Advanced mathematical formalisms (operator theory) for quantum field theory

7:30 What happens is that for each possible, if you like, each omega in here is a history, an xt of omegas of that particular history, the value of xt. Now you can spread that please, it's a function of t, a continuous function. The particle doesn't suddenly go from here to here. It may not be a differentiable particle. And so transcribe that around, this is a continuous function of t. And the second is that if the measure is not concentrated,

10:00 You can get some very funny results. For example, let us call m i omega, thinking again, perhaps, of this example, between, by definition, the supremum over all t, containing i, some finite integral of x t of omega, in other words, for each omega we take this supremum over all t. What we're doing is taking the supremum of the family of logical functions, but if this is a continuous... We cannot guarantee that this is actually going to be a measurable function, i.e. a random derivative. This is slightly bizarre, because from a very physical point of view, you would indeed expect that you would sort of look at the positions of the parties before trying to take this improvement. That would be something which is a random derivative. That's not a probability law. But in point of fact, from a technical point of view, that may not be possible, because of course you remember that the treatment of an arbitrary family However, if mu is concentrated on a continuous puncture, even if that shows this thing can be computed miserably with puncture, as in the case of TAMS, this may not be measurable. Again, purely physical considerations suggest that the systems we've got, it may be too rare. So, right on the other hand, we define m i twiggle of omega to be the suit over all e k non-i's contained in T.

12:30 Again, I'm using the Brownian motion cases, where the Tn is a countable dense set in I. So, I stands for IEB. If you take a countable dense set, then of course you can always do it. Now, M I quibble is measurable. It's the supreme mode of a countable function. And what's more, M I quibble in M I is made on the continuous function, which is defined... And this therefore means that if mu is concentrated on such functions, we can use them by history. It should be fairly obvious, of course, that if you, you can always chuck away sets of measure zeroes, mind you, and they're not going to affect anything. But if mu is concentrated on such functions, we use them for class-based, and then m-i becomes measurable, because basically it just isn't very much visible. So here again, it's very suggestive that in the case of Brownian motion, what you would certainly expect would be a physical random interval whose results you could actually write down on a piece of paper, you would expect this to be a measurable function, and that is guaranteed, provided the measures concentrate on the continuous function. This again suggests that in general, this space may be much too large for our purposes. In the case of Quantum field theory, let me say what the underlying of this is, is that it will turn out that omega is in fact the algebraic dual of certain topological spectrums, and that in a sense is sort of bad, in the same way that this thing is bad. And the nice part of the algebraic dual is the topological dual, the minus the sentence, the continuous functions here, and the nice part of this. And a lot of the effort in the end of the course will be geared towards showing that measures on the topological, on the algebraic duals will be shrunk from measures on the topological duals.

15:00 Which is the exact analogue of thinking as to this, and in fact you can prove this theory from that thing, if I get the time I will do that. So the string theory from algebraic duals and topological deals ultimately, in this particular case, comes down to the string theory of mystical internal structures. So that's the sort of thing we have in Rome. That last statement is that the space aggregate is in general too big. And unfortunately the situation is not just that. Sigma is also too small. The sigma algebraic sigma is in general... Again, I use words like much smaller in a sort of emotive sense, I put them in commas in the case, than the Burrell-Sigma algebra, on, over here, generated by the positional product, apology. Those of you who are related to this, actually, it turns out, and this is actually a major defect. The actual track, if an actual track, you certainly can't feel it in the case, but you do want to be able to talk about it later on in the answer. It's a distinction between talking about measures on some of the sets in the algebraic draw, which I could probably come forward to immediately, and talking about measures on the real sets in the cohomological theory. The space gets smaller, and in some sense the number of sets gets bigger, so you can prove that they don't get erased. The more multiple sets you have to measure the structure, the smaller the space, the more handled it is. So both of these two things go together, and both of these are regarded as fairly major details, but this one is in fact elementary still, by the standards of what's going on. Well, as I say, it takes literally 18 pages of my notes to go through it, but I'm going to do it in a couple of minutes. It is a couple of afters. Everything in the afters seems really pretty easy, but also later we're going to prove them.

17:30 Most of the questions are actually wrong by now. You know, they're just badly spated, but they're going to prove what they say they're going to prove. But I won't do that this term. That'll be next term, I think. This term will be... I should concentrate this term on... Just using a common go-off there, which we're all already known for, we're not going to get into this section. Right, coming to the two, question number three is that if U, T, and N are bare measures, then there exists an analogous theorem for the sigma algebra generated by a cylinder set which is based on bare sets. And remember that the bare measures are automatically written down. In the particular case, it's relatively compact, single-threaded, that's why it's finite bare measures. In actual practice, of course, the typical lowly compact, single-compact equations you'd choose would usually be a metric space, in which case the Bayer measures and the Grohl measures are the same things anyway, so this is a rather fine description, which in fact is why we're very much delighted with it. There's nothing in there special about choosing the Grohl set, despite the fact that we've chosen the Gauss set. Right, well that's, that's called Grohl's theorem. That's the end of section 3.5. Now, the next section, 3.6, is a very long one, and it runs right into the quite active quantum field theory section, and this is called generalized stochastic processes, and these are really stochastic processes in which the index set is itself a vector space, and typically an inter-dimensional vector space, and indeed, more or less an inter-dimensional logical vector space, although I shall somewhat play down the logical aspects. So, section 3.6 is called generalized stochastic processes.

20:00 And of course the point is that at a quantum field we can have a generalized stochastic process of a certain type. Here's the pre-field, and here's the ways to improve it. Now, before I actually give the definition, let me give an intuitive idea of what one means. Let's x be, well, we're saying x plus, and this is run by the derivation. Let's consider this stochastic field on the probability space. So, it's a family of random variables or multiple functions that you can find and then only get. I'm assuming they take their values in the real life, it's something like the position of a particle in the ground in motion and that is taken in positive times as the index. Now, let E be some space of, I'm going to call them test functions. I'm not using that in any technical sense, I'm completely in test functions first, the same thing in functions of complex, or with the certain homology of mathematics, I've got a record of space, so either John or I will be doing that at some stage or another, but at the moment, I just mean test functions, well, everybody knows what I mean, I don't mean to pretend that nobody knows what distribution theory is, I know what I mean, I like to speak about test functions anyway. There's some space of test functions on our class, so there's some... Some smooth functions on our class are probably the complex form of, say, C-infinity. Nice, nice function. And then I've tried applying a chain. Phi of f of omega is equal to the interval between north and infinity of f of t,

22:30 x of t of omega of t of t. Put that in inverted commas, it'll happen. And the rest belongs to these fixed functions. In other words, for each fixed omega, It's clear now that we're going to set it to two. Of course you can do that. Whether or not you can do one of the only things in some sense, you'd grab this as integral over the families around the world, which is somewhat Julia's nature of course at the moment. What can we say about this? We can say then that F is the final F. There's a linear math on E to the random variables. Well, I'm going to go to Sigma Mu. Now, a sort of example one has in mind, there's a nice discussion of this in the book by Girlfriend and Belenkin, the volume four of their great six or seven volume of these days treatise, which discusses generalised classic approaches and the measures of them. The example they give is that if this were Brownian motion, for example, If you actually make a measurement of Brownian motion, you can't really make a measurement of a single instance of time. I mean, the measuring procedure takes a certain finite amount of time, and the amount of time is determined by the measuring equipment. So in some sense you smear out the actual values in some sort of very common way, with something which depends on the equipment. And this F is meant to represent the effect of that equipment. That's the idea they give in Brownian motion. So they say in actual practice, what you always measure is not really... X-tier variables that allow this sort of smeared quantity, because that's the motivation they give for introducing things like this, which are linear maps from vector spaces to random variables of a certain variable space. Now, in the case we're concerned with, it's wise to say that this phi of f is precisely a quantum field. I mean, in the sense that you normally smear quantum fields of functions that are transpired, you can regard them precisely as being linear maps from vector spaces to random variables. So that's our motivation. The mean for reducing the capital processes, the motivation for this, is rather more Latin than the alphabet and the code.

25:00 There, let's now formalise this and give the actual definition. Why do you think the inverted commas are being used? Well, that's all right. It's that which would be an inverted commas problem, because it's not clear that this is in any sense a measurable function, and it's a question of weak integration, how you define this integral, without you watching me. It needs to be defined, and it's not a topic, but that makes sense, but Also, of course, I haven't, well, nowhere have I said that this thing is a measurable function of t, which is, I mean, this is a measurable function for fixed t of only, but I have no reason to suppose I can integrate this thing in any sense, even for fixed t, okay? I might do it, this is a continuous function of t, okay, in the case of Brownian motion, but in general we don't know about this, and so the whole thing needs to be examined properly. So let's have a definition of 3.6.1, and this is the formal definition of this, that in The real vector space is convenient to use real vector space and then mainly what occur, in practice, as long as there's no difference between the two. To give you a real vector space, an omega sigma mu, a full vector space, then a linear stochastic process on omega sigma mu, that's the terminology, a stochastic process over the vector space and the measure space, is a linear, say a real linear phi from E to a random variable. That's the definition of a linear stochastic. That's not quite the same thing as a generalized stochastic process. I've chosen to make the distinction. I'll just say it was what the distinction is, but in the particular case that E is a topological vector space, and therefore carries a topology, you require an addition of this linear mapping continuous. In other words, it's like a continuous linear function.

27:30 So, of course, you have to apologise for that in an appropriate sense as well, but essentially you put some sort of continuity on. When you do that, then you put a generalised stochastic process, but since at the moment I want this to be clear, we'll just stick with this. Now, once we have the notion of equivalence between stochastic processes, in some sense you would not expect two stochastic processes to be different if... The finite dimensional distributions of any finite collection of variables there have the same probability distributions. Let's write that up. The two linear logistic processes are said to be equivalent. They have the same probability measures. What I mean is that i of f1 is pi of f2 up to pi of fn. That's the same JPM side. If we just think any finite collection of x which in some sense code the stochastic process, I mean each f gives us a random variable, so we choose a collection of finite collections and code the stochastic process with it, if this finite collection is the same during probability distribution in the usual sense of collections of random variables, then we say these two things are equivalent. Now the terminology here differs from one place to another.

30:00 Here, these terraces. Equivalence class under this equivalence relation. Oh, I see. Yes, sorry. Of course, you should show that this is the equivalence relation. Now, note this very carefully. In the literature, you will find vast amounts of work by Stiegel, and also... Gross, which is, you know, it's excellent work, and it's very high-powered, and it's very useful, and in many ways is fundamental to the whole construction of physics, under the title of things called weak distributions. It's not entirely clear what the connection is between that and the conventional integration of mathematics theory. It requires some careful reading to see the connection, but the connection sometimes comes through this definition here, that Groves and Siegel have had their work somewhat different from the way I'm going to do it. I don't mean to say you shouldn't consult their work if you're in any sense interested in this, but it's certainly historically started to fall wrong in many respects. Okay, so Siegel calls it a weak distribution, and that word ought to be used by Groves. The notion of the mean of the expectation are generalized in Kessel's process, perhaps defined in the way that you might have expected. The mean of the expectation of phi of F, and remember now, you really can't think of phi of F as being a quantum field, because I'm explaining life, so keep that in the back of your heads. And of course, as always, one should say, if it exists, I mean, the function of the random variable and the measure of the function of the random variable, of course, may be integral, or may be infinite, or may not be integral at all.

32:30 Now, in the quantum field theory sense, that would be precisely the fact that you have to take the value of t. On the other hand, the covariance of phi of f, phi of g, where f and g belong to e, is... You can find, I call it the C of F and G, you can look back at the definitions I gave for complex covariance for finite classes of random rows and make sure the definitions I'm giving aren't insensible and complex. You see, phi has been the expectation value of the product of phi of X, phi of G, and mu. I won't keep on writing of omega or omega, but do remember that this is phi of F of omega. I don't want to keep writing this, but remember that's really what that means. It's this minus the expectation value of phi of X. And again, the turn-out of quantum field theory, that's what seems to be called the connected 2.3 function, that notation goes in the path to 2. At least that's true for the three skills, it's a lot easier to quantify than the other. Well, no, in effect, they're the definitions of the old definitions. I'm just emphasising the fact that they now depend on pairs of functions, so I'm calling them functionals, which is really a five-linear functional on E cross E, whereas before it was just a finite function, so is E. And so the same thing is true for E, it's the expectation functional. But of course the definition actually is the same as the finite function. As indeed it should be, because that would have stopped E being a finite dimensional vector space, or just a finite collection of things. Are you considering these as maps from each of your numbers or something? Well, E of this thing is of course a real number, because phi of f... We'll assume now that random... When I talked about random variables before, I did say they took a function of real life. And I am assuming now that phi of f takes its place within the real world.

35:00 And of course you can do derivations of factors, particularly for this in the art of physics, but I haven't done that. So let's assume the private number's the real one. So this is a real number. So E is certainly a map from random variables that connect all of the IEs into the real numbers. Because that's what comes to the set, and this is all, the expectation value is just that. It's a number which is associated with the integral variable. It's better to think of it as E of F, and you can see phi is fixed. Phi is fixed once and for all. It's like you could call this E and F if you want. I mean that's an alternative notation. I call that C of F and G, not C of Phi of F and G. And if Phi is fixed, the stochastic process of Phi is really fixed after the Witten-Globinian consideration. It's the F which we're going to vary. So this in that sense is a function of F. Now, comment. Linear. The analogue, which has proved, because that was 3.5 times 1. And it's not just a straight sort of analyte, it uses the proof of conducts to understand the point, but it differs, actually, from the format it's done. And also, and most importantly, what we're seeing, and this last step will be absolutely crucial, which in my terminology is 2.451, which John, of course, also did a hint version. But in both cases, they were five-dimensional systems. Was yours like a compact group? So that would exclude an interdimensional vector space, a quantum vector space. So what I'm driving will be a version of Kolmogorov's theorem which is true of an interdimensional. I mean E is typically interdimensional, I'm not really interested in getting into this part of it. So these will be central features. Now I'll start setting up some terminology and now I'm going to be much more careful with projection maps and this sort of thing

37:30 because what I'm working up to ultimately will be a general theory of objective systems and measures. And Hilbert's limits of projective systems. And you simply can't do that without his definitive terminology reasonably correct. So I want to start off by giving some radiometric definitions of projection maps and what they mean. And this is just to familiarise you with the terminology for it. So the interesting thing is this went to. Let's f and g, the arbitrary subspaces of the real... I'm going to define the projections. Now the main thing is to note that the term modeling is actually used throughout the whole of the course. P subscript F matches E onto E squared F in the usual sense that that typical generic element goes into the equivalence class. So this is the usual mapping onto the quotient space. And if G is contained in F, then P will match E squared G into E squared F by the equivalence class under G. This is the unit definition, and this is well defined because that's contained in that, so two of those which I put there, modulo g, are certainly put there in modulo f.