Minding quanta

0:00 There, running, okay. No, even, you can't actually disturb it by, you know, them making four. Yeah, probably would actually, maybe. Yeah, I'm feeling well aware of what it would. Sorry about that. And if we, um... I know it's a gentleman. I think there'll be a beer. I've got to do this. Thank you. what are you going to talk about who me yes you that's a good question does that mean silence or something about non-communativity I can't remember the title it'll be quite a bit it'll come up when I get my phone questions that's it well I don't know curiosity Curiosity. Well, that actually was the, don't panic, remember. Right. That's right. You know. That's the problem. Okay, well, we can stop then. We're having this fake handshake because we do that with their nose. rather than applause. But it doesn't mean applause. You know what you want to use, right? And so this is John Cramer's paper, and we'll pull up the rules of your hand in the half an hour talk and then discussion after that. Well, 15 minutes. Thank you. organizers for inviting me, and it has to be a very interesting topic so far. I'm going to start with a metaphor, and then I'll talk about the relationship between

2:30 quantum theory and interpretation, and I'll introduce the transactional interpretation and apply it to various quantum paradoxes. I'll say a little bit about time, pseudo-time, So the metaphor is a 19th century poem by John Godfrey Sachs that most of you are published in the book. I won't recite it all together, but I'll say it doesn't help to do the first part anyway. There were six men of Hinduism and to learn the much implying who went to see the elephant, though all of them were blind that each my observation might satisfy his mind the first approached the elephant happened to fall against his brawl on a certain side at once we gave the brawl God blessed me but the elephant is very like a wall and then it goes on the next guy thinks the elephant looks like a spear and the next guy thinks the elephant looks like a snake and the next guy thinks the elephant looks like a tree and the next guy thinks the elephant looks like a van and the last guy grabbed the elephant's tail like a rope. And the poet concludes, and so these men of understand disputed loud along each in his own opinion exceeding stiff and strong, though each was partly in the right and all of them were wrong. And then this is actually a moral thing about theology. So often theological words are disputed inside wean and grayed on another in ignorance of what each other mean and prayed about the help of not one of them seen. Okay, what's the relationship to quantum mechanics? Well, I'll modify the verses here a little bit and say that the moral to some extent applies to the interpretations of quantum mechanics. There are lots of people who have ideas about how quantum mechanics should be interpreted and like the blind man and the elephant talking about something that we can't really get a very good grip on. Okay, so let's talk a little bit about quantum theory and interpretations. Quantum mechanics is a theory that our current standard model for describing the behavior and energy on the small scales for things like photons, atoms, nucleotide quarks, luons, electron, and so forth. And like all theories, it consists of two parts. there's a mathematical formalism

5:00 and an interpretation of that formalism. The interpretation is largely language, words in terms of the word we were talking about yesterday. however product science differs from most other physical theories in a number of ways. Basically, most physical theories are formulated by somebody having a mental picture of what's going on in the universe and then translating this into mathematics and formulating it and testing it and so forth, but for quantum mechanics in a rather interesting way, Heisenberg was utterly opposed to having physical pictures of things. One of his friends related that when he was a student, an undergraduate, there was a chemistry book that showed valences of little hooks and eyes attaching atoms together and Heisenberg was outraged at this idea that somebody could actually use a picture like that to represent something that was going on with an atom, which he thought you were going to be mysterious for the picture that it rendered and appropriate as a picture. So he did a lot of pictures, and when he put together his matrix mechanics, it was mainly, I guess one would have to say, by dumb luck, since Max Born had explained to him that it was matrices he was actually using, and not something he'd made up out of whole plot. And Schrodinger, on the other hand, did have a little picture, he was trying to make a model of trying to do something like the electromagnetic wave equation. Rob just had mass, and he thought he was making a wave equation that was like the electromagnetic classical wave equation, which had visibility surrounding space. After he had done it, Bohr and Heisenberg pointed out that he had problems with looking at it that way and he was sort of discouraged about the fact that his physical picture hadn't worked. So, as a result, quantum mechanics has this formalism that's accepted and has been used for 80 years and works very well. The interpretation remains a matter of controversy and debate. Like the opinions of the six blind men, there are many rival interpretations of quantum mechanics. However, today I'm going to be talking about transactional interpretation, but I had time about some of the others, but interpretation of, what does an interpretation do to the formalism? First of all, without the interpretation, we just have symbols on a small, small page

7:30 as we're set. It provides links between the mathematical symbols of the formalism and the elements of the physical world. Secondly, there may be paradoxes that emerge in mathematics, places where things like peculiar or counter-intuitive and so forth, and one of the roles of the interpretation should be to neutralize the paradoxes, and it should really neutralize all of them rather than only addressing a few of them. Some of the interpretations focus on one particular aspect of interpretational problems or some of the others, checking this on this article. It should provide tools for visualization, for speculation, and for extension. And it should not have its own subformalism. If you do, it's perfectly legal to revise final mechanics if you want, but you want to admit that you're revising it rather than interpreting standard final mechanics. And it should not, in general, make its own testable predictions because the formalism makes the predictions, not the interpretation. Interpretation may be falsified, but it's found to be inconsistent with the moralism of the experiment. Here's an example of an interpretation, F equals NMA. Without an interpretation, those are just letters on the screen. They don't necessarily connect to anything. So the interpretation is that F means a vector force, which is applied to a body, and results in a product that's proportional to the scalar mass of an object, and the second time-derivative A of its definite position. What this does is it relates the formalism to the physical observables. It avoids the paradoxes that arise. If you have a mass less than zero, a negative mass means the acceleration is moving one way and the force is going the other way. It ensures that they are parallel. Okay, now let's talk about the transactional interpretation. First of all, a quick overview. We interpret We interpret the wave function psi of quantum formalism as a returative wave offer that goes out into the universe offering to form a quantum event involving an exchange of energy and momentum and so forth. We interpret the complex conjugate of the psi star as an advanced wave going the other time direction and acting as a confirmation to proceed with the quantum event.

10:00 The mini-sci-sci-star combinations we see in the quantum formalism are interpreted as indicating the formation of a forward in time, back in time, standing wave, a sort of handshake that transfers the energy and momentum in certain quantities. In this interpretation, observers do not have any special role. The transactions that connect with observers are not really different from any other transactions. since the picture is intrinsically non-local, the paradoxes associated with non-locality are resolved, and as it turns out, essentially all the other paradoxes I'm aware of are resolved too. And it's something that philosophers like to say it's economical in that Bohr's statistical interpretation is usually a separate assumption, but it's intrinsic in the transactional interpretation. With that, let me go into a little bit of detail. Here we have a quantum matrix element, which involves a side star and a side. That's what it's making, so it's a sandwich that allows us to get an expectation value. So the question is, what does it suggest? And the hint is that the Wigner operator for time reversal is complex conjugation. quantum mechanical quantity and take a complex conjugate, you're effectively reversing the time direction. So if you do that to a quantum mechanical wave function then it makes it a retarded wave in an advanced wave. In the simple example of a plane wave, we have an E to the I to K R minus M to the T plane wave. When you take a complex conjugate of it, it changes the sign of I. It makes the line go in the other direction and makes time go in the other direction as well. A retarded wave carries positive energy to the future, so an advanced wave carries negative energy to the past. The transactional interpretation at some level is based on the old Wheeler-Lyman treatment of the classical electromagnetism. The wave function that we get from Maxwell's equations, the electromagnetic wave equation, in space and time, and because it's second order of time, it means there's two solutions, and we can identify one of these solutions as a retarded way, as a way of carrying positive energy into the future, and the other one we can identify as advanced ways going in

12:30 the other time direction, so we have a future light cone and an advanced light cone. Now, the usual way of dealing with these advanced solutions is to throw them away, basically put them in garbage. That was what I learned in graduate school, that you pay no attention to the bad solutions behind the curtain and focus on the return solutions because they could violate something that my professor called the causality maverick condition. And I did a lot of trouble about that because I don't know what maverick conditions are and causality isn't one of them. But it does seem to be a law of physics. But Wheeler and Kleinman had another idea about how to deal with this, which they got actually stole from Dirac, I think. which was that instead of throwing away the gas solutions, you can reformulate electromagnetism, and with any irradiator process, produces half retarded waves and half advanced waves, and it's time symmetric rather than building in an arrow of time as you would if the advanced solutions were gone. Okay, so with that, you can consider Wheeler planning We have an emitter here, which is trying to bring the energy, it sends retarded waves along the positive light coming this way, and it sends advanced waves in the other direction. Since it's sending out positive and negative energy at the same time, it doesn't really lose any energy by doing this at this initial point, because, but if we have an absorber over here that sort of catches the ball, so to speak. receives the end. One way of saying that it absorbs the wave is that it makes a wave with the opposite phase. And so you could create a wave with the opposite phase that essentially stops the retarded wave by canceling it out on this side. And according to the Wheeler-Funding boundary condition, it makes an advanced wave which goes back down the track that the initial wave came back to the emitter at the instant when it was emitting the radiation, and beyond that, the negative time direction cancels out the advanced wave too, so the net result is that the advanced waves over here are gone, the return waves over here are gone, and all we have is the back and forth between the emitter and absorber, which accounts for the energy transfer, transfer momentum, recoil, source, and so forth and so on.

15:00 I wonder why we can find it did this, and it's for sort of peculiar reasons having to do of the electron, and that part of the paper is essentially a failure, because one needs a self-energy electron for other purposes, but the mathematics associated with it, their electromagnetism is perfectly valid, and one can use it. So this is classical physics, this is what inspired the transactional interpretation. We basically use the same protocol for a quantum mechanical process in which we say that the emitter, an emitter is trying to initiate an electromagnetic wave sends an offer wave psi somewhere out of this space. Over here is the absorber. The absorber sends this confirmation wave psi star back to the emitter and this process recycles back and forth and back and forth and transferring energy and momentum, building up a standing wave in space-time, which in some circles is called wave-function collapse. This embodies the uncertainty principle because only that part of the off-the-wave that has been reinforced by the confirmation process becomes part of the completed transaction. and so a transaction can project out only one or two localized complementary variables. It cannot be localized in position and momentum, for example, at the same time because either you confirm an I can say the momentum or you confirm an I can say the position, but you can't be voted at the same time because of the way the transaction works. And this accounts for Heisenberg's uncertainty principle. The board probability law, you can have an emitter which sends out this psi over here, then the strength of which the absorber is going to respond will be proportional to psi. It sends out a confirmation wave coming back in the other direction, psi star. So there's sort of an echo that the emitter receives a light-a-sonar wave going out and coming back, which has magnitude psi, psi star. And And the emitter among the possible transactions which could be formed presumably has a number of these size-high stars echoes coming back and on a statistical basis chooses with the

17:30 higher probability for a big size-high star that were low in it, which leads to the more probability loss. So you can sort of see it emerging in the transactional interpretation. In the Copenhagen interpretation, observers play the special role of the lapsers of wave functions. This leads to various problems, for instance, in quantum cosmology where there are no observers present. You have to ask the question how this happens. In the transactional interpretation, transactions involving an observer are the same as any other transactions, and so the observer-centric aspects of the Copenhagen interpretation are Now let's talk a little bit about the question of the issue of whether the transactional interpretation can be tested. The simple answer is no. Push the button, I shouldn't have. The simple answer is no. As I said earlier, it's the formalism of quantum mechanics that makes all the testable predictions. So as long as an interpretation like the transactional interpretation is consistent with the formalism that makes the same predictions as any other valid interpretation. And so you cannot denote the experimental test if you distinguish one interpretation from another. However, if you can find interpretations that are inconsistent with the formalism of its predictions, then things are a little different. If this is true, interpretations can be falsified. I'm going to talk in a minute about an experiment which I can't say falsifies the Copenhagen and many worlds interpretation, but at least it falsifies some versions of those interpretations in which people like to talk about the way measuring a particle kills their experience. The transactional interpretation follows the quantum formalism very closely and it does not have any problems in this particular area. Other interpretations may. It's sort of a matter of taste what we mean by this sort of thing. Okay, now let me talk about quantum paradoxes and how the transactional interpretation deals with them. The first paradox I like to call the Einstein's whole paradox. At one of the early Solveig conferences, the one before the conference involving the Einstein's clock paradox, Einstein raised some of his first objections to quantum mechanics. The objection was, let's imagine a situation where a photon is emitted from a source having no directional reference,

20:00 system is zero. Therefore, the wave function expands like an inflating bubble and progressively larger and larger and larger as time passes until it reaches some detector, which detects the light. At that point, the bubble pops, and the wave function that had it out here in space, part of Einstein's description, at least disappears. And so Einstein's question hey, how does the wave function of the advent over here at B and C know that it's supposed to go away? How is the information of the detection transmitted across space-like distances to these other places? How is it avoided that you can't simultaneously detect a single locon that's been emitted here at two different detectors? How does that happen? Heisenberg, who was in the audience, answered by using his biggest knowledge interpretation of how we're going to it at the moment, This is the paradox. It's a little as if you make a bottle of beer and throw it into Boston Harbor and it disappears in quantum ripples that spread across the Atlantic getting further and further away from Boston and going off in all directions until they reach Copenhagen over here and suddenly a bottle of beer jumps onto the dock and the ripples disappear everywhere else. at least in certain versions of the interpretation of quantum mechanics that's what quantum mechanics says happens to electrons and photons when they move from one place to another I notice this is Carlsberg here which funds the Neal's Moore Institute the transactional interpretation explains this by saying that the transaction develops between the source and the detector transferring the energy and momentum there since there's a one-poton boundary condition of this transaction, this is the only transaction that can form, and although there were way functions at the other places, they're not allowed to form competing transactions because this is the one that wasn't contested. The transactional interpretation handshake acts non-locally to answer Einstein's question, and this effect is in effect an extension of the pilot wave, I guess, with Broglie. The offer wave going out is essentially the Broglie's pilot wave.

22:30 Okay. I think I'll skip the Schrodinger's cat paradox. I don't want to spend too much time on talking about interpretation, so let me just look through here and go directly to non-locality. This is the Einstein-Polosky-Rosen experiment. The basic idea is that you have a source here which produces a pair of entangled photons in an unigual zero state. they're constrained by momentum conservation so they have to have the same polarization state, either circular or linear or whatever and if you have a pair of cross polarizers here, you observe that the angle between one of the polarizers and the other polarizers based on Alice's law with the detected intensity goes like the cosine squared of theta it says if you took this polarizer and moved it over in front of this the same arm because of the correlation of the two potons and so the curve you get was like this. Now a lot of people were saying, well, why is this so interesting, couldn't the sources be sending correlative potons out in a different polarization state? And Ferberi, a farmer, answered that question by saying, okay, suppose we consider that possibility. If we put an extra polarizer in the system connected to a motor on a random number generator, which ultimately gives the polar photons a definite state of polarization, say horizontal one time and vertical the next, and sideways and various angles again and again, you get a different correlation. It looks like the red line here. So what's happening here is not that the two photons are going out of a definitely correlated the polarization state, but they're going out at an arbitrary polarization state, but as soon as that polarization state is measured over here, you have to be required to measure the same polarization state over here, and then you do the black curve. Apparently, the measurement on the right side of the apparatus causes, in some sense of the word, cause, the photon on the left side to be the same quantum mechanical state, and this does not happen until well after they've left the source. The Einstein-Pelosti-Mosem influence across space-time works even if the measurements for kilometers have been demonstrated in Geneva or even light years apart. And this raises the question, when people first hear about it, is could this be used for faster than life signaling? And the answer was the play by Philippe Everhard, who said to show that whatever measurements

25:00 you make on the two ends of an experiment like this, the quantum mechanical operators who have two measurements commute with one another, and then we cannot use these to send signals from one another. It turns out that Eberhard's theorem gets broken if the quantum mechanics is slightly non-linear. As long as quantum mechanics is linear, it works, and you can't send any signals. The transactional interpretation of this is that since the two photons are entangled, you cannot just make a transaction between one of these and the source. You have to make a transaction between both of these and the source at the same time. and the advanced retorted handshakes are such that the photons go out on the green lines and the conformations come back along the red lines so that the communication paths are not across space-like distances between here and here between the two detectors, but there's a path going back along the light-like line going back up here and then out to here for both of them. So therefore, the kind of communication that's going on, A, prevents you from sending signals, and B is completely consistent with relativity because these are the rest invariant quantities. Now let's talk about Wheeler's delayed choice experiment. This is an experiment invented by John Wheeler in which we have a classic two-slit experiment. Light from some source back here is passed through two slits through a collimator and reaches a screen over here where interference may form. And so we should get an interference pattern like this, indicating that light is passed through both slits and makes maximum minimum of it appear. Alternatively, we can flip the screen down and run it through the lens, which refocuses the image of the slits on the screen back here. And by putting detectors at these two slip points, we can tell whether the photon passed through slip 1 and appeared at 1 prime, or whether it passed through slip 2 and appeared at 2 prime. Now, Wheeler's trick is that you take a graduate student who has very fast reaction time, and you tell him not to decide which of these two experiments to do until after the light is already cast between the slips and commit itself. And if you do that, you get just the answer you would expect from the problem, because it doesn't matter when you decide which experiment to do. As a matter of fact, people have actually done a Wheeler's Delay Choice experiment this way and demonstrated that it works. In the best wheeler's account,

27:30 the photon does not decide if it's a particle or a wave until after it's passed through this list, even though the particles must pass only one slide by a bottle of wave must pass through both. Wheeler associated with the mathematical choice to determine whether the photon is a particle or wave retroactively. How do we deal with this in the transactional interpretation? Well, the screen is up, the transactional bars between and the waves involving both slits. So we've got a photon detected on the screen made of a transaction which passes through both. However, if the screen is down and we have a detection event here, the boundary condition enforced by the lens means that the light going back and only you have it go to slit two. And so the transaction is warranted involving the slit So if the screen is hot, the transaction parts between sigma-1 and the source through both splits, but if the screen is down, it parts only between the detection point and one of the splits. In either case, when the measurement decision was made is irrelevant because the transaction parts all along the flight path to the hotel from the beginning to the end. Now let me tell you about a variation of this experiment called the AFSHAR experiment. These are the same experiment as before. We play one new trick. We place opaque wires at places where the interference minima would occur. In other words, every place where there's a dip in the interference pattern, we put a wire there. and we use wire such that they have about 6% of opacity for light that has no shadowing to do for a white light that has no regular collimation. We place a detector over here and we measure the transmission of light from this point to this point with those slits open and we ask how much light is transmitted. The question is, what fraction of the light is not transmitted from 2 to 2 prime because of these wires? This is the same as asking whether there's an interference pattern there when we're managing particle behavior. We're told by a lot of interpretations of home mechanics that when you measure a particle light behavior, you should see no interference.

30:00 And so the question is whether this measurement of the light going from 2 to 2 prime kills the interference or not. This is AppChart's results. This is with both slits open and no wires. So here's the light light going through both slits. There's no wires in here. And this is the image detected here of the light going through the system. And we use that as a calibration to measure the strength of the regular light. So no grid and two slits, there's no loss. If we block one of the slits and measure only the light going through measure the light going through only one slit, we see that it's reduced by 6.6 plus or minus 0.2 percent. So there's a grid one slit, there's about a 6 percent loss. But if we open both slits, the ratio of the loss is actually slightly negative with the error bars, so it's essentially zero. So what actually I've demonstrated is that in this situation, the interference minimum must be there as all the light goes through, and that was getting intercepted by the wires in this situation. Now, this experiment was criticized by the fact that the wires can form a diffraction gradient, and you can get an image of the diffraction gradient at the other slip, so this sort of compromises the idea that this is a measure of particles coming from slip to exclusively. However, he did another version of it where he used only one wire, but he certainly can't called unwirdered fraction rating. So this is the wire press at one slit open. You can see the wire here. I mean, you can see the image of the slit here. You can see the airy frame around it. And you can see all this junk out here from the photons that hit the wire and scattered off in different directions. This is the CCD image of what's coming through. If you now open both slits and have no wire present, you see the image of one slit, you see the image of the other slit, and you see none of this stuff here because now there's no wire in the inner curve to hit. But now if you put in this one wire in the inner curve position, which is just like this, these two curves are almost indistinguishable, and so again it demonstrates that when both slits open and had a wire present. None of the photons were hitting the wire because otherwise we'd have this stuff here going on. So the conclusion is that interference is still present

32:30 even when an eye unambiguous which wave experiment is being performed. Measuring particle-like behavior does not suppress wave-like behavior until terrible non-interactive measurements are made. And it appears that light light was passed through both slits to create interference, only one slit. The transactional interpretation explanation of this is that the offer way certainly passes through both slits on its way to this part of the experiment, and when it reaches the wires, you cannot have transactions formed on the wires because of the destructive interference there. So no matter what's going on downstream, there will be no line in the from the wires. Therefore, the absorption by the wires should be very much less than 6% because of the destructive interference, and this is consistent with what's observed. I should also say that the normal quantum mechanical calculations give you just what offshore is observed, so it's not going to be a mystery if this happens, just as the interpretations. Some of the interpretations are somewhat confusing. Another thing I wanted to say about the transaction interpretation is that you can use it to visualize processes, I don't have time to go into it, but there's this illicit vitamin interaction-free photon-bomb experiment that was published about eight years ago. And I have a paper in Foundations of Physics that's going to be published in the near future describing the use of the transactional interpretation that allows you to draw migraines like this, indicating what's happening and showing where the offer weights are going, where the confirmation weights are going, and how this object, which is intercepting light is being probed without actually having any photons strike it. Okay. Finally, in conclusion, let me say a little bit about time and so forth. I don't have time to talk much about this, but a person named Modlin was arguing that the transactional interpretation had a problem because the early transactions can create conditions that set up later competing transactions whether or not the wife might not be there, and he argued that you can't have all her wives for both at the same time. This basically demonstrates that one has to be a little more careful about describing these echoes I was speaking of.

35:00 They have to be hierarchical with the source deciding about the URLs of the ones that are at close intervals to the emitter first, and then successively to the ones that are further on. But the knife is also building an emergence of the future in the pseudotime transaction scenarios that I was talking about a little earlier. And one can ask if the transaction interpretation is deterministic. If it is deterministic, there's no problem with Ronald McDonald's argument, but my view is that it's not. The current constraints of the transaction interpretation do not determine the future of whether they place constraints on physical laws. It's rather like when we go to the grocery store and get a cashier of your debit card, there's a handshake between the counter and the bank, which determines that you have enough money in your bank account to pay for the groceries you bought, but does not determine what you decided to buy. You decided to buy milk or beer or whatever. And this is where, but what's going on here is that the law of conservation of money is leading force. It can't be created or destroyed. I actually can destroy it. It's pretty hard to create it. Anyway, the demonstration is that the emergence of the future from the present is rather like a frost forming a windowpane. You have long fingers or causal handshakes that probe the future. The present is not determined by them. It's only constrained. I don't think I'll talk about the error of time. I have some things to say about the error of time in transactional interpretation, but let me just go to the conclusions since I'm running out of time. The transactional interpretation provides a way of understanding the counterintuitive aspects of quantum mechanics, as advanced retarded handshakes provide a way of understanding the intrinsic non-locality of quantum mechanics while preserving the constraints of special relativity. Among the quantum mechanical interpretations, the transactional interpretation is unusual in providing a graphic way processes, including quantum computing, I might add, and also provides insights into the nature of time and the emergence of the future from the present. The references are here. The original paper was published in the 1986 issue of Reviews of Modern Physics. I have a book chapter that was published in 2001 that's on the web here, and I will put I talked about a version of this talk on my website, I believe it's already on the conference website, and that is the end.

37:30 Thank you for that very clear conversation of your ideas, but because it was so clear, it leads me to... Yes, that it's disturbing, I mean, it's dangerous to be clear, right? No, not that it works, it's that I have to do exactly the same, and have done exactly the same, with Bowman's interpretation. All the paradoxes he has claimed... Except that, I'm wrong, but the Bowman's interpretation is inconsistent in general, but it's special relativity, isn't it? It's not true. Everybody's missed the paper that Adrian and I wrote way back in the 80s, where we actually showed exactly how it fitted into the special environment. I think this is what makes me depressed, though, because you've got these different interpretations which seem to be consistent, because people are really broken to try and find out what is wrong with that, if there isn't. And in the end, people just have to make up their mind what they prefer. And we don't seem to have learned anything new and that's... My feeling is that some of the interpretations, in particular the Mini Worlds and the Copenhagen interpretations, have problems with the Afshar experiment, although I think people are probably declining wires to work in... Well, I'm sorry, I'll just use that because I've actually answered Afshar in the Copenhagen interpretations. Because I'm very interested in all these interpretations. I know a lot a lot of people think I'm a bummer, and I do get ever so angry when I say that. Oh, you're a bummer. You can't say that that's the only one I believe. In fact, I don't believe it. I'm just saying that it is a way that you can understand that he's removing the paradoxes that you can also resist in. And that's I think it's a problem. I would say it wasn't a bad idea. I'll explain you before. Let's see, you mentioned something about the ice forming in the window. You went kind of fast. I wonder if you could take us back to that slide. What I was saying was that these transactions were leaking to the future, placing constraints on the press.

40:00 That was it, you just passed it. Placing constraints on the present, but they're mainly constraints that involve conservation laws, like you don't want to emit more photons than are received, and you don't want to have energy and momentum constraints, and so forth. The last paragraph there. Is there a present? Long fingers and console handshakes probe the future, but the present is not determined only constraint. Let's see. The handshakes probing the future, going into the future. That's what the question is. Is it all local in time? Well, that's the thing is is the present a point or an interval. The issue here is whether it requires a block universe in which things are sort of frozen for all time and we're just sort of dancing through it. without thinking you have free will, without realizing that everything is predetermined. My argument is no, that these connections with the future are not determining the present from the future, but only constraining the present and the future to follow conservation laws and so forth. What the handshaking is about is sort of making sure that things come out right and hold into the transaction. Is the president a point or an interval? It's a fuzzy. It's a fuzzy interval. Are you comfortable talking about Aeronasta yet or not? His interpretation of quantum... You're talking about post-diction and prediction? Yeah. I'm sorry to be honest. I mean, I think, you know, it's very, the future constraints do affect what one measures. It sort of imagines the future interfering with the past to produce the present. Yeah. Now, the question is, is the present that he defines by imagining it being this interference between a prediction or a retrodiction

42:30 different than the presence you're constructing as being a fuzzy interval? My inclination would be to say no, except that in the 1986 paper, there's an analysis of one of his things in which he's claiming, maybe it's not what he would say today, it was what he was saying in 1985 or something like that, but basically saying that creating a paradox solving post and prediction, which was analyzed by the transaction interpretation, showing that things were a little different than what they were saying in that particular paper. It was Aronoff and two of the other people, I forget exactly. I asked him about your interpretation about the transactional stuff, and he's claimed it's different than what he's doing. Now, he wasn't able to explain why it was different, But when you ask the question, he's adding normalism to quantum mechanics, and this is simply talking about standard quantum mechanics. So they're a little different in that regard. Yeah? I don't know if I was giving a talk. I used to follow an example of what this pre-diction and post-diction is about. My wife and I is a calendar, and sometimes she makes appointments that I don't know a lot. And so my future is already determined by what she does. Now, if she tells me about it, then that doesn't work. So there has to be some uncertainty in the constraints. But you can make those constraints, not let me know. That's a very nice example. I actually wonder if there's any application of your ideas in biology. The notion of biological systems somehow anticipating the future and the possible presence of quantum effects somewhere at the core of biological systems I suggest that if you had a way to... I think that requires actual transfer of information, though, right? And as I was saying, the way this... There's a lot of discussion about causality in my review of my research paper.

45:00 And I was making the distinction between micro or weak causality and strong causality. The weak causality would say you can't have influences coming back in time, even at the microscopic level. And the weak causality says you can't have observer-to-observer communication across time-like intervals. I think what you're talking about would involve, effectively, observer-to-advertence, observer communication over time-like intervals, wouldn't it? I mean, if you have biological systems being influenced by the future, then you could use that to send permission. Well, you do, but it's within the system. As though the system in a future state sent the message back to itself earlier. Don't go there. You don't want to go here. Yeah, Henry Staff has published some papers in which he's investigated, sort of started with Steve Weinberg's investigation of what happens when you introduce slightly non-linearities in the quantum mechanics, and it turns out that under those circumstances you can't have those kinds of influences you're talking about, but as long as quantum mechanics is linear, they seem to be blocked by what I was talking about, Evermer's theorem, which is fairly rigorous as long as quantum mechanics is linear. It's linear. Yeah. Yeah. Well, I'm testing it. Memory is predictive. In other words, more memory is . Because we know that something's going to happen. It might happen. Because it's similar to what's already happened. Yeah. But that's not really communication with the future. Thank you. Other questions? Thank you. Thank you.

47:30 Other questions? Thank you. Do you have a slide? Do you have a slide? Good. Better? Better? Time. Yes, that we call the time. We call it time. Whoa. So, this one will protect us on the screen, right?

50:00 I'm going to thank the organizers first, Andre, and my family, and Benrich. and others who have been involved. It's a very nice, it's a lot of work, and it's been very good. I also have learned things about Charles Erisman at this conference, and in the previous one I attended here, I think the category theory is in decline in the United States. I don't think that's true. I think it was more fashionable a few years ago, but it's part of our world now. And there are lots of people in the United States, of course it's a big country, who are interested in it. And classrooms will come and go, but it's here to stay for sure. And it uses. Now, getting to my talk, I'm talking about what Iraq's whole theory. We should come before the last talk, because I'm perhaps working my way towards what John talked about. And I think, actually, in a way, what I'm going to talk about is some of the motivation to be a leader, a leader, a leader, a leader, a leader, a leader. On the other hand, thinking about one of the talks yesterday, somebody mentioned how category theory is used superficially by some people. This could be a little category theory perhaps today, but it will be in the superficial variety. there will also be some physics John said yesterday something about the interesting versus the new there will be interesting things in the talk there will be new things in the talk however, you may have any trouble detecting new things, this is old physics and it's certainly, it's part of

52:30 the motivation for the kinds of things that this transformation of the vision that we have finding ideas however, I'm not there I strongly believe We're inclined to believe in causality and resist the zigzags of we have inclined as mathematical artifacts but we may have no choice but to accept them when it's trying to make sense out of the picture. So this is a little ancient history. I should talk about motivation, this great book, the paperback, the collected papers on quantum electrodynamics, also the book of Sylvan Schwerer, which has lots of good stuff in it. Many other books I have trouble with, just cannot follow what's in them. And some of the papers in here I find quite unreadable, but a few are very readable, although you just have to keep reading and reading and reading, or at least I do so I'm going to talk about single particle theory multiple particle theory the Dirac theory, the next steps and then stop again, this talk could have been given let's see, 80 years ago 75 years ago, and maybe in most of the was, probably let's see okay, single particle theory, so I'm giving you a setting This is, I'm a mathematician, so you're going to get in a mathematical framework. A separable Hilbert space. A state is a point or an equivalent class of points in non-zero vectors in the Hilbert space. We have a one-parameter family of unitary operators, which update the states to time t from time zero. And these operators satisfy the Schrodinger equation, or a version of it. You take the time derivative of u is equal to minus ihu, where h is the Hamiltonian operator. I'm taking it to be time-dependent, but potentially time-dependent. However, those of you who are mathematically inclined don't feel it have to interpret anything. We can even think of some of this as finite dimensional, and this is just mathematics.

55:00 H of T is some bounded operator. That's one feature of my talk, is I'm not going to assume the Dirac equation and the complexity, the difficulties associated, but I'm just talking in general terms, maybe 20% in some respects. Okay, so this is a one-particle sentence. There's a mistake in my abstract. One sentence starts out saying, electromagnetism is seen to be blah, blah, blah. What I mean by that is a theory of a single particle in an external electromagnetic field. So this is how that theory developed. Well, this is my version of how it developed. I shouldn't take much credit because this is pretty straightforward. Okay, so now I'm going to pass to the multi-particle theory, where we move to a tensor algebra. That is, I'm trying to get this through if you can see it. We have the Hilbert space itself, but the tensor product of two copies, tensor product of three copies. We have the vacuum, no particles present in the complex complexes. We take a direct sum of all of these things, and we have another Hilbert space, a grand Hilbert space. I'm not going to tell you what the inner product is, it'll just take a few minutes of time. And we can think of a vector in here as a tensor product of one particle state, x1 up to xn. Or we can take any of your combinations of such things which are mixed states, not product states. However, we're going to pass immediately to what I'm calling the anti-symmetric talk space, probably what everybody else calls it too, a subspace of this Hilbert space where we permutations of these product states corresponding to fermions. If x1 equals x2 the state disappears you can't have two particles in the same state. This mathematical notation comes from French mathematicians By now, very classical. You're assuming particles are fermions, right? You're assuming baryons.

57:30 You probably have both of us in the same state. That's right. No, no, I meant baryons. Now, let's see. I'm trying to state the transparency, so I put some of my stuff at the bottom here. Let's see. Okay, so we've got, so these are our states. Linear combinations of fermion states, n variable, including n equals zero. Is that unreadable? I don't want you to read too much. All right. Hold it. All right. Anyway, okay, now at the bottom. We're back in the original Hilbert space. We might have a bounded linear operator. Again, take this to be a very simple operator so we don't have to worry about unboundedness and problems. Such an operator in the original Hilbert space induces an operator in the tensor space or space, and it's given by the formula, third line from the bottom, acts as zero on the vacuum space, it acts as a on the one particle state, it's a tensor identity plus the identity tensor a on two particle states, et cetera, and the last two lines show you, it just acts individually on the firewalls and that's even in the in the product space before you move the fermions or either way so this is just a way of inducing and it's really categorical if you think about it. Operators working on one space you take a pump for and go over somewhere else. Okay, onward. Now, we're still in the multi-particle theory. I'm going to take an orcanormal basis of the Hilbert space, and now I'm assuming I guess separability, since I'm indexing the basis of positive integers. So, we've got the infinite dimensional Hilbert space, and we're going to do the standard creation and destruction operators. Okay, so these go from the anti-symmetric space to itself and take the n-fold systems to m-minus-1-fold systems. Standard construction, this is the way mathematicians describe stuff like this. You can say loosely the operator destroys the state e sub k

1:00:00 Okay, and then it has an adjoint. There's an inner product around here, and this is an adjoint with respect to the inner product, but I'm not going to tell you what it is, and I'm not bothered with it. So the adjoin starts with a vacuum and joins the state EK, or takes a state consisting of particles of X1 up to Xn and joins EK. If EK is present, though, already in the list, this is going to give you a zero, because we don't allow termions to, two particles to be in the same state. Now, here's a theorem that the operator we just talked about, the one induced on the clock space, can be written as a combination of these destruction and creation operators using the inner product back in the original Hilbert space. Somehow you can take an operator on a bigger space and, with the presence of the creation and destruction operators, reduce what it's doing to some inner product back in the original Hilbert space. of Feynman equals to the Vader. Furthermore, now if we want to talk about evolution on this new space, once we've got this, if we take the Hamiltonian on the original space, move it up to the multi-particle system, and then apply the exponential law to it, we'll get the evolution on the new space. And it's no great surprise that the evolution of X1 up to Xn, all simultaneously. They're interacting, the particle is interacting with an external field. Okay? Multi-particle here. Furthermore, you can, if you don't like to solve exponential, the symbolic exponential equation, H of T, depending on time, you can do approximations, linear expansions, and more sophisticated expansions. But I'm giving you a sense that this is computable. Okay, now we're going to turn to the D-Rack theory, a way of presenting it. Again, most people have seen this.

1:02:30 Throw in something else, we have a particular operator back on the original Hilbert space, which I will call, for lack of a better name, the free particle Hamiltonian. I will tell you the Dirac's Hamiltonian at this spectrum right here, positive energies and negative energies in a gap of 2nc squared. We don't really need that. Instead, I'm going to assume that the space decomposes into two subspaces, one of which I'll call h plus and the other h minus. the negative energy states. And that's all you need to know. We don't need to know what this projection value measure is associated with the pre-particle halitone. We don't need to know what the spectrum is, just that they're two orthogonal subspaces. I think, for my purposes, I need to assume that they're both infinite-dimensional, and that's all we need to know. Then I will say H decomposes as the direct sum of these two spaces. It needs to be understood that H0 are invariant under H0 and under the evolution that goes to H0. You're saying that the invariance under H0, that's an assumption. Yes. I mean, it follows. It's certainly true. I can't look at the deer. I can't tell me. The three-part plan. I'm just talking about a single particle. Okay. Now I'm going to pick an organ on a basis in another assumption. I don't know if you all can see this. We have an orthonormal basis, but I'm now going to make it subordinate to the two subs basis. That is, all of the orthonormal basis elements are either in H-plus or in H-minus. And again, I'm going to assume that both of these are separable. And so you just decompose the positive integers into two sets. Positive integers going with basis elements in H-plus, positive integers going Okay, that's just a little educational thing. Now, what am I going to do? I'm going to digress. The main part of my talk is the next slide. This is a digression I'm inserting before I give you the next slide. I just want to remind people of some mathematics that you can form, if you have any set whatsoever,

1:05:00 you can form little L2 of that set. It's all functions that go from the set and the complex numbers with the property, but the functions are square-summable to a finite value. The set can be arbitrarily large, and when you start summing it, it will be countably non-zero. It must be zero, except the countable number of members of the set. So this is little L2, and the set is a... This is a Hilbert space. It's not necessarily a separable Hilbert space, though. It is, at best, as countable, and it's not as separable, at best, as unenountable. This is just mathematics, as we say. I'd like to consider three cases where the set we have in mind is the power set of the positive integers, which is the sigma-algebra, closed under countable unions and intersections, arbitrary unions and intersections, and complements for that matter. And I would also like to mention another set where we're only interested in finite subsets of the positive integers. And that set, of course, is closed under finite unions and finite intersections. And then thirdly, a set which I will call P sub D in honor of D-Rack. A set of all subsets of the positive integers which intersect the set N-plus in the finite set, and these complements intersect the set N-minus in the finite set. So these are finitely many positive integers from the set N-plus present, and finitely many numbers of the set N-minus are not present. All but a finite number are present. This also is another algebra source closed into finite unions and finite intersections. I want to consider these three as candidate sets S. Now, all of them are subsets of the first one, the power set of N. So if I take one of these things, one of these sets, a subset of the integers, I can associate with it. And John used bras and kets. You see I'm using the ket notation here. So we have a set I1, I2, et cetera, perhaps finite, perhaps infinite. So I'm going to use the slash I1, I2, dot, dot, dot, bracket to refer to a particular

1:07:30 member of little L2. Now, these members of little L2 are functions which associate with each member of the power set of N a real number. Well, this particular one associates, it gives you one if the set T is the set inside of the cats. It's zero otherwise. It's actually a basis element for little L2 of P of N. These are, this is a vector which has one N is equal to one and all other N is equal to zero. It's indexed by sets. Okay, well that's more mathematical nonsense. And now I guess I'm going to give you my main slide, which I claim in a way summarizes one of DRAC's great ideas. I'll show you part of the slide. Okay, sitting over here on the left, we have the thought space and the disruption and creation operators mapping that space to itself. However, this space is actually isomorphic. of the set of finite subsets of N and in fact the isomorphism the standard isomorphism is the one right down here just take the state consisting of the the product state consisting of fermions having states EI1 up to EIN I've normalized by root N that coil which is related to the inner product and you go over to the vector I just mentioned in little L2. So you can rewrite the space as little L2 space. This little L2 space is contained in a non-separable little L2 space, namely little L2 of the power set of the integers. And the destruction and creation operators rewritten in the little L2 notation you have a very natural extension to the non-sufferable L2 up there. So that's at the top of the picture. Natural extension, you can write down the formula, and I will write it down on the next slide. However, it's also true that you can look at little L2 of this Dirac-type power set,

1:10:00 which is another subspace of little L2 of the power set of N. and when you're out there in the middle column you can restrict to this subspace over here and the creation and destruction operators go down in a particular way. Then you can take an isomorphism back to the fox space a very natural isomorphism which is written down at the bottom there perhaps a little hard to read and it It induces some new creation and destruction operators on the, on the, uh, top space. Now what in the world is going on? Well, I want to say what the isomorphism J is, which takes anti-symmetric clock space over to the second little bit of L2, uh, on the non-standard set piece of D of N. This is, I'm just dressing up a mathematical notation about what Dirac did, I believe, and make it understandable for a mathematician and therefore aliens of the rest of its species. What happens is, when you're looking at the clock space now, and you look at a product state, EI1 of the EIA, EJ1 of the EJB, where some of these are in H+, and some of them are in H-, them over to this element of the piece of D of n, where I went up to my A are present, and all of the integers, all of the other integers are present. Let's just say N minus. All of N minus is present, except the integers J1 up to Jb. So these are the holes. Now, Dirac is working over in a little L2 space, as I do believe, but we're working back in the earlier L2 box space, what we described earlier, except now when we see these negative energy electrons, Ejy up to Ejz, we now think of them as something else. We're going to call them positive energy positrons. And there's a little more interpretation in this view that I'm not giving you. It's appropriate to think that the charge is reversed, the electromagnetic field acts in such a way that the net charge of your system doesn't change.

1:12:30 I find this is a picture describing, in some sense, or motivating Dirac's adjustment to creation and destruction operators and his whole theory. Okay, I'm close to being done, and now I'll just draw a few conclusions from it. Well, first of all, in this picture, when you move to the non-separable Hilbert space, the creation and destruction operators induced at these formulas here. If you start with the states indexed by I1 and I2 present, you end up with a state indexed by I1, I2, and J, provided J wasn't present there already, zero of J was present. That's a creation operator. I'm looking at the operators on the L2 spaces, the L2 spaces. The destruction operator, you start with the state corresponding to I1, integers I1, I2, etc., and you remove K. April was present. Now, K was not present. You kill everything in sight. I mentioned that the middle index minus 1 to the J plus 1. Don't look at it too closely. You do remember that points in the building space are up to a scale factor representing the same state, but I'll come back to that in a moment. I'd also like to point out that the following this diagram, that the new DERAC creation and destruction operators agree with the old ones on if K belongs to N plus, the indices that go with positive energy states. They reverse if k belongs to N minus. Now, then what do we do? Well, we have lots of things to be done. Interpretation, I'm skipping some things, but I will just talk about a Hamiltonian. I'm going to use the same formula that I used earlier as going to a multi-particle operator. Same formula, but this time taking the Dirac operators. Instead of the creation and construction operators So you have four sets of terms, and you can interpret the terms as terms that permit

1:15:00 transition to an electron or an ordinary positive entity of electrons from one stage to another. Terms that correspond with the destruction of an electron in a electron, which is of of course, conserved charge. Terms correspond to the creation of electron positron pair, also conservedly charge. And terms smeared by transition of a positron from one state to another, also a charge . Now, if you accept this, there are a few footnotes here. But then you can go and write down the evolutionary law, or at least an approximation for it here, times, small times. Okay, so that's it, except for a number of details of interpretation and next steps leading towards the trans-transactual interpretation, Weaver-Bynum, and many other things. First of all, there was a little lie back there. I offered you a Hamiltonian, but there's some difficulties with the Hamiltonian, H sub D that I looked at in the last slide. The difficulties can be illustrated if you say, take a special case, Hamiltonian. Then if you sit down and write down what the DERAC formulation gives you, it gives you this thing. And the first set of terms of projections on the positive energy states and the negative energy states, now being interpretive. We just get the new name for the Hildred Space, which is the same Hildred Space. The first set of terms is a nice nice bounded operator. The second set of terms is a nice bounded operator too, assuming H0 is bounded, which is not even true, but this is sort of a model case. I haven't shown you the Dirac equation. Remember, the Dirac equation is quite complicated. It was a great achievement, but we're trying to avoid some of the things that are focused in on a few others. The third set of terms, unfortunately, the trace of h0 pi minus, which in almost any reading is going to be infinite. So we have a problem. This is just a constant term as a scalar, a trace of h0 pi minus, just a number. Unfortunately, if h0 has a spectrum which is infinite and a symmetry between pi plus and pi minus,

1:17:30 then this number is going to be infinite. So we modify the thing on the previous screen. One solution, one possible solution is to drop this scale of the term, which is a scale of the trace, the zero, by minus, times the identity. And if there's a scale of the times the original state, we drop it because, as most of us know, that constant adding to the Hamiltonian, in most circumstances, can be ignored. It doesn't affect anything physical. Okay. However, there's another problem. What to do when you replace H0 by H of T, where this is a true, there's a true external field. Well, if the field is not too large, you can probably do the same thing above and you won't get into much trouble. So I'm going to say you perturb the previous days, you know, just one little phrase here, use findings methods. That's several more talks, and after you've had those talks, then you would go to John's talk, the Wiener planning and formalism. One thing I should say, going back to the last slide, is that even the Dyrite formulation, if you look at this thing back here, you notice that the evolutionary operator, H of D of T is defined in terms of what's going on in the Hilbert space, a single particle. Again, we have an inner product, H of T, EK, comma, EJ. It's back in the Hilbert space for one particle. We all start to throw in these creation and destruction operators, but we don't have to work in this vast Hilbert space, built up by sensor products and symmetrizing. We can work that and finally exploit that in a big way and give you many wonderful things with it. Okay, back to finishing up here. So we've returned to the previous case, the H0 case, throw out this trace, maybe, and use Flangman's methods. Now, the question is how justified Flangman's methods are to a mathematician at times. And certainly, in some cases, if we're looking at things like standard operators, all of this is completely justifiable. The third set of problems, which is perhaps somewhat smaller, but is disturbing to a mathematician, and I think disturbed the number of instances, too. You look up there, you see the C, C, H0, C, high minus.

1:20:00 That C is a conjugation operator. It depends on the choice of a basis for this whole business. And it only affects the negative energy parts, that is the new positrons. It's a coordinate system dependent. And in an earlier slide I showed you a minus one to the J plus one, which is somehow depending on the ordering of the positive integers. So there's something in there that doesn't have this intrinsic coordinate-free aspect to it. What's the solution to that? One solution is just ignore it and keep moving forward. Okay, so that's, again, in many... I'd like to give a couple more talks to the way I shall prepare them, leading towards a full discussion. My point of view is that I'd like to hold on to causality as tightly as I possibly can and only take what steps I'm forced to to the other point of view. I believe that what you said is completely consistent with quantum mechanics, But I'm just concerned that there may be a way that these intermediate tests are a certain vocution that Feynman and Wheeler invented, that people are attributing reality to some of the interactions forward and backward when it's weak of the convergence of the final sum that's needed. However, I'm working my way, I think, towards your point of view. Okay, that's it. category here to play with some of this, because I never said anything much about H0 or HFT, or the other states. Question? Why are you so stuck on deficient causality? I'm a reductionist. That is, I believe in trying to reduce that. It's not that I, you know, that's my religion. It's just that I think a way to advance is to try and insist on things and see if you can force them to work. And if they don't work, they don't work. But abandoning them too quickly, I think, is a mistake. And I think causality is a great principle in science. It's much greater than the evening's test of this conference.

1:22:30 Causality. I mean temporal causality. And in fact, there have been discussions of violations of causality here. And boy, those are the most interesting things going on in science. Anybody find something that's really a good violation? What? What? Categoric theory is 100%. I have no objection to formal causes, blind causes, et cetera, but as a scientist, This paradigm was invented very useful mathematically in things to try to make it work. If it doesn't work, fine. I'm going to push it as hard as possible. Well, now it comes to ontology. Now it comes to reasons and the question. Yeah. In other words, the Aristotelian point of view is perfectly wonderful. but this scientific and mathematical point of view one of the wonderful things they're always talking about predicting the future and parapsychology and this and that my god scientific law which takes the present state and the law and tells you what's going to happen in the future is a wonderful thing that dare die may reach outlived its usefulness but I'm one of those like pushing make sure it's outlived its usefulness I think it probably has and that's why your talk should have come after mine Well, I have to do that. I haven't given my talk. I'm before yours, because I've lied to the phone. I agree. I'm in another way. I'm looking at some things that were done 80 years ago, still trying to puzzle out what they mean. I think a mathematician has a lot to contribute in this theory by reworking the ideas and finding out some of the gaps, Maybe in a logical way where you might do it intuitively and get the answer much quicker, but we'll help you with certain kinds of building blocks. I felt the clear is that C-Stylogy would say about it. What does C-Stylogy mean to action? They use all kinds of stuff. No, no, no, it's true. But generally, the mathematical approaches I see tend to be non-logistic. I'm kind of pushed towards relativity, although you may not see it here. One thing that's always bothered me about the idea of the the graphite of having negative energy electrons in the vacuum and pulling them out by positrons is that it seems interesting like that. A, it seems confusing to answer that there are private electrons in their own rather than positrons. B, why doesn't the vacuum have negative charge if there's all negative electrons in there? B, why doesn't, if there's negative energy stage,

1:25:00 why doesn't the vacuum have negative energy? You didn't say anything that would tie in very closely to any of those points, but what would you have to use? I agree with you, because I've been bothered by these things too. The one slide, if I can find it, it's probably right under me, no wonder I can't see it. Right here, if you're in the next to last column, you're talking about the questions you asked. If you're in the last column, things have disappeared. You just interpret everything you see. positive energy electrons and B positive energy positrons and all of the infinite C's disappeared. I read an article by Digner, Dyson or somebody where he said that the infinite C negative energy now stands as a wonderful metaphor, blah blah blah but unnecessary. Okay, so that's one approach. The other approach is the primitive mind. You transform it away. It's just motivated some madman to do this. A great anything to come up with this stuff, that it would be transformed away at the end here. That's right. Maybe it's the motivation and it's wonderful. But now, a friend of ours, John of mine, who said, says that he found that it's actually rather convenient when you're doing the kinds of computations and stuff, depending in terms of the native energy seed. But it's more intuitive than working over here where the seed is gone. You just have positive energy part And all positive energy, electrons, and positive electrons. The other thing that I was going to cover too was that you're talking about very honestly, positive energy is very nice to have positive parity and negative parity. How does the parity get flipped in the process of one from one to the other? And that's some of the interpretation that I didn't get. I can't answer that question either. in plenty ways, plenty ways going on mathematically that are sort of forced by aesthetics. Okay, now, are you ready to go with that? Okay, I can solve an equation, the wave equation, without any reference to all the things you can do or space. Without any reference to anything you can do or Hilbert says you're this or that. Go and develop earthly adequate solutions. Right. I don't have any space to talk about it.

1:27:30 And I can figure out the properties of the solution. And the properties of the solution are going to know what interests me, as well as... Why are you not? Because you might make a... You might make a time zero. You need something in the future. Once I've got a solution, I've got a function. Something f equal to t. You usually have a f plus g, where one is going... No, no, no, no. I have a slide at 5, I mark time at P0, and I proceed forward, march forward in 5. Okay, well, if the solution satisfies what I hesitate to call the causality boundary condition, the initial time condition of a given state, then 5. Okay, but what I know is I know certain things at a certain position, I know things at a certain on it, and then I proceed with an issue. Physics does not give you boundary conditions. Boundary conditions. I don't have a precedent. Yeah, but the precedent is I have a fixture here, and proceed forward. Now I can't talk about having a fourth fixture. Well, you can do the exact same strategy works with your application. You can proceed of an equation of a source, which is what the Iraq did, say assert that I know these things at such and such time proceed forward. I don't have to make any reference to these strange states you're talking about where I get the paradox of space intents or non-intentities depending on whether I do a transformation. What if you have a final time to write down the rest of the time zero? Well, that's a difficult connotation, but in principle, beyond the particles, you probably can't do it. But the point is, are you telling us something that's mathematics about the inadequacies of the physicist's mathematics, or are you telling us something that is actually science, something that relates to physics? I'm paraphrasing what Dirac said in a mathematician's type language and trying to understand it. In paraphrasing, is the physics going away? Are you taught us there's something wrong with the physics?

1:30:00 I hope the physics isn't going away. I mean, mathematicians always have a strange way of looking at things, perhaps, but I'm sure through most people in the audience that you can bear with our in other words, I'm certainly not trying to get away from the physics coming into it. I believe in mathematicians, even with Newtonian mechanics there's hundreds of years of mathematical development to make sense with Newtonian mechanics, the beautiful theory that it is wrong or not. And I think the mathematicians have something to contribute with human formalism and, you know, disturbing it when you look at a few different angles, the elephant. You know that, too. Well, I want to know, you know what a really hard time about, is that infinity you've got, that this is the third term, the scale of infinity to the trace, is that an indication of something that's all related? The solution for what HD was, it's sort of arbitrary, too. I mean, I based it on the other case, but it's an arms class, you decide how you get here, and trying to examine explicitly what decisions are made. Thank you.