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Question:
Thanks to all those who answered to "Suggestions on Hartree-Fock." I am an undergraduate working on a problem in Quantum Hadrondynamics. I am trying to understand how the Hartree-Fock theory is formulated in terms of the infinite dimensional Grassmannian. If anyone is able, please explain this to me in detail. Or, please suggest some titles that I may want to read. Please understand that this is only my third semester out of high school. My mathematical background is composed of the usual calculus sequence. I have also completed a one year equivalent in Fouriere analysis and complex analysis. And recently I have completed a semester of study in advance differential equations(systems and stability theory terms of Lyapunov functions). Please suggest what other mathematical knowledge I must obtain to understand the Grassmannian formulation of the Hartree-Fock theory. Thank you in advance.



Replies:
Grassmannians??? That sounds un-necessary. You can find descriptions of them in quantum field theory books though. But Hartree-Fock is much simpler than that and can be derived from basic quantum theory. Although I did not think even simple quantum theory was generally taught before the junior year in college... Basically the mathematical knowledge you need is a thorough understanding of the Schrodinger and Heisenberg formulations of quantum mechanics, the concept of a quantum operator, Hilbert spaces, -- and the physics behind the Pauli exclusion principle which is what the Fock part of Hartree-Fock is needed for... plus the concept of Grassman variables...

I just noticed the beginning of your note again. Are you doing research work with a professor at your university? Did your professor refer to infinite dimensional Grassmannians? Also, I am not sure what quantum hadron-dynamics is - is this another name for the "chromodynamics" of the strong force with quarks and gluons? I am no expert in this area, and you may need to get hold of a real one... However, some other suggestions on areas of mathematics you probably ought to investigate to understand what you are doing:
1. Group theory, symmetries, particularly the applications to physics
2. The concept and methods of "second quantization" (assuming you already know something about regular quantization).
3. Perturbation theory
4. The mathematics of arbitrary kinds of spaces: topology, differential topology, etc.

A book that might be readable for you is Itzhikson and Zuber's book on quantum field theory (which I am pretty sure does discuss the Grassman techniques). Good luck!

Arthur Smith



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