Functional Integration and Quantum Physics (eBook)

Front Cover 1
Functional Integration and Quantum Physics 4
Copyright Page 5
Contents 6
Preface 8
List of Symbols 10
Chapter I. Introduction 14
1. Introduction 14
2. Construction of Gaussian Processes 21
3. Some Fundamental Tools of Probability Theory 30
Chapter II. The Basic Processes 45
4. The Wiener Process, the Oscillator Process, and the Brownian Bridge 45
5. Regularity Properties—1 56
6. The Feynman–Kac Formula 61
7. Regularity and Recurrence Properties—2 73
Chapter III. Bound State Problems 101
8. The Birman–Schwinger Kernel and Lieb's Formula 101
9. Phase Space Bounds 106
10. The Classical Limit 118
11. Recurrence and Weak Coupling 127
Chapter IV. Inequalities 132
12. Correlation Inequalities 132
13. Other inequalities: Log Concavity, Symmetric Rearrangement, Conditioning, Hypercontractivity 149
Chapter V. Magnetic Fields and Stochastic Integrals 161
14. ltô’s Integral 161
15. Schrödinger Operators with Magnetic Fields 172
16. Introduction to Stochastic Calculus 183
Chapter VI. Asymptotics 187
17. Donsker's Theorem 187
18. Laplace's Method in Function Space 194
19. Introduction to the Donsker-Varadhan Theory 211
Chapter VII. Other Topics 224
20. Perturbation Theory for the Ground State Energy 224
21. Dirichlet Boundaries and Decoupling Singularities in Scattering Theory 237
22. Crushed Ice and the Wiener Sausage 244
23. The Statistical Mechanics of Charged Particles with Positive Definite Interactions 258
24. An Introduction to Euclidean Quantum Field Theory 265
25. Properties of Eigenfunctions, Wave Packets, and Green's Functions 271
26. Inverse Problems and the Feynman–Kac Formula 285
References 292
Index 306