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Group Theory for Physicists (SS19)

Lecturers

Prof. Dr. Stefan Dittmaier
Dr. Philipp Maierhöfer

Dates

  • Lecture: 4 hours, Mon & Wed 14-16, HS II, start: 24.04.2019
  • Exercise class: 2 hours, Fri 14-16, HS II, start: 10.05.2019

Target Audience

  • BSc students ("Wahlpflichtmodul")
  • MSc students ("Elective subject")

Content

  • Basic concepts and group theory in QM
    (Symmetry transformations in quantum mechanics, group-theoretical definitions, classes, invariant subgroups, group representations, characters, (ir)reducibility, Schur's lemmas)
  • Finite groups
    (unitarity theorem, orthogonality relations, classic finite groups, applications in physics)
  • SO(3) and SU(2)
    (basic properties, relation between SO(3) and SU(2), irreducible representations, product representations and Clebsch-Gordan decomposition, irreducibletensors,Wigner-Eckart theorem)
  • SU(3)
    (basic properties, irreducible representations, product representations, applications in the quark model of hadrons)
  • Lie groups
    (basic properties, Lie's theorems, Lie algebra, matrix representations and exponentiation)
  • Semisimple Lie groups and algebras
    (basic concepts, Cartan subalgebra, Cartan-Weyl and Chevalley bases, root systems, classification of complex (semi)simple Lie algebras, Dynkin diagrams, finite-dimensional representations, a glimpse on applications in theories of fundamental interactions in particle theory)
  • Lorentz and Poincare groups and algebras
    (basic properties, finite-dimensional and infinite-dimensional representations, method of induced representation, application to particle states)

Prerequisites

Linear Algebra I & II, Quantum Mechanics

Literature

  • R.N. Cahn, "Semi-Simple Lie Algebras and Their Representations", Dover Publications.
  • R. Campoamor-Stursberg, M. Rausch de Traubenberg, "Group Theory in Physics", World Scientific.
  • R.W. Carter, "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters", Wiley Classics Library, Wiley.
  • J. Fuchs, C. Schweigert, "Symmetries, Lie Algebras & Representations: A Graduate Course for Physicists", Cambridge University Press.
  • H. Georgi: "Lie Algebras in Particle Physics", Westview Press.
  • R. Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications", Dover Books on Mathematics.
  • B.C. Hall, "Lie Groups, Lie Algebras, and Representations", Springer.
  • M. Hamermesh: "Group Theory and Its Application to Physical Problems", Dover Publications.
  • P. Ramond, "Group Theory: A Physicist’s Survey", Cambridge University Press.
  • W.-K. Tung: "Group Theory in Physics", World Scientific.
  • B.G. Wybourne, "Classical groups for physicists", Wiley.
  • A. Zee, "Group Theory in a Nutshell for Physicists", Princeton University Press.

Requirements for Course Achievement ("Studienleistung")

Active and regular participation in the tutorials, including solutions to 60% of the homework problems. Further details will be given in the lecture/ exercise class.

Lecture Notes

Problem Sheets

  1. Problem sheet: groupth19-01 (discussion: 10.05.2019)
  2. Problem sheet: groupth19-02 (discussion: 17.05.2019)
  3. Problem sheet: groupth19-03 (discussion: 24.05.2019)
  4. Problem sheet: groupth19-04 (discussion: 31.05.2019)
  5. Problem sheet: groupth19-05 (discussion: 07.06.2019)
  6. Problem sheet: groupth19-06 (discussion: 21.06.2019)
  7. Problem sheet: groupth19-07 (discussion: 28.06.2019)
  8. Problem sheet: groupth19-08 (discussion: 05.07.2019)
  9. Problem sheet: groupth19-09 (discussion: 12.07.2019)
  10. Problem sheet: groupth19-10 (discussion: 19.07.2019)
  11. Problem sheet: groupth19-11 (discussion: 26.07.2019)
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