Category Theory & Physics (contd.)

0:00 For instance, this is the case for the Einstein tensor or for the energy-momentum tensor. And you see that the pure exigence of different morphisms constrains the possible form of the action. just to end this digression what can be reality and in general activity so following to what I just mentioned any real object if you prefer any physical object must be covariant and the best way to have a covariant object is to describe it as a tensor including and so on. But this reality, in general, we have no access to it. We cannot know, for instance, the totality of the energy momentum tensor of a system. But we can observe it and have some partial information. How to do that? We have to know what is an observer, like us. An observer is also real, so it's also a covariant entity, and in fact, it is defined as a time-like line in space-time, and this line has a tangent vector at each point, and if we normalize it, it is unique, and we call it in this case velocity. So the velocity completely represents the observer, the velocity at each position along this wall line. So what is the measurement? What can be the result of a measurement of a system by an observer? The system is described by some covariant quantity, the observer also. So, the result must be covariant, because the result of an observation is something real, it should not depend on the choice of a system of coordinates, for instance. This must be a scara, because when you make an experiment, in general, the result is a number. And so the only way to define the measurement is to define a tensor contradiction of the tensor associated to the object by the tensor associated to the observer. For instance, if you want to measure the energy of the system,

2:30 you will write the tensor contraction of the energy momentum tensor by the velocity of the observer. This is called usually with indices like that, but you don't need indices completely covariant expression which is defined independently of any choice of coordinates so you see that you don't have access to all information in that you don't have access to the world reality but you have only a partial access which is defined by this contraction which involves your position and your properties in space-time Well, if you have a differential manifold, you can have many objects, especially the metric. And what is interesting is if you have a metric, this allows you to define very new objects which live in the natural models associated to the differential manifolds. And, as I told you, general relativity in general relativity space-time is defined by a metric, but since the metric will define many other objects, you can also consider that general relativity is defined by these other objects. and each choice in this list and you can also enlarge the list will give you a different point of view of general relativity and if you want to generalize general relativity if you want to search for a new physics the possibilities will depend on the choice of the point of view you will adopt So, for instance, if you have a metric, this allows you to select some specific connection among all the connections, and you will define metric connections and also the Levit-Civita connections that come in one minute. This allows also to define, to precise the duality between vector and form, for instance, because the space of vector fields and the space of one form are dual to each other, but there is no canonical duality.

5:00 And if you have a metric, this defines a canonical duality, and this is very interesting. This is used all the time. This is what is called rise over the indices. In fact, this is a duality in use by the metric, which extends to all tensors, multilectors, and forms. In the differential manifolds, you have exterior algebra. But when you have a metric, this exterior algebra acquires a new structure, it becomes a Clifford algebra. And especially in the case of space-time, this Clifford algebra is the space-time algebra. So you have a Clifford bundle, and each fiber of the Clifford bundle is the Clifford algebra. And what is interesting is that this Clifford algebra indicates you that many important structures are present. Because in the Clifford algebra, you find representation of the spin groups. So this is, of course, there is nothing quantum here, but you directly find the spin representations. And you will find also the conformal algebra, which is the lead algebra of the conformal groups. So very naturally, when you have a metric, you have the spin structure, you have the conformal structure which is upper. When you have a metric also, this allows to reduce the group to one of its subgroups, which is the Lorentz group. This is due to the fact that when you have no metric, if you change from one frame to another, you can use any element of the general linear group. But if you insist to consider only the autonormal frames, which is possible when you have the metric, so this group is reduced to the Lorentz group. And this is very important because these group reductions can be generalized to the fiber-bundle reductions. And for instance, this allows you to reduce the frame bundle to the bundle of autonormal frames, which are called tetrads. And autonormal frames are related by the lens group.

7:30 And this is very important for the theory of connections. And you have also many other things. So very quickly, in this diagram, you see that everything which is in red is the improvements which are brought by the metric on the natural bundles. For instance, if you consider the frames, which are well defined on a differential manifold, if you have a metric, you can define the OED frames, the autonomal frames, and also, after that, the spin frames. If you have the connections which are well defined on a differential manifold, the metric allows you to select metric connection, levici-vita connection, and so on. And you have many new structures which are brought by the metric very quickly on connections. I will not go into the details, but the connection allows you to construct the covariant derivative of any tensor and in particular of the metric. So when the covariant derivative of the metric is zero, the connection is said to be metric. only one connection which is metric and has not option, and this is precisely the Levitivita connection, which is so important in general volatility. It's so important that you can see generativity as based on the Levitivita connection associated to the metric, rather than the metric itself. And in fact you will say that the connection is the gravitational field. And this is at the basis of the quantization that you have seen this morning. The loop quantum gravity is based on the quantization of connections, but this also opens the door to other theories, which generalize general relativity by considering other types of connections, which can be non-metric. and this is, for instance, the case of the vague connection, which is not linked to a metric, but rather to a class of metric, which is called a conformal structure. Or we can also consider the Dirac or Carton theories, which are based on the connection with torsions. And torsion is interesting because it is linked with spin, so this gives you some tracks to consider new theories.

10:00 Group production is very important because the linear group may be reduced to its subgroup which is the special autonomic group, the Lorentz group but in turn this group has a universal covering which is called the spin group and so you have a natural way from the linear group to the Lorentz group to the spin group And this way can also be applied to fiber bundles. And then you first can have a Lorentz reduction. So the frame bundle, which has this group, is reduced to the autonomic frame, also called tetrad bundle, and the linear connection, which is the Levit-Civita connection, is reduced to a Lorentz connection. and this is very well used and also you can improve the if some conditions are verified on the many forms you can improve this universal covering as a bundle universal covering and so you will go from the orthogonal group to the spin group and from the tetrad bundle to the spinner and this is the way which allows the quantization of general activity, so this is very important. Very quickly, just I will mention other view. One view very important is that of the conformal group, and this group is very important and I claim natural, because this is a natural enhancement of the Poincare group, which is the isometric group of Minkowski's pastime contains the Lorentz group also linked to other very important groups in general relativity the conformal group leaves invariant the conformal part of the metric if you want to leave invariant the metric this is the Lorentz group but if you want to leave only the conformal part which is the causal structure this is the conformal group so this is very important And there are some nice results that if you consider the Clifford algebra and you anti-symmetrize it, you obtain the B-algebra of the conformal group. And also if you start from the algebra of observables, which by deformation gives the algebra of quantum mechanics,

12:30 if you add the metric and then you deform, then you obtain again the conformal algebra. All these results indicate that the conformal group is probably very important, because you can go in all directions from space-time and you find the conformal group. And I will finish, but just to mention, this is the same with symplectic structures, because you can also go in almost any direction, starting from space-time, and you will find symplectic structures. And why, for instance, the universal governing of the Reutz group, which is so important, is this group, which is the symplectic group in two dimensions over a complex number. And this means what? This means that you have a very fundamental symplectic structure, which is, in some sense, the square root of the metric. so you start by considering the metric and then you find that there is something more fundamental which is the symplectic structure this is the square root of the metric exactly in the same way that spinors are the square roots of vectors and the Dirac equation is the square root of the time-level equations so this allows also to generalize general relativity because usually when one seems to enlarge the number of dimensions One thing considering larger autogonal groups, but this suggests, and this is the work which is presently done, to consider not larger autogonal groups, but larger symplectic groups, insisting on the importance of symplectic structure. last what is also very important are the complex structures and complex group are group acting in complex spaces are occurring very naturally from the spin structure from the lower end structure and from the conformal structure and if you want to define spinors twister and circular objects which is very natural and which is very useful for interpreting general activity and to quantize it then this requires complexification this suggests that maybe mikovsky spacetime is just the real part of something more large and complex which is

15:00 is the complexified Mikoski space-time, and there is an isomorphism between this complexified Mikoski space-time and the two by two complex matrices, and real Mikoski space-time would just represent the Hermitian part of that. And this is, you cannot avoid that if you want to consider spin-offs, twisters, and so on. So what I suggest also is that if you If you want to generalize physics, you should not forget complex structures. So just to conclude, the basis of our present physics, I mean space-time, the basis of our conception of space-time is differential manifolds. Differential manifolds admit many canonical natural models, vector models and so on. Any structure you want to add to that leaves in this bundle. So you can choose the point of view of leaving if any of this bundle. So you will select a peculiar tool like algebra, connection and so to make physics. and all this structure allows you to modify, to generalize, to enrich or to quantize our view of space-time for instance I have not mentioned non-commutative geometry but this is based on the generalization of the algebraal function on a minimum first you have a very very big set of collection of possibilities And, of course, the main question, which remains in my sense, and it will be, of course, discussed here, is how to trade covariance when you do that. And we have met this morning the question in the case of quantization of connections. But any kind of generalization, any kind of quantization you want to make, you will have to deal with this question of how to trade covariance. covariance in your new theory, which is one less background in independence. Thank you. Can you get back to the BNCM service slide? Yes, I will try.

17:30 There you see that any real object is covariance, meaning described, as a tensor. I think I don't understand this statement probably for two reasons. The first one is, if you measure the number, it's actually a tensor. So, if you mean by real, something which is, I think, memorable in an operational sense, then I don't see the point. And the second thing is, is a tensor field observable? I think not. I think any local quantity in GR is not observable. So, what do you mean by any real physical objects? Right. Okay. When I mean real here, is some kind of reference to that of independent reality. So real doesn't mean measurable. There are plenty of real objects that I cannot measure. For instance, if I have a very simple example, if I have systems behind the horizon or in my future and so on, there are real in this meaning, which I'm discussing, but I cannot measure them. You're completely right. The result of a measurement should be a scalar. And in fact, I am in some position in space-time, at some point of space-time. So what I do is to take the contraction of this tensor, this tensor field, by my tensor field, and take the value at the point where I am. In general relativity, you can only measure things where you are at the moment of your measurement. So this is exactly the scalar. But this is constructed only from covariant real quantities. So the result of a measurement should only involve real quantities. If not, this has no meaning. And it should be as you said, stellar. And it should be local. There are no non-local measurements in generativity. For instance, if you observe very distant galaxies, what you do in fact? The galaxy emits light, a photon, and this photon has a full velocity, which in this case is normalized to zero, but you don't mind, and the photon reaches you, so at your position this photon has a full velocity, and what you will measure, for instance, the wavelength of the light coming from the galaxy is given by contraction of your velocity by the velocity of the photon.

20:00 at your position and there is something different which is the contraction of the velocity of the photon with the velocity of the galaxy when it emitted the side signal and if you make the ratio of 2 you get exactly what is called the ratio but you can only measure local things and contractions of tensors my question if it's just a short comment Yeah, maybe a philosophical interest. Just to say that, I'm maybe puzzled by the word covariance, which sounds to me like formal counseling, something which is a phenomenal word. And reality is more like mythology. So I'm puzzled by the association of reality to covariance. Well, this is my claim, yes, but I don't know what is reality, and there are many meanings of reality, but my claim is that if you want to define any kind of physical or independent reality in relativity, it has to be described by covariant object. If not, this would be not a covariant object. Okay, we have a question. yes can i sort of press press you from the other direction i mean suppose the tensor in question is the electromagnetic field sure so according to the way you're setting it up that counts as a real object because it's described by a yes tensor yes in this case you have additional get freedom but this is what you have in mind? No, it wasn't actually. I was thinking we think that at a point we can in principle measure, it's going to be a frame dependent thing, but we can measure we can get our hands on the electromagnetic, the electric and magnetic field. only at 20 points we get out at a single point more than one number but the way you set it up with the TUU makes it look like for any tensor no matter how complex no matter how many independent components, a measurement is always

22:30 going to spit out one number well if you have a tensor with say 10 indices then you have to make a concentration with 10 times my velocity maybe you can have different ways of contracting and you can also use the metric which allows you to lower the indices and you can make symmetrization and dissent symmetrization but if you have a tensor with many indices only possible results are something like that and then you will But you can also here have, for instance, the metric or something, and you can have some anti-symmetrization or six-symmetrization. But the result should be a scalar, so you should contract this. You should have the right number of contractions in order to get the scalar quantity. And it has to be local, of course, because it's impossible that an observer measures something which is not at his position. for instance why the concept of velocity between two objects which are not at the same place has absolutely no meaning so it should be local i should have at the here local this this is because it's local that gives you only a number maybe follow up on this question maybe the confusion is if you measure electric electromagnetic field at a point not only measure the strength which is a number but you also measure the polarization was that the question and somehow if you say that the outcome is always one number then you know how does that apply to say electric field also have the polarizations polarization is defined by an angle it's a number also it's some peculiar contraction i i don't remember exactly how you do that but But you just have your u mu at your disposal. You have u mu and you have the metric. And as soon as you have u mu and the metric, you can construct many things because you can... In fact, u mu, u is a vector. And by duality of j, the metric duality, you have one form from that. J is a two-form, and you can have this, you know, you subtract from the metric the tensor product of the velocity one-form by itself, and this gives you a tensor, because this is a tensor, this is a tensor, which is exactly the spatial tensor at your position.

25:00 So the spatial part of space-time for an observer is perfectly well defined, at least positionally of course, not in the whole space. So you have also this sensor, and this is what is called, usually we put here, spatial indexes. And you can also, you are allowed to make contractions with that, because this is tensorial and perfectly defined. so maybe we can discuss in more details but I'm sure there is no exception well if you take the electromagnetic tensor that's anti-symmetric so if you simply contract it like this you would just get zero if you knew well if you knew it's anti-symmetric if you contract it with UV you knew you'd just get zero so you need at least two different this is zero So you need two independent vectors to make a measurement of the components of that. Well, an isolated observer cannot measure electromagnetic field, you think? Yes, or maybe you need more, but certainly you need that. well I think there must be a way to measure well it is a way to measure polarization of electromagnetic not just the modulus or the size but also the direction I will give you the answer in a moment was there another maybe issue I had a question on a query to answer some of the points raised this in Oliver's talk this morning about taking the equivalence class of the metrics and fields to represent the real space and how things really are. And I'm wondering if I could sort of present an analogy, sort of trying to pick one might hesitate just a little bit so it could say that that's the mathematical object which represents how things are. if we consider just

27:30 the electromagnetic vector potential and gauge transformations of that, and we say, okay, look, different A fields represented, connected by a gauge transformation, these represent the same state of affairs. Now we usually don't say that it's the equivalence class of manifolds that represent the real state of affairs. We say something like it's the loop phases, the holonomies, so the phases around the loops. And so we go to a gauge invariant, so the proper gauge invariant representation as the loop representation is. And now that seems to be what we don't have in the case of general relativity, because we have the equivalence class of the manifold with their different metric fields on them. But we don't have a completely other way of stating what the content of the theory is that we do have in the electromagnetic case. So that might give us grounds for hesitation to say that it's the equivalence class that represents the mathematical object that represents how things are. Really what we'd like is something analogous to be that gauge-invariant theory which tells us how things are. Because otherwise, the temptation might be to say, well, okay, so if the physically real is everything that's invariant across the equivalence class, or one thing that's invariant across the equivalence class is that there's a manifold in each and there's a space-time point in each. So it looks like you might have to say, in fact, space-time points in the manifold are real things, but particularly you need to deny that. But if you take the connection, in some sense, the dual of the connection are one forms, with some value in the algebra of the Rendsblum, for instance. and you have holonomies also associated to that because you can integrate this one forms over any curve and then you will have an integration of exponential of something which is in the Lie algebra so this is a group element and so you have the holonomy group which is exactly an holonomy is exactly if you define a connection This will associate to any loop a group, which is a group of holonomy, and this will play the same role as your holonomy group with other connections, because in fact the interest to consider connections is that in filtering gauge fields are connections.

30:00 There are one forms with values of some algebra, but this algebra is the Lie algebra of the group with internal action. Here the connection is also one form with value in the Lie algebra, but here the Lie algebra is the Lie algebra of, for instance, the Lorenz group, whose action is not in an internal space but in space itself. This makes the specificity of general relativity. so you have exactly the same tools at your disposal if you want but what is important in general relativity I think is that you already have a covariant formulation a guess invariant formulations if instead of writing the Einstein equation j'ai menu is this is not a covariant form because but if you write tensor, Einstein tensor is proportional to energy momentum tensor this is already covariant you don't need more of course this may be not easy to make calculations but in some cases it's also easy to make calculations without coordinates but you have these gauge independent formulations in gauge theory usually you have no way to for instance if you try to write Maxwell equations, well, you can write precisely this way or so, but this is not gauge-independent. The gauge is somewhere and you have not this gauge-independent formulation that you have here. You used essentially differential geometry. So what about discrete structures in non-commutative geometry there is a possibility to generalize all these structures to discrete spaces but what is your own position concerning the problem if you start from the algebra of functions on a mini-fold

32:30 if you want to construct a non-commutative version of that, you have to transform this algebra into an algebra of operators with some commutation rules. And there are some canonical ways to do that, which are false. So maybe non-canonical ways, I don't know, but you have many ways to do that. You have some kind of, you have deformations, you have current state quantization, this is the method I mentioned you, you have also more standard methods so I think this is not difficult to obtain the fuzzy structure of the manifold then you have to define your differential structure on it and I think the difficulties lie in this part you have some trivial differential structures that you can define on it but this will not help you to make physics so after that I don't know. I think I will ask you the answer because I'm not a specialist in non-commutative geometry, but I know that, for instance, if you take the work of MADOR or people like that, they claim to have found nice differential structures on non-commutative space-time. then they have some kind of Dirac operator and they are perfectly happy with the metric they construct and that and they find many of these nice properties I don't know exactly if they find all of them or not this is presently this is not completely finished this one but so there are some claims that you can find everything but the truth is to find the good differential structure and I don't know the generality of the possibility of finding it so any more questions yes oh there's one last question one last question but perhaps you could say something about the fact that the frame bundle has a particular kind of relationship to the base manifold that other bundles associated with principal bundles do not this is called a solver form, yes and this is exactly what in some sense, if you consider the frame bundle

35:00 This is the principal fiber bundle on space-time. When you are engaged in arrays, you have also principal fiber bundles of space-time. As I mentioned to you, the difference is that the fiber is more or less a group in each case. In the case of Youngmin or Gage Therese, the group acts on some internal space which is independent of the basis. In the case of the Frangonal, the group is not independent of the basis. This is the Lorentz group, and this dependence is expressed by a tool which is called the solder form. Right, and I thought that was the answer that you should have given to Chris. Maybe. Maybe. So this is important because what I claim, and there are also other arguments linked to spin, for instance, that I have no time to present, is that the real physics of space-time is not in the space-time manifold, but probably in the frame-gonder. And in the frame-runner, you have the solder form, but you have also many things. In some sense, particles with spin, they don't live in the manifold, but they live in the frame-runner. And to see that, you see that a particle without spin follows a geodesic of the manifold. A particle with spin doesn't follow a geodesic of the manifold. you have a parapetru equation which is some kind of zero limit of the deviation geodesic deviation equation but in some sense the particle with spin follows not the geodesic but follows a rational principle in the frame model so from the point of view of spin the frame model or, of course, the spin below is much better, is the spins where the things really happen. This really has nothing to do with the content of your talk, but you raised the question at the very beginning of the problems. Do we, after all, have to quantize the theory of gravity or general relativity? I have no idea to what extent this still applies, but there is this very old argument of Bohr and Rosenfield

37:30 that you can't have a classical and a quantized theory in any interaction. My memory is right. The way the argument goes is if you could, one's quantized, the other isn't, then you use the connection to generate violations of the uncertainty principle. This was written this morning. This is classical and this is quantum, this is your point, right? Well, a particular argument is to why that gets you in trouble. It's interesting to think about why it gets you in trouble. Like, why can't it come to the expectation value? Very often people write that. You try that, then you get yourself. You're opening a can of worms, too. Well, certainly you shouldn't want to, but you still could have to point out Okay, it seems like I have all questions, probably a little bit over time, so let's thank Mark once well.