Spin Networks & Topological Quantum Computing (contd.) / Galilean Theory & SR

0:00 A state separated geometrically would be related to some kind of topological connection between them. It just means, you know, it's very crazy. I feel one knows what a tangles tree is, what a tangles scale, that is. Are we just using entangled, in that case, in two really different ways? They're not really the same. I know you feel they ought to be. I appreciate I feel they ought to be as well. There's a strong hint about what's missing as soon as you look at this. and I put that back just because it was on the previous slide, to ignore the rest of the slide, because we can do the mathematics of calculating things about entanglement without having the least idea about the spatial relationship between these two observers, the left and the right observers. So we proceeded to ignore the entire spatial context. So that tells us something. It tells us that quantum mechanics is incomplete in that sense. I mean, you know, the way you're using it is incomplete. You're not taking into account how the information got carried out to those two places and all sorts of things. So if you took it back into account, then at least the geometry ought to be... And in any case, there might be quantum mechanical experiments which weren't spatial but still produced the same formalism. Yeah. I have no idea what they would be. And there are experiments where things are close together, but it's basically the same phenomenon like a double-slit experiment. Well, anyway, let me continue on here. Permuting my slides. As long as you don't start braiding. Braiding slides is not so easy. I don't know how to do it. So here are some solutions to the An-Baxter equation for you to look at, the ones that I happen to be interested in here. This matrix, if you square it, is equal to the identity.

2:30 But that little minus one, that little extra phase in there is sufficient to make it entangled. So this is an example of what we started to suggest, that we might have a topological gate, square equal to the identity this is just swapping over here and it's got a little phase and it's enough you can use that instead of CNOT that's very close to being more or less like CNOT it's just a little extra phase is all it needs in order to generate enough unitary transformations you're far from generating enough unitary transformations if you're not putting a little minus sign this one's not interesting to me as a topologist because of the fact that it squares to the This one, on the other hand, if you check it out, we're not square to the identity usually. And it is entangling whenever a squared is not equal to b squared. These are unit complex numbers in there. And it satisfies the braiding relationship. And if you take it over into topology and analyze it, you find out that it can detect the linking numbers of links, things like that. So it's an okay gadget. It's an amphibian. It lives in quantum computing and it lives in. When you say detect, you mean it separates out, it projects it in some way? Separates out. Detects. Oh, when I say detects. It has to do with something I'm not going to tell you, but I'll tell you. Suppose I have a curve here and another one which is going around it like this a number of times, right? Let's draw it. So those are both closed curves and one goes around the other one? if I give them some orientation so that I could count things then I would you and I would both agree that this one goes around that one twice and that's its linking and there's a there's a little algorithm which I wanted to hide which which lets me compute linking numbers by using this matrix you could imagine in the case of two strands that just the powers of it are going to be keeping

5:00 You can track of how many times one curve winds around the other, and that's basically what current is. So that one detects linking. And this one is not one that you would normally think of if you were a topologist, unless you had the right context. But it's one that everybody writes down in quantum bets. It's called the Bell basis change matrix, because you see, you can think of it as a change of basis from the standard basis to various EPR type states. And this represents 0, 0, minus 1, 1. And this represents 0, 1, plus 1, 0. And this represents minus 0, 1, plus 1, 0. And this represents 0, 0, plus 1, 1. And that's a basis. So that's a basis of entanglement states. This matrix is very entangling, so obviously it's good for the Berlinski theorem. But it also is a solution to the Yang-Bachs equation. And, in fact, it can detect subtle things. It can detect, for example, that these rings are linked using the algorithm that I'm hiding. It can detect that these rings are linked without any linking numbers happening because, you see, there are no linking numbers there. So this has some subtlety. So there are subtle relationships between the topology and the quantum theory entanglements, know how to make a theory out of that. I like Ted's question, but I don't know how to answer it. Anyway, these are some examples. And then you might say, well, okay, fine. Why don't you use any one of those? And you've made the computing or the quantum processes topological. Well, almost, but not quite. Because you see, you could still have unitary transformations on the lines. I mean, what does the quantum process look like? You have these little black boxes, operating operations, right, they are little black boxes. We can't go inside and make them into some continued, well maybe we could, but that would be putting in more structure. But each one in this picture is a little topological black box transformation, one of these matrices. And then there will be some unitary transformations that are formed on single qubit lines as well. And if you allow both of those, then you can write a general unitary transformation in terms of that. But those little green dots don't interact in any particular way with the black boxes.

7:30 So I get an extension of the braid group, which is very obscure. I get an extension of the braid group by the quaternions, if you like. But it's obscure and not patently topologically. So one should continue thinking. right, haven't solved the problem by doing this. It's just partly solved. And it leads, and of course it leads back to the question, well, how topological can you make the elements a view of two, right? Did you have a question? Well, yeah, I'm just going to question you. I'm seeing you just saying that they satisfy the same relationship like this. Yeah, how do they? What was on the face? However, you could come to, like, this is post-time press, so they satisfy one great relation. They don't satisfy those elements which are removed from each other. No, they satisfy that too. What happened to Marlon? So they satisfy only one great relation. Yeah, let me clarify that. Let me clarify that. You see, the R that we're talking about is this little R here. It operates on two strands, right? But then consider the following, that I could have an R. And then I'm doing four strands. And then on these two strands, I could do an R. This is just going down fairly. two commute with one another, don't they? These two operations. I only operated on the last two tensor lines here, and on the first two tensor lines here, and then it was R in each case, but it didn't matter in which order I did it. Okay, so you say you could extend it to the... Yeah, the intent was to extend it to any number of lines like that. So the R will occur... In general, the R occurs like this. There's some lines going on, and then suddenly we decide for a pair of adjacent lines to do an R and then the lines continue on for a while like that. And that's what I mean by using it. I'm allowed, even for CNOT gate,

10:00 when I said CNOT is general, I mean the same thing, that I can choose a pair of adjacent tensor lines and do CNOT on them and then I get to do unitary transformations on single lines. That's the picture of the quantum computer from the usual point. It's still confusing us on that point. No, but I'm just thinking about the grade group. Well, this corresponds to one of the generators of the grade group, right? Yeah, actually the identity n corresponds to one strand. Doesn't this picture tell all for you, right? But if I put the bit of algebra in here like this, that is the representation of this bit of braiding. The other bits are braiding just here. Yeah, this is one generator in the braiding, right? So if I was looking at somewhere there is the usual braiding relationship perhaps happening like this, right, then that could happen in, you know, in the context of a lot of lines, and then it's just R, R. Yeah, what I'm saying, I will try last time, if it doesn't come across, that it's in the interpretation where you take tensor product of this kind of identity. Yeah. This tensor product of those two items represents what's done in a way. Because if you consider... Yeah, well, when I wrote, maybe this is all you're talking about. If I write, let me translate something from algebra to pictures and back, okay? tensor I, tensor R, tensor I, right? The I is one-dimensional, or acts on a single strand. This I acts on a single strand. The R acts on two strands, and the other I acts on one strand. So that's the translation between the algebra and the representation, right? When I wrote an I, I meant it to act on a single strand. I'm still not answering your question.

12:30 but doesn't that mean that in a way the eyes are different because the strands are not the same strands well the eyes are good this is one of the usual argument about indistinguishables I mean the eyes involved are i tensor 1 tensor 1 tensor 1 tensor 1 and the next i is 1 tensor i tensor 1 tensor 1 and so on right there are a whole bunch of different i's each one distinguished by its place I think that's all I meant was that in a way you should distinguish the i's in your notation because they actually refer to different places. But that's like saying that I should distinguish 231 by saying 2 sub 3, 3 sub 1, 1 sub 2, 1 sub 1, right? And I could, but I don't. But you don't want to. I understand that. Yeah. So the next theme here that I thought would be good to take up What can we do with SU2, right? Could we make braiding in SU2? And then we'll see that we can, we've got an interesting story there. So let's remind ourselves about SU2 a little bit. SU2 is related to physics, and topology is related to SU2. What I've drawn here is something that we have probably all done at one time or another, where you have a 720-degree twist on a belt, and here's a sphere hanging in mid-air, and it won't rotate. It's just fixed in mid-air, and we're allowed to move the belt around. So we relax it a little bit, and we take the belt down underneath the sphere, front and back underneath, like that. And you see that this twist turned into the reverse twist, and now it cancels the remaining twist, and so 720 degrees went away. What will happen with 360 degrees is the same thing, except that you won't see this top part, and you will have turned a plus twist into a minus twist, but you haven't gotten rid of it.

15:00 So this is a situation where a 360 degree twist is not the identity, but 720 is, and it corresponds formally to the fact that if you look at the wave function for a fermion, and then you walk around the fermion and come back to where you started, So it changes the sign, and topologically, or group theoretically, it corresponds to the fact that S3, SU2, the unit of platernions, double covers the rotation. If you go around in a rotation and lift to an element in platernions, then you will, if you go fully around platernions, 60 degrees, you'll get the negative of the element. So this is an example of a topological relationship of entanglement in the topological sense and entanglement in the physical sense where the entanglement here is not exactly the same as the other entanglement, but it is the entanglement of an observer with what he observes mechanically. And we understand that it's coming from the necessity of doing things in unitary groups. Why do you use superfix 3? Oh, because the SU2 is actually the three-dimensional sphere. It's equivalent to the set of vectors in four space at unit distance from the origin. I'll remind you why with the next slide, because the thing I want to calculate here, I want to do a little exercise, which turns out to be very interesting. I'll just put this side up. The exercise, in order to do the exercise, I have to remind you about how pleturnians work, or how elements of SU3 work. There's the information we were just talking about anyway. There's less superfix . There we go. I'm eliminating a few things. There we go. So here's quaternions. Here's SU2. I just wanted to compare some language. So an element in SU2 is pairs are complex numbers. It's going like this. So the determinant is the absolute square of z plus the absolute square of w, right?

17:30 And if you wrote it out, a plus di and c plus di, and you multiply the right-hand one like j, and you get a corresponding determinant. And this is another way of saying polynotices. It could correspond the element of su2 to a determinant. And if the determinant of this is equal to 1, then it says that the sum of the squares of A, C, and D is equal to 1, and that puts it as a unit vector in force-based cell to 3-sphere. Oh, I see. Okay. I'm going to work with the quaternions, but anything I do with the quaternions is, by this translation, something you could just as well say for SU2. And quaternions multiply like this. The quaternion can be written as A plus, pardon the B, because it's not that big. a plus b times the unit vector in three spaces by jk. Okay, we just write it that way. And then you have a squared plus b squared equals one. So in other words, a typical element in there is of the form e to the u theta. Okay. u squared is minus one. So that's a typical element in su too. Yeah, S super three was just a shorthand for the three-sphere. Yeah, I'm sorry. It's just something that I always think, but it isn't necessarily relevant to the rest of the discussion. You know, like in math grad school, people often say to people, if they're topology students, what's your favorite sphere? And they're expecting to answer things like, Well, my favorite sphere is the 3-sphere because it's illegal. Okay, so a quaternion is of the form A plus BU, and then the pure quaternions, the ones that are just elements of S, are 3. S2 is the 2-sphere you see. Those guys are elements of the 2-sphere, right? They're unit directions in 2-sphere. The product is minus the dot product plus the cross product. Okay? So then you know everything about the quaternions from that. And then the way you represent the rotations is as follows. If you start with a quaternion in the form of 8 plus VU,

20:00 like that, G, this is what G is. Then you conjugate by, you multiply G times the vector times G inverse. And that rotates the vector v around the axis u by theta if this is cosine theta over 2. That's the famous formula of Hamilton or Drew Gritz. So here's what it looks like explicitly. You multiply it out. Think about what that did. That's what it does. find out how rotations compose, you can just multiply the quaternions. So this is how the quaternions work. And why am I worrying you about this? Well, I want to solve the braid relation. So let's do this exercise and see if we can. I'm not going to eliminate a couple of bits of calculation. I want to solve the braid relation. So what does that mean? It I want one quaternion G to represent one braid, and the other quaternion is going to represent the other braid, where these are three strand braids, I'm thinking, right? So they're, they're neighboring braids. And I want the braiding relation to be satisfied. And the braiding relation is G H G is H G H, right? That's what it was. All right, G, and H next to it, and G is the other order. Now, that's exactly the same as saying H inverse GH is GH G inverse. What have I done? I have multiplied on the left by H to get H inverse over here, and I've multiplied on the right by G. This isn't how I did this the first time. The story of how I did this the first time is I just wrote this all out So I had three rather messy looking equations, and then I gave it to Mathematica. And then Mathematica, and then I punched the button, you know, it says solve. I punched the button, and instantly Mathematica gave me the answer. So I thought that's pretty interesting. Mathematica is very similar. There must be a reason why it gave me the answer so fast, so I went back and thought about it a little bit.

22:30 Okay, so what does this say? It says H inverse GH is GHG inverse. It says that rotating G via H inverse is somehow the same as rotating H via G, right? That's what it says. And in fact, in terms of the axes of G and H, that's exactly what it says, because these are scalars, and they don't actually have anything to do with these equations. So what it actually sees, if you take GHG inverse, the A remains the same. And if you take H inverse GH, the C remains the same. It's just this thing that's operated on. And I wrote U upper G for GH, G U, G inverse. Okay, U operated on by G. V operated on by H inverse. So it says that U operated on by G is equal to V operated on by H inverse. In other words, you have the two axes, and the angles are going to be the same because of this. Okay, so it says there's one angle of rotation. and if you take one of them and rotate it by angle theta around the other one it should be the same as taking the other one and rotating it by angle minus theta around the first one so that's our gradient relation it's actually a kind of nice geometric relation that you'd like to be happening among these rotations and then you do a little more algebra and you find out that that tells you that the dot product of the two axes has to be cosine of two theta divided by two signs where theta and theta are cosines in science. So that means there are lots and lots of representations of the three-strand gray group into SU2. Lots of them. And if you make these angles not very, well, not very, if you just keep these angles away from the orders of the finite subgroups of SU2, then these representations are going to be dense. So you get lots and lots of dense representations of the braid in SU2. So this is another piece of the puzzle. It says that if you just restricted your attention to SU2, you could generate it by braiding. So those little dots in my previous picture could be braiding, but then is the braiding that you do there going to be related to the braiding you did in the large at the level of the two-strand, at the level of the two-cubit gates. Not obviously. But this is definitely another piece of the puzzle, okay?

25:00 And so we're probably running out of time here. Have we? Long or go run out of time? Almost run out of time. So I'll tell you a little more and you'll see what happens. But this is, as long as we went this far in the quaternium thing, Here's an example of what I was doing. I could have written one of them as just a complex number, so it's just a rotation related to i itself. And then I can take h in this form. So this is just trigonometry for the cosine of the choice of angle. And then I need some relationship between this and that, that's a dot product relationship. Then in SU2, getting back to SU2, the first one is just this, simple complex number rotation. And the other one, if you play it out, is the conjugate of the first one by a little unitary F here, like that. So the two, so you can make lots of examples of this type where one of them, it's in some sense the generic example, one of them is just phases and the conjugate phase on the diagonal, and then the other one is a conjugate of that by some little unitary matrix, and those two, those two, H and G, are braiding one another. Well, that's a picture of how to get braiding in S or two. So now where are we in this problem? The braiding operators are still arbitrary elements of... Oh, I'm sorry. This is prior to having written that slide. Prior to having done what we just did, the braiding operators still need arbitrary elements of U of two. But we now know that the elements of U of 2 could be produced by braiding, but may be of a different sort than the braiding that I started with up here at the two-cubic gate level, right? And then I want to show you how this kind of fits in with suggestions which weren't coming directly from this kind of elementary thinking by Friedman, Kataev, Larsen, and Wren, who suggested using topological quantum field theory to make models that would be coherent across the board, okay?

27:30 And topological quantum field theory is a long story. It started with Schwartz, who figured out how to use functional intervals to represent some topological invariance of three manifolds, and then Witten and Attia and other people started doing it in the late 80s, and it became a popular subject. But I'm going to jump, since we don't have a lot of time, to one of the answers, and then you'll see what the pattern of this is and like topological quantum field theory is useful. So this is a re-expression of a model due to Ketaya. And I have a couple of levels of re-expression of this model. One of them, in a moment, in a little while, in a few minutes, we'll show you how it's related to spin networks. But the model itself is very simple and amounts to making some quantum patterns out of a particle theory, which is about as simple a particle theory as you can imagine. It has one particle, really, and one empty particle. Maybe you want to think of those as two particles. That is one particle and two possibilities for charged and uncharged, marked and unmarked. And I'm using Spencer Brown's mark with malice aforethought because it's really very close to his formalism in some sense. So this particle can interact with itself to either produce itself or not. And that looks like you aren't going to get very much, but in fact you start getting some interesting patterns out of this. For example, you can think of spaces of multiple interactions. So suppose that I have three of them, and I ask them to interact in such a way that I get nothing. Then these two could interact to produce something, but then these two have to interact to produce nothing. If these two interacted to produce nothing, nothing interacts with something to produce something, so it wouldn't work. So this space of interactions, possibilities, in here is one dimensional. Next up. Now I have four and I want to go from four to nothing in the order of these interactions that I've indicated with the tree. So the first two could interact to produce nothing, then nothing interacts with something to produce something,

30:00 and something interacts with something to produce nothing. That's okay, that's possible. And then the other possibility is something, something, something, something, something, something, and then nothing. But the last one has to be something in order to get something. So the dimension of this space of processes is two. Let's see. How much time would you like to have? How much time would I like to have? Well, there's a few times right now. Quite right. But I'm sure that people want to go on here. Yeah, yeah, I know. How about 10 more minutes and call it quits? Would that work? I don't want to kill you. But I do want to show you the things that are interesting. So, sorry. Here's the next one. Five with five. And here are the possible processes. And the dimension is three, and in general, the dimension of one of these process spaces is a Fibonacci number. The reason it's a Fibonacci number is because, you see, you can't have two unmarked processes in a row. It's not possible. And so if you were to make a diagram of such processes like this, I mean just the strings. You see, I'm not allowed to put two stars in a row, and otherwise I'm constrained. You will generate a Fibonacci number of things. So these spaces have Fibonacci dimensions. The reason I brought in laws of form, just quickly, is because, you see, in laws of form, there's one logical particle, the mark, and two forms of self-interaction. The mark could make a distinction on one side of itself or the other. In that case, this one becomes one and this one becomes nothing. So it's exactly the laws of form interaction. And so the epistemology of this little particle theory fits laws of form. Laws of form, in Spencer Brown's hand, never got quantized. He didn't think of a probabilistic situation where the two could go one way or the other, but that's what we're doing. In his case, he thinks of this as an underpinning for patterns of logic.

32:30 So the next thing that we're going to do is we're going to think about changing the form of one of these process spaces. You see, suppose that I'm looking at a process space like this, and I decide that I'm interested actually in this process space. I can think of processes that start here and end up in an unmarked state, or I can think of processes here. And these are well-defined processes, it just means that these two interact, and then these two interact, and then they interact with each other and come out, right? And I'll get a basis for this space. There should be a transformation from this basis to this basis, and I hope to have one that's so beautifully coherent that everything will work very well. Now, I haven't told you what rules the coherency should be, but at least it ought to be linear, otherwise we're very far away from this sort of thing. So now I'm going to jump to the answer. If you change bases in this way, So this is telling you that if it had been in the middle of the interactions, it looked like this. You can rewrite it like this, okay, or vice versa here. These are the correct coefficients to take if you want this thing to really work well. Tau is the inverse of the golden ratio, and this is a little unitary matrix. And then, why am I doing this? I know this is on one of my slides but it's probably easier to just show you there's going to be a braiding in a moment you'll see there's going to be a formula for a braiding like that that's like the two anions right near one another going around one another it's just going to multiply by a complex number but suppose that you wanted to know about abrading between two things that are farther apart in the process space like that. Should there be abrading? Well, there could be because you see, it could be that you have this recoupling change of basis which takes you back over to something like this. You see, that's between this

35:00 and this. That's that recoupling matrix. And then there's a known multiplication by some root of unity, which takes you back to this, and then you could go by the inverse of the recoupling transformation back here. So this looks complicated, but this is actually a way to get braiding representations. It really works. If you analyze how to make such things, you can create braidings and recouplings that interact in the right way so that you get global braiding. Global braiding is what we were looking for. and this actually works in this case that I'm showing you if you use these roots of unity which are related to the golden ratio again so this is what this particular model looks like there's a basic three strand rate group representation given by this R and the F on the other thing it works exactly like the one I was showing you in general And it generalizes to grading on arbitrary kinds of strands by means of this recoupling theory. So then the question is, how do you know that such a recoupling theory is going to be consistent and that it will work? And one way to do this is to verify a whole slew of identities that such things have to satisfy if they're going to work, which I won't bother you with because we're out of time. But another way is to build the recoupling theory in a topological way to begin with. And it turns out that one way to do that is to use a generalization of Penrose's original spin mark theory. So I'm going to give you just a quick glimpse of that and we'll stop. So, how quick a glimpse can I give you a Pemeris to spin that? Well, I'm eliminating everything, but Pemeris, Pemeris had this beautiful idea a long time ago

37:30 of taking some of the basic information about the representation of SU2 and turning it into a theory about how to form combinatorial networks. There are rules for counting loops and things in these networks, which end up corresponding exactly to spin angular momentum recoupling in standard physics. And some of gadgetry in his network theory looks like this, that when you have some lines, they add up like this. And there are generalizations of adding up overall permutations that look like this. And that's what this means. Like here, I've added up the two lines and subtracted off the switch and divided by two factorial. that's a, that's a projector in this theory. And then there are the vertices in the networks and they are combinations of projectors. Corresponds to the representation theory of SU2. But you can take this as the basis of your network theory and then he, and then he used it to prove a theorem which looks like this, that evaluations of these networks, according to some combinatorial rule, can be used to define angles between the three ends if the ends have large spins. So the network appears like a kind of gadget with spinning particles sticking out of it, and they have angles between them. And the angles fit into three-dimensional space. So that's a spin network theory. The remarkable thing about this theory is that it can be generalized, and then it's coming straight out of topology. It gets generalized in this instead of that funny identity that I showed you for a moment of hemorrhosis, you make it into a braided identity like this. So there's a braid. It's not permutations anymore. And it gets expanded into lines like this. And this turns out to calculate a knot in there. And the A is the matrix, is it? Excuse me? The A? The A is an indeterminate bound. Oh, right. It's just an but it could be you think there's some complex number, whereas in Penrose, it was minus one.

40:00 And in a certain sense, you can regard this as a starting point that happens to fit into SU2. I mean, from topology, this is a combinatorial topology, it's a natural starting point. So it tells the story in the Penrose kind of way that there ought to be a way to start that doesn't assume the three space, and this isn't assuming it. topological ideas about three space. Anyway, you then build the network theory in analogy to the way Penrose built it, but it's got braiding in it. And because it's got braiding in it, it means that when you have a vertex and you have a little twist, there's braiding happening. And this model that I just showed you is actually one of the simplest possible starting models in that period. So So rather than do the details, let's just draw you a picture of the palm brush. Skip. So where are the Fibonacci anions? Here they are in this deformed version of the Penrose theory. A deformed version of the Penrose theory, what corresponded to taking two lines minus their switch way. In the Penrose theory, it does two with delta equal to minus two, so it's just a topological deformation of that. This is our mark, our particle. Two particles can interact, you see there, iconic, sorry. The two particles can interact by sharing lines and ending up with a copy of the particle, or they can interact in such a way that no lines are shared and nothing happened in the output. And then there's one other thing that could happen, but I don't to allow 2 plus 2 to be equal to 4. It's forbidden. And the forbidding of 2 plus 2 equal to 4 is what fixes it into the Fibonacci numbers and the Goldman ratio. It fixes it into those roots of unity. And then you go, by that kind of focusing, you can, oh, it's built topologically to begin with. So the fact that those things give you a great group representation is a tautology in this context. You don't have to verify a whole And then you can do the calculations and see that you get the matrices that I was showing you. So this is a topological approach to this form of topological computation.

42:30 But what it's really saying is that it's actually possible to coherently represent the braid group into the unitary groups in such a way that quantum processes can be represented by braiding in a global way. And that, whether it turns out to be useful for quantum computing, is certainly an interesting result. Of the Penrose kind, in fact, right? Like the spin geometry theorem. It's telling you that you can go underneath the unitary groups to something more combinatorial, namely, grade group representation, and pick up everything you need in principle. But of course, in practice, you'll have to take limits of products of things in it. And you get away from having to dig it all into space, spaces of some kind. That's right, yeah. Which makes me feel that if only we could push it a little bit further, we could actually grow the spaces out of all this. I think that's probably what Penrose wanted to do. I wanted to grow space and space-time out of the simple underlying rules, the simple underlying processes. I don't think anybody is quite there. People use what they call spin networks, or spin phones and so on, and they usually mean that they have their favorite group that they like and they've labeled networks with elements of the group, and then they're going to see interactions and so on. and they aren't trying to do this apparently tougher job of basing everything on something very simple underneath. Rightly so, I mean, for them, because they would mix up their motives if they were trying to do that at the present time. But anyway, I think this seems to me to be part of the story of trying to actually do the Penrose spinning. It's part of the story, which is still incomplete. So thank you for listening, Bjorn. Thank you very much indeed. Thank you. It's the end of Luke,

45:00 of one's delivery. Now we have Cynthia Whitney talking on Galilean theory and special relativity theory of Rosetta Stern. Okay, I've got the mention of Rosetta Stone in my title, both because it's kind of handy and nearby, and also it actually bears on the issue that I'm going to talk about. You see, the whole problem here that I want to address is a matter of communication. I think we've got a situation in physics that's quite similar to the situation that was being faced by these scholars of the ancient civilizations before they found the Rosetta Stone, namely that there are really some non-commensurate views that don't talk to each other and one way or another way, and kind of reject whichever one they didn't choose. And I would like to see that situation change. And so if I can do anything to remedy that, that's what I want to do. We certainly have a number of different viewpoints in evidence in physics today. We've had historically a number of different views about space-time, starting from the rather empty abstract idea and advancing through various situations where there's something physical associated with space and time and culminating at the present time with Einstein's idea that whatever space and time are, they're really the results of measurements, which you can juxtapose with Copenhagen's idea that all measurements are uncertain and that certainly leaves you very much adrift in regard to space and time. There's another fellow that I have told about this conference and maybe someday he'll come to it too in Greece, a scholar in Greece named Nasikas who talks about stochastic space-time which I think kind of completes the loop here. So maybe we'll hear about that one of these years.

47:30 We also have a variety of different assumptions that people make. We had ether very much in evidence at the time when Maxwell was working, and he really was a very intuitive, physical kind of guy and was definitely thinking inside his head about stresses and strains inside of a physical ether. Lawrence's ether was kind of close to that, but a little bit different because he was very interested in the responses to all of the stresses in the ether and in particular manifest in terms of the actual physical contractions of things or actual physical slow in the clock and so on. And then we have Einstein's view on the matter, which just simply disregards all that, and we don't even speak of ether anymore at this point in history. So any people who are still retaining other ideas and discussing them are very much put on the fringe of the technical society. We also have quite different mathematical languages in current use. We have familiar old Galilean vectors that we can do vector addition on. But to go along with that, we also have in place of that, in relativity theory, matrix multiplication instead of vector addition. When you add up velocities, you multiply matrices instead of adding up little vectors. And the rules are completely different, and the consequences are completely different. And it's a whole new ballgame that way. And then, of course, we also have tensor analysis, which is an extension on the idea of matrix analysis. And probably has essentially the same kinds of increasing richness in one of the things that can happen when you use it. So, here I stand in the middle of this mess because I'm a journal article and I get all these different people submitting papers to me, arguing quite different viewpoints and not communicating with each other because they're just really living in different worlds. So, I see people trying to crack into this situation, first of all, by reexamining the assumptions, which I think is a very important exercise, but to which the mainstream community never seems to pay any attention, however carefully people go into it. I also see lots

50:00 lots of people arguing for an ether and then lots of other people arguing against an ether and it's an argument that can never come to a conclusion and i see lots of papers about presumed mathematical errors in the in the development of special relativity theory which i believe may in fact be there because that you know the first cut that einstein did was pretty much off the cuff, and there are a number of places in it where you might stumble. But they do have a tendency to copy each other. They do that without a doubt, yes. And especially when you get to the level of textbooks. It'll go on for decade after decade with people copying each other. So I'm pretty sure, yes, there are errors there, and nobody cares. So again, communication simply doesn't happen. So So I have to conclude that we've got a big problem here about not having a common language. Now, I want to try to construct a common language, a sort of a basic or Fortran or something of the physics world. And as you know, I'm starting over with a different postulate as a basis for doing that. And I recall again the slide that I showed yesterday, defining what my postulate is. And now I'm going to give a little bit of background of what preceded that, going back to the days when Einstein was formulating his postulate. And when he first did this, it was before he knew about anything like the photoelectric effect. And so he was pretty much conditioned on Maxwell's theory, and he built it to be consistent with Maxwell's theory. And so if you try to draw a picture about how light propagates, you would have spherical waves emanating from a source. are indicated on there with little comments about how this thing expands with the wavefronts going at speed c, and that speed was relative to the ether for Maxwell, but it was really relative to anybody who might be looking, so far as Einstein was concerned. And it had a constant amplitude profile in the radial direction, and it was spreading out in angles,

52:30 so it had a constant angular profile, but it was getting more and more dispersed as it went out in radius. And somehow it got to a receiver, and when you confront the photoelectric effect, right away you run into a big question here about how did all the energy end up in one place when it was being scattered like that. Do you mean the energy of one quantum? Well, he was having this idea before there was such a word. I mean, with scintillating screens, it can actually count them. Well, I'm a political. Count quanta in a very low-light situation where the quanta are falling on a scintillating screen. Exactly. But it does have this philosophical problem. How did it boil down to that when we thought it was a Huygens wave? Exactly, yes. So you redraw this picture after the photoelectric effect and although it looks rather similar, you can't have all those arrows on it anymore. There used to be arrows going every which way. Now there can really only be one arrow. So how does that work? How did it get to be like that exactly? Now, it's kind of a quagmire, and people have, in fact, been investigating this for years and years, and I have a good friend in the Boston area who is now an older lady who's the widow of a professor she collaborated with and ultimately married, who started introducing alternative postulates back in the 1950s. And I want to give a lot of credit to these folks because they did something that had not been done before, which in retrospect has proved very, very productive and has, in fact, reconciled a lot of experimental facts that are otherwise kind of hard to put together. So this is Moon and Spencer's postulate, which basically has light from a source. They speak of spheres, but of course, the light finally arrived somewhere, so they haven't really confronted

55:00 that problem about whether it truly is spheres or is it a needle-like structure. But the point is, and what's important about it is that they postulated that the speed of this signal was key to the source forever and ever. This is not a Ritzian postulate, because a Ritzian postulate Let's fly, and the photon simply travels at C plus whatever the V of the source was at the time of emission. And their postulate is different and has proved a lot more successful, where they basically maintain a contact with the source on an ongoing basis until the light finally arrives at the receiver. Now, because this has been rather successful, I think it's an idea that's worth developing further. So maybe I'm starting on the second 50 years of development on this, but I'm going to need help, I think. You do young slips. Pardon me? You do two slips. I believe so, although I would have to dig in the literature to find the answer to that specific. They have looked at a lot more engineering types of questions about how homopolar generators work or how welding arcs work and other things that people have a temptation to disregard. You have several departments at MIT, for example, and there will be the physics department and there will be the metallurgy department. And there's stuff that gets taught in the metallurgy department. They have up the herring furnace, which is used for processing molten metals. And it's not supposed to work so far as the physics department is concerned. To stand back and look at this situation are the folks in the electrical engineering department who are willing to look at any dancing. Someone was in the electrical engineering department, and that's why he was rather more liberated than anyone else around, as was his student Spencer. So they had this one I think was a very good idea, but I think it needs some further development The great goal that I've added to it is to not leave all of the responsibility for the control of this light signal with the source, but instead to let that be seated to the receiver at a point halfway through the scenario so that I've got more symmetry here. That would agree with radar.

57:30 Yes. I believe, yes, you're the second person within a week who's mentioned to me. Well, I used to work for a radar firm. Yes. And the other guy did too. I wasn't counting dollars per millisecond. I was actually doing radar work. Yeah. There was a guy in the United States by the name of Hughes in exactly the same situation, and he said the same thing to me within the last week. that there's a resonance here with antenna theory, that this, when you look at what goes on, you're receiving antenna would support this kind of idea. Can you just make the postulate as clearly as possible again, please? That a sequence of two stages occurs in a propagation process. That's the basic postulate. And in the first part of it, the light is expanding from the source, tail end tied to the source, front end progressing at 2C, set for it, if you will, proceeding at speed c from the source, but always relative to the source. So if the source moves, if it has a velocity, if it has an acceleration, whatever dance it does, this thing always stays relative to that source. So the whole of this motion is relative to the source only? Just like it was. Because that is quite easy to test, in fact, isn't it, in an optical or radar set up? It might be testable, but I haven't worked out how you would do that. So same question to you, then. Can you do down the slicks like this? I believe that you can, and the kinds of things that I have looked at have not specifically been that, but the ones that I've looked at are stellar aberration, which again is an interference effect. Sanap effect, which again is an interference effect. And all of those interference effect type ones that I've looked at, yes, you can make some sense of with this kind of postulate. and so I believe the answer is yes to your question to the particular case. That's the one to do. Right, I'll put that one on the list. I think I'll outline a pretty simple laser interference experiment. I'll test that for you with some pretty fairly simple gear. Good. He wants to test if it's really like that, not if this is a theory that works. He wants to know if it's the actual theory. Well, give me a break. Pretty difficult, actually. to define what the word actual means in this context. Because you only have a few contacts with reality.

1:00:00 You send it off and the pulse of radiation on it. And then you sit and wait. And as Professor Moon pointed out 50 years ago, what happens in between, you do not know. No, you don't know. But in fact, the way to do it would be to make interference close up and interference far away and then watch the phase differences. and they will only solve one way or the other in that sort of situation. And then, of course, you wait for the target. The target reflects some of it back and there will be a change in frequency. And that change in frequency goes in two ways. Firstly, in the time which you send a pulse off, it gets to the target. The target has moved away. And secondly, the target then reflects a moving object. Why reflect? Why doesn't it just absorb? Well, it partially absorbs and partially reflects. If it absorbs at all, you don't see it. No, it's a detector. All we're testing is that you're transmitting. No, no, no. We're talking about radar. We're talking about radar. Okay, well, you still do the same thing. We use a mirror there. There are practicabilities about moving receivers around as fast. You don't have to do, because this proved it just to move the transmitter. Maybe, but I was going to say you get twice the Doppler effect because it spreads out, and then you reflect from a moving object. And so you have to pull the back from two. Also, there's some dispersion as well. That's right. Yeah. I don't know about that. put a factor of two in everything when you work out the Doppler speed of recession. Now, my experience is with stationary radars, or radars on ships, which are virtually stationary, not radars moving around in aeroplanes, so that part of this doesn't really come up. Yes, you've got the moving source and moving target, or if it's an active receiver transmitter, yes, and it's still the same idea, except that you have to realise that you have to relate and think. The truth is you don't know what happens in between unless you actually, as you say, you put up things in front and then the wise acres will say, ah, well, you disturbed it.

1:02:30 exactly you did there's no doubt about it you did yeah well okay this if the picture only went this far and if the front end were traveling at c instead of to c that would be the moon spencer scenario but i've got my scenario had the second half where the collapse phenomenology is just like the emission phenomenology only playing backwards in time. Now, my claim is that with that being postulated as the actual mechanism whereby life gets from one place to another place, that it becomes possible to link up a lot of things that we previously thought were completely unrelated to each other. the last one if you're going to use a reflective beam do you then set the velocity to match that of the reflecting entity or what yes when it's in the control of the so it follows the velocity the velocity is c plus the whatever whichever one is in control entity that does this yes c plus the velocity whatever i might have to rig that up i might have to rig that up we'll talk about that some more well when when the light envelope manages to make contact with a candidate receiver and that receiver takes control of it at that point the collapse begins which means that the contact with the source is relinquished and the envelope starts to collapse into the receiver the question is it's a simple reflector that's the point and it does that take control there is actually a discussion about a reflection type of scenario in the reflector or a scatterer that's the thing so there's some discussion that's pertinent to your question that's in the blue book as well reflection oh refraction let us suppose for simplicity that we're working in a vacuum would certainly add a layer of complexity. There's a question about control. Yes, you can't set these things up without having an odd lens in the line.

1:05:00 Oh, you're right about that. That's a bit nasty. You might have to vibrate the lens or something. That problem comes up actually in the stellar aberration scenario, which I mentioned as being one of the test cases for any of these postulated alternatives to Einstein's relativity theories. Is that covered in the notes? Actually not. it is covered elsewhere. So you'll be sending me emails or whatever, ping me on that particular question. So you know what to do with the moving line? Oh, yeah. One of the things which is deeply confusing, I think, in this game, is that certain kinds of particles are said to have coordinates, even if you have to have a wave probability waving up. But some other kinds of particles and photons, which are among them, are said not to have coordinates. You can't really assign coordinates to them even though you say they're moving and so on and so forth. And this that I'm speaking in quantum speak now, that is really confusing when you try to make sense of these arguments. It speaks to what you mentioned a moment ago about you don't really know what's going on in between. You know about the photon when that energy was in the possession of the source, and you know about the photon at the end when the energy is in possession of the receiver. Where it was or what it did in between, you don't know. And the only way you can find out anything is to put some other receiver in there, which is a different problem. So you don't know. No, as long as they are at least the same result. I mean, we can't complain. They're just good or bad models from a convenience point of view. People often ask me if I believe in waves or particles in regard to light. And I do everything I can to resist answering that question. As far as I'm concerned, that information is basically private-secret information. It's folded up on a piece of paper inside the envelope, and I'm only talking about the envelope. And I don't ever want to talk about anything but the envelope. Can you tell me what causes the friction? Not right now. Not in this presentation. But it is an interference phenomenon, and it's just like focusing a lens or anything.

1:07:30 They're all really the same. Well, basically, in general terms, such things come about because if the light really works this way, but you're modeling it with a constant C, you really, in order to get the correct results have to change that c and make it a function of the velocity of the receiving optics and when you do that and then recompute where things would come to a perfect focus that moves and that comment applies to any any combination of rays that you might want to interrogate so the trajectories don't change well it doesn't really have trajectories and have that in classical optics either it doesn't indeed no but there are certain ones that do which work extremely well well yes it's a way of thinking about the problem but it isn't i mean if you have a spherical wave wave front i'm sorry i'm probably being naive but if you have a spherical wave front and you assume the compound's propagation whatever you if you assume about the velocity, it will still produce interference. Yes. So if you introduce a little bit different assumption about the velocity, you get a little bit different interference patterns. Yes, the question is whether you can detect it within the experiment as a stat. Yes. About the one that I'm most familiar with, because it is the simplest one, is this dollar aberration issue, which is not adequately handled by special relativity theory, despite claims to the contrary. And the problem with it is that the results in that experiment do not depend on the velocity of the stars. And the problem was first articulated at a time before people even knew that the stars were not fixed in the heavens. And they certainly didn't know anything about rapidly rotating binary systems. And you throw that into that mix, the existing explanation simply doesn't make any sense. Am I being a bit mean to ask you to clarify what is this? You just say that the velocities of the stars seem not to appear in them. Yeah, because the only thing that actually affects what comes out on the ground is the velocity of the Earth in its orbit around the sun. And the only way you detect stellar aberration is to look now and look six months later.

1:10:00 Yes, but I descend from that view because astronomers estimate the rate of rotation of stars. What does that mean? as solid bodies by looking at the splitting of lines because part of the stars coming towards you are going away. Oh, that's quite true, but that's not the same problem at all. The problem I'm alluding to is that if you've got a binary system and you've got a tremendous velocity difference between the two paired stars, that if the result for stellar aberration were to be as special a function of the velocity difference between the source and the receiver, this would create very dramatic dancing around in the image in the focal plane, which does not happen. It's the dancing you're talking about? You're not talking about simply a line shift? Oh, I see. It's dancing. Okay, thank you. Oh, I didn't realize that. So that is a well-known problem in the dissonant community. It's the modulation of the light by sidebands that the motion is not just the shift but the outlet. So the angle that appears on the floor ought to be very dynamic in response to the very dynamic system out there that's creating light. Just to conclude my conversation, so this is all wave pathic collapse, no trajectories at all? Yeah, yeah, that's right. All right, I said that I had a different postulate from what's currently in use. And you can compare the consequences of that and figure out what kinds of discrepancies are going to creep up because of that. Obviously, if reality is one thing and your model is another thing, you're going to have your estimates of any kind of result of any experimental observation somewhat corrupted by that mismatch. And you can figure out in a number of ways just what that discrepancy is. The first time I did this, I figured it out empirically by looking at the Sinac effect

1:12:30 and figuring out what it had to be. That was before I even had this alternative postulate. I've since worked it out again from the postulate. Well, however you go at it, you get the same result for what the skewing of your data is going to be. And bottom line is that there's what I call a desynchronization parameter that creeps in everywhere. Every time you try to estimate something, the amount of screw up is captured by that delta. Now, there starts to be a pattern that develops here that you can develop relationships between the real velocity that you're trying to look at and the ones that we actually operate with in Einsteinian theory and it turns out that this corrupted estimate has just the properties needed for the Einsteinian velocity so familiar in special relativity theory. So now you've got two things in hand. You've got the Einsteinian velocity that has been around for a hundred years and then you've got the Galilean velocity which has been around for like 400 years I suppose or possibly And there's a relationship between them. And you can write it either way you prefer to write it. You can, you know, if you know it one direction, you can invert it and get it in the other direction. And so they're kind of nailed together here. So that prompts you to ask additional questions because there's another character that appears in the special relativity books. The point is that there exists such a thing as the covariance velocity and it too can be expressed in terms of this delta parameter and all that's involved in getting it is to write down this thing as it's defined and then express it in terms of the Galilean velocity that you already know about and clean up the algebra so there's another relationship now

1:15:00 i find it exceedingly interesting that when you work out all these possible relationships that they make something that is formally very neat you will notice on this chart here we've got But everything here that you could want to know, everything here that you might know instead, and the functions to map from what you actually do know into what you want to know, that anything you want to know about is a function of something else that you do know about. And if the functions all have the same form, have a look at these denominators. that they are all of the form 1 plus or minus something, and the pattern of the somethings in the denominator vary from 0 here, minus 1 here, plus 1 here, 0 here, minus 1, minus 2, and over here is 0 and plus 1 and plus 2. And this is very, I think, tantalizing and interesting little algebraic relationship. And I would love to be able to engage the interest on some mathematicians in this thing. Let me just point out that it is recursive in nature. If you take a variable x and define a function of the form 1 plus x delta, and then define another function of the form y over 1 plus y. And then you substitute in the y, you will find the recursion developing. It's just adding an energy to it. So these things are all recursively related to each other, and I would invite any commentary that this mathematical community can give me about that. From an engineering point of view, however, you'd like to get more explicit and get that delta parameter expressed in terms of the actual variables that you have at hand, which

1:17:30 you can always do. There's nothing more complicated than a quadratic equation, and all you have And you've got the delta expressed in terms of any variable you might wish to have it expressed in terms of. And then instead of just having a formal table, you have a table with actual functions in it, many of which are quite familiar. Here's some usual stuff from special relativity theory, which you've known for a hundred years. There's some stuff from Galilean theory, which people have known and forgotten about. And there are all sorts of connections spelled out between the Galilean variables and the Einsteinian variables. So where this table, without this understanding with the new postulate, would be completely disconnected and nobody in the table would have a neighbor, missing information is filled in, and once you know one of these variables, you can compute any other one. Furthermore, you can put it all on the computer and make plots of it, and what you have displayed here is the Galilean velocity, velocity that results from having the wrong kind of model to process the data that you're seeing when you try to estimate velocity. And you have also the proper velocity, which is also known as the covariant velocity. That's kind of the punchline of the talk. So now Now it's a time for me to hand out my handouts, of which you are the favor of the acceptance because everybody else is gone. And it's got the tables on the back. So, the option of all of this is that two worldviews that were previously thought to be completely disconnected are, in fact, not disconnected.

1:20:00 And you can translate from one to the other, and that's the Rosetta Stone. Now, that means that you're relieved of the burden of having to completely discard something that's been familiar and reliable for hundreds upon hundreds of years. You can keep them both, and you can translate from one to the other, and hopefully come to a better understanding of what things really mean in the more mysterious modern special relativity theory. When people find problems in relativity theory, they tend to look for it in the algebra or some deep philosophical issue, and the problems are never there. The problems are really in the language, describing what the equations mean. And now that you've got a good dictionary, you may be able to come up with better language for describing what happens in special relativity theory. So, when you're dealing with very rapid objects that are giving out light on their own, such as remote galaxies... Correct. I understood that the, it depends which way around they're going, but I understood that the doctorship was something, diffractionally speaking, something like root of 1 plus C over V over root of 1 minus C over V, that sort of thing. So when, sorry, V over C. When V over C is small, that reduces to the usual formula. I don't really understand what the relationship between this is and that. One thing that's going on in current science is to look at these signals from distant galaxies and try to infer the speed that they're traveling at as a function of distance. And people are coming up with the impression that the expansion of the universe must be increasing with time,

1:22:30 which they don't understand because you'd think it would be the other way around because gravitation would act to pull things back in. Are there other things called gravitation, you see? That's what they don't understand. Who knows what's out there, but let me point out here that if you process data using the formulas from special relativity theory, you are always going to get a velocity estimate that is a little bit lower than the actual velocity of the source that emitted that signal. And that could certainly contribute to the impression that those fellows out there are not going at the speed that you expect, and that there must have been some increase in the expansion rate of the universe in the intervening time. Of course, what they have been using to try and describe that is Einstein's lambda that they could put it the other way. Maybe we have a contribution to that. Anyway. I thank you for your attention. Thank you. Thank you for coming. Good. Well then, look, we can talk some more. You were amusing about how to make an experiment. Well, indeed, I'd have to give it a bit of thought for a while, I think, and just to get the most convenient set up because, you know, practicalities will take things a bit. But, yeah. But, of course, the problem, as I see it at the moment, we need to have a clear idea of which elements of the path affect it. I mean, what about the light's got to pass through air and the air has got a tiny refractive index shift from back on. of course the degree is of importance here what do you mean by a vacuum in the context of the question needs to be answered so yeah all right but certainly the basic idea of I would have thought that, I don't know, I'm very ignorant about these things, but I would have thought that quantum field theorists have some sort of model of this collapse process, you know, when the wave front hits the screen, and so you actually make an observation, therefore, the usual government observation.

1:25:00 You would think, wouldn't you? They don't make a lot of observations, but it's simple. There are one or two models around. I don't know the names, famous ones, three letter acronyms, non-linear. They're not particularly set in concrete. I agree. No, they're hypotheses. I mean there's no measures, there's no empirical data in their class as well as others are. I hesitate to mention this, but you know this, the set of gravitational field equations we purport to have got from quantum mechanics, you can write down the low gravity version, the Newtonian version, if you like, quite easily. And it differs from the usual del squared pi equals naught, or del squared omega e, or where omega is the potential, and it's del squared, del squared, omega equals naught, you see, sorry, it's fourth order. So that when you solve it for a spherical solution, yes, you do get the inverse square term, but you get two others. And the two others seem, I have done some rather wriggly things with initial conditions, which I must look at again, seem to produce a kind of Hubble recession. In other words, you have a massive point source here, and you do a spherical solution, and any test particle that happens to be around is saying, providing it's far enough away, and of course it has to be a long way away, otherwise the inverse square gets in, it will move away at the speed approximately at a rate proportional to the distance.

1:27:30 this doesn't go on forever because of course after a while you work up a speed where the you need some form of relativity anyway and the whole sort of pseudo newtonian calculation collapses anyway so uh there's a limited range over which it's valid of course we hope to do the real thing the hard way but it's hardly a universe, I mean it's just source and lots of test particles whereas what you've actually got is loads of galaxies all over the place so it could be complete nonsense but at least it produces its own Hubble effect without any further ado, you don't have to put it in There is, coming back to the original point you made, of how does the light all get into one place, there are the makings of some models that will support that on a wave basis, though they need testing. People don't spend any time with them, and certainly it applies all right, I think, with silicon diode detectors and things like that and as the scintillation phosphor is a bit of a different matter because i don't know what the crystal sizes are or what the nature of the crystal is we know quite a bit about semiconductor crystals of course but the the model which seems to me to work with the wave model that works with silicon crystals would be that the you take each whole crystal that has got its electron modes. These are the valence and conduction electron modes spanning the whole crystal. These are coupling from... We're talking about coupling energy across the bandgap by the irradiation. And all it does is inverts the thing gradually. It puts more energy than you would expect at the temperature in the higher two energies. And then, because it's multimodal, you've got a noise going on. chamber and there's no loss that's the whole point about making this crystal very low loss until some phenomenon occurs in the case of a silicon crystal

1:30:00 it's a phenomenon near the boundary that that causes a quantum change because you are you're creating stress which is which is beyond what is tolerable and this is a storage effect sort of thing so the energy is picked out one place at stresses are high, because it's a local phenomenon, but the energy in feed is a large phenomenon in that type of model. Now, there's probably a similar model available for phosphorus, but of course a lot of them are things like powder phosphorus, and I very much doubt whether there's any single crystal model. I'm dead sure, but I don't think there will be any one that bridges across the particles of the powder, for instance. So one would need to check the energy storage time requirements to get that model to make some of this. I don't know. I think some of those phosphorus see dots really at countable rates, don't they, Dan? Yes, they do, if it's low enough. I was just going to postulate this kind of arm-waving model for that sort of thing. It may be, you see, that it's probabilistic in the following sense. You know that if you count the scintillations, you eventually map out the classical intensities, the patterns, interference patterns, and so on. But the picture one has in one's mind is of little bullets arriving at a certain rate. And okay, there's a big fluctuation in the rate, but eventually it settles down to a certain average. It may not be like that. It's not like that. It may be more like that the scintillating matrix is in a kind of anticipatory mode, and the wave comes along, the classic wave, and sometimes the matrix responds and sometimes it doesn't. intense the wave is the more likely it is to respond or something like that but even then you see you get into trouble by saying well what is this is it I mean

1:32:30 that that that would would have the light as a classical way which we don't believe in anymore then you can hardly believe in a photon which is a great sort of spread out thing with no real coordinates producing a scintillation here pinpoint thing and I think it's all probabilistic there is still a question about the pinpointedness of that