Topics in Mathematical Physics
M393C
#54642
Fall 2017
Instructor: Thomas Chen
Email
Office: RLM 12.138
Office hours: Th 3:15-4:00 PM
Lectures: TTh 2:00 - 3:15 PM
Location:
RLM 12.166
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Syllabus
The purpose of this graduate course is to provide an
introduction to mathematical aspects of Quantum Mechanics and Quantum Field Theory,
and to make some fundamental topics in this research area
accessible to graduate students with interests in Analysis, Mathematical Physics,
PDEs, and Applied Mathematics.
No background in physics is required.
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.
Symmetries and conservation laws, Noether's theorem.
Legendre transform and Hamiltonian formalism.
Symplectic form, Poisson bracket.
Liouville theorem, Liouville equation.
Integrable Hamiltonian systems, Arnol'd-Jost theorem,
action-angle variables.
Constrained Hamiltonian systems.
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Quantum mechanics.
Link to classical mechanics via path integrals and Wigner transform.
Stationary phase estimates.
Derivation of classical Liouville equation.
Ehrenfest theorem.
Spectral theory, selfadjointness.
Point spectrum, bound states, Birman-Schwinger principle.
Continuous spectrum, scattering states, wave operators, asymptotic completeness, scattering operator.
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Manybody systems, bosons, fermions.
Kinetic energy estimates, Lieb-Thirring estimates.
Electrostatic inequalities,
Stability of matter of 1st and 2nd kind.
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Bose gases. Electron gases.
Mean field limits, macroscopic scaling limits.
Derivation of nonlinear Schrodinger,
Boltzmann, and Vlasov equations.
Quantum de Finetti theorems. Positive temperature formalism.
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Fock space and second quantization.
Field quantization.
Renormalization and spectral theory. Feynman graphs.
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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 S. Klainerman, which include an introduction to Harmonic Analysis.
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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, and Klainerman).
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