Is Mathematics Invented or Discovered? | Carl Bender | Escaped Sapiens #67
May 17, 2024
auto_awesome
Carl Bender, an applied mathematician and physicist, discusses complex numbers, PT-symmetric quantum systems, and the link between mathematics and reality. He explores the history and practical applications of numbers, the evolution of quantum mechanics, PT symmetry, Lorentz transformations, and collaborative research with Feynman. The conversation also touches on beauty in mathematics, decision-making strategies, dating dilemmas, and partnerships in mathematics and physics.
PT symmetric systems challenge traditional energy stability concepts in quantum mechanics.
PT symmetry introduces new class of systems with real energy levels, diverging from hermitian principles.
PT symmetric systems showcase complex behavior, allowing for new applications in quantum physics.
Experimental setups validate theoretical foundations of PT symmetry in physical systems, opening new research avenues.
Deep dives
PT Symmetry Connects Stable Systems to their Environment
PT symmetry, a concept in quantum mechanics, reveals a unique connection between systems and their environment. Unlike traditional isolated systems or those in contact with the environment, PT symmetric systems exhibit real energy levels while maintaining dynamic equilibrium between energy flowing in and out, akin to a water system in balance. This surprising feature challenges conventional notions of energy stability and system interactions, posing new possibilities for stable systems connected to their surrounding environment.
Implications of PT Symmetry in Quantum Mechanics
PT symmetry, discovered around 2000, stunned physicists by allowing systems to have real energy levels despite not adhering to the conventional hermitian condition in quantum mechanics. This breakthrough introduced a new class of systems where stability and interaction with the environment coexist, delineating a novel understanding of energy behavior in dynamic equilibrium.
Challenging Hermitian Energy Expectations in PT Symmetric Systems
Traditional quantum mechanics teachings emphasize hermitian systems as the basis for real energy levels, denoting stability and unchanging properties over time. In contrast, PT symmetric systems showcase the paradoxical nature of having real energies while diverging from hermitian principles, paving the way for complex variable applications and complex behavior in quantum and classical systems.
Experimental Verification of PT Symmetry in Physical Systems
Experimental setups, such as light propagating through intertwining optical fibers, exemplify the manifestation of PT symmetry in physical systems. By demonstrating light transfer between coupled fibers following PT symmetric principles, researchers validate the theoretical foundations of PT symmetry and its application in real-world scenarios, opening avenues for exploring the dynamics of stable yet interacting systems in quantum mechanics.
The Concept of PT Symmetry in Physics
PT symmetry, a fundamental concept in physics, involves examining the behavior of systems under parity and time reversal. The podcast delves into how PT symmetry can impact the acceleration and expansion of the universe, proposing a theory where gravity behaves attractively at short distances (galactic scales) while exhibiting repulsive behavior at larger scales (universal distances). By viewing gravity through a complex plane lens, the hypothesis suggests a shift between attractive and repulsive gravitational forces based on distance scales.
Optimal Dating Strategy and Mathematical Approximations
The episode explores a thought-provoking analogy between dating strategy and mathematical problem-solving efficiency. By referring to the optimal dating approach of encountering a percentage of potential partners before committing to the next best option, the speaker illustrates a strategy that mirrors a mathematical principle in a game scenario involving selecting the highest number from a set of values. This strategy highlights the statistical likelihood of success when applying a specific selecting threshold in such scenarios.
Utility of Applied Mathematics in Physics
The conversation shifts to the significance of applied mathematics in physics, emphasizing the necessity for mathematical theories to be practically useful and apposite in understanding real-world phenomena. The speaker advocates for the exploration of approximate mathematics for tackling complex problems in physics, aligning with the perspective that mathematical solutions should provide insightful approximations even when exact solutions are unattainable. This approach underscores the pragmatic application of mathematics in addressing intricate physical challenges.
Carl Bender is an applied mathematician and mathematical physicist holding positions at Washington University in St. Louis, the University of Heidelberg, Imperial College, London. He was also one of my own favorite lecturers. He taught me about perturbation theory and asymptotic series which are powerful mathematical tools for solving difficult problems in physics. I invited Carl on to the podcast to discuss complex numbers, and their application in physics. At a conceptual level Carl talks about the link between mathematics and reality, the history of complex numbers and what they are good for, his research into weird new quantum systems known as PT-symmetric quantum systems, his interactions with Richard Feynman, and the role that beauty plays in Mathematical discovery.
►You can find out more about Carl Here:
https://web.physics.wustl.edu/cmb/
These conversations are supported by the Andrea von Braun foundation (http://www.avbstiftung.de/), as an exploration of the rich, exciting, connected, scientifically literate, and (most importantly) sustainable future of humanity. The views expressed in these episodes are my own and those of my guests.
Remember Everything You Learn from Podcasts
Save insights instantly, chat with episodes, and build lasting knowledge - all powered by AI.
