Time, Relativity Theory, Quantum Physics / Dimensionality of Time

0:00 The Centre of Mathematicalists, who are investigating classifications of Fondheim and Alibras with fractures, and eventually Alain Connes succeeded in applying these techniques for a complete classification of Fondheim and Alibras, for which he got the field mill. It is a remarkable series of papers. So, the origin is certainly Gapha, Gapha and Gapha. Subsequently, sometime in the early 90s or maybe late 80s, I will go along with a physicist called Locke in this country, forward it is possible to formulate known physics per the standard model in the language of operator algebras. And that paper attracted the attention of many physicists and subsequently it has evolved into a public which is a sub-characterist and I will describe what it is. So, to tell you that this stuff is not really so revolutionary Let me go back to how we normally do, say, GR, general relativity, or how to deal with manitons. So that's the standard way that people teach in GR in the class, in the physics classes, is that they will start with some differential geometry. where we leave it, it manifests one time. So, we give a manifold, say points on the manifold, can you get me?

2:30 Zero points. Points on the manifold and various structures on the manifold, for example vector fields, differential forms, and perennial adjustments. So the focus of the addition here is a manifold, and a manifold itself, for example, a two-sphere or a torus, or a rapid tube, or whatever it is and from here we try to define different geometry structures and try to formulate physics. So, the standard way, for example, in elementary physics at least, we consider dynamics is as of a point moving in space of a particle. However, when we go to, already in the laboratory mechanics, something changes. Something changes. What we do here is not really mindfulness. What, well, we start with a mindfulness cube, which normally is called as a punctuated space. Then, when we go into the hematonic mechanics, we go to the phase space, which the standard rotation for this, for the pressing case, is a potential environment. So, we go into the phase space. But we don't really, in hematonic mechanics, we don't really deal with this, but what we do is take a space and we consider the smooth functions of, say, in this space. So, these are functions of this space. So what we have done is replace the manifold, which in this case is the J-space, by what I can call as a dual of this manifold, namely the space of functions of this manifold. So we have already changed the focus, from then we go from here into these functions and the direct causal brackets are not between points on this space, but the causal brackets are between functions, between functions. So, we have already departed from our original notion of a manifold by witnessing it by smooth functions on this manifold, but there is also There is also an additional structure that we deal with here, namely this space here is actually an algebra,

5:00 let me call it some algebra A. What do we mean by an algebra? There is a natural way of multiplying points here, namely we can take a pair of these points, say, pair of these points, these are two functions, pair of AUS in this space here, in the pair of functions, and we simply multiply with YUS. So, we have G at the point P, is F of P here. So, we have replaced here, this algebraic structure is, when we do ordinary classical mechanics, we do this routinely, namely we make point-based multiplication of solutions. So, what has happened is that the only phase space, which is your underlying classical metaphor, has actually got replaced by the kind we come to, the Hamiltonianianianian when the algebra of functions in the initial structure even better for something like this. So you can already ask at this level that this is what I am doing and since I am dealing with functions, they are not dealing with a manifold. And you can think of this structure here. Wait, this structure, I think. I'm going to support that occasionally. something else. Since I have been dealing with this, that structure abstractment in the algebra of functions with that multiplication rule, is it possible, that is what I am going to plus 3 mechanics, can I tell that there is an underlying phase space, okay? This is by no means a trivial question, because what I have given are the functions, not the underlying manifold. So, how do I get that underlying manifold from that phase space, from that algebra? This class essentially is the theory of alpha and alpha. They were not interested in phase phases, but they have proved that if you give me an answer to a vector, where the multiplication log is as I have given you, point based multiplication. So, this is a complicated algebra. That is the point based multiplication means that Mg is Gf. So, point based multiplication. So, because it is computing, and there is a topology in the input, so that it becomes what they call a C star topology, then Gelfan and I have to tell you how to recover the underlying

7:30 manifold, well as a house drop, technically as a house drop topology as well. And the reconstruction theorem along with the topology is unique, unique up to homeomorphisms of the underlying, homeomorphisms of the manifold. Namely you can take that topology and there So, here we start from the free space and when we do classical mechanics and phosphoperectics, unknowingly we are going to a dual, namely the space of functions. And Kelkon and my mark are telling you how to recover back the manifold, at least in a restricted sense, from this unplane answer. Okay. Now, this dual thing, namely replacing the manifold by the dual, namely the space of functions, be called the stuff of non-combined geometry. When core talks, then of course you put additional structures, people will say geometry means there is a metric, there is a very important metric in this space. Alec Cohn in particular has studied it, but then may not become very typical. So, basically when we go through this dualizing, we are going from and deal with algebra whose underlying algebra is in terms of functions, we are going into a description or non-committed with the argument. Now, when we go into quantum mechanics, we do equal twice things. What do we do? We replace each of these functions. We are now dealing with functions are in face space, and when we do part of the hands, we are represented by operators. So, the original functions that we are dealing with have got replaced by operators, and in particular, we find that the operators do not commute. This is the process, for example, the position operator and the momentum operator is equal to momentum operator, position operator plus IH1. So, when we go to quantum mechanics, we have taken this commutative algebra of functions on the phase phase, give it to an abstract leaflet and replace it by a non-commutative non-commutative algebra. This has very dramatic consequences because once you do this deformation,

10:00 so this is a deformation for the origin algebra by a small parameter called x square and when you do this deformation, from this algebra there is no longer a reconstruction theorem that you can recover the phase space. It is no longer true that there is a way to reconstruct So, the situation has dramatically changed and what we can recover in some sense is that what we get is a positive phase space. What I mean by this, you cannot localize points by this is the standard uncertainty principle you can no longer localize points on the phase space, because of this uncertainty difference. That is what I mean by deficit phase space. So, the situation has now taken two steps. So, you can see that quantum mechanics is a non-committed geometry. I can say it as a slogan, it is non-committed geometry. on this position space. I could say that. It is actually the essential points that I am trying to make. Namely, we are now dealing with nine tools and junk leather, but we have Let me say immediately something that will be, when you are dealing with manatures, the kind of structures you can write have certain limitations. For example, let me say what can be proved is that now it is possible that when I even So, even though I have reformed this original geometry to a non-conflict geometry, so there is no underlying manifold, it is never that possible to describe dynamics, it is possible to describe Laplacian, it is possible to describe a part of TG on this case, okay. Whereas, if you try to describe this in terms of your original manifold, it is quite impossible, because there is no original manifold which is underlying. So, in other words, we have actually made a conceptual heap when we go from this, once

12:30 of the manifold, into the duals. It's a very profound heap. Now, what has all this to do with our problem? With what I will be talking about, say, later on. Hopefully in colour, because God has promised to use colour in the slight presentations. Very good. But anyway, let me go. Now, why did I bring up this exam? Well, if I go to quantum gravity, there are at least semi-classical reasons to expect that there will be limitations on face dimensions. What do I mean better? Well, the typical scale of gravity, say length of the A's length scale is something like 3 to the power of minus 3 to 2 centimeters. This is roundly small, something like 10 to the power of 19 orders of magnitude smaller than the size of an electron. The electron having typically the size of electron or weakly interacting particle having typically the size of 10 to the power of minus 15 centimetres. So, this is 19 orders of magnitude smaller than a nuclear space. and so because the nutrient size is 3 to the power of minus 30 centimetres, so it is extremely small size and to probe such as short distance, we need a content wavelength. If you want to probe this, we need a content wavelength or the pipe wavelength which will be smaller it must have this inequality problem. So, that the wavelength in the particle is roughly on this order or at least smaller. So, this would say that m is larger than or equal to plank mass, which is for the order of it is about a 19 degree. It is a huge mass and And what we are doing is, when we try to do this probe, we are putting 10 to the power

15:00 of 90 g in 10 to the power of minus 32 whole cube centimeter cube. We are trying to put this much energy into this much volume. So, it is a trivial termination that classical yellow relativity will say that this will lead to black hole formation and the horizons. So, if you put this much of energy in this much of volume, just to see what is happening, if you apply classical relativity, you will find the value of black holes and the horizons. So, the horizons will prevent you to screen this area, this volume with this point that we are trying to examine and they will go over the ocean. So, we would expect that because of these considerations, there should be quantum gravity cannot allow us to probe distances or times much shorter than of this kind of scale, okay. Similar arguments can be made for time as well, namely if we try to localize times also for very short time distances, we will have to use an enormous amount of energy because of the relative time relationship which was mentioned here. So, because of that again so much energy, we are going to expect to create black holes and they will do all kinds So, to model this. So, we are therefore expecting an uncertain relation of this kind and how do I model this? Well, we remember our quantum mechanics. We know that in quantum mechanics, the limitation on the measurement of position in space was modeled by saying that the computation relation of position and momentum was i h bar. So, we modeled the limitation on the of position and momentum by this equation. So, the suggestion there is that we model, this is in J space, but now we are finding limitations on measurements in space-time. So, by analogy we can imagine that position, now of course we are talking the domain of So this is the position operator with new component, position operator with a new component.

17:30 And normally we would say they compute, but to take care of this limitation on measurements, we can say, we can write like an equation like this, whereas this t-curve value is adequate of this blanks constant. So we have left there but the algebra of position operators including It's kind of a computation relation, but theta is an antiseptic weighting with dimension and length square, but that typical length is coming, it's plank length. So, we are getting an algebra like this. This is how we end up with, so this is the algebra, which is of some, so this leads to, this algebra, the corresponding algebra is called, So, what is happening now is that space time has become fuzzy. And what we have to understand is that even though space time has become fuzzy, can I do quantum mechanical? Can I do quantum field theory on it? And what are the experimental predictions that come from this kind of a model? And can I check these experimental predictions against experiments? I mean, can I, are we doing, if I use metamorphosis in my way, not in the way of which we hear, by metamorphosis I mean those explanations about the physical world which are out of our possibility of checking in our present historical time. So, maybe hundred years later somebody can do it. So, in that sense, it's purely a metamorphosis speculation, totally out of the experimental or is there some discipline in what I am doing. So, in this case, in fact, people have understood more of this how to actually formulate part of field theories on these places, carry out the normalization theory and actually extract predictions of various kinds including those which can be at least they are definite predictions and they can be tested for example in LSE and you can put limits on data. And there are other dimensions which I give a shot.

20:00 So now, I want to say that there is a concrete model. That is to say, there is actually, this, maybe they are lucky, that there are a class of rather concrete models where this kind of algebra is actually realized. People are doing experiments on it right now and this corresponds to this qualitative, quantum qualitative. Quantity, I remind you, so this is something that a huge number of experiments are going on. It's a very incredible system, but the system itself is very simple. I believe Landau was the first one who thought about it. So, what we are not here is a plane geometry where the electrons are confirmed in this plane and there is a very strong magnetic field in the third direction. So, the magnetic field is here and the electrons are moving in this plane. And already in the north, because of this extra magnetic field, what happens is the dynamics of the electrons on this plane are strongly changed. So the electrons are moving in this plane and classically they are orbiting around some center. On the plane in each, there are a lot of orders which are going around the center, some center. So the center is called the guided center coordinate. It is described by what people call as guided center coordinate. called it X A and there is also another quantity, these levels, energy levels called X A. So, the guidance center is what corresponds classically to this center of the orbit. So, the quantity is this operation that is seen. And then it's expectation that it's telling you where the

22:30 orbit is located. And what happens is that because of the magnetic field and it is velocity dependent something to coordinates, it turns out that this guidance is a coordinates no but it is something like a hyaluronic. So, it's a multiple, it's minus 6.1, and the other is this one. So, the guiding sector coordinates is actually behaving like a, is the algebra of the guiding sector coordinates, which is describing to you where the electron is organized, in the plane, is actually describing a buoyant plane. Whereas the internal body is giving you, it turns out that it is also true because the internal body also has a similar algebra, but they compute, namely X and Y compute. So, in this model, these kind of structures we are speculating above are actually experimentally realized. So, and we have these electrons, these are strongly carving electrons moving in this plane. This quarter is already at the level of the Landau model for the free particles, but by that we finish, all kinds of funny things happen in the quantum heart system as you might have heard. And therefore, you can now imagine that these are the basic dynamical variables. Can I construct field theories on this thing, on field theories corresponding to multiple excitation of this, of the electrons in this way, of the levels of the electrons possible in this plane and all these experimental predictions can be varied or they can be proved if you can. So, there is a this kind of work is just not starting. It is no doubt true that because of our experience in the applications are not committed to geometry in terms of that there are new features which come in the description of the quantum hole electron of the standard ones and it will be very interesting to know or check whether we can compare our speculations or the models we have built of quantum field theories on the Moya plane

25:00 that they recheck in the context of the quantum quantum electron. So that kind of work is just not starting. I thought of giving you a concrete example of how even though you you can describe your space tank truly and recursive the space tank, but since it's exactly 25 minutes, I will stop suddenly. Thank you. So I think we have time for a couple of questions. Before I ask you to pose your question, I might point out that Kbala Chandru and his collaborators have just published a book dealing with these topics on fuzzy physics, world scientific press. He wasn't going to tell you himself. It's called Fuzzy and Fuzzy is Sousy physics. Sousy is supersymmetric, as Amy Brute knows. Good, we have any questions, sir? So, if you take space-time and you perform it, this moyal term, you're going to violate isotropia, aren't you? Are there going to be preferred directions in space lines? No. I guess what happened there, I understand. The question I can put that he is asking is that will you violate isotropia? Correct, the question I guess. Because of this object that is coming on the right hand side, you look at this thing here, and then I make the Lorentz transformation naively on this, say, rotation of Lorentz transformation. You change the theta, but theta is a fixed anti-symmetric matrix. So what happens to Lorentz invariance, for example? Okay. The naive application of the, the naive application of Lawrence's transformations here indeed changes theta. So, until let us say three years ago, people were taking this as a model for checking Lawrence's Indian violation, because it is no longer, I mean these are some issues which are not privately decided, but are experimentally decided. So, So people were inferring certain numbers and writing check.

27:30 But then there appeared a very interesting set of papers, which organizes again with the Russians. In fact, the Russians seem to have taken over this film, namely Drinfeld. Drinfeld has quite well. There is something about awkwardness. I don't know whether. Drinzell pointed out some time ago in the context of the Hobbes and generalizations of group theory. Groups have got introduced as an algebraic structures. And Drinzell pointed out that even if you have things like this, there is a way to make So, this is a remarkable statement which was subjected by a group around Julius West, Pao Vaske and a bunch of people and also by a group in Finland, Chai Chian, Masur Chai Chian and a couple of others and they proved that once you know it is quite obvious that there So, there is a way to define the action of the Pochrane group on this even though it is an anti-semitic matrix on the right hand side. So, subsequently there have been a number of papers especially by the group of Julius West and also by us where the entire literature geometry including gravity, Riemann-Tex and everything has been recast in the algebraic language and in the context of form-algebras. This I think is very, very interesting and shows already that the standard differential geometry is not necessarily compelling for doing this. So in that context, even though it looks anaesthetoric, nevertheless you will not be able to detect it. Okay, I guess I'll... Actually, Balachandran is going to give us a popular-level account of this very same topic. In just a few moments, we do have a moment to take a coffee break. We're supposed to have coffee available for those of you who are interested and trying something to go with it. Exit this door to your right and you're moving right at the end of the hallway where you can find and become your precious.