Particle creation by black holes

0:00 A particular lecture imperfectly qualifies as an article of two sorts. The first one is a human, so-called ordinary quantum mechanics, which concerns quantum theories that are allowed to assume unproblematically at the outset that the observables pertaining to the systems of which they're theories are the self-adjusted part of some set of bounded operators on some particular field workspace. And you can think of the odyssey of quantum interpretation of these theories as consisting in part of trying to characterize determinate sub-algebras of that algebra in a way that spurts, navigates between the syllab of no-goal results, like the cell inequalities of the quotient-specker theorem, like the crib is of the measurement problem. And Rod's contribution to the literature, the contribution of The second class of things Rob worked on, well, there are other classes as well, concern what I'm calling quantum mechanics of infinity, which addicted to quantum theories where this assumption at the outset the observables are the bounded self-adjoint operators on some covert space is for one reason or another problematic. One kind of reason it's problematic is for quantum systems with infinitely many degrees of freedom such as quantum fields or quantum statistical mechanics and thermodynamic limit. Representations of the canonical connotation relations constituted the physics of those systems aren't unique optimitary equivalents. That raises questions like what the problem we represent are available on. Rob's work on this family of theories consists of many sorts that a true honor to draw attention to. One sort is extending problems and solutions encountered up here in the interpretation of ordinary quantum mechanics to the setting of QM infinity. So, for instance, it's Haas doing our bell correlations with arbitrary local algebra in the quantum field theory. But the kind of thing that grew out of that was an appreciation of the isolation of characteristic features of the QM infinity setting or characteristic and surprising features of the QM infinity setting. and many of them had to do with who came to it. And what I want to talk about today, I want to tell you about some real cool work Rob did, is something he and Hans called the 2001 paper,

2:30 an intrinsically mixed state. It's a kind of mixed state that's unprecedented in ordinary quantum mechanics. Here's a first pass on what kind of state it is. It's a kind of mixed state that's orthogonal, that's in scare quotes, to every pure state of the system of which it's a state. And there's a sense in which, or I'm going to try to suggest, there's a sense in which the prevalence of such states in quantum mechanics of systems makes it difficult to transpose the interpretation of ordinary quantum mechanics to that set. Now these articles here, they're written in a language that's like the language of angels. The sentences, each sentence is laden with significance. Each sentence is at the same time perfectly precise. And each sentence is, to a large extent, incomprehensible to mortals, like myself. So what I'm going to do today is issue paragraph after paragraph after paragraph with mortals speak in an attempt to make it more accessible to people who are already fluent in angels speak, what in principle fixed states are. those who are fluent in angel speak might hear my utterances, my paragraphs as crocophonous. I want to apologize to them ahead of time and tell everyone else this is an excuse or an apology yet. This isn't going to be a fully rigorous presentation. This is an intrinsic mistake. here's the plan. I'll be using a lot of words and the ones I'll be using repeatedly that matter, I've got a little a picture of here. I'm going to start by talking about a general framework for quantum theories, a framework meant to accommodate ordinary quantum mechanics and quantum mechanics sub-infinity in one view. I'm going to talk about the instantiation of this framework in ordinary quantum mechanics, where the seven observables, the self-adjuvant power systems, and not an operator of the appropriate space, and point out something about mixed states encountered The something is something like this. Mixed states and ordinary quantum mechanics can be understood to assign quantum probabilities to elements of the observable algebra, the one-dimensional projectors that correspond to pure states. Non-zero probabilities. Now I'm going to talk enough about how to go about QM sub-infinity to explicate what an intrinsically mixed state is.

5:00 It's going to be a state where that home truth, of which that home truth in ordinary quantum mechanics doesn't hold. It's going to be a mixed state where there's no straightforward way to understand the probabilities it assigns as probabilities assigned to pure states that are not zero probabilities. And then, it's going to be a final section, I hope, where I talk a little bit about implications of the availability in the most general setting of intrinsically mixed states for familiar interpretive postures encounter. including, I hope, a bit about Rob's work on the modal interpretation of Paul I want to emphasize again that these intrinsically mixed states are familiar to anybody who converts at the 90-grade approaches. Others have remarked on their implications for the foundation's questions before and I've been talking a lot to philosophers. Neither of us here now. I want to acknowledge Hans Holverson for leading me away from some mistakes I was going to make. And John Irwin for probably leading me into mistakes. So, general framing of the form of theories. We'll start with C star algebra. Let me talk about where it came from and then say a little bit more about what it is. Here's a way to Symmetric operators on a Hilbert space satisfying canonical commutation relations or canonical anti-commutation relations for a system of interest. Build polynomialism and then add to the algebra the limit of every sequence of elements of things already in the algebra that converges where your criteria of convergence is very stern. What you wind up with is a structure called a C-star algebra which is just a bunch of elements that form a vector space over the complex numbers. You can define on them a multiplication operation that's not necessarily commutative. There's an adjoined operation. It's got some structure. There's also a norm. And finally, the operator is closed in that norm. That's the pattern of

7:30 Is it crucial that it's a linear space over C? Probably not. In the quantum theories I'm talking about, it's going to be. And there's going to be a lot of questions you could ask like that where the reason I'm talking about a special case is that the point three is I'm talking about our problems. Okay, the reason I really asked the question was I wanted to get a handle on whether this framework was the same one that Jeff Groot and Hans Albers and Rob Clifton assumed in the talk earlier this morning are not. Jeff? Yes, it is, except that it doesn't have to be over complex times. So let me say scalars and then observe. First, I'll talk about the scalars. A shock-borne example of a C-star algorithm that we all have met and know about is a set of bounded operators on some Hilbert space, the Hilbert space with the Hilbert space adjoint serving as the involution, the star operation. And the Hilbert space norm, operator norm, which is defined in this way, serving as the C star algebraic norm. Observe not at all in passing that you have a Hilbert space. You have other criteria for when sequences of Hilbert space operators converge. There's a strong topology, a strong topology, a weak topology, we'll find them there. The important thing is these aren't necessarily equivalent topologies. Sequence of operators, if it converges in the large topology, it converges in the weak topology, but not vice versa. Not every sea star algebra is a similar picture, you know, with the body of an unhelpment space. But it's always the case that if you've got a C-star algebra, you can find something called a Hilbert space representation of it. It's a map from the algebra into a set of bounded operators on some Hilbert space. A map that preserves relevant algebraic structures. Yeah. So what you get is the image of the algebra under the map of some subset of bounded operators on the Hilbert space. That's of the projection operators in the Hilbrick space. The representation need only to contain the zero operator and the identity operator.

10:00 They're the identities for the multiplication and the addition in the algebra. They need to contain a whole host of other projectors that are in the set of bounded operators. So to get yourself one of these, start with the C-stone algebra, get yourself the representation. And it's a count of the temptation, which is there when you're in the Hilbert space, of starting with the representation, the set operative, and the image of the map, and closing it in a different topology, in a weaker topology, in the weak topology than the topology that's closed in. What are you going to wind up with if you add limit points of sequences of guys in this, in a weaker topology than the norm topology, the seastore-emptible, It's something called the von Neumann algebra. And the von Neumann's got a famous theorem, a double commutant theorem. It says the von Neumann algebra you wind up with when you start with the representation and close in a new topology is the same as the double commutant theorem. A commutant of a set of operators acting on a Hilbert space is the set of all operators on the Hilbert space that commute to the original set. A shockworn example of a von Neumann algebra is a set of bounded operators on a Hilbert space. Again, not every von Neumann algebra is an algebraically isomorphic. So the picture I'm encouraging you to keep in mind is the general approach to quantum theory is this. You've got algebra, I think they can be abstracted into one of those ways of representing the algebra and sets of body operators on healthy spaces where the representation proper is a subset, can be a proper subset of the von Neumann algebra arising from it, which in turn can be a proper subset of this set of object operators. And moreover, these von Neumann algebras can be different. algebra can be able to be isomorphic. One symptom of the difference is if you've got something that's a state on this bond-liminal algebra, it might not, it won't define, necessarily define, a state that's countably additive on this bond-liminal algebra. Obviously, to define a state on a representation is how the expectation does is the algebra. But there's no way to extend the state to the full-bondonial algebra.

12:30 So these different monogamy algebra host importantly different sets of states. States have to be countably additive with probability measures for close-up spaces. So that's all very subtle. But the thing about what I'm calling ordinary biomechanics, none of this matters. None of these distinctions between the algebra, it's a representation, it's double cognitant, which is a non-women algebra, a set of bounded operators on the Hilbert space, the representation and the non-women algebra affiliated with the subsets of, and that is the same as the matter, because you just start, and there's good reasons for doing this, you just start with your algebra observables being the self-adjoint part of a set of bounded operators on several, several. And now a bunch of things that I think are familiar about what the admissible quantum states are. So what's a state on set quantum operators on a half of the space? Well, linear functional from VH to the complex numbers, that's norm, it's the signs of identity It's positive and all that's positive. It's linear. And finally, it's countably added any set of paralyzing document projectors. A state maps that set to the same number as the state maps elements of that set to. And that's what a state is. We know that, at least for observable algebras that are acting on Hilbert's face as a dimension that there's a one-to-one correspondence between states on those observable algebra. And density matrices, trace class, salvage of operators, trace one, where via the trace description, the density matrix encodes the expectation value assignment of the state. These density matrices in general may be expressible as convex combinations of one-dimensional projectors. in the cases that the density matrices are maximal. In any special resolution in terms of a parallelizer It also follows that states and ordinary quantum mechanics form a convex set. You have two things that are states and you make a

15:00 convex combination of them. It is a state. Extreme elements of this convex set, guys that can't be expressed as convex combinations of other guys, distinct other guys, are called pure states, the rest are called mixed. So the pure states correspond to the one-dimensional projectors, the mixed states correspond to the rest of the density matrices, and states, well, states in general correspond to countably additive probability measures over closed subspaces of the Hilbert space, including the closed subspaces that correspond to pure states. So one of the things, the density matrix assigned probabilities are pure states in its range. Among the quantum probabilities, the state-pointered quantum mechanics assigned are probabilities to elements of the observable algebra that are associated with pure states, the one-dimensional projection. So this is the first, this is the home truth of ordinary quantum mechanics that I'm going to suggest. It doesn't hold, it doesn't hold in a straightforward way, in the quantum mechanics in the immediate systems. Now, it's not just a truth, nobody tries to make any pay out of this fact about density matrices in ordinary quantum mechanics. It's a, we've appreciated, it's a, quantum probabilities are weird, they're hard to make sense of. They interfere, they violate bell inequalities, they're strange things. And there's something about the way mixed states are clear, suggests that there's at least some quantum probabilities we can get a handle on. The mixture resolved as a convex combination of pure states. It's tempting to adapt one or two different interpretations of the mixture that take these probabilities assigned by the quantum state to be probabilities we're familiar with from classical physics. One's a kind of ensemble interpretation of proper mixtures where you have a bunch of different preparations going on and a slew of systems prepared by one of the preparation devices. And the mixture lengths between quantum probabilities just the relative frequency in that mixture prepared in particular ways. For single systems, it's tempting, again, trying to find some

17:30 fundamental probabilities you can understand in a model of classical probabilities. For single systems, it's tempting to give an ignorance interpretation according in which the system really is in one of its pure states. And these probabilities are found by the state are just epistemic probabilities about which pure state is there. And this hope that mixtures hold out, that we can understand mixture weights as probabilities that although they're assigned by quantum mechanical states are classical in a familiar way. Also, from some views, how much to afford a way to understand those most significant of quantum probabilities or world probabilities. So here's the setup. It's going to be a unitary, ideal, non-disturbing measurement of an object observable over my inventors, O sub I, before my system initially in a superposition by a plunger which is raised. And this entanglement of the composite system is going to imply uniquely a mixed state for the component system, that's the planar system, is the state that we produce as well. And the idea of both collapse and modal interpretations, To exploit the fact that these probabilities, these quantum probabilities assigned by the mixed state, in this special case, admit interpretation as the kind of probability of the motor. So on the collapse interpretation, after a measurement of the unitary model would be given by this, one of these states, one of these pure states, is the actual state of the pointer. And this mixture, it's an ignorance interpretation where the coefficients, the weights just tell us, reflect that we don't know which transition took place. Model interpretation is more subtle. Model interpretation, at least as originally put together, it doesn't think there's

20:00 So this is an improper mixture, it's not really a state of the system, but still, it's useful to identify which value attributing propositions are true, which observables have a true values. According to the modal interpretation of the systems in that release state, once again, And again, observables that share the spectral resolution of that least are the termite. And these mixture weights, these quantum probabilities, give a probability distribution over which of the possible values of those observables are their actual values. So at the end, we're explaining this fact that mixtures assigned to quantum probabilities that can be interpreted as probability assignments to pure states to make sense of moral probabilities. And now I'm going to say, in quantum mechanics of infinite systems and the presence of intrinsically mixed states, you won't always be able, well, you won't for intrinsically mixed states be able to understand them as assigning probabilities to elements of the observable algebra that correspond to pure students. So insofar as that assumption is something on which those interpretational maneuvers I just talked about rests, those interpretational maneuvers are available in that right later than G.O. 17. Okay, so, here's an example, sometimes we can talk about field theory, but there are many of them that it prompts this to recognize with the number of systems you can't be able to find and occupy, go to infinity. Here's where it's going to start to matter, the set of distinctions I made in the beginning. Relevant algebras, C-star algebras and obscurables, are going to emit in-equivalent representations, in particular representations whose binorman algebras aren't isomorphic, on such that the set of things in the counts of states on this Monday night, if I'm not aware of this presentation, coincides with the set of things in the counts of states on the other. So the general motion of state appropriate to the ab step, the C star algebra, is as follows. It's a linear map that's null and positive. Where's the count of latitude requirement gone?

22:30 It's gone away with the projection . Sure, I mean, algebra isn't going to contain projection operators other than 0 and 1, and if it doesn't, there's a little bit of interest in further require the behavior of those projections under a state. Algebraic states, like the set of states, they form a common set, extremal elements of which are pure. The rest are mixed. A useful device for talking about states on the algebra Is there GNS representation? So you're giving this to start out with a state on an algebra on the abstract algebra. You can always, a theorem, tells you that you can always find for this abstract state a representation of the concrete Hilbert space. Because you can always find a map from the algebra of bounded operators acting on some poker space. And a vector in that element space such that the vector acts on the operators in the representation in such a way that it mimics the expectation value the algebraic state assigns to elements of development. This GNS representation is more or more unique. If you find one and the other GNS representation you find, that can be transformed into it. So first useful result. We say a GNS representation arising from a state is irreducible if the only subspaces of the representing pulpit space invariant under the action of the representation are the null subspace and the whole pulpit space itself. So reducible representations are something like representations with our super selection rules. The first useful result, if an algebraic state is a pure algebraic state, if and only if, when you make its GNS representation, you find out it's irreducible. The only varied subspaces are the zero subspace in the whole healthy space. Otherwise, it's mixed with the GNS representation by non-tributing varied subspaces.

25:00 You might ask this question of yourself about the set of algebraic states. If I take one, make it GNS representation, I've got a Hilbert space, I've got a von Neumann algebra, the ability of von Neumann algebra. Which other algebraic states are well behaved in that Hilbert space? which other algebraic states define countably added to probability metrics over that non-Liman algebra. The answer is the set of states that Fennstein was creating are states in a Mavis folium. They try not to be exactly the algebraic states that can be implemented as density matrices in the GAS representation, the omega. They are the states that are well-hamed or countably added on the projectors that you're feeling on non-Liman algebra. foils like a kinship relation between algebraic states, they kind of all live together in each other's GNS representations. Two algebraic states are called quasi-equivalent if they pick out the same folia. If every other, well, a magus quasi-equivalent to phi, every algebraic state that can be represented as a density matrix in a magus GNS representation can be represented as a density matrix in the phi's. other extreme two algebraic states are disjoint if their folia have null intersection if any algebraic state representable on the first GNS representation isn't representable as a common probability measure on the second and vice versa these aren't in general mutually exclusive but for the kinds of states I'm talking about And now, intrinsically mixed states, my first pass, what they are, what they were, is a kind of mixed state on a system that's orthogonal to every pure state on that system. No, I can say it without the scare clothes. An intrinsically mixed state is a mixed state. Omega, that's disjoint from every pure state. So it's a state on an algebra. That's a mixed state. That's disjoint from every pure state on an algebra. That's such that it can't live in the GNS representation. There's a well-being probability measure for any pure state on that algebra. And the next thing I want to do is try to give you some sense physically mixed states arise. I'm going to begin by a very rough guide to how to type

27:30 different kinds of von Neumann algebras. The typology I'm talking about applies to algebras that are factor algebras. What's a factor algebra? It's a von Neumann algebra such that and its intersection with its commutant consists of multiples of an identity. It just happens to simplify the classification. Remember, von Neumann algebras are distinct from C-star algebras in that the former abound in projection operators. So it's with respect to the behavior of the projection operators in our boundary we now do that. These classificatory principles are not a typical. So, production operators are acting in a public space, or essentially the subspaces of that space. Which are just the range of the production operators in question. So the first notion we'll need is that of one projection operator being a sub-projection of another. That's pretty easy. F is a sub-projection of E in S-ray, which the subspace is associated with, is a subspace of E's. So I can easily switch the E to F here. E is a sub-projection of F in the picture. Moreover, it means a proper sub-projection. F's a proper sub-projection would be a subspace associated with F. Sub-space, that's F's range, is a proper subspace of the subspace size. Now we have enough on hand to say when a projection operator is minimal with respect to a non-nombin algebra. It's minimal if the only sub-projections it has in the algebra are zero in itself. Examples of minimal projections in the isomorphic desensified operators on the helipot space are the one-dimensional directions. And here's an intuition I want to talk about. There's a correspondence between pure states, the theory of online algebra, instead of on the document of the open space, and you know the projections. The intuition goes by the way of this idea about states.

30:00 The purer the state is, the more uptight it is, the more things it tells you are impossible. So a pure state is going to be the state that tells you more things are impossible than any other state. And that's going to correspond to the states associated with one-dimensional projections. the mixed state of W and the pure state jumps on to the subspace I expand by alpha. W is going to declare impossible the subspace span by gamma. T sub alpha is going to declare impossible that, but also the subspace span by data. So it's going to declare impossible everything the mixed state W because impossible, plus it costs more. And so it's going to spoil W's claim to purity. And nothing's going to spoil a piece of Elvis' pipe, purity. So in virtue of being the most tight kind of stay-around, a piece of Elvis appears to be. And it's the most tight kind of stay-around in account of the minimal projection. Projection. Yeah. The last bit I need to put in place is a notion of equivalence for projection operating instead of an urban analogy. This is defined by way of partial isometers from H to H. Think about partial isometry this way. an x on an arbitrary vector, say this blue one, as follows, with the partial axometry from the subspace S e to the subspace S f. First thing it does is it projects the blue vector into the subspace family. So it annihilates the complement of the component of the blue vector orthogonal to its initial subspace. And then it maps the result isometrical in the subspace. And it turns out So, something's a projection operator, if and only if it's a product of an isometry, a partial isometry. So, we can peel these partial isometries not only to identify which things are projection operators, but also to say that when projection operators are equivalent. So the definition is they're equivalent if there's a partial isometry such that V star V equals V and V V star equals F.

32:30 That's going to be the case just in case the subspace that's E's range is isometrically imbeddable in the subspace that's X range. Which is a nice equivalent image. and finally armed with the notion of equivalence we now have enough material on hand to say when a projector in a von Neumann algebra is infinite just like a set is infinite when it can be put in a window of course by the proper set of it itself a projector is infinite even only that there is some other projector in the von Neumann algebra and this proper self-rejection that we're going to do. Renal projections are symphony projections. That's enough to characterize different types of boundary and factor that we use. A type one factor is the type we all know about is the type that frames ordinary quantum mechanics. It's a factor correspond to one-dimensional subspaces. Every type one factor is an isomorphic to some sort of . A type two factor, it's got no minimal projections, but it's got finite projections. So put that in a type of snippet. The type three factor, which is the kind I'm going to be talking about, has no non-zero minimal projections, all its projections are infinite and equivalent. And so a natural thing to ask given the disassociation I motivated intuitively between one-dimensional minimal projections and pure states on an algebra. It's given that a type three factor algebra has no minimal projections, does it have any pure states? Are there any states that are extremely the set of states on an algebra? This is a case where intuition sanction

35:00 conclusion that's true. Type 3 factors don't have pure states. It's got to do with the lacking minimal projections. Any state you care about, you're going to associate it with some projection in the algebra. That projection is going to have a proper sub-projection. And that proper sub-projection is going to be associated with a state that spoils the first state's plan of pure. So type 3 having the pure states. This is a wish of a weirdness. We have a state of a mixed state, but it's not a mixture of pure state on the algebra. There are no pure states on the algebra. It's a kind of mixture, unprecedented, of the ordinary. And now you might say, well, that's a very, that might be a very interesting mathematical possibility, but isn't it idle? This has got to do with physics. The answer is it's got a lot to do with physics. If Hans weren't becoming a father, he would have talked about some of the ways Types II and III algems emerge in region physics, but no matter what. So suppose that you're doing the Kline-Gordon field and its vacuum state on an empty space-time. And you happen to care about an open-bounded region for space-time as well. An open-bounded region such that it has a non-empty space-like complement. So, the vacuum state, the global state, is going to define on the subalgebra of observables associated with the urban modern age, a state just by restricting the state on the whole algebra to the subalgebra. And it's the vacuum state restricted to the algebra of observables corresponding to the zero. This is just a state on a C-star algebra that can make its TNS representation. And having made its TNS representation, you can take the double cognitive of a von Neumann algebra. That's our recipe for making a von Neumann algebra. It's physically connected to a state of affairs in the modern region in constant space. If we do this in all known models, the cognitive von Neumann algebra we obtain is a type 3 factor.

37:30 the kind of algebra that just claimed doesn't admit any pure states. So open-mounted regions of Minkowski's basement are associated with learning algebras, for which there's only states of the sort unprecedented learning There's another example, a similar sort. This time, we're considering that C-star Algebra is a verbal associated with an open, unbounded unit of spacetime, like the regular wedge there. Again, we're considering, say, I say that Karnkorn can be able to find the end and It's in a vacuum state, a global state, so we can restrict it to the algebra solution with the wedge. We can take the GNS representation of the state of the wedge algebra, take the double-cognitive to get a binomial algebra, put the binomial algebra and take the time turning back. It's in addition, so was the previous one, but this example is more interesting. It's, in addition, this algebraic state, a state that satisfies the condition for equilibrium that people think are appropriate to algebraic systems, to systems, so it's an equilibrium state with respect to a set of time translations, equilibrium states with respect to evolution. evolution. And the set of time translations with respect to which this is an equilibrium state. The time translations between the surface as a constant time for a family of uniformly accelerated observers. So the time translation can correspond to isometry for a Kowski metric. So not only is this Kowski-Dakking state restricted to the Rimblin edge, a state that gives rise to the top, type 3 Ron Reumann algebra, a factor of Ron Reumann algebra. It's also a state that's an equilibrium state that you expect to . And that's, and this is the next class of situations in which type 3 factor Ron Reumann algebras

40:00 arise in physical applications. Pretty generically, in the thermodynamic limit, equilibrium states of systems at finite temperatures, are states such that C-Sci algebra observables for the system, are states such that if you take the genius representation and know the affiliate von Lennon algebra, what you get is a type 3 factor. So the von Lennon algebra is associated with typical pristine states of thermodynamic systems, are von Lennon states. If intrinsically mixed states bother you, you might hope that if you retreat to Al de Bragwell, that everything's going to be okay. Somehow, it's just an artifact of of these representations that are making things go weird. That if you deal with everything algebraically, we're going to reestablish a pleasing sort of connection between the probabilities of quantum state of signs and elements of the algebra of its own of the signs probability that corresponds to pure states. We will regain understanding of a fixed sign and non-zero . I'm going to consider a state omega, a C-star algebra, and I'm going to call it type 3 The following things all fodder. The first is, that phenomenon that will be built doesn't contain projectors qualifying to pure states on that phenomenon. That is to say, the algebraic state I started with can't be understood to assign non-zero probabilities to pure states that will be filling in the . Well, maybe it didn't be understood just how to assign non-zero probabilities to pure algebraic states on the C-star algebraic A and not on these representations. Next point. If Rho is a pure state on the C-star algebraic

42:30 this type 3 factor is this joke from Rho. Another way of putting that association between pure states and irreducible GNS representations. There's no irreducible representation all three vectors to be made that can be represented as a density matrix. Here's a plus. You can build von Neumann algebras that contain projectors corresponding to pure states. But, the intrinsically mixed state of Baylor with which you started can't be extended in a well-behaved way, can't be extended to the countably additive state in those von Neumann algebras. So in the first case, you haven't got an algebra that has elements corresponding to pure states. In the second case, my hook and my crib got an algebra with elements corresponding to pure states. The intrinsically mixed state you started with doesn't define probabilities. Here's another way to express it for weirdness. There's representations in which both intrinsically mixed data and pure role on the abstract can be realized as vector states. But in those representations, the vectors are . So again, by helping my crook, you can get a representation in which they cohabit, but they don't cohabit in a way that's intrinsically mixed data and a pure state that you can understand the intrinsic state is assigning the pure state around zero probability. So all these facts I take to further gloss. Omega is intrinsically mixed. Omega is a kind of state not president on their time. How am I doing in my time? I'm feeling much. Ten minutes. Okay. So I'm going to skip apart where I argue for the claims. So here are some coal-hearted facts about how representations and the affiliated binominal algebra work. The first is that the row and abstract algebra A is pure.

45:00 The affiliated binominal algebra is a factor. The next is that factor states are either quasi equivalent or disjoint. There's no other option. So, you can see that it follows from that approach the pure state. It's disjoint from intended to be missed. As follows. Assume that it's not disjoint. The pure state of the transatlantic states aren't disjoint. Therefore, the pure state row can be represented as a density matrix. This row is going to define a pure state because it's a low effect on density matrix state. that pure state pi extends uniquely to be affiliated by and the extension is also pure. So if you can represent rho, omega c and s representation, you can have pure state type 3 factor algebra. But we know type 3 factor algebra don't have pure states. So no, you can't represent pure rho on the GNS representation of the Type III faculty state. And it follows that Kalmeva and Roe must be disjoint. They must, either they have the same volume or they have no intersection. Sheldon can have the same number one as well. The ones who are representing the GNS representation, therefore, they are disjoint. I hope I've given you some sense of this weird kind of state that you encounter as physical and the quantum mechanics of linear systems. And that's returned to where I started, the kind of problematic and efficient that one encounters

47:30 in our very quantum mechanics and see to what extent they can be transphosed to setting infinite systems. There's one comment. I think there's a theory there. How are we going to give an ignorance interpretation of it? How are we going to give an ensemble interpretation of it? This joint from every pure state of the system could really be in. We can't understand any quantum probabilities, omega signs, quad-linear functional, as probabilities for any pure state that are different from zero. So we don't even know how to begin to find an ignorance and an ensemble interpretation of the intrinsic state. So Wald asks in a 1999 paper, are statistical probabilities truly distinct from quantum probabilities? It seems that in the case that quantum probabilities are assigned by intrinsically mixed states. They are. That answers a resume. Collapse interpretations. First observation about intrinsically mixed available. It describes a system without determining values according to the eigenvector eigenvalue. Why? It's not an eigenvector of anything in the algebraic syllabus of our non-algebra. Moreover, it's stuck in this predicament according to the collapse interpretation. According to the collapse interpretation, you collapse to a pure state with the probability that you assign to that pure state, but intrinsically mixed omega can't be straightforwardly interpreted to assign non-zero probability to any pure state. Collapse of any pure state has probably been zero. In my fantasies, I imagine supremac-type dialogues with very naive advocates of Sailor Bonesy or GRW who are led into finding out that their interpretations don't work either by granting a bunch of assumptions in the files. I want to know that. Modal interpretations. So a modal interpretation wants a density matrix because it wants to use its spectral resolution to identify determinate observables and use the weights of the

50:00 of the projectors and the spectral as a key to the sign of the probability of these changes to. What's a model interpretation going to do with one of these? The projection operators aren't in the abstract algebra. If you build the affiliated von norman algebra, you take a projection operator, they're not the kind you need, they're all infinite dimensional. You build a von norman algebra to have the kind of projection operators you need, you can't understand the epithelic state as a sign of the probability to those projection operators. model interpretation, at least as initially formulated, can't wrest from an intrinsic state any account either of what observables are determined or with what kind of probabilities. Enter Clifton 2000. So what Robert observed in this paper about the model interpretation of the quantum field theory is alive to this problem with type 3 algebras, and there are apparent incompatibility with the role in play of her own interpretations. Well, what really takes down a modal interpretation of a system that's a specification of a subalgebra of the algebra observable that can be assigned to determine the values, dispersions or dispersions. And he cooks up a recipe for doing that and shows both that, for safe and cold faithful states, that that positive operator is zero only if it is zero. The recipe he cooks up identifies the biggest algebra identifiable using only the state in Washington and the population but he also worries.