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.
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
9. The Segal-Bargmann transform for compact Lie groups 44
10. To infinity and beyond 52
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.
Mathematics Subject Classification.
Primary 81S05, 81S30; Secondary 22E30, 46E20,
Supported in part by an NSF Postdoctoral Fellowship.
2000 American Mathematical Society