What does is mean for the universe to have a wave function? How does Hawking’s “no boundary” proposal mean that the universe comes from itself? And is it really the final answer? I discuss these questions and more in today’s Ask a Spaceman!

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EPISODE TRANSCRIPT (AUTO-GENERATED)

All you need to do to figure out the mystery of the beginning of the universe is to take your general theory of relativity and run the clock backwards to see what happens, you know, at the beginning. Except you can't. You can't because of one teensy teensy little problem, and that's the problem of the singularity. It means that the equations we use to describe everything else in the universe, like black holes and the expansion of space, just give up. They quit. You ask them to solve this one more problem and they look at their watch and they say, hey, I gotta run. I think I left the toaster on at home. And they never come back. So if we can't actually solve what's going on at the beginning of the universe, can we at least get a sense of it? Get a feel for it? Most people just shrug their shoulders and move on to other problems. But if you're Stephen Hawking, you just dive right in. Yeah, your approach is only half-baked and you're almost certainly wrong, but it's worth a shot in the dark, right? Today we're going to talk about Hawking's proposal for the beginning of the universe, in which he calmly and flatly states that the universe has no beginning.

And not like it's been here forever, no beginnings, but as in, that question doesn't even make sense, no beginning. Like yes, you can string the words together to form a grammatically complete English sentence, but that particular combination of words has no useful meanings. It's like if I were to say, what flavor of dishwasher did you use for your cell phone plant? Yes, it's a sentence. No, it doesn't make sense. That's the kind of territory we're in right now. Or not. I mean, Hawking was smart, but he wasn't infallible. So let's crack this one open to see what all the fuss is about. And even if we end up being wrong, well, at least we'll enjoy the journey. And our journey to the beginning of the universe doesn't start with Hawking. No, we need to rewind back a bit. Not to the Big Bang, to the 1960s. That's when John Wheeler, perhaps the most important physicist most people have never heard about, was hard at work poking at all things general relativity and quantum mechanics. In fact, he was one of the few people in history to be able to confidently talk about both fields with relative ease because that's the kind of guy he was.

Back in the 1960s, quantum mechanics was all the rage. I mean, it still is, but it was back then too. Physicists had found great success quantizing all sorts of tiny, high-energy things. And once you got a quantum theory of a tiny, high-energy thing, you could do all sorts of cool stuff, like predicting new particles and understanding how stars work. So here's Wheeler, who really, really gets quantum mechanics. He lives and breathes it. And then he also knows general relativity like nobody's business. Honestly, probably better than Einstein did. And we had recently uncovered the fact that the universe was once a Big Bang, and it used to be a lot smaller, hotter, and denser. And it stands to reason that at one point, the universe was so small, so hot, and so dense that it was just like one of those tiny high-energy particles sitting in the lab. It was ripe for a quantum description. How do you quantize the universe? Well, Wheeler wasn't one to beat around the bush. He didn't play games.

He went straight to business. We had already had a standard procedure for quantizing something. You take your normal, non-quantum theory of a thing, you find the important bits, and you promote them, yes, that's the technical term, to these little things called operators that, and I'm skipping over half a book of math here, that allow you to speak in terms of wave functions and probabilities and all the wonderful entangled fuzziness of the quantum world. The key idea is you start with your normal, non-quantum theory, then you find the important bits and you say, those are quantized. Those involve wave functions and uncertainties. So, Wheeler and another guy named Bryce DeWitt did that. They took the equations of general relativity, found the important bits, specifically the ways that space could bend and the kind of stuff to inhabit space, and made them fuzzier. Instead of one single description, you instead now have a variety of possible descriptions. It's just treating the entire universe the same way we treat an electron, and taking that idea seriously.

For an electron, we replace something like its position with a wave function describing where the electron might be the next time we go looking for it. And for the universe, we replace one specific universe with galaxies over here and some bends and wiggles in space over there with a whole cornucopia of options that contain every single legally valid arrangement of space and matter that is allowed by general relativity. Now, the attentive listener, and don't feel bad if this wasn't you, whenever a professor in class would use that line, it never referred to me, would notice that I have deliberately said space instead of the usual space-time. That's because this quantum mechanical description of the universe explicitly does not include time. It's just space, not space-time, just space. That's because we're dealing with quantum probabilities. The wave function of an electron also does not include time. We use another equation, the Schrodinger equation, to tell us how that wave function evolves and changes and moves around in time.

If we replace an electron with a fuzzy blob of probabilities, that fuzzy blob of probabilities doesn't know on its own about time. It's the Schrodinger equation that keeps it on track like a choreographer. It tells the wave function where to go and when and what beat to land on. In quantum mechanics, this is fine. We just assume that time exists as a normal part of the functioning universe. And we know that as time goes on, electrons do all sorts of electronic-y things. So we can just stick time evolution in to make our math work. But the wave function itself doesn't know about time. But now we're talking about the universe. The whole entire stinking universe. There is no external observer. There's no one there to watch the wave function of the universe evolve. There's no laboratory. There's no measurement device. There are no timers or stopwatches or metronomes. The Wheeler-DeWitt equation, which it came to be known, does not tell us about the evolution of the universe. Instead, it's more like a box.

The equation says, here are the allowed configurations of space and matter. They do not say how those configurations will evolve, what they'll do, when they'll pay back their student loans, any of it. In fact, those equations are, well, not exactly useless, but also not exactly informative. They tell us what's allowed, but they don't even give us a single wave function of the universe. In other words, they don't even tell us which solution to the equation is our universe. They tell us what wave functions can exist in the sense of being compatible with general relativity. It's like the Wheeler-DeWitt equation isn't even a blueprint for the universe. It's a machine for making blueprints. But you need to feed this machine information to get it started. For a blueprint for a house, you need to know the lot size, the building material, local codes, how many bathrooms you think you need, two on every floor, sure, you got it. Once you have that information, then you can turn the crank on your machine and make a blueprint, a wavefunction for the universe.

In physics, we call this extra information the boundary condition. This is stuff you know or observe or measure, and they're all needed to get physics going. You need to know the position and velocity of the ball. You need to know how long your guitar string is. You need to know the temperatures or pressures of a star. Once you fill that in, then the equations can get to work and you get useful information out. So all we need to do to make use of the Wheeler-DeWitt equation is to add extra information about the beginning of the universe. Wait. Oh. Well, the only possible solution is, of course, contributing to Patreon. That's patreon.com slash pmsutter where you can get this podcast universe started. Thank you so much. I appreciate your contributions. I thought the whole point of this program was that we couldn't just Get to the beginning of the universe. And now, thanks to the magic of Wheeler and DeWitt, we have the precise machinery we need to solve the beginning of the universe. If only we knew one thing, just one tiny piece of information, one measly morsel, and we could do it.

If you can feed one piece of information to the Wheeler-DeWitt equations, if you can provide a boundary condition to it, some external piece of information, Then you can turn on the machine and generate a blueprint of the universe. Hawking did it. Well, he had an idea, which is more than anyone else had at the time. Two decades after Wheeler and DeWitt, Stephen Hawking comes along and connects, as he usually does, several different lines of thought. And he realizes that the problem for one thing is actually the solution for another. Check this out. Hawking is staring at the Wheeler-DeWitt equation. It's a puzzle that reveals the universe, but all the puzzle pieces are scattered around and we don't have the picture on the front of the box. If we know how to place just one piece, we can put together the quantum wave function of the universe. We just need that first piece. The boundary condition that defines the beginning state of the universe, but we can't measure it. We can't read any device or look through any telescope to figure it out.

Nothing gives us access to the first moment of the Big Bang. And we have no theory of quantum gravity around the corner to just ask that theory, what is the beginning of the universe like? We don't know. If we can make some solid statement about the beginning of the universe, then we can establish a quantum picture of the universe through the Wheeler-DeWaite equation. But because we can't make a solid statement, that just leaves us with taking a wild guess and seeing if it sticks. Hawking decided. that his wild guess would be as grounded as possible. He argued that the best boundary condition of the universe, the best statement that you can make about how it all gets started, had to be self-justifying. That means the guess about the beginning of the universe couldn't come from anywhere else. You can't point to God or Wheeler or anything to just hand you the answer. Because the universe is every single thing to ever exist in totality, and you can't reach outside of that. There's no hidden corner of the cabinet that exists outside the universe to just give us the boundary.

And the most self-justifying statement Hawking could make about the beginning of the universe is that it had no beginning. In other words, what if the reason you can't find the boundary at the beginning of the universe, the start of the universe, is not because it's hidden or inaccessible, but because it genuinely isn't there? Now, this is a very lovely thought to have, but we're not here for lovely thoughts. We're here for down and dirty physics. It's one thing to say something crazy. It's another to turn that into a working theory of nature. Thankfully, Hawking had exactly the tools he needed. One of the defining features of the Wheeler-DeWitt equation is that it doesn't involve time. It doesn't know or care that the universe evolves, expands, does interesting things, heads out to dinner on a Tuesday night just because it's wild like that. The key that can unlock the Wheeler-DeWitt equation is a solid statement about time. Specifically, the most important time of all, the beginning of the universe.

So we need to involve time somehow in all this mess if we're going to make progress. So Hawking involves time. Instead of just looking at space, He stitches together these geometries back to back like frames in a film. He makes a sequence of them, representing an evolution to the history of the universe. These frames tell the story of the cosmos. Now, we don't know what that story is. I mean, from Wheeler and DeWitt's machine. We can observe it, and so we know it, but we're trying to explain it from first principles. So Hawking constructs all these paths. Possible histories of the universe, trajectories, evolutions, stories. In some stories, the universe gets really big really fast and fizzles out to nothing. In others, it never even expands. In still others, there's nothing but matter. In others, nothing but nothing. Now, all of these paths, all of these potential histories of the universe, all share one thing in common. They're in the usual space-time that we know and love. Cause and effect, past and future, speed of light, all that.

They all have a beginning, a first frame in the movie of that particular version of the universe. So Hawking thought, what if we just made time behave differently? Now, I'm going to share a term with you. And when I say it, it's going to sound really wrong, like icky, deep in your gut, wrong. But I'm going to say it and then I'm going to explain it. It will still feel icky, but at least it will have an explanation. Here goes. imaginary time. Yeah, imaginary time. Time, but imaginary. Listen, I don't know how much you know about imaginary numbers, but they're really cool and fun and definitely worth bringing up in your next workplace all-hands meeting. The core idea behind imaginary numbers is to pretend to take a square root of, I don't know, negative 4. The square root of regular 4 is 2. But the square root of negative 4 is, um, is a what? In normal grade school math, this is where your teacher scoffs at you and says you can't take the square root of a negative number. But this isn't grade school.

We're not going to take the square root of a negative number. Instead, we're going to say that the square roots of negative numbers are an entirely new kind of number. A brand new category. Like you have whole numbers, rational numbers, negative numbers. Now you have imaginary numbers, which are all the square roots of the negative numbers. I swear I'm going somewhere with this. The trick... Hawking-Pole was that he took all these histories of space-time, these potential paths, trajectories, stories of the universe that normally evolve in time, and he replaced time with imaginary time. He multiplied the passage of time by the square root of negative one. Now, we actually do this in quantum mechanics all the time, or should I say all the imaginary time, as a trick. You see, sometimes when we get equations that are really, really, really hard to solve, we replace time with imaginary time and the equations just get easier. Then we solve them and then we swap back to normal time to get our answer out.

Just a little reshuffling in the back end to work through some thorny problems. But when Hawking does this to the spacetime of the universe, he gets a bonus. It's not a party trick anymore, it's a statement. You see, in normal spacetime, the universe has a beginning. But when you replace time with imaginary time, the beginning goes away. This procedure actually puts space and time on equal footing. Makes them all creatures of curvature and geometry with no separate identity. Like we know that time feels different than space. You can go anywhere you want in space, but you can only go towards the future in time. But if you swap out regular time for imaginary time, then space and time are identical. They're the same. It makes the beginning of the universe no special or unique time at all. It becomes like the South Pole, which is really just like any other point on the globe. You reach the South Pole and you keep walking. You don't run into like a singularity or an impenetrable wall. You just keep walking.

And once you reach the South Pole and keep walking, then it's only ever north from there. You reach the beginning of the universe and it's not special or unique. Maybe it's a little bit hot. But all you have in front of you is more future. No beginning, no start, no boundary. A universe that justifies itself. By making this switch to imaginary time, Hawking could encode his the universe has no beginning idea and he could crunch through the math. Voila. A key that unlocks the Wheeler-DeWitt equation and the know-how to run the mathematical machinery. And what do you get for all this work? Nothing less than a wave function for the universe. Let's start a little smaller than the cosmos. How about back to our old friend, the electron? When we say that the electron has a wave function, we mean that we never really know where to find the electron until we go looking for it. Just loves a good game of hide-and-go-seek. The wave function is spread out all over space. Technically, it fills up the entire universe.

And it tells us the chances of finding the electron in any one spot. Where the wave function is especially dense or it has peaks, we have a good shot of finding the electron there. Where it's a little thinner, eh, we still have a chance, but don't bet a lot of money on it. The wave function tells us the betting odds, but it takes a measurement to find the winner, where the electron actually is. So the wavefunction of the universe is just that, but for the universe. The wavefunction encodes all sorts of different possibilities for the universe, different kinds of matter arrangements, different kinds of expansion histories, different kinds of dark energy behaviors, the whole deal. It doesn't tell you exactly which universe you live in, because by itself, it contains so many possible cosmologies. And yes, that means all of Hawking's machinations when it came to the Wheeler-DeWitt equation didn't even give him a prediction for the universe. It gave him many possible universes because that's how quantum mechanics works.

You can only deal in probabilities. But there is progress here because it gave Hawking a wave function. It really narrowed down the possibility. And the wave function lets us, or originally Hawking and his co-author James Hartle on this work, read off the most likely universe. Again, it's not a guarantee, just like the wave function of electron never gives guarantees. But we can take the wave function of the universe and read off the betting odds. And we can ask, is our universe special? Is it the most likely universe? Is it something in between? Right away, Hawking's result gave a surprising answer. The peak of the wave function that he derived gets inflation. That's right, inflation for free. It predicted that the most likely universe was one that started off small and smooth, got big really, really quickly, then settled back down into a more sedate expansion phase. That's our universe. Yeah, we don't know what inflation is or what powered it or why it shut off, but we strongly, strongly, strongly, strongly suspect that something like inflation happened in the very early universe.

Check out the episodes on inflation if you're curious. But it was originally proposed as this hammer to break through some nagging problems with the standard Big Bang picture. And here comes Hawking's proposal. Starting with the assumption that there's no such thing as a beginning to the universe, working through the mathematical requirements of that, reading off the most likely cosmos, and we get inflation for free right there. It's like inflation and no boundaries go hand in hand. Once you have inflation, you have all the rest. Flatness, homogeneity, seeds of structure, CMB fluctuations. All the rest of cosmology just falls out. Oh, and you get another bonus from Hawking's analysis. This universal wave function also predicted that the cosmos started in a smooth, low entropy state, which is exactly what you need to kick off an arrow of time and give the whole thing a future. That's nice. This is a lot. And the implications are tremendous. If the no boundary proposal holds, it means that the cosmos is entirely self-contained.

The singularity just dissolves. It's not explained away, but it tells us that it's geometrically meaningless. The South Pole is a singularity of sorts. It's a place where all the lines of longitude intersect. But if you march down there, you don't get sucked into a black hole or see the face of the creator or disappear into oblivion. You just take one more step and start making your way back north. Be sure to pack a jacket, though, just in case. This makes the question, how did the universe begin, meaningless. There's no answer, not because we can't formulate an answer, but because the words in the question lose their proper grammatical meaning. I know your teacher once told you that there are no stupid questions, but here's Hawking saying that there is one stupid question. It also makes the universe entirely self-contained. There's no spark. There's no flash. There's no bang. The universe just is. Consequence of the laws of physics. It exists because it exists. In this view, you don't need a creator, and Hawking was very big on pointing this part out.

It's all perfect and complete and logical and powerful, except for all the places where it goes wrong. And there's a reason that while interesting, the Hartle-Hawking no-boundary proposal isn't exactly taught in grade school textbooks yet. Actually, there are a lot of reasons. And in usual fashion, just when you felt like we are on the verge of a triumphant explanation of all of space and time, I'm going to yank it all away and tell you, not quite. First, we need to start with the elephant in the room, which is that we do not have a working theory of quantum gravity. If we did, we wouldn't need to debate about this because we could just read off the answer about the beginning of the universe. For Hawking to play his games, he needed to make an absurd number of approximations and guesswork as to what a full theory would tell him without actually knowing the answer. As while the choices he made were relatively reasonable, they were also choices and assumptions that don't necessarily have to hold in the real universe, which is the one we particularly care about.

And even if we take his assumptions at face value, or at least begrudgingly allow them, then the most likely universe, the peak of the wave function, isn't exactly our universe. It's slightly different. The most likely universe, the most probable one, is actually smaller than ours, with less inflation. In other words, our particular universe has just a bit too much inflation. Again, that's not ruled out. We're part of the wave function too, but it's hard to swallow when the most likely universe is one that is decidedly not us. The problem has a name, by the way, called for various reasons Boltzmann's babies. It's... Let's just say the concept of Boltzmann brains needs its own episode, so please just ask, and this argument is just a cute nickname for the fact that Hawking's most likely universe is smaller and younger than ours, so any random brain should be a cosmological baby. Listen, it's stupid, but I feel like I have to mention it, and I can do a whole episode on the broader concept without cringing too hard, I promise.

I think. Please, just ask. Anyway, Hawking also played some funny games with the math. He had to play some tricks, like the real-to-imaginary time switcheroo to solve the equations. But people have attempted the real deal with fewer mathematical games, and there are results that this no-boundary, not-a-beginning is not a smooth and easy thing. That instead of giving you an easy way to pinpoint the nature of the universe, it's actually crazy chaotic, and it's essentially impossible to read out, this is what our universe is supposed to look like. Which is generally a bad thing if you're trying to predict what our universe should be like. And then there's the whole thing with probabilities. In normal quantum mechanics, we have a procedure called the Born Rule for transforming wave functions, which we can't see or measure, into actual probabilities, which is what we care about in our experiments. We have a translation mechanism between wave functions in the math and probabilities of measurements in the laboratory.

This makes sense and is super well tested for... electrons. But the universe... All possible universes? Wave functions of the universe? How do you extract probabilities? How do you turn the wave function of the universe into betting odds of which universe we live in? Hawking just followed the normal procedure, but are you seriously telling me I can treat the entire universe the same way I treat an electron? That seems shaky at best. And the fact that it gives us an almost but not quite correct answer might be telling us that it's not quite working the way Hawking claimed. And excuse me, but it makes sense to observe or measure an electron and get its wave function to collapse and the electron to do something interesting. We put it in a box. We let it hit a screen. But how do we take the measurement of the universe? Aren't we, and I'm just saying this based on the last time I checked, inside the universe? How do we observe the universe if we're a part of the thing we're trying to observe? I know quantum mechanics has a whole big problem when it comes to measurements in general that we largely ignore, but it doesn't seem like we can ignore it here when we're talking about observations of the very thing that we are participating in.

So again, it's another problem of leaping from a wave function into a measurement. Oh, and that arrow of time thing where the smooth, chill, low entropy universe just popped out of this analysis like it was no big deal? Well, Roger Penrose pointed out that Hawking assumed smoothness from the beginning. he assumed that reaching the beginning of the universe was just as easy as walking across the South Pole. And that we shouldn't be surprised that the initial universe was smooth because that's exactly what Hawking baked into his assumptions from the start. Like I said, he made a lot of assumptions about what the math should tell him without really knowing the math of quantum gravity. So Penrose pointed out that you don't get to wave your victory flag saying you predicted the era of time because it really just snuck in through the back door when you weren't looking. Speaking of time... We have a big fat problem with the nature of time that Hawking's proposal doesn't really solve. The no boundary proposal says that there is no time before the universe.

That time itself emerges from the geometry after the no boundary transition. But he also spoke about the universe emerging or nucleating or coming into being. These kinds of words smuggle in a temporal notion. The universe exists. We know it exists. And you can say no boundary this and no beginning that. But to say that time emerges implies that there is a state of existence in which there is no time. So how does something, I don't know, start if there is no time to give it a moment of ignition? I know it sounds like I'm just twisting myself in word knots, and I kind of am, but so did Hawking. And so is anyone who's ever talked about this. It could just be a failure of language in our monkey brains when the math says one thing and we can't put it into words. Fair enough, that's happened before. or it could be an irreducible failure of the whole idea. Maybe it's that statements that don't make sense are actually wrong. I'll leave you with this. The no boundary proposal is intriguing, but it has some serious flaws.

And unless or until we achieve a quantum theory of gravity, we can't wrap it in rigorous enough mathematics to decide if we're on the right track or not. While we all generally agree that the Wheeler-DeWitt equation is the correct machinery, We don't agree on pretty much anything else, up to and including if wave function of the universe is even a valid concept. But even if we did, even if Hawking was right, even if time emerges with space, even if there is no such thing as the beginning, your work isn't done. In Hawking's view, the universe simply exists as a consequence of the laws of physics. But why do the laws of physics exist? And on that, Wheeler, Hawking, And all the rest are silent, as am I. Thank you to Vito G. and Sophia K. for the questions that led to today's episode. Thank you so much, everyone, for sharing this series as far and widely as you can and leaving reviews on your favorite podcasting platform really helps show visibility. And please keep sending me questions. That's askaspaceman at gmail.com or you can find a form to fill out at askaspaceman.com.

And of course, thank you to all of my Patreon contributors. I couldn't do this show without you. I'd like to thank my top contributors this month. They are Justin G, Chris L, Alberto M, Duncan M, Corey D, Michael P, Nyla, Sam R, Joshua, Scott M, Rob H, Scott M, Louis M, John W, Alexis, Gilbert M, Rob W, Jessica M, Jules R, Jim L, David S, Scott R, Heather, Mike S, Pete H, Steve S, Lisa R, Kevin B, Eileen G, Stephen W, Deb A, Michael J, Philip L. and Stephen B. That's patreon.com slash pmsutter. And assuming that there are no boundaries, or maybe if there are, I will see you next time for more complete knowledge of time and space.

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