11th UK Foundations of Physics Conference — Day 2

0:00 Thank you. Here is the title, and the work is done in collaboration with Oreste and Virginia at the University of France. Now, what are reduction theories? As you know, the reduction principle in the standard formulation of quantum mechanics allows to interpret the two-way function as describing a linear system. What is then the trouble with the reduction principle, it is that it can be formulated only, it is accompanied by a distinction, by a distinction between the quantum system on one side and the measuring apparatus on the other side, and on the other side, and this is a on the other side reduction theories try to describe reduction by a definite mathematically precise correction to the Schrodinger equation and it is supported this connection is buried in any circumstance how is this done a stochastic process is introduced which induces a suitable set of quantities when I say quantity I mean what most people call an observable I don't like this word so I call it a quantity and the operator which represents it this induces the quantities of a suitable set to have definite values what does this mean? the stochastic process shows the wave function towards the eigenstates of these quantities or the common eigenstates of these quantities which are saying that these quantities are stochastically sharp Now, the quantities which can be stochastically sharpened are defined microscopically, are defined quantum mechanically, but have a microscopic character which is in short time. And second feature, they are always related to position.

2:30 What is the result of the process? the result of the process is a localization of the microscopic objects all successful theories are based on a spontaneous localization process which becomes effective only when the macroscopic level is split and what about the standard reduction principle there is nothing acting in a strong way on microscopic systems the reduction is obtained reduction is obtained via via the micro macro correlations which are settled by the by the measuring apparatus what is the difference between non-relativistic and relativistic spontaneous localization in the non-relativistic case in the non-relativistic case as we shall see the definition of these microscopic quantities involves an integration of a tiny space region which is a sphere and the region why it is a sphere and such spherical regions cannot be reduced to a point because of two reasons first, the quantities would lower their microscopic character this means that they would become sensitive to the microscopic system structure of the this system and furthermore the processes would amount to the focalization of the constituent particles Okay, in the relativistic case, the difficulty, what is the difficulty? The difficulty is that these spherical regions are not, of course, Here I have listed them and I shall return on this. But let me say just now that they cannot be considered as fully satisfactory.

5:00 Now, how does it take place this sharpening? Well, consider this stochastic differential equation in the intersides. We have the usual Schrodinger term and a term which we call stochastic term which contains this stochastic process B of T here this operator H psi is a nonlinear operator here is its expression and this particular expression is very important that we shall see and the linear process is the usual and it is normalized in this way well it is easily shown using that this equation conserves the norm of psi now what is the fundamental equation of property of this equation it is that drop the Schrodinger term for a while and then one can show that the limit for t equal to infinity of psi of t is just the projection on an eigen space identified by this parameter epsilon of A of the initial state vector which is normalized, of course, and the probability of ending up just in the eigen space epsilon is given by the square knot of the projection. This result can be proved formally, but it is not at all surprising. Why? It is not surprising because as you can easily see if you apply this non-linear operator to psi and psi is an eigen state of A then you get zero. So that the stochastic process stops when you reach an eigen space of A. Well if both terms are kept then there is, if it is there, there is a competition between the two processes. The same thing can be done for a set of compatible quantities, there is one term for each quantity,

7:30 the stochastic process has this property and this causes the reduction to the simultaneous of the eigenstates of the operators Now, I'll describe the non-relativistic the non-relativistic this is essentially the only one which has been considered apart from some minor modifications. Now, in this equation here, replace the distinct line by x which is the running point of space, in space. sum of i becomes integral of the x of course and to make this particular choice of this ai lx is the of course which represents the mass contained in a sphere of radius a in the center of the i take the ti independent of x because of reasons of translation and and then replace variance, and then replace the stochastic process by one by a set identified by these stochastic variables, by a set identified by a continuous index with this problem. For example, the operator NX can be defined in this way it is the integral of the sphere, I was mentioned before, of a operator, which can be defined, for example, in this way, in the non-alarmistic case. The sphere is macroscopic but small, and the suggested value is the radius, for the radius of the sphere is something of this kind, intermediate between micro and, well, already macro, but small. Why macroscopic? Well, it is clear, because otherwise this these operators here becomes sensitive in a microscopic structure, and why small? Because otherwise, the process when the quantities of X would not would not discriminate between different between different microscopic distributions of mass.

10:00 Well, the important properties of this quantity is, as I was saying, that they are able to discriminate and microscopically distinguishable mass distributions. Of course, one must choose the value of this strength constant, and it can be shown that G can be chosen so that the resulting theory has all the . Now, we need now a relativistic, well, let me say a few more words. The previous proposals of relativistic theories, reduction theories, as I have already said, they are not fully satisfactory, and I have a criticism, a soft, but I have a soft criticism against it, and it is that the problem is considered mainly as a technical problem of theory, and Waters proposes to, in a sense, to go the other way around, that is to say to start by trying to identify a suitable set of microscopic quantities to be inserted in a stochastic equation of the type I have shown before. Now, we need an relativistic framework, and the most natural one is, well, better, since it is a manifesto covariant, and such a framework is provided by the Tomon-Hagas-Binger equation, which is an interruption with the equation, it is written here, and where sigma is a space-like surface which replaces time x is a point of this sigma here is a point of x and this delta sigma of x which appears here is the full volume of a small bubble of a small bubble around the point x which produces a new sigma sigma prime i is a superscript here which indicates that we are in the interaction picture and this is the

12:30 interaction picture interaction Hamiltonian density provided it is probably defined this is the Lorentz scale so this is manufactured one there is a problem of integrability what does this mean that we should get the same result for example if you consider two different powers we You get the same result if you first advance the hyper-surface by means of this bubble and then do things and vice versa. You get the same result and this is a consequence of the of these operators for space-related points. Okay, you can choose a particular reference frame and write the usual interaction feature in Newtonian and you can also go to the selected feature if you want these two equations of course are not managed to covariant but they are covariant provided that the ingredients are properly performed now I propose a relativistic stochastic equation which I write in this way it has the same step there is a Newtonian term and there is a stochastic term which is written here which is the same structure as we have seen before this Gaussian random variables, this delta beta appearing here we have these properties and they are independent for different space line bubbles so that one has done this, this equation is, if this is a scalar, this is a scalar, this is manifest in covariance, and I can also observe that this prescription here is compatible with the arbitrary smallness of the space-time factor. There is also here a problem of integrability, gravity but I shall discuss it oh the idea here is that these operators these operators SI which appear here should be suitable should represent suitable macroscopic here also one can

15:00 one can choose a particular reference frame write down the interaction feature equation and also the Schrodinger feature equation and the relation uh this s i is related this s Schrodinger feature s is related to the interaction like this by this formula and the problem is now to identify the operators s i of x which have been I shall try to do this job in two steps. First, the operators SI are defined in this way. They are the integrals of the suitable integration domain, d of x, of operators, small, lowercase, s, i of x. And the support that d of x has a Lorentz invariant definition, and this s i of x is also a Lorentz scalar field, and it is of course at the end it shall be defined with field operators at space-time point x bar what about the domain? the domain is defined in this way there are other ways of defining it but things do not change very much You know, this is the humane which is described by this inequality. The microscopic operators shall have the following interpretation. They will be defined as representing the spatial density of a quantity of stuff. and this stuff is related to the presence of measured particles and therefore the capital S operators represent have the meaning of a time-integrated amount of stuff because this is a four-dimensional domain

17:30 now, what are the problems? well, there are problems coming from the definition of this domain integration because this domain of integration extends to infinity in space and therefore the states the values type of all types of states in which we apply it in principle must be intended as states of the universe and this has an important consequence because there is a contribution to the values of these quantities coming from remote staff this constitution is overwhelming so that these quantities could become insensitive to locally these different institutions of staff what is the situation we are confronted with it's a situation of this kind we have a superposition one state for example, one state this glass is here, in the other state this glass is here, but all the rest is the distribution of stuff is the same. so that these operators could become insensitive to the local distribution of stuff, which is the type in which we are interested. Now, how do we define stuff? We propose that the density of stuff is the quantum analog of the classical quantity invariant trace of the energy momentum change. 10 minutes including questions 10 minutes including questions I skip I must skip the discussion of I must keep the discussion of things go in the right in spite of this particular shape of the domain deofix nevertheless the equation is sensitive to the local distribution of strength unfortunately i have no time for this this is done in two steps

20:00 first one considers a local system and one shows that that the quantities are sensitive to the distribution to the different distributions and one goes to the state of the universe and one can show that things going the right way because of the particular structure of the operator which appears in the stochastic equation. which is S minus psi S psi, so that the contribution from, and this has the consequence, that the contribution from the remote staff cancels between the two terms. Ok, I will come now to the open problems. Of course there are open problems. The first problem is, do the time-integrated amount of staff operators as I have defined them, do they really describe classical microscopic quantities? This problem arises from the fact, again from the particular shape of the domain of integration. if one, you know, if one fixes time then the space shape of the domain for large time difference is that of a spherical shell of very small thickness when the time difference t bar minus t becomes large so it's a spherical shell very very very thick And this would make these quantities sensitive to the microscopic structure of the system. And this would be bad. However, we have seen, well we have not seen actually, but we should have seen But the sensitivity of those quantities to the distribution of stuff, for the distribution of stuff, for this sensitivity, tails are not so important.

22:30 So I hope that we shall be able to prove that this problem is not there. Another problem, this is more subtle in a sense and more dramatic. What does it mean? It means that we should get the same result if we first advance the space-time surface in a region and then in the other region of bicycles. And the problem comes from the fact that because of the presence of the case in that domain then these operators as I of x do not commute exactly, at least they do not commute exactly, neither, even if the separation is spaced, right? They do not commute exactly, neither among them, neither with the Hamiltonian And this, of course, could give rise to different results in the two ways of advancing the space but we must take into account that the stochastic term does not cause a time evolution in the usual sense if the whole business works what is the consequence of the presence of the stochastic term the consequence is just reduction so that in the sense the wave function is given the state vector is given and it is reduced to one of its terms in the correct way and we have no reason to believe that the statistics of this reduction will not be the right one then where is the problem well unfortunately the problem is still there why let me come back to the to the non-relativistic case in the non-relativistic case this here once these quantities are given for any

25:00 once these are given the result is determined one gets a certain reduction one gets the reduction to a certain state it should be it should be happening the same thing here it is to say that the quantities I have indicated before as delta beta, once these are given, not given if you accept this expression once they are given, the result of the reduction, the reduction should be determined the final result should be a specific term in that certain position we are considering. And, of course, if we get different results in this reduction process, it will always be a reduction process, but if we get a different reduction, a reduction to a different term, to be first advanced here and then here or vice versa that would be let me say that the appeal of this kind of theory would dramatically fall down i have a final consideration still if if you take the c going velocity of light going to infinity limit of this model here, you get another interesting model I have described before. And on the other hand, and this kind of model is, apart from minor modifications, this kind of model is the only working one, is the only existing one, I can say. and that which I have presented here is a so natural generalization of that non-relativistic model that if it doesn't work in the sense I have just described I would become very pessimistic about the possibility of a relativistic reduction thank you Antonio, I think there's a problem with relativistic reduction theory

27:30 if they give the same predictions as ordinary formal quantum minimum theory. The ordinary formal theory predicts the form of non-locality because of the violation of the Benelty positive. And this appears to rule out any rates invariant reduction theory. I don't know whether the difficulty you are mentioning is related to the integrability problem I mentioned before. Maybe it is. However, let me note that in the non-eratilistic case, you have a vibration of the pair inequality. the web, all that, and well, apart from violation of relativity, that is from a certain point of view non-palatable because of this action, at least dance, even apart from relativity it is difficult to swallow. However, it is there so that they... I'm saying this is a problem with the theories, not the problem with the world in which we live. We have time for one more brief question. I have a question about your density of stuff. You suggested that the trace of the energy momentum might be a good relativistic definition. As I'm sure it's clear, if you have, say, massless photons, then that means you're not going to have any stuff, which... Could you repeat, please? The last sentence. If you're talking about photons where the mass is zero, then you will have no stuff because of the tracelessness. Is that something... Electromagnetic field, which is the only field, the only classical field which has a direct field, the trace is zero. The trace of the energy momentum is zero.

30:00 I have no problem with that. I think that if stuff is related to the presence of massive partners, that is okay. May I follow up? Another possibility that comes to mind would be some sort of eigenvalue decomposition of the stress tensor, in terms of, say, some eigenvalues and eigenvectors. Does that seem like it might also work or would there be disadvantages? Thank you. I have no answer. Let's thank Alberto again. Our final speaker of this session is Andrew Whitaker from Queens University of Belfast, who will give us another scheme on implementation. This is work done with Pt-Panferl. This will be largely in the quantum teleportation setting. Of course, it's well known in quantum teleportation that we need both quantum interaction and, of course, then a following classical signal. It's easy to say, of course, this is an obvious requirement to avoid fuss and the night thing, the fact that we need this classical signal coming through. But it still may be an interesting question to ask, well, what elements can go in a non-local way with the quantum interaction? What has to wait for the classical means? Of course, if we start off with EPR bone, bell kind of context, there are very standard answers to this. We'll say any non-locality relates to correlations between measurement results in the two wings of the experiment. Now, in Schmochmann, it's terms outcome independence, or what Jarrett called completeness. And it's true there's non-locality, but we'll say observation of results in one of the wings, but one wing does give non-local information, usually probabilistic in nature, on measurement results in the other wing, but we'll also say, of course, since these results are not in control of the experimenter, They aren't used to send signals, so there's no problem with relativity.

32:30 On the other hand, it's usually maintained that what's called parameter dependence in Chimini's terms or locality in Jarrett's terms is upheld in these experiments. And this means that the decision on which quantity measure in one wing cannot influence the results obtained in the other. And this, of course, will mean we can't send signals faster than light and have this able to live with relativity. As I said, this is what is conventionally certain. Certainly, if one interprets this, what we've just said, as the expectation value of the results, it certainly must be true. Otherwise, you could have faster than light signaling. However, there are cases where certainly faster than light signaling is prohibited, it, but it may still be plausibly argued that something, which, I wouldn't perhaps want to give a name if I certainly don't want to call it information, but something about experimental procedures in one wing does reach the other non-locally with the quantum interaction. And here, we look at these ideas in the context of quantum teleportation. Okay, I'm sure that all or nearly all of you are very familiar with quantum teleportation. I'll just run through it, just as I say, to set the scene. Alice and Bob, they share a pair of particles in an EPR state. So this is, two goes to Alice, three goes to Bob, this is our EPR state. That's particles two and three, particle one is in the possession of Alice, and the state vector of this is the one which is to be teleported. any state vector for particle one. So we put this and this together, and this is our initial state of the whole system. We now like to write this in a different basis. This is the bell basis, of course. The bell basis is the states, complete set of states of maximally entangled states. This, again, is our EPR state. These are the psys with up, down, plus or minus sign. These are our fys, up, up, down, down, again, plus or minus. So these are our set of bell states. And just mathematically, we can then rewrite the psi 1, 2, 3 I had just a moment ago

35:00 in terms of the bell states for particles 1 and 2 and states for particle 3. And we see we have this correlation between the Bell states and the interesting states of particle 3. This is what we started off with, apart from the minus sign, which obviously doesn't matter very much. These other three are slightly different. Here, one of the signs have changed. Here, in fact, the A and the B have swapped round in those two. So this, again, is just maths coming from the previous statement. Now we've come to really the physics here. Alice now does a Bell-State measurement on particles 1 and 2. Of course, we have to say now, what does Bob and C? As we've said, each measurement result correlates to a particular result of Bob's particle 3. Alice will get a Bell-State result, and Bob will get a measurement on particle 3. Bob's statement is left in one of the four states, the probability of each being a quarter. And these four states, these are more or less what we just had on the previous sheet, just rewritten. That's the one we noticed, which was AB, just as we started off from, apart from this unimportant phase factor. The other three, well, we have these simple matrices coming between them. The second one is they give us an original state only by an irrelevant phase factor. The other three actually are related to the original state by rotations of pi about the x, y, and z axes. So, Bob has got one of these. He's very close there, if you like, just with the quantum signal. But he needs to know which of the four is correct. So, to recover the original state, Bob has to apply the appropriate rotation, or in the case of the second one, just leave unchanged. So, of course, he needs to know which action to take, which rotation, or whether to leave unchanged. This information, which Alice must provide over the classical signal, over the classical channel, on the basis, of course, of her result for the Bell measurement. So Alice now has, of course, there can be no cloning. Alice has no trace of the original state. She just has the Bell measurement. Her system is left in the Bell state. But Bob will have a copy, if you like, of the state we started off from. Just a couple of points about the quantum teleportation.

37:30 Of course, quantum teleportation is not a flow of information. Bob actually doesn't have any information at the end. It is only a state vector at one place, transformation of a state vector at one place, where Alice has it, to Bob at another place. Alice can know psi, and there's no particular reason why Alice can't know psi. She can know how the state of the system is set up, but Bob cannot know it. he in his possession. Again, we can ask, does this violate local realism? The answer seems to be yes, from Zukovsky, and then Clifton and Pope. So, that, as I say, is just a reminder of quantum teleportation. In contrast, in the scheme we're going to be looking at, in fact, we will be sending information from Alice to Bob. We now then, this is what we're We now consider a series of trials in which Alice takes the succession of particles as particle 1, each with the same state vector, AB. We set up the state vector in the same way. We can achieve this by measuring an appropriate component of spin with a Stierengel. And we just take, we extract spins of a merge in a particular beam. So we can, essentially what we've started off by doing is what we've done so far, we do it four times. so we imagine Alice does a Bell state measurement on a group of four such particles, let's say the whole thing is done four times just to make that a little bit clearer we consider here only trials where she gets a different result in each measurement, she'll get four different results in the four measurements of course, there'll be lots of times we won't, she won't in fact I'm going to be 32 times she does this, 29 times she won't get a set of four different results, but three she will concentrate on those. Obviously, in the end, you'll have to let Bob know whether this criteria, I'll call it criterion Q, is obeyed, and this is our classical, this constitutes our classical sequence. So we're looking at these cases where she gets different answers for her four measurements in the four different cases. Now we come to what Bob measures. Bob has measured, But in each case, he does exactly the same thing. And of course, he may get a set of different answers. He measures S set 3 in each measurement.

40:00 H-bar over 2, of course, or minus H-bar over 2 in each case. So he then sums his answers. And of course, he may get different answers with different probabilities. If they're all plus H-bar over 2, he'll get, of course, 2H-bar. The probability of that, which means we work out, is A to the fourth, B to the fourth, also for measuring 2H-bar. Or, of course, you may get three up and one down, if you like, or three down and one up. We can work out the probability of those. You may, of course, then get the sum from summing to zero, and we'll get that probability. We'll put these probabilities, of course, adding to one. Now, if we look at the expectation value, and this takes us right back to the beginning, if we look at the expectation value, then everything is very dumb. averages to zero. But we don't do that. If we look at S z squared, in fact, this will depend on A and B. It is not going to be zero. This will give us some sort of variance of S z. This is zero only in the rather boring cases in which A and B are zero. The additional state, remember, was A up plus B down. But we now consider something different. That was the cases where Alice did do these Bell measurements. Now we consider the case where she does a different measurement. And what we hope to transfer from Alice to Bob is what measurement she does. She may do a different measurement, measuring SZ1 and SZ2. And again, we'll consider cases where she obtains four different pairs of results. She'll get four different sets of results. And again, from the state vectors, we can show, in fact, that in this case, she's bound to get the probability of SZ total being zero is one. She's bound to get that. So that is different from what we've had before. Now, of course, we can quite happily say that at the level of expectation value, of course, it doesn't matter what measurement she did. There was no effect on Bob's results of what measurement was made by Alice. But at the level of variance or standard deviation of Bob's results, there is an effect. And we've got to work this through in a standard deviation way, although this is not, in fact, particularly necessary, as I shall say in a few minutes. We can work through this. We can, in fact, normalise a sense divided by 2h-bar. We can get a variance. We can get a standard deviation for the results.

42:30 Whereas, of course, if she did the other measurements, the standard deviation would have been bound to be a zero. Well, I don't want to discuss what we can introduce from this, but first I'll just look at two slightly different cases. Now, we said initially that the fact this is only going to work three out of 32 times. four strike rates, and we can easily improve that by saying rather than Alice having to get four different results, which is rather unlikely to happen, we'll just say she'll measure, when she does the results, she'll do two pairs, she must get one of the psys and one of the fives. This she will do, in fact, in half the occasion, so we improve our strike rate, if you like, from 330 seconds to a half. We can go through the sums, I won't to bother to go through all the sums, they come out more or less very much analogous to what we had before. So, if we're worried about that low strike rate, we can certainly improve that. So, I won't go through the details there, which are very straightforward. I'll just refer briefly to one other point, though. We may, in fact, go the other way. Rather than measuring just one group of four particles, as we have done so far, we may choose to measure, in fact, N groups of four particles for N experiments altogether. Again, we can just go through these sums, which I won't labour. It's a little bit more tricky, obviously, but straightforward. I'll just dwell on the point that, in fact, our standard deviation, which we're discussing, this is now proportional to n, or rather, the standard deviation, in fact, would be proportional to root n. And I just dwell on this, because occasionally people have said, when talking about this, This, they talked about it as a scheme for finite ensembles. The implication being that it will disappear for infinite n, for an infinite ensemble, according, as one thinks about these things, I think, according to the central limit theorem. Now, the reason I think we're inclined to think about this way is that we do tend to normalise things. by dividing by 2n, we do that, our results turn out to be root n. And we say this is some sort of narrowing according to the central limit theorem.

45:00 However, I would stress again that it's the root n which is the experimental result. This kind of effect was discussed a long time ago by Girard, Eurimini, Weimer, and they were kind of criticised by Newton for essentially saying that these things matter because the movement said, in the infinite ensemble, these things disappear. GRW came back and argued very conclusively that, in fact, that wasn't the case at all. They got worse, more striking in the infinite ensemble case. I'll move on to experimental application. I mean, the experimental application is fairly straightforward in principle. A series of entangled pairs of particles are sent to Alice and Bob. For each pair, Alice also receives a particle in this state from the Stern-Gerlach. Now, just technically, it may be preferable to aim with a single substantial run with N large. This is because, in fact, if we take the first case, N equals 4, one set of results is not very helpful. We're bound to get zero if we do the second measurement, but we may get zero if we do the Bell state measurement. So one set of results is not very helpful. Whereas if we go to large N, in fact, if N is large, the probability that Bob obtains the result of zero when Alice does a Bell state measurement becomes very small. So a single satisfactory run almost certainly settles the case. So we don't need to go to the standard deviation. A single result will be satisfactory. Let's now turn to the theoretical implications. Of course, there is no suggestion that FTL signaling is involved. There's the need for a classical signal. This is this criterion. We have to say this criterion was obeyed, that ALICE did get the correct spread of results. So there's a need for a classical signal to establish that ALICE had equal numbers of results in the different categories. And neither, again, to come right back to the very beginning, neither made it say that the type of measurement carried out by Alice affects the expectation value of Bob's measurements. But it is also very difficult to maintain, as you might like to do, that the type of measurement performed by Alice has no effect on Bob's results.

47:30 I mean, let us say, if and when Bob receives the message, Alice said my results were okay, you've got some genuine information. So if and when Bob receives the message from Alice that her set of results obeyed the required criteria, criteria in a particular value there, that message is mutual as regards to two possibilities. That gives no information on whether she's done measurement A or measurement B. So in a sense, Bob feels that that information must have come with the quantum signal. may cause Bob to understand he'd done Bell state measurements or measurements of SZ1 and SZ2. So it's clear that, well, what I shouldn't really call information, but you can perhaps call it potential information latent information, must have been with Bob before the classical signal. It must perhaps have followed a capital faster than light in a non-local way. Non-local if one ignores here the possibility of hidden variables. And just to spell this out a little bit, Bob knows he will end up, at the beginning of the experiment, Bob knows he's going to get one result at the end. A or B, it was a Bell-state measurement, or the other measurements, or he has to abort, because Alice's results were not suitable. So there were three measurements to start with. On receipt of the EPR particles, Bob is able to say, well, it was A or abort, or B or abort. Only on receipt of the classical signal, he'll say it was A or B or abort. But it's rather striking here, I think, that we have three results before the quantum signal, after the quantum signal, only one, of course, after the classical signal. It's now about 10 more minutes. It's a minute to break. Yeah, thanks. Just one detail again. The scheme has been set up for maximal entanglements. In fact, it will work quite nicely with non-maximal entanglements. If you put in here, this is our initial state. If you put in that with a C1, for example, there with a C2. You do, the sums, obviously, the more complicated the result goes through in a very satisfactory way. Okay, just to finish off with, then, we'll say what we've got, because it is actually a transfer of information. In fact, although we put it in a quantum teleportation context, it is really much more like quantum cryptography. As a means of passing information from A to B, the scheme is obviously a candidate for quantum key distribution. We've got information

50:00 and Bob can have shared information. Because as noted, the classical signal, as we just said, on the classical channel, the yes-no contains no information of benefit to Eve at all. So it's already looking quite a good candidate for quantum cryptography, but the only problem is to ensure that any eavesdropping will be detected by Bob and Alice. As we've said about at the moment, Eve can be hammering away, it won't actually help her very much, but she may be interfering with things Bob and Alice won't work. We have to adapt this somewhat to make sure that we can detect each problem. Here, very much, we've kind of blended the well-known Bennett Brassard protocol into what we've already done. What we had previously was Alice did a Bell measurement or this measurement, A or B, Bob did B dash. We have to extend these, in fact, so Alice may do A or B, but she may, instead of measuring SZ1 and SZ2, SX1 and SX2. Bob may measure S said 3, but he may measure SX3. And it is fairly clear that if Alice does that, B, and Bob does B dash, that's just what we had before, of course A is no problem at all anyway, if Alice does C, and Bob does C dash, that again is fine, but there may be a mismatch. B, C dash, C, B dash. Now if Bob makes the right choice, B, B dash, or C, C dash, he'll always get the correct result when Alice has done the correct measurement B or C. But in other cases, if he makes the wrong choice, this is BC dash or CB dash, he may not. So that is how things are set up. By the public channel, Alice then says whether she chose B or C, if, in fact, she had chosen either. If she had chosen B, she says W. If she chose C, she does Y. Now, of course, the last thing we want to do is to tell Eve whether it was B or C or A. So if it was A, we have to adopt a subterfuge. If it was A, she just puts W or Z over randomly. so this is the kind of thing we all have this was Alice's measurement whenever she had B she puts W whenever she had C she puts Y whenever she had A she puts W or Y

52:30 randomly Bob of course independently of course with no knowledge of what Alice has done has done B dash C dash C dash B dash of course completely at random now some of these WB will be right, that goes forward as a tick. YC dash will be right. Some will be mismatches. WC dash will be wrong. YB dash will be wrong. Those are dots. That information now has to be discarded. This information is good information, shares now by Bob and by Alison Bob. So incorrect matches are discarded. WC dash, YB dash. The remainder forms are key. Bob's measurements now say that the weather is up. it. This is the results you've got, and this is A or B or A or C. So we'll write this as this. These are the results you've got. A now becomes 1, B or C becomes 0, and this will give us a shared key between Alice and Bob. What about Eve? Well, the best Eve can have done would have been to measure SZ3 or SX3 more or less at random. Sometimes, of of course, Alice and Bob will have been in tune, and she'll be in tune with them, if nothing happens. Sometimes Alice and Bob will be out of tune, so whatever Eve does is irrelevant. But there'll be cases, of course, where Alice and Bob measured the same thing, Eve was out of tune. Now, so Alice and Bob should agree, but Eve will have done a wrong measurement and disturbed the system. And in the standard way, then, Alice and Bob can check this by comparing publicly a portion of the key. This portion When they should be, obviously things are fine. If on the other hand, perhaps one out of eight cases, the result is wrong, then that shows there has been an eavesdropper. So this scheme is capable of, as they say, checking that the eavesdropping has been going on. So actually, just to summarise this scheme, I think it's kind of an interesting scheme, I think. I'm not particularly putting forward as a, perhaps, a terribly useful scheme, one doesn't really know. What I think is pretty interesting, though, as I said, to start off with, is how it spells out what, at least in this case, has gone with the quantum signal. Certainly, it does seem something has gone with the quantum signal, and what then has to follow on in the classical signal. So that's all. I'll finish at that point. Thank you very much.

55:00 I suspect that the conclusion about the quantum non-locality of the latent information would look rather different in the Everett interpretation, as you might expect. In Everett, when you perform a local operation, in the presence of entanglement, it's going to have an effect on a global state. We can argue that's not particularly necessarily a non-local effect. But it's not going to have any direct effect on anything else, and it's only when you lug some system which is necessarily a quantum system over to the other side, doing a perfect measurement there, that the information is going to get across, and you can make that explicit, as though what you've had been doing in their interpretation. So I just expect that what flows well will look different when you've got the average interpretation, well I haven't thought quite how it's going to look. But I mean, really, in a sense, nothing is ever non-lobal in sight and effort interpretations. You could always wait and do your world splitting at an appropriate time. Would that be correct? Thank you. Since whatever Bob actually gets really came from the source and not Alice, why is this called teleportation? I said which isn't really a comment on your presentation sorry, going back to what I did or the original quantum teleportation it's a question more on the industry than on your presentation Yeah, I'm sure it was called quantum teleportation to attract the maximum interest and money. I mean, it does seem to me that it is sort of very much what one means by teleportation, isn't it? Something is here, we duplicate that state over there. If one wanted to transport study from A to B, you would have to wait until the universe and both sources produced two studies, coincidentally, or something like them. all the rest now that seems to be a long haul from teleportation to my mind i i thought it is i i say i'm not responsible for that name i thought it was the start obviously it's i'm not star trek here i must have met a little bit of a loss here but i agree actually a scottian and a blank scottian you fill in the fit in the scottian yeah okay i

57:30 I think it's obviously much easier in this federal to have a two-state system here and a two-state system there. Coincidences are much more likely with a two-state system than the Scottish. Okay, right, yeah. If there are no more questions, then let's thank the speaker. So I think we're on break now until 11.15. That's a generous break, so let's be back. No, no, no. to give a talk on a configurative philosophy of the event of the nation. Thank you very much, Jeeva. Can I thank the organizers, Nabi Brown, and Jeeva, for all the work they've done? So I could be pretty quick in the beginning part of my talk, but I do want to concentrate on more of the software and the software. So I do want, equally, I want to rapidly go through hope we all agree on. I think there's quite a lot that we agree on, which is what makes the topic interesting. So my favourite way of formulating the present situation is in terms of the consistent history score, which I hope most of you are familiar with. So the most simple and straightforward formulation of this in terms of projection operators, as I've indicated here, hasn't there been a picture of the actual functions. Even connecting onto face-based cells, that could also be configuration space, can be other, I've got other information, at least the pointer appears. And that's going to be relevant to some of the functions of this program. And what one finds is that, given that one's using something like face-face cell configuration space, then typically, if I delete a projection from this string here, this is describing a history, alpha zero is one, face-face cell, alpha one will be another, alpha two another, and so on.

1:00:00 and it has, as it were, a sequence of snapshots, and these are going to be some coarse-grained description of the total systems cosmologically of the universe as a whole. But if one deletes one of these projections, then the probability for the history, as it were, one time not specified, would just equal the sum of probabilities for the histories with all the different possible states of affairs at that time that the problem is missed out. Okay, and that would hold, if and only if we had the consistent solution, it would hold to each k, if the problem is satisfied. And this is a very direct and straightforward interpretation. Many of these economists have tried to understand this as some sort of analysis of what probability means. I think that's quite the curious way of doing it. I think in a simple way to see what this is about, it's a sequence of state reduction, the sequence of state reduction. So what one would have on a standard formula is that we could perform sequence of measurements. Now, the thing is, what happens if you miss out on measurements, not only sequence of measurements? Can you just derive the probabilities where you sum over intervening states of affairs where you didn't, in fact, perform a measurement. The opposite of the pan is not getting any interference effects insofar as the state reduction is not actually eliminating any phase relationships and so on. So the point about this condition being satisfying is that you can compute the probabilities whether or not you apply the measurement postulates at the various times, T0 up to Tn, onto the corresponding projections. Now, what you also find, and this is much more important, of course, is that for these projections onto cells and phase space, the problem with it is a sharp repeat over the history of satisfying local quasi-deterministic laws. so one has effective Hamiltonian flows one has effectively quasi-classical equations of motion and this is where one recovers

1:02:30 an approximately classical world so using that with such restrictions of the identity of each K one obtains what is usually called a quasi-classical history space for a quasi-classical domain and the most important example is Brownian motion where one has a massive particle in a thermal bath usually one has models this in terms of harmonic isolators one can do this in the judicious formalism, also in the one gets very similar results one can also do it using identical formalism this goes back to final environment, this is the So here's the, one's looking, one's expressing similar ideas actually, in terms of intervals over the history space, but now it's a completely fine-grained, so we use this expression one that contains some over-paths for the result. Coarse graining now, but one obtains an expression like this of the way one makes this sort of approximation for the action, modeling the gravity motion. One has one term representing dissipation, The second term here, which is the coherence term, the idea given here, is due to exponential damping for parents of the coherence function now, which differ, q prime, q, and q, when this quantity is large, then obviously it's going to be much coming down. So when this quantity impacts of the order of one, which is when this is satisfied, one or more, then one expects to see a very rapid damping of half diagonal terms. And that takes place, well, for example, is massive of the order of grams, temperature, room temperatures, centimetres, then we have deco-earance times of about 10 to the minus 40 as a ratio of the relaxation time. And I think actually the more important thing to make here is that it's an order one for OK, so this is really telling us how far down this model is going to apply, giving us quasi-classical

1:05:00 behaviour to apply all the way down to small molecules. OK, so pretty well everything in our terrestrial environment can be modelled in this way. as long as we're looking at temperatures in the order of 10 keldens up we're going to do the same thing in terms of functions, the Fogler-Planck equations it's a cramice type, they're just written down as standard expressions for face-to-face density functions, in the high temperature limits this is the usual expression relating the density matrix to the Ligna function and in terms of it we have now the equation just substituting into this for the density matrix we have exactly the same sorts of expressions here for the off-diagonal elements, this is the very equation derived by the Calvary-Legget model Now, other sorts of models, diffusion of foreign bodies in fluids and gases. Here, foreign bodies, well, it's now talking about really something like dust. So we're going to view some interpenetration of the fluids. Hydrogen average equations for integrals over local densities. studies of behaviour of these and what happens is continuity equations and these are the usual multiplicative operators in which the quantum mechanics is the quantum of these densities and if you can only second quantise them then you get the usual expressions of quantum field theory number of densities momentum densities and so on and you find again that you recover classical hydrodynamical equations And now you have no preferred coordinates, as you will, in the actual fields. The point about this sort of work is that we've always known that classical equations are approximately satisfied by expectation values, for case case variables, in SGM. But what we're finding here is that the probabilities for history is different from the classical ones,

1:07:30 the probabilities of managing is small. for the sorts of systems here considered thermal baths, fluids histories concerning these hydrodynamical variables the point about them is because of the continuity equation they're approximately conserved and exactly concerned quantities we know what exactly decoherent well so here's a couple of references which will review the sort of material anyone who wants I would recommend anyone has a couple and this I think is the end of the agreement I don't think anything that I've said is dispute, I hope you all agree that we have achieved a great deal understanding classical dynamical processes can be very much say implicit somehow in the state, let's just say these are structures or dynamical structures within the literature dynamics and are publicated by the literature dynamics but is that the whole story what happens at this point in the literature Now, apart from those who may have stick their necks out, most physicists really have little to say from this point on. So people like Elmer and Hartle will, at this point, are noncommittal as to precisely how this accounts for classicality. Precisely how we can understand these sorts of results. And Halliwell is another example of someone who is very diffident about, say, any more. And it's easy to see why. The two straightforward questions that one wants to pose, one is that there may be other kinds of effective dynamics, and you'll be entirely different kinds of issues. Another is that, even given a unique set, how are we to select out a single one?

1:10:00 so they bear on one another is there a unique history if you say yes to this and the various alternatives to Simon von Mechanics insist that there is only a unique history but then the problem is rather focused on this question on what is the digital space Of course, these programs offer ways of selecting out a unique history space. Now, if one says no, one's in the realm, clearly, of the average interpretation, which is what I'm giving a defense of. The answer would also be no on many lines theories. But for most of these people, there's a fact for the matter as to which history is ours. So there's no real problem with probability. But it makes it almost still sharper now, well, what is precisely the history space? The idea is that there's a unique history, every possible history is real. In that case, what exactly is the history space? It's on this version where one says, no, there's no unique history. But further, there's no factor in the matters to which history is ours. The real world has just the superposition of the histories, and therefore one can represent that same superposition in accordance with different bases that one makes the problem of interpretation of competency very acute. I had some names up there of people that roughly take the line that I would defend. if there's no unique history, all histories are accepted, and further one has a genuine superposition that seriously is then one has no uniqueness on the third basis. Now, how bad is that? Well, the functionalism of consistency alone, I think it is pretty bad, because you can deform one consistent basis into another. You can have a history space consistent up to a certain point, modify the basis beyond that time, and we'll then do it in more or less arbitrary ways, and then subject to consistency. Consistency is too weak. So what one wants is something much stronger, someone wants something like the structures that we need to pick out need to be more robust than merely consistent ones. And that's what we have with quasi-classical domains. Now, are there other kinds of quasi-classical domains, other characteristics?

1:12:30 Well, maybe there are. It doesn't really matter. The point is, we, our normal environment, is constituted out of one of those dynamical structures. So, well, how expanatory is this sort of thing? What I want to just draw attention to is that this is a very general problem in philosophy. One is all the time dealing in philosophy with, as it were, levels of explanation appropriate concepts to some given phenomenology. The great standard case of it is in philosophy of mind, going from neurobiology to folk psychology, how appropriate is it to include intentionality of the agents and so on? If the underlying system is realizing those intentions is a neurobiological one, what about propositional attitude psychology? How is that to be, as it were, reduced to neurobiology? In the set theory, if one takes propositions in practicality seriously, as an exact, descriptive set of sentences and so on, and applies standard, formal methods, model theory, and so on, what that's really doing is looking at the statements that we can actually make in terms of set theory, and what has then led to very familiar problems of vagueness, sororities, paradoxes, and so I need to go back to the beginnings of philosophy. Tim Williamson here at Oxford is famous for insisting that we must have, as it were, a precise, realistic struggle of ordinary language in terms of its separate terms, which leads into the view that there is a fact for the matter, which is precisely how you hear, so you must remember somebody's head, if that person is to come, but not bored. This is taking realism too far. Further, that someone very famous in philosophy of mind, taking part of the subject of aptitude psychology to, or existing, but for every concept that we use or are capable even of learning, there must have disillusioned some innate competence to use that term. The more relaxed attitude is to say that when patterns or forms are explanatory and predictive

1:15:00 and useful, then we can take them as perfectly real. and that is the view I think that fits best with the errors of interpretation I'd like to say there, as it were the preferred basis problem ends, there is no mention of mind, there is no mention of observers there is no mention of experiments there is no mention of measurement the many-minds of interpretation I think has done a great disservice to the error approach in, as it were mixing in those sorts of elements that really derive more from and the Copenhagen Interpretation. All right, now, the one place where there is an issue about the landfall and so on is with the probability, because what now of the interpretation of probability in this scenario? I'm getting very worried about time, actually. You have got five more minutes. I have five more minutes. Very sorry to say, I don't think I've had much time for the discussion. Now, let's see. The issue, probability, in a certain sense, I think, is also vast. There's nothing about the mind, nothing about combination, nothing about the mind. And that is at the level of what masters usually call objective chance. Let me just give you a general framework for thinking about this, which is extremely influential. this is due to david lewis here in supervenience the doctrine of all the rest of the world is a vast mosaic of local matters a particular fact just one little thing and then another and it is no part of the thesis that these local matters are made we have geometry, a system of external relations of spatio-temporal distances maybe points in space in itself, maybe point-sized bits of matter or fields, maybe both we have at least points in a global of qualities, a perfect piece of an arrangement of qualities, which mean nothing bigger than the point at which to be substantiated. For short, we have an arrangement of qualities, and that is all. There is no difference without difference in the arrangement of qualities, or else supervenes on that. This works with the means process, and we see this. Okay, and what is he talking about? Well, laws of nature, counterfactuals, causation, persistence through time, mind, language, and so on. This is a very austere program,

1:17:30 the great majority, I think, of philosophers have their respect for this program, although they may not believe that it can actually be carried through. Lewis himself is pessimistic that it can be carried through in the case of chance. Okay, now, why is chance posing a problem? Well, Lewis puts it in terms of the principle principle, the principle namely that our credences, our rational expectations, should track objective chances. so roughly he was certain that Cornelius Fair must give equal credence to handsome tales Lewis' problem, I can see dimly how it might be rational to conform my credences about outcomes to my credences about history symmetries and frequencies I haven't the faintest notion how it might be rational to conform my credences about outcomes to my credences about some mysterious unhuman magnitudes there's more to say in this thing And it's worth studying his views on this. My point, though, is that the ratios in amplitude squares in the branching structures that we're familiar with, the deodorant interpretation, the claim is that chances supervene on those. And these are perfect, even in the magnitude. The theoretical qualities, arrangements of qualities of the sort that Lewis is concerned with. He doesn't list amplitudes, quantum-mechanical amplitudes, he's concerned mostly with the classical picture, in which spatio-temporal relations are the fundamental ones. But with quantum mechanics, I think we must have made additional fundamental relations in the whole space and all. So the claim is that this pattern, this structure state is just what our ordinary understanding of chance objective chance in the world supervenes upon and I see the difference here I see a rather little difference here with say what does things supervene on randomised emotion, what do thoughts supervene on, neural processes and so on what colours supervene on spectral reflectances of surfaces we learn in a chance to supervene on these constructions in common mechanics.

1:20:00 But it can't be denied that there's a further problem. Roughly speaking, how can there be possibility in the face of certain knowledge? Now, I have two examples here of contemporary debates going on in the philosophy literature. I don't really have time to go through them but I will present what I see as a kind of a proof that there must be some sense to the notion of uncertainty in the face of certain knowledge and it's something to do with self-vocation which is the first of these conferences that is a mention now I ask you to consider a simple classical example, and this is even performable, involving the severing of the corpus callosum, such that one ends up with effective division of the mind. and one can take this to the extreme where the body is divided as well so this becomes then somewhat a science fiction-y scenario but I think one that is realisable in classical physics is to act very seriously the question is what does one anticipate in the face of such a division and one's going to have a banquet on one successor, and bread and water on the other, what is one going to anticipate in the face of this division? And one can make all of the usual sorts of caveats, the process is paddeless, the process is perfectly symmetrical, and so on. It seems to me that one only has options, one either anticipates oblivion, or one anticipates some sort of telepathic experience, whereby you both eat the banquet and then the water, or you have the normal sort of expectation that we have in the face of a chance of event. One expects one or the other and one doesn't know which. One can drive this point home. The one case where I would grant me know exactly what to expect is where the experiences are identical to. Now just after

1:22:30 division the experiences are identical and at that point each does not know what room he's in if you like. Is it the banquet room or is it the cell with the present water in? I know that I will achieve the state of ignorance about what is my immediate surroundings so here is a notion of uncertainty in the face of knowing everything that there is to know okay I'm sorry to have gone on so long thank you very much I'm sorry I must say this is really to advertise now a talk that we will be given tomorrow David Wallace is going to speak on how theory can be developed to analyse specifically why is it the amplitude that are of relevance now in evaluating this kind of Thank you very much Thank you Any questions? I'll come in Just a quick question picture of some of the many minds pictures, because there's a fact about which histories are, there's no problem with probability. Could you explain why that is, because it's not so obvious? Well, if all histories are realised, people often say if all histories are realised, they haven't translated any sense, because if all we know is which history we have up to a certain time in the present, we don't know what our history actually consists in from this point on. If we have a probability measure over a space of histories, then the measure of those histories which agree on the one that is ours up to this time and which have some future in the future will then give us a probability for that future event. Now, in fact, that event may or may not occur, which is a fact of the matter already. We just don't know whether our history is the one that contains it or not. So it's a perfectly conservative and straightforward ignorance of interpretation probably will be available in that case, so long as we have a measure over the space of history. And then the situation It's a part of a theory, and any of the usual has just been carried on the theory. Could you just go back to the example we've finished with, just enlarge it a little. In what sense, then, does this...

1:25:00 What sense is it meaningful to me to say that there's a probability of having a banquet or having a bread of water, or indeed having a probability of being in one state of ignorance or another? I think this is a key point here, and I think you're running out of time. I'd like to hear more about it. I agree, and I'm sorry not to have been able to say more about this. The claim is that chance is just this physical structure of division. This is a classical monitoring process. That is what objective chance is. Now, the issue is, how is that consistent with uncertainty? it ought to our credences, credences have ended up with uncertainty so the problem is not to give an account of one objective chances the problem is to give an account of uncertainty how can there be uncertainty when we know anything that there is unknown that is what I'm claiming I'm not getting a theory of this I have theories to offer as to how to analyse the situation so as to make more sense of uncertainty is rather, I have much more confidence in the argument that it must make sense than the particular theory that I would offer to articulate apparition of uncertainty. I say it must make sense that there is uncertainty because I see these three options as exclusive and exhaustive, and I see one and two as simply not panable. But I can use only three. Can you give quantitative probabilities? No, I can't, not with this sort of analysis. I can only use the principle of indifference The principal rejection is defeasement. Now, to give a quantitative analysis, one needs much more. David, well, I hope, will convince us that it is the aptitudes and not some other features of the statements of the state that is relative to the quantitative analysis of the model. Okay. I have a question. I mentioned the syntagmals, certain knowledge. from a physical point of view is the kind of reaching the objective too I'm sorry the point here is that one knows everything there is to know how can there be ignorance how can there be uncertainty one knows exactly what's going to happen one knows each of the two successes what they're going to be doing one could know everything there is to know about this process with brain surgery and whatever that's going on here

1:27:30 what is unknown and yet something is unknown and it is bound up with something like first-person expectation Is there a case that I can't believe denying in that case together with revising our ideas of time normally uncertainty about the future or assumes that there is a future is the idea of time that we usually have. But an older idea of time would be to say, there is no future. The future hasn't happened yet. And that is precisely the situation of your man before the split. For that person, there is no future about what will be their experience. And therefore, there is nothing about it to be uncertain. I would deny that actually in that situation we know that everything there is to know. Well, it's not my view. What I think would be my view in a quantum mechanical case is that the future relative to the person prior to the quantum mechanical experiment is objectively a superposition. In that sense, the future is not the end of the year. The result is really compelling time. We want to see the structures in that future state that will be patterns, propagated patterns, representing people. But a superposition is no experience, so there is no future plan about what is the experience of an absence of service. Well, I do disagree with that line. Fair enough. So, I mean, we wondered if we would have agreement. All right, we should go to the next talk. I'm very pleased to invite Lucy and Harley, who's my colleague at Charlie and I, of the world of faith, and the thoughts that we're going to show, we're going to talk to ourselves, the white quantum theory.

1:30:00 Okay, so I'm currently at Oxford, but I'm very shortly going to be at the Brayford Institute in Waterloo, in Canada. This work has been funded by the North Society Research Fellowship. A number of people have probably heard this talk before, so I apologise for that. The basic question that I'm going to ask here is why quantum theory? So let me put that question in context. When you look at the normal Paulism of quantum theory, it's quite an abstract formulation. It involves all sorts of elements that nobody would really anticipate until they get along. And the way it was built was through this long-running interaction between theory and experiment. And it doesn't really answer the question why contemporary, but this situation is similar, perhaps, to the situation with the Lorentz transformations. And these Lorentz transformations existed. They were apparently theoretically adequate. But it wasn't until Einstein came along and showed that these natural transformations followed from some very simple principles that we had a sense that we understood why those were the right transformations. So the question is, can we do something similar in the case of constant theory? What I'm going to do is I'm going to present five reasonable principles or actions for quantum theory. What I mean by quantum theory here is quantum theory for discrete dimensional novel spaces. And I'm not going to recover any particular evolution, I'm not going to recover any particular homotomy, just the general structure of quantum theory in Hilbert's space with unitary evolution or evolution guided by super operators. And after that, I'm just going to tell you the axioms. I won't have time to go into the duration. I'll give you an indication of how it works. The axioms have this feature, which will become clear.

1:32:30 Four of the axioms, four of five axioms, are consistent. In fact, they're obviously consistent with classical probability theory and also quantum theory. And it's the remaining axiom which gives us quantum theory. This axiom, which I explain in context later, says that there exists a continuous reversible transformation between any two geostakes. In fact, the word that characterised quantum So, this acting is applied to a particular physical setup, but I'm going to describe the sort of physical setup that I'm interested in, and it's this. We have a situation where we have three types of device, a preparation device, a transformation device, and a measurement So the preparation device is used to prepare systems in some particular state. It has a knob on it which you can vary to vary the particular state that you're preparing the system in. It also has a release button on it, which I'll explain in a moment. The transformation device transforms the state that infringes on it, and it has a knob on it which you can use to vary the particular transformation that you affect. the measurement device measures the state. So this has a knob on it to vary the particular measurement that you make. And then it has outcomes labeled by L equals 1, L equals 2, L equals 3, etc. It also has a special outcome which I call the null outcome. So this is a bit like, imagine if you had an array of detectors and the detectors could be labelled by L, C, F, O and A, etc. If none of the detectors clicked, you could say that was the null outcome. The point of the release button is that if you have a transformation device here, which is just the form of, say, a hemistic transformation, then if you press the release button, then certainly you will get one of the null outcomes. So for example, if you release a particle onto your detectors, certainly one of them will fire. If you don't press the release button, then certainly you'll get the null outcome. Now, the reason I put this null outcome in is really for mathematical

1:35:00 convenience. It's not necessary, it just helps. It makes the mathematics much neater. so in quantum theory in this kind of theory in general we can certainly measurements of probability any quantum theory can ultimately be regarded as measurements of probability for example when expectation value is just a probability weight of sum so when I refer to measurements in this talk what I really am talking about is measurements of probability so by measurement what I mean is where we make that the outcome, so when we measure the probability that the outcome belongs to some given set of non-north outcomes. So for example, you could measure the probability that the outcome is L equals 1 or L equals 2. So that's the basic setting which is familiar to us from quantum theory anyway. Now let me define a critical notion, which is the notion of state. So here I'm taking a very basic definition for state. The state is defined to be that thing represented by any mathematical object that can be used to predict the probability associated with any measurement. This is an essentially a non-ontological definition of state. It's a definition of state which just basically tells you what the probability of a second-plane measurement might perform. So once you've got this definition of state, which incidentally is consistent with the quantum state, the quantum motion of state, once you've got this definition of state, One obvious way, one obvious mathematical object that you can write down, the particular state, is simply a list of all probabilities, all conceivable measurements that you could possibly perform. Certainly this object would specify the state. However, in general, in quantum theory, there is some structure, so this would, in general, be too much information. I think in general in any physical theory you have some structure so this would usually be too much information so one might instead consider

1:37:30 some subset of measurements which are just sufficient to list their probabilities to specify the state so this is a set of measurements probabilities associated with measurements which I call additional measurements which are just sufficient to specify the state I'll give an example of this in the context of quantum theory later. For example, in quantum theory, you also establish the state of a sphere of heart particle. You can measure the probabilities associated with spin along the plus and minus set directions, and also the probabilities associated with spin along the plus x and the plus y directions. Those four probabilities are sufficient to determine the state. This fiducial set of measurements isn't necessarily unique to choose many different sets of measurements, but we can fix on one particular set of measurements and put in the fiducial set. So here we get an integer, k, which is the number of measurements required. And this is a particularly important integer in this work. So k, I call it the number of degrees of freedom, it's the number of measurements required, the number of probabilities required in this list. A second important interjection is panic. dimension. I call it dimension because n is actually incompatible. So n is defined as basically the maximum number of distinction states. So I want to be a bit more technical about that. Imagine you have various preparations here by varying this null system onto measurement operators, and you can vary this measurement operators. We're looking for the maximum number of preparations that can give rise to this joint set's outcomes,

1:40:00 which means that they can be distinguished in single-shot measurements. And then we notice there's two relationships in classical and quantum theory. In classical probability theory, k is equal to n. In quantum theory, k is equal to n squared, which is quite suggestive, at least. And it also suggests the whole hierarchy of theories, which have k equal to n squared, and so on and so forth. What I'm going to do now is go through classical probability theory and explain how the structure of that looks from my present point of view. And then do the same with quantum theory. And then I'll take the actives and I'll explain roughly how those actives give rise to quantum theory. So here is classical probability theory. So consider, for example, the ball, which can be in one of n boxes, or the ball can be missing. Remember we have this null event, so we're not imposing normalization. We can associate a probability in each of the boxes, and then these probabilities specify the state. So the state is simply given by writing down the probability for the ball being in each of the boxes. So we see here that k is equal to n because the number of probabilities required to specify the state is n instead of the k is equal to n. An example is a bit where n is equal to 2, so here the state is representing the probability of the ball being in the first box and the second There are some constraints on these probabilities, some of the constraints are represented in this diagram here. There are some special states, one special state is where the ball is definitely in box 1, so we have probability 1 there and 0 here. The other is when the ball is definitely in box 2, so 1 here and 0 here. and the other is when the ball is definitely missing. I'm going to define

1:42:30 pure states to be those states, except the null state, which cannot be regarded as mixtures of other states. So it turns out in this space the pure states are these states. And then we see a crucial feature which is true in general for discrete which is the pure states form a discrete set, the classical probability theory, pure states form a discrete set, and this will contrast with the situation in quantum theory. So I'm trying to describe classical probability theory in a form which is as similar to quantum theory as possible. So let's think about measurements that we perform. So one measurement we perform is just when we look to see whether the ball is in box one. And we know that the probability associated with that measurement is P1. And we can write that down as this vector dot positive with this vector, R1 with the vector dot B. So in this situation, we can say that associated with this measurement, where we look in box one, is the vector r sub one. And of course we can do the same looking in box two or box two and so on. But this isn't the most general sort of measurement we could make. A more general measurement would be the following situation. We could toss a coin. If the coin can look in box 1. If the coin comes up tails, we can look in box 2. In that case, the probability associated with finding the ball is given by this expression, the convex sum of the R1 vector and the R2 vector. So we see that the vector associated with this particular measurement is this one here in brackets. We can think of any sort of measurement in classical probability theory and it always turns out that you can associate it with a vector like this. So in general the probability associated

1:45:00 with any measurement is given by some vector which is associated with that measurement dot with the state, with a vector which specifies the state. So this really is a sort of basic summary of the classical probability theory. We have classical probability is characterized by K D equal to N. It's characterized by the state that we have some particular set of allowed states, which has the feature that the pure states in this set form a discrete set. Also, one can calculate the structure of the allowed moments in the end-of-the-optic set. It's not difficult to work out in the properties of that set. So now, we'll see the quantum theory. So here's an example for a spin-half system, a qubit, on the diagonals we have the probability of spin up and spin down along the z direction, and on the off diagonal we have A, A star, and A, you can calculate, is given by this expression, where this is the probability of finding a particle having spin up along the x direction, this along the y direction. So this quantity is linear in all the various probabilities. So this object is linear in these four probabilities. And we can specify, rather than representing the state by this object, we can equally well represent the state just by writing down these four probabilities. So we see that in this situation, when we have n equals 2, there are two distinctable states, k is equal to 4. It's interesting to look at the shape of this set of lab states. Well, there are four degrees of freedom here, four dimensions, and that's too many for draw diagrams. But we can impose normalisation, we can impose that pz plus c minus is equal to one, and then we just have three pieces of freedom left over, px plus, py plus, and pz plus. And then you can show that the pure states, so then you can show that the allowed states are those states which are inside this sphere, which is inside the unit of Q. This is basically the block sphere in the different coordinates that you should use. So the allowed states fall inside this ball.

1:47:30 and the fewest states are states on the surface of this wall. So then you see an essential property of constant theory which distinguishes it from classical probability theory, which is that fewest states form a continuous set. We also see in this particular case that k is equal to n squared and in fact that's true in general. That's obvious if you just count the number of real parameters in the density matrix. Well, in quantum theory, the probability is given by the trace of some positive operator associated with the measurement times the density matrix associated with the state. Now we know the density matrix is linear in probabilities, and therefore this formula is going to be equivalent to this formula, where we take the state represented by a row of probabilities associated with the traditional measurements, and then the measurement is given by some R-type vector. So we see that in Watson's theory we have also this formula We can also look at transformations, and it's easy to show that the transformations in constant theory are given by P going to Z times P, where Z is at K times K real matrix, K P here is equal to N squared. The final difference is here. Okay. So quantum theory is characterized by k being equal to n squared, p being equal to, so the state's p belonging to the center allowed state, which is characterized by the fact that the fewer states form a continuous set, and some sets also for the measurements and the transformations.

1:50:00 So before I actually state the axons, let me just introduce one more concept, and this is the concept of subspaces, some names because of the analogous concepts in quantum theory. So let's consider first the situation in classical probability theory. Imagine you have five boxes, so you have A equals five, but actually the ball is never found in his last two boxes. so the ball is only ever found in one of his first three boxes or missing well then in this situation actually actually the system behaves like one which has n equals 3 rather than n equals 5 one can one can generalize this concept by imagining that you have the log set the distinguished set of n distinguishable states it could be the case only outcomes associating with m of the n-distinguishable states actually occur. And in this case, we would say that the state is actually constrained to an m-dimensional subspace. So in quantum theory, there is a similar concept. If the state belongs to some subspace of the lower space, then it's a smaller dimensional lower space. Okay, so here are the axioms. The first axiom concerns probability, and here I'm taking a relative frequency approach of probability, but one can change that. In fact, Ridic-Shack is considered a different approach, taking the Bayesian view of probability. It doesn't really affect this overall picture, so just for convenience I'm taking the relative frequency approach. So we say that the axiom states that if you take the relative frequency in the limit of n tends to infinity, the number of times you get a given outcome divided by divided by the total number of x, then this should tend to the same value p for any ensemble prepared by the same preparation. This is basically a stability requirement. Axiom 2 concerns subspaces. So this says, well, first of all, let's surely just systems which n equals 1, n equals 2, etc, etc. And furthermore, that all systems which are of

1:52:30 As I mentioned n, all systems constrained to an n-dimensional subspace should behave in the same way. So we know that's true in quantum theory, we know it's also true in classical theory. If you have a system which is a dimension 3, or which is constrained to a 3-dimensional subspace, then those systems have the same properties. Axiom 3 concerns composite systems. So it says that for composite systems, N should be equal to NA times NB, and K should be equal to KA times KB. It's quite easy to justify the assumptions, although I don't really have time to go into that. Axiom 4 is the continuity axiom this is the crucial axiom it says that there should exist a continuous reversible transformation between a pair of pure states and this relates to the fact that there exists in quantum theory a continuous set of pure states so you can have this continuous reversible transformation Axiom 5 is a simplicity axiom which I'll explain more properly in just a moment. Let me just read it. It says that for each n, you have the situation that k would take the minimum value consisting of the hyperactins. This is my least favorite of the five axins, and I've been trying to get rid of it so far without success. Let me just explain briefly how the proof works that we get quantum theory. One minute. In one minute, you can show that the state is represented by some feedbacks and you can obtain these equations that I referred to previously. You can show that k must be equal to n to the power r. Here's a nice, simple proof, which essentially follows from the fact that k is equal to ka times ke, composite systems. We can show that the k equals n case is ruled out by the continuity axiom. If you have k is equal to n, then necessarily you'd have a discrete set of pure states. So that implies that k is equal to n squared. Then you consider the simplest non-trivial situation, which is where n is

1:55:00 equals 2 and k equals 4 and you can recover the quantum theory of a qubit. Then you use this case to construct the general n-case just by imposing the constraint that each two-dimensional subspace should be this one. You can also obtain the correct quantum rules for composite systems, for transformations, and for state updating. So just a few conclusions. Why the quantum? Well, the answer I'm giving here, ironically I suppose, is that it's because we don't want to have these passable jumps. Okay, these are these diagrams. I guess I should tell you, these transplants were originally prepared for a conference in honour of John Wheeler. So I wanted to try and represent the philosophical position I feel is being taken, or is suggested by this approach. So this is Wheeler's diagram. I think I'm probably slightly misusing it but somehow the idea is that you have an eye attached to a body which is part of the universe and it's observing the universe. The point I wanted to make was actually we're not just observers in the world, we're also actors. So we have hands which change things in the universe and we have classical information going both from the universe to us but also from us into the universe to the actual universe. and actually furthermore there's not just one of us there's lots of us and physics is constructed in the context of many observers and so maybe this is a more appropriate diagram and also give you this representation of how it is a bit like okay, thank you applause applause applause applause applause applause Can I just say, the only reason you have those damn classical jumps is that you have a damn discreet classical state space, and that is not the space of classical physics, Newton, La Cage, Hamilton, Einstein in this classical world, etc.

1:57:30 It's a good job to prepare to move beyond classical physics. I'm not talking about classical physics, I'm talking about classical probability theory. Let me answer your question though. There's too lot of questions coming. Let me answer the comment. In the sort of classical paradigm, we have two basic choices. We can either impose this continuity all the way down, so we can say that actually this continuous nature of classical quantities, which might be regarded as coarse training, actually continues all the way down. We can say that, and that might be suggested by the the usual approach in classical physics, or we can say, actually, at a certain level, we would like to impose discreetness. And there's good reasons for wanting to do that. If you don't want this to exist structure all the way down to zero, then you would want to impose discreetness at some level. If you do impose discreetness, then you have real problems with taking a system which has only a few distinguishable states and allowing it to evolve in a continuous way. What I think is neat is that just by really adding one additional concept, in addition to the usual classical paradigm, gives you continuity? Well, the usual classical paradigm has continuity. Not a particular type of continuity that I'm referring to. You're referring to it. My problem is directly related to what's your... Assume the spin one-half party with the polarization vector. The classical state space would be the space of all measures on the sphere. And the pure states in classical statistical mechanics are just the point measures on the sphere, the delta functions on the same sphere. There is a continuous set even for the pure states in classical statistical mechanics. So, it will depend on any dynamics. It's just the state space structure also in classical. I think you're not taking to heart the approach I'm trying to take. and working within a discrete where you have a discrete number of then you cannot take a quantum system

2:00:00 your statistical operator sort of says the usual statistical operator of over C2 there the classical counterpart really is prescribed then you have a choice which is not the classical There are different ways of thinking about I think this approach I'm taking is motivated by the quantum information approach. In quantum information, normally we compare classical bits with quantum bits. Actually? Can I ask about the axiom about composite systems? Does it involve some notion of locality? One way to motivate the absence of composite systems is to demand that if one of your two subsystems is in a pure state, then joint probabilities should factorize. if you impose that then that that kind of relates to localities but it's not I think it's a much weaker statement n is equal to nA times nB maybe I haven't said yeah you want to say that associated with the composite system there are more there are more distinguishable states than the product you get from the two subsystems so that would imply the existence of some kind of non-local additional states I'm wondering about if you look at a hidden variable theory for non-equilibrium distribution where we get non-quantum where, where, where and one wonders which axiom my opinion is that it's the continuity axiom because if you just take a trivial case of a qubit then already there, just the continuity axiom is pretty much enough to get you the blocks here and yet already non-equilibrium quantum theory could differ but I'm not so sure Well, just on that point, the question is, as a part of my theory, whether you do have discrete probability at all, and it's always the position variable.

2:02:30 Well, let me say you're just talking about spin measurement. Yeah, how do you analyze spin? But that wasn't my question. I mean, my question is, if you don't have it, what happens then? You end up with a classical model where you've got continuity satisfied? Which one, so the simplicity axiom? So the reason for the simplicity axiom is you can use the subspace axiom you can use the composite system axiom more or less to show that k is equal to n to the power r And then the simplicity action is essentially saying that you should take the single value of R which works. Well R equal to 1 doesn't work because of the continuity action. So the next one that does work is R equal to 2. If you didn't have the consistent simplicity action then maybe you could consider R equal to 3 or R equal to 4. So the problem isn't so much in ruling out the classical system, it's in obtaining the quantum. Yeah, it's ruling out all the more complicated information. And my gut feeling is that these higher quantum theories won't work, they'll have some problems. But I haven't been able to show that. It might be that they do work, you could construct them, and it could be that quantum theory is actually a restriction of one of these, in the same way that classical probability theory is a restriction of quantum theory. And that would be beautiful, because then there will be some empirical questions. You don't have any sense of what such a theory would look like to you? No, I have some sense, but I can tell you what I... In constant theory, density matrices are essentially quadratic in some objects. I think in an enqueue theory, it would be cubic and so on. It has some very strange features. Okay, let's thank the speaker again. Thank you.

2:05:00 as pre-computed mechanics. Thank you. All right, what I'm going to talk about is a draft of a book. I put the draft out as a preprint. Automechanics is our most successful physical theory. So I think there are only two possibilities. One, it is exact with, perhaps, because that's a subject of great debate, a need for reinterpretation at a foundational level. The other is that it is non-exact, but is a very accurate asymptotic approximation to a deeper level theory of pre-quantum mechanics so let me give some motivation some motivations for considering the second possibility the first is what I'll call the riddle of canonical quantization the standard approach is that we write down a classical theory and then quantize it Poisson brackets by minus i over h bar times commutators. But this raises a question, if quantum theory is more fundamental, why can't it be obtained directly from an operator theory? And what is the origin of the Planck constant h bar? It seems to me that canonical quantization looks very much like an algorithm to invert the classical limit of quantum theory and suggests that perhaps there should be some other route to get quantum theory directly in an operator framework. The second is the measurement problem. Quantum mechanics is a linear theory with a unitary time evolution. Quote, measurements involve a nonlinear state vector reduction induced by a classical apparatus. There are many other reformulations of this. But basically, unitary evolution by itself in a single

2:07:30 Hilbert space doesn't give you everything there's also the preferred basis problem a question that one can state is how do probabilities become actualities why should one have probabilities without an underlying sample space then there are certain peculiarities of quantum theory non-locality of the Einstein-Pedulsky-Rosen and quantum field theory next I'll come to unification of quantum theory with gravitation the focus here has been on pre geometric approaches theories like string theory but the underlying theory that unifies quantum theory with gravitation can in fact be pre-quantum mechanical as well and then we come next to the notorious cosmological constant problem that all attempts within the standard quantum theory framework to explain why the cosmological constant is nearly zero and not quant mass to the fourth have failed. So I've given some general reasons for considering that possibly the quantum theory might not be exact and that one should look and we're a barotial reason and that is I have a concrete idea as to how one might proceed so I've been working on developing it let me now state in outline form what this idea is and then I'll sketch some details the idea is this in classical mechanics the variables are all commutative they're all proportional if you regard them as matrices to the unit matrix if we go one level down to quantum mechanics chronological quantization or backup, you have the classical limit, h bar equals to zero. The variables are infinite matrices. There are very special infinite matrices, which obey the Heisenberg algebra. Commutator QLPM is ih bar delta LM. Commutator QLQM is zero. In particular, the different indices L and M label space-like boxes. Variables are commutative at space-like separation

2:10:00 what I'm going to suggest is that there's another level down which I turned first generalized quantum dynamics or in my book I turned to determined trace dynamics in which the variables are totally general matrices, there are no a priori to commutativity properties or anti-commutativity properties for fermions, so QL commutator QM is non-zero and there's total mal-locality I make this one basic structural assumption and then some other more technical ones that I'll describe. The basic assumption that everything falls from is that the dynamics is assumed global unitary invariant. And I'll explain what that means. And this is the basis for the emergence of quantum mechanics. And then what I'm going to argue is that the thermodynamics or equilibrium statistical mechanics of this class of matrix models gives rise to quantum mechanics. because it is the statistical mechanics of the deeper level theory and then the Brownian motion corrections look very likely to give stochastic Schrodinger equations which have been widely studied as a phenomenology for objective state vector reduction so at a very low level there are corrections to the Schroding revolution which are the Brownian motion in this underlying trace dynamics that that you don't get just by looking at the thermodynamics. Okay, so let me give a brief overview of trace dynamics. It's a non-commutative generalization of classical, Lagrangian, Hamiltonian dynamics. And this is how it works in the bosonic case. The QR be a set of non-commuting coordinates. You can think of them as matrices on a finite dimensional hill where space much more over space. QR dots are high derivatives. Now I can write down a Lagrangian that's a function of the QRs and the QR dots, but clearly ordering of the factors is important if they don't commute. So there isn't the well-defined derivative, the L by the QR. bar. But now let's make use of the cyclic property of the trace. So I'm going to assume that I'm always getting the operators or matrices that are trace class. Let's define a

2:12:30 trace Lagrangian L, when we underline, in the book it's L boldface. So L boldface is trace of L. Let's now form the variation of the trace of L and we secretly reorder so that all variation QRs and the variation QR dots of each term stand to the right. That's just a convention. And then I make a definition. The variation of the trace is a trace of sum over r of the l by the qr variation qr plus the l by the qr dot variation qr dot this now defines derivatives of a number which is a trace with respect to an operator and i've taken care of the factor ordering problem by using cyclic invariants but now it's very easy to show that you get a complete analog of every first order or tangent So, for example, if I define an action function, or a trace action, as boldface s is integral dt of boldface l, and require its variation to be stationary, then you easily deduce the condition for this, the necessary condition of the order of the operator order of the branch equations, the l by the qr minus e by dt, the l by the qr dot is zero. so we've gone directly now to operator equations without canonical quantization the problem now is that these are of course too general and so we have to find a way from this very general set of operator equations of getting back to quantum mechanics so we have a dynamic that is more general than quantum mechanics and now the idea is to recover quantum mechanics let's consider the equilibrium statistical mechanics dynamics. Now, to do statistical mechanics, what you have to do is to look at generic conserved quantities, because they are what determine the canonical ensemble. So, if you look at the general trace dynamics, first, there's a trace Hamiltonian, as always conserved. This is well-known in field theory for people who study matrix models. If you define PR as the L by the QR dot, and the trace Hamiltonian as the Legendre transformer of the trace Lagrangian, trace sum over R PR PR dot minus L, then you easily deduce

2:15:00 that the H by the QR is minus PR dot, the H by the PR is QR dot. This is the Hamilton form of the operator equations of motion. These are again operator equations. And now So d by dt of the trace Hamiltonian is traced by my definition, the h by dqr, qr dot, plus the h by dpr, pr dot. When I substitute these, I find it's minus pr dot, qr dot, plus qr dot, pr dot, but using significant variance inside the trace, that's zero. So the trace Hamiltonian is always conserved. a second generic conserved quantity and this was a discovery of my graduate student Andrew Millard and originally in the very specific context of wild wild wild order have a coming but then we found in fact it's much more general let's consider the operator c tilde which is a sum over r of the commutators QR with PR. So if one were in quantum mechanics, this would be the sum over R of I, just repeat it, and the conservation would be trivial. But since these are general non-community matrices, the conservation of this operator is no longer trivial. And for bosons, it's anti-self-adjoint. When there's time reversal, there may be self-adjoint directions, which will be important later on. With fermion, C tilde generalizes ozone degrees of freedom of the commutator QR with PR minus the summation over all the Ferriand degrees of freedom of the anti-commutator of the Grassman's QR with PR. And now I'll introduce an assumption. Suppose that H is global unitary invariant. In other words, it involves no non-commutative constants. It simply is constructed by taking the P's and Q's and taking monomials and then use just ordinary numbers, one-by-one matrices as the coefficients. In that case, you can show the corresponding to the global unitary invariance, because you can then replace each Q and P and H by UQ, U dagger, UP, U dagger, and the cyclical invariance of the trace implies that all the U's and U daggers cancel around the trace. There, therefore, is a corresponding other current,

2:17:30 and another current is just C tilde, t is zero. Then finally, there's a natural phase space measure, d mu, which is a product over all canonical pairs R and all matrix elements M, Q, R, N, because remember the Qs are matrices, and over the real and the imaginary parts which are indexed by A. So you just take every real quantity in sight and take all the product. You can show that invariant under canonical transformations, and PR is minus GDQR, and QR is DGDPR, and therefore there's a generalized Eugle theorem, because the Hamiltonian dynamics has this form. If G is a Hamiltonian, then these just give you the time derivatives. So therefore D mu is invariant under Hamiltonian flows, or more generally under canonical flows. That, once you have a generalized Eugle theorem, you can start to use statistical mechanics, It means that if you postulate that phase space is equally populated at one instant of time, it's populated with equal probability at all later times. Whereas if the phase space elements could change in volume, you could postulate uniform occupation at one time, and that would no longer be preserved at later times. So you can't even do statistical mechanics without a Leuco theorem. all right so now we have the basis to look at the canonical ensemble and the canonical ensemble is always formed there's a maximization of enter the argument but the result is always that you have rho is the inverse but that's the normalization constant times e to the minus a sum of constant lagronic multipliers for each conserved quantity so there's a tau times the trace Hamiltonian and then since c tilde is an operator trace of a matrix lambda tilde times c tilde and the normalization integral d mu rho equals 1 implies that z is just integral d mu so lambda tilde and tau are the ensemble parameters Now, let me look at ensembles in which the average is C tilde

2:20:00 Let's consider the average of C tilde This is an anti-self-adjoint matrix So I can write it in the canonical form I-effective times D Where I-effective squared is minus 1 I-effective adjoint is minus I-effective or I-effective commutes with D canonical form for anti-self-joined matrices. And now I'm going to introduce an assumption. I'm going to consider only ensembles where D, the magnitude, is a multiple of the unit matrix. So I'm assuming, in effect, that each direction, that there's no favored direction in the underlying matrix Hilbert space. I'll call with some force ID equals H-bar times the unit matrix, because it will play the role of the Planck constant. right now a little bit of technicality comes in c tilde average is a function of lambda tilde therefore lambda tilde has to be i effective times some scalar and the result of this is that the canonical ensemble only partially breaks the global unitary invariance the canonical ensemble invariant under any unitary global unitary transformation which leaves I effective invariant. Proof if I take trace lambda tilde c tilde and make this restricted this restricted global unitary transformation it goes to trace lambda tilde u effective dagger c tilde u effective. I then use the cyclic property to bring the u effective around but if that commutes with I effect if it commutes with lambda tilde and you get back the trace value to c tilde so therefore if we form an average using the full measure d mu, we actually average too much because of this invariant so I'm going to define a restricted measure which is d mu with one overall unitary rotation frozen, a unitary rotation that commutes with I effective and one other bit of notation now let X be any variable U R or P R let X effective be the part of X that commutes with I effective so X effective is a half

2:22:30 X minus I effective X I effective because now just check I effective, X effective is symmetric so it's the same as X effective, I effective Okay, now I can state the correspondence with quantum field theory, which is that averages over this restricted measure with one global unitary invariance frozen, one that can use with I effective, of the parts of the phase space operators that commute with I effective correspond to the operators in the quantum field theory with a role of I replaced by I effective. And this is actually the major reason I decided to write a book because I had missed the point that you have to do essentially a fixing of the measure when I wrote about this a few years ago. I could never get the correspondence with field theory to work out quite right. Okay, now what you can do is derive equal partition theorems which are really word identities of quantum mechanics for these averages. And what you find, I won't go through any technicalities, is that when the coefficient of the trace Hamiltonian, which has a dimensions of one over a mass, is very small. And my conjecture is that that mass should be identified with the plant mass. And when you replace C tilde by its average inside these word identities, so C tilde is replaced by I effect of H bar, then the word identities have exactly the structure of quantum mechanics. So in other words, inside these thermodynamic averages with these two approximations, you get effective canonical commentators, commentator QR effective, PS effective, is I effective, H bar delta RS, which have now, so if these are now spatial labels, you get effective locality coming from equal partition. That's the second statement, QR effective, QS effective, Q to get zero, up to of tor errors so the locality here is an immersion property and you also get the heisenberg the the correct equations of motion x dot effective is i effective over h bar commentator h effective with x effective you get the complete structure of quantum mechanics coming out with these two approximations Now, I'll make a few remarks and then conclude.

2:25:00 First, this works because when you vary the averages, a term involving lambda comes down from varying the row, the canonical ensemble, but it miraculously cancels out. And the reason it cancels out is that when I take the effective projection, remember I said I was taking projections of all my x's. The anti-commutator of I effective with a commutator of lambda tilde with an x, and that's what comes down from varying the canonical of the row. When you vary C tilde, you get this commutator. It's proportional to lambda tilde, since I effective for use of lambda tilde, the commutator of lambda tilde with a projection x effective. but that commutator is zero because lambda tilde is just i effective times a constant and i effective commutes with x effective so that the term that might mess up the quantum mechanical structure just cancels out automatically and therefore that's the only thing that I have to assume to get quantum mechanics out is that the tau term the coefficient of the trace Hamiltonian decouples at least her very low soft modes but the conditions for that are non-trivial they're discussed in detail in the book first you need boson fermion balance you need something that smells a bit like it's not super symmetry you need equal numbers of bosonic and fermion degrees of freedom for it to be consistent and a sufficient condition also is that x dot effective and c field effective if they have this joint support on operator phase space that's enough for it to be sufficient for one to have a consistent approximation of the thought terms to be neglected. This is discussed in detail in the manuscript. Alright, now let me briefly discuss why they're around in-motion corrections. Remember I said you get quantum mechanics when you replace C tilde by its average, which is I-effect of H-bar. But there will be a correction which is rapidly fluctuating over the operator phase space. Here let me make an onselt that delta C, the effective projection is what's important, is I-effect of H-bar times a complex constant, which of course means that I'm now saying the C-tilde is no longer exactly anti-sulfate joint, and that can happen when there's time reversal violation. So in other words, H-bar then becomes H-bar times one

2:27:30 plus a constant K-naught and another constant I, K-one. So you have a rapidly fluctuating real and a rapidly fluctuating imaginary part in the effective plot constant. What happens then is that the Schrodinger equation that you derive is modified. You find the psi is minus, which is originally minus I H effective psi to T in the I effective equals one sector and now setting H bar equal unity becomes minus I H effective psi to T plus the term I signifying H effective psi Brownian motion, and then another term, sigma h effect psi, times other Brownian motion. This one comes from the real part of the fluctuating minus constant, and this one comes from the imaginary part. And now if you go through standard renormalization arguments that were first given by Girardi-Weber and Girardi- Remedian-Pearl, if you renormalize, what you get in this case is the energy driven version of the stochastic Schrodinger equation where the stochasticity is driven by sigma prime which comes from the real part of fluctuating the plant's constant times H effective side and then H effective minus its expectation times the constant that came from the imaginary part of fluctuating the plant's constant and this equation has a lot of remarkable properties state vector reduction with the precisely born rule probabilities onto energy eigenstates you need to have sigma e non-zero in other words an imaginary part of the fluctuating Planck's constant to get a reduction if sigma is zero and you only have the real Planck constant fluctuating the energy variance remains constant and when sigma squared is taken of order of 1 over mplunk the reduction rates are satisfactory energy fluctuations are taken into account. All right, this might not be bad. If you write, do phenomenology and write sigma squared as 1 over m, where 1 over m is a mass that might be smaller than the Planck mass, it turns out the best bound you can get from experimental data

2:30:00 is that m is bigger than 10 to the minus 1 GeV. This comes from looking at Charm-Meson-D case. And that's still 10 to the minus 20th of the clock mass So there's a lot of room In what we know experimentally For there to be fluctuating terms In the Schrodinger equation The reduction times are typically T reduction is M over delta E squared Where delta E is the energy variance If I use Units that atomic physicists might like to use eV in seconds. 10 to the 19 GEP is 10 to the 13 eV squared seconds. So with Amelor, the clock mass, there's no threat to foreseeable quantum computers because the kind of energy differences that are superimposed in devices are much smaller than the eV, and the coherence lines in devices are, you don't even reach fractions of a second. Well, just as an example, in the recent squid experiments of lucans where they see tunneling between superconducting currents moving in opposite directions in a squid ring. The energy difference in the superposition is 10 to the minus 6 eV and the coherence time is 10 to the minus 6 seconds. So this parameter is 10 to the minus 18 eV square seconds and you start seeing spontaneous reduction effects at 10 to the 13 pb square seconds. So again, you're very far away from confronting this experimentally. Okay, well, two final remarks. One I have down. This, my pre-print gives a general framework for an immersion quantum theory, but not the specific theory that makes it work. That's a task for the future, to find the theory of everything which might fit into this pre-quantum mechanical framework. The other is that my timetable for the book is I'm going to try and give some lectures at the Institute for Van Stoney on it this fall. So I probably will do two draft revisions, one in October and one at the end of the first semester. So if any of you read it and have comments, they would be most welcome. Just email me at the Institute. Thank you. Thank you. Okay. Thank you, Nate. No, so I'd like to ask. The view seems to be that Haughton theory is a theory of a sort of thermal equilibrium distribution, some underlying matrices.

2:32:30 That's right. So, would you have any views about, and to my mind that kind of idea suggests that while there must have been a time in the past where there was not equilibrium, there would have been some kind of relaxation, that's a year in the universe. so I wondered if you yeah I don't know what epoch that might have been I suspect since we look back at the Big Bang and it looks pretty always still pretty homogeneous it would have been I think it's reasonable that there would have been such an epoch and it would have presumably been very very early on right after whatever initial tunneling event that led to the universe but I at this stage obviously I don't have an estimate of when it might have been I think there would have been one Yeah, in the orthodox picture quantum mechanics is obviously taken as a fundamental theory. Yeah. It seems to me that your picture you're relying very heavily on statistical mechanics, so is it fair to say that the underlying ontology of the world in your picture is a classical type of ontology? If you look at the dynamics of the matrix elements here, they obviously look like classical equations in motion. But in fact, even in quantum theory, it's well known that if you take the expectation of the Hamiltonian as a classical Lagrangian, that you can rewrite all the Schrodinger dynamics as a set of classical equations. So in one sense, it's classical deterministic But in another sense, there's still the structure, the operator structure of quantum mechanics is still there. I'm really looking at the dynamics of more general matrix operators. And I don't want to commit whether this is going to end up being a theory of finite dimensional things like matrices of the monster group, of some super big group, or whether it will be ultimately be an infinite dimensional operator space of some trace class operator space. I think although the matrix elements are still basically an operator dynamics at the lowest level because it's the operators that each Q corresponds to a full operator to our matrix and I'm not giving the

2:35:00 interpretation, I'm not taking the matrix elements as something to quantize for example when you do conventional matrix models you start with a matrix and then each something that you apply the croissant bracket to commutator recipe to that's not what i'm doing here yes you stress repeatedly that your variables could be more general matrices or operators do you at least assume that they're self-adjoint to normal well the answer is in the bosonic case i can obviously always make them self-adjoint because any major can be split into a self-adjoint an anti-self-adjoint part. For the fermions, it's a bit complex what the adjointness assumptions are made, and that's discussed in the book. In other words, for fermions, the psi isn't self-adjoint, there's a pair psi and psi dagger for the Grossman case. For the bosons, I simply assume that all the two are self-adjoint, but that's no restriction, because I can always take, as long as I'm working in a complex silver space, remember I said my coefficients in forming the Hamiltonian have to be complex C numbers. So then there's no restriction assuming that the Q's are self-adjoint. I assume that in the book that they're either self-adjoint or anti-self-adjoint and then that's... So they're normal? Yes, yes. They're either... they're normal. That's right. They're either self-adjoint or anti-self-adjoint. Yes? I probably should say in the book that they're primarily just assuming that they're normal. That's a good point. Yes? Do you think that reduction to energy at the end states will do the job to reproduce reduction in the ordinary sense? Yeah, I think the only place where I can see a possible path is in the far reaches of intergalactic space where the energy fluctuations from accretion are very small. And there it gets just more. I wrote a paper in General of Physics 8 where I followed up on Lane-Huson's suggestion that accretion of molecules makes a difference if you have a pointer left pointer right if you just look at the energy of the mass of the pointer they're the same but if you look at the interaction with the environment is fluctuating you only need to accrete one nitrogen molecule to cause reduction in 10 to the minus eight seconds and then you start looking at the ambient fluxes within the solar system it's more than enough to account for it, even with a Planck mass m. It's enough to account for it within the

2:37:30 galaxy. When you get into intergalactic space, it becomes a bit borderline. But you can look at the estimates. Of course we can't ever, we're not gonna be able to get out there and do the experiments, and maybe there are weird superpositions, macroscopic superpositions out there, I am. i can see why you solve the cosmological constant problem but you're doing it in a way that creates problems for phenomenology i mean you said to get the word identities because you have to have equal fermionic and corresponding degrees of freedom yeah no that alone doesn't solve the cosmological problem the only statement i can make about the cosmological constant problem is this that in this theory there's a possible route to the cosmological constant problem coming from view theory and solve scale invariance, use scale invariance. It gives you a condition T mu mu equals zero, which is an operator equation. It gives you an infinite number of constraints and makes all the particle masses be zero. In this theory, the only conserved quantities are trace quantities. So, scale invariance says that the trace of T mu mu is zero, which is a single number condition, and that's precisely matched to the single number cosmological constant so at least the counting looks better here because you don't have at the underlying level you don't have concert operators you only accept for c tilde you have the the invariant the global invariances the invariances like Lorentz invariance or scale invariance will give you trace invariances it's a trace energy momentum tensor and a trace momentum so in fact it looks more promising than field theory for solving the constant logic constant but I haven't claimed that I have the rest of the data I guess I don't understand in standard field theory if you do have the core you expect cancellations yeah but not when but not when you break supersymmetry for example in other words with supersymmetry you still have equal numbers of bosons but the supersymmetry breaking always gives you back the cosmological process and the problem there is that once you go the supercurrent is an operator and the condition you get too strong a condition from supersymmetry basically so you hear

2:40:00 the operators would be coupled from the traces and my idea here for getting the cosmological constant zero is that you have a sum rule trace of T mu mu equals zero or traces and super current equals zero but then t mu mu is non-zero but that's clearly compatible with its trace being zero in your list of motivation for your approach you mentioned this thing about our problems of explaining the origin of H-bar yeah I think I didn't see an explanation in what you were giving it didn't you just sneak it in? no what I said was that I'm looking at the ensemble that c-tilde is a conserved quantity and has some ensemble expectation if I assume that that ensemble doesn't favor any direction in Hilbert's space then it has the canonical form of the unit anti-selfage right operator times the magnitude that magnitude plays the role of h-bar here one more question from Peter Harland Yeah, in your trace Hamiltonian dynamics, have you investigated what must I imagine is a trace Hamilton-Jacobi theory? No, I haven't looked at the Hamiltonian equation. Is that alternative room to the Trudeau equation? It could be. Yeah, that would be interesting. No, there are a lot of things I haven't done. I haven't looked... Anything in class 1 mechanics that can be done with first variations carries over here. Things like the Yappinness exponent, stability theory, which require second variations. vary twice, the non-commutativity will cause modifications, and they may be interesting to explore.