@phdthesis{Zeiner2007,
  author    = {Zeiner, J{\"o}rg},
  title     = {Noncommutative Quantumelectrodynamics from Seiberg-Witten Maps to All Orders in Theta},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-23363},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2007},
  abstract  = {The basic question which drove our whole work was to find a meaningful noncommutative gauge theory even for the time-like case (\$\theta^{0 i} \neq 0\$). In order to be able to tackle questions regarding unitarity, it is not sufficient to consider theories which include the noncommutative parameter only up to a finite order. The reason is that in order to investigate tree-level unitarity or the optical theorem in loops one has to know the behavior of the noncommutative theory for center-of-mass energies much greater than the noncommutative scale. Therefore an effective theory, that is by construction only valid up to the noncommutative scale, isn't sufficient for our purpose. Our model is based on two fundamental assumptions. The first assumption is given by the commutation relations \eqref{eq:ncalg}. This led to the Moyal-Weyl star-product \eqref{eq:astproduct2} which replaces all point-like products between two fields. The second assumption is to assume that the model built this way is not only invariant under the noncommutative gauge transformation but also under the commutative one. In order to obtain an action of such a model one has to replace the fields by their appropriate \swms. We chose the gauge fixed action \eqref{eq:actioncgf} as the fundamental action of our model. After having constructed the action of the NCQED including the {\swms} we were confronted with the problem of calculating the {\swms} to all orders in \$\tMN\$. By means of \cite{bbg} we could calculate the {\swms} order by order in the gauge field, where each order in the gauge field contains all orders in the noncommutative parameter (\cf chapter \ref{chapter:swms}). By comparing the maps with the result we obtained from an alternative ansatz \cite{bcpvz}, we realized that already the simplest {\swm} for the gauge field is not unique. In chapter \ref{chapter:ambiguities} we examined this ambiguity, which we could parametrised by an arbitrary function \$\astf\$. The next step was to derive the Feynman rules for our NCQED. One finds that the propagators remain unchanged so that the free theory is equal to the commutative QED. The fermion-fermion-photon vertex contains not only a phase factor coming from the Moyal-Weyl star-product but also two additional terms which have their origin in the \swms. Beside the 3-photon vertex which is already present in NCQED without {\swms} and which has also additional terms coming from the \swms, too, one has a contact vertex which couples two fermions with two photons. After having derived all the vertices we calculated the pair annihilation scattering process \$e^+ e^- \rightarrow \gamma \gamma\$ at Born level. By choosing the parameter \$\kggg = 1\$ (\cf section \ref{sec:represent}), we found that the amplitude of the pair annihilation process becomes equal to the amplitude of the NCQED without \swms. This means that, at least for this process, the NCQED excluding {\swms} is only a special case of NCQED including \swms. On the basis of the pair annihilation process, we afterwards investigated tree-level unitarity. In order to satisfy the tree-level unitarity we had to constrain the arbitrary function \$\astf\$. We found that the series expansion of \$\astf\$ has to start with unity. In addition, the even part of the function must not increase faster than \$s^{-1/2} \log(s)\$ for \$s \rightarrow \infty\$, whereas the odd part of the \$\astf\$-function can't be constrained, at least by the process we considered. By assuming these constrains for the \$\astf\$-function, we could show that tree-level unitarity is satisfied if one incorporates the uncertainties present in the energy and the momenta of the scattered particles, \ie the uncertainties of the center-of-mass energy and the scattering angles. This uncertainties are not exclusively present due to the finite experimental resolution. A delta-like center-of-mass energy as well as delta-like momenta are in general not possible because the scattered particles are never exact plane waves.},
  subject      = {Raum-Zeit},
  language  = {en}
}