Gauge Theories & Loopy Locality

0:00 This is, in part, a semantic issue, but it's something we need to address at the beginning. Do you want to use this as a weight? That would be an idea, yes. At least when they're on the screen, I don't know. I'm not sure. We'll certainly hold them down. You should do that. Okay, so here's two takes on what a gauge theory is. One by Schrautmann. I'll give you the both there. For me, a gauge theory length is a physical theory of a dynamical variable which at the classical level may be identified with a connection on a principal bundle. The structure group of the bundle is a group of gauge transformations of the first kind, and this other group of gauge transformations of the second kind, local gauge transformations essentially, may be identified as a subgroup of the group of ultramatism of the bundle. In this sense, gravitation is a gauge theory. I'll come back to that remark in a minute. On the other hand, John Irwin, as many of you may know, has been pushing a different blind on how to understand what a gauge theory is. And he distinguishes this way of thinking of gauge theories from a way according to Wichang Niel's theories, at least parallel examples of gauge theories. And according to Berman, what counts is whether within a constrained Hamiltonian formalism, something counts as a gauge theory. That's to say, when you try to give a Hamiltonian formalism, formulation of a dynamical theory, constraints show up. I'll say more about that later. If they do, you'll be able to gauge theory. If they don't, then they're not. He also ends up with some different takes on what makes something gauge theory, including the general theory of gauge theory. I guess we have a Yang-Mills gauge theory if Troutman's structure group were a Blee group of the right kind, like the one where a B2 or a B3. General relativity did not be Yang-Mills, the Yang-Mills theory, because the structure group didn't be like that.

2:30 So, I'm going to focus on Yang-Mill's five gauge theories, which would count as a gauge theory both for Erwin and Troutman. Richard, could you say in a slightly less cut of the term? Well, if you've gone off with Erwin's take on what a gauge theory is, you look at the constraints which make something gauge theory. You look in particular at the first class constraints. It doesn't mean anything to you, just don't worry about it. And if the algebra formed by those constraints is positive, if it's a Lie algebra, then you have a Jan-Mills type gauge theory, and then you can give a title on the formulation of it. If the algebra doesn't fall, then you still have a gauge theory for the development, but you don't have any other gauge theory. That's the difference. That's how you take it. Is that helpful? So, in general, the theory of the case, is this thing for the logic? Yes. Right. Okay. Okay. What about locality? I'm moving fairly fast now to get on to the interesting stuff, so please do interrupt me if I'm going too fast. As far as locality goes, I want to distinguish a kind of causal notion of locality, which would be something like local action, which is a principle that I've more or less quoted from the picture by Einstein, time, from a kind of semantic locality, which I call separability. The idea being that separability is telling you that whatever physical process you're dealing with is determined by what's going on locally, at each point in space time. If that fails, then there wouldn't be non-separability. you have a visual process which was probably the non-term locally, non-term by what's going on each state-time point in the region in which the process is termed. Okay.

5:00 Now, if that's what a gauge theory is, for me, today, I'm talking primarily about the annual gauge theory, and that's what locality is. Then we can proceed to look at the particular gauge theories and see what to say about locality in those theories. Okay, so practical electromagnetism of the gauge theory. Although I'm interested in non-abelian gauge theories today, the annual type of theories. I'll frequently just use classical electromagnetic system as a simple example of such theories, but then I could give it a few more details without getting into the complexity. So, classical electromagnetic system as a gauge theory, this is all pretty familiar from John, I'm sure, but let's go through it anyway. If you write Maxwell's equations in empty space in Moran's variant form, they look like that, where the electromagnetic field tensor is expressible in terms of the four vector potential A mu, defined on the space-time manifold in that way, and of course the A mu is determined by that equation, too, only after a local gauge transformation, where it's an end-of-anything function of the coordinate. Now, in classical physics, purely classical physics, the electromagnetic field is all derived from the electromagnetic field, represented And in that context, the potential A-new, defined implicitly, like, that's for agents 2, can be regarded as a new mathematical convenience. Okay. Um... You see, that's because this means that's like, that's the charge part of this. Um, that's important, because they're going to go into a conscious judge, of course, and in that context, what I said there should not be true. Yeah. Okay. Okay. Now, that electromagnetism generalizes to non-Abelian theories. I'll just give you a little indication of how that works. This time, I'm writing the field equation in the current, not in the state. And for a particular non-Abelian gauge theory, the analogous equations would look something like that, where in this time we've got a vector over S and A and over J. This would be, I guess, an example of O3 non-Abelian gauge theory. You see, interestingly, that not only is it the case that the vector potential is gauge dependent,

7:30 but in this case the field itself is also gauge dependent. That's just one example of a non-abelian gauge theory to give you the idea. Nothing that I say from now on will depend on that example. Okay, so now, following up on the last comment, what happens if we look at quantum mechanics? Well, there are two stages of that. The first stage would be to treat electromagnetism classically and see what happens to classical electromagnetism acting on quantum mechanically described particles. That's where the Harnoff-Bohm effect becomes important, as I'm sure you're all familiar with. If you look at how electromagnetism enters into the quantum mechanical particles, it does so directly via the four-factor potential A mu, indirectly through the tensor f mu nu. And a consequence of that is that there are circumstances, as pointed out by in which electromagnetism has effects on charge particles, quantum mechanical charge particles, which can't be understood. So in terms of the local action of the field, the classical electromagnetic field tensor f mu nu, or the electric and magnetic field that you can be deposed into in some frame. So, basically, either the electromagnetic field ends at the distance, or in violation of local action, all is more to electromagnetism than the electromagnetic field, and in particular the potential seems to acquire some kind of physical reality, even as gauge dependence, which is a little puzzling. Now, it turns out that we can get around this gauge dependence by noting that all of these effects accounted for by a certain function of the gauge differential, a mu, and that function is itself gauge invariant. The Dirac phase factors, things like that, where we take that integral around the quote of current space time, those are entirely sufficient to account for, for example, the Haramon-Bohm effect. And as I say here, these are gauge invariant unlike

10:00 the four vector potential itself. Now, there's something a little uncomfortable about dealing with a theory in which this gauge dependence of the four vector potential is always appearing, where the phenomena themselves are gauge invariant. So it would be nice to somehow rewrite the gauge theory, in this case electromagnetism, to somehow allow for or eliminate some such gauge dependence. And that's one way of trying to see the motivation for a fiber bundle formulation. In such a formulation, you still deal with something you call the gauge potential, But it's no longer thought of as an object, a vector, or one form defined on the space-time manifold, but the connection on the fiber bundle. And depending on how you slice up the fiber bundle, depending on how you take a section of that bundle, you can represent the intrinsic connection of the bundle on the space-time manifold by one or another of these four vector potentials. So, that's one motivation for going to the cyberbundle formulation, the formulation that Trapman took to be definitive of a gauge theory, remember, at the beginning. So, don't take all the details of this picture too seriously, but I hope this will give you something of the idea of what's going on in a cyberbundle formulation of a gauge theory. Perhaps all of you will be familiar with this picture, and I took it from Sonny Aoyan's appendix, in a book, and it's going to be a theory possible. The idea is this is an illustration of actually two fiber bundles. Here's the space-time manifold. Sitting above it, there are fibers of this bundle, the so-called principal fiber bundle, and this other one, an associated vector bundle. The idea being that the fiber, remember, going back to Troutman, has the structure defined by the group appropriate to that gauge theory. U1 in the case of electromagnetism, which would be SU2 or SU3 or whatever in the case of non-veiling gauge theories. And each point in the bundle represents a particular configuration, if you like, of that field.

12:30 the connection on the bundle defines what it is that's going to count as horizontal rather than vertical motion within the bundle. The four vector potential is what you get if you project that down onto the manifold connection, that's to say, and depending on how that projection goes, depending on which point of the fiber you take at which point nearby it, as you move along this curve in the manifold and track it above in the fiber, that will determine which particular vector potential down here is going to stand in for a more intrinsic connection of the bundle. So now, from this perspective, you can think of a change of gauge as corresponding to simply an arbitrary choice of one section of the bundle rather than another, leaving the intrinsic bundle structure, and the connection on the bundle, invariant. So the gate changes look very much like coordinate changes, and it looks like the intrinsic structure representing electromagnetism is going to be more like the connection on the bundle than like any one of these objects defined now in the nanopause. I'll come back to this thing later. Okay. And I will go to explain in more detail the difference between principles and associated bundles, because that'll be important later, but not for now. Okay. Now, let's go on then to see more about fiber bundles. And I want to introduce the notion of a phalonomy at this point, but that'll be important later on. And for that purpose, I have a different picture, which is understand, given what I'm trying to get across. Think of this sheet here as the fiber above these points along this curve. And now I'm going to give you a different picture. We have a different picture of the fiber. The fibers here are represented by little spheres above points along the curve. So now point P, now the manifold, above it is this little p, and above different points along this curve cp in the manifold, there are different spheres, okay?

15:00 So what happens if you start off from a point in the fiber above p and you trace around this curve and come back to p? Well, in the fiber above p, actually we start off at u, we go all the way around the curve, we come back, we end up at a different point in the fiber. that corresponds to a halonymy around this curve. To be more precise about that, the halonymy of a closed curve beginning and ending at the base point P is going to be an element of the structure group. In the case of electromagnetic U1, this would correspond to AC2 or O3 or something. An element of the structure group isomorphically onto itself. So, here we've got the view, and the element of the structure group gives us that V. If we start up at some different point, the same element would match at some different point, something to do. And the whole one of these tells them what element of the structure group is doing the mapping in each case. Okay? That would be associated, then, with the curve in the manifold down here. Don't worry about all the other details. These figures are not produced explicitly stored, and they had other purposes in other places. But that's the only point I need to get across. Okay, so now clearly if the connection that's going to determine the holonomy, the connection tells you what it is to move horizontally as you go from one point in a fiber above a point here to a point in a fiber above a neighboring point here. So the connection will determine the holonomy. Okay? And given the connection everywhere, you have the holonomy, all of the holonomies and all of the closed curves. There's also a kind of converse result. The Hoanamese of a gauge theory relative to an arbitrary base point like P also determine the connection, but not uniquely. They determine the connection only up to what we can think of as a vertical automorphism. That's to say something that moves around everything in the fiber about every point in the same way. So we don't have unique determination of the connection by holonomies, but we have essentially the determination of the connection by the holonomies.

17:30 Okay. Now that's good, because the observable interference effects, like that holonomic effect, depend only on the holonomies. That's where the direct base factor came in, essentially. Therefore, distinct connections with the same holonomies predict the same observable interference effects. Okay, so now you're beginning to see that the connection on the bundle is still apparently not the most faithful representation of electromagnetism, because distinct connections give the same holonomies, and the holonomies are what seem to be one-one correlated with the interference So, in our attempt to get rid of gauge dependence, perhaps we haven't yet succeeded. Okay, a few more transparencies to catch up with what I've been saying here. Okay, so we go up to about here. Just notice that the phylogeny itself doesn't have this factor E in there, but sticking the factor E in there shows why it is that the interference effects depend not just on the phylogeny, but also on the electric charge of the particle, which is after all what you'd expect. to respect the different churches. engaged leaders in the non-community destructive groups, like the one in that little picture I showed you, the philonomies themselves are not gauge invariant. But their traces are. And the traces of philonomies are called wolfs and moots. And there are lots of rather nice theorems that show that in the cases that I'm interested in, and don't ask me for details because they're helping with my fingertips now, but they're there. All of the gauge invariant information contained in the Philanomys, indeed, the whole theory can be reconstructed from the Wilson looks. So that gives you all of the physical content, in some sense, of the gauge theory. In the case of electromagnetism, the Philanomys yet is a Wilson look because it's a scale that trades equal to itself in the problem. but the generalization is important to that. Okay, so now we're beginning to see some candidates for representing the intrinsic properties of things

20:00 in a classical gauge theory. Quantum gauge theory that can come later. And what are those candidates? Well, we started off with the potential decline on the manifold. That didn't look like a particularly good candidate because gauge changes could change the potential arbitrarily within, you know, limited by the gauge transformation itself. Then the connection of the bundle, that looked like a candidate too, but I hope to have given you reasons to think that the holonomies, rather than any particular connection that generates those holonomies, is a better candidate for faithfully representing electromagnetism in a classical U1 gauge theory and analogously to representing some other gauge interaction in a non-Abelian gauge theory. But let's go further. What are these holonomies? Where do they come from? Well, given the principal fiber bundle with a connection, the holonomies obviously fall right out because the connection defines the holonomies and almost, as I explained, vice versa. But where does the fiber bundle come from in the first place? We could just postulate it, but there are actually deeper structures here that we should look to. And I'll start off now with a quote from this nice book by Gambini and Pullen called Loops, Knocks, Gasteers, and Poins and Grappets. Don't get scared about the knocks and the poins and gravets. Okay, here's a quote. I thought I simply quote it rather than paraphrase it, because they put it so nicely. A generalization of the notion of philonomy may be defined intrinsically without any reference to connection. See, so far, it was the connection that generated the philonomy, and the notion of philonomy didn't make any sense if there was no underlying connection. But they're saying one can take deeper. Polonomies can be viewed as homomorphisms from a group structure defined in terms of equivalence class of closed curves onto a Lie group. Each equivalence class of closed curves is what we technically saw a loop, and the group structure defined by them is called the group of loops. The group of loops, they claim, is the basic underlying structure of all the non-local formulations of Gauss The word non-local there, given what I said earlier, I would translate as non-separable.

22:30 They're not talking about causal non-locality, they're talking about descriptive non-locality. So what is this loop group and what's going on here? Well, let's start off with the idea of a closed curve on the manifold. A closed curve on the manifold is typically understood to correspond to a function from some open interval of the real into the manifold, an appropriately continuous function. if that's the case then it will come equipped with a parameterization which tells you what number to assign to each point on the image of the curve in the manifold but one can consider unparameterized curves where an unparameterized curve is an equivalent class of parameterized curves where you get one member of the class by reparameterizing the curve but that's still not a loop for a start we have to be talking about closed curves to be talking about loops And loops, the way we're understanding now, are still not closed, unparameterized curves. There's something a little different from that. Maybe I will use it short this time. Okay, suppose that, for example, somebody said, okay, here is a closed curve in a manifold. We think about that as, quote, an equivalence class of parameterized curves. Is that a loop? Well, not yet, because there will also be things like this. And that will differ from the original included class of unchramaturized curves. There will be the same loop. Why will it be the same loop? Well, either by definition, there will be defined a loop to include within its superfluous class Not just all of the uncraneturized curves corresponding to that, but also all other curves is determined only by little, you know, twigs, little bits of loops that enclose no area. So that's how we would define the loops. The point is, essentially, to make sure that anything that happens as a loop has, in the end, the same kilometers or any possible connection. That's the idea. And that's why we define it this way. Although the difference itself may not explicitly include that motivation, of course, to define it that way.

25:00 So the curves only need to be continuous. They don't need to be differentiable. I didn't say. I think that they need to be smooth. Well, smooth means continuous and differentiable, except at some finite number of points, I think. This would still be okay, but it couldn't be really that. That's my understanding. But I may be wrong about that tool, but I'm pretty sure that's right. Okay, so now we've got loops. How do you make them into a group? Well, I think you can sort of get the idea of how this is going to work. Suppose we had this loop here, and we had this other one here, And we could make them into another loop, just by taking this composition into the middle plane, essentially. Provided we were going the right way around the loop at the beginning. So we have composition of loops, just sticking them together and seeing what big loops you make out of them. We need an identity so we can get a group structure. The identity is just going to be a general loop that sits right where it is. Inverse is going to be a loop going backwards. Now we thought that we need to get a group structure. That was the aim of the game. So now we have a group of loops. Okay. Now we can think of representations of that loop group by homomorphisms from the loop group into some Lie groups. That's all we've got to start with. And provided the homomorphisms have the right nice properties, and I don't know the details about this. If you don't require any properties, then you don't quite get the result we want. Provided you have nice homomorphisms, you can think of these homomorphisms from a loop group into a loop group as carried by a principal fiber bundle. In which case the loop group which the homomorphisms are taking you into from the loop group count as a structure group for the fiber bundle. The group that defines the fiber structure. And when you've done that, the homomorphisms you start off with turn out just to be the homonomies to the bundle connection. So that's where homonomies come from at a deep level. You don't come in there armed with a principal fiber bundle.

27:30 You show that if you start out with the basic object, these loops, and the lead group, then the Lee group, give you all of the structure which is necessary to generate the principle by the bundle. So now I think we're getting deeper. We're getting closer to a faithful representation of the underlying structure which these theories are supposed to be describing. Okay. We're still talking about classical books, sorry. so any single representation of this type ipso facto gives you a connection on the fiber bundle that the representation gives you that's the claim that then obviously along the lines we were discussing before induces holonymies and true that these holonomies are in one-to-one correspondence with the homomorphisms by which you first set up this principal fiberbundle representation. So that tells you, so to speak, what is the A to integrate around each loop. It was given to you by when you first wrote down this representation. Right. And so are all these representations classified? Don't have time. for this good question. Somebody can tell me. Great. Okay, so with all of that sort of technical background, we can start asking the interesting philosophical questions about classical Yang-Mills gauge theories. And here's a question. What intrinsic properties are represented by the various mathematical objects that are involved in a formulation of classical Yang-Mills gauge theories? What object do we have going for it? We started off with a four-vegged potential and a field, say in the case of electromagnetism, which I'll simplify to in a moment. Then we started thinking, well, maybe it's the connection and the curvature of the principal phylobundle with appropriate structure groups. Maybe those are the appropriate objects. By the way, I didn't say this before, but the appropriate phylobundle representation for the field

30:00 is the curvature of the connection on the bundle. Or maybe it's the holonomies of such a connection, where we assume the existence of a fiber bundle with a connection, and then we say, oh, it's the holonomies that are the objects that we should be focusing on. Or maybe, in my last transparency, we should be looking at the holonomies, now not viewed as defined the way holonomies normally defined, assuming some bundle structures to start off with, but rather, it defines simply as one of one-person from the loop group, one of those appropriate structure group, in this case, U1. So I think that this last category of objects that we should be focusing in on if we want to try to understand the intrinsic, the most intrinsic description of the physical reality underlying classical electromagnetism. The idea is that each nice loop with scare quotes, I'm not talking technically now, in space-time has certain electromagnetic properties. Either they are ascribed directly to the loop if you want to be some sort of a substantivalist, or they're ascribed to something or other which is on that loop if you want to find something other than space-time of these properties. And these are directly represented by the homonymy of the corresponding loop in the technical sense. Moreover, these properties don't sit with me on any intrinsic process of space-time points composing the nice loop. Hence, classical electromagnetism is an inseparable theory. That's right. Now, let me be careful. I wrote that a bit too hastily. Remember, way back, when we were looking at classical electromagnetism as just a theory governing instructions from classically described particles, there was no need to introduce any of this stuff with vector potentials and bundles and all that so I think I want to restrict or qualify this claim in the context of classical electromagnetism acting on quantum mechanically described particles one should think that classical electromagnetism is non-separable theory but the theory does conform to local action because any influence over all by such a nice loop has no need to influence anywhere else I'm making this claim much too fast. Sorry about that. I should have stopped and thought that through. I didn't have time. I believe that to be the case, but it wouldn't be very hard for you to come up with good objections at this point that I'm not ready to respond to. Okay. many people disagree with me about this. I've had extensive emails of correspondence with the Nordlin and

32:30 for example, and they think, no, look, it's the bundle connection. the most intrinsic description, the physical reality underlying electromagnetism in the context which I've outlined. I think that's wrong. I think it's wrong for essentially the same reasons that it's wrong to believe in Newton's absolute space even in the context of Newton's own theory. You don't need it. In both cases, you're postulating the excess physical structure with no additional empirical consequences. So, there's an epistemological reason for going away. go here, maybe there is some unique bundle connection out there, which is generating the least alone. But we could have no possible reason to believe that there is. In the absence of such a reason, why don't we just say there isn't such a reason? Or at least not go on to believe there is. We don't need it in order to do everything we need to do good with it. Richard, could I ask before you get on to the second paragraph? In the Newton case, there is a traditionally much-emphasized aspect about the freedom to identify which inertial frame is at rest with respect to absolute space. A freedom meaning it makes no difference. Now, is there here also a kind of parameterized class of different bundle connections? which make no difference, which you choose? I'm tempted just to say yes, but I think the issue is a little more complex than that because we seem to have some sort of intuitive grasp on what it would be for one inertial frame to be the one at absolute rest. And when I say intuitive grasp, it seems that that notion looks like it should have operational content somehow. We can think about that being in that frame, moving with respect to it or something. In this case, that's harder to make sense of. How can you make sense of your being in a, quote, frame corresponding to a choice of one rather than another bundled connection? It doesn't seem to be any link between your representation of yourself and your representation of the connection, in terms of which you can make sense of the idea of your hooking up to the world via that connection rather than

35:00 another one. So that still seems to me to be a similarity. I'm not quite sure what to make with that. And is this class of bundle connections exactly caught by what you said way back about the vertical isomorphism? Yes. That's the claim. And I stand by it. So when you go, when you come at these things from the loop group representation approach, you get and you therefore have all these loops and you get exactly that freedom in the bundle connections arising from all these different groups. Again, that is the claim, but it's the sort of thing that should be backed up by more than claims. I'm pretty sure that the theorems are out there, but what I would need to do is to find them, quote them, give you reference. Now, everything I said about electromagnetism would also apply to a classical non-reguling gauge theory, SU2, SU3, whatever, But such theories are kind of boring because they don't have any interesting physical applications. It's only electromagnetism, as far as I know, that has any interesting applications at the classical level of gauge theory, the annual gauge theory. Non-abelian gauge theories, I'm perfectly successful, of course, in describing strong and electro-weak interactions, but only at the quantum mode. So, if I'm right about the deep structure of classical gauge theories, does that generalize the contact gauge theory? But that's, as I said earlier, where I'm at at the moment in my own thinking, although your questions were already indicated that I think it's harder about things like that. I don't know. Okay. We're not totally out of time yet. So that raises the question, how do you quantify the gauge theory anyway? It turns out there's an amazing number of different ways of proceeding, which makes life a little complicated. it. I'm going to sort of smooth the path a bit by neglecting lots of them from the beginning and focus on a kind of tried-and-true method for quantizing gauge theories due to Dirac in lectures he gave in 1967. We start off with essentially Ehrman's understanding of what a gauge theory is. A gauge theory is now understood as a classical dynamical theory because Hamiltonian formulation involves constraints. What is a constraint? Constraint is an identity by the canonical variables. For example, if you try to formulate classical

37:30 electron mechanism as a Hamiltonian theory with configuration variables, the components of the four vector potential, then the momentum canonically conjugate to A0 sends out the guidance lead to the zero. That's a constraint. There's another constraint that you get by requiring that that constraint be preserved over time, and that is the Gauss rule. So those that sort of things that mean by constraints, that there's some function of the canonical variables that is identically equal to zero. Okay. Now, you can formulate Yang-Mills theories in this way, and more problematically, general relativity. Again, we won't worry about that. That's where the problem of time and such like comes from in quantum gravity. So, how do we quantize such a theory? You start with an algebra satisfied by the basic canonical variables of the theory, which is typically the Poisson algebra generated by taking the Poisson brackets. You just treat this as an abstract algebra. You replace the variable by operators, and you replace the algebra, or the Poisson-Bracket relations by commutation relations in the usual way, just like you do to get the Heisenberg relations in non-optimistic pointy mechanics. Okay. Now, the abstract algebraic now represented by operators acting on some space of function all, where q itself is a function, say, of space-time coordinates. that he might be a scalar for you. And the functionals are thought of as candidate wave functionals to the quantized theory. You restrict the space of functions for those that are annihilated by the constraint function that operates it, where a constraint function is the thing that gets set to be zero by the constraint. Finally, you define an inner product on this space, make it into a Hilbert space. Well, not finally, Having done that, we write down a Schrodinger equation for the wave function. So that's basically direct method to quantize the gauge theorem. And there's lots of little wrinkles on all that, which I don't have time to go into. Okay. Now suppose that you do that for electromagnetism. That's one way to get a quantum theory of electromagnetism.

40:00 if you start off with canonical variables vector, free vector, A and E for electromagnetism whose quantum operates that kind of space as functional to the full connection, then you can show that the resulting theory has the usual thought representation you go back to standard quantum electrodynamics as far as the Maxwell field goes but there are other ways for applying direct method to electromagnetism that proceed somewhat differently And this is interesting, because this is where loops come in again. There's these two nice papers by Aishem and Aishem and Rivelli that I have here, if anybody wants to reference. And they show that you can get so-called loop representations of pre-quantified electromanicances, in which the wave functionals are of this form, where gamma is one of our loops, this time, interestingly, on a three-manifold, not a fourth-manifold. Three-manifold corresponds to a time-swice of a space-time manifold. Also, interestingly, if you generalize a non-abealing derivative theory, these functions will sometimes define over more than one loop. That doesn't happen for electromagnetism. Now, I say there are alternative ways. There are lots of different loop representations that you can get by this procedure. Some of which are unitarily equivalent to the usual clock representation, some of which are not. I think I should get that quote from this actually from an Ishmael paper. Okay. One can construct several Poisson algebres on the Maxwell phase space, which seem to be reasonable candidates for quantization, a la De Rapp. Each of them gives rise to a vial algebra. However, in general, these vial algebras are inequivalent. Some of them can be represented in the standard clock space, while others cannot. The textual treatment of the quantum Maxwell field is the conventional vial algebra. However, there's no prior reason for preferring it to the vial algebra of loops and one-forms, or of loops and surfaces. On the contrary, in non-abelian theory, the conventional algebra is less natural. hence it is the algebra based on closed loops the algebraes based on closed loops that seem to admit the most useful extension to the non-abelian context so this is really

42:30 fascinating but puzzling we thought we had a fairly simple question let's look at quantized Young-Mills-Gage theories, and see whether the sorts of things I was saying about the underlying ontology will go through for them. But now we see that there is no unique quantized electromagnetism, let alone quantized non-Abelian-Gage theory. And depending on which of these different theories you look at, it seems that you're going to end up with rather a different answer. can extract an answer from any one of those theories, which itself is not going to contribute. At least as much as I've managed to keep up, we can take the standard crude approach of complying electromachronism and we end up with the usual block representation, and that, whether we agree with all the steps on route, it certainly works. You're saying that there are lots of other approaches that are not just other approaches to the same thing, but generally it seems to end with other things. There's also one approach which ends up being a good one. Yeah, but does that mean we have to accept the fact that these other approaches are fully capable of leading to the wrong thing, in the sense that they lead to something specifically different to the right, or not? Again, that's a good question. I only came across this quite recently, and I can't give you a straight answer to that question. It looks at first sight that such approaches are not leading you where you want to go, because what you want to be able to retrieve is the folk representation. And you can't do it. On the other hand, you might have a different attitude to these sorts of things. I mean, I think of a paper by where they talk about whether they raise the question, are Rindler quanta real? and they show that there are unequivalent representations of I guess it's in a scalar field in that case one of which gives you sort of standard representation for the scalar of the particle and the other of which gives you the Rindler quanta and the two are unequivalent And one might say, well, that just shows the Rindl representation wrong. But they say, no, both representations are right,

45:00 but they're complementary to one another. So you could try playing that trick here. You could say all of these different ways of quantizing electromagnetism, even when they yield inequitable representations, are somehow telling us another part of the whole truth. It's just that they don't fit together. So we don't yet know what the fundamental ingredients are in these other approaches. And once we knew what those ingredients were and how they related to the fog, the ingredients, we may still discover that these different things are the same thing in disguise. The world is the way it is, and there's only one world. And these different representations of it, on this picture, which I don't think it's the right one, on this picture, they're revealing different complementary aspects of the world. so that the additional ways of quantizing are telling us more of the truth, which is in some sense compatible with what we already know but just doesn't mesh with it. I'm sounding like Laura, I don't like that. But, yeah.