Mathematical Physics 3003 Lecturer: Dr. Max Lohe Level: III Points: 2 Duration: Semester 1 Contact Hours: 2 lectures per week; 1 tutorial per fortnight Prerequisites: 9786 Mathematics I (Pass Div I) or 9595 Mathematics IIM (Pass Div I) Assumed Knowledge: 9600 Classical Fields and Mathematical Methods II or equivalent; 7243 Differential Equations II and 2958 Complex Analysis II or 2187 Vector Analysis and Complex Analysis; 5807 Algebra II

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It is suitable for intending mathematical physics students, but should be useful also for all physics students, as well as those mathematics students interested in the way analysis and algebra are used in physics.

A sound knowledge of linear vector spaces and abstract algebra (mainly group theory) is assumed, as is available in 5807 Algebra II, but for those students who have not completed this course I will provide help such as extra lectures, tutorials and handouts, in order to cover the required assumed material. I strongly recommend, however, that students take 5807 Algebra II first.

This course covers most of the following topics:

1. Tensor Algebra

Finite-dimensional vector spaces, bases and change of basis, linear operators, linear functionals, the dual space, dual basis, contravariant and covariant vectors. Inner product spaces, indefinite metrics, Gram-Schmidt orthogonalization, tensors, tensor products and contractions. Exterior algebra, r-forms, exterior products, dual vectors and forms (time permitting).

2. Algebras

Finite and infinite groups, groups of transformations, symmetry groups; the rotation and unitary groups, the Euclidean and Galilean groups. General properties of matrix Lie groups and Lie algebras, infinitesimal generators, one-parameter subgroups, the structure constants. Quaternions, Clifford algebras, Grassmann algebras (time permitting).

3. Distributions

Infinite dimensional vector spaces, topology, convergence. Spaces of test functions, continuous linear functionals, distributions, delta-functions and derivatives of distributions. Fourier transforms of distributions, tempered distributions, calculation of Fourier series and Fourier transforms. Green functions and linear partial differential equations, solution of the wave equation and Poisson's equation.

No single book covers all the material of this course although there are several books in the library which do cover most of the topics at a suitable level. Here is a selection:

C. J. Isham, ``Lectures on Groups and Vector spaces for Physicists'' (World Scientific, 512.54 I79l).
S. Axler, ``Linear Algebra Done Right'' (Springer, 512.64 A9691 .2).
C. W. Curtis, ``Linear Algebra, An Introductory Approach'' (Springer, 1984) 512.64 C978A.
S. Roman, ``Advanced Linear Algebra'' (Springer, 1992) 512.64 R758a.
Wu-Ki Tung, ``Group theory in physics'' (World Scientific, 1985) 517.986 T926g
R. F. Hoskins, ``Generalized Functions'' (Wiley, 517.51 H826 g).
R. P. Kanwal, ``Generalized Functions, Theory and Technique'' (Academic, 1983) 517.9824 K16g
T. Schucker, ``Distributions, Fourier Transforms and some of their applications to physics'' (World Scientific, 1991) 517.98243 S384d

R. Geroch, ``Mathematical Physics'' (University of Chicago, 515.14 G377m).
Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, ``Analysis, Manifolds and Physics'' (North-Holland, 1981) 517 C5495a