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

0:00 Tn is supposed to kill this great function for arbitrary n. So, imposing this infinite set of constraints with arbitrary n, we can explain about this algorithm. Does that make infinity easier than something else? n doesn't have to be constant, that is expected, yeah. n is arbitrary, so therefore we just get a functional equation which just involves... at each point we get an equation. So what happens in the linear idea? This is a hope and has not been realized in the full theory and I don't have any positive or negative results, it just doesn't look that sufficient. What happens in the linearized limit? The statement is that this idea actually does work in the linearized limit. Now, the Hamiltonian operator that I had up here in the full theory can be written in this holomorphic representation, just transcribe it literally in the linearized limit, and it just becomes, so to say, the creation operator multiplied by the annihilation operator. c times delta by delta c, this is the typical form of the Hamiltonian in the Bergman representation, plus, in fact, this term just reduces to delta by delta t of x. The T of X, the time in the linearized theory, can just be taken to be the imaginary path of the trace of C. So this, in fact, is the second order of the relativistic equation, exactly done in Carroll's paper. And, in fact, we get the isolation of time. I can again go into this in detail. So that is one example of this application of the new variables. The next thing that I would like to do... What was the time in that? The time in that was the state of the magnet. It was the imaginary. The wave functions depend on the linearized connection sequence.

2:30 In this decomposition, the transverse, transverse part, longitudinal part, and the lateral part. And the sine is just equal to the imaginary part of it. There's a hole inside that holds everything. It's a global statement. Exactly like your statement. C is the linearized connection and Ct is the trace part of it. That's right. That's exactly why it works. Now let me go back to the main program, and the main program had to do with, you introduce the operators, then you choose the representation. What I would like to do in this third part of the talk is to tell you a little bit about these representations, where there are two representations, there is a sound zone representation and the loop space representation. Now I would like to tell you how they look like in the linearized limit. A more complete story will be in the talk that will follow. So, in the subdual representation, you first of all define C to be the linearized connection, just take the linearized connection and convert the internal induction to space-time, and now you can impose the quantum constraints. Up here, the program was, you choose a representation, impose the quantum constraints. Now, of course, as everybody knows, the key simplification of the linearized theory is that you can actually solve these equations explicitly, the quantum constraints. And finally, symmetric, transverse, traceless part out of the CAD. To begin with, CAD is not necessarily symmetric, because we have, basically, because we have an internal index up here, we have the freedom to make arbitrary internal rotations. And then, however, when you impose all the constraints, the wave function just depends on that. And I want to emphasize that, of course, this is a key simplification, absolutely essential simplification, which is not available in the full theory. There is no doubt about it now. But nonetheless, what I would like to do is...

5:00 Exploit this simplification and try to carry out the program that I outlined above, so to say, step by step. In particular, I would like to focus on going about choosing the inner product using the reality conditions and the unitary evolution as my guideline, just to see how it works in the leading experience. It is the whole of mathematics. It has information about that. Now, again, all these things are sort of fool's talks in their own right, so what I'll do is just to tell you the outline of ideas. The ideas are the following. If you want to incorporate something like hermeticity conditions or reality conditions, in other words, if you have some functions in the space, think of that here, and you want to sort of make the corresponding operator Hermitian in the context. There is one natural way, in other words, a very nice way available, if you can do it, and that nice way comes out of geometric quantization. Geometric quantization proceeds in two steps. The first step, which is sort of, say, in some sense trivial, namely, if you are given a phase space, then you can construct so-called pre-quantum theory in which, given any function whatsoever on the phase space, you can promote it to an operator in this pre-quantum Hilbert space. The second step is a non-trivial one, when you introduce a certain polarization. And it is in the second step that not all classical functions go to well-defined quantum operators, you get all the sacroiliac problems. However, if you work with one kind of representation, one kind of polarization, which is called a Kähler polarization, then you have the advantage that most of the basic, elementary, simple classical functions go... First of all, they go to precondom operators and those precondom operators, just as they are, without any further work, go to condom operators. So this experience suggests that, in order to incorporate these conditions, which are basically hermeticity conditions, perhaps what we should do is to use a scalar polarization. And what I would like to indicate in the next transparency is that you can do that and it

7:30 actually does give us the right result in the linearized theory. Let us take the usual linear phase space, the two degrees of freedom, a simplification that is available in the linearized theorem. So we have got two degrees of freedom, this is a linearized metric, this is basically the linearized extension curvature. Then basically what one is doing is introducing a complex chart on this real phase space, like Q plus IT kind of thing. And then this complex path is just the transverse traceless path of the CAD, of the linearized connection, self-dual linearized connection, which involves the linear derivatives of the linearized metric plus i times this bit up here. Now, since this is transverse traceless, I could actually work entirely in the three-dimensional real space. It is just for writing transparency, it is convenient to just go to the momentum space, which is available in the linearized theory, and write the equations in the momentum space. So let me do that. In the moment of space, we can introduce a triangle, a complex triangle, which is a unit radial vector k, and you have got a vector on the 2-sphere, m and m-bar. This is a standard Newman-Pelrose type of triangle up here. When I do that, since this is transverse traceless, it has only two parts in this expansion. There's the m-n part and the m-bar m-bar part, and these are, so to say, the two helicity states. So the idea now is that I would like to use these as the convenient complex coordinates on the real phase space. So in other words, we have CAD as complex coordinates, but I have taken the irreducible information in CAD and put it in two functions on the phase space, two complex functions in the phase space. Then you can take the symplectic structure. The notation is that wherever you see these double things up here, they're all in three-dimensional objects. So I can take the symplectic structure, which is just gh, gh, gl. And I just can now plug everything in and express it in terms of these coordinates in a phase space. And I just get it to be equal to dz raised to dz bar. Imagine a part of that. We just kind of think that happens in harmonic oscillators. You start with dp raised to dq, e raised to g, and then you just get the simplistic structure to be dz raised to dz bar. The precondom Hilbert space, which just exists as a phase space, is given by... Now you introduce a polarisation and this is a key ingredient and this is the one that I would like to choose to be Kähler polarisation. In this context what it means is roughly

10:00 I would like to use a polarization which is in the anti-holomorphic direction, delta by delta G bar A. And this actually we can do in the linearized theory without any problem, rigorously. And now, if I say that the polarization is delta by delta G bar A, in other words, roughly I am asking the wave functions to be holomorphic functions of G's rather than anti-holomorphic functions, All these terms, which are polarized, which satisfy the polarization condition, are precisely of this form. Arbitrary holomorphic functions of z times an exponential factor. And I want to emphasize here that all this d3k by k etc. came out straight out of the polarization condition, which came out of our desire to work in the holomorphic representation in the chart that is provided by the self-dual connection. So if I substitute this form of the wave function and here, then we just obtain the inner product between two quantum states, which is dz by dz bar times the Gaussian factor, the standard Gaussian factor, times psi of zr. You can work out the Hamiltonian operator. I'm not very surprised that the Hamiltonian looks like an assembly harmonic oscillator. These are the multiplication of the creation operators multiplied by the annihilation operators, and that's a frequency operator. And automatically, the reality condition is satisfied. And again, one could have seen this already. If the scalar polarization of a required type existed, then the reality condition would automatically be satisfied that was something guaranteed by the general framework, and you can check that it is in effect. Just to give a feeling for these states, the vacuum state is represented by a function just 1. Psi of z up here is equal to 1, is a vacuum state. If you want a 1-grammit of state, you consider the following function. Suppose you want a guy return with momentum k0 and say helicity plus. Then you consider the following holomorphic function. Given any g of k, psi gives you the number which is the value of g1 at k0. If I had g2 here, I would have just obtained the helicity 2. The final results are just old results in a new bottle. There is nothing new. But what was nice and instructive was the procedure.

12:30 The only input is the choice of polarization. This was motivated by self-duality. Of course, in this whole prescription, one can go and check that since you get the right answer at the end, it is equivalent to doing decomposing fields into positive and negative frequency parts, but that has not been done explicitly. The second remark up here is that there is a certain amount of universality here, and I'm just going to use it, that's why I'm stating it. Supposing we are interested in a Maxwell theory rather than Then the Maxwell linearized connection or Maxwell connection, I'm just denoting it by c just to make the analogy of it. If I take its Fourier transform, it is of the type g1 of m plus g2 bar of m bar. I just have m m bar terms up here. Once I do this, the Maxwell Hilbert space is given exactly by this formula up here. That's the Maxwell linear product. This is the Maxwell-Hamiltonian, and this interpretation is all true where the word graviton is replaced by a photon. All that has changed is really the spin rate of these objects. In other words, in the case of gravity, we have m a, m b. In the case of Maxwell theory, we only have m a. This is the self-dual representation, and it's not sort of profound or in some sense known in the literature. Because one has already seen the Bergman representation, this self-dual representation has been obtained somewhat differently from what had been done before. The new representation is really the so-called loop-space representation, and this is the work of Rovelli and Kahneman. In their exact theory, what they have done is, they have introduced a certain class form. Remember, in the exact theory we have a representation in which wave functionals are holomorphic functions of A's, at least we aim at finding a representation in which wave functions are holomorphic functions of A's. Then you find the following kind of transform. Not first that, this blue thing is an exact theory, then I start with the linear theory.

15:00 Not first that, if I consider any connection, and if you give me any curve gamma, I can construct the holonomy associated with this gamma and this connection. It is just the path-order exponential of a dot dx. This holonomy can be thought of as a function of two variables, namely it depends on the connection and it depends on the curve itself. Now, what they have proposed is to perform a transform along these following lines. Given any function of a, I would like to obtain a function of gamma in the root space. You try to integrate this function against a kernel, and the kernel is just provided by the holonomy. The reason why this is interesting is the following. J. Kessel and Smolin have shown that if I look at this holonomy by itself, fix the gamma and just look at A, they have a certain sense of, with the suitable regularization that they have used, they have shown that this function of A, which has support just on the, so to say, on that curve up here, actually satisfies the Hamiltonian constraint equation. And the Gauss-Waugh constraint equation. It does not satisfy the diffeomorphism constraint, but it satisfies these two interesting constraints. Therefore the idea is that if I look at this transform up here, no matter what I plug in here, this function up here on the right hand side will automatically satisfy the Hamiltonian constraint and the Gauss-Waugh constraint. The analogy is a bit like the forward. Supposing we are doing a Klein-Gordon theory, then I can consider arbitrary functions of momentum and I can integrate them against I can integrate them against this particular kernel. Out pops a particular function and that function automatically satisfies the Klein-Gordon equation. That is the analogy up here. Now what I would like to tell you is how this loop space description works in the linear field. Here you can do everything rigorously and there is a nice representation of photons and gravitons on the space of loops. And it works as follows. First of all, when you do a linear theory, where you have this linearized connection C, and you can check easily by looking at the transformation against Gauss law, that in fact what you have is an abelianization. When you do small fluctuations, you have basically a triplet of abelian connections. So what you have to do is, I would like to work out this transform for the triplet of this abelian connection. Now in order to, already I have many indices up here, and in order to simplify life,

17:30 Instead of looking at a triplet, let me look at just one abelian connection, A. That is to say, let me look at Maxwell theory and let me describe this transform for you in the case of a Maxwell theory. In the case of a Maxwell theory, the holomorphic representation says the wave function is a holomorphic function of these functions Z of A, I mean, Z1 and Z2 of K. Wave function is a holomorphic function of these two objects. That's what this is. And I would like to perform this transform. The amazing thing is that you can actually perform this transform for all states in the Fox space. And what is the answer? The answer is the following. First of all, let me give you the first answer. The function of one under this transform, the very small transform in the linearized case, actually goes to one. One goes to one, I think. So this actually integral exists in general and has its nice property. Furthermore, more generally, all the information in the loop space is captured in certain things, which one might call form factors associated with the loop. You give me any closed loop in the physical space-time and I can define a certain form factor associated with that, which I defined up here. And then the story is the following. One photon state up here is like a holomorphic function, sorry, one photon state up here is going to be given by the following, psi of z equal to z1 of k0 is a function which represents a one photon with momentum k0 and with a helicity 1. That is just mapped to a of gamma, the function of gamma, which is equal to the first form factor.

20:00 These are all evaluated at K naught. Here we have got a functional on the space of connections, here we have got a functional on the space of loops. And similarly, n particle states go to n particle states. Not incidentally that although this was not the original motivation, what this transform does in the linearized theory, at least, is that it gives us a precise correspondence between quantum states of spin-1 fields in physical space-time and quantum states of a closed string, because these are just functionals on the string here. And perhaps actually, and this work is being pursued at Syracuse, perhaps this existence of this transform may shed some new light again on all problems associated with the role of all number-votal strings in QCD, which also had to do with the relation between, in that case, the non-abelian case, gauge connections, and what happens to the quantum state. Well, the short answer is yes, I believe that we can actually write down in order to... Let me finish this talk. That's right, it is all complete. I'll just come back. Well, so to summarize that, I just want to stop here. To summarize then, what new variables have done is really provided new tools in order to analyze some of these old questions up here, and in some sense... We'll see in the next two talks that they actually have given us new concepts also in order to understand perhaps what the microscopic structure of space-style geometry is. I have run out of time, but I have prepared some small remarks on what light it shapes on the role of edge spaces and, again, some particularity possible connections with strings. If anybody is interested, I can talk to them privately about these issues.

22:30 Now, I believe there is a universe transform. I do not have it. Now, what is the general idea? The general idea is the following. Say, in the linear case, first and then more vague ideas about the non-linear case. Where I got this, given any loop, where I got these things called the form factors of the loop. As you said, this seems to be over-complete in the same sense that the coherent states are over-complete. What one would like to do is, one would like to obtain, one would like to be able to obtain... A complex structure on the loop space, this is the conjecture, in which all these f's are holomorphic functions. It's an over-complete set of holomorphic functions. So if I just say that, well, so in other words, here are the holomorphic functions. Any function of f's is all holomorphic, but there should be a nice way of saying what the complex structure is, which will look at, so to say, irreducible complex coordinates. Then the conjecture would be that there will be a one-to-one correspondence, an invertible thing, between holomorphic functions on the loop space and holomorphic functions of a's. In the non-linear case, there are further conditions, but we just learn from Karlo, there is work by Girasoro on precisely this kind of question, what is the relation in general, given any function of connection, or can you represent it as a function of the loop space, given the function of the loop space, can you drag it back to the connection? So this is because this complex structure sort of knows about photons about gravitons and so on so this is a new so it's not actually two ones and one can talk about even an odd function if you want it you can talk about just uh if you you know look at the parameter and just decompose things to positive negative frequency is it similar to these or is it completely different kind of context

25:00 No, no, no. This is decomposing to the... that's not true. I mean, this is just the talk of the day. Okay, but by metaphors you mean with a metric or without a metric? Oh, without a metric. You mean complex terms on a loop space? Yeah. Or parameters? Yeah. Parameters. Parameters. Yeah. I see. Okay. But it's not decomposing functions on the loop space in a quadratic way. It's just a parameter. In most of the things I've read, which is not everything on your work, you usually talk about asymptotical class, which is a very good thing to do, but are there any essential differences or interesting things that come up when you do the non-compact classes that you're interested in? Okay. I mean, everything that, sort of, the general program I outlined just goes through. The issue of time doesn't go through, because I really use this energy to give me delta by delta t. So the general program goes through in that spatially compact case without any problem at all. If the topology is non-trivial, for example, of the manifold, then you have got this new effect. In general, even if the topology is just R3, then because I'm talking about wave functions of the connection, you have got things like the theta-vacuum structure of Yang-Nills theory. These are different from the theta-vacuum that Raphael and John and Don Bates and Jim Hartley have been talking about. They are because they have to do with the internal groups of the system. But what has happened is that basically now, because of the inclusion of the internal degree of freedom, the kinematic symmetry group has been enlarged from the three-dimensional diffeomorphism group to a semi-direct product of the three-dimensional diffeomorphism group with the internal index of freedom. There is a much richer structure topologically. That is being analyzed. It is far from being fully analyzed.

27:30 In fact, Chris and Avis is a writer. I wrote a paper quite some time ago about trying to look at the theta-vacuum structure of Yagnir's theory in the presence of non-trivial topology. And amazingly, we can just take what there is literally and transport them up here. Yes, I would like to return to the question of the reality condition. Is that the maintenance nice lecture and that there may be a real difficulty in imposing the reality condition in the full theory? What are the commutation relations between the reality condition and the constraints, if one tried to impose it initially? In points and brackets, sort of? For example, yes. The reality condition imposed initially is preserved intact. That means that the commutation relations are just... That this unstructured functions. I believe. It's a long time ago. I have been on the north side because it's a long time ago. It probably is structured functions, yes. So if you quantize, you must somehow ensure those commutation variations as well as the commutation variations in the constraints. Absolutely. But that, I think, is an extremely important and difficult problem. How far is this realized in those reject representations which we have? How much attention did they pay to the reality condition and their commutation? What I do know is that when we talked about it some time ago, I do not myself know the details of this work because this is quite recent, but when we talked about it some time ago, in fact, surprisingly on the loop space it seemed that the reality condition, which I was always afraid of in a sense because non-polynomial and all that, seemed to have a nice form. I mean, a really simple form. It was telling you something geometrically. I am more optimistic than I used to be, but that absolutely is, you know, an enormous important problem in the whole program and that has not been yet addressed in any degree, I mean, any real degree. Since you wake up to spend heavily on self-fueled equipment, you must expect to have some exercise in the future, and you'll wake up to spend that money only on self-fueled medicines. Do you think that's a possibility?

30:00 One is really doing real gravity and with the... it doesn't depend on metric at all, right? I mean if you talk about quantum mechanics, metric or the triads are the canonical conjugate variables. So... But on the other hand, I'm not sure that... that's true, but I'm not sure that there is much importance to that remark. I mean that's true. But if I just want to understand, just because there's not much dynamics and I do not know how to evolve, I cannot do much physics other than just asymptotic physics. So the statement is that if you really are looking at asymptotic quantization, that remark is true. But I don't know how it helps in terms of actually doing quantum dynamics on constraints or this kind of thing. I don't think so. I think this is not really self-dual gravity. Because one of the things was to look at just self-dual gravity. And the structure that you get there is a completely different kind of mathematical structure. But there's no solution, right? Their function just depends on self-dual connections. They're not solutions to any equations. Yeah, at one point, I think, early on in this work, as I understood it, there was a question as to how one factor ordered the dimensions of strength. And one factor ordering was preferred to get formal closure, and another factor ordering was used to solve the strength equations. Is that still an issue? No. It turns out that if you work in the loop space, which is it? So this is the most satisfactory statement about solution of the constraints. This is in the loop space. In that case, in fact, because it's transformed, and when you take the transformed, roughly speaking, things conjugate, it turns out that, in fact, the fact-ordering in which it closes formally is precisely the fact-ordering that is relevant. Except that, you know, this is formal closure, and why should that be important? I'm using the last function for the different density of weight. I mean, I realize that you can write down the scalar density of weight minus one all day long without it being physically dependent on the metric or anything else. You just say it's the scalar density of weight one and there's an end on it.

32:30 On the other hand, and then you proceed to solve your constraints and carry out your final program. But on the other hand, if I want then to see how classical space-time comes out in some classic limits, Then the customary interpretation of the last function is it has something to do with proper time between two times like this and I can see in linearized theory how you can get around this way but in the general theory I don't see how you can get around the problem of saying that now the proper time depends on the choice of coordinates on the slice because no matter how you represent it you will have And if you change the three-dimensional coordinates, all your objects are perfectly well-behaved, including the last one, because it picks up a Jacobian in front. And so the proper tone between the two slices then depends on the three-dimensional coordinates. I don't understand how that works. Dr. First of all, it is really, as you said, it is true that one can do everything consistently with just density-weight one. In order to reconstruct a space-time you really do have to solve an equation in the sense that you have to do something non-polynomial. You have to sort of extract the scalar lapse out of the density. In other words, I really have to de-densitize n. So I have to multiply it by dead q to the minus 1. And so the physical... The time that you are talking about is the dedensitized object, definitely. It is not the density object. Therefore, that doesn't have any coordinate dependence. No, of course it does. But one does have to do this work. So, at the end of the day, one does have to do this work. If you want to consider the four dimensional geometry, you do have to do that work. It's the square root of g. It's actually. Which way is it? Is it the square root of g which is polynomial? No, g is polynomial. Thank you. g is polynomial. Can we return once more to the problem of foliation fixing? In the case that you have an asymptotically flat space time which is also linearized inside, you have of course an almost flat size which you interpolate between beta at infinity and inside. What is the way in which you can see...

35:00 There is no, I mean, if you take this canonical point of view very strongly, then when you go to quantum theory, there is no four manifold with a four metric and so on. And so, I'm not I have to solve the Schrodinger equations, I have to isolate what I mean by time, and the time is coming, I mean if this is true, then the time is coming out of that imaginary part of CT, you know, the full thing, that's what I mean by time. Then I don't have to worry about under what conditions can I reconstruct a four-dimensional space-time, are there some nice wave functions which can dynamically or actually correspond to the four-dimensional space-time, that's absolutely an important question to be addressed. It's not an abstract question of the formalism. It's a question of what particular solutions to the equations. No, but I would say that the state of the field is different on a slide which looks like that because there are different internal degrees of freedom included. So, we are talking somehow only about the data at infinity. You can say that only if you have space-time. If we come to the compact space and this discussion, I think that this question will reappear. Oh yeah, I mean, in fact, the prescription I was giving, at least on the face of it, doesn't work in the compact space. I mean, maybe somebody can do something to it. So, I know... I don't know the answer to that question. Charles Torey has done the thing for the, in general, things of the type of self-dual representation, how to do propagators and how to do final path integrals and so on in that particular case. It's an open problem to actually implement, use his general formalism.

37:30 To actually do the calculations, for example, to see what happens, if you did a Hawking-Hartwell proposal or something of that sort, and then, it's just an open question, I do not know, I mean, just Jim and I were just talking about it last night, there are some conjectures, but it's an open question, I do not know what happens, largely. Nobody else can hear what you're saying. Right, yeah, I was just saying, full dimensional action. But yesterday we just heard from Jim that... It was kosher to be complex. No, no, I understand. I don't know anybody who has learned this in the form of that notion. Yeah, but on the other hand, one could ask that question in stages. One could do, one should. I mean, one can ask that question just in Euclidean domain. I mean, if you don't believe in Euclidean domain, it is an exercise to go to the Lorentzian domain. And there, everything is real, and one should be able to do this. If you just, yeah. That's right. And then one should be able to do this, and then the... It could be that... One could actually do it in the Euclidean domain, and the conjecture was something like, by doing it on the Euclidean domain, you might get a wave function, which is a function of real age, because it's just SU value. The complex, in the Lorentzian domain, the function is a holomorphic function of complex And it could be that there is analytic continuation of those which satisfies automatically the other equation. I do not know. I mean, these are all conjectures. I mean, you know, if you look at the final law in a product, then it is exactly the same in a product that you would get in another case. No, that has to do with choice of the polarization.

40:00 If I use, for a harmonic oscillator, for example, you fix a symplemic structure once and for all. Then if you fix a symplemic structure, then naively one would think that d by dz or d by dz bar are equally allowed polarizations. It turns out that's not the case. If you have fixed the convention for the sign of the symplectic structure, d by dz will give you an, say for example, d by dz may give you an exponential of minus dz bar, whereas d by dz bar will give you an exponential of plus dz bar. So your choice of the symplectic structure sort of determines one or the other. So here you have fixed your symplectic structure. I mean, of course you could just do the other, but that doesn't matter. Here you have just fixed your symplectic structure. That decides. Whether you are going to use D by DZ as a holomorphicity condition or D by DZ bar as a holomorphicity condition for both Z1 and Z2, you don't have freedom to twiddle this one leaving this invariant, and that freedom fixes positivity of conditions. Well, basically, this was the action, but now this action depends on both A and E. It's exactly like in Yang-Nil's theory, so I can add a topological term. That's a perfect example. And then, the usual thing is true that the topological term leaves a few equations invariant. It doesn't change the equations. Here, you need a dimension-full constant because the dimensions are different in Yang-Nil's theory here, so you have to do things. Now, the point is that this is similar to the Yang-Nil's theta terms. But there are differences. One difference is that here A is complex, in Yang-Mills case A is real. On the other hand, here wave functions in the canonical quantization program, for example, are holomorphic functions, and there they are real. It turns out that these two differences just compensate each other. And you can just repeat the Yang-Mills analysis about going to the space of connections on which the wave function is defined, looking at the topology of that space of connection, look at the Chern-Simon forms and so on. And actually find that in fact the topology is non-trivial. The topology of the configuration space is non-trivial. And what that means is, when you've done it all, that you've got the canonically conjugate momentum is no longer E, but it has a BPC. Now in the canonical quantization program, this is what is replaced by delta by delta E. Therefore E is replaced by this whole thing. Now I go to the constraint equations.

42:30 The constraint equations of the theory turns out that the Gauss law constraint doesn't change at all. It turns out that the vector constraint doesn't change at all. But the only thing that changes is the scalar constraint. And in the scalar constraint, previously, the scalar constraint just looked like theta was 0. It just looked like h bar by i delta by delta a, h bar by i delta by delta a, and psi is equal to 0. That's how it looked like. But now, we have got these theta terms. The terms which are linear in theta are pseudo-scaleless. I mean, yes, they are pseudo-scaleless because of the presence of D up here explicitly. And therefore, any solution to this Gauss-Lacke constraint, I mean, if I choose to solve the thing for theta, some value of theta, and then I make a parity transformation. By that I mean, in all this business, you have to fix spatial orientation right from the beginning. If I flip my spatial orientation, then these terms change sign. That's because you've got that epsilon symbol in your topological term. Yeah, sure. So these things will change time, and then it is no longer true that the parity would leave any one theta sector invariant other than theta equal to zero. To map the theta equal to zero sector, theta equal to theta naught sector, theta equal to minus theta naught sector. Notice that this is different from the old things, all of the things, in which one just added this term to the action for the following reason. It is true that on solutions, the numerical value of these terms does reduce to that term, but this is just a function of A, connection. It is not a function of the metric or the triad, etc. Conceptually, this is important for the following reason. This theta vacua has this other interpretation which has to do with mapping from the distant past vacuum to the distant future vacuum in the Yang-Ming scheme. If you are using the metric representation or the triad representation, then this configuration space is really disconnected. And therefore there are no transitions, they are just disconnected. There is no continuous path whatsoever between, from one sector to another sector. So you don't expect those things to happen. Whereas in the A representation, the configuration space is not, it's connected. It's topologically non-serial. So I think this sort of sheds light about, well, the work that you and Chris and some sort of other things have done before, a strong series of problems.

45:00 One announcement to remind everyone that the program for tomorrow morning and tomorrow afternoon has changed from what you originally got in the mail. Remember that the problem of time, half of the discussions have been moved to the morning with Carol Kukosch, Lee Smolin, and Jim York. In the afternoon, the other half of that program and then Sidney Coleman is speaking at 5 o'clock instead of in the morning as was originally announced. The second thing was that I cut off the discussion on proceedings earlier this morning. I don't know whether anyone wants to continue discussion on proceedings, whether they should exist, etc. No desire. All right. Dinner's at 1 o'clock.