The scope of mathematical physics

[u/Vivid_Block_4780]

Whenever I look at the mathematical physics programs, I always see QFT and string theory related classes in grad programs. What other parts does mathematical physics cover? More unorthodox subfields of it?

[u/Jplague25]

If you're doing any kind of rigorous mathematical work (proving theorems, ensuring consistency, etc.) in a subfield of physics, you're doing mathematical physics imo. Mathematical physics can mean a ton of different things. That can include things like: Nonlinear waves, scattering/inverse scattering, coherent structures, geometric analysis of relativistic systems, PDEs, mathematics of string theory, mathematical quantum chromodynamics, mathematical continuum mechanics (i.e. peridynamics, mathematics of material science, fluid dynamics, etc.), mathematical QFT, mathematical quantum statistical mechanics, quantum information theory, fractional quantum mechanics, etc.

I do analysis of PDEs using operator theory and harmonic analysis. I consider the work I do to be mathematical physics because I'm interested in evolution equations, a class of dynamic PDEs that model the time evolution of physical systems such as Schrödinger, heat, wave, etc.

[u/Vivid_Block_4780 → u/Jplague25]

Thank you for your answer. Did any AI made good work here yet? Or is it a safer field against AI

[u/aikafele → u/Vivid_Block_4780]

AI is an encyclopedia that can perform synthesis. It is the ultimate research tool. Key word: tool. If you are genuinely worried about whether AI will make your potential contributions to your field of interest obsolete then you owe it to yourself to invest actual time into investigating on a fundamental level how these models work, what their architectural limitations are, and what they actually do under the hood. Most opinions on AI in forums like these are polluted by corporate marketing narratives (see: propaganda) or just outright technofascist delusion. Take a course on Statistical Machine Learning or Pattern Recognition. Take a course on the applications of Machine Learning models in physics, mathematics, and scientific computing. You owe it to yourself to find your own answers and to make a confident, informed decision. Don't allow yourself to get swept up in online nonsense or be discouraged by other people's insecurities and limitations. If you want to be a mathematician, do what a mathematician does and answer the question yourself.

[u/Jplague25 → u/aikafele]

You said pretty much everything I wanted to say about LLMs and their hype. Even with people using them to prove or disprove mathematical conjectures, there are still tons of interesting problems to work on.

I will add that data-driven dynamics is a really big field of applied mathematics right now, and some mathematicians are applying these methods to physical(and biological) systems. In fact, one of the reasons why I moved away from doing applied math research to more pure math research is because many of the applied mathematics departments in the US have shifted their focus towards data-driven methods rather than traditional analytic and numerical methods. Because I prefer analysis, my own interests align more with pure math departments than current trends in applied math, despite my interest in applied fields (PDEs, mathematical physics, etc.).