Don Page ｜ Preferred Operators Instead of Preferred Basis States

All right, it's my honor now to introduce your next speaker. The next speaker is Professor Don Page. He's a professor of physics at the University of Alberta. He's worked on quantum theory, cosmology, and black hole physics. In particular, he has collaborated with Stephen Hawking on the thermodynamics of black holes. He has also helped create the renowned Page-Wuthers model, a model in which time is not a fundamental concept but emerges from entanglement in quantum systems. So, Professor Page, please, the floor is yours, and we're really happy to have you on this topic. Well, thank you very much for organizing. It's really great to have this Zoom conference. I wish I could meet all of you in person, but at least it's good to see the faces of you and other people and so on. So, yeah, I'm going to talk. My slide here is of Marine Lake, which is in the Canadian Rockies, and the Valley of the Tent Peaks on the top of Mount Temple, which is kind of the highest mountain in that area. So, I'm going to talk about preferred operators instead of preferred basis states. So, Everett showed that one need not have wave function collapse in quantum theory. Instead, one can have a single quantum state, in the Heisenberg picture, that never changes. However, if one decomposes the state into basis states, there would generically be non-zero amplitudes for many different basis states, sometimes called Everett many-worlds. And many have sought to answer the challenge of finding a preferred set of basis states in order to get a preferred set of Everett worlds. So, I'm arguing that the search for a preferred basis state, at least for the whole universe, is misguided, and what we need instead is a set of preferred positive operators. So, these would be the operators whose expectation values I propose would give the relative measures for observation. I should say, I do find there's an awful lot viable in the decoherence program, but I don't think it necessarily gives you a unique preferred basis for the whole universe. I mean, maybe it can for some subsystems and so on, and it'd be interesting to see how maybe it can be helpful for finding these operators that are useful for observation. But I'm saying, well, I'm saying, well, it's a more important problem is to find the operators whose expectation values give the relative measures for observation, but that's more important than finding some basis of the quantum state. So, okay, my view, a complete theory of the universe would not only give the complete set of dynamical laws, what's been at least by often by particle physics, misleadingly called a theory of everything, but also the quantum state of the universe, such as what Hartle and Hawking have proposed and Blinken and other ones, that's still not not, there are still problems, technical problems with some of the particular proposals. And thirdly, the so far unknown rules for extracting the probabilities of observations from the quantum state. So, I shall propose a basic framework for these rules that the measure for each observation be the expectation value in the quantum state of the universe of a corresponding positive quantum operator for each possible observation, which I should call an awareness operator. Now, one needs to say what observations are, and I propose that most basically, they're sentient experiences or conscious perceptions. It's all, each being the entirety of a conscious awareness, all that one's consciously aware of at once. Then I shall explain why for a large universe with indistinguishable copies of observers, the quantum awareness operators cannot be projection operators, as one might have assumed from Born's rule. So I'm saying Born's rule does not work for giving probabilities of observations. Okay, now to introduce, I would say the bare quantum theory describes the universe in terms of quantum states, operators, expectation values, wave functions, amplitudes, paths, integrals, etc. But does not by itself connect these observations. So bare quantum theory is not directly testable. Well, to make comparison with observations, Max Born suggested the wave function gives probability. Later was proposed that probabilities are given by the absolute squares of amplitudes, and these can be taken the expectation values of projection operators. So in this, well, I should change paper talk in this talk, I shall take the mathematicalization of Born's rule to the idea that the probabilities of observations are given by the expectation value of projection operators. And then I'll show that that doesn't work and propose alternatives. So now I'd like to do a Bayesian analysis for theories. So consider a theory T, theories T sub i, that for each possible observation, O sub k, they each give a probability for the observer, probability of O sub k given T sub i. So, so that's the conditional probability, if the theory were correct, what's the probability gives to the observation. Then if to get to get posterior probabilities to the theory, it seems that we do need to assign prior probabilities, T sub t i to the theories T sub i. As far as I can see, these are subjective. Though in science, we generally adopt Occam's razor of favoring simpler theories and giving simpler theories higher prior probability. Then we use the observation to test the theory, T of O given T, it's then what's called the likelihood of the theory. And by Bayes' theorem, we can calculate the posterior probability of the theory as the product of the prior probability of the theory times the probability observation given the theory, and then normalize that by dividing that by the sum over, over all possible theories. So we would like to get the posterior probability as high as possible by choosing a simple theory, which if we adopt Occam's razor would have high prior probability, and that gives a good statistical fit to one's observation, O sub k, so high likelihood. So there's, there's the two factors in there and there's the tradeoff. So now there's the question in a complete theory, what are these observational probabilities? Since a complete theory for a universe should completely specify all properties of that universe, one might think it has no room for probabilities between zero and one. So all likelihoods, T sub i would be zero or one. I've actually, as an aside, collapse theories of quantum theory, I don't, they, in my regard, they would not be complete until they specify exactly what collapse actually does occur. So whereas Everts, you don't have a collapse, so you don't have to, you know, you don't have to specify what the collapses are, because there just aren't any collapses, and I'm taking the Everts view of no collapse. Okay, then, if you did say that all the likelihoods are zero one, the posterior probabilities of these theories, given the observation, would just be proportional to the prior probabilities of theories that do give this unit probability for that. So there would, so except for this, except for this excluding theories that don't predict that the observation occurs, all the posterior probability would just be given by the prior. So it would be sort of completely theory, no, no, no, no real impact for observations, except this minor one of excluding theories that don't predict the observation at all. So for observations to play a bigger role in science, which, as a theorist, I admit that experimental physics and observational sciences are very important. And so we would like to get a way of getting likelihoods between zero and one. Now, suppose the theory can get normalizable measures for the observations, even if all observations occur, but if they all have measures and they're normalizable, then the normalized measures I'm proposing can be interpreted as if they're these likelihoods, O sub k given T sub i, that can vary between zero and one. They're not really probabilities in the sense that they're not propensities or potentialities to become actualities, they're really actual measures of the observations that are determined by the theory. But if they can be between zero and one, then I'm proposing that they could be used in a Bayesian analysis as if they were probability. I mean, they would be in a hypothetical thing where imagine somebody outside the universe picking an observation at random, and using this measure as the weight for that, then you could say if an observation were picked at random by some outside, then that would be a probability in the more traditional sense. But I'm not proposing, I'm proposing they all do exist, that there's nothing that picks them out. All of them with positive measures do occur, but with different measures. Okay, now what could these measures be? In a classical theory, the measures could be functional to the phase space trajectory, such as the times that the system spins in different regions of the phase space. You can imagine one region corresponds to one observation, another region corresponds to another, and so on and so on. And then the trajectory passes through all this, and then you can see that the times that they spend in there could be these relative measures. And then you can normalize it, you can interpret them as if they're probability. On quantum theory, the measures could be certain functionals of the quantum states or maps of the quantum states and non-negative numbers. So here's another argument against all likelihoods, zero or one. If a specific theory leads to the definite existence of more than one observation, one might be tempted to say that probably the observation given the theory is one, because it definitely occurred. And for all of these observations, that would give unit likelihood for any theory predicting the definite existence of a particular observation used to test the theory. But then this procedure would mean you could construct a theory of maximum likelihood, namely unit likelihood, no matter what the observation turns out to be, just by having the theory predict that all possible observations definitely exist. But this seems too cheap, it's supposedly explaining everything, it says everything has unit probability, therefore, it's unit probability that your observation occurred. But it's not distinguishing yours from mine. I mean, it's really explaining nothing. And that's indeed, you know, some restricted, some complaints about multiverse theories are of this nature. And I think even a true multiverse theory should not predict all possible observations of equal measure. So I'm proposing the normalization principle, or you might say meta-theory, that each theory T sub i gives normalized probabilities for the different possible observations. So for each theory, the sum of the probabilities of the observation for that theory is unity. Well, it's just the usual thing, if you want the probabilities to add up to unity, and this is conditional probability, so conditional on the theory, probably the observation should add up, add up to one. So, okay, there's this trade off between prior probabilities and likelihood, the prior probabilities, piece of TI for the theories, those are, those are intrinsic plausibilities for the theory before you consider any observation, of course, it's very hard for humans to take because we're, our existence is conditioned on things. So it's pretty hard for us to really be that, but, and as I say, that seems to be subjective, I don't know of any, any way that you can deduce that, you know, from, from something, but, but, you know, if you apply Occam's razor, you might say simpler theories give higher, but even if, even if you can order the theories, there's still a question of how much weight to give to each theory. Okay, then there are these conditional probabilities for observations, or the likelihoods for the theory with a fixed observation, and then the posterior probabilities are proportional to the product. So prior probabilities are subjective, usually higher for simpler theories, and the highest prior probability might be for the theory that nothing concrete or contingent exists, only mathematical truths or just pure logic exists, but abstract, but nothing concrete, but then since observations are regarded as concrete, the probability that would be zero, so the posterior probability of that theory would be zero. Then maybe the next simplest theory might be the theory that all observations exist equally, but then surely there's an infinite number of possible observations, so then if all exist equally, the likelihood of any particular one would be one over infinity would be zero, normalized. So this would be my criticism of theories like David Lewis, although, you know, he didn't put a measure on it, but if you, he said that basically anything logically possible actually occurs, maybe, you know, it's always different world or different, I don't know what to call it, the world's supposed to be everything, philosophical thing. But anyway, he didn't really put, but if you if you just said all of these different things have equal measure, then then the likelihood would be zero. So that would be the way I would rule out something like David Lewis's modal realism theory, if you if you augment it by putting measures on these theories and, you know, say are on these different possibilities and saying that, you know. Now, the other extreme would be a maximum likelihood theory giving the probability of unity for one's observation. And of course, if that's normalized, it means you're giving zero probability for any other observation. So that would be sort of like an extreme solipsism, not only that only I exist with my observation, but only my present observation. And anything in my memory, I mean, anything I'm aware of now, my memory would be part of my present observation. But these memories will not be really referring to any previous observation. So this would be this would be a maximum likelihood. So, but this seems to require a very complex theory, if you had a complex theory that just produced one observation and no others. And so I, although it's a bit subjective, I think that because it's complicated, I think it'd be very plausible to assign this an extremely tiny prior probability. And so you'd get a, even though the likelihood is maximum unity, you'd still get a very low posterior probability. So, so I think there is this trade off between prior probabilities and likelihoods between intrinsic plausibility and fit to observation. So now let's relate a little bit to cosmology. The major problem of cosmology is how to obtain probabilities of observations from the quantum state of the universe. And this is particularly a problem when internal inflation leads to a universe of unbounded size, so that there are apparently infinitely many realizations or occurrences of observations of each of many different kinds or types, making the ratios ambiguous. And there's also the danger of domination by Boltzmann brains, observers produced by thermal and or vacuum fluctuations. So it seems this major problem is related to the measurement problem of quantum theory, how to relate quantum reality to our observations that appear to be much more classical. So I'll take this approach that observations are fundamentally conscious perceptions or sentient experiences, being all the ones consciously aware of at once. So, I shall take an Everettian view that the wave function never collapses. So in the Heisenberg picture, there's a single fixed quantum state for the universe, which could be a multiverse, and we could have many different parts with different coupling constants and so on, but it's all a single quantum state. And I shall assume instead of many worlds, there are instead many different actually existing observations, many different sentient experiences, your mind, people in the past, people in the future, people in other branches of the wave function, perhaps people in other branches of the multiverse, where there may be a different coupling constant that allows consciousness and so on. So there's many of these, but they have different positive measures, mu sub k, it's the measure of this observation, which are in some sense how much the various observations occur, but these measures are not determined by just the contents of the observation. And for simplicity, I shall assume that there's a countable discrete set, O sub k of observations, that the total measure is normalized unity for each possible theory that gives the normalized measures mu sub k for all possible observations. So then in a Bayesian analysis, I shall interpret the normalized measures of the observation that each theory gives as the probability of the observation given the theory, even though it's not really probability in the in the sense that there's any randomness or uncertainty or indefiniteness. Well, there's uncertainty for individual observations within the universe, but from a view of the whole, if you could view the whole quantum state, there would be no uncertainty about it. But it's, I'm interpreting as if there are these probabilities, these likelihoods that you can put into the Bayesian analysis. So now, now I'm, I have a particular framework for this called, I call it sensible quantum mechanics, or in the popular thing I wrote called mindless sensationalism. So the map from the quantum state to the measures of observations could be nonlinear. I mean, an example of a nonlinear thing is when the, in, in decohering histories, you have sets of histories that are decoherent, those sets depend nonlinearly upon the, upon the quantum state. And so then the, the, and then the, the measures for these sets or the, for each element of the set are probably of a particular history within a set. Then of course, it depends on the set, which depends on the quantum state in a nonlinear way. However, in my mind, it's somewhat simpler to assume a linear relationship. I've not, there's no way to prove that that's right, but I'm assuming a linear relationship in the sensible quantum mechanics. And I've also mindless sensationalism because it proposes that what's fundamental is not minds, but conscious perceptions, which crudely might be called sensations, although they include more than what one is consciously aware of, e.g. memories and called sensations. So that's a little bit of a misnomer to say it's just sensations. But, but, but I am staking, what's fundamental are the individual conscious perceptions and not any unique sequences, because the sequences, you know, might, might effectively split, you know, if you, in everything. Okay, so the measure, this mu of this observation I'm proposing is the expectation value of a corresponding operator A, A of O, I'll call that A sub k, that's, that's a positive operator. And this sigma denotes the quantum state, which is a positive linear functional of quantum operators. So, okay, so the measure is the expectation value of this awareness operator, or the expectation value of that, or, or I could say it's of A sub k, which is the same as A of O sub k. So sigma is the quantum state of the universe, a positive linear functional of quantum operators, and A sub k is a non-negative, what I call an awareness operator corresponding to the observation or sentient experience, or conscious of perceptions, another phrase I use synonymously, O sub k. And the quantum state might could be a pure state, it could be a mixed state given by density matrix, or it could be a C star algebra state. I'm being agnostic as to what the quantum state of the universe is. And the awareness operators, along with this linear relationship above, and also complete description of what the observation is, they're all supposed to be given by the theory, the complete theory in my mind, if it has this form, it will have these elements. So now let me talk about Born's rule. Traditional quantum theory uses Born's rule with the probability of observation being the expectation value of the operator that's a projection operator. So applying it twice, you get the same one back. If you apply the same one twice, if you apply two different orthonormal projection operators, after that you get zero. And so in that case, the probability observation would be the expectation value of a projection operator. So Born's rule works when one knows where the observer is within the quantum state, e.g. in the quantum state of a single laboratory rather than the universe, so that one has definite orthonormal projection operators. However, Born's rule does not work in a universe large enough that there may be identical copies of the observer at different locations, since then the observer does not know uniquely the location or what the projection operators are. So let's spell this out a little bit. Why does the Born rule die? Suppose there's two identical copies of the observer at locations B and C, that each can make the observation O1 and O2, and let's suppose these observations do not reveal the location, they don't show whether you're at B or C. So Born's rule would give a probability piece of 1B, the expectation value of the projection operator at location B, and piece of B2 for the projection operator to the other observation, if the observer knew that it were at location B, with the projection operators there being P super B for the location and sub 1 for the first observation, and P super B and sub 2. Similarly, I'd give the probabilities P superscript C for C for the different location, the expectation value of that projection operator, and piece of P2 for the other operation at the location C, if the observer knew it were at location C, with the projection operators being there. However, if the observer is not certain to be at B or C, and without loss of generality, let's suppose that the probability of the observation 1 at location B is less than the probability than the projection operator expectation value at C. If you're uncertain, then the probability should be between these two. But there's no state-independent projection operator that gives an expectation value with this property for all possible quantum states. So if the projection operator is supposed to be chosen before you imagine applying it to different quantum states, then you can't get that. So no matter what the orthonormal projection operator piece of 1, piece of 2 are, there's an open set of states that give expectation values that are not positively weighted means of the observational probabilities of two locations. So the Born rule fails in a cosmology when it's large enough to have two observers that can't tell where they are. There's two copies of two different locations, and their observations don't tell them which of the observations they are. So the failure of the Born rule means that in a theory piece of I, these awareness operators whose expectation values in the quantum state sigma sub I, the I telling which theory of the universe, it gives the probabilities a normalized measure for the observations as probability of observation given T sub I, it's the measure of that O sub I, and that's the expectation value of the operator A sub I of O sub K. In other words, the expectation value of the awareness operator in that theory. So they cannot be projection operators. However, the awareness operators could be weighted sums or integrals over spacetime of localized projection operators, O sub K comma X, at locations denoted schematically by X. I can include location in time also. Say onto brain states there. So suppose you have a localized projection operator on some region of spacetime onto what would be a brain state at that location. That's what it could be, and then you would weight these and integrate over the spacetime. Now, this does lead to a problem, because in the Boltzmann brain problem, in local quantum field theory with definite globally hyperbolic spacetime, any positive localized operator is localized to be just within with act non trivially within some finite region, such as a localized projection operator will have a positive expectation value and a non-pathological quantum state. So there's always some probability of finding some non-zero probability of finding something within this small region. So for example, the charge, you can find the charge positive, there's always some probability to find positive charge within a finite region. Of course, if you go all the way out to infinity, you can have a state with zero charge and the probability is then zero finding the total charge of the whole universe to be positive, but for any, any, any small region, the probability of well, the expectation value of an expectation of a projection operator onto non-zero charge in this finite region could be positive and similarly for any other projection operator. So if such a prediction localized operators integrated with uniform weight over spacetime with infinite for volume, it will give an awareness operator with an infinite expectation value. You won't be able to normalize it. Now you could say, let's cut it off. If we take the integral only up to finite cutoff time TCC and normalize the resulting awareness operators, then for a universe that continues forever with a three volume bonded below by positive value, the intervals will be dominated by times of the same order of magnitude as the cutoff time. If at late times and probably prefer volume drops very low for ordinary observers, that is ones that, you know, they're all from, from stars and planets and evolved like we did. If they just come from the, from these expectation values in something that's near the vacuum, then most of the measure for observations will be contributed by thermal or vacuum fluctuations, so called bolts and grains. That is bolts and grains will dominate the measure for observation. So if bolts and grains dominate the measure for observations, one might say, so what? Couldn't it just be that our observations are that of ordinary observers? Or couldn't it be that our observations really are bolts and grains that happened to have the ordered observation? But in this Bayesian analysis, that doesn't work. Since bolts and grain observations are produced mainly by thermal or vacuum fluctuations, it would be expected that only a very tiny fraction of their measure would be for observations. So ordered is our observation. This tiny fraction, plus the even smaller fraction of ordinary, ordered, ordinary observer observations, in comparison with the dominant disordered bolts and grain observations, if you put the cutoff, if you put the cutoff far beyond the lifetime of stars, then way, way beyond, sufficiently far beyond that the bolts and grains dominate, then the ordered observations would be only a tiny fraction of the measure of all observations. So that's the normalized probability of one of the ordered observations, which we use as the likelihood of theory, we're taking one of our one of our ordered observations test theory, then that would be highly diluted. And it would be much smaller than those of alternative theories in which bolts and grains do not dominate. So if these alternative theories do not have prior probabilities are too small, they would dominate the posterior probability. In other words, you might need to complicate the theory a little bit to get rid of bolts and grains. But if you don't, if you don't do that, then the likelihood of our order observation would be extremely low in a bolts and grain theory. And so we would get a low posterior probability. So in summary, bolts and grain domination, which is predicted by many simple extensions of current theories, e.g., with the awareness operators or their equivalent being obtained by uniform integration over spacetime up to a cutoff that's n taken to infinity. This gives a reductio ad absurdum of such theories, it makes their likelihoods very small. So surely there are alternative theories that avoid bolts and grain domination without such a cost of complexity, that their prior probabilities would increase so much as the gain in likelihood for not having the normalized probabilities of our ordered observations highly diluted by disordered brain output. So I view that this bolts and grain problem is analogous to the ultraviolet catastrophe of the late 19th century classical physics. So this is Stan did not believe that an ideal black body of thermal equilibrium would really emit infinite power. And most physicists now do not believe that bolts and grain really dominate observations. But it's a question of how to get rid of them. And there may be some cost of complexity, but I'm arguing that surely there must be some way of getting rid of them, that the cost is not as much as the gain in likelihood. So, okay, so I've made some partial attempts on this. These are all sort of schematic and also they're not terribly elegant. So, although I think they add less complexity, then I think they would definitely increase the posterior probabilities, but they might be increased more by finding more elegant solutions. So the approach that gives awareness operators the universe integrals over spacetime and localized projection operators, or equivocally, many people don't even, they don't talk about these operators, they just count all observation occurrences equally, no matter when and where they occur in spacetime. That gives it especially severe bolts and grain problem in spacetimes with a positive cosmological constant, as ours seem to have, with the spatial hypersurfaces having three volumes that asymptotically grow exponentially, as in the k equals one or in other words, closed slicing of the disinterface. At each time, counting the number or measure of observations is growing with the volume is called volume weighting. Well, in 2008, I proposed the alternative of volume averaging, which gives a contribution to the measure for an observation from a hypersurface that's proportional to the spatial density of the occurrences of the observation in the hypersurface. So that would be rewarding the spatial frequency of observation occurrences rather than the total number that would diverge in internal inflation if the hypersurface volume is taken to infinity. So I would say the spatial density is surely much higher now when we have stars and planets and ordinary observers, the bolts and grain thing, I mean, the probability per four volume may be very low, you know, e to the minus 10 to the 42 or something, it could be exponentially tiny, but the problem is it's not zero. And so if you go to infinite volumes, it diverges. Okay, now that's this volume average ameliorates the bolts and grain problem by not giving more weight to individual spatial hypersurfaces at very late times when bolts and grains might be expected to dominate. But you still have to consider all integrate over all the time over the sequence of hypersurfaces. And if you do that with it, measure uniform in the element of proper time, you get a linear divergence if time goes to infinity, because at very late times, the density of bolts and grains just goes to a constant. They need some suppression at late times to meet this divergent. So in 2010, I proposed agnesi weighting, which admittedly is ad hoc, but I replaced dt by dt over one plus t squared, where t is measured in blank units, which of course, the integral of that converges. Of course, you see it's ad hoc, but it's, you know, it's a simple function that the integral converges. So if you had weight, the weight would go down with time. And more recently, I proposed new measures depending on the space time average density or sad observation occurrences within a proper time for the Big Bang or bounce. And when these measures are combined with volume averaging, and a suitable quantum state, such as my symmetric bounce one that I won't have time to talk about here, but as the universe having a bounce, when it was really small, they appear to be statistically consistent with all observations and seem to get much higher likelihood than current measures using the extreme view that the measure is just given by the quantum state. So now there may be other questions that we can try to hope to be answered once we find more about these awareness operators. And that's, why do we not observe ourselves to be ants? You know, other observations that at least naively seem difficult to explain by Born's rule include our observation, they were humans rather than ants, like the much greater number of ants on earth. So if you get up, you said it was proportional to the projection operators on to some sort of sentient, some sort of sentient being and let's allow the ants to be sentient. If you just have simple projection operators, then the ant might count for just about as much as a human. Well, there may be more than there may be a certain number of more protection operators, because the size of the humans, but anyway, it might be more comparable if you just had if the probabilities were just given by projection operators. So, but there's ways you can, the modifications of Born's rule can explain the probabilities of observation by the way that it can explain by copies, the fact that we apparently do not observe ourselves from both brains. And by modifying considerably from the Born rule, we can also explain the fact that we did not observe ourselves to the ant. However, it's not yet known when the correct modifications are. So there's many possibilities for probabilities of observation. So I have a framework, but I don't have a detailed theory because I don't have a detailed theory of what the awareness operations are. So here, I'll not give an exhaustive list of possible replacements for Born's rule, but just describe a few of the basic possibilities. The simplest set of possibilities seem that the probabilities of observations are given to the expectation values of positive operators, one for each observation that are not projection operators, as I've discussed. And then if angular brackets denote the expectation value of an operator, then the probability observation is given by this expectation. And here, I will assume we haven't necessarily normalized the operator, the a sub k, so that maybe the expectation value is some of them for all a k's, they may depend on the theory, they may depend on the quantum state. So here, I'll allow that, but then I'll just normalize. So the probability observation in a particular theory is just normalized expectation value of the awareness operator. So then the main open questions in this SQS, Sensible Quantum Mechanics framework, in which the probabilities of observation are the expectation values of positive awareness operators, one for each operator, is what these awareness operators are. So this SQM framework does not assume that there's a space time, so it would be valid even in formulations of quantum gravity in which space time is not fundamental. However, if the part of the quantum state that gives the dominant contribution to the expectation value of the awareness operators can lead to the approximation of an effective space time, or of a quantum superposition of space time, one idea is that the contribution to the expectation value of an awareness operator for each space time in the superposition is the expectation value of the existence of this space time multiplied by the weighted sum or integral over the space time of a localized projection operator for each space time region. So here, I'm ignoring possible interference between different space time, but assuming that there's some quasi-classical space time, there's a whole set of them with expectation values, and then in each one, we do an integral. And then to, okay, to avoid golf and brain, we'd have to do this integral in a way to avoid things. Let's see, I'll think I'll skip over that, I kind of said most of that, but time. And then to explain why I observe a set of humans, despite the greater number of ants, it seems plausible to postulate that human brains are much more efficient in producing sensing experiences than ants. So the total probability of human conscious observations is not far below that of ants to make human observations much less probable. So I mean, the thing is, if the ants dominated less, since the total has to add up to one, it would dilute the human observations and make our less, you know, less probable. Therefore, one might postulate that the localized projection operators for human brain configuration should be given much greater weights, WCK, than the localized projection operators for ant brain. So it's weighted projection operators, and part of the weight could have to do with the complexity, or that, well, something about the human brain. Perhaps some part of this weighting should be the complexity of the corresponding brain, since human brains seem to be much more complex than ant brains. But I am skeptical of the hypothesis that simply getting high complexity or high information processing is by itself sufficient for getting a large weight multiplying the corresponding localized projection operator for corresponding awareness operator. As an example, it's it's not at all obvious to me that even if you've got some computer that is eventually is much more complex than human brain, that it would necessarily be more, more, more, have more observation, more measure of observation, and then humans, there may be, you know, there may be other particular facets about humans that make their observations, you know, have higher weight. And then, you know, it could be that we find ourselves to be humans is because of the selection principle, where we find ourselves to be some of the more probable, some of the some of the systems that give higher measures for awareness. So, okay, so a schematic formula, if you suppose that the expectation value of a given space time is S sub L is a E sub L. And at the space time, in this space time, the expectation by the localized projection operator, corresponding observation is, is NKL. So, it's sort of like a number density of the of the observations in this space time as a function of space time. And suppose this space time, the weight factor for the integration over four volume is WI depends on the location, for example, it could be one over the volume of a hypersurface at a certain time in the denominator multiplied by one plus t squared. Anyway, so you get some you get some integration factor that that converges. And if the intrinsic weight factor for the observation is W sub K, so that would be higher for human observation, then the relative probability would be, you would have the weight for the observation WK, then you'd have a sum over space time, so the expectation by the space time, and then you're going to have an integral over the space time volume with this weighting, volume weighting factor, and then times the, the density of the expectation value of the localized operator. So that's that's sort of a schematic form you could have, if indeed, it's dominated by, by parts of the wave function where there, where there is space time. And then, okay, this shows there's a lot of freedom in the modification of Born's rule in cosmology, even within the restriction of sensible quantum mechanics in which the relative probabilities of the observation are expectation wise of the positive operators that are the awareness operators corresponding to the observation. There would be even more freedom if one abandoned this linear relation of SQM and allowed the relative probability to be non-linear in the expectation value, such as these expectation values to a power S that's different from unity, or be the more general function. S0 would be sort of just counting the observation, but if S is, if S is not too far from one, it'd be very hard to tell the difference. So if you go beyond sensible quantum mechanics with this linear relation, and then goes to this non-linear relation, you might ask how that can be consistent with observations that have S significantly different from unity. The key is to remember that piece of K is the relative probability of an observation itself, which I'm taking the ascent in experience. It's not the fraction of results reported one way rather than another for some unconscious reporting device. It would give you, if you observe these results, it would give you the relative probabilities of the observation of the, of the, say of the total results or whatever. And so if S, if you had some Gaussian, then the shifting S away from one might shift the, the Gaussian for, you know, what you'd expect to see for the number of heads or tails, but if S is pretty close to unity, then the Gaussian would not have, the spread would be pretty much the same. And so anyway, you can't, it's, you're not, it's not the frequency of, of, of things that are reported. Okay. So that I think includes my talk and grateful, well, discussion with, oh darn, I should have had a lot of other people. I forgot. I asked the organizers to send me a list of people here. I was going to acknowledge other people I had talked about here, but when I wrote a paper based on this, I just did as a John Carroll in Caltech. Sorry, I forgot to add in the, there's many of you that I'm grateful for conversations with. Okay. So that'll end my talk. Thank you very much, Professor Page. I invite everyone with me for a round of applause. Great. So we are good to have some questions. I see a hand rise by Charlie, Charlie Bennett. You want to unmute yourself and please ask your question. Well, I don't think there's the ant problem is very big. Maybe most of the conscious awareness in the, in the world is, is by ants and they don't, they don't think there's a people problem because most they're, they're, they're people are, are, are big and destructive, but, but, but, but don't seem to be thinking very much. And really I would say the ant problem is a little bit like the problem until recently that there, that the cosmologists seem to be almost entirely male. And it's not because the females don't think it's just because they weren't cosmologists until recently. Right. It's just, I guess the ants probably wouldn't dilute. I mean, there's so much uncertainty in what this measure is that if the answer is equal, if it was diluted by maybe a factor of a million, but I mean, I think we're talking about enormous factors in this, in these posterior probabilities. So it's, it's perhaps true that just diluting the, reducing the, the, the likelihood of the theory by a factor of a million, maybe it wouldn't do much. If say, if, say if the ants had a million times the. It's just that, that you should be considering not the total number of ants, but the number of ants that are holding zoom conferences about cosmology. Well, I I'd rather get a theory that has all possible observations and, and, and normalize the measure for all of them and not, you can, of course, once you've got those, you can, of course, do conditional probabilities, additional probabilities of lots of people attending zoom where they think they're attending zoom conferences and so on and so on. But I mean, I would prefer that the theory give the, you know, give basically that the probability for an observation itself, and that you could then use that better to test theory. Once you start taking conditional probabilities, it becomes more, you know, it becomes more arbitrary, which conditional probabilities you take. So I prefer to stick with all conscious observation and normalize the total measure for all of them. Is there any, I see another question from Ding. Ding, if you want to unmute yourself and ask your question. Yes, thanks. Thanks for the nice talk. I just, I just wonder about conscious observations. Do you think it's possible for some conscious being to observe superpositions? And if so, how do you incorporate that into formalism? Yeah, that's, I don't know, that's, that is something I guess, how is this? You know, what, what properties do these have that normally they don't see that? I mean, decoherence may help with that, but I'm a bit testified, mystified. And maybe I want to ask Wojtek Durek, you know, or some other thing about the fact that, you know, he talks about the, they talk about decoherence, that the, the, that the density matrix of this system that becomes decoherent, it becomes diagonal or becomes close to being diagonal in the pointer basis. But if you ask the basis in which it's exactly diagonal, at least in the very simple models of Yosemite, this, this density matrix, most of most of the eigenstates of it have, well have the spread of the of the original thing. So in other words, if there was a huge spread in the position of very big, so exactly why we don't, you know, why we don't see a big, a big spread in position, it's not yet completely obvious to me, I hope maybe decoherence can help with that. But I'm trying to learn more to see, does that, does it really, does it really help with that? Should I interject? Yes. I want to ask a question, particularly, many places you talk about the, the, the, the, the diagonality of the density matrix is a symptom of decoherence. But isn't it, are you, are you saying that it's a symptom that generally occurs? Or are you just saying it can, in some cases, it can occur, because it, it certainly doesn't seem to occur in the, say, the Yosemite model, where you just have this particle that's, that's moving, say, in one dimension and interacting things. If you calculate the density matrix of that, then the eigenstates of that density matrix, you know, basically, basically, if you have a Gaussian, and you've got to pull the, the, the spread, the variance of X, is this, all of the, all of the, all of the eigenstates have the same expectation value of X, say, the middle of the thing. And so the variance of the, of the, you know, if you average the variance over all the eigenstates, that's just the same as the variance of the original thing, which is very big. So is that? Yeah, so it sounds to me like an invitation for another talk that I should give, but I won't. So let me just keep it to something very quick. So first, coming back to what Ding asked, can you observe superpositions? You do observe superpositions all the time. The point is, if you have a vector in a certain basis, it has superposition in another basis. So if you have states which are sufficiently localized, they have superpositions of momentum, for example, right? So, you know, it's not so much do you observe superpositions, you do, because you have complete freedom in, in, in, in adjusting the basis. Now, the, the, the point that I think addresses some of what you've asked, has to do to some extent with decoherence. So indeed, diagonalization is a symptom, which occurs after some time, and which is imposed by the interaction Hamiltonian or generally by Schrodinger equation, but you need Hamiltonians to run the Schrodinger equation. But in a way, the real question is not so much what are the states of the systems, but what will you perceive, right? And, and the business of, of, of quantum Darwinism is answering precisely this question. So that means that there are these stable states, and only stable states can get transcribed in many copies on the environment. And the moment you have non-linearity, and you need non-linearity to go from one original to many copies, you have to perform selection of what can be copied. So this was, you know, the first part of, of, of my talk yesterday, if you still remember, it's a repeatability, right? You need to have something that survives. And, you know, the nice thing that, that comes out from both decoherence and quantum Darwinism is that the requirement of survival, which is what gives you eventually a diagonalization and pointer state, is also the requirement for being able to make multiple copies. So I think these things work together. And I, I mean, I, I, yeah, I better stop here, because I'm going to run over time. Let me just ask you about this. I'm a bit bothered by this example. I mean, of course, it's an extremely simple example. And maybe that maybe the simplicity of it is what's misleading. But as I understand it, and I did some calculations on that, that you use the same model, you have this, this particle, it's moving. Yeah, I actually, I actually extended to a baseball, which I said that suppose... You mean the dust grains, the dielectric sphere? Well, it wasn't, I was just using their thing where they had some, they did, they just had some cross section for the interactions. Keep going, maybe I'll, maybe I'll get the context. But in the very, in the very simple models, suppose I imagine, okay, let me just, let me just say it's a baseball as an example. It turns out that the baseball with the plank momentum has going at 100 miles per hour. Anyway, let's suppose that we started with a state that's a minimum uncertainty state at the beginning. But the uncertainty in the momentum, it's got zero expectation value for the momentum. But the RMS value of the momentum is the plank value. So it's RMS value of the velocity is 100 miles per hour. And we'll have a minimum uncertainty. So the position spread is a half of a plank length for the center of mass. Okay, so it's got big momentum. So it's very, you know, it very rapidly, classically, very rapidly spreads out. Now it interacts with the, with the environment. And this, the Yosei thing leads to, if you calculate the density matrix now of this baseball, you get, you get the, the eigenstates of the density matrix basically have the same form, except for some phase thing that oscillates. It's the same as, as the, as the energy eigenstates of a harmonic oscillator, even though I'm not putting as a harmonic oscillator there. But, but, but the basic claim is that each of these eigenstates has zero, has zero mean for the, for the position. So that the center of each of these eigenstates is, you know, at position zero, but then, and, and the grounds, well, the one with the highest probability is, is indeed sharply peaked about that. But, but there's an enormous number. I mean, I mean, it's like having a temperature much, much higher than the frequency of a harmonic oscillator. So it's sort of like a thermal state in a harmonic oscillator, but it's not a harmonic oscillator. So the idea is that, that the, that most of these, most of these eigenstates have a variance that's, that's, you know, comparable to, to what you would get from this hundred miles per hour for however, however long you take. So in other words, these, these states that seem to arise from the, of course, this is a very simple model. With this, with the interaction of the environment, it doesn't localize, it doesn't localize the baseball. I mean, each of the, in the, in the basis in which the baseball is, is, its density matrix is diagonalized. In that basis, the baseball doesn't get localized. It's, you know, it's, it's got the, on average, the variance is the same as the full, as the full variance. So is that, I don't know, is that, I mean, in other words, and that's a state, I mean, that density matrix is what happened a long time. So it's, it's surviving. Now, you know, I don't know what happens if you send photons of this thing, and what happens with that, maybe that would bring in something new. But, but it seems a little worrisome. I mean, in other words, this is kind of a lacuna in my understanding of decoherence, because it seems like at least at this very first step, when I just have the baseball decohering by having the, the atoms of the air and light and so on scatter off of it, that that doesn't localize the baseball, the eigenstates of the density matrix are just as delocalized as they were without the interaction. So I'm, I'm sorry to interrupt. My, my own example. Excuse me, guys. I would just like, this discussion needs to be pursued, but I would just like to officially terminate the session of Professor Dompage, and maybe we can stop the recording and give him a last round of applause.