmetries is actually Hamiltonian, in the sense that it has something of a Hamiltonian function, which will give rise to that particular symmetry. So for instance, if we have group G, And let's take the one parameter subgroup, I'll label the parameter as t, so we have elements g of t. If we're going to act on classical phase space, we can first form the corresponding vector field and examine what gt inverse does to a point, and then differentiate between eta acting on f, actually the same as with f. The existence of this function I suppose I'd better just start with some location. I'm going to suppose that in classical mechanics, what we have is

7:30 A syntactic manifold, that is, we have differentiable manifold x, structure omega, and in case of proof acting on this, then in order for the x-contact transformations, the condition is that if we look at omega, evaluated with gx on two vectors, which are themselves acted on by g, as looking at omega at this point here, x. Now, in fact, since I'm going to be dealing with Lie groups, what I want to do is to go back and look at what happens at the Lie algebra level. So, think about group elements in an intuitive way as corresponding to the exponentials elements in the Lie algebra. And I could think perhaps of the things I wrote previously as g of t as being the exponential of tx. I mean, if I go through this kind of construction, I can construct a corresponding eta. Here, which is going to depend on what the element x in the algebra was, in the e-algebra, correspondingly, we can set the spectra field, and the Hamiltonian condition means that eta x acting on f has to correspond to take the Poisson rapid of f with some function, which I'll write as phi of x, and this is the sort of situation that's normally happening.

10:00 For instance, if one looks at the case of quantum mechanics in one dimension, where one has the real line being one's configuration space and a term being the base space, then one has a group of translations in configuration space, what they do, they take a point, a cube, in the base space, and they shift it across into... Q plus A, A. And then if we look at the corresponding action on the little bit somewhere in the range of groups, I'll take something like this, Q plus T, A, as translation through T, A is being my group element corresponding to T.

12:30 If I differentiate that, T equals naught, if I'm in the corresponding vector field, then I get out this A, E, Q on group A. So the vector field would be in this case a, b, c, q, and the Hamiltonian function would be something which gave a to the square root of q when we took a class on that for them. And a function which does that is a to the square root of t. In this case of translation we have this particular Hamiltonian function here. And this is something which is the generator of the group of translations in one dimension. One can also get examples of systems which look perfectly good from a classical point of view, where there's a genuine base-based classical system, for instance R2, and then one can produce a group which acts on it as contact transformations, but which nonetheless doesn't have the right Hamiltonian property, where when one looks at the vector fields, they turn out not to have any corresponding functions. Or another thing going on, and something which I haven't mentioned as yet, is the following. It may happen perfectly well if we do have a good Hamiltonian function corresponding to the algebra, a phi of x. But it may happen that when we look at the Poisson bracket corresponding to x or corresponding to y, we get something which is different from the function corresponding to the Lie bracket of xy.

15:00 That is, x going through phi x is not a homomorphism of the algebra that we consider the ordinary algebra-graphic theory, the class-graphic algebra-geography. It's clearly desirable, and generally happens in the, there is an equality there, and this math theory is actually a homomorphism of the algebra. So, we'll put that on as an extra requirement. Then the inspector feels there doesn't mean locally. Yeah, locally. Yeah, that's right. Always locally there is a function, but it may be that it's not suitable. The other thing is that, in some sense, this identity here never fails to be one very, very much. There's a sort of well-defined way, which is, and I think it's a little bit easier to see in this class, if I've got the synthetic manifold with a group action, If the group G satisfies all these requirements, then what I'll say is that I've got a panel-forming G-space. Clearly, one of the things, again, which we'd like to do is to be able to classify what sort of panel-forming G-spaces are available. And there's some great problems among those ordinary G-spaces. So, first of all, we would hope that the G-space would break up into portions, each of which was transitive, but on each of which we'd move back to transitive.

17:30 There, there is a theorem, and the theorem is that if x only can, then there exists a covering map to O, so I've specified a particular plane as the normal attitude that we look at the dual of the linear, this is the set of linear, the space of linear functionals on the algebra, then there will be a transposed action as we defined in the usual way.

20:00 The dual of the algebra, G-star, is also going to be a space on which to move out. ...in general and ...G-star. So that G-star will break up into orbits and what we do is pick one of them. We'll take, and this map, which enables us to project down from the space we first thought of onto this orbit, is such that we, when we look at what this omega here corresponds to in the new situation. To get omega maps to omega tilde, you find omega tilde, t, evaluated at the pair eta x with eta y, t, omega, evaluated at that pair, is the same as omega tilde, evaluated at the time of the world.

22:30 I'd better slip back into a computer back on this board and explain what's happening. We've had G-stars and G-space. Now, if we take an element, a point, let's say, we take a key, then we can look at the way the linear fractional key, the bracket of X and Y, that's something which is anti-selective in X and Y. It doesn't have any kind of non-singularity properties, which we may hope, but the only thing, obviously, is the answer of a fairly well-defined word. In particular, if we take a point in our orbit O, and if we look, the group is acting, so we can find corresponding vector fields, and let me call these particular vector fields e to the tilde of x and distinguish them from the other ones, that is, they're formed by the same kind of process, except that here we're forming the vector fields with the group acting on O rather than acting on F. By saying that omega tilde and eta tilde x, eta tilde y, t and y, that turns out to be a good definition of a syntactic structure on the orbit.

25:00 It turns out that omega O, together with its omega, is actually a syntactic kind of problem. And that was basically the observation material. When we do that, we can also produce what I call theta over x, a function on the orbit. So it's saying that the playtilde x of p is just a plain development of what p does to x. p being an equational on the algebra gives us a value here and we can evaluate this at each point. That produces an omega tilde and a playtilde. It turns out that all this hinges together and that these three things actually provide us with a syntactic space on which g acts. All of this is covered by a Hamiltonian transformation, in the sense that we just talked about it over there, and what I can then say is that this covering map is such an omega, maps to the omega tilde that's over the phone there, so that in some sense this gives a classification of the possible Hamiltonian G-spaces. It says that they all get spaces which cover orbits in the duality of the algebra.

27:30 Of course, one thing to be admitted there is that this has to be a practical. You can string a number of these things together to get a new one. If you're just looking for the practical ones, then this will be the purpose. Now, it makes the best use of code. The linear algebra, I think, is having the three angular momentum of j1, j2, j3 as the basis. There's a normal killing in a product one, which means it's normal in a product space that can be identified with its two. And so we can also think of this as being a basis for SQ2 star. Now the action of the algebra is simply to rotate these three things in the same way as it would rotate the axes in the ordinary space, so that the orbits, so we've got this situation of a three-dimensional dual of the algebra, and the rotation is just going to be the spheres, just the linear functions, but because SQ2 is an integral of the space, we can identify it with its dual. I should say that this seems to be a thing which very often causes confusion in the early stages, that, in some sense, using SU is not a very good idea, because it just leaves them. It tends to make one think that a lot of things which actually happen in duality are the original happening for the other. Galileo's group. Galileo's group. Galileo's group. You could improve them in two dimensions, because that's simple enough to do, without the attempt of showing that it's difficult.

30:00 Or again, perhaps a better one still is the Heisenberg group. It turns out in the case of collation proofs, again you've got three dimensions here, there's no natural way, but again, in the dual, what you claim is that you want to consist of planes, but together with, in fact you get a picture, which you have any one of these axes, and then if you look at the organ which contains the other two axes, then each point of that is an organ itself. See, one of the interesting things which the Cosmonaut Theory gives you is that these orbits always have to be two-dimensional, even-dimensional, because they're synthetic manifolds, and synthetic manifolds have to be even-dimensional, so in this case we have a sphere, in this case over here we have a dimension two or a dimension zero, and then if you follow through the rest of the Cosmonaut Procedure, then corresponding to each of these planes, you get a complication, the complication... This is one in which Pax constant is given by the distance there, and these individual points give you something that's a bit like a quantization, except it's a sort of classical quantization with Pax constant equal to zero. Can you say, for instance, we order an example, but the only way to detect the structure is to... Well, we take, say...

32:30 In this case, we would have to take on the order, so we have that might be something in the form of 0 and 0 and n. What we would have to do is to see how the linear functional represented by 0 and 0 and n, what that did to an element of the algebra, which was actually the Lie bracket of the two elements of the algebra. Now, since in this case the linear functionals are inner products of things, we would have inner product of 0 and 0 and n with this Lie bracket. A similar example is when supposing one had J1 and J2 now. D-wrapped with J1 and J2, J3. I'm using mathematicians convention which all the I's disappear. I get J3 and then this thing here acts on J3 just to give us F. It would be such an introvert. The J1 factor field is one which is sort of infinitely, that would be something that would sort of be a field like that. The J2 one will be one which goes that way. And this will be saying that if you take the syntactic form with those two that sign, you get the... If G-star is the representation space of G, can I think of a better way? G-star is a representation space for G. It's because G acts both what's called the co-adjunct action. That is, it acts where the transpose of the action does. I'm just telling you the physical picture of it. I create definitions of... I mean, the definition...

35:00 Here you can see you have a linear function on the table. If you look at our g-adds, that gives you the... and that means that we do have an action of the group on the dual realtor. I mean, as it stands, this is not quite an action. What we have to do is to take it on the table. That's just a general action of the group on the dual realtor. ... the corresponding functions. I mean, for instance, if we want the function corresponding to j3... The evaluation of point P, sorry, point, say, 0, 0, that was just the definition, but then you come to 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, There are many wonderful set of squares you need to make the thing invariant. In fact, since the group acts as contact transformations, since the sphere has a unique syntax which is invariant under rotations up to multiples,

37:30 you know from our earlier reasoning that it must just be some multiples. The only extra thing that the construction gives you is the actual molecule itself rather than something else. You see, in this sphere, it's just the axis. Would G start the Newton-Hawkins list? No. No, because if G is a contrary group, then the algebra is ten-dimensional, and the deal with the algebra has to have the same amount of ten-dimensions, so it couldn't possibly be. In fact, it's quite a complicated object. It would have to be this big of a problem, but it doesn't exist. Well, it's a big uptake of twistor space and that sort of thing, but rather an awkward thing. It's known what it is, but the trouble is that there are a lot of other things.

40:00 There are basically only two planes of orbit. There's the origin and there are spheres. But in the case of the Planck-Rae group, there are these two planes of orbit, but they are in a very simple relationship. And in the case of something like this... I think one interesting point to make here is that many people have been so minded to study root symmetries before quantum mechanics came along that I suppose in principle one could have given a description of the elementary particles by using this theorem, one with the analogous thing of saying that one was looking for an irreducible representation of the Poincare group would be to say that one was looking for a robot. Can you go back to that theorem you were writing down? Yeah. Can you give some feeling for the kind of covering that you've got and how much? Professor, I don't want to say too much about it except that in the sort of cases one's interested in orbits rather than just covering maps and covering these spaces which of interest are actually the orbits themselves.

42:30 Could you give us an idea? The method of proof of the theorem? Yeah, it's diapolically simple actually. You're given, to start off with this function, that at each point, at each point of this complex paraphore, There is, given an element of the theory of algebra, this function and the evaluation of this function. Use that as the definition of what else the evaluation on capital X of this theory of algebra is going to do. Now what I've got here is that something which takes points of my original synthetic manifolds and turns them into linear functions on the linear algebra because it's very easy to see that this thing is a linear function because the matrix took y to the y-axis of the homomorphism and therefore that sort of thing is linear. So this is linear in y, so we have a linear function on the linear algebra. If we just consider the math x goes to px, that is the government math. x was transferred.

45:00 And we can also see that this, if we follow through the definitions, we find that the, if we act with the group element in here, has the same effect as acting with the group element in the co-adjuvant action over here, and so, as opposed to putting it in another way, you could say, P, the point of the geodex, the x-corresponding orbits in the geodex of the algebra, the synthetic manifold is not as itself, so we go over onward. It's a matter of... Constructingly, the syntactic structure itself goes over in the right. And that's just a matter of figuring out a way in which these functions here are defined relative to the vector field. Calculation rather than the things out in the two cases, you finally can make it work. The base code here, there's no problem at all. It's just one of these tricks of identification. And then in the non-linear, it's a covering. It's a particular way in which it goes, if you find all that comes out. If anybody wants to see details of the proof, they're in the screen on the left hand side.

47:30 This is a sphere of distance that can be set up by quantization. The procedure of quantization of a classical system consists partly in building up a Hilbert space, which normally consists in the ordinary quantization. The operators are normally given quantum mechanics to be solved by doing them, but there's no reason why we shouldn't let them be skewed in to be multiplied by i's, and since that's a convenient way to do things, that's a different version for those who use the quantum mechanical operators to be skewed in. So one of the first things you want to do is to decide what the silver space of wave functions is going to do. If we look at the functions on x, if we look at mod psi-squared and integrate that against omega to the n, then that is plane 1. x being a symmetric manifold has dimension 2n. Take the exterior product of w itself n times and I end up with, and if the manifold x is oriented then I can actually integrate against this.

50:00 I can check out and see which particular functions here give me a plane 1 answer. The only thing is that it turns out that's not altogether satisfactory, or at least it's not always satisfactory. It's a little bit too restrictive. So what we will do really is not so much look at the strict functions, but some lame numbers, and then look at the second ones. Because in some cases, the copies of C are a little bit twisted up, and the flavors of this thing are probably the same. Now that raises another question, which we could perfectly work in a reasonably unambiguous way, in the case when we go to London, it's not quite so clear what we should be using there, and what we want really is some notion, if we've got two things in here, say we've got this side of that, and then we've got another section beside all of that, we want to know how we should multiply those together in such a way that we can then go on and integrate some notion.

52:30 I'm in a particular phase when I'm taking the same thing each slot. We've got to do that. And that gives us some idea of how we should be able to translate the... So we want something notionally parallel transport in this line bundle. So we want not just a line bundle plain and simple, but a line bundle with connections. And I'll take that connection to be... Some things like mathematicians and physicists, I would ask them, why are they doing all this? Because there are examples known where the language isn't previous or... I don't know if there are many examples where mathematicians and physicists can have this type of activity. But certainly there are language others can have. No, I mean there are language others which are of physical importance. That sort of thing which we used in describing the massive particles transforming into a comparator of language others can have. So, you know, it's not a generalization for generalization state, it's something which one really needs. Okay, so one has to learn, and one hopes to have, these other two concepts of a sort of metric in the playbook of parallel transport. And just as in general relativity, one connects the connection to the metric in the tangent space by saying, oh, one wants the metric to be invariant in parallel transport, one does the same thing over here.

55:00 The parallel transport, this possible function over here, and in that particular game, I should say that supposing that I take some kind of basic section, S0 locally, then any other section will be described by some function times that locally, so I make that a function, and then if I want to say what this does for this thing, then I'll take that that actually gives me X on theta times... I'm going to show you an example of a formula called S0, where alpha is a one-point. So that would be the way my alpha here actually get out of the formula for the algebra here. In fact, what happens is that if I take S0x, what the value of S0x squared would be. Well, it turns out that if I differentiate the natural logarithm, divided by 2 pi, and I actually get the difference between, if I take this formula... Here, I look at the difference between this complex conjugate, which of course is imaginary, is actually the same form as this one here, and that's the sort of the link between the favorites.

57:30 Incidentally, one sees that if part of it is real, then this thing here vanishes, and what one actually has is this thing vanishes, so one has a constant of them. And so on. The cost is exactly the other way round. We start off with a curvature, in a way which I'll explain in a minute, and then we find a parallel transport, and then we come back and find a method. And the way in which I've worked out is that the curvature of the form which is given by this alpha is just d alpha. I mean, in general, in a general environment, when you have something like that, the alpha is actually an orderly one form. There we can compute selector and roto, and we can actually de-alpha as a curvature of the connection, and de-alpha as a two-form of the one-form, and the constraint which we put on is that de-alpha is in fact exactly the syntactic two-form. And so what we started off by doing is looking for lane bundles which have a connection such that the curvature of the connection is actually the syntactic two-form.

1:00:00 Is there any one? Well, I mean, there's been a particular group, the whole of the non-zero complex, non-colon flavours. I mean, actually associated with that one. Well, I must confess here that I'm using a slightly different notation.