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Department of Physics
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Physics 251 Home Page---Spring 2019


This page contains copies of the class handouts, and other pertinent items of interest for the U.C. Santa Cruz Physics 251 group theory and modern physics class.


SPECIAL ANNOUNCEMENTS

new!!! Solutions to Problem Set 4 have been posted to Section IV of this website.

Final project presentations have been posted to Section V of this website.

Final grades have now been posted. Have a great summer break---you earned it!!!


Table of Contents

[ I. General Information and Syllabus | II. Disability Statement to Students in Class | III. Problem Sets | IV. Solutions to Problem Sets | V. Final presentations of the student projects | VI. Other Class Handouts | VII. Textbooks available on the Web on Group Theory, Lie Groups and Lie Algebras and their Applications in Physics | VIII. Articles and Books of Interest | IX. Lecture Notes on Group Theory, Lie Groups and Lie Algebras and their Applications in Physics | X. Websites of Interest]



I. General Information and Syllabus

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
e-mail haber@scipp.ucsc.edu
webpage scipp.ucsc.edu/~haber/

Class Hours

Lectures: Tuesdays and Thursdays, 9:50--11:25 am, ISB 231

Recommended Textbooks

There is no required textbook. However, I can recommend a number of textbooks that you may find useful during this course. The first textbook listed below comes closest to covering the syllabus of this course. Apart from the first two textbooks (which were released in 2018), all other textbooks listed below served as the required textbook for this course at one time or another in years past.

Theory of Groups and Symmetries , by Alexey P. Isaev and Valery A. Rubakov
Group Theory in Physics: A Practitioner's Guide, by Rutwig Campoamor-Stursberg and Michel Rausch de Traubenberg
Group Theory in Physics, by Wu-Ki Tung
Groups, Representations and Physics, Second edition, by H.F. Jones
Group Theory in Physics: An Introduction, by J.F. Cornwell
Group Theory for Physicists, Second edition, by Zhong-Qi Ma and Xiao-Yan Gu
Lie Groups, Lie Algebras, and Some of Their Applications, by Robert Gilmore
Lie Algebras in Particle Physics, Second edition, by Howard Georgi
Group Theory: A Physicist's Survey, by Pierre Ramond
Symmetries, Lie Algebras and Representations, by Jürgen Fuchs and Christoph Schweigert     (Errata)
Group Theory in a Nutshell for Physicists, by Anthony Zee

Recommended Review Articles

R. Slansky, Group theory for unified model building, Physics Reports 79 (1981) 1--128.

Course Requirements

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 Thursday June 6 from 3--6 pm and on Friday June 7 from 2:30--5:30 pm in ISB 231.

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.

Course Syllabus

  1. Introduction to Abstract Group Theory
  2. Fundamentals of Finite Groups
  3. Group Representation Theory
  4. The Symmetric Group and Young Tableaux
  5. Introduction to Topological Groups and Lie Groups
  6. SU(2) and SO(3)
  7. Global and Local Properties of Lie Groups
  8. Lie Algebras
  9. Representations of SU(2) and SU(3)
  10. Complex Semisimple Lie Algebras and their Representations

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II. Disability Statement to Students in Class

UC Santa Cruz is committed to creating an academic environment that supports its diverse student body. If you are a student with a disability who requires accommodations to achieve equal access in this course, please submit your Accommodation Authorization Letter from the Disability Resource Center (DRC) to me privately during my office hours or by appointment, as soon as possible in the academic quarter, preferably within 1 week. I also am open to and want to encourage you to discuss with me ways I/we can ensure your full participation in this course. If you have not already done so, I encourage you to learn more about the many services offered by the DRC. You can visit their website (http://drc.ucsc.edu/index.html), make an appointment, and meet in-person with a DRC staff member. The phone number is 831-459-2089 or email drc@ucsc.edu. UC Santa Cruz is committed to creating an academic environment that supports its diverse student body. If you are a student with a disability who requires accommodations to achieve equal access in this course, please submit your Accommodation Authorization Letter from the Disability Resource Center (DRC) to me privately during my office hours or by appointment, preferably within the first two weeks of the quarter. At this time, we would also like us to discuss ways we can ensure your full participation in the course. We encourage all students who may benefit from learning more about DRC services to contact DRC by phone at 831-459-2089 or by email at drc@ucsc.edu.

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III. Problem Sets

Problem sets and exams are available in either PDF or Postscript formats.

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IV. Solutions to Problem Sets

The problem set solutions are available in either PDF or postscript formats.

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V. Final presentations of the student projects

Students are required to give half hour presentations on a project involving an application of group theory to physics. These presentations will be collected here.

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VI. Other Class Handouts

Class handouts are available in PDF (and in some cases also in postscript) format.

1. Robert Gilmore provides a very nice introduction to the the basic building blocks of algebra---groups, fields, vector spaces and linear algebras. I have scanned the first ten pages of his book, Lie Groups, Lie Algebras, and Some of Their Applications, and provide it here.   [PDF]

2. 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 (denoted as Q6 by Ramond), 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.

3. Thanks to Sandra Nair who came across the periodic table of finite simple groups. Check it out here.   [PDF]

4. Chapter 1 of the book by Alexey P. Isaev and Valery A. Rubakov entitled Theory of Groups and Symmetries (World Sicentific, Singapore, 2018) is provided free of charge by the publisher. It can be found at this link.

5. Chapter 1 of the book by Zhong-Qi Ma and Xiao-Yan Gu entitled Group Theory for Physicists (World Scientific, Singapore, 2007) provides a brief review of linear algebra that is often used in group theory. This chapter is provided free of charge by the publisher and can be found at this link.

6. 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].

7. A collection of important results involving the matrix exponential that are especially useful in the theory of Lie algebras (together with a proof or derivation of each result) are provided in this class handout. Included is a statement of the Baker-Campbell-Hausdorff (BCH) formula and its proof. For completeness, I also include a collection of corresponding results involving the matrix logarithm.   [PDF | Postscript].

8. I am providing a table of the real Lie algebras corresponding to the classical matrix Lie groups, taken from Group Theory in Physics: An Introduction, by J.F. Cornwell (Academic Press Inc., San Diego, CA, 1997). The table provides the definition of each matrix Lie group and the corresponding Lie algebra, along with its dimension.   [PDF]

9. In this class handout, I describe details of the local properties of a Lie group using the techniques first introduced by Sophus Lie. For example, I show that the existence and the properties of the structure constants can be derived by studying the local properties of the group multiplication law. I also discuss the concept of generators of an infinitesimal Lie group transformation, which can be applied to an action of a Lie group on a manifold or to an action of a Lie group on the group manifold itself. These concepts are then illustrated in the case of SO(3).   [PDF | Postscript].

10. In this class handout, I discuss the properties of the Cartan-Killing form on a real Lie algebra and related theorems. Some applications are presented: the construction of real forms of a complex Lie algebra and the construction of a completely antisymmetric third rank tensor related to the structure constants of the Lie algebra.   [PDF | Postscript].

11. On problem 5 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].

12. 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].

13. There are many useful relations that can be derived involving the generators of SU(n), the structure constants fabc and the symmetric tensor dabc defined in the preivous handout (and generalized to arbitrary n).   [PDF | Postscript].

14. 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]

15. 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]

16. 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].

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VII. Textbooks available on the Web on Group Theory, Lie Groups and Lie Algebras and their Applications in Physics

1. Group Theory in Physics, by Wu-Ki Tung is available for online reading.

2. Lie Algebras in Particle Physics, Second edition, by Howard Georgi.

3. Quantum Theory, Groups and Representations: An Introduction, by Peter Woit (final draft version)     [PDF]

4. Semi-Simple Lie Algebras and Their Representations, by Robert N. Cahn.

5. Group Theory: Birdtracks, Lie's, and Exceptional Groups, by Predrag Cvitanović.     [PDF]

6. Lie Groups, Lie Algebras, and Representations, Second edition, by Brian C. Hall.   (Errata)     A preliminary version of this book, which was subsequently published by Springer, can be found here.

7. 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.

8. Classical and quantum mechanics via Lie algebras (draft version), by Arnold Neumaier and Dennis Westra provides numerous physics applications of the theory of Lie algebras.     [PDF]

9. Geometric Mechanics, Part I and Part II, by Darryl D. Holm. These two books provide numerous applications of the theory of Lie groups and Lie algebras in classical mechanics.     [PDF-I | [PDF-II]

10. Lie Theory 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.


VIII. Articles and Books of Interest

1. One very nice treatment of finite group theory from a physicist's point of view can be found in Chapter 10 of Frederick W. Byron and Robert C. Fuller, Mathematics of Classical and Quantum Physics (Dover Publications, Inc., New York, 1992), originally published by the Addison-Wesley Publishing Company in 1970, but has now been reprinted in an inexpensive paperback edition by Dover Publications. For your convenience, I am providing a link to Chapter 10 here.   [PDF]

2. Why is the cross product only defined in Euclidan spaces of 3 and 7 dimensions? Check out the following papers for some insight:

A slightly more sophiticated mathematical analysis is provided in See also Chapter 7 of Pertti Lounesto, Clifford Algebras and Spinors.

3. 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.

4. The only real division algebras are the real numbers, the complex numbers, the quaternions and the octonions (the latter is non-associative). For more information on these issues, have a look at a fantastic book entitled Numbers, by H.-D. Ebbinghaus et al. The connection to the possible vector products in Eucliean spaces is mentioned on pp. 278--279.

5. The geometry of quaternions and octonions is discussed in a very readable book entitled The Geometry of Octonians, by Tevian Dray and Corinne A Manogue. This book also includes very nice material on related group theory topics.

6. 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.     [PDF]

7. An elementary introduction to equivalence relations can be found in A Course on Algebra, by Ahmet Feyzioglu.     [PDF]

8. 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.

9. Related to the previous item, and introduction to Monstrous Moonshine can be found at arXiv:1902.03118  [PDF]. Although this subject seems highly mathematical with no connection to physics, the mathematics of the Monstrous Moonshine has recently found applications in theoretical physics as the following INSPIRE search reveals.

10. Conjugation and conjugacy classes in finite group theory is treated in a very nice set of notes by Keith Conrad, entitled Conjugaation in a Group. In these notes, you will find a proof of the statement that elements of Sn with the same cycle structure belong to the same conjugacy class.   [PDF]

11. 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]

12. The hook length formula provides a very simple way to compute the dimension of the irreducible representations of the permutation group. An unusual but simple proof of this formula is given by Kenneth Glass and Chi-Keung Ng, "A Simple Proof of the Hook Length Formula," American Mathematical Monthly 111, 700 (2004)     [PDF]

11. 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 yields a simple proof 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]

12. 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].


IX. Lecture Notes on Group Theory, Lie Groups and Lie Algebras and their Applications in Physics

1. Gregory Moore is teaching a course at Princeton University entitled Applied Group Theory. For this class, he has written an extensive set of lecture notes. Links to these notes (in PDF format) are provided below. These notes are still under construction and will be continually updated over time.


X. Websites of Interest

1. Confused on how to multiply two permutations together? Check out the following step-by-step instructions.

2. A list of all groups of sixteen or fewer elements along with some their properties and additional links can be found at Wikipedia.

3. GroupNames.org is a database, under construction, of names, extensions, properties and character tables of finite groups of order 500 or less.

4. Groupprops, the group properties wiki, provides links to over 7000 articles, including most basic group theory material. It is managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago.

5. 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.

6. 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.

7. A very nice set of notes on the symmetric group (also called the permutation group) can be found in an article on Wikipedia. Note that this article also provides references that treat symmetric groups on infinite sets, although the main focus of this article is on the group Sn where n is a positive (finite) integer.     [HTML]

8. For further details on the Monster group, check out the Wikipedia article, which also provides links to the classification of the finite simple groups.

9. A musical piece entitled Finite simple group (of order two).

10. nLab is a wiki-lab for collaborative work on Mathematics, Physics and Philosophy. Here is a link to the nLab group theory webpage, which contains a lot of useful information and interesting links.     [HTML]
From the nLab grop theory webpage, you can find a direct link to the the nLab Lie group webpage, with many useful links to related topics.     [
HTML]

11. From the nLab grop theory webpage, you can find a link to the the nLab Lie algebra webpage, with many useful links to related topics.     [HTML]
You can also find a link to the nLab Clifford algebra webpage.     [
HTML]

12. The connection between symplectic matrices and unitary quaternionic matrices is very nicely explained in some notes by John Baez that can be found here.

13. A Table of Lie groups and their associated Lie algebras can be found in Wikipedia.     [HTML]

14. In part (h) of problem 4 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).

15. 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]

16. 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]

17. The Japanese chalk that I use in my class lectures is produced by Nihon Rikagaku Industry (check out this link); it is better than any chalk available in the USA. However, the world's best mathematicians have sung the praises of another brand of Japanese chalk that is unfortunately no longer available. For more details, check the following link.   [HTML]

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haber@scipp.ucsc.edu
Last Updated: June 16, 2019