7.1 Jones index

Jones index quantifies the size of subfactors within von Neumann algebras. Introduced by Vaughan Jones in 1983, it revolutionized operator algebras and connected them to knot theory and statistical mechanics.

The index takes values in a specific set, including some discrete points and a continuous range. It provides insights into subfactor structure, influences statistical dimensions in physics, and has applications in quantum field theory and knot invariants.

Definition of Jones index

• Jones index quantifies the relative size of a subfactor within a larger von Neumann algebra
• Introduced by Vaughan Jones in 1983, revolutionizing the study of operator algebras and their applications
• Plays a crucial role in understanding the structure of von Neumann algebras and their subfactors

Historical context

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  Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original

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  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

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  Local Nets of Von Neumann Algebras in the Sine–Gordon Model | SpringerLink View original

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  Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original

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  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

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Top images from around the web for Historical context

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  Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original

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  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

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• 

  Local Nets of Von Neumann Algebras in the Sine–Gordon Model | SpringerLink View original

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• 

  Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original

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  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

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• Emerged from Vaughan Jones' work on subfactors in the early 1980s
• Built upon previous research in operator algebras by Murray and von Neumann
• Resulted from attempts to classify II₁ factors and their subfactors
• Led to unexpected connections with knot theory and statistical mechanics

Motivation and significance

• Addresses the fundamental question of how "large" a subfactor is within its parent factor
• Provides a numerical invariant for classifying subfactors
• Bridges gap between operator algebras and other areas of mathematics and physics
• Sparked renewed interest in von Neumann algebras and their applications

Properties of Jones index

• Serves as a measure of the "size" or "complexity" of a subfactor inclusion
• Connects abstract algebraic structures to concrete numerical values
• Exhibits surprising restrictions and patterns, leading to new mathematical insights

Fundamental properties

• Always takes values in the set ${4\cos^2(\pi/n) : n = 3, 4, 5, ...} \cup [4, \infty]$
• Finite index implies both factor and subfactor are II₁ factors
• Satisfies multiplicativity: $[M : N] = [M : P][P : N]$ for intermediate subfactor N ⊂ P ⊂ M
• Invariant under isomorphisms of subfactor inclusions
• Lower bound: Jones index is always greater than or equal to 1

Relation to subfactors

• Measures the "relative dimension" of a subfactor within its parent factor
• Smaller index indicates a "tighter" inclusion of subfactors
• Provides information about the structure of intermediate subfactors
• Relates to the minimal index in subfactor theory
• Influences the possible values of statistical dimensions in conformal field theory

Calculation methods

• Involve techniques from various areas of mathematics, including linear algebra, representation theory, and functional analysis
• Require understanding of trace properties in von Neumann algebras
• Often utilize diagrammatic methods developed by Jones and others

Basic techniques

• Use of basic construction and Jones projections
• Computation via Pimsner-Popa basis
• Trace method: $[M : N] = \text{tr}_M(e_N)^{-1}$, where $e_N$ is the Jones projection
• Calculation through statistical dimensions in certain cases
• Application of Temperley-Lieb algebra for some subfactors

Advanced approaches

• Utilization of planar algebra techniques
• Employment of Ocneanu's paragroup theory
• Analysis of principal graphs and their growth rates
• Use of subfactor homology and cohomology theories
• Application of free probability methods in certain cases

Jones index theorem

• Establishes a surprising restriction on possible values of the Jones index
• Connects abstract algebraic structures to concrete numerical values
• Has far-reaching implications in operator algebras and related fields

Statement of theorem

• For a II₁ factor M and a subfactor N, the Jones index [M : N] takes values in the set: ${4\cos^2(\π/n) : n = 3, 4, 5, ...} \cup [4, \infty]$
• Known as the "Jones set" or "Jones' discrete series"
• Excludes all values between 1 and 4 except for specific algebraic numbers
• Proof involves intricate analysis of Temperley-Lieb algebras and representation theory

Implications and consequences

• Reveals unexpected structure in subfactor theory
• Leads to classification of subfactors with small index
• Connects to representation theory of quantum groups
• Influences development of quantum invariants in knot theory
• Provides insights into conformal field theory and statistical mechanics models

Applications in mathematics

• Demonstrates the far-reaching impact of Jones index theory beyond operator algebras
• Illustrates unexpected connections between different areas of mathematics
• Provides new tools and perspectives for solving long-standing problems

Knot theory

• Jones polynomial derived from Jones index theory
• Subfactors associated with knots and links provide topological invariants
• Planar algebra techniques used in both subfactor theory and knot diagrams
• Connection to quantum groups and their representations in knot invariants
• Application in the study of 3-manifolds and topological quantum field theories

Operator algebras

• Classification of subfactors with small index (≤ 4)
• Study of automorphism groups of factors
• Investigation of amenability and property T for II₁ factors
• Development of fusion categories and tensor categories
• Analysis of infinite-dimensional Lie algebras and their representations

Jones index in physics

• Demonstrates the relevance of abstract mathematical concepts in physical theories
• Provides a bridge between operator algebraic methods and physical phenomena
• Offers new perspectives on quantum systems and statistical models

Quantum field theory

• Describes statistical dimensions of superselection sectors in algebraic quantum field theory
• Relates to fusion rules and operator product expansions in conformal field theory
• Connects to anyonic statistics and topological order in condensed matter physics
• Influences the study of boundary conditions in quantum field theories
• Applies to the analysis of defects and interfaces in topological phases

Statistical mechanics

• Describes critical behavior in lattice models (Potts model, Ising model)
• Relates to transfer matrices and partition functions in exactly solvable models
• Connects to integrable systems and Yang-Baxter equations
• Applies to the study of phase transitions and critical phenomena
• Influences the analysis of entanglement entropy in quantum many-body systems

Generalizations and extensions

• Expands the applicability of Jones index theory to broader contexts
• Addresses limitations of the original theory in certain situations
• Provides new tools for analyzing more complex algebraic structures

Higher dimensional cases

• Generalization to higher-dimensional subfactors and planar algebras
• Development of higher-dimensional quantum invariants
• Study of subfactors in type III von Neumann algebras
• Investigation of index theory for inclusions of C*-algebras
• Application to higher-dimensional conformal field theories and topological phases

Non-factor extensions

• Extension of index theory to inclusions of von Neumann algebras that are not factors
• Development of relative entropy techniques for general von Neumann algebra inclusions
• Study of index for inclusions of W*-categories and 2-categories
• Investigation of index theory in the context of Hopf algebra actions
• Application to quantum groupoids and weak Hopf algebras

Relation to other concepts

• Illustrates connections between Jones index and other important mathematical and physical quantities
• Provides different perspectives on the meaning and significance of the Jones index
• Offers new avenues for applying index theory in various contexts

Jones index vs coupling constant

• Jones index as a generalization of coupling constants in physics
• Relationship to Kac-Moody algebras and their levels
• Connection to central charge in conformal field theory
• Influence on the study of quantum groups at roots of unity
• Application in the analysis of integrable systems and exactly solvable models

Connection to entropy

• Relation between Jones index and various notions of entropy in operator algebras
• Connection to entanglement entropy in quantum systems
• Application in the study of quantum information theory
• Influence on the development of free entropy dimension
• Relationship to amenability and hyperfiniteness in von Neumann algebras

Open problems and research

• Highlights ongoing areas of investigation in Jones index theory and related fields
• Identifies key challenges and potential directions for future research
• Demonstrates the continued relevance and vitality of the subject

Current challenges

• Complete classification of subfactors beyond index 5
• Understanding the structure of subfactors with infinite index
• Developing effective computational methods for Jones index in complex cases
• Clarifying the relationship between Jones index and quantum dimensions in general
• Extending index theory to more general algebraic structures (e.g., tensor categories)

Future directions

• Application of Jones index theory to quantum computing and quantum error correction
• Investigation of Jones index in the context of non-commutative geometry
• Development of index theory for inclusions of vertex operator algebras
• Exploration of connections between Jones index and geometric group theory
• Study of Jones index in relation to higher-dimensional topology and geometry