Emergence of Properties and Non-Separability of Hamiltonians in QM

0:00 So, we might make a start. So, welcome to this last session of the Sigma Countries for this year. And it's a great pleasure to welcome back one of our fairly regular summer physicists, Biddy Crouch of Orson, Texas. He's going to talk to us today about the emergence of quantum mechanics. That sounds a fascinating topic we look forward to very much. Great. I'm happy to be back, and I hope you'll like what I have to present for you. So what I'm going to do is develop a proposal by Paul Humphreys in a fairly recent paper, How Properties Emerge, which was published in 97, and he develops this notion of fusion. Now, his paradigm for fusion is non-severability of states, and what I'm going to suggest is that's maybe the best way to think about fusion, especially in connection with some notion of emergence. So what I'll do is argue that it might be better to think of emergence in terms of the non-separability of quantum Hamiltonians, and I'll try to make a case for that after giving some simple examples of non-separability. Okay, so first what I'll do is give you some sort of background on Humphrey's point of view, and then we'll go to some simple quantum cases and look at it in that context. So to characterize this notion of fusion, he introduces assumption L, which says that there's a hierarchy of property levels, L0, L1, Ln, and maybe beyond, of which at least one distinct level is associated with the subject matter of each special science, and Lj is irreducible to Li for i less than j. And what I'm going to argue is that at least one is a bit of an understatement. probably a rather large number of levels that you would associate with each special science. Now he also introduces the following notation. He says PI is to be, so the superscript I

2:30 corresponds to an I-level property and this is the case of I is the first level in which the instance of a property occurs. The set of properties which is associated with each level i is denoted by this set right here. So the m, the subscripts are just sort of labeling the various properties to be found at that level. And you do the same sort of thing for entities. And in general, Humphrey says, i-level entities may have j-level properties for i So a little bit more background on this point of view and then on to quantum mechanics. So the fusion operation, he introduces with this sort of bracketing expression with the star in between the two dots of placeholders for properties. And this is supposed to be a process that combines two I-level properties to form an I-plus-one-level property. And he suggests that this ought to be regarded as an I-level operation, although I'm going to suggest that I think it's better construed as an I-plus-one-level operation and that you can get some sort of downward from this. So I think this is an interesting claim. So what he suggests is that interactions that give rise to entangled states in quantum mechanics lend themselves to the fusion treatment. he says that the result of fusion is the non-severability of states and the non-severability of states leads to the emergence Alright, so some preliminary comments. That's the view. And we'll be looking at non-separable states, non-separable properties, Hamiltonians in particular, and time evolutions in just a little bit. Alright, so as I already said, it's a non-separable Hamiltonian at the I plus one level that gives rise to non-separable states. So this is why I think the fusion is an I-plus-one level operation, so I'm disagreeing with, let's see, this plane right here.

5:00 The I-plus-one level of Hamiltonian turns I-level states and properties into I-plus-one level states and properties, and what I want to suggest is this is a type downward causation. You have the property at the I-plus-one level pulling things up from the, affecting things at the eye level and pulling them up to the eye plus one level, and it looks like a kind of normalization. Now, the crucial feature for understanding emerging phenomena is an ongoing non-separable interaction, and what I want to suggest to you is that this is what goes on in quantum chemistry. This is precisely what you have in quantum chemistry, and it causes very substantial problems in solving problems in quantum chemistry. And I'm hoping that I'll have time to get into some of that because I think it's interesting. Ultimately, not the most relevant feature, but certainly a very relevant feature, especially in light of the history that goes back to the British emergentists and their view about emergence and how emergence comes about. All right. And when the non-separable interaction ceases, non-separable states persist. This is something to keep in mind for later. So you can have, even in space-like separated systems, this non-separability of the states called EPR correlation persist despite the fact that the interactions cease. Okay. So to sort of help you to develop some sense for what I mean by non-separability of states and non-separability of properties, I thought it might be good to just kind of focus on simple two-state systems and pairs of two-state systems. Okay, so before we get into that, let me just provide you with a little bit of basic information. That's this point right here. In quantum mechanics, there are basic three elements, three basic elements of a quantum system. That's its state, its properties, and its evolution. state as the condition of the system with respect to all of its properties, properties

7:30 you might think of as quantities that can be measured on the system, although that sort of operational claim some people might object to, but just to have a handle, I think it's a good thing to have a handle on it, and the evolution is how the state changes with time, which characterizes how the properties of the system also change with time. So what I want to do is just focus for the moment on two-state systems, and typical two-state systems are spin-and-a-half systems or spin-one systems that have no mass, like photons. These are very nice systems. The simplest, non-trivial kind of system that you can talk about in quantum mechanics from the standpoint of pedagogy, whenever I teach a course on philosophy of quantum mechanics, again I really try to get the students in to understand this model and then you generalize to the larger cases. Okay, so we want to look at states, properties, and evolutions. So for a two-state system the state is represented by a vector. It It has two components, A and B, and what you require is that AA star plus BB star is equal to 1. All the star means here is the complex conjugate, so if A is equal to C plus ID, then A star is just equal to C minus ID. If you multiply these two together, what you'll see is that you get a positive real number coming out of that. And the idea is that you want this, the sum of the products here to sum to 1. The reason you do that is you give state vectors a probabilistic interpretation in quantum mechanics. All right, the general form of property matrix is this in two dimensions. More generally, you require this sort of symmetry feature right there, that Aij is equal to Aji star. So, corresponding components here are complex conjugates with respect to one another, and for AII equals AII star, that means that the components along the diagonal are real.

10:00 The general form of evolution in a two-state system is e to the iat, where a is a property matrix. If a is diagonal, then the evolution, this should be u here, has a simpler form. It's just e to the iat, e to the ic, setting b to the star equal to zero. All right, so that's a whirlwind tour of two-state systems. And now we'll go on to pairs of two-state systems and two non-separability. living. By the way, if you have any questions of clarification as I'm talking, it's fine. If it's an objection, please hold it until the end and during the discussion. But if you need me to clarify something, I don't mind. You can just speak up if there's a pause and I'll try my best to try to clarify for you. Okay, so when you're talking about compound systems, the key operation for correctly representing compound systems is the tensor product operation. And the feature of non-separability that we're interested in, Humphrey in the case of states, me in the case of time evolutions, has to do with the fact that you take sums of tensor products of states and properties, and that's what gives you a non-separable state. Not all sums, but many of them are non-separable. Okay, now what I want to show you first is an example of non-separability having to do with states, and I tried to make it as possible. So if I have two vectors, phi 1 and phi 2, phi 1 has components A and B, phi 2, C and D, then their tensor product is defined as this four-dimensional vector. So basically what I'm doing is I'm taking A here, I'm multiplying it by C, D, and I'm taking B and also I'm multiplying I get by CD. So I get ACAD, BCBD. Hopefully that helps you to see the pattern. All right, so

12:30 what do we mean by a non-separable state? It means simply a state that can't be put in this form. Can't be put in tensor product form. And this is a nice, simple example of a non-separable state. And it's pretty easy to see that you can't put that vector into this form. The reason is that if AD has to be equal to zero, that means that either A is zero or D is zero. If A is zero, then the first element is zero, which of course it's not. And if D is zero, then this element has to be zero, which of course it's not. So what this shows is that this vector, phi 1 plus 2, cannot be represented in tensor product form. This is an example of a non-separable state. And for property matrices, the situation is very similar. This is the definition of a tensor product of properties. Basically, what you're doing, you can think of it this way. I have the A element here, I'm multiplying it by E, F, G, and H, that gives me this little block here, and I take B and multiply it by the same C and D, and that's my tensor product. Okay, and if you have a 4 by 4 matrix, it's separable if you can put it in this form, if you can find two matrices such that when you take their tensor product, you get this path. So, what I want to show you now is that this matrix cannot be put into tensor product form. And let's see why that is. I tried to pick very simple cases here, so it's pretty easy to see. So here I have AH must be equal to zero, but if A is equal to zero, then this component's zero, so that's no good. And if H is equal to zero, then this would be zero. Again, that's no good. So what we have here is an example of a non-separable property matrix.

15:00 And if you wanted to, it's sort of an exercise, you could sort of see that this is just a superposition of two tensor product matrices. So you would use this rule to form this tensor product for each of these and then some component-wise and you would get this vector, I'm sorry, that matrix. So this non-separable matrix has this form right here, superposition of tensor product matrices. So, what I'm going to do here is try to connect what I've said so far about non-separability with what we began with, which is this notion of fusion. All right, so Humphreys characterizes fusion as a process that combines two I-level properties to form an I-plus-one-level property. And my example, I think, shows that what the fusion operation basically corresponds to is sort of taking the tensor product of matrices and summing them together. Okay, so the idea is this, because it's in tensor product form, would be at the I level. Actually, in this case, it would be at the lowest level. with this, and then when you take their sum, you get a property at the next level. Alright, so fusion can be understood as taking tensor products and then taking their sum, and if the sum leads to something that's non-separable, then you have formed a higher level property. And as I showed you in the last slide, you can't put the sum of these two in tensor product form, so it's a level two property. Alright, and this just generalizes it to triplets, quadruplets, etc. and I think it's relatively straightforward, so I won't dwell on that.

17:30 So this is why you have lots and lots of levels in social science. Yeah, that's right. I'm coming to that, but that's right. Okay, so for evolutions, the situation is very similar to that for properties. So I consider this a diagonalized matrix. I ask the question, can this matrix be put in this form? And what I argue is that in general it can't. You would have to satisfy these four conditions right here. And you can easily show that these four conditions are satisfied if and only if alpha plus delta is equal to beta plus gamma, which need not be the case in general. So what I would say is that separability is an exceptional case, non-separability of the Hamiltonian is typical. It's a typical case. All right. Now, you may be wondering, what does this have to do with emergence after all? So what I want to do now is sort of tie into the historical literature from back, say, around the end of the 19th century and the beginning of the 20th century. There were a group of philosophers known as the British Emergentists who tried to capture some sort of notion of emergence. And one of the first was Mill, and one of the last, who was the last in that group, C.D. Broad is probably the last person who wrote in this mode out of this particular tradition. And what I want to suggest to you is that this notion of fusion that I'm pointing to here, not just in connection with properties but also in connection with time evolution, I think sort of fits into what they said in a certain way. Anyway, the way in which drew a division between kinds of properties, resultant properties, meaning those that are non-emergent, and the emergent properties was in terms of additivity. So if you had

20:00 properties that were additive, mass is an example, then what you would say is, ah, that's not emergent, it's just resultant. Now, what's interesting is they took scalar and vector addition, especially in connection with forces, as being paradigms of additivity. But what's interesting is that we know now that additivity fails in the case of mass. So if you take four hydrogen atoms, you combine them to form one helium atom. The helium atom weighs less than the sum of the four hydrogen atoms. So the question is, well what happened to the mass? Where did it go? Well, Einstein's theory tells us that it was converted into energy. So energy was given off in the process and what you can do in making the shift from classical mechanics to relativity theory is to say that the fundamental quantity is not really mass but it's mass energy and if you treat mass energy as the invariant then you have additivity back again so you can restore additivity by in this specific case by going to a more general theory now also Newtonian mechanics many quantities position force these are represented by three vectors so they have three components in this case if you have a force right and it's pointing in that direction let's say and what I can do is institute a coordinate system X Y and Z and there will be components of the force going in each of these three directions so that's the way to think of this is sort of the components of the force now if you take two forces F and G, and you say, what's the resultant force? It's just this. You just sum the components together to get the force. But unfortunately, additivity also fails for vectors in the classical situation. We know that velocity can't be additive, because if velocity were additive, then it would be possible for massive bodies to go faster than the speed of light, which, of course, they can't. So how do we change this? Well, you do so by instead of

22:30 using three vectors, you use four vector quantities. So examples would be position space-time, and momentum energy. These are quantities that behave additively. All right, so additivity does seem to be a nice thing to focus on, as being a very simple way of taking the properties of the systems and combining them. And what was of particular interest of the British emergentists were those properties that had to do with causal properties, and so additivity forces was their paradigm. Now there's another type of additivity that you get in classical mechanics that I think is worth mentioning, and the reason is that it contrasts very nicely with the tensor product operation, which is what you use in quantum mechanics. So this is the counterpart to the tensor product operation in classical mechanics. It's called the Cartesian product. It's also known as the direct sum. And it's another type of additivity having to do with the way in which you form composite systems out of smaller systems. Alright, so if you're talking in classical mechanics, the usual way of representing a system in that context is to talk about phase space, and phase space will have three position components and three momentum components, and that's for a single particle. Now, if you have two particles, then you form a larger phase space. So you'll have three position components for each particle. So X1, Y1, Z1, three position particles. It's the same for two, and then three momenta particles for a total of 12 distinct components. If you go to the more general case, and you have of n components, then that will be represented in a six times n dimensional phase space. And this is to be contrasted with the situation in quantum mechanics. If the system consists

25:00 of n m-state systems, we consider two two-state systems, but if you were to generalize this to n m-state systems, then your tensor product space would be of dimension m to the n. So So this grows linearly, this grows exponentially, a very different kind of construction. Now, another point of contrast, there are no non-separable states in classical mechanics, and this has to do with the fact that you're using the direct sum. There are, however, non-separable Hamiltonians in classical mechanics, And what's interesting is this is the result of nonlinear terms in the equation of motion. So the reason I point this out is that in quantum mechanics, you don't have nonlinear terms in your equation of motion. So you might be inclined to think, ah, well, how can you get non-separability in that context? Well, it has to do with the fact that the way in which you form composite systems uses the tensor product rather than the direct sum. So it plays an extremely important role in quantum mechanics. All right, so now what I want to do is I have three slides that tie into the more general literature. So I'm coming back now to this notion of levels. And then after that, I want to raise an objection to my characterization of emergence and suggest two ways in which it's possible to respond. So, if you've been keeping up with the literature and philosophy of mind, then you're probably familiar with an argument by Kim called the exclusion argument. This diagram is supposed to capture the essence of that argument. So what Kim says is this. He says, let's suppose that we have the causal closure of the physical. And let's suppose that the mental supervenes on the physical. The multiple lines here are supposed to indicate multiple realizability. So this is why it's a supervenience relation.

27:30 Kim says, if you have the causal closure of the physical the mental supervening on the physical, then you have to regard this process, one mental state leading to another, as being merely a pseudo-process. Pseudo-process. All right, so let me give you an example of a pseudo-process. All right, so if I had a little laser light in my hand and I sort of flashed it along the wall, the dot that moves along the wall is not a process, it's a pseudo-process, okay? The process involves photons leaving my little laser light and striking the wall, and then they're bouncing and reflecting off the wall to your eyes. That's the process. The apparent motion of a dot down the wall is a pseudo-process. So what Kim is saying is that my thinking that I ought to scratch my head, and then I shouldn't do that right now, because it's a little embarrassing situation. I don't know. My thinking that thought, and then doing the action, there's no causal relationship between the two. It's a pseudo process. So Kim is not happy with this. With this conclusion, he likes the idea of mental causation. So now, it's very interesting, he's giving up the notion of supervenience, and now he wants a stronger notion of identity, and he's moving, sort of, I would say regressing, back to a reductionist, a reductionist way of thinking about things. You like this? All right. Okay, so the reason I bring this up is because I thought it might be a nice way to sort of tie in to the broader context and give you a very different picture of the situation in this very special context of the quantum mechanics. So that's what the next slide will do. All right. Okay, so we have at least one level associated with each special science. This goes back to assumption L. And he does indicate, I should point this out,

30:00 that this is merely a heuristic, he says, Humphrey says, and likely will need to be given up, I want to suggest that, yeah, there are serious shortcomings in it, but it's a nice heuristic, it's a nice starting place. Alright, so the picture is something like this. In general, you'll have different levels physics, chemistry, biology, and higher up, let's say, psychology And what you can have are systems evolving at a lower level, and then you have a non-separable Hamiltonian coming into play, and suddenly you have a more complicated system. It might be a molecule consisting of several atoms. Where you draw the line is a little bit tricky. I don't know where physics ends and chemistry begins. It's a tricky question. But the idea is that you have a physical system consisting of independently evolving components. Suddenly they interact by way of a non-separable Hamiltonian, and you've got a chemical system. And that could evolve under this Hamiltonian. Maybe other systems come into play, and you have evolution to yet a higher level, emergence to a higher level. And it may be that once this takes place, that with respect to the systems of interest, you no longer have anything happening done on this level. It all takes place on this level. So it could be that when something emerges, nothing is left at the lower level to correspond to that. So this picture is very different than this. This picture, so you're supposing that the physical is the primary level, and anything sort of higher than that, well, it's going to be, in this case, a pseudo-process. And we're saying, no, we think of it very differently. It may be the case that the lower levels go out of existence when the higher levels come into existence. So it's a very different picture. All right, so Humphreys suggests that quantum non-separability is the hallmark of emergence. If so, then the number N of levels in the hierarchy is likely to be very large, as Carl already indicated.

32:30 So just to sort of drive that point home, I have another diagram. So, in this case, instead of thinking in terms of the special sciences, in thinking of levels in a very coarse-grained sort of way, what I'm doing now is I'm looking at the levels in a very fine-grained sort of way. So here I have three distinct Hamiltonians, and there's no interaction term. So everything stays at level one. And then an interaction occurs. So systems one and two have an interaction component. System three stays relatively independent. And so now things are evolving at level two, and you have the initial, one of the other systems of the triplet evolving at level one. So you have multiple levels where something's happening. Now let's suppose that all three of them are interacting, and what happens is there's nothing happening at levels one and two. everything happens at level 3. So, what happens is you get more and more entanglement by way of this non-separable Hamiltonian. You get higher and higher emergence of properties. Now, as I said, there are some shortcomings in this picture. So, let me sort of draw those out a bit. It's not clear whether there are distinct boundaries between domains. Physics and chemistry, this is clearly the case here. My guess is that probably you could generalize this to higher levels. There are no distinct boundaries. How can you say, this is where physics ends, this is where chemistry ends? You can't. The analysis thus far offers nothing by way of clarifying the distinctive characters of chemistry or the other special sciences. So if you were really going to implement this picture, you'd have to add more to it. This sort of appealing to non-separable states and non-separable Hamiltonians is very suggestive, but to get the special sciences, more analysis is going to be necessary.

35:00 All right, and the last two points are just to elaborate on that point. Now, here's an objection I think is very interesting. Somebody might look at the British emergentists and say, yeah, they focused on this notion of additivity, but you shouldn't take additivity so literally. What they really meant was what we need to show that properties are resultant and not emergent is that there's a simple algorithm for combining causal factors. So additivity of forces like vector addition, it's just a paradigm. what's required for emergence. Now, if it's a simple algorithm that's what's needed, then this project, it looks like maybe in trouble. Because the fusion of quantum Hamiltonians is a relatively simple algorithm for combining causal factors. It's relatively straightforward. In this context, what you do is you take the kinetic and potential energy terms of the components and the interaction terms involving the components, and these are the causal elements in the quantum context. You combine them using tensor product operation and additional linear operators. This is just matrix addition in effect. You might say, well, what about wave mechanics or anything you do in wave mechanics, you can always find a matrix counterpart. And the mathematical operation is relatively simple and simplicity is the key feature. So despite the substantial differences between the quantum and classical algorithms for combining

37:30 causal factors with compounded systems, it seems that both sets of algorithms, that used and that used in quantum mechanics satisfy the emergentist condition for what constitutes a resultant property. Alright, so what I want to do is develop two responses to this objection. All right, so the first point is more of an epistemic point, and I don't regard the epistemic point as being the deeper sort of point. I think the deeper point is the second one involving some sort of ontic feature that brings in the causal features of the system into play. I think it's important to be aware of the epistemic problems because I think that if you read the British emergentists, there is oftentimes some sort of a kind of propensity to blur the distinction between the epistemic and the ontic. And, you know, thinking how could we possibly explain how chemicals combine to form more complicated things, we just don't have the tools to explain, for example, how hydrogen and oxygen, two gases, can be combined to form a liquid. And it looks like we'll never be able to explain that. So they regarded water and the liquidity of water as being emergent from the gaseous qualities of hydrogen and oxygen because they thought it would just be entirely too difficult to get an account of this. But what I want to suggest to you is that's an epistemic issue and really the key issue is the ontic one when you're talking about things like emergence. I think that somebody who wanted to probe this issue, the epistemic issue, takes that side of things seriously, and it should be taken seriously. I'm just saying that the

40:00 object should be taken more seriously. There are extremely difficult problems that arise in this context. Even though the formula for combining properties to form higher level properties is a simple one, it doesn't follow that the system that you're studying in these terms will be correspondingly simple. In fact, it turns out that even for very simple real quantum systems, things get incredibly difficult very quickly, and I want to try to capture that. I want to talk a little bit about quantum chemistry, and I think I'll have time for that and time for this. So, I think the simplicity of the algorithm is misleading. I mean, what I did was I used very simple toy models, so you might be inclined to think, ah, it's simple, right? It's just algebraic manipulation is all that's involved here. But see, that's a highly specialized case. You're working with finite number of degrees of freedom, and each of those degrees of freedom is finite. Very special case. And it's not the typical case in quantum mechanics. But for these two models, it's very simple to solve Schrodinger's equation. In realistic models, dimensional, or the number of degrees of freedom are infinite, things get much more complicated very quickly. It turns out that such models cannot be solved exactly in most cases. Reason has to do at bottom with the non-separability of the Hamiltonian. You can use approximation methods, but you need to introduce significant ad hoc assumptions, even get results, substantial results in relatively simple cases. Okay, so that's sort of an overview of the epistemic issue, and what I want to do is say more about this. But before I do that, I want to make make some brief comments about the ontic issue and then there too I'll say more about that. Now the key

42:30 ontic feature that I regard as being crucial for emergence is that the systems are continually interacting. And what this does is that it produces an inextricable link between the component systems. And these links between systems are so robust that it's no longer meaningful to talk about components that are parts of a compound. The simplicity of the algorithm and the complexity of solving the resulting equation of motion are primarily epistemic matters. To my mind, the simplicity or complexity of the algorithm is really not crucial, although I think it's important. What is crucial are the robust lengths between causal features of the subsystems. So, what I want to do is try to explain what I have in mind here by these lengths. All right, so I have four, as it turns out, I have four slides on the epistemic issue and two on the ontic, and I think I should be able to, well, actually, I'm not sure. So let me do this. I'm going to do the ontic, turn to the ontic issue, and see how things are going, and then make a decision, because I'm going to watch it. So this is the ontic aspect of... Alright, so what we do is we consider a compound system consisting of two components. and the initial state of the compound is a superposition. And what we do is we evolve the system using a unitary time evolution operator. Now, if you can put the time evolution operator in tensor product form, then you can derive

45:00 a density operator from this state expressed in density operator notation. You can derive a density operator that corresponds to just one component of the compound system. Now, if the Hamiltonian is separable, then what you'll see is that all the time evolution takes place within this region right here and this number, it's a positive real number all that's evolving is the first component of the system. The second component is stationary So in this situation where the Hamiltonian is separable, each component of the compound system evolves independently, and you can see that here. Now if you derive the corresponding expression for a system in which the Hamiltonian is non-separable, again all the evolution takes place here, but you'll notice that the difference is that the evolution here involves both components of the system not just one component, okay so in effect, the most complete expression of the state of a component system involves appeal to the time evolution of the entire system okay, so the part cannot be understood independently of the whole Let's see how it's shot. Now, it's not real clear at the moment that identical particles are involved here, but they are because really the systems that I have in mind are systems that you consider in quantum chemistry, so multi-electron atoms and molecules, so you're thinking in terms

47:30 And in that context, the emergent holes, like a molecule, they're produced by an essential ongoing interaction of its parts. And in almost every relevant case, I can't think of one where it's not relevant, identical particles are involved, and special care must be taken as a result of this. And the way I would put it is like this That identical, interacting identical particles Cannot be identified with those that existed Prior to the interaction Due to Pauli's exclusion principle So if I take, let's say, the simplest possible molecule I take two hydrogen atoms And I allow them to interact Before the interaction before there's any overlap between the wave functions of the two hydrogen atoms you can say there's an electron here it belongs to this atom there's an electron here it belongs to this atom when you allow them to interact the electrons are identical in the sense not only can you not distinguish which atom brought which electron with it the question doesn't make any sense to even ask. You know that you have two electrons there, but you can't say which electron was brought by which atom. And the same with the protons. I mean, so what happens when these two hydrogen atoms combine is that what I would say is that two systems, Sometimes the two electrons go out of existence, and two new electrons that are identical come into existence. Some very curious things follow. Density operators for identical particles are identical except for the label, which merely assists in indicating how many identical particles are present in the system. It does not denote. Margano put it this way, number becomes an observable despite the indistinguishability of the numbered entities. Now, what I want to do is point out that you're not really

50:00 forced to such a radical ontological thesis, right? With regards to identical particles, you can, there are alternative positions. Von Frassen is one that I take as sort of an example here. Von Frosten says that a boson may be individuated by its history, but the history is empirically vacuous, and it's a non-quantum feature. So I don't think of that as particularly physical. A fermion can have a value state that involves only quantum features, and what you do to make that work as you break the eigenvector eigenvalue link. Now, this interpretive approach, which involves hidden variables, runs counter to what most philosophers of physics think, so I'm not so concerned about that here. You could do it, that's right. I think originally when he published papers on this topic in journals, I think he did advocate the position, but in the book, it looks like there's a bit of a conceptual shift. He becomes more of a pluralist, right? His attitude is, ah, the more interpretations, the merrier, so let me give you this one, right? But I think in the papers, he was advocating it, but then there's sort of a withdrawal. At least that's, that's, Jude Bree is... Right, but I didn't look it, but it's just books. Yeah, yeah, yeah. That's my sense. But you're right. He does have sort of a pluralist view in the book and does not strictly speaking advocate this view yeah but he does indicate that this could adopt if you wanted yeah all right i think i will not go into the epistemic issue if you want question period i can we can look at those slides so i want to stick with the stay with this metaphysical theme. I've been wavering up here about whether to do that or not, and I think it's a good thing to do. I'll justify my wavering by saying this is my first time giving this presentation. So anyway, let me draw a conclusion by way of contrast. What I'm going to do is

52:30 is look at three different ontologies. One is sort of the British emergentist ontology. The second is what I take to be Humphreys, and the third is the one that I'm putting forth here in this paper. So to get at these pictures, I talk about the notion of an independent characterization of an entity. Now, by characterization of an entity, I mean a list of properties is possessed by the entity, and what I say is that a characterization is independent if the elements of the list make no essential reference to any other entity. All right, so with that, with those two notions in mind, the notion of characterization, the notion of independence, we can characterize three distinct ontologies. So the first one says this, there are genuine parts and wholes that may be independently and there are some suitable criterion for distinguishing between part-whole relations that are emergent from those that are resultant. As an example, the British emergentists introduced the criterion of additivity of forces to distinguish between emergent and resultant holes. Holes involving additive forces are resultant, and those involving non-additive or configurational forces are emergent. So the key here is that you do have genuine part-whole relations, So Humphreys has the more radical position of the remaining two. He wants to give up the idea that there are these parts there. So in the second ontological view, we have only resultant holes have genuine parts. Strictly speaking, emergent holes do not. Emergent holes are produced by a fusion of certain entities that can in some ways be likened to parts. It is because these part-like entities cease to exist upon fusion that they cannot be viewed as genuine parts of the emergent whole. The parts only exist when the whole does not, and vice versa. The view that Humphreys sketches is a case in point. Fusion involves the formation of a non-central state. All right, and here's the view that's being indicated here.

55:00 Emergent holes have genuine parts, but these parts cannot be characterized independently from those associated holes. Emergent holes are produced by an essential ongoing interaction of its parts. These are the central features that a new view sketched above. The non-several Hamiltonian constitutes an essential ongoing interaction. As I already indicated, this view falls somewhere between the first and the second. And emergent holes do have parts, as in the first, but not in the second. But there's no characterization of these parts that is independent of that hole to which it belongs, unlike the first view. The third is less extreme than the second in that an essential interaction of the parts causes them to go out of existence, as in the second. But new parts emerge that are dependent on the hole, unlike the second. by adopting the third view we can say that it does not make sense to talk about reducing an emergent hole to its part since the parts are in some sense constructs of our characterization thank you very much indeed very interesting and deep and vital issues if I could just raise one or two I think the first thing about emergence, particularly related to problem mechanics, is whether you're going to take what you might call an orthodox view, a kind of eigenvalue, eigenvector link kind of view, and then you can make some other strong statements about emergence. And on the other hand, it's the kind of invaluable view, which then tends to be a bit more murky and things are not so straightforward. So let me just concentrate on two or three questions about the orthodox views. We're just going to... That makes my life easy. Invaluable people who have representatives in the audience make their points about how they would interpret the situation. It seems to me that there are two sort of puzzles about emergence. The first one is the sort of puzzle you get. If you just take a simple example of a single state of two spin-and-a-half particles, you've got a definite property for the total spin, and you want to, if you're a reductionist or some sort of supervenience notion, you want to try and say that this total angular momentum is somehow explained by the separate

57:30 properties of the individual part. Well, now, on orthodox interpretation in such a entangled state, individual electrons don't have any properties. You can't explain properties on the basis of properties that don't exist, I think. So the whole enterprise of, you know, the emergence seems very clear and straightforward. Yes, you've got a lot of emergent properties coming from entangled states. Well, then there's the other sort of case where this is a problem chemistry case, which gave all the different quantum chemistry. The way quantum chemistry used to be done, and still is done, in fact, is to assume that what we call the Born-Oppenheimer approximation, you decouple the electronic and the nuclear motions. So roughly what you've got in mind is you've got the nuclei at fixed positions, and then in those locations you work out what the energy of the of course the decoupling term doesn't really work, and many people pointed out that it's only an approximation, and they've got a philosopher, you should take them dead seriously. So really, molecules, the nuclei, do not have definite positions. The positions are entangled with the electronic motion. And what this means is that molecules don't have shapes. And this is a terrible shock to the chemists, because most of chemistry is about the shapes of molecules. If you open a book on chemistry, you see benzene, green, all these things. They have shapes, and roughly an explanation of chemistry is how one shape locks onto another shape. was that sort of subject, so you're completely torn away, the whole underpinning of most of chemistry by telling the chemists that molecules, after all, they don't have shapes, so after all, they talk about the shapes and how molecules fit together, all that, all that, all that. And so, the way that chemists tend to respond, there's not a lot of, there's a question coming, how do you really make sense of this? They say, oh, well, yes, you You can't build up the idea of shape from the individual molecules, you know, in that sense it doesn't ride on there, they don't have shapes, but nevertheless they must have shapes because chemistry is very practical, important, full of application and so on.

1:00:00 Therefore, shape must be an emergent property, not in the sense that it emerges in the quantum mechanical treatment, because we've just proved it doesn't. It's the obvious case, generally, of a similar state case, where the thing has the complex property, but you have a lot of simple properties. It's another, because having a case of quantum mechanics, it doesn't have this property of the whole system, the chain. And so the chemists say, well, we must have a chain. And so they kind of sort of feed it in by hand. This is sort of getting back to this reputable way that the quantum chemist proceeds. And they say, oh, it is a chain, but it gets about the borough, not the high, I don't think. So they once again hog down the nuclei and solve it. The end part of referring to the electronic questions and get out the head edges and talk about the oscillations or the rotation term. Now, I think there's a basic difficulty and inconsistency right at the heart of what you get is centered around this question of shape. Now, I've observed my second point, an interesting healthy comment you make about that, but it's an opposite case in the first one. And then the final point is about the identical particles. Well, now, you said that your view was that you had these two separate atoms, and then the electrons, you couldn't tell which atom, and when they fused them together with a molecule, you said, well, maybe the electrons will give out of existence, and then we create it. Well, I think that there is a sort of formalism to doing n-particle quantum mechanics, which is sex-departisation, which does roughly work like that. In fact, we introduce these creation-analysis operations, so in the context of n-particle quantum mechanics, which really, I think, a pass-on-de-parlay, I don't think you should take it too seriously. Of course, what it does mean is the two separated atoms, the electrons have got to be anti-symmetrized, so you can't tell which electrons and which atoms. And when the whole thing is mixed up with the molecules, you can no longer tell which is which, you've got to anti-symmetrize, both before and after. But if you wanted to translate this into a quantization form, you could get a sort of count in terms of creation.

1:02:30 I mean, that is something that is very useful when it comes to quantum field theory, in the context of just doing, you know, elective quantum mechanics. And I thought it was a very convenient way of testing half-tree, half-tree, so on. So those are sort of three comments. Well, obviously, I have to respond to this. I want you to think about those three points. Yeah, the issue of shape, and along with this is the notion of molecular orbitals. Yeah, these are extremely problematic notions. If you look at the way in which, I mean, what do you mean by, what was originally meant by an orbital? It was just a wave function, right? That you ascribe to one electron, you ascribe a different wave function to each electron in, let's say, the atom, a multi-electron atom. See, that's just the first starting point. That's just the starting point. What you do is, those are the organs, and then what you do is you say, all right, of course, these aren't the wave functions of the electrons of the atom, so we want to develop a procedure that will get us the appropriate functions. And this is where the Hartree-Fott method comes into play. And basically what you do is you say, all right, let me take one electron. This is the one I want to calculate the state for. And in order to do this, what I'll do is smear all the other electrons. So I take the average spatial distribution, and thereby I get a field associated with each electron. And then what I can do is I can solve the equation of motion for the nucleus, the electron I chose, and then this field that I obtain by averaging all the other electrons into some sort of field. I solve that and I get a new wave function for this chosen electron. And then what I do is I say, all right, now I'll take another electron and I'll smear the fields of all the others using the same average quantity

1:05:00 wave function for this one. And you iterate this process, I don't know, I think I was talking with Hans Primas, he said, oh yeah, these computer programs, they're basically doing millions of iterations of this process. And what you're trying to do is basically to come to a fixed point where the smearing doesn't make a difference on the equation, the function that you get for this specific function or for this specific electron so it's sort of like a fixed point problem except much a bit more complicated because of all the When you ask, what is the best minimum you can get in the space of all polaric wave functions, and this is really what Hartley meant. But it still assumes that the particles move independently in the average field. But of course they don't move independently, it involves the polarization. And this is another way of putting Hartley's- And this is where the Fock part comes in. Yeah, where you're anti-symmetrized. That's right, you see anti-symmetrized to take into account the exclusion principle. I mean, by this point, to talk about orbitals just gets crazy. I mean, I don't even know how you make sense of orbitals in that context. And this is what some quantum chemists who are more critically minded, like Woolley, I don't know, it's Woolley and Ogilvy. these guys are much more careful I forget the first initial are much more careful in the way that they talk about chemistry and what they their view what they say is that we're much more pessimistic about whether quantum mechanics can really take us much beyond simple atoms and molecules and even there it has difficulties One of the ones that Woolley likes to talk about is C6H6.

1:07:30 And it turns out that C6H6 has, I forget, either six or eight isomers. You know what I mean by an isomer? Okay, so the benzene ring was the first form discovered for C6H6. But we now know that there are other forms, so you can have that. So at each of these vertices, you have a carbon atom. Now, this is another form of C6H6, a third is that, and there are, I think, three others. I can't think of them offhand, but it doesn't matter. The point is that if you were to start with the Schrodinger equation for C6H6, there's nothing, There's no way that quantum mechanics can tell you what C6H6 looks like. And it can't distinguish between these three. There's no way for it to do this. So how do quantum chemists typically work? They work this way. They say, all right, I know experimentally that the molecule has this shape, and now what I'll do is I'll use, say, the Hartree-Fock procedure to determine what the angles are, what the bond lengths are. So what quantum chemistry does is elucidate structure. It doesn't, you can't derive structure from it. It only elucidates structure. So when people say, you know, yeah, everything, quantum chemistry can principle handle. I mean, even for a relatively simple molecule like this, take you very far. You have to put a lot in before you get anything out. So I would agree with you about shape. I mean, you have to put in, you have to treat shape classically, in effect, if you think of shape as being the position of the nucleuses of the atoms that are brought in to form the

1:10:00 molecule. Now, what these guys also say is that it doesn't, it also doesn't make sense to talk about so you said shape they would say it also doesn't make sense to talk about atoms in molecules there are no atoms in molecules right then of course they're their nucleus nuclei of course there are electrons but there are no atoms and I think part of the reason has to do with the with the poly exclusion principle and the identical particles problem so yeah shape is a huge problem and quantum chemistry does not give you shape you have to put shape in to get more refined shape out so that's, I would agree with you now there is one way that people do try to talk about shape in quantum chemistry that I think is worth mentioning and that's Dater's way of treating shape and his idea is when you look for are surfaces of constant charge density. And that's what describes the shape of the molecule. Is this the notion of shape that chemists typically use? I don't know. But I do know that this is also a classical way of thinking about shape, because you're not talking about superpositions, you're talking about charge densities. So, yeah, I think shape is a real problem. And for chemists, you're absolutely right, about chemical properties and it's something that quantum chemistry looks like can't give you it's it's I think they've been made yeah well Well, let's see, I can, let's see, yeah, that's, yeah, that's, yeah, that's, that's a double bond, let's see, where is the, shoot, I thought it was in this paper. yeah it's very nice because he also gives you the the artifact in mind gives you the years that

1:12:30 And this was discovered in the 1860s. And this, I think, was 63. So these are very recent developments, relatively speaking. I'd love to end this, but then we. Sure. I wanted to ask you about the characterization of emergentism as I think it applies to part of biology. In population genetics, it's standard to talk about theories of additivity with respect to fitness so for example the question if a heterozygote is not exactly halfway in between the two homozygotes in terms of fitness we've talked that's an interaction term it's a non-failure of additivity and similarly epistatic interactions between genes of different loci give rise to non-additive fitness effects um and i guess in your framework that would be an instance of emergence and well I don't know I mean it gets tricky see because what I'm focusing on here are causal features and of course these are causal features you're pointing to as well because you're saying the heterozygote because it has the two distinct genes you know the dominant recessive rather than you know the pure dominant this makes a regards to its survival, and it may be that, I mean, there's some cases where the heterozygote is actually better with regards to, say, malaria case. Right, that would be a failure of additivity, of course. Right, complete failure of additivity in that kind of... But I was thinking, as soon as you move away from its being precisely intermediate and thickness, you might be a failure of additivity, just moving way far away. It's kind of non-linearity. Right, right. Oh, yeah, yeah, but to tie it up to the kind of, see what, yeah. So what you're saying is your proposal, I mean, just to start with that, I thought that was what you're suggesting, that that's the way to think about when you have an immersion property. Is that wrong? That's not what you're saying? Well, what I'm saying is the key is the fact that you have a non-separable Hamiltonian, which means that the parts are interacting in such a way that it's impossible, well, it's impossible to think of them as parts anymore. It seems to me when you're talking about the genes, that these parts aren't necessarily interacting.

1:15:00 Well, in a certain sense, they're on the chromosome, right? What I was thinking, I mean, it was that the genes are interacting in the development of the organism regardless of whether the heterozygote is precisely intermediate or somewhere else. I mean, the notion of additivity as a quantitative feature here doesn't, it's not the same as the question of whether the two genes are working independently or are interacting. In terms of physical process, they're all interacting. Yeah, that's true. Another fact about them is to where the fitness value is. My sense is, what I would say is that, yeah, this probably fits into the model that I have in mind. It's going to be much more complicated because what's going to happen is you're going to have a shifting of levels. So it may be that these two genes interact in a way, they're separate, on their separate chromosome, or what is it, separate parts of the chromosome, right? Different chromosomes. And what happens is they're giving rise to different processes, right, within the cell, I would say, or more general entity, biological entity. And eventually those interactions, I think, are probably going to be at the molecular level. So I would say you're in the domain of quantum mechanics, and that non-separable Hamiltonians will be coming into play here. So I think that there is a kind of emergence at play here, but I would say to trace it all out would be a very complicated... So just to understand your idea here, I mean, the question of whether it's an emergent process of property or not, I mean, your proposal is in a sense a reductionistic way of analyzing that question, because you're saying, look at the processes way down there at the micro level, see how they work, and then that'll give you the answer. So I simply can't look at what biology tells me about genes development and even answer the question. Right, right. I can't even do it for, you know, simple molecules. You see, that's the difference between what you're saying, I think, and the British emergence. It would just sort of take additivity as one can analyze, say, at the level of biology and say, well, this is an emergent property.

1:17:30 but you figure that the story down at the microbe is extremely complicated and it's less interactive but you don't have to go there to get an answer but for them are you you're not trying to make this an exclusive kind of oh absolutely not no no this is this is this is one way of making sense of emergence that i think will have applicability to the biological context there's no reason for it not to not to yeah i'm i'm i'm confused about why why you're giving that So I think I can easily come up with examples of things that we would want to call emergence that have nothing to do with the quantum level of technology or economics. So, I mean, supposing we go, you know, supposing there are some aliens or, you know, we discover our planet two billion years ago, and they ask the question, you know, two billion years in the future, can you name the thousand most prominent species? and what criteria govern their fitness. And I think one can argue that they would be unable to do that no matter how much they measure about the distribution of different species of bacteria, which is all that exists at that time. And therefore, the properties by which, properties of things, even simple things like being an animal or being a plant or any property associated with fitness and multicellular creatures must be emergent from the point of view of what a biologist 2 million years ago could study. Another example would be, supposing we ask somebody in 1900 to name the 100 most profitable businesses and most businesses that do the most volume in the year 2000. I think it's hard to speak on it, but they might have been able to guess airlines, But they certainly would not have guessed on mobile phone sellers. And I think that you would probably make an argument that systematically that's a question that they would be unable to answer and therefore a mobile phone is an emergent, being a mobile phone is an emergent property of both the plastic and the metal and the economics. It's an emergent property of that kind of business. Yeah, now, see, this is why I'm inclined to stay away from the epistemic notion, because when you start to get into that context, things get a lot more fuzzy. The reason I'm inclined to stay at the ontic level is that I can capture in a very precise way

1:20:00 how it is that the causal property of the parts can only be defined with respect to the causal property as a whole. Now, I pointed out these other features that are often connected with emergentism, right? We couldn't possibly predict this, right? There's no way we could use... Well, the same thing goes here. We can't use quantum mechanics to predict these shapes. There's no way to do it. You have to have a sense for what the shapes are, and then you can use quantum mechanics to refine, tell you what the inner nuclear distances are and the angles between... which are important things for quantum chemists. say, yeah, the prediction element is something that a lot of emergentists point to, but they also talk about a calculational archangel, right? I forget who talks in this way. I think it's broad, but I'm not completely sure. But one of the British emergentists turned the epistemic issue, to my mind, into an ontic issue by pointing to a calculator, something like a laplacian demon which is often used to turn the the epistemic issue of cannot predict that a lot of philosophers fall into when they talk about the determinism issue popper and particular when when you think of it as as an ontic notion then you're pointing to something like a laplacian demon and that brings it back to the level of ontology as opposed to epistemology so limitations on predictability i'd say yeah that's that's a mark of the emergent but what mark of the emergent, I want to know what's going on causally that gives rise to the difficulties, the predictive difficulties, and I think that's what this sort of inextricable interaction does. But there is something in these, I think in these examples that I gave, which is causal and which is different from the causal sorts that you identify modern mechanics, I think it's the combinatorial explosion of possibilities that either in biology or in economics, it for all types of services and possible. Feedback loops, basically, is what's involved in these combinatorial . Not just feedback loops, but just the central rapid growth of the possibilities. And I think that also, here you see things that people seem to be pregnant. It's clearly a backward causation. It's clear that, for example, this backward causation on silicon would not be

1:22:30 nearly so much silicon around the services, in homes and things like that, whether or not for certain things that happened in economics that were unpredictable because so many things could have been discovered or invented, and the number of things that were are a very small subset of those things that might have been. Right, right. Can you put back up the slide that was the last one? I just wanted to ask you about the role of considerations about identical parts and inclusion in your kind of emergentism here, I think. So, you might have earlier, but maybe this will do the trick. That's the obstacle. So, emergent holes have parts, but these parts can only be characterized in them. Yeah, we might want to go back, but your idea that when you had the step diagram, the heuristic diagram on physics going up to chemistry and stuff, and you said that the lower level, in a sense, disappears, illustrate, I mean, the way I thought you were supporting that was with the considerations of the identical particles. No, that was just non-separability of Hamiltonian and the background. And then when you go into the context of, say, multi-electron atoms and simple molecules, then this additional feature of indiscernible particles comes into play that serves to complicate the picture. But to my mind, the key is the non-separability of the Hamiltonian. Yeah, maybe it's too strong. Maybe your point is it's too strong to put it that way because it does seem that the summary of my ontological position implicit in that is the identical particle stuff, right? Is that maybe part of your point? No, I wasn't yet ready for this. I would just want you to get a handle on how the considerations about identical or indiscernible particles play a role in your defining your form of emergence. Yeah, the idea basically is that when the atoms interact to form a molecule, you no longer have the atoms anymore.

1:25:00 They're gone. You might be inclined to think, ah, no, two hydrogen atoms combined to form a hydrogen molecule. Are the hydrogen atoms still there? No, they're not there anymore. They're gone. It no longer makes sense to talk about, you know, the electron that this hydrogen atom brought with it, and the electron that this hydrogen atom brought with it, the proton that this, you know... So, but to the extent that you wanted, if you wanted to use quantum mechanics to talk about the two hydrogen atoms before they joined together and you still wanted to define a two-particle state, you'd have to anti-symmetrize. And in a sense, quantum mechanics is saying, therefore, yeah, you already can to say there's, well, of course there's two, but I mean, Teller likes to say that, no, there's two quantum, there's not two individuals. Yeah, see, I'm not sure, I, yeah, maybe I'm mixing context, maybe this is the problem, because when you, when you go to Teller, what you're talking about is field theory, you're going to the field theory context, and I'm saying, no, I'm not interested so much in field theory here, I'm just trying to capture what you get in quantum mechanics, and I'm trying to abstract the ontology out of that. So in that context, you're not anti-symmetrizing electrons, or I'm sorry, yeah, the electrons of hydrogen atoms that aren't interacting. It's only in the context that they're interacting, where the wave functions are overlapping substantially, that the anti-symmetrization comes into play. So I'm trying to keep it in the quantum context, and there I think that this is the only, when you have Okay, and the field theory context is different. Well, that's not right. That's not right? If you want to anti-symmetrize the knowledge, it just doesn't actually make any difference. You're just going to add the probabilities that they're not interacting. I mean, certainly, I mean, if it was an example of the facts, it's in the anti-symmetrize the knowledge. And that is not the right thing to do. Everything is, you know, it just happens that they're not interacting, then they become an additive to the probabilities, and so on. Well, I've talked before about the possibility of a combination of the way I could pass. I think a very interesting question, if I could get Anthony to answer it. Because, you see, it's a bohemian of the mechanics. On the first sight, people tend to think that there can't be any emergence

1:27:30 because they don't modulate these things about non-mediarity and non-attentivity. I mean, many people don't think that's going to be narrowing on the question of whether you can explain holes in terms of their past, in terms of complication, it's not really a fundamental issue. But now, the focus, and maybe you'll think that can't be of any sort of emergent aspect. But for both, as I have said, there are many aspects of emergence of holism. Would you like to comment on that, actually, as I'm ready to do next first? Well, in one thing you've been highly pointed out about Freud-Bohm theory is that you can't really think of a system of particles as a mechanical system in the following sense, that we have parts that interact according to a specified and fixed rule, like there may be some fixed potential energy function between particles, some fixed expression of forces between the partner, as in each other mechanism, whereas in this theory, there's no pre-specified function to tell them how they interact. The way this partner moves depends on what the way it functions or what the system is. They see that as being distinctly non-mechanical. In the sense, you can't think that the system is being made up in tracking points of pre-specified books. But what that has to do with the emergence, I'm not sure. One might possibly... You know, I should subscribe to a particular historical view of this theory, which goes back to looking at how this theory emerged from de Broglie's work in the 1920s. and it might be an interesting sort of emergent story to start there. But Brewer was looking initially at articles as similarities in the fields, due to the planning of re-space,

1:30:00 and he had some complicated nonlinear field equations that he couldn't solve, and the hope was that there would be one that would have these couple nonlinear field equations we'd have singularities in the fields and to get a consistent solution the singularities would have to move in a certain way which turns out to be the velocities given by the gradient phase of what we call the Schrodinger wave in other words, the Freud-Bone-Gadden equation now he couldn't solve these equations of course, these complicated nonlinear equations couldn't prove that this really happens, that these singularities end up moving according to this simple rule, guided by the gradient of the phase of some complex function. So what he sort of does, he tries to argue that, well, maybe this would happen. And in his paper in 1927, which is at the end of it, he says, well, actually, we could drop that sub-function and just postulate that we've got this complex wave of configurations With these particles moving according to the simple rules, we forget about the substructure. One might view that as conceptually a sort of emergence, but in the same sort of way that one might think of the concept of electromagnetic field emerging conceptually from some of the primitive notion of elastic waves in the ether or something, and there's a sort of scaffolding that you then drop at a certain point in the theory and then Maxwell had this sort of view about the concept of emerging physics. The theorem is what we love, sort of exceptional emergence. I'm not sure what it's been happening to you. I'm going to make a comment on that because you mentioned it in value. It was a long time. I mean, did you think there? The sort of way as well as how you had to explain that realistic kind of view is they don't think of themselves as a poor reductionist or anti-reductionist. And yet many people think it's going back to classical physics because it's quantum potentials, it's getting validation, atomic potentials, and so forth. They're really far from the way they think about it. I just wondered how you fit that programme in your...

1:32:30 Yeah, I'm inclined to think that that program will fit in very nicely with this. Because, I mean, if you think of the quantum potential, it's, again, it will be a, when you're talking about interacting systems, it's going to be a very holistic kind of influence of that potential on the particle, which is the hidden variable in this context, something that has a definite position at all times. So what I'd be inclined to say is that the causal feature, you can't understand the time evolution of the particle without appeal to the whole thing so that sort of holism I think is a good part of this picture if you start to talk about the causal properties of the system as sort of being independent independently describable no appeal to the other system the other systems involved being necessary, then in that context, I would be inclined to say, well, that sounds like it could spell trouble for this point of view. But I like the holistic aspect of the Bohmian theory, the fact that it's the potential as a whole that's guiding the particle, and I think that does mesh rather well with this picture. I think that actually the terminology there, I mean, it's often called the core to this I think this is not, doesn't at all do justice to the sort of way that Bergenheim developed particularly their understanding of the theory, but we can agree with that actually. I mean, the quantum potential is not something that transmits bits of energy across from one place to another. I mean, you think of causality as a causal process, you know, the so-called quantum potential is not like that. It may pass with information in some sense, it's not energy. The first thing is to me, what a big... It's common in the sense of being deterministic. I think it's a more robust philosophical sense of course. Yeah, that's a bit more robust. We're going to say, we certainly want to deny.

1:35:00 I mean, if you talk about... I mean, we'll talk to Heidi. You know, he would say, It's not that sort of thing at all, and then they get that in the state of the law and all that kind of thing. It's a very different picture, I think. The community is leading to talk about it. But James, you've got to talk about it. I'm going to finish, because this is kind of the same answer. Yeah, no, no, fine. It's just quite a clarification, really. So, you said that you think that emergence comes from non-Septal more Hamiltonians rather than non-Septal estates. But do you, I mean, but you do think that you can locate non-separability at the level of states as well as at the level of Hamiltonian? Oh, yeah. Absolutely. You get it with state. And the reason I'm inclined not to go with the states is because you can get a... If it's just the state but not the Hamiltonian of the system that's non-separable, then you can get an expression by way of the partial trace for the components and the time evolution of those expressions do not involve appeal to the whole, right? They evolve separately for all intents and purposes. So I see that as, I mean, this sort of relates to Michael's point. I want something more robust that serves to connect these systems together, right, than merely these quantum correlations. And that's what I think you get when you start talking about the Hamiltonian by contrast with just the states. So your thought is that you can start off with states, two systems that are separable states, but if you have a non-separable Hamiltonian... Then you get non-separable states. You get non-separable states. Right. And isn't it also the case that if you start off with non-separable states and you have a separable Hamiltonian, you still get a non-separable state? That's right, too. So why the asymmetry of what you say about that? I mean, just because you can work out what happens to the non-separable state

1:37:30 by running a separable Hamiltonian on each bit, as it were, I mean, just because the time evolution works separately, I mean, that doesn't mean that the whole thing isn't evolving together, because what you end up with is still non-separable at the end, right? right. But as I said, that doesn't seem to have carried with it the right sort of causal connection. What you want is some sort of transformation of energy between the systems that serves to connect them together. And that's not what you're going to get if you have a non-separable state governed by a separable Hamiltonian. The parts don't really interact. They're just evolving separately. There's no real connection between them except the quantum correlations and what I want is something more robust than just merely the quantum correlations. I want there to be a flow of energy and the flow of energy is such that you know the time evolution of this component depends essentially on the time evolution of the whole and that that seems to me to be a more robust form of a tie and an inextricable tie between these two systems. I mean unless you think that with a non-separable state you don't really have Well, I want to say you have parts in a certain sense, right? See, Humphreys wants to say you don't have parts. I want to say, no, you do have parts, but these parts depend, make essential reference to the whole. But just because there is this essential reference to the whole doesn't mean that there's nothing there that can be called a part. It seems to me that the theory allows for you to deduce some, at least formally, some density operator that's describing something. And what I would say is that it's describing a dependent part of the whole. Something that's non-autonomous, but nevertheless exists. That's what I'm thinking about. It's quite dangerous to start taking powerful traces, these are improper mixtures, not really a big state, not really a big state. I'm going back to this a few years ago. I think saying, well, if you can just take a powerful trace and get this lentil matrix

1:40:00 I would say it depends on what context you're in. If you're in a Fox space context which allows for the creation and destruction of particles, I would say that's absolutely right. But in the context of quantum mechanics where I'm talking about creation and destruction, in terms of what the theory allows, but in terms of the underlying metaphysics, so to speak, then I think that this kind of talk is okay. But in a different context, you're right. There might be serious problems for this approach that would need to be thought through. And I looked at the literature some, there's very little talk of using, say, relativistic quantum mechanics or field theory in talking about the properties of molecules. It rarely comes out. It's very interesting. They're not so... I mean, you have a lot of computer programs. It's amazing how many approximation schemes are out there for doing this. But my sense is most quantum chemists are not interested in... Most chemists are not interested in these programs because it doesn't do for them what they really need to do, which is, as you said earlier, understand what the shape of the molecule is. Yeah. Yeah, it's just a speculation. It seems to me that you pointed out one possible way of getting emerging properties, basically from this quantum entanglement, and quantum entanglement in a more serious way. It's not just . But there is a big research program in which tries to bridge between quantum mechanics and classical mechanics in a harmonious way. And if they are right, this decoyance, unfortunately, if they are right, this kind of entanglement, either on the level of space or on the level of penitonium, will go away or disappear smoothly as the number of components in a very macroscopic object goes very large.

1:42:30 and what they want is the world prediction, because we have wonderful classical model in the many-body physics, and we have also wonderful quantum model in a few-body physics, and we want kind of continuance between these two. If we accept that what the decoyance theorists say is true, does that mean your kind of particular emergence won't appear on the level of macro-object? Because always quantum effect is washed away and everything will be classical and there we have a complete encoupling and this kind of thing. Because then it's very strange because what you want to say is something not just about quantum mechanics but something about special science which usually tends to deal with macro objects or more and more complicated macro objects. Yeah, I don't think so, because I'm inclined to think that what will happen at that level, and this is something I've tried to understand in the past in working on quantum chaos, how quantum mechanics can give rise to classically chaotic behavior. Now, classically chaotic behavior depends on the non-separability of the classical Hamiltonian. So what I'm inclined to think is that it's going to arise in that context too. But the problem is explaining how the classical non-separable Hamiltonian arises out of the quantum non-separable Hamiltonian, and that I don't have a clue. I don't think anybody has a clue how to do that. It seems to me maybe one of the best ways to approach the problem would be to find a common framework in which to carry out the research project, and there are two natural frameworks that suggest themselves. One is the phase-based formulation of quantum mechanics, and I'm planning to look into that. There's a really nice book by Frank Schreck. Do you know this book? It's on the phase-based formulation of quantum mechanics. That holds more promise than looking at, say, Hilbert space formulations of classical physics,

1:45:00 because it seems to me that Hilbert space formulations of classical physics only connects up with classical statistical mechanics and not point particle mechanics. So I think what you would need to do is work within that sort of framework. framework, and what would be nice is if you could see how the tensor product operation dovetails into the non-linear terms. I don't have a clue as to how that would work. Decoherence is another way, but there you're working primarily within the quantum framework, I think. My sense is that at least from the standpoint of quantum computation, they want to push away the decoherence They want to keep it as beta as long as possible so the quantum computation can finish. But yeah, I think decoherence is a very important process. But I think it's just a small part of a much larger, more complicated picture. That's the F-sharp. Well, I'm going to do more of the impact. Does anybody else want to make a title of time? Well, I think we should think through a game very much indeed. Thank you. Thank you.