Eva Miranda: The Mathematical Bridge Between Classical and Quantum

Theories of Everything with Curt Jaimungal

Curt Jaimungal

Physics, Philosophy, Society & Culture, Science

4.6 • 606 Ratings

🗓️ 26 January 2025

⏱️ 124 minutes

🔗️ Recording | iTunes | RSS

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Summary

Eva Miranda explains how hidden geometric structures can unite classical and quantum physics, exploring integrable systems, Bohr‑Sommerfeld leaves, and geometric quantization. The conversation reveals a promising path toward bridging long‑standing gaps in theoretical physics.- 00:00 - Introduction- 06:12 - Classical vs. Quantum Mechanics- 15:32 - Poisson Brackets & Symplectic Forms- 24:14 - Integrable Systems- 32:01 - Dirac’s Dream & No‑Go Results- 39:04 - Action‑Angle Coordinates- 47:05 - Toric Manifolds & Polytopes- 54:55 - Geometric Quantization Basics- 1:03:46 - Bohr‑Sommerfeld Leaves- 1:12:03 - Handling Singularities- 1:20:23 - Poisson Manifolds Beyond Symplectic- 1:28:50 - Turing Completeness & Fluid Mechanics Tie‑In- 1:35:06 - Topological QFT Overview- 1:45:53 - Open Questions in Quantization- 1:53:20 - ConclusionSPONSORS:- I personally subscribe to The Economist. TOE listeners get 35% off the annual subscription. No other podcast has this! https://economist.com/TOERESOURCES:- Eva Miranda's website: https://web.mat.upc.edu/eva.miranda/nova/- Roger Penrose on TOE: https://www.youtube.com/watch?v=sGm505TFMbU- Curt's post on LinkedIn: https://www.linkedin.com/feed/update/urn:li:activity:7284265597671034880/- Join My New Substack (Personal Writings): https://curtjaimungal.substack.com- Listen on Spotify: https://tinyurl.com/SpotifyTOE- Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join- Support TOE on Patreon: https://patreon.com/curtjaimungal- Twitter: https://twitter.com/TOEwithCurt- Discord Invite: https://discord.com/invite/kBcnfNVwqs Theories of Everything with Curt Jaimungal features long-form, technically detailed interviews with leading researchers in physics, mathematics, consciousness, and philosophy, exploring topics at the level of active research. For academics, graduate students, and anyone seeking depth beyond popular science. Learn more about your ad choices. Visit megaphone.fm/adchoices Learn more about your ad choices. Visit megaphone.fm/adchoices Learn more about your ad choices. Visit megaphone.fm/adchoices Learn more about your ad choices. Visit megaphone.fm/adchoices

Transcript

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0:00.0	So you're unveiling something for the first time.
0:03.0	The audience is in for a huge treat.
0:04.8	I had a chance to preview it fortunately, and I'm so excited to go through this, or for you to go through it.
0:10.8	Thank you.
0:12.3	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.
0:21.4	So what is quantization?
0:23.2	Why is it important?
0:24.5	And how did you even get into the field of Poisson geometry initially?
0:29.0	Yeah, let's talk about that.
0:31.5	Let's think about the world as we see it.
0:35.3	This would be classical mechanics.
0:41.3	The world that it's, that it's following, you know, Newton's law, right? The force is, it's related to the acceleration, okay? And this is the world
0:51.3	of classical mechanics. We are used to the movement of trajectories of celestial bodies follow this pattern.
1:00.0	And, well, there is a step forward from, you asked me about POSON, right?
1:07.0	So to go from Newton to Hamiltonian dynamics, which is more or less a change of coordinates.
1:12.9	And then we can formulate the equations of movement of particles in something called a cotangentium bundle.
1:21.2	This sounds very mysterious, but essentially it's formed by pairs of position and momentum.
1:28.3	Okay?
1:29.3	And then the principle that guides the movement of the particles is the conservation of energy.
1:35.3	And we think of the energy of the Hamiltonian of our system.
1:39.3	And then our system just follows these two equations here, which are Hamilton's equations.
1:47.0	This is just a system of differential equations, and as the movement of the particle evolves,
	...

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