Mathematical PhysicsUniversity of Chicago Press, 01.08.2015 - 358 Seiten 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. |
Inhalt
| 1 | |
| 3 | |
| 16 | |
4 Subgroups | 24 |
5 Normal Subgroups | 29 |
6 Homomorphisms | 32 |
7 Direct Products and Sums of Groups | 35 |
8 Relations | 39 |
30 Compactness | 165 |
31 The CompactOpen Topology | 172 |
32 Connectedness | 177 |
Dynamical Systems | 183 |
34 Homotopy | 188 |
35 Homology | 199 |
Relation to Homotopy | 211 |
37 The Homology Functors | 214 |
9 The Category of Vector Spaces | 44 |
10 Subspaces | 53 |
11 Linear Mappings Direct Products and Sums | 58 |
12 From Real to Complex Vector Spaces and Back | 62 |
13 Duals | 65 |
14 Multilinear Mappings Tensor Products | 71 |
Minkowski Vector Space | 79 |
The Lorentz Group | 87 |
17 Functors | 90 |
18 The Category of Associative Algebras | 97 |
19 The Category of Lie Algebras | 104 |
The Algebra of Observables | 111 |
Fock Vector Space | 114 |
General Theory | 120 |
23 Representations on Vector Spaces | 125 |
Summary | 132 |
25 Subsets and Mappings | 134 |
26 Topological Spaces | 136 |
27 Continuous Mappings | 147 |
28 The Category of Topological Spaces | 154 |
29 Nets | 160 |
38 Uniform Spaces | 217 |
39 The Completion of a Uniform Space | 225 |
40 Topological Groups | 234 |
41 Topological Vector Spaces | 240 |
Summary | 248 |
43 Measure Spaces | 249 |
44 Constructing Measure Spaces | 257 |
45 Measurable Functions | 259 |
46 Integrals | 262 |
47 Distributions | 270 |
48 Hilbert Spaces | 277 |
49 Bounded Operators | 285 |
50 The Spectrum of a Bounded Operator | 293 |
Finitedimensional Case | 302 |
52 Continuous Functions of a Hermitian Operator | 306 |
53 Other Functions of a Hermitian Operator | 311 |
54 The Spectral Theorem | 319 |
55 Operators Not Necessarily Bounded | 324 |
56 SelfAdjoint Operators | 329 |
| 343 | |
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Häufige Begriffe und Wortgruppen
A₁ abelian groups adjoint associative algebra bounded operator category of sets Cauchy Cauchy net collection commutes compact complex number complex vector space condition consider consisting continuous mapping converges coset define definition denote diagram of figure direct product direct sum discrete topology eigenvalue entourage equivalence class example Exercise finite number forgetful functor given group G Hausdorff Hence Hermitian operator Hilbert space homologous homomorphism homotopic integrable intersection intuitive isomorphism K₁ Let G Lie algebra linear combination linear mapping loop mapping of sets measure space monomorphism Mor(V morphism n-cycle neighborhood nonzero normal subgroup Note object obtain one-to-one open sets open subset Proof properties Prove real line real number real vector space representation satisfied self-adjoint sequence space H space of reals square-integrable step function structure subspace superset theorem tion topological group topological space topological vector space uniform space unique V₁ vector h whence zero