Finite Canonical Commutation Relations

   You can never solve a problem on the level on which it was created.

	-- Albert Einstein (1879-1955)


   "Silence will save me from being wrong (and foolish), but it will also
   deprive me of the possibility of being right.

	-- Igor Stravinsky, composer (1882-1971)

A working paper on FCCR nxn matricies as local kinematical replacement for CCR, and construction of REPS of CCR by pxp matricies over Galois fields.

Some remaining text of the chapters is still being translated to hypertext. An abstract and a brief summary of the essentials are available. The central object of interest here is a chain of algebras given in terms of nxn complex matricies that connect the quantum mechanical Canonical Commutation Relations (CCR) for n unbounded with the Canonical Anticommutation Relations (CAR) for n=2, which seem to provide a local, finite, discrete quantum theory. Finite dimensional representations of CCR by matricies over Galois fields are constructed in appendix J.

The detail of mathematics, physics and philosophy presented here is far more than what would be usual in an any professional journal. I believe it to be sufficient that any mathematician/physicist should be able to reproduce and confirm (or correct!) every detail in the exposition.

   Copyright Notice

• Abstract
• A summary of FCCR
• Notational Conventions
• Symbols
• FCCR Table of Contents
• About Author
• Email: Author Bill Hammel

                                CONTENTS

• I. Motivation and Introduction

• II. Truncated Creation and Annihilation Operators

• III. QM and Limits of unbounded n

• IV. Higher Commutators

• V. Algebra of Observables
  Polynomial functions of the generators Q(n) and P(n)
  Canonical Transformations

• VI. P(n) and Q(n) as Algebra Generators
  Irreducibility
  Solvability v. Unitarity

• VII. Discrete n-Fourier Transforms
  Fr(n), connecting Q(n) and P(n) and UPSILON(n) relating
  N(n)-eigenbasis with the eigenbasis of the phase operator F(n)

• VIII. FCCR Structural Theorems
  and Matrix Element Calculations

• IX. Eigenvalue Problems
  P(n), Q(n) and Sine and Cosine Operators

• X. Diagonalizing Transformations
  XI(n) and PI(n);
  Diagonalizing UPSILON(n)

• XI. Invariance Group of G(n)
  Right and Left G(n)-Hermiticity.
  Adjoint and Coadjoint actions - Orbits.
  Analogy to the Lorentz Group.
  

• XII. Homogeneous Complex Spaces
  Complex Lorentzian hypercones.

• XIII. Uncertainty Relations
  Comparison with CCR,
  Exceptional cases where proof method fails in FCCR and CCR.

• XIV. IRREPS of SU(2), SU(1,1) SL(2,C)
  Quantized Noncommutative Geometry

• XV. Relativstic Structure

• XVI. Elementary Systems

• XVII. More Limits

• XVIII. Conclusions

• Appendix A:
  Some Topological and Algebraic Definitions

• Appendix B:
  Lie Groups and Lie Algebras
  Clifford Algebras and Spin Representations
  of Orthogonal and Pseudo-orthogonal Groups

• Appendix C:
  Lie Algebras and Lie Groups
  su(2), su(1,1) and sl(2,C)

• Appendix D:
  G(n)-Hermiticity
  G(n) Invariance Group
  Hyperbolic Complex Manifolds

• Appendix E:
  Some Polynomial Formulae, Hermite and related polynomials,
  their roots and algebraic approximations of them

• Appendix F:
  Analytic Functions,
  Fourier and Hilbert Transforms and
  Dispersion Relations for Causal Functions

• Appendix G:
  Hilbert Subspace Calculations

• Appendix H:
  Asymptotics of Rotation Matricies
  Proof of Theorem 14.5.

• Appendix I:
  Trace Formulae

• Appendix J:
  FCCR Algebra over Finite Galois Fields and
  Finite dimensional Representations of CCR
  in non-normable algebras over Galois fields.

• Appendix K:
  Necessary Multiple Concepts of "Time".
  

• References