 But good afternoon, before we start, as I told you, the first day, this series of lectures have been founded by the Kuwait Foundation for Advancement of Science. We have the owner today to have the president of the foundation, Professor Shibab Al-Din. So I will ask him to say a few words. We just signed a new agreement for collaboration, so I would like to invite him to say a few words to the audience. Please. Good evening, everybody, and I'm delighted and honored to be here once again just for your information. The first time I visited Trieste and the center was 1975. I'm sure only a handful here were born by that time. But I was there to sign an agreement with Abdesalam, the late Professor Abdesalam, a great friend. And that agreement has been sustained. It was signed by Kuwait University at that time, but then KFAS Kuwait Foundation for the Advancement of Science took it over. And I'm delighted that had the opportunity to sign a renewal of that agreement with some expansion of some of the activities. Not only the Salam lecture series, but a few other activities, KFAS supports some of the diploma students and some of the PhD students. But we also pride ourselves in allowing many young scientists from the third world to have an opportunity to come and benefit from this great center. I was fascinated by science and physics. I studied nuclear engineering. I couldn't quite study physics. But I'm fascinated by physics. And I'm here to listen to the third lecture by Professor Alan Gooth about inflation. I followed what you lectured in Kuwait before I arrived here, so I know about the first and second lecture. And I know that the audience has been so excited, so I'm not going to stand between the audience. But thank you for the Nardu. And it's a great opportunity to see so many young faces among us here that still have the love for science and the love for physics. And that's a great thing to do. Thank you very much. Okay, so the first day I gave an introduction to Alan, saying all the great discoveries he had made and the process he has won. But let me just, since today is the last talk, I will say a few more personal things about Alan. And you don't mind. One is that all the prizes that he has won, he has won one other prize that people don't mention too much, which is the messiest office in all Boston, I think. And we were talking with Antonio Skardike recently, apparently you can barely walk within the office because it's full of papers. However, I remember we organized the 25th anniversary of the famous Newfield Conference a few years ago in Cambridge. And Alan gave his talk, and after he managed to find the slides of the talk he gave 25 years before in the same office, he managed to find his papers. So somehow I think he cares very much about the second low thermodynamics. And I think he's touched with your conference today. The other point, he has been very supportive of SDP activities. He was a member of the direct medal panels, and he was thinking very much for supporters for several years. And the last thing is something I learned recently. We were in a conference together one year ago in Stanford. And there was a talk by a young collaborator of Alan. And the talk was a very heavy computational subject and a lot of numerical calculations and so on. And at some point people were asking questions to the young collaborator. And at some point he said, sorry about this. The person who did all the computing was Alan. So he was the senior person doing a collection for the young person. So that's something that gives us some high level to see. And again, you can see the arrow of time going in the different direction in this case. So that's the subject of today's lecture for Alan. So let's welcome him again. Yeah, so today's lecture, I'm going to be changing gears significantly. The main focus will not be inflation, although I'll get mentioned in the end. I have to admit I don't know how to give a talk that doesn't mention inflation, at least. But today, the focus of attention will be the arrow of time. What is it that makes the past different from the future? This is a question that physicists have been talking about and debating for over a century. It's always a little dangerous to give a talk about a topic that physicists have been debating for over a century, because it's hard to say anything new under those circumstances. But in this case, I think I really do have something to talk about that's new. It doesn't really originate with my own ideas. The work that I'll be talking about is work that's in collaboration with Sean Carroll and a former graduate student of his, Jin Yao Tseng. And the original idea really came from the paper from 2004, if I remember right, by Sean Carroll and Jennifer Chen, and what we've been doing is elaborating on that idea, clarifying how the mechanism works. The key question is one that is very fundamental to the nature of the world around us. The key point is that when we see things in nature, for example, this sequence of pictures, it's absolutely clear without anybody captioning those pictures in what order they occurred. There's a very clear arrow of time that we expect in all the events that we observe, and for many kinds of events there's just no ambiguity about what order they happened. If anybody told you that this was the first picture and then it evolved to that and then to that and that, you would know not to believe that person because we just never see things happen that way. So among the things we observe, there's a very, very clear arrow of time. On the other hand, if we look at the laws of physics as we know them, they are completely time symmetric. The underlying microscopic laws of physics make no distinction between one direction of time and the opposite direction of time. Now I should clarify here because probably a lot of you know about CP violation. It was discovered in 1964 in the Nobel Prize-winning work of Fitch and Cronin that CP, a symmetry that's the combination of charge conjugation and parity, charge conjugation means to replace every particle by its antiparticle. Parity basically means to reflect things in a mirror. It was discovered in 1964 that CP symmetry is violated in certain kinds of decays of K mesons. In addition, it's known that a symmetry called CPT, where the extra T there is time reversal symmetry, is guaranteed to hold in any Lorenzen variant local quantum field theory. So we interpret this violation of CP symmetry as implying a very tiny violation of T symmetry, at least in the K meson system. So T symmetry, as is defined by particle physicists, is not quite exact. However, the point of view that we are taking is that it's still true that the laws of physics makes literally no distinction between the past and the future. And the reason we say that is that from our point of view CPT, the entire combined symmetry, is also a time reversal operator. It does still mean as long as CPT symmetry is exact, it does mean that there exists for every state a corresponding time reverse state which will evolve along exactly the same trajectory as the original state, but backwards in time. Another way of describing what we're saying here is that in the old days, before 1964 when T was thought to be a good symmetry, it was thought that all you had to do to construct the time reverse of a given state was to reverse all the momenta and all the angular momenta. And then if you let things run, it would just follow the same trajectory as the original trajectory, but in the opposite direction of time. Since 1964, the statement becomes more complicated. Now, in order to construct a time reverse state of some given state, you have to, first of all, reverse all the momenta and all the spins. You still have to do that. But then you have to also replace every particle by its antiparticle and also reflect the whole thing through a mirror so that you're executing this parity symmetry. But if you do all that, the state that you end up with will evolve exactly backwards from the evolution of the original state. So in that sense, time reversal symmetry is still absolutely exact. For every state of nature, there is a corresponding time reverse state that will just evolve backwards along exactly the same trajectory as the original state. So complete time symmetry from that point of view, which is our point of view. I want to introduce this topic by quoting Feynman. I hope you guys like Feynman quotes as much as I do. I think this is a wonderful quote. In the set of lectures called the character of physical law, Feynman has a lecture about the hour of time. And he begins by describing how obvious it is in our everyday lives that there is an hour of time. The future is very different from the past. And then he goes on to say that the most obvious interpretation of this evident distinction between past and future and this irreversibility of all phenomena would be that some laws, some of the motion laws of the atoms are going one way. There should be somewhere in the works some kind of principle that Uxels only make Wuxels and never vice versa. If you're not sure about the English, those are made upwards. They're not in the dictionary. And so the world is turning from Uxley character to Wuxley character all the time. And this one-way business of the interaction of things should be the thing that makes the whole phenomenon of the world seem to go one way. He goes on to say, we have not found this yet. That is in all the laws of physics that we have found so far there's not seem to be any distinction between the past and the future. That's the basic issue that I want to talk about today. Now even though we don't know what really causes the distinction between being past and future that we observe, we do know how to characterize it. And Feynman did go on to describe this. The evolution that determines the hour of time is the evolution of entropy, the evolution of order. That is ordered systems tend to evolve into disordered systems. This basically is Feynman's Uxels and Wuxels. So the world was turning from an ordered state to a disordered state. And we have a quantitative way of measuring this order called entropy. We don't need the technical definition here. But entropy is a measure of disorder. And one way of describing this universal behavior of entropy is what's called the second law of thermodynamics. The entropy never increases and can either stay the same or decrease. I'm sorry I said that backwards. It's written here right. We always say entropy increasing. This order always increases and we never see it decreasing. Under some circumstances it can be more or less constant. But it never goes down. Okay and a classic example to illustrate the behavior of entropy that I think puts things in a way that makes it just common sense understandable is to imagine a gas in a box where we start the gas in the corner of an otherwise evacuated box. And if we let that go starting with the gas in the corner, the gas will rapidly spread to fill the entire box. And that's an example of how this disorder increases, how entropy increases. And then once the box is completely filled with the gas, the gas will come to a state of equilibrium. It will continue to fill the box. You could wait essentially forever and you'll never see it go back to the corner of the box although in principle it might if you waited long enough but it's vastly longer than the age of the universe. But what you will see is that the gas will remain in the state of equilibrium with small random fluctuations about the equilibrium state. That will be the behavior that you would observe. The opposite motion, the return of all the molecules to the corner of the box is possible but it's highly unlikely that the positions and velocities of the particles will ever have exactly the right combinations to bring all the gas particles back to the corner of the box. So what I want to conclude from this is that statistically we do understand how this process works, how disordered systems, how ordered systems tend to become disordered in time. The growth of entropy is something that is a natural feature of thermodynamics as long as you imagine starting the system in a low entropy state like the gas in the corner of the box we understand how the gas spreads out to fill the box. Hold on, a little bit of water here, my voice is drying up. Okay but if this gas in the box is going to be a metaphor for the universe then it immediately leads to the question of what caused the gas to start in the corner of the box? What is it that started that caused the universe to start in a state of relatively low entropy so that the entropy could be increasing ever since? And that's really the aspect of the question that will be the main focus of what I'll be talking about for the rest of today's talk. How did the early universe or why did the early universe end up in this low entropy state? So let me first discuss some history here. Statmeck is a very old subject statistical mechanics and some of the primary work was done by Ludwig Boltzmann and he considered the question of could the low entropy initial state perhaps have just been a rare fluctuation from a huge universe that was in thermal equilibrium? Every now and then there'd be a huge thermal fluctuation because all kinds of things can happen and maybe that explains how our universe arose. Boltzmann wrote a paper about this and he said yes our universe could be just a rare fluctuation from the state of equilibrium. He pointed out that assuming that the universe is great enough and then like now we don't really know how great the universe is the probability that such a small part of it as our world should be in its present state is no longer small. He wrote this in 1895. From a modern point of view Boltzmann was partly right but basically wrong and Feynman talks about this. Feynman says the answer to this question could the low entropy state be a rare fluctuation from equilibrium is just plain no and the reason is that in order for the if the order in the universe were a fluctuation in equilibrium then it would not extend so far as what we observe. Boltzmann was right that we don't know how big the universe is so no matter how improbable it may be for a galaxy like ours to form as a fluctuation it may still be likely that it happened somewhere in a very large universe. But Feynman argued we still know something very clear about relative probabilities which are very important here. So it will be vastly more likely to form one isolated galaxy as a thermal equilibrium surrounded by just thermal equilibrium gas then it would be to form two galaxies and the probability of forming 10 to the 11 galaxies which is what we now observe they didn't know this in 1895 on it but when then we had a large universe even in 1895 the probability of saying 10 to the 11 galaxies all of which formed as just random thermal fluctuations is too small to even contemplate. So because of this issue that small systems are far more likely than large systems and what we see as a large system Boltzmann's suggestion that it can all just be a fluctuation from thermal equilibrium absolutely does not work. So we take Feynman's point of view Boltzmann was wrong our universe cannot be simply a thermal fluctuation in thermal equilibrium. So what's usually accepted these days is that there's some kind of a cosmological solution to this question well one which is not terribly well defined but for lack of any other explanation it's usually assumed that the low energy initial state was fixed by some unknown set of laws of physics that describe the actual creation of the universe and we don't really know how to describe the creation of the universe. And these unknown laws would violate time reversal symmetry because they would apply to the initial state of the universe but obviously not to the final state of the universe. So the idea is that there might be specific laws of physics that we don't know about yet that describe the creation of the universe that would set it into a low entropy state and this is essentially as far as I know the current law that is we don't really have a better theory of this but I will propose to you where I think is a better theory than this but this basically is the current view on this topic. So I've been talking to you for two days about inflation as a phenomena that did not create the universe but controlled a lot of the properties of the very early universe so it's natural to ask can inflation explain the arrow of time and I included here a wonderful diagram of the history of the universe that was made by the WMAP satellite team. WMAP was the satellite before punk that was launched specifically to measure the cosmic microwave background radiation. There's a little picture of the WMAP satellite on the right side of this picture. So everything starts with quantum fluctuations and inflation. It says that after 375,000 years I said 380,000 years yesterday the cosmic background radiation decoupled and it ceased to interact with matter and the photon started traveling on straight lines. There's a period that's called the dark ages until about 400 million years into the history of the universe when the first stars form the development of stars and galaxies and planets proceeds for billions of years. The universe starts to accelerate about five billion years ago and that's the history of the universe. The number they give here is 13.77 billion years for the age of the universe. The number I gave you earlier was 13.82 which is a slightly revised number coming from the punk satellite which doesn't make as cute pictures as this but it's more recent data. Anyway that was just for show. So does inflation explain the arrow of time from very early on in inflation people talked about this question. The first paper that weighed in on it was a paper by Paul Davies. There's Paul Davies and he wrote a paper in as a letter to nature in February 1983 the title of which was inflation and time symmetry in the universe and he wrote in his abstract that the recently proposed inflationary universe scenario explains several of the mysteries of modern cosmology. I argue here that it also provides a natural explanation for the origin of time asymmetry times arrow in the universe. That paper stood unchallenged for a whole several months until Don Page wrote an anti-article also as a letter to nature published in July 1983. The title of his paper is that inflation does not explain time asymmetry and quoting from part of his abstract he wrote that Davies has argued that the inflationary cosmological scenario provides a natural explanation for the time asymmetry of the universe. Here I dispute this argument by noting that the inflationary scenario implicitly invokes time asymmetry with the assumption of the absence of initial spatial correlations to be understood I don't know exactly what that means but never mind. He went on to say that no scenario based on CPT invariant dynamical laws can explain time asymmetry apart from postulating or explaining these special initial conditions as Henry has emphasized. So what Don Page was basically saying is that inflation can't explain the arrow of time because inflation really assumes that an arrow of time has been set up as the inflation starts. As I mentioned I'm not sure exactly what he meant I'm not sure exactly what he meant when he talked about the absence of initial spatial correlations as being crucial for inflation but I do actually agree that he's right basically with that point that is inflation as it had been described up to that point and as it's described today when one talks about how inflation started one is invariably using a language and a set of assumptions that are based on our understanding of things like the second law of thermodynamics that is when we describe how inflation starts we've already made assumptions that break time reversal symmetry. Well I do want to highlight that was Don Page's other next statement the statement that no scenario based on CPT invariant dynamical laws can explain time asymmetry apart from postulating or perhaps explaining these special initial conditions. That statement we definitely disagree with we being Sean Carroll and me and Chen Yang sang and the rest of the talk I'll try to explain to you why we disagree with that statement. We think it is possible to actually develop an arrow of time in a system which has CPT invariant laws and which makes no special assumptions about initial conditions and that's what I will try to explain. So the proposal we sometimes call the spontaneous two-headed arrow of time and as I mentioned at the very beginning it goes back to a paper by Sean Carroll and Jennifer Chen from the 04 here it means it's from 2004 called spontaneous inflation and the origin of the arrow of turn. There's been some related work in the literature about how the arrow of time works in physics that I thought I should cite. Rafael Busso wrote a paper in 2012 rather closely related to what I'll be talking about are actually two papers by Julian Barber and collaborators. What they're talking about is very similar but also differs in some important ways from the version that I'll be describing and Hong Liu and Paolo Glorioso wrote a paper about the dynamics of the second law which is somewhat different from what I'll be talking about but also about it certainly about a related topic and there should be some time in the not-for-distant future a paper by Sean Carroll, Chen Yau Zhang and me although I have to admit we've been very slow about this we've been talking about this paper for I don't know it might be five years and somehow we've never gotten around to actually writing it although both of us have been giving talks about it. The key idea is actually very very simple and can be summarized in one sentence and then I'll explain how to put flesh on the sentence but the one sentence is that if the maximum possible entropy of the universe is infinite we will assume it is then any state of finite entropy is a state of low entropy obviously because it's small compared to infinity and that really is the secret. Now I should say here that we don't really know whether the physical system that's called the universe or the multiverse is described by a system that could have an infinite entropy. I think for all we know the maximum possible entropy could be finite or it could be infinite. We're going to hear be assuming that's infinite and show how that leads to an hour of time. Later I'll talk about the alternative possibility that the maximum possible entropy is finite and I will argue that that leads to very significant cosmological problems. So on the grounds of cosmology not on grounds of fundamental physics but on grounds of cosmology it looks like the idea of an infinite maximum entropy it's much much better with what we observe. So here goes. Okay the important point is that if the maximum possible entropy is infinite the entropy can increase from any given starting point and then if we go back to this metaphor I've been talking about of the gas in the box the gas in the box is a finite maximum entropy but if you take away the walls and let the gas expand throughout all space then there is no upper limit to the entropy and that becomes our new metaphor for the universe gas without a box and that in fact is exactly what the toy model that I'm going to tell you about is a gas without a box. What I'm going to show you here is a toy model and by a toy model I mean a mechanical model that has some of the basic properties that we think the universe does but which is very simple and by no means represents the universe but which nonetheless shows that a physical system with these properties can achieve what I've been claiming can achieve the creation of an arrow of time starting with fundamental laws that are completely time symmetric and starting from an initial state which is arbitrary. So I just said that. So here's the model we're just going to consider a gas of n some large number of non-interacting particles just moving in empty infinite space so these particles just move on straight lines at constant velocities. I want to describe how we're going to imagine setting up our initial conditions. We're going to choose our initial conditions by making up a probability distribution for the positions and the velocities and I claim that whatever we make up will still allow us to show what we want to show so it's completely arbitrary what probability distribution is used for positions and what probability distribution is used for the velocities and then we imagine using a random number generator to turn these probabilities into a particular choice of initial positions and velocities for the n particles and then we just let it evolve where the evolution is just the particles move at constant velocities. So there is an important point here which some may think is obvious some may think is controversial we think it's obvious that is we're going to insist that the probabilities be normalizable that means you have to be able to add up the probabilities to some finite number that you can then reweight things and make that number one probability should add up to one. Now note that this rules out what we consider to be ill-defined options such as having a uniform probability to be anywhere in space. There's no way you could have a uniform probability everywhere in space and have the probabilities add up to one because whatever the probability is in one little cube it's the same as all others and there's an infinite number of those cubes. So we want to claim that a uniform probability throughout space is simply not logically possible. So to argue that a uniform probability is not logically possible I'd like to show you a logical contradiction that you get into if you try to believe that it is possible. Mathematicians always insist that probability distributions be normalizable as part of the axioms of probability theory but we want to argue that it's not just an axiom that's convenient for mathematicians but it really is necessary for logical consistency. So to illustrate that point I'm going to consider a thought experiment. It'll be kind of approved by contradiction. We'll assume that it is possible to have a probability distribution that's uniform one dimension isn't enough. We'll assume that it is possible to have a probability distribution that's uniform on the real line and I'll show you that you immediately get contradictions. So the current tradition comes about by imagining that we had a random number generator that used this probability distribution generating numbers that are equally likely to be any place on the real line and we're going to use that random number generator to generate two numbers A and B in succession and then we want to ask what is the probability that the magnitude of B is larger than the magnitude of A. That's the question we want to ask and what I want to show you is that there are two ways one can answer that question each of which are completely logical but which get totally different answers and I claim that the only place where the error can come from is the original assumption that this probability distribution makes any sense at all. So here goes. The first argument simply says that A and B should be equally likely to have to occur in either order if you have a random number generator is the definition of a random number generator is that each generation of a random number is independent of what happened before. So the numbers A and B if they can be generated in the order of AB they can equally likely be generated in the order of BA and if that's true if they equally likely to have been generated in either order the probability that B has a larger magnitude than A has to be a half because if they're generated in the opposite order the statement would reverse between being true or false. So that's one of the two answers. On the other hand you can ask yourself how would I look at this if A was generated first and I asked after A was generated but before B was generated what would be the probability that B would have a larger magnitude then once you know what A is the only way that B can have a smaller magnitude is if it's somewhere between minus A and A and that's a finite part of the real line. On the other hand if B ends up anywhere else on the real line and there's an infinite amount of everywhere else then B will have a larger magnitude than A. So from this point of view the probability that B has a larger magnitude than A is one because this infinite segment has infinitely more probability than the finite segment. So it can either show that the probability is a half or that the probability is one and I think that nothing's wrong with either of these arguments except the initial assumption that makes sense in the first place to talk about a probability distribution evenly spaced on the real line. So we claim that contradictions like this show that a uniform probability distribution is simply not conceivable not logical at all and I might just add to give credit that I learned this particular paradox from Aaron Wall. There are other paradoxes as well but this is the simplest I think and most elegant I like it. Okay going back now to our toy model we're going to insist that the probabilities be normalizable as we just said and elaborated on. Normalizability in some sense implies that the distribution is localizable in some sense and I'll explain exactly how. If the probability distribution space is normalizable I can consider spheres centered around the origin of my coordinate system and the origin of my coordinate system doesn't have to have anything to do with where the particles are actually most likely to be. It doesn't matter but if I start with spheres centered around the origin of my coordinate system and make them bigger and bigger and bigger the definition of a normalizable probability distribution is that as those spheres become infinitely big the probability that all the particles are inside will approach one that will be the sum of all probabilities which has to be one and if it's going to approach one it means if I make the sphere big enough I can arrange for 99.99 percent of the particles to be inside the sphere while the sphere is still finite. So that's what I want to do. I want to talk about what I'll call the 99.99 percent sphere which is the sphere which has the probability yeah it's the sphere which has the probability that all n particles are inside the sphere is 99.99 percent is what I'm saying. Some number arbitrarily close to one. Okay so for all practical purposes we can think that we have the sphere and all the particles are inside it with very high probability and now what I want to do is let that system evolve. Initially we could draw our 99.99 percent sphere with essentially all the particles inside it. Particles are moving all random directions it certainly is no obvious hour of time and we really did construct this so that there is no hour of time. I might just add that in order to be really sure that we have not produced an hour of time in the way we constructed our initial state we can insist the only thing we need to insist on is that the probability of any velocity v is equal to the probability of the corresponding minus v and that guarantees that the whole thing is is time symmetric. So we construct our initial state and notice that it has no hour of time but then we ask what is this going to look like some very far some very late time in the future. So we want to fast forward to a much later time and the way we're going to choose the time that we're going to use is to use the fact that the particles losses do not change with time so we're going to choose a time which has the property that if we take an average magnitude of the velocity and multiply it by that time we'll get a radius which is very large compared to the original radius of this 99.99 percent sphere. So in other words if we went long enough we're guaranteed that all the particles will be far away from the sphere itself. The average distance will be very large compared to the size of the original sphere and then if we want to draw a picture of that the original 99.99 percent sphere will be a small dot in the center of the picture and all the particles would be moving outward from it. All the particles started inside this little sphere at time zero so they're all necessarily moving outward and in fact we even know to good accuracy what the velocity of each particle will be in terms of where it is it's just the velocity it needs to get from the dot to the present location in the time t that is. So all the particles will be moving outward and one has what you might consider a Hubble distribution of velocities like in cosmology where the velocity of each particle is proportional to the distance from the dot at the center. So I claim first of all visually this clearly has an hour of time all the particles are moving outward the time reverse would have all the particles moving inward. This picture is very different from its time reverse and that means that it has a very distinct arrow of time. Earlier I told you that time is related to entropy and I'm not going to go try to go through the details here but what I can define what's called a coarse-grained entropy which does grow indefinitely as this gas expands so it does have an increasing entropy just like it ought to. I might mention one fine point here you'll notice that I wrote coarse-grained entropy. It turns out that if one takes the statistical mechanics definition of entropy and asks how does it change with time the answer technically is that doesn't change it with time at all it's a constant and that's a consequence of what's called Neuville's theorem of classical mechanics. When we say that entropy increases what we're really doing is talking about something which is more technically called coarse-grained entropy which is why I put that here. The distinction let's see how do I describe it. We have to think of phase space which is a space where what has six dimensions for every particle if we're describing particles the three positions and the three momenta three components of the position vector three components of the momentum vector are put together to make one six dimensional space for one particle and if we have many particles we get six times the number of particles we just put together all those numbers and think of them as one big space. The statistical mechanical system is described as some kind of a blob in that space and what Neuville's theorem tells us is that when that blob evolves its volume never changes and the volume basically is what's the entropy is defined in terms of but what happens as the system evolves is that this blob in phase space gets chopped up into many kinds many tiny threads which more or less fill regions but only fills them in the sense that there are threads near every point but the threads don't actually fill the space and to see entropy increase one has to do this operation called coarse-graining which corresponds basically to blurring your vision saying that there's a maximum resolution that I have and that if the distance between these threads is smaller than what I can resolve it will look like these threads are really filling the volume and not just having a lot of threads running through the volume and it's only after your coarse-grained that you actually see this rise of entropy which is so crucial to our understanding of what the world looks like. If you didn't follow that don't worry about it but I thought I would say it. In any case the entropy of this system does rise in the very same in the very same sense that we talk about entropy rising in any statistical mechanical system. So we've achieved our goal we've gotten an arrow of time from a system which has completely time irreversible laws of evolution and which starts from a state which had no arrow of time. So as I said if we evolve the system forward in time entropy will start to grow approaching its maximum value of infinity means it means it just keeps growing forever and we have an arrow of time so we started with a state with no arrow of time that's what the question mark means and developed a state with a well-defined arrow of time. Now the interesting thing and this is where the word two-headed that I mentioned earlier comes from we could ask what would happen if we take that same state that we selected by running our random number generator and ran that state backwards in time. The equations of evolution are perfectly simple things in this case they just move with constant velocities so we could follow them backwards on their velocities and ask where these particles would have been at earlier times. And what you see is that for earlier times the story is exactly the same as the story we just told. So if we go back to very early times the entropy will start to get large in this direction for exactly the same for exactly the same reason the particles will spread out further we go back in time the more spread out the particles will be in space. So the bottom line is that we end up with this two-headed arrow of time system where there's a time period in the middle where the arrow of time is undefined but that period is only a finite duration of time but then if the system goes on forever there'll be an incident period of time in the future where the arrow of time is well defined and entropy will be growing and an infinite amount of time in the past where the arrow of time is well defined where entropy will be growing toward the past. So the arrow of time will point the opposite direction but there will be a well-defined arrow of time. I should maybe point out that for the folks who are living at the bottom here they have no way of knowing that the arrow of time is pointing backwards from your point of view to them earlier versions of further down on this diagram will correspond to the future. Their existence will seem perfectly normal to them. To them the past is going up on this diagram and the future is going down but they don't know anything about my drawing this diagram and there's nothing that they can perceive in their own world which is any different from what we perceive. We simply perceive that there's a direction of time in which the entropy is larger in a direction in time in which the entropy is smaller and the direction in which the entropy is larger is what we call the future. Okay now of course this ever-expanding gas that we've been talking about in our time model is intended as a metaphor for the real universe and I have in mind a real universe which is undergoing eternal inflation. So instead of this gas moving outward with particles moving with constant velocities I want to imagine an eternally inflating universe which is exponentially expanding and producing more and more pocket universes as time goes on and I claim that the story fits that the eternally inflating universe can act in exactly the same way as this toy model and again the eternal inflation simply means that in many inflationary models once inflation starts it never completely stops. It stops in places and where it stops it produces what we call a pocket universe but it's always continuing in other places. So I'm going to switch gears a little bit now and talk about some related topics but so let me summarize what I've said so far which is really the key parts concerning the hour of time. I've tried to show that time symmetric laws of evolution with time symmetric initial conditions can nonetheless produce an hour of time. A key requirement for this to work is that the system has to have an infinite maximum possible entropy and the hour of time that gets produced is two-headed pointing to the future and the future and to the past in the past. Let me point out while we still have this diagram on the slide if we take this diagram and turn the whole diagram upside down you'll notice that nothing changes. The arrow is still pointing the same direction as they used to point. So part and parcel of this viewpoint is that there is no global breaking of time symmetry. The global system is exactly time symmetric but nonetheless if you're living anywhere up here you would see a very clear hour of time. If you're living anywhere down here you would see a very clear hour of time only if you were living somewhere in this finite region in the middle would the hour of time be confused and since this is only a finite extent in time and the others are infinite if you would just plump down in a random place in this picture you would see an hour of time. The finite interval in the middle has zero measure on the whole space. Okay um I want I want yeah but it's the microphone not working is that what? Possible I didn't turn it on? I thought we couldn't turn it on. It confused us. In any finite time evolution of the universe this argument would say that you can I mean if there is a if the system evolves only for a finite time and if I just reverse suppose the gas has expanded to some if I reverse everything and start that as the initial state there isn't another state where entropy is decreasing right and the so hour of time is the universe's finite has has been around only for finite time right so how how does this explain the hour of time in our universe? Well it suggests that the universe has not been around for a finite time I mean I want to I want us to believe that this is the picture of the entire multiverse and I'm claiming it has under this picture is right it's been around for an infinite period of time and we would be living someplace up there or down there. Oh I see how you really want to you want to claim that they want to believe that this is the right picture. Okay we certainly don't know I mean we have no idea what we would see if we trace the universe back before our big bang. Okay what Fernando is asking about is what about the famous theory that I wrote with Valenkin and Borda which said that inflation cannot be eternal into the past. This is all consistent with that I would not tell you something that violated my own theorem that would be pretty stupid of me but it's worth explaining why it does not violate my theorem. My theorem said that if you follow inflation backwards you cannot have a situation where the Hubble expansion rate exceeds a certain value arbitrarily far back in the past but what happens here is you cross this border and then the Hubble expansion rates turns around and has the opposite sign so it does not violate the theorem because you do not have uniform expansion as the person in the future would define it that was arbitrarily far back. So this is completely consistent with Borda, Guth, Valenkin theorem. Okay so I want to change gears now and talk about two related issues related in the sense that they talk about phase space and cosmic evolution and which are crucially related to the underpinning of what kinds of things can happen or cannot happen in cosmology. So the first thing I want to deal with is the question of can fine-tuning be explained in the context of Hamiltonian evolution and by Hamiltonian evolution I mean the evolution of a system that obeys Hamilton's equations which as you know are one way of writing the equations of mechanics. The very form of Hamilton's equations though do restrict the kinds of mechanics that can happen but we do believe that all the physics that we know of can be described as Hamiltonian evolution and I want to assume that everything in the universe can be described by Hamiltonian evolution and ask whether this what the properties of that statement are. So one issue is whether or not fine-tuning can occur and in fact there have been strong claims made by Roger Penrose that inflation can't really work because system that underlies inflation is a Hamiltonian system and Penrose claims that as long as the system is a Hamiltonian system there's no way you could ever explain fine-tuning and Penrose's argument is basically the argument which I think is valid for finite phase spaces. For a finite phase space we have this theorem by Leeuville going way back to the 1800s whatever it was that says that if you follow the evolution of a system in phase space phase space being this space that combines momentum and position that the volume that describes the uncertainty about that system will never change. So evolution just means moving around blobs in phase space and the assumption for finite systems is always that the probability of a system having a given form just depends on the volume in phase space that has that form. So Penrose argues that can never change. Evolution can move this blob around in phase space but it always has the same volume in the same density and if something is improbable because it only corresponds to a tiny fraction of phase space it will always correspond to a tiny fraction of phase space no matter what kind of evolution you've undergone. So that argument has been raised very clearly by Penrose and perhaps others but I claim it's clearly wrong if the available phase space is infinite which is what we what is the case for classical models of inflation and I'm suggesting that we have pretty strong evidence that for the full system that describes the actual universe I think we have good reason to assume that the phase space is infinite in any case I'm going to assume it and and discuss it in those terms. So I claim that the system has an infinite phase space then it's clear that Hamiltonian systems can fine-tune parameters and there's a simple example of it just to show that it can happen we could let the Hamiltonian be equal to minus p times q. This is a system that just has one degree of freedom you can think of as a particle moving on a line where q is the position along the line and p is the momentum and the standard Hamilton's equations say that the time derivative of p is equal to minus the partial of the Hamiltonian with respect to q so in this case that's p and the time derivative of q is equal to the partial of the Hamiltonian with respect to p which is minus q and I think we all know how to integrate these equations in our head q decreases exponentially being fine-tuned to zero that is no matter where q starts if this is the evolution the evolution will drive q to the value of zero at the same time p gets larger and larger exponentially and the fact that one who knows while the other falls is really a requirement of newville's theorem while q is being fine-tuned to zero p is being spread all over the map and that's essential to keep this phase space volume fixed it also makes it clear that you can only do this if you have an infinite phase space if p could not get arbitrarily large it would not be possible for q to get arbitrarily small I should add in case that people didn't notice that this already has an infinite phase space because p and q can have an infinite range of values even though it only corresponds to one particle moving on a line it's not hard to get an infinite phase space now if you wanted to modify this example you could easily change it to fine-tune p instead of q and by doing things a little bit more in a slightly more complicated way you could fine-tune any linear any combination of p and q that you want so this is just intended as a simple example now when I some years ago I got involved in an email shootout on on this question which happens sometimes among physicists and physicists are very stubborn so when I pointed out this example some of the one of the people on the email chain said well maybe that works for your Hamiltonian but it probably doesn't work for real Hamiltonians because real Hamiltonians always have a lower limit to the energy energy cannot become arbitrarily small and the formula I just gave you minus p times q has no lower limit if you allow p and q to both get large and positive the Hamiltonian which you interpret as the energy can become arbitrarily negative so they said maybe they didn't say maybe they said it only works because you use the Hamiltonian that was not bounded from below so I scratched my head a bit and after scratching my head a little bit I asked my son Larry so I'm fortunate enough to have a son Larry Goose who's a fabulous mathematician he's now in the faculty in the math department at MIT and he immediately came back and said well why don't you just put an inverse tangent in front of your minus pq and that that makes it bounded from below it doesn't do much to change the equations of motion the equations of motion using this Hamiltonian are these you get a p squared q squared plus one in the denominator p squared q squared plus one looks a little complicated but actually it's not because the Hamiltonian itself is conserved by Hamiltonian evolution so h is constant as the system evolves and that means that p times q is constant as the system evolves and that means that p squared q squared plus one is a constant as the system evolves so it just puts a constant into the problem it's still true that p is going to grow exponentially and q is going to shrink exponentially with time giving exactly the same qualitative results as we had in the other case um one question one might want to ask is can one quantify how good the fine tuning is uh and uh i attempted to do that by using the epsilon delta kind of thing that we learned about when we learned about limits uh what i claimed as you could actually prove it's not hard to prove uh that basically it works as well as you want uh as long as you're willing to wait in arbitrary long time that's that you don't have control over but you can decide uh how small you want to insist that q will end up uh so we could choose an epsilon and insist that the magnitude of q when we're finished should be less than epsilon um now we imagine starting with some probability distribution in pq space in phase space uh so we'll always end up with probabilities you can always arrange to have some probability that q is larger than any amount you might choose but we can specify a probability delta so that uh we will end up with the magnitude of q being less than epsilon with probability of one minus delta so we're going to think of both epsilon and delta as small numbers that are free for you to choose you could choose how small you want to make q and you can choose how large the probability should be that q actually ends up in the desired range and then you can show uh that if you wait long enough you can always arrange uh so that the probability that p that q is less than epsilon uh is bigger than one minus delta that is you've achieved the goal of making q as small as you want with the probability as high as you want uh and the only restrictions that you have to be willing to wait the amount of time that depends on the nature of the Hamiltonian uh but you can always make the fine tuning as strong as you want okay now i guess i want to talk about one more topic and then we'll stop uh and this is the topic which i mentioned yesterday as well but uh it's important in this context as well and i think i'll i guess we're talking about again it's a question of considering the alternative to the hypothesis that i've been making throughout the talk i've been assuming throughout the talk that the universe is a system which has no maximum entropy the maximum entropy is infinite there's no limit but now i want to consider the alternative to show you what problems you get into if you imagine the alternative uh so suppose uh instead uh the maximum possible entropy of the universe uh was finite uh suppose uh that reality could be described by some quantum system with the maximum possible entropy uh then if you let that system run it will be like the gas in the box as a prototypical case uh if you let that system evolve it will approach equilibrium to get closer and closer to equilibrium uh and in the equilibrium situation uh the gas in the box or the system that describes our universe uh will undergo what are called plurae recurrences that is if you wait some phenomenally long period of time the point in phase space that describes the actual universe will meander around and come back to at least be arbitrarily close uh to its initial position um and in the course of that all microstates will occur with equal probabilities uh all microscopes that are available given the energy and any other conserved quantities uh so what i want to argue is that this is grossly in contradiction with what we observe for the universe uh even if we only think of this happening sometime in the far future uh the key point is that even if this thermo-equilibrium system only occurs in the far future it's important to understand that life will not actually terminate once the universe reaches the thermo-equilibrium phase uh one might think that life can't exist in a state of thermo-equilibrium and i think many people believe that um but what's been emphasized recently in some quarters of cosmology is this concept of what's called the Boltzmann brain uh the idea is that in thermo-equilibrium essentially anything can happen uh we know how to calculate the probability of anything to happen and the the important factor in all these calculations is what's called the Boltzmann factor it's e to the minus the energy of the thing you're trying to produce uh divided there's the exponent is divided by Boltzmann's constant times temperature uh so things that have a large mass are very unlikely to materialize as a thermo fluctuation but anything can materialize uh it's just that the probability gets incredibly small uh if the mass is large so if we imagine this thermo-equilibrium phase of the universe if we're talking about the universe behaving this way in the thermo-equilibrium phase of the universe which we imagine is in the far future someplace um it will still be possible for anything to materialize and that includes a brain b r a i n brain uh which might be an exact copy of my brain or your brain or your friend's brain and by exact copy I mean it's possible not probable but nonetheless possible uh that a particular brain will have exactly all of your memories and all of your thought processes and would really be absolutely indistinguishable from your own brain uh now the appearance of such a brain would be enormously improbable in any finite space-time volume but nonetheless if our universe reaches a thermo-equilibrium it will stay there forever and will have an infinite period in thermo-equilibrium and infinity is a big number uh so infinity times a very small probability is still infinity uh infinity times any small number is still infinity so there'll be uh therefore an infinite number of brains that will materialize in this thermo-equilibrium phase that will exactly like my brain or your brain or your friend's brain um and then if you ask well who cares we're here and they're there so why do we care but the point is that we don't really know we're here uh what we know uh is the memories that we have in our brains uh so when we ask what do we expect to see next uh the proper way to ask that question would be as a conditional probability in the technical sense the proper way to ask the question would be given what i know about my memories and and thought capabilities given those things what do i expect to see next and the point is that if there are infinite numbers of brains that have exactly those memories and thought processes and only one that existed during the approach to equilibrium phase then the probability is dominated totally by the Boltzmann brains which would outnumber us infinitely um and the conclusion would be that we would with probability one be a Boltzmann brain uh and it would mean that within the next second we would probably see the whole world around us dissolve um because the whole world never existed for these Boltzmann brains they just have these memories uh ingrained as just a purely random choice in a thermal fluctuation uh the world that the Boltzmann brain thought it remembered uh never existed so in a a tiny fraction of a second the brain would discover that it was really just surrounded by a thermal equilibrium gas of some kind uh and of course we don't see that uh so the conclusion uh is that we must not be living in a universe that's dominated by these Boltzmann brains and therefore I would say we must not be living in a universe which reaches a thermal equilibrium this is a related thought um if the another way of seeing that we don't live in a world that's controlled by thermal equilibrium is uh was pointed out in a paper uh by Dyson Kleben and Suskind in 2002 uh they were basically comparing the historical view of understanding our world to the thermal equilibrium view of trying to understand our world uh historically everything depends on what's happened in the past and that's the way we normally look at things we think of our universe as being a product of the Big Bang and we think of our lives as being a product of the choices we've made since we were born um but in thermal equilibrium probabilities are completely determined by just counting states uh any macroscopic description that corresponds to a large number of microscopic states has a high probability uh any macroscopic description that corresponds to a low number of microscopic states has a very small probability uh and what uh Dyson Kleben and Suskind pointed out uh is that if our world really were governed by state counting uh then we can imagine a state which looks exactly like the world that we see uh except that the cosmic microwave background uh has a temperature of 10 degrees say is the number they use instead of the 2.7 degrees uh that we actually measure uh their point is that this state would have vastly more micro states that would correspond to it and therefore be vastly more probable uh than the 2.7 degree state that we actually see and this is another another way of arguing uh that our world cannot be governed by thermal equilibrium statistics um and finally I want to point out that eternal inflation gets around these problems uh as we wanted it to it does act like the gas expanding into an infinite space and eternal inflation more and more of these pocket universes are constantly being created and that means that new regions of this infinite space space are constantly being explored uh there never are Poincare recurrences in the internally inflating system uh and that means that if we are living in an internally inflating universe we have good reason to trust our historical basis uh for describing what we expect to see and I think that basically finishes what I want to talk about let me just summarize what I tried to tell you uh I tried to tell you first of all uh that it is indeed possible for time symmetric laws of evolution with a time symmetric initial conditions uh to still lead to an hour of time and which entropy is uniformly growing throughout the space that was the main point really of what I tried to describe in today's lecture uh and then secondly I claim to have shown that Hamiltonian evolution uh as long as the available phase space is infinite uh can lead to fine-tuning of dynamical variables uh which is what happens during inflation and finally I claim to have shown that if the universe is described by underlying physics which has only a finite available phase space then there would be a maximum possible entropy and I claim that that leads to a situation uh which is just untenable uh as a possible cosmology so from the cosmological point of view I claim that the notion of an infinite phase space with an infinite maximum possible entropy uh seems to be exactly what we need to describe our universe and I'll stop there thank you thank you very much in the island questions if you have finite available phase space and we have evolved only for finite time why would we reach thermal equilibrium why would we reach thermal equilibrium well um okay if you if you think there's something that's going to stop the universe in some finite time then I agree you can avoid this thermal equilibrium phase uh but Hamiltonian evolution does not do that right if you have something the evolution of a system is governed by a Hamiltonian then it does go on forever um and you do have this infinite thermal equilibrium phase and my claim was that if you if the evolution of the universe that's the part that you like uh is followed by an infinite period of thermal equilibrium uh then you do get into trouble because if you ask the conditional probability statement what would it be with my memories and my thought processes predicts should happen next uh you'd be predicting that you should be a Boltzmann brain and should see quickly that you're actually just a brain surrounded by a thermal equilibrium gas future right I mean I suppose like if you imagine that the universe has expanded only for 13 billion years I don't care about what happens in zillion years later right well an important difference between the way you're thinking and the way I'm thinking uh is that uh I believe that when we ask what should happen next we should only make use of information we actually know as the conditions for this conditional probability statement and we don't know that we're living in the first 30 billion years of the history of the universe all of the Boltzmann brains that are just like mine that are going to be living in the thermal equilibrium phase will think the same thing they will also think uh that they're living in the first 30 billion years of the history of the universe so if we believe that when we ask conditional probability statements like that we can only take into account what we actually know then all the Boltzmann brains count as contributing to this conditional probability thank you for a very thought-provoking lecture so I'm not convinced by your proof that uniform distribution is not acceptable okay yeah your second argument that if you choose an A then the probability that B will be smaller than A is equal to zero it's right but of course the probability that A is finite is also zero to start with no no no I mean the probability that A is smaller than any finite number is zero okay that's true but who cares then A must be infinite no there are no infinite numbers they're just numbers to go from they extend from minus infinity to infinity but with the curly brackets and not closed brackets minus infinity is not a number what I mean is no my argument is I mean if you draw a number from a uniform distribution on the infinite line yep then um you can show that any uh this number will be bigger than any number in absolute value that just prespecify and so it's infinite well infinity is not an option a random number generator has to generate a number from the set which is supposed to be defining a random probability distribution on the random number generator cannot generate infinity that's not that's not allowed it has to generate some number for this to I mean that's that's what a random number generator is I agree it's possible to build a random number generator that generates infinity and minus infinity with equal probabilities but that's just a binary random number generator that's not a random number generator that generates numbers that are uniform on the real line so do you think there's a distinction between your infinity minus infinity two option random number generator and a random number generator that generates numbers randomly on the real line um maybe i'm not being clear i'm sorry um but yeah so so what I mean is that there is an infinite there is a mass equal to one at plus infinity minus infinity when you take this uniform distribution well so all the numbers that you draw are infinite by definition well okay well I guess we have a different idea of well infinity I mean you can define it you can take a finite box and uniform distribution of finite box and then uh draw from finite box and then let that box go to infinity and then uh you will in this case I mean there is no paradox well I think the limit doesn't exist so uh I guess we need to agree on what we mean by a random number generator that can generate a number uh throughout the real line with equal probability I certainly agree with you that we could build a random number generator that either spits out plus infinity or minus infinity but that's not what I'm talking about when I talk about a random number generator that generates numbers evenly spaced on the real line a real number there's a set of real numbers does not include infinity or minus infinity I think I think there's a well-defined meaning to the set of real numbers and it does not include those so a random number generator that I'm talking about has to generate one of those a real number and I think you agree that you're not talking about that kind of random number generator maybe we can continue okay we can continue later probably there's another question about this over coffee later another question here so it's about the same issue uh as far as I understood the second argument uh in the second argument you compare the size of the segments basically from zero to modulus of a with the size of from modulus of a to the infinity right but if we're talking about the real line in principle both segments are isomorphic so you have equal numbers of numbers or points that you're disposed of because r is dense so I wouldn't say at least I don't say it that way that both segments are really different well I think it's important that isomorphic is not what's relevant what would be relevant is isometric and they're not isometric that is the probability distribution that we're trying to show does not exist is the probability distribution which has equal probabilities for every length in the world on the real line uh if you don't care about lengths and just say I'm going to talk about mapping points into points arbitrarily then what you're saying is right and you can construct a normalizable distribution uh doing what following what you're saying but that's not what we're talking about we are talking about a definite metric on the real line which says equal probabilities for equal intervals and that's what I was claiming is not possible and I still claim it's not possible okay there is another question this right um all right uh I think there's something fundamental that I'm still not understanding about what prevents us from reaching thermal equilibrium in the future you know especially because you know with uh lambda you know with dark energy eventually we will have um um gravitationally disconnected regions and so we would effectively have a finite regions of space that could reach thermal equilibrium so that's true okay so that's a clarification so all that we're saying is that we don't leave in a thermal equilibrium world right now and but but in the future we could reach such a state uh if um dark energy continues and basically uh regions of the universe become isolated one from the other well what I'm saying is that if all we had was our pocket universe and it reached thermal equilibrium uh that that would be an impossible cosmology a cosmology that would predict things uh that are absurd because under those circumstances uh there would be for every brain that's exactly like yours or mine an infinite number of copies in the thermal equilibrium phase that would outweigh us enormously and as long as we believe that the right way to ask the question of what do I expect to happen next is to think of it as a conditional probability question uh given what I know what do I expect to happen next that condition given what I know would include all the Boltzmann brains and you'd predict that you should see what a Boltzmann brain would say which would be very different for what we actually see uh so uh what I just described about what would happen if we only had our pocket universe is very different from what happens in the internally inflating universe when more and more new young universes are constantly being created as long as more and more new young universes are constantly being created then you can arrange uh for the normal brains brains that really did evolve historically from a big bang to outnumber the Boltzmann brains and I claim that that's what's necessary uh for the world that we see to be the world that we predict okay but the new worlds would not uh would not affect um basically some regions of space that are I don't know some virilized region that in the far distant future will not be able to see um all the other regions of the universe so suppose that you were born in such a distant future supposing that still we would have galaxies and systems in which we would exist but in which you could not see anything beyond uh where you're gravitationally bound so uh what would prevent you from reaching thermal equilibrium in in that case given the fact that as you said uh if by inflation you will create new universes uh the universe in which your universe new universes will create um do not do not basically interact and are not able to uh to communicate to one with the other or perceive the the presence of each other okay now that's an important point and it's it is controversial there certainly are a number of physicists who would agree with the thrust of your question um the way I look at it uh it doesn't matter whether these different okay let me sorry let me start that sentence again um uh when we asked the question of what do we expect to see next uh I insist that that's a conditional probability question um what we're saying is given what I know what do I expect to happen next and I believe that to ask that question in a valid way you should be rigorous in distinguishing what you actually know uh and I claim that we don't know that we live in the universe that's 13.82 billion years so we've learned that uh so we know that our brains have those memories of being told that the universe is 13.82 billion years and being told that people have done measurements of the cosmic microwave background radiation and this and that uh but all we know is that we have those memories in our brain we don't know that those are the reality uh so if you accept that uh then when you ask this conditional probability question given what I know what should I predict will happen next uh you should be averaging over all the brains that ever exist that have exactly those memories and I claim that in under some circumstances most of those brains actually a fraction one of those brains uh would be Boltzmann brains brains that only form disthermal fluctuations and only have those memories as pure coincidental chance uh and that if you're asking your conditional probability question uh the answer then is totally dominated by the Boltzmann brains even though they might be causally disconnected from us and may not affect us but by us I now mean the the one of us that's real uh even if the Boltzmann brains don't affect the one of us that's real in any way even if there's no way that light signals could be sent from one to the other it doesn't really matter uh I I believe for asking this probability question uh and therefore it's important that when we consider the whole multiverse uh we can get out of this problem and the way the problem is can be evaded is to uh make sure the physics has the property that uh real uh normal brains let me call them and a normal brain is one that evolves from the big bang uh you have to arrange the physics of the multiverse so that normal brains outnumber the Boltzmann brains hopefully by a lot and there are ways to do this although it doesn't involve some technical issues that are not completely settled another question well uh used in the second point it is said that you claim that the Hamiltonian evolution can lead to fine-tuning of dynamical variables and you gave an example right but I think that the sample you gave is not fair because you selected the Hamiltonian that is not even on the time reversal so it it gives a question of motion that are not symmetric on the time reversal that time internal that you gave both let me think um it's just a just a remark yes yes I don't think the remark um I don't think the remark actually is quite true um when you write down an arbitrary Hamiltonian if you have a if you have a system of well-defined sort of standard variables we have particles and velocities and so on uh then you you know what time reversal should look like uh but in this case I'm really thinking of this as an abstract Hamiltonian where we could have made a canonical transformation from the original variables and in fact secretly I didn't tell you uh the Hamiltonian I showed you really is a canonical transformation of an upside down harmonic oscillator um and an upside down harmonic oscillator is time reversal and variant uh so I think if one wants to allow for canonical transformations one has to allow a more general definition of time reversal and variants and in particular one should allow undoing this canonical transformation and getting back the upside down harmonic oscillator which manifestly is time reversal and variant so I don't think that's really an obstacle um maybe I should re-explain it in terms of the upside down harmonic oscillator and its original coordinates and an upside down harmonic oscillator and that just means I mean it should mean a particle that's moving in a potential energy function which is minus k x squared uh the particle will roll down the hill uh and the fine tuning is that if it rolls very far down the hill the variable p minus q with appropriate constants in front of each will get fine tuned to zero that is the momentum will become almost exactly what it needs to be for a given q uh and the Hamiltonian I gave you was really just a canonical transformation where this p is the old p plus q and this q is the old p minus q uh and one of those gets fine tuned p minus q gets fine tuned if you do the upside down harmonic oscillator but you shouldn't write down the equations when you get home and send me an email if it's great very good so let me ask you something myself um so just say at the beginning that is uh this hour of time issue has been people have been thinking about it for a couple of centuries or so and it hasn't been solved so you will you agree that uh so the issue about infinite entropy can be something that people will have thought in the past but the real new ingredient is that you have to use the multiverse to make the whole thing sense because I mean thinking about uh in the past people could have thought of infinite entropy something that people could have thought of 50 years ago or something what is the multiverse that the new ingredient that makes the whole thing makes sense is that is that a fair argument I think that's right I mean I think what my toy model shows is that you don't need eternal inflation to develop a system with entropy grows indefinitely but I think you're right if you want to not only have the entropy grow indefinitely but to also have life continue indefinitely then you probably need eternal inflation if you get away with that and just had a conventional big bang with no inflation I think it might still work with life only living for a finite amount of time but if you well it would work if you had zero cosmological constant because then the far future would be just empty Minkowski space and not a thermo equilibrium state. One thing which I didn't mention in today's talk which is crucial to the answer I just gave is that the vacuum energy of the universe also known as the cosmological constant as a peculiar thermal property and the equilibrium space of space with the positive cosmological constant is a space which goes on exponentially expanding forever it's called the sitter space and it was shown back in the 1980s by Gibbons and Hawking that when quantum mechanics has taken into account a decider space a permanently exponentially expanding space actually has a finite temperature the temperature being equal to the Hubble expansion rate divided by two pi it's an effect that's similar to black hole radiation that comes about because of the changing of the metric with time so if you ended up in a decider space from my point of view you'd be in trouble because of the Boltzmann-Rein problem but if you ended up in a space with zero cosmological constant then you'd be approaching thermo equilibrium at zero temperature where these problems would not arise. Very good, very good. Just a minor point at the end. You mentioned yesterday the major problem for the multiverse is the the measure problem. Yes. Will that affect any of this discussion you had? Yeah, it certainly affects at least the answers to some questions I've given. I've tried to say things vaguely to not not tread on the measure problem but what I've been a conclusion from what I've been discussing is that to build a viable model of the multiverse you need to arrange for normal brains to outnumber Boltzmann brains and that means counting and as I mentioned yesterday counting in the multiverse involves this measure problem. How do you count exactly? The problem is that if you count everything there's an infinite number of both and you can't really compare infinity to infinity and decide which is bigger. So to define counting in the multiverse one needs to adopt a particular recipe and that's called a measure to measure in the sense that the mathematicians talk about probability measures and given a model of the multiverse whether or not the normal brains outnumber the Boltzmann brains is a question which cannot be answered until you assume a measure. So in fact the Boltzmann brain question actually became a topic of conversation in the cosmology community because of this. It can be used and has been used as a way of distinguishing between different measure proposals and some measures clearly do die because they lead to a situation where the Boltzmann brains dominate so those measures are ruled out to be acceptable a measure has to have the property that the normal brains at least have a chance of dominating over the Boltzmann brains in some cases we don't know and those measures are provisionally acceptable but could be ruled out later if we learn how to calculate that ratio better. Okay so before we finish again please remember everybody has to leave the room for the refreshments and for everybody means I mean everybody except only for the diploma students just that this room was broken and then then I don't want that to happen in this case and we're responsible to ask questions to Alan. So I think please I would like you to join me to thank Alan for this wonderful set of lectures.