Düz ve eğri uzaylarda değişmeli olmayan alanların çift üretimi üzerindeki etkisi

Our objective in this thesis is to compute the pair production rates for both bosons and fermions under the influence of non-abelian gauge fields on the manifolds $\mathbb{R}^{3,1} \equiv \mathbb{R}^2 \times \mathbb{R}^{1,1}$ and $S^2 \times \mathbb{R}^{1,1}$. We will compare the pair production rates of the spherical cases with the flat ones, and also compare the non-abelian cases with the abelian ones to see effects of both curvature and non-abelian field strength on the pair production. We first review the pair production process, i.e. the so-called Schwinger effect using the path integral formalism for bosonic spin-$0$ i.e. scalar fields and for fermions, spin-$1/2$ i.e. spinor fields, and also subsequently review the recent results obtained in the literature on $\mathbb{R}^{3,1}$ and $S^2 \times \mathbb{R}^{1,1}$ with abelian orthogonal uniform electric and magnetic fields. We then move on to generalize these results by the inclusion of a uniform non-abelian magnetic field due to an external $SU(2)$ gauge field. In doing so, we find the opportunity to compare the pair production rates on $\mathbb{R}^{3,1}$ and $S^2 \times \mathbb{R}^{1,1}$ with non-abelian field switched on, and also compare its influence to previously obtained results without the non-abelian field. Novel effects of the presence of the uniform non-abelian magnetic field together with the effects of constant positive curvature of the $S^2$-submanifold are emphasized.