Eva Miranda: The Mathematical Bridge Between Classical and Quantum
episodePreviously titled “Quantum Physics Missing Link Discovered... [Geometric Quantization]” — renamed by the publisher on Aug 3, 2026
Transcript
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What is the main topic discussed in this episode?
So you're unveiling something for the first time. The audience is in for a huge treat. I had a chance to preview it fortunately and I'm so excited to go through this or for you to go through it.
Thank you.
Firstly, I think it would be great to talk about what quantization is, as most people know about the path integral quantization or canonical quantization. So what is quantization? Why is it important? And how did you even get into the field of Poisson geometry initially?
Yeah, let's talk about that. Let's think about the world as we see it. This would be classical mechanics. The world that it's uh that it's following, you know, Newton's law, right? The force is um it's it's uh related to the acceleration. Okay. And this is the world of classical mechanics. We are used to the movement of trajectories of celestial bodies follow this uh this pattern. And well There is a step forward from you asked me about Poisson, right? So to go from Newton to Hamiltonian dynamics, which is more or less a change of coordinates, and then we can formulate the equations of movement of particles in something called a cotangent bundle. This sounds very mysterious, but essentially uh it's formed by pairs of position and momenta.
Okay, and then the principle that guides the movement of the particles is the conservation of energy. And we think of the energy as the Hamiltonian of our system. And then our system just follows these two equations here, which are Hamilton's equations. This is a uh this is just a different a system of differential equations, and as the movement of the particle evolves, it follows these equations, these simple equations here. But there was a surprise long time ago, there was an experiment, the double slit experiment, that showed that uh not only light but also electrons uh have this wave particle duality, right? Uh the experiment throughout electrons through a double slit. And while we could expect, if they were particles, that we would say see this double slit again projected on the second screen, there was an effect of interference pattern.
So we see uh on the screen, we see a pattern that corresponds to wave. So this was quite a surprise. So maybe everything every equation we had been using was not totally correct. And well here we have Niels Bohr lecturing about quantum mechanics on Iowa, precisely uh showing uh explaining this this experiment. So we are at the beginning of a new area. We are jumping from the classical to the quantum uh reality. And but this quantum reality looks like something very modern, almost science fiction, but we could think it's quite old. It all started in the n in the in the n in the in the past century with Planck. Who introduced the concept of energy quanta to explain blackbody radiation, right? And he already proposed that energy could be emitted in discrete units.
And this idea of having energy in discrete packages or units was around also, of course, in the theories of Einstein, of Bohr. Who developed the atomic model with quantized electron orbits? Ah, De Broglie, who proposed the wave particle duality for matter And Heisenberg, Schrödinger, and so many people. We can see many of these people In this congress in Solway in nineteen twenty seven. We can see most of the main characters of this uh revolution, which is nowadays it l it looks a bit old, right? And here, but still we we maybe don't understand completely, right? We see the Sherning Earth cat, uh and well the cat is here or is not here. What's the mystery? So we can make some jokes about it, about your cat.
I have good and bad news. So Indeed, maybe the truth is that we are not we are a bit in both world, right? Classical and quantum. Already this was observed by Feynman. Nature isn't classical and if you want to make a simulation of nature, you'd better make it quantum mechanical. It's a wonderful b uh problem because it doesn't look so easy. And here we are, still trying to understand it. So there is a lot of uh at the beginning. There was a lot of discussions about classical versus quantum world.
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Chapters
8 chapters
1
What is the main topic discussed in this episode?
0:00–7:47
2
What is the fundamental difference between classical and quantum mechanics?
7:47–19:37
3
How do Poisson brackets connect classical observables to quantum commutators?
19:37–29:56
4
What are integrable systems and why are they important in symplectic geometry?
29:56–38:23
5
What was Dirac’s dream for a universal quantization map and why did it fail?
38:23–46:18
6
How do action‑angle coordinates simplify the study of Hamiltonian systems?
46:18–1:03:07
7
What is geometric quantization and what choices does it require?
1:03:07–1:13:28
8
How do Bohr–Sommerfeld leaves arise in the quantization of the cotangent bundle?
1:13:28–2:03:02
Speakers
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