Abstract
The 2022 Prizes from the International Mathematical Union featured two focusing on mathematical physics: a Fields Medal to Hugo Duminil-Copin, and the Carl Friedrich Gauss Prize for Applications of Mathematics to Elliott H. Lieb. Duminil-Copin (and his teacher, Stanislav Smirnov, 2010 Fields Medal) developed rigorous ways of thinking of a lattice at its critical local connection probability that leads to a path through the lattice (Fig. 2.1). They employed a discrete complex analysis. Then, they develop ingenious ways of describing that lattice at the critical point, ways amenable to rigorous mathematical proof. In a lifetime of work, Lieb has employed mathematics as the usual physicists’ starting point to reveal the physics. For example, Lieb and collaborators (1964) showed how that Ising lattice might be thought of as a quantum field theory (with annihilation and creation operators). Themes are: Technique is physical. Rigor is revealing. Tricks are physical. And, mathematical physics often leads to deep mathematics.
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Notes
- 1.
J. Cardy and A. Zamolodchikov had derived the relevant feature—conformal symmetry, perhaps not so mathematically rigorously. It turns out that that mathematical rigor was not only for show, to justify what the physicists had uncovered but also led to deeper physical understanding.
- 2.
Subsequent to the Gauss Prize (“for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory″): Lieb received the American Physical Society Medal for Exceptional Achievement in Research (for ″major contributions to theoretical physics through obtaining exact solutions to important physical problems, which have impacted condensed matter physics, quantum information, statistical mechanics, and atomic physics″); the Kyoto Prize in Basic Sciences (“Pioneering Mathematical Research in Physics, Chemistry, and Quantum Information Science Based on Many-Body Physics”); and, with Joel Lebowitz and David Ruelle, the Dirac Medal from the International Center for Theoretical Physics (“for groundbreaking and mathematically rigorous contributions to the understanding of the statistical mechanics of classical and quantum physical systems”).
- 3.
Again, the earlier physical accounts by J. Cardy and by A. Zamolodchikov did directly explore the conformality by approaching and thus suggesting (albeit correctly) what it looked like.
- 4.
The Ising model has a rather more extensive life than I have indicated here. Duminil-Copin has written, “100 Years of the (Critical) Ising Model on the Hypercubic Lattice” as his Fields Medal presentation, where connections with probability theory, percolation, and the “conformal bootstrap” are developed.
- 5.
Which are the microscopic degrees of freedom excited in the conformally symmetric critical system? How are they related to the particles at the critical point having zero energy, in Schultz, Mattis, and Lieb’s account?
Bibliography
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Krieger, M.H. (2024). Why Mathematical Physics?. In: Primes and Particles. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-49776-6_2
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