xiv CONTENTS

X : SELF-ADJOINTNESS A N D THE EXISTENCE OF DYNAMICS

I. Extensions of symmetric operators Appendix Motion on a half-line, lintit point-limit circle

methods 2. Perturbations of self-adjoint operators 3. Positivity and self-adjointness I : Quadratic forms 4. Positivity and self-adjointness 11 : Pointwise positivity 5. The commutator theorem 6. Analytic vectors 7 . Free quantum fields

Appendix The Weyl relations ,for the free jield 8. Semigroups and their ganerators 9. Hypercontractive semigroups

10. Graph Limits 1 I . The Feynman- Kac formula 12. Time-dependent Hamiltonians 13. Classical nonlinear wave equations 14. The Hilbert space approach to classical mechanics Notes Problems Reader's Guide

List of Symbols Index