abstract: Chapter 1 introduces some key elements of important topics such as; quantum mechanics, representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´ tivistic wave equations that will play an important role in the work to follow. In Chapter 2, a complex covariant form of the classical Maxwell’s equations in a moving medium or at rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´ netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used. Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´ operators of the Poincare group. A connection between the spin of a particle/field and ´ consistency of the corresponding overdetermined system is emphasized in the massless case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨ evolution of exact wave functions of the generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the methods introduced in Chapter 5 a model for the quantization of an electromagnetic field in a variable media is analyzed. The concept of quantization of an electromagnetic field in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode of radiation for this model is used to find time-dependent photon amplitudes in relation to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the uncertainty relation, are explicitly given in terms of the Ermakov-type system. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

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.