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Eric Carlson's final course written presentation has been posted to
Section IV of this website.
A link is posted to a pedagogical treatment of root and weight diagrams of Lie algebras by Aaron Wangberg and Tevian Dray. Their paper can be accessed from Section VII of this website and links to an attendant webpage are provided in Section VIII of this website.
A link to some notes by Terence Tao on the classification of complex Lie algebras has been posted in Section VIII of this website.
The solutions to problem set 4 have been posted to Section III of this website.
All the final class presentations are now available and have been posted to Section IV of this website.
I have also added two links relevant to Andy Elvin's presentation on the point and space groups to Section VIII of this website, along with a reference and link to a marvelous book entitled Continuous Symmetry.
The class handout entitled Notes on the Casimir operator, second-order index and the dual Coxeter number of a simple Lie algebra, which is posted to Section V of this website, contains some material that was covered in the last lecture and the bonus 30 minute presentation that preceded the class presentations.
The connection between symplectic matrices and unitary quaternionic matrices is very nicely explained in some notes by John Baez that have been linked to Section VII of this website.
Have a great summer!
The General Information and Syllabus handout is available
in either PDF or Postscript format
[PDF | Postscript]
Some of the information in this handout is reproduced here.
General Information | ||
|---|---|---|
| Instructor | Howard Haber | |
| Office | ISB 326 | |
| Phone | 459-4228 | |
| Office Hours | Mondays 2--4 pm | |
| haber@scipp.ucsc.edu | ||
| webpage | scipp.ucsc.edu/~haber/ | |
Lectures: Tuesdays and Thursdays, 2:00--3:45 pm, ISB 235
Group Theory in Physics, by Wu-Ki Tung
Lectures on Advanced Mathematical Methods for Physicists,
by Sunil Mukhi and N. Mukunda
The basic course requirements consist of four problem sets, which will be handed out during the quarter, and a term project. (There will be no exams.) Due to the limited time in a quarter, it will be impossible to do more than sketch some of the most basic applications of group theory to modern physics. To encourage students to delve deeper, all students will be required to complete a term project based on their reading of a particular topic in group theory and its applications to physics. The project may be presented orally or in written form at the end of the term. Oral presentations are encouraged since they will benefit all members of the class. Please follow the following schedule:
The oral presentations will take place on Monday June 10, 2013 from 4--8 pm in ISB 235.
All projects should include a one page bibliography (containing references pertinent to the project). Copies of this bibliography should be made available to all students in the class. For those projects presented orally, a xerox of transparencies and/or a readable set of notes should be made available to the class. If an oral presentation is not possible (not the preferred option), a full written version of the project is an acceptable substitute.
Problem sets and exams are available in either PDF or Postscript formats.
The problem set solutions are available in either PDF or postscript formats.
Students are required to give half hour presentations on a project
involving an application of group theory to physics. These
presentations are collected here.
1. John Sullivan provided a nice handout with a
table of groups of
order 15 or less. You can find the table in
PDF format. Note that in this table,
V stands for the Klein group (Viergruppe in German), also called the
Klein 4-group. It is the smallest non-cyclic group and is isomorphic
to the dihedral group D2. The group T, also called the
dicyclic group Dic3, can be defined as the order-12 group
generated by two elements a and b such that a6=e (where
e is the identity element) and b2=a3=(ab)2.
2. These notes provide a review of some important concepts in linear algebra that are relevant for the theory of matrix representations of groups. Some of the topics included are: changes of basis, similarity transformations, and matrix diagonalization. [PDF | Postscript].
3. These notes provide a detailed treatment of the properties of the most general three-dimensional proper and improper rotation. The general form for the corresponding 3x3 orthogonal matrix is derived and is used to provide a simple method for determining the axis and angle of rotation and the equation for the reflection plane, if present. In an appendix, the Euler angles are introduced and the Euler angle representation of a three dimensional rotation is explicitly given in terms of the corresponding angle-axis representation. [PDF | Postscript].
4. I am providing a table of the real Lie algebras corresponding to the classical matrix Lie groups, taken from Group Theory in Physics, Volume 2, by J.F. Cornwell (Academic Press Inc., San Diego, CA, 1984). The table provides the definition of each matrix Lie group and the corresponding Lie algebra, along with its dimension. [PDF]
5. A collection of important results involving matrix exponentials that are especially useful in the theory of Lie algebras are provided in this class handout. Along with each result, I also provide a proof or derivation. The notes end with a statement of the Baker-Campbell-Hausdorff (BCH) formula and its proof. [PDF | Postscript].
6. On problem 4 of Problem Set 3, the pfaffian was introduced. I have expanded the treatment of the pfaffian to a set of notes, which are provided as a class handout. In these notes, I first show that any even-dimensional complex invertible antisymmetric matrix is congruent to a block diagonal matrix consisting of multiple copies of iσ2 (where σ2 is one of the Pauli matrices). Using this result, I prove that the square of the pfaffian is equal to the determinant. [PDF | Postscript].
7. The Gell-Mann matrices are a set of traceless hermitian matrices that generate the Lie algebra of SU(3). The properties of the Gell-Mann matrices, along with an explicit list of the structure constants fabc and the totally symmetric tensor dabc are provided in this class handout. The dabc can be used to construct a cubic Casimir operator in the SU(3) Lie algebra. [PDF | Postscript].
8. The root diagrams of the rank-two semisimple Lie algebras are nicely presented in Brian G. Wybourne, Classical Groups for Physicists (John Wiley & Sons, New York, 1974). I have scanned in two pages from this book which exhibit the root diagrams of the rank-two semisimple Lie algebras. [PDF]
9. In Brian G. Wybourne, Classical Groups for Physicists (John Wiley & Sons, New York, 1974), you will also find a very nice treatment of the Dynkin techniques for analyzing semisimple Lie algebras, albeit with a number of typographical errors. In this handout, I have scanned in three tables from Chapter 7 of Wybourne's book. The first table provides the Dynkin diagrams and root structure of the semisimple Lie algebras (with the typographical errors corrected). The second table lists the scalar product of the roots, and the third table provides the Cartan matrices of the semisimple Lie algebras. [PDF]
10. I have written up a set of notes on the quadratic Casimir operator and second-order index of a simple Lie algebra. The relation of these quantities to the dual Coxeter number is clarified. [PDF | Postscript].
11. A very nice application of Lie group theory to particle physics arises in the mathematical description of spontaneous symmetry breaking. I have written up a set of notes on the group theory of spontaneous breaking of SU(N) and SO(N) via a second-rank tensor multiplet. In the past few years, these group theoretical techniques have been especially useful in the study of little Higgs theories. [PDF | Postscript].
1. Group Theory in Physics, by Wu-Ki Tung is available for online reading.
2. Lectures on Advanced Mathematical Methods for Physicists, by Sunil Mukhi and N. Mukunda is available for online reading.
3. Groups Representations and Physics, by H.F. Jones is available for download via CRCnetBASE. Access is available to UCSC students and staff from a UCSC IP address.
4. Lie Algebras in Particle Physics, 2nd edition, by Howard Georgi.
5. Semi-Simple Lie Algebras and Their Representations, by Robert N. Cahn.
6. Group Theory: Birdtracks, Lie's, and Exceptional Groups, by Predrag Cvitanović.
7.
Lie Groups, Lie Algebras, and Representations, by
Brian C. Hall.
A preliminary version of this book, which
was later published by Springer, can be found
here.
8. A detailed elementary treatment of various topics in abstract algebra, including the theory of groups, rings, vector spaces and fields, can be found in A Course on Algebra, by Ahmet Feyzioglu.
9. Symmetry Groups and Their Applications, by Willard Miller Jr.
Access to pdf files of this out of print book are provided by
the author and can be found
here.
10. Lie Thiery and Special Functions, by Willard Miller Jr.
Access to pdf files of this out of print book are provided by
the author and can be found
here.
1. Are you interested in learning more about quaternions? Check out Visualizing Quaternions, by Andrew J. Hanson (Elsevier, Inc., Amsterdam, 2006), which provides a very readable account of their origin, mathematical properties and applications in visual representations.
2. Since quaternions are non-commuting, it is not clear how to define the determinant of a quaternionic matrix. Over the years, many people have given differen definitions. For a very clear introduction to this subject, see Quaternionic determinants, by Helmer Aslaksen in The Mathematical Intelligencer, 18 (1996) pp. 57--65.
3. An elementary introduction to equivalence relations can be found in A Course on Algebra, by Ahmet Feyzioglu. [PDF]
4. If the largest finite simple sporatic group, a.k.a. The Monster, intrigues you, then check out Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, by Mark Ronan (Oxford University Press, Oxford, UK, 2006) for an exciting, fast-paced historical narrative that describes the quest for the classification of the finite simple groups.
5. Schur's first lemma applies to the case where the representation space is taken over either the real or complex numbers. But, Schur's second lemma as stated in class applies in general in the case where the representation space is taken over the complex numbers. It can fail if the representation space is taken over the real numbers, as in the case of the representation of SO(2) by 2x2 real orthogonal matrices. A paper entitled A Basic Note on Group Representations and Schur's Lemma by Alen Alexandrerian discusses the modification to Schur's second lemma in the case of a representation space over the real numbers. This paper is written in a scholarly mathematical style, which may appear too abstract to physics students at first glance. However, the basic results are quite easy to extract, and I encourage you to do so. [PDF]
6. The group SU(2) is isomorphic to the group of quaternions of unit length. This isomorphism provides a simple framework for understanding the isomorphism SO(3) ≅ SU(2) / Z2. It also provides an easy way to prove that the automorphism group of SU(2) is isomorphic to SO(3). For further details, see Molecular Symmetry with Quaternions by Harald P. Fritzer, Spectrochimica Acta Part A 57 (2001) 1919–1930. [PDF]
7. The connection between symplectic matrices and unitary quaternionic matrices is very nicely explained in some notes by John Baez that can be found here.
8. Aaron Wangberg and Tevian Dray have provided a pedagogical introduction to root and weight diagrams of Lie algebras. In particular, they show how to construct root and weight diagrams from Dynkin diagrams, and how the root and weight diagrams can be used to identify subalgebras. Have a look at Aaron Wangberg and Tevian Dray, Visualizing Lie Subalgebras using Root and Weight Diagrams. [PDF | HTML].
1. A list of all groups of sixteen or fewer elements along with some their properties and additional links can be found at Wikipedia. A more extensive list of groups with thirty or fewer elements, with additional details provided for each group, can be found at the open source mathematics website.
2. Space groups in two dimensions are called wallpaper groups. You can find very nice computer images of these groups here. I can also recommend a marvelous book by William Barker and Roger Howe, Continuous Symmetry: From Euclid to Klein (American Mathematical Society, Providence, RI, 2007). In particular, Chapter VIII of this book provides a superb introduction to the wallpaper groups. For the three-dimensional case, one can find a very useful summary table in the Wikipedia article on space groups.
3. If the origin is fixed, then the space groups reduce to point groups. For further details, see the Wikipedia article Point groups in three dimensions. Point groups in other dimensions, along with many useful tables, are discussed in the Wikipedia article on the concept of a Point group.
4. For further details on the Monster group, check out the Wikipedia article, which also provides links to the classification of the finite simple groups.
5. A free copy of a very nice textbook by Sergei Winitzki, Linear algebra via exterior products, published by lulu.com, provides a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. I have taken Winitzki's treatment of the pfaffian and have translated it into a language that is more familiar to physics graduate students (see class handout 6). Nevertheless, there is something to be said for the more powerful coordinate-free approach.
6. In part (h) of problem 3 on Problem Set 3, you were asked to establish the topological equivalence of the real two dimensional sphere S2 and the complex manifold CP1. This is most easily understood via the representation of the extended complex plane (which includes the point of infinity) by the stereographic projection of the Riemann sphere. The two Wikipedia links just given will provide you with a good starting point for further study. In particular, the Wikipedia article on the Riemann sphere discusses the relation between the Riemann sphere and the complex projective line (i.e. CP1).
7. Tevian Dray at Oregon State University has created an interesting website that provides a nice introduction to root and weight diagrams of Lie algebras and Dynkin diagrams. [HTML]
This website is part of a larger effort associated with a paper by Aaron Wangberg and Tevian Dray entitled Visualizing Lie Subalgebras using Root and Weight Diagrams. [PDF | HTML].
Access to the complete website can be found here. [HTML]
8. Notes on the classification of complex Lie algebras have been provided by Terence Tao at this website. [HTML]
Additional notes by Terence Tao on other topics in the theory of Lie groups and algebras can be accessed here. [HTML]
9. In his term project, Andy Elvin mentioned the space groups in two dimensions, also called the wallpaper groups. You can find very nice computer images of these groups here. I can also recommend a marvelous book by William Barker and Roger Howe, Continuous Symmetry: From Euclid to Klein (American Mathematical Society, Providence, RI, 2007). In particular, Chapter VIII of this book provides a superb introduction to the wallpaper groups. For the three-dimensional case, one can find a very useful summary table in the Wikipedia article on space groups.