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preprint postscript copy: preprint math.DG/0208190

Stud. Appl. Math. Volume 113 Issue 1, 31-55 (2004)

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preprint postscript copy: preprint math.DG/0304083

published in Lett. Math. Phys. 64, Issue 3, 229-234 (2003) (2003)

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preprint postscript copy: Sfb288 preprint #420

published in Comm. Math. Phys. 212, Issue 2, 297-321 (2000)

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preprint (pdf)

published in "Discrete Integrable Geometry and Physics", ed. A. Bobenko, R. Seiler, Oxford Science Publications, Oxford 1999

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preprint (pdf)

published in "Discrete Integrable Geometry and Physics", ed. A. Bobenko, R. Seiler, Oxford Science Publications, Oxford 1999

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postscript copy: Sfb288 preprint #231

published in Comm. Math. Phys. 204, 115-136 (1999)

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postscript copy: Sfb288 preprint #138

published in Phys. Lett. A

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postscript copy: Sfb288 preprint #101

published in Phys. Lett. A

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postscript copy: Sfb288 preprint #42

(the file has a missing titlepage and is partially without pictures)

published in Phys. Lett. A

The content of this PhD thesis is centered around the sine-Gordon equation and in particular around a space and time ("doubly") discretized version of it, which arises in discrete analogs of surfaces with constant negative Gaussian curvature (socalled discrete K-surfaces). In the present work symplectic structures for a class of discrete spacetime systems which contain the sine-Gordon system are derived with methods from Lagrangian mechanics/symplectic geometry. The derived symplectic structures are used for quantizing the corresponding sine-Gordon system. The approach via differential geometry allows to show that central elements in some of the involved algebras arise from gauge freedoms of the corresponding differential geometric frame. Quantum integrals of motion are derived. A reduction of the quantized discrete sine-Gordon equation, the quantized discrete pendulum equation (a discrete version of the nonlinear harmonic oscillator) is investigated in detail, the spectrum of the corresponding quantum integral of motion (the "quantum pendulum hamiltonian") is described in terms of Bethe Ansatz equations. Some parts of this spectrum can be derived explicitly (the quantum pendulum hamiltonian is a square of the Hofstadter's butterfly, which is related to the quantum Hall effect). Finite dimensional representations of the corresponding algebras are constructed. Explicit constructions for the R-matrix at roots of unity are given. This allows to draw connections to models in statistical mechanics and in particular to free fermions on a lattice. Connections to the 2 dim billard in an ellipse and to other sytems are given, partially as an outlook on quantized versions of the underlying discrete surfaces and/or their normal maps (the normal map of K-surfaces provides a sigma model).

arxive.org abstract, see also Mimirix project webpage.

slides on slideshare.net work in progress

arxive.org abstract

german original (pdf)

slightly extended english version (pdf)

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