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Metodos matematicos em quantizacao canonica de espacos de fase nao triviais

By Jose Manuel Pe-Curto Velhinho


This thesis is devoted to several aspects of the problem of canonical quantization of non-trivial phase spaces, both in finite and infinite dimensions. We will consider quantum representations of the type of the coordinate Schrodinger representation. In part I we discuss general aspects of the problem of quantization of non-linear finite dimensional phase spaces. Besides reviewing concepts that will be generalized in parts II and III, we discuss and criticize a particular quantization of the torus proposed by Gotay. We argue that algebraic relations among classical observables should be taken into account at the quantum level. Part II is devoted to measure theoretical methods in the canonical quantization of scalar field theories. We review the theory of cyclic representations of the Weyl relations and the quantization of the free scalar field with non-zero mass. We study the support of the free field measures, using a sequence of variables which test the field behaviour at large distances, thus allowing to distinguish between the typical quantum fields associated with different values of the mass. Part III is devoted to mathematical methods in the canonical quantization of gauge theories of connections. The use of projective methods together with the formalism of groupoids allows a systematization of the construction of the space of quantum connections introduced by Baez, clarifying the relation between this space and the quantum configuration space introduced by Ashtekar and Isham. We also study properties of the Ashtekar-Lewandowski measure in. We show that this measure is ergodic with respect to the action of the group of diffeomorphisms. We also prove that a typical quantum connection restricted to an analytic curve leads to a parallel transport discontinuous at every point of the curveAvailable from Fundacao para a Ciencia e a Tecnologia, Servico de Informacao e Documentacao, Av. D. Carlos I, 126, 1249-074 Lisboa, Portugal / FCT - Fundação para o Ciência e a TecnologiaSIGLEPTPortuga

Year: 2001
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