Mathematical Physics (Chicago Lectures in Physics)

Mathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space. Geroch uses category theory to emphasize both the interrelationships among different structures and the unity of mathematics. Perhaps the most valuable feature of the book is the illuminating intuitive discussion of the "whys" of proofs and of axioms and definitions. This book, based on Geroch's University of Chicago course, will be especially helpful to those working in theoretical physics, including such areas as relativity, particle physics, and astrophysics. Contents 1 Introduction 2 Categories 3 The category of groups 4 Subgroups 5 Normal subgroups 6 Homomorphisms 7 Direct products and sums of groups 8 Relations 9 The category of vector spaces 10 Subspaces 11 Linear mappings; Direct products and sums 12 From real to complex vector spaces and back 13 Duals 14 Multilinear mappings; Tensor products 15 Example: Minkowski vector space 16 Example: The Lorentz group 17 Functors 18 The category of associative algebras 19 The category of Lie algebras 20 Example: The algebra of observables 21 Example: Fock vector space 22 Representations: General theory 23 Representations on vector spaces 24 The algebraic categories: Summary 25 Subsets and mappings 26 Topological spaces 27 Continuous mappings 28 The category of topological spaces 29 Nets 30 Compactness 31 The compact-open topology 32 Connectedness 33 Example: Dynamical systems 34 Homotopy 35 Homology 36 Homology: Relation to homotopy 37 The homology functors 38 Uniform spaces 39 The completion of a uniform space 40 Topological groups 41 Topological vector spaces 42 Categories: Summary 44 Constructing measure spaces 45 Measurable functions 46 Integrals 47 Distributions 48 Hilber spaces 49 Bounded operators 50 The spectrum of a bounded operator 51 The spectral theorem: Finite-dimensional case 52 Continuous functions of a Hermitian operator 53 Other functions of a Hermitian operator 54 The spectral theorem 55 Operators (not necessarily bounded) 56 Self-adjoint operators