Mathematical Quantization: 1st Edition (Hardback) book cover

Mathematical Quantization

1st Edition

By Nik Weaver

Chapman and Hall/CRC

296 pages

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Hardback: 9781584880011
pub: 2001-05-31
$175.00
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pub: 2001-05-31
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Description

With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics.

Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras.

In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups.

For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field.

Table of Contents

QUANTUM MECHANICS

Classical Physics

States and Events

Observables

Dynamics

Composite Systems

Quantum Computation

HILBERT SPACES

Definitions and Examples

Subspaces

Orthonormal Bases

Duals and Direct Sums

Tensor Products

Quantum Logic

OPERATORS

Unitaries and Projections

Continuous Functional Calculus

Borel Functional Calculus

Spectral Measures

The Bounded Spectral Theorem

Unbounded Operators

The Unbounded Spectral Theorem

Stone's Theorem

THE QUANTUM PLANE

Position and Momentum

The Tracial Representation

Bargmann-Segal Space

Quantum Complex Analysis

C*-ALGEBRAS

The Algebras C(X)

Topologies from Functions

Abelian C*-Algebras

The Quantum Plane

Quantum Tori

The GNS Construction

VON NEUMANN ALGEBRAS

The Algebras l8 (X)

The Algebras L8 (X)

Trace Class Operators

The Algebras B(H)

Von Neumann Algebras

The Quantum Plane and Tori

QUANTUM FIELD THEORY

Fock Space

CCR Algebras

Realtivistic Particles

Flat Spacetime

Curved Spacetime

OPERATOR SPACES

The Spaces V(K)

Mstiex Norms and Convexity

Duality

Matrix-Valued Functions

Operator Systems

HILBERT MODULES

Continuous Hilbert Bundles

Hilbert L8-Modules

Hilber C*-Modules

Hilbert W*-Modules

Crossed Products

Hilbert *-Bimodules

LIPSCHITZ ALGEBRAS

The Algebras Lip0(X)

Measurable Metrics

The Derivation Theorem

Examples

Quantum Markov Semigroups

QUANTUM GROUPS

Finite Dimensional C*-Algebras

Finite Quantum Groups

Compact Quantum Groups

Haar Measure\

REFERENCES

About the Series

Studies in Advanced Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT037000
MATHEMATICS / Functional Analysis
SCI040000
SCIENCE / Mathematical Physics