11th UK Foundations of Physics Conference — Quantum Real Numbers

0:00 The space of the continuum that we have in ordinary sort of Newtonian continuum continues right down to the very lowest level. They talk about distances between particles in terms of those things. And there seems to be no discussion, or else I'd say it isn't the space of mathematics. So the sort of questions I think that do arise, and I'm standing in the way of the pointer. Is that does a subatomic particle have a position at all times? It seems to me this answer to this will depend on what you mean by having a value for its position. What is a real number? If you just take the ordinary real numbers then of course it won't, but it's not necessary that they are the real numbers associated with quantum. Does it always have a momentum and such? What is the composition of the spatial continuum of subatomic particles? Now, to answer this question, we have to build a model which is larger than just the standard continuum that we've all learned to grow up and love from our undergraduate mathematics. But there are other developments. What is the geometry, I call it, the sub-atometry of this space? Is it Euclidean? Is it locally Euclidean? What does it look like? I mean, people seem to assume those things without ever asking questions about them. What is the distance metric in this continuum? One of the things I want to show you later on is that you can get some surprising results when you use the example I'm going to give. The distance between entities, atoms, electrons, whatever they are in the system, need not be the same as the classical distance between the apparatus which you're measuring. So you have to be, again, it's something you have to look at and see what's really going on. I call it the intrinsic distance here, but I don't do it anywhere else, so you don't have to worry about it. Can a subatomic particle have a position at all times and yet pass through two classically separated slits in the screen? And what I think I'm going to show you is that in the model that I have, it can. There's no problem with that. It has a position at all times and yet it passes through two. It could pass through two classically separated slits. Of course, that's not the position that most people were talking about.

2:30 And what's the relationship between quantum real number value, which is the thing I'm going to talk about, and classical real numbers? When is this called a measurement? And I pinched from Einstein, who says the theory that determines what we can observe, when talking to Heisenberg, and I'm also saying it's the theory you have that decides what values things can take. One of the things I would like to think about, one of the things I think is amusing to think about, is the first measurement problem the human race came across, which was the incommensurability of the side of a square with the diagonal of the square. But the Greeks, you can imagine, this is not historically correct probably, but you could imagine them thinking, damn it all, we can't at the same time measure both the sides and the diagonals. So what did they do? Well, they didn't think of it in terms of numbers. Eventually there was something about the theory of cats involving the theory of ratios, but they never ever got to the idea of irrational real numbers. So they never really solved the problem. And I think it's the same sort of thing in quantum mechanics. We may be saying that a particle can have both a position and momentum at the same time. And we're not looking for the right class of numbers to find their position and momentum values in. One of the things that I'll hope to... I put this on now because I probably won't get to it otherwise. It's one of the things I've learned from the model I want to talk about. You have a non-relativistic subatomic system described in the Hilbert space. All my Hilbert spaces are infinite dimensional. Sorry, they're just huge, okay? Countable, but infinite. We can construct quantum real number systems such that the real number magnitudes of the positions of the particles satisfy classical dynamic and differential equations globally and classical in the sense that they are either Hamiltonian in such a way that for each infinitesimally small region of classical space, Heisenberg's quantum equations of motion appear to be an average in a certain way, accelerations to a particle in that region are closely approximated with quantum real number values.

5:00 So what I'm trying to say is that the Heisenberg equations of motion are an approximation to a classical-like system of dynamical equations in which, rather than using ordinary numbers, we use these quantum numbers to represent the values of positions, momentums, energy, etc. This is what I haven't really finished doing, but the quantum real number values of the magnitude of line segments in the region are closely approximated by classical real numbers, so the local geometry is Euclidean to a high degree of approximation, so what happens is that these quantum numbers that I'm going to define locally are well approximated by classical real numbers and so when you come to define Distances between points on lines, if you're trying to develop Euclidean geometry in the way of Euclid, sorry, if you're trying to develop, in Euclid and I was reading a wonderful book by Robin Hartshorn, if you haven't read it, it's a marvellous book on Euclidean geometry, it's written for advanced undergraduates, so it's okay for me, and it is, he talks about Hilbert's, 1899, There's a book on the Euclidean geometry and he gives a historical development of it as well. It's a beautiful book, I think it's well worth it. What's it called? It's called, I don't know what it's called now, something Euclidean, but I can't remember. It's called Euclidean geometry. He's keen on geometry and not on numbers. He puts off on, he says that the Greeks did not use numbers, they used magnitudes. One of the weaknesses in our theories nowadays is that we tend to The other thing I would have to say, even though I think it's a bit naive, I do believe there's a physical reality, but it's probably only approximated to greater or lesser extent by the physical theories that we have. It's scary sometimes as it seems that people seem to think their physical theories are the reality and not approximations to it.

7:30 The trouble is that they aren't. We can never be quite certain how good an approximation they are because the only way we can talk about them is through our theories and through measurements of quantities that arise in our theories. So therefore, I'm always, even though I'm going to put this forward, I don't want you to think that I think this is... This is just another model which I hope will give us insight into what the quantum world is like. So just let me quickly remind you of what a real number system is. It's got to contain as subsystems the natural numbers, the counting numbers, the integers and the rational numbers. And then you can either complete it by Daedrican cats to get the Daedrican rules or by Cauchy sequences to get the Cauchy rules. Now, in the logic that we'll be using, I'm using a sheaf theory, the logic is intuitionistic, and so you can't prove that the Koshy rules are the same as the Dedekind. They are not the same because you can't use the law of exclusion or the axiom of choice. However, you can define additional multiplication, you have to have the binary relations of, we'll say, less than and less than equal to, So there's certain properties that I've just taken from what we normally take to be a real number system and that's what we use to construct this. What I find surprising is that the people who developed quantum mechanics didn't take this extra step. See, in the usual Hilbert space representation of a quantum mechanical system, take a self-adjoined operator in a state which I'm going to assume here. I don't put hats on operators, I'm sorry. People try to make me do it but I keep forgetting, so they're hapless. But these are all operators and the states are just trace classes of the joint bound with operators in trace one. Then instead of interpreting trace row A as an expectation value available in the state row, we want to consider the functions A sub Q from open subsets of state space to the ordinary rules as our quantum number real numbers.

10:00 Now it turns out that they are real numbers in the same sort of sense as I gave on the previous slide, provided you interpret that in the terms of the sheaf of continuous functions on state space. These are in the category. This gives us a topos and these are the real numbers in the category of sheaves on a topological space. The real numbers are given by the continuum. I have all the properties except I have to explain the slight differences. The thing to be careful about is that I am dealing with unbounded operators. I'm getting with, I take with the common, I take the CCRs, the canonical computational relations, take irreducible representations, not number-throwing representations, and then you have to adjust the state space, you can't use all states, but nevertheless you can do it, it's doable, and then the real numbers look like this, trivial, that in the sheaves on state space, For any open set, you get these, the natural numbers, which are just locally constant functions, locally constant, the integer, locally constant functions, rationals, again locally constant functions with values and integers, the Cauchy rules are locally constant functions with values and integers, and the Dedkin rules are continuous functions, and these are... Now these are the ones that we'll be taking as the numbers, the real number system that we're dealing with. All of these other, this is dense, and this When we put the metric topology on it. So it's got the same sort of density, the rationals are dense. This means that, for example, here I define this real value function and, of course, not all data can be used in this form.

12:30 The other thing which is important to know is that The quantum rule number, if you look, if you call the continuous functions of this form, developed by taking traces, if you call them block A, then they inherit orders from, actually they inherit the orders from the rationals, but they also, it's the same order that's on the delicate rules, and these orders are equivalent to the operator orders on software drug operators. So if A is, if the operator is A, If the operator A is greater than or equal to 0, then the corresponding quantum number will be greater than or equal to 0. If the operator A as an operator is greater than or equal to B as an operator, then the quantum number A will be greater than or equal to 0. Again, that's sort of useful. And then you can define the distances, etc., but I don't want to skip that. So what we do is we have a basic postulate which says that the values of quantities are given by quantum real numbers of this form or functions of, continuous functions of an operative operator U is an open subset, I haven't bothered to tell you how to work out the topology if you're interested I'll tell you, and these real numbers are sections of the subsheefs, the subsheef of A of E and S, and we're going to assume that the physical quantities always have quantum real numbers but it's even when we don't know what they are. The other thing is that we assume, as you notice, that the open sets, the only thing you need to know about them, is that the topology is such that a singleton is not known in a set. It's not good enough just to know the value of the operator, a trace row by one row. You have to know that some neighborhood of a row. So in a sense, I assume that you don't know a single state. But since the state itself is a self-adjoined operating, you can also think of it as a number, but I don't want to go into it.

15:00 Now, but under certain circumstances, the values of the quantities may well be well approximated, only approximated though, by classical constant real numbers. Remember the, or you could put in here the rations, it doesn't really matter. There's approximately classical if for W belonging to open subset. This, this, this is a, I'm sorry, this, these two bars here indicate the distance between these two numbers in the metric. I apologize if you can define that. So you take the, you take the quantum number If you take the quantum number associated with the square of A evaluated at W, then you take the quantum number associated with A at W squared, and you look at the distance between those two numbers, and if that is small, then we say that they are approximately classical. In fact, I'm not being really careful enough, we did it a lot better. Carefully than that, and this was some of the work I did with Thomas on the sharp values, suppose you had two globally defined numbers in, as I said, Cauchy rules, but they could also be, they could be just irrationals, and they have a slit as defined by this closed interval, and you can define closed intervals in the sheaf, there's no problems, and you let u be an open subset. Then if the value Z view of the Z coordinate of a particle, then it is, I'm sorry, is epsilon-sharp, I'm sorry I've left out the definition of epsilon-sharp, but there's a definition of epsilon-sharp on an interval which implies this, and for our sake we will just take this as the definition, it takes too long to go through the other definition, where this, this is a number, Here I'm giving it pointwise, it's defined pointwise, but this is a Dedeckin rule number, because it's not of the form trace rho a, but it's a continuous function, so it's a Dedeckin rule, and it's of course just like the square of the deviation between trace rho z squared and a, so that what we're saying here is something is absolutely sharp, is absolutely sharp,

17:30 In an interval, essentially it means that the values of Z at a view lie within this interval, Z1 less than Z at a view less than Z2, but in fact they lie well within the interval, so well that it is essentially the length of this thing is much less, the square of the length of Z at a view is much less than the square of the length of Z. You should be able to prove some results. If the slit is narrow and that slit is small, then here, so that for each row in that set, s z of rho squared is small, so that we can approximate, well, we can now approximate z hat of u by a constant real number in this integral. The question we would like to answer is that does that number have to be in the spectrum and unfortunately I haven't been able to completely prove that. I believe it should be in the spectrum of the operator, but I haven't been able to do it in the continuous spectrum. Anyway, so you have this idea of epsilon sharpness of numbers. Giving rise to real numbers approximating the given quantum number. You also have a Heisenberg type inequality which says if you have a second stage which is such that the values of a pair of commuting the quantum real numbers associated with a pair of conjugate variables like P and Z And if both of them are epsilon-sharp values for particular intervals, then the product of the length of those intervals is bounded below by 2h cross epsilon.

20:00 So it doesn't say that the particle doesn't have a position at the same time, it just says that if you want to make them, if you, as an experimenter, want to sharply focus them both at the same time, then the steps will have to be wide enough, the product of them has to be wide enough. But it says nothing about whether the particles do or do not have values at all times. Okay, the other thing I should have said is that the distance I meant to define this before, which is the value of x minus y is the max of x minus y of y minus x, with this distance rd is complete metric space and separable, rd is a fence, it's filled in the sense that if a is not invertible, that is not, break it around, not a belonging to rd, then a is equal to 0. Then, if we have a real number a without an inverse, then it's identically zero. Unfortunately, you can't deduce the commonness from it, because of the logic, and in fact it's not true. If a is not identically zero, it could still not have an inverse on some interval, on some open set. It could be like the function Z hat, which the spectrum runs from minus infinity to infinity, so if you have any open interval so that the value of Z hat contains zero, then it won't be invertible. I won't talk much about truth patterns. In this particular case, we'll probably have about five minutes left, won't we? Anyway, let me just give you a couple of quick examples to show you. This is a trivial one. This is an example where you have a particle. There are two slits. Alpha is supposed to be the center of this slit, and beta is supposed to be the center of this slit, and the z-axis is up this way, and you take two open sets, u sub alpha, u sub beta, in state space, defined in this way, and this way, where epsilon alpha and epsilon beta are two small numbers.

22:30 You could use quantum numbers here, but I usually just used standard rules, which are contained in the quantum numbers. So the quantum real number that Z hat on the open set, which is just a union of those two things, is just a value, it's a single number, okay? So I have an open set which is somehow localizing the part of it around here. Another part of it localizes around here, but I can't have a... Since all I need is an open set and the union of two open sets is open, then this is still defined as an open set, and that's a single number. And it's localized in the sense that you can find numbers z1 and z2 hat, such that z1 hat is less than this number is less than that, and the distance between these things is finite, so it is localized for a reason, in fact you can do the obvious thing with it. Let me just show you, one of the things I haven't done properly is the dynamics. One of the important things about this is with the dynamics, you shouldn't be able to bring in nonlinear dynamics. For the underlying quantum numbers and that what the hope was that we should be able to do something about the measurement problem with those numbers that hope hasn't been realized yet it's just because of my lack of ability that's all I just need to work out maybe there should well there could be that anyway and that's not good but here is one interesting little A side is that if you prepare a particle 1 either to pass through the slit I1 or I2 in the screen, the particle 2 is prepared on the other side with a position which is approximately classical, that means, I meant that this holds, sorry about all the nasty language, so on some other than said. If 1 passes through I1, we assume that this number becomes epsilon-sharred. And then, if the path is derived to, and this of course is what, I haven't explained how this happens, this is magic, this has to be, there should be some dynamical way of this occurring, and I haven't been able to find it yet, but if you, anyway, I want to tell you what I would like to do, and then this becomes a term chart, then those things are obviously.

25:00 When one passes through the screen, there is an impulsive von Neumann interaction between 1 and 2, which is just the old standard von Neumann interaction which is used for measurement. And then, as a consequence of this interaction which only is impulsive, the value of particle 2's position changes either by this amount, proportional to x1 or w1, Or by something proportional to x1 and w2. And the resulting value of 2's position remains approximately classical. And hence it's recordable. But what's interesting is that if the particle actually went through the two slits simultaneously, and you try to work out what happens, when you work out what the new value, after it's interacted through the one-nomen interaction with x2, The position of particle two is no longer approximately classical and therefore if this is right, and this is only a model, if this is right, it would say that the particles that pass through both slits are not observed by this particular type of interaction and therefore it doesn't notice them, so it never tells you if they've gone through both, it only tells you if they've gone through one. Anyway, I should, perhaps I have to finish now. The other thing I would just like to say is that if you do the calculation using distances, if you take the standard and method way of describing Bell's theorem with the potential to spin a half-part, which was prepared at the source, which we take at the origin of the coordinate system, if you do the standard calculation, I've already done it, so I won't show it to you, What's surprising is that if you look at the distance, if the particles are identical, then the distance between them on one of these sets, this particular set, which is the set of states which closely approximate what you would have expected from the quantum mechanical calculation of what the states are, is very small.

27:30 So now it works, even though the two pieces of apparatus are well separated, but the distance between the identical particles, the point about it is just the fact that the particles are really identical. Clearly they're identical in the sense that we don't count distinctions, but they can't distinguish each other, and so therefore, if this is, again, so even though the apparatuses are a long way apart, The fact that the parameters are distinguishable means that the distance between now and time t, when they just come into the apparatus, is... Well, what ramifications that could have for anything else. Anyway, that's all I wanted to say. Any questions? So you start with a Hilbert space and a state space, and then your quantum numbers are functions from that space into reals. So it seems like you'll get very different number systems depending on which physical system you start with. But I guess I'm wondering how those different number systems that are induced by different physical setups would be. I don't know. Yes, I'm not quite certain how to compare them except by... No, I'm not quite certain. It's a good question in the sense that I don't know how to compare them. They obviously are different because you will have different state-space, different topological space. But you do get a system of real numbers. What I'm trying to say is that for each system, it carries with it its own system of real numbers and therefore its own continuum and therefore its own underlying geometrical structure of space.

30:00 And then what happens if two systems previously isolated begin to interact and you have a completely different state space again? Sure, and then you have to rebuild it. You have to have a bigger number system which contains all of those, right? Sure. I'm sorry, I didn't talk about what happens when you have, with molecules. Here I was mainly talking about single particle systems, but you're right, it becomes quite, perhaps what you, what one really needs is a sort of universal state space from the beginning, which contains everything. Okay, thanks very much. Interpreting the state effects of a quantum, geometric algebraic quantum mechanics. Yeah, my talk is perhaps somewhat complimentary for that, because I want to present a way of doing quantum mechanics without complex numbers entirely, General objects and vectors, this is largely not. I am heavily relying on the work of others, mainly a group of mathematical physicists working in Cambridge, and I'm making more reference to the classical world. I am approaching this way to do quantum mechanics from the viewpoint of a pure philosopher trying to interpret quantum mechanics and I'm taking interest in this because I think it is of the utmost interest to us when we hear that quantum mechanics can be done in a radically different way and it's entirely equivalent to the usual Hilbert space problems which we all know.

32:30 I'm going to change the title of my talk, which says with instead of in, and I will explain this in a minute. Let me recount a little bit of history for those of you who haven't heard of geometric algebra at all. This is a kind of vector calculus which was developed mainly in the 19th century by people like the school teacher Glassman and by Hamilton and in a unified way by William Clifford. The theory is particularly or was meant to be suited to represent three-dimensional rotations, and geometric algebra as it is today is a kind of elaborate version of this 19th century calculus of Clifford algebra. This affords an alternative formulation of quantum mechanics, and this is why we call it GQM. And also more general quantum theories, relativistic quantum mechanics and even quantum field theory and general relativity all can be done in this very same formalism. Just start out with a different kind of calculus and you can develop a unified simple language for the whole of physics. That's the program. And I found this so amazing that I started to work my way into this. I have to, for those of you who haven't heard of this, give you an idea of what this is about. It defines the geometric product of two vectors as simply a product which is associative, distributive or addition, and gives a scalar for every square of a vector, so that we will be able to... We calculate the length of vectors and then we define an inner product and outer product in terms of this more general product, so if there is nothing in between the two vectors a and b this means this more general geometric product and the dot signifies the inner product which will return Scala and the wedge product will stand for the outer product as a result

35:00 The geometrical product can be written in this way. More interesting, we now have a symmetric part of every geometrical product and an anti-symmetric part, the wedge product. And this will have the following intuitive interpretation. It's an element of the plane spanned by A and B. It's an area element and you can bring it about that you have a kind of unit element and it has a direction so that this object will be facing in the opposite direction A, which B will face in the opposite direction. As does the wet A. But you define this facing in one direction or the other just inside the plane, without leaving the plane. So you have no need for the cross product or Gibbs product in this calculus. You have a natural replacement for this. If you have more than two elements in a geometric product, there will in general be the possibility that an element like this A, which B, which C. This is a trivector. Well, this is usually called a bivector. And it can be geometrically interpreted as an oriented volume element. Here you can bring it about that you have an oriented unit volume element. The last product of these three unit vectors will be a distinctive element of the whole algebra we are about to construct and this is suggestedly called capital I because it will be a natural reflectance of the imaginary unit because

37:30 If we square this new object we have we find immediately that this becomes minus one and for any yeah for these three basis vectors and the circular permutations of the we find that they have this property so For example, sigma 1 wedge sigma 2 gives i sigma 3, and this is immediately recognizable as the property which is the defining property of the Pauli algebra, which can be alternatively represented with complex 2x2 matrices, so we immediately Being introduced to a very simple calculus have a replacement for the complex numbers which we use to represent, for example, the quantum mechanics of one spin-half particle, very quickly and very easily. The general object, which is constructible in this way, if we confine ourselves to the three, Basis vectors will be of this kind and it will contain up to eight different elements namely one scalar, three different vectors, three different bi-vectors and one tri-vector. The space R3 which is called usually the algebra of space and I must simply tell you that this procedure generalizes easily to the four-dimensional case and then you have what is called the space-time algebra which is generally sufficient to do relativistic quantum mechanics or relativistic physics in general.

40:00 Okay, how can we do quantum mechanics with this? Let me return to how one does quantum mechanics with this. In order to do quantum mechanics, and I'm always confining myself to the single spin-half particle, except I will make a short reference to two particle states. What do we have to find is equivalence of the state vector and of operated action on the state vector and one natural mapping which apparently is easy to see but I haven't invented it is this one, if you have this spinor So if you have a pure state represented by two complex numbers, you should just translate it into this, in this way, into an element of this algebra. For example, z plus state, this is made exactly to have these two as simple spaces states, becomes just a scalar, one, and z minus becomes minus i, so two. And the other ones you will see in a minute, because I want to bring out something of a geometrical content of this procedure, which up to now is just an abstract replacement of quantum mechanics by another calculus, but with added geometrical content. That was the original advertisement. Now, one has to reflect a little bit on the next step. That operator action on a state has to be transferred or translated into geometric algebra in this way. The sigma 3 has to make sure that the objects we arrive at in this kind are still of the same kind as these objects.

42:30 And this has to be made sure by right multiplying the basis vector. Otherwise, as you can see already, this object does not contain any vector part and no tri-vector part, but only a scalar part and a bi-vector part. It's an element of the even subalgebra of the algebra of space. And in order to make sure that the result of acting with an arbitrary operator on this All of this is still in this even sub-algebra. We just need to write, multiply one of the basis vectors and that we choose sigma 3 is simply an artifact or it's comparable of choosing the simple basis. It's a purely formal procedure we should carry with it any content. Left multiplication with the basis vector. Multiplication with the imaginary unit now becomes, this is the consequence of these two assumptions, this is put in, this is put in, this comes out, this pops out, so to speak. Multiplication with the imaginary unit becomes right multiplication with this element, which is the unit bivector symbolizing the plane orthogonal to sigma 3. Calculate this. Let me show you how the expectation value looks in geometric algebra if you have the general geometric product of two vectors a and b and let these points and brackets Let's denote the scalar component of this and let kb tilde equals ba be the reverse of a and b, so the reversing of the order in any product, say this element tilde would be sigma 3 sigma 1, if you make these two abbreviations, then the application value just comes out like this, it's sigma k sigma 1.

45:00 Geometric project sigma, oh excuse me, sigma k, psi, geometric code project sigma 3, geometric code sigma tilde, and the scalar component of this. For future reference, very briefly, I will refer to the geometric code at the back of my hands for 2 spin half, 2 pixels. There you will be, at least I was in the beginning, disappointed because the nice geometrical feature that we are operating on normal intuitive three-dimensional space is lost because you have to glue together two of these spaces. But one can imagine, one can understand why this has to be done. Actually we're operating in kind of spaces which are relative to any of the single particles and normal quantum mechanics will end up with a 16-dimensional algebra and normal quantum mechanics where two particles operate in an eight-dimensional space or has eight independent degrees of freedom and what we introduce here to achieve this is an identification an operator which helps to identify the imaginary unit in the book in both of the spaces. From the start they are not identical and this correlator gives us the equivalent of the usual This is an orthodox two particle quantum mechanics and I just want to introduce this. I will confine myself to the one particle case in the rest of my talk except at the very end when I want to make a reference briefly to the singlet state which is written like this here. The complete basis for the two particles would be given by this transformation. Where this correlation operator has always to be right multiplied to every element of this algebra and if you work this out you get the spin-singlet state and you get this multi-vector.

47:30 Okay, now since I'm a philosopher I'm trying to squeeze out what kind of intuitive content this might have. We find that the equivalents we have in geometric algebraic quantum mechanics, who are the ordinary spinos, are rotation operators. We find, totally ignorant of quantum mechanics, that this operator, r equals m, geometric product m, where m and m are unit vectors, Then you will find that this object is a rotor, meaning that if you left-multiply this to A and right-multiply R tilde to A, then you will arrive at A prime, and if you do this to a unit triple, this will just be a rotation in space. Probably you will see that B is a unit bi-vector, defined by this, and this is, make sure that it has norms, yeah, this one, and you will see that this exponential expression, written out, is just again, contains a scalar part and a bi-vector part, and this is Actually, what all these guys here, these rotors, it's their form, and we find that our translation has led to nothing but translation of these spinors into rotation operators. z plus to this vector and then right multiplying z plus tilde the equivalent of this is of course no alteration at all and if you work this through in this way you will just see that you you're encoding in this way in the

50:00 The spinor equivalent in geometric algebra and quantum mechanics is nothing but rotations from a given fixed point, which is just arbitrarily taken to be the point where Z plus hits the North Pole, where Z plus hits the Unisphere, which is in the North Pole, and in this way you You just encode rotations in ordinary space. So what you are doing and what you are doing basically is doing quantum mechanics on the unit sphere and this is I think a well-known approach for the spin-off case since it is equivalent to doing quantum mechanics on The Bloch sphere, when you're confining yourself to pure states, you're working on the Bloch sphere, and if you're adding mixed states, you're also working inside the Bloch sphere, but the additional content you have here will probably best come out when you look at phase variance of the states, And you will see that what geometrical algebra has in terms of additional content is that these phase variants, so states which are equal to, for example, y plus up to phase, that these are equivalent to rotation of the plane orthogonal to y plus. So this is an interesting case. If you would like to do the hidden variables theory for this system, which would assume that the hidden components

52:30 Here the two components in the plane orthogonal to Z plus have values then you would from the start see that they should transform if you have any combination of states or so that they should transform in this way for example when you have the state that you have not only Y plus but Y plus rotated 180 degrees if any of the normal quantum mechanical calculations If this forces you to use, say, Iy plus instead of Y plus, then the AQM equivalent is turn this data around two times by this rotor, and this is like two times on the left. And any invariant theory which you would like to do from the start, from the quantum mechanical structure itself, would have to obey at least these laws. Probably what quantum contextuality for such an invariance theory would amount to would become more apparent than in this calculus. So let me, I'm sorry that I've used up my time already, but let me just try to dive a little bit more into interpretation. 1930 book, where he tries to characterize what superpositions are, or how they should be viewed, you know, superposition process is a kind of additive process, and it implies that states can be in some way added to give new states, the states must definitely be connected to the mathematical quantities of the kind which can be added together to give new quantities of the same kind. The most obvious of these such quantities are vectors. This is meant to motivate the following principle, which actually isn't verbatim from this book, but this is what he tries to convey to us, this well-known superposition principle.

55:00 Now, in order to have a little bit of time for more questions, I just say summarily that I took an example, took any two states, tried to work out a simple solution, and found another state, which happened to be i ket y-minus Now if you translate this into geometric algebra, you find that this is one of these phase invariants, it's just y-minus turned 90 degrees around. So what you see is, but this expression is more telltale actually, it's, you see that this factor, the complex factor has, as its equivalent, in fact a rotor, a rotation operator, so there's, so the translation reveals that You have to transform the states first into other states before you can add them together to get the superposition. So the superposition principle, if you look at it in geometric algebraic quantum mechanics, takes a little bit different form. This is, I think, revealing that v1 and v2 be any pure states and v3, blah, blah, blah. In the sum such that v3 times v tilde, the geometric program of course, is 1, is a pure state, this would be the superposition principle if you would have tried to work through this or any other example from the spin-off case from the start of the geometric algebra, then you would have to use this principle, but now this condition says that The V3 must be a rotor. It has unit length, and this is the definitional property of a rotor, and we said that any rotor corresponds to a pure state, so there is a kind of, so this additional condition here, which we saw in the orthodox

57:30 This principle now turns out to be that V1 and V2 be any pure states, then any V3 such that V3 is a pure state is a pure state, so there is kind of doubling, so you see a thing which you didn't see in the usual formalism, namely that you're putting in the condition that V3 is a... It's a pure state or is a rotor so there is a it's superposition principle in general as seen from this viewpoint takes a slightly different form it's not true that you can use any combination and then just try to make sure that the norm is one rather this translation reveals that If V1 and V2 are any pure states, any rotors, then there are pure states C1 and C2 such that V3 is a pure state. So what GAQM reveals is that the superposition principle, as we know it, selects some special states. And it does so here by relying on complex numbers, and if you translate this into geometric algebra and quantum mechanics, you see that the complex numbers come out as additional rotors. So this gives a different flavor to the superposition principle, and let me defer what this means to the discussion. Well, just two questions. Very quick questions. Yeah, two. Well, I mean, this is all new to me, but it looks as if the revised version doesn't get the full force because it says there are pure states, whereas the original version had an any B3. And the supersubstantial principle and the geometric algebra approach certainly should get something of that.

1:00:00 So there is many, many ways to superpose P1 and P2, which are, at least, possible. Yes, but doesn't it make clearer that the superposition principle in quantum mechanics is different from the classical superposition principle in the sense that it selects states, certain sums of, Well, yes, in the ordinary QM, it's a very mild restriction, it's just using norm, and I know that your complex numbers become pure states, i.e. rotors, but there must be a bit more than just there are pure states C1 and C2, because that is compatible with just a pair. There's got to be some freedom there. Just a reminder, the similar work was realized for Dirac-Spinoz, also for dimensions. You ask or you say, so... No, I say so. People of the De Broglie school, I go shock. And later Boudet from Marseille. And also Heston in the 60s. And they did such an approach that they used the Clifford Algin one. If I didn't mention David Heston as his name, that's just by chance. Actually, these people I mentioned from Cambridge are very strongly relying on David Heston's approach. Thank you. For the last book, Bob, is it Kirker or just Cooker? Cooker. Cooker. Cooker. Oxford University, Probability without Numbers and Time as a Measure of Contents. Oh, very interesting. So actually I want to present very new technical results, actually, which I think might have an importance in how we would look at probability in the context of maybe physics and it essentially brings Domains of study like statistical physics, quantum mechanics, quantum information theory, in a circular way closer together, from a point of view of probability theory. So it's joint work with somebody called Key Martin.

1:02:30 Well, first of all, it probably will raise more questions than answers, what I'm going to tell about, because it's a very different perspective on probability than is usually given, both technically and conceptually, I think. And, well, then again, one of the lessons of quantum mechanics is that different perspectives are sometimes important, so I think that's one of the main lessons. One of the other lessons we got from quantum mechanics is that there is a clear separation between an ontological and epistemological perspective on the description of a system. And one of the interesting things that this approach will reveal is that although we will start off very epistemologically, something which is usually considered as an ontological given thing in the study of quantum mechanics will come out automatically in a structural way. So, just to go over... What I'm going to try to do, so what we are going to do is we give a qualitative count of probability, both classical and quantum, although essentially most of the thing I'm going to do, I will illustrate on a three-dimensional classical system, which is very simple of course. This means like just three states, three possible pure states. In a sense, I would say that we follow Ramsey's idea, which says that probability is the logic of partial belief, but we are going to put it slightly more informatically, which means that probability is the logic of partial knowledge. And this will then introduce the notion of partiality or physical states and properties and actually the real number, because a lot of what we saw already talks about numbers, the real numbers will really only be brought in at the end. The usual probability theory starts with giving numbers to certain states. At the end, the numbers normally come in at the end and you can do whatever you want, you can replace it with the best thing you want. In a sense, one could say that actually the immersion-trivial coordinate could be even timed or something like that, if one wants to give the more physical dimension to this, or entropy. And the result is actually a unified view of qualitative and qualitative information. There has been some development about qualitative information very much in people in the computer science community in terms of domain theory. So, usually it's called scum domain and they usually do data typing. It's a very qualitative approach to information. And in our approach this thing will be kind of married with the more physical way of looking at information in terms of data theory. So, the content of the stuff is essentially a bunch of theories and constructions, and it will, as I told you, be illustrated on three-dimensional classical states, which is essentially trine, I think it's three symbols.

1:05:00 And the methodology will be that we will operationally derive a partial order of distributions. Just we take probability distributions and we will see that actually there lives a very natural partial order which in a sense satisfies this equation. The space of all probabilities equals what I call the Bayesian cosine, that is partial order, and a coordinating measure that is actually bringing in a little bit of the real numbers at the end. So, I'll start with learning this one. So states, essentially everything I'm going to do is classical but you can do the same thing quantum. I'm just going to illustrate at the end how to do it. So, to make things simple we conceive states as representations of knowledge, for example of an observer. And so these are classical probabilistic states. You have a number of, you have boxes, for example, you have three boxes. And you assign a number to the chance to find the ball in the first book. The first box, in order to find the ball in the second box, in order to find the ball in the third box, obviously they have to add up to one, and that's what I call a probabilistic state. So, the three-dimensional case, as such the state space, is a triangle. We have pure states here, and since the linear condition we put, summing up to one, we find, for example, the pure state here. The maximally mixed states somewhere in the middle of a triangle, which would be one-third, one-third, one-third, and here on the side we find, for example, state one-half, one-half, zero. How do they look in terms of boxes? One-third, one-third, one-third, we have no clue where it is. One-half, one-half, zero, we know that's not there. No clue where it is here or there. Okay, we know it's not there, we know it's not there, so... These are key terms. One feels that there is a natural order on these three examples.

1:07:30 But to say that this one is below, this one is below, this one is below, this one in the following sense, in state terms, that this one is less informative than that one. Or that it's only part of the information of that one. Because there is no information at all, we have no key. There is more information. We know that the object is not here. So there is a natural order in this thing, and in the observable terms this would be that this one has less knowledge than that one, for example, or has only part of the knowledge than the other one. this is a partial order which obviously is very easy to define or something like that. Can we define this on the whole set of bits in any natural way? Assumes from that, would it exist is that if X has less knowledge than Y and they are the same, I mean by less knowledge, I don't mean that they know that for example X knows less things, no, I mean that whatever X knows, Y knows too, and probably more, and if they obtain the same new knowledge, then X still has less knowledge than Y. It's very natural. So if I put this in a kind of an explanation, if x would be below y, you give some new information to both of them, and the sum, well, I shouldn't say sum, adding up the information would still be below that. And actually, he says that state of data for a joint observation should preserve this order, so we get new information, we update our knowledge, and so we want to preserve the order. Actually, does mathematics allow such an order on the state? Yes, it does. It's non-trivial, and it has very strong uniqueness properties. How does it work? This is Bayesian update. This is what people call Bayesian update. What does it mean? It means... I've got a value here, a value there, and a value there of finding a ball in the box. I open this box, I don't find it, 2 to 0, and this will be updated essentially according to that rule.

1:10:00 This must be, so actually you have to go to A, 1 to 9 to be, and this becomes C, not 1 to 9, like any normal answer. That's a Bayesian update though. I wrote it in two fashions. Here in a fashion that I put the middle to zero and there I actually just forget about the middle. I can think about this of going from three-dimensional space to two-dimensional space. And actually this little trick of forgetting about the zero and that this is essentially going from a three-dimensional situation to a two-dimensional situation gives us a full definition inductively for a partial order of states from the following method. So we assume that a state is below another one, if for any possible experiment I can perform, which is looking in the box, in the lower order, it's still below the other one for all possible observations that I can make. So it's a very natural, it's putting the condition I put in the previous page, of like making an observation and then preserving the order, putting it in mathematical terms, using this notion of algebraic, which is the usual thing used in probability theory, and this provides us with an inductive definition. Obviously an inductive definition needs a base case. Actually, I put down a base case here. It's kind of postulating. Is it very harmful? Well, I'll explain what it is first. So what we do actually, this is the two-dimensional state. Two-dimensional state is a pure state here, a pure state there. And the other ones are the line in between. Here is the maximally mixed state. And we say that this is the least in our order. These ones are above. And then if we follow this line, we increase in the order there, if we follow this line. That line we increase in the order here, and so this we say, this we postulate as a two-dimensional thing. Luckily, we have some theorems. For example, there is a unique partial order on the two stakes, such that this thing is indeed the bottom, and that satisfies mixing, what means that if x is below y, and we take a complex combination of x and y, The result should be below 2, which is a very natural assumption you make, and this forces you to end up with the same problem.

1:12:30 There is a lot more, there are a lot of other uniqueness theorems, which by making some assumption, but one of the most striking things is probably this little ground. This is Shannon entropy. We put it upside down and we forget about the slope. This already essentially indicates that Shannon entropy in a two-dimensional case will be monotone and essentially all the reversion and destruction. Actually, it does it for all the resulting partial orders. And obviously, in all the resulting partial orders, pure states will be maximal. And the thing we obtain is truly a pose. But it's actually much more. It's something called a scolomane, which is almost a complete lattice and has a lot of topological properties in terms of approximation and things like that. It's like the mathematical structure people use in information science to study data structures and evolution of data types and processes. And so when I write here, channel entropy is a measure of confidence, I cannot go into detail. This is an extremely strong statement. It's much stronger than saying that it's moment-run or something like that. It's much stronger. It has some topological topography. It has a very strong relation in terms of partiality of elements. Because the way to think about this is essentially that this is complete information and this is partial information. This is actually nothing. This is essentially no information at all. And the more you increase, the less partial your information becomes. So, so far we build up this partial law in an inductive manner. I just mentioned that there is a purely algebraic definition of the order as well, so it doesn't refer to the inductive of the order, so we say x is smaller than y. If there exists a common permutation of conditions for which x and y both increase, remember we can define them as maps from 1, 2, 3, to 0, 1, 2, to 0, 3, to 0, 1, so we look for a common permutation if they both increase and these equations are satisfied for all e, then this defines actually also the partial order, so it's purely algebraic way of defining it.

1:15:00 You can define a partial order in states by saying nothing relates to nothing, except to itself, that's a partial order. You can also define trivial things, do everything relates to everything, but that's not a partial order, that would be a pre-order. Something which is considered in physics, for example, the majorization order, but that's not a partial order, because there all maximal elements relate to each other. The question I pose now, how strong is this partial order? How much do I capture it? The idea is to find and to capture as much as possible, and this is like the first theory, but I'm going to elaborate on that further, that the only orderized morphism on this partial order, which is identical on pure states, and actually that's something else than fixing the symmetry of the pure states, that the pure state 1 goes to the pure state 1, 2 goes to 2, because otherwise… It would just give this theorem up to the symmetry, which is identical to pure state and which preserves Shannon entropy as the identity. So kind of fixing the entropy and fixing the maximum normals fixes everything. So this order really captured a lot and actually this is too much of a strong condition as we will see further. The next part of the talk I will show that actually from pure logic we can derive this partial order. So this was a very, what I did so far was kind of a very informatic way of thinking about it and letting it emerge in a very natural way. But you can actually get it from pure logic. Without even too much reference to all these informatics perspectives. But first, let me say how this thing looks for quantum mechanics. States are density matrices. We have three equivalent definitions. First one is just by comparing diagonals in density matrix after diagonalization by means of common unitary generalization. In terms of these equations, the analog of the above, and we also have an inductive definition, and the funny thing about the inductive definition is that you replace Bayesian update by quantum state update. So, quantum state update will take the structural part in the development of this partial order, there where in the previous case we used Bayesian state update.

1:17:30 And actually all the theorems which I showed before are two power values from the quantum case. The only thing I have to do is the relational entropy, but it just looks a little bit more complicated, but essentially it's all the same. So actually we defined a partial order, which has a clear significance, because we defined it in terms that something that accused a certain state was more informative than another one. It was not obvious that we would obtain even some partial order or something like that, so we do. And now we are going to kind of push things beyond the limit like... How much structural can we recover from this partial order? Actually we can get a lot. So, just think now about this partial order as a purely abstract partial order. We forget about the fact that the elements actually represent probabilities and distributions. There are three important things on defining something which I call irreducible elements. Sorry, I'm not going to read them. These are purely all the theoretic ways of walking in the partial world and identifying special elements. For example, the greatest load bound of the maximal elements above any given one. The list of the bounds of the maximal chain of coordinates and quantified coordinates below any element with a chain as elements way below. Way below is actually the topological notion of domain theory, so it's not important. But these are purely order theoretic characterizations of elements of the following form. This you can prove. It actually turns out that they correspond to the probabilities of this one. You have zeros for a series of numbers and non-zeros for the other one. We have the following theorem. And actually, these irreducible elements, starting with partial order, in the classical case, give us a power set, which is the algebra of classical properties, in the quantum case, give us the lattice of closed subspaces, starting from the pure ortho-theoretic system. So I forget about the fact that my elements of probabilities I obtain with lattice isomorphic to the lattice of closed subspaces. Actually, except for one element. In the classical case it could be the antiseptic of the power step, in the common case it could be like the trigonal subspace.

1:20:00 So in a sense these are usable encode physical properties. So you can just get them out of this partial order. What does the star mean on the left hand side? Upside down. Conceptually you have to put them upside down because the state, it's just the way the order is defined, the states of pure states are up, while here the pure states are down. So it's upside down, but it's not important. It's actually more conventional that people in this kind of domain theory think they use orders where they put maximum elements, pure states as maximum elements. So anyway, we covered physical properties. Now I'm going to define another notion, which I mentioned before. I define, again, purely or theoretically, the notion of coordinates. A coordinate is an element with a chain as a downside. So, complete or as the answer. Or an irreducible. An irreducible doesn't have a chain as the answer, it's a supreme of elements, so it's kind of a special coordinate. And it turns out that these correspond with elements of the form. Now these elements are actually quotients of this one. Any sort of complement, i.e. value a, for example this, value b to its own complement, which would be these are zero and these are non-zero, so it's kind of the weight of a property with its own complement, kind of balanced way. Now, we have a theorem with respect to these coordinates, otherwise I wouldn't have defined it. Every element, partial order, even classical or quantum, admits a unique and useful representation as a joint of coordinates. This is logical decomposition. This is logic. So I've got a kind of primitive, the notion of primitive ingredients within my partial order and I can write any other element there as a joint. I'm still talking purely over theoretically here. So I say that you can decompose states over the partial logical components. And actually the main theorem with respect to all this is that order isomorphism, that preserves you. Maximum, again, maxima is used to get rid of the symmetry-spying-magic correspondence with 0-1 value of coordinating maps, which actually gives me the probabilistic space, as opposed to plus-coordinating maps, and the same for quantum.

1:22:30 I'm going to illustrate this now with a little drawing. This is Boolean algebra, logical classical system, Boolean algebra. You made things upside down. So the brown spots go up, the orange go down. This would be pure states. This would be the maximum image state. I throw away the zero. And so you should think of this as like the representation of the partial order. And it corresponds with the triangle, of course, because we constructed it from it. So it has all the points of the triangle in this partial order. We connect all these points which come from our logic with coordinating lines, and then we are going to start writing any element or just joints of them, just like in this three-dimensional cage, just as pairs. What I'm doing is purely abstract construction. And actually, if we do this, we end up with a full triangle. So this is purely ortho-theoretical construction. Where do we bring in grilles? Actually just by saying that these lines are the ones. You could take these lines just as a continuous set of points, and this all theoretically doesn't make a difference. It's just when I say, okay, this is a real interval, that I bring in the reals. Before that, there is no reals there. So it's purely all theoretical construction which essentially captures the full structure of the probability space, but you only bring in the reals at the end. And so here, for example, the one I construct here would be the state I construct here. One last slide. This actually is a bunch of additional commands with respect to some applications. So, we construct this concept completely or theoretically starting from the propositions on a classical system or a classical hardware system. Minimal propositions we can construct in a purely or in a form of theoretical manner. You don't have to refer to the numbers or something like that, you just logically reconstruct it. Obviously, I can't talk about it because I don't have time, but a way physically to think about these lines,

1:25:00 exactly because they are, for example, coordinated by Shannon entropy, is that they encode a flow of time. For example, the second law of thermodynamics would say that you go from a pure state into the maximum order and so that your evolution would be going down in the order, so the order encodes the flow of time in terms of the second law of thermodynamics. So, okay, obviously partiality for states and properties is very important when you look at compound systems, one system in a Bell state, which initially, if you look at one or two systems, which initially doesn't have any definite properties except for the trigger ones, so the process creates properties, and all this extends to convergent, countable, the real continuum, I mean, as supports, rather than discrete sets of supports, you can do this for continuance, then you have to bring in some additional This is just a little illustration of how you can do this for all the distributions on the real line. Another very important remark, which I cannot illustrate, is that the Bayesian update rule is actually not even important. I can use all the maps and I always end up with the same poster. Which is very striking, and obviously here the category theory comes in and we understand that this thing, this category, this post, this category is very universal in a certain sense. It has a bunch of universal properties. Now, just a final remark I want to make is that a couple of months ago I was at Jeremies, I turned together with Sorkin, and he made a remark there in context to this. Space-time structures. I asked him, like, okay, you want to get rid of the reals in terms of space and time, but you get, you maintain probabilities. And then he answers, like, yeah, I know that, but I'm not going to wait until somebody comes up with a way to get rid of probability, of real numbers with respect to probabilities before I am going to push my program further. And I think, actually, This would be a very valuable candidate for a program like that, and maybe all the programs in quantum gravity and things like that. So, whatever. And that's a remark that it reaches with a very avant-garde topology and approximation study method. Two papers are available. This is like a monster of 90 pages, very unreadable, very painful. This is unfinished. Sean, any questions? Yeah?

1:27:30 I'm not sure that I fully understood that. Maybe you can tell me if I'm on the right track. You're trying to tell us that everything we understand qualitatively about probability theory can be understood as resulting from the fact that a half a half zero contains more information than a third third third. Essentially from that update rule. You can reconstruct the space of probabilities from this operational idea of a data. Is there a way of understanding in those terms why we should consider ourselves as having more knowledge in the case two-thirds, one-thirds, zero, than we have in the case half-half zero? What is the consequence of your update rule without having to resort straight back to Shannon? You want to see this in terms of the update rule, that this is above? Not just that half-half-zero is above third-third-third, but that two-thirds-one-third-zero is above half-half-zero. Well, two-thirds-one-third-zero... It's actually used because you use zeros you're already talking in the two-dimensional subspace so and this this this obviously is the only tricky point but we have a bunch of uniqueness theorems there for the one which i say if you will if you want to have mixing and this one has a bottom then this is comes up as the only candidate now To prove we do get a partial order, and all the rest I did is just starting from the abstract partial order, so I can reconstruct my whole probabilistic space up to this. So what I did here on the slide, I went very quickly over it, but it is just starting from this partial order, from this idea, and I obtain it back again, completely. I just have to do this. I just have to say which... Point on the line corresponds to each number in this ordinary line. These are quotients of the physical property on the scalpel. So I'll start from the logic. You'd say that all the two-thirds, one-third tells us is that it's on the line between half-half and zero-one. And it's further along than... It's closer to a pure state.

1:30:00 I was wondering if you could say a little bit more about this emergence of time. You're sort of saying that time emerges this somehow, but you're also talking about that you're updating information, that you're updating knowledge, you're getting new information. That already seems to be the most time-honored, isn't it? Oh, yeah. I would say that at least these update rules you obviously have to approach in a counterfactual way. It's kind of, it's like an a priori thing to develop. I did, I went through a lot of stages here. First, I started from probability theory, and then I kind of constructed this post-it and proved things about it that was actually a strong thing. Then, given this post-it, what can we do with it? So from that moment, I assumed that this thing is fundamental. It seems to be fundamental, I constructed it with significance, and mathematically it seems to be something very universal, which is very strong universal property. For example, it doesn't depend on the explicit update rule at all. It doesn't depend that I can change this map with this real normalization in a bunch of other maps and always end up with the same... You mean not the rule, but there's still an order, isn't there? Oh yeah, the order, it's... It's time order. Well, the order, that's the way... I want to read it as time order because of the second rule of Terminal X, for example, because I... And if you, for example... Oh, this was all for... Final sports, you could say, you could abstract this setting to model for space-time for studying statistical mechanics or thermodynamics or something like that, and obviously then you want to see the order definitely as a flow of time and emergence, for example, of disorder, whatever, but this is, you know, this is also from last spring, so I was hoping, we are working like on a more theoretic version of the Schrodinger equation. Now, we didn't finish, so I was hoping that I might have been able to present it here, but it's not so. Well, this is a question about the theorem that you stated just after you defined coordinates. I think you said that any distribution is the join of a set of coordinates. Yeah. There is a trivial theorem that any probability distribution is uniquely defined by its assignments to every event of the number that it gives it.

1:32:30 I mean that's just the statement of what the distribution is. It would look to worryingly to me as if your theorem was a statement. There is no doubt of that triviality because it looked like the drawing of distributions ought to be the minimally informative augmentation of each of those distributions and that the minimally informative augmentation of a group of coordinates which are just lambda and one minus lambda for various events You see, it seems obvious that the distribution that's the minimally informative augmentation of a collection of coordinates should be precisely the distribution that assigns those probabilities lambda to each of the events in the so-called sample space. Actually what you do is obviously, you go from a state which has n numbers to a state which actually has n-1 quotients. It's all consecutive quotients. And these quotients, which are n-1 numbers, are actually these coordinates. So, this actually is indeed, like you say, a version of an event here and an event there. The difference here is purely all the theoretical structure, which kind of brings it back to effect, without actually starting with probability theory. When I do this, I don't have probability theory. It's a partial rule. So, obviously, that's why it's sad because proof, starting from probability theory, doesn't make solutions. So you recover that fact actually in all theoretic terms. Yeah, just one last question, please. One question, short question. When you define the update of the partial order, you say update of new information preserves the... Because that's the order relation, right? Isn't it possible to skip the word new in the definition, because if you update information, and that would be information that the other already would have had, they would become equal, but they would still have a partial order. Okay, the new one can coincide with the old one, obviously. Yeah, just one person tells the other what they know. For example. They can obviously come together, but they'll not never... This is actually something which happens with the... If you would verify this rule on the majorization of the data, they constantly swap, just to give a comparison.

1:35:00 Okay, I think we'll finish the session then. Thank you.