A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

by Peter Szekeres

• ISBN13: 9780521829601
• ISBN10: 0521829607
• Format: Hardcover
• Copyright: 2005-01-17
• Publisher: Cambridge University Press

Summary

This book provides an introduction to the major mathematical structures used in physics today. It covers the concepts and techniques needed for topics such as group theory, Lie algebras, topology, Hilbert space and differential geometry. Important theories of physics such as classical and quantum mechanics, thermodynamics, and special and general relativity are also developed in detail, and presented in the appropriate mathematical language. The book is suitable for advanced undergraduate and beginning graduate students in mathematical and theoretical physics, as well as applied mathematics. It includes numerous exercises and worked examples, to test the reader's understanding of the various concepts, as well as extending the themes covered in the main text. The only prerequisites are elementary calculus and linear algebra. No prior knowledge of group theory, abstract vector spaces or topology is required.

Table of Contents

Preface	ix
Acknowledgements	xiii
Sets and structures	1	(26)
Sets and logic	2	(3)
Subsets, unions and intersections of sets	5	(2)
Cartesian products and relations	7	(3)
Mappings	10	(3)
Infinite sets	13	(4)
Structures	17	(6)
Category theory	23	(4)
Groups	27	(32)
Elements of group theory	27	(3)
Transformation and permutation groups	30	(5)
Matrix groups	35	(5)
Homomorphisms and isomorphisms	40	(5)
Normal subgroups and factor groups	45	(4)
Group actions	49	(3)
Symmetry groups	52	(7)
Vector spaces	59	(39)
Rings and fields	59	(1)
Vector spaces	60	(3)
Vector space homomorphisms	63	(3)
Vector subspaces and quotient spaces	66	(6)
Bases of a vector space	72	(9)
Summation convention and transformation of bases	81	(7)
Dual spaces	88	(10)
Linear operators and matrices	98	(28)
Eigenspaces and characteristic equations	99	(8)
Jordan canonical form	107	(9)
Linear ordinary differential equations	116	(4)
Introduction to group representation theory	120	(6)
Inner product spaces	126	(23)
Real inner product spaces	126	(7)
Complex inner product spaces	133	(8)
Representations of finite groups	141	(8)
Algebras	149	(29)
Algebras and ideals	149	(3)
Complex numbers and complex structures	152	(5)
Quaternions and Clifford algebras	157	(3)
Grassmann algebras	160	(6)
Lie algebras and Lie groups	166	(12)
Tensors	178	(26)
Free vector spaces and tensor spaces	178	(8)
Multilinear maps and tensors	186	(7)
Basis representation of tensors	193	(5)
Operations on tensors	198	(6)
Exterior algebra	204	(24)
r-Vectors and r-forms	204	(2)
Basis representation of r-vectors	206	(2)
Exterior product	208	(5)
Interior product	213	(2)
Oriented vector spaces	215	(5)
The Hodge dual	220	(8)
Special relativity	228	(27)
Minkowski space-time	228	(7)
Relativistic kinematics	235	(4)
Particle dynamics	239	(5)
Electrodynamics	244	(7)
Conservation laws and energy--stress tensors	251	(4)
Topology	255	(32)
Euclidean topology	255	(2)
General topological spaces	257	(7)
Metric spaces	264	(1)
Induced topologies	265	(4)
Hausdorff spaces	269	(2)
Compact spaces	271	(2)
Connected spaces	273	(3)
Topological groups	276	(3)
Topological vector spaces	279	(8)
Measure theory and integration	287	(21)
Measurable spaces and functions	287	(5)
Measure spaces	292	(9)
Lebesgue integration	301	(7)
Distributions	308	(22)
Test functions and distributions	309	(5)
Operations on distributions	314	(6)
Fourier transforms	320	(3)
Green's functions	323	(7)
Hilbert spaces	330	(36)
Definitions and examples	330	(5)
Expansion theorems	335	(6)
Linear functionals	341	(3)
Bounded linear operators	344	(7)
Spectral theory	351	(6)
Unbounded operators	357	(9)
Quantum mechanics	366	(44)
Basic concepts	366	(13)
Quantum dynamics	379	(8)
Symmetry transformations	387	(10)
Quantum statistical mechanics	397	(13)
Differential geometry	410	(37)
Differentiable manifolds	411	(4)
Differentiable maps and curves	415	(2)
Tangent, cotangent and tensor spaces	417	(9)
Tangent map and submanifolds	426	(6)
Commutators, flows and Lie derivatives	432	(8)
Distributions and Frobenius theorem	440	(7)
Differentiable forms	447	(34)
Differential forms and exterior derivative	447	(4)
Properties of exterior derivative	451	(3)
Frobenius theorem: dual form	454	(3)
Thermodynamics	457	(7)
Classical mechanics	464	(17)
Integration on manifolds	481	(25)
Partitions of unity	482	(2)
Integration of n-forms	484	(2)
Stokes' theorem	486	(7)
Homology and cohomology	493	(7)
The Poincare lemma	500	(6)
Connections and curvature	506	(53)
Linear connections and geodesics	506	(4)
Covariant derivative of tensor fields	510	(2)
Curvature and torsion	512	(4)
Pseudo-Riemannian manifolds	516	(6)
Equation of geodesic deviation	522	(2)
The Riemann tensor and its symmetries	524	(3)
Cartan formalism	527	(7)
General relativity	534	(14)
Cosmology	548	(5)
Variation principles in space-time	553	(6)
Lie groups and Lie algebras	559	(28)
Lie groups	559	(5)
The exponential map	564	(5)
Lie subgroups	569	(3)
Lie groups of transformations	572	(6)
Groups of isometries	578	(9)
Bibliography	587	(2)
Index	589