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Note the slight modification of the last problem of Problem Set 1.
My co-author has requested that the textbook should be password protected. I will provide the username and password in class. The class textbook is often being updated with corrections and additions. Be sure you are looking at the most recent version.
Three new class handouts have been posted to Section VI of this website. These include a nice handout with a table of groups of order 15 or less, a fascinating table entitled The periodic table of finite simple groups, and a link to Chapter 1 of the book by Alexey P. Isaev and Valery A. Rubakov entitled Theory of Groups and Symmetries.
In Section VIII of this website, yo will find links to numerous articles that are closely connected to the topics being covered in this class.
I have provided links to many websites of interest to the students of Physics 251 in Section X of this website
A slightly revised Problem Set 1 has been posted to Section III of this website. There is a new part (c) in the second problem.
The General Information and Syllabus handout is available
in either PDF or Postscript format
[PDF | Postscript]
along with comprehensive bibliography
[PDF | Postscript]
The general course information and syllabus are provided
below. Fur further details check out the General Information and
Syllabus handout from the link above.
General Information | ||
|---|---|---|
| Instructor | Howard Haber | |
| Office | ISB 326 | |
| Office Phone | 459-4228 | |
| Office Hours | Mondays 2--4 pm | |
| haber@scipp.ucsc.edu | ||
| webpage | https://scipp-legacy.pbsci.ucsc.edu/~haber/ | |
Lectures: Tuesdays and Thursdays, 11:40--1:15 pm, in ISB 231
From Finite Groups to Lie Groups and Lie Algebras: A Guide for the
Perplexed Physicist, by Howard E. Haber and John Terning
This textbook is a work in progress. The latest version can be
obtained here: [PDF]
Suggestions for improvement would be gratefully appreciated. Please
let me know of any errors or typographical errors that you encounter.
A Physicist's Introduction to Algebraic Structures, by Palash
B. Pal
Continuous Groups for Physicists, by Narasimhaiengar Mukunda and
Subhash Chaturvedi
Theory of Groups and
Symmetries--Finite Groups, Lie Groups, and Lie Algebras , by Alexey P. Isaev and Valery A. Rubakov
Theory of Groups and
Symmetries--Representations of Groups and Lie Algebras, Applications , 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
R. Slansky, Group theory for unified model building, Physics Reports 79 (1981) 1--128.
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:
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, an electronic version of the transparencies and/or a readable set of notes should be made available for posting on this website for the benefit of the other students. 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 will be collected below.
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. Check out the periodic table of finite
simple groups. [PDF]
This was created by Ivan Andrus in 2012, and he posted a detailed
article explaining how and why he made it here:
[HTML]
3. Chapter 1 of the book by Alexey P. Isaev and Valery A. Rubakov entitled Theory of Groups and Symmetries (World Scientific, Singapore, 2018) is provided free of charge by the publisher. It can be found at this link.
1. Lie Algebras in Particle Physics, Second edition, by Howard Georgi.
2. Quantum Theory, Groups and Representations: An Introduction, by Peter Woit (Revised and expanded version, under construction) [PDF]
3. Semi-Simple Lie Algebras and Their Representations, by Robert N. Cahn.
4. Group Theory: Birdtracks, Lie's, and Exceptional Groups, by Predrag Cvitanović. [PDF]
5. Lie Groups, Lie Algebras, and Representations, Second edition, by Brian C. Hall. A preliminary version of this book, which was subsequently published by Springer, can be found here.
6. 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. [Volume 1 | Volume 2]
7. 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]
8. 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]
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:
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 on pp. 7--14 of 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.
In 2023, Gregory Moore taught a course at Princeton University entitled Applied Group Theory. For this class, he provided an extensive set of (unfinished) lecture notes. Links to these notes (in PDF format) are provided below.
1. A course on visual group theory appears
on YouTube, which can be accessed here:
[HTML]
This course has been inspired by a book by Nathan Carter entitled
Visual Group Theory.
2. Confused on how to multiply two permutations together? Check out the following step-by-step instructions.
3. An elementary introduction to the group of units in the integers
mod n (denoted by Un) can be found
here [HTML]
A table that lists the groups Un for all
n ≤ 30 can be found here: [HTML]
4. A list of all groups of sixteen or fewer elements along with some their properties and additional links can be found at Wikipedia.
5. GroupNames.org is a database of names, extensions, properties and character tables of finite groups of order 500 or less.
6. Groupprops, the group properties wiki, provides links to over 8000 articles, including most basic group theory material. It is managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago.
7. 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.
8. 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.
9. 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]
10. For further details on the Monster group, check out the Wikipedia article, which also provides links to the classification of the finite simple groups.
11. A musical piece entitled Finite simple group (of order two).
12. 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]
13. On my Quora feed a few years ago, the following question appeared: why do the sporadic simple groups exist? Alon Amit provides the following answer here: [HTML]