Canonical group quantization and boundary conditions

In the present thesis, we study quantization of classical systems with non-trivial phase spaces using the group-theoretical quantization technique proposed by Isham. Our main goal is a better understanding of global and topological aspects of quantum theory. In practice, the group-theoretical approach enables direct quantization of systems subject to constraints and boundary conditions in a natural and physically transparent manner -- cases for which the canonical quantization method of Dirac fails. First, we provide a clarification of the quantization formalism. In contrast to prior treatments, we introduce a sharp distinction between the two group structures that are involved and explain their physical meaning. The benefit is a consistent and conceptually much clearer construction of the Canonical Group. In particular, we shed light upon the 'pathological' case for which the Canonical Group must be defined via a central Lie algebra extension and emphasise the role of the central extension in general. In addition, we study direct quantization of a particle restricted to a half-line with 'hard wall' boundary condition. Despite the apparent simplicity of this example, we show that a naive quantization attempt based on the cotangent bundle over the half-line as classical phase space leads to an incomplete quantum theory; the reflection which is a characteristic aspect of the 'hard wall' is not reproduced. Instead, we propose a different phase space that realises the necessary boundary condition as a topological feature and demonstrate that quantization yields a suitable quantum theory for the half-line model. The insights gained in the present special case improve our understanding of the relation between classical and quantum theory and illustrate how contact interactions may be incorporated.

Investigations on formation and specification of neural precursor cells in the central nervous system of the Drosophila melanogaster embryo

Investigations on formation and specification of neural precursor cells in the central nervous system of the Drosophila melanogaster embryoSpecification of a unique cell fate during development of a multicellular organism often is a function of its position. The Drosophila central nervous system (CNS) provides an ideal system to dissect signalling events during development that lead to cell specific patterns. Different cell types in the CNS are formed from a relatively few precursor cells, the neuroblasts (NBs), which delaminate from the neurogenic region of the ectoderm. The delamination occurs in five waves, S1-S5, finally leading to a subepidermal layer consisting of about 30 NBs, each with a unique identity, arranged in a stereotyped spatial pattern in each hemisegment. This information depends on several factors such as the concentrations of various morphogens, cell-cell interactions and long range signals present at the position and time of its birth. The early NBs, delaminating during S1 and S2, form an orthogonal array of four rows (2/3,4,5,6/7) and three columns (medial, intermediate, and lateral) . However, the three column and four row-arrangement pattern is only transitory during early stages of neurogenesis which is obscured by late emerging (S3-S5) neuroblasts (Doe and Goodman, 1985; Goodman and Doe, 1993). Therefore the aim of my study has been to identify novel genes which play a role in the formation or specification of late delaminating NBs.In this study the gene anterior open or yan was picked up in a genetic screen to identity novel and yet unidentified genes in the process of late neuroblast formation and specification. I have shown that the gene yan is responsible for maintaining the cells of the neuroectoderm in an undifferentiated state by interfering with the Notch signalling mechanism. Secondly, I have studied the function and interactions of segment polarity genes within a certain neuroectodermal region, namely the engrailed (en) expressing domain, with regard to the fate specification of a set of late neuroblasts, namely NB 6-4 and NB 7-3. I have dissected the regulatory interaction of the segment polarity genes wingless (wg), hedgehog (hh) and engrailed (en) as they maintain each other’s expression to show that En is a prerequisite for neurogenesis and show that the interplay of the segmentation genes naked (nkd) and gooseberry (gsb), both of which are targets of wingless (wg) activity, leads to differential commitment of NB 7-3 and NB 6-4 cell fate. I have shown that in the absence of either nkd or gsb one NB fate is replaced by the other. However, the temporal sequence of delamination is maintained, suggesting that formation and specification of these two NBs are under independent control.