Finite Canonical Commutation Relations
You can never solve a problem on the level on which it was created.
 Albert Einstein (18791955)
"Silence will save me from being wrong (and foolish), but it will also
deprive me of the possibility of being right.
 Igor Stravinsky, composer (18821971)
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
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 nFourier 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 Pseudoorthogonal 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 nonnormable algebras over Galois fields.

Appendix K:
Necessary Multiple Concepts of "Time".

References
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Created: 1996
Last Updated: May 28, 2000
Last Updated: November 28, 2002
Last Updated: February 24, 2004