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Emerging Jordan forms, with applications to critical statistical models and conformal field theory

Lawrence Liu

Abstract

Two novel frameworks for handling mathematical and physical problems are introduced. The first, the emerging Jordan form, generalizes the concept of the Jordan canonical form, a well-established tool of linear algebra. The second, dual Jordan quantum physics, generalizes the framework of quantum physics to one in which the hermiticity postulate is considerably relaxed. These frameworks are then used to resolve some long-outstanding problems in theoretical physics, coming from critical statistical models and conformal field theory. I describe these problems and the difficulties involved in finding satisfactory solutions, then show how the concepts of emerging Jordan forms and dual Jordan quantum physics are naturally suited to overcoming these difficulties. Although their applications in this work are limited in scope to rather specific problems, the frameworks themselves are completely general, and I describe ways in which they may be used in other areas of mathematics and physics. Several appendices close the work, which include improvements to a widely used computational algorithm and corrections to some published data.

Emerging Jordan forms, with applications to critical statistical models and conformal field theory

Abstract

Two novel frameworks for handling mathematical and physical problems are introduced. The first, the emerging Jordan form, generalizes the concept of the Jordan canonical form, a well-established tool of linear algebra. The second, dual Jordan quantum physics, generalizes the framework of quantum physics to one in which the hermiticity postulate is considerably relaxed. These frameworks are then used to resolve some long-outstanding problems in theoretical physics, coming from critical statistical models and conformal field theory. I describe these problems and the difficulties involved in finding satisfactory solutions, then show how the concepts of emerging Jordan forms and dual Jordan quantum physics are naturally suited to overcoming these difficulties. Although their applications in this work are limited in scope to rather specific problems, the frameworks themselves are completely general, and I describe ways in which they may be used in other areas of mathematics and physics. Several appendices close the work, which include improvements to a widely used computational algorithm and corrections to some published data.
Paper Structure (10 sections, 4 theorems, 23 equations)

This paper contains 10 sections, 4 theorems, 23 equations.

Table of Contents

  1. Background
  2. Mathematics: basic concepts
  3. Inner products
  4. The spectral theorem for normal operators
  5. The Jordan canonical form
  6. Representation theory of algebras
  7. Mathematics: representation theory of various Temperley--Lieb algebras
  8. Generators and relations for Temperley--Lieb algebras
  9. Cellular algebras
  10. Standard and co-standard modules

Key Result

Theorem 1

If $A$ is a normal operator on a finite-dimensional complex inner product space $V$, where the inner product is positive-definite, then $V$ has an orthonormal basis consisting of eigenvectors of $A$. Therefore, a normal operator on such a space is diagonalizable.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1: Spectral theorem in terms of eigenvectors
  • Theorem 2: Spectral theorem in terms of projection operators
  • Theorem 3
  • Theorem 4: Jordan canonical form
  • Definition 2
  • Definition 3
  • Definition 4: GrahamLehrer1996
  • Definition 5: GrahamLehrer1996