Old problems in quantum gravity in the light of new loop space variables (last part) / subsequent discussion (contd.)

0:00 Carlo Rovelli will speak first. He'll speak for approximately 25 minutes and we'll allow about five minutes for brief questions and then Lee Smolin will speak another 25 minutes and then we'll have a period for longer discussions. I will report here on work done in collaboration with Lee Smolin on The definition of your approach to quantum gravity, which is the loop representation approach. The central idea, as already produced in this slide, is to represent states and operators of quantum gravity in terms of function over the space of loops. Now, representation is used here in the sense of Dirac. That is, we think we have an abstract state space with an operator algebra, and this abstract space may be realized in concrete by specific choices of functional spaces, that is, choice of basis. Now, the idea of the loop representation for quantum gravity follows from... It is a direct development of two previous results. One is hastical reformulation of general relativity, and the second one is Jacobson-Rismolin's discovery of a set of solutions, a large space of solutions to the Miller-DeWitt equation when you're looking at the hastical variables. Now there are two central results obtained in the loop representation. The first is that it is possible to define a regularized algebra of observable and constrained with regularization that preserves the diffeomorphism, the three-dimensional diffeomorphism invariance of the theorem.

2:30 And the second and most surprising result is that it is possible to find solutions of the constrained equations, of the quantum constrained equations, and more precisely It is possible to write the general solution, closed form, the general solution of the different Morpheus constraints and a large class, large space of solution of the entire set of quantum constraint equations. Now, these turn out to be related to Knott and Link. Now, there are two ways to arrive at the loop representation. One is the one that Abai sketched this morning, that is starting from the self-dual representation, In which the state is represented by an holomorphic function of the connection and then defined from that transform to a space of function of loops. This is somehow equivalent, as I said, to define the momentum representation for one dimensional quantum mechanics. By first defining a coordinate representation and then defining Fourier transform and usage to bring the observer and the dynamical operator to the loop representation. This is the way through which the loop representation was found and is more intuitive, but essentially because of the little control that we have of this multidimensional integral is not particularly rigorous. However, there is a second way to arrive at the loop representation, the following, which consists in... forget... Now, let's talk about new representations and these loop solutions and start from the beginning, from scratch, from the classical theory, from the new variables, and quantize a closed algebra of observables that we call T-observable, that is, look for a representation of this algebra in terms of linear operators.

5:00 It is not canonical Poisson algebra that is not done by coordinate and momentum and this is therefore in the framework or at least in the spirit of what Aysham has been saying many years and this morning that The quantum reality has not to be searched in Fox space, Hilbert space, that is, the fundamental observable algebra that we have to quantize has not to be a canonical algebra. So that's the algebra. Now, this second way of arriving at the loop representation will be described in the second... In the next talk by Liz Morling, he will describe this algebra and its quantization and he will discuss the regularization of this algebra as a constraint and somehow he will re-obtain these results that I will get here waiting in the context of a more well-defined regularization scheme. Okay, let me flash this transparency of the self-dual representation. The self-dual representation states a homomorphic function of the connection, the fundamental observable of the connection and its conjugate momentum given by this operator, and we obviously have three There are also quantum constraint equations which express the SU invariant. This is the new constraint in the invariable theory and the old constraint, the homomorphism, and the new version of the Leibniz equation.

7:30 Now, this object, under suitable conditions on the loop Gamma, is a solution of the Hamiltonian Moreover, since it is clearly gauge invariant, its whistle loop is also a solution of the gauge constraint. Now this is a result that can be obtained by a formal calculation as well as in the context of a suitable regularization of this video. Now I will use also, and Apple would be crucial in what follows, multiple loops. I will use this notation that the holonomy for multiple loops is simply the product of the holonomy of the similar loops. Now also this object here is a solution of this equation under again suitable conditions of the loops. Now these conditions are that the loop or the multiple loops has to be differentiable and without self-intersection. To the loop basis, that is, introduce this object that already Abai discussed this morning, and try to represent the states of the theory, not in terms of this object here, but in terms of this object here. So what we need is a mapping from this to this object, and this is what we want to define. This mapping that we call a format is defined in two steps. And is the following, is we go from the central representation to its conjugate representation, that is, we consider instead of the set of the C, the set of the linear function of the C, physically we go to the graph from the K, the distribution on the side.

10:00 Now, on these... On these are naturally defined the conjugate operator of the only operator of this theory, which give again a representation, a better representation of the observable art. Now, the mapping that I defined goes from this object to the set of the linear function of the, of the function of multiple loops. And how is it defined? It is defined in the following way. That is, it is defined in terms of the anomaly that was introduced in the past. This is the main object in this story. I have here multiple loops. I'm not requiring it to be differentiable or everywhere or not self-intersecting and actually the space of loop that I am considering is the space of piecewise differentiable multiple loops on the theorem. Let me say immediately before commenting that this, because I'm considering multiple loops, This object has a kind of pock-like structure, because there is a component of one loop or two loops or three loops. Now, the transformers define this way, that is, given one of these distributions, the corresponding loop functional will be the value of this distribution on the holonomy. Now, this is a rather abstract definition, so we won't... In order to do that, I introduced a measure, mu, on the space of the connections, and I stressed the fact that this measure is not necessarily related to scalar problems in the space of unconstrained states, so we are not required to be engaging variants or to Or our fundamental operator to be self-adjoint, in this respect, in the structure defined by you. If I have a measure, then this distribution, or better, some of these distributions, suitable subspace of them, is expressed in terms of function, function of the connection, in this way.

12:30 And I can write this formula in the following way, as an integral. And now the analogy with the Fourier transform is clear, where the analogous of the exponential is given by this analogy. This is rather formal in the sense that we are not addressing any question related to the actual support of this measure or to the actual space of integrable functionals with respect to this measure. However, let me just stress that if I change this method with another measure which is not singular with respect to this one, in particular if I make some diffeomorphic transformation or a gauge transformation, The transform itself does not depend from the choice of the main theory, since it is defined... Pardon? Absolutely, yes. Let me... I mean, this is related to how do we actually choose the space of this object here. I mean, there is... The measure has to be the one in which these states of space are integral. There are these... Right, yes. This is the... Yeah, you are asking if there is an alternative kernel. You are asking if this is invertible. I think Chris' question is that there's lots of measures and then, in the first... whatever you're getting at the first class... There are different types of functions that represent the same distribution. This is...

15:00 What is the question, sir? Are you saying that that measure used only to find the technical facts about that and it's not very profitable? No, absolutely. This is a mapping that you give me a file and I give you back a new principle. I think the statement is that the transform is defined independently of the place of measure from the dual space to the root representation, because the bias is defined, the transform is just defined by that top equation in red, but if one wants to represent it not from the dual space of itself to representation but from elements of itself to representation, one must choose a measure and then it depends on the measure. Well, there are specific cases in which one can choose such a measure and actually compute this integral, and I give one example this morning. In the linear, in the linearized theory, we have a specific choice of this, the set of this function, so we have control over the set of functions that we have, and there is also natural measure to be chosen, and we can perform this integral. I mean, this is exactly the point. I mean, we have two ways to arrive at the loop representation. This is exactly the point in which this way is formal, as I said in the beginning. And the reason is, we don't know about this matrix because we don't know what exactly is the space of function we are starting from. It's a self-dual absolute. My statement about the fact that this does not depend on the choice of the measure doesn't mean that any measure has the same result. It means that if I make a gauge transformation, the way in which the measure transform and the gauge transformation can be absorbed is accurate.

17:30 It does not depend on this type of treatment. Now, the functional phi that represent this given distribution depends on the particular measure that we choose. Now, what is the sense of introducing this transform? What we really want to do is to use it to bring the entire theory to the loop space. Essentially, what we have to bring to the room space is the observable and the constraint. So, thinking at the Fourier analogy, it's clear that what we mean is the definition of equivalent operators. If we have a loop operator, or tilde, and a self-rule operator, or hat, I say that they are equivalent if they transform into rings between the two. Transparency, this is equivalent to saying that the self-dual operator and the loop operator act in the same way on the anomaly. Well, I remember this one acts on the connection and this one acts on the loops. And the analogy here... I think it's quite clear. In the Fourier transform, I say that this is a momentum operator in the coordinate representation, and this is a momentum operator in the momentum representation of the atom in the same way, in the kernel of the Fourier transform. Now, not every self-dual operator has a loop equivalent. In particular, the connection and the conjugate moment in the connection have no loop equivalent. However, and this is the first and anterior result, there is a big enough number of observable,

20:00 well, big enough in the sense that Chris was saying this morning, that can be brought to the loop representation. In next talk, Li will describe a large class of them, so let me just give one example here. And finally, a class of operators that I call T0, that actually can be carried to the loop representation, and this T0 is defined the following way. I'll use again this holonomy. Now, I think A as an observable, a function of T space, so for any alpha, this object can be seen as an observable. The corresponding operator is a diagonal operator, and I want the corresponding operator in the loop representation, so I have to use this formula here. Now, it is direct to see that... This is satisfied if I define a loop operator T tilde for any alpha in the following way. T tilde alpha acting on a loop functional in the point gamma, gamma is a multiple loop, is the loop functional in the point multiple loop gamma union alpha. The action of this operator on the low limit is equal to the action of this loop operator on the low limit. So this will be a self-dual operator corresponding to this classical observable. Okay. Now, not only I can bring a class of observable to the loop representation, but I can bring the constraint to the loop representation. Let me discuss the constraint and the solution of the constraint. Well, on the SU , the gauge constraint simply disappears in the loop representation, since everything is gauge invariant.

22:30 The volume is gauge invariant, so we are transforming it into the kernel of the C2 operator. Now, the diffeomorphism constraint in the loop representation is essentially the derivative of diffeomorphism on the connection. Now clearly if I act on the autonomy by transforming, by moving in the free space both the connection and the loop I get zero, so not surprisingly the loop equivalent of the diffeomorphism constraint is again the generator of diffeomorphism, that is the generator of the action of the group that comes around in free space. I want to solve this equation, so I want the functionals that are annihilated by this generator. That is, I want the functionals that are constant along the action, along the orbits of the diffeomorphic group on the loop space. Now, these orbits are... Objects quite known and very well studied in the mathematical literature because two loops are in the same orbit of the diffeomorphism group, that is, they can be brought one to the other by a diffeomorphism if and only if they are knotted in the same way, and two sets of loops if and only if they are knotted and linked in the same way. Therefore... These orbits are what, in mathematical literature, is called the knot and link classes of the... yes? I was going to say something about . Are you saying that ? Yeah, there is an . Yeah, yeah, most obviously. Yeah, right. And the orbits of this group are given by these link classes. Actually, here, our loops have... These are differentiable and there are intersections, so we need a slight generalization of the linked classes in order to include the present of this corner and these intersections. The generalization needed in order to include the corners is quite trivial, the generalization needed in order to comprise the intersection as, I mean, there is some literature on that, a little bit. Now, so what I've said, I've said that the general solution of this constraint is given by a loop function which depends only on the linear, the generalized linear class of the multiple loops, or in other words that we have one independent solution for every generalized linear class energy of sigma. Now let me come to the Newtonian constraint.

25:00 Now, what I'm wondering is, if I go back to the transform idea of this function, do you know what function of that is? Do we have the property that we use the function of the ? No, because I don't have an explicit inverse transform. If I had an explicit inverse transform... I mean, I can think of one, for example. If I took the function of a, it would be 0, except for the name is black. There are a lot of examples of these things, but they tend to measure in the sense that one is not necessarily different from what is in the other. If this exists, and it would probably take some time to exist, the transform doesn't have to exist, but if the transform happens to exist, there's a particular measure that means it's not going to be different from what is in the other. The inverse image of it is different from what is in the other, and it's not going to be different from what is in the other. It doesn't have to be the same thing. Is that correct? No. But maybe we should discuss this after. But it doesn't have to be like that. Okay, now, um... The information of n is kind of a counter to the question of n? Right. Absolutely. Absolutely. This is a basis. I mean, this is one independent solution. I mean, any Hilbert space has a counter to the basis. So, let's continue with that. But it's enough to do a certain form of work, right? Okay, the Hamiltonian constraint is actually the one that the transform was supposed to make simpler, and in fact, it is possible to write a loop equivalent of the Hamiltonian constraint, but it is not difficult to do, thinking of the Klein-Gordon Fourier transform analogy that I said this morning, that if I have a loop state that has support only of non-self-intersected, differentiable loops.

27:30 This intuitively is just a linear combination of those particular anomalies that do solve the Hamiltonian constraint, the self-dual representation, then it is a solution of the Hamiltonian constraint. Now, this actually happening can be calculated after regularization of this operator. Now we have one set of common solutions of these things which are given by loop functional which has support only on this loop sphere and depends only on the linked classes and the linked classes that satisfy this are the ordinary linked classes of the three manifolds. So we have one sector of solution of the entire set of constraints which is given by the ordinary linked classes of C. I don't remember if I mentioned, in the self-dual representation there are other solutions of the Hamiltonian constraint which are related to intersections, so here corresponding there are another class of solutions related to intersections and I don't want to enter the claim here, and we don't know if this is a complete set of solutions or in which sense if there are other solutions. So let's jump to the conclusion and summarize. First... The state of cerebral and constraint of quantum gravity may be represented in terms of functional over the space of these tri-differentiable multiple loops, too. This representation may be obtained In this representation, the constraint equation can be solved. The general solution of Newton-Wolfe's constraint is given by a function of the generalized domain classes of sigma and a space of solution of the entire set of constraints can be defined in class of form and contain the ordinary class of sigma. Now I want to stress this blue thing here, that these results are obtained without any additional input besides general activity and standard quantum mechanics.

30:00 There is a great number of open problems, obviously, and let me mention two of them, because the last one is that there are solutions besides this. Another one that I would like to mention is the following. Our expression for the loop constraint operator is in regularized form. It is a complicated expression in terms of a limit of operator. However, if we formally consider the unregularized operator, this is given by an infinite factor multiplying... Now we cannot use this operator since it is divergent. However, I think that this is something quite interesting. This has to be understood, because in some sense this which is maybe the simplest geometrical object or space, they are coded inside the entire dynamical state, and therefore the high-secretion, the dynamics of general relativity. And finally, let me conclude by saying that the structure, the general picture of this mid-space representation is Quite rich and complex, however, is far from being a theory for quantum gravity, because there are two crucial elements which are missing, and one is the Hilbert space structure of the set of these solutions, and the second one is the definition of physical observable, to be defined on this. In the wing class, you have a solution that satisfies all those constraints, and that's the function, which satisfies all the constraints. If it satisfies all the constraints, now what I want to know is, if I do the inverse transform, and dive back to the wave function on the A field,

32:30 isn't it necessary that that also satisfies all the constraints, and therefore it must be the key function to be right, and therefore the transform must be the key function to be right? Yeah, provided that you're able to... Well, I don't understand, but the order, well, it's a complicated issue and the loop space doesn't give you an advantage because of the difficulty of asking yourself, but the measure, well, the ordering on how, if one takes the operators back and forth, one would expect to get an operator, when one acts on the space of self-dual functional, which depends on the measure. That takes you between the consciousness of space and the space that looks like space. The idea is that bonding your space with your self-representation, you can get things that look like this in your face and generate this naively. And to expect that when you take them over from the dual space to the space itself for representation, that will depend on the measure. The measure is not expected to be between what they're doing, right? Nobody knows, and they may stop, but there aren't any between what they're doing during the measure of any use on the space that's mentioned on the manifold. So those operators will depend on the measure, and therefore their solutions will have to be from what they're doing. So on the loose space, or on the conjugate space, things will look simple, but on the space of functional debate, things will not look simple, and you won't be able to recognize the functions as being... When I approach them directly into the test, it takes the air for the test, which will also be different. However, it is somewhat difficult to exhibit these things, and I will give you a number here. If I work in the traditional variable, which is the three metrics and the three terms, let's forget about any terms, let's just talk about them. Then I could have worked either with the principle, which is represented, or with the metric representation. With the metric representation, it is rather easy to write down a whole bunch of functions that you cannot prevent.

35:00 To write the same function, let's say, the curvature of the statement, I don't know, let's say, the function, let's say, the function, let's say, the curvature of the statement, let's say, the curvature of the statement, the statement there is that also there is a duality, something manifesting in the model, and they can be transformed back to what they look like as well. Are they? What is our determinants of KAB? If you have a metric that is easy to construct, it's impossible to do that. If you don't have a metric at your disposal, you can't construct something, you cannot put And if I write down for you some obvious functions that you should know about, here the curvature squared times the volumetric, just like this. And you ask me what does it look like, you see what it looks like. The form of it, I should be able to make a transform and go to the curvature of it. You can ask me what that transform should look like, I have no idea how. That's what I'm going to do. It's not transparent. But perhaps we should say that that's the discussion we'll actually do soon. Well, this discussion is actually a very good starting point for my talk. The way in which we originally came upon the loop representation is the way that Karloff has indicated. However, exactly these issues arose and disturbed us and are clearly subtle, are clearly issues that involve subtle issues of measure theory and so forth that are complicated I certainly don't understand, but the resulting The system of equations and algebra and so forth on the loop space was simple and could be understood without these difficulties, so it occurred to us that there must be some other way of getting at this result and establishing the distance of the loop representation, the algebra of operators on it, the action of the constraints and so forth, without making any reference to the self-dual representation, and then it becomes a question for further work.

37:30 What the nature of the relationship between the two representations is. So it's the task of my talk to do this, to present this alternative approach to the loop algebra. Abai did to me what Jim did to him and chose my title, but it's an appropriate one, luckily. And so this is what I'll do. To the extent that I have time, some of the conceptual issues that arise in approaching quantum gravity in this fashion, and I am going to try to balance my remarks between the first task and the second task, I both have to show you enough detail about this construction to convince you that it's worth looking at further, and that it's well defined, and... I will not show you so much detail that I have no time left for some remarks about what the meaning of all this is. My program is the following. We're following very much, as Carlo said, the point of view set out by Chris for many years, which I only understood after you had done this. Algebra of observables, which is an algebra under Poisson brackets, which is closed onto itself, and which is not canonical. It's not based on the local a, sigma, tilde, or q and p, but nevertheless it's a closed Poisson algebra. We're going to quantize, not in a canonical manner, but in a different manner, by finding a space of linear functionals. Which will be these functionals of loops such that there is an algebra of linear operators on the space of functionals which have a certain commutator algebra which is a deformation of the algebra of the classical algebra of observables. Meaning of the term deformation is equal up to terms of higher order in h bar. Of crucial importance in this whole construction are two issues that I want to stress all the way through. One of them is regularization. None of this will make any sense if we have operators whose commutator has coincident delta function singularities that aren't well defined, so I will show you how this whole thing can be done in a manner which is regularized, which is to say the operators are defined so that intrinsically no coincident delta function singularities can arise from their commutators.

40:00 The second issue, which is closely related to this, is how to do this without breaking the spatial dimorphism invariance. And I'll give you the clue in advance. The clue is to use the parameterizations on the roots. But I will show you how we can do this. After setting the whole thing up, which will take most of my time, I will go back and discuss... The way things behave under spatial dickeymorphisms and the character of these solutions that Carlo discussed, and I'd like to stress some features of them. I think it's Chris this morning said that it would be very interesting to know something about the space of three-metrix model dickeymorphisms, and I would like to suggest that in this representation... Can say something about the structure of the analogous problem and the reason we can is that the mathematicians have been so busy and fruitful with knot theory, so I'd like to make some comments about that. If I have time, I'll discuss how the results on the Hamiltonian constraint are recovered in the context of the regularization that I introduced. So we want to have a classical algebra of observables. We'll work on any compact pre-manifold. The extension of all this work, by the way, to the asymptotic and flat case is crucially important for getting an algebra of physical observables that one understands, as I'll say in the end. And that's in progress, but for the moment it's compact. And the space of curves we'll be interested in are piecewise differentiable, non-degenerate, that is, their tangent vector is never zero, and parameterized loops. We're going to be interested in finding an algebra on the gauge invariant sub-manifolds, and let me just tell you, without further ado, how this algebra is constructed. Well, this algebra is a graded algebra of observables that depends on loops. So there's a zero part, a one part, a two part, and so forth. The zero part is what has already been introduced. It's the trace of the holonomy of the loops. It depends only on the a and not the sigma tilde. So that is a complete set of commuting observables under Poisson graphics, but it's certainly not enough to parametrize the phase space.

42:30 The way in which we extend this to observables that have parametrized the whole phase space is we need something that depends on the sigma tilde, but we'd like them to engage invariant observables, and so what we do is we draw loops and we insert. The sigma tildes and then take the parallel transport around the loops and then trace. So these observables of the simplest one is called T1 which has one insertion of a sigma tilde and I'll explain this rather arcane notation but this is the best one we've so far found. They depend on a loop gamma and a point x at which the insertion takes place and an index a which is a spatial index. They are proportional to the trace of the parallel transport around the loop. And the sigma tilde a at that point. But of course, this has to be a singular operator because sigma tildes live at points and loops are one-dimensional structures, so we have an integral ds around the loop of a three-dimensional delta function, which tells us that the point x is at the point gamma of s. So this formalism is full of distributional type singularities. And although I don't have time to show it, I can comment on it later, there is an extension of the formalism in which one eliminates all of these things by integrating over suitable test functions and essentially one has to go to ribbons of operators. But that's a long technical thing, if you want I could discuss that. But it's important that the only singularities that arise are distributional singularities and they can be gotten rid of with suitable test functions. So this is the definition of the T1 algebra. By the way, it's a density of rank 2 at the point x, because that's where we'll put the weight of the delta function and the sigma tilde is the density of weight 1. No, no, no, we're in the classical theory. This is a classical observable in the phase space of general relativity. It's connected to the sigma tilde, so that goes there. So these are rather complicated things. But let me show you how to work with them. The two that we've defined so far, the t0 and the t1, have a closed algebra, which we call the small algebra. We're eventually going to be defining tn for n greater than 1, but let me show you these. The t0 commute with the t0. What about the commentator of the t1?

45:00 Well, the essential thing is that the T1 has a sigma tilde that when you, if it happens to have, to live at the same point, let's say that, well, here is a picture. This curve is gamma. This curve is eta that the T0 depends on. If the point X happens to be at this point, which is the one point where they meet, then in their Poisson bracket, the sigma tilde at X. The term has a non-vanishing term with the A's in here and the action of that is exactly to break the two loops, eliminate the sigma tilde, and rejoin them in two different ways. So one gets something which is proportional to the trace of the holonomy around these two loops where I've broken and joined in two different ways. Okay, and the fact that there are two ways to do it, the fact that there's a minus sign there, all come from the SL2C two component spinor algebra, and the- Well, there's a rule for this, which is that these are parameterized, so they have orientations, and the number of times you have to switch the orientations to the minus one, minus one to that turns out to be the rule. You have to check that. These are multiplied by a singular distributional function, which I'll comment on if people want me to later. So this commutator, in the end, is written as the singular distributional function times These T0s, because there's no longer any sigma-tildes here, and gamma, sort of sharp eta, means break gamma and go around eta, and what the inverse means, go around gamma and go around eta the inverse way. So a T1 with a T0 is a T0. A similar calculation shows you that a T1 with a T1 is a T1. Yes. Absolutely. Absolutely.