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41: Reality Is More Than Complex (Group Theory and Physics)
574 Plays5 years ago
Children who are being taught mathematics often balk at the idea of negative numbers, thinking them to be fictional entities, and often only learn later that they are useful for expressing opposite extremes of things, such as considering a debt an amount of money with a negative sum. Similarly, students of mathematics often are puzzled by the idea of complex numbers, saying that it makes no sense to be able to take the square root of something negative, and only realizing later that these can have the meaning of two-dimensional direction and magnitude, or that they are essential to our modern understanding of electrical engineering. Our discussion today will be much more abstract than that. Much like in our discussion in episode five, "Language of the Universe", we will be discussing how math and physics draw inspiration from one another; we're going to talk about what different fields (such as the real, complex, and quaternion fields) seem to predict about our universe. So how are real numbers related to classical mechanics? What does this mean complex numbers and quaternions are related to? And what possible physicses exist?
Update: Dr. Alex Alaniz and the Breaking Math Podcast have teamed up to create a new youtube show called the "Turing Rabbit Holes Podcast." We discuss science, math, and society with spectacular visuals. Available at youtube.com/TuringRabbitHolesPodcast and on all other podcast platforms.
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Transcript
Understanding Negative and Complex Numbers
00:00:00
Speaker
Children who are being taught mathematics often balk at the idea of negative numbers, thinking of them to be fictional entities, and only learn later that they are useful for expressing opposite directions of things, such as considering a debt and an amount of money with a negative sum. Similarly, students in mathematics often are puzzled by the idea of complex numbers, saying that it makes no sense to be able to take the square root of something negative,
00:00:21
Speaker
and only realizing later that these can have the meaning of two-dimensional direction and magnitude, or that they are essential to our modern understanding of electrical engineering.
Interconnection of Math and Physics
00:00:28
Speaker
Our discussion today will be much more abstract than that. Much like in our discussion in episode 5, Language of the Universe, we will be talking about how math and physics draw inspiration from one another.
00:00:38
Speaker
We're going to talk about what different fields, such as the real, complex, and quaternion fields, seem to predict about our universe. So how are real numbers related to classical mechanics? What does this mean complex numbers and quaternions are related to? And what possible physics exist? All of this and more on this episode of Breaking Math.
Guest Introduction: Alex Alanis
00:00:56
Speaker
Episode 41, reality is more than complex. I'm Sophia. And I'm Gabriel.
00:01:07
Speaker
And this is Breaking Math with this again, we had on Alex Alanis, who's on the episode syntax matters, syntax matters. And Alex, thank you for being on the show. You're welcome.
00:01:18
Speaker
And Alex is one of my coworkers. By the way, hi, everyone. It's, it's great to be back. I know I've been gone for quite a long time. I've been very, very busy at home. So, so it feels great to be back. I've been working behind the scenes, but now I'm back in front of a mic and boy, it feels good. So, um, yeah. And as you also know, Alex is one of my coworkers, uh, properly, Dr. Alex or Dr. Alanis. Uh, if you will, Alex, will you tell us a little bit about your pedigree, uh, in, in, in STEM specifically?
00:01:46
Speaker
Well, I have a bachelor's degree in physics, a master's degree in mathematics, that's all classes, no thesis. I was heading towards a doctoral degree in math, decided to go back to physics to stick to the real world. So I'm a particle physicist by training. Awesome, awesome. And you've worked at a few national labs, is that right? I did eight years at Los Alamos National Laboratory, working as a nuclear weapons, what they call a designer.
Episode Inspiration and Listener Engagement
00:02:10
Speaker
Okay, very good. And then for the sake of this episode, which is on mathematics and physics, will you tell us a little bit about how your background influenced your contributions to this episode?
00:02:21
Speaker
Um, roughly for 10, maybe 15 years after I finished my doctoral degree, I wanted to dot every I and cross every T and just codify physics in as simple a way as possible, which is what is inspiring this episode. Awesome. And also part of what inspires this episode are a lot of, I guess you'll call them water cooler conversations that Alex and I have at work where we just chat in between cubicles, you know, during our downtime.
00:02:44
Speaker
And there's been a very, very strong desire to make something for people studying physicists that help them to see the big picture. Would you say that's probably a pretty good representation?
00:02:57
Speaker
It is. And to back it up, I've written probably a thousand pages. Most of them are available online through a Google Drive that are step by step that go and derive quantum mechanics every inch of the way, every I dot or every T cross. So if our listeners want to actually read the derivations of some of the stuff that we talk about, please send us an email.
00:03:16
Speaker
And you can get a hold of us at breakingmathpodcast.gmail.com. In fact, if you want to talk to us, you can go to patreon.com slash breakingmath. We have the tensor poster available there for $23.46 per poster, plus $15 shipping in the United States. We were doing free shipping for a while, but it just doesn't make sense. It was taking a little bit more time for us to actually mail out the posters than it was to have the posters help us help you bring you the show.
00:03:46
Speaker
Then that's the Tensor poster, which talks about the tensors which are used all over math. They're used in quantum mechanics. They can be used in electrical engineering, not electrical engineering, but Maxwell's equations.
Revisiting Math-Physics Relationship
00:03:59
Speaker
In Maxwell's equations. And you could use them in general relativity, which the last part of the poster talks about.
00:04:06
Speaker
Yeah, and general relativity does not always mesh very well with quantum mechanics, which this episode will be very heavy into quantum mechanics, so... Yeah, and we're not going to totally get into exactly why, because it's going to be a long episode, but we're also on Twitter at breakingmathpod, facebook.com slash breakingmathpodcast, where we have that poster available for, oh wait, no, it's not $15 shipping, it's $12.57 shipping.
00:04:31
Speaker
But we have the poster available there, and you can go check out some applets at breakingmathpodcast.com, and that's all of our plugs. So one of the cool things that I like about this episode is that it touches on this theme that we've visited a few times, I think, in breaking math, physics and mathematics informing each other in this almost spooky way.
00:04:53
Speaker
So on our physics episode, what do we use as an example? Oh my goodness. Wow. That puts me on the spot a little bit. It's been almost two years now since we've done the original physics episode, Language of the Universe. I know that I talked a lot about Richard Feynman, who is very vocal about the relationship between mathematicians and physicists.
00:05:19
Speaker
What were some of the examples we used? My goodness. Guys, it's been so long. Well, if you want to hear those examples, put a language of the universe, episode five, available where you're listening now. Yes, yeah, exactly. Yeah, but of course, mathematics is the language of the universe. And the wonderful thing is that you can have discoveries in mathematics that don't show up in physics until later on. And I think a couple of those examples are things like imaginary numbers that were part of mathematics before they were part of physics.
00:05:48
Speaker
Oh, yeah. And basically what this episode is also going to be about, right, is what physics are possible in, right, something like that? Correct. Mathematicians and physicists together have figured out that they just can't go willy-nilly creating theories. There are rules and restrictions, and they're very simple, actually. And those are known for being willy-nilly.
00:06:12
Speaker
There's going to be a lot of physics that's discussed in this episode, but it all accumulates in talking about something called the Dirac equations and something called Lorentz invariance. So those are the punch lines. We're going to talk about that at the very end. Leading up to that, we talk about conservation. We talk about the Schrodinger equation. We're going to talk about Maxwell's equations. What else? What else is on the plate?
00:06:36
Speaker
Klein-Gordon equation, Klein-Gordon equations, and how it's all wrapped up with algebraic topology in fiber bundle theory.
Physics in Different Dimensions
00:06:44
Speaker
Alex, can you tell us what kind of universes can you have physics in the way that we understand what physics, like what can we apply our current formulations of physics to what dimensions and stuff?
00:06:56
Speaker
So currently we understand the universe is a four-dimensional space-time manifold. Where quantum mechanics- And real quick, can you explain for our audience what a manifold is? A manifold is just a structure, a geometrical structure, the surface of a plane, a three-dimensional space, anywhere where you can have vector fields and tensor fields and things are differentiable, etc.
00:07:16
Speaker
Yeah. And so basically a vector field, you could say that, uh, this is, uh, just like a, an area kind of of virtual space, like, and some kind of dimensional space map, a map for fold out map could be a manifold. Yeah. That's what manifold means literally too. So I was trying to literally think of, yeah, thank you. Yeah.
00:07:37
Speaker
And of course, for our listeners, we're going to do our best. And Alex, we appreciate your patience as we interrupt you so many times and say, wait, stop. Can you give a concrete example of that? All in the name of making this information very accessible. So we appreciate your patience with that ahead of time.
00:07:53
Speaker
Certainly. So anyway, can you have this manifold? So essentially, what's inspiring this episode is the progression from quantum mechanics, which was experimentally foisted on us, so we couldn't deny it. And then the addition of the weak nuclear force and how we figured methods to unify it as electroweak theory and the strong nuclear force. And now we have the unification of these three forces called the standard model.
00:08:17
Speaker
There is a progression of mathematical ideas that seemingly you could test out more alternate universes, richer universes, such as you talked about, grand unified theories. In parallel to that, during the 1930s and 1940s, mathematicians were putting together these structures called fiber bundle theory in algebraic topology. And it turns out that they are the same structures that general relativity uses and that the standard model uses.
00:08:46
Speaker
And the value of the mathematical approach is that they have studied which manifolds are not nice, don't have nice properties like differentiability. So they inform us as to what types of physics, what types of hypothetical universes may exist out there, as opposed to just being carte blanche. Keep going.
00:09:07
Speaker
so so an example of that and i always look for any excuse to mention rick and morty in the universe of rick and morty any universe at all exists they have like the blender dimension and like you know all kind of dimension so that's not actually possible there used to be an idea that all possible universes exist like there's a universe that exists where we are 50 feet tall or whatever um one thing that uh
00:09:33
Speaker
One thing right away about those kind of universes that you constrain in Alex, correct me if I'm speaking out of turn, is that there is an infinite amount of values between 0 and 1, but there's also, for example, an infinite amount of values between 0 and 1 that are
00:09:51
Speaker
There's an infinite amount of values between zero and one-third and between two-thirds and one. So even though you have an infinite amount of possible universes, you can constrain those just within the universe itself by what can actually happen. Because there's not going to be a universe where things just kind of break the laws, I think, of statistics and stuff. I mean, maybe to clarify this, how do you define a physics?
00:10:18
Speaker
How what is recognizable as a physics like what are we working with in this episode? So just to clarify there is the many world's theory of quantum mechanics that every possible universe exists. That's loose language. It means variations of you know, we're famous movie stars in some other universe and that doesn't violate
00:10:39
Speaker
What I mean by other types of universes would be, for instance, physicists often study gravity in two-dimensional manifolds. It makes gravity theories much more amenable. Superstring theories have 11 dimensions because of mathematical reasons to make the math work. And that's what I mean.
00:10:58
Speaker
So would you say that these are just, we take the core expressions of, which we'll be talking about, I think on this show today, like for classical mechanics, I mean, for like quantum mechanics, we have the Schrodinger equation, which we'll be talking about.
Classical to Quantum Mechanics Transition
00:11:15
Speaker
And just you take these basic equations and you just extend them to more dimensions or fewer dimensions, correct?
00:11:23
Speaker
Well, the path that people discovered in the 20s was you can take, you can sum the potential energy and the kinetic energy and algebraically, and you can turn that into a differential equation over the complex numbers. So when you look at a conserved system, say a spring mass that's just sitting there oscillating forever, if you turn that into the Schrodinger equation with the right transformations, you will now have a quantized spring, much like a carbon-carbon atom, and it will have a finite spectra.
00:11:52
Speaker
which we'll get into in the next section. Awesome. So without further ado, here is reality is more the complex. We're going to start off by talking about the conservation of energy and how it could lead to the Schrodinger equations and then how we can talk about higher dimensional and then we could talk about more involved fields like the quaternions and how the Schrodinger equation turns into the Klein-Gordon equation.
00:12:21
Speaker
We're also going to get into Maxwell's equations that describe electromagnetism. We're going to talk about how they can be written in more than one form and what that applies to. We're also going to talk about the Dirac equation, and that relates to everything that we've spoken to up until that point. Following that, later in the episode, we'll talk about tools like Lorenz transforms and what that means both mathematically as well as in the world of physics.
00:12:48
Speaker
And then finally, we're going to end up with talking about a little short discussion of symmetry and fiber bundles and how it ties a whole discussion together. This will be a fun episode, very much related to language of the universe, like part two, just with more robust proofs. Very much.
00:13:05
Speaker
Now, you're telling us off air, Alex, that there's a wave formulation of classical mechanics. But before we get into that, we have to talk about exactly what classical mechanics is. I mean, classical mechanics is a ballpark example. I mean, it's just with physics without relativity, just in three-dimensional space plus time, right? Indeed, classical mechanics, I think of it as Newton's laws.
00:13:34
Speaker
Yeah, and Newton's laws have... Let's talk about real quick what a closed system is. Now, a closed system in classical mechanics and really all physics is a system where you have... It's a system of energy, right? You have energy that is your potential energy in your system plus the kinetic energy in the system, right? If it's closed and it's a constant, yes.
00:13:59
Speaker
So one example that I have always liked personally is basically like if you go down a slide at the very top of the slide you have a lot of potential energy but very little kinetic energy and once you get to the bottom of the slide you have a lot of kinetic energy but your potential energy for the system at which
00:14:22
Speaker
Uh, potential is zero. When you get to the bottom of slide, uh, you have kinetic energy that increases. In as much as that decreases, correct? Yes. There's, there's tons and tons of examples like this. I mean, you know, there's roller coaster examples. I mean, all of the middle school experience experiments, basically, you know, from dropping things, you know, um, just a simple demonstration of gravity or a spring in a box or things like that. So, so these are all the classical examples.
00:14:50
Speaker
And the reason why I like these examples too is because with these springs or roller coasters, you have friction and you have the heat released in the spring, which radiates into the universe. So that's why if you do it in a lab, it's not going to be a closed system, right? You can't physically, perfectly model a frictionless bowl. Frictionless is, unless you're talking about superconducting materials, no, you're not going to get that in the lab.
00:15:11
Speaker
Yeah, so the interesting thing to do about potential energy is that kinetic energy, correct me if I'm wrong, is always related to the velocity, at least in classical mechanics, the velocity, well actually it's quantum mechanics too, right? Momentum squared divided by 2? 2m, yes.
00:15:35
Speaker
Yeah, which is the same thing as one half mv squared, which you might have learned in high school physics.
00:15:43
Speaker
Yeah, yeah. I noticed here there's an example. I don't know if I'm jumping ahead of myself here, but we talk about how one thing that is not an example is a hairdryer in an aluminum box. Did you want to talk a little bit about that? Oh, I just chose that one because the heat is obviously just escaping and you can't model the system without modeling the heat escaping and stuff. And why an aluminum box? Oh, it just arcs to this one XKCD. It's one of his what ifs. It's what if number 35.
00:16:13
Speaker
Okay. And that is like, what is the what if question? Oh, this is what if you put a hairdryer in an aluminum box? And the answer is basically the box gets very hot because the energy they expend within the box radiates outward because the box is not a closed system at all. Okay. Okay. I mean, but it would be a closed system if we had a perfect thermos, right? Yes. I think we got to just across what a closed system is. Okay.
00:16:36
Speaker
Now can you talk about what a wave formulation of classical mechanics would mean? So as I said earlier that I think of classical mechanics as Newton's laws. And by that I mean differential equations. And by that I mean we have three spatial variables and time. So we have vectors and we have tensors.
00:16:56
Speaker
And we talk a lot about those real quick on Navier-Stocht, which is one of our episodes on vector calculus. You can also reformulate classical mechanics in terms of conservation of energy. You can replace Newton's laws with kinetic and potential energies. And there are several theories that people learn in third-year school, Lagrangian mechanics and Hamiltonian mechanics.
00:17:22
Speaker
And those are a reformulation of physics in terms of momentum and energy and potential energies. In 1834, Hamilton and Jacobi explored a wave mechanics version that is essentially equal to all the other versions of mechanics. There are different flavors of it. And so they had in 1834 a wave equation which is equivalent to the Schrodinger equation.
00:17:48
Speaker
And if I can add to that, I was just thinking, in a room with light from a lamp, we think of optics as a ray physics. However, when you start looking at these shadows of sharp objects, say a shadow from a razor blade, you will notice a fuzziness. And that's because you are probing the microstructure of light, the wave nature of it.
00:18:09
Speaker
That was forced on physicists when they started shooting electron beams through crystals and suddenly became obvious that Mechanics has a wave nature
Wave Functions and Quantum Mechanics
00:18:20
Speaker
to it. That is absolutely wild. I love I love that story I've read that story many many times and even though it's well established physics at this point I still find it fascinating So with Isaac Newton, you know a shadow like if you have if you're flash if you have a flashlight and you do shadow puppets those Generally speaking are pretty
00:18:36
Speaker
I mean, not entirely, but pretty. The shape is very close to what you would think the ideal would be if you just projected it, right? Yeah, exactly. Like if you made the object out of soot and you just kind of put this soot into the paper, ground it into the paper, and had that exact shape.
00:18:57
Speaker
Yes, yes, and that very much just from the what the looks of it intuitively supports the corpuscle model of the bullet the photon the That model but you said that if you shine a light on a shoot sharp razor you say or make it simpler Just a double slit. Yeah. Yeah, and you will get the fraction patterns that only waves can do
00:19:18
Speaker
Yeah. Yeah. And the diffraction pattern is, uh, like, let's say you have a pawn, you throw two stones in, uh, in the, say the two, uh, the, the two ripples, uh, create like this, uh, interplay of like troughs and peaks. And that's how you have to add a light. Uh, and, um, and this is pretty much how we know that wave, let the way of light that has a wavelength. Correct. Exactly. The double slit type stuff. Exactly.
00:19:47
Speaker
And these kinetic and potential energy stuff, all this stuff that treats energy is a very general thing, which is kind of like spooky sometimes, like there's this famous problem with a roller coaster in physics. And the loop-de-loop problem, the way that it really works is you calculate the amount of potential energy
00:20:05
Speaker
Basically how much centrifugal force happens in a loop-de-loop and you just do some pretty simple stuff and it turns out to be two-fifths of the height. So you could do some cool stuff with energy and that doesn't change when you go, as our next section will talk about, into quantum mechanics.
00:20:28
Speaker
So now we're going to talk about the Schrodinger equation and exactly what it means and why it's so important. So you had a point that you're going to bring up. So high school kids aren't aware that they take high school physics and they're seeing Schrodinger's equation. Anytime they write conservation of energy, you are then given magical rules in third year or second year physics in college to turn it into a differential equation over the complex numbers. That is how the Schrodinger equation comes about.
00:20:56
Speaker
It is foisted on students and never questioned, but again its origin goes all the way back to 1834 and there's a reason why we turn a classical equation into a differential equation So there are any time you see a conservation of energy equation You are looking at a Schrodinger equation be it for an orbiting mass an orbiting electron oscillating spring whatever the system is and
00:21:21
Speaker
Yeah. Um, I like that. Um, one of the biggest frustrations I think in education is, is, as you said earlier, uh, the, um, quote unquote magic formula where you just do it like you're not, you don't really understand why. So it's really important for appreciating why you're doing what you're doing to understand the reasons behind it and not just arbitrarily do it.
00:21:41
Speaker
Yeah, that's one thing that I hope to get across in this episode especially is I know myself as a student of any time I've been a student and I haven't understood the reason why the equations exist, I don't remember it at all. And it is good. That's one of the things that we appreciated about your papers was that you seem to really go into everything in a really straightforward sort of building things on top of things sort of way.
00:22:10
Speaker
So yeah, so you were saying that the fact that we can use a wave equation at all for kinetic, for mechanics in general, is something that had been established like almost 200 years ago, in 1834. Right, almost 200 years ago. They had no reason to use the wave formulation, so they took the limit of it in geometric optics. So where optics are nice and linear rays.
00:22:34
Speaker
Yeah, it's funny. So yeah, just from like an experiential standpoint, just from what people have observed through, you know, experiments, there's no reason yet. That doesn't mean it doesn't, you know, necessarily work. Can you talk about exactly what you mean, optics in this context? In the world we live in, the everyday world, we think of optics as, optical phenomenon as rays. But when you really start to pass those rays through double slits, you begin to see the wave nature of it.
00:23:00
Speaker
And so the formulation of mechanics in a waveform, it can be taken to the limit where the wavelength becomes negligible and everything is a ray. And that's what they were doing in 1834. Formulating mechanics as a wave equation and taking the limit to where everything becomes rectilinear.
00:23:17
Speaker
If I may, piggyback on the point that you just made, and also our previous point that a lot of people just don't consider this aspect of it. There was a physicist, I believe, Louis de Broglie. He was the one who said that, okay, so if light
00:23:34
Speaker
you know, has both a wave and a particle do things that we recognize as particles also have a wave like property. That is to say, not only protons and neutrons, but even like complex structures such as DNA. And he was absolutely correct. I believe they've done the double slit experiment with things such as DNA strands, and those also have wave properties.
00:24:00
Speaker
Yeah, it's really crazy what happens when you get down to that sort of level. And real quick, though, when we talk about the Schrodinger equation, we're going to be talking about this variable slash function slash whatever you want to call it, psi. And psi is in the Schrodinger formulation, a complex number, correct? Psi is a wave function that is complex valued. It has typically three positions, x, y, z, and t with complex valued values.
00:24:28
Speaker
And so let's say we have this up psi and it's defined over space, right? So if we have like a certain like, you can imagine psi like maybe next to your elbow, like this isn't a good example, like just to show you what site could sort of look like a, it's like zero at your elbow and it's like,
00:24:45
Speaker
Sorry ranges from negative from the wave function itself is from negative infinity to positive infinity in all dimensions time and or space and it's integral over that is one so you are somewhere.
00:25:01
Speaker
Okay, cool. And if you multiply Psi by its complex conjugate, which if Psi is a complex number is just, you could see, we've discussed this before, complex numbers, if you just, it's something like 2 plus 3i, which means 2 plus 3 times the square root of negative 1.
00:25:17
Speaker
And you can model that as 3 on the x-axis and 2 on the y-axis, and then the length of that is that number times its complex conjugate, which in this case would be the square root of 14 or something, or 13.
00:25:34
Speaker
But in psi times its complex conjugate at some event is a probability that the particle that we're considering can be measured at that event. Yes, psi multiplied by its complex conjugate gives you the probability.
00:25:52
Speaker
Yeah and remember in the double slit experiment you fire light through these two slits and you get this diffraction pattern on the other side which is like you have these lines are weak and then it goes like weak and then no light and then stronger than no light stronger and then the middle it's the strongest and then it just it's like
00:26:11
Speaker
But the psi times psi is complex conjugate when you consider the plane onto which it's being projected, the wave function would look like what is graphed out when you look at the light, right? My favorite example of the wave function is the orbitals around a hydrogen atom. You have the s orbitals that are spherical, the p orbitals which look like dumbbells. That's just my mental image of wave functions. Of course, that's just for that particular system.
00:26:41
Speaker
I want to do a quick little tangent on that if I may real quick. That is something that I learned in chemistry. I remember I understood the beauty of orbitals in gravity, but when it comes to the shape of orbitals around the atom, why? So I took my kids to the kid museum here in Albuquerque many times where they have vibrating plates of different shapes and you can pour sand on them and then you can change the frequency of vibration and you get these beautiful geometric patterns. Those are the wave functions.
00:27:09
Speaker
Got it. Okay. Wow. And I just, I have not experienced that that often. So, so basically, um, we have the, this museum here in Albuquerque, New Mexico called Explorer. And I'm sure these museums exist in many places where you've got demonstrations of physical science. And as you said earlier, you have a plate, a vibrating plate with sand, and you can, uh, change the frequency of the vibration. But as you change the frequency, what are some of the shapes that you see the sand make?
00:27:36
Speaker
They, um, I don't know. I can't, they look, they kind of look like you might, what you might find on like, uh, linoleum squares, right? Like it's kind of like almost like squiggly, weird patterns, like beehive patterns. You know, that's where the interference that it maps out the nodes.
00:27:53
Speaker
OK, OK. Yeah, and the reason why this works, I mean, not the whole reason, but when you vibrate this plate, each part vibrates a little bit, and the parts that aren't moving, they collect sand while the ones that do move, the sand kind of moves away from them. But so that's what we mean by siamine wave function in the complex space. Awesome. So then in an atom, it's the same thing. It's just that instead of sand, you have electromagnetic
00:28:23
Speaker
They're the orbitals or the probability shapes of the electron clouds.
00:28:28
Speaker
Wow. So now, if we consider E to be... Well, before we talked about the energy of a total closed system is equal to the kinetic energy plus the potential energy. And we could write that as E equals P squared over 2m plus V, where E is energy, P is momentum, m is mass, and V is potential energy. And the reason why kinetic energy... Kinetic energy is always going to be measured the same way, correct?
00:28:55
Speaker
Yeah, kinetic energy is always measured the same way. V, the potential, if we're talking about an orbiting system, then it's proportional to one over R, which, if it's derivative, gives you one over R squared Newton's law. If it's a spring mass system, it's one-half kx squared is the potential. So it takes many forms.
00:29:15
Speaker
So yeah, and so we're talking about the Schrodinger equation, which again is... The Schrodinger equation just talks about... It takes up a formula for energy in terms of kinetic energy and potential energy, right?
00:29:31
Speaker
And then you put psi into this. You have a few substitutions which we'll talk about in a second. But you get this equation that has psi in it. And if you solve for psi, you get something that can tell us about the probability space, right?
00:29:49
Speaker
Correct. So in a simple spring mass system energy conserved, one half kx squared would be the potential energy term. And then you'd have p squared over 2m would be your kinetic energy term. That's one half mv squared in a different form. And then you have e and e being constant. What you do is you convert e into a partial derivative with respect to time.
00:30:10
Speaker
So what we have is a substitution where we turn E into a derivative differential operator on time and P into a differential operator on space.
00:30:20
Speaker
Yeah, and the substitutions are basically that E is equal to the change in psi, which is our wave function, over time. And that P, which is our potential, the way we're using for our momentum, is equal to I times h-bar times the gradient of psi over space.
00:30:42
Speaker
And remember, the gradient of something is, as we talked about in our vector episode, if we're on a mountain, the gradient points towards the direction we need to go to climb the quickest. So saying at the moment, is it at all physical to say or correct to say?
00:31:02
Speaker
Saying that the momentum is equal to the gradient of psi over space times h bar. Is there any physical explanation? If you look at the equation, P is squared. It's the gradient operator squared. It doesn't no longer mean the gradient.
00:31:26
Speaker
That's right, because e equals p squared over 2m. Right. And so what we have is an equation, a differential equation in time on one side and space on the other. That's a separable differential equation. And if you solve the space part for, say, 1 over r potential, you get the orbitals of hydrogen.
00:31:46
Speaker
Yeah, which at the time was I heard that at the time people didn't appreciate how difficult this would be to solve for like in general and thought that physics would be like just relegated to solving equations for like the foreseeable future of those of you that some people had. That that has been a view and probably still is within a certain small school.
00:32:08
Speaker
Just out of the interest of our listeners, why is that not a view that everyone has now? What informs the view that exists or doesn't exist?
00:32:19
Speaker
What pushes that is that there are people who believe we have approached the end of physics and by that I mean What just happened at the Large Hadron Collider? No new physics was found so it could be that it you know The next thing we do is we build an accelerator so large that we create black holes Which can't be explored that is the end of physics
00:32:43
Speaker
experimental uh physics at least right i mean right so there's no more theory to be had because there's nothing more to explore and if i'm correct though we we use i in p equals i h bar times the gradient of uh psi we just put the i in there so that when we square it we get a negative right to make the signs work out or does that have a physical reason it it has a physical reason in that the equation on the right hand side for the spring has p squared and it has x squared
00:33:12
Speaker
And so we're going to get a wave equation. The solution of this differential equations are sines and cosines, which can be formulated in terms of complex numbers. The sine theta is e to the i theta plus e to the minus i theta over 2.
00:33:27
Speaker
Yeah, and basically what this is referring to is, and we'll be talking a little bit about this in the symmetry of the circle, is like if you take E to a complex number, you just take the direction that the complex number is going and then...
Complex Numbers and Their Role in Physics
00:33:45
Speaker
That is you take e to that to that to that real number and that's how long the vector will be and then you just and then to rotate it around you just You plug the rest into you just I can't remember the geometric interpretation But it's like you just keep rotating when you multiple Yeah, you keep rotating when you multiply this complex number so and it's also kind of the reason why e to the i pi might I mean It's definitely the reason
Maxwell's Equations and Electromagnetism
00:34:15
Speaker
why
00:34:15
Speaker
e to the I pi minus 1 equals 0, which is a very famous equation by Euler. So now we're going to talk real quick about Maxwell's equations. One more. I think we've talked about him in the past. Do you remember? Did we talk about him in the Black Hole series? I don't think, not in the Black Hole series, no. If anyone listens to the podcast and wants to tell us what episode we talked about him on. We're only the hosts. I'm just kidding. I'm just kidding.
00:34:43
Speaker
But we have four, well, first we're going to talk real quick about the formulation with the variables E and B, which is the electric field and the magnetic field. And then there's going to be another formulation, correct, based on potentials.
00:35:00
Speaker
Yeah, potentials, the same as what we were talking about before, basically, and we'll get into that. But the first law that was discovered a while ago, we're not going to go too in-depth, is Gauss's law, which says that the divergence of Eve, the electric field at a point, is equal to the density of electric charge at that point.
00:35:21
Speaker
divided by some constant that makes the units good and it's a ratio constant called the electric permittivity constant. I liked your phrase there's some unit that makes the equations, what was it? Some unit that makes the equations good. Like that's what physicists do when you have a constant, right? It's sort of like spackle, just like stuff it in there.
00:35:44
Speaker
Yeah, sometimes I try to challenge myself to think about what the units of G mean. So basically, the Gauss's law is just saying that... So we have Gauss's law, which is saying that if you have an enclosed volume of space, so you just cup your hands together and you look at that volume, or you'd put a balloon or a neon sign, anything that can... Any shape.
00:36:08
Speaker
The amount of flow in and out of that space is in proportion to the amount of electric charge within that space. Then we have the same one for magnetism, which says that the flow in and out of that space for the magnetic field is always zero.
00:36:27
Speaker
Then we have Faraday's law of induction which talks about how the curl of E, which is basically kind of like the axis around which the vectors of E swirl, is in proportion and in the same place as the opposite shift in direction of the magnetic field over time.
00:36:47
Speaker
So that's if you have a current flowing through a wire and put a compass next to it, it'll kind of move in the direction of the wire. It'll point towards the wire, right? Correct. There's a right-hand rule. If you wrap your hand around a wire, your fingers are pointing. Your fingers, which are curled, they have a circulation, are indicating the magnetic field lines.
00:37:17
Speaker
And finally we have the Emperor's Circular Law, which is kind of the opposite of the other one that we just talked about, but we have some fudging constants. I don't know that law too well. Do you know Emperor's Circular Law in a way with Maxwell's edition?
00:37:35
Speaker
So Maxwell's equations were four equations that were constructed experimentally. And Faraday, back in the early 1800s, was good at using magnetic iron fillings to map out the fields. And over time, people discovered that if you open up a circuit, it creates a magnetic field. If you close a circuit, it creates a transient magnetic field. And so all four Maxwell's equations were derived that way experimentally.
00:38:03
Speaker
Of course, all these formulations are talking about the magnetic field and the electrical field, which are really easy to – well, not easy – well, they're easiest enough to measure in a lab, right? You can make the manifest in elementary school with a magnet and some iron filings, and you put that together with a piece of paper that separates out the filings, and you can actually see, quote-unquote, see the field lines.
00:38:26
Speaker
Yeah, because the little shavings, they form little tiny little oblong blobs that point in the direction of the magnetic field. Or it's kind of like if you ever seen a magnetodoodle, the little lines. But what does it mean to write these in terms of magnetic vector potential and electrostatic potential?
00:38:44
Speaker
So just as Newton's equations give us classical mechanics, F equals ma is really a differential equation for three components, i.e. a vector. The vector formulation of electricity and magnetism with three dimensions for magnetic fields, three dimensions for electric fields,
00:39:01
Speaker
can also be transformed into a formulation that uses potential energy versions, a vector potential and a scalar potential. And that is a very convenient form when we get into relativity. It is natural for relativistic formulations of electricity and magnetism. Yeah, it makes the two equations that we just talked about, Maxwell's equations and the Schrodinger equation, as we formulated it, kind of easy to talk about together, correct? And that will be the next section? Correct.
00:39:29
Speaker
All right, now we're going to talk about the Dirac equations. And to talk about that real quick, we're going to have to talk about the conservation of energy and momentum. And all that says is we have Einstein's famous equation E equals mc squared. But how does that relate to the conservation of energy and momentum?
00:39:47
Speaker
So in classical mechanics, we have conservation of energy.
Conservation Laws and Relativity
00:39:51
Speaker
Say if I keep you going back to mass spring, you have E, p squared over 2m, and k squared over 2. There's a four vector version of conservation of energy. It's just a generalization in relativity. And so then we apply the same rules to turn that into a quantum mechanical equation. It's known as the Klein-Gordon equation.
00:40:12
Speaker
Yeah, and so we have the conservation of energy momentum equation itself is E squared minus P squared C squared is M squared C to the fourth. And so in the same substitution P equals I h bar times the Laplacian of psi gives like what you were saying the Klein-Gordon equation.
00:40:34
Speaker
And real quick, let's talk about a property called Lorentz invariance and how it relates to transformations. Would you like to talk about that? Okay, so in classical physics we're familiar with Galilean transformations and it just means that if you have Newton's equations are valid in one coordinate system and you move to another coordinate system, which is a Galilean transformation, you should still have the exact same laws.
00:41:01
Speaker
Special relativity these generalized to so-called Lorenz transformations. Yeah, and that's basically where you have the rotations as normal but then you have a boost and the and a boost is like It it's it's where you get it that weird like Einsteinian like two people on a train kind of stuff it uh It's where somebody's going in the runs. They have to be going right towards you or at least um, I
00:41:25
Speaker
Essentially, the two different frames are separated in space and there's a constant velocity between them.
00:41:34
Speaker
Yeah, so we have this Lorentz invariance which in special relativity basically let you turn any set of coordinates into any other ones given that we know the relationship in real space-time between the two, right? Correct. And the equations of relativity should remain valid no matter what coordinate system you change to.
00:41:55
Speaker
Yeah, and so what does it mean to say that the Dirac equation... Well, real quick, before we talk about exactly that, this Dirac equation, the Dirac factor, the equation that we just talked about, which is where we took E squared minus P squared C squared equals M squared C to the fourth, which it itself says that basically energy is like the hypotenuse between momentum and mass.
00:42:22
Speaker
is that you take the substitution, you give us the Klein-Gordon equation, so you take the Klein-Gordon equation and you factor it and you get beta k pk plus mc times gamma lambda p lambda equals... So what you do is you take the Klein-Gordon equation, which is a differential equation, and Dirac, for reasons that are historical, factored it.
00:42:48
Speaker
into two differential equations, each one of them being the Dirac equation. It happens that one describes the electron and the other one describes the positron. The way you factor it is it forces you to have coefficients which are complex valued.
00:43:03
Speaker
Yeah, so in the same way that if we do x squared equals negative 2, you can't solve that using real numbers. You have to use complex numbers.
Quaternions and Symmetry in Physics
00:43:12
Speaker
These require quaternions, right? Correct. And so each individual Dirac equation in the factor is only using complex on totem. It's for a quaternion, which is i equals square root of negative 1, j, same thing, k, the same thing. It's a vector system.
00:43:27
Speaker
Yeah, isn't the famous equation i squared equals j squared equals k squared equals ijk equals negative 1? Isn't it on a bridge somewhere? It's on a bridge. Yeah, so he was thinking about it one day and he had a nail in his pocket, I think, and he scratched it onto the side of the bridge and it's been, like, immortalized ever since. He being Hamilton.
00:43:46
Speaker
Oh, that was Hamilton? Okay, I can never remember people, but thank you. And so what does it mean that the Dirac equation is not the Rund's invariant? So if you consider the Maxwell's equations and you transform them and you do Lorentz transformations, in the new coordinate systems you recover back Maxwell's equations.
00:44:07
Speaker
Dirac's equations when you transform them a Lorentz transform you do not get you get some gobbledygook math and so what people did is they they started adding correction terms just guessing and It turned out that the correction terms are Maxwell's equations. In other words Dirac equation plus Maxwell's equations combined are now fully Lorentz invariant and that was a major discovery of
00:44:35
Speaker
And is this Schrodinger equation Lorentz invariant? No, it is by default a classical equation. It takes classical potential and kinetic energies. Okay, and classical obviously can't be Lorentz invariant because they have different ideas of what simultaneity is. So let's talk about symmetry, which we have to have our own episode on sometime. We've been wanting to do for a while, right? That's right. Symmetry and breaking symmetry.
00:45:03
Speaker
Yeah, so symmetry is obviously, well maybe not so obviously, well somewhat obviously, when you can do an operation on something and get the same thing back, right? Correct. So you have symmetry between you and your apparent image in a mirror because all you have to do is rotate it through the fourth dimension and you get back the same thing you had before, just flip it. What does it mean to say that a circle is symmetric?
00:45:29
Speaker
Well, rotationally. Yeah, the symmetry, and I'll talk about it in terms of e to the i theta. It means that if you rotate a circle by an arbitrary angle, you still have the circle. It still looks the same. And so wave functions, you can multiply them by e to the i theta. You're rotating them in complex coordinates.
00:45:51
Speaker
It has no effect on the physics when you take the wave function, multiply it by its complex conjugate. So there's an inherent symmetry of the circle, an S1 symmetry. So in the Dirac equation, you have inherently the symmetry of the circle. In other words, you can multiply the wave function by e to the i theta, and you don't change the physics of it. It happens that it describes an electron-positron photon.
00:46:18
Speaker
As physicists began to discover more particles, they took the S1 symmetry, the e to the i theta, and began to explore others. A richer symmetry would be the symmetry of all rotations, the group of rotations through theta, phi, and some other angle psi, say.
00:46:35
Speaker
And that became a model to build further theories. In fact, it built the electroweak theory, which combines weak force with electricity and magnetism, and ultimately the standard model, which now has the strong, the weak, and the electromagnetic force as a set of symmetries growing richer. And gut theories are even richer group theories.
00:46:56
Speaker
This relates to the fact that you can't have physics in every dimension using physics as we understand it. Correct. From a physicist's point of view, you would think, well, I'm going to explore lots of different group theories as possible extensions of what we haven't observed yet. And you do that because there's still a lot of problems with the standard model. So we know it's not correct.
00:47:21
Speaker
What the mathematicians did is they were doing exactly that, but they were generalizing it and using algebraic topology in a theory called fiber bundle theory.
Manifold Theory and Differentiability
00:47:31
Speaker
And they've already explored what fiber bundles, what structures are nice and differentiable.
00:47:39
Speaker
And I think I've tried to come up with an example of a case when it is not friendly. So here it goes. Imagine a Mobius strip, which is a piece of paper that you can draw rungs on, so it looks like a ladder. Twist it and glue it, you have a Mobius strip.
00:47:57
Speaker
And that is a half twist, correct? Half twist, yes. So it's like if you took a ring, you just cut the ring, and then you twisted it halfway and then glued it back together. Correct. And so it's one-sided. If you actually put your finger on it, you stay on one side.
00:48:14
Speaker
Yeah. We travel twice the distance, obviously. Correct. And if you imagine projecting light from above onto the table below, you will imagine you'll see a circle down below. And we're imagining kind of like maybe an overhead projector where we have a Mobius strip on the projector slide. And we're looking at from above, we just see a circle, right? That it was projected.
00:48:39
Speaker
The circle is nice. It itself is a manifold. You can take the rivets on it. It's a nice object. The mobius strip, as you go around the circle once, the rungs and the ladder, as you come back around once, they have flipped. They're now pointing in the opposite direction.
00:48:55
Speaker
And just we're talking about a really, really thin mobius to write where we're not we don't care about the shape of obviously the width of the what would be the window to strip and you've drawn lines on it, you know, but what I mean is like, let's say this in physical space, let's say we made this out of paper, the shadow that it would have would be kind of like a weird wobbling. Well, no, I'm just imagining the center line. Just just imagine the center line of the mobius strip being projected down as a circle.
00:49:18
Speaker
Cool, just to clarify everything. Anyway, sorry, continue. So the rungs, in order for the rungs to come back to their original orientation, you have to go around twice. The rungs are discrete. They're either correctly oriented or oppositely oriented. That is not differentiable. That is a case of a fiber bundle structure where you shouldn't bother with physics. Physics is like differentiation. Well, the Harry Ball theorem,
00:49:46
Speaker
uh he said theorem theorem that goes back and it essentially says you can't comb a hairy ball there's always going to be a um how would you call it uh a cowlick or a part yeah right and what it means for the earth save we're talking about wind velocities is there there must be from math a proof there must be a point where the where there is no velocity
00:50:11
Speaker
And to talk about exactly what combing a ball means, you could comb a doughnut if you had a hairy doughnut because you could either just comb it around the ring or you could make all the combs going round towards either the bigger radius of the big ring or the smaller radius of the cross section, right? Correct.
00:50:34
Speaker
But yeah, so at that point where there's no velocity, you can't define differentiation in a vector sense. That's an S2 manifold. That's the surface of a sphere. So the Harry Ball theorem demonstrates that you can't have physics in S2? 2S2. So it turns out that the ones that are friendly are S1, S3, and S7, and that's it. Yeah, those are the only spheres from 1 to infinity, right?
00:51:01
Speaker
right, that are rich enough to be good for physics. And so that's, a physicist would be ignorant of that, the mathematicians are
Group Theories and Educational Challenges
00:51:08
Speaker
not. And so that's, essentially you can codify physics in a very compact form, whether you're talking about general relativity or theories, these are theories are so-called algebraic group theories. Essentially, it's about differentiation. In general relativity, it's about differentiation in space-time.
00:51:29
Speaker
In group-based theories, the standard model, it's about differentiating the group itself. As I said, the circle is a continuous group. You can do derivatives on that manifold. It's a group thing because you can rotate arbitrarily and still get the circle back. So there are group theories and geometric theories. They have the exact same fiber bundle differentiable structure.
00:51:52
Speaker
All this weird... It's so weird that at the beginning of this episode, we started with real numbers in classical mechanics and just we have all these... Anybody who studied physics, it's just bizarre how we have all these... How simple physics really is given the data, it seems like.
00:52:13
Speaker
It is, and it is a shame that at the senior level there is no finishing course that just covers this, given a semester, and then perhaps a more advanced finishing course for masters and PhD students. The vast majority of physicists and mathematicians out there are ignorant of this, and I think it's a sad state.
00:52:31
Speaker
And we have some papers by Alex and if you want them you could just go to breaking math podcast comm papers that HTML that's where we that's our Now like we've only had one paper so far on episode one, but now we'll have some of Alex's papers there, too That's right episode one and episode 41
00:52:52
Speaker
And I will be giving you the three books in which chapters the key ideas are most compactly and clearly presented. This excites me because I know that this episode was thick. You guys saw how I was so quiet this episode. I didn't want to intimidate anybody. You know what I mean by getting too technical. So I just, I'm just kidding. But like everything in this episode is very, very well cited for that. We thank you, Alex. Those citations are going to be available also. Uh, what, uh, what do you think on the, on the page? Like,
Conclusion and Acknowledgments
00:53:17
Speaker
Yeah. Just go there. There'll be probably like a little link there on the, uh, it'll be just go to breaking math podcast.com slash papers.html. And, uh, you'll find the papers. Cool. And all the other 39 papers are coming, right? You feel bad now? I feel pressure.
00:53:41
Speaker
Physics is a study of the simplest natures of physical systems, and this simplicity sometimes becomes apparent in the relationship between math and physics, and how deep connections are found between concepts where those relationships are not very physically obvious. We have explored how the nature of the fields we use to approximate physics shape the nature of that system, vice versa, the possible future realms of physics, and how these concepts and others are intertwined.
00:54:07
Speaker
I'm Sophia. And I'm Gabriel. And this has been Breaking Math with us. We had on Alex Alanis and Alex, is there anything else you'd like to say about what we've discussed? No, just thank you for this opportunity. Yeah. And if you have any feedback or questions about Alex's papers, please send them. We'd love to further this discussion. Absolutely. I'd be glad to do that.
00:54:30
Speaker
And a special thanks to Ian Harris, who did some voice work on the episode. He has a podcast, Critical and Thinking, with Ty Barnett, and he has a TV special at Amazon called Extraordinary, and a website, skepticmedian.com, so thank you.