Physics has set itself quite the task. The job is to explain the whole universe, everything inside of it, and all of its doing and throwing. Usually we think of that in terms of the matter, the stuff, and the forces how that stuff interacts. So when we count our progress towards the big goal of physics, we sometimes think about the number of particles and the number of forces.
We reduce that number when we break open matter to see what it's made out of, or when we unify forces like we be merged electricity and magnetism into electromagnetism and then later added the weak force to make the electroweak force sorry magnetism. Eventually, we hope we'll get down to one particle and one force that explains everything that happens.
But what if there there is a better way. Maybe forces are intuitive to us, part of the stories we tell about what we see with a simple intuitive analog in our experience. But what if they aren't a natural fit to the real mechanisms of the universe itself? What if forces emerged from something else. Today on the pod, we'll talk about something we've mentioned many times in many contexts.
It's time to dive in and talk about the mysterious quantity that has no analog in our intuitive experience, but seems to dominate in the events of the universe, action, and specifically, why the universe seems to minimize it. Welcome to Daniel and Kelly's Extraordinary Minimal Universe.
Hello.
I'm Kelly Wintersmith.
I study parasites and space and it's winter right now, so it is my season of least action.
Hi. I'm Daniel. I'm a particle physicist. It's winter here in southern California, though I can hardly tell.
So we're.
We're talking about potential energy and kinetic energy a little bit today, which reminds me of my intro physics class, which makes me wonder, what is the coolest demonstration you have done for your students as a teacher.
Ooh, that's a good question. I think the coolest demonstration I've ever done doesn't actually relate to kinetic or potential energy. I know, there's like those famous ones where you like push a bowling ball away from your nose and then you stand there all cool as it comes back and doesn't crunch you in the face. But I think the most interesting ones are the ones that reveals something about the universe you can't otherwise see. So I once brought
in a homemade cloud chamber. This is something you can actually build yourself in your garage with pretty simple ingredients that could show you muons flying through the air. It reveals to you that the world around you is filled with these invisible particles flying at high speeds, and you can build the device using fairly simple ingredients to reveal them. These cloud chambers have like super saturated air, and as the muons fly through, they create these droplets which become visible.
You probably have seen them at science museums. I just think that's awesome because it shows you that the world is so much more complex than your senses can reveal.
Did we discover muons pretty early on? Because they're so easy to see?
We discovered the muon years and years ago using cloud chambers actually, and then you can spot them also using big blocks of emulsion. Emulsion is sort of like a three dimensional bit of photographic plate that you later like slice up and develop as photographs. And we put them on the tops of mountains and we saw a lot of particles shooting through them, and that was the first clue about muons.
Yeah, who did we know what they were immediately or did it take a while to work that out?
No?
Actually, it's sort of a famous surprise because we had the idea of like atomic structure and it was all very nice and neat. We had protons and neutrons and electrons, and then we discovered the muon which wasn't part of the atom. And a famous physicist I Robbie said, who ordered that, like, we don't need this. We're you know, we have a pretty good thing going, like get out to hear universe with your pesky facts.
Yeah.
No, I've been there running experiments and you're like, no, please, don't do that, universe.
But it does it anyway exactly.
And you know, the universe is under no obligation to make sense to us, or to do what we expect, or to follow rules that sort of connect with our intuitive way of thinking. Which is why I'm so excited today to finally get to talk about this concept we've been mentioning and referring to for months and months and months now, and we're going to do a deep dive
and an explanation of it today. And what is it called called action and the principle of least action, and it's a completely different way to think about motion in the universe and why things fall down, or why things slide the way they do, or even how quantum fields oscillate. Really, it's the most basic principle we've discovered, but it's also kind of counterintuitive, so it takes a little bit of a mental mind shift to think about the universe in
terms of action. It's also a terribly terribly named word because it has nothing to do with the word action we use in English.
Well, it is a physics concept, so it would have to have a bad name.
Why did they do this? You know, they should have just come up with a new name, plasma, cagical or something and use that so that when you hear it, you're like, Okay, this is something new. I have to make a new space in my brain. But if you use an existing word, that has to like share that room in your brain with that other concept which has the same word.
Well, I think somebody needs to hold on to plasmacajigle or whatever it was that you said. Thing needs to be named that, and you should be in charge of naming things in physics from here on out.
All right, Well, I guess I just need to discover something and then I can name it flasmaca jiggle, and then we'll be all set. You know, I've done the hard part already, and you know, just the pesky bit about actually discovering something in the universe.
Yeah, right, yeah, that tiny bit.
And so we asked the extraordinaries, why does the universe flazmaca jiggle?
No, we didn't. No.
I went out there and I asked folks what they knew about action, because this is something we get requests about a lot, and we've mentioned occasionally on the podcast and people ask us, ooh, would you go into more depth about that? And it's something I love doing on the pod, is explaining something on a depth beyond what's typically out there in popular science, but I hope in an actual, approachable way that'll make people really appreciate the
way physics is done on the cutting edge. But of course, before we dig into it, I wanted to know what people already knew about this concept, So I asked, folks, why does the universe verse minimize action as usual? If you would like to join this crew of people who respond to these questions without the opportunity to google them. Please write to us questions at Danielankelly dot org. We would love to add your voice in the meantime. Think about it. Why do you think the universe minimizes action?
Why does it flasmacajicle.
Here's what people have to say. Since I have no idea what this means, I'm forced to fall back on a template answer of.
Entropy, because the universe is just bound to follow laws with no shortcuts and no long cuts. So I think that minimizing action is probably another way of saying minimizing divergence from laws.
For every action, there's an equal and opposite reaction, So I guess the universe minimizes reactions as well.
To safe energy and to minimize possible outcomes.
To know, is the universe lazy? Is it just chilled out?
It is the world likes to be in its lowest energy state.
Because actions speak louder than words, and there's far too much noise pollution.
Because anything that happens, or any force that exists, has a counter force pressing against it.
Maybe it's lazy, that's my excuse. I suspect that it has something to do with entroview, though.
Too optimiz for efficiency, kind of like natural selection.
It's just something we observe, but we don't know why. Do individual processes independently tend to pars the money or is there an underlying unified model that polls all behavior to a deeper minimization.
Seems like it would be the only way that it could go, because if there was something at a higher energy, there was always something that could come lower.
I was pleased to see I wasn't the only one who didn't know the answer, and went with sarcasm.
I never know what to expect from these responses. I always learned so much. Sometimes bill know exactly what I'm talking about and get very insightful answers. Other times people are confused by the physics word and as is not their fault at all, and it seemed to have happened
in this case. But this is exciting to me because it means that not a lot of folks out there are familiar with this concept of action being minimized, which means we get to teach them all something very very cool about the universe.
Yes, we are helping to make you the most interesting person at the party.
So let's say, Kelly, I'm trying to assess the level of sarcasm there, because I am one hundred percent sincerely excited about action, and I don't talk about it at parties very much, but it is something super duper nerdy and cool to discover, like an organizing principle of the universe, especially when it's counterintuitive, when it's not the way we typically think about stuff. That's like one hundred percent sincerity. And I'm assuming you feel the same way I do.
I do.
I don't know what kind of parties you're thinking of, Daniel, but the parties I go to we talk about things like pleasma.
Okay, all right, Well we're not explaining plasmaka jiggle today because I haven't discovered it yet. But we are going to talk about action. But let's start by thinking about a simple motion of an object, and then we'll talk about how we typically understand it, how people probably think about its motion, and then we'll switch and think about it in another way. So the classic example, of course,
is a ball flying through the air. You have a ball, you're playing catch with your kid or your dog or whatever. You throw it and it goes across your yard and hits the ground or gets caught by your dog or whatever. And from a traditional physics point of view, this isn't too complicated. You can understand it by thinking, like you throw it with your arms, so it has some initial velocity. And we know that things in the universe, if they have velocity, they just keep moving unless some force acts
upon them. And in this case, for example, the force is gravity pulling it back down to the Earth. And so we can use those various elements to understand the motion of the ball. We have velocity, we have acceleration that changes its velocity. We can put that together to get an equation that describes where the ball is at any time, and it comes out to be a parabola. Right, It goes up and it comes back down and it hits the ground.
It's also a great song by Tool. Do you think everybody hum.
A few bars of it for us? Kelly?
No, no, no, no, no. Oh, hey, I'm not gonna go anymore.
So do you think everybody knows what the shape of a parabola looks like?
Oh? It's hard for me to remember a time when I didn't know what the shape of a parabolo looks like. So I should ask you that question. Kelly, do you can you imagine a parabolo?
Yep, I can, but I also do a fair bit of modeling.
Yeah, okay, so that's a good point. Let's help people visualize a parabola.
It's just a you. You're making it too complicated.
A parabola is sort of like an upside down you. Right, it's a little bit tighter at the top, but basically a parabolo is well defined by what happens when you throw a ball across your yard. Right, It's going to go up, and then it's gonna turn around, it's gonna come back to and that's the parabolic shape. It's defined by an equation that has like an x squerginit for people who like to think about equations, but for visual people, it's basically an upside down you. And so that's what
a brable looks like. Okay, So that's sort of the traditional way to think about how things move. You have matter, and you have forces. Forces act on matter to provide acceleration. A few rounds of calculus and you get your equation of motion where the thing is at any time. We're good with that.
So far, we're good, right, And so in this model, we're not paying attention to things like friction or wind or anything like that.
That's exactly right, because those things are really hard to add using this way. Like wind is a force, but it's not constant. Friction is a force, but it depends on velocity. So we're going to stick with a simple calculation because that's what this approach is good at. Simple calculations, what we call conservative forces with constant acceleration. Very easy to do those calculations. But there's something interesting about this motion if you look at it from another point of view.
This is going to sound a little bit weird and random, but stick with me. So what you do is you think about the kinetic energy of this ball. Kinetic energy is just the energy of motion, and you usually calculate it like one half m v squared vias the velocity. So you calculate the kinetic energy of the ball as it moves through the air. Cool Now calculate also the potential energy of the ball. Potential energy here is just from gravity, so it's like mgh just the height of
the ball. As it goes up it has more potential energy, as it goes down has less potential.
Energy, and m in both of those was mass.
Yeah, yeah, exactly. And so you calculate these things kinetic energy and potential energy. Now do something weird, subtract them. Usually we add these things to make total energy. But this time, let's subtract them. Let's do kinetic energy minus potential energy. So now we're calculating the difference between the kinetic energy and the potential energy. You might think, well, what does that mean? Intuitively, Like, if you add them, it may sense because it's okay, that's total energy. What
does the difference mean? It means nothing intuitively, It's just this weird number we're calculating. Okay, okay. The cool thing about this number, The reason we're talking about this number is that the path the ball took minimizes that number. Like we have the path the ball took. It was a parabola. We calculate the kinetic energy minus potential energy. That number is smallest for the path that the ball took.
If you change the path a little bit, you add a little divit or middle higher, a little lower, that path would have a higher value of kinetic energy minus potential energy. So the path that the ball took is the one that minimizes this difference between kinetic and potential energy.
Okay, and so is that just because kinetic energy would always be greater whenever you add like a divot, because you're making the parabola longer, anytime you're adding a divot or something.
I think you're asking, why is the parabola the path that minimizes this difference?
Right?
Yeah, and we're gonna explain that. The answer is yes. We're gonna explain that in detail in a minute, but for now, let's stop talking about kinetic energy. Money is potential energy because it's a lot of words, and let's just relabel it with a new word, right, so we don't have to use so many words. And we're gonna call it action. Why we don't call it plasmaca jiggle,
I don't know, but we're gonna call it action. And so this turns out to be a general principle in physics that the path of things take you can figure it out by finding the path that minimizes the action this weird difference between kinetic and potential energy. So we started out just like I'm going to do this calculation. I'm not gonna tell you why or what it means or anything, and now we did it, and we discovered something that's actually physically useful. The universe seems to pay
attention to this. It's not just a number, Daniel calculator. I could calculate anything like the number of walnuts times the temperature outside right. I can make up whatever quantity I want, But that doesn't help me solve a problem.
It just seem to be important to the universe. This number, the action, the difference between kinetic and potential energy is all so an arbitrary, random thing that we constructed, but it seems to be respected by the universe somehow, because the universe, when the ball is flying through the air, chooses the path that minimizes this weird number, this thing we call action.
So you can like tell the future if you know.
So let me it's been a while since I've taken physics, and so so I'm imagining you've released the ball, and so to do this is it like, you know, you stop and you take a picture of the ball at every second, and you redo this calculation at every second and then you can predict where it's going to be the next second by doing this calculation.
No action, It's exactly the opposite of that, which makes it a great question. So the way you described it in your mind is the way we typically think about the universe, that the past controls the future. We have like frame by frame, what happens now affects what's going to happen in a moment.
Right.
The ball moves this way because it has velocity and it has a force pulling on it. That's sort of like a computational way of thinking about the universe that you calculate it like frame by frame, and that's actually called the Newton schema. Newton came up with this idea that like the future of the universe is determined by the present, and he thought about like universal clocks and so many fun digressions we could go on there. And
that's the way that the typical force story works. You think about velocity and forces and you cycle through frame
by frame. That's not the way the action works. You can't use action to predict the future the same way because to calculate the path of the ball using action, you have to know the initial position and the final position already in order to find the path that goes from So there's a weird thing here which seems to imply maybe that the present depends on the future, you know, like where the ball goes now depends on where it lands, and so you have to know the future to predict
the present. Not quite that. We're going to dig into it in a minute, but that's actually the fundamental misunderstanding at the heart of the movie Arrival.
WHOA, yeah, I know, all right, stick with us to the end.
Yeah, exactly. So we have this other way of thinking about motion where if you know the initial point A and the final point B, you can figure out what path something took just by calculating this weird thing called action and finding the path that minimizes the action. So let's work through another example, just to make sure we have this in our heads, because it's really important. Let's take an even simpler example, a ball with no gravity.
Right now, we're out in space. Okay, we've ignored Zach and Kelly's advice, and you've gone to space and built a colony. We've made all sorts of mistakes, but now we're out in space and we get to play zero G catch with our kids or with our space dog. Right, So what happens when you throw a ball in space?
You die?
How's it? Going to move you die, Kelly. Doesn't happen immediately. I mean, we got lots of radiation and eventually we die, but we do we to have one nice afternoon of playing catch with our space dog.
Okay, right, right, So if you're in deep space in your space suit, you're gravity's probably negligible. Right, So does that mean potential energy is zero?
Exactly right. So let's think about the motion of a ball when there's no potential energy. So the force method is really easy here. You throw the ball that has velocity. There's no forces, So what's gonna happen. It's just going to keep having that velocity. Very intuitive. It's going in a certain direction. It keeps going in that direction every moment in time. You can predict the future because you have its velocity and that tells you exactly where it's
going to go. Very simple. Okay, Now let's try to do that with the action method. So the action method says, you know where the ball started, and you know where the ball ends up, and you know how much time it took to go from A to B. Find the path that it took from A to B. And so to do this, what you do is consider all the possible paths, a straight line, a wiggle, a sinusoid, you know, any other sort of crazy path that goes from A
to B in the same amount of time. Okay, now, going from A to B in a specific time, that already specifies the average velocity because you know how far it's gone and how long it's taken, So you're already kind of restricted. So the simplest path, the straight line. This one has constant velocity, and so it's going to have the smallest action because it has the smallest integrated
kinetic energy across that path. If you imagine doing something else like going super fast first and then slowing down at the end, or going super slow for a while and then speeding up to get there at the right time, all of those things have more deviations in kinetic energy, and because kinetic energy has velocity squared in it, it's going to end up with the larger kinetic energy integrated
over the path. So the way to have the smallest integrated kinetic energy is to keep your kinetic energy constant. That's why straight line with constant velocity is the path that minimizes the action. Any deviation in a direction is going to require more kinetic energy, which increases your action or any chain, the velocity is going to give you larger kinetic energy overall. So in the simplest case, action also predicts that the ball will go between you and your space dog in a straight line.
Am I being pedantic by thinking like, well, it couldn't have sped up halfway because you threw it, And wouldn't it just stay the same speed the whole time with no friction and no gravity, And so what do we gain by thinking about it that way? Because it couldn't have done any of those other things?
You're absolutely right. So you're thinking about this still in the Newton force scheme, which makes total sense because it's easy to understand why a particle would need a force to deviate from the line it's moving on, and so you're looking for a force to explain any other kind of motion than a straight line. Makes perfect sense. But right now we're thinking about this in another way. We're
not bound to forces as explanations. We're just considering the action of various paths, and in this case we don't gain anything. In fact, it feels like more complicated to do with the action way right, And the action is going to make things simpler when we get into Hairer situations. But I just want to sort of like line up a simple situation so we can get practice thinking about the action.
So you've got us all on the edge of our seats. So we're going to take a break, and when we get back, you are going to give us a Hairer situation where the curse mack amiggle.
I said, oh, I'm so sorry. We'll get more complicated.
All right, We're back, and Daniel's going to give us a more complicated situation where this concept of action is going to help us understand what's going on better.
Maybe because it's physics.
It does make it easier to analyze it's really hairy problems. It's also the way physicists think about motion, you know, when we get to quantum field theory and all that stuff. It's fundamentally based on action. And so I really want to move people out of the mindset of thinking about forces and thinking about action, because like, welcome to the cutting edge of physics. This is how we do it, all right. So we consider the very simple case you're
playing catch with your space dog. There's no gravity, things moving a straight line, Kelly was bored by that. She's like, why do we even do this? Fair point.
I wasn't bored.
You were underwhelmed a little.
You know, probably death was imminent, which is interesting.
Don't think about the death of my space dog. That's not nice.
Sorry, I do like space dog.
I'm sure we'll give lots of shielding in our space dog spacesuit. Right, Yes, take care of our critters. Yeah, because they didn't get to choose whether they go to space.
Right, No, like I didn't get to choose. Didn't go well Forlika.
No, exactly. All right, so let's get back to other scenarios. Now, let's get back to your question about why a ball in gravity moves under a parabola. Right, So we're going to add gravity back into our situation. Bring our space dog back to Earth. Toss a ball to the space dog in the backyard. It doesn't move in a straight line.
It moves in a parabola. The first picture tells us why, because you have an acceleration, and acceleration is a second derivative of position, and so if you integrate that constant acceleration twice, you get an X squared term. You get a parabola. It all makes sense. So things move in a parabola because there's a constant force. What about the action point of view, Why do we move in a parabola, Well, we want to minimize this difference between kinetic energy and
potential energy. Potential energy increases as we go up, right, So because we want to minimize action, and action is kinetic energy minus potential energy. That means we want to have lots of potential energy. So potential energy increases as we go up. So we don't want to go up fast to get high potential energy. Right, So instead of going in a straight line, we go above our target. We go higher up in order to get more potential
energy to minimize our action. But you don't want to go too far from the straight line because going too far from the straight line requires high kinetic energy. We're increasing our speed on the way up and our speed on the way down. So the parabola is a perfect balance between these things. You get more potential energy without
getting too much additional kinetic energy. So the parable is the path that minimizes the difference between the kinetic energy and the potential energy when you have both at play. It's this really fascinating harmonic balance between these two very different things. And you can think about what the universe
is doing in two ways. You can think the universe is doing the Newton thing like frame by frame, it's thinking, the ball is here, I have this force, so I have that velocity on it, so it's going to move this way. Or you can think about it from the action point of view, and you can say, the ball started here and went there. What path between those two and this amount of time minimizes the difference between kinetic and potential energy, and this is the path that does it.
And it's fascinating because this turns out to be a general principle, not just in this one case where you're playing ball with your dog, but in every case. The least action tells you what the universe does, how things get from A to B. And you can start from this principle least action, and you can derive F equals MA, so you can derive the force equations from it. Like Newton just like writ it down. He's like, hey, this works. And you might ask, well, where does that come from?
And the answer is action. Right from action, you can derive classical mechanics. Now classical mechanics doesn't then explain like, what is action? Why does the universe minimize it? To get any insight in that, we're going to have to go deeper into action in quantum mechanics and in philosophy. But it's fascinating because it seems like underneath all of these things, this force picture of the universe, there is a deeper principle from which you can derive F equals MA WHOA.
Okay, So one thing I think I'm still stumbling on a little bit is that I guess I still feel like, Okay, so you throw a ball in the air and you're subtracting the potential energy, I still feel like you should be getting the least path, not necessarily because you're subtracting the potential energy, but because any other path would require you to put additional energy into the ball at some.
Point as it goes through that path.
And am I just thinking that because I live in this world and that's my expectation and I don't realize it doesn't have to be my expectation, or am I misunderstanding something. No.
I think you're saying this is the only path it can take, because to take any other path something would have to do that. Yeah, like you need to go and push it or something. And yeah, you're thinking about the universe in terms of forces. You have Newton's ideas so deeply ingrained in your mind that like a ball is going to move it constant velocity unless something changes that, right, you have that so deeply ingrained it's obvious to you. But that comes from least action. All of Newton's laws
are derived from the principle of least action. And so it turns out that's not the fundamental way the universe works.
That comes out of insisting that everything minimizes action. And it's actually super cool because, as we mentioned earlier, in many cases, the force approach is simple and it works, and it's very intuitive and connected to our experience because this is how our world works, right, Like, we notice that you've got to push stuff to get it moving, and you've got to push it to turn it and
all this stuff. We think about the world in terms of forces, but sometimes that picture is hairy and it's hard to use to do calculations, Like it's easy in the examples we talked about, But what if you add wind, or what if you add friction or what if, for example, you take your ball and you attach a string to it, and the string is now tied to a squirrel, and the squirrel is on a roller coaster. How are you
going to do that calculation? Well, you still could in principle, right, there are still forces there on the string, and the rollercoaster has some applies to force, and the wind is a force which now weirdly depends on velocity. And what you discover is that it's a mess. Like many things in physics, we can only solve a few very simple
examples because everything else is totally intractable. So the reason introductory of physics students usually calculate the path of a ball without wind resistance and without friction is because otherwise it's a huge rain. And so this force picture is good for simple examples, but it becomes totally intractable for anything realistic.
That.
The amazing thing is the action doesn't The action remains fairly simple because you can still just write down the kinetic energy and you can write down the potential energy. In many of these situations, it's not hard to calculate the kinetic energy or the potential energy, and then you
just find the path that minimizes them. And that's because it relies on these concepts of energy, which helps you ignore a lot of the complicated details of what happens between moment A and moment B. People who have done physically problems know, for example, if you start, for example, with a roller coaster and it starts from a certain height and it goes through all sorts of crazy loop to loops, and then you're asked to like calculate the
velocity of this roller coaster car halfway down the track. From a force point of view, you're like, oh, my gosh, how am I going to calculate that it's done? All these crazy maneuvers have to know the forces and the acceleration every moment. It's a nightmare. But from an energy calculation point of view, you just have to know, oh, it's gone down a certain distance. I know the potential energy that gets turned into kinetic energy. Boom, I have
the answer in one step. This action approach is sort of analogous to that because it also relies just on these energy quantities which are sort of deeper and more fundamental, and don't worry so much about these little details.
Okay, I like that.
So when you get to like upper division physics, you learn about Lagrangen and Hamiltonian mechanics, from which you can derive F equals M. But they start out by calculating this difference between kinetic and potential energy, minimizing that and use to so many really hairy problems you would never imagine trying to calculate with Newton's method that you can just pop out in a few lines and get the answer.
It feels like magic when you first do it. But what's happening here is you're accessing a deeper rule of the universe.
Well what is the rule? Because now we're getting into the good party tricks.
This feels like a deeper rule in the universe because it works not just for classical mechanics like the cases we're talking about, you know, squirrels and strings and wind and all sorts of stuff. It also works in every scenario. So this action formula that we talked about kinetic energy minus potential energy, that's the action for classical mechanics with
conservative forces. But you can also do this, for example, for quantum mechanics, and instead of writing down kinetic energy minus potential energy, which require things you don't always know about particles, you write down something called the lagrange and density. In the case of like quantum field theory, this is just the fields that exist and how they interact with each other. And so, for example, the standard model of
particle physics. If you google, like what is the equation of the standard model, it shows you this thing that starts with L equals and then a bunch of terms. That's the lagrangeon of the standard model, and the action is the integral of that. And so the way we specify what's in the universe, what's out there, all we have to do is say what the action is, what are the pieces of the action, and how do they interact?
And everything else falls out of that, Like literally all the equations of quantum mechanics and quantum field theory say, if you have this action, then all these things happen. So the game of particle physics these days, and of all physics is to say, well, what is the action of the universe. We try lots of different things. We try this, we try that, we try the other thing, and the ones that turn into rules that align with what we see out there in the universe, those are
the ones we go with. So the standard model of particle physics is just a description of the action of the universe, not in terms of kinetic energy of a ball, but in terms of the fields, their motion, and actually in terms of their oscillation, their kinetic energy, and their potential energy.
Okay, so this seems pretty exciting because you know, you and I have been recording for over a year now, and I don't feel like there have been a lot of instances where we've talked about things that play nice with both classical mechanics and quantum mechanics. And that could be because my memory is.
Kind of like a sieve, but I don't think so. So is this pretty rare?
This is pretty rare, And it's actually a beautiful way to think about the connection between quantum mechanics and classical mechanics, and it gives you some insight into like our classical world, because when you're calculating the quantum mechanical action, you don't just think about an individual path. You think about all the various paths perhaps people have heard about, like the fine mean path integral approach to quantum mechanics. We imagine
like an electron goes from here to there. You don't just think about one way for it to go from here to there. You think about all the well, what's happening there is You think about all the possible paths, and in classical mechanics, you say, oh, I'm to choose the one path that minimizes the action. In quantum mechanics, what you do is you take each path, and that path has a complex number multiplied by it that depends
on the action. So you have all these paths and each one has this complex number in front of it, and the paths near the least action all interfere constructively with each other. These complex numbers allow them to contribute together, and the paths where the action varies a lot interfere destructively, so they cancel out. And so what's happening here is that all the paths are contributing, but just like in famous quantum mechanical experiments, the ones that are far from
the least action destructively interfere with each other. They cancel themselves out, and so what happens is that you end up with this like envelope of paths around the least action. And this is so fascinating because that interference depends also not just on the action, but on planks constant hbar, and as h bar gets bigger, you get more contributions from paths near the least action, and as h bar gets smaller, you get fewer contributions from paths near the
least action. So, now here you have a knob that you can very smoothly you say, I'm going to crank h bar all the way down to zero. What happens is that you have only a single path contributing. That's classical mechanics. Classical mechanics is quantum mechanics with hbar set to zero. If you crank hbar up to some number, then you start to get quantum effects where you have contributions from things near the least action path. And that's
the quantum effects that we see. If you cranked up hbar beyond what it is in our universe, quantum mechanics would be more obvious.
Right.
And so this shows you that there's like not just a smooth continuum where classical mechanics sits on like one edge of it, right, But also that classical least action emerges from quantum interference. The reason it seems like we live in a classical world is because HBr is so small that it looks almost like it's zero from our point of view, and you can understand that very clearly using this least action approach and thinking about all the various paths that the particle would take.
And so does thinking about things from a least action approach make you feel more confident that we're going to be able to one day sort of marry these approaches a little bit more effectively or not necessarily, because.
No, it's the most sensible way to think about the relationship between quantum mechanics and classical physics. Essentially thinking about classical physics as like a zoomed out version of quantum mechanics. And we talk a lot about how, you know, classical physics emerges from quantum mechanics, and we say, that's kind of a mystery because we don't know how to go
from the laws of quantum mechanics. You know Schroeninger's equation for an electron to zoom out to get f equals ma for a ball flying through the air, And that's true for the force picture, but in the action picture, we actually do kind of know how to do that, which is amazing. And so there is this connection between classical physics and quantum physics, where classical physics is an
extreme version of it. So that's very beautiful, and unfortunately that doesn't solve like the bigger problems in physics, which is like how to integrate gravity into quantum mechanics. But amazingly, you can also think about gravity from an action point of view. It's a very natural way to think about general relativity.
And when we get back from the break, that's exactly what we're going to do, all right, So now we're going to talk about what least action can tell us about gravity.
Did I say that right?
You totally?
Did?
You sound like a physicist?
Oh?
Oh that's great. Now I'm going to get invited to all the parties.
And I meant that, of course, if it wasn't obvious, one hundred percent as a compliment, and.
I took it one hundred percent as a compliment.
I'm glad there wasn't any doubt, No, never would be all right. So we've been talking about least action and the way that helps us reimagine classical physics as stuff moving from A to B not because of the forces, but because it's the path that minimizes the action. And then we talked about how you can even use this in quantum mechanics, and it provides a nice bridge between classical physics and quantum physics. But folks out there are of course aware that one of the biggest struggles in
physics is to unify quantum mechanics with gravity. They're classical theory of space time and how things move. And so is action something we can use to understand gravity? And the answer is yes. In fact, it's quite natural because general relativity already has very similar minimal principles.
Right.
The principle of least action says you find the path that minimizes the action. Well. Already, in general relativity, light, for example, follows the path that minimizes its travel time, right, And in general things in general relativity follow paths called geodesics. Geodesics are the shortest distance from A to B, right, And you can calculate the shortest distance by calculating the action across that path. Where in this case, what is
the action. It's the curvature, the curvature along your path. So to figure out in general relativity how something will go from A to B, you find the path where, along with you minimize is not kindemic energy minus potential energy, but the action of general relativity, which is just the curvature of space. And so you can say, all right, my action is the curvature of space, and then you say find me the paths that minimize the action, and boom,
all of general relativity pops out from there. All of Einstein's equations just flow from specifying the action to be the curvature of space. It's really kind of incredible.
So I have a question.
So the curvature of space is influenced by gravity or is reflected by gravity.
Gravity is the motion of things under the influence of the curvature.
Yeah, okay, So the equations that we were talking about were kinetic energy minus potential energy and potential energy has gravity in the equation. But when we think about general relativity, are we not using those equations or like, how does it mess things up that the term for gravity is in the equation.
Yeah? Great question. So when we were talking about our simple scenario of a ball moving under as a parable, we were doing classical physics, but we weren't thinking about general relativity. We had a very simple model of objects with kinetic energy and potential energy, and we were assuming gravity with constants and G was just a number. Now we're totally generalizing that and we're going to solve for any sort of motion of any object through space and time,
and the same equations do pop out. But we're doing Einsteini and gravity now instead of Newtonian gravity. You know, the previous version was like thinking about gravity as a force. Now instead we're doing general relativity, and so we're not calculating action as kinetic energy and minus potential energy. That's the Newtonian way. We're calculating in the Einsteinian way. And the incredible thing is that you can just specify the
action and all of general relativity comes out. This is sort of why action is super amazing, because you specify it and you have this one set of rules, find the path that minimize the action, and that tells you all of the physics. All the physics comes out of.
That tell me more physics that comes out of it.
Danie, Well, the challenge now to like unify quantum mechanics and gravity is to find an action which has both gravity and quantum mechanics in it. And we know how to define quantum mechanical action, and that gives us all quantum field theory. We know how to define general relativity action that gives us all of Einstein's equations. We don't know how to find a unified action for both of those things. So that's sort of the question of quantum
gravity now expressed in terms of this action principle. What is the action that includes both curvature and quantum mechanical fields. We don't know that.
But if we could figure that out, then we would know what exactly.
If we could figure that out, then we would have the laws of quantum gravity. Okay, yeah, absolutely, and that would be amazing. Yeah, but we can also like zoom in on one part of this to give another example, like Maxwell's equations. Maxwell's equations are these four equations. They're kind of complicated. They have all these differential terms in them, you know, changing electric fields costs, magnetic fields and vice versa,
all this stuff. It was a huge stroke of genius when Maxwell unified these, and then Heapicide clarified all of them a century or so ago. And now it turns out that all of Maxwell's equations pop out if you apply this minimal action principle to a vector Fiel. We know that the electromagnetic field is a vector field, meaning that at every point in space is not just a number but like a little arrow. So it has a
direction and a length. And if you say I have a vector field and I'm going to make some simple requirements about symmetry, like there's gonna be no preferred frame of reference or there's no ether for example, then that really constrains the kind of action that you can have. And there's basically only one way to write it. And from that way of writing it, boom, Maxwell's equations pop out. And so you might wonder, like, where do Maxwell's equations
come from? Well, this tells you if you have a vector field that respects some basic symmetries, there's only one set of equations that minimize the action of that field. Those are Maxwell's equations. And so it tells us something maybe deep about the universe, that maybe everything that happens in the universe comes because the universe is minimizing action and only actions that respect some of these symmetries.
Okay, so I'm still feeling grumpy at whoever called it action, but it feels like this is so fundamentally important that this is like Nobel Prize territory, So like, who figured out this action thing?
Yeah, action has been around for a little while. Hamilton and Lagrange both came up with this alternative formulation of classical mechanics more than one hundred years ago before the Nobel Prize even was conceived of, and sort of generalized classical mechanics. And then it was Nuther who really gave us some deep insight into it because Nuther's theorem is this really deep theorem of physics that relates symmetries with conservation laws. It says like, well, where does conservation of
momentum come from? Why does that happen in our universe? Well, it happens because of a symmetry. The symmetry is translational symmetry. That doesn't matter where you do your experiment. If you build a large hadron collider here or around Jupiter or in another galaxy, you should always get the same answers. Space is the same everywhere. That's where conservation and momentum comes from. And check out our whole episode on Nuther's symmetry. But there was a crucial detail we glossed over in
that conversation. Those symmetries that we talk about in Nuther's theorem, those are symmetries of the action, and we didn't dig into it in that episode because, as you see, action is a whole set of baggage to introduce. But it's only symmetries of the action that give conservation laws. So that means that, like you write down your action, maybe it's kinetic energy, mutters potential energy, or it's the curvature
of space, or its quantum field action. If that action is the same here and Jupiter and Alpha Centauri, then you get the conservation of momentum. And so it's symmetries of this action specifically that Norther showed us give us conservation laws. So Nother's theorem is a deep insight not just into where conservation laws come from, but why action is so fundamentally important.
It seems like she clearly should have gotten a Nobel.
She should definitely have. And these prints really guide the way that we do particle physics, Like when we devise the action of the Standard model, we have to have those symmetries because we know those symmetries will give us the kinds of conservation laws we see in the universe, like we see momentum is conserved, and so we can't build actions that don't respect the symmetry that gives us conservation of momentum. And it's a bunch more symmetries that
we know we have to have in our action. So when you're devising an action, there's not a lot of choices to make, right because you're really restricted by all of these symmetries. You can't just add terms willy nilly because they would break these symmetries. And so if you wanted to start from scratch and devise a new action for quantum field theory, there's really only a couple ways you can do it. The respect those symmetries, we're super
duper limited. And so it feels like those symmetries tell you what can happen in the universe, right, what's possible,
and the action tells you what does actually happen. And so like all of physics is basically that, right, what are the symmetries of the universe, and then what actually happens is well, what minimizes the action on those symmetries, And so it's a completely different way of thinking about the universe instead of like the way you were describing, which I think is very intuitive and Newtonian of like thinking about the universe frame by frame, what's happening, Let's
update things in a sort of like computational simulation sort of view. Instead, it's thinking about how things go from A to B to minimize the action along that path.
So when you were saying, devise a new action, I'm having trouble wrapping my head around what that means because when every time you've said action, in my head, I have subtracted two different kinds of energy. So what does it mean to devise a new action? Is that like think of a different path that something could take, or well.
Everything comes from the action, right, So you want to describe the universe, you write down an action, you apply this minimal principle. It predicts how things will move, and you compare that to what you actually see out there. So, in the case of a ball flying through the air, if I want to describe that, it works if I write down kinetic energy minus potential energy. If I write down kinetic energy minus three times potential energy, doesn't work.
If I do kinetic energy time seven minus two potential energy, that doesn't work. It doesn't describe what I'm seeing out there in the universe. So if I want to describe the universe, I have to find some way to describe the action, so that minimizing that action gives me the things I see out there in the universe.
You were using the word action the way people people actually use the word action, and the word action the way physics people use the word action, and that's what was do Okay, I get it now.
You were using it both ways.
Oh oops. And so kinetic energy, by the potential energy is a good approximate action for that one scenario we were talking about throwing a ball in your backyard. The action of the universe we don't know, because we have an action for quantum field theory and action for general relativity. We don't know how to put them together. But the goal of physics now is to find what is the action of the universe, because once we know that, we
know what it does. And this is a different way of thinking about the universe, and it relies on knowing where something was and where something will be and figuring out the path between them. And that's a very different way of thinking about sort of computationally, frame by frame, and it has this sort of feeling this flavor to it. As we were talking about earlier, that like the present
depends on the future. Like if you imagine a photon going from here to there, and we know the photon follows the path that takes the least amount of time
from here to there. But when the photon is like halfway along the path, we don't know where it's going to go yet, Right, it hasn't gotten there yet, So how does it decide where it's going to go in order to minimize the total path when we haven't figured out yet where it is going to go, Right, it feels like sort of backwards or it's like depending on
the future in some way. And this was the vague feeling that inspired ken ching short story which inspired the film Arrival, the idea that like, maybe aliens come and they don't see time and the way we do flowing from the past to the future. They experience all of time like a book, and they can rifle forwards and backwards.
And you know, it's inspired by this exact concept, but it's a little bit of a I don't know if it's a misunderstanding or a misrepresentation, because there is no actual retro causality here, you don't have to know the whole history of the universe to solve these problems. It is true that the path depends on the final destination, but you can't change that final destination. It's not like
you can influence the present using the future. There is a dependence on the future outcome, but it's not like you can control that. So you can't like change the photon's path halfway through. So there is a subtlety there which is crucial for the movie Arrival to work as a story, But fundamentally it's at odds with the actual physics of it.
I like seeing you be a wet blanket to.
All right. So we started this whole thing with the question why does the universe minimize action? So now we've explained what action is and how minimizing action gives us all of the physical laws we already discovered in other ways, and turns out to be a very general principle. It's connected to symmetry and helps us think about quantum mechanics and classical mechanics together and maybe tells us that our concept of time is a little bit archaic and needs
to be updated. But what is the answer to the question why does the universe minimize action? And the short answer is, we don't actually know. It's hard to think about action philosophically because it doesn't have a simple intuitive analog like forces. They make sense, right, we see forces happen. It's easy to think about the universe as fundamentally a set of matter and forces acting on that matter, because that's sort of the way our minds work.
Right.
Whatever action is, it seems to be vitally important to the universe, even if it doesn't have a natural intuitive analog in our minds. That doesn't mean it's not the fundamental principle of the universe. And so philosophers, of course have latched onto this and they have raging arguments about
what action is and what it means. Some suggest that what it means that the universe minimizes action is this constructive interference principle that like all possible histories do actually exist, it's just that most of them cancel each other out, and the universe minimizes action because the least action path is the only one that survives constructive interference. Between all of these paths.
What happens to the other kellys.
They cancel themselves out No, yeah, I know there's a plus Kelly and a minus Kelly out there that end up.
In zero, equally valuable Kelly's.
Well, you know, in the end, the universe minimizes action, not because it's lazy or because it prefers simplicity, but maybe just because only self supporting things can exist. Things that persist and be observed as law governed are things that you know, don't interfere with themselves, that reinforce themselves.
But it's still an open question. You know. Our job as scientists and as curious beings in the universe is to be open minded about how the universe works, figure out those laws, and then just take a step back and be like, hmm, what does that mean about the universe that it does this instead of that? Why is this thing so important to the universe? And that's the juicy part, and that's the part we have not deeply understood.
All right, friends, So put another tick in your tally of dKu episodes that end with us saying we don't know.
So we've done a deep dive into action, which I always wanted to do. But even still, there are a few places where I was a little bit imprecise to simplify things. So for those of you listening very closely, and I love all of you, there are some additional details I want to mention. First, I said often that action is the difference between kinetic and potential energy. Technically, that's the lagrong and density, and the action is the
integral of the difference between the kinetic and the potential energy. Similarly, for gr the action is not just the integral of the curvature, its volume integral of the Richy scaler, which is something that measures the curvature, but it also includes
the determinant of the metric tensor and a couple other things. Secondly, I talked a lot about the principle of least action, implying that the action is always minimized, but the actual statement is that the action should be stationary, which in calculus means its derivatives vanish. That means that it can be a minimum, and it often is, but it could also be a maximum or even a saddle point. In the case of geodesics, for examples, there are several different
kinds of geodesics. Spacelike geodesics minimize the distance, cool timelike geodesics actually maximize the proper time, and then null geodesics for light have no proper time. Finally, remember that the future does not depend on the past. The path we calculate using action between the past and the future depends on the boundary conditions including the future and the past ones. So knowing where and when the object starts.
And ends and we look forward to seeing you next time.
And I hope this has given you another way to think about how things happen in the universe and why some things happen and other things don't. In the end, it all minimizes the action.
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