Univalent Functions in Quantum Probability Theory

Sebastian Schlessinger (Autor)

Buch | Softcover

268 Seiten

2025
American Mathematical Society (Verlag)
978-1-4704-8180-3 (ISBN)

Noch nicht erschienen (ca. Dezember 2025)
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Delves into the interplay between complex analysis and quantum probability theory, this text bridges classical concepts with non-commutative processes. It explores univalent functions via Loewner chains, diverse independence notions, central limit theorems, and higher-dimensional challenges through guided exercises.

This book reveals how univalent functions appear in quantum probability theory. Building upon the recently established one-to-one correspondence between Loewner theory and the theory of non-commutative additive processes, the author invites readers to explore the interplay between complex analysis, classical probability theory, and quantum probability theory. Monotone independence and its relations to classical, free, and Boolean independence underpin the development of ideas. Beginning with essential concepts from classical probability theory and complex analysis, the book goes on to define a quantum probability space and introduce five notions of independence. From this foundation, the central chapters explore convolutions and their respective central limit theorems; univalent functions; classical Loewner chains on the unit disk; slit mappings; and the relationship between free hemigroups, Loewner chains, and nonlinear resolvents. The final chapter offers an outlook on higher dimensional generalizations, including several open problems. Exercises with solutions invite readers to engage with the material throughout. Univalent Functions in Quantum Probability Theory is an essential resource at the intersection of previously distinct fields. Intended for graduate students and researchers alike, it assumes a solid foundation in real and complex analysis, with basic knowledge of classical probability theory and Hilbert spaces.

Sebastian Schlessinger, University of Wuerzburg, Germany.

Introduction
Classical probability theory
The complex toolbox
Quantum probability theory
Convolutions and additive processes
Univalent functions
Radial Loewner chains revisited
Slit mappings
Free hemigroups, Loewner chains, and nonlinear resolvents
Graph products as quantum random walks
Outlook on higher dimensional generalizations
Compactness of the class $S$
Continuous extension of univalent functions
Bibliography
Index

Erscheint lt. Verlag	15.12.2025
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	1-4704-8180-4 / 1470481804
ISBN-13	978-1-4704-8180-3 / 9781470481803
Zustand	Neuware