Contemporary Mathematics

Volume 260, 2000

Holomorphic methods in analysis and mathematical physics

Brian C. Hall

Dedicated to my "father" Leonard Gross, and to the memory of my "grandfather" Irving Segal.

CONTENTS

1. Introduction 1

2. Basics of holomorphic function spaces 2

3. Examples of holomorphic function spaces 7

4. A special property of the Segal-Bargmann and weighted Bergman spaces 12

5. Canonical commutation relations 16

6. The Segal-Bargmann transform 21

7. Quantum mechanics and quantization 30

8. Toeplitz operators, anti-Wick ordering, and phase space probability

densities 38

9. The Segal-Bargmann transform for compact Lie groups 44

10. To infinity and beyond 52

References 58

1.

Introduction

These notes are based on lectures that I gave at the Summer School in Math-

ematical Analysis at the Instituto de Matematicas de la Universidad Nacional

Autonoma de Mexico, Unidad Cuernavaca, from June 8 to 18, 1998. I am grateful

to Salvador Perez Esteva and Carlos Villegas Blas for organizing the School and

for inviting me, and to all the audience members for their attention and interest. I

thank Steve Sontz for corrections to the manuscript.

The notes explain certain parts of the theory of holomorphic function spaces

and the relation of that theory to quantum mechanics. The level is intended for

beginning graduate students. I assume knowledge of the basics of holomorphic

functions of one complex variable, Hilbert spaces, and measure theory. I do not

assume any prior knowledge of holomorphic function spaces or quantum mechanics.

2000

Mathematics Subject Classification.

Primary 81S05, 81S30; Secondary 22E30, 46E20,

Supported in part by an NSF Postdoctoral Fellowship.

©

2000 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/260/04156

Volume 260, 2000

Holomorphic methods in analysis and mathematical physics

Brian C. Hall

Dedicated to my "father" Leonard Gross, and to the memory of my "grandfather" Irving Segal.

CONTENTS

1. Introduction 1

2. Basics of holomorphic function spaces 2

3. Examples of holomorphic function spaces 7

4. A special property of the Segal-Bargmann and weighted Bergman spaces 12

5. Canonical commutation relations 16

6. The Segal-Bargmann transform 21

7. Quantum mechanics and quantization 30

8. Toeplitz operators, anti-Wick ordering, and phase space probability

densities 38

9. The Segal-Bargmann transform for compact Lie groups 44

10. To infinity and beyond 52

References 58

1.

Introduction

These notes are based on lectures that I gave at the Summer School in Math-

ematical Analysis at the Instituto de Matematicas de la Universidad Nacional

Autonoma de Mexico, Unidad Cuernavaca, from June 8 to 18, 1998. I am grateful

to Salvador Perez Esteva and Carlos Villegas Blas for organizing the School and

for inviting me, and to all the audience members for their attention and interest. I

thank Steve Sontz for corrections to the manuscript.

The notes explain certain parts of the theory of holomorphic function spaces

and the relation of that theory to quantum mechanics. The level is intended for

beginning graduate students. I assume knowledge of the basics of holomorphic

functions of one complex variable, Hilbert spaces, and measure theory. I do not

assume any prior knowledge of holomorphic function spaces or quantum mechanics.

2000

Mathematics Subject Classification.

Primary 81S05, 81S30; Secondary 22E30, 46E20,

Supported in part by an NSF Postdoctoral Fellowship.

©

2000 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/260/04156