Methods of Mathematical Physics
M393C
#54289
Fall 2024
Instructor: Thomas Chen
Email
Office: PMA 12.138
Office hours: Th 11:00 AM - 12:00 PM via Zoom
Lectures: TTh 12:30 - 2:00 PM
Location:
11.176
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Syllabus
The purpose of this graduate course is to provide an introduction to mathematical methods of Quantum Mechanics and Quantum Field Theory, with connections and applications to neighboring research areas, which tentatively include kinetic equations and deep learning. No background in physics is required, but some preparation in Analysis/PDE is useful.
Updated course information will be posted here and on
Canvas.
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Topics
This list of topics is tentative and will be modified frequently.
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Basics of classical Lagrangian and Hamiltonian dynamics.
Principle of least action, Lagrangian formalism.
Legendre transform and Hamiltonian formalism.
Symplectic structure, Poisson structure.
Liouville theorem, Liouville equation.
Symmetries and conservation laws, Noether's theorem.
Integrable Hamiltonian systems, Arnol'd-Jost theorem,
action-angle variables.
Constrained Hamiltonian systems, non-holonomic systems.
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Quantum mechanics.
Derivation of classical mechanics and action principle via path integrals.
Stationary phase estimates.
Derivation of classical Liouville equation via Wigner transform.
Ehrenfest theorem.
Bound states, point spectrum, Birman-Schwinger principle.
Dispersive solutions, continuous spectrum, scattering, wave operators, asymptotic completeness.
Nonlinear Schrodinger equations. Strichartz estimates.
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Manybody quantum systems, bosons, fermions.
Ground states, kinetic energy estimates, Lieb-Thirring estimates.
Stability of matter.
Mean field limits, derivation of nonlinear Schrodinger equations. Quantum de Finetti theorems.
Kinetic scaling limits, derivation of Vlasov and Boltzmann equations.
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Quantum field theory.
Fock space and second quantization.
Field quantization.
Renormalization, Feynman graphs.
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Applications in Deep Learning
Underparametrized DL networks, explicit global cost minimization.
Overparametrized DL networks, links to non-holonomic systems.
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Texts
These are some recommended texts.
- V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer.
- R. Abraham, J.E. Marsden, Foundations of Mechanics, AMS.
- T. Cazenave, Semilinear Schrodinger Equations (Courant Lecture Notes), AMS.
- L.C. Evans, Partial differential equations, AMS.
- S. Gustafson, I.M. Sigal, Mathematical concepts of Quantum Mechanics, Springer.
- J. Glimm, A. Jaffe, Quantum Physics from a functional integral point of view, Springer.
- R. Haag, Local quantum physics: Fields, particles, algebras. Springer.
- T. Kato, Perturbation Theory for Linear Operators, Springer.
- L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Elsevier.
- E.H. Lieb, M. Loss, Analysis, AMS.
- E.H. Lieb, R. Seiringer, The stability of matter in Quantum Mechanics, Cambridge.
- J. Moser, E.J. Zehnder, Notes on Dynamical Systems (Courant Lecture Notes), AMS.
- C. Muscalu, W. Schlag, Classical and multilinear harmonic analysis, Vol. 1, Cambridge.
- M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vols. 1 - 4, Academic Press.
- E. Stein, Harmonic Analysis, Princeton University Press.
- T. Tao, Nonlinear dispersive equations, AMS.
- G. Teschl, Mathematical Methods in Quantum Mechanics, AMS.
- T. Wolff, Lectures on harmonic analysis, AMS.
Online resources:
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Lecture notes by T. Arbogast and J. Bona for Methods of Applied Mathematics.
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Lecture notes by W. Schlag on Harmonic Analysis.
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G. Teschl's book is available for download
here.
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Lecture notes by T. Wolff on Harmonic Analysis.
If you would like to prepare ahead over the summer break, the following will be useful: Topics in Functional Analysis, including distributions, Hilbert spaces, spectral theory of selfadjoint operators (book by Teschl, lecture notes by Arbogast-Bona). Methods of Harmonic Analysis (lecture notes by Schlag, Wolff).
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