Mathematical Physics
Description
"A remarkable achievement . . . an informing sight of the sort of mathematics that underpins much of the present day research in theoretical physics." -C. J. Isham, Contemporary 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.
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Content
- Intro
- 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
- 43. Measure Spaces
- 44. Constructing Measure Spaces
- 45. Measurable Functions
- 46. Integrals
- 47. Distributions
- 48. Hilbert 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
- Index of Defined Terms
