Non-Linear Dynamics as Framework for Understanding Subquantum Structure — Part II

0:00 ...to values, kind of, the only way we can have this choice, but two argument functions, not very two. There are very few values. These are the values. Right, so this is a brief outline of what I want to do, colour-coded, as it were, into three bits. The first bit, in some sense, we've done, so please have that. Just to remind you, we had a discussion about Bell's theorem and the whole issue of non-locality, which course is what Einstein hated about quantum mechanics and well he had some ideas about non-linearity might play and Basil and David wrote a paper in 8081 expanding on that and I've tried to sort of develop that further with some ideas from let's say sort of modern day theory about chaotic dynamics or deterministic non-periodic dynamics as I called it the basic idea being to just remind you to think if the universe were somehow governed by a system of this type, if the equations of physics describe one of these systems on its invariant state And because these are basically zero measure, low dimensional objects in state space, any type of, you know, to use Michael's word, counterfactual perturbation, which might be one where you say, well, given that I've measured the position of this particle, what would have happened if I'd measured the momentum? any question of that sort is basically meaningless because with probability 1 that type of perturbation that type of thought experiment or the perturbation associated with that type of thought experiment would take you off this invariant set into a part of state space where by construction your physical system is undeclined so I think it presents a perspective on

2:30 problems of non-locality. However, what it doesn't do is tell you how you can kind of marry quantum theory as we understand it today, which is clearly a successful theory of physics. You know, it doesn't tell you how you could marry, in some sense, quantum theory with this type of system. So that's really what I want to do in this talk. I'm conscious that you know my level of mathematical sophistication is nothing compared with Freddie and Maurice's so I'm kind of groping in a sense for trying to express ideas that I have in my head into mathematically kind of coherent, into a mathematically coherent framework, but it's very much a, you know, at a sort of somewhat rudimentary level at the moment. But nevertheless, maybe I can convince you there might be something in it. Now, the black bullets are trying to just continue with the discussion about non-linear dynamics, and I want to talk about this thing called symbolic dynamic and introduce a very fundamental thing in dynamical system theory about a kind of duality between the conventional state-space representation of dynamical systems as things like differential equations and what's called a symbolic or symbol-space representation where you represent dynamics by strings of symbols, often just binary symbols. And indeed, many of the properties of dynamical systems are based on, are certainly most easily proven in using these symbolic representations of the monolimial dynamics. and I mentioned perhaps briefly yesterday things about synchronization of oscillators and again symbolic dynamics provides a very kind of simple way of thinking about that and then I want to just refer briefly to some linear structures which actually exist in symbol space notwithstanding the fact that the underlying system is non-linear. I then want to go on to try

5:00 to take kind of conventional quantum theory and see to what extent it can be developed in this symbol string representation. So to what extent we can think of quantum states as symbol strings. And again, I'm going to very much draw on this duality between the symbolic dynamic representations and the state-space representations here. So we'll go through a bit of that. And then the final bit to kind of marry these two bits together and and come back to this issue about entanglement and this will lead me to um think about entanglement as a dynamical process and in particular as a as a as a as a reduction as a dynamic reduction in the dimension of this uh underlying attractor which is somehow governing the dynamics and then to think about that as gravity and Then we'll finally finish with some of the links to Penrose's ideas. So I would say I am fairly sympathetic to a lot of Penrose's speculations, but I've kind of turned the logic of some of his arguments on their head to some extent. So I have a different take on his ideas, And it's to do with actually thinking about entanglement, large-scale entanglement, as a manifestation of gravity as a large-scale manifestation of entanglement. Anyway, so that's the outline. So, symbolic dynamics. I mean, it's probably easiest to think of a specific example. And a very familiar and very rudimentary example of a system which has chaotic behavior is just the logistic map. So you just iterate on the real line some real number x with this function f, which is just a quadratic. the whole books and libraries have been written about this equation and there are various and some of the sort of Tom catastrophes can be viewed

7:30 in terms of bifurcations as mu increases from 0 and above I just want to consider just for illustrative purposes this map where mu is greater than 4 When that happens, if you think about the real line, and imagine this is the interval between 0 and 1, basically anything outside that interval 0 to 1 gets mapped to minus infinity under repeated applications of f. So minus infinity is a sort of a attracting point in some sense. But what I want to consider is within this unit interval between 0 to 1, what are the, what is the set for f. In other words, what points, what points in the real, in the unit interval, keep within the unit interval under repeated methods of f. And so for example if you took f equals half, then you'll get a half times a half is a quarter so multiplying by mu which is greater than four will give you something bigger than one and then that will map to something negative and then eventually go off to minus infinity so this bit in fact anything in here will get mapped off to minus infinity so if there is an invariant subset it lies somewhere on these two and then just by you know just following the analysis it turns out that what you actually end up with is the invariant set is the Cantor set where you just take away the middle third and then remaining the two thirds, you take away the middle third of that, take away the middle third of that. You can represent a point on this Cantor set by just writing, considering its expansion in base three numbers. And if that base 3 expansion only contains 2s and 0s, then you know that you're on the counter set. If you take a general point on that unit, in the unit interval, then in general it will have 1s as well as 0s and 2s.

10:00 And those won't be on the set. So, in fact, since almost all points on the real, on that unit interval will have kind of mixtures of zeros and twos and ones, you can say that this Cantor set has measured zero. On the other hand, if you just think that any ternary decimal you can map in a sort of one-to-one way by just replacing a two with a one, and by just noting then any value of x can be written in binary with 1's and 0's you can immediately see that the Cantor set is also an uncountable set so I think Cantor originally sort of came up with this as just showing you illustratively an example of a set which is uncountable which has measured 0 on the real line but at the time it was just a curiosity but the more interesting thing perhaps now it can be shown to be the invariant set of this quadratic map okay now so we consider so the idea of a symbolic or a symbol sequence is to think of a of some sort of partition of this invariant set and a natural partition would be to consider all the points sort of in this half of the invariant set, use this symbol, and this is an orange. By the way, I originally had apples and oranges because apples and oranges are really very different. You should never mix apples and oranges. But I was then very surprised to find at breakfast, George can tell me this, I got totally confused because when I tried to find orange was something, what was it called, apple juice or something, and then next to it there was another one which was apple something, sin juice. Is that right? So I decided, oh well. Apple sin is orange. Well that's right, I thought there's different types of apple juice, so there's no, where's the orange juice? So I decided that, obviously in Sweden, apples and oranges aren't so different, so I decided not to use an apple here just to get to produce it.

12:30 It's got to be arrived. Applesine comes on Chinese apple. It means it's linguistic. Is it Chinese apple? Plain Chinese. But in Spanish it means apple without juice. Anyway, so the point being, there's obviously no point using apples here because they're all the same thing. So I'm just using upside-down oranges there. So the point is these are just symbols, okay? And they represent, they denote different, this partition, if you like, of the Cantor set. So we can, so what we do is, consider another way of representing a point in the invariant set. So X is a point in the Cantor set and phi is a kind of mapping to a symbol sequence so each of these A0s, A1s are either up oranges or down oranges and the nth symbol is an up orange if the nth iterate of this quadratic logistic map applied So you map, you're down in the cancel set, and this f, you're applying it n times, so it's mapping you around. After you've got it n times, you can ask, are you in the up-orange part of the set, or in the down-orange? So if you're in the up-orange, then that symbol is up-orange. And after n iterates, you're in the down-orange, the symbol is down-orange. So, yeah, actually this seems to be a bit missing here. If X was, say, my .220 internally, so that's a point, dot, dot, dot, these are 2s and 0s, then under this mapping, the symbol sequence, at least for the first three elements of the symbol sequence, would be down orange, down orange, up orange. Well, OK. Oh, OK. Oh, OK. um well I let let me just use zero and one now instead of up orange and down orange so so you know we have this duality the point is we have this duality we can either think of the state X as a point in the particular point in

15:00 the invariant set or we can describe X by a essentially you could think of it is a time sequence of symbols. So one, one, zero, this is now one. This is the one symbol, instead of the down orange, and zero is instead of the up orange. So these are two sort of equivalent ways of describing the system, both the state as a point, as a particular state, and the sort of time sequence under repeated iterations of the dynamical logistic map of where, if you like, to this kind of course binary partition of the invariant set so and in fact yeah in fact in in in so in simple space the the quadratic logistic map F actually corresponds to a left shift. So suppose I had a state and its symbol sequence was that and then I operate on that state little x with f, my little f. The equivalent mapping in symbol space is just to shift the sequence, or if you like, if I had a pointer there, the pointer would go to there. So you can always think of the whole system being shifted and then dropping that one, so you're just left with that one, one zero ones. Or if you like, you had a pointer which just moved you along. So this is this idea that the nonlinear dynamics in symbol is actually just a rather simple kind of shift operation with respect to that partition. So this is an important sort of thing I want to get over, this kind of duality. There's this state-space description and the symbol-space description. The symbol-space you can just think of, if you like, as a time sequence of the position of that state with respect to the binary partition under the action of the non-linear dynamics.

17:30 and then you have the original non-linear dynamics in state space and then this corresponding operator the shift operator in symbol space and they're doing the same thing and in fact, more generally and this is what people, when we get into the mathematics of all this talk about, is kind of homomorphisms between the original dynamical space and the symbol space and this could be for a one-dimensional system completely, generally, when you have, say, a multi-dimensional Cantor set, for example, some Cartesian product of Cantor sets, and then you have a kind of n-dimensional set of symbol sequences. And one way of thinking about this could be you know when you're describing n chaotic oscillators so I mean there's a lot of work for example goes on in in looking at coupling of these logistic maps so instead of just having one logistic map you might have several of them and they're coupled together through through these sort of coupling terms. And you can represent that through N, if you like, of these symbol sequences. Now, if the oscillators are completely unsynchronized, they're just kind of going on independently of each other, then obviously these symbol sequences will be uncorrelated. And one can think then of the effective dimension of the total attractor. If you think of this as a system with a sort of total attractor, then in the case where everything is unsynchronized, really you're just dealing with n independent attractors, and the dimension will just be the sum of the dimensions of the individual attractors. On the other hand, when those coupling terms are such, for example, as to bring the system into complete synchronization, and that is a very common feature of chaotic dynamics under suitable circumstances, you can have complete synchronization. Even though these time series of 1s and 0s are completely non-periodic, the individual oscillators can synchronize. And then effectively you're just dealing with n copies of a single system. So the dimension of the system is then just the dimension of one of those oscillators.

20:00 so in some sense you can think of coming into synchronization as being associated with a dynamically induced reduction in the dimension of the underlying total system attractor now one point to notice is that there is this interesting despite the fact that f the little f itself can be a very non-linear dynamic there is actually a simple linear structure exists in symbol space. So, for example, if you had two of these symbol sequences, and suppose we thought of this left shift operator as operating on them, so just shift everything to the left and remove the one on the left and so on. If you defined plus as just mod addition so so if you sorry if you if you so let's just say again if you took these symbols and treated them as zeros and ones and then just thought of these zeros and ones as the as this the binary expansion of some real number and then added these together let's say not one so point one zero one plus point zero one 1 would be plus 0.10 and then and then just put that back into symbol space so 010 plus 011 equals 101 then basically that's the action of F that the shift sequence is kind of linear on that on that thing so F applied to SA defining this addition of symbols by going into binary reels add mod 1 and then back SA plus SB, it's just F acting on SA plus F acting on SB. And a simple way of saying that, it's just this F, if it's a left shift, it's like doing a multiplication by 2, and then doing mod 1. So simple, multiplication is distributive over addition, you have this kind of linear structure. So even though F itself, little f, is a highly non-linear operator in state space, this capital F does actually contain linear structure, which I think is rather interesting. anyway, right

22:30 what's this got to do with quantum theory so I just want to go to go to a rather different direction and just think about symbol sequences and how possibly you could ever get to the notion of a complex Hilbert space of a n-cubit vector, n-cubit state vector Two points I want to make about Hilbert spaces is that, you know, as people often say, Hilbert space is vast. So an n-qubit state vector has a 2 to the n-dimensional Hilbert space. And the second thing is complex Hilbert spaces are complex. They have imaginary numbers. So how to sort of encode, at least, or incorporate these two kind of key ideas into this notion that we could possibly write a quantum state vector by a symbol sequence. So, again, I want to return to this idea of thinking about a symbol sequence. And again, if you like, we can have up oranges and down oranges. and I'm just going to define symbol, which looks like a minus sign as something which just transforms an up-orange into a down-orange sorry, it's actually been a down-orange into an up-orange anyway, so it just transforms one into the other and so the, let me call it negation of the sequence will just be this operator applied to all elements So in particular, the negation applied to the negation So if you go for an upper orange to a down orange Then minus brings it back to an upper orange Minus minus equals the original Well again, just not to bother with oranges all the way through Let me use zeros and ones But just remember that I'm writing minus zero equals one But this minus sign is just inverting the symbol Now, the simplest thing is just to think oh yeah, I should say, I'm considering a symbol sequence which has 2 to the n binary symbols So n is going to be some very large number, but to start with, let's just consider n equals 1, two symbols and I'll define this i operator as

25:00 as this. So you take the second element and move it to the front and apply the minus operator. Then you can very quickly see if you apply i again to this, you take that second one, move it to the front with a minus you get minus a1 minus a2 and that was minus of s. So I've got an operator which is a bit like a square root, well it is a square root of minus 1. Apply it to s. if you want to, you can write this as a kind of a, you know, as a kind of like a little matrix operation, thinking this is a row vector, where this matrix will be a 2 by 2 matrix, a bit like the... Yeah, you asked, in fact, your file matrix, yeah, 1 minus 1, yeah. Okay, if we continued a little bit further with n equals 2, so now we have four elements, 2 to the power 2, It's now very straightforward to define square roots of minus 1, which have some interrelated structure between them. So, again, this is just a permutation sort of operation. Take the second element, move it to the front, put a minus sign in. Take the fourth element, move it to where the third one was, but don't have the minus sign. So, if you do that twice, you'll see that E1 squared gives you minus 1, and similarly these, this, and this. But if you actually compound E1 and E2 together, and E2 and E3 and so on, you'll find that these actually satisfy quaternionic multiplication. So, by these sort of sets of permutations, you've got something entirely equivalent to quaternionic multiplication. Now, again, that's not a total mystery. if you use this matrix notation and these E's now are well they're basically 4x4 matrices so I'm using the fact that I is a 2x2 matrix and this 1 is a 2x2 identity matrix and now these matrices which if you like affect the permutations can be written in this way and these in fact are nothing other than the essentially the Pauli spin matrices Remember, I is a two-by-two permutation matrix. And again, we have the vial matrix now, again,

27:30 but with this as a two-by-two unit matrix. Well, we can continue this kind of trick, if you like, on by invoking a kind of self-similarity principle. So if we go to n equals 3 now, so 2 to the 3 is 8, define, so we had with n equals 1 we had one single square root, with n equals 2 we had the three quaternium square roots of minus 1. With n equals 3 we get seven square roots of minus 1. And what I've done is basically to take, let's see, to take that structure but now to substitute in, instead of the i, to substitute in e1. So it's a kind of and E2 and E3. So on this top row, you've got that I minus I matrix, but now instead of I, I've got the E1, the 4 by 4. So I've got three of these. There's now my new file symplectic matrix, and then that's the other term in the quaternionic thing, but instead of I, I've got E1, E2, and E3. So I've now got seven square roots of minus one. With the identity matrix I've got sort of eight all together which sounds a bit like octone octonians now I didn't think this had anything to do with octonians but I remember last year Freddie you mentioned this thing called a smash product which might transform these into octonians I'm not this is something I haven't I confess followed through so I'd like to talk with you a bit a bit about this again to see if we can make some linkage there because But anyway, there's nothing now stopping you just carrying on for arbitrary n. For arbitrary n, the essential ingredient is to use self-similarity. And again, this is the whole kind of fractal stuff coming in to some extent, where you can define 2 to the n minus 1 square roots of minus 1. And there are a number of these quaternionic triples embedded within them. And in particular, if you take these 2 to the n minus 1, square root of minus 1,

30:00 and consider the identity as well, and apply them, say, to just sets of zeros, you'll get sequences, 2 to the n sequences. And the correlation, they're pairwise orthogonal, so the correlation between any two members of this sequence will be 0. so I'm calling them an orthogonal well, orthogonal set they're orthonormal in a sense but that's not all of these will have equal numbers of zeros and ones so they in some sense have a if you define a norm in terms of the numbers of ones in the sequence then all of these have a same number of ones but they're all orthogonal to each other now this is one thing is the number of these square roots of minus one increases exponentially so the number of my square roots of minus one permutation operators increases exponentially with n and again just to remind you that's sort of how the Hilbert space goes in terms of the dimension of Hilbert space with the number of qubits So, what I want to do is, in some sense, think of these sequences as both, like in the symbol space, state-space duality, think of them as both representing the state of a system at an instant in time in state space and also in the symbol using that phi homomorphism think of them as also representing a sequence of time, a time sequence with respect to some partition, some binary partition of this underlying attractor which the non-linear dynamics is giving. Now, between total unsynchronization, between lack of correlation between pairs of sequence and complete synchronization, you can obviously get partial synchronization. And indeed, you know, in non-linear dynamics, it isn't just total or unsynchronized states.

32:30 You can get states of partial, you can get dynamical systems which are partially synchronized. In particular, one can represent, one can think of two sequences, let's call the first one S1, and the second one SI, such that the synchronization, the degree of synchronization lies somewhere between 0 and 1. And one could think of a sort of one-dimensional array of these oscillators where the degree of synchronization sort of, as it were, decays away in this cosine or sinusoidal state. So in other words, we can start to build in wave-type dynamics. So let's consider our sequence S, say, here. minus S is where things are perfectly anti-correlated so every 0 symbol goes to a 1 symbol so the correlation is minus 1 if I choose any one of my 2 to the n minus 1 square roots of minus 1 then this produces a symbol sequence which has no correlation it's orthogonal to S I can do that there and then minus one of those square roots of minus 1 gives me something which is uncorrelated with this guy sorry, is anticorrelated with this guy and uncorrelated with either of these two and more generally one can define sequences where as I say, the correlation is varying by cosine phi, where phi is some phase angle. This is technically how one would define S of phi in terms of symbol sequences, but I won't dwell upon that as well. The thing is, you can end up writing a kind of two-sphere where a point on the equator is some sequence of zeros and ones, equal numbers of zeros and ones, As you go along the equator, each point has a symbol sequence, but the correlation with S is decreasing until you get to here, where the correlation is zero. And this is the square root of minus 1 operator acting on S. And as you go round back to the antipodal point,

35:00 then you get to minus S, where the correlation has gone to minus 1. Or you can go up in this direction, where the sequences, again, start to decorrelate from S. But where you go towards just a sequence with nothing but zeros, or in this direction, nothing but ones. And again, this will be uncorrelated, this sequence will be uncorrelated with this one, which has an equal number of zeros and ones. and in particular if you get to stop at this point here at at co-latitude theta then by construction the number of zeros will be equal to cosine squared theta upon two. So when theta is zero it's all zeros and when theta is pi by two it's a mixture of zeros and ones. So this is a bit like the block sphere for a qubit quantum theory. So this is the block sphere for a sort of standard Hilbert space description. So my picture then is to try to think of the wave function in some sense not in the kind of usual way of some sort of superposition which is hard to know what it means, but actually as this symbol sequence. And thinking again about the symbol sequence as a kind of duality between a set of possible outcomes, a time sequence of outcomes, or a state, I am thinking now of going away from this sort of somewhat indeterminate sense of the state vector in quantum theory to something rather definite. a definite state with a definite value given by the symbol sequence but which if one thought of it as a in terms of a time sequence of perhaps measurements it would give you a completely random set of zeros and ones and this guy here would be another definite symbol sequence where this was well this i in a sense now becomes one of my 2 to the n minus 1 square root of minus 1 and is uncorrelated it has an uncorrelated symbol sequence with the symbol sequence here

37:30 and there is a kind of a parallel with the Born rule in quantum theory which is that if I were to define a norm on my symbol sequence as in this form then when a is 0 AI is 1, this is 1. So this is basically giving me the, if you like, the probability that if I chose a random, if I randomly chose an element in my symbol sequence, would it be a 0 or a 1? So by defining this norm on the symbol sequence, I can interpret this norm as giving me the probability that any one of the elements AI is equal to zero. In other words, it gives me the frequency of occurrence of, say, the zero symbol over the one symbol. So in other words, the norm of that would be cosine squared features So, I have, with these symbol sequences and with these representations of square root of minus 1 as permutation operators, I have something which, in some sense, seems to encode wave dynamics. And if you remember, I talked about this linearity of f, the shift map, having some, even though the underlying state space dynamics was non-linear, There was some linearity in the shift sequence allowing me to define a kind of addition. So I have something which seems to me starts to get something like wave-particle duality. The particles being the individual symbols in some sense and the waves being these square root of minus one representations plus this linearity of the shift map. me in a vague philosophical way about the pilot wave the bone sort of stuff that one really is talking about both at the same time not one or the other the duality of the symbol sequence versus state vector approach you know they're both they're both there at the same time that's just a philosophical remark. So this is really kind of the main point here. So when we come

40:00 to kind of think about, if you like, tensor products and so on in quantum theory, well I'm, I guess the corresponding thing in my own picture would be of where the state vector is a kind of set of these symbol sequences and if the system is entangled it merely means there is some correlation some correlation between the symbols so from a kinematic perspective entanglement in this picture would just be kind of denoting some correlation, partial correlation or complete correlation between symbols and incidentally these correlations are invariant under permutation transformation. So I could apply my square roots of minus 1 operators. If I applied them to all of the symbol sequences in a kind of global sense, it wouldn't change the relative correlations at all. So there's an invariance of the correlations under, say, my square roots of minus 1, or indeed, if I added another sequence using my definition of addition which I gave you before, we just leave invariant these correlations. Excuse me. Yeah. Can you just say the symbol sequence represents a state, and maybe we could go back one more time. You want to add a state psi as the collection of 2 to the n symbol sequence. No, n symbols, well, n symbol sequences, each of which, let's say it's 2 to the n, each one, so you could think of it going down here, each one is 2 to the n. But how could you possibly represent an arbitrary state, an arbitrary direction, an arbitrary point on the sphere, like this, you have n times 2 to the n possibilities to define psi, but there are infinite amount of psi, obviously. well in a sense the the only the points well a condition I suppose has to be that theta it's in a sense okay it's not actually true because I'm defining this

42:30 norm is defined in a sense the theta here is only defined for points on the sphere where this is a rational number in fact it's a binary rational it's a dyadic rational number I mean this has to be this is giving me the this is giving me the frequency of zeros in my symbol sequence So it's, using this definition, these thetas are only defined for values theta for which cosine theta is a diated rational number. Okay, I think this will pick out... So in fact, it's a bit of a discrete sphere, in this sense. Yeah, so you have to slide, so to say, on the latitude part, still along each of these serves that you get now, you have to do many points. other kind of discreetness also that's the only discreetness i mean i'm i'm i'm putting a discrete finite number of points on this on the sphere on this sphere at least such that um uh the so that the correlation between the symbol sequence at the point here and the point here has to be a dyadic rational. So it's not defined for angles where cosine is not rational. So, is it true that you have here an action of a Clifford algebra where your roots of minus one are the specific ones that appear in the Clifford algebra except the trivial one and that it's acting on the sequences up to the triviality of having it from a certain point on a translation because you have this S prime there. So is it true that you can see everything as action of a clipper term? I'm not sure about that, to be honest. Because you have two N of these. But let's see if it's... And so the organs of a clipper term given by the minus root. There seems to be linkages with clipper term. It's much more like we're thinking about some non-associative thing, but in fact it is the clipper term which is already acting, I think, relation where the translation of it is as prime you assume that this is the same thing or you can put them in the same class yes and then you have two to

45:00 the end elements there in the end that they represent something with the orbits of the Clifford algebra. Well I guess the question is can you build up this Clifford algebra with these kind of self-similar constructions because that's what I'm not sure about I mean when I've looked at Clifford algebra they might see it in that way but maybe it can be. Yeah but I just look at the roots of the minus roots there you have the minus one roots they come from the those if you want, or tensor product of quaternins, or just matrix rings if you want. Yeah, these are just matrices. And then you have the axons on your sequences, and you have an equivalence relation that you say, okay, these are the same, if they're different by such an S prime. So then you identify them in a sense. Is that right? Well, it's all going slight. You say that there's just an S prime there, and so... Well, I'm just saying if you add an S prime to all of them, It doesn't affect the correlation structure between any pair of pairs. And there you have this action, I think, from the flipper till the brook. And there you see that it doesn't harm the... Yeah. Well, this doesn't harm it either. This doesn't harm it either. Neither of these harm it. Yeah. So I believe that. Yeah, it looks like that. Yeah. Okay. Well, okay, so the last bit really is to, you know, try to bring the two bits together in some sense I'm thinking of imagine a process where we have these oscillators and they come into synchronisation through some, maybe some intermittent coupling so E denotes E denotes some perhaps intermittent, so some sort of unit interval of time where some coupling is added. And this may be occurs several times, and during the period, what happens here is you're going from a totally uncorrelated, unsynchronized state, gradually bring, as the coupling is kind of continually added into complete synchronization now from this dynamical perspective the what you would see is or what you would you would describe this process of bringing the states into synchronization in terms of

47:30 some underlying reduction in you like your invariant set from sort of just multiple copies of the invariant set down basically onto just one single copy. So there's a reduction in the dimension of the attractor, if you like, associated with this synchronization. So the picture I'm trying to paint here is of a possible linkage with quantum theory where this F operator, the shift operator, is acting a bit like the Schrodinger equation, so giving you what would be equivalent to a time sequence. Well, so you have to remember all the time, this could represent a single state in state space or a time sequence with respect to some binary partition of the attractor. And then every now and again, something extra, some extra dynamics comes in, into synchronization and eventually it's in this case it's completely synchronized and I'm saying that if one thinks of this synchronized sorry the correlation as a as an entanglement then one then in tank that in this sort of analogy well maybe it's not more than an allergy this entanglement really is a dynamical process. So I'm trying to bring in, perhaps go towards the picture where people see state reduction as something additional to Schrodinger evolution. So my picture, it's not so much state reduction, if you like, but entanglement that is bringing in some additional non-linear dynamics, which is leading to synchronization. So the entanglement itself process which the underlying non-linear dynamics is bringing systems into synchronization. The thing here is that and then F itself, as I say, although F state space to some nonlinear operator, F itself has in Stimble space some linear structure to it. So when I see, you know, so this is where the maths becomes a little bit difficult,

50:00 I haven't quite sorted my head out here, but if one wanted to try to describe an equation which both included, in my way of thinking, Schrodinger dynamics and something where systems were becoming entangled I might require an equation which was sort of both a symbol space equation and a state space equation together and I kind of see that in some sense when I look at the Dirac equation, I see parts of this you know, these gamma matrices and things as, in my way of thinking of things, as representing these symbol sequence dynamics but the interaction of the electron as being something where, which is giving you kind of entanglement and interaction, which is actually occurring, if you like, more in the more conventional sort of state-space way. So maybe this is why these equations look so mysterious, because they're actually a hybrid for symbol-space, state-space dynamics. You're not going to one or the other, but you need to somehow have both together. That's just a speculation. Well, a lot of this is speculation, because the next bit is macro-speculation. So, yeah, so I do want to, I mean, I want to kind of agree with Bell that we, you know, he didn't like this word observing, observation and measurements and things. So let's just think of what we call measurement as a macro-scale, very substantial increase in the degree of entanglement of our quantum, the quantum system we're trying to study. So it's entangling with the measuring apparatus, entangling with some macro scale system. So in my way of thinking of it, then measurement is associated with a kind of macro noticeable synchronization. So even when two little quantum particles would entangle, there would be some it would represent some sort of synchronisation and therefore some small reduction in attractor dimension. But when it becomes really big with the measurement system then one is talking about a really big reduction in attractor dimension.

52:30 Okay. Now, the second point here is to bring in the work of Roger. Now, he's argued by order of magnitude things, that during measurement, the sort of timescales for state vector collapse seem to be kind of consistent with gravitational effects becoming important. Now people sort of, you know, I know this is very controversial and so on and so forth, but just at face value, the back-of-the-envelope calculations, it seems to be consistent that gravity is not totally negligible during measurement processes. Now, Roger has argued that, you know, we have this unitary sort of process, the Schrödinger process going on, and then when gravitational effects become important, we should add on an extra term to the Schrödinger equation. in fact his view is that kind of kills entanglements that sort of destroys entanglements but I want to kind of turn this argument on its head because I want to sort of propose that maybe there is this underlying non-linear theory and maybe these symbol sequences provide a way to represent the non-linear dynamics incorporate the sort of notion of Schrodinger dynamics, but basically there is this underlying non-linear theory, where entanglement actually is a kind of a new, as associated with it, a new dynamical mechanism, which is synchronization, non-linear synchronization. And so when this becomes, when this non-linear synchronization becomes really big, because we're synchronizing we start to notice it. And if we use Roger's order of magnitude calculations, then what we could say is we notice, what we're noticing is a phenomenon that we call gravity. Now, the reason I'm saying it's gravity is because this reduction in attractive dimension is a kind of a universal thing. If you have, it's a couple of nonlinear systems,

55:00 this degree of synchronization is a kind of an irreversible process systems come into synchronization once they're in synchronization they stay there an irreversible process similarly with gravity and in some sense you could think of this as clumping in state space just as gravity as a kind of universal clumping mechanism so this kind of notion of a tract of reduction a kind of universal clumping. So this is the kind of speculative remark, really, that with respect to a possible non-linear theory in which quantum mechanics could be embeddable, in which states are really something very definite. In terms of state space, the state has a very definite position, and there's a very definite, it corresponds to a definite symbol sequence of outcomes, there's nothing indeterminate about this then macro entanglement actually is something we notice and as I say we call it gravity and we don't experience it as a non-local force because clearly on the macro scale any local entanglement will completely overwhelm non-local ones it seems to me this is kind of in keeping with the spirit of general relativity, because in general relativity, we don't think of gravity as a force, and it doesn't contribute to the energy momentum tensor, but it's an intrinsic property, i.e. the curvature, of what I call the state, if you like, of space-time, which is the metric. And it seems to me by comparison, we shouldn't really think of trying to, you know, describe gravity by some clever contribution to the Hamiltonian, which conventional quantum gravity theories do. But again, it's some intrinsic property of the quantum state, which is now not curvature, but this entanglement. So providing we can recast quantum theory in a slightly different way, it seems to me gravity could have the same sort of relationship to the state vector as it does in general relativity. So, I don't know, I'm going a bit over the top here, but anyway, so in that sense, this notion of quantising gravity

57:30 just seemed to be a misguided idea. It's me talking rubbish. Again, just to say, I mean, I think this is quite in keeping both with Penrose's notions that gravity it. And as I mentioned, if you go to state space, these attractors are, you know, this question of whether you lie on an attractor or not, seems to be uncomputable in exactly the same way as knowing whether you're on the Mandelbrot set is uncomputable. Although I have to say, I haven't seen any rigorous proofs of this for the sort of dynamical systems I'm thinking about, so this is not perhaps a rigorous fact, I can say. The only point kind of agree with Penrose is to think that somehow it's just this is an effect that's only an add-on to the Schrodinger equation when when you do measurement this is something which I I can see I see this sort of nonlinear framework as fundamental all the time and not just something that's an add-on when you're doing measurement and at that level I would say you know it's the same as Newtonian physics I mean I don't think we would want to say Newtonian physics is okay, except when relativistic effects are important. You know, you don't get any of the conceptual revolution of GR by just sort of adding post-Newtonian corrections to Newtonian physics. I mean, Newtonian physics is fine for calculating things, but even when relativistic effects are unimportant, we don't consider it to be correct in any sense. And I similarly think that there's quantitative theory as well. So, yeah, so it's my absolutely last slide on this. I'm trying to present, I mean, these are very jumbled ideas in my head, as you probably realise, but sort of, and I'm struggling a bit, you know, to get the mathematics right. It's driven by trying to express ideas in my head rather than a pre-existing mathematical formula, I have to say. And the motivation is this coming back to this Bell theorem to try to understand non-locality. So in a way I could explain to my grandmother, and I see that's the kind of challenge to explain. And really the idea is that in these types of non-linear theory,

1:00:00 the notion of counterfactuals which are kind of key to proving the Bell theorem the fact that I could have measured this even though in practice I'd measured the Bell inequalities lie on the fact that these experimental parameters are free variables but positing the existence of some non-linear theory for the entire system, that is to say the entire universe, then these dynamically unconstrained perturbations really become unallowable, they take you out of the set of dynamically allowable states that's the kind of key idea I think turning everything if I were to put all my speculations into one, as it were, basket and make some sort of completely perhaps wild claim, but maybe something that Einstein would have been happy with, is the idea that maybe actually gravity is the thing that allows us to make sense of Bell's theorem in terms of ideas we can comprehend. In other words in ideas that don't require non-local causality nor disimplausible conspiracy. So that's where we came in. I'm not sure we're actually close here. Do you have a picture in your head of a picture to go with this phrase, the non-linear dynamics of the universe, then? Well, it's a picture like that Lorentz attractor, okay? But it's obviously something much bigger than that. But as I say, we're the ants. You know, we're the ants. It's like the ants on the sphere. You know, we're the ants on the attractor. And we can't see, because we're not at liberty to see off the attractor. So it's a dynamic we are not familiar with. It's definitely the dynamics that we're not familiar with on a day-to-day basis. And any of these counterfactual perturbations are things that would pull you off. The things that say, you know, what would have happened if I'd measured momentum, whereas actually I'd measured position, would have required, if you'd expressed that mathematically, it would have been asking the ant to walk off that set that it wasn't allowed to walk off, and there is no existence, in some sense, of that attractor. I mean, that's the picture I have, if you like.

1:02:30 and what I'm trying to stress is the conventional way people think about non-linear dynamics is just in terms of the differential equations and of course that doesn't carry with it this additional structure of the invariant set of the differential equations that's why Penrose and Deutsch and others kind of dismiss chaos theory as being irrelevant to this whole question but that's because they only think of it in terms of the differential equations If you only think of it in terms of differential equations, you can clearly walk off the attractor without any problem, and you can reintegrate with your differential equations and point off the attractor, and then you'll just wander back onto the attractor. But if you specify that your states have to be on the attractor, so you're looking at the system plus the boundary conditions, then you're not allowed to do that. And the simple dynamics is a simple way of trying to encode that, because the symbols are representing symbols or they're representing a partition of the attractor and only of the attractor. Do you envisage that it's something that could be verified observationally or is it purely in the domain of a theory of, you know, an interpretation of quantum mechanics? Is there some additional... The only thing I've thought of, and this is, this is sort of, this kind of baby goes to some algebraic stuff, is that when people talk about entanglement, I've seen papers that use these, what do they call them, vibrations of the sphere to talk about entanglement, and they're equivalent to, you you know, the existence, the algebraic equivalent is the existence of the quaternions and the octonions. And then it seems, you know, when they go to, say, four qubits, this decomposition associated with vibrations of the sphere doesn't exist because there aren't any of these norm, what do they call them, division algebras? Well, division algebras, but they are not. Yeah. You know, the sort of structure I've got here doesn't have any such restrictions on it. I mean, these similar things just go on forever. So it had sort of struck me there may be something which could, in principle, at least lead to some potential experiment where the way I would describe four or more entangled qubits

1:05:00 could actually lead to something distinctly different from standard quantum theory. maybe it's fortuitous that four cubits is spin two which is gravity whether that's a coincidence or not I don't know but something vaguely suggestive about that It's quantum gravity which you don't believe in I don't believe in it but I believe it's sort of at some eventual level it's a spin two excitation There was one place where you seem to be kind of similar I was thinking about with this idea of entanglement coming into entanglement and then going into entanglement. I see it as an irreversible process they're not going into or coming out of the state as a whole, you know when you start off you have zillions of oscillators the synchronisation is very, because you have to remember in a sense that these equations do have a direction of time I mean they are I mean the Lorentz equations have kind of an energy dissipation so the synchronisation tends to be one so that means everything is getting more and more entangled becoming more and more entangled, that's right and that does seem to be the way people see the world but the whole point is that's the same with gravity I mean gravity is you start off with a uniform ball of gas And it starts clumping, you know, and the clumpiness carries on forever. I'm not thinking about the gravity side of things, I'm thinking about the pure quantum mechanical side of things. And I wonder whether I'm allowed to do that in your picture. Well, you're not, because you're not, because when you say pure quantum mechanics, I mean... I mean, people talk about, you know, entropy of entanglement. You know, entanglement seems to be a thermodynamic-like quantity. And this would be entirely kind of consistent with that, that the degree, just as the entanglement of the universe seems to be something which is a thermodynamic quantity. So this attractor, if you like, reduction is a one-dimensional quantity.

1:07:30 Of course, locally, you know, local bits. you just, you know, if you just take a chunk of the universe, then obviously that doesn't have to be entangled under that one bit. Well, locally it would do, but there's a kind of global sense in which it isn't. And I am thinking about, you know, the unit, I'm trying, I mean, in a sense, my system is the whole universe this attractor this reduction in dimension is is a reduction in dimension of the total you know attractor of the whole universe as considered as some gigantic dynamical system I think it much isn't it much the same way as you and David you're thinking of the whole of course locally you know literally it can come in and out of synchronization I think what the What we were thinking about specifically was local, what you're calling local entanglement because we wanted the apparatus and everything that was coming from there to be in this entangled state so it would be no non-local. There that's the picture that agrees with I think with what you're saying. It's just a more general process where one would like to see unentanglement going on as well, the possibility of unentanglement. as well as in time because they seem to be these two processes going on in well again i can imagine that uh in some sense in some sort of state space sense if you took a subset of state space and measured a kind of house door type dimension calculation on a subset that wouldn't necessarily have to be uh increasing in the one dimension so it could increase and decrease and fluctuate in some completely, you know, non-monotonic way. But there would be some global measure of the system where that dimension was decreasing. So one of the experiments that we did at birth rate was to see whether we could see this untangling going on just simply by moving our detectors further and further apart so there was some natural spontaneous localisation occurring. And we didn't see any. and what I would like to see would be the disentangling as well going on spontaneously by some underlying

1:10:00 nonlinear process because that's what I like about this you do have an underlying nonlinear process which is actually entangled I'm trying to build in some new physics as well as redescribing things so that's the question is whether you've not gone too fast It turned the other way around. Yes, it's a crazy idea. You're working with the strings, and then you define operator I, right? Yes, by flipping and minus sign. But you can also, I think one can also think about this. Just a very simple thing that you do without minus sign, you just flip them around. Right? and then you get, of course, this, right? Yes. And then you can, then you can play around with another class structure. It means, in the end, you can derive hyperbolic structures, because you can define, I guess, for the sake of this E, right? An e-axis. And you can think about defining a number, let's say, whatever, you say it's a normal number, just an ordinary number, and then you have an e, and then you can define a conjugate, if you want. And then you have a notion of a model, right? And then you can make... This is constant then on the hyperbola, right? This is constant.

1:12:30 And then you have, then you can, I don't know, just an idea. Yeah. Yeah. Then you will have to tip it through. Oh, really? Yeah. Yeah. Well, let me, I don't know. Just, should I get, I mean, I don't want to, I've talked probably. Should I get the last bit of what I want to do? What's the best thing to do? I don't want to hug all the time. As long as there's a happy ending. This will only take 10 minutes, maximum. Where do I get to? Remember? Yeah. There's my fish. There's my caravan. There's my caravan. There's the pendulum. So what we're trying to do is run our climate models. In fact, what I've done is some calculations using models from the UK, France, Germany, Japan, America, Russia, China, so on and so forth, into this ensemble, they all have slightly different, they basically are all, you know, the same underlying equations, which are Navier-Stokes thermodynamic equations and so on, but these tin cans, which are these representations of the unresolved scales, vary, people have different ideas, because there isn't a kind of unique solution to that, so we're trying to develop kind of this probability forecast system much in the same way as we do weather forecasting. And I just want to give you some examples. So temperature is an easy one. So what we're doing is looking, so we run the models over the 20th century and again over the next hundred years using a scenario for how CO2 is likely to increase. And it's based that by sort of about 2070, 2080, we'll have doubled pre-industrial revolution values of CO2.

1:15:00 So pre-industrial CO2 was about 285 parts per million. We're now at 381 parts per million. These are based on sort of the last 20 or 30 years of this century where it's estimated CO2 will be up at around 560 parts per million. and if you don't believe that China is building a gigawatt power station every week and burning it, burning coal so that alone will have substantial impact on CO2 in the opposite. Anyway so this is a kind of an example, I'll just show you this temperature, I'll show you some rainfall stuff as well so this is asking a year a summer, northern summer would only occur once in every 20 years in the 20th century. How often would we expect that type of, you know, one in 20 year event to occur towards the end of this century? So this is showing you a kind of probability based on these ensemble predictions of that type of thing. So you can basically see, apart from, say, regions in the North Atlantic and maybe, I don't know, Reptartica pretty much every single year will be have the type of warmth that we saw once every 20 years in the 20th century this will just give you a sort of feeling for what we're talking about so the implications for that are you know, are these the north the polar latitudes are definitely a problem area because polar bears are going to have a tough time because it's going to be very likely on this basis that there will be no summer sea ice in the Arctic And this is actually, actually you can see already that this is a satellite imagery, you can already see that sea ice is going. So this is the sea ice margin in 1979. And it's actually just running through. These are satellite pictures of the sea ice. So you can see how that margin from 79 is already shrinking. We're going through winter-summer, winter-summer spacing fluctuations. That's where we are now. So the prediction, both types of probability forecasts applied to the Arctic would lead to pretty much no summer sea ice at all. You may not worry too much about polar bears, but one worry is Greenland.

1:17:30 It was thought about 10 or 20 years it would take 500 plus years to melt Greenland but it's realised that melting is not so much the issue it's this disintegration of the ice sheet this much more dynamical process I'm sure you've seen these pictures of these big pools of water now forming on the surface of the ice cap and then they sink, they're these boreholes where the water sinks down lubricates the bottom between the ice and the bedrock. So there is a worry that perhaps even within 100 years, much of Greenland will disintegrate. Now, Greenland has the equivalent of 7 metres, 23 feet of sea level rise. So if Greenland disintegrates, sea level will go up by about 7 metres. in West Antarctic there's a similar worry there so that's 14 feet so it's a caramel in a flat core this time you go to Fortran people run by a flat by the boat the rainfall the rainfall I would say tricky actually you've got these again, these probabilities so how often would a 1 in 20 year dry summer occur interestingly Europe we had that big hot summer, hot dry summer in 2003 wasn't it so we're looking maybe at once every three years at that sort of thing one of the things about that Europe was that hot summer And in fact, it turned out that the plants were so stressed in the summer that they completely shut down all the photosynthesis that normally occurs, which takes in carbon. And the respiration, which is what they, you know, they have to use up energy to stay alive, releases carbon into the atmosphere. So the vegetation, which we normally think is a sink of carbon, became a source of carbon. now what's more worrying are these well this region here of Amazonia because we had a dry year a few years ago

1:20:00 where massive well a lot of these were man made but a lot of burning the rainforest became tinder dry the rainforest really is a big source of carbon, sink of carbon rather conditions. And if, you know, if these droughts occurred once every three years, then I think this does become a worry. Now this is another, this is a biosphere feedback. So this is where Gaia, if you like, is not helping. It's, you know, the biosphere is not going to save us. If we destroy the biosphere, we exacerbate the greenhouse problem, because then we don't have this natural sink of carbon. And this is a map showing the other thing. Will it get wet anywhere? Because rainfall is a tricky one. And some regions will get wet. The monsoon, for example, nobody lives much here, but this is the monsoon region. And the worry there is that with, you know, Bangladesh is a country which is close to sea level floods a lot when you have heavy monsoon rain, so you get the Brahmaputra and the Ganges which flow down into Bangladesh and flood and then you get some monsoon storm surges and such like. Coupled with seven metres of sea level places like Bangladesh which have hundreds of millions of people are going to become uninhabitable. So the other side of the problem is flooding and the environmental refugees. Are the monsoons still going to be monsoons? They're not going to dry up? at all. No, actually the prediction is the other way round for the monsoons. The monsoons actually will become more intense, apparently. But apparently there's some drawing at the moment, sir. Well, I think there's no, at the moment, to be honest there's no obvious trend yet detectable in monsoons so these predictions may be wrong but such as they are, the predictions the probabilistic predictions towards more likely more intense monsoons. So the rainfall question is incredibly complicated depends very much on the region. Our prediction for Europe is wetter winters and drier, hotter summers. At the moment we're in a dry, yeah. No, this is actually all, this is a 5% probability going to about 25-30% probability, so an increase in the

1:22:30 probability of wet winters but I have to say that there are issues to do with models which you know this is the best predictions we have at the moment so we had a couple of wet winters recently I mean if you're interested there's an interesting Oxford project which a project in Oxford with some colleagues of mine where you can download a pretty good climate model onto your PC and they're trying to produce you know really large sized ensembles just slightly changing parameters in the model so it's an interesting kind of project you can get involved with but the key to the problem is is this issue about resolution these are the next generation of models this is a simulation of a storm over the Gulf of Mexico in a computer model so becoming they're looking more realistic compared to those quantilist pictures I've showed you before. The trouble is, to run this a day, you know, takes about a, you know, a day of simulated time takes a day of more plot time on a computer, so, you know, it's not much good for a hundred year simulations. That's why, you know, a lot of us are trying to, yeah. I mean, what we're looking forward to is very, in the next couple of years, we'll be at what's So this is the progress in computing in floating-point operations per second. So in the 70s, we could do about 100 million floating-point operations per second, and we're not to about 10 to the 15 floating-point operations per second. And what I find bizarre is that this is, you know, there are as yet no plans for... I mean, most of these were going to be used by the military for testing bombs and things like that. And it's, I mean, I go to Brussels and things and try to get them to, I'm part of a group trying to promote a thing called the European Virtual Earth Laboratory to actually put, you know, make a centre in Europe like CERN is for particle physics for these kind of petaflock, dedicated petaflock computing for climate. I mean, climate is such a big issue. But trying to get Brussels... Europe is that we don't have a manufacturer of supercomputers you know

1:25:00 unlike the space industry there's masses of aerospace industry in in Europe and so Brussels puts just tons of money into into aerospace but we have no supercomputer manufacturers they all come from either Japan or the United States and the bastards just won't give give us any money to do this you know If I'm not correct. So CO2 is a problem. It's amplified by water vapor. Increases are due to fossil fuels. We use these ensembles with complex models. Melting of ice sheets is a problem. Feedbacks of the biosphere is a real problem. Accuracy is available. It's limited by computing. Oh, yeah, I just wanted to say one thing. Most of these changes we see in climate models to the increase in carbon. So if we were able to cap emissions so that CO2 was 400 and something parts per million instead of 560 parts per million by the end of this century, then the effect on climate would be fairly proportional. So there is everything to play for here. I don't think we should be too fatalistic and say that this is hopeless, we can't do anything. Any effect we can have, and I see the primary one as trying to persuade China and India to put carbon sequestration technology into their coal-fired power stations, if there was one thing that could be done to try to reduce this problem, that would be it. then it will have a proportionate effect on climate and all these changes I'm showing you will be smaller by proportion so lobby your politicians to meet with the Chinese and Indian maybe we have to subsidise it's a rather unpalatable thing to say because China is a growing economy and so on, but of course they say the West has created the problem so far not driving around in forward rises It's okay, that's fine. But the real problem is persuading China and India to use carbon sequestration technology for their power stations. Isn't there also a dimming factor? There has been a dimming factor in the past

1:27:30 because, you know, industry and, you know, cars and things have emitted aerosols, which actually have tended to reflect sunlight. And actually that, to some extent, particularly during the 50s, offset some of the, if you look at the global temperature, there is actually a sort of a slight level off. And I believe that's due to a kind of compensating effect from the fact that the air is so rich and particulate from pollution. The thing about aerosols, though, is that they have a very short lifetime. And if you've stopped, if you've cleaned up industry, aerosols get washed out almost immediately by rain. of weeks, you can go from a dirty atmosphere to a clean atmosphere if you don't. CO2, on the other hand, is around for hundreds of years. It's a very well-mixed gas, and once you've emitted it, you know, it doesn't come out of the atmosphere for a very long time in general. A molecule of CO2 will be around for a few hundred years in the atmosphere. So, we have actually cleaned up industry, although, to some extent, China's got a problem still, but in general the global levels of particulate aerosols have come down and so in some sense we are unmasking the global warming signal. But it is important to see the two as quite distinct problems because the pollution in the standard form of pollution is a very transitory thing. You can get rid of it overnight and it'll wash out and that's fine. This is something you can't get rid of overnight. yeah it's very likely you know the calculations suggest if there was no water in the atmosphere if you just double co2 it would probably only change global temperature by maybe a degree or maybe one and a half degrees because of the interaction with water vapor and this is this is there's uncertainty the the it's likely to be somewhere around four or five degrees in some models it's as much as 10 degrees warming and if because of this interaction between water and the co2 so water is an amplifier and it's because the reason I said if you increase the simplistic well it's The simplistic one is very simple.

1:30:00 It's just that the fact that if you warm through CO2, the air then can hold more water vapour before the water condenses out. And that accelerates the problem. The problem is water vapour is very heterogeneous, if you like, in the atmosphere, unlike CO2. So doing the calculation exactly is difficult. but I think if CO2 were the only greenhouse gas then we wouldn't really we'd be looking at a bit of a problem but it wouldn't be anything like the type of problem so I try to say that it appears to be so the primary feedback is with water vapor but as I say there are these other feedbacks particularly this biosphere one If you kill off the rainforest, then again, you're killed off. Incidentally, I mean, warming the oceans, you know, the oceans absorb, also absorb CO2. And, you know, like a bottle of Coca-Cola, if you have a cold bottle of Coca-Cola and you open the top, it doesn't fizz over. If you have a warm bottle of Coca-Cola, you open the top, it fizzes over. That's because as you warm Coca-Cola, its capacity to hold carbon dioxide in dissolved form decreases. So again, there's lots of these depressing feedbacks. As you warm the ocean, their propensity to take in CO2 also goes down. So this is the difficulty with the whole problem. There are these feedbacks. Unfortunately, almost all the feedbacks people come up with seem to be positive ones. things that make the problem worse and there are very few negative feedbacks the only negative feedback is actually low level cloud um if you can increase low level cloud enough then again it acts a little bit like the um this is liquid now i'm talking about liquid water uh low level cloud again reflects sunlight as you obviously know what about high level cloud by the way high level cloud traps heat more than reflect sunlight algae yeah If it increases, you know, if it changes the albedo to absorb more sunlight, then it's a positive feedback. Bioness, I mean, you know, the growth spares, I don't know. Yeah, I mean, there's issues that algal blooms in the oceans, which is, and the change in

1:32:30 feedback there. But that would have catastrophic environmental implications quite apart from the feedback. The other problem with CO2 in the oceans is that they have started to measure increases in carbonic acid due to an increase in CO2, and that can, I mean there is this worry that crustaceans who build shells, you know, they will just literally dissolve away in the future. so you know eat your lobsters now because they may not be around in countries well they will be easier to eat there'll be a little window of opportunity when they're easy yeah sorry this is a bit depressing but that's the way it is well at least that's the way the science says it is course but you know whether they always get these things wrong so who knows which option. So what do you love? I don't know I mean I think in general and I don't think it's my particular skill but I think