Derivation of the quantum formalism form a perspectival principle

0:00 I was only able to explain how the addition of velocities becomes much more understandable to a velocity if we change the distance with time, and if time is not absolutely certain, the addition of the classical addition of velocities should be considered effective. But the main point is that starting from this principle and the experimental facts, together with the fact that it's local, so it's a linear transformation. He was able to actually derive the Lorentz transformations. So he actually derived the formalism from this principle. So what I propose to do in this talk is to try something similar to quantum mechanics, to start from a principle and some experimental facts and derive the formalism of quantum, of elementary quantum mechanics. By this I mean that the observable Symmetries should be given by unitary operators and unitary operators. And finally, that the dynamics is given by the unitary operators. So we want to derive these things as general principles. So let me get started right away on this. So I'm going to start with a totally elementary idea that makes sense.

2:30 An experiment, and it doesn't matter whether it be an experiment, it has some elementary outcomes, so we have these experiments, and they have elementary outcomes, this could be a detection screen for a Stern-Gilak experiment, it could be a classical experiment, okay, so here we have this experiment, there are elementary outcomes, and the point of making spots on the screen. I'm going to use the language of classical probability theory. We form a sample space to see all the elementary outcomes, but in general we want to be able to say not just elementary events, but suppose I want to say either S1 or S2 is the case, or maybe S3 is not the case. So what we do is we form key events. These are compound events which we can think of like S1 or S2 and so on, and they're simply represented by subsets of the elementary outcomes of the sample space, and this forms a very simple structure, an algebra of subsets, that all these subsets of intersection and complement are responding to a negation, and I need to talk about this slightly more, as that was already done by Boole in the 1840s. We need only one more axiom.

5:00 We say that every element, that is to say, A times A, which is A into section A, into itself, every element is highly potent, and then this completely captures the original idea because there's a classical theorem from the 1930s by Stone which says that every Boolean ring in algebra is in fact isomorphic. It's the same thing as an algebra because there's simply a Boolean ring. In order to deal with the possibility of an infinite number of elementary events, an infinite boolean sigma algebra, which will mean that boolean sigma algebra simply is obtained as follows. We start with boolean algebra, and it also forms a partial order set by saying that A is less than or equal to B. If A times B is A in our model, A is the section B equals A. And if we assume that every countable infinite set is v, that the greatest lower bound is this, which is called sigma. And that's the basis of probability theory, by the way. And there's a theory in the 1940s by Loomis, who gave a detail which also said that every sigma can be realized. It's a bit more complicated. You have to take off small sets of measures here. And given such a Boolean algebra, we can talk about truth values, namely in our original model if we say a particular outcome S1 obtains, then any set which contains S1 is true, all the other ones are false, and if we go to our abstract Boolean algebra, that turns out to be correct. There may be reasons why we don't have enough information. It doesn't matter whether from the point of view of useful mathematics, whether it's epistemological, we don't know enough, or because...

7:30 The universe doesn't know its future, but in general we have to have a probability assignment rather than a truth value assignment, and Kolmogorov in the 30s was able to define the idea of a probability by essentially one axiom. He says that if you have a sequence of elements from the sigma l equal to b, which are disjoint in the sense that beta is a intersection and the theta product is zero, i-hat equal to j, then the probability of the union, or i equal to sum, is the sum of the probabilities. Essentially that's the only axiom together with these two obvious ones. And that's the whole basis of modern probability theory, is the sigma l to work together with such a probability function. So that's classical, but it's also quantum mechanical. There's nothing different between this and the classical. The same homework will also work on mechanical experiments. I want to discuss a particular experiment which I'll use in a moment as an example. So let's, we could take a Stern-Gerlach experiment, say, for a spin-1 system, for example, if we pass, but I want to do a slight variation on it, I'll do a type of Stern-Gerlach experiment, an s equals 1 system, in which I'll only measure whether the spin is zero or not, or another way to say it is I'm measuring the square of the spin, and the reason I want to do this, to be using this, is that in the case of the square of the spin, It turns out the square of a spin for a spin in one system in two perpendicular directions, and I measure the square of the spin, they commute as operated in Hilbert space, and that means when I do an experiment, I can do an experiment in which I can measure one without disturbing the other, I can measure either simultaneously or one after the other in two and then actually in three directions a lot as long as they're perpendicular. And you can actually do such an experiment.

10:00 For one experiment, you can't do the experiment by measuring the spin and squaring it, because if you measure the spin, then it will disturb measuring the spin in a particular direction. If I directly measure, by what I'll call a minimal measurement, the square of the spin, and that was done about five years later by a physicist called Grader in 1927. He used an electric instead of an magnetic field. An electric field is given by a polar vector, so it doesn't distinguish between the plus and minus of the metric in that sense, and he actually was able to show that you get, in this case, a spin-1 system just that breaks up into two parts, s equals zero and s squared equals zero and s squared equals one. The sigma algebra that you get if you measure all three of them is a very simple one, just consists of eight elements, and then you measure these three things, their complements, and zero and one, and that's already a form of algebra. Now, so this is just an eight element theory and algebra. I want to mention now that although Raider did this experiment with an electric field, there are more modern experiments using interferometry, which I'll be coming back to, because I'll be using that with experiments. In which we directly, we can measure the spin and then bring together with, let's say, with certain crystals, we can, by chromatography, put together and measure the square of the spin by simply coherently putting together these two beads. What happens is, instead of measuring a single experiment, doing a single experiment, I now move over to two experiments, several experiments, and I do another one.

12:30 For instance, in this case, I rotate my three axes to a new axis, and I do a new experiment. Well, in that case, I'll simply get a new sigma algebra with x prime, y prime, z prime here, and, in general, with only zero or one. And in that case, I'll have a new sigma algebra, a new probability function, but I can also do something else that's a little bit more complicated. I can simply rotate it around a fixed axis, and in this case, I'll have a 4-album Boolean algebra by fixing this, let's say, with the axis. I can keep rotating in this way. I'll get a series of such Boolean algebras, 3, 1, b, 2, and so on, which either have 0, 1 in common or the 4-albums. And this already forms a fairly complicated scheme of room in algebra. Here we come to a major difference, namely in classical physics, the basic assumption is that when I do an experiment in algebra, what I'm measuring is certain aspects of the underlying that the system itself has, and all the properties together... All of these can be put together into a system of Boolean algebra or sigma algebra of the properties of the system itself, and therefore that each one of these, this whole union, actually embeds into, in more detail, what we do is we set up in that classical system a space omega, say for a gas it's r6 to the n, but where did you have constraints, say for a solid, will be generalized in certain manifolds and mathematical terms, and the algebra... Part of these is of this space-space. It's not arbitrary subsets, you don't want non-measurable subsets, but what you do is you simply take the open subsets, the simplest ones, the intervals and so on, and you form the signal to regenerate it, which is, they call it the L-algebra, or measurable, the algebra of measurable sets, so it's the simplest algebra that's formed by the open sets. And, in general then,

15:00 This algebra, the union of the original PIs, which I guess the measurements are imbeddable in this particular signal. What we find, in fact, is that in the, so this is the case for classical physics. If we go to our experiment on the measurement of squares or spins, these three directions, we find that We can't embed it into a single boolean algebra, and that's even true for a very small number of them. There exist 40 such friends, x, y, z, different friends based on 32 directions, such that the union of those 40 is itself not embeddable into a single boolean algebra. That's a mold result, basically, that Specker and I proved more than five years ago, and the reason for it is as follows, that if there were... Such an embedding into a single Boolean algebra. A Boolean algebra has many, many homomorphisms such as zero-one, or homomorphism zero-one, or two-thousand assignments to be, but given any such, I can find it through this intuitive embedding to then give a map of this family, zero-one, and that's what the old result shows in the government system. So there's a basic difference in this, at least for these particular properties of the quantum mechanical system with the classical ones. What it says is that you can't make this naive assumption that properties exist intrinsically in this system. The properties of squares and spins simply are not there as intrinsic properties of this system. They're expectant. Terminology, they are extrinsic. They are relational properties. In this case, we have the structure of the job of the right. Rocky is saying that the properties are not intrinsic, they are relational. So we want to make use of a very similar idea as the case of relativity, that time and therefore also length and mass are not intrinsic properties so much, but they can be functions in that case.

17:30 Here too, properties are not considered to be intrinsic but are related, so we'll make use of this relaxation of this practical term. So that's what I want to use. Incidentally, I use the word here, relational or perspectival, not in the sense that it was used perhaps here in this conference. And that several, a number of people in the 80s, including myself, I had this idea in 1978 and then in 85 of having a perspectival thing which is much stronger than what I'm using here. There, the idea was that you actually go beyond quantum mechanics, use the polar decomposition or the phyto-partinal composition of two interacting systems to give states To give states persistence at every instance beyond what's done in the papyrile or modal interpretation, and that was to deal with the measurement problem. I'm not concerned with that question here. I'm in a certain sense finessing that question and measurement because I already have reduction in these experiments. All I'm doing is, the idea of relational theory is simply that The very modest part of ordinary quantum mechanics we use, and that is that if I haven't had an equal here, an interaction or a measurement, I can't say anything at all about it. I don't go beyond ordinary quantum mechanics. So that's what I mean here by string theory. So we can't use a Boolean algebra anymore as our basic structure. What is it we can use in its place? What is the structure of its properties? In a certain sense, it's obvious. It's what I've already said. You replace the Boolean algebra by the union of these Boolean algebras or sum algebras that occur in each experiment. So you simply take the union of the two.

20:00 We'll use that as our basic structure for our quantum mechanical reading structure. So that's what we'll have. The properties forms that you're using in the boolean algebras. Of course, the one we'll use, and you won't forget to, is what we'll call the standard one, is the one that occurs By starting with a complex Hilbert space and then applying and then using the projection operators or the closed linear substrates to be the properties of different systems to which we associate a complex Hilbert space in which we define the basic operation of product line sections that only relate to mu and also the complementarity. So that's the structure we have for Hilbert space. Boolean algebras consist simply of taking a bunch of commuting operators, looking at the sigma algebra they generate inside the complex Hilbert space. And that's an example of our B.I. It's not even the case that the two elements, I can form that conjunction. I can only do that when they sit in the same boolean algebra. It's much weaker, I'll come back to, than something like a lattice or an algebra operator. So let's start going to get a very weak structure, and then we'll be able to reconstruct the general formulation.

22:30 The way I'm going to do this is... I can do this for this week the hard way or the easy way. So I'll start with the easy way to show you how to experimentally get the structure from a large number of experiments and then indicate how to actually get down to a very small number of experiments that's still successful. So I'm going to use the idea of interparometry here, and the particular thing that I mentioned earlier by the red tiling, and we can use the interparometry on cohomology, manifolds, and mathematics, but the logic could have used the logic... One number of experiments, successful experiments have been done on this, and the basic structure here will be to use the spectrometer. I'm going to act like I'm working with photons, but it works just as well with other particles like protons and atoms. And there, in this case, will be a mass center spectrometer. It consists of two mirrors. And who leaps with it? Articles of Millers. And who shapes it? So this is the basic structure that's used in this paper by Siler, Yerdao, together with Yuyi and Sintiak. And it's been known now for a long time that we can essentially deal with any unitary operator and we can measure any two-dimensional... The basic idea, I'll go rather quickly here, is that we use these two forms, thermal forms, to reconstruct, if I start from, for instance, 1, 0, then I'll end up... From here, I'll end up with the vector dpi by time on the cosometer, which is 0.1, and in general, for a vector, the initial vector, k1, k2, I'll get an output vector, which is the output of this unitary operator, k1, k2, and that will be x. We'll be getting one of these strips. So that's the basic idea. We sometimes call the pre-measurement operator and the result of the passive detector is to measure any two-dimensional thing in an operator.

25:00 And now, what Seibinger and so on do is give us a very simple theorem from... Mathematics about unitary operators in higher dimensional spaces, namely every unitary operator in n-dimensional space can be written as a product of two-dimensional operators. So I'll write this thing in this very simple form, where ucw is this unitary operator, and u would be one, only the one principle. So, this is an operator, an n-dimensional space, which just acts on a two-dimensional subspace, leaves the rest of it alone, and the way to take it mathematically is to start with the identity operator, and in the The way you do it is you can start with a momentary unitary. And by applying the appropriate one of these unitaries to it, we can make one of these elements three rows, and we successfully make everything three rows down to the left. There's nothing of a diagonal unitary. Therefore, by applying the unilateral operation, we get a diagonal unitary. The important thing is that we can realize the product of these unitaries by taking the output of one of them and putting it into the other one. So for instance, let's look at the case of a three-dimensional unitary. We can actually realize any three-dimensional unitary by means of stars. We choose two, and therefore three of these three dimensions are one, using this sequence of input and outputs.

27:30 So this is the three-dimensional case. You give the diagonal, you call it the H-string. This is F. This is the H-string. Therefore, the three-dimensional case. Now, I want to mention one other thing about this that we have to add to this. Remember, we wanted to make sure this will actually, as it stands, give you a multi-generous, three-dimensional operation operator. We can also get degenerate ones by just ignoring the fact that they're different, but we want, if you remember, to have minimal measurements. And you get this by simply using extra mirrors, or in the case of Newton's crystal, to bring together, as I mentioned earlier today, to bring together a complete suite, where we have a generator, and we can operate under the command of Newton's crystal, or Newton's other ones. So to give you an example, the actual... If we look out for that x squared, x squared will correspond to the, comes out there as the following three items that correspond to x squared and y squared, so we, so these three, so these three, from, so this is the unit error operator I was talking about. I break it up in this way, and I end up with a box here, which I get these three outcomes, but I put these two together upwards, and what I'm doing is measuring everything squared, if I put together...