Special Subjects in Mathematical Physics 2
| Numbering Code | U-SCI00 44221 LJ57 | Year/Term | 2022 ・ First semester | |
|---|---|---|---|---|
| Number of Credits | 2 | Course Type | Lecture | |
| Target Year | 4th year students or above | Target Student | ||
| Language | Japanese | Day/Period | Wed.2 | |
| Instructor name | FUKUMA MASAFUMI (Graduate School of Science Associate Professor) | |||
| Outline and Purpose of the Course |
***Lectures are offered in Japanese*** We discuss the representation theory of Lie groups and Lie algebras with applications to physics in mind. If time permits, we will briefly discuss differential geometry and its applications. |
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| Course Goals |
- Thoroughly understand the basic concepts - Be able to apply and use Lie groups and Lie algebras correctly |
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| Schedule and Contents |
Lectures will cover the following: Chapter 1. Lie groups (Weeks 1-3): 1.1 Review of group theory 1.2 Linear Lie groups and examples 1.3 Lie transformation groups 1.4 Invariant integrals Chapter 2. Basics of representation theory (Weeks 4-7): 2.1 Equivalence between representations 2.2 Irreducible representations and direct sums of representations 2.3 Tensor product representations and contragredient representations 2.4 Unitary representations of compact groups Chapter 3. Representation theory of Lie algebras (Weeks 8-12): 3.1 Lie algebras of linear Lie groups 3.2 Representations of Lie algebras 3.3 Representation of SU(2) 3.4 Semisimple Lie algebras 3.5 Roots and weights 3.6 α-string through β 3.7 Fundamental system of roots 3.8 Dynkin diagrams 3.9 Representations of simple Lie algebras Chapter 4. Explicit representations of simple Lie algebras (Weeks 13-15): 4.1 Representation of SU(N) and Young diagrams 4.2 Representation of SO(2l) and spinors 4.3 Representation of SO(2l+1) and spinors |
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| Course Requirements | Students are assumed to understand the basic contents of "Quantum Mechanics 1 and 2" and "Special Subjects in Mathematical Physics 1". | |||
| Study outside of Class (preparation and review) | Students should review the content of the class eac time. A report will be requested at the end of each unit (approximately once every 2-3 weeks). No special preparation is required before the class. | |||
| Textbooks | Textbooks/References | No textbook will be used | ||
| References, etc. |
- Kazuhisa Shima, 連続群とその表現 [Continuous Groups and Their Representations], Iwanami Shoten. ISBN: 978-4007300974. - Tetsuro Inui, Yukito Tanabe, and Yositaka Onodera, "Group Theory and Its Applications in Physics", Springer. ISBN: 978-3540191056. - Howard Georgi, "Lie Algebras in Particle Physics: From Isospin to Unified Theories (2nd ed.)", Westview Press. ISBN: 978-0738202334. - Robert Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications", Dover Publications. ISBN: 978-0486445298. |
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