Classicality in QM — Discussion

0:00 Thank you. functions, and this is always 0 in every point in the . Therefore, one usually says that observables are always They cannot think of that way at all. However, this only holds at an on-tick perspective. In an engineering perspective, things become different. Let us first speak about quantum physics again. Quantum mechanics, we represent our intrinsic observables by operators on these devices. So we can define what an eigenstate is. An eigenstate is a eigenvalue equation here. And as a result, we get that two observables, A and B, are compatible, exactly if they are simultaneously diarizable in the Jirvatskate age. Here's a shot two, which I skipped. How can we now reveal the analogies between quantum system and quantum system? We need an appropriate language. This language is supplied by the framework of algebraic quantum theory. In algebraic quantum theory, we have on the observable from the particular algebra. We have states, which are expectation value functional. Therefore, they are elements of the dual of this algebra. And in the final step, we can represent the observable by operators acting on this stage, which is trying to contextually emerge in the systemic observables, which are now part

2:30 of a larger W star algebra. This is W star algebra because it has the state space as a dual and a free dual. Scrooge. Scrooge. And the battery. If you turn it, it focuses. Ah, it focuses. Ah, nice. Very good. Thank you. Okay. And so, using this language, yeah, you can now derive the analogies between quantum mechanics and classical physics. In quantum mechanics, we start with the non-combinative C-star algebra. algebra, we construct the states by the expectation where you say your function is by forming star color in the different states. And in the final step, we get the von Neumann algebra of bounded operators from the different states. Classical physics, as I have shown before, we start with the common statistics by algebra, the algebra of continuous functions over a phase phase. And now, it has also been mentioned by Simon Trotten before, the states, statistical states of probability measure assigning an expectation value to an observable over a phase phase. And the corresponding construction is the W star algebra, which is the set of essentially bounded functions over the phase space with respect to a particular reference measure, to a particular reference measure. It can be arbitrarily chosen. So this is exactly this context reality arising here. We have this set of states, we choose one particular reference state, and with respect to this particular reference state, this larger W star algebraic emergence connects with it. Which is the same you do in quantum mechanics. The states on the C star algebra are not automatic. No, of course. We choose one particular reference state in the Hilbert space, and then by the that Gell-Fernay-Mann-Siegel construction reach arrive at the Iger-Bow found it.

5:00 Our aim is now to define the notion of an eigenstate for a plastic system. This can be done by defining the dispersion of an observable in a particular state row, which is nothing else than the variance of this observable But now things are different, of course, again, in quantum mechanics. We start from the eigenvalue equation, and from the eigenvalue equation, we obtain rho of R is A, rho of A is a square, and therefore dispersion of the observable A in an eigenstate rho is zero. So we have to importantly write that eigenstates of observables in quantum physics are dispersion And a similar construction can be now done in classical physics. Here we can choose particular observables which are constant over some regions in phase space and non-constant and, for technical reasons, also neither 0 or 1 in the remainder of the phase space. And from that, we can easily construct particular statistical states, namely uniform distributions as the reference measure used for the GNS construction. And continuing the dispersion, again, here is zero. That means these particular classical states are dispersion-free with respect to this observation. And so now we have a concept of an eigenstate in a classical dynamical system. And as I mentioned before, two observables, that means they share all eigenstates together, and all the eigenstates spend the rule face to face. So, as a first ... So, summarizing, we can say we have ontic observables, which are continuous functions over the face-to-face in classical systems, And we have contextually ,, which are essentially bounded functions over the space space And likewise, we can now define an ontic state as nothing

7:30 else than an individual point in space space, which are the pure states . And the space space can now be identified with the support of the uniform . And the first attempt could be now the following. We call two classical observables f and g incompatible if they are not simultaneously ionizable, which means they do not have all eigenstates in common, and their eigenstates do not span the new space. However, this gives a problem because every on-dig state, every point in phase space, is an eye state of every observator. And of course, all the points bend or generate through phase space. So this is exactly the typical, as is the very known size, classical observators are all the time. How can we arrive at an epistemic quantization? And in order to do this, we have to analyze the classical measurement process. Let us first define an instantaneous measurement. So given an epistemic observation, a function that could be constant over some regions and phase phase, We arrive at the following fact, that if f is constant over f in phase phase, all points x in phase phase are assigned to the same value of a by f. And this means, by measurement, so a is the type of instantaneously measuring the observable of x. And this means by instantaneously measuring f of x, we cannot tell whether we are in the state x or we are in a different state y, which belongs to the same set where f is constant. And therefore, we introduce the notion of epistemically equivalence, we call two points x and y in phase phase epistemically equivalent if they cannot be distinguished by measuring f alone. There's an equivalence relation on the phase space and equivalence relation to partitions

10:00 of the phase space. And so we know that the classes of epistimacy equivalent states partition the rule phase space. For the sake of simplicity, I would like to consider only finite partitions in the following. So the partition F is given by a family of sets A1 to An. A1 and F2 sets are pairwise destroyed and all together they cover the same phase phase. This is the partition induced by an epistemic observable by means of this epistemic vector And now there are two interesting limiting cases, mainly the first case, of course in this case we do not have the finite partition anymore, we call the partition the identity partition if it contains only every point in each set here, each of our singular sets. And on the other hand, we call it a partition, the trivial partition, but it contains only the rule of A to A. Another important notion is that of a refinement of two partitions. So we can choose one partition here, for example, a partition of the rectangle x into rectangular subsets, A1 to A4. We can use another partition, namely the triangular subsets, This is the so-called product partition, which is nothing else than the set, generated by all the intersections of the set of both partitions here. This partition is finer than the both other ones, and therefore is the product of the Feynman. So far, I have discussed for the instantaneous measurements. In order to describe continuous measurements, we have to take the dynamics of the system explicitly into account. So we consider a general dynamic system, maybe given by a space base, given by a map of x onto x, usually described by a field of fism. And this map can be iterated, so for an invertible time discrete system, we have the following, that state x at time

12:30 t is given by the t iterate of the map phi applied to an initial state of 0. So here, again, start with a given state of t, we apply the map phi, and it brings us to a successor of this. Taking the dynamics into account, we can now define continuous measurement. First, we make an instantaneous measurement, and we have f in a state x0. And because these states belong to classes of epistemically equivalent states, we know by measuring f in the initial state x0, nothing more than x belongs to a class A i of 0. Now we let the system void, and in the successor we measure f again. And now we know that the succeeding state x1 belongs to another set, ai1. This means the same as that the math applied to h0 is an element of ai1. And we can simply rephrase this expression by telling that x1 has 0 is also a member And so, we can describe the information gained by this continuous measurement simply by computing the intersection, X0, known to be in AI0 from the first measurement, and in 5, 3 image of like one after the second session. And for the second description, we introduced the preimage of the rule partition by the set of preimages of all elements of the partition. And now we can define what is called the dynamic refinement of the partition. Namely, as before, we know from two measuring measurements that x0 belongs to Ai0 intersected with the preimage of Ai1, which is an element of the partition product of the original partition

15:00 and the preimage of the partition in the middle name. And so here is an illustration again. Let us assume that this shaded area here is mapped onto A1, this rectangle A1 by the dynamic. And of course, we know that this shaded area is a preimage of A1 and the dynamics. Here, we have this rectangular partition. Here, we have this preimage. And now the product partition is another partition, given by this set, this set, this set, this in this position, which is the refinement of . So in this sense, dynamic refinement is refinement of partitions. And formally, we can introduce the finest refinement operator simply by computing the product partition of all the time iterates of the original partition, beginning in the infinite past and ending in an infinite future, which, so to say, describes an ever-lasting continuous measurement. And one important notion is now that we call a partition generating if the dynamic refinement restores in the end the rule phase phase. That means the finest refinement operator applied to a partition gives the identity partition. In this case, we assure that during everlasting measurement, we have access to every individual on the stage, to every individual point of the stage. I understand. You call it a continuous method. Yeah. So you're thinking, you're iterating a discrete method. Yeah, the time dynamics must not be continuous. Yeah, the time, and so you're thinking of a sort of a unit, let's say a second law. Yeah. And they're directing that. Exactly. But it does depend upon that choice of scale. No, there's no scale involved here. One simply defines the discrete time dynamical system, which is described by a representation of . Of course, in general, one would like to deal with differential equations, and then one is continuous . In our case, we restrict it to the .

17:30 On the other hand, any non-generating position has a residual cost strain. That means that the finest refinement of the position is not the identity position. In this case, it's now particularly interesting for our aim. But first of all, the finest refinement continuous measurement. How can we restore measurements with finite duration? And this is done in exactly the same way as said, by considering the four-partition algebra, which is the closure of this partition under the Boolean set operation. And strictly speaking, one has to consider four-partition algebra here. So let's just be an illustration of the finest refinement of a partition. This region here formed by unions of elements of this partition is now a member of the partition I refer. And this can be applied in the following case if we measure an observable app only used for finite duration, for example, starting here during measurement and then the system along this graph here, we know that for the finite measure rate of time, the system remains in this set of the partition algebra. And using the partition algebra, we can now define what we call epistemically accessible, which has also been done by Actually, we call it epistemic state, and we If it's directly accessible by an observable belonging to the w star algebra, if this set S belongs to the partition algebra generated by the final refinement operator, which is due to the partition here induced by our observable S, namely by the equivalent relation introduced by S. And again, for generating partitions, we know that every state is extremely accessible.

20:00 For non-generating partitions, on the other hand, the only accessible, the only extremely accessible states are two states which belong to the finest grain in this finest refinement. And so we cannot obtain more information than that is given here by the finite refinement. And in particular, that means that quantity states are generally, namely with respect to non-generating positions, that is generally non-accessible. And this means that we can use these finite strains of the finite refinement to define particular statistical states, which are not necessarily eigenstates of any observable. And so we can now define compatibility for a classical system. We call it two epistemically observable F and G compatible, if they are, exactly as in the quantum case, simultaneously, but now restricted to epistemically accessible states. And that means that f and g have all eigenstates in common, and these eigenstates must also be effectively accessible with respect to f and g. And of course, these states, , namely by the partition . And an important conclusion is that for generating partitions, so if two observables f and g both induce generating positions F and G, these observables are always competitive. And now, we can speak about the epistemic quantization of a classical system, because we have observables F and G incompatibles if they are not simultaneously diargonificed by epistemically accessible states, which, in fact, means that F and G do not have all lying states in common. these eigenstates are not epistemically accessible by F and G. And in the literature, it has been suggested to speak about complementary observables if they are maximally incomprehensible. And so now we can also define complementarity for statistical systems because of epistemical observables

22:30 F and G complementary if F and G do not have any epistemically accessible . And here are two examples who assume that the partitions F and G are not generating, because for the generating case, we know that we have compatibility. Let us first assume that there is a set belonging to the final requirement of F. The final requirement of F is that f is not the same as the finite refinement of g. In particular, that f does not belong to the finite refinement of g, but let us assume that the set f is epistemically accessible by means of measuring g, which means that s belongs to the partition algebra of the finite refinement of g. And in this case, we know by construction that f is constant over f, and f is there for an i set of f. However, g cannot be constant over s because s does not belong to the final component of It belongs to the partition algebra of g, but as a join of sets of the partition algebra of g cannot be constant. And therefore, s is not an eigen set of g. And so we have proven that to observe f and g for a class of the system must not necessarily share all ion states and they are and they are competitive. Let us further assume that f and g, two positions, are not generating and that their finite refinement are disjoint. In this case, no eigenstate of S of F is epistemically accessible with an epistemically accessible eigenstate of G and what the last one. And that means that F and G are maximally incompatible. F and G cannot have any epistemically accessible common eigenstate, which corresponds to our notion of complementarity. And these examples now illustrate that we can generalize the idea for the concept of compatibility, incompatibility, and incompatibility, and then complementarity to rule partitions. So I would like to speak about quantum partitions here. We call two partitions F and G now compatibles if they are both generating, which means

25:00 they are both, despite their finance refinement, are both identity partitions. partitions F and G incompatible if they are not identical. This is in particular the case if one of them is not generating. And we call it partitions F and G complementary if the final requirements are destroyed. And as a consequence, generating partitions are already But for non-generating partitions, we can have both cases that might be incompatible, that might be incompatible, that might be even complementary. And so I come to my conclusion. From a quantitative perspective, we have seen that classical observables are always competitive with each other. But this means that we have to consider individual positive states, they're and these are accessible by continuous measurement of quantum observables. On the other hand, from an epistemic perspective, we know that epistemic observables partition the phase space into equivalent classes. And taking the dynamics into account, we arrive at the final dynamic requirement, which provides us the epistemically accessible states. And this residual cost strain in an epistemic description gives right to what we call a epistemic concentration. So far, we have a couple of applications for these ideas. And because my main working field is in the cognitive neuroscience, in the computational neuroscience, we have provided several examples from this perspective. So for computational neural science, I'm going to show that epistemic descriptions of the phase phase of the neural network can be applied to incompatible observations. This is work based on a paper of Amari, and we are also currently further working on this project. In the cognitive neural science, I have shown that one can speak about incompatible implementations of cognitive forces, of physical systems. In the philosophy of mind, David Shalmers has introduced the neural, the idea

27:30 of neural correlated consciousness. And in particular, he defines so-called phenomenal families. These phenomenal phenomenally partition of the mental state space. And by our construction, we can map these partitions of a mental state space of a neural state state. And therefore, we can speak about emergent incompatible description for neural correlates of consciousness. And this point and the first point together give now rise to an account for the psychophysical problem, which is due to ,, namely that the mental domain is related to the first person's perspective, The material domain is related to a third-person perspective. And the first-person perspective, of course, is related to neural consciousness. The third-person perspective is related to some epistemic measurements, which can be done on neural networks. And so all together, we have here an idea how to speak about the complementarity of first-person and third-person perspectives accounting for the psychophysics problem. But on the other hand, I have to see that there are still some open problems with our approach. The first one, it is very known that generating partitions of a chaotic dynamical system give right to a so-called K-flow, which is a dynamic with non-vanishing Colmogoxina entropy, and And these both papers, for example, introduced or proved the existence of incompatible descriptions for the case. So we have a discrepancy here. In our setup, we say that generating partitions always get right through compatible descriptions and related to the approach, one knows that there are incompatible descriptions as well. How can we relate these both? There's a very known construction, the so-called cross product, which can be used to embed a commutative algebra observables into a larger algebra where the dynamic is concluded as well. And these larger algebra, these cross product algebra are non-commutative as well. question is how can we relate our findings with these incompatibilities arising from

30:00 these larger . And finally, so far, we do not have any non-trivial example for incompatible observables. And one way to provide them could be, for example, to use the idea of misplaced a partitioning for all these participants. And that's it. I thank you for your attention. Thank you. I would not say that we can predict something, but we have now a formal approach which can you to exactly speak about this notion of first-person and third-person perspective could be incompatible at all. Just for clarification, perhaps a question has a comment. So the epistemic observables you consider are step functions, so you are saying in practice we've got some limitations on measurements, we cannot really measure continuous functions, we can measure step functions, and that's what motivates the practice. the results of that. And I think that maps quite closely on to Bacon's work.

32:30 There's a question here. Well, I was actually, similar to Guido, I suppose, interested in distinction. I was wondering whether you have a motivation for restricting attention to black distributions The motivation was to introduce the notion of ambiguous measurement. So if a function, if an observer would be a monotonic, for example, there wouldn't be any ambiguity in measuring the value of an observer in two different points. And the second point was, how can we speak about eye states of such observables? And the construction I have used here in the very beginning was exactly in this way. The construction here is what exactly in that way that we define a function which is constant over some region in phase phase, which is non-constant over the remainder in phase phase. We can construct a uniform distribution over the set. And then we can speak about the fact that this distribution here, the statistical state, is an eigenstate of this particular function here. I guess maybe to answer my question is that ultimately you're only interested in the sharp measurements and you're trying to semi-projective measurement blind for them. Yeah. Thanks. This may not be possible, but in a few minutes, you can give us some kind of an intuitive feel for how this formalism applies to differentially describe first person and first person perspective would be wonderful to get better. It's because I'm not connecting the formalism with the intuitive idea at all. understand what it might be used for. Can I? Yeah. Leave it there. Leave it there.

35:00 The idea is that David Chiamat speaks about phenomenal families. So we have, in a very abstract sense, we have a phase-based mix of a neural system, and we have something like a mental phase-based. I don't know what this means exactly, but Schallmann's idea was to consider a particular partition of this metaphase space, which is then described by there was disjoint phenomenal content. So these are the phenomenal families. And on the other hand, we have this idea of supervenience, that two different points in phase space could divide to the same phenomenal content. So if we have here, phenomenal content A, and here we have this And now we can simultaneously assign to this phenomenon content A a particular observable label called FA. And every time if the phenomenon content A is present, we say that we have a mapping from the mental phase space to this phenomenon content. And now you see that these four points, six and y in the mental phase phase will be neural. You mean the neural. You mean the neural. Yeah. In the neural phase phase. I think these four points in the neural phase phase cannot be distinguished by means of measuring f. And therefore, they belong to an occurrence class in the neural phase phase. And in total, because we have a partitioning of the mental phase phase, this is right to a corresponding partitioning of the neural phase phase. And that's it. So this is, so to say, the first person has selected. On the other hand, we can do, for example, neural imaging measurements or EEG measurements on the neural phase phase. And here, again, the same situation is present. There might be two different neural activation states which give rise to the same measurement for the large test. For example, if we measure the EEG, we know EEG is something like the sum

37:30 of all the microscopic neural activations over a large area in the brain. And these sums here give rise to ambiguity in measuring F. And so again, this microscopic observable is just rise to a partitioning of the neural phase but again, and so we have, let me write it down this way, We have two different partitions, the black one is used by the first person perspective, the red one is used by the third person perspective. And if we now determine the finest refinement of these partitions, they could probably not be the same. And in fact, most likely they wouldn't be both identity partitions. And in the case, people speak about the compatibility between the mental and the material for men. Yes, please. In particular, you can take your mental stage in the field, or the kind of physical field, and look at the neurons and detect them. I mean, there have been several people who looked at this. There was the mathematician Chris Leeman, who wrote a paper on this. with the fact that the neural partitions in particular, the metric there, which is the ordinary metric, is very different from the metric individual here. The metrics don't correspond at all. All I remember is he did get a result, as a result of certain things, that intersections do correspond to intersections, And that allowed them to define a certain thing, a check cohomology, and show that there was an invariant. Nevertheless, the first check cohomology group is actually an invariant in going from the visual field to the neural spectrum. So there is something there that's somewhat connected. Yeah, really, and this is a very interesting point, because in dynamic systems, there's another notion of a Wack partition. We know that all micro-partitions are generating partitions, and for micro-partitions, we have exactly the case that boundaries along the extending directions are mapped off to each other. So this could be related to this one. Yes, I wanted to make a remark concerning a few of those questions.

40:00 You know, really, how can this stuff be applied to the distinction between mental states and neural states, something like that. And I think when you look at this figure, Peter, which you've illustrated there, I think the crucial part of the whole business is to identify purely on the basis of the neural data points, the EEG data or something like that, MEG, to identify and generate a partition. And then on the basis of that generating partition, the idea is to predict certain well-defined find distinction between metal spaces, not defined in the sense of the whole methodology. And you said already that this business to identify generating partitions is non-trivial, but it can be done and it can be approximated. I think this is the main difficulty in the experimental procedure. And in the second application which you mentioned, we actually have applied this methodology already to experimental data, and this is quite promising, quite successful. Thank you. This is Jeff. I know a little piece of experimental data which might be interesting to you in this topic. There's a rather elderly neurologist who now studies among other things that he has degenerations of age. And one of the examples he has in his own experience is that he has some sort of any visual field which is filled with a kind of good shape and he's come up with an explanation for this in terms of the physical properties and that's how it is made but I know how finds remain and if he doesn't find a copy Oh, I'm sorry Could you explain another very last comment you made that you don't have any non-trivial examples of thinking pastoral partitions? Yeah, so far we have only trivial examples. What is the notion that trivial or non-trivial? Ah, okay. The idea is, which is related to this notion of a misplaced partition, If we have a chaotic system, and if we know that there are generating partitions, one can even try to choose arbitrary partitions, which are then not generating, and therefore they are caught misplaced. And our idea is to construct such misplaced partitions for chaotic systems, which are

42:30 known to have generating partitions, and then to look whether the finest refinement generated by these misplaced partitions could provide to these incompatible descriptions that I was talking about. So we only know from the conceptual point of view that if there is a cross-drain in the finest refinement, we can explicitly construct these incompatible observables on these sets simply by choosing functions which are constant over the sets and by by assigning uniform lenses to these sets, and that's it, it's only conceptual work at this point. If you want to have any, yeah, non-trivial example how to achieve a partition for concrete physical systems, for example, for, I don't know, the . Now we do have a quick question. Thank you very much. I guess we have a tiny break it back. Yeah, take five. Thank you very much. Thank you. The actual, I know what the interaction and what they look like is essentially...

45:00 Thank you. Thank you. Thank you. Thank you. Thank you.

47:30 Thank you.