Daniel and Kelly’s Extraordinary UniverseWhat did Emmy Noether reveal about the Universe?
Daniel and Kelly talk about the life and underappreciated contribution of mathematician and accidental physicist Emmy Noether
The universe is like a big puzzle. It's a mystery for us to unravel. But it's not just a heaping pile of unorganized facts. There are patterns to the madness. There are clues that suggest a deep underlying organization. For example, if you multiply mass and velocity together something we call it momentum, you get a number which the universe seems to respect. You calculate the momentum of a bunch of rocks, You get some number. Then you bang them into each other, change their directions, break them into pieces. Whatever you calculate to momentum again, same number. The universe respects this number, conserves this weird combination of mass and velocity. Why does it care about mass times velocity? Does it not care about mass times acceleration or mass times ice cream or chocolate times velocity. For a long time, we had no idea. We just saw that it did respect this until a genius came along, a mathematician who, during a brief dabt in physics, made a connection that explained it all. Emmy Kneather pulled back a layer of reality to show us the deep underlying mechanism behind these conservation laws and not just about momentum, about anything the universe respects and conserves. Without her equation, we wouldn't have understood the beautiful mathematics, the foundations of the Standard model. We wouldn't have discovered the Higgs boson. So in honor of Emmy kne there's one hundred and forty third birthday, we're going to be diving into her life and her work. Welcome to Daniel and Kelly's extraordinarily momentous universe.
Hello.
I'm Kelly Wiener Smith. I study parasites and I get less metrical with it.
I'm Daniel. I'm a particle physicist, and I'm getting rounder and rounder. I'm not sure what was happening over there.
I might have both of those things happening concurrently, but it's okay.
You know.
The nice thing about getting older is that I care way less than I used to, so you know, freedom in that regard.
I like to think that I'm approaching a spherical physicist, which is the approximation most physicists would make about themselves eventually. Anyway, it's a.
Very efficient shape. I like that, and I like the idea that there's a goal in mind. Yeah, so today we are talking about a woman scientist who many people don't know, who they probably should. Do you remember when you first learned about Emmy whose last name I'm going to try to avoid saying.
I do remember I learned about Emmy Nuther in graduate school when I was taking quantum field theory, because that's when you're thinking about the fields and how they oscillate and their symmetries very importantly. And I remember my professor Lawrence Hall writing this equation on the board, Nuther's theorem, and feeling like, oh, my gosh, that is so deep and powerful because it just told you so much about how the universe worked and connected these two ideas. And the equation itself is so simple, and when he derives it seems so obvious. But it's one of these things that it's so brilliant, it's so insightful that it's hard to take yourself back to before it was understood. It's like it changes the way you look at the universe so deeply that it's hard to remember how much of an insight it was.
Oh that's beautiful, And I bet you have totally hooked our audience on learning all about this equation.
I hope so, and it was in that same moment that he told us that this was an equation by a female physicist. And you know, that's unfortunately rare thing to hear, especially somebody contributing to physics in the early parts of this century. You know, physics is still not a place of gender balance, unfortunately, and in the early part of this century much much less so. And so hearing about the deep contribution nw they're made to physics is especially impressive given how long ago she did it.
So I'm under the impression that she was a mathematician who jumped into physics to just dazzle everyone and then went back to math. Was math comparably bad about sex balance on faculty back then? I'm guessing the answer was yes and still is yes.
The answer was yes and still is yes. Unfortunately, math is no better than physics. Like if you look at a breakdown of faculty in the United States, it's like twenty ish percent female in physics and in math departments. Yeah, absolutely, we still have a lot of the same problems that we did. Things are better it used to be one percent. Twenty percent is progress, but we have a long way to go. Not that we need exactly fifty percent female, exactly fifty percent male. But we know that this is emblematic of issues we have in our department that make it harder for women to succeed.
Yep, all right, Well, lots of progress to make, and so we decided that we wanted to have a series of episodes about women's scientists who are amazing who you perhaps have never heard of, in response to actually a couple different listener questions that we got. And so this month we're going to have a few episodes of that type, and this is the one we're starting with today.
Yeah, Emmy Nuther is maybe one of the greatest, most impactful geniuses you've never heard of. She made contribution to physics that impact you, impact our understanding of the universe. But she hasn't talked about nearly well enough. And so to get a sense for like, how much did people know about her? Have people heard about her? Do people understand the contribution she made? I went out there and asked people, Hey, do you know what Emmy Nuther revealed about the universe? So here are a bunch of answers from our volunteers, and if you'd like to participate for a future episode. Please don't be shy. We want to hear your voice on the pod. Write to us two questions at Danielandkelly dot org. So you've had a few clues already from our introduction, but do you know what Emmy Nuther revealed about the universe. Here's what our listeners had to say. That there's a simple relationship mathematically with symmetries in the universe. She revealed things that Noeita could.
The one who came up at the idea of using type one A supernova as a way to measure the expansion rate of the universe. Amy Nota have revealed something about the translational symmetry in the universe, and that all positions in the universe are somewhat indistinguishable.
There was a standard candle in the universe might have been a certain type of supernova.
I have no idea, but I'm eager to find out mathematician, maybe back in Einstein's time, who came up with a really important concept but never got full credit for it.
I'm sorry, I have no idea who that is.
Emy unfortunately did not reveal anything about the universe to me, specifically, because I don't think i've ever heard the name.
Where there is a conservation law, there will be some symmetries. And this changed the way that we approved which can bring together physical properties.
Emmy North, another woman who deserved a Nobel price but did not get it, showed that symmetries in the universe were related to conservation laws.
Why is anyother.
Gonna be the name for the momentum?
Oh, I guess she did something about momentum.
I'm gonna have to go look this up.
Ah, they might have proved that there's no either, just kidding. I think it had something to do with.
Neutrinos or was it symmetry breaking something?
Oh no, I have no idea who this person is at all, So it must be a woman and she did something really important and then took the credit, and that's why we have no idea who she is. Or maybe it's just me and everybody else does.
So we had a handful of listeners who gave like spot on answer, which makes me wonder if we had asked a similar number of people who weren't listeners to our show, like a random draw from the population, what the answer would have been. Because I think we've got a population of listeners who are more likely than average to know the answer, but still there were plenty who didn't. And the last answer was just like spot.
Oh man, yeah, I know. And there's some great guesses in here, like the no Ether love that joke.
Nice yep yep epic. Let's jump right in. Tell us about all right, and I'm gonna try to say the name. Tell us about Emmy Noether.
Neuter neuter.
Yeah, that's not what you were saying earlier.
My pronunciation of her name is probably going to vary a lot. There's not a lot of symmetry or conservation in my pronunciation of this name. It's a little uncomfortable for me, but I'm gonna do my best. I mean, Luther comes from a really interesting family. Her dad was a mathematician. She was born late eighteen hundreds, and so she comes from an academic background. You know. Her family valued thinking and education and sort of higher intellectual pursuits. Some of her brothers became scientists, another one became a mathematician. So her father is a well known mathematician, Max Nuther. Though eventually, of course, Emmy totally eclipsed him, so instead of Emmy being Max's daughter, now Max is like Emmy's father. Way to go, Emmy, you know, like the way and you go to pick up your kid at school. I'm just like Hazel's dad. I don't have a name. I'm just Hazel's dad.
You know What's funny. I was the older sibling, but my brother George was way cooler than me. So even though I came first, everyone I met at our school was still like, oh, man, are you George's sister? And I was like, no, that's not supposed to work that way. But anyway, he was way cooler than me.
So that's what happened exactly. Well, I'd be very proud to go down in history as Hazel's dad. I hope that happens anyway. Back then, though she comes from an academic family and her brothers were encouraged to pursue higher intellectual degrees, it's not an option that women had. So she was actually trained to be an English and French teacher at a finishing school, but she was like, man, I don't want to do that. I want to study math. She had like this deep appetite and this aptitude for math at a very early age. But the problem is in Germany, where she grew up, women were not allowed to enroll in universities, so she like tried to enroll in university to get a degree at Gottingen and they were just like no, So she just went anyway. She's like sat in on the lectures and she's like, I'm just gonna listen. Like they're talking about math. I'm here, I'm learning about it. And so she picked up a lot of math.
That must have required some buy in from the faculty members right to not kick her out. Is there any sense for like, were the members of the maths department, you know, feeling differently than the members of the admissions group. Were they happy to have her there?
Yeah, they certainly didn't kick her out right, and so she could sit in the back and she wasn't disturbing anybody. But you're right, people could have been like, this is deeply offensive. You must leave, and that didn't happen, and times were changing. It was only nineteen oh four, a couple of years later when erling In University decided to allow women to enroy. So she was sort of on the cusp of that, and then she immediately enrolled and she got a degree in math, and then she was fortunate in some way that she was hanging out in Gottingen right around the time when it was the center of the universe in terms of math and geometry. So like, obviously this is right around the time of Einstein and relativity, right, And he's a German professor, and like Germany was definitely the center of physics and math at the time. You got guys like David Hilbert Klein, Minkowski, Shortsiled. All these guys who made huge contributions and we're thinking deeply about the physics of space and time and relativity were all there like in Gottingen at the time, and so she worked with them. She did her PhD. And she got a PhD in nineteen oh seven, right, so like more than one hundred years ago. She was the second woman to ever get a PhD in math. So like really cutting edge. This is not an easy track, you know, somebody like me, I went into physics. I followed a path. It was laid out for me. You know, I knew exactly what to do and where to go. I'm not a trailblazer in any respect. You know, there's like a very clear track for me to follow.
I think you're under selling yourself.
But okay, I mean sure, I had to do a lot of hard problem sets, but you know, the track was there, it was clear what I needed to do. But she not only had to do the hard problems, she had to make her own path. You know, this is not something an established trajectory for folks. She was really resilient. When she got her PhD. She couldn't get a job. People just kept rejecting her because she was a woman. And so for the next eight years she did research in math and she taught, but she wasn't paid. She didn't have an official position. She just like hung around. She was just like a volunteer. But you know, she came from a wealthy family which could support her, so she didn't need to like go work as a seamstress, and she just followed her passion. In nineteen fifteen, for example, she applied for a position at Gottingen and she was rejected and the review committee said, quote, I have had up to now unsatisfactory experiences with female students. She's an exception, which is like, you know, thanks for grudgingly admitting that she has.
Talents here with faint praise.
Yeah, yeah, And then David Hilbert, who was like one of the greatest mathematicians in history and already recognized at the time, you know, clearly a leading mathematician of his age. He wrote and complained. He said, quote, I do not see that the sex of the candidate is an argument against her. After all, we are a university, not a bathhouse.
That's great, I know, go Hilbert.
But they weren't persuaded, and they did not invite her onto the faculty. So instead Hilbert signed up to teach a bunch of classes that he thought she would be good at teaching, and then he just had her teach in his place. So like, yeah, she got to teach, but she didn't get the job. So like, you know, there's pluses and minuses there, and maybe as sort of like a middle finger to the faculty, she ended up swimming a lot at the men's only pool.
Ah, good for her. Do we happen to know if Hilbert paid her for her teaching labor or did he get the pay while she taught.
He hired her as a teaching as, so she was paid sort of at the gradual student level even though she was doing faculty level work. And I don't know if he supplemented it privately.
I mean, that's still more than she was going to get otherwise. And it's a chance to show everybody, screw you, guys, she can totally do this.
Yeah, exactly. And you know, she just stuck around even though officially the institutions didn't want her there. Her colleagues valued her, like Hilbert knew that she was a genius. And all these guys were thinking about physics and thinking about Einstein's new theory of relativity, which is a very mathematical way to think about the universe and space and time. And you know, these are very new ideas mathematically, like Remont had just figured out how to think about higher dimensional spaces and surfaces in those spaces and to do this math, and Einstein had learned about it and used it to describe gravity, and so this was all very new and very exciting, and so there was a lot of talk between the mathematicians and the physicists. Emmy was mostly interested in math, like she was really excited about what we call algebra, which is much more than like you know, x equals seven plus y and manipulating equations like we learn in middle school algebra is a broader field that thinks about, you know, relationships between objects and symmetries. You have abstract algebra, linear algebra, you know, thinking about like vectors and matrices and rotations and symmetries, and it's a really fascinating and deep area of mathematics and very closely related to general relativity and what Einstein had done and thinking about the laws of physics. So this is the kind of thing she was very excited about, and so she was working mostly in abstract algebra, but she did spend an afternoon or two thinking about the problems in physics, which is when she came up with her famous Nuther's theorem.
Oh man, I mean, algebra is great as long as it's not the word problem. I'm helping my daughter with those right now, and those are no fun. But algebra in general I could do for fun all day as long as it's not the word problems.
You know, word problems are really challenging. Translating a paragraph into equations is really hard, and we have a whole class on just that here because a lot of students show up and they can do the math, but they don't know how to get to the math. You know, you have this problem and there's like a train, and there's times, and there's balls, and there's inclined planes or whatever. They don't know how to translate that into the math. And you know that's actually the core physics, knowing how to go from like here's a problem in the real world. How do I build a mathematical model that captures the essential bits of it that lets me calculate the answer? How do I convert this word problem into math? That's what physics is.
Like.
When you want to solve a problem in physics, you need to turn it into a mathematical model. But that's not easy, right, It's hard, and you got to teach it to people. And so a lot of people show up here and they struggle with that and they feel bad about it because they don't know how to do it. But like, the whole field of physics is basically that, like, turn this into a math problem. That's what we do, and so it's overlooked, I think and undertaught.
This is though, what got me into science. Like I hated word problems in high school. But when I took an ecology class and our job was to think about a system like a predator prey system and we were asked to describe it with equation M. Like the first time I did that was the first time I thought, I'm going to spend the rest of my life as a scientist. This is so much fun, I think. I just don't like the like, you know, the train leaves it five and it's like, oh, this is boring, it's not real. But when you're trying to describe like the universe, that's much more.
Inciting to me. Well, I remember this contrast my first year of college between my physics and philosophy classes, because in physics, you had a question and you're like, Okay, this electron is oscillating. What's going to happen over here on this attena over there? And you want an answer, But there is an answer, and you can get the answer by turning the problem into a math problem. And then math is either right or wrong, you know, and like there's rules about math and there's no arguing, Whereas in my philosophy class, which I also deeply enjoyed, you know, there's a question of like you have to pull a lever on this train to kill one person or let ten people die? What's right what's wrong, and man, we could argue forever we make any progress, and people could disagree, and you couldn't definitively say who was right and wrong. You couldn't turn it into a math problem. People have been arguing about that question for thousands of years and still nobody knows the answer. So to me, that was frustrating, anisode of a relief to get to, like turn something into a math problem. And yeah, I think that's probably why I became a scientist and not a philosopher.
I like the comfort that math and statistics brings.
Yeah, I agree, I agree.
All right, So on that note, let's take a break, and then when we get back, let's figure out what Emmy revealed to the world. Okay, we're back. We're done geeking out about how much we love equations that describe animals in the rest of the universe. So now let's talk about what Emmy's amazing equations reveal about our universe. I guess that's still geeking fine.
Geek, Yeah, it is good. That's what we're here for. This geek out deeply. Yeah. So, I mean, if there was an expert in algebra and thinking about equations and thinking about symmetries of those equations, you know, like very simply, if you have an equation like X equals ten, you could add two to both sides and it doesn't change the answer, right, X plus two equals twelve still reveals the same X. Right. There's no change there in the fundamental solution to the problem. And so this is the kind of thing she was thinking about, though, of course, lots of other kinds of changes and more dramatic things about the universe, because there's lots of fascinating symmetries in the universe. Like you know, take for example, a sphere, right, a perfect sphere, you rotate it, you still have a sphere. Right, There's no change in the sphereness of the sphere. And if you have a sphere in the universe and you spin it, it's going to interact with the universe the same way. It's going to bounce stuff off of it, it's going to gravitate, like, there's no fundamental change in how the universe treats it once you rotate the sphere. It's an example of a symmetry, and that's like the purest, cleanest kind of symmetry. You could also have other kinds of symmetries. Like let's say you have an object that's not a sphere, you know, like a cube, and you rotate the cube. Now the cube looks a little bit different, right, but in lots of situations it still acts the same. Put that cube in orbit around the Sun, and it doesn't matter how you've spun it, It's still going to have the same orbit. It doesn't change the orbit of the cube around the Sun. Dynamics of its orbit around the Sun don't change if it's a cube, or if it's a circle, or if even if you squeeze it and make it like a rod like, it only matters what its total mass is. Nothing else matters. So there are symmetries there to gravity. You can change this object, you can transform it, and not change the fundamental physics. So there's a symmetry to gravitation. You can transform the objects, but the equations don't change.
So I assume that before Emmy came along, we probably understood that spheres in cubes were symmetrical. So what did she add to that? Did she add the understanding that that's going to be the case anywhere in the universe, or have we not gotten to her contribution.
Yet we have not yet gotten to her contribution. But what she did is show us what these symmetries mean, the consequences of these symmetries. But first, let's get a little bit more comfortable with the kinds of symmetries that we're talking about here. Another really important and fundamental symmetry in the universe is location. We call this translation invariance, which is a fancy way of saying that if you do an experiment anywhere in the universe, you should get the same answer. Like, let's say you're measuring the speed of light. For example, you could do your experiment out in space near Jupiter. You could do your experiment in the space between the Milky Way and Andromeda. You could do it some other random place in the universe. This speed of light is just a number, and as long as you have perfect vacuum, doesn't matter where you are. You do your experiment anywhere you should get the same answer.
So, say you were to measure something and it was different when you were near Jupiter, perhaps because Jupiter is massive, What does that tell you? Does that tell you that you're measuring something that's not symmetrical, and thus isn't related to what Emmy came up with.
Yeah, that's a great question because you might think, oh, Daniel specifically was talking about measuring in a vacuum. But we know the speed of light isn't always the same if you're not in a vacuum, So what's going on there. The answer is a little technical, it may be unsatisfying, which is that you're measuring a different thing in that case, and you have to include Jupiter in your experiment. So, for example, if you're measuring the speed of light near a massive object, where you're measuring the speed of light through a glass crystal, then you're measuring something different than if you're measuring the speed of light in a vacuum. And so it doesn't matter where you are in the universe. If you measure the speed of light near Jupiter, like you want to know the speed of light through the atmosphere Jupiter, you got to bring Jupiter along with you and then do the experiment out in the middle of deep space or over here or over there. You should get the same answer. It doesn't depend on the actual space, right And what this tells us is that like space has no fundamental markings, there's no way you can identify where you are in space, which is something we kind of knew, all right, there's no like zero to the origin of space that means something deep. There's no way you can like figure out where you are in space. You can't like look up your fundamental coordinates in space because there aren't any. All we have in space are relative distances. I'm a meter from here, I'm ten meters from there, because all of space is the same.
Okay, got it.
So that's one example of a symmetry of the universe that's really important. And another is direction, Like it shouldn't matter what direction you're pointed in when you're measuring fundamental constants about the universe or doing an experiment, because the universe has no preferred direction. Right like here on the surface of the Earth, obviously there's an up and there's a down, and there's one definition of that that makes sense, right because we have gravity, and so we're near a gravitational field, and so it makes sense to call one direction up and one direction down. But if you're far from a planet, there is no direction that's like more up than down. If you mentally imagine the galaxy, you probably put it in your mind is a flat disc, and you imagine things above that disc and blow that disk, and you organize it in your mind. But you could also spin that galaxy at an angle and tilt it or look at it from another direction, and those are just as good. They're equivalent. Right. There's no like fundamental updirection out in the universe. There's no way to look at the milky Way that's right or wrong, and there's no way that the universe prefers any angle over another.
Yeah, astronauts report finding this very confusing when they get up there, but they learn it quickly.
There's yeah, And that's really countertit because it's not part of our experience because we live in a place where there is a defined up and down. And so in that same way, you might complain and say, well, your experiment can have a different outcome in the middle of deep space and near the surface of the Earth, And that's true, but you've got to include the Earth in your experiment, right. So if your experiment is like, how much do I weigh on a scale when there's no masses around me, the answer is going to be zero. No matter where you are in the universe, there's no part of space that depends on it. No angle matters in the universe. But if you're including a planet in your experiment, you know then you're going to measure the same value on the scale no matter where you and that planet are and the orientation of that planet. So you got to include the planet in the experiment in order to really get the symmetry.
Got it, Okay? Are there symmetries we should talk about?
There are so many fascinating symmetries. Like another important symmetry is time. We don't think that the laws of physics depend on time. We think, for example, the speed of light is the speed of light, and it's the same today as it's going to be tomorrow and a thousand years ago and a billion years ago. We think the speed of light is a constant. And this is sort of a fundamental assumption we make in science all the time. That's sometimes unspoken, that you can measure something and then come back twenty years later and measured again and you should get the same answer, right. I mean, you're measuring the same universe. It's the same thing. But it's not something that necessarily has to be right. It could be we live in the universe that is evolving and changing. We just sort of assume that the universe has physical law, and then we can do experiments to reveal those laws, and we can do it whenever we like. It doesn't matter if it's Christmas or if it's summer or whatever. And that's fascinating and very very useful. But that's like another symmetry of the universe.
So I'm not really used to thinking of time as a thing that you can call symmetrical, like you know, for me, if something symmetrical are out, like the one side of my head has a lot more gray hair than the other. Thing is I'm thinking of like a shape that is not exactly the same.
Yeah, So the way to think about it is this. It's like imagine applying a transformation and then seeing if there's a change. So, for example, you're thinking, the left half of my face is the same as the right half of my face. The transformation you're doing there is you're reflecting it, right, So it's called a reflection symmetry. If your face is symmetrical, then take your left side and reflect it onto your right side and they should look the same. Most people faces actually not that symmetrical. And so if you do that, you look kind of weird. But that's reflection symmetry, right, And so the concepts always have a transformation, and then a question you ask, like has this changed? So in your case, we're saying reflect your face through a mirror, and we're asking do you look different. In the case of time, for example, we're asking shift time forward by ten seconds or by a billion years? Do the laws of physics make any difference? Do your experiments get the same results or not? And so that's the sort of connection. You always have a transformation you're applying, and then you're asking is something changed or not?
Okay, so what kind of transformations could you apply that are not symmetrical? Oh, I think that'll help me understand why time is symmetrical.
Yeah, that's a great question. Something the universe is not symmetrical to is acceleration. Like take your experiment, and now take your experiment and put it on a rocket ship so that there's a thrust. It's like accelerating through space. You can get different answers, and you can think about this very simply, like say you're just in a box and your experiment is I have a ball on the floor, and I'm just watching it. If you're not accelerating, you're going to get one answer. If you are accelerating, then the ball is going to roll to the back of the box, right because it's not being accelerated and the box is being accelerated. It's like a bowling ball in the back of a truck. And so you're going to get a different answer. And so the universe is not symmetric to accelerations. Like, you accelerate your experiment, you're going to get different answers. You move your experiment, you shouldn't get a different answer. You spin your experiment to a different direction, you shouldn't get a different answer. You delay your experiment by a week, you shouldn't get a different answer. But if you accelerate it, you're definitely getting a different answer. So the universe not symmetric to acceleration.
And does Emmy's equation predict what should be symmetrical and what shouldn't or that just requires some intuition.
Oh yeah, that's another great question. No, it doesn't tell us what the symmetries are, but it does tell us that if you find a symmetry, it has a big consequences for how the universe behaves, and some of these symmetries are easy to think about, like or location or direction. They sort of make intuitive sense. By some of the most important symmetries turn out to be weird quantum mechanical quantities that we never really think about. We talked recently about like electrons. Electrons are quant mechanical objects, and we think of them as having like physical properties mass and direction and spin and this kind of stuff, that they also have weird quantum mechanical properties, like their wave functions can have a complex piece to them. By complex, I don't mean complicated. I mean like you know five plus four I the way some numbers are real numbers and some numbers are complex numbers. They have an imaginary component. Electron wave functions can have an imaginary component, and this isn't something you can measure, because whenever you measure the wave function, you always take its amplitude square. The imaginary part goes away. But it is a feature of the electron, and so you can think of it as like an angle, like you can rotate this thing through complex space. So electrons have these weird complex internal quantum mechanical angle that you can can't measure, and it turns out that the universe respects that you can rotate electrons through this angle and it doesn't change anything in the equations. So that's like another weird symmetry that the universe has. And if people want to read more about this, this is called you one symmetry of the Lugrongen. So it's not just like obvious physical symmetries that the universe has. We also have these weird internal quantum mechanical symmetries that have very deep consequences thanks to Nuther's theorem.
We were talking about these angles on the episode where we were talking about charge and force. Isn't that right? And what context did they come up in? I guess they just came up in the wave function context.
Yeah, Well, we were talking about why we have photons, and it turns out that you can't respect this weird electron angle without having photons. Photons are the things that make this weird electron angles symmetric in the equations of physics, and that has the big consequence thanks to Nother's theorem. But you have to have photons. You can't just do this in the universe with just electrons, and so some people like to say that the reason we have photons is to respect this electron symmetry.
Okay, thanks, I'm expecting an honorary degree in physics one of these days, so I have to make these connections between episodes.
I'm gonna give you a pod in physics. Eventually.
This is the O stand for I went over my head. Oh God, oh man for not catching that they're going to take away the PhD I already have. I don't know who they is, but someone's taken it away. The man. All right, let's all take a break and ponder which university should be giving me my honorary PhD. And when we get back, we'll dive into the details of Notether's theorem. All right, we're back, Daniel. You've been sort of giving us tantalizing. It's about symmetries and what Notre taught us. Let's talk more about what her theorem's all about.
Yes, so Notthern's theorem is really important. It tells us what it means that there's a symmetry in the universe. It tells us what the consequences are like. You might say, okay, so I can do my experiment here, or I can do it in the middle of deep space. I get the same answer, who cares? What does that mean about the universe? Well? Notther's theorem tells us what it means. Specifically, it tells us that any symmetry implies a conservation law. It means that there's something out there that the universe respects that if you calculate it, that number never changes, you know. So to think about what a conservation law is, like, let's say you have a little economy and people are spending money in trading on eggs for dollars or whatever. You can spend those dollars, but this dollar is still go somewhere, and they still are somewhere, right, So, like the number of dollars in a closed economy is conserved because you have a certain number of dollars and they're just going to move around, but the number of them doesn't change, right, So the total amount of money is conserved there. And I know that in real economics the total amount of money isn't conserved and value can be created in whatever. But like say you have a simple economy with a few dollars and they're just moving around, the dollars can flow. There's a current of money, but overall, there's a conservation of those dollars.
Right, Okay, assume a spherical economy exactly.
So another theorem tells us that all the symmetries we talked about previously and will go through each one, imply a conservation law in the universe. There's something that the universe respects these Every symmetry has a conservation law, and any conserved quantity implies that there's a symmetry there.
We start with that sphere example.
Yeah, so let's take a sphere and say we move it somewhere else. It's still a sphere, right, that hasn't changed. So what does that translation symmetry imply? It implies conservation of momentum. Right. The reason that the universe conserves momentum is because there is no special place in the universe. There are no markings in space. It doesn't matter where you do your experiment or where you take your measurement that you should get the same answer. Noother's theorem says, if that's true in your universe, then momentum is conserved. And that's kind of like a big leap. You're like, what, like I get there's a connection there. We have positions. But now we're talking about momentums and velocities, and like it's kind of crazy, right, Like what is momentum conservation anyway? It just says that if you calculate the momentum of stuff, which is mass times velocity, and then you let physics happen. Like you have two balls and they bounce off each other, and you measure their mass and their velocity beforehand, you add it up, you get a number. You measure their mass and velocity afterwards, and you calculate momentum and add all up. You get the same number. It's like if I buy eggs from you, I trade you dollars, the total number of dollars in the universe hasn't changed. Well, if we bounce balls off of each other, the amount of momentum in the universe hasn't changed. And that's because of notother's theorem. Another's theorem says we have momentum conservation because space is the same everywhere. It doesn't matter where you do your.
Experiment, okay, and so if you do your experiment at different times, you get the same results. Because what is conserved.
Time symmetry means that there's a conservation of energy. The reason we have conservation of energy is because of time symmetry, and Nuther's theorem connects these things. It tells you exactly what is conserved if you have a certain symmetry. And so there's this pairing between energy and time. There's a pairing between position and momentum, and Noutherar's theorem tells us that, like translation symmetry, position symmetry gives us conservation momentum, time symmetry gives us conservation of energy, and in fact it goes even deeper than that. That's a really useful definition of energy. Energy is the thing that's conserved if you have time symmetry. That's the way a lot of physicists think about energy now in terms of Nuther's theorem and these conservation laws, and it's hard to sort of wrap your mind around, like why is this? Can we understand how Nutherar's theorem connects these things?
Can you give me an example of, like, if I do the double slit experiment, maybe I'm making it at two different times, how does that tell me about conservation of energy?
Let's get to conservation of energy, because it's tricky, because it turns out energy is not actually conserved in the universe, which tells us something else about the universe. But let's go back to simple momentum and thinking about balls and thinking about what it tells us about the universe and how space is the same. Right, it's interesting that you could do the same experiment anywhere in the universe and get the same answer. Why does that tell us something about momentum. Well, imagine that we had a universe that didn't have the same kind of space everywhere. Let's say we had a universe that had an edge to it, like a wall, you know, like a place where if you tried to go there, you just bounced back. Right. Now, we have a universe where you have most of the same kind of space, normal space, but then you also have an edge bit, and an edge bit has special properties that if you hit it, you bounce back. Right. So do we have a momentum conservation in that universe? No, we don't, because what happens if you throw a ball against the edge of the universe it bounces back, it changes its momentum, and there's nothing to compensate for that. Usually, if I want to change the momentum of something, I bounce it against a wall, that wall pushes back and it absorbs that momentum. The walls like pushed in the other direction. But if I have the edge of the universe, it's like special magical bit of space that can just turn things around and change their momentum. Boom, I have violated conservation of momentum. So there's an example of how having the space not be the same everywhere will violate conservation momentum. You can't have conservation and momentum in that universe. And to me that's really deep because it tells us, hey, if momentum is conserved, that means space is the same everywhere and there isn't an edge to the universe. Boom. It's like amazing these moments when you can do an experiment here on Earth and from that concludes something about deep space super far away. Like what I love it when like the math clicks together and has these literally far reaching consequences, is about what may be happening super far away. You got moment That doesn't mean the universe is infinite, right, It means there's no special space in the universe. You could still have a finite universe, just not one with an edge. If we have a finite universe has to like wrap around it stuff. You can have a finite universe where every bit of space is the same.
I know just enough about physics now to get my own way. So that's where we are in my learning career.
All right.
So we have talked about space bending or is it space time bending? Okay, but that doesn't change momentum, and so we still have conservation.
Yes, you can still have conservation of momentum even if space time bends. Yeah, exactly, okay. And this is also really fascinating because it connects us with quantum mechanics, Like this is a general theory about the universe, but it tells us there's a deep connection between position and momentum. And we kind of already knew that because quantum mechanics says you can't know position and momentum simultaneously, that these two quantities are linked in a deep way, you know, Like why is it that if you measure position, you can't know momentum instead of if you measure position you can't know like angle or energy or something. Because there's a pairing between these two quantities, position and momentum, they're deeply connected with each other. And if you know something about like Fourya transforms, it's very natural to understand how you transform something from physical space to momentum space. You understand they're connected by these fourya transforms. They really are not independent quantities. They're two different sides of the same coin, and so Notther's theorem reveals that as well. So these are deep connections. So now if you want, we can talk about energy conservation and time translation invariance.
First I want to ask did she have like one equation from which all of this popped out or was it a different equation for each one of these connections.
It's a single equation. It tells you that if your laws of physics are invariant under some transformation, then it tells you what quantity is conserved. So you put in your laws of physics, which we usually describe in terms of a lagrangein, which is basically like kinetic energy minus potential energy, and then you do your transformation. You say, well, what happens if I change it by shifting it to the left or shifting it to the right. And if the answer is that you get the same lagrange in out the same fundamental laws of physics, then it tells you what quantity is conserved and as a single equation, so you can change the symmetry you're exploring and it will tell you what the conservation law is.
Okay, all right, awesome, Well let's move on to energy.
Yeah, so energy is super fascinating. And we were saying earlier that if the universe has time translation symmetry, meaning it doesn't matter when you do your experiment, then energy is conserved in the universe. Cool, And everybody thinks energy is conserved in the universe. You're taught energy is conserved in the universe. Energy conservation is fundamental, and people think about energy the way sort of we talk about dollars earlier, Like, yeah, I have energy here and I can transform it, but you can't create it or destroy it. Right, Like if I'm going really really fast and I slam on the brakes, where's my kinetic energy go It goes through friction into the heat of the break pads, you know. Or if you have a book on a shelf that has potential energy you knock it off, that's transformed into kinetic energy. Energy and mass can be transformed back and forth into each other. All this kind of stuff. It's really intuitive for people to think that energy is conserved in the universe. But that's only true if the laws of physics actually are invariant to time. And it turns out they're not quite.
Oh no, how are they not quite? Everything you know is a line.
Well, what we include in the laws of physics are the behavior of space time, and the expansion of space breaks that. So the universe is not the same as it was a billion years ago or five billion years ago. The universe is expanding. Space is expanding. So it turns out that space has to be static to satisfy this time invariance. Because like the amount of dark energy in the universe, the fraction of dark energy in the universe is changing, the fraction of matter in the universe is changing. So because we don't have time translation in VERYOCE, it's because the universe isn't actually the same fundamentally as it was five billion years ago. Energy is not conserved in our universe. Wow, And we can understand that very easily. But just by looking at the dark energy, like the universe is expanding, that makes more space. Dark energy is constant density, which means that as the universe expands, it doesn't get diluted. It's not like matter, where you make more space and you still have the same number of protons. The density has gone down, the amount of matter has stayed fixed. Dark energy is weird. As you make more space, because the density is constant, you get more dark energy. So dark energy is just increasing. Where is it coming from? It doesn't have to come from anywhere because energy not conserved in our universe. That's what Nother's theorem tells us. It tells us energy doesn't just flow. It can be increased, it can be created, and it can be destroyed. So Nother's theorem is all about flow in currents. It says that when you have a symmetry there's a current that's conserved, things can't be created or destroyed. They can only flow. But because time is not symmetric quantity in our universe, energy is not conserved. It doesn't have to flow. It can be created and destroyed. And that's a really deep thing to understand about the universe, because boy, do we have energy problems. And if we could just create energy, Wow, that would change our experience, right, that would be a big deal.
And was this a revelation when the theorem came out or was this something we kind of had to handle on, but now it was proven.
It was really troublesome actually, because Einstein didn't like to imagine that there was something in the universe that violated time translation in variance, and so he famously like rejected adding this to the equations partially for that reason. And it's only later when we discovered that the expansion of the universe was accelerating that we realized we needed this, and so yeah, that sort of blew up our whole thinking about how the universe works. I know we've said this on the podcast before, but it's hard for people to think in because they still write in and say, are you sure as an energy conserved? Like, it's not conserved in our universe, folks, unfortunately, And Another's theorem tells us what that means. It goes back and forth. Right, if you have a symmetry, it tells you what's conserved, and if you have a conservation, it tells you what the symmetry is. And like that's deeply important because the game of physics is to like figure out what is the puzzle of the universe, and recognizing the symmetries or the conservation laws is equivalent. It's like saying this is something that's important to the universe. It's fundamental, it's deep. Momentum is an important quantity in the universe for some reason, and the reason is connected to the reason that space is the same everywhere, the way that like the number of podcasts in the universe not an important quantity. The universe does not care. You can study podcasts and you will not learn anything fundamental about the universe, except if you listen to this podcast, you will learn a lot about the universe.
The universe cares about this podcast, probably.
But it's not like every time a podcast is created, another one is destroyed. If you discovered that that there was like a conservation law podcast that would tell you this some symmetry in the universe that's related to podcasting, I don't know what that would be. And you're really inside into how the universe works.
So was Emmy appreciated for this contribution in her time or no? Because big people like Einstein were not willing to accept the results.
People in the field definitely appreciated Nuther. Einstein said, quote Nuther was the most significant creative mathematical genius thus far produced since the higher education of women began, so there's a qualifying there. He's like, she's the smartest lady I ever met, and a lot of people since were like, she's just the smartest mathematician period. You don't have to put lady ye.
Well, and also qualifying since women were allowed in academia, which was like seven years earlier or something like that. Right, that's a pretty big quality.
It's a pretty big qualif But you know, Hilbert and all these folks, they knew what they were dealing with. They understood they were in the presence of genius. And she made a lot of important contributions in abstract algebra, like all huge contributions there. Like this is a small part of her legacy, but it literally underlies all of physics. Everything in physics is Nuther. There's a physicist, Ransom Stevens who said, quote, you can make a strong case that her theorem is the backbone on which all of modern physics is built. Like boom. You know, Maxwell, he did a little piece of it. Higgs, he found a little piece of it. But all at rest on the foundation of Nuther's theorem. All of our lagrongens. Every time we talk about particle physics and how it works. It always has to have symmetries and like symmetries and the group theory, which is another kind of abstract algebra. It's like so interwoven into the foundation of modern physics. It's like we're all playing in Nuther's backyard.
So do you feel like she's being taught about in classes more often?
Now?
Like are we making up for the fact that we talked about you know, Einstein a lot, but those they are not really at all? Or is she still I mean, based on the audience responses, there's still plenty of folks who don't know about her.
There's still plenty of folks that don't know about her, So she's not talked about enough. And I think that, you know, these things take time for somebody to get to Einstein's level of like cultural impact. They have to be celebrated in their time or there has to be like a campaign after they die to resuscitate their legacy and write books about them. And there just hasn't been as much attention paid. And you know, it takes effort, Like somebody's got to go out there and do the research and write the books and make this argument in order to have it rise above the level of noise in this chaotic media landscape. So it's not easy to sort of change the trajectory. But I think in this case it's very well deserved.
Yep, sounds like there's a book project in there or a movie.
Yeah, definitely, we need the Nether biopic for sure. Who's going to plant her? Maybe Rachel Weiss or something.
Maybe I can't say I know what she looks like. I should look it up.
And so, you know. Another's theorem tells us about the symmetries in the universe, and it makes a lot of sense when we think about translation symmetry. It gives us momentum conservation. Rotation symmetry that gives us angular momentum conservation, like the reason an angler momentum is conserving the universe, the reason galaxies are discs, the reason the Earth still spins is because the universe doesn't have a preferred direction. Time translation symmetry gives us energy conservation that weird internal electron angle symmetry we talked about earlier. Nuthu's theorem tells us what that means, and what it means is conservation of electric charge and that's pretty important in the universe. Like you can transfer electric charge, but you can't create it or destroy it. We have looked like we have tried to create a destroy electric charge. Nobody's ever seen a violation of that. That means that the universe really, really really wants this weird internal electron angle to be preserved. You know, it's fascinating, and this is like really deeply interwoven into the way we structure all the quantum field theories in physics.
All right, so we should all get Emmy on a T shirt for starters, and we all need to have a poster with things that Emmy maybe didn't actually say. Yeah, because I heard a bunch of if Einstein posters do have like quotes that you probably never actually is that true?
Yeah, And unfortunately, these days, if you google Einstein and look for images, there's a bunch of AI generated nonsense. It's not actually pictures of Einstein. So yeah, the information landscape is deeply polluted. But I think somebody out there who's like a Netflix executive should commission at like seven part Emmy Nuther biopic miniseries. And I promise you if you do that, you're going to win an Emmy for your Emmy series.
Oh nice, nice, and we're available to hire to help out.
Absolutely, Yes, I want to be on the set consulting.
Yes, that sounds good. I'll play Emmy. I don't know what she looks like, but probably probably so did Emmy get to live a nice long life.
So there were ups and downs, you know. She finally got a position after World War One, though she wasn't paid, and so she was working in Germany for a while. In nineteen thirty three, when the Nazis came to power, she, like many Jewish scientists, left. She went to Brent mahar and worked there and at the Institute for Advanced Studies, so in New Jersey. But later she died of complications of surgery on an ovarian syst So she got a very warm reception and she was very pleased to be in New Jersey. But she didn't last much longer. And so it's always a tragedy when somebody so smart dies so young, because I always imagine, like, what could they have contributed? What do we not know about the universe because this person died. You know, it's like thinking if Mozart had lived to seventy five, what music of his could we be enjoying today? We just don't know. It's just been deleted from the human experience.
We're going to experience a similar feeling on our next episode when we talk about Nettie.
Stevens, another female scientist.
Another overlooked female scientist who tragically died young.
All right, well, thanks to all the women out there making contributions to science. We hear you, We support you. We look forward to your biopic on Netflix, and.
I'm available for hire.
For those two.
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