Mathematical Physics

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University 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.
 

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Inhalt

1 Introduction
1
2 Categories
3
3 The Category of Groups
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
Index of Defined Terms
343
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Über den Autor (2015)

Robert Geroch is professor in the Departments of Physics and Mathematics and at the Enrico Fermi Institute at the University of Chicago.

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