Transcription
What to you is the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in in beauty is the simplicity. So if you look at E= MC², so when when a few concepts come together, that's why the Euler identity is often considered, uh, the most beautiful equation in mathematics. Do you do you find beauty in that one, in the Euler identity?
Yeah. Well, as I said, I mean, what I find most appealing is is connections between different things that, um, so the if ei= minus one, um, so yeah, people oh, uses all the fundamental constants, okay, that that's I mean, that's cute, um, but but to me, so the exponential function was interested by Euler to measure exponential growth, you know, so compound interest or decay, anything which is continuously growing, continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function. Um, whereas pi, uh, comes around from circles and rotation, right? If you want to rotate a needle, for example, 180°, you need to rotate by pi radians, and i, complex numbers, represents the swing between imagine axis of a 90° rotation, so a change in direction. So the exponential function represents growth and decay in the direction where you really are. Um, when you stick an i in the exponential, it now it's it's instead of motion in the same direction as your current position, it's the motion at right angles to kind of position, so rotation. Um, and then, so e pi equals minus one tells you that if you rotate for time pi, you end up at the other direction. So it unifies geometry through dilation and exponential growth or dynamics through this act of of complexification, rotation by by by i. So it connects together all these tools mathematics, um, yeah, dynamic structure and complex and complex and, um, the complex numbers, they all considered almost, yeah, they were all next door neighbors in mathematics because of this identity.
Do do you think the thing you mentioned is cute, the the the collision of notations from these disparate fields, um, is just a frivolous side effect, or do you think there is legitimate like value in when the notation, all the our old friends come together, night?
Well, it's it's it's confirmation that you have the right concepts. Um, so when you first study anything, um, you you have to measure things and give them names. Um, and initially, sometimes you because your your model is again too far off from reality, you give the wrong things the best names, and you only find out later what's what's really important. Physicists can do this sometimes. I mean, but it turns out okay. So, actually physics. Okay. So, E= MC². Okay. So, one of the the big things was the E, right? So, when when Aristotle first came up with his laws of of motion and then and then, um, Galileo or Newton and so forth, you know, they saw the things they could they could measure. They could measure mass and acceleration and force and so forth. And so, Newtonian mechanics, for example, F= ma was the famous Newton's second law of motion. So, those were the the primary objects. So they gave them the central building in the theory. It was only later after people started analyzing these equations that there always seemed to be these quantities that were conserved. Um, so particular momentum and energy. Um, uh, and it's not obvious that things happen energy, like it's not something you can directly measure the same way you can measure mass and and and velocity so forth. But over time, people realized that this was actually a really fundamental concept. Hamilton eventually in the 19th century reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object. Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics, like what happens to to all the states, like it's, um, it it really was a central actor which was not obvious initially. Um, and this, uh, helped actually, uh, this change of perspective really helped when quantum mechanics came along. Uh, because, um, the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking because everything was a particle and so forth to to to quantum mechanics, you know, because I think because it was a wave. It just looked really really weird. Um, like you ask, what is the quantum version of F equals MA? And it's really really hard to to give an answer to that. Um, but it turns out that the Hamiltonian, which was so, um, secretly behind the scenes in classical mechanics, also is the key, uh, object in, um, um, in quantum mechanics. There's there's also an object called Hamiltonian. It's a different type of object. It's what's called an operator rather than than a function. But, um, and, um, but again, once you specify it, you specify the entire dynamics. So there's something called Schrödinger's equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. So side by side, they look completely different objects, you know, like so one involves particles, one involves waves and so forth, but with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. For example, in classical mechanics, there's this thing called Noether's theorem. Every time there's a symmetry in a physical system, there is a conservation law. So the laws of physics are translation invariant. Like if I move 10 steps to the left, I experience the same laws of physics as if I was here. And that corresponds to conservation momentum. Um, if I turn around by by some angle, again, I experience the same laws of physics. This corresponds to conservation angular momentum. If I wait for 10 minutes, um, I still have the same laws of physics. Um, so this time translation invariance, this corresponds to the law conservation of energy. Um, so there's this fundamental connection between symmetry and conservation. Um, and that's also true in quantum mechanics, even though the equations are completely different, but because they're both coming from the Hamiltonian, the Hamiltonian controls everything. Um, every time the Hamiltonian has a symmetry, the equations will will have a conservation law. Um, so it's it's it's it's once you have the right language, it actually makes things, um, a lot a lot cleaner. One of the problems why we can't unify quantum mechanics and general relativity yet, we haven't figured out what the fundamental objects are, like for example, we have to give up the notion of space and time being these almost Euclidean type spaces and has to be, um, you know, and you know, we kind of know that at very tiny scales, um, there's going to be quite fluctuations of space space-time foam, um, and trying to to use Cartesian coordinates xyz is going to be, it's just it's it's a non-starter, but we don't know how to what to replace it with. Um, we actually had the mathematical, um, um, concepts, the analog Hamiltonian that sort of organized everything.