Scattering Theory 29:172
Mathematical Methods II
- Room: 618, Van Allen Hall
- Time: 10:30AM - 11:20AM
- Days: Monday, Wednesday, Friday
- Text: Mathematics for Physicists,
Philippe Dennery and Andre Krzywicki, Dover, 1995(1967)(primary text)
- Text: Lie Algebras in Particle Physics, Howard Georgi, ARP (1999),
- Instructor: Wayne Polyzou
- Office: 306 Van Allen Hall
- E-mail: email@example.com
- Phone: 319-335-1856
- Office Hours:W 1:00-2:00, Tu 12:30-1:30, F 12:30-1:30 or by appointment
- Grader: Maleki Sanukesh, Mehdi
- E-mail: firstname.lastname@example.org
- Office: 302 Van
Possible final grades are A+,A,B,C,D,F. The grade of A+ is for
performance that is a full grade above an A. Grades are based on
homework scores 15%, hour exam scores 25% x2, and the final exam 35%.
Exam dates will be determined by the instructor after
consultation with the students.
Homework assignments and important announcements will appear
on the web version of this syllabus
solutions, exam solutions, and lecture notes will be posted on
the class website
My lecture notes are normally written during
the evening before each lecture and posted on the morning of my lecture.
I do not proofread the notes so be warned that they may have errors.
If you do not understand something from the posted lecture notes, check
with me after before or after class.
I will try to correct errors as I go so expect changes in the
latter parts of the notes.
This is the second half of a two semester course on mathematical
methods in physics. The purpose of this course is to expose students
to the type of mathematics that is used in intermediate and advanced
physics classes. I plan to teach the material as a mathematics
course. Deductive reasoning is an important part of physics research,
and a complete development of mathematical methods, including proofs,
helps develop deductive reasoning skills. I also believe that a
careful treatment of the mathematics increases a student's confidence
in the methods, and provides the student with the skill to critically
apply these methods and the proper background to apply new methods. I
will also discuss examples and assign problems that illustrate the
appliaction of these methods to physics problems.
The course has two texts. The primary text is by
Philippe Dennery and Andre Krzywicki. It is an excellent book
and it is available in an inexpensive Dover edition.
The primary text shares my point of view about how to approach the
subject of teaching mathematical methods. It has some gaps and does
not have any exercises. The second is an excellent reference on
group theory for physicists, which is not covered in the primary textbook.
For this semester I plan to follow the organization of the
primary text with additional lectures on group theory.
In addition to these texts there are a number of excellent
references on specific areas of mathematics that are used in physics.
The references listed below go deeper in many of the subjects that I
will cover in this class and cover some relevant areas of mathematics
that will not be covered in this class; I have chosen them because
they are the books that I have personally found to be useful both as a
student and researcher.
- Functional Analysis,
Frigyes Reisz and Bela Sz.-Nagy, Dover, 1990(1950).
Readable treatment of functional analysis.
- Functional Analysis and Semigroups,
Einar Hill, Ralph S. Phillips, AMS Colloquium Publications, Vol. XXXI, 1957.
The best book on analytic properties of resolvents and semigroups.
- Methods of Modern Mathematical Physics, Vol 1-IV,
Michael Reed and Barry Simon, Academic Press, (1972,1975,1978,1979).
This is a four volume set of books that cover almost all aspects
of functional analysis that are relevant for physics. Contains excellent
- Real and Complex Analysis,
Walter Rudin, McGraw Hill, 1972.
first year graduate reference on analysis.
- Real Analysis,
H. L. Royden, Mac Millan, 1968.
Main competitor to Rudin. Used this semester in the math
- Generalized Functions, V1-5,
I. M. Gelfand, G. E. Shilov (V1-3), I. M. Gelfand and N. Ya. Vilenkin (Vol 4),
I. M. Gelfand, M. I. Graev, N, Ya. Vilenkin (Vol 5),
Academic Press (1964,68,67,64,66).
Readable and well written - the definitive reference on distribution theory,
harmonic analysis, infinite dimensional integration. One of my favorite
- Linear Operators (Parts I,II and III),
N. Dunford and J. Schwartz, Wiley, (1957,1963,1971).
Comprehensive three volume work on linear operators.
- Methods of Mathematical Physics, Vol I and II,
R. Courant and D. Hilbert,
Wiley, 1989(1937)(v1) , 1962 (v2).
Comprehensive reference written by leading mathematical physicists.
- Functional Analysis,
Has useful material on semigroups of operators and material that is
important in quantum mechanics.
- An Introduction to Probability Theory and its Application, V1,2,
W. Feller, Wiley, 1950.
Standard reference on probability theory, proof of the law of large numbers
and central limit theorem.
- Differential Equations, Dynamical Systems, and Linear Algebra,
M. Hisrch and S. Smale, Academic Press, 1974.
Modern treatment of linear algebra and differential equations -
nice emphasis on qualitative methods that are important in dynamical
- Ordinary Differential Equations,
V. I. Arnold, MIT Press, 1981.
Clear and concise treatment of differential equations from a modern point of
- Transversal Mappings and Flows,
R. Abraham and J. Robbin, Benjamin Cummings, 1967.
Nice treatment of generic properties of dynamical systems that is
hard to find elsewhere.
- Singular Integral Equations,
N. I. Muskhelishvili, Dover, 1992 (1953).
One of the earliest references in treating a class of integral equation
that are important in scattering theory and imaging.
- Perturbation Theory for Linear Operators,
T. Kato, Springer, 1966.
Contains important material on time dependent scattering theory,
also illustrates many important concepts using finite-dimensional examples.
- Scattering Theory by the Enss Method,
P. Perry, Harwood, 1983.
The first section gives a beautiful introduction to functional analysis, and
uses Weiner's theorem to give a neat geometrical characterization
of spectral properties of linear operators that has applications in scattering.
- Foundations of Modern Analysis,
J. Dieudonne, Academic Press, 1969.
Elegant work on analysis by one of the Bourbaki
contributors; formulates many theorems of elementary calculus in a
geometric manner that applies equally to infinite and finite dimensional
- Complex Variables,
R. Redheffer and N. Levinson,
Holden Day, 1970.
Clear elementary reference directed at physicists, mathematicians and
- The Theory of Functions,
E. C. Titchmarch,
Classic reference, contains much of what is now called
mathematical physics. Easy to read.
- Applied Analysis,
C. Lanczos, Dover, 1988(1956).
Contains practical material relevant to mathematical physics.
- Mathematical Physics,
Robert Geroch, University of Chicago Press, 1985.
A unique abstract treatment of mathematical physics that starts from
category theory. Good job of motivating why certain abstract mathematical
structures are important.
- A Survey of Modern Algebra,
G. Birkhoff and S. Maclaine,
Mac Millan, 1965(1941).
Standard introductory reference on algebra. Written by two
S. Lang, Springer, .
Standard graduate test on algebra.
- The Theory of Groups,
H. J. Zassenhaus, Dover, 1999(1958).
Clear and compact reference that focuses on group theory.
- Theory of Group Representations,
M. A. Naimark and A. I. Stern, Springer, 1982,
Nice treatment of group representation theory.
- The Theory of Lie Groups
C. Chevalley, Princeton, 1999(1946)
This is a classic and well written reference. It deals
with some of the fundamental properties of Lie Groups.
- The Classical Groups - Their Invariants and Representations,
H. Weyl, Princeton, 1939.
A classic references which includes material on the classification of
- Representation Theory of Semisimple Groups,
A. Knapp, Princeton, 1986.
Readable and useful.
- Group Theory and its Application to Physical Problems,
M. Hammermesh, Dover, 1989(1962).
One of the earlier references on group theory written by a physicist.
- Lie Algebras in Particle Physics,
H .Georgi, Benjamin Cummings, 1982.
Nice treatment of Lie Algebras relevant to symmetries of particle physics.
- Gian-Carlo Rota on Combinatorics,
G. C. Rota, Birkhauser, 1995 .
Collection of Rota's papers.
Has useful material on Mobius functions, Zeta functions and partial
orderings that is important in theories involving many degrees of
- General Topology,
J. L. Kelly, D. Van Nostrand, 1955.
Clear treatment of the branch
mathematics that is used to properly formulate convergence.
- Topological Groups,
L. S. Pontryagin, Gordon and Breach, 1966,
Stands out as one of the most readable book on advanced mathematics -
contains excellent treatment foundation material for topics that are
important in Lie groups.
- Differential Geometry, Lie Groups, and Symmetric Spaces,
S. Helgason, Academic Press, 1978.
Best treatment of symmetric spaces, pretty good for differential geometry
and Lie groups as well.
- A Comprehensive Introduction to Differential Geometry (V1-5),
M. Spivak, Publish or Perish, .
Readable treatment of differential geometry
that is relevant for general relativity and gauge theories - contains
some fun historical material.
- The Topology of Fiber Bundles,
N. Steenrod, Princeton University Press, 1951.
One of two treatment of the formal mathematics behind gauge symmetries
- Fibre Bundles,
Dale Husemoller, Springer, 1966.
Covers same material as Steenrod.
- Measure Theory
P. Halmos, Springer 1974(1950).
The subject is becoming more important in physics, expecially for
problems in dynamical systems, statistical physics, and quantum
- Foundations of Differentiable Manifolds and Lie Groups,
M. Warner, Scott Foresman, 1970.
Nice compact treatment of Differentiable Manifolds and
- PCT Spin Statistics and All That,
R. F. Streater and A. S. Wightman, Benjamin Cummings, 1964.
Clear treatment of SL(2,C), finite dimensional representations of the
Lorentz group, representation analytic functions of several variables,
distribution theory with applications to quantum field theory.
- Algebraic Topology,
E. H. Spanier
The first comprehensive textbook on the subject. Most
contemporary mathematicians learned the subjec from this
- Gravitation and Cosmology,
Steven Weinberg, Wiley, 1971
Nice treatment of Reimannian Geometry in Chapter 6, nice treatment of
the Weyl Tensor.
- The Quantum Theory of Fields, VI
Chapter 2 contains a nice treatment of the
representation theory of the Poincare group, projective representation
and central extensions of groups,
- Operator Algebras and Quantum Statistical Mechanics V1,2,
O. Bratelli and D. Robinson, Springer 1979,1981,
Mathematics of systems of and infintie number of degrees of freedom.
Homework Assignments and Calendar
- Week 3
- Monday, February 3
- Lecture 6:
- Wednesday, February 5
- Lecture 7:
- Friday, February 7
- Today only - office hours moved to 11:30-12:30
- Lecture 8:
Homework #3, due, Friday, Febuary 14, solutions (NOTE CORRECTION ON PROBLEM 6)
- Week 4
- Monday, February 10
- Lecture 9:
- Wednesday, February 12
- Lecture 10:
- Friday, February 14
- Lecture 11:
Homework #4, due, Friday, Febuary 21 - note errors on this homework
- problem 4 -2n --> 2n; problem 5 -n --> n , (1-x**2)n --> (1-x*x)^n - see corected problem sheet.
- SPRING BREAK!
- Monday, March 16
- No Class
- Wednesday, March 18
- No Class
- Friday, March 20
- No Class
- Week 15
- Monday, May 4
- Wednesday, May 6
- Lecture 41:
- Friday, May 8
- Final Exam (Note correction AM --> PM)
- Monday, May 11, 8:00PM-10:PM, 618 Van
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