Diese Dissertation behandelt ein Problem aus der Deformationsquantisierung: Nachdem man die Quantisierung eines klassischen Systems konstruiert hat, würde man gerne ihre mathematischen Eigenschaften verstehen (sowohl die des klassischen Systems als auch die des Quantensystems). Falls beide Systeme durch *-Algebren über dem Körper der komplexen Zahlen beschrieben werden, bedeutet dies dass man die Eigenschaften bestimmter *-Algebren verstehen muss: Welche Darstellungen gibt es? Was sind deren Eigenschaften? Wie können die Zustände in diesen Darstellungen beschrieben werden? Wie kann das Spektrum der Observablen beschrieben werden? Um eine hinreichend allgemeine Behandlung dieser Fragen zu ermöglichen, wird das Konzept von abstrakten O*-Algebren entwickelt. Dies sind im Wesentlichen *-Algebren zusammen mit einem Kegel positiver linearer Funktionale darauf (z.B. die stetigen positiven linearen Funktionale wenn man mit einer *-Algebra startet, die mit einer gutartigen Topologie versehen ist). Im Anschluss daran wird dieser Ansatz dann auf zwei Beispiele aus der Deformationsquantisierung angewandt, die im Detail untersucht werden. / This thesis discusses and proposes a solution for one problem arising from deformation quantization: Having constructed the quantization of a classical system, one would like to understand its mathematical properties (of both the classical and quantum system). Especially if both systems are described by ∗-algebras over the field of complex numbers, this means to understand the properties of certain ∗-algebras: What are their representations? What are the properties of these representations? How can the states be described in these representations? How can the spectrum of the observables be described? In order to allow for a sufficiently general treatment of these questions, the concept of abstract O ∗-algebras is introduced. Roughly speaking, these are ∗ -algebras together with a cone of positive linear functionals on them (e.g. the continuous ones if one starts with a ∗-algebra that is endowed with a well-behaved topology). This language is then applied to two examples from deformation quantization, which will be studied in great detail.

University, Diss., 2007--Frankfurt (Main). / Zsfassung in dt. Sprache.

To study coisotropic reduction in the context of deformation quantization we introduce constraint manifolds and constraint algebras as the basic objects encoding the additional information needed to define a reduction. General properties of various categories of constraint objects and their compatiblity with reduction are examined. A constraint Serre-Swan theorem, identifying constraint vector bundles with certain finitely generated projective constraint modules, as well as a constraint symbol calculus are proved. After developing the general deformation theory of constraint algebras, including constraint Hochschild cohomology and constraint differential graded Lie algebras, the second constraint Hochschild cohomology for the constraint algebra of functions on a constraint flat space is computed. / Um koisotrope Reduktion im Kontext der Deformationsquantisierung zu betrachten, werden constraint Mannigfaltigkeiten und constraint Algebren als grundlegende Objekte definiert. Wichtige Eigenschaften verschiedener zugehöriger Kategorien, sowie deren Kompatibilität mit Reduktion werden untersucht. In Analogie zum klassischen Serre-Swan-Theorem können constraint Vektorbündel mit bestimmten endlich erzeugt projektiven constraint Moduln identifiziert werden. Außerdem wird ein Symbolkalkül für constraint Multidifferenzialoperatoren eingeführt. Nach der Entwicklung der allgemeinen Deformationstheorie von constraint Algebren mithilfe von constraint Hochschild Kohomologie und constraint differentiell gradierten Lie-Algebren, wird die zweite constraint Hochschild Kohomologie im Fall eines endlich dimensionalen constraint Vektorraums berechnet.

Mannheim, Univ., Diss., 2004.

Freiburg i. Br., Univ., Diss., 2006.

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold (\\\"phase space\\\"). His algorithm gives a non-commutative, but associative, product (a so-called \\\"star-product\\\") between smooth phase space functions parameterized by Planck\\\'s constant ℏ, which is treated as a deformation parameter. In the limit as ℏ goes to zero, the star product commutator goes to ℏ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov\\\'s method which applies to the infinite-dimensional symplectic \\\"manifolds\\\" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold (\\\"phase space\\\"). His algorithm gives a non-commutative, but associative, product (a so-called \\\"star-product\\\") between smooth phase space functions parameterized by Planck\\\''s constant ℏ, which is treated as a deformation parameter. In the limit as ℏ goes to zero, the star product commutator goes to ℏ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov\\\''s method which applies to the infinite-dimensional symplectic \\\"manifolds\\\" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.