And I should say I'm very grateful to the organizers,

in particular to Jens for having encouraged me to say a few words here. It doesn't really fit into the program, because although I guess there were times when I had a sort of passing interest in matrix models,

but I don't remember what they are. But I still keep a very lively interest in matrix mechanics. So this is about matrix mechanics. And it will sketch a view of quantum theory

that will result in the insight that quantum theory is not about one algebra. It's about the filtration of algebras, many algebras. So that's very important, to anticipate something. So let's see how it goes. This is somewhat slimmed down version, Jens,

of what I showed you, so that you won't be worried. I think I should get through in an hour. So here is a table of contents. I will briefly tell you what I'm going to tell you. Then I want to recall a very, very simple standard argument

why physical theories are never predictive. There is not a single physical theory that can be used to predict the future. Then I want to briefly recall why relativistic quantum theory

should satisfy locality in the sense of Einstein causality. Then I will present to you the ETH approach to quantum theory. Then I briefly have to talk about what

an event is in quantum theory and how to detect one. And then I would like to sketch the relativistic version of the ETH approach. This is a somewhat recent development. I'm not pretending it has already reached its end point. And then we will summarize.

Here are credits. I had many discussions with many people. And some several years ago, I had a last PhD student who was a very pleasant partner for discussions and efforts in this direction.

But this juvenile, he's actually French, but he had the misfortune to get married to a Swiss wife who refused to gypsy around. And so he gave up his academic career in spite of the fact that I think he might have been very talented for one.

So what this lecture will be about. I will outline some new foundations for quantum mechanics. It is called the ETH approach to quantum mechanics. E stands for events, T for trees, and H for histories. The approach enables us to introduce a precise notion

of events into quantum mechanics. That's something that Rudolf Haag always emphasized. He said we have to know what events are, otherwise we won't understand what quantum mechanics is about. And so I've tried to sort of follow up on his proposal.

Then I explain what it means to observe an event by recording the value of an appropriate physical quantity. And I will then exhibit the stochastic dynamics of states of isolated open systems. States do not evolve according to the Schrodinger equation,

as is mistakenly claimed in almost every course in quantum mechanics. Then I want to focus on how quantum theory could be reconciled with relativity, and what it tells us about the fabric of space time. And that's about it.

So the specific topics that I should address, and of course I will not be able to talk in depth about all of them, are as follows. Foundations of quantum theory, so why are physical theories never fully predictive? That will come just next.

Why is quantum theory intrinsically probabilistic? What are events? How do we measure physical quantities and detect events? What is the role of time in quantum theory? Unfortunately, I had a flu, and so I missed almost all the talks at this meeting very regrettably,

but those that I heard didn't feature time. But they were, we did feature time. Oh, you did, very good. Yes, so, good. Locality and Einstein causality, I already mentioned that.

So, what are the basic problems in coming up with a framework that unifies quantum theory with the theory of space and time? Of course, I wish I knew the answer. I'm not quite sure what to say about it, but we will say a few superficial things about it.

I tend to believe, and I will argue why, that a consistent quantum theory of events must necessarily be relativistic. I think non-relativistic quantum mechanics ultimately doesn't really make sense, okay?

And then, of course, we want to speculate a little bit about whether space and the causal structure of space-time might, in some sense, emerge from quantum theory. So, what prevents theories from being fully predictive?

This picture is less nicely visible than I was hoping. You see, the black line is my world line, and I'm at the moment at this space-time point that is labeled present. I have access to some of what is inside my past icon.

I may have received signals from events that happened inside my past icon, but I have absolutely no access to events that happen outside my past icon, and these events might be so drastic

that they might kill me, maybe in five minutes, but they might also kill you. And so, whether you think classically or quantum mechanically, it is obvious that because of a lack of knowledge of initial conditions,

we can never predict the future with certainty. So, the past is a history, the past for a creature like myself is a history of events. These are these blue little things that you see here, and the future is an ensemble of potentialities

of which I cannot exactly predict what they will be. Now, I believe this fundamental difference between past and future is an essential element that should be retained in a sort of reasonable formulation of quantum theory. You have a question or comment.

So, the problem is I had this flu, and I don't hear very well, so, yes. So, if you take any conventional physical models, you have equations which are reversible in time. You can run them forward and backward.

Yes. So, what distinguishes on this diagram future from the past, if we just make evolution and relabel future? So, you see, the past is something I have experienced. It's in my book of photographs and my diaries and so on. Book of photographs and et cetera,

it's a space-time event which is located in that point. Noted by present is what you have now. And it's just the current state of moment. It's not necessarily, you see what I'm trying to say? You don't, in that photograph, you have a print of the past, as you call past.

But in reality, what you do, you trace the equations, the physical models and the experiences to the past, and you make a conclusion that what is drawn in the photograph happened some time ago. But it's just in the present moment. It's located in that point, in that black.

Yeah, yeah, yeah, I'm listening, I'm following. You can try to run physical equations, not to the past, but to the future, make prediction for the future. Yes, you see, the problem is you have the wrong idea what the physical equations are. But you're in good company, with my exception,

I think almost everybody in this room has the wrong idea about what physical equations are. But you have to wait a little bit till you understand why I'm saying that. And of course, I will, Jens, I will add three minutes at the end because I already answered a long question. Could you please add four minutes because I would like to make a small comment?

Yes. I'm very, very happy that you put the black dot in the middle of the line. But it doesn't mean that I will grow up to an age of 150. All right, so that explains why even classical theories

are not really predictive. Now let's go to quantum theory. Can this be, somehow my sketches are not so visible, that's a shame. So this is a Gedanken experiment which I worked out with Schubnell and with Jeremy Fupman from MITS actually.

So let's look at the system that is the composition of two subsystems. A subsystem confined to Q and another subsystem which in fact just consists of a particle P whose orbital wave function is prepared

in such a way that it will propagate into a cone opening to the right. The cone is unfortunately not visible anymore. And the point is now that very reasonable time evolutions, quantum mechanical time evolutions have the property

that the evolution of P is essentially independent of the evolution of the degrees of freedom of Q. Even if Q has very many degrees of freedom simply by cluster properties of the time evolution. Now we look at the concrete realization of this setup.

Q consists of a spin filter that measures the component of the spin of a particle in the Z direction. And it consists of a particle P prime whose orbital wave function is prepared in such a way

that it propagates into a cone opening to the left and hitting the spin filter. And now I prepare particle P and particle P prime in an initial state that is a spin singlet. And then I let time evolution set in.

Let's make the following assumptions. Let's assume that the Schrodinger equation describes what will be seen in the experiment. Just the Schrodinger equation. Debo has probably left.

He's an advocate of the many worlds interpretation and these people claim that the Schrodinger equation describes everything. Now I want to show that this cannot possibly be correct. So let's make the following assumptions. P and P prime are spin one half particles

prepared in a spin singlet initial state. The spin filter that will measure the spin of P prime in the Z direction is in a very poorly known initial state which is not entangled or at least not necessarily entangled with the initial state of P prime and P.

The dynamics of the state of the total system, everything together, is assumed to be fully determined by a Schrodinger equation. In particular, it follows then that the initial state of the spin filter must determine whether P prime will have its spin upwards or downwards.

If it has the spin upwards, it will pass through the spin filter. If it has the spin downwards, it will be absorbed. By the filter. I also want to assume that the correlations

between the outcomes of a spin measurement of P prime and of a subsequent measurement of the spin of P are as predicted by the standard quantum mechanical assumptions that have been tested in the experiments by Asper and Chisler and so on. So let's assume that these correlations are the way

the experimentalists tell us they are. Now there is the following fact that I already mentioned. The Heisenberg picture dynamics of observables such as the spin referring to the particle P that remember propagates into this cone

opening to the right, this Heisenberg picture dynamics is essentially independent of the dynamics of the degrees of freedom of Q. So this follows from our choice of the initial conditions of the orbital wave functions of P and P prime

and of cluster properties of the time evolution. Here one has to do some analysis. It's not a totally trivial statement. Well, it then follows that the spin of particle P is essentially conserved up to very slow, small corrections before it will eventually be measured,

say, in a Stern-Gerlach experiment. But P and P prime were initially in a spin-singlet state, so the expectation value of the spin of P will be the way it was at time zero, namely close to zero for all later times.

However, this contradicts the third assumption because if P prime, for example, is seen to have spin up then P should have spin down in order to satisfy the usual correlations.

But apparently the Schrodinger equation doesn't predict that. So it follows that the Schrodinger equation can never predict the outcome of experiments in quantum mechanics. This, of course, may sound trivial, but people seem to tend to forget it.

So it also suggests that the predictions of quantum mechanics are all probabilistic. Now we have to briefly address the problem of locality of quantum theories. So let us assume that the Copenhagen heuristics

is correct in the sense that if the spin of particle P prime has been measured to be sigma prime, which is up or down along the z-axis, and the spin of P has been measured to be sigma, which is also up or down, but along an axis n,

then the state of the system rate after these two measurements is a simultaneous eigenstate of the two projections, P pi of P prime sigma prime ez. It's the projection that measures on spin sigma prime in the z direction for particle P prime,

and the projection pi P sigma n, which projects onto a state of particle P with spin sigma in the direction of the axis n. And the state is a simultaneous eigenstate

of these two projections corresponding to the eigenvalue plus one. That's the usual Copenhagen heuristics. Wait, wait, when you were saying that the two parts of you are not equivalent, you are saying the Heisenberg picture is okay for isolated systems?

You are saying the Heisenberg picture is okay for isolated systems? The Heisenberg picture in contrast to the Schrodinger picture is a totally safe building block for quantum theory. You see, you might even agree, although Schrodinger was probably a much better mathematician than Heisenberg,

but Heisenberg had a much deeper understanding of what quantum mechanics was. All right, so now, so it is possible that these two spin measurements are made in space like separated regions of space-time,

so that the localization regions of these two projections are space-like separated. The order in which the two measurements then occur depend, of course, on the rest frame of the observer, who records the data of both measurements. But in order for the prediction

of the correlations to be unambiguous, it cannot matter whether you first apply pi P and then pi P prime or in the other way. And that suggests that they should commute. Of course, this implication is logically a little bit too strong.

They might just commute on all states on which such measurements can be made. But let's be a little generous and say they have to commute. This is really the right way of understanding local commutativity in relativistic quantum theory, I think.

All right, so that's much about locality. Now, I would like to sketch the ETH approach. So in fact, the basic problem I want to tackle is to clarify the notion of an event featured by an isolated system.

I will always look at isolated systems. Why is that? We have a very clear idea about Heisenberg picture time evolution for isolated systems. If we look at the system that potentially interacts with an environment, we don't know how

to describe time evolution in a conceptual way. To say that the system is isolated doesn't mean that it is closed. It can still release signals to the outside world. So I look at open, isolated systems. All right, so as I already mentioned at the beginning,

I think time is certainly an absolutely fundamental quantity, and so I want to involve it into my discussion as much as possible. Let's suppose the present time is t naught. And let i be an arbitrary interval of future times.

Here is a definition. I consider an isolated open system S. Then potential future events in S are described by certain orthogonal projections labeled by time intervals in this interval t naught

infinity after the present. The star algebra generated by all potential future events by such families of orthogonal projections and located in a future interval i of times

is denoted by e sub i. I mean, this is your understanding of time, but you are not saying that all events are just labeled by the time. This is a filtration by time. Yes, I mean, you will see that this picture will then be refined.

So that's e sub i, the algebra generated by potential events occurring at times inside this interval i. If I take the algebra generated by all e sub i's

where i is contained in the interval t infinity, I get an algebra that I denote by e greater or equal to t. And e is the one generated by all of them. Temporarily, these closures are closures in the operator norm.

But here in the definition of the e greater or equal to t algebras, it will become crucial to pass to wick closures once representations are chosen. All right. Now, just by definition, it is obvious that ei contains or is equal to ei prime whenever i contains

or is equal to e prime, e greater or equal to t contains or is equal to e greater or equal to t prime if t prime is larger than t. Now, quantum mechanics of an isolated system is defined by a filtration e greater or equal to t

of algebras of possible potential future events. That's sort of a good picture about how to introduce a quantum mechanical system. So of course, not usually what we do in our courses. You see, most people believe, and it's in fact a fact,

that these algebras e greater or equal to t are independent of t. If you look at the system with finitely many degrees of freedom, then the potential events are projections that can be viewed as functions of the momentum

and the position operator, and then e greater or equal to t is independent of t. Yes, but no, no, but the algebra, really, if you look at these algebras, they don't change. They're all the same, right?

And that's, of course, a problem. Most of what we do in our courses on quantum mechanics is to talk about quantum mechanical systems that do not feature any events. It's like talking about black holes during the entire course. That's a little disappointing.

So here is a principle that will guarantee that events will be possible. And I call it the principle of diminishing potentialities. It says that e greater or equal, for a realistic model of quantum theory, e greater or equal to t contains,

but is not the equal to e greater or equal to t prime. The greater or equal is missing here. Whenever t prime is larger than t. So this you should try to record. I think this is important. Unfortunately, people have troubles believing

that this is a reasonable principle, because in our discussion of simple examples of quantum mechanical systems, it's never true. It's unfortunately the e greater or equal to t's are all the same algebra. OK, now as I said, it's important to actually

imagine that these are von Neumann algebras. So we should introduce states and representations and so on. The state on these algebras is defined in the usual way. And now let's call it omega, say. Then I set omega sub t to be the state

omega restricted to the algebra e greater or equal to t. OK, now it is perfectly possible that omega is a pure state on the C star algebra e. But since e greater or equal to t sits inside e properly,

it might be a mixed state on e greater or equal to t, just by entanglement. In fact, for realistic models, the algebras e greater or equal to t will be type 3-1, I believe. And then there are no pure states that are continuous.

OK, so in fact, there's another mistake we do in our courses on quantum mechanics. We always do as if pure states were sort of what we should talk about. But in fact, pure states never appear in nature.

Only mixed states appear. And here you now understand why. All right, so this observation that states restricted to e greater or equal to t tend to be mixed, opens the door towards a clear notion of what might be meant by events

and to a theory of direct measurements and observations of events. So I defined the notion of potential future events. And I now want to render it a little more precise.

So we say that the potential future event that might happen inside at some times greater or equal to t is given by a family pi sub xi of disjoint orthogonal

projections contained in an algebra e greater or equal to t that add up to the identity. OK, that's a potential future event. Now in accordance with the Copenhagen Mambo Jumbo, it appears natural to say that the potential future

event actually happens in the interval t infinity of times if the state omega restricted to the algebra e greater or equal to t looks like an incoherent mixture of states labeled by these projections pi sub xi, right?

Because if something, if an event happens, you say that the off-diagonal, the interference terms should disappear, and so they're reduced. I should maybe have written here a t,

but in fact, all these operators are in e greater or equal to t. I'm sorry for a few misprints. So omega t, namely omega restricted to e greater or equal to t should look like an incoherent mixture of states that are in the range of these projections pi sub xi, OK?

Is this clear? At least for people who know a little quantum mechanics, I guess it's pretty clear that this reasonable mathematical expression of the Copenhagen stuff.

So for an operator x in e greater or equal to t, we define the adjoint action of x on the state omega t to be the linear functional given by minus omega t, but by the expectation of the commutator of x with a in the state minus omega t.

OK, so you mean this was saying that they are sort of diagonal in the same basis, and this is the computation that you're writing. Yes, so now of course, 5 implies, and is implied by saying that rx omega t vanishes for all operators x generated by these projections.

OK, all right. Now in the following, we consider some stratum of physically important states. This is, in fact, I think the big point

of all these discussions. We have major problems in general quantum theory, also in quantum field theory, to specify what we mean by physically important states. And I will leave this a little bit vague. I cannot, first because I don't know much about it,

and second because in one hour, one cannot clarify everything anyway. All right, and then with respect to the stratum of physically important states, we always look at weak closures at von Neumann algebras, but they don't change my notation.

So here is a slightly abstract version of what I just went through here. If m is a von Neumann algebra and omega is a state on m, and x is an operator. In m, we say that rx omega is a linear functional, which when evaluated on a is given by the expectation

of minus the commutator of x with a in the state omega. The centralizer of a state on the algebra m consists of all operators x in m, whose adjoint action on the state omega vanishes.

This is a subalgebra which is very easy to see. It's in fact a star subalgebra. And omega is a normalized trace on the centralizer. And so we know the possible structures of these centralizers completely.

The center z sub omega of the centralizer is defined to be all operators in the centralizer c sub omega, which commute with all other operators in the centralizer, as usual. So now we are prepared to introduce

the notion of what an actual event is. So again, let S be an isolated open physical system. Here is a definition. Given that the state omega t is the state of S on the algebra e greater or equal to t,

an event is happening at time t or later if the center z sub omega t contains at least two non-zero orthogonal projections, pi 1 and pi 2, that are disjoint and that have a strictly positive

expectation value with respect to omega t. An event is happening if this condition is true, and I will tell you what it means for the event to happen.

So but that's so far a definition of what I mean by an event is happening, and I will then interpret it. So since I'm a simple-minded person, I would like to assume that the center z sub omega t

of e greater or equal to t is generated by a discrete family of orthogonal projections. It could also be a continuous family, but it's a little easier to argue with discrete families of orthogonal projections, where labeled by points xi and x sub omega t, x sub omega t is simply the spectrum

of the centralizer. I assume this spectrum of the center of the centralizer is a countable set. Here is an axiom that will clarify what I mean by saying that an event is happening. Suppose that the cardinality of the spectrum

of the center of the centralizer is at least two, and there are two projections, pi sub xi and x sub omega t, for at least two different points xi that have non-zero expectation in the state omega t.

Then to say that an event happens means that the state omega t must be replaced by one of the states omega sub t and xi that are given by sandwiching everything between pi xi.

For some xi in the spectrum of the center of the centralizer is the property that omega t of pi xi is non-zero. The probability for choosing a certain xi, meaning that I choose the state omega t xi

to predict, to make predictions about the future, the probability of picking a xi is given by Born's rule. So you mean this is the reduction of the wave function? This is the reduction of the wave function. But here it is, you see here, it sort of comes naturally.

You don't have to say that because Henri has made a measurement, the wave function collapses. It is coded into this filtration of algebras. But then it generates a t, of course. Pardon? It generates a t of the ETH. Absolutely, yeah.

I didn't realize that these pictures come out so poorly. It's terrible. So apparently the time evolution, you know, if you believe that this is a good point of view, you conclude that the time evolution of states of an isolated physical system S

is described by a stochastic branching process. This branching rule is as determined by this axiom and Born's rule. And the possible futures, given the present, the present here is denoted by this constant,

the system is in a state rho, the future is a tree-like structure. And the future that we will actually experience will be one path from this tree. And that's called a history. Now this shows of course that the evolution of states

in quantum theory, as soon as you talk about the quantum theory with events, the evolution of states is not given by any Schrodinger equation. It's given by a stochastic process like that. Although the Heisenberg picture evolution of observables

is the way it always was. Now I would like to emphasize, some people believe that this is just a somewhat abstract version of the Decoherence Mamba Jumbo. But it's actually totally different. I could go into that. The Decoherence Mamba Jumbo attempts to impose

some kind of Markovian structure on the sequence of observations or measurements. This is very non-Markovian. Although we would have to clarify what we mean by Markovian in quantum theory.

Anyway, it's really very different. Now, pardon? It looks close to a path integral. No, it has nothing to do with a path integral because I'm not talking about amplitudes. No, I understand. Maybe it might remind you.

And then, you know, I cannot prevent you from being reminded. But in a path integral you don't have to talk about just amplitudes. You have the wave function, the square modulus is given by two paths which should track close together

for a physical system. We have to discuss that at the end. Alright, so how would you record an event or detect that an event has happened? Let's see. Well, you see, that depends now on who I am.

Whether I have good eyes or good ears or not so good ears or whether my brain still works or not so much. Depending on people who record events they have a certain list of physical quantities available that they are able to measure or observe.

The physical quantities available to a certain observer is denoted by O sub S. This is neither a linear space nor an algebra so it's just a list of abstract self-adjoint operators representing physical quantities that, let's say, Gurop can measure.

Alright. So, then of course for any abstract physical quantity y hat and the given time t we should specify a concrete self-adjoint operator y of t inside this algebra E greater or equal to t

that represents the physical quantity at time t. For an autonomous system the operators y of t and y of t prime are conjugated to one another by the propagator of S. S first understood by Heisenberg.

So, suppose that at some time t an event happens meaning that there is some kind of maximal family of this joint orthogonal projection pi sub xi contained in the center of the centralizer of the state omega t given this algebra E greater or equal to t

and this family contains at least two elements with positive probability of occurrence. Let's look at the operator y t that represents y hat at time t.

Here is its spectral decomposition. The etas are eigenvalues of y and the pi sub eta of t are the spectral projections of these operators y of t. We then say that y of t can record the event happening at time t if basically the spectral projections

of y of t are inside the algebra generated by these projections little pi of xi. Now, unfortunately there is very little chance that they are exactly in the algebra

but they should have a small distance and what the distance is can be clarified using conditional expectations. You mean the distance in log or in conditional expectation? You see there is always a conditional expectation of every operator in E greater or equal to t on z.

Okay, using the state. Yes. And then, yes. So you need both the filtration E of t and omega. Everything depends on the choice of a state plus the filtration. Absolutely, yes. Well, a state, you know, is not always the same state.

The state sort of branches whenever there is an event. Okay, so well in that case we say that the quantity y hat can be used to sort of detect an event of this sort.

This is a little bit brief but in fact this can be made totally precise. I just don't want to go into further details for reasons of time. Now let's pass to the relativistic setting. So this is again my worldline, worldline of JF. I do not indicate here my death, Jens.

I'm at the moment at this space-time point P sub t. But there used to be a little while ago at the space-time P t zero. That was in main station in Zurich. The E greater or equal to t should now probably be identified

with all operators, functionals of say fields, electromagnetic field and so on, that are localized inside the forward-like cone erected over the space-time point P t.

The E greater or equal to t naught that was available to me when I was still back in Zurich are all fields localized inside the forward-like cone V plus P t naught. That obviously contains the red algebra.

It is also now clear why this principle of diminishing potentialities might be true and how this might depend on dimension. You see, suppose my theory has massless modes

such as the photon and the graviton and so on. Then it could have happened that some, for example, that I don't know who, but probably Jens wanted to call me at some point in the future of P t naught, but in the past of P t.

And unfortunately my cell phone was turned off. You never had one. I have one, and I can be called, but I don't use it very often. So if it was turned off, then his message zoomed out along the surface of a light cone in between the red cone and the blue cone

because electromagnetic waves travel at the speed of light. I will never be able to detect his message because I cannot catch up with photons. They are always faster than whatever I can do. They will never penetrate the photons

that Jens created here in this little double diamond. These photons will never come into the inside of the red forward light cone, so they are lost forever. This means that the relative commutant of e greater or equal to t inside e greater or equal to t naught

is a big algebra. It consists of all asymptotic electromagnetic fields located in this region here. So it's an infinite dimensional algebra. In fact, this was something that Bucholtz first analyzed in fairly great detail

in connection with a scattering theory for photons. So I think I'm not quite sure when he formulated the theorem, but it goes as follows. In a relativistic quantum field theory, in even space-time dimension,

with massless particles, the algebra e greater or equal to p t of all physical quantities, observables potentially measurable in the future of the space-time point p t is of type 3-1, and the relative commutant is type 3-1-2, actually.

Now this is a result of what might be called Huygens' principle. This will not be true for relativistic theories in odd dimensional space-times. So you see, probably the fact that our space-time is even dimensional has a fairly fundamental significance.

It is one of the ingredients that enables us to understand what events are. Okay, now you see, I did as if relativistic quantum theory

depends on my presence. I would now like to take myself out of the picture and make it independent of individual agents. And so that goes as follows.

This is now a little bit stupid, but I don't know how to do it better for the time being. Probably at some point this will be superseded by more reasonable formulations. Let's suppose that m is some manifold, say topological space is enough for the moment. I consider a fiber bundle,

quantum mechanical f, whose base space is given by m, and the fiber above a point p in m is given by an infinite dimensional phenomenon algebra that I denote by e greater or equal to p. We assume that all these algebras are isomorphic

to a given one, namely for example to this type 3-1, hyperfinite type 3-1, say. Okay, now we say that the point p-naught is in the past of a point p,

written as p-naught by seeds p. If there exists an injection map, yotta, that maps e greater or equal to p into e greater or equal to p-naught, it enables us to identify e greater or equal to p with a subalgebra of e greater or equal to p-naught,

and with the property that the relative commutant of yotta e greater or equal to p intersected with e greater or equal to p-naught is actually an infinite dimensional non-commutative algebra. Pardon? Sorry, what?

Oh, okay, thank you. You looked critical, Alain. Yeah, I am critical in every one, because I think that such an injection can be found in all cases. I mean, I don't think it's precise enough. You see, it's not that there exists

abstracting an injection, because I think this will always exist. I think you want it to satisfy some person. Yeah, yeah, I agree with that. I agree with you, yes. In fact, I'm not quite sure how to say this more precisely, but maybe you can. You should say it more precisely, because as it is, I think it's empty.

Yes. Well, you see from the picture, we sort of know what we mean. So we have a point p-naught here, and then this is the forward lacon, and the p is here, and then obviously this algebra is a sub-algebra of this one. But I mean, you want to say it better

than what you want us to make. All right. So this relation will introduce a partial order, an m. No, in fact, I mean, this is part of the structure. I mean, giving these maps is part of the structure. You cannot just say they exist, you see.

Part of the structure. Absolutely, yes. All right. And then it could happen that two points p-naught and p are neither in their, that neither p-naught is in the past of p-naught, p in the past of p-naught, and then we say that there are time, space-like separated,

written like this. So these relations will determine the causal structure on m. So it's to take it in the beginning. Pardon? You see, they determine the causal structure, but what I want is to have it from the beginning. Yes.

Okay. So now we have to reintroduce events in this setting. Let sigma be some space like hypersurface contained in m that contains a point p.

e greater or equal to sigma is then the algebra generated by all the e greater or equal to p primes, where p prime belongs to the surface sigma. Definition. We say that an event happens in the point p if the center z sub omega sigma

e greater or equal to p of the centralizer c sub omega sigma is non-trivial and contains at least two projections with positive probability of occurrence, just like before. But you have a choice. Pardon? You have a choice between these two projections.

Yes. Well, you know, in the end it will be probably infinitely many projections, but you need at least two, otherwise no event happens. So the axiom is the this is now an important axiom. Compatibility locality. If two points p and p double prime of m

are space-like separated and events pi p sub xi and pi p double prime sub eta actually happen in the points p and p double prime, then they should commute. This, of course, is somewhat related

to the locality discussion I gave at the beginning of this talk. Yes, Jens? Yes. Good. So they should commute. This, I believe, will have something to do with introducing geometrical structure on m.

But I don't understand this so well yet. So you see here is a graphical representation, an event happening in p, and an event happening in p double prime. But the two people in p and p double prime cannot see each other's event yet.

All that p can actually see are events that have happened in the past-like cone of p. So now I would like to describe histories of events. I choose a space like surface sigma

with the property that some bounded subset of sigma lies in the past of a certain point p as shown in the following figure. Here is sigma, here is p. And if I intersect the past-like cone in p with sigma,

I get this disk-like region. Now, in the past of p, all kinds of events may have happened. I would like the prediction of whether an event happens in point p not to depend on anything

that is space-like separated from p. That would be terrible. It should only depend on what has happened in the past-like cone of p. Okay, and so let's see why this works.

So we would like to say that the events, the initial condition chosen on sigma and the events that happened in the past-like cone of p actually determine the state omega p

that I have to consider to decide whether an event happens in the point p or not. Okay, and so here is why this works. Here is an inductive hypothesis.

p1, p2, et cetera, all the points in the past of the point p, but not in the past of any point on the initial condition surface sigma. With any of these points, we can associate an orthogonal projection,

pi, super p i sub xi i, where xi i is in the spectrum of the center z, super p i sub omega p i. And this projection actually represents the event that happened in the point p i or its future.

We define so-called history operators. History operators are ordered products of these projections. Now we have to make sure that this ordered product is well defined. Well, if p i is in the past of p i plus one, then in fact pi p i will stand to the left of pi p i plus one.

But if they are spaced like separated, the order doesn't matter because they commute. So this shows that this order product is totally unambiguously defined. It's a time-oriented product? Yeah, right.

So then we said we define the state omega sub p on the algebra e greater or equal to p to be given by... But you mean it's no-go projection, of course. Pardon? No, no, this is not the projection. It's a product of projection.

But omega p is now given by this particular state. You see, this is just iterating the rule we had before, going through all the events that have happened. And the important thing is we only consider events in the past like on of p.

And okay, here is the generalized Bohr rule. Now we can do the induction step. So we can now answer the question whether an event happens in the space-time point p or not. Well, an event will happen if the center z sub omega p

of the centralizer of the state omega p given the algebra e greater or equal to p contains at least two disjoint orthogonal projections with strictly positive probabilities in omega p. Then we can go on and...

That's the induction step. I think it's quite clear that the compatibility locality axiom can be expected to yield non-trivial constraints on the geometry of space-time in the vicinity of space-like separated points in which events happen.

But this is something I don't really understand, where I evaluate, so I don't want to confront you with half-baked ideas about this matter. It's time to stop. I think I'm pretty much within my given slot. So here is a summary and a few conclusions.

So I think, as in the genesis of special relativity, the electromagnetic field, or other fields describing massless modes, will actually play a key role in the genesis of...

system quantum theory that talks about events and solves what people call the measurement problem. I do not believe that this has been appreciated. I mean, I'm an old man, and it is unlikely that old people have very original ideas, but I think this is an idea that has been overlooked,

as far as I can say. In particular, we have understood that apparently, if we believe that my notion of events is reasonable, then presumably spacetime will have to be even dimensional,

because Huygens' principle is wrong in odd-dimensional spacetimes. I believe that, as in the genesis of general relativity, the causal structure of spacetime will play a key role in the functioning of a relativistic quantum theory.

The noncommutative nature of quantum theory and the compatibility locality axiom governing the relations between events that determine the causal structure on spacetime. In fact, the events in this sense in which I've introduced

them are really what weaves the fabric of spacetime. You see, eventually I would like to give up any prejudice about spacetime and just reconstruct it from this notion of events.

Yes, and so, OK, well, here. There is obviously a very natural arrow of time. We always wonder why is there irreversibility, but in fact, the right question is why should we have expected anything to be reversible?

I think nature is simply not reversible, and you cannot formulate a reasonable quantum theory that describes measurements and facts and events that is reversible. And in this approach, the reversibility

has automatically been eliminated by this kind of stochastic process that describes the evolution of states. So this is all I wanted to tell you, and I thank you for your attention. OK, any questions or comments?

Martin? Of course, I like very much that you put the observables in front and states. Pardon? I like very much the idea to put the observables in front and the states. But I did not do that. You said Heisenberg is better than Schrodinger.

No, no, but yes. But not that the word observable is ill-chosen. You see, it's really the potential events that play. That's a better notion than the notion of observables, because the notion of an observable

only makes sense when there is an observer. But most of the time, there are no observers. But anyway, yes, good. I just have a question with the word propagator. Yes. So you said when events and their detection, so you said an autonomous or something like that.

So do you specify when you just give the axioms for these, say, filtered algebra? Will there also be among this, say, unitary continuous unitary curve which

resembles a propagator? So there was the word propagator. Yes. So you see, in an autonomous system, it sort of makes sense to say that an operator, say, a function of fields that is labeled by a space time point.

And the same observable in another space time point,

they should be related to each other by conjugation. And the operator that conjugates them is the propagator. This is clear, but do you give this in advance? I mean, if you make a list of axioms, so the ETH.

You see, I mean, I wish I could do computations like all the other people did in the other talks I heard. This is very abstract. I cannot do computations. But ultimately, the basic object is this filtration.

OK? If in the non-relativistic regime, there is no problem because time labels everything. In the relativistic regime, it's these e greater equal to p's. Then it becomes a little more delicate. Now, you see, as many of us were somewhat indebted

to Buchholz and his clear views of algebraic quantum field theory, he wrote the paper with the late John Roberts about QED. And there, they looked at these.

In fact, they didn't know about that initially, but Buchholz explained it to me. They looked at these algebras of fields located in forward like cones. And they said time evolution can be reconstructed from the way these algebras are embedded into one another.

They have a discussion of that. So that has already been studied, that problem. But maybe it was a little vague, or is it OK? Maybe we can render it more precise afterwards.

Yes? I have two questions. The first one, I think we actually what's the meaning of probability in the framework of the Copenhagen application of quantum mechanics. But at some point of your talk, you wrote something probability in index t. What does it mean operationally?

Well, I wish I understood what you are referring to, but I can try to guess. It was not probability t, but probability omega t.

The probability of seeing the event psi happening. Maybe I missed it. I didn't remember the omega. But I think that's what was written. So that's simply omega t of pi xi.

Now, you see, I could be kind with you and interpret your question in a sort of deep way, namely, what does it mean for theories where things cannot be repeated arbitrarily often to be probabilistic?

And that's a somewhat tricky question. I have thought about it. If you now ask me to summarize my view in two minutes, I'm going to fail. I'd be happy to discuss that with you. But I think it does make sense. You see, it makes sense in a similar way

as it makes sense to apply notions of probability theory to interpreting the CMB, for example, the fluctuations of the CMB, right? So if you believe you understand that, you should also understand this.

But you had two questions. Yes, the similar one. What does Eigen's principle play a fundamental role in it? Pardon? Eigen's principle plays a fundamental role. Absolutely. But we know that even in classical background and the generic classical background, curved background,

the Eigen principle is not valid for the octet inside the rifle. So would that mean that on a classical geometry, but not flat, all your confliction will fail? No, you see, I only need Huygens' principle in the, Huygens' principle is the same as the principle

of diminishing potentialities. What I need, and I wish I could replace it by a weaker formulation, is that if P is in the future of P0,

say, then I would like the commutant of E greater or equal to P intersected with E greater or equal to P0 to be non-trivial. That's the way in which I need Huygens' principle.

And, you know, I'm sort of fairly confident. We speak of the working system. OK? Maybe one last question. Yes, I'm not sure I understood the meaning of event in quantum. You have not understood it?

I'm not sure I understood it. Well, I'm also not sure. One can never be sure. But can you be? But is that your question? Yes, this is my question. And what is an event in quantum? Well, then I have to repeat my talk, which doesn't make too much sense, probably. I can just give you the definition once more,

if you like. Can you give a description in words of what is on an end? Without formulas?

Yes. Yes. So we saw that in a reasonable quantum theory, these algebras, E greater or equal to T, let's talk about non-relativistic for the moment. It's a little easier. Get smaller and smaller.

In other words, what I can still learn about, possibly learn about life and the universe and so on now, is really less than what I could learn 10 years ago. That's the basic principle.

If you accept this principle and you prepare a system in a state that may be perfectly pure as a state on the entire big C star algebra of everything, then when you restrict the state to one of these algebras

E greater or equal to T, it will look mixed. Is that clear? That's what entanglement is about. You have to talk louder because I don't hear so well. But I assume that it is clear. All right.

So if omega t is a mixed state, then you can write it always. If it's a mixed state, you can write it like this.

That just says that it is an incoherent position. Now, the point is that for an event to happen, I would like these pi xi's to be somewhat special projections.

Namely, I would like, you see, if you form any operator out of the pi xi's, let's look at the x, which is some kind of function f of xi pi xi.

Then it follows from this equation that the adjoint action of x on omega t vanishes. Now, for an event to happen, I would like this operator x to belong to the same algebra

as the operators on which I evaluate my state. So I would like x to belong to e greater than or equal to t. That sounds reasonable. Now, then you conclude that apparently from this condition

and from this condition, you conclude that x belongs to the centralizer of the state, omega t, on this algebra. Is that also clear?

No, I mean that's just the definition of the centralizer. If that's not clear, you just believe that this is the right definition. Can I summarize what you are saying by saying the following? Please. But you have to speak louder, otherwise the lady doesn't hear it. You see, normally you hear when you

hear the quantum mechanical formalism. You hear something about pure states, reduction of the wave packet, and so on. What Jo is proposing is a formalism which is more involved in which he has these algebras which are filtered, and in which what he proposed is that the evolution of the state

actually can only be of the type which is here. Mainly, when we're talking about the reduction of the wave packet, in fact, Jo is asserting that what you are doing, you are taking reduction with respect to the center of the centralizer, and this occurs in a certain specific manner.

And so he's asserting that this is the way things occur in practice, and that somehow this is a correct conceptual understanding of the quantum mechanical formalism. Yes, this I understand, but my question was more basic, what is an event, and can one imagine a continuous sequence of events in the evolution of a system,

or how should I imagine? So, you see, the idea is the following. But I mean, you know, at some point we will probably have to stop because you want to go and have dinner. But let's maybe make a comment on this.

I'm sorry, I'm a physicist, so I would like to have a more intuitive understanding of the notion. Well, we all would like, you know, if things were easy, then we would all understand it. But it's unfortunately not such an easy thing,

although the mathematics is basically still absent. And once we add the mathematics, it will become even more difficult. If you want, we can make it a more private discussion, I don't know. Oh, no. Well, maybe this can be answered very quickly. You see, suppose some event happened at time t.

Then at, let's discretize time. It's a little more, since you are a physicist, this is more intuitive. Then I march from time t to t plus delta.

And I ask, is there an event happening? Well, with extremely overwhelming probability, the answer is yes. So there will be several projections. Pi xi one of t plus delta,

pi xi n of t plus delta, describing the event that may happen at t plus delta. But only one of them will have an overwhelming probability of occurrence.

If delta is small, then maybe omega t plus delta of pi xi one t plus delta will be a one minus a very tiny quantity. And all the other projections will have an extremely small probability of occurrence.

So then what happens usually is that you just pick the projection with the overwhelmingly large probability. Then you would say, well, then the whole picture becomes almost the same as the Schrodinger evolution. Because whether I now choose the state omega t

plus delta or the state omega t plus delta pi xi one t plus delta pi xi one t plus delta normalized,

it makes essentially no difference. These two states are almost the same. Then I go to the next step. Again, there will be an event happening, but one of the possible projections will have an overwhelmingly big probability of occurrence. So you could say, well, that this picture

is just a more complicated version of what you learned in school. The problem is if you do this n times and you always choose the projection with the biggest probability and ask what is the probability of the history,

it's exponentially small. So every once in a while, you have to pick one of the projections that is unlikely to appear, just for, if you like, entropic reasons. And then you can say, these are the times at which, as observed with not entirely sharp eyes,

you really see that an event happens. Now, there are toy models illustrating these concepts. You could look at the Limplatian evolutions

of density matrices and implement these ideas about events in conjunction with Limplatian evolutions. This leads to some interesting mathematical problems. I could give you a full story in the example of two-by-two matrices,

which is maybe not so exciting. If you go to m-by-m matrices, and you want to know exactly how these trees of possibilities and the histories and so on look like just in the example of Limplatian evolution,

it's already very difficult. And one of the reasons why it is difficult is that we don't quite know how to parametrize density matrices in an efficient way. So in fact, I have no very explicit results to offer about that, but I think that's something that could now be attacked by people

who know more mathematics than I do. Let's thank our speaker. Okay.