Functional Integration and Quantum Physics (eBook)

eBook Download: PDF

1979 | 1. Auflage
295 Seiten
Elsevier Science (Verlag)
978-0-08-087402-9 (ISBN)

Infos

Functional integration and quantum physics

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It is fairly well known that one of Hilbert's famous list of problems is that of developing an axiomatic theory of mathematical probability theory (this problem could be said to have been solved by Khintchine, Kolmogorov, andLevy), and also among the list is the "e;axiomatization of physics. What is not so well known is that these are two parts of one and the same problem, namely, the sixth, and that the axiomatics of probability are discussed in the context of the foundations of statistical mechanics. Although Hilbert could not have known it when he formulated his problems, probability theory is also central to the foundations of quantum theory. In this book, I wish to describe a very different interface between probability and mathematical physics, namely, the use of certain notions of integration in function spaces as technical tools in quantum physics. Although Nelson has proposed some connection between these notions and foundational questions, we shall deal solely with their use to answer a variety of questions inconventional quantum theory.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 16.11.1979
• Mitarbeit: Herausgeber (Serie): Barry Simon
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• ISBN-10: 0-08-087402-9 / 0080874029
• ISBN-13: 978-0-08-087402-9 / 9780080874029