Structuralist Approaches to Quantum Gravity — PSA 18th Biennial Meeting

0:00 The first talk, two parts. The first part, me and the second part, Dean. I'm going to talk about applicability of category theory to support of quantum field theory, TQF team, which is closely connected to the program of canonical portal gravity. And I have to say that in the spirit of this workshop, I'm going to be somewhat provocative and speculative about these issues. Some preliminaries about category theory. Very briefly, it's a unifying conceptual framework to talk about various concepts are found across the boundaries of mathematical domains, typology, logic, theory theory of categories, and a category consists of both objects and morphisms between those objects, but what really counts are the structure-preserving morphisms and not the structured objects. The notion of an important one here. Informally speaking, a function is a structure-preserving map between categories, mapping objects to objects, morphisms to morphisms in a structure-preserving manner, preserving composition and identity morphism. Category theory also provides an alternative notion of mathematical structure, but I have to say that that's not exactly the sense of in this paper. For me, structurism is defining or characterizing objects, not intrinsically, but by some set of relations. And here I'm going to talk about TQFT because it's an example of categorical background independent physics. TQFT, very very briefly, I think John Biot is going to talk more about TQFT and its relation to quantum gravity. So basically just one picture, my understanding of TQFT.

2:30 We have a schema from Atai's axiomatic canonical formulation of TQFT as a puncture between two mathematical domains, one intuitively having to do with global space-time description or topological manifolds and one having to do with quantum description or Hilbert spaces. On the left hand side we have a category of coportisms. In this picture we have a two coportism which is basically a two-dimensional manifold taking a one-dimensional manifold, a union of two circles, to another one dimensional manifold, one circle, describing topology chains of one dimensional spaces, as it were. Generally speaking, in a category of n-cobortisms we have oriented compact n-1 dimensional manifolds as objects and n-dimensional proportions as morphisms between these manifolds. On the right hand side, actually this is the wrong way around, on the left hand side, this time, we have a category of Hilbert's faces, as objects. That's not my right. Pardon? It's not my right. You were right. It's on your right as well. It's relational. I'll continue on with two minutes of time. Gilbert's space is objects and linear operators is morphisms. TQFT is a puncture. Mapping objects from the left-hand side to objects on the right-hand side. And cobordisms to operators. And this connection is meant to give us quantization of topological spaces in some sense. And regarding this scheme, John Bice has put forward a couple of challenges concerning the interpretation of TQF-T and especially the role that category theory plays here. The first challenge has to do with interpretation, but here I'm going to focus on the second challenge namely the applicability of category theory in setting out this axiomatic framework

5:00 so the question is also second challenge is what i'm focusing here the question is to understand more deeply the role that category plays here And there are many possible answers to this question. In the spirit of this workshop, I'm going to adopt a very optimistic premise. That's how physics hooks onto the world in a way that renders the category-stirable formulation of TQFT intelligible and well-motivated. And despite the way I put it there, it's meant to be neutral between realism and non-realism. And I propose to take a structuralist perspective on this matter and begin by considering some symmetries in physics. There are a couple of good reasons to begin by looking at symmetries. One hint comes from, again, from Jung Baez, who begins one of his papers, by the quote up there, basically saying that TQFT shows us that describing dynamics using group representation is only a special case of describing it using category representation. To explain this remark very quickly, I'll just say this. From category-thewable perspective, a group can be viewed as a one-object category, a groupoid, and group representation can be viewed as a puncture between such one-object category and the category of vector spaces. So this makes it possible to generalize category representation is a puncture from a suitable category not necessarily a groupoid to the category of vector spaces and in these in these terms TQFT is a kind of representation of category of cobordisms so in terms of category theory we have a mathematical generalization from a

7:30 particular kind of one object category to suitable many object categories so that a group representation is a special day special case of category representation and the question is is there some some metaphysical generalization to go along with such mathematical generalization and what I What I aim to do here, I aim to give some heuristic backup to Paes' remarks about this mathematical generalization. And I do this by looking at symmetry groups from a distinctly structural perspective. That is, I want to focus on the relations that the group elements stand for, the symmetry transformations, rather than on the objectual systems related by these And I claim that, from this perspective, the moved category theoretical framework is a natural one when taken together with a simultaneous move to background independent physics. So I want to appeal to such a structuralist characterization of space-time symmetries in order to motivate a generalization of group representation to category representation. In terms of philosophical characterization of symmetries, we could take Van Frassen's characterization as a starting point. According to Van Frassen, the essence of symmetry lies in structure-preserving transformations. where the notion of structure-preserving transformations are transformations that leave all the structure intact where the structure refers to partitioning of a domain of individuals into equivalence classes with respect to some equivalence relation of relevant similarity and that partitioning structure just amounts to the group structure of transformations But from the structuralist perspective, symmetric transformations are not only structure-preserving

10:00 in one process sense, but they can also be structure-defining. and more significantly they can be system defining and to explicate to explain this a bit are an example from Castellani and Middlestad we talked about group theoretical constitution of objects. They start from Wigner's classification of three relativistic elementary particles by labeling all the a representation of the Bunkera group. Starting from that inner result, Castellani and Middlestead characterize or constitute objects by their transformation properties as described in the state space under a space-time symmetry group. So symmetry here is not only a structure-preserving, but also a structure-defining, in the sense that theory theory theoretically there is nothing more to the elements of the structure domain but they transformation properties we can also consider defining a kind of objectual system by space-time symmetries more generally and so as to include the classification above and this is what actually does. She begins by defining a Galilean system, for example, as a physical system whose state space forms a representation space for the symmetric group in question. That is a homogeneous Galilean group in this example. And then a Galilean particle is a Galilean system picked out by irreducible And with this program of Castellani in mind, I want to propose a generalization. follows that we generalize this picture of system-defining space-time symmetric transformations to make sense of the mathematical generalization of category

12:30 representations. And there are two points here, the role of category theory and the significance of background independence in the scheme. About the role of category theory first. Remember that in terms of category theory if the generalization is one object category to many object category. In one-on-one object categories corresponding to symmetry groups, we have invertible morphisms as symmetry transformations. What I'm proposing here is basically to understand space-time cobordisms, the morphisms of cobordism category, as a natural generalization of symmetry transformations they define the objectual system under study and category theoretical language then fully captures the idea there is nothing more to an entity but its transformation properties the group theoretical approach to constituting objects which already operates very much in a structural spirit naturally generalizes to category theoretical approach to constituting topological The group elements, categorically theoretically, correspond to morphisms, just like cobordism operators taking one space to another, or to itself. There are two major differences here. First, in this generalization, we have generalization from one object category to many object categories. And secondly, there's a difference in the ontological status of morphisms, and this has to do with background independence. The move to category theoretical framework is significant from this structure's perspective, because it brings the role of transformations as morphisms to the fore in defining objectual systems. And this makes room for the possibility of background independent generalizations of the scheme in which transformations are taken seriously as part of the system under consideration.

15:00 And the significance of background independence in all this is that in background independent physics, we simply cannot have symmetries with respect to some background geometry. So the constitutive role of space-time symmetries, for example, must then be replaced by something else. What I'm suggesting, obviously, is that there is more general notion of transformation working in category theory that could do that. But there's a difference here regarding the theoretical status of these transformations. In physics with a fixed background geometry, primitive transformations do not contribute directly to theories ontology. But here we have transformations which are part of theories ontology, maybe even the most fundamental part that is space-time proportions category theoretically define spaces and are part of the system's ontology as space-times and very quickly there's an interesting level dependence to the scene as well namely the suggestion to use a theory of n-categories to capture the fact that the objects of n-dimensional cobordism category can be further sort of as morphisms in a category of n-1 dimensional cobordisms at a lower level. So in a sense, the objects and morphisms are on a par in an important structural sense. Objects are defined by morphisms which themselves can be objects at another level. And presumably a theory of categories would be suitable for describing all this. this is the second half of the double act can you sort of start shouting at me if I overwrote I will, don't worry I'm going to hear John Byers and Smollin Stachel talking about him.

17:30 Okay, so originally I was going to develop some of the things that you, Hart, was talking about. But I decided instead to say more general remarks about what I think Lee Smollin and John Stachel will be talking about. I won't be mentioning John Byers as much as you, Hart, I'll cover that a bit more. And you can see the differences in talks by kind of the nice latex slides and then the scruffy scribbles and all over it as well. Okay, so really what I want to do is connect some of the issues that Juhal was talking about on structuralism and category theory, and some of John Byers' ideas, with the kind of issues that Smoling and Stacey were talking about. And I think they'll be speaking about these issues today as well. And the uniting theme throughout all this is going to be a structuralist tendency, like Steve mentioned. And a further unifying theme is that category theory would provide the most natural framework for this kind of structuralism you see running through all of the work. Okay, so in particular, I won't be arguing because I won't have enough time, and I'll be trying to gesture towards the view that structuralism makes natural many of the kind of aspects we would like our quantum theory of gravity to have. So things like background independence and diffeomorphism invariants. So I'll be trying to show that it makes these things kind of more natural. So the plan is, well, begin by saying a little a bit about the problem of quantum gravity, and say something about what would constitute a structuralist approach to quantum gravity, and try and discern three aspects of structuralism running through the work of Bayer's synonym and Stachel. Okay, the structuralist tendencies that you mentioned. And then I'll briefly go over each of these kind of aspects, not really focusing much on the Johns and Bayer's form, but on the other two in particular. And I'll be concluding with a kind of a question which maybe we can discuss afterwards. Okay, so problem of quantum gravity, I think Joy Christian puts it quite nicely here.

20:00 The conflict between the two foundational theories, general activity and quantum field theory, has primarily to do with the axiomatically presupposed fix of alternate structure underlying quantum dynamics. of such a fixed monodynamical background structure in the general relativistic picture of the world. Okay, so the idea is that so far all our quantum field theories are essentially background independent. Okay, so the kinematics and the dynamics are all defined with reference to a fixed metric. So we speak of the fields being defined at certain space-time points and then the dynamics the dynamics of these fields, defined at these points, or in the algebraic approach, certain regions, okay, but the same point goes through, whether you use regions or points, you're all, well, when we go to general relativity, points or regions, as if you're morphism invariant. Of course, the point of general relativity, the big, the lesson of general relativity, if you like, is that the metrical structure should be dynamical, okay, it's part of the dynamics of the theory rather than the kinematics. And when we accept that the metrical is dynamical, then it becomes very problematical to see how we can get a quantum field theory at all. We certainly can't get a nice causal structure because we've got no metric to define the canonical connotation dimensions. So we're missing one of the fundamental axioms of quantum field theory in general relativity. because we can't get these CCRs set up. So we can sum all that up quite quickly by saying QFT is a background-dependent theory, whereas general relativity is a background-independent theory. And just a little bit about what I mean by background structure, because it's generally just assumed and not really gone into any detail. Now, by background structure, I kind of mean some structure, S, that acts upon or is dynamically connected to some other structures, R, such that none of the R's don't affect the S.

22:30 and so on. So if the theory requires that condition, requires some S along those lines, then it's background dependent. If it doesn't require any of those, any such S, then it's background independent. Obviously the background structure par excellence is the metric in special relativity. It affects the dynamics it affects the trajectories of the particles but it's not actually affected by the dynamics of the particles itself. So that's a very basic characterisation of the problem of quantum gravity. We need to somehow construct a quantum field theory on a non-metric manifold, or on a differentiable manifold. There are many other ways of characterising it. So now, what would constitute a structuralist approach to quantum gravity? Well, generally, you characterize it by the fact that the objects of the theory, theoretical entities of the theory, are essentially relational. They're not intrinsically defined, but they're only defined relative to some other structures. So the points in general relativity are not intrinsically defined, they're only defined relative to some metric field, which gets provided when you solve the dynamics of the theory. So I think each of the other speakers can really be seen as a spousing form of structuralism, whether they know it or not. They do seem to kind of concur on each of these issues in some way, but they generally focus on one more than another. So in John Baiti's case, we have the kind of stuff mentioned by Juhar. But there's also a kind of focus on mathematical similarities. And these similarities have to do with the relational structure rather than the objects. So clearly the objects in the mathematics of space-time and the mathematics of quantum theory are very different. So Hilbert spaces and Bex's in Hilbert spaces are very different from manifolds, points on the manifolds. If you look at the relational structure, then you see that they're not really so different. And this is basically kind of the relational topological quantum field theory.

25:00 So I think Bayes is kind of espousing a structuralism in that sense. Smalling kind of focuses on background independence, and the need for background independence in our theories. So our theories should be free of absolute background structures, which is the metric in special relativity. Why is that structural? Well, because structuralism generally rules out any kind of absolute structures, okay, things that are intrinsically defined, okay, so obviously background structures, absolute background structures are intrinsically defined, they don't depend on any kind of connection to the other entities of the theory, okay, so I think Smollin's a structuralist in that sense. Stachel, kind of similarly, Smollin emphasizes the relational aspects of general relativity, okay, and the big focus really is on the whole argument, the lessons of the whole lessons of general covariance. And he thinks that this, that the whole argument, a nice end solution of the whole argument, points to some kind of relationism, okay, in general relativity. And also a kind of, a loss of individuality concerning the status of space time points. I'll be saying a little bit about both of those, the last two, the last two things. And I think, I mean, this is a sideline, but I think all of these three aspects can be kind of connected by a category theory, which we can sort of see in this fact. So this is all I'll be saying about John Bayer. So structuralism emphasizes big points of relational structure of things, and objects are considered to be constituted in terms of these structures. And then we can see that category theory really captures this idea Okay, so I have this quote here, what matters about an object is it's more than to and from other objects. So Category Theory encourages a relational worldview in which things are described not in terms of their constituents but by their relations to other things. Okay, so it seems that Structuralism and Category Theory are really kind of identical or really after the same kind of thing. Okay, so how much time have I got?

27:30 Five minutes. Five minutes? Five minutes. I don't have to be very quick. Speak faster. Okay, so John Stachel, first then. Stachel, as I said, lies with concern with general covariance and diffeomorphism invariance in the whole argument. and what diffeomorphism implies ontologically and metaphysically. So I was going to introduce the whole argument here. I'll put it like this. The whole argument is really kind of a generalisation of Leibniz's shift argument. So you consider two Leibniz shifted universes. This universe and the one that's shifted five metres to the right something. We'll call these diffeo-lots. The question is, do these represent distinct possible worlds or not? If you're a substantivalist about space-time points, then you're going to have to say that they do represent distinct possible worlds. Because the objects in the universe have been shifted onto a different set of points, so there's different relations is holding, okay? So you'd have to say that they're distinct. If you're a relationist, then you would say, no, they represent one and the same world, because the relational structure between the objects is exactly the same. There's no measurement you can make to decide which universe you're actually in. There is another option, which I mentioned, which is sophisticated substantively, to still say that the point is individual. John, I thinks that the relational viewpoint points to the fact that space-time points are non-individuals. Is that right? I'm not sure that that follows through. Maybe we can talk about that in the discussion. But he does say that I'll talk about my paper. Oh, well, there you go then. I don't need to talk about it. So then, a little quote from the physical points of space-time are a secondary derivative role in the theory and cannot be used in the formulation of part of the answer and not part of the question. So the point there is that the points have no intrinsic physical meaning before you specify the dynamics of a theory, before you're given a metric.

30:00 Clearly very relational, clearly very structural. And I won't talk about that. You've got about two minutes. Two minutes, okay. I've already said a bit about background independence so the point is and this is part of the problem of quantum gravity it really kind of, I think boils down in some cases in the lines of some people to the problem of background independence so like I said before all the QFTs in field theory are background dependent space given by field on space connection structure. The basic feature of general relativity is that this structure is dynamical. It varies in response distribution of matter fields. So quantum gravity becomes true. It's a problem when constructing quantum field theory is differentiable no metric. As Juhal mentioned, topological quantum field theory is one way of going and one way of constructing such a theory. There's no metric appearing. You've just got topological spaces. Another one is loop quantum gravity, which is Lee Smolin's main interest. And, well, I thought until last night that loop quantum gravity was still background-dependent, because you needed to make a fixed kind of topological structure. You need to fix the topological structure, okay? But because it's a canonical approach, you need to make a 3 plus 1 split. And I thought that still held, but apparently Lee tells me that that's been solved now, right? Okay, well that's good, that's good. Okay, I'm not, well I remain to be convinced and I hope I am convinced. Okay, I suppose the option is to go, I think this is what we might do, to go towards a covariant kind of theory and introduce spin thumbs, okay, so you're implicitly covariant so there is no two plus one split. Okay, I hope we can talk in the discussion period as well about the condition of background independence because while it seems intuitively well motivated the idea that we shouldn't have any of these unmovers in our theories it's intuitively well motivated but I can't is it a methodological principle or is it an ontological principle I don't know, so maybe we can address that

32:30 but I should say from the structuralist perspective background independence is a natural condition it outlaws such structures from the outset because only things which are relationally defined are allowed I'm going to finish that but that's what's to come background independence and individuality hopefully a solution of that thank you sorry about the time constraints but we need to move on the next speaker is John Staple from Boston University on structural individuality and quantum gravity. Thank you. Thank you very much. I want to thank Steve for organizing this and for inviting me. I have to be brief, of course, and brevity is the mother of dogmatism, so you'll forgive me if I just sound very much more optimistic than I want to be, but anyway, I have some papers which I explain these ideas much greater detail, I've got copies, and I can take your name and send it to you if you want to hear it, so here's the outline of my thoughts. Well, what is structural, really? I think it's going around a lot. Structural individuality, which is the main point that we just need to lose. I don't think a lot of times we talk about the geometry and algebra stuff, but that's in a paper of mine called The Brief History of Spacetime. I suppose humorous homage to Hawking's book. Critique of various formalisms, you see there, And then, what structures do quantize, which is a vital question. Okay, so, first point. What is structural realism? There's apologies here to the Greek scholars in the audience. I'm using their names Pythagoras, Plato, and Aristotle just to give you a quick handle on the three points of view. I'm not claiming these exactly reproduced of philosophers' positions, although I did get a nihil abstat from our departmental Greek philosophy scholar. The term structural realism can be and has been interpreted in several quite different ways, but I think all of them are really variants of the following three broad categories.

35:00 Only structures are real, nothing else really exists. The world that appears to be given by our perceptions is an illusion, and to matter, matter does not exist. I mean particulate matter, fields are just as much matter in this sense as particles and I symbolize that by the word Pythagoras the second point of view structures impose themselves upon an otherwise unstructured reality, only formless chaos would remain if the structures were not giving shape to the world, and I remind you of Genesis 1-2, and the earth was without form and void so the Bible is point two about the structure matter is inherently formless and passive and has to have structured matters imposed And I take Plato as my motto for that point of view. The third point of view, the world is composed of entities of various kinds, their essences, essentialism, that's why I bring in Aristotle, and is inherent in their nature to be structured in various ways. It is impossible to separate these entities from their essential structural properties and relations. Matter, then, is inherently property, structured, and dynamic. And it's this third version of structural realism that I espouse, and it forms a philosophical background for my further comments. may not be adherent in this third version, most adherents, myself included, assume that the entities of which the world is composed are of many different kinds, that's different essences in other words, and that these kinds are organized into a structural hierarchy, not necessarily linearly sequential, but rather more like a lattice of structures. Now I'm going on to my second point, structure and individuality. I don't have slides for every point unfortunately, except for a while we'll be without slides. it seems that as we probe deeper and deeper levels of this structured hierarchy of the world the property of inherent individuality that characterizes higher entities, such as human beings, cells stones, crystals, is lost to use a long established philosophical terminology we have reached a level at which the entities have quiddity they're this and not that this is an electron and not a proton, but they don't have hexiety This particular electron, as opposed to that particular electron, doesn't seem to make sense, whereas this particular person, as opposed to that particular person, certainly doesn't. And I quote Runes' philosophical dictionary. Hexeity is a term employed by Duns Scotus to express that by which a quiddity or general essence becomes an individual or particular nature or being.

37:30 The lower identities, insofar as they have any hexiety, inherit it from the structures in which they are enmeshed. You can speak about the 2p spin-up electron in an atom, for example, individuality to the electron, insofar as it does, but only a version of the structure with its connection. According to general relativity this property of inheriting this characteristic of inheriting its individuality only from the relational structure holds to the points of space-time. And I remind you that philosophers, at least since Kant often tried to use the points of space-time as a principle of individuation otherwise indistinguishable entities, so that fails if general relativity is correct. It holds the elementary particles of matter, which some relatives have tried to use as a principle to do individuation with points of space-time. They sort of reverse the Kantian argument. Let's use matter to individuate the points of space-time. Rebellion, before him, do it. But this, evidently, doesn't work either. Since the thought-for theory or theories of quantum gravity, because we have to realize there may be different ways of approaching the problem, or with certain progress would probably presumably underpin the existing theories of general relativity and quantum field theory at a deeper level it's hard to believe that the individuality of the fundamental entities of the theory lost already at the level of general relativity and quantum field theory will reappear, this property of individuality will reappear at the fundamental entity, at the level of the fundamental entities of quantum gravity whatever these entities turn out to be that's what led me to try to look for a way of generalizing the principle of general covariance to a principle which could hold whatever the fundamental entities are. And this is what I call the principle of maximal commutability. The way to implement this indistinguishability in any fundamental physical theory is to demand that the theory be observed under all possible permutations of its basic entities of the same kind. In general relativity, this is implemented for points of space-time by demanding general covariance under all the theomorphisms of the base manifold. And that was referred to in the previous talk. mechanics this is implemented for the elementary particles of a given kind electrons let's say by demanding that the Hamiltonian of the theory be a variance under all permutations of each kind of particle which occurs there at the level of quantum field theory it's implemented for quantum by the use of a clock space description and Teller's book on quantum field theory gives a nice discussion of this without the need for the surplus structure of the non-relativistic description now I'm hitting the discussion on geometry and algebra so I'm going on to my

40:00 What point, particular various points of view, so now I have this principle of maximal permutability which is whatever the fundamental entities are you better demand permutation out of all of these entities. In other words there's no rigidity to the underlying structure. All of the relationships between all of the properties of the individual entities emerge from the relational structure in which they're embedded. So first I turn to string theory. Protervative string theories immediately seem to fail this test, since the background space-time of whatever number of dimensions you choose, whatever your favorite is, it's only invariant under a finite parameter at least a group of the group of all possible diphtheromorphisms. And this means you only need to fix a certain number of points in the background space-time, and then the relations of all the others follow immediately from that. So you don't have maximum permutability, you have minimal permutability, if you like, and therefore there's an absolute element, which I just referred to before. Many string theorists coming to the field with a special role for the quantum theory program initially found it difficult to accept this criticism. So I find it encouraging that this point now seems to be widely acknowledged in the string community. And I'll quote from two recent review articles by Michael Green and Thomas Banks. And my principle here, as you'll see, will be, quote, as much as I can for the people who work in the field. My opinion is worth a lot less than their opinion. So, speaking of the original string theory, Michael Green says, the description of string theory is wedded to a semi-classical perturbative formulation in which the string is viewed as a particle moving through a fixed background geometry. Although the series of super-string diagrams has an element of the description in terms of two-dimensional surfaces embedded in the space-time, this is only the perturbative approximation to some underlying structure that must include a description of the quantum geometry of the target space, as well as the strings propagating through it. And a little later he says a conceptually complete theory of quantum gravity cannot be based on a background dependent perturbation theory. In a complete formulation, the notion of string-like particles would arise only as an approximation as would the whole notion of classical space-time. And speaking about more recent developments, which is summarized by the name M-theory, whatever M stands for Mother Matrix I'm sure it doesn't An even worse problem with the model is that the formalism is manifestly background-dependent.

42:30 This may be adequate for understanding M-theory in specific backgrounds, but it's obviously not the fundamental way of describing quantum gravity. Next, I'll quote from Thomas Banks. A new article, 1998. String theorists have long fantasized about a beautiful multidisciplinary principle which will replace Einstein's marriage of remaining in geometry and gravitation. Matrix theory, most emphatically, does not provide us with such a principle. Gravity and geometry emerge in a rather awkward fashion, if at all. You're a very honest man, obviously. Truly, this is the major defect of the current formulation, and we need to make a further conceptual step in order to overcome it. And of course, it's my hope that the emphasis on the importance of the principle of dynamic individuation of the fundamental entities, with its parallel requirement of the invariance of the theory under the entire permutation group acting on these entities may contribute more contribution to help mainstream theorists make the split conceptual set which they are coming to realize next i'll turn to uh my b canonical conversation general relativity is inherently a four-dimensional theory treating processes as the basic element and remember a space-time point is really or an event in the physical language, is just the classical limit of a process as you let the dimension shrink to zero. And one doesn't have to take the points as seriously ontological. I want you to just say that there's no limit to the smallness of the size of the extreme space-time region in the classical theory. And of course, we would expect quantitatively to impose a limit. The breakup into a three-form, three-plus-one form, by introduction of a frame of reference is represented geometrically by the of the vibration and foliation of space-time and this has something artificial about it and I'll show you a couple pictures to show you what I mean here I've just skim out the diagram the process this is obviously a two-dimensional perspective on a four-dimensional space-time I'm just one of those little circle sort of things that give you an image of more dimensions and that's

45:00 meant to be literally cross sections because this tube then this world to represent a process which is limited in space, unlimited in time, although you could cut it off at those two space-like sections I've indicated there, and then you have a process which is limited in space and time. All right, so that's what I mean by process. Here's what I mean by a vibration and a foliation. A vibration is a family of time-like curves, which fills the space-time up, and foliation is a slicing of the space-time by apparently space-like hyperservices. So you have the vibration and the foliation represented by the green and red. Well, you usually see why I don't make my living as an artist, but my best attempt. And now, if we have a process, which is the basic thing we have in general relativity, and we impose a vibration and a foliation on that, you see we can get space-like cross-sections of the process and we can have an evolution in time quotes along the vibration but you see as I shift that around I get different slicing and this is only one possible vibration formation but just moving around gives you the idea what you call a space-like hyperservice depends on the vibration for the action you take it's all relative to that Now, Lee Sparland has, more eloquently than anyone else, emphasized the significance of process in our approach to physics that General Rotherly Lee's was too, and let me give a quote from Lee's wonderful book, Three Rotes of Quantum Gravity, which I recommend very strongly to everyone to read and second mind thinking about many questions, and I will have scavenged many beautiful diagrams in the book, as you'll see later on. Motion and change are primary. Nothing is, except in a very approximate and temporary sense. How something is, or what's, what its state is, is an illusion. Well, I would, that's a little bit too strong for me. I would say it depends on a vibration and affiliation. If you reduce a vibration and affiliation, you can give meaning to the state and its evolution in time, but only relative to the vibration of H. It's not an illusion so much as the concept which is relative to a frame of reference. That would be the nice physical word for the vibration of H, the corresponding physical idea is a choice of frame of reference. It may be a useful illusion for some purposes, but if we want to think fundamentally, we must not

47:30 lose sight of the essential fact that it is an illusion. So to speak the language of the new physics, we must learn a vocabulary in which process is more important than and prior to stasis. And it's that last sentence I want to emphasize. vocabulary which process is more important than prior to stasis. Yet the canonical formalism, of course, just reverses those priorities. It breaks up three plus one break up, gives us stasis first, and then tries to recover process from stasis. And that seems to be the main problem. Raphael Sorkin, in a wonderful review paper, forks in the road on the way to quantum gravity again summarizes the some of the drawbacks of the kind of the formalism very well and I quote him one of the major problems he introduces as he says what has been called the problem of time so the problems would probably be more appropriate word in three such problems. One concerns the temporal meaning of a logical order required by a projection postulate. One writes the projections in a definite sequence determined by the order of observation in time. But how can such a rule avoid leading to a vicious circle in a theory in which time itself is one of the things being observed? In other words, this is all well and good if you have fixed background space time. So the sequence of measurements has a natural temporal order attached to it. But if you're trying to define time in the theory you're dealing with, what does the sequence of measurements, a lot of the sequence, have A second closely related difficulty concerns the frozen formalism. In consequence of the Hamiltonian constraints, the physical observables are necessarily all time independent, and one seems forced into an attempt to fix the time gauge in order to recover a semblance of space-time from the disembodied space like hypersurface to which the formalism directly refers. oh, there, yeah finally, in a framework based on observables rather than beables and this is more Sorkin's way of approaching this problem how are we to speak about, say, the early universe

50:00 if there were no observers then and none in the authoring for a long time to come I would rather put this point in a different way since I'm not as wedded to the idea of beables So I'd rather put the last point in an observer of free language. Since we need macroscopic devices capable of registering irreversible marks to interpret the usual quantum mechanical formalism, how are we to interpret the formalism at a stage of the evolution of the universe before such devices, which need not be of human origin, I'd point out, have formed? In other words, if the original stage of the universe in terms of the structural hierarchy, I suggest, is not only diachronic, but synchronic, Emerged at a later stage how we interpret upon the formless at a stage before the individual IE has emerged Not for us to talk about measurements of macroscopic devices, which Macroscopic entities of any kind with irreversible marks can be made. That's the generalization of the measurement system And he points out, Sorkin points out Okay, he points out that none of these problems of time would seem to exist for the sum of our history's space-time approach. This is the final approach. Time itself doesn't need to be recovered because it is there from the very beginning as an aspect of the space-time So to summarize, the canonical formalism, by its very nature, reduces a vibration and a foliation, which reduces the different morphism group to the subgroup that preserves the vibration and the foliation. Namely, you must take points of a fiber into points of a corresponding fiber, and they take points of a leaf of a foliation to corresponding points of a leaf of a foliation. And there certainly must be a price to pay for this reduction from the full difiermorphism loop. And now we'll talk a bit about loop quantum gravity. What's happened? Okay, well, a vein of oil. Is there an extra ball? That's a good question. I'm not the one who knew me. I'll try to turn it over to you just by chance. Tahiti. Fantastic. Thank you. That's the reason why I was here for non-experimental.

52:30 Sorry, just anticipating Lee's territory, mind you, there's a spin network. It's just essentially a graph of a space like that, a spatial hypersurface. It's a graph of the edges on the edges. I'm sure we'll go in a bunch of detail, so I'm just going to leave it at that in the interest of time. price that we pay for introducing this vibration and foliation is that instead of a quantum of process which is what we'd be led to expect from Lee's beautiful encomium to process thinking we get a quantum of process by the way would be something we expected the order of the plant length to the fourth power in its magnitude and there's a quantum of four volume the loop quantization procedure produces quantum of three volume and two area in other words of the order of Planck length cubed and Planck length squared, respectively. But such quantic, as I indicated by my previous pictures, can only be defined relative to a frame of reference, which in the classical case is implicitly defined by the initial data and its evolution along the foliation and vibration. The initial data, remember, is the first and second among fundamental forms of the initial hyperservice, if you like, the metric and its firstly derivative in the direction, in the unnormal direction. So I can make quite good sense of what it means by saying I have a particular hyperservice data but of course in a quantum theory speaking crudely you can only give you half that day because you're the first form of the second one but for something from our quantum analog of that so I don't know what frame of reference it is I'm defining these content of volume and area with respect to so I say what this frame of reference is in the case of loop quantization is not clear to me which doesn't mean that it's not clear and I hope to hear from Lee whether it's good him but even more seriously by introducing a foliation one is defining time, the evolution of which must be continuous. So one precludes the possibility of quantization of time. Oh, I'm sorry. Here's how the loop quantization begins with causes. So we were taking this picture. This is how the area and volume are produced in the loop quantization procedure. I'm looking to talk more about that. So what I want to show you is the next picture. in the spin form approach

55:00 when one tries to set the spin network there's three dimensional entities which we just talked about we try to set this into motion by means of the spin form one has to introduce finite time steps by hand with no intrinsic indication of their size you have to just build it in of course dimensional considerations suggest that the unit of time should be of the order of the plot length but this has to be built in by hand so to speak uh you see here we have this spin foam network developing that jumps from here to here but why here to here and why is it developed continuously and what's the size that's there well that has to all be built in by hand it seems to me it's not given to you by the theory and indeed when i turn to uh fortini markupulo who has worked and thought deeply about these problems oh here's another picture the spin foam involved in the quantum sphere network. She's thought deeply about these problems, and in her most recent paper, she says, the first spin foam models were based on the predictions of loop quantum gravity. A major result of loop quantum gravity is that quantum operators for special areas and volumes have discrete spectrum. Discreteness is central to spin foams, which are discrete models of space-time at part length. Since, based on results of other approaches to quantum gravity, okay, that's just her way of, I mean, she's saying better than I, what I was trying to say before. Then she says, thus, a spin-form model will be a good candidate for quantum theory of gravity, only if it can be shown to have a good lower energy limit, which contains the known theories, namely, general relativity and quantum field theories. In other words, this has not been done, this is just a proposal. It doesn't fall automatically from the canonical form of the press. and not to slight John Bias, I have a quote on the two spin foams are close analogs of Feynman diagrams and indeed they are Feynman diagrams in the group field theory approach this means that as with the Feynman diagram theories in spin foam models we can condition any expectation value by limiting the class of spin foams to be summed over or weighting them with a suitable factor so this is of course too much freedom more in this case. It's not a fully, the theory is not fully given to you by the attempt to take

57:30 the canonical formalism and dynamize it, so to speak, because once you shoot something, it's a lot harder to revive it and bring back a life than if you hadn't killed it in the first place. so now I'll think about the causal set approach the approach favored by the smaller and I'm just Again, I just have to very briefly give you a picture. This is taken from a paper by Fattini-Marcopolo. Gives you some few little pictures of what Fattini-Marcopolo said. Yeah, you met Raphael, sir. Pardon me? You met Raphael, sir. This picture is taken from Fattini's. Right, no, no, but he favored it. He, I'm sorry, what did I say? I'm sorry. I'm also a fan of it. Yes, okay, no, I did, I'm sorry, I apologize. Everybody's also a soul of error. Okay, so the idea is you just give a set of points or entities which have causal connections among them. They form a partially ordered set. And we have some pictures here. The very simple one at the bottom, the more complicated one at the bottom. But you have two A-causal cross-sections of the causal set, circle things, those that would take the place of space like hyperseptics, points which have no causal relations among them. So those are the eight causal sets. This approach seems to have already adopted the viewpoint suggested here. And again, it's Raphael Sorkin, I'm going to quote from his most recent paper. After introducing the idea of a labeled course set, it's short for causal set. After introducing the idea of a labeled course set in which it is introduced, Sorkin comments. After all, labels in this discrete setting are the analogs of coordinates in the continuum. And the first lesson of general relativity is precisely that such arbitrary identifiers must be regarded as physically meaningless. The elements of space-time or of the cross-set have individuality only to the extent that the pattern of their relations to the other elements it is therefore natural to produce

1:00:00 a principle of discrete general covariance according to which the labels are physically meaningless but why then have labels at all well maybe given uh given time to read the rest of the quote later with get the point he's he's recognized this principle already so much for my critiques now I'll say a bit about this problem of what structures in the general theory relativity there are a number of space-time structures that occur The papers were discussed in much more detail. Anybody who liked the references or copies, I can raise them. The chronogeometry, what rods and clocks tell you, is represented mathematically by a pseudometric tensor field on a four-dimensional manifold. The inertial gravitational field is represented by a symmetric affine connection on this manifold. Then there are compatibility conditions between the two previous structures, which essentially says the covariant derivative of the metric with respect to the connection must vanish. And physically, this means that rods and clocks gravitation field, they read correctly. There's no distorting effect of the inertial gravitation field on the rods and blocks. That's essentially the physical field of the compatibility conditions. There are a number of ways of formulating the relation between these three entities. Traditionally, both historically and in most textbooks, it's possible to start with the metric field and derive from it the unique connection, the Christophel symbols that automatically satisfy the compatible conditions however it's also possible to treat the metric and the connection as initially independent and allow conditions a compatibility is to emerge from Palatini type variation principles and this was able to prove very important in the Astakar approach although then get associated with a but it's a very important point for a lot of the two to be independent and then use compatibility conditions that's an important alternative way of thinking, and much more physically intuitive, because you see why connection is not going to do with the inertial gravitational field. And the curvature is really about how geodesics approach or move away from each other, whether you can accelerate or unaccelerate in a way, otherwise you don't understand what curvature has to do with gravitation.

1:02:30 And it's possible to combine either of these approaches with tetrahead formalism, in which you introduce the tetraed, the one-dollar of the metric, and you combine this with various mathematical representations of the connections. For example, the connection forms have been used very much. They're treated as differential forms, the connections. But also you can use tetraed components of the connections, which the connections, the tetraed components are scales, and that's very useful for physical interpretation, as I've shown in some other work. So there's various possibilities here, but there's also another way of approaching the whole problem, I know nobody has yet used this in the quantum-gravity game, although the Sorkin approach, by emphasizing causal sets, comes closer to it. Namely, one can abstract from the volume-defining property of the metric, the four-volume-defining property of the metric, and which results in a causal, in a conformal structure on the matthew. Those two metrics differ only by a factor, have the same causal structure, they define the same null construction, so to speak, classically. And this is all that's needed to represent the causal structure of space-time. On the other hand, it's also possible, starting from a connection to abstract, the connection carries with it an inherent proper time, if it's a time like geodesic, or at any rate, some unique affine parameters associated with the connection. But you can abstract from that affine parameter and say, I only want to define the paths. The geometry of paths results, if you do make that abstraction, that's called the projective structure in the mathematical sense. So you have a projective structure which defines the class of preferred and most importantly is the time-like paths in space-time results. And then compatibility conditions between the causal and the projective structure can be defined, which guaranteed the existence of a corresponding compatible metric and connection. And that's the work of Ehlers, Pirani, and Schild. Now, it seems to me that just as it was so important in historical development of quantum mechanics to choose an appropriate formulation of classical mechanics to apply some quantization technique, not that you can't use different formalisms, but the quantization technique and the formalism seem to have a very close relationship to each other. So I suggest that it may well be the case that one or other of these formulations of general relativity will be more helpful in solving the quantum gravity in one or more of the various ways in which it has been or will be posed. I'll start with this one. Thank you.

1:05:00 Okay, we're going to take a break and get first in line for the coffee. Let's try and meet back at 25 past for Lee Smolin and John Byers' papers. Okay, thank you. Let me pause. Can I pose this? Thank you. Would you like me to give a yellow color in the blue room because of the quantum gravity system? If you can get John Byer I can start slowly I was reading his paper on the phone and on the interactive screen he had a whole variety of music and one of the albums was the greatest hit So if you don't want to take a picture, the night they draw a vote takes you down. What would happen to John by his marriage homeboy? We can sing along to quantum gravity. Okay, well, sorry to drag you away from the coffee and bagels. Hopefully we'll have a bit more time for discussion as we start a little early. Our next speaker is really someone else who really needs a great introduction, and Lee Smolin from the Perimeter Institute for Theoretical Physics talking on quantum gravity at the background independent level.

1:07:30 I'm very grateful for the invitation. Let me say that the way that I see the search for quantum gravity, it's a search for new languages, conceptual, mathematical, and new principles to be expressed in those languages, and all the different approaches that people have spoken about are heuristic, that is, are ways to get there. Quantum general relativity, which is another name for loop quantum gravity, is one way to get there, and we have learned a lot from it, and I will be talking about some of that, and some people already have, but really what's important also is to talk about the questions that puzzle us, and so I want to split my time more or less equally between a report of those things that we really understand and that is that we have learned from this approach and therefore stand as ingredients to be put together in a theory and I very much agree with Sean's point of view and maybe we'll have a discussion and address it but it's also equally important to talk about the things that really puzzle us and really confuse us and I meant to of three-quarters of one and a few lines of two, but talking yesterday, I was encouraged to switch it, so I'll start with a prelude about two of the things that are most puzzling of the moment, maybe at least to provoke people and to get everybody away after the coffee, and then I'll talk about some of the basic structures that came out of Luke Bottom Gravity, although John has already mentioned them and the other speakers mentioned them, so I won't to say as much and then i'll come back to some other puzzling points um and i'll start with the thing that for me is the most puzzling and i'm aware that these are issues that philosophers have talked about for millennia but um and i agree and i do i am sure that anything that i have to

1:10:00 say is naive but the issue which to some of us is most puzzling has to do with the easy division has been made for a long time between laws on the one hand and initial conditions on the other hand, and in the laws between kinematics, which is the language of what kind of stuff do we talk about, and dynamics. And both of these distinctions break down in different aspects of cosmological theorizing. And what we're doing in quantum gravity is, if nothing else, is an attempt to make cosmological theories. So, for example, the relationship between law and initial conditions is breaking down in all the ways in which the issues of the parameters of the standard model of elementary particle physics and the parameters of the standard models of cosmology, those parameters are migrating from the law column to the contingent column, the initial conditions column. And there are lots of examples of that migration. I could talk about them, you know, one by one, but certainly the fact that the electron mass depends on the Higgs field, which depends on what phase the vacuum of the universe is in, and so forth. And it has to realize a, you know, a traditional reductionist agenda by finding one true platonic mathematical theory that, from which everything will be calculable, leading from the standard model to grand unifying theories to string theories, has gone, I think it's fair to say, the opposite way that it was expected. That is, the more unification and the more symmetry, the more free parameters, and the more things that were possibly law-like migrate to contingency. So the boldest way I know how to put this question, and I'm just going to raise this a question and then go on because I don't even know a language to talk about this so I'll give first my language and then somebody else's language. My language is what is the status of internally true laws of nature what is the meaning of things that transcend time if A, time has been reduced to a network of causal relations, which I'll argue in a little bit at least in lukewarm gravity

1:12:30 is being realized Two, if we accept the hypothesis that the Big Bang was really the beginning, so the past of any observer has a finite number of causal relations or events in it, then what is the status of the idea that there's a clear separation between laws on the one hand and history on the other hand? And a political philosopher, Roberto Unger, puts it this way, And I know there's an easy answer to this, but it still provokes me. He says, if time is an illusion, then so are our causal judgments. Fine, that's the answer to Julian Barber. However, if time is for real and the universe has a history, then our causal judgments lack a secure basis in general laws because the laws will also have a history. Now, you might say, well, no, the laws won't also have a history, but it's not that easy. And I just want to leave that hanging there for the discussion. Here is an apparently unrelated issue, which is also on the forefront of what's puzzling us. But I'm raising them together because I suspect that they are related. You know, a lot of things in a field like Plum of Gravity, where there's a lot of stuff that is really technical, people may suspect that a lot of the technicalities are irrelevant principles in the real physics, and I think that that's true. So here is a question that has been raised from time to time, but never dealt with, I think, until within the last year. What is the status of the claim that there is this special scale in physics, the Planck length, which is a combination of fundamental parameters, Planck's constant, gravitational constant, and the speed of light. There are lots of claims that this length has a certain special role. It's a threshold on scales longer than this. Classical geometry is the right description of space-time, a la general relativity. On scales smaller than this, some new description. Quantum space-time is supposed to hold sway. And every once in a while, somebody, a student, or somebody in a bad mood, asks the question, a minute, lengths transform under transformations between different observers. So if we're two different inertial observers, and I see a phenomenon that's smaller than the threshold,

1:15:00 so I would describe it in a quantum mechanical, quantum geometry language, it might be larger than the threshold for another observer. So in whose reference frame is this the threshold? And the question can be raised about in terms of lengths or quantum mechanically in terms the same issue. Now, this question has been ignored by most of us because it made us too uncomfortable. One might notice that actually Lorentz invariance is not a principle of general relativity. It happens to be an accidental symmetry of a solution which is in common use in coursework, that is Minkowski spacetime, but that doesn't correspond to anything in nature. And also, nasty people like to point out that the universe actually is not Lorenzenberry and has a preferred reference frame. And this goes back to the history issue. So this makes us properly nervous, and so nervous I think that most of us ignore this issue. But in fact, there was a resolution of this question, there's a possible resolution of this question on the table which is elementary in the sense that anybody could have invented it knowing high school algebra and reading the basics of relativity and in fact was invented and reinvented several times but has become to be taken seriously and this is the idea some people call doubly special relativity that you can actually modify Einstein's principles from his 1905 paper and and I was shocked when I realized this this was actually known by Falk in the 1930s and it's Giovanni Milino Camelia who really revived it as well as Jean Miguel and Andy Albrecht and here are to cut through a lot of discussion. Here are some possible other axioms for relativity. The first axiom, as Einstein gave it, as Galileo gave it, the equivalence of inertial observers. The second axiom modified to say all we really know about the invariance of the speed of

1:17:30 light is in the limit of light photons which have relatively low energies compared to this fundamental scale upon scale. And once you have this fundamental scale, you can ask, maybe you can weaken this to the velocity of a photon, which might be a function of the ratio of its energy and that fundamental scale, in some new theory, only the limit that the energy is very low should go to the universal constant. And if that's going to be coherent, then this scale should be universal. So in contrast to the paradox that I raised, maybe the Planck scale is universal, is the same for all observers. So if you have one threshold where you see new physics, I see new physics at the same time. Now, is that possible? Is that nuts? Well, it turns out to be actually elementary. And here's just one of an infinite number of possible resolutions of it. Take your Lorentz algebra. There's things that generate boosts, and I'm thinking momentum space, but you don't need to, and add to your generator of boosts a new object which is the Planck length times the energy times a generator of changes of scale of energy. And what you're doing is you're saying energies transform as well as momenta but I'm going to add a term that just stretches the energy again and I'm going to arrange the coefficients so that there's one energy such that for one particular energy the action of the boost is cancelled by this action and you end up finding that you have another version of special relativity you have I'm trying to get this all there so everybody can see it you have a new nonlinear. You can check if you're bored by the rest of what I have to say that the algebra of Lorentz transformations is left unchanged by that addition. Again, something that Fock noticed in 1928 or something, but has been not noticed since then. There's a new invariant which is nonlinear. So this is now the relationship between energy, momentum, and mass. That means there's an energy-dependent speed of light satisfying that principle. And why are we thinking about quantum gravity are very excited about this because if you didn't

1:20:00 know anything about string theory or loop quantum gravity or any fancy stuff and you just went around into the laws of physics sprinkling terms that have Planck length in them you would say hey maybe I should modify the relationship between energy and momentum with some term that has the Planck length in it or maybe I should modify the formula for the speed of life and you quantum gravity phenomenon, and this one, the energy dependence of the speed of light, is detectable now, well it's boundable, up to a factor of a thousand in existing data with timing of signals that come from very far away. And an already planned gamma ray observatory, the glass observatory, is going to be able to see this effect in 2004, 2005. To what order? Order one. So, just because the electronics has gotten so much better. Okay, so that's my prelude. And the prelude is, the point is, we're still worried about the simplest things. We're still worried about basic principles. We've learned a few things, and now I'll go and tell you what we think we've learned. But on the one side, basic issues that are hard to think about still plague us. And on the other side, however, experiment is coming around the corner. So maybe all this stuff has a chance to become real science. Now, just three transparencies or so about loop-bottom gravity, And then I'll, in whatever time I have left after that, I will mention a couple of other issues. But those are the two big ones that face us. Various things were said about loop quantum gravity. To make sense of it, I distinguish between, borrowing from philosopher talk, between loop quantum gravity 1 and loop quantum gravity 2. Loop quantum gravity 1 is the quantization of the Einstein equations. In a certain, is that five minutes? Half time. Ah, okay. In a certain, using a certain set of variables. Okay. And I think we can claim that it is an honest quantization of the Einstein equations.

1:22:30 We no longer very much distinguish between the canonical approach and the path that enroll a spin-phone approach that I guess John will talk even more about than I do. I hope a little. A little. Okay. because we think that this is an honest combination of quantum theory with the principles of general relativity, and any quantum theory can be expressed either in a Hamiltonian language or a sum over a history's language, and it's no different here, as far as we can tell. But I'll tell you first about the Hamiltonian language and then a minute about the space-time language. And loop quantum gravity 1, anyway, is the quantization of the Einstein equations or some extension like supergravity or couple of thematic. Loop quantum gravity, too, is what you get when you get imaginative people coming into the research program who think about things like why is there still some background dependence about topology, and let's see if we can muck about with the mathematical and conceptual language that we got by quantizing Einstein equations to find something more plausible on various grounds. So, loop quantum gravity 2 is a family of ideas and theories, none of which are properly, can be derived by quantizing the Einstein equations, but which use very similar language, generally with less structure. quantum gravity 1 the states are diphtymorphism classes, what is a quantum geometry what is the state of quantum geometry quantum geometries are labeled by a basis in a certain Hilbert space and the Hilbert space is labeled by and this is a mouthful of diphtymorphism classes in that work embedding so we assume contrary to good advice that you fix the topology and differential structure and a spin network is a graph with some labels on it coming from category theory and that's all I'll say about it and for every way of embedding the graph you have a different stage well I say a little bit more on the slides the edges come from representations of SU2 or maybe some other algebra and the vertices come also from representation theory Then, there's the notion, I'm just really repeating myself here, of an embedded spin network, where you fix the manifold possibly with the boundary, you embed the spin network in the boundary, and you distinguish those up to 50 morphisms.

1:25:00 and those are the states and there are results concerning those states like the most important so far are the discrete character of area and volume that is these states are eigenstates of observables like the volume of the universe or the area of that spatial boundary and it's that discreteness by the way that gives rise to predictions for corrections to the energy-momentum relations. So that there are actually predictions from this work that relate to what I said about special relativity, although that work is new and is in flux. Another key result, key kind of result, of which there are many examples, is that the dynamics acts at nodes generally to create nodes or to eliminate or move around nodes. Okay. And there are a lot of results to talk about, but I'm just trying to give you the picture and then we can sum up. So what is a quantum space-time geometry? I'm going to define it abstractly because John will probably draw more beautiful pictures. No? We'll also define it abstractly. A causal spin foam is an object that represents a history that goes from one initial spin network to one final spin network, and the idea of the history is it consists of a series of events, and events are local changes of, say, these characters here in the spin network. And one can think of, there are various ways to represent it, but one can think of a causal spin foam history as a combinatorial structure which has boundaries, which are the spin networks, events, which are local changes in the spin networks, and partial orders between the events. and as a way to propagate the labelings from the spin network graphs into this combinatorial structure it's sometimes realized in terms of triangulations of the space-time and sometimes realized in terms of extending the edges of the spin networks to two surfaces

1:27:30 and those two realizations are equivalent I think what's important from the point of view of the issues like relational substantival is that each history is something which has structures abstracted from the structures of a Lorentzian manifold. It has events, a discrete set of events. The events have a partial order, borrowed from Raphael Sorkin's causal structure approach. The a-causal sets also have a structure which is the structure of the spin networks. Now, what do we do with them? There's beautiful pictures. What we do with them is we behave completely stupidly and conventionally. That is, we go back to Feynman. And this I want to say because I think that this is a weak point which needs And we say, okay, each history has an amplitude, F is a spin-foam history, and the amplitude is, roughly speaking, a product of amplitudes for each of those events where there was a local change in the spin network. and there are formulas, candidate formulas for those which have been deduced by different reasonings from quote, quantization of general relativity but in each such event there's a set of labels which enter and you need some function of the events of those labels which is a complex number which we say is the amplitude of the events and then we say that the total probability for a physical state to evolve from one initial spin network final one in a total time t is given by the total amplitude squared conventionally where that's considered to be the sum over all of those spin network histories where there's a measure of time given by in the simplest case by just the number of events you would consider that a measure of time and this is john was quoted john bias was quoted by another john is saying you can weigh and consider these amplitudes differently, this is just one thing that you can compute. Okay, so skipping a lot of stuff,

1:30:00 that's the basics of neutron gravity one. And I think one can say regarding, and I would use the word structural if I really knew that I could use it in a professional way. regarding the relational versus substantival debate, and substantival was already a big step for us physicists. I think what one can say is that this picture is relational, like spin-foam history by spin-foam history. Each spin-foam history is a network of relations. There are two kinds of relations. the causal relations and the relations of incidence that is where the nodes are tied together the nodes of the spin network are tied together by edges and so there are two kinds of relations and they build each one of these space-time histories and we have interpretations of those relations in terms of quantities taken from classical general relativity so the picture is relational on the other hand it is about something it is about certain kinds of relationships and so this is not as extreme as Raphael Sorkin's idea that everything should be described by causal relations but it is a reduction of the notion of a quantum space-time geometry to a network of relations which are about something that is about areas and volumes in terms of the spatial relations and in terms and they're about causal relations now um i think i could say more about lupon gravity but i think i go on to the problems as we see them now oh i wanted to say something about lupon gravity 2. well lupon gravity 2 could be seen as the response to your criticism that why do these structures need to be embedded classical topological manifolds, and the answer is they don't. You can just erase that part of it. When you erase that part of it, that is, the spin networks are no longer embedded in a three-dimensional topology, and the spin foams are no longer embedded in a four-dimensional topology, you greatly simplify

1:32:30 the structure, but you lose the claim that it was derived from a quantization of general relativity. Okay, so in the last few minutes, I want to describe what the conceptual problems are. Five minutes. Perfect. So, one, this is a candidate quantum theory of gravity. And let me just be strict and say luke quantum gravity 1. There's a description of what the states are. There's a description of what the histories are. There is proposals in the literature, and there are different ones under different names for what the evolution amplitudes are. And the relationship to physics is supposed to be made conventionally by the sum over histories, sum over all the spin-phone histories, to get an amplitude to go from an initial state to a final state. the first question that we worry about is what is the right measurement theory for these beasts and there is an approach to that that I'll just discuss which has been discussed by some philosophers and criticized which is this idea which is called relational quantum theory here are some of the key ideas the basic idea came from criticism of the many worlds interpretation, taking the point of view that the many worlds interpretation fails and the coherent histories interpretation which tried to follow it fails. So we have to really do something new. And the idea here is to somehow take Bohr seriously and formalize and promote Bohr's insight to a genuine mathematical structure. Bohr's insight is that the conditions for using the quantum state description are made by defining a split of the universe into a system to be described quantum mechanically and the rest. And the basic idea of these points of view is to relativize that in the sense that for every way of splitting the cosmology and other, you should have a Hilbert space description, and a Hilbert space description in a sense is concerning with the information that can be measured or conveyed

1:35:00 across a boundary. Lewis Crane wrote about that. Carlo Rovelli has some papers writing about it. Cotini Marco Pulo proposes that the divisions be made using the causal structure itself, that is roughly speaking for the causal past of any event, there is a Hilbert space description, which is a partial quantum mechanical description that is a description in terms of an open system, in terms of density matrices rather than Hilbert space states, and that the whole quantum cosmology is a collection of all the density matrices corresponding to information that would be seen coming from the causal past of any local observer, and these are tied together by a certain structure, which is proposed, a certain mathematical structure, and Jeremy, who is here, Mr. West here, and Chris Isham actually preceded this with the proposal for a mathematical structure for the consistent history interpretation in terms of Topos theory, and there is a sense not completely developed that that structure is the right structure for this proposal in which the regions are divided up in terms of causal set. And that's a very nice idea that various people are developing, but it works nicely as long as the structure that the quantum theory is hung on, which is the causal structure, is given once and for all. And that's the context in which But the whole point of general relativity is that the causal structure is dynamical and is the result of dynamical evolution. So now you have to imagine something very wicked in which the structures that Jeremy and Chris studied are generated dynamically. And to my knowledge, unless Jeremy knows something I don't, nobody knows how to do that. And he's being very quiet. She'll know what that is. And a related issue is, it's fine to say that we take the conventional quantum mechanical view and sum over space-time histories, but again, this is an old issue that people raised a long time ago, but we see it concretely here. When the structure of quantum theory is hung on causal

1:37:30 structure, but the causal structure is dynamical and to be summed over, we have an issue. We can And I think that's the news, is that we can do it formally. But what is the measurement theory? And one begins to wonder things like, is the sum over histories itself relational? That should each local observer deduce aptitudes for what they may observe from their past by summing over possible histories in their past light cone, rather than imagining a sum over histories of, quote, the whole space-time? No one knows. That's why I'm mentioning these. And so a related issue that we worry about, maybe John will mention also, is that there's the old dirty little secret about summing over histories in final gravity, which is that the instruction is to sum over a set of, quote, histories with complex amplitudes. Now, the same-form amplitudes don't generally look like E2DIS, but they are complex numbers. and we don't know no mathematician can tell us how to define those sums sums over complex numbers are hard what people know what physicists and mathematicians know how to sum is sum over probabilities probability averages as in statistical mechanics so there's an old trick where you notice that your action here comes from the integral over time of some Lagrangian and you say what if time was purely imaginary so it was the square root of minus one times some time called Euclidean time then the sum that we have to do would be over e to the minus some objects which we could identify with a Hamiltonian and so now we're doing classical statistical mechanics with probability measures one dimension up that is we're doing the classical statistical mechanics of the histories and that's what quantum field theorists do all the time Now, the problem is that there is no way to ensure that the procedure involves a choice of a time variable. And there's no way known to ensure that different choices of time variables do not lead to different results for computations done this way. and therefore we have a problem either in fact there's a way to define these

1:40:00 so that these sums can be defined in a way that's independent of the choice of time variable or the issues about preferred time variables etc. are coming back in in an ugly way and this worries us a great deal I should say that there is very beautiful and important work to really do these sums in an important class of models, Byron, Anilol, and Jan Onborn and collaborators, and that beautiful picture I showed comes from their work, and they just commit this thing. They just pick a preferred time variable, or they define a preferred time variable in each history that they're summing over, and they just do this operation with that preferred time variable, and they get very intriguing results, but results that seem to not be Lorentzian variant, to come back to the thing. And so we're caught between that, and if that doesn't work, if none of this that I mentioned works, then we're really back to square one with quantum theory. So just to close, one way to view the project of quantum gravity is really as the way to attack the foundations of quantum mechanics. That is, quantum mechanics, quote, works very well for everything else in nature when we're not incorporating space and time. But it's obvious that the issues in quantum mechanics interact with space and time in confusing ways, like Gall's theorem and a-causality and so forth. So one way to attack them is to try to combine general relativity with quantum theory and see what happens. The, to me, surprise of loop quantum gravity is that one can say, I want to follow a certain route to making a quantization of general relativity, and an answer does come out. And at the level of mathematical structure, the answer seems okay. There seems to be a thing which is called the quantum theory of general relativity, and then there are various generalizations and extensions of it. and that's intriguing that is there really seems to be an answer to that homework problem what is the quantization of general relativity at the same time trying to really get real physics out of it and connect to experiment seems to push us back to worrying about the same kinds of conceptual issues

1:42:30 that started the whole project so I'll stop there thank you Okay, our next speaker is John Baez from University of California, Riverside. And he's going to tell us about higher dimensional algebras and quantum gravity. Higher dimensional algebra. So, when I was preparing to give this talk, I was preparing to start off with a joke, and the joke is one that was all, the first part of the joke is all maybe familiar to you, which is this joke that a scientist at a philosophy of science conference feels like an elephant at a zoology conference. But in fact, that joke isn't really appropriate now that I've actually been listening to these talks. I think it's more like going to a zoology conference and noticing that everyone there is acting like an elephant. So I did my best to try to pass myself off as a philosopher by reading from some prepared notes and wearing a suit and everything. But then it turns out that everyone for your clothing, and more importantly, they're using transparencies and drawing pictures of spin networks and bordisms, throwing around words like FUNKER at someone. So, it's very interesting, but I think it's a very good thing that both sides are trying to come together, and maybe philosophy and physics will actually meet at the quantum scale. So, I've been working on an approach to quantum gravity called spin-bone theory. I'm a mathematician, but I dabble in physics pretty intensively. And what I'd like to do here is reflect on some of the underlying ideas of spin-foam theory, in particular ideas concerning the notion of equality. So equality is extremely basic to mathematics and therefore also to physics. I think almost anyone who knows anything about mathematics and physics has the impression that these people write equations a whole lot. And an equation is a sort of peculiar thing where you have two different things on both sides of the equation.

1:45:00 And there's a little symbol in the middle that says that they're the same. And the funny thing is, of course, that they're different, but you're asserting that they're the same. And the one true equation, those of the form X equals X, are the ones that you never use, which is peculiar. So X equals X appears in, say, first-order logic with equality. basic axioms, but I've never seen any proof in mathematics that ever used the fact that x equals x for anything. You can't use it for anything. So there's something peculiar about equality, and it's interesting that it's getting problematized more in recent work on quantum gravity, and also in work starting in the 1940s on category theory, which has taken a new turn in the theory that I like to call higher dimensional algebra, as n-category theory. And in this n-category theory, what's going on is that the concept of equality is getting dethroned from its important role in mathematics, and it's being replaced by the notion of isomorphism, or more generally, the notion of equivalence. So, going way back to the beginning of this story, this rumination on the mystery of identity and its relation to physics and the concept of change goes back sort of into the misty history of Greek philosophy, the Presocratics. So as you probably all know, Parmenides was puzzled by the very fact that there could be change because he was asking, how could something both be and not be? And Heraclitus, in this, well, no one knows what Heraclitus really said, of course, But pericletes is rumored to have remarked that you can never step into the same river twice. And that's led to lots of speculations of various sorts. So the basic interpretation of that, of course, is that it's a different river the second time you step in it. All the water molecules in there are different ones. And so what do you mean that you're stepping into the same river? Actually, I really like how the writer Borges points out that there's an extra twist, you are different when you step into the river twice. So the question here is, how can something be different and also the same enough to say that it's happened twice? And so the modern answer,

1:47:30 of course, at a naive level, invokes the idea of similarity, that a similar you steps into a similar river at a later time. Mathematically, we try to sharpen up that notion of similarity formalize a version of it. It's not really powerful enough to capture the Heraclitus problem, actually, but we sharpen it and replace it by the notion of isomorphism, which means similar, exactly similar in some respect. And so we would like to say that an isomorphic U steps into an isomorphic river, but that's not exactly how it's in the real world, because we don't know what kind of isomorphisms we're talking about there exactly. Anyway, all these a lot more focused when we start doing mathematical physics and as we try to make physics more relativistic or relational. So what keeps on happening is that we remove background structure, various things that we tuck to be equations or identity. We realize we're actually just isomorphisms. So let me just go through some famous examples. So in Galilean physics, when we when we learned that you can do a Galilei boost to a physical system, and that's a symmetry of the laws of physics, that amounts to the realization that it doesn't make any sense to talk about being at the same place at a different time, so that you can't naively identify points in space at different times. Then in special relativity, when we go to the Lorentz boosts, we learn there's another twist to that, which is that you can't speak of the same time at a different place either. And in general relativity, it gets yet another layer of caution where we learn that it doesn't even make sense to speak of two particles moving at the same velocity at different points in space-time, which you can do in Galilean physics and special relativity. So in all of these cases, naive equations are getting replaced by something else, which I'll claim as an isomorphism, or a choice of isomorphism. Just to mention two other examples, so in quantum mechanics, we learned that it doesn't make sense to talk about the same electron at a different time.

1:50:00 And so actually when I said that the periclitus puzzle was partially caused by the fact that the water molecules in the river were all different at a later time, actually a lie according to modern physics. According to modern physics, it doesn't even really make sense to talk about the same water molecule at a different time. So it's in a way that the puzzle becomes even more intense. And this idea of being careful about when you say that different things are the same really gets heightened the most of all, I I think, in gauge theory, so in gauge theory, you don't want to identify things at different points in a naive way. So, for example, it doesn't make any sense to talk about the phase of the wave function being the same here as it is here, or, say, the color of a quark being the same here as it is here. Because to say such a thing, what you really need to do is you need to pick a choice of a path, a way of carrying the electron or the quark from one point to the other. Then you can compare them when they're at the same point, but the answer you get as to whether they're the same or not depends on which path you took. So you can see, I hope, what's going on here. In any case that I mentioned here, what we're doing is we're relativizing concepts by removing background structure that allowed us to blithely treat as equal things that aren't actually equal. So by a background structure, I assume you all are familiar with this concept by now, since everyone's talking about it here, I would say that it's a structure in our model of the physical world which is independent of the state of the world. And I just want to emphasize that one of the things that you use background structures for all the time is to choose what you might call a default isomorphism between things. That is, for example, the constant time slices in Newtonian physics allow us to say what we mean by two things at different points occurring at the same time. And when we remove that background structure, for example, going to special relativity, instead of having this fixed default isomorphism that we can always blithely use, we have to specify the isomorphism as part of making a valid statement. these two events occur at the same time in this particular frame of reference in special relativity.

1:52:30 So the frame of reference is providing us with a specific choice of an isomorphism, and someone else might make a different choice. So the big difference between equation and isomorphism is that there's only one way for things to be equal or not, but there are lots of different ways that they could be isomorphic, and so you need to specify the choice of the isomorphism. So, as we make physics ever more background independent, we're having to be ever more cautious about asserting equations between things, until apparently, in the limit, all you can say as an equation is x equals x, which is completely useless. So, to do anything useful, you need to specify isomorphisms between things, rather than assert their identity. So nobody really knows where this will all lead us. It seems to be moving in the direction of an ever more background independent physics, but nobody really knows how you could possibly formulate physics without any background, partially because we don't even really know what you should count as background or not. For example, in string theory there are ideas that you might want to treat the choice of not as background, but as part of this particular state of the universe that we are in. Or, for example, even the dimension of space-time might be just a part of the state we are in. But the problem in string theory is that nobody really knows how to formulate string theory in a background-free way. Instead, they're able to analyze one of these states or phases, or they may call them vacua, at a time, and they see relationships between them that they haven't really been formalized into a big meta-theory. Also, of course, it's not quite clear what you're supposed to do with this big meta-theory that describes all the possible space-times of different dimensions and different particles. Presumably, for us to get any good out of that, we need to know where we are in this big thing. So, anyway, I'd like to suggest that a good framework for studying these issues is sitting around in mathematics and it's N-category theory, which started out in 1945 with the work of Eilenberg and McLean on category theory.

1:55:00 So let me say, let me focus on that. So, let's see, what am I supposed to stop, by the way? I'll be too much happier if I know. Well, about 11.30. Okay, okay, I'll just, I can keep track on myself then. Yeah, so a category is a very simple thing, so I might as well give you the full definition. So a category consists of a collection of objects, which if you wanted to draw, if I had transparency, I'd draw a bunch of dots. They're completely featureless things. Also a collection of morphisms. For any pair of objects, there will be a set, say X and Y, there will be a set of morphisms going from X to Y, and you want to draw those as arrows if you want to visualize them. And then there are a couple of axioms, one of which says if you have an morphism from X to Y and a morphism from Y to Z, you can compose them to get a morphism from X to Z. Then one that says that composition satisfies the associated law. So if you do this and this and this, you've got a well-defined unique answer, regardless of how you parenthesize that triple product there, the triple composite. And then finally, that every object has an identity morphism going from it to itself, which acts as the identity for composition. So that's all there is to it. It's a very simple notion. But the interesting thing is, is that the passage from set-based mathematics to category-theory-based mathematics is very profound and changes our attitude towards all sorts of mathematical questions. as well as being a very powerful technical tool in many problems in mathematics and these days also in physics. So in set-based mathematics, the fundamental predicate is membership. And the main thing you do, one of the main things you do with membership is you check to see if things are equal. So to see if two sets are equal, you peek inside them, look at all the members of one and see if they're the same as the members. See if those are also members of the other. And conversely, But in category theory, you can't do that. You deliberately forbid yourself from peeking inside the objects and seeing if they're the same by looking at their constituent parts. Instead, all there is about an object is its morphisms to and from other objects and the ways of composing those. So you tend to think of objects as being defined or obtaining their individual characteristics not by their composition, their constituents, but by the relationship with other objects. And it's very interesting that the sort of first non-trivial theorem you learn

1:57:30 if you're studying a texting category theory, it's called the UNEDA lemma, and I won't attempt to state that lemma, but if you want to write that down, you can look it up, and you'll see that what that UNEDA lemma does is it makes this concept very precise, that you can reconstruct everything there is that you want to know about an object It's morphisms, say, from other objects or alternatively to other objects. So I think that to the extent that I understand structuralism, the philosophical concept of structuralism, that this is pushing mathematics in that sort of direction. Now, the funny thing is, in category theory, there still is a notion of equality of objects. I said there's a collection of objects. And normally, although not necessarily, we formulate category theory within the context of set theory, just because that's the easiest thing to do. It's been already pre-established as a backdrop in mathematics. And so you certainly can talk about equality of objects in the normal formulation of category theory. However, there's sort of an informal layer to category theory, which is that certain things are regarded as sins that you're not supposed to do. And if you do them, you don't really understand category theory. And one of them is talking about equality of objects. So it's sort of a principle in category theory that you should never have any theorems that say that two objects are equal. Instead, what matters is isomorphism of objects. So an isomorphism, what's that? Well, in category theory, it's just a morphism from x to y, which has an inverse going back from y to x. Inverse meaning that if you compose this way, you get an identity morphism. or alternatively those the other way, this way and this way you get an identity morphism and so the idea is that if you have two objects that are isomorphic any sort of structure or property about one can be transferred via this isomorphism to the other one so they're just as good in every respect and so you should never try to say that equality is better isomorphism. You try to state things just in a way that if they're true for one object, they would be true for any other isomorphic object. So for example, category theorists feel very comfortable about talking about certain things which set theory-based mathematicians

2:00:00 become very squeamish about. So a category theorist will talk in the category of sets, for example, about the one element set. So that's the expression you hear quite often. So a set theorist would immediately say, which one element set? There are lots of different one element sets. To see which one I have, I look inside and see what that element is. But in category theory, the idea is, yes, there are lots of one element sets, but it doesn't matter. They're all isomorphic, and they're all isomorphic, in fact, in a unique way. There's a unique function from one element set to any other. So, in fact, in category theory, this rumination on the notion of identity is carried to the point where we're really even using basic words like the in a new way. And, in fact, I wrote a paper with my collaborator, James Dolan, where we talked a little bit about the concept of the generalized the. So in set theory, you're allowed to say the if you know that you're referring to a unique thing. Any other thing with that property is equal to it. In category theory, you're allowed to use the if you're specifying something which is unique up to a specified isomorphism. So if you have any other thing with that property, there will be a specified isomorphism between it and the one you're pointing at. So this shows that some very fundamental issues are being called into question here. And what's really fascinating is that it's not just sitting around talking about this stuff. It winds up getting used in real-world mathematics and these days even in physics. So I want to not continue talking about the delightful philosophical aspects of category theory that just urge you all to learn about it because it's lots of fun, but instead emphasize how it gets used in modern physics. So one wonderful example is the theory of Feynman diagrams Now Feynman probably would have laughed if I said that Feynman diagrams had anything to do with category theory But luckily he's not around anymore, so I can say it I'm getting ridiculed So he was studying a concept of particles interacting In quantum field theory, and he started using diagrams to draw these Where you have little edges which represent particles And little vertices which represent interactions But the interesting thing is that those edges have no relationship, well, they have some relationship, but you cannot interpret them as being, say, a fixed world line of a particle in Minkowski's space-time.

2:02:30 They're much more metaphorical than they were before. And what really turns out to be the best way to think about them from a mathematician's point of view is that it's a branch of category theory in which you have a category where the objects are collections of particles. For example, the edges coming into your Feynman diagram, the collection of all those edges would be an object in your category. And then the Feynman diagrams themselves are the morphisms, and you compose them by sticking one on top of another. So you're doing one process and then another process and formalizing that in a pictorial language. So by now there's, in fact, a mathematical, well-worked-out discipline which allows you to understand what you're really doing in this sort of diagrammatic reasoning. that is to make this diagrammatic reasoning as rigorous as normal symbolic reasoning with equations. And this is a branch of category theory which is called the study of symmetric monoidal categories. Those extra buzzwords I don't want to explain except to say that they relate to the fact that in addition to being able to stick one Feynman diagram on top of each other, you can also do other things like stick two of them side by side and have two processes going on in parallel. And so that's one place where category theory is very present. You're studying the category of representations of a group, actually, by means of these Feynman diagrams. A group could be the Poincaré group or something. But also, somewhat independently, although I'm sort of motivated by this, Penrose invented the theory of spin networks, which are actually Feynman diagrams where the group is SU2. But he was thinking of them very differently. not so much as processes involving particles. He sort of was, actually. But nowadays, in loop quantum gravity, we think of those spin networks, as Lee was showing them, as things which compose space. So you think of a quantum state of space in loop quantum gravity as a spin network. It's like a Feynman diagram where the edges are labeled not by particles, but just by abstract spins.

2:05:00 And what's interestingly happening there we start using those pictures to represent states of a system instead of processes, we then, the first thing we do after that is we start talking about processes going between them, which are these spin phones. So loop quantum gravity suffered and still suffers to some extent difficulties with understanding the space-time aspects of quantum gravity as opposed to the purely spatial aspects. And so people have turned towards developing an extra level of it where you think of these two-dimensional surfaces interpolating between spin networks. And these surfaces have certain labels by spins. They look like a bunch of soap bubbles with the faces labeled by spins. So they're called spin foams. And those now represent processes, and you can compose those. That is like doing one process and then another, as some of these pictures showed, but what's curious about this, you see, is that we are adding an extra level to the story. We had thought of these Feynman diagrams or spin networks as being morphisms, processes, but now we're thinking of processes between processes, and this is where n-category theory comes in. So in n-category theory, you develop a formalism where you allow yourself to talk about objects, morphisms between objects, but don't stop two morphisms between morphisms, and so on. And a beautiful thing is that in spin-foam theory, we're really using that language of n-category theory. And it's been best worked out, I'd say, for quantum gravity in a toy model in three space-time dimensions. It's been very rigorously worked out that you can use the language of three categories to formulate the theory. And the passage of time is the three morphisms, the third dimension, basically. And mathematically, there's no reason to stop at 3 or 4 or any specific number. So, in fact, Mackay has taken this to the extreme by developing a language of infinity categories in a formal language that doesn't even have equality as part of the language. So that means it's not just a sin to use equations. It just prohibits you from ever making that sin by not having it be in the language. and when you do this then you can talk not just about objects being isomorphic but also morphisms

2:07:30 being isomorphic by means of a two-morphism and so on so you're allowing yourself to be very consistent in this demand that instead of just asserting that things are equal you specify a way to get from one to the other so i think there's a lot of interesting stuff for philosophers to think about it in this, because it's getting back to some very basic and very old questions, but it's being applied to some very new physics. So I wish that more philosophers would write about the foundations of category theory and uncategory theory, and I know of one new example. David Corfield, who's a philosopher of mathematics, is now at Oxford, has written a book on higher dimensional algebra and pictorial reasoning, which is very much part of higher dimensional algebra. And I guess that book isn't out yet, but it should be coming out soon. So if you want to read about this from a philosopher's perspective, I think that would be a good place to start. I also have a bunch of stuff about it. Okay, I'll stop here. Thank you. Okay, we're going to have questions and discussion really for any of the speakers, and I'd like to keep things fairly informal if possible. So, Nick, you were moving first. Thanks. We've been talking mostly about canonical approaches to gravity, and I wanted to ask what people thought a bit about some string theory approaches. So John Stachel put up a few quotations about on string theories, seeing string theorists in particular in relation to the existence of the background structure and space-time. It seems to me that it was something that is assumed and built into string theory approaches. So it seems to be, we got to talk to Don Merrill about just this kind of question in the spring. He had a story about string theory and M-theory that sort of addresses the nature of the assumption of a background structure. I think John Byers was hinting at the same kind of picture that he was suggesting. Namely, there's this allegedly M-theory sitting out there. And that's a theory that doesn't have any background space-time in it, and in fact there are sectors of the theory where there's maybe no space-time at all, there are some that looks like ours, there are various sectors that have different kinds of space-times, and the way to think about string theory is there's some kind of idealization, some kind of approximation in certain sectors of the theory to what this background-free theory looks like.

2:10:00 Does that seem like a plausible picture? Well, I think Lee Smalling is getting... So the first thing that everyone should know is that it's good that you said allegedly about M-theory because no one knows what that theory is. So that's the crucial point. So the string theorists are, I think, very wisely seeking to find a background-free formulation of string theory which might reduce to known string theories limits, and that's, I think, the right thing to try to do. But they haven't done it. So far, they've found lots and lots of relationships between the various hoped-for limits, which points at something misty in the middle there, which is supposed to be M-theory. But I just like to warn everybody, if you want to read a book and see the fundamental equation of M-theory or something like that, you won't find it, because it doesn't exist. So at least there is this story where there won't be a .. It's not background-independent, so far, is it? Well, this non-existent M-theory is background-independent. Well, then I'll say it's all property. If I can comment, I think that we do have to, I mean, string theory is a very, very interesting structure, a very, very interesting set of developments. I think we also have to be hard on them, and I think that nobody in the scientific and philosophical community has been hard enough about them, because you use the right words, and those are stories, those are not even conjectured at the level of well-defined, either mathematical conjectures or physical hypotheses. And if one wants to have a useful discussion about the future of string theory, I think one has to start by writing down the list of results and separating them from the list of stories and conjectures and speculations. and when one does that I think I at least even though I'm somebody who's worked on string theory

2:12:30 and I think Don and I are among the few people who've worked on both approaches for example the commonly stated claim that string theory is consistent to all orders of string perturbation theory is also a conjecture and in fact the proof at second order was only done last year even though you can find lots of an infinite number of claims by string theories that just go by that point and it's very far from clear when one sees the technical difficulty of the proof of second order whether the theory is unambiguously defined at that point which doesn't mean that string theory per se is not an important set of conjectures I think it's important to say what it is and where it may interface with experiment. And I think the most important thing to say in the context of what I said about Lorentz invariance is that there is no atriori reason to presume that naive Lorentz invariance is true in the quantum theory of space climate when correctional order L-Punk. Lorentz invariance is not a symmetry of general relativity. it's not a symmetry of the universe, etc., string theory is a development, is an approach to chronography that takes it asymptomatic that the theory is well-fined, consistent with exact Lorex invariance under the standards of Lorentz transformation in textbooks, and that's its strength that succeeds in doing that, at least a second-order perturbation which is by itself an achievement, even if it fails after that. But it also is, it's one vulnerability to experimental tests. One of the issues with stream series is that there's very little vulnerability of falsifiability, and the one place where it is falsifiable is on that. We will have to add it. I think John's stage one. First, I'm trying to make a terminological point. He was just using contrary anal integrants. In general, it is not part-haring, but it is a rest-haring to manage its space for each one. And that's very important. It's terminology, because people use a rest-haring to mean each of those things. If we use part-haring in there for the world of translations, then we don't get it yet. And on the string theory, precisely the reason I quote it from the string theory is because if they say it,

2:15:00 the string theorists say it's much better to show it to another string theory than you and I say it when they won't believe us because they think we're prejudiced here. Also, very often, the background matrix introduced surreptitiously, they felt like sums in differences of squares of things, and they introduced the background matrix there without saying it. But anyone with the relativity realized that that's what they did. But the picture I sketched is compatible, I think, with all the quotations that you put up. I don't think it is. The one margin of n3, sort of, reasonable, like matrix theory, is not background independent, and it's in a way, it's in a certain gauge, you gauge it as well. The only way it might be background independent is that it's got kind of non-commuting coordinates in the deep brain. That's the only way I can say that it might be background independent. Yeah, well, Banks was the one who depended that major theory in that review article. He said it's very, very good. I think Don Howard was next to it. I wanted to make a very general historical and philosophical comment that I made the statue of Steve French break, that was the audience who didn't react to it. And it's on this issue of trying to get peculiarity out of structure alone. There's an old history to this ambition. This ambition is very much alive at the end of the 19th century and beginning of the 20th century, so much so that it even had a name back in that time. It was called the German Das Gesetz dahindeut, the principle of univocalness. and it was a lot of methodology literature that was in effect the idea that a necessary condition on an adequate theory is a kind of categoristic-like demand that it determined a unique model for itself. And this idea that in some such faction you can get the peculiarity out of structure was important in various philosophical quarters, very important in the Marbury real content. My sense is that this dream dies in the late 20s, early 30s who are accumulating some really solid non-categoricity results. And Vile, for instance, is someone who writes very interestingly about the frustration that we encounter when we run up against this limit of the failure of what do you think of monomorphism?