300 | Solo: Does Time Exist?

0:00:00.1 Sean Carroll: Hello, everyone. Welcome to the Mindscape Podcast. I'm your host, Sean Carroll. Welcome to a new year by the conventional calendar, of course, 2025. But also, this first podcast episode of 2025 is Episode 300. And looking back on previous episodes that ended with 00, that is to say 100 and 200, I did some solo episode of some sort to celebrate the little milestone. It's a pretty big milestone. We have a lot more than 300 episodes because many episodes don't get numbers, the AMAs don't get numbers, holiday messages don't get numbers, special episodes, etcetera. But still, having 300 even numbered episodes is pretty darn good. We've been doing it for a while, longer than I might have guessed at the very beginning.

0:00:45.0 SC: So, what to do for a solo episode to start things out? In some sense, I've already been talking too much. I did the hits and misses holiday message, which went on for a long while. I did a previous solo episode about emergence not too long ago. But okay, there's a tradition, and when there's a tradition, we might as well stick to it. But I do worry about doing too many solo episodes because I'm not an expert in that many things. I don't wanna do a solo episode just on me ranting about things. I think it's okay to admit that one is not an expert on everything, or even one doesn't have interesting things to say about every issue out there. Sometimes it's better just to listen to other people, which is why we have the format of Mindscape as we have it.

0:01:25.0 SC: But I do have some things that I know something about, so I couldn't decide. So, what I did is I went on to Patreon where we have our wonderful Patreon supporters for Mindscape, you yourself could be a Patreon supporter if you just go to patreon.com/SeanMCarroll. And of course, Patreon supporters get ad-free versions of the podcast. They get to ask questions for the AMAs and occasional other benefits like, I ask them, what solo episode should I do? And I did have some ideas and I took suggestions also. And the idea that I had that eventually won out by the vote, and I think it's a perfectly good one is, is time real? Now, just to not keep you in suspense, yes. The answer is yes, time is real.

0:02:12.0 SC: But there are people in modern physics and philosophy who question whether time is real. In fact, I once wrote a paper almost as a joke, but... I mean, the title was a joke, the paper was not a joke, I don't like to think. But the title was, "What if Time Really Exists?" Because it has become so common in physics to not only say, "Well, maybe time is not fundamental, maybe time isn't even real, maybe it's an illusion," but to also pat yourself on the back for thinking this is a profound thought, even though many, many other people have already had this thought also.

0:02:43.5 SC: I don't think time is an illusion. I think it is absolutely real. It might not be fundamental. As we've often stressed on the podcast, one should think of things that are emergent as nevertheless real. The table in front of me is real even though there are no tables in the standard model of particle physics. The idea of a table is an approximate higher-level description that has a lot of causal and explanatory power. Maybe time is like that. But sometimes people want to label things that are just emergent as illusions or somehow not real, right? This becomes much more popular in the popular imagination when it comes to things like consciousness or free will or what have you. They wanna say, "Well, it's just an illusion, it's not what you think it is."

0:03:29.9 SC: To me, look, who cares about the definitions, right? You can call it that if you want as long as you're clear about what you mean. I think it's a bad choice though, if you want to actually get across to people what it is you're trying to say. I disagreed with Dan Dennett about this. He said he's an illusionist about consciousness. And I said, "Does that mean you think consciousness is an illusion?" He said, "No, of course not. I just think it means that it's not what you think it is." And I just think that it's a very bad marketing strategy to call that illusionism about consciousness. Because an illusion has the connotation, the word "illusion," of being a fake, being something that isn't really happening. When a magician saws a person in half in the box, and it looks like that is what is really happening, the reason why that's an illusion is because nobody actually did get sawed in half. [chuckle] It's not just there's an emerging of the sawing in half. It's like no one really got sawed in half at all; it's just fake. And I don't think that that's what's going on with consciousness or free will or time, for that matter.

0:04:30.2 SC: But there's still plenty of good questions, forgetting about the nomenclature or whatever we're gonna call it; plenty of good questions about the fundamental nature of time. Is it fundamental or is it emergent? 'Cause when you have a table in front of you or when you have the air molecules in this room and you say, "I have a higher-level emergent description, you know what the microscopic description is, right? There's some full microscopic description in terms of atoms or molecules or quantum fields or what have you. And you can talk about how the higher-level emergent thing is related to that. If you're telling me that time is emergent, what is it emergent from? What is the more fundamental thing? What is the microscopic theory? What is the nature of this thing we call time? Why does it exist? Is that even a question that we have any right to ask or any hope of an answer for?

0:05:18.5 SC: So, that's what I wanna talk about today. There are absolutely indications in fundamental physics that time is not fundamental, but it's very, very uncertain. So, the good news is, the short answer to "Does time exist?" is, "Yes, it does." Okay, but if it's emerging, what is it emerging from? That we don't know. We don't know whether time is fundamental or emerging. If it is emerging, we don't know how that works. I think some people have convinced themselves that we're pretty close to knowing how it works, and I think that they're not quite being honest with themselves. So, we're gonna dig into that today.

0:05:52.9 SC: This is something I'm actually thinking about myself at a research level these days, and either for that reason or just independently, the podcast will be a little bit more technical than usual. I'm gonna try to keep it soft and gentle, try to explain everything, but we're gonna get into some of the weeds. And "technical" I mean in a physics sense here. You know that I'm a big believer that philosophy has a lot to offer physics and vice versa, but in particular questions, in my individual humble opinion, either physics or philosophy is doing a better job at taking certain questions seriously. Things like the measurement problem in quantum mechanics, the ontology problem in quantum mechanics, the nature of the time asymmetry in the universe, etcetera; I think that philosophy and foundations of physics have done a better job than good, old conventional physics.

0:06:42.9 SC: The nature of time, I don't think that's true. I think that philosophers have thought a lot about the nature of time. I think that they have not kept up or put any emphasis on the best ideas we have from physics about whether time is fundamental or emergent. There's plenty of room for philosophy to offer some help to the physicists because the physicists frankly are very confused, but there's not a lot of activity in actually doing that. So, maybe if some philosophers of physics are out there listening, this is a nudge in that direction. So, even though I'm not gonna give you a nice, compact, happy moral of the story at the end of the day because we don't know what the answer is, there's plenty of food for thought, I think, in this subject.

0:07:29.0 SC: And before we move on, let me just mention one of the things that I should have be mentioning all along. I don't know, I've been bad about this, but many of you know we still have the Mindscape Big Picture Scholarship. This is a college scholarship for either high school students or current college students can apply at bold.org, that's B-O-L-D dot org. You can just search within "BOLD for Mindscape" or go to bold.org/scholarships/mindscape. And the idea is, we give money to college students who want to pursue these academicy, intellectual, less practical side of deep questions; the big picture is what we're interested in.

0:08:09.2 SC: And huge thanks to all of the Mindscape supporters out there who've been giving money to the scholarship fund. We've been doing very, very well. This year, we're planning to give two scholarships worth $20,000 each, which is considerable. That's gonna help some students quite a lot. And I'm gonna ask for even more contributions because we're gonna keep doing this every year; that my goal. And the deadline for contributing for this year is January 20... Oh. Sorry. The application... You can always contribute; there's no deadline for that. The deadline for applying, if you are an actual student who wants the money, is January 20th. So, we're still looking for applications, and the winners are gonna be announced February 20th. So, that's very, very exciting. I deeply appreciate all the efforts out there from the Mindscape listeners to help the next generation ask these big questions and hopefully even answer some of them. So, with that, let's go.

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0:09:22.7 SC: Time is one of those ideas that we've had around long before we had physics, right? We have used time, we've measured time, but it has been adapted into physics. Physics, when I say "modern physics," sometimes people mean modern physics as in 20th century; I mean like Isaac Newton and after, okay? I think that Isaac Newton, classical mechanics, and the people around him like Galileo and Descartes and so forth, they really codified physics in a way that was very, very different than what came before. And the notion of time came along. We had time before Isaac Newton and we still have time after, but the meaning of it changed a little bit. And I think that because a lot of people who are not professional physicists still think about time in the everyday sense, the pre-Newtonian sense, one of the things you have to do to understand time is to reconcile what physicists mean when they use the word with what ordinary people mean when they use the word.

0:10:19.0 SC: One way I'm thinking about it is, what exists? We're asking, does time really exist? Well, if you didn't know anything about physics or philosophy, etcetera, and someone said, "What exists?" what would you say? I think the typical answer, people would include different things, whether you believe in angels or something like that, might be controversial, but most people would say that the world exists. The world, in some sense, consists of space where things are located, and there's a whole bunch of things in space: Tables and chairs and planets and people, etcetera. Space, if you wanna be a little bit physicsy about it, it's a three-dimensional manifold. And the things that are in it, physics tries to understand. It describes them in terms of mass and motion and location and what have you.

0:11:07.1 SC: And then in that pre-Newtonian picture, the reality of the world is that three-dimensional space plus everything in it, and time is something that happens. And I had to be vague here because it's not very specified. The world changes from moment to moment, and time is how we talk about that change. Here is what the world is at this particular time. At a certain later time, the world is something different. You could imagine saying words like "time is change" or "time brings about change." It's, again, unclear exactly what the best set of words to use is, but conceptions like that are evolved. The world is three dimensions plus stuff; time tells us about how that stuff changes, evolves, becomes different from moment to moment.

0:11:58.0 SC: Now, Newton comes along, Isaac Newton, classical mechanics, Principia Mathematica, etcetera, and he invents these equations, right? The equations are very strict. It's a deterministic, clockwork universe, and the equations at their most fundamental relate the configuration of stuff in the universe at one moment in time, to the configuration of stuff in the universe at a different moment in time. So, there's a relationship between what the world is at one moment and what the world is at all other moments. Of course, there always was a relationship. The world is not re-arranged to scattered and completely randomly constituted from moment to moment in time. There's always, long before Isaac Newton, some notion of continuity and even some rules of thumb about how things change from moment to moment. But when Newton comes along, you really have this rigidity, this deterministic aspect to the evolution of the universe over time.

0:12:56.3 SC: Of course, the most vivid version of this is in the Laplace's demon thought experiment from Pierre Simon Laplace much later, circa 1800, where he tries to say that... Well, he imagines, saying that if a vast intelligence knew everything about the universe, the complete state of the universe at one moment of time, that is to say more specifically the positions and velocities of everything in the universe at one moment of time, then given Newton's equations, you could tell exactly what the universe would be in the future and what it had been in the past. Laplace's demon doesn't exist. None of us is Laplace's demon. It's just a way of making vivid the idea of determinism, of making it very clear what it implies.

0:13:36.1 SC: And even though it's not necessary, this existence of deterministic laws of evolution suggests thinking about the universe in a new way. It suggests a four-dimensional perspective. Because what the universe is at any one moment is so clearly, unambiguously, and rigidly related to any other moment, you can think of time as a location: As a way of finding yourself in the universe that you could think of as four-dimensional. Of course, Einstein and relativity made that almost necessary to think of the universe as spacetime, as four-dimensional, because the way that you slice up spacetime into space and time is not absolute as Newton assumed it was, but rather relative to how you move through the universe. But even before relativity comes along, the existence of deterministic equations of motion lets you imagine that the whole of reality should be thought of as the four-dimensional thing, the four-dimensional system. This is the eternalist point of view on time rather than the presentist point of view on time, which says that only the current moment is actually real.

0:14:47.7 SC: For today's purposes, we're not gonna debate presentism versus eternalism. I'm actually quite... I'm an eternalist by inclination, but I don't really care. [chuckle] I think that whether you're a presentist or eternalist about the universe, that is to say, do you believe that only the present moment is real or that all moments are equally real, is in a very plausible sense, a matter of taste, right? There's no experiment you can do to show, even in principle, to show that eternalism is right or presentism is right. But it's an attitude. It is a way of thinking about all of the universe. And attitudes matter. They suggest ways forward. This is always true about theories of physics. It's very often true that different theories of physics can be formulated in completely equivalent ways, but when you formulate them in one way or another, they might suggest ways of going beyond those theories. And what's super important about that is that we don't know the final answer yet. So, going beyond the theory we have right now actually matters, and therefore, stances you take towards different existing things in the universe might suggest different ways of going forward.

0:16:00.9 SC: So, in the post-Newtonian world, it at least makes sense to think of time in a different way than we thought of it before, before we thought of time as playing some role in the reality of the universe, right? The universe evolves with time; that's the most fundamental thing about the universe. If there were no time evolution, if the universe just existed, it's almost as if the universe is not something you could talk about, right? The universe wouldn't have any sensible thing to say because you say, well, you look at something. Well, looking is something that unfolds in time, isn't it? You didn't look before, now you're looking. [laughter] So, we're so embedded in the evolution of time that it's necessary somehow to how we talk about and how we conceptualize the universe.

0:16:49.4 SC: The shift then is from thinking of time as something that really has to do with the unfolding of reality, to time which is more like a coordinate, a label, a way of finding out where you are, specifying where you are in this four-dimensional reality. And so this raises a new question that didn't exist before, and some of you long-time listeners will know where I'm going here. The arrow of time, the directionality of time becomes important. The arrow of time was not an issue for Aristotle or for pre-Newtonian thinkers in general, and it took quite a while for post-Newtonian thinkers to realize that it was an issue. I believe that the phrase "the arrow of time" is due to Arthur Eddington, who's a 20th century physicist and astronomer, astrophysicist; mathematical astrophysicist, let's put it that way.

0:17:41.9 SC: Because Newton's laws, these wonderful rules that govern the rigid motion of the universe, the evolution of the universe over time, don't pick out a direction to time, right? And this is one of the reason why in the 19th century, there was so much struggle to come to grips with the Second Law of Thermodynamics, which does have an arrow of time built in. The second law says that entropy doesn't decrease with time, and either increases or remains constant in closed, isolated systems. That clearly distinguishes between the past and future. Entropy was lower in the past, will be higher in the future. That makes a difference between the past and future, even though the underlying Newtonian laws don't. So, how do you reconcile those two facts? This is a new problem. It's a problem that didn't exist before Newton's equations came along.

0:18:31.8 SC: I'm not gonna dwell that much on the arrow of time here. Some of you who are not old, old-time listeners might not know that I wrote a whole book about the subject. My first trade book was called "From Eternity to Here: The Quest for the Ultimate Theory of Time." So, I talk about the arrow of time, cosmology, presentism, eternalism, time travel, all of those things. I'll give you the very short version right now because it will actually become relevant later on. Because today we're interested in "Is time itself fundamental?" not "Why does time have an arrow?" Maybe these questions are not disentangleable, right? Maybe they have something to do with each other.

0:19:07.3 SC: So, entropy is defined by Ludwig Boltzmann in the following way. He says, you look at a system, either you literally look at it or you imagine knowing something about it, and you get incomplete information. If you think that some system is actually made of atoms but you don't personally see the atoms, you see, "Oh, I have a table, it has a certain size and mass and things like that," there could be many, many different arrangements of the atoms that look like that table or whatever other macroscopic system you're looking at. And the entropy is a way of counting how many ways there are to organize or arrange the microscopic constituents into the macroscopic thing you're looking at. When the entropy is low, that means that the macroscopic information you have is enough to pretty much pin down what the microscopic arrangement is.

0:19:57.4 SC: There aren't that many ways to arrange the underlying atoms. If you have all the air in the room around you but you knew that all the air for whatever reason was confined to a cubic centimeter over there in the corner and it was vacuum everywhere else, that is a low entropy configuration. You know a lot about where the atoms are, I just told you. Whereas if I tell you that the air is uniformly spread throughout the room, that doesn't tell you a lot about the individual atoms. Individual atoms could be here or there; there's many different ways to arrange them to make the whole thing look uniform overall. So, that's high entropy. And the second law says the universe starts with low entropy and it gets larger as time goes on. That is an arrow of time. There's a huge question for cosmology, which is, why did the universe start with low entropy? That's gonna be a question that cosmologists should keep in mind. We're not gonna primarily worry about that right here and right now.

0:20:52.0 SC: Now, there's a big leap that can be made, and I am in favor of making it, which is the following. The laws of physics at the microscopic level, as far as we know them, do not have an arrow of time. The macroscopic world has an arrow of time. In fact, it has apparently various different arrows of time; there are various different ways in which the past and future are distinguishable from each other. Most obviously, you remember the past; you have records of it. You predict the future, and in fact, you can affect the future by the choices that you make right here in ways that you cannot affect the past by the choices you make right here and right now. So, those are two arrows, the arrow of memory or recording, the arrow of causality and influence. There's also the psychological arrow of time, there's an aging arrow of time, etcetera. The big claim that I'm going to endorse is that all of these arrows of time are basically just different manifestations of what we call the "thermodynamic arrow of time." Thermodynamic arrow is just the fact that entropy tends to increase over time.

0:22:00.9 SC: So, there's a huge amount of work to be done by scientists and philosophers to understand whether or not that claim is correct, and to understand why entropy increasing has anything to do with things like why we're all born young and then grow older, right? Now, the thing, it can be done, but it's more than of a work to be done. I'm not gonna do it actually today. I talk a little bit about in "From Eternity to Here" and elsewhere. It's something I'm fascinated by in many deep ways. But for our present purposes, let's just take it on board. The reason why there's an apparent arrow of time is because entropy is increasing. So, the reason why you have a feeling that you're moving through time is because entropy is increasing.

0:22:44.0 SC: You can kind of see if you squint, if you're not too hardcore about it, you can imagine why that would be true. Why do you have memories of the past and not the future? Well, the entropy was lower in the past, that means that we know that in the early universe there were fewer possible configurations of the underlying stuff the universe is made out of, therefore you can use information in the present universe to draw conclusions about the path you took to get from the past to here in ways that it's not possible to do with the future, because there's no future boundary condition of either high or low entropy that anyone really knows about. And given that memory aspect, you can also imagine understanding the causal aspect. Because we have memories and records of the past, we can't change them. We think that we can actually infer what did happen in the past. But because we can't do that towards the future, we can imagine different possible actions that I take right now having different possible consequences in the future.

0:23:43.9 SC: So, psychologically, you're always carrying with you not only who you are in the here and now, but a memory of who you were just a moment ago, and also a set of possible people you can be in the future. And those are different, right? There's an imbalance there because of the arrow of time, and so that's what gives you the feeling of moving through time, or if you wanna put it that way, time moving you along. It's because you're constantly updating your impression of who you were and your guesses about who you're going to be next in this asymmetric way.

0:24:17.5 SC: Obviously, there's a lot more to be said about that. We've said a little bit before in the podcast, the conversation we had with Jenann Ismael talked about that and others; Dean Buonomano, it's a rich, rich subject. Okay. Not what we're talking about today. We wanna talk today about time itself; not the arrow of time, right? Two separate things, very, very important. When I say that entropy underlies the arrow of time, sometimes people mangle some words together and claim that I'm saying that entropy is time or something like that, right? Entropy explains the arrow of time, which is a feature that time has. It doesn't explain time itself. Time is a thing in the Newtonian view, which is more or less inherited by subsequent theories. Time is a label, a coordinate, etcetera. There's some updates there as we'll talk about in a second.

0:25:08.0 SC: But time could exist without an arrow, right? There's a fact about our observed universe that time has an arrow, but you can still talk about time and time evolution and things like that, even if the arrow didn't exist. If you just have a pendulum swinging back and forth in a frictionless universe so it swings back and forth forever, time is passing, the pendulum is changing, but there's no arrow. If you took a movie of that and you played it backward, it would look perfectly sensible; there's no difference between the evolution toward the past or toward the future. So, it's that that we're trying to think about now: The actual existence of time itself, not the fact that it has an arrow.

0:25:47.8 SC: Anyway, two things do come along in the 20th century that are relevant to this question: Relativity and quantum mechanics. You shouldn't be surprised that both relativity and quantum mechanics have something to say about the fundamental nature of time. Obviously, relativity has a lot to say about the fundamental nature of time. It's what says that time is not universal. In Newton's world, the existence of space and the existence of time were separately given to us. In Einstein's world, all we're given is four-dimensional spacetime, and how we divide that up into space and time is up to us, to some extent.

0:26:26.0 SC: Now, in special relativity that comes along in 1905, spacetime itself is still flat. Minkowski space, there's no gravity; there is only one actual spacetime that you're supposed to be thinking about. And so there's a set of different ways to divide it up that are all different but all kind of natural. This is what we mean when we talk about "reference frames" in special relativity. When you have just a piece of paper and you're doing two-dimensional, spatial, Euclidean geometry, there's a natural set of coordinates you can use. Cartesian coordinates, you draw perpendicular axis, you label them X and Y. Now, you could rotate them or move the label of the origin or whatever, but perpendicular Cartesian coordinates are natural things to use when you have a flat piece of paper.

0:27:14.8 SC: Likewise in flat Minkowski, special relativity spacetime, there's a natural set of ways to divide spacetime into space and time. Different observers will do it differently, but for each observer there's a natural way to do it: Inertial frames, global reference frames, whatever you want to call them. When general relativity comes along, 10 years later in 1915, now you're saying that spacetime is curved and it's curved in wild ways. It's curved in ways that can change arbitrarily from point to point. Not arbitrarily, but by a lot, from point to point, from moment to moment. And now this was also true in special relativity but you didn't care as much. Now in general relativity, you cannot help but face up to the fact that you could slice spacetime in an infinite number of ways into space and time.

0:28:04.7 SC: Sometimes in general relativity, people talk about "many-fingered time." I think this is a wheelerism. I think this goes back to John Wheeler, but I'm not exactly sure; it could have been somebody else. The point being that if you take the universe in general relativity and you take a slice, a slice through the universe that you label "Here's what I'm gonna call one moment in time," and you say, "Alright, how do I march it forward in time?" you are allowed in general relativity to march it forward faster in some places than others. Many-fingers is the different rates at which time is moving in some sense, according to your coordinate system, which is entirely arbitrary. Clocks, physical clocks that actually measure time, always measure one second per second.

0:28:49.0 SC: But in general relativity, since there is no natural coordinate system that everyone can agree on, the freedom to choose how you measure space and time is in your phase, and therefore you can choose whatever coordinates you want. No one can blame you, right? Certain coordinates might be more convenient than others, but you're allowed to do whatever you want. So, in general relativity, the freedom to divide spacetime into space and time is quite unmissable. You cannot help but talk about it. And you know what? Honestly, I'm gonna say, that doesn't have any huge ramifications for the fundamental nature of time, other than the very, very huge ramification that time is part of spacetime. But in terms of time being a label, being a coordinate, being part of the fundamental description of reality, it's still all there. You would say it's really spacetime not time, but other than that little update, the basic way of thinking that you inherited from classical mechanics still works pretty well.

0:29:51.7 SC: Now, let's be honest, if it weren't for quantum mechanics, I wouldn't be saying that. I'd be saying that the relativity revolution was absolutely huge, [laughter] it was conceptually very, very important to how we think about reality, etcetera, etcetera. But as much as I love relativity, in terms of the amount of conceptual updating you need to do, relativity is nothing compared to quantum mechanics. So, it's still a classical theory. You need to improve your way of thinking a little bit. But many of the basic ideas that you inherited from Isaac Newton can just be translated and you can still use them in relativity. The notion of time, the nature of time itself, goes through more or less unchanged. Not everyone agrees about that. I think they should agree, but famously people like Julian Barbour, physicist, wants to say that general relativity changes our notion of time so much that time is an illusion, really, in general relativity.

0:30:48.7 SC: As I said before, I'm not sympathetic to that point of view. I'm not gonna explicate here why he says that or why he shouldn't say that. But putting it on the table, he wrote a book, [chuckle] you're welcome to read the book about that; that's not where we're going right here. We are going into quantum mechanics because of course we are. Quantum mechanics is where it gets tricky, the notion of what time is. It needing to get tricky, but where's really, really gonna get tricky is where we try to marry together quantum mechanics with gravity to do quantum gravity. Sometimes people will say, "Well, wait a minute, we don't understand quantum gravity. How can we say anything about quantum gravity?" It's true, we don't fully understand quantum gravity, but we understand a lot about quantum mechanics and we understand a lot about gravity. So, there's a lot that we expect to be true about quantum gravity. And either it is and we should try to understand it, or the reality of the situation is even more profound and we need to try to understand that. So, I think it does absolutely pay off to try to think about how the combined implications of quantum mechanics and general relativity tell us something new and interesting about the nature of time.

0:32:00.0 SC: Okay. So, at the most fundamental level, you probably know, I've talked about this a lot before. So, quantum mechanics is tricky because you need to not only say, "Oh, there's a wave function and evolves with time," but you also need some rules for what happens when you measure the wave function: That it collapses, there's probabilities, etcetera, etcetera. I'm gonna advocate something like the many worlds approach, where that collapse of the wave function is only apparent because observers live on different branches of the wave function; they don't see the whole thing. Other people will advocate different points of view. I'm, again, not interested in that particular conundrum right now. I will just put on the table, it is easiest for me to talk for the rest of this episode as if many worlds is true, not because of the worlds.

0:32:47.9 SC: I don't know how often I've said this, but I do wanna emphasize, I don't care about the worlds, [laughter] many worlds. I mean, they're there. I think that it makes absolute perfect sense to believe that they exist and to think about them, but that's not why I believe in it or don't believe in it. It's just a prediction of the Schrödinger equation and the formalism of quantum mechanics. And I take that the lesson of Everett to be that if you properly interpret what the wave function is, it can just obey the Schrödinger equation and it fits all the data. Okay? So, that's where I go for many worlds, is that it fits all the data without adding anything to either the ontology of the theory or the dynamics of the Schrödinger equation. So, we'll talk that language. We'll talk the language that what quantum mechanics is, is some quantum state evolving with time according to the Schrödinger equation.

0:33:41.0 SC: Now, when you say the words "the Schrödinger equation," you have to be a little bit careful here, because sometimes people think that means literally the equation that Schrödinger first wrote down in 1925 or 1926, which is now sometimes called the "non-relativistic" Schrödinger equation. So, Schrödinger was literally interested in the wave function of an electron in atom or something like that, right? He knew about relativity as my old quantum field theory professor Sidney Coleman liked to say, "Schrödinger was no dummy," [laughter] and he actually invented a relativistic version of his equation first, but it didn't fit the data; it didn't work. You needed to be a little bit more clever than they were at the time. So, the non-relativistic version than he came up with worked better in the non-relativistic regime. So, that's what we know as the Schrödinger equation. Of course today we know you can also have a relativistic version of the Schrödinger equation, and there's complications involved with quantum field theory, etcetera.

0:34:42.3 SC: But that's not what we're going today either. And what basically all I wanna do is to say that the Schrödinger equation, as far as time is concerned, isn't that different from Newtonian physics, right? It's a nicer equation, in many ways that the technical reason that it's nicer, is because it is linear. What does that mean? That means that the fundamental thing that is evolving over time according to the Schrödinger equation, is the wave function or the quantum state. For today's purposes, you can call either one; you can use either terminology to talk about the state of a quantum mechanical system. So, the wave function for a single particle is just a complex number at every possible position it could have. And you square that number to get the probability of observing it there. For more complicated things with many particles or many fields or whatever, you assign a complex number to every possible configuration of all those different things. And you square that to say, what would the configuration look like probably if you were to observe it?

0:35:46.1 SC: Or you could just more abstractly talk about a vector in Hilbert space. There's all sorts of different mathematically equivalent ways of talking about the quantum state of a system. But it's precisely analogous in this way of thinking to the classical state of a system. The classical state of a system is the position and velocity of every object, every constituent of whatever system you have, and it tells you everything you need to know. Likewise, the quantum state tells you everything you need to know. The crucial difference is, you can't measure the quantum state without disturbing it. So, that's a whole thing to go into, maybe some other... I think I have gone into that, right? In previous podcasts, previous solo episodes. Okay. But the linearity is what I'm trying to get to here of the Schrödinger equation. The linearity means that you can think of the Schrödinger equation as a box. You feed into the box the current state of the quantum system, the wave function or whatever you want to call it. And the Schrödinger equation spits out the rate of change of that quantum state, okay? It's a way of saying, given what the state is, how fast is the state changing.

0:36:56.0 SC: And the relationship, that function you give me the state I tell you how fast it's changing, is linear. If you gave me two states and you added them together and you plugged into the function, it would give you the sum of the two different changes, okay? This reflects the fact that the quantum state is a vector. You can add states together in ways that you can't add classical states together. Okay? If I have a classical system of two particles... No, forget that. Let me say, one particle, okay? Let me imagine I have one particle, and so I have one particle in three-dimensional space. The classical state of that particle gives me its position and its momentum or its philosophy. If I take two different possible states of a classical particle, okay? So, two different instances of here's the position, here's the velocity. It doesn't mean anything to add them together. I can't add together two different positions of particles.

0:37:56.1 SC: You might think, well, I could just draw axes and make little vectors and add them together. But the resultant vector would entirely depend on where I put the origin of my coordinates. Points in space do not naturally have a vector structure. You cannot add two points together in space. This is a deep, deep, super important, frankly underappreciated feature of quantum mechanics, that when you have two wave functions, you can add them together, you can scale them by real numbers, they have all the properties of vectors. And the reason why that makes sense is because the Schrödinger equation is linear. If you have one quantum state evolving over time and you have another quantum state evolving over time, they both solve the Schrödinger equation individually. Then the sum of those two quantum states also solves the Schrödinger equation. This is a wonderful feature that we will take advantage of in just a second.

0:38:52.1 SC: But for the moment, all I care about is, what the Schrödinger equation is telling you is something very, very analogous to what Newton's laws of motion are telling you. Newton's laws say, "You give me the classical state of the system, I will tell you how it changes in time." The Schrödinger equation says, "You give me the wave function of a system, I will tell you how it changes in time; there's no difference between those two things." But this linearity honestly makes things much easier. So, Newtonian mechanics is not necessarily linear. You might have one particle being affected by all sorts of other particles in all sorts of complicated ways, right? The Schrödinger equation has less freedom in what can happen because it's this linear system, which means that it's very, very restricted in what can happen.

0:39:38.8 SC: Let me just pause as an aside here. You've heard about chaos theory, right? Chaos theory says that if I have two different possible states a system can be in that are almost the same but a little bit different, then they can evolve in ways that are very, very different. You can step on a butterfly and completely change the weather patterns a month later, okay? Tiny perturbations of initial conditions can lead to huge differences later. One practically synonym for chaos theory is non-linear dynamics, because the whole point of chaos theory is those tiny little initial differences can be amplified. They can grow bigger. That's what chaos theory is all about. That's a feature of non-linearities. Non-linearities let you amplify tiny little changes. But linear systems, which are much simpler, have the feature that tiny initial differences become tiny later differences. The differences don't grow exponentially or anything like that. There is no chaos in the evolution of a quantum mechanical wave function. Quantum chaos, which is a topic that people talk about, absolutely exists, but it only appears in the classical limit of the quantum mechanical evolution. The Schrödinger equation itself doesn't have any chaos in it.

0:41:02.0 SC: Okay? Why am I telling you this? Well, because there is a feature of classical dynamics, especially when you think of general relativity as an example of classical dynamics, that is kind of important to the nature of time, which is that in classical mechanics, time can end. You can hit something called a singularity, right? In general relativity, for example, the universe could collapse in the future. We don't think it will, but maybe it will. We used to think that it was quite plausible. Now it's less plausible now that we know that the universe is accelerating, but it still could happen. The future is hard-to-predict part of the arrow of time, right?

0:41:41.0 SC: So, if you just follow the equations of Einstein's general theory of relativity, you could reach a moment of time in which the theory breaks down, in which the physical quantities representing reality become infinitely big and the equation stop making sense. That's what a singularity is. And so time itself can end. It's a boundary to spacetime and therefore to time. And singularities are not actually these rare things that it's hard to make in general relativity; they're kind of generic. There seems to be a singularity in our past, if you believe what general relativity is telling you. You shouldn't believe what general relativity is telling you, of course, because general relativity is a classical theory. But that's my point. In classical theories like general relativity, it would make perfect sense to talk about the beginning of time or the end of time because you can reach singularities. If you look at quantum mechanics as given to you by the Schrödinger equation, time cannot end. The Schrödinger equation says time keeps marching on, okay? I once cheekily labeled this fact the "Quantum Eternity Theorem" because, well, it's a long story, but I did this in the context of my debate with William Lane Craig.

0:43:01.7 SC: But the point is, if all you knew about the universe was the Schrödinger equation and there was any time evolution at all, okay, so you picked an initial condition that would change over time, then it's easy to prove that it changes forever into the future and it was changing forever into the past. That's the Quantum Eternity Theorem. There are no singularities in the Schrodinger evolution of a wave function because everything is linear. There's no way for things to blow up and become infinite. It's a theorem. Okay? Now, if you talk to people, even very, very good professional physicists out there, they won't always say those words. They won't always agree with what I just said. They should agree, but you have to be very, very clear about what assumptions you're working with, because we're casually introducing the idea that someday we will include gravity into our quantum theory.

0:43:56.6 SC: So, what I said was 100% absolutely correct. If you believe that some version of the Schrödinger equation is correct and you believe that the universe is evolving in time at all, then the Schrödinger equation predicts it will evolve forever to the past and to the future. But if you instead start with some classical picture and imagine quantizing it, then you can get yourself into the belief state, [laughter] where you think your classical variables are representing singularities. They can't, if you think purely quantum mechanically. But of course, since we don't perfectly well understand quantum gravity, anything is possible, right? So, there's a theorem that if the Schrödinger equation is true, time evolution happens forever. If it happens at all, it happens forever. But maybe the Schrödinger equation or even a relativistic version of it is not true. Maybe quantum gravity needs something beyond that. Okay?

0:44:52.8 SC: Those are all on the table because we don't fully understand quantum gravity. That's fine. Nevertheless, since we do understand the Schrödinger equation, even if we don't understand quantum gravity... And remember the Schrödinger equation in its... I say "remember." That's a bad habit that physicists have, telling people to remember things they might never have heard before. So, let me back up. Schrödinger equation, you do remember, 'cause I already said it, the Schrödinger equation is just the quantum version of Newton's laws of motion, okay? So, it doesn't matter what the system is. All different systems have a version of the Schrödinger equation, just like all different systems for F=ma, Newton's second law of motion, all different systems have some specification of what are the forces, what are the masses. And then you do the math. Likewise, for the quantum mechanical version, every system has some version of the Schrödinger equation.

0:45:44.8 SC: That box, you give me the state, I tell you how fast it's changing. The way that that is accomplished, according to the Schrödinger equation, is by calculating what is called the Hamiltonian of the system. So, you giving me the Hamiltonian in quantum mechanics is exactly analogous to you telling me what forces there are in classical mechanics. If I know the Hamiltonian, which is basically a way of thinking about the energy contained in the quantum state, the Schrödinger equation tells me how fast it will evolve, okay? So, even though Schrodinger himself was interested in electrons in atoms, there is a version of the Schrödinger equation for every system we can think of in conventional quantum mechanics. For systems with many particles, for quantum field theories, there's a version of the Schrödinger equation 'cause there's a version of the Hamiltonian. You can have versions of the Schrödinger equation representing perfectly relativistic Lorentz invariance dynamics, okay? The Schrödinger equation has a time variable in it. It tells you how the system is changing with time.

0:46:49.9 SC: And therefore people think, "well, it can't be compatible with relativity because relativity says I can choose different time variables." That's fine. There's no actual incompatibility there. The Schrödinger equation is just saying, "Yes, you can pick different time variables, so pick one [chuckle] and then I will tell you how the system evolves with time." So, you pick a time variable, the Schrödinger equation says, "Here's the Hamiltonian and the state with respect to that time variable. I will tell you how it evolves with respect to that time variable." So, I'm just emphasizing here that talking about the Schrödinger equation is in no way a restriction. It's a perfectly general idea, whether you're single particles, quantum fields, gravity, whatever it is. No current version of physics has some improvement to the Schrödinger equation that it needs to contemplate. Maybe there will be someday; that's always possible. But the Schrödinger equation is the state-of-the-art right now in terms of quantum dynamics, okay? And people in fact have thought about improving the Schrödinger equation. I don't think they've actually succeeded, but people definitely thought about it. Okay.

0:47:57.0 SC: So, anyway, where does that put us? In quantum mechanics, if all we're caring about is the Schrödinger equation in quantum states, that is the many world's perspective. Even though I don't care about the worlds, like I said, it's just the perspective that says that's all there is to quantum mechanics: States and the Schrödinger equation. All we need to know is what is the state, what is the Hamiltonian that makes it go, and then we can do it, okay? And because of this feature, the time lasts forever, once you have Schrödinger evolution. You have two options to what can happen to the quantum state over very, very, very long periods of time; the two options were it eventually comes back to where it started or it doesn't. [laughter] Those are the two options. And this is also because the Schrödinger equation isn't that different from classical mechanics; this is also a feature of classical mechanics.

0:48:55.2 SC: This is something called the "Poincaré recurrence theorem." Henri Poincaré, mathematician and physicist contemporaneous with Einstein, did a lot of work that was very relevant to relativity, but also practically invented chaos theory, arguably; did a lot of work in classical mechanics generally. And Poincaré was thinking about the solar system, where you have a bunch of planets orbiting the sun. And forget about the fact that the sun itself can move. Fix the sun as if it were infinitely massive. Let all the planets go around it. The planets go around at different rates, right? Mercury's year is shorter than the earth's year which is shorter than Saturn's year, etcetera. So, from moment to moment, you can look at the positions of the planets. They'll all be different relative to each other. And one question you can ask is, if they're in a certain configuration relative to each other now, will all the planets return to the same configuration sometime in the future?

0:49:53.0 SC: One Earth year later, they certainly won't, 'cause mercury will have gone around several times and Saturn will not even have gone around once. But can you wait long enough so they all line up once again? And a moment's thought tells you, yes, that should eventually happen basically because there's only a finite number of choices, right? It's an infinite number of choices for where the planets can be because they move on smooth trajectories, we're talking about classical mechanics now, but it's a bounded set of trajectories, right? Every planet is moving on an ellipse, okay? We're also ignoring the influences from one planet to another. So, they're moving on perfect ellipses like Kepler would've liked, okay? So, the planets are moving on ellipses. They just do the same ellipse over and over and over again. Even if they're different orbital periods might be incommensurable, in other words, one orbit is not some integer multiple of the other, Poincaré was able to show that because...

0:50:51.6 SC: There might be irrational numbers relating the orbital periods of the different planets. So, being a clever mathematician, he said, "What I really care about is not will they line up exactly where they started, but if you give me a tiny number epsilon, will all the planets line up to within epsilon of where they started? And will that be true no matter how tiny you made your number epsilon?" And what he's able to show is, yes, indeed, for a very, very tiny number epsilon, the planets will line up in exactly the same configuration within a precision of epsilon eventually in the future basically 'cause their orbits are bounded. There's not that many places they can go. This is called the "Poincaré recurrence theorem." I believe, although I'm not 100% sure, that the Poincaré recurrence theorem came after Friedrich Nietzsche started talking about a universe that eternally recurred. Nietzsche did talk about that, but he talked about it for different reasons. I used to know this; it's in "From Eternity to Here." If you read that book, the stories are in there and in the correct chronological order. Okay.

0:52:00.0 SC: So, under certain circumstances, a classical mechanical system will evolve for a long time, and it will return to where it started. It might take a very long time. Poincaré showed that typically it takes a very long time. That's absolutely going to be true, but it will eventually happen. You'll return again and again and again infinitely often to where you started. However, what if you have a comet? [laughter] What if you have a comet that is not on a bound orbit, right? A comet that is not on an ellipse, but on a hyperbolic orbit that is just zooming into the solar system, passing by and then zooming back out. That case does not recur. If you include the comet in your coordinates that are telling you what the solar system looks like, the comet doesn't recur all by itself. It just goes from minus infinity to infinity. It never is in the same place twice. Okay? So, that's obviously an apparent exception to the Poincaré recurrence theorem.

0:52:55.9 SC: What's going on? What's going on is that, the space of states is not bounded. So, Poincaré, again, being a careful mathematician, showed that if you have a bounded phase space, a bounded space of states under the rules of classical mechanics, you will eventually recur. But if you have an unbounded phase space, then you don't necessarily recur. So, an unbound orbit like a comet moving on a hyperbolic trajectory, the system will not return to where it started again. Why am I telling you this? Well, because, the same thing is all true. All of the same things are true in quantum mechanics. If the Schrödinger equation is telling you all that is going on, you can have either recurrent evolution of the quantum state or non-recurrent evolution. Does the state eventually come back to where it started? Or does it wander off into ever different places, configurations, whatever you wanna call them, toward the infinite past and the infinite future?

0:53:54.6 SC: And in fact, you can figure out fairly simply, just like in classical mechanics, the question is, is phase space bounded or unbounded? It becomes basically the same question in quantum mechanics, but now it is, is Hilbert space bounded or unbounded? Hilbert space is the space of all possible distinct quantum wave functions. And really what we mean by "bounded" or "unbounded" in this sense is, is Hilbert space finite dimensional or infinite dimensional? Remember, Hilbert space is a space of vectors, so it's a space of things that are, well, it can be described by axes that are orthogonal to each other. The number of axes is typically very large. Very, very often in quantum mechanics, we think about infinite dimensional Hilbert spaces, but sometimes we think about finite dimensional Hilbert spaces, and those finite versus infinite have nothing to do with the dimensionality of space as we know it.

0:54:52.7 SC: So, a single spin like the spin of an electron, is described by a two-dimensional Hilbert space. That is a small number, finite number. The position of a single electron is described by an infinite dimensional Hilbert space, simply because there are an infinite number of different dimensions... Sorry. [chuckle] An infinite number of different locations that the electron can be in. There's only a finite number of spin measurements you can do. It's either spin up or spin down, that's it. There's only two of them, thus two-dimensional Hilbert space.

0:55:26.0 SC: As I have talked about before in various places, so I don't need you to have heard it before, but there are reasons to think that we should be taking seriously the possibility that Hilbert space of the real world is finite dimensional. This is not all obvious; it's not even necessarily true; that's why I'm phrasing it carefully. There are reasons to think it. The reasons are basically because there is gravity, right? If it weren't for gravity, if gravity didn't exist, if you just had like the standard model of particle physics or quantum electrodynamics or whatever, quantum field theory generally says that the dimensionality of Hilbert space is infinite. Like I said, even for a single electron all by itself, the dimensionality of Hilbert space is infinite.

0:56:13.2 SC: A quantum field theory is basically, you can think of as describing not just one particle but zero particles, N particles for any N, numbers of particles changing back and forth to each other. So, of course, you could have an infinite dimensional, you generally will have an infinite dimensional Hilbert space there as well. One way of thinking about that is, take a little box, take a region of space; not a physical box with walls but just imagine boundaries that are like a cube or something in some region of space. And ask yourself, how many possible things can happen in that region of space, okay? It sounds like an infinite number of things. I put one particle in there, two particles in there, three particles. In quantum field theory language, I can excite the field a little bit or a lot or really a lot, and so forth. There's an infinite number of things I could think about.

0:57:01.0 SC: So, without gravity, Hilbert space would very naturally appear to us as infinite dimensional. An infinite number of things can happen inside a box. Hopefully you can kind of predict where this is going. With gravity, there's a cutoff. There's a new limitation on what I can do, which is if I try to put too much stuff in my box, it collapses into a black hole. And the black hole has a size, and if that size is bigger than the box, then I have violated the rules of the game, okay? There's only a certain amount of energy I can put into any region in a theory with gravity, while it is literally still fit inside that region that I have just invented. So, that's a very hand-wavy argument, but it translates into thinking that in quantum mechanics, there's only a finite dimensional Hilbert space describing any region of spacetime.

0:57:52.2 SC: Now, this is where it gets, again, more subtle, okay, there could be an infinite number of regions of spacetime. So, the Hilbert space of the real world, the quantum mechanical space of states of reality could be infinite dimensional. Even if there's only a finite number of things that can happen in any region, as long as there's an infinite number of regions, then there's still an infinity there. But we think that in our real world there are plausible reasons to think that there is an observable universe that is finite in size; not just finite right now, but finite forever. There is a horizon around us because of the acceleration of the universe that's a finite size. In fact we know what the size is, of order 10 billion light years across, where you can never see what's happening outside that horizon around us because of the acceleration of the universe, because of the dark energy pushing everything apart. That makes the observable universe finite in size. So, it is very natural to imagine, although we don't know any of this for certain, but it's very natural to imagine that the quantum mechanical Hilbert space, the space of states for our observable universe, is finite dimensional.

0:59:09.8 SC: Now, is the whole universe finite dimensional or not in quantum mechanical description? We honestly don't know. Sorry to be vague about that. I'm just trying to be... I'm trying to both tell you what the plausible things are, but also to be clear about what we don't know yet. So, basically what I'm trying to say is, when you're thinking about the Schrödinger equation and the evolution of the universe, it's perfectly respectable to think about either finite dimensional Hilbert spaces or infinite dimensional Hilbert spaces. That is to say, recurrent evolution as in Poincaré, a quantum system that will come back to itself infinitely often; or non-recurrent evolution, evolution that will just keep evolving forever and ever. Hopefully the implications are sinking in.

0:59:54.9 SC: If you pause the podcast right now and think about what I just said for the last 10 minutes, there's an implication here. I said two things. One is, the Schrödinger equation says that if the universe evolves with time at all, it will evolve forever into both the past and the future. And then I said it is plausible that the Hilbert space of the universe, the space of quantum mechanical states, is finite dimensional. And finally, in-between saying those two things, I said that if the Hilbert space is finite dimensional, there will be recurrences eventually. It will last a very, very long time. For the reasonable numbers, given the size of our universe, our observable universe, etcetera, there is something called the Poincaré recurrence time. And classically it's very straightforward to define; quantum mechanically it depends on how close you're defining your states to be to each other.

1:00:48.2 SC: But roughly speaking, for realistic numbers, the Poincaré recurrence time for our observable universe is of order 10 to the power, 10 to the power, 122 years. Where do we get that from? It's a long story involving the cosmological constant and the horizon and the entropy that Hawking calculated, etcetera. But the point is, this is a very, very big number. The universe right now is approximately 10 to the 10 years old. The recurrence time for observable universe is supposed to be 10 to the 10 to the 122 years. That's very, very much longer than that. But it's finite. [laughter] 10 to the 10 to the 122 is still a finite number, okay? And therefore the recurrence theorem says you will eventually come back to where you started. And this remarkable fact was pointed out not long after, it was emphasized, I guess, not long after we realized the universe is accelerating in a paper by Lisa Dyson, Matthew Kleban, and Leonard Susskind; Leonard Susskind, former Mindscape guest. They pointed out that the universe has a finite recurrence time. That's bad.

1:02:01.0 SC: Why is that bad? Well, one way of thinking about it is to go back to entropy again. Remember Boltzmann said that low entropy states are those that have a tiny number, relatively tiny number of arrangements of constituents that look that way; high entropy states have many, many arrangements, correspond to many, many arrangements? And in some sense, what the Poincaré recurrence theorem is telling you, this is not exactly the same result; I should be more careful here. There is a further thing that one might want to talk about which is ergodicity. Ergodicity is the idea that when you take your initial state and you set it off to follow the equations of motion, it will eventually fill up all of the allowed configurations or possibilities or states or whatever it is. That can't be exactly true because for example, energy is conserved, electric charge is conserved. There are conserved quantities that limit what are the allowed states you can be in.

1:02:57.9 SC: But ergodicity is within those constraints. The state wanders all around this space of possibilities and fills it up. And quantum mechanically, there are also analogous constraints there, but they're sort of a similar thing. Given all the allowed states, you will eventually reach them, and for very similar reasons that the Poincaré recurrence theorem says. So, the recurrence theorem says, given any individual state you start with, you eventually come back to it. The ergodicity idea is that you will get to all the possible states, okay? So, if the entropy tells you how many states look a certain way, and you eventually go to every possible state, if you're in a finite bounded space of possibilities, then you spend the vast majority of your time in what look like high-entropy states. Because most of the microstates are in high-entropy, macro configurations, right? That's kind of what it means to be high entropy.

1:03:55.8 SC: So, if you just have a system that evolves forever but in a bounded space of possibilities, most of the time it will look like high entropy, that is to say thermal equilibrium. In the case of the universe, that corresponds to empty space, what we call "de Sitter space," nothing going on whatsoever, the universe is completely emptied out. That turns out for subtle quantum gravity reasons to be the high-entropy configuration of the universe. That's not surprising, by the way, because that is where we are going to, right? Entropy is increasing, the universe is expanding, it's emptying out, eventually it will be perfectly empty. That is, the high-entropy state that we're eventually getting to.

1:04:37.0 SC: But then, because you're evolving forever, and there's only a finite number of things you can do, you don't stay high entropy forever; there will be occasional fluctuations downward in entropy. This is exactly what Boltzmann himself pointed out in the 1890s when he was trying to solve the problem of the arrow of time. It wasn't called that problem yet, but he was trying to understand why the second law had a direction. And he suggested, if you start in equilibrium and you just wait long enough, you will eventually fluctuate downward into a state of very, very low entropy, and then you will relax back upward into a state of high entropy again. Maybe that, Boltzmann said, is our universe. Actually, what he said was, is our Milky Way galaxy. He didn't even say the word "Milky Way." He said, it is "our galaxy." We didn't know at the time in the 1890s that there were any other galaxies. But Boltzmann was smart enough to know that in thermal equilibrium, life could not exist. So, he used what is essentially anthropic reasoning to say that you would have to wait around in a finite universe until you fluctuate into a low-entropy state and then back.

1:05:46.0 SC: Now, long-time listeners certainly know where this is going and where this was... It was pointed out in a correct but not the most elegant way by Dyson, Klebman, and Susskind, that there's a lot of thermodynamically crazy evolutions that you will eventually see if you go forever in a universe with a bounded phase space or Hilbert space or whatever. It was sharpened in a follow-up paper by Andreas Albrecht and Lorenzo Sorbo that coined the term "Boltzmann brains," the idea being that if all you were doing was waiting for an intelligent observer to appear, you don't need the rest of the universe to appear. Among all those possible states you could be in, and remember you will eventually visit all of them because there's a finite number of states effectively and an infinite amount of time for you to try all of them out, the most observers will find themselves in an otherwise empty universe. Indeed, maybe not even with the body, right? Just their brains, enough to look around and go, "Ha, thermal equilibrium," and then die, okay?

1:06:53.0 SC: That's the Boltzmann brain problem. And succinctly stated, the Boltzmann brain problem is, if you are in an eternal universe, which is what the Schrödinger equation predicts, right, the universe will last forever to the past and the future, and the universe evolves forever in some bounded space of possibilities, then when you look for any particular configuration you care about, either a person or a galaxy or universe or whatever, you will find the lowest... Sorry, the highest entropy possible version of that thing you are looking for. So, what is the highest entropy possible universe that can have an observer in it? An observer that is in thermal equilibrium other than a universe that is in thermal equilibrium other than just that one observer, or maybe even just the brain of that observer. That's the Boltzmann brain problem. Boltzmann himself did not point out this problem; maybe he could have.

1:07:46.8 SC: It was actually Arthur Eddington that first explained a version of it. So, it's Eddington's brains, really, that we're talking about here, but that was understood to be a problem. I'm not gonna talk about the Boltzmann brain problem in any great details here, but this is basically what I... So, sorry, let me just finish the story. So, Albrecht and Sorbo said, following Dyson, Klebman, and Susskind, they said this is clearly a worry about our favorite cosmological model. [laughter] The cosmological model suggested to us by the data, and it's absolutely not set in stone but it's just the simplest interpretation of what we see, is that there is a cosmological constant, it will last forever, do a little quantum mechanics to that, there's a finite number of states we can have in our universe, you should cycle through them in perpetuity forever and therefore most of us should be Boltzmann brains. That's a problem.

1:08:48.0 SC: And so indeed in that paper that I mentioned very quickly earlier, the paper I wrote called "What if Time Really Exists?" I simplified that issue down to its essence, forgetting about the cosmological constant and real cosmology or anything like that. If you have time evolution governed by the Schrodinger equation in the most straightforward normal way with a finite dimensional Hilbert space, you will have some version of the Boltzmann brain problem. Now, you can argue about whether or not the Boltzmann brain problem is a bad problem or not. I think it is a bad problem, and therefore I think that observers would not have any right to trust their own deductions about what the universe is like in that kind of large-scale cosmology. And that's problematic for all sorts of reasons. Okay?

1:09:42.3 SC: So, therefore I said in the "What if Time Really Exists?" paper, we know something deep about the universe. It can't be a finite dimensional Hilbert space and ordinary Schrödinger evolution. There you go. Even though we don't know much about quantum gravity, right, even though we don't know lots of different things, that is apparently not a viable alternative for a fundamental picture of how things work in the universe. I still think that's right. I mean, that narrows it down. So, remember, I want to say it again very, very specifically so you know what the assumptions are here. Ordinary Schrödinger evolution, finite dimensional Hilbert space. Those two things all by themselves give you a Boltzmann brain problem, irrespective of any details about quantum gravity, etcetera. So, one of those two things has to go, okay? It could be either one, we don't know. But that is an important implication for the nature of time. The simplest way for time to exist is for time to be eternal. And the implication of the cosmological constant is at least our observable universe has a finite dimensional Hilbert space. So, you might have thought that was a nice model. It seems not to quite work.

1:10:54.3 SC: Okay. And this is what led me to... So, the Boltzmann brain problem is something that I think was very important. And Kim Boddy, who was a graduate student at Caltech at the time, she and I worked on a paper about the Higgs boson, believe it or not. You might know that an interesting thing about the Higgs boson is, given its mass and given other things we know about particle physics, the thing about the Higgs boson is it's a field, the Higgs field, filling all of space. There is symmetry breaking because the Higgs field gets an expectation value, as we call it. A zero expectation value is what most fields have. There's a value in empty space. On average, if you measure the value of the field, it will be zero. The Higgs field is not zero. Because of its potential energy, it gets a value 200 and some GeV, I think it is, even in empty space. So, that is a vacuum state of the theory that has a lower energy than if the field were at zero. That's what causes the spontaneous symmetry breaking, gives masses to other particles, all the fun that is associated with the Higgs boson.

1:12:00.0 SC: People have asked, is that value of the Higgs boson the only allowed vacuum state? And of course, we don't know because there's a lot of things we don't know. But you can extrapolate, and under very, very reasonable circumstances it turns out that we're at least on the edge of having another vacuum state for the Higgs boson with a much larger expectation value and even lower energy, okay? So, what that means is, it's hard to visualize in your brain without being able to plot the graph, but it could be that we right now in the real world live in what is called a "false vacuum state." Vacuum state in physics just means state of lowest energy. False vacuum is a state that appears to be the lowest energy, but somewhere else out there in field space, in the space of possibilities, there's an even lower energy state. And then you can imagine a transition, it might be very slow, very rare, but it would eventually happen, where a bubble of the true vacuum forms, and then it grows and takes over the whole universe.

1:13:05.0 SC: As Sydney Coleman, again, my quantum field theory professor said long ago in a paper that he wrote, this would be the ultimate ecological catastrophe. If a bubble of true vacuum appeared in our universe and grew and hit us, we would all die. Because from our point of view, all the local laws of physics would change. So, the current knowledge of the Higgs boson puts us on the ragged edge of disaster [laughter] in terms of that. It suggests that maybe there could be another vacuum out there, and maybe we could transition into it, and maybe that would be the end of all life on Earth and indeed in our observable universe. Okay? Now, that would be bad, I think, because I like the universe, at least compared to the alternatives. But it would get us out of the Boltzmann brain problem. [laughter] That is to say, if what you're worried about is fitting the data, not the future of humanity, then the fact that we are ordinary thermodynamic observers, not Boltzmann brains, could be explained if the universe doesn't stay in this false vacuum state forever, but instead transitions to a better vacuum state.

1:14:17.0 SC: So, Kim and I wrote a paper, can the Higgs boson save us from the menace of the Boltzmann brains? And the idea would be, if the Higgs would eventually undergo a transition to a better vacuum state, that would make our current state of the universe only have a finite lifetime not long enough to make Boltzmann brains. And then we'd be safe from it. It'd be bad for other reasons, but at least we'd be safe from that empirical problem. Okay. We submit the paper, the paper was good, we did a good job. We submit the paper, and the referees, instead of worrying about our calculations dealing with the Higgs boson and the false vacuum, worried about the Boltzmann brain problem. They're like, "Yeah, I don't believe that problem. Not a big deal," [laughter] which is frustrating. Like, okay, you can believe that, but other people think it's a problem.

1:15:07.0 SC: But that made us think, we should just write another paper saying... I think maybe I thought this first when I shanghaied Kim into joining me and then Jason Pollack, who was another student, also joined us at some point. And I said, "We should just write a paper saying, 'Here's why the Boltzmann brain problem really is a problem.'" Okay? And eventually I wrote a paper by myself called "Why Boltzmann Brains are Bad." But that was only after we finished this paper, Kim and Jason and I, because step one in saying "Here's why Boltzmann brains are bad" was to say "Here's why there are Boltzmann brains, let us rehearse the problem," right? Rehearse the idea that there can be quantum fluctuations that would create brains, that would be observers, that are cognitively unstable, etcetera, etcetera.

1:15:55.2 SC: And what we realized is that the traditional argument for Boltzmann brains was not convincing. It depends on what you mean by the traditional argument for Boltzmann brains. If you go back to Boltzmann or to Eddington, pre-quantum mechanics days, and you just have a box of gas or some equivalent of a box of gas, then the Boltzmann brain argument is entirely convincing in my mind. In quantum mechanics, it's more subtle, okay? And the reason why people were talking about Boltzmann brains, the hand-wavy argument is that the equilibrium state that you get for de Sitter space, for a space with a non-zero vacuum energy, the state of the universe that we think that we are headed toward, as we said, that has a horizon, a la Hawkin, just like a black hole has a horizon, and has a temperature just like a black hole has a temperature.

1:16:50.0 SC: And so empty space has a temperature, which is not zero, and therefore there should be thermal fluctuations. There should be random motions that give rise to brains and planets and universes and whatever, if only you waited long enough, much like a classical box of gas. And so what Kim and Jason and I realized after banging our heads against it for a while is, actually that's just wrong. [chuckle] And that's just not thinking correctly about how quantum mechanics works. You can have a quantum mechanical state that you say is a thermal state, okay, a state of thermal equilibrium. But the quantum mechanical thermal state is a very different kind of thing than a classical thermal state. A classical thermal state is based on your ignorance. You have a box of gas and you say, "Oh, it's in a thermal state, it's in thermal equilibrium. What does that mean?"

1:17:40.9 SC: It means that in the box of gas, all the different molecules and atoms and whatever, they each have some specific position and velocity, but you don't know what they are. So, you assign them a probability distribution, and it's the maximum entropy probability distribution, and that's the thermal probability distribution, okay? But that's not what happens in quantum mechanics. In quantum mechanics, you have a wave function, or more generally, you have a mixed quantum state, which can be like a probabilistic sum of different wave functions, okay? So, we have all the technology to deal with that; that's what you learn in your quantum mechanics classes. So, you know what a thermal state looks like in quantum mechanics.

1:18:21.6 SC: But it's not the same character. It has the same observational consequences. If you have a quantum mechanical thermal state and you measure it, you will see different outcomes. Because it's quantum mechanics, right? You will see a distribution of possible outcomes for the positions and velocities of the particles just 'cause it's quantum mechanics. But when you're not measuring it, it's a very different kind of thing than the classical thermal state. Because classically, there is an actual microstate where things are moving. The atoms are moving around. You might not know what the actual microstate is, but it exists.

1:18:58.9 SC: Quantum mechanically, that's not the story. Quantum mechanics isn't like statistical mechanics. The quantum state corresponding to a thermal state is completely static. It is not changing over time. If you measure it over and over again multiple times, you will get different answers, again, because quantum mechanics. But in an empty universe, there's no one there to measure it. [chuckle] You need to think about the state for its own sake, not what measurement outcomes you would get if you were to look at the state. And the state for its own sake isn't changing over time. It's static, it's stationary, nothing's happening. So, we made the argument that in fact when the universe does expand and empty out and become empty de Sitter space with a temperature, so it is a thermal state, but it is not a fluctuating state. There's nothing happening from moment to moment that could pop into existence, right?

1:20:00.0 SC: When we talk in quantum field theory about virtual particles popping into existence, that's colorful language; that's not what actually is happening. In the proton, for example, people show these pictures of what it looks like inside a nucleon, and you see all these quarks and gluon fields fluctuating in and out. That's completely wrong-headed. That's not what is actually happening. The quantum state of a proton is exactly the same from moment to moment in time, and the same thing is true with empty space. If no one's measuring it, there's nothing happening, there's nothing fluctuating, there's nothing coming into existence, and certainly not Boltzmann brains or whole civilizations or whole universes, anything like that.

1:20:40.8 SC: Now, how does this get reconciled with the recurrence theorem? Well, the idea is that the universe is not a closed system in this case, right? There's a horizon around us and therefore features of our universe leak out; they leave us, right? Our universe is not a closed system. It is finite dimensional Hilbert space, but it is connected to an apparently infinite dimensional Hilbert space representing the entire rest of the universe. Now, again, that may or may not be true, so Kim and Jason and I were very, very clear. If what is going on is that our observable universe is described by a finite dimensional Hilbert space that is embedded in a larger infinite dimensional Hilbert space, then it can just settle down. Our universe can go into a completely, can asymptote to a completely stationary state where there are no dynamical fluctuations.

1:21:37.7 SC: And so we distinguish between dynamical fluctuations and Boltzmann fluctuations and things like that. But nothing would happen; no actual brain would ever pop into existence. And therefore under those circumstances, there is no Boltzmann brain problem. If the whole Hilbert space is infinite dimensional, Boltzmann brains would not dynamically pop into existence. Okay? That paper, we also got published. We did get that one published. It also went through referee problems 'cause people just don't like the Boltzmann brain problem, but eventually it sailed through.

1:22:11.2 SC: Okay, so, let's take a step back and see where we are because we're trying to think about the nature of time here, right? So, we've been talking about the possibility that time is fundamental; that it's right there in the Schrödinger equation, taking it seriously. And what we're finding is, an interesting thing that will continue to be true, which is that whether or not time acts in a certain way in a way that is compatible with the universe as we know it, also depends on how space behaves, or more generally, the space of possibilities. If you just have the Schrödinger equation and time is fundamental, then time is eternal. And then if the space of possibilities is bounded, aka Hilbert space is finite dimensional, then you run into troubles with the Boltzmann brain problem with recurrences and fluctuations and so forth.

1:23:00.0 SC: If the space of possibilities is unbounded, then time evolution can go eternally in a way, that at least in principle maybe, plausibly, could be compatible with the universe we see. In some ways, the paper I wrote with Jennifer Chen back in 2004 on a double-headed arrow of time, a universe that was statistically symmetric to the far, far, far past and the far, far, far future, is an implementation of this kind of idea. We were semi-classical in the way that we talked about it with baby universes and so forth, but it's the same basic strategy. And the reason why it works is because in the specific version of the... In the specific understanding of the more or less realistic idea that we live in an accelerating universe which will eventually settle down into a static state, you don't have fluctuations in that static state, so you don't need to worry about the Boltzmann brain problem.

1:23:51.8 SC: Okay. So, that's good. That's a helpful connection between the space of possibilities in space and matter and energy and those things, and the possible behavior of time. But guess what? Not everyone agrees. [laughter] So, in particular, we wrote the paper saying, "Look, you're free of the Boltzmann brain problem if you settle down to a stationary state, because quantum mechanically it's not like a thermal state in classical mechanics where secretly things are moving beneath your notice, it's that things truly are static." Other people say, "Well, no, even if the quantum state is static, quantum mechanics is very subtle, and things can still happen in the universe even if the quantum state is not changing over time." Okay. That's a dramatic thing. And I still to this day have not correctly wrapped my mind around it. I don't even know if it's true or not. This is exactly the thing that is sort of on the table, whether we can understand this correctly.

1:24:52.0 SC: In the specific example considered here was Seth Lloyd, a quantum information person, a very successful physicist at MIT, who wrote a paper responding to ours saying, "No, no, no, even if the state of the universe is static, there still can be things that happen." So, what does that mean? Well, remember, and I'm gonna try to simplify the technicalities here a little bit, the actual way that Seth talked about it was using the decoherent histories formalism. This is a way of thinking about the evolution of quantum systems by talking about under what circumstances can we think about an evolving quantum system as describing a set of individually coherent histories, "coherent" in the informal sense, not the formal sense. Actually, you want them to be decoherent in the formal sense. You want them to separate from each other.

1:25:46.7 SC: The standard example here is the double-slit experiment. When a single electron passes through slits and interferes on the other side, then that is an example of two histories: The electron goes through one slit, the electron goes through the other slit, that don't decohere from each other. They will interfere later on, they still are part of the same world, okay? And therefore, people like Griffiths and Omnis and Hartle and Gell-Mann have developed this formalism to say, you cannot actually separately calculate probabilities for the question "Did the electron go through the left-hand slit or the right-hand slit?" They both are there. They're not separate probabilities. They interfere with each other later on. Whereas, if you observe which slit the electron goes through, then decoherence happens. And now you don't have interference on the other side of the double-slit experiment, and now you can attach probabilities to whether or not the electron went through the left-hand side or the right-hand side.

1:26:50.2 SC: Okay? So, there's a whole formalism built up about this, and this is what Lloyd used when he talked about his objections to our paper. But I don't think you need to really dig into it, although we'll bring it up again. The point is that, once again, quantum states have this feature that two different states can be added together to get a third state, and if the two states you started with solve the Schrödinger equation, then the state that you get by adding them will solve the Schrödinger equation, okay? So, very roughly speaking, what Seth pointed out is that even if your overall quantum state is static, you can think of it as the sum of various states that are not static, that are changing with time. If you think about waves going up and down, you can imagine two waves, each of which individually are changing with time, but their sum remains constant, okay? If that's not quite obvious to you, think about a constant function, okay, a function between zero and one that just takes on the value one, and subtract a wave from it.

1:27:57.2 SC: And so take another function that is wiggling and then changing with time, okay? Take one minus that function, so then your wave function plus one minus your wave function is just one, which is just a constant. So, you have two individually changing things that can add up to a constant. And what Seth points out is, I can take a static quantum state and I can divide it into a set of histories of things happening, things changing over time. And when I add them all up, I get that static quantum state. But he says, "I'm allowed to think of it as a superposition of many things actually happening dynamically," including Boltzmann brains popping into existence, etcetera. And then there's a lot of math involved with, okay, showing that that's realistic and so forth.

1:28:46.8 SC: In fact, this is a version of something that Fay Dowker and Adrian Kent pointed out a long time ago, and was a inspiration for the paper that I wrote on quantum mereology with Ashmeet Singh, and Dowker and Kent called it the "set selection problem." So, mathematically, what Seth Lloyd says is completely correct. You can't argue with the equations in this case; it's trying to understand what the equations are saying. Okay? So, it is simply a true fact that you can take a quantum state that is not changing over time and express it as a set of histories of things that are changing over time. The question is, should you consider those changing histories as really happening? And I will give you two arguments that you should not. Seth says you should. I'm gonna give you two arguments that you should not.

1:29:40.9 SC: One argument is, it would be terrible [laughter] if it were true. When I expressed the original version of the Boltzmann brain problem, that came about through work of Dyson, Kleban, Susskind, then Albrecht and Sorbo, what they were thinking was that in the thermal state that you get at the end of cosmic evolution or the long-term limit of cosmic evolution in the presence of vacuum energy, there's a temperature. And the temperature, they were thinking that the temperature implies fluctuations just like classically it would, and therefore there's a probability for things happening that depends on that temperature. And what Kim Boddy and Jason Pollack and I pointed out is that that's not true in the precise way they were thinking of it, because it's a different kind of temperature, okay? There's not actual thermal fluctuations associated with it. And so what Lloyd is saying about even in a static state, you can have dynamical histories that sort of constitute that static state. That would be true whether or not there was a temperature. That's true even in the vacuum state of empty space without a cosmological Minkowski space, right?

1:30:56.0 SC: You can have literally the emptiest possible quantum state, a universe with nothing in it, no vacuum energy, no particles, no vibrations of any sort; everything is absolutely static; and you can express that as the sum of all sorts of things happening, changing over time. That's just a feature of quantum mechanics that follows from exactly the same logic that Seth Lloyd used. You can think of it this way, and he actually pointed this out in his paper. The simple harmonic oscillator, right? One little quantum system that we think we understand, the ground state to the simple harmonic oscillator. An oscillator that is sitting there doing nothing to the extent that it possibly can in quantum mechanics, can be decomposed as the sum of various things changing over time. And what that means is, that if you take this attitude that all those histories that you're inventing to sum up to make an overall static quantum safe, if they count as really happening in some sense, and this is where the philosophy comes in, what counts is really happening, right? If they count is really happening, then essentially everything you can conceive of is happening even in empty space all the time.

1:32:09.0 SC: Civilizations are arising and falling in the vacuum of quantum field theory. Okay? So, is that what you want to believe is true? This is a terrible argument; this isn't a very convincing technical argument, but I'm just trying to push forward the implications of this view. If you say, I can take a static quantum state and decompose into a sum of dynamical, changing quantum states, and I treat every one of them as really happening and truly real, then that has wild implications for what you think is truly real. Then you sort of get time evolution, even though you're in just empty space and nothing is naively happening at all. So, maybe that should bother you, maybe it shouldn't, but you have to come to grips with it anyway.

1:32:57.1 SC: A slightly more respectable technical objection is, yes, I can take a static quantum state, I can divide it up, express it as many different dynamical states that just happen to be dynamical in the right way to add up to no dynamics, that's what he's proposing doing. But there's more than one way to do that. In fact, there are an infinite number of ways to do that, and this is what Dowker and Kent pointed out. There's a set selection problem. You can, in the context of the decoherent histories formalism, if you have a set of histories, a single set, you're saying, "We'll look at this history, that history, this other history, and there's a complete set," it includes everything that could possibly happen, then the point of decoherent histories is, if those histories decohere from each other, if they don't interfere, okay, then you can assign probabilities to them and you can say, "Yes, 20%, this will be the history, you see 70% something else," and so forth. But you can't assign probabilities between different sets of histories.

1:34:00.2 SC: So, if you take the static wave function and express it as a sum of things, you can do that in many, many different ways, and there's no guidance whatsoever as to what is the right way. If you can do it one way, you can do it any number of other ways, and they should all by the rules that you've just invented, be counting as equally real. Do you really want that? [chuckle] It certainly removes any possibility that quantum mechanics ever predicts anything or that assigns a probability to anything, 'cause literally almost everything happens all the time in almost every quantum state. I don't think that's what we want quantum mechanics to do. So, somehow I firmly believe that can't be the right interpretation; that can't be the right way to think about what these equations are telling us. But we need to come to grips; we need to figure out. This is not an unanswered question. This is something I'm thinking about, other people are thinking about. How do you decide what actually happens in that non-evolving quantum state, okay?

1:35:03.0 SC: I will tell you my guess. My guess is that it has something, once again, to do with the arrow of time. I think it has to do with emergence, in other words. When do things really happen? I think that, again, this is very hand-wavy 'cause we don't understand it completely, but there is a classical limit of quantum mechanics, right? And sometimes you're taught the classical limit is just like when things are big, they behave classically. That's leaving out a little bit of the discussion. In quantum mechanics, when things are big like a planet, they "can" behave classically, but they don't "have to" behave classically. I can take the wave function of a planet like the Earth, and I can mathematically construct a wave function that puts the Earth in a superposition of being here and on the other side of the sun, and some of it is in the Andromeda Galaxy, and so forth, okay? Highly non-classical behavior, even though the Earth is a very big thing.

1:36:00.3 SC: What of course happens, at least from this many worlds perspective, is that the Earth keeps interacting with the rest of the universe and it decoheres and branches; and on each individual branch, the Earth looks more or less like a classical object like we know and love and live on. So, you get the branch where the Earth is here, the branch where the Earth's on the other side of the sun, the branch where it's on, in Alpha Centauri, and so forth, okay, or the Andromeda Galaxy. And so that process of decoherence and branching relies on the arrow of time, relies on dissipation, relies on the fact that the whole state is very far from thermal equilibrium. And so my rough feeling is that that aspect, the aspect of entropy increasing, dissipation happening, decoherence occurring, is missing from this point of view that says, I can take a quantum state that is perfectly static and just divide it up into many different histories and then consider them all to be real. I think that those histories that you're considering are not ones that are living in any classical emergent reality. And maybe that's a quasi-anthropic argument, I don't know. Maybe the point is that you can't get observers in those trajectories. I honestly don't know. That's what I'm saying. We need to think about it.

1:37:20.2 SC: And so now I can get into the even deeper reason why we need to think about this. So, the issue that I've just raised is, do things happen in static quantum states? And one way of saying this is, can time emerge when it is not fundamental? So, so far, we've been talking about ideas or theories or frameworks in which time is truly fundamental. It's right there in the Schrödinger equation and the quantum state is changing over time. And we ran up into this issue that, okay, maybe if there's an unbounded space of possibilities, even though there's time evolution, the local, visible, observable part of your universe settles down to a static quantum state. And then we ask, "Well, okay, are things still happening in that static quantum state? Are Boltzmann brains or whole civilizations randomly fluctuating into existence?"

1:38:16.0 SC: So, okay. This is an important question, but it is right up next to another important question, which is, what if time isn't fundamental in the first place? And why would you ever think that? Well, why would you think that is because of quantum gravity. So, again, we don't understand quantum gravity perfectly well, but we understand a little bit about quantum mechanics, a little bit about gravity. We can lump them together and see what happens. So, you just follow your nose and imagine that general relativities like any other classical theory, let's try to quantize it, see what happens. And this was done years ago, right? In fact, the famous equation that pops out of this perspective is called the Wheeler-DeWitt equation, named after John Wheeler and Bryce DeWitt, who cooked up this equation, I don't know, in the '60s or '70s or something like that.

1:39:08.2 SC: It was literally just take general relativity as a classical theory, quantize it the same way you would quantize anything else, and what you find is that those features of general relativity that we talked about before, that time is not absolute, right, that time is a little bit loosey-goosey in general relativity, that you can slice your universe in all sorts of different ways, this is something that the fancy language for is called "diffeomorphism invariance." You can choose to coordinatize your spacetime manifold in an infinite number of possible ways, and they're all equally good. That's the diffeomorphism invariance. And what that means is that, time kind of has a different feeling, a different status in general relativity. And like I said, at the classical level, that is a technicality that you have to be aware of, but it's not truly changing your view of the theory: You solve Einstein's equation, and you get on with your life. At the quantum level, it becomes really important, precisely because the ordinary Schrödinger equation treats time as fundamental. The ordinary Schrödinger equation says, you give me the quantum state, I apply the Hamiltonian to it, and what that tells me is the rate of change of the quantum state with respect to time.

1:40:32.9 SC: In general relativity, this symmetry, this diffeomorphism invariance takes that statement and says, what actually happens in quantum gravity is you give me the quantum state, I apply the Hamiltonian to it, and it doesn't change with time. That is to say, the ordinary Schrödinger equation says Hamiltonian acting on quantum state is the time derivative of the quantum state. The Wheeler-DeWitt equation says, Hamiltonian acting on the quantum state equal zero. So, there is no time variable in the Wheeler-DeWitt equation. The Hamiltonian "annihilates the state," is the way that we technically say it, H psi equals zero is the Wheeler-DeWitt equation. So, time has disappeared. In the Schrödinger equation, we have d by dt, the derivative with respect to t, time of the quantum state. And according to the naive version of quantum gravity, that should vanish. There's technicalities there, it's strictly speaking true if you have a compact to spatial universe, but versions of it should be true even if you don't. So, we're not gonna get into those details.

1:41:48.1 SC: So, this feature... So, in other words, just to let it sink in, when we're taking the attitude that time is fundamental, that it's right there in the Schrödinger equation, and we weren't worrying about specific things that we might think are true about quantum gravity and so forth, we said, "Well, maybe you eventually evolve to a state that's not evolving with time," okay? Maybe the universe sort of approaches asymptotically some equilibrium state and it just stays there. And now we're saying, okay, now we are taking seriously some implications of the straightforward naive version of quantum gravity, and it says that there isn't any time evolution at all; not just eventually, but just as a fact. This seems naively to be in contradiction with the world that we live in, right? I look around, there's clocks. I think that time is part of my life, and the Wheeler-DeWitt equation says, "No, it's not, not really."

1:42:41.8 SC: So, this is called the "problem of time" in quantum gravity. The problem with time is different than the arrow of time. The arrow of time exists even classically, and there's the question of the arrow of time: Why is it there, where did it come from, etcetera, etcetera. The problem of time is specific to quantum gravity, and it just is the question, why is there a time at all if the fundamental equation of quantum gravity doesn't have any time in it? That's the problem of time. And of course, obviously, for obvious reasons, people have thought about this a lot. This is not something that we're newly approaching here. And guess what? The motto is, the motto is, well, time must be emergent. This is something that has been stated by people thinking about quantum gravity for many decades now, that if the Wheeler-DeWitt equation is on the right track, then time should somehow be emergent. Is the Wheeler-DeWitt equation on the right track? It's hard to say. I think that there's arguments either way.

1:43:44.2 SC: Clearly, quantizing general relativity in the most naive way is not the entire story of quantum gravity. It doesn't work. It blows up. It gives you nonsensical results. But maybe in some big, infrared, effective field theory kind of sense, every version of what's going on at small scales, whether it's strings or loops or whatever, should smear out and look like general relativity quantized, at the end of the day, and therefore something like the Wheeler-DeWitt equation should hold. After all, it just really depends on the symmetry that the theory has, this diffeormorphism invariance symmetry. So, maybe that should be robust even if we don't understand all the details of quantum gravity. But because we don't understand the details, you have to keep an open mind. That's why I'm laying out all the different possibilities. Maybe time is fundamental, as we were just talking about before, but maybe ala the Wheeler-DeWitt equation it is not, and we have to get it to be emergent.

1:44:39.0 SC: So, what would that mean? How could time emerge if the fundamental equation of science, of physics, is the wave function of the universe never evolves with time? Well, happily, quantum mechanics giveth and quantum mechanics taketh away. And so it took away time from us, but it gives us something that we already discussed: This fact that quantum states are vectors and can be added together. And by itself, that might not seem like a big help. But the point is that, you can imagine that in quantum mechanics everything is relational. In other words, that time is not out there, it is not absolute, it is not an external parameter saying how things are evolving from moment to moment. Rather, in this point of view, time is just a statement about the relations between different things going on in the universe. It's, in other words, we might try to...

1:45:38.8 SC: Sometimes people glibly say, "Time is what clocks measure," right? This point of view that I'm trying to explain here is an operationalization of that. It's really taking seriously the claim that time is what clocks measure. You can, in other words, have a quantum state that is not changing with respect to any absolute out there existing time. But within the quantum state, there can be things that you call clocks, and they can be in superpositions of many different possible clock readings, and those superpositions of different possible clock readings are maybe entangled with the rest of the universe. So that when the clock reads something, or in the part of the quantum state where the clock has a certain reading, that counts as, that is what time it is for the rest of the universe. That's the basic idea for time trying to emerge somehow, and the question is, can you make it work?

1:46:39.0 SC: So, the way I like to think of it is this way, the way I like to warm up to why this might be a plausible thing. In quantum mechanics, states are vectors; quantum states can be added together. If I take the universe as a whole or any sufficiently big quantum system, imagine I take two different configurations of the universe, two different quantum states the universe might be in, and I add them together. You might think, from your experience adding vectors together, if you add a vector that is pointing in one direction and the vector is putting in another direction, you get a third vector, the sum of them, but you can't go backward, right? You can't say, "Okay, here's the sum of two vectors, what are the two vectors I added to originally to get them out?" right? If I say I have two and I have three, these are vectors in one-dimensional space, right, the vector two and the vector three, I add them together to get the vector five. That's fine, but if I say I have the vector five and I added two vectors together to get it, what were the two vectors? You have no way of knowing. Was it two and three? Was it six and minus one? There's an infinite number of possibilities.

1:47:47.3 SC: However, it turns out that our intuition is not very good on this score, because our intuition is built up in one-dimensional, two-dimensional, three-dimensional vector spaces. In the huge, huge, huge dimensional vector spaces that describe quantum states, it turns out that when your vector spaces get very big, almost all vectors are perpendicular to each other. You give me two random vectors, I take their dot product, their overlap, and it'll be very, very, very close to zero unless I worked really, really hard. And this is something that is a little bit counterintuitive. Let's put it this way. If I take a box of gas with a bunch of atoms in it or a bunch of photons or something like that, and I consider it as a true quantum mechanical system so I have the quantum state of that, here is a true fact. If I add a single particle like a single electron or a single photon or whatever to that box of gas, there's many, many, many, many particles in it, I add just one particle to it, classically, I have not changed the state very much, right? The overall mass and pressure and things like that have not been changed very much 'cause I just added one particle.

1:49:00.8 SC: Quantum mechanically, the quantum state that I get by adding that one particle is perpendicular to the state that I started with. They are orthogonal to each other. So, quantum states very, very rapidly and easily become perpendicular to each other; "orthogonal" as we like to say in quantum mechanics. And so that at least gives us hope for picking them out from each other, for reverse-engineering from the sum of two states back to the original ones we added together. If we have some basis, some set of preferred directions in this vector space, and I know that the states I'm adding together are sort of perpendicular to each other, then I can try to reverse-engineer what they were. Okay. So, that's a bit of math, sorry about that, that I wanted to let you know about. So, you might think when you add together two different quantum states of the same system, you just lose a lot of information, it becomes mush. But at least in principle, maybe under the right circumstances, you can figure out what are the two states I added together.

1:50:02.0 SC: Especially if you have some extra criteria, right? Like, you're looking for classical states. If I take two quantum states that are behaving relatively classically, like a well-localized, sensible wave function, if I generally take two well-localized wave functions, the wave function I get is not well-localized. If I take a wave function where a particle is located near x and a different wave function where a particle is located near y far away, those two are separately, classical-looking particles localized. But I add them together, now the particle is not localized. There's part of it near x, part of it near y, okay? So, if I say, well, I know that the two states that I added together both look classical, and here's what the result looks like, a little lump near x and a little lump near y, I can go, oh yeah, okay, you added together a classical looking state with a particle near x and a classical looking state with a particle near y, I can figure that out. I can disentangle them, okay?

1:51:01.2 SC: Good. Keep that in mind. And now consider an ordinary time-evolving system, okay? So, imagine a quantum system that is changing with time. Forget about emergent time, etcetera, just have the ordinary Schrödinger equation, take a quantum state, plug it in, let it evolve with time. And now imagine recording, solving the Schrödinger equation and writing down what the state looks like at t equals zero, and t equals one, and t equals two, etcetera. Maybe one, two, maybe these are measured in nanoseconds or something, right? Like very, very quick snapshots of the quantum state of the system at slightly different moments of time, right? So, I could do that. Here's what I could do in quantum mechanics. I could take all these different quantum states, which represent this big system at different moments of time, and I can just add them together. I'm not saying that the universe does this. I'm saying I can do this. I could imagine that time-evolving state, take snapshots of it at discrete moments of time, every nanosecond, and then I add them together, and I get one state. Okay? 'Cause I can add them together; I'm allowed to add quantum states together. I'm allowed to add, have one state, and they're all distinguishable because all the different states are gonna be more or less orthogonal to each other.

1:52:16.9 SC: I've basically taken all of the information that was contained in a single quantum state evolving with time, and encoded it in a single quantum state that is not evolving in time, right? By taking snapshots of the time-evolving state at different moments and then adding them together into one big state. And if I'm in a nice situation where things are more or less classical looking, etc., maybe I can even go backwards. Maybe I can even take that one big state and extract all of the behaviors of the state at different moments of time and then arrange them in the right order, okay? So, I have basically encapsulated time evolution into a single state that itself is not evolving with time. And all I'm saying here is that, I'm trying to convince you that we can imagine in quantum mechanics in a way that you can't imagine in classical mechanics. You can imagine time being emergent in the sense that I could have a quantum state that is not evolving with time, but really I can think cleverly of a way of expressing that quantum state as if it were a quantum system that was really evolving in time because it's really the sum of those time steps in that evolution. The question is how to make that work, how to operationalize it, okay?

1:53:42.0 SC: And again all this is stuff people have been thinking about for a long time. So, one of the classic papers here was written in 1983 by Don Page and William Wootters. It's well enough known that it's now called the "Page and Wootters mechanism." And they did a very simple thing; very, very, nice simple thing. They basically said, okay, imagine I have a quantum system... They were inspired by the Wheeler-DeWitt equation but they didn't look at that explicitly. They just said, imagine I have a quantum system that doesn't evolve with time. It's, psi is just static, psi is the wave function of the universe; the quantum state, it's not changing over time. But imagine that I can decompose the quantum Hilbert space, the space of all possibilities; I can break it up into two subsystems which is very, very plausible; very, very easy to do in a wide variety of circumstances.

1:54:36.0 SC: And they say, so I have one subsystem that I'm going to call the "clock subsystem" and the other subsystem is the "rest of the universe" subsystem, okay? So, the rest subsystem. And basically if the whole system is not evolving with time, then they can set it up so that the clock subsystem has different possible values. When you observe a quantum system, you get different possible answers. And so, let's imagine that whatever this subsystem is, it's clock-like. It's something that we can measure, like literally looking at your wristwatch and looking at what time it is. And they show that what very naturally happens is that there is entanglement between the clock subsystem and the rest of the universe with the property that there is an emergent time evolution to the rest of the universe; that if you think about arranging the clock states, the states of the clock subsystem in order, zero, one, two, three, etc., then those are separately entangled with states of the rest of the universe...

1:55:45.0 SC: The state of the rest of the universe at time zero, the state of the rest of the universe at time one, the state of the rest of the universe at time two, where this is not real time; this is emergent time; it's not fundamental time; it's real, as we said at the very, very beginning. But it's emergent. It's emergent because it's literally what is being read off from measuring the clock subsystem, okay? So, at every value of the clock subsystem, the rest of the universe looks like it has evolved to that time. And there's some technical details there. It really looks like Hamiltonian evolution, etc. It's kind of marvelous, it's kind of wonderful. So, it at least is plausible that time emerges in quantum mechanics in exactly that way. There's a whole 'nother tradition of, rather than taking this abstract quantum mechanics for its own sake point of view, taking quantum gravity seriously and giving variables that describe the universe that might be the scale factor of the universe or the value of some scalar field and using those as clocks.

1:56:48.2 SC: So, the game to play in quantum cosmology, the study of quantum mechanics applied to the whole universe, is to pick out a clock and let that be an emergent time variable. Okay? And this is something that in principle is doable. And I think that's the state of the art, honestly. There's been improvements over time, as it were. People have done some technical details. The Page and Wootters paper is pretty simple. They make a lot of assumptions to simplify their lives, which is completely legit, and people have complicated those assumptions over time and looked at it more carefully, and great. And this is still, I think, the cutting edge of how we think about the emergence of time in quantum mechanics to this day.

1:57:36.0 SC: Now, but I don't... What can I say? People haven't looked at this seriously enough. It's one of those cases where we live in a universe that is very classical. We know what the universe is like in some ways, so we know what answer we want to get when we look at quantum mechanics and we say, "Please correctly describe the universe." And what that leads to is a bit of carelessness, a bit of sloppiness. We say, "Well, maybe if you think about it this way, you get the right answer," and people go, "Oh, yeah, okay. Yeah, that looks plausible," they move on with their lives. 'Cause there's other things about the universe we don't understand, right? Why does the Higgs boson have the mass it does? Why is there more matter than antimatter? And those seem like pressing problems. Whereas "How does time emerge?" seems like, well, it must happen. And here's a somewhat plausible story about it, so we'll accept that and move on. And I don't think they've really come to grips with it as seriously as they should.

1:58:33.8 SC: One exception is Andy Albrecht, the same guy who with Lorenzo Sorbo wrote about Boltzmann brains back in the day. Andy and I are very sympathetic in terms of what we think are interesting physics and cosmology problems. So, we end up working on the same thing very frequently. So, Andy wrote a paper with... It was Albrecht and Iglesias on what he called the "clock ambiguity." And there was a previous paper he wrote, and I'm not gonna get the title right, but it was something like "a theory of everything or a theory of anything." So, here's the issue. Don Page and Bill Wootters said I can get time to emerge in a quantum state that is not actually evolving in time by dividing the quantum system into the clock subsystem in the rest of the universe, then the rest of the universe appears to evolve with respect to the reading on the clock. And Albrecht and Iglesias say, okay, sure. What if I divide up the space of the quantum states differently? So, you chose a way to divide up the space of quantum states into clock plus rest of the universe. I could take a different subsystem to be my clock. I could make the clock subsystem be part of yours, but part of something else, right? There's literally an infinite number of ways to divide up...

1:59:55.2 SC: It's literally like slicing a pie, right? I can slice any one pie, in principle, in infinite number of ways. And what they argued is... And here, this is something one could agree or disagree with. People have not taken it seriously enough, so I don't think that the dust has settled even though the first paper was, I don't know, over 20 years ago. What they argued was, that they could essentially slice up the Hilbert space of possibilities in enough different ways that they could get the clock and the rest of the universe to do whatever you want. It's reminiscent of this thing where in the static state where Dowker and Kent are complaining about; I could express a static state as a set of time-dependent histories in an infinite number of ways; I can get whatever I want. And Albrecht and Iglesias are saying same thing with the sort of Page and Wootters emergent time mechanism: I can get time to emerge in any way I want. And what that means is, I can basically get the effective laws of physics for the rest of the universe to be whatever I want.

2:01:04.0 SC: And they're not changing the laws of physics. In this picture, the laws of physics are supposed to be given to you and all you're doing is taking a solution to the Wheeler-DeWitt equation to h psi equals zero. This quantum gravity replacement for the Schrödinger equation or specific version of the Schrödinger equation, I should say. And you're supposed to say, here's what this means in terms of an emergent time evolution, and Albrecht and Iglesias are saying, I can get what... Well, time evolution means laws of physics, I guess is what I'm trying to say here is, right? It's not just that things change with time; they change in a particular way, and that way is supposed to be a reflection of the underlying laws of physics. And Albrecht and Iglesias says you can get time to emerge in a sufficiently large number of different ways that in exactly the same quantum state, you can talk about it as if the laws of physics or whatever you want them to be.

2:02:01.3 SC: That's a problem, right? That's what they call the "clock ambiguity." This freedom of dividing the quantum space of possibilities, the Hilbert space, into subsystems is of course exactly the problem of quantum mereology that I've talked about many times. But I was talking, always talking about it in the vastly simpler case where you imagine that time really is fundamental, that the Schrödinger equation is describing time evolution, and we're using, in the paper that I wrote with Ashmeet Singh about quantum mereology, how to divide Hilbert space up into subsystems, we absolutely took advantage of the fact that time evolution was real and is fundamental, and we could use that to say, okay, certain ways of dividing the Hilbert space up makes sense and certain ones don't, okay?

2:02:49.1 SC: Here, where you don't have real time evolution, your freedom is vastly greater, and you can just divide things up however you want. So, that's it; that's the state of the art right now. This is what I'm trying to get you to. I don't know whether time can be emergent or not. There's other wrinkles here. As you might tell, my voice is running out. I'm at the tail end of a cold. It's that time of year. I can't talk for too much longer. But there's other wrinkles here. One other very interesting wrinkle is an idea called the "thermal time hypothesis," which was proposed by Ellen Kohn and former Mindscape guest Carlo Rovelli. They're saying, well, what if the universe is in a thermal state and they can use features of the thermal state to get an emergent time parameter? But I think it has exactly the same sets of issues here.

2:03:42.0 SC: So, here's the upshot of this. And if time is supposed to be emergent in the sense that the real, true quantum state of the universe is not changing with time, but we think that we can divide the universe into clock subsystems in the rest of the universe, there is the worry that there's too much freedom in that, and that as a result of all that freedom, there's actually no answers, no sensible answers to how time really emerges in that particular system. Okay. So, that is the state of the art right now, and I will tell you my guess. Now, this is what I'm doing for a living when I'm not doing podcasts and things or writing books. When I do physics research, this is one of the things I'm thinking about very seriously, which is again, I've already given it away, but it's that we need to take decoherence, classicality, and the arrow of time more seriously. So, the game that is being played here is, quantum mechanics gives us a lot of freedom to divide up the space of possibilities in different ways, and that freedom is too much, say Albrecht and Iglesias. That's the clock ambiguity. And my suggestion is, well, yes, but some of those ways are not going to give rise to nice, emergent, classical descriptions of space and the things in it and the evolution of those things in it. Okay?

2:05:11.0 SC: So, again, the lesson being, maybe the way that time emerges depends on the way that space emerges, and in particular in a way that allows you to get a classical or semi-classical description. And this, by the way, opens an interesting possibility. I don't know if that's true. That's like a guess, okay? Or let's call it a conjecture; that sounds more sciencey. It opens an interesting possibility. I said before that if time is fundamental, then you have two choices. The Hilbert space, the space of possibilities, is finite dimensional or bounded. And in that sense, you're in trouble because of Boltzmann brains and recurrences and fluctuations. If it's unbounded, you have at least a possibility of getting an empirically adequate description, a description of cosmology that matches the universe we see, because you can have time evolving forever but not come to the same place over and over again. That's one possibility, I think, is absolutely alive and on the table. Maybe time is just fundamental. Maybe it's just there. That's absolutely worth taking seriously.

2:06:19.0 SC: The other one is that time is emergent. It is not fundamental. And then I think the hope would be the following. What if the space of possibilities is bounded? Okay? So, the Hilbert space is finite dimensional. The problem before was that if you have real time evolution and that time evolution is eternal, if you have a bounded space of possibilities, you will explore all of it and you'll explore all of it again and again, and that's bad. But if time is not fundamental, then the space of possibilities can be finite dimensional. And maybe the way that time emerges basically says there's only a finite number of ticks of the clock. Okay? The emergent clock doesn't tick forever. It doesn't go forever. There really would be a beginning to time and an end to time.

2:07:11.0 SC: Or maybe it's cyclic, that's also a possibility, but at least it's finite in scope, okay? So, maybe you can escape the menace of the Boltzmann brains or recurrences, etc., if time is emergent and there's only a finite number of ticks of the clock. And maybe that helps explain why the Big Bang is so special, right? It's a boundary. Maybe this is a way of having the Big Bang truly be the beginning of the universe. And it raises the specter or the possibility or the prospect, however you want to put it, that not only was the Big Bang the beginning, it was time equals zero. But there is also a last moment of time. Because time is only emergent anyway, and the whole space of possibilities is finite dimensional. So, somewhere in the future, we will reach the end. And you might go, "Well, what will that be like? Will we hit a wall? Will we hit a singularity?" And the conjecture would be, no, that's not what it is like.

2:08:08.9 SC: Rather, we just approach equilibrium, right? It's very much a fading away rather than a burning out. The universe ending with a whimper, not a bang, in that sense. So, in that case, if that's all anything close to true, you don't notice anything 'cause nothing is happening. The set of things that happen gradually cease to become interesting as you approach the later moments in the history of emergent time. So, that's the other way I think it can work. There's basically two possibilities I think are alive. One is that time is fundamental and eternal, and the space of possibilities is infinitely big, and we seem to live in a universe with time evolution because the universe is always evolving and there's always an arrow of time. The other possibility that makes sense to me is that the space of possibilities is finite, time is emergent, and for some reason that I have no understanding of whatsoever right now, the way that it emerges begins with a very, very special low-entropy state, and then sort of expands and cools the way that our Big Bang did to some finite period of time in the future.

2:09:19.0 SC: Maybe the reason why it does that is because that's the way to get semi-classical and properly allowed descriptions or something like that. I honestly don't know, but that is something we're thinking about. So, this is not connected very strongly to how we would teach quantum mechanics to undergraduates. You teach the simple harmonic oscillator and you treat it as a spherical cow, right? You imagine the simple harmonic oscillator is all by itself in the universe. But the lesson, the moral of this story is, the rest of the universe matters; that we need to take into consideration the way that you measure things, the way you experience things; what you mean by time passing and all that stuff, and all that relies on the idea that there is a classical limit, that there are individual branches of the wave function in which we can coherently talk about stuff happening that more or less resemble classical evolution.

2:10:17.0 SC: Is that really necessary? Is it the only way to go? I truly don't know. I think these are very, very good open questions. So, time, something we don't have a complete 100% understanding of. Is it real? Yes. Does it exist? Yes. Is it fundamental? Is it emergent? Even if we think we completely understand the equations of quantum mechanics, we don't yet know the answers to those questions. That is both kind of sad because we've had 100 years to figure it out since the advent of quantum mechanics and we haven't done it yet, but it's also heartening. It's always fun when there are big, unanswered questions yet to be addressed. So, that's what we're trying to do. If I do it, I will let you know. I will update you. Thanks for spending this time with me, whether it be fundamental or emergent. And here's to a hopeful 2025. Take care, bye-bye.

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