**Fields Institute Communications**

Volume: 30;
2001;
451 pp;
Hardcover

MSC: Primary 81; 46;
Secondary 47

Print ISBN: 978-0-8218-2814-4

Product Code: FIC/30

List Price: $145.00

AMS Member Price: $116.00

MAA Member Price: $130.50

**Electronic ISBN: 978-1-4704-3054-2
Product Code: FIC/30.E**

List Price: $145.00

AMS Member Price: $116.00

MAA Member Price: $130.50

# Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects

Share this page *Edited by *
*Roberto Longo*

A co-publication of the AMS and Fields Institute

The beauty and the mystery surrounding the interplay between mathematics and
physics is captured by E. Wigner's famous expression, “The unreasonable
effectiveness of mathematics”. We don't know why, but physical laws are
described by mathematics, and good mathematics sooner or later finds
applications in physics, often in a surprising way.

In this sense, mathematical physics is a very old subject—as Egyptian,
Phoenician, or Greek history tells us. But mathematical physics is a very
modern subject, as any working mathematician or physicist can witness. It is a
challenging discipline that has to provide results of interest for both
mathematics and physics. Ideas and motivations from both these sciences give it
a vitality and freshness that is difficult to find anywhere else.

One of the big physical revolutions in the twentieth century, quantum
physics, opened a new magnificent era for this interplay. With the appearance
of noncommutative analysis, the role of classical calculus has been taken by
commutation relations, a subject still growing in an astonishing way.

A good example where mathematical physics showed its power, beauty, and
interdisciplinary character is the Doplicher-Haag-Roberts analysis of
superselection sectors in the late 1960s. Not only did this theory explain the
origin of statistics and classify it, but year after year, new connections have
merged, for example with Tomita-Takesaki modular theory, Jones theory of
subfactors, and Doplicher-Roberts abstract duality for compact groups.

This volume contains the proceedings of the conference, “Mathematical
Physics in Mathematics and Physics”, dedicated to Sergio Doplicher and
John E. Roberts held in Siena, Italy. The articles offer current
research in various fields of mathematical physics, primarily concerning
quantum aspects of operator algebras.

Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Readership

Graduate students and research mathematicians interested in quantum theory and functional analysis.

# Table of Contents

## Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects

- Cover Cover11
- Title page v6
- Dedication vii8
- Contents ix10
- Preface xiii14
- Programme of the conference xv16
- Frontispiece xvii18
- Letter by Connes xix20
- Letter by Kastler xx21
- Letter by Haag xxii23
- An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center 124
- Modular invariants and subfactors 1134
- On the PCT-theorem in the theory of local observables 3962
- Transplantation of local nets and geometric modular action on Robertson-Walker space-times 6588
- Classification of subsystems, local symmetry generators and intrinsic definition of local observables 83106
- From local perturbation theory to Hopf- and Lie-algebras of Feynman graphs 105128
- The flat tube theorem for vector valued functions 115138
- Point interactions 139162
- Perturbative algebraic field theory, and deformation quantization 151174
- Sum rules for the free energy in the mean field spin glass model 161184
- Fractals in noncommutative geometry 171194
- What I woud like to understand 187210
- The Rohlin property for automorphisms of 𝐶*-algebras 191214
- Environment induced localization and superselection rules in a gas of pyramidal molecules 207230
- Connes-Moscovici-Kreimer Hopf algebras 219242
- The structure of the automorhpism group of an approximately finite dimensional factor 249272
- Braiding and extensions of endomrophisms of subfactors 261284
- Bicategories of operator algebras and Poisson manifolds 271294
- Notes for a quantum index theorem introduction 287310
- Conformal field theory and Doplicher-Roberts reconstruction 297320
- On the distance between MASA’s in type II₁ factors 321344
- Recent results concerning Eₒ-semigroups of 𝔅(ℌ) 325348
- Locality and modular invariance in 2D conformal QFT 341364
- Tensor products of Banach spaces and the Stone-Weierstrass problem of 𝐶*-algebras 355378
- Perron-Frobenius theory for positive maps on trace ideals 361384
- Space- and time-like superselection rules in conformal quantum field theory 379402
- Finite quantum groupoids and inclusions of finite type 393416
- On generalizations of the spectrum condition 409432
- Algebraic orbifold conformal field theories 429452
- List of participants 449472
- Back Cover Back Cover1477