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This work is principally concerned with the operator approach to the orbifold compactification of the bosonic string. Of particular importance to operator formalism is the con formal structure and the operator product expansion. These are introduced and discussed in detail. The Frenkel-Kac-Segal mechanism is then examined and is shown to be a consequence of the duality of dimension one operators of an analytic bosonic string compactified on a certain torus. Possible generalizations to higher dimension operators are discussed, this includes the cross-bracket algebra which plays a central role in the vertex operator representation of Griess's algebra, and hence the Fischer-Griess Monster Group. The mechanism of compactification is then extended to orbifolds. The exposition includes a detailed account of the twisted sectors, especially of the zero-modes and the twisted operator cocycles. The conformal structure, vertex operators and correlation functions for twisted strings are then introduced. This leads to a discussion of the vertex operators which represent the emission of untwisted states. It is shown how these operators generate Kac-Moody algebras in the twisted sectors. The vertex operators which insert twisted states are then constructed, and their role as intertwining operators is explained. Of particular importance in this discussion is the role of the operator cocycles, which are seen to be crucial for the correct working of the twisted string emission vertices. The previously established formalism is then applied in detail to the reflection twist. This includes an explicit representation of the twisted operator cocycles by elements of an appropriate Clifford algebra and the elucidation of the operator algebra of the twisted emission vertices, for the ground and first excited states in the twisted sector. This motivates the 'enhancement mechanism', a generalization of the Frenkel-Kac-Segal mechanism, involving twisted string emission vertices, in dimensions 8, 16 and 24. associated with rank 8 Lie algebras, rank 16 Lie algebras and the cross-bracket algebra for the Leech lattice, respectively. Some of the relevant characters of the 'enhanced" modules are determined, and the connection of the cross-bracket algebra to the phenomenon of 'Monstrous Moonshine' and the Monster Group is explained. Algebra enhancement is then discussed from the greatly simplified shifted picture and extensions to higher order twists are considered. Finally, a comparison of this work with other recent research is given. In particular, the connection with the path integral formalism and the extension to general asymmetric orbifolds is discussed. The possibility of reformulating the moonshine module in a 'covaxiant' twenty-six dimensional setting is also considered.

We study point particles to illustrate the various symmetries such as the Poincaré group and its non-relativistic version. In order to find the Noether charges and the Noether currents, which are conserved under physical symmetries, we study Noether’s theorem. We describe the Pauli-Lubanski spin vector, which is invariant under the Poincaré group and describes the spin of a particle in field theory. By promoting the Pauli-Lubanski spin vector to an operator in the quantized theory we will see that it describes the spin of a particle. Moreover, we find an action for a smooth spinning bosonic particle by compactifying one string dimension together with one embedding dimension. As with the Pauli-Lubanski spin vector, we need to quantize this action to confirm that it is the action for a smooth spinning particle. / Vi studerar punktpartiklar för att illustrera olika symemtrier som t.ex. Poincaré gruppen och dess icke-relativistiska version. För att hitta de Noether laddningar och Noether strömmar, vilka är bevarade under symmetrier, studerar vi Noether’s sats. Vi beskriver Pauli-Lubanksi spin vektorn, vilken har en invarians under Poincaré gruppen och beskriver spin hos en partikel i fältteori. Genom att låta Pauli-Lubanski spin vektorn agera på ett tillstånd i kvantfältteori ser vi att den beskriver spin hos en partikel. Dessutom finner vi en verkan för en spinnande partikel genom att kompaktifiera en bosonisk sträng dimension tillsammans med en inbäddad dimension. Som med Pauli-Lubanski spin vektorn, kvantiserar vi denna verkan för att bekräfta att det är en verkan för en spinnande partikel.